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(************************************************************************) ( Copyright (C) 2013 - 2015 ) ( Author C. Cohen ) ( DRAFT - PLEASE USE WITH CAUTION ) ( License CeCILL-B ) (**********************************************************************) From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq. From mathcomp Require Import choice path finset finfun fintype bigop. Require Import finmap. (**************************************************************************) ( This file provides a representation of multisets based on fsfun ) ( {mset T} == the type of multisets on a choiceType T ) ( The following notations are in the %mset scope ) ( mset0 == the empty multiset ) ( mset n a == the multiset with n times element a ) ( [mset a] == the singleton multiset {k} := mset 1 a ) ( [mset a1; ..; an] == the multiset obtained from the elements a1,..,an ) ( A `&` B == the intersection of A and B (the min of each) ) ( A `\|` B == the union of A and B (the max of each) ) ( A `+` B == the sum of A and B ) ( a \|` B == the union of singleton a and B ) ( a +` B == the addition of singleton a to B ) ( A `\` B == the difference A minus B ) ( A `\ b == A without one b ) ( A `` B == the product of A and B ) (* [disjoint A & B] := A `&` B == 0 ) ( A `<=` B == A is a sub-multiset of B ) ( A `<` B == A is a proper sub-multiset of B ) (***************************************************************************) Set Implicit Arguments. Unset Strict Implicit. Unset Printing Implicit Defensive. Lemma sumn_map I (f : I -> nat) s : sumn [seq f i \| i <- s] = \sum_(i <- s) f i. ( Goal: is_true (implb (P (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) (@eq_op nat_eqType (E (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) O)) ) Proof. by elim: s => [\|i s IHs] in f ; rewrite ?(big_nil, big_cons) //= IHs. Qed. Lemma sumn_filter s P : sumn [seq i <- s \| P i] = \sum_(i <- s \| P i) i. Proof. by rewrite -big_filter -sumn_map map_id. Qed. Lemma sumn_map_filter I s (f : I -> nat) P : sumn [seq f i \| i <- s & P i] = \sum_(i <- s \| P i) f i. Proof. by rewrite sumn_map big_filter. Qed. Delimit Scope mset_scope with mset. Local Open Scope fset_scope. Local Open Scope fmap_scope. Local Open Scope mset_scope. Local Open Scope nat_scope. Definition multiset (T : choiceType) := {fsfun T -> nat with 0}. Definition multiset_of (T : choiceType) of phant T := @multiset T. Notation "'{mset' T }" := (@multiset_of _ (Phant T)) (format "'{mset' T }") : mset_scope. Notation "[ 'mset[' key ] x 'in' aT => F ]" := ([fsfun[key] x in aT => F] : {mset _}) (at level 0, x ident, only parsing) : mset_scope. Notation "[ 'mset' x 'in' aT => F ]" := ([fsfun x in aT => F] : {mset _}) (at level 0, x ident, only parsing) : mset_scope. Notation "[ 'm' 'set' x 'in' aT => F ]" := ([fsfun[_] x in aT => F] : {mset _}) (at level 0, x ident, format "[ 'm' 'set' x 'in' aT => F ]") : mset_scope. Identity Coercion multiset_multiset_of : multiset_of >-> multiset. Notation enum_mset_def A := (flatten [seq nseq (A%mset x) x \| x <- finsupp A%mset]). Module Type EnumMsetSig. Axiom f : forall K, multiset K -> seq K. Axiom E : f = (fun K (A : multiset K) => enum_mset_def A). End EnumMsetSig. Module EnumMset : EnumMsetSig. Definition f K (A : multiset K) := enum_mset_def A. Definition E := (erefl f). End EnumMset. Notation enum_mset := EnumMset.f. Coercion enum_mset : multiset >-> seq. Canonical enum_mset_unlock := Unlockable EnumMset.E. Canonical multiset_predType (K : choiceType) := Eval hnf in mkPredType (fun (A : multiset K) a => a \in enum_mset A). Canonical mset_finpredType (T: choiceType) := mkFinPredType (multiset T) (fun A => undup (enum_mset A)) (fun _ => undup_uniq _) (fun _ _ => mem_undup _ _). Section MultisetOps. Context {K : choiceType}. Implicit Types (a b c : K) (A B C D : {mset K}) (s : seq K). Definition mset0 : {mset K} := [fsfun]. Fact msetn_key : unit. Proof. exact: tt. Qed. Definition msetn n a := [mset[msetn_key] x in [fset a] => n]. Fact seq_mset_key : unit. Proof. exact: tt. Qed. Definition seq_mset (s : seq K) := [mset[seq_mset_key] x in [fset x in s] => count (pred1 x) s]. Fact msetU_key : unit. Proof. exact: tt. Qed. Definition msetU A B := [mset[msetU_key] x in finsupp A `\|` finsupp B => maxn (A x) (B x)]. Fact msetI_key : unit. Proof. exact: tt. Qed. Definition msetI A B := [mset[msetI_key] x in finsupp A `\|` finsupp B => minn (A x) (B x)]. Fact msetD_key : unit. Proof. exact: tt. Qed. Definition msetD A B := [mset[msetD_key] x in finsupp A `\|` finsupp B => A x + B x]. Fact msetB_key : unit. Proof. exact: tt. Qed. Definition msetB A B := [mset[msetB_key] x in finsupp A `\|` finsupp B => A x - B x]. Fact msetM_key : unit. Proof. exact: tt. Qed. Definition msetM A B := [mset[msetM_key] x in finsupp A `` finsupp B => A x.1 B x.2]. Definition msubset A B := [forall x : finsupp A, A (val x) <= B (val x)]. Definition mproper A B := msubset A B && ~~ msubset B A. Definition mdisjoint A B := (msetI A B == mset0). End MultisetOps. Notation "[ 'mset' a ]" := (msetn 1 a) (at level 0, a at level 99, format "[ 'mset' a ]") : mset_scope. Notation "[ 'mset' a : T ]" := [mset (a : T)] (at level 0, a at level 99, format "[ 'mset' a : T ]") : mset_scope. Notation "A `\|` B" := (msetU A B) : mset_scope. Notation "A `+` B" := (msetD A B) : mset_scope. Notation "A `\` B" := (msetB A B) : mset_scope. Notation "A `\ a" := (A `\` [mset a]) : mset_scope. Notation "a \|` A" := ([mset (a)] `\|` A) : mset_scope. Notation "a +` A" := ([mset (a)] `+` A) : mset_scope. Notation "A `` B" := (msetM A B) : mset_scope. Notation "A `<=` B" := (msubset A B) (at level 70, no associativity) : mset_scope. Notation "A `<` B" := (mproper A B) (at level 70, no associativity) : mset_scope. ( This is left-associative due to historical limitations of the .. Notation. ) Notation "[ 'mset' a1 ; a2 ; .. ; an ]" := (msetD .. (a1 +` (msetn 1 a2)) .. (msetn 1 an)) (at level 0, a1 at level 99, format "[ 'mset' a1 ; a2 ; .. ; an ]") : mset_scope. Notation "A `&` B" := (msetI A B) : mset_scope. Section MSupp. Context {K : choiceType}. Implicit Types (a b c : K) (A B C D : {mset K}) (s : seq K). Lemma enum_msetE a A : (a \in A) = (a \in flatten [seq nseq (A x) x \| x <- finsupp A]). Proof. by transitivity (a \in enum_mset A); rewrite // unlock. Qed. Lemma msuppE a A : (a \in finsupp A) = (a \in A). Proof. ( Goal: @eq (list (Choice.sort K)) (@EnumMset.f K (@msetn K (S n) a)) (@nseq (Choice.sort K) (S n) a) ) rewrite enum_msetE. ( Goal: Bool.reflect (@eq (@multiset_of K (Phant (Choice.sort K))) (@seq_mset K s) (@seq_mset K s')) (@perm_eq (Choice.eqType K) s s') ) apply/idP/flattenP => [aA\|/=[_ /mapP[x xA -> /nseqP[->//]]]]. ( Goal: @eq bool (@msubset K (@msetU K B C) A) (andb (@msubset K B A) (@msubset K C A)) ) exists (nseq (A a) a); first by apply/mapP; exists a. ( Goal: is_true (implb (P (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) (@eq_op nat_eqType (E (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) O)) ) by apply/nseqP; split=> //; rewrite lt0n -mem_finsupp. Qed. End MSupp. Section MSetTheory. Context {K : choiceType}. Implicit Types (a b c : K) (A B C D : {mset K}) (s : seq K). Lemma msetP {A B} : A =1 B <-> A = B. Proof. exact: fsfunP. Qed. Lemma mset_neq0 a A : (A a != 0) = (a \in A). Proof. by rewrite -msuppE mem_finsupp. Qed. Lemma in_mset a A : (a \in A) = (A a > 0). Proof. by rewrite -mset_neq0 lt0n. Qed. Lemma mset_eq0 a A : (A a == 0) = (a \notin A). Proof. by rewrite -mset_neq0 negbK. Qed. Lemma mset_eq0P {a A} : reflect (A a = 0) (a \notin A). Proof. by rewrite -mset_eq0; apply: eqP. Qed. Lemma mset_gt0 a A : (A a > 0) = (a \in A). Proof. by rewrite -in_mset. Qed. Lemma mset_eqP {A B} : reflect (A =1 B) (A == B). Proof. exact: (equivP eqP (iff_sym msetP)). Qed. Lemma mset0E a : mset0 a = 0. Proof. by rewrite /mset0 fsfunE. Qed. Lemma msetnE n a b : (msetn n a) b = if b == a then n else 0. Proof. by rewrite fsfunE inE. Qed. Lemma msetnxx n a : (msetn n a) a = n. Proof. by rewrite msetnE eqxx. Qed. Lemma msetE2 A B a : ((A `+` B) a = A a + B a) ((A `\|` B) a = maxn (A a) (B a)) * ((A `&` B) a = minn (A a) (B a)) * ((A `\` B) a = (A a) - (B a)). Proof. (* Goal: @eq (Equality.sort nat_eqType) (@fun_of_fsfun (prod_choiceType K K) nat_eqType (fun _ : Choice.sort (prod_choiceType K K) => O) (@msetM K A B) u) (muln (@fun_of_fsfun K nat_eqType (fun _ : Choice.sort K => O) A (@fst (Choice.sort K) (Choice.sort K) u)) (@fun_of_fsfun K nat_eqType (fun _ : Choice.sort K => O) B (@snd (Choice.sort K) (Choice.sort K) u))) ) rewrite !fsfunE !inE !msuppE -!mset_neq0; case: ifPn => //. ( Goal: is_true (implb (P (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) (@eq_op nat_eqType (E (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) O)) ) by rewrite negb_or !negbK => /andP [/eqP-> /eqP->]. Qed. Lemma count_mem_mset a A : count_mem a A = A a. Proof. ( Goal: @eq (list (Choice.sort K)) (@EnumMset.f K (@msetn K (S n) a)) (@nseq (Choice.sort K) (S n) a) ) rewrite unlock count_flatten sumn_map big_map. ( Goal: @eq (list (Choice.sort K)) (@EnumMset.f K (@msetn K (S n) a)) (@nseq (Choice.sort K) (S n) a) ) rewrite (eq_bigr _ (fun _ _ => esym (sum1_count _ _))) /=. ( Goal: @eq (list (Choice.sort K)) (@EnumMset.f K (@msetn K (S n) a)) (@nseq (Choice.sort K) (S n) a) ) rewrite (eq_bigr _ (fun _ _ => big_nseq_cond _ _ _ _ _ _)) /= -big_mkcond /=. ( Goal: @eq nat (@BigOp.bigop nat (Choice.sort K) O (@enum_fset K (@FinSupp.fs K nat_eqType (fun _ : Choice.sort K => O) A)) (fun i : Choice.sort K => @BigBody nat (Choice.sort K) i addn (@eq_op (Choice.eqType K) i a) (@iter nat (@fun_of_fsfun K nat_eqType (fun _ : Choice.sort K => O) A i) (addn (S O)) O))) (@fun_of_fsfun K nat_eqType (fun _ : Choice.sort K => O) A a) ) have [aNA\|aA] := finsuppP. ( Goal: is_true (implb (P (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) (@eq_op nat_eqType (E (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) O)) ) by rewrite big1_fset // => i iA /eqP eq_ia; rewrite -eq_ia iA in aNA. ( Goal: @eq (list (Choice.sort K)) (@EnumMset.f K (@msetn K (S n) a)) (@nseq (Choice.sort K) (S n) a) ) rewrite big_fset_condE/= (big_fsetD1 a) ?inE ?eqxx ?andbT //= iter_addn mul1n. ( Goal: @eq (list (Choice.sort K)) (@EnumMset.f K (@msetn K (S n) a)) (@nseq (Choice.sort K) (S n) a) ) rewrite (_ : (_ `\ _)%fset = fset0) ?big_seq_fset0 ?addn0//. ( Goal: is_true (implb (P (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) (@eq_op nat_eqType (E (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) O)) ) by apply/fsetP=> i; rewrite !inE; case: (i == a); rewrite ?(andbF, andbT). Qed. Lemma perm_undup_mset A : perm_eq (undup A) (finsupp A). Proof. ( Goal: is_true (@perm_eq (Choice.eqType K) (@undup (Choice.eqType K) (@EnumMset.f K A)) (@enum_fset K (@FinSupp.fs K nat_eqType (fun _ : Choice.sort K => O) A))) ) apply: uniq_perm_eq; rewrite ?undup_uniq // => a. ( Goal: is_true (implb (P (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) (@eq_op nat_eqType (E (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) O)) ) by rewrite mem_undup msuppE. Qed. Section big_com. Variables (R : Type) (idx : R) (op : Monoid.com_law idx). Implicit Types (X : {mset K}) (P : pred K) (F : K -> R). Lemma big_mset X P F : \big[op/idx]_(i <- X \| P i) F i = \big[op/idx]_(i <- finsupp X \| P i) iterop (X i) op (F i) idx. Proof. ( Goal: @eq (list (Choice.sort K)) (@EnumMset.f K (@msetn K (S n) a)) (@nseq (Choice.sort K) (S n) a) ) rewrite [in RHS](eq_big_perm (undup X)) 1?perm_eq_sym ?perm_undup_mset//. ( Goal: @eq bool (@eq_op nat_eqType (@BigOp.bigop nat (Equality.sort I) O r (fun i : Equality.sort I => @BigBody nat (Equality.sort I) i addn (P i) (E i))) O) (@all (Equality.sort I) (@pred_of_simpl (Equality.sort I) (@SimplPred (Equality.sort I) (fun i : Equality.sort I => implb (P i) (@eq_op nat_eqType (E i) O)))) r) ) rewrite -[in LHS]big_undup_iterop_count; apply: eq_bigr => i _. ( Goal: is_true (implb (P (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) (@eq_op nat_eqType (E (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) O)) ) by rewrite count_mem_mset. Qed. End big_com. Lemma sum_mset (X : {mset K}) (P : pred K) (F : K -> nat) : \sum_(i <- X \| P i) F i = \sum_(i <- finsupp X \| P i) X i F i. Proof. (* Goal: @eq bool (@eq_op nat_eqType (@BigOp.bigop nat (Equality.sort I) O r (fun i : Equality.sort I => @BigBody nat (Equality.sort I) i addn (P i) (E i))) O) (@all (Equality.sort I) (@pred_of_simpl (Equality.sort I) (@SimplPred (Equality.sort I) (fun i : Equality.sort I => implb (P i) (@eq_op nat_eqType (E i) O)))) r) ) rewrite big_mset; apply: eq_bigr => i _ //. ( Goal: is_true (implb (P (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) (@eq_op nat_eqType (E (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) O)) ) by rewrite Monoid.iteropE iter_addn addn0 mulnC. Qed. Lemma prod_mset (X : {mset K}) (P : pred K) (F : K -> nat) : \prod_(i <- X \| P i) F i = \prod_(i <- finsupp X \| P i) F i ^ X i. Proof. by rewrite big_mset. Qed. Lemma mset_seqE s a : (seq_mset s) a = count_mem a s. Proof. by rewrite fsfunE inE/=; case: ifPn => // /count_memPn ->. Qed. Lemma perm_eq_seq_mset s : perm_eq (seq_mset s) s. Proof. by apply/allP => a _ /=; rewrite count_mem_mset mset_seqE. Qed. Lemma seq_mset_id A : seq_mset A = A. Proof. by apply/msetP=> a; rewrite mset_seqE count_mem_mset. Qed. Lemma eq_seq_msetP s s' : reflect (seq_mset s = seq_mset s') (perm_eq s s'). Proof. ( Goal: Bool.reflect (@eq (@multiset_of K (Phant (Choice.sort K))) (@seq_mset K s) (@seq_mset K s')) (@perm_eq (Choice.eqType K) s s') ) apply: (iffP idP) => [/perm_eqP perm_ss'\|eq_ss']. ( Goal: is_true (implb (P (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) (@eq_op nat_eqType (E (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) O)) ) by apply/msetP => a; rewrite !mset_seqE perm_ss'. ( Goal: is_true (implb (P (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) (@eq_op nat_eqType (E (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) O)) ) by apply/allP => a _ /=; rewrite -!mset_seqE eq_ss'. Qed. Lemma msetME A B (u : K K) : (A `` B) u = A u.1 B u.2. Proof. (* Goal: @eq (Equality.sort nat_eqType) (@fun_of_fsfun (prod_choiceType K K) nat_eqType (fun _ : Choice.sort (prod_choiceType K K) => O) (@msetM K A B) u) (muln (@fun_of_fsfun K nat_eqType (fun _ : Choice.sort K => O) A (@fst (Choice.sort K) (Choice.sort K) u)) (@fun_of_fsfun K nat_eqType (fun _ : Choice.sort K => O) B (@snd (Choice.sort K) (Choice.sort K) u))) ) rewrite !fsfunE inE; case: ifPn => //=. ( Goal: is_true (implb (P (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) (@eq_op nat_eqType (E (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) O)) ) by rewrite negb_and !memNfinsupp => /orP [] /eqP->; rewrite ?muln0. Qed. Lemma mset1DE a A b : (a +` A) b = (b == a) + A b. Proof. by rewrite msetE2 msetnE; case: (b == a). Qed. Lemma mset1UE a A b : (a \|` A) b = maxn (b == a) (A b). Proof. by rewrite msetE2 msetnE; case: (b == a). Qed. Lemma msetB1E a A b : (A `\ a) b = (A b) - (b == a). Proof. by rewrite msetE2 msetnE; case: (b == a). Qed. Let msetE := (mset0E, msetE2, msetnE, msetnxx, mset1DE, mset1UE, msetB1E, mset_seqE, msetME). Lemma in_mset0 a : a \in mset0 = false. Proof. by rewrite in_mset !msetE. Qed. Lemma in_msetn n a' a : a \in msetn n a' = (n > 0) && (a == a'). Proof. by rewrite in_mset msetE; case: (a == a'); rewrite ?andbT ?andbF. Qed. Lemma in_mset1 a' a : a \in [mset a'] = (a == a'). Proof. by rewrite in_msetn. Qed. Lemma in_msetD A B a : (a \in A `+` B) = (a \in A) \|\| (a \in B). Proof. by rewrite !in_mset !msetE addn_gt0. Qed. Lemma in_msetU A B a : (a \in A `\|` B) = (a \in A) \|\| (a \in B). Proof. by rewrite !in_mset !msetE leq_max. Qed. Lemma in_msetDU A B a : (a \in A `+` B) = (a \in A `\|` B). Proof. by rewrite in_msetU in_msetD. Qed. Lemma in_msetI A B a : (a \in A `&` B) = (a \in A) && (a \in B). Proof. by rewrite !in_mset msetE leq_min. Qed. Lemma in_msetB A B a : (a \in A `\` B) = (B a < A a). Proof. by rewrite -mset_neq0 msetE subn_eq0 ltnNge. Qed. Lemma in_mset1U a' A a : (a \in a' \|` A) = (a == a') \|\| (a \in A). Proof. by rewrite in_msetU in_mset msetE; case: (_ == _). Qed. Lemma in_mset1D a' A a : (a \in a' +` A) = (a == a') \|\| (a \in A). Proof. by rewrite in_msetDU in_mset1U. Qed. Lemma in_msetB1 A b a : (a \in A `\ b) = ((a == b) ==> (A a > 1)) && (a \in A). Proof. ( Goal: is_true (implb (P (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) (@eq_op nat_eqType (E (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) O)) ) by rewrite in_msetB msetE in_mset; case: (_ == _); rewrite -?geq_max. Qed. Lemma in_msetM A B (u : K K) : (u \in A `` B) = (u.1 \in A) && (u.2 \in B). Proof. by rewrite -!msuppE !mem_finsupp msetE muln_eq0 negb_or. Qed. Definition in_msetE := (in_mset0, in_msetn, in_msetB1, in_msetU, in_msetI, in_msetD, in_msetM). Let inE := (inE, in_msetE, (@msuppE K)). Lemma enum_mset0 : mset0 = [::] :> seq K. Proof. by rewrite unlock finsupp0. Qed. Lemma msetn0 (a : K) : msetn 0 a = mset0. Proof. by apply/msetP=> i; rewrite !msetE if_same. Qed. Lemma finsupp_msetn n a : finsupp (msetn n a) = if n > 0 then [fset a] else fset0. Proof. by apply/fsetP => i; rewrite !inE; case: ifP => //=; rewrite inE. Qed. Lemma enum_msetn n a : msetn n a = nseq n a :> seq K. Proof. ( Goal: @eq bool (@msubset K (@msetU K B C) A) (andb (@msubset K B A) (@msubset K C A)) ) case: n => [\|n]; first by rewrite msetn0 /= enum_mset0. ( Goal: @eq (list (Choice.sort K)) (@EnumMset.f K (@msetn K (S n) a)) (@nseq (Choice.sort K) (S n) a) ) rewrite unlock finsupp_msetn /= enum_fsetE /= enum_fset1 /= cats0. ( Goal: is_true (implb (P (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) (@eq_op nat_eqType (E (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) O)) ) by rewrite msetE eqxx. Qed. Section big. Variables (R : Type) (idx : R) (op : Monoid.law idx). Implicit Types (X : {mset K}) (P : pred K) (F : K -> R). Lemma big_mset0 P F : \big[op/idx]_(i <- mset0 \| P i) F i = idx. Proof. by rewrite enum_mset0 big_nil. Qed. Lemma big_msetn n a P F : \big[op/idx]_(i <- msetn n a \| P i) F i = if P a then iterop n op (F a) idx else idx. Proof. by rewrite enum_msetn big_nseq_cond Monoid.iteropE. Qed. End big. Lemma msetDC (A B : {mset K}) : A `+` B = B `+` A. Proof. by apply/msetP=> a; rewrite !msetE addnC. Qed. Lemma msetIC (A B : {mset K}) : A `&` B = B `&` A. Proof. by apply/msetP=> a; rewrite !msetE minnC. Qed. Lemma msetUC (A B : {mset K}) : A `\|` B = B `\|` A. Proof. by apply/msetP => a; rewrite !msetE maxnC. Qed. ( intersection ) Lemma mset0I A : mset0 `&` A = mset0. Proof. by apply/msetP => x; rewrite !msetE min0n. Qed. Lemma msetI0 A : A `&` mset0 = mset0. Proof. by rewrite msetIC mset0I. Qed. Lemma msetIA A B C : A `&` (B `&` C) = A `&` B `&` C. Proof. by apply/msetP=> x; rewrite !msetE minnA. Qed. Lemma msetICA A B C : A `&` (B `&` C) = B `&` (A `&` C). Proof. by rewrite !msetIA (msetIC A). Qed. Lemma msetIAC A B C : A `&` B `&` C = A `&` C `&` B. Proof. by rewrite -!msetIA (msetIC B). Qed. Lemma msetIACA A B C D : (A `&` B) `&` (C `&` D) = (A `&` C) `&` (B `&` D). Proof. by rewrite -!msetIA (msetICA B). Qed. Lemma msetIid A : A `&` A = A. Proof. by apply/msetP=> x; rewrite !msetE minnn. Qed. Lemma msetIIl A B C : A `&` B `&` C = (A `&` C) `&` (B `&` C). Proof. by rewrite msetIA !(msetIAC _ C) -(msetIA _ C) msetIid. Qed. Lemma msetIIr A B C : A `&` (B `&` C) = (A `&` B) `&` (A `&` C). Proof. by rewrite !(msetIC A) msetIIl. Qed. ( union ) Lemma mset0U A : mset0 `\|` A = A. Proof. by apply/msetP => x; rewrite !msetE max0n. Qed. Lemma msetU0 A : A `\|` mset0 = A. Proof. by rewrite msetUC mset0U. Qed. Lemma msetUA A B C : A `\|` (B `\|` C) = A `\|` B `\|` C. Proof. by apply/msetP=> x; rewrite !msetE maxnA. Qed. Lemma msetUCA A B C : A `\|` (B `\|` C) = B `\|` (A `\|` C). Proof. by rewrite !msetUA (msetUC B). Qed. Lemma msetUAC A B C : A `\|` B `\|` C = A `\|` C `\|` B. Proof. by rewrite -!msetUA (msetUC B). Qed. Lemma msetUACA A B C D : (A `\|` B) `\|` (C `\|` D) = (A `\|` C) `\|` (B `\|` D). Proof. by rewrite -!msetUA (msetUCA B). Qed. Lemma msetUid A : A `\|` A = A. Proof. by apply/msetP=> x; rewrite !msetE maxnn. Qed. Lemma msetUUl A B C : A `\|` B `\|` C = (A `\|` C) `\|` (B `\|` C). Proof. by rewrite msetUA !(msetUAC _ C) -(msetUA _ C) msetUid. Qed. Lemma msetUUr A B C : A `\|` (B `\|` C) = (A `\|` B) `\|` (A `\|` C). Proof. by rewrite !(msetUC A) msetUUl. Qed. ( adjunction ) Lemma mset0D A : mset0 `+` A = A. Proof. by apply/msetP => x; rewrite !msetE add0n. Qed. Lemma msetD0 A : A `+` mset0 = A. Proof. by rewrite msetDC mset0D. Qed. Lemma msetDA A B C : A `+` (B `+` C) = A `+` B `+` C. Proof. by apply/msetP=> x; rewrite !msetE addnA. Qed. Lemma msetDCA A B C : A `+` (B `+` C) = B `+` (A `+` C). Proof. by rewrite !msetDA (msetDC B). Qed. Lemma msetDAC A B C : A `+` B `+` C = A `+` C `+` B. Proof. by rewrite -!msetDA (msetDC B). Qed. Lemma msetDACA A B C D : (A `+` B) `+` (C `+` D) = (A `+` C) `+` (B `+` D). Proof. by rewrite -!msetDA (msetDCA B). Qed. ( adjunction, union and difference with one element ) Lemma msetU1l x A B : x \in A -> x \in A `\|` B. Proof. by move=> Ax /=; rewrite inE Ax. Qed. Lemma msetU1r A b : b \in A `\|` [mset b]. Proof. by rewrite !inE eqxx orbT. Qed. Lemma msetB1P x A b : reflect ((x = b -> A x > 1) /\ x \in A) (x \in A `\ b). Proof. ( Goal: @eq (list (Choice.sort K)) (@EnumMset.f K (@msetn K (S n) a)) (@nseq (Choice.sort K) (S n) a) ) rewrite !inE. apply: (iffP andP); first by move=> [/implyP Ax ->]; split => // /eqP. ( Goal: is_true (implb (P (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) (@eq_op nat_eqType (E (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) O)) ) by move=> [Ax ->]; split => //; apply/implyP => /eqP. Qed. Lemma msetB11 b A : (b \in A `\ b) = (A b > 1). Proof. by rewrite inE eqxx /= in_mset -geq_max. Qed. Lemma msetB1K a A : a \in A -> a +` (A `\ a) = A. Proof. ( Goal: forall _ : is_true (negb (@in_mem (Choice.sort K) a (@mem (Equality.sort (Choice.eqType K)) (multiset_predType K) A))), @eq (@multiset_of K (Phant (Choice.sort K))) (@msetB K A (@msetn K (S O) a)) A ) move=> aA; apply/msetP=> x; rewrite !msetE subnKC //=. ( Goal: is_true (implb (P (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) (@eq_op nat_eqType (E (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) O)) ) by have [->\|//] := altP eqP; rewrite mset_gt0. Qed. Lemma msetD1K a B : (a +` B) `\ a = B. Proof. by apply/msetP => x; rewrite !msetE addKn. Qed. Lemma msetU1K a B : a \notin B -> (a \|` B) `\ a = B. Proof. ( Goal: forall _ : is_true (negb (@in_mem (Choice.sort K) a (@mem (Equality.sort (Choice.eqType K)) (multiset_predType K) A))), @eq (@multiset_of K (Phant (Choice.sort K))) (@msetB K A (@msetn K (S O) a)) A ) move=> aB; apply/msetP=> x; rewrite !msetE. ( Goal: @eq bool (@msubset K (@msetU K B C) A) (andb (@msubset K B A) (@msubset K C A)) ) have [->\|] := altP eqP; first by rewrite (mset_eq0P _). ( Goal: is_true (implb (P (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) (@eq_op nat_eqType (E (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) O)) ) by rewrite max0n subn0. Qed. Lemma mset1U1 x B : x \in x \|` B. Proof. by rewrite !inE eqxx. Qed. Lemma mset1D1 x B : x \in x +` B. Proof. by rewrite !inE eqxx. Qed. Lemma mset1Ur x a B : x \in B -> x \in a \|` B. Proof. by move=> Bx; rewrite !inE predU1r. Qed. Lemma mset1Dr x a B : x \in B -> x \in a +` B. Proof. by move=> Bx; rewrite !inE predU1r. Qed. Lemma mset2P x a b : reflect (x = a \/ x = b) (x \in [mset a; b]). Proof. by rewrite !inE; apply: (iffP orP) => [] [] /eqP; intuition. Qed. Lemma in_mset2 x a b : (x \in [mset a; b]) = (x == a) \|\| (x == b). Proof. by rewrite !inE. Qed. Lemma mset21 a b : a \in [mset a; b]. Proof. by rewrite mset1D1. Qed. Lemma mset22 a b : b \in [mset a; b]. Proof. by rewrite in_mset2 eqxx orbT. Qed. Lemma msetUP x A B : reflect (x \in A \/ x \in B) (x \in A `\|` B). Proof. by rewrite !inE; exact: orP. Qed. Lemma msetDP x A B : reflect (x \in A \/ x \in B) (x \in A `+` B). Proof. by rewrite !inE; exact: orP. Qed. Lemma msetULVR x A B : x \in A `\|` B -> (x \in A) + (x \in B). Proof. by rewrite inE; case: (x \in A); [left\|right]. Qed. Lemma msetDLVR x A B : x \in A `+` B -> (x \in A) + (x \in B). Proof. by rewrite inE; case: (x \in A); [left\|right]. Qed. ( distribute /cancel ) Lemma msetIUr A B C : A `&` (B `\|` C) = (A `&` B) `\|` (A `&` C). Proof. by apply/msetP=> x; rewrite !msetE minn_maxr. Qed. Lemma msetIUl A B C : (A `\|` B) `&` C = (A `&` C) `\|` (B `&` C). Proof. by apply/msetP=> x; rewrite !msetE minn_maxl. Qed. Lemma msetUIr A B C : A `\|` (B `&` C) = (A `\|` B) `&` (A `\|` C). Proof. by apply/msetP=> x; rewrite !msetE maxn_minr. Qed. Lemma msetUIl A B C : (A `&` B) `\|` C = (A `\|` C) `&` (B `\|` C). Proof. by apply/msetP=> x; rewrite !msetE maxn_minl. Qed. Lemma msetUKC A B : (A `\|` B) `&` A = A. Proof. by apply/msetP=> x; rewrite !msetE maxnK. Qed. Lemma msetUK A B : (B `\|` A) `&` A = A. Proof. by rewrite msetUC msetUKC. Qed. Lemma msetKUC A B : A `&` (B `\|` A) = A. Proof. by rewrite msetIC msetUK. Qed. Lemma msetKU A B : A `&` (A `\|` B) = A. Proof. by rewrite msetIC msetUKC. Qed. Lemma msetIKC A B : (A `&` B) `\|` A = A. Proof. by apply/msetP=> x; rewrite !msetE minnK. Qed. Lemma msetIK A B : (B `&` A) `\|` A = A. Proof. by rewrite msetIC msetIKC. Qed. Lemma msetKIC A B : A `\|` (B `&` A) = A. Proof. by rewrite msetUC msetIK. Qed. Lemma msetKI A B : A `\|` (A `&` B) = A. Proof. by rewrite msetIC msetKIC. Qed. Lemma msetUKid A B : B `\|` A `\|` A = B `\|` A. Proof. by rewrite -msetUA msetUid. Qed. Lemma msetUKidC A B : A `\|` B `\|` A = A `\|` B. Proof. by rewrite msetUAC msetUid. Qed. Lemma msetKUid A B : A `\|` (A `\|` B) = A `\|` B. Proof. by rewrite msetUA msetUid. Qed. Lemma msetKUidC A B : A `\|` (B `\|` A) = B `\|` A. Proof. by rewrite msetUCA msetUid. Qed. Lemma msetIKid A B : B `&` A `&` A = B `&` A. Proof. by rewrite -msetIA msetIid. Qed. Lemma msetIKidC A B : A `&` B `&` A = A `&` B. Proof. by rewrite msetIAC msetIid. Qed. Lemma msetKIid A B : A `&` (A `&` B) = A `&` B. Proof. by rewrite msetIA msetIid. Qed. Lemma msetKIidC A B : A `&` (B `&` A) = B `&` A. Proof. by rewrite msetICA msetIid. Qed. Lemma msetDIr A B C : A `+` (B `&` C) = (A `+` B) `&` (A `+` C). Proof. by apply/msetP=> x; rewrite !msetE addn_minr. Qed. Lemma msetDIl A B C : (A `&` B) `+` C = (A `+` C) `&` (B `+` C). Proof. by apply/msetP=> x; rewrite !msetE addn_minl. Qed. Lemma msetDKIC A B : (A `+` B) `&` A = A. Proof. by apply/msetP=> x; rewrite !msetE (minn_idPr _) // leq_addr. Qed. Lemma msetDKI A B : (B `+` A) `&` A = A. Proof. by rewrite msetDC msetDKIC. Qed. Lemma msetKDIC A B : A `&` (B `+` A) = A. Proof. by rewrite msetIC msetDKI. Qed. Lemma msetKDI A B : A `&` (A `+` B) = A. Proof. by rewrite msetDC msetKDIC. Qed. ( adjunction / subtraction ) Lemma msetDKB A : cancel (msetD A) (msetB^~ A). Proof. by move=> B; apply/msetP => a; rewrite !msetE addKn. Qed. Lemma msetDKBC A : cancel (msetD^~ A) (msetB^~ A). Proof. by move=> B; rewrite msetDC msetDKB. Qed. Lemma msetBSKl A B a : ((a +` A) `\` B) `\ a = A `\` B. Proof. ( Goal: @eq (@multiset_of K (Phant (Choice.sort K))) (@msetB K (@msetB K (@msetD K (@msetn K (S O) a) A) B) (@msetn K (S O) a)) (@msetB K A B) ) apply/msetP=> b; rewrite !msetE; case: ifPn; rewrite ?add0n ?subn0 //. ( Goal: is_true (implb (P (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) (@eq_op nat_eqType (E (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) O)) ) by rewrite add1n subn1 subSKn. Qed. Lemma msetBDl C A B : (C `+` A) `\` (C `+` B) = A `\` B. Proof. by apply/msetP=> a; rewrite !msetE subnDl. Qed. Lemma msetBDr C A B : (A `+` C) `\` (B `+` C) = A `\` B. Proof. by apply/msetP=> a; rewrite !msetE subnDr. Qed. Lemma msetBDA A B C : B `\` (A `+` C) = B `\` A `\` C. Proof. by apply/msetP=> a; rewrite !msetE subnDA. Qed. Lemma msetUE A B C : msetU A B = A `+` (B `\` A). Proof. by apply/msetP=> a; rewrite !msetE maxnE. Qed. ( subset ) Lemma msubsetP {A B} : reflect (forall x, A x <= B x) (A `<=` B). Proof. ( Goal: Bool.reflect (forall x : Choice.sort K, is_true (leq (@fun_of_fsfun K nat_eqType (fun _ : Choice.sort K => O) A x) (@fun_of_fsfun K nat_eqType (fun _ : Choice.sort K => O) B x))) (@msubset K A B) ) apply: (iffP forallP)=> // ? x; case: (in_fsetP (finsupp A) x) => //. ( Goal: is_true (implb (P (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) (@eq_op nat_eqType (E (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) O)) ) by rewrite msuppE => /mset_eq0P->. Qed. Lemma msubset_subset {A B} : A `<=` B -> {subset A <= B}. Proof. ( Goal: is_true (implb (P (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) (@eq_op nat_eqType (E (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) O)) ) by move=> /msubsetP AB x; rewrite !in_mset => ?; exact: (leq_trans _ (AB _)). Qed. Lemma msetB_eq0 (A B : {mset K}) : (A `\` B == mset0) = (A `<=` B). Proof. apply/mset_eqP/msubsetP => AB a; by have := AB a; rewrite !msetE -subn_eq0 => /eqP. Qed. Lemma msubset_refl A : A `<=` A. Proof. exact/msubsetP. Qed. Hint Resolve msubset_refl. Lemma msubset_trans : transitive (@msubset K). Proof. ( Goal: @transitive (@multiset_of K (Phant (Choice.sort K))) (@msubset K) ) move=> y x z /msubsetP xy /msubsetP yz ; apply/msubsetP => a. ( Goal: is_true (implb (P (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) (@eq_op nat_eqType (E (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) O)) ) by apply: (leq_trans (xy _)). Qed. Arguments msubset_trans {C A B} _ _ : rename. Lemma msetUS C A B : A `<=` B -> C `\|` A `<=` C `\|` B. Proof. ( Goal: forall _ : is_true (negb (@in_mem (Choice.sort K) a (@mem (Equality.sort (Choice.eqType K)) (multiset_predType K) A))), @eq (@multiset_of K (Phant (Choice.sort K))) (@msetB K A (@msetn K (S O) a)) A ) move=> sAB; apply/msubsetP=> x; rewrite !msetE. ( Goal: is_true (implb (P (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) (@eq_op nat_eqType (E (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) O)) ) by rewrite geq_max !leq_max leqnn (msubsetP sAB) orbT. Qed. Lemma msetDS C A B : A `<=` B -> C `+` A `<=` C `+` B. Proof. ( Goal: is_true (implb (P (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) (@eq_op nat_eqType (E (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) O)) ) by move=> /msubsetP sAB; apply/msubsetP=> x; rewrite !msetE leq_add2l. Qed. Lemma msetSU C A B : A `<=` B -> A `\|` C `<=` B `\|` C. Proof. by move=> sAB; rewrite -!(msetUC C) msetUS. Qed. Lemma msetSD C A B : A `<=` B -> A `+` C `<=` B `+` C. Proof. by move=> sAB; rewrite -!(msetDC C) msetDS. Qed. Lemma msetUSS A B C D : A `<=` C -> B `<=` D -> A `\|` B `<=` C `\|` D. Proof. by move=> /(msetSU B) /msubset_trans sAC /(msetUS C)/sAC. Qed. Lemma msetDSS A B C D : A `<=` C -> B `<=` D -> A `+` B `<=` C `+` D. Proof. by move=> /(msetSD B) /msubset_trans sAC /(msetDS C)/sAC. Qed. Lemma msetIidPl {A B} : reflect (A `&` B = A) (A `<=` B). Proof. ( Goal: @eq bool (@msubset K (@msetU K B C) A) (andb (@msubset K B A) (@msubset K C A)) ) apply: (iffP msubsetP) => [?\|<- a]; last by rewrite !msetE geq_min leqnn orbT. ( Goal: is_true (implb (P (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) (@eq_op nat_eqType (E (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) O)) ) by apply/msetP => a; rewrite !msetE (minn_idPl _). Qed. Lemma msetIidPr {A B} : reflect (A `&` B = B) (B `<=` A). Proof. by rewrite msetIC; apply: msetIidPl. Qed. Lemma msubsetIidl A B : (A `<=` A `&` B) = (A `<=` B). Proof. ( Goal: @eq bool (@msubset K A (@msetI K A B)) (@msubset K A B) ) apply/msubsetP/msubsetP=> sAB a; have := sAB a; rewrite !msetE. ( Goal: is_true (implb (P (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) (@eq_op nat_eqType (E (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) O)) ) by rewrite leq_min leqnn. ( Goal: is_true (implb (P (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) (@eq_op nat_eqType (E (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) O)) ) by move/minn_idPl->. Qed. Lemma msubsetIidr A B : (B `<=` A `&` B) = (B `<=` A). Proof. by rewrite msetIC msubsetIidl. Qed. Lemma msetUidPr A B : reflect (A `\|` B = B) (A `<=` B). Proof. ( Goal: @eq bool (@msubset K (@msetU K B C) A) (andb (@msubset K B A) (@msubset K C A)) ) apply: (iffP msubsetP) => [AB\|<- a]; last by rewrite !msetE leq_max leqnn. ( Goal: is_true (implb (P (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) (@eq_op nat_eqType (E (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) O)) ) by apply/msetP=> a; rewrite !msetE (maxn_idPr _). Qed. Lemma msetUidPl A B : reflect (A `\|` B = A) (B `<=` A). Proof. by rewrite msetUC; apply/msetUidPr. Qed. Lemma msubsetUl A B : A `<=` A `\|` B. Proof. by apply/msubsetP=> a; rewrite !msetE leq_maxl. Qed. Hint Resolve msubsetUl. Lemma msubsetUr A B : B `<=` (A `\|` B). Proof. by rewrite msetUC. Qed. Hint Resolve msubsetUr. Lemma msubsetU1 x A : A `<=` (x \|` A). Proof. by rewrite msubsetUr. Qed. Hint Resolve msubsetU1. Lemma msubsetU A B C : (A `<=` B) \|\| (A `<=` C) -> A `<=` (B `\|` C). Proof. by move=> /orP [] /msubset_trans ->. Qed. Lemma eqEmsubset A B : (A == B) = (A `<=` B) && (B `<=` A). Proof. ( Goal: @eq bool (@msubset K (@msetU K B C) A) (andb (@msubset K B A) (@msubset K C A)) ) apply/eqP/andP => [<-\|[/msubsetP AB /msubsetP BA]]; first by split. ( Goal: is_true (implb (P (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) (@eq_op nat_eqType (E (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) O)) ) by apply/msetP=> a; apply/eqP; rewrite eqn_leq AB BA. Qed. Lemma msubEproper A B : A `<=` B = (A == B) \|\| (A `<` B). Proof. by rewrite eqEmsubset -andb_orr orbN andbT. Qed. Lemma mproper_sub A B : A `<` B -> A `<=` B. Proof. by rewrite msubEproper orbC => ->. Qed. Lemma eqVmproper A B : A `<=` B -> A = B \/ A `<` B. Proof. by rewrite msubEproper => /predU1P. Qed. Lemma mproperEneq A B : A `<` B = (A != B) && (A `<=` B). Proof. by rewrite andbC eqEmsubset negb_and andb_orr andbN. Qed. Lemma mproper_neq A B : A `<` B -> A != B. Proof. by rewrite mproperEneq; case/andP. Qed. Lemma eqEmproper A B : (A == B) = (A `<=` B) && ~~ (A `<` B). Proof. by rewrite negb_and negbK andb_orr andbN eqEmsubset. Qed. Lemma msub0set A : msubset mset0 A. Proof. by apply/msubsetP=> x; rewrite msetE. Qed. Hint Resolve msub0set. Lemma msubset0 A : (A `<=` mset0) = (A == mset0). Proof. by rewrite eqEmsubset msub0set andbT. Qed. Lemma mproper0 A : (mproper mset0 A) = (A != mset0). Proof. by rewrite /mproper msub0set msubset0. Qed. Lemma mproperE A B : (A `<` B) = (A `<=` B) && ~~ (msubset B A). Proof. by []. Qed. Lemma mproper_sub_trans B A C : A `<` B -> B `<=` C -> A `<` C. Proof. ( Goal: forall (_ : is_true (@msubset K A B)) (_ : is_true (@mproper K B C)), is_true (@mproper K A C) ) move=> /andP [AB NBA] BC; rewrite /mproper (msubset_trans AB) //=. ( Goal: is_true (implb (P (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) (@eq_op nat_eqType (E (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) O)) ) by apply: contra NBA=> /(msubset_trans _)->. Qed. Lemma msub_proper_trans B A C : A `<=` B -> B `<` C -> A `<` C. Proof. ( Goal: forall (_ : is_true (@msubset K A B)) (_ : is_true (@mproper K B C)), is_true (@mproper K A C) ) move=> AB /andP [CB NCB]; rewrite /mproper (msubset_trans AB) //=. ( Goal: is_true (implb (P (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) (@eq_op nat_eqType (E (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) O)) ) by apply: contra NCB=> /msubset_trans->. Qed. Lemma msubset_neq0 A B : A `<=` B -> A != mset0 -> B != mset0. Proof. by rewrite -!mproper0 => sAB /mproper_sub_trans->. Qed. ( msub is a morphism ) Lemma msetBDKC A B : A `<=` B -> A `+` (B `\` A) = B. Proof. by move=> /msubsetP AB; apply/msetP=> a; rewrite !msetE subnKC. Qed. Lemma msetBDK A B : A `<=` B -> B `\` A `+` A = B. Proof. by move=> /msubsetP AB; apply/msetP => a; rewrite !msetE subnK. Qed. Lemma msetBBK A B : A `<=` B -> B `\` (B `\` A) = A. Proof. by move=> /msubsetP AB; apply/msetP => a; rewrite !msetE subKn. Qed. Lemma msetBD1K A B a : A `<=` B -> A a < B a -> a +` (B `\` (a +` A)) = B `\` A. Proof. ( Goal: forall _ : is_true (negb (@in_mem (Choice.sort K) a (@mem (Equality.sort (Choice.eqType K)) (multiset_predType K) A))), @eq (@multiset_of K (Phant (Choice.sort K))) (@msetB K A (@msetn K (S O) a)) A ) move=> /msubsetP AB ABa; apply/msetP => b; rewrite !msetE. ( Goal: is_true (implb (P (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) (@eq_op nat_eqType (E (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) O)) ) by case: ifP => //= /eqP->; rewrite !add1n subnSK. Qed. Lemma subset_msetBLR A B C : (msubset (A `\` B) C) = (A `<=` B `+` C). Proof. apply/msubsetP/msubsetP => [] sABC a; by have := sABC a; rewrite !msetE ?leq_subLR. Qed. Lemma msetnP n x a : reflect (0 < n /\ x = a) (x \in msetn n a). Proof. by do [apply: (iffP idP); rewrite !inE] => [/andP[]\|[]] -> /eqP. Qed. Lemma gt0_msetnP n x a : 0 < n -> reflect (x = a) (x \in msetn n a). Proof. by move=> n_gt0; rewrite inE n_gt0 /=; exact: eqP. Qed. Lemma msetn1 n a : a \in msetn n a = (n > 0). Proof. by rewrite inE eqxx andbT. Qed. Lemma mset1P x a : reflect (x = a) (x \in [mset a]). Proof. by rewrite inE; exact: eqP. Qed. Lemma mset11 a : a \in [mset a]. Proof. by rewrite inE /=. Qed. Lemma msetn_inj n : n > 0 -> injective (@msetn K n). Proof. ( Goal: @transitive (@multiset_of K (Phant (Choice.sort K))) (@msubset K) ) move=> n_gt0 a b eqsab; apply/(gt0_msetnP _ _ n_gt0). ( Goal: is_true (implb (P (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) (@eq_op nat_eqType (E (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) O)) ) by rewrite -eqsab inE n_gt0 eqxx. Qed. Lemma mset1UP x a B : reflect (x = a \/ x \in B) (x \in a \|` B). Proof. by rewrite !inE; exact: predU1P. Qed. Lemma mset_cons a s : seq_mset (a :: s) = a +` (seq_mset s). Proof. by apply/msetP=> x; rewrite !msetE /= eq_sym. Qed. ( intersection ) Lemma msetIP x A B : reflect (x \in A /\ x \in B) (x \in A `&` B). Proof. by rewrite inE; apply: andP. Qed. Lemma msetIS C A B : A `<=` B -> C `&` A `<=` C `&` B. Proof. ( Goal: forall _ : is_true (negb (@in_mem (Choice.sort K) a (@mem (Equality.sort (Choice.eqType K)) (multiset_predType K) A))), @eq (@multiset_of K (Phant (Choice.sort K))) (@msetB K A (@msetn K (S O) a)) A ) move=> sAB; apply/msubsetP=> x; rewrite !msetE. ( Goal: is_true (implb (P (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) (@eq_op nat_eqType (E (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) O)) ) by rewrite leq_min !geq_min leqnn (msubsetP sAB) orbT. Qed. Lemma msetSI C A B : A `<=` B -> A `&` C `<=` B `&` C. Proof. by move=> sAB; rewrite -!(msetIC C) msetIS. Qed. Lemma msetISS A B C D : A `<=` C -> B `<=` D -> A `&` B `<=` C `&` D. Proof. by move=> /(msetSI B) /msubset_trans sAC /(msetIS C) /sAC. Qed. ( difference ) Lemma msetSB C A B : A `<=` B -> A `\` C `<=` B `\` C. Proof. ( Goal: is_true (implb (P (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) (@eq_op nat_eqType (E (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) O)) ) by move=> /msubsetP sAB; apply/msubsetP=> x; rewrite !msetE leq_sub2r. Qed. Lemma msetBS C A B : A `<=` B -> C `\` B `<=` C `\` A. Proof. ( Goal: is_true (implb (P (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) (@eq_op nat_eqType (E (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) O)) ) by move=> /msubsetP sAB; apply/msubsetP=> x; rewrite !msetE leq_sub2l. Qed. Lemma msetBSS A B C D : A `<=` C -> D `<=` B -> A `\` B `<=` C `\` D. Proof. by move=> /(msetSB B) /msubset_trans sAC /(msetBS C) /sAC. Qed. Lemma msetB0 A : A `\` mset0 = A. Proof. by apply/msetP=> x; rewrite !msetE subn0. Qed. Lemma mset0B A : mset0 `\` A = mset0. Proof. by apply/msetP=> x; rewrite !msetE sub0n. Qed. Lemma msetBxx A : A `\` A = mset0. Proof. by apply/msetP=> x; rewrite !msetE subnn. Qed. ( other inclusions ) Lemma msubsetIl A B : A `&` B `<=` A. Proof. by apply/msubsetP=> x; rewrite msetE geq_minl. Qed. Lemma msubsetIr A B : A `&` B `<=` B. Proof. by apply/msubsetP=> x; rewrite msetE geq_minr. Qed. Lemma msubsetDl A B : A `\` B `<=` A. Proof. by apply/msubsetP=> x; rewrite msetE leq_subLR leq_addl. Qed. Lemma msubD1set A x : A `\ x `<=` A. Proof. by rewrite msubsetDl. Qed. Hint Resolve msubsetIl msubsetIr msubsetDl msubD1set. ( cardinal lemmas for msets ) Lemma mem_mset1U a A : a \in A -> a \|` A = A. Proof. ( Goal: forall _ : is_true (@in_mem (Choice.sort K) a (@mem (Equality.sort (Choice.eqType K)) (multiset_predType K) A)), @eq (@multiset_of K (Phant (Choice.sort K))) (@msetU K (@msetn K (S O) a) A) A ) rewrite in_mset => aA; apply/msetP => x; rewrite !msetE (maxn_idPr _) //. ( Goal: is_true (implb (P (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) (@eq_op nat_eqType (E (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) O)) ) by have [->\|//] := altP eqP; rewrite (leq_trans _ aA). Qed. Lemma mem_msetD1 a A : a \notin A -> A `\ a = A. Proof. ( Goal: forall _ : is_true (negb (@in_mem (Choice.sort K) a (@mem (Equality.sort (Choice.eqType K)) (multiset_predType K) A))), @eq (@multiset_of K (Phant (Choice.sort K))) (@msetB K A (@msetn K (S O) a)) A ) move=> /mset_eq0P aA; apply/msetP => x; rewrite !msetE. ( Goal: is_true (implb (P (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) (@eq_op nat_eqType (E (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) O)) ) by have [->\|] := altP eqP; rewrite ?aA ?subn0. Qed. Lemma msetIn a A n : A `&` msetn n a = msetn (minn (A a) n) a. Proof. ( Goal: is_true (implb (P (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) (@eq_op nat_eqType (E (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) O)) ) by apply/msetP => x; rewrite !msetE; have [->\|] := altP eqP; rewrite ?minn0. Qed. Lemma msubIset A B C : (B `<=` A) \|\| (C `<=` A) -> (B `&` C `<=` A). Proof. by case/orP; apply: msubset_trans; rewrite (msubsetIl, msubsetIr). Qed. Lemma msubsetI A B C : (A `<=` B `&` C) = (A `<=` B) && (A `<=` C). Proof. ( Goal: @eq bool (@msubset K A (@msetI K B C)) (andb (@msubset K A B) (@msubset K A C)) ) rewrite !(sameP msetIidPl eqP) msetIA; have [-> //\| ] := altP (A `&` B =P A). ( Goal: is_true (implb (P (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) (@eq_op nat_eqType (E (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) O)) ) by apply: contraNF => /eqP <-; rewrite -msetIA -msetIIl msetIAC. Qed. Lemma msubsetIP A B C : reflect (A `<=` B /\ A `<=` C) (A `<=` B `&` C). Proof. by rewrite msubsetI; exact: andP. Qed. Lemma msubUset A B C : (B `\|` C `<=` A) = (B `<=` A) && (C `<=` A). Proof. ( Goal: @eq bool (@msubset K (@msetU K B C) A) (andb (@msubset K B A) (@msubset K C A)) ) apply/idP/idP => [subA\|/andP [AB CA]]; last by rewrite -[A]msetUid msetUSS. ( Goal: is_true (implb (P (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) (@eq_op nat_eqType (E (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) O)) ) by rewrite !(msubset_trans _ subA). Qed. Lemma msubUsetP A B C : reflect (A `<=` C /\ B `<=` C) (A `\|` B `<=` C). Proof. by rewrite msubUset; exact: andP. Qed. Lemma msetU_eq0 A B : (A `\|` B == mset0) = (A == mset0) && (B == mset0). Proof. by rewrite -!msubset0 msubUset. Qed. Lemma setD_eq0 A B : (A `\` B == mset0) = (A `<=` B). Proof. by rewrite -msubset0 subset_msetBLR msetD0. Qed. Lemma msub1set A a : ([mset a] `<=` A) = (a \in A). Proof. ( Goal: @eq bool (@msubset K (@msetU K B C) A) (andb (@msubset K B A) (@msubset K C A)) ) apply/msubsetP/idP; first by move/(_ a); rewrite msetnxx in_mset. ( Goal: is_true (implb (P (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) (@eq_op nat_eqType (E (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) O)) ) by move=> ainA b; rewrite msetnE; case: eqP => // ->; rewrite -in_mset. Qed. Lemma msetDBA A B C : C `<=` B -> A `+` B `\` C = (A `+` B) `\` C. Proof. ( Goal: is_true (implb (P (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) (@eq_op nat_eqType (E (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) O)) ) by move=> /msubsetP CB; apply/msetP=> a; rewrite !msetE2 addnBA. Qed. Lemma mset_0Vmem A : (A = mset0) + {x : K \| x \in A}. Proof. ( Goal: @eq bool (@msubset K (@msetU K B C) A) (andb (@msubset K B A) (@msubset K C A)) ) have [/fsetP Aisfset0 \| [a ainA]] := fset_0Vmem (finsupp A); last first. ( Goal: is_true (implb (P (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) (@eq_op nat_eqType (E (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) O)) ) by right; exists a; rewrite -msuppE. left; apply/msetP => a; rewrite mset0E; apply/mset_eq0P. ( Goal: is_true (implb (P (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) (@eq_op nat_eqType (E (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) O)) ) by rewrite -msuppE Aisfset0 inE. Qed. Definition size_mset A : size A = \sum_(a <- finsupp A) A a. Proof. by rewrite -sum1_size sum_mset; apply: eq_bigr => i; rewrite muln1. Qed. Lemma size_mset0 : size (mset0 : {mset K}) = 0. Proof. by rewrite -sum1_size big_mset0. Qed. From mathcomp Require Import tuple. Lemma sum_nat_seq_eq0 (I : eqType) r (P : pred I) (E : I -> nat) : (\sum_(i <- r \| P i) E i == 0) = all [pred i \| P i ==> (E i == 0)] r. Proof. ( Goal: @eq bool (@eq_op nat_eqType (@BigOp.bigop nat (Equality.sort I) O r (fun i : Equality.sort I => @BigBody nat (Equality.sort I) i addn (P i) (E i))) O) (@all (Equality.sort I) (@pred_of_simpl (Equality.sort I) (@SimplPred (Equality.sort I) (fun i : Equality.sort I => implb (P i) (@eq_op nat_eqType (E i) O)))) r) ) rewrite big_tnth sum_nat_eq0; apply/forallP/allP => /= HE x. ( Goal: is_true (implb (P (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) (@eq_op nat_eqType (E (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) O)) ) by move=> /seq_tnthP[i ->]; apply: HE. ( Goal: is_true (implb (P (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) (@eq_op nat_eqType (E (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) O)) ) by apply: HE; rewrite mem_tnth. Qed. Lemma size_mset_eq0 A : (size A == 0) = (A == mset0). Proof. ( Goal: @eq bool (@msubset K (@msetU K B C) A) (andb (@msubset K B A) (@msubset K C A)) ) apply/idP/eqP => [\|->]; last by rewrite size_mset0. rewrite size_mset sum_nat_seq_eq0 => /allP AP. apply/msetP => a /=; rewrite msetE. ( Goal: is_true (implb (P (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) (@eq_op nat_eqType (E (@tnth (@size (Equality.sort I) r) (Equality.sort I) (@in_tuple (Equality.sort I) r) x)) O)) *) by have /= := AP a; case: finsuppP => // _ /(_ _)/eqP->. Qed. End MSetTheory.
(************************************************************************) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Copyright (C) 2013 - 2015 ) ( Author C. Cohen ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( DRAFT - PLEASE USE WITH CAUTION ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( License CeCILL-B ) (**********************************************************************) From mathcomp Require Import ssreflect ssrbool ssrnat eqtype ssrfun seq. From mathcomp Require Import choice path finset finfun fintype bigop. From mathcomp Require Import bigenough. (**************************************************************************) ( This file provides representations for finite sets over a choiceType K, ) ( finite maps with keys in a choiceType K and the values in an arbitrary ) ( type V, and total functions from K to V with finite support. ) ( The domain (resp. support) of a finite map (resp. fintely supported ) ( function) is a finite set, and so is the codomain (resp. image) when V ) ( is a choice type. ) ( ) ( {fset K} == finite sets of elements of K ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( {fmap K -> V} == finitely supported maps from K to V. ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( {fsfun K -> T for dflt} == finitely supported functions ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( with default value dflt : K -> V outside the support ) ( ) (****** finite sets ****************************************************) ( ) ( In the remainder, A and B are of type {fset K}. ) ( ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( - {fset K} is provided with a canonical structure of predType, in order ) ( to enable the notation "a \in A" ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( - There is a coercion from {fset K} to Type in order to interpret any ) ( finset A as the subType of elements a : K such that a \in A. ) ( Because of this coercion, writing a : A makes sense. ) ( ) ( The following notations are in the %fset scope ) ( fset0 == the empty finite set ) ( [fset k] == the singleton finite set {k} ) ( A `&` B == the intersection of A and B ) ( A `\|` B == the union of A and B ) ( a \|` B == the union of singleton a and B ) ( A `\` B == the complement of B in A ) ( A `\ b == A without b ) ( A `` B == the cartesian product of A and B ) (* [disjoint A & B] := A `&` B == 0 ) ( A `<=` B == A is a subset of B ) ( A `<` B == A is a proper subset of B ) ( #\|`A\| == cardinal of A ) ( fincl AsubB a == turns a : A into an element of B ) ( using a proof AsubB of A \fsubset B ) ( fsub B A == turns A : {fset K} into a {set B} ) ( f @` A == the image set of the collective predicate A by f. ) ( f @2`(A, B) == the image set of A x B by the binary function f. ) ( ) ( In order to support the following notations, we introduce three canonical ) ( structure that reflect the finiteness of a predicate, in the following ) ( notations, p (resp q) are such finite predicates, which are ultimately ) ( represented by elements A (resp B) from {fset K}. ) ( ) ( [fset x in p \| P] == the set of all x of type K, such that ) ( x \in p and P x where P is a predicate on K ) ( [fset x in p \| P & Q] := [set x in p \| P && Q]. ) ( ) ( [fset E \| x in p] == the set of all the values of the expression E, for x ) ( drawn from the collective finite predicate p. ) ( [fset E \| x in p & P] == the set of values of E for x drawn from p, such ) ( that P is true. ) ( [fset E \| x in p, y in q] == the set of values of E for x drawn from p and) ( and y drawn from q; q may depend on x. ) ( [fset E \| x in p, y in q & P] == the set of values of E for x drawn from ) ( p and y drawn from q; such that P is true. ) ( [fsetval x in p] == the set of (val x) for x in the finite predicate p ) ( [fsetval x in p \| P ] == the set of (val x) for x in p, such that P ) ( [fsetval x in p \| P & Q] := [fsetval x in p \| P && Q] ) ( ) ( For each notation above, there is an additional one with ':' instead of ) ( 'in' which is used to range over the finite type A instead of the finite ) ( set A, and the optional predicate is over A instead of K ) ( For example: ) ( [fset x : A \| P] := [fset x in {: A} \| P] ) ( == the set of all x of type A, such that P x ) ( [fset E \| x : A] == the set of all the values of the expression E, for x ) ( drawn from the finite type A ) ( ) ( For each [fset ...] or [fsetval ...] notation, there is a keyed variant ) ( written [fset[key] ...] or [fsetval[key] ...] for locking ) (**** finite maps ******************************************************) ( ) ( Operations on finmap: ) ( The following notations are in the %fmap scope ) ( ) ( f.[? k] == returns Some v if k maps to v, otherwise None ) ( f.[p] == returns v if p has type k \in f, and k maps to v ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( f.[k <- v] == f extended with the mapping k -> v ) ( domf f == finite set (of type {fset K}) of keys of f ) ( codomf f == finite set (of type {fset V}) of values of f ) ( k \in f == k is a key of f ) ( := k \in domf f ) ( [fmap] == the empty finite map ) ( [fmap x : S => E] == the finmap defined by E on the support S ) ( f.[& A] == f restricted to A (intersected with domf f) ) ( f.[\ A] := f.[& domf `\` A] ) ( == f where all the keys in A have been removed ) ( f.[~ k] := f.[\ [fset k]] ) ( f + g == concatenation of f and g, ) ( the keys of g override the keys of f ) ( ) (**** finitely supported functions *************************************) ( ) ( Operation on function with finite support, i.e. finmap with default value ) ( for elements outside of the support. Contrarly to finmap, these are total ) ( function, so we provide a coercion to Funclass ) ( ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( {fsfun K -> T for dflt} == finitely supported functions with default ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( value dflt : K -> V outside the support ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( {fsfun K -> T of x => dflt} := {fsfun K -> T for fun x => dflt} ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( {fsfun K -> T with dflt} := {fsfun K -> T for fun=> dflt} ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( {fsfun K -> K} := {fsfun K -> T for fun x => x} ) ( ) ( [fsfun for dflt] == the default fsfun ) ( [fsfun of x => dflt] == the default fsfun ) ( [fsfun x : A => F \| default] == the fsfun which takes value F on A ) ( x has type A : {fset T} ) ( [fsfun x in A => F \| default] == the fsfun which takes value F on A ) ( x has type T, where A : {fset T} ) ( we also provide untyped variants and variants where default is ommitted ) ( e.g. [fsfun x : A => F] [fsfun x => F \| default] [fsfun]... ) ( and many variants to give the possibility to insert a key : unit ) ( to prevent conversion from diverging, e.g. ) ( [fsfun[key] x : A => F \| default] and [fsfun[key] x in A => F \| default] ) ( ... ) ( (f \o g)%fsfun == composition of fsfun ) ( fsinjectiveb f == boolean predicate of injectivity of f ) (***************************************************************************) Set Implicit Arguments. Unset Strict Implicit. Import Prenex Implicits. Reserved Notation "{fset K }" (at level 0, format "{fset K }"). Reserved Notation "A `&` B" (at level 48, left associativity). Reserved Notation "A `` B" (at level 46, left associativity). Reserved Notation "A `+` B" (at level 54, left associativity). Reserved Notation "A +` B" (at level 54, left associativity). Reserved Notation "A `\|` B" (at level 52, left associativity). Reserved Notation "a \|` A" (at level 52, left associativity). Reserved Notation "A `\` B" (at level 50, left associativity). Reserved Notation "A `\ b" (at level 50, left associativity). Reserved Notation "{fmap T }" (at level 0, format "{fmap T }"). (* Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Reserved Notation "x .[ k <- v ]" ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) (at level 2, k at level 200, v at level 200, format "x .[ k <- v ]"). Reserved Notation "x .[~ k ]" (at level 2, k at level 200, format "x .[~ k ]"). Reserved Notation "x .[& k ]" (at level 2, k at level 200, format "x .[& k ]"). Reserved Notation "x .[\ k ]" (at level 2, k at level 200, format "x .[\ k ]"). Reserved Notation "x .[? k ]" (at level 2, k at level 200, format "x .[? k ]"). Reserved Infix "`~`" (at level 52). Reserved Notation "[ 'fset' k ]" (at level 0, k at level 99, format "[ 'fset' k ]"). Local Notation predOfType T := (sort_of_simpl_pred (@pred_of_argType T)). Section extra. Lemma mem_remF (T : eqType) (s : seq T) x : uniq s -> x \in rem x s = false. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) Proof. by move=> us; rewrite mem_rem_uniq // inE eqxx. Qed. Definition ffun0 (T : finType) (X : Type) : #\|T\| = 0 -> {ffun T -> X}. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) Proof. by move=> T0; split; rewrite T0; exists nil. Defined. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Definition oextract (T : Type) (o : option T) : o -> T := ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) if o is Some t return o -> T then fun=> t else False_rect T \o notF. Lemma oextractE (T : Type) (x : T) (xP : Some x) : oextract xP = x. Proof. by []. Qed. Lemma Some_oextract T (x : option T) (x_ex : x) : Some (oextract x_ex) = x. Proof. by case: x x_ex. Qed. Definition ojoin T (x : option (option T)) := if x is Some y then y else None. Lemma Some_ojoin T (x : option (option T)) : x -> Some (ojoin x) = x. Proof. by case : x. Qed. Lemma ojoinT T (x : option (option T)) : ojoin x -> x. Proof. by case: x. Qed. Section AllSigs. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Variables (S : Type) (T : S -> Type) (R : Type) (f : forall x, T x -> R). Implicit Types (s : seq S) (t : forall x, seq (T x)). Definition allsigs s t := foldr (fun x => cat (map (@f x) (t x))) [::] s. Lemma size_allsigs s t : size (allsigs s t) = sumn [seq size (t x) \| x <- s]. Proof. by elim: s => //= x s IHs; rewrite size_cat size_map IHs. Qed. Lemma allsigs_cat s1 s2 t : allsigs (s1 ++ s2) t = allsigs s1 t ++ allsigs s2 t. Proof. by elim: s1 => //= x s1 ->; rewrite catA. Qed. End AllSigs. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma allsigs_comp S (T : S -> Type) R R' (f : forall x, T x -> R) (g : R -> R') s t : allsigs (fun x y => g (f x y)) s t = map g (allsigs f s t). Proof. by elim: s => //= x s ->; rewrite map_cat map_comp. Qed. Prenex Implicits allsigs. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Notation "[ 'seq' E \| i <- s & j <- t ]" := (allsigs (fun i j => E) s (fun i => t)) (at level 0, E at level 99, i ident, j ident, ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) format "[ '[hv' 'seq' E '/ ' \| i <- s & '/ ' j <- t ] ']'") : seq_scope. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Notation "[ 'seq' E \| i : T <- s & j : U <- t ]" := (allsigs (fun (i : T) (j : U) => E) s (fun i : T => t)) (at level 0, E at level 99, i ident, j ident, only parsing) : seq_scope. Section EqAllSigs. Variables (S : eqType) (T : S -> eqType). Implicit Types (R : eqType) (s : seq S) (t : forall x, seq (T x)). ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma allsigsP R (f : forall x, T x -> R) s t z : reflect (exists p : sigT T, [/\ tag p \in s, tagged p \in t (tag p) & z = f (tag p) (tagged p)]) (z \in allsigs f s t). Proof. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) elim: s => [\|x s IHs /=]; first by right=> [[p []]]. ( Goal: @eq bool (@fsubset K A (@fsetI K B C)) (andb (@fsubset K A B) (@fsubset K A C)) ) rewrite mem_cat; have [fxt_z \| not_fxt_z] := altP mapP. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by left; have [y t_y ->] := fxt_z; exists (Tagged T y); rewrite mem_head. ( Goal: is_true (negb (@eq_op (fset_eqType K) (@fsetD K C B) (@fsetD K C A))) ) apply: (iffP IHs) => [] [[x' y] /= [s_x' t_y def_z]]; exists (Tagged T y) => /=. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite !inE predU1r. ( Goal: Bool.reflect (@ex2 (@finset_of K (Phant (Choice.sort K))) (fun S : @finset_of K (Phant (Choice.sort K)) => is_true (@fsubset K (@FinSupp.fs K (Choice.eqType K) (fun x : Choice.sort K => x) f) S)) (fun S : @finset_of K (Phant (Choice.sort K)) => and (@prop_in2 (Choice.sort K) (@mem (Choice.sort K) (finSetPredType K) S) (fun x1 x2 : Choice.sort K => forall _ : @eq (Equality.sort (Choice.eqType K)) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f x1) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f x2), @eq (Choice.sort K) x1 x2) (inPhantom (@injective (Equality.sort (Choice.eqType K)) (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f)))) (forall a : @fset_sub_type K S, is_true (@in_mem (Equality.sort (Choice.eqType K)) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f (@val (Choice.sort K) (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S) a)) (@mem (Choice.sort K) (finSetPredType K) S))))) (fsinjectiveb f) ) have [def_x' \| //] := predU1P s_x'. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by do [case: _ / def_x'; rewrite def_z map_f] in s_x' not_fxt_z . Qed. (* Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma mem_allsigs R (f : forall x, T x -> R) s1 t1 s2 t2 : ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) s1 =i s2 -> (forall x, x \in s1 -> t1 x =i t2 x) -> allsigs f s1 t1 =i allsigs f s2 t2. Proof. move=> eq_s eq_t z; apply/allsigsP/allsigsP=> [] [p fpz]; exists p => []; by move: fpz (fpz) => [???]; rewrite eq_s eq_t //= 1?eq_s. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma allsigs_catr R (f : forall x, T x -> R) s t1 t2 : allsigs f s (fun x => t1 x ++ t2 x) =i allsigs f s t1 ++ allsigs f s t2. Proof. ( Goal: forall (_ : is_true (@fdisjoint K (@domf K V f) (@domf K V h))) (_ : is_true (@fdisjoint K (@domf K V g) (@domf K V h))) (_ : @eq (finMap K V) (catf h f) (catf h g)), @eq (@finmap_of K V (Phant (forall _ : Choice.sort K, V))) f g ) move=> z; rewrite mem_cat. ( Goal: Bool.reflect (@ex (Choice.sort K) (fun x : Choice.sort K => @eq (option (Choice.sort V)) (@fnd K (Choice.sort V) (@domf K (Choice.sort V) f) (@ffun_of_fmap K (Choice.sort V) f) x) (@Some (Choice.sort V) v))) (@in_mem (Choice.sort V) v (@mem (Choice.sort V) (finSetPredType V) (codomf f))) ) apply/allsigsP/orP=> [[p [sP1]]\|]. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite mem_cat; case/orP; [left \| right]; apply/allsigsP; exists p. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by case=> /allsigsP[p [sp1 sp2 ->]]; exists p; rewrite mem_cat sp2 ?orbT. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma allsigs_uniq R (f : forall x, T x -> R) s t : ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) uniq s -> (forall x, x \in s -> uniq (t x)) -> ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) {in [seq Tagged T y \| x <- s & y <- t x] &, ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) injective (fun p : sigT T => f (tag p) (tagged p))} -> uniq (allsigs f s t). Proof. ( Goal: forall (_ : forall (x : Choice.sort T) (px : is_true (@in_mem (Choice.sort T) x (@pred_of_finmempred (Choice.eqType T) p))), P (@FSetSub K (@Imfset.imfset key T K f p (Phantom (mem_pred (Equality.sort (Choice.eqType T))) (@pred_of_finmempred (Choice.eqType T) p))) (f x) (@in_imfset T f p x px))) (k : @fset_sub_type K (@Imfset.imfset key T K f p (Phantom (mem_pred (Equality.sort (Choice.eqType T))) (@pred_of_finmempred (Choice.eqType T) p)))), P k ) move=> Us Ut inj_f; have: all (mem s) s by apply/allP. ( Goal: forall _ : is_true (@fsubset T (@fsetD T U X) U), P (@fsetD T U (@fsetD T U X)) ) elim: {-2}s Us => //= x s1 IHs /andP[s1'x Us1] /andP[sx1 ss1]. ( Goal: @big_rel_class_of (@finset_of K (Phant (Choice.sort K))) (@fsubset K) ) rewrite cat_uniq {}IHs // andbT map_inj_in_uniq ?Ut // => [\|y1 y2 ]. (* Goal: @eq (@finset_of V (Phant (Choice.sort V))) (codomf (@restrictf K (Choice.sort V) f A)) (@Imfset.imfset imfset_key (Finite.choiceType (@fset_sub_finType K (@domf K (Choice.sort V) f))) V (fun k : Choice.sort (Finite.choiceType (@fset_sub_finType K (@domf K (Choice.sort V) f))) => @FunFinfun.fun_of_fin (@fset_sub_finType K (@domf K (Choice.sort V) f)) (Choice.sort V) (@ffun_of_fmap K (Choice.sort V) f) k) (@mem_fin (Choice.eqType (Finite.choiceType (@fset_sub_finType K (@domf K (Choice.sort V) f)))) (simplPredType (@fset_sub_type K (@domf K (Choice.sort V) f))) (@subfinset_finpred (Finite.choiceType (@fset_sub_finType K (@domf K (Choice.sort V) f))) (@mem_fin (Choice.eqType (Finite.choiceType (@fset_sub_finType K (@domf K (Choice.sort V) f)))) (predPredType (@fset_sub_type K (@domf K (Choice.sort V) f) : predArgType)) (@fin_finpred (Choice.eqType (Finite.choiceType (@fset_sub_finType K (@domf K (Choice.sort V) f)))) (pred_finpredType (@fset_sub_finType K (@domf K (Choice.sort V) f))) (@sort_of_simpl_pred (@fset_sub_type K (@domf K (Choice.sort V) f) : predArgType) (pred_of_argType (@fset_sub_type K (@domf K (Choice.sort V) f) : predArgType))))) (fun k : @fset_sub_type K (@domf K (Choice.sort V) f) => @in_mem (Choice.sort K) (@val (Choice.sort K) (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) (@domf K (Choice.sort V) f))) (@fset_sub_subType K (@domf K (Choice.sort V) f)) k) (@mem (Choice.sort K) (finSetPredType K) A)))) (Phantom (mem_pred (@fset_sub_type K (@domf K (Choice.sort V) f))) (@mem (@fset_sub_type K (@domf K (Choice.sort V) f)) (simplPredType (@fset_sub_type K (@domf K (Choice.sort V) f))) (@SimplPred (@fset_sub_type K (@domf K (Choice.sort V) f)) (fun k : @fset_sub_type K (@domf K (Choice.sort V) f) => andb (@in_mem (@fset_sub_type K (@domf K (Choice.sort V) f)) k (@mem (@fset_sub_type K (@domf K (Choice.sort V) f)) (predPredType (@fset_sub_type K (@domf K (Choice.sort V) f) : predArgType)) (@sort_of_simpl_pred (@fset_sub_type K (@domf K (Choice.sort V) f) : predArgType) (pred_of_argType (@fset_sub_type K (@domf K (Choice.sort V) f) : predArgType))))) (@in_mem (Choice.sort K) (@val (Choice.sort K) (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) (@domf K (Choice.sort V) f))) (@fset_sub_subType K (@domf K (Choice.sort V) f)) k) (@mem (Choice.sort K) (finSetPredType K) A))))))) ) apply/hasPn=> _ /allsigsP[z [s1z tz ->]]; apply/mapP=> [[y ty Dy]]. suffices [Dz1 _]: Tagged T (tagged z) = Tagged T y. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite -Dz1 s1z in s1'x. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) apply: inj_f => //; apply/allsigsP; last by exists (Tagged T y). ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by have:= allP ss1 _ s1z; exists z. suffices /eqP: Tagged T y1 = Tagged T y2 by rewrite eq_Tagged => /eqP. apply: inj_f => //; apply/allsigsP; by [exists (Tagged T y1) \| exists (Tagged T y2)]. Qed. End EqAllSigs. Lemma big_allsigs (R : Type) (idx : R) (op : Monoid.law idx) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) (I1 : Type) (I2 : I1 -> Type) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) (r1 : seq I1) (r2 : forall i1, seq (I2 i1)) (F : sigT I2 -> R) : ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) \big[op/idx]_(i <- [seq Tagged I2 i2 \| i1 <- r1 & i2 <- r2 i1]) F i = ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) \big[op/idx]_(i1 <- r1) \big[op/idx]_(i2 <- r2 i1) F (Tagged I2 i2). Proof. ( Goal: @eq R (@BigOp.bigop R (@sigT I1 (fun x : I1 => I2 x)) idx (@allsigs I1 (fun i1 : I1 => I2 i1) (@sigT I1 (fun x : I1 => I2 x)) (fun (i1 : I1) (i2 : (fun i2 : I1 => I2 i2) i1) => @Tagged I1 i1 I2 i2) r1 (fun i1 : I1 => r2 i1)) (fun i : @sigT I1 (fun x : I1 => I2 x) => @BigBody R (@sigT I1 (fun x : I1 => I2 x)) i (@Monoid.operator R idx op) true (F i))) (@BigOp.bigop R I1 idx r1 (fun i1 : I1 => @BigBody R I1 i1 (@Monoid.operator R idx op) true (@BigOp.bigop R (I2 i1) idx (r2 i1) (fun i2 : I2 i1 => @BigBody R (I2 i1) i2 (@Monoid.operator R idx op) true (F (@Tagged I1 i1 I2 i2)))))) ) elim: r1 => [\|i1 r1 IHr1]; rewrite !(big_nil, big_cons)//= big_cat {}IHr1. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by case: (r2 i1) => [\|i2 r21]; rewrite /= !(big_nil, big_cons)//= big_map. Qed. Section NatHomomorphism. Variable T : Type. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma homo_ltn_in (D : pred nat) (f : nat -> T) (r : T -> T -> Prop) : ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) (forall y x z, r x y -> r y z -> r x z) -> ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) {in D &, forall i j k, i < k < j -> k \in D} -> ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) {in D, forall i, i.+1 \in D -> r (f i) (f i.+1)} -> ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) {in D &, {homo f : i j / i < j >-> r i j}}. Proof. move=> r_trans Dcx r_incr i j iD jD lt_ij; move: (lt_ij) (jD) => /subnKC<-. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) elim: (_ - _) => [\|k ihk]; first by rewrite addn0 => Dsi; apply: r_incr. ( Goal: forall (_ : is_true (@fdisjoint K A B)) (C : @finset_of K (Phant (Choice.sort K))), is_true (@disjoint (@fset_sub_finType K C) (@mem (Finite.sort (@fset_sub_finType K C)) (predPredType (Finite.sort (@fset_sub_finType K C))) (@SetDef.pred_of_set (@fset_sub_finType K C) (fsub C A))) (@mem (Finite.sort (@fset_sub_finType K C)) (predPredType (Finite.sort (@fset_sub_finType K C))) (@SetDef.pred_of_set (@fset_sub_finType K C) (fsub C B)))) ) move=> DSiSk [: DSik]; apply: (r_trans _ _ _ (ihk _)); rewrite ?addnS. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by abstract: DSik; apply: (Dcx _ _ iD DSiSk); rewrite ltn_addr ?addnS /=. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by apply: r_incr; rewrite -?addnS. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma homo_ltn (f : nat -> T) (r : T -> T -> Prop) : ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) (forall y x z, r x y -> r y z -> r x z) -> ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) (forall i, r (f i) (f i.+1)) -> {homo f : i j / i < j >-> r i j}. Proof. by move=> /(@homo_ltn_in predT f) fr fS i j; apply: fr. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma homo_leq_in (D : pred nat) (f : nat -> T) (r : T -> T -> Prop) : ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) (forall x, r x x) -> (forall y x z, r x y -> r y z -> r x z) -> ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) {in D &, forall i j k, i < k < j -> k \in D} -> ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) {in D, forall i, i.+1 \in D -> r (f i) (f i.+1)} -> ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) {in D &, {homo f : i j / i <= j >-> r i j}}. Proof. ( Goal: forall _ : is_true (@fsubset K (@fsetU K A B) C), @eq bool (@disjoint (@fset_sub_finType K C) (@mem (Finite.sort (@fset_sub_finType K C)) (predPredType (Finite.sort (@fset_sub_finType K C))) (@SetDef.pred_of_set (@fset_sub_finType K C) (fsub C A))) (@mem (Finite.sort (@fset_sub_finType K C)) (predPredType (Finite.sort (@fset_sub_finType K C))) (@SetDef.pred_of_set (@fset_sub_finType K C) (fsub C B)))) (@fdisjoint K A B) ) move=> r_refl r_trans Dcx /(homo_ltn_in r_trans Dcx) lt_r i j iD jD. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite leq_eqVlt => /predU1P[->//\|/lt_r]; apply. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma homo_leq (f : nat -> T) (r : T -> T -> Prop) : ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) (forall x, r x x) -> (forall y x z, r x y -> r y z -> r x z) -> ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) (forall i, r (f i) (f i.+1)) -> {homo f : i j / i <= j >-> r i j}. Proof. by move=> rrefl /(@homo_leq_in predT f r) fr fS i j; apply: fr. Qed. End NatHomomorphism. End extra. Module Type SortKeysSig. Section SortKeys. Variable (K : choiceType). Implicit Types (k : K) (ks : seq K). Axiom f : seq K -> seq K. Axiom perm : forall s, perm_eq (f s) (undup s). Axiom uniq : forall s, uniq (f s). Axiom E : forall (s : seq K), f s =i s. Axiom eq : forall (s s' : seq K), s =i s' <-> f s = f s'. End SortKeys. End SortKeysSig. Module SortKeys : SortKeysSig. Section SortKeys. Variable (K : choiceType). Implicit Types (k : K) (ks : seq K). Definition f (s : seq K) := choose (perm_eq (undup s)) (undup s). Fact perm s : perm_eq (f s) (undup s). Proof. by rewrite perm_eq_sym chooseP. Qed. Fact uniq s : uniq (f s). Proof. by rewrite (perm_eq_uniq (perm _)) undup_uniq. Qed. Fact E (s : seq K) : f s =i s. Proof. by move=> x; rewrite (perm_eq_mem (perm _)) mem_undup. Qed. Lemma eq (s s' : seq K) : s =i s' <-> f s = f s'. Proof. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) split=> [eq_ss'\|eq_ss' k]; last by rewrite -E eq_ss' E. ( Goal: @eq bool (@fsubset K A (@fsetI K B C)) (andb (@fsubset K A B) (@fsubset K A C)) ) rewrite /f; have peq_ss' : perm_eq (undup s) (undup s'). ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by apply: uniq_perm_eq; rewrite ?undup_uniq // => x; rewrite !mem_undup. ( Goal: is_true (@in_mem (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f (@fsval K (@FinSupp.fs K (Choice.eqType K) (fun x : Choice.sort K => x) f) a)) (@mem (Choice.sort K) (finSetPredType K) (@FinSupp.fs K (Choice.eqType K) (fun x : Choice.sort K => x) f))) ) ( Goal: @ex2 (@finset_of K (Phant (Choice.sort K))) (fun S : @finset_of K (Phant (Choice.sort K)) => is_true (@fsubset K (@FinSupp.fs K (Choice.eqType K) (fun x : Choice.sort K => x) f) S)) (fun S : @finset_of K (Phant (Choice.sort K)) => and (@prop_in2 (Choice.sort K) (@mem (Choice.sort K) (finSetPredType K) S) (fun x1 x2 : Choice.sort K => forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f x1) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f x2), @eq (Choice.sort K) x1 x2) (inPhantom (@injective (Choice.sort K) (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f)))) (forall a : @fset_sub_type K S, is_true (@in_mem (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f (@fsval K S a)) (@mem (Choice.sort K) (finSetPredType K) S)))) ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) rewrite (@choose_id _ _ _ (undup s')) //=; apply: eq_choose => x /=. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by apply: sym_left_transitive; [exact: perm_eq_sym\|exact: perm_eq_trans\|]. Qed. End SortKeys. End SortKeys. Hint Resolve SortKeys.perm. Hint Resolve SortKeys.uniq. Hint Resolve SortKeys.E. Notation sort_keys := SortKeys.f. Notation sort_keys_perm := SortKeys.perm. Notation sort_keys_uniq := SortKeys.uniq. Notation sort_keysE := SortKeys.E. Notation eq_sort_keys := SortKeys.eq. Section ChoiceKeys. Variable (K : choiceType). Implicit Types (k : K) (ks : seq K). Lemma mem_sort_keys ks k : k \in ks -> k \in sort_keys ks. Proof. by rewrite sort_keysE. Qed. Lemma mem_sort_keys_intro ks k : k \in sort_keys ks -> k \in ks. Proof. by rewrite sort_keysE. Qed. Lemma sort_keys_nil : sort_keys [::] = [::] :> seq K. Proof. ( Goal: Bool.reflect (@ex2 (@finset_of K (Phant (Choice.sort K))) (fun S : @finset_of K (Phant (Choice.sort K)) => is_true (@fsubset K (@FinSupp.fs K (Choice.eqType K) (fun x : Choice.sort K => x) f) S)) (fun S : @finset_of K (Phant (Choice.sort K)) => and (@prop_in2 (Choice.sort K) (@mem (Choice.sort K) (finSetPredType K) S) (fun x1 x2 : Choice.sort K => forall _ : @eq (Equality.sort (Choice.eqType K)) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f x1) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f x2), @eq (Choice.sort K) x1 x2) (inPhantom (@injective (Equality.sort (Choice.eqType K)) (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f)))) (forall a : @fset_sub_type K S, is_true (@in_mem (Equality.sort (Choice.eqType K)) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f (@val (Choice.sort K) (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S) a)) (@mem (Choice.sort K) (finSetPredType K) S))))) (fsinjectiveb f) ) have := sort_keysE ([::] : seq K). ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by case: sort_keys => //= a l /(_ a); rewrite mem_head. Qed. Lemma sort_keys_id ks : sort_keys (sort_keys ks) = sort_keys ks. Proof. by have /eq_sort_keys := sort_keysE ks. Qed. Definition canonical_keys ks := sort_keys ks == ks. Lemma canonical_uniq ks : canonical_keys ks -> uniq ks. Proof. by move=> /eqP <-; exact: sort_keys_uniq. Qed. Lemma canonical_sort_keys ks : canonical_keys (sort_keys ks). Proof. by rewrite /canonical_keys sort_keys_id. Qed. Lemma canonical_eq_keys ks ks' : ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) canonical_keys ks -> canonical_keys ks' -> ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ks =i ks' -> ks = ks'. Proof. ( Goal: forall (_ : is_true (canonical_keys ks)) (_ : is_true (canonical_keys ks')) (_ : @eq_mem (Equality.sort (Choice.eqType K)) (@mem (Equality.sort (Choice.eqType K)) (seq_predType (Choice.eqType K)) ks) (@mem (Equality.sort (Choice.eqType K)) (seq_predType (Choice.eqType K)) ks')), @eq (list (Choice.sort K)) ks ks' ) move=> /eqP; case: _ /; move=> /eqP; case: _ / => eq_ks_ks'. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by apply/eq_sort_keys => x; rewrite -sort_keysE eq_ks_ks' sort_keysE. Qed. Lemma size_sort_keys ks : size (sort_keys ks) = size (undup ks). Proof. exact: perm_eq_size. Qed. End ChoiceKeys. Arguments eq_sort_keys {K s s'}. Section Def. Variables (K : choiceType). Structure finSet : Type := mkFinSet { enum_fset :> seq K; _ : canonical_keys enum_fset }. Definition finset_of (_ : phant K) := finSet. End Def. Identity Coercion type_of_finset : finset_of >-> finSet. Notation "{fset T }" := (@finset_of _ (Phant T)) : type_scope. Definition pred_of_finset (K : choiceType) (f : finSet K) : pred K := fun k => k \in (enum_fset f). Canonical finSetPredType (K : choiceType) := Eval hnf in mkPredType (@pred_of_finset K). Section FinSetCanonicals. Variable (K : choiceType). Canonical fsetType := Eval hnf in [subType for (@enum_fset K)]. Definition fset_eqMixin := Eval hnf in [eqMixin of {fset K} by <:]. Canonical fset_eqType := Eval hnf in EqType {fset K} fset_eqMixin. Definition fset_choiceMixin := Eval hnf in [choiceMixin of {fset K} by <:]. Canonical fset_choiceType := Eval hnf in ChoiceType {fset K} fset_choiceMixin. End FinSetCanonicals. Section FinTypeSet. Variables (K : choiceType) (A : finSet K). Lemma keys_canonical : canonical_keys (enum_fset A). Proof. by case: A. Qed. Lemma fset_uniq : uniq (enum_fset A). Proof. by rewrite canonical_uniq // keys_canonical. Qed. Record fset_sub_type : predArgType := FSetSub {fsval : K; fsvalP : in_mem fsval (@mem K _ A)}. Canonical fset_sub_subType := Eval hnf in [subType for fsval]. Definition fset_sub_eqMixin := Eval hnf in [eqMixin of fset_sub_type by <:]. Canonical fset_sub_eqType := Eval hnf in EqType fset_sub_type fset_sub_eqMixin. Definition fset_sub_choiceMixin := Eval hnf in [choiceMixin of fset_sub_type by <:]. Canonical fset_sub_choiceType := Eval hnf in ChoiceType fset_sub_type fset_sub_choiceMixin. Definition fset_sub_enum : seq fset_sub_type := undup (pmap insub (enum_fset A)). Lemma mem_fset_sub_enum x : x \in fset_sub_enum. Proof. by rewrite mem_undup mem_pmap -valK map_f // fsvalP. Qed. Lemma val_fset_sub_enum : map val fset_sub_enum = enum_fset A. Proof. ( Goal: @eq (Equality.sort V) (@fun_of_fsfun K V default (of_ffun key) x) (@Option.default (Equality.sort V) (default x) (@Option.map (@fset_sub_type K S) (Equality.sort V) h (@insub (Choice.sort K) (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S) x))) ) rewrite /fset_sub_enum undup_id ?pmap_sub_uniq ?fset_uniq//. ( Goal: is_true (@in_mem (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f (@fsval K (@FinSupp.fs K (Choice.eqType K) (fun x : Choice.sort K => x) f) a)) (@mem (Choice.sort K) (finSetPredType K) (@FinSupp.fs K (Choice.eqType K) (fun x : Choice.sort K => x) f))) ) ( Goal: @ex2 (@finset_of K (Phant (Choice.sort K))) (fun S : @finset_of K (Phant (Choice.sort K)) => is_true (@fsubset K (@FinSupp.fs K (Choice.eqType K) (fun x : Choice.sort K => x) f) S)) (fun S : @finset_of K (Phant (Choice.sort K)) => and (@prop_in2 (Choice.sort K) (@mem (Choice.sort K) (finSetPredType K) S) (fun x1 x2 : Choice.sort K => forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f x1) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f x2), @eq (Choice.sort K) x1 x2) (inPhantom (@injective (Choice.sort K) (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f)))) (forall a : @fset_sub_type K S, is_true (@in_mem (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f (@fsval K S a)) (@mem (Choice.sort K) (finSetPredType K) S)))) ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) rewrite (pmap_filter (@insubK _ _ _)); apply/all_filterP. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by apply/allP => x; rewrite isSome_insub. Qed. Definition fset_sub_pickle x := index x fset_sub_enum. Definition fset_sub_unpickle n := nth None (map some fset_sub_enum) n. Lemma fset_sub_pickleK : pcancel fset_sub_pickle fset_sub_unpickle. Proof. ( Goal: @big_rel_class_of (@finset_of K (Phant (Choice.sort K))) (@fsubset K) ) rewrite /fset_sub_unpickle => x. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite (nth_map x) ?nth_index ?index_mem ?mem_fset_sub_enum. Qed. Definition fset_sub_countMixin := CountMixin fset_sub_pickleK. Canonical fset_sub_countType := Eval hnf in CountType fset_sub_type fset_sub_countMixin. Definition fset_sub_finMixin := Eval hnf in UniqFinMixin (undup_uniq _) mem_fset_sub_enum. Canonical fset_sub_finType := Eval hnf in FinType fset_sub_type fset_sub_finMixin. Canonical fset_sub_subfinType := [subFinType of fset_sub_type]. Lemma enum_fsetE : enum_fset A = [seq val i \| i <- enum fset_sub_type]. Proof. by rewrite enumT unlock val_fset_sub_enum. Qed. Lemma cardfE : size (enum_fset A) = #\|fset_sub_type\|. Proof. by rewrite cardE enum_fsetE size_map. Qed. End FinTypeSet. Identity Coercion finSet_sub_type : finset_of >-> finSet. Coercion fset_sub_type : finSet >-> predArgType. Hint Resolve fsvalP fset_uniq mem_fset_sub_enum. Delimit Scope fset_scope with fset. Local Open Scope fset_scope. Notation "[` kf ]" := (FSetSub kf) (format "[` kf ]") : fset_scope. Lemma fsetsubE (T : choiceType) (A : {fset T}) (x : A) (xA : val x \in A) : [` xA] = x. Proof. by apply/val_inj => /=. Qed. Notation "#\|` A \|" := (size (enum_fset A)) (at level 0, A at level 99, format "#\|` A \|") : nat_scope. Definition fset_predT {T : choiceType} {A : {fset T}} : simpl_pred A := @predT A. Coercion set_of_fset (K : choiceType) (A : {fset K}) : {set A} := [set x in {: A}]. Arguments pred_of_finset : simpl never. Section SeqFset. Variable finset_key : unit. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Definition seq_fset : forall K : choiceType, seq K -> {fset K} := locked_with finset_key (fun K s => mkFinSet (@canonical_sort_keys K s)). Variable (K : choiceType) (s : seq K). Lemma seq_fsetE : seq_fset s =i s. Proof. by move=> a; rewrite [seq_fset]unlock sort_keysE. Qed. Lemma size_seq_fset : size (seq_fset s) = size (undup s). Proof. by rewrite [seq_fset]unlock /= size_sort_keys. Qed. Lemma seq_fset_uniq : uniq (seq_fset s). Proof. by rewrite [seq_fset]unlock /= sort_keys_uniq. Qed. Lemma seq_fset_perm : perm_eq (seq_fset s) (undup s). Proof. by rewrite [seq_fset]unlock //= sort_keys_perm. Qed. End SeqFset. Hint Resolve keys_canonical. Hint Resolve sort_keys_uniq. Canonical finSetSubType K := [subType for (@enum_fset K)]. Definition finSetEqMixin (K : choiceType) := [eqMixin of {fset K} by <:]. Canonical finSetEqType (K : choiceType) := EqType {fset K} (finSetEqMixin K). Definition finSetChoiceMixin (K : choiceType) := [choiceMixin of {fset K} by <:]. Canonical finSetChoiceType (K : choiceType) := ChoiceType {fset K} (finSetChoiceMixin K). Section FinPredStruct. Structure finpredType (T : eqType) := FinPredType { finpred_sort :> Type; ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) tofinpred : finpred_sort -> pred T; ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) _ : {mem : finpred_sort -> mem_pred T \| isMem tofinpred mem}; ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) _ : {finpred_seq : finpred_sort -> seq T \| ((forall p, uniq (finpred_seq p)) forall p x, x \in finpred_seq p = tofinpred p x)%type} }. Canonical finpredType_predType (T : eqType) (fpT : finpredType T) := @PredType T (finpred_sort fpT) (@tofinpred T fpT) (let: FinPredType _ _ mem _ := fpT in mem). Definition enum_finpred (T : eqType) (fpT : finpredType T) : (* Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) fpT -> seq T := let: FinPredType _ _ _ (exist s _) := fpT in s. Lemma enum_finpred_uniq (T : eqType) (fpT : finpredType T) (p : fpT) : uniq (enum_finpred p). Proof. by case: fpT p => ??? [s sE] p; rewrite sE. Qed. Lemma enum_finpredE (T : eqType) (fpT : finpredType T) (p : fpT) : enum_finpred p =i p. Proof. by case: fpT p => ??? [s sE] p x; rewrite sE -topredE. Qed. Lemma mkFinPredType_of_subproof (T : eqType) (pT : predType T) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) (fpred_seq : pT -> seq T) (pred_fsetE : forall p, fpred_seq p =i p) : forall p x, x \in fpred_seq p = topred p x. Proof. by move=> p x; rewrite topredE pred_fsetE. Qed. Definition mkFinPredType_of (T : eqType) (U : Type) := ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) fun (pT : predType T) & pred_sort pT -> U => fun a mP (pT' := @PredType T U a mP) & phant_id pT' pT => ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) fun (fpred_seq : pT' -> seq T) (fpred_seq_uniq : forall p, uniq (fpred_seq p)) (fpred_seqE : forall p, fpred_seq p =i p) => @FinPredType T U a mP (exist _ fpred_seq (fpred_seq_uniq, (mkFinPredType_of_subproof fpred_seqE))). Definition clone_finpredType (T : eqType) (U : Type) := ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) fun (pT : finpredType T) & finpred_sort pT -> U => fun a mP pP (pT' := @FinPredType T U a mP pP) & phant_id pT' pT => pT'. Structure is_finite (T : eqType) (P : pred T) := IsFinite { seq_of_is_finite :> seq T; _ : uniq seq_of_is_finite; _ : forall x, x \in seq_of_is_finite = P x; }. Lemma is_finite_uniq (T : eqType) (P : pred T) (p : is_finite P) : uniq p. Proof. by case: p. Qed. Lemma is_finiteE (T : eqType) (P : pred T) (p : is_finite P) x : x \in (seq_of_is_finite p) = P x. Proof. by case: p. Qed. Structure finpred (T : eqType) (pT : predType T) := FinPred { pred_of_finpred :> pT; _ : is_finite [pred x in pred_of_finpred] }. Definition enum_fin (T : eqType) (pT : predType T) (p : finpred pT) : seq T := let: FinPred _ fp := p in fp. Lemma enum_fin_uniq (T : eqType) (pT : predType T) (p : finpred pT) : uniq (enum_fin p). Proof. by case: p => ?[]. Qed. Lemma enum_finE (T : eqType) (pT : predType T) (p : finpred pT) : enum_fin p =i (pred_of_finpred p). Proof. by case: p => ?[]. Qed. Canonical fin_finpred (T : eqType) (pT : finpredType T) (p : pT) := @FinPred _ _ p (@IsFinite _ _ (enum_finpred p) (enum_finpred_uniq p) (enum_finpredE p)). Definition finpred_of (T : eqType) (pT : predType T) (p : pT) (fp : finpred pT) & phantom pT fp : finpred pT := fp. Structure finmempred (T : eqType) := FinMemPred { pred_of_finmempred :> mem_pred T; _ : is_finite (fun x => in_mem x pred_of_finmempred) }. Definition enum_finmem (T : eqType) (p : finmempred T) : seq T := let: FinMemPred _ fp := p in fp. Lemma enum_finmem_uniq (T : eqType) (p : finmempred T) : uniq (enum_finmem p). Proof. by case: p => ?[]. Qed. Lemma enum_finmemE (T : eqType) (p : finmempred T) : enum_finmem p =i p. Proof. by case: p => ?[]. Qed. Definition finmempred_of (T : eqType) (P : pred T) (mP : finmempred T) & phantom (mem_pred T) mP : finmempred T := mP. Canonical mem_fin (T : eqType) (pT : predType T) (p : finpred pT) := @FinMemPred _ (mem p) (@IsFinite _ _ (enum_fin p) (enum_fin_uniq p) (enum_finE p)). End FinPredStruct. Notation "[ 'finpredType' 'of' T ]" := (@clone_finpredType _ T _ id _ _ _ id) (at level 0, format "[ 'finpredType' 'of' T ]") : form_scope. Notation mkFinPredType T s s_uniq sE := (@mkFinPredType_of _ T _ id _ _ id s s_uniq sE). Section CanonicalFinPred. Canonical fset_finpredType (T: choiceType) := mkFinPredType (finSet T) (@enum_fset T) (@fset_uniq T) (fun _ _ => erefl). Canonical pred_finpredType (T : finType) := mkFinPredType (pred T) (fun P => enum_mem (mem P)) (@enum_uniq T) (@mem_enum T). Canonical simpl_pred_finpredType (T : finType) := mkFinPredType (simpl_pred T) (fun P => enum_mem (mem P)) (@enum_uniq T) (@mem_enum T). Canonical fset_finpred (T: choiceType) (A : finSet T) := @FinPred _ _ (enum_fset A) (@IsFinite _ _ (enum_fset A) (fset_uniq _) (fun=> erefl)). Program Canonical subfinset_finpred (T : choiceType) (A : finmempred T) (Q : pred T) := @FinPred _ _ [pred x in A \| Q x] ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) (@IsFinite _ _ [seq x <- enum_finmem A \| Q x] _ _). Next Obligation. by rewrite filter_uniq// enum_finmem_uniq. Qed. Next Obligation. by rewrite !inE !mem_filter andbC enum_finmemE. Qed. Canonical seq_finpredType (T : eqType) := mkFinPredType (seq T) undup (@undup_uniq T) (@mem_undup T). End CanonicalFinPred. Local Notation imfset_def key := ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) (fun (T K : choiceType) (f : T -> K) (p : finmempred T) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) of phantom (mem_pred T) p => seq_fset key [seq f x \| x <- enum_finmem p]). Local Notation imfset2_def key := ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) (fun (K T1 : choiceType) (T2 : T1 -> choiceType) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) (f : forall x : T1, T2 x -> K) (p1 : finmempred T1) (p2 : forall x : T1, finmempred (T2 x)) of phantom (mem_pred T1) p1 & phantom (forall x, mem_pred (T2 x)) p2 => seq_fset key (allsigs f (enum_finmem p1) (fun x => enum_finmem (p2 x)))). Module Type ImfsetSig. Parameter imfset : forall (key : unit) (T K : choiceType) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) (f : T -> K) (p : finmempred T), ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) phantom (mem_pred T) p -> {fset K}. Parameter imfset2 : forall (key : unit) (K T1 : choiceType) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) (T2 : T1 -> choiceType)(f : forall x : T1, T2 x -> K) (p1 : finmempred T1) (p2 : forall x : T1, finmempred (T2 x)), ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) phantom (mem_pred T1) p1 -> phantom (forall x, mem_pred (T2 x)) p2 -> {fset K}. Axiom imfsetE : forall key, imfset key = imfset_def key. Axiom imfset2E : forall key, imfset2 key = imfset2_def key. End ImfsetSig. Module Imfset : ImfsetSig. Definition imfset key := imfset_def key. Definition imfset2 key := imfset2_def key. Lemma imfsetE key : imfset key = imfset_def key. Proof. by []. Qed. Lemma imfset2E key : imfset2 key = imfset2_def key. Proof. by []. Qed. End Imfset. Notation imfset key f p := (Imfset.imfset key f (Phantom _ (pred_of_finmempred p))). Notation imfset2 key f p q := (Imfset.imfset2 key f (Phantom _ (pred_of_finmempred p)) (Phantom _ (fun x => (pred_of_finmempred (q x))))). Canonical imfset_unlock k := Unlockable (Imfset.imfsetE k). Canonical imfset2_unlock k := Unlockable (Imfset.imfset2E k). Notation "A `=` B" := (A = B :> {fset _}) (at level 70, no associativity, only parsing) : fset_scope. Notation "A `<>` B" := (A <> B :> {fset _}) (at level 70, no associativity, only parsing) : fset_scope. Notation "A `==` B" := (A == B :> {fset _}) (at level 70, no associativity, only parsing) : fset_scope. Notation "A `!=` B" := (A != B :> {fset _}) (at level 70, no associativity, only parsing) : fset_scope. Notation "A `=P` B" := (A =P B :> {fset _}) (at level 70, no associativity, only parsing) : fset_scope. Notation "f @`[ key ] A" := (Imfset.imfset key f (Phantom _ (mem A))) (at level 24, key at level 0) : fset_scope. Notation "f @2`[ key ] ( A , B )" := (Imfset.imfset2 key f (Phantom _ (mem A)) (Phantom _ (fun x => (mem (B x))))) (at level 24, format "f @2`[ key ] ( A , B )") : fset_scope. Fact imfset_key : unit. Proof. exact: tt. Qed. Notation "f @` A" := (f @`[imfset_key] A) (at level 24) : fset_scope. Notation "f @2` ( A , B )" := (f @2`[imfset_key] (A, B)) (at level 24, format "f @2` ( A , B )") : fset_scope. ( unary ) Notation "[ 'fset' E \| x : T 'in' A ]" := ((fun x : T => E) @` A) (at level 0, E, x at level 99, only parsing) : fset_scope. Notation "[ 'fset' E \| x 'in' A ]" := [fset E \| x : _ in A] (at level 0, E, x at level 99, only parsing) : fset_scope. Notation "[ 'fset' E \| x : A ]" := [fset E \| x : _ in {: A} ] (at level 0, E, x at level 99, only parsing) : fset_scope. Notation "[ 'fset' x : T 'in' A ]" := [fset (x : T) \| x in A] (at level 0, x at level 99, only parsing) : fset_scope. Notation "[ 'fset' x : T 'in' A \| P ]" := [fset (x : T) \| x in [pred x in A \| P]] (at level 0, x at level 99, only parsing) : fset_scope. Notation "[ 'fset' x 'in' A \| P ]" := [fset x : _ in A \| P] (at level 0, x at level 99, only parsing) : fset_scope. Notation "[ 'fset' x 'in' A ]" := [fset x : _ in A ] (at level 0, x at level 99, only parsing) : fset_scope. Notation "[ 'fset' x : T \| P ]" := [fset x in {: T} \| P] (at level 0, x at level 99, only parsing) : fset_scope. Notation "[ 'fset' x : T \| P & Q ]" := [fset x : T \| P && Q] (at level 0, x at level 99, only parsing) : fset_scope. Notation "[ 'fset' x : T 'in' A \| P & Q ]" := [fset x : T in A \| P && Q] (at level 0, x at level 99, only parsing) : fset_scope. Notation "[ 'fset' x 'in' A \| P & Q ]" := [fset x in A \| P && Q] (at level 0, x at level 99, only parsing) : fset_scope. ( binary ) Notation "[ 'fset' E \| x : T 'in' A , y : T' 'in' B ]" := ((fun (x : T) (y : T') => E) @2` (A, fun x => B)) (at level 0, E, x at level 99, A at level 200, y at level 99, only parsing) : fset_scope. Notation "[ 'fset' E \| x 'in' A , y 'in' B ]" := [fset E \| x : _ in A, y : _ in B] (at level 0, E, x at level 99, A at level 200, y at level 99, only parsing) : fset_scope. ( keyed parse only ) Notation "[ 'fset[' key ] E \| x : T 'in' A ]" := ((fun x : T => E) @`[key] A) (at level 0, E, x at level 99, only parsing) : fset_scope. Notation "[ 'fset[' key ] E \| x 'in' A ]" := [fset[key] E \| x : _ in A] (at level 0, E, x at level 99, only parsing) : fset_scope. Notation "[ 'fset[' key ] E \| x : A ]" := [fset[key] E \| x : _ in {: A} ] (at level 0, E, x at level 99, only parsing) : fset_scope. Notation "[ 'fset[' key ] x : T 'in' A ]" := [fset[key] (x : T) \| x in A] (at level 0, x at level 99, only parsing) : fset_scope. Notation "[ 'fset[' key ] x : T 'in' A \| P ]" := [fset[key] (x : T) \| x in [pred x in A \| P]] (at level 0, x at level 99, only parsing) : fset_scope. Notation "[ 'fset[' key ] x 'in' A \| P ]" := [fset[key] x : _ in A \| P] (at level 0, x at level 99, only parsing) : fset_scope. Notation "[ 'fset[' key ] x 'in' A ]" := [fset[key] x : _ in A ] (at level 0, x at level 99, only parsing) : fset_scope. Notation "[ 'fset[' key ] x : T \| P ]" := [fset[key] x in {: T} \| P] (at level 0, x at level 99, only parsing) : fset_scope. Notation "[ 'fset[' key ] x : T \| P & Q ]" := [fset[key] x : T \| P && Q] (at level 0, x at level 99, only parsing) : fset_scope. Notation "[ 'fset[' key ] x : T 'in' A \| P & Q ]" := [fset[key] x : T in A \| P && Q] (at level 0, x at level 99, only parsing) : fset_scope. Notation "[ 'fset[' key ] x 'in' A \| P & Q ]" := [fset[key] x in A \| P && Q] (at level 0, x at level 99, only parsing) : fset_scope. Notation "[ 'fset[' key ] E \| x : T 'in' A , y : T' 'in' B ]" := ((fun (x : T) (y : T') => E) @2` (A, fun x => B)) (at level 0, E, x at level 99, A at level 200, y at level 99, only parsing) : fset_scope. Notation "[ 'fset[' key ] E \| x 'in' A , y 'in' B ]" := [fset[key] E \| x : _ in A, y : _ in B] (at level 0, E, x at level 99, A at level 200, y at level 99, only parsing) : fset_scope. Notation "[ 'fset[' key ] E \| x : A , y : B ]" := [fset[key] E \| x : _ in {: A}, y : _ in {: B}] (at level 0, E, x at level 99, A at level 200, y at level 99, only parsing) : fset_scope. ( printing only ) Notation "[ 'f' 'set' E \| x 'in' A ]" := [fset[_] E \| x in A] (at level 0, E, x at level 99, format "[ '[hv' 'f' 'set' E '/ ' \| x 'in' A ] ']'") : fset_scope. Notation "[ 'f' 'set' E \| x : A ]" := [fset[_] E \| x : A] (at level 0, E, x at level 99, format "[ '[hv' 'f' 'set' E '/ ' \| x : A ] ']'") : fset_scope. Notation "[ 'f' 'set' x 'in' A \| P ]" := [fset[_] x in A \| P] (at level 0, x at level 99, format "[ 'f' 'set' x 'in' A \| P ]") : fset_scope. Notation "[ 'f' 'set' x 'in' A ]" := [fset[_] x in A] (at level 0, x at level 99, format "[ 'f' 'set' x 'in' A ]") : fset_scope. Notation "[ 'f' 'set' x : T \| P ]" := [fset[_] x : T \| P] (at level 0, x at level 99, format "[ 'f' 'set' x : T \| P ]") : fset_scope. Notation "[ 'f' 'set' x : T \| P & Q ]" := [fset[_] x : T \| P & Q] (at level 0, x at level 99, format "[ 'f' 'set' x : T \| P & Q ]") : fset_scope. Notation "[ 'f' 'set' x 'in' A \| P & Q ]" := [fset[_] x in A \| P & Q] (at level 0, x at level 99, format "[ 'f' 'set' x 'in' A \| P & Q ]") : fset_scope. ( binary printing only) Notation "[ 'f' 'set' E \| x 'in' A , y 'in' B ]" := [fset[_] E \| x in A, y in B] (at level 0, E, x at level 99, A at level 200, y at level 99, format "[ '[hv' 'f' 'set' E '/ ' \| x 'in' A , '/' y 'in' B ] ']'" ) : fset_scope. Section Ops. Context {K K': choiceType}. Implicit Types (a b c : K) (A B C D : {fset K}) (E : {fset K'}) (s : seq K). Definition fset0 : {fset K} := @mkFinSet K [::] (introT eqP (@sort_keys_nil K)). ( very transparent, to ensure x \in fset0 is convertible to false ) Fact fset1_key : unit. Proof. exact: tt. Qed. Definition fset1 a : {fset K} := [fset[fset1_key] x in [:: a]]. Fact fsetU_key : unit. Proof. exact: tt. Qed. Definition fsetU A B := [fset[fsetU_key] x in enum_fset A ++ enum_fset B]. Fact fsetI_key : unit. Proof. exact: tt. Qed. Definition fsetI A B := [fset[fsetI_key] x in A \| x \in B]. Fact fsetD_key : unit. Proof. exact: tt. Qed. Definition fsetD A B := [fset[fsetD_key] x in A \| x \notin B]. Fact fsetM_key : unit. Proof. exact: tt. Qed. Definition fsetM A E := [fset[fsetM_key] (x, y) \| x : K in A, y : K' in E]. Definition fsubset A B := fsetI A B == A. Definition fproper A B := fsubset A B && ~~ fsubset B A. Definition fdisjoint A B := (fsetI A B == fset0). End Ops. Notation "[ 'fset' a ]" := (fset1 a) (at level 0, a at level 99, format "[ 'fset' a ]") : fset_scope. Notation "[ 'fset' a : T ]" := [fset (a : T)] (at level 0, a at level 99, format "[ 'fset' a : T ]") : fset_scope. Notation "A `\|` B" := (fsetU A B) : fset_scope. Notation "a \|` A" := ([fset a] `\|` A) : fset_scope. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( This is left-associative due to historical limitations of the .. Notation. ) Notation "[ 'fset' a1 ; a2 ; .. ; an ]" := (fsetU .. (a1 \|` [fset a2]) .. [fset an]) (at level 0, a1 at level 99, format "[ 'fset' a1 ; a2 ; .. ; an ]") : fset_scope. Notation "A `&` B" := (fsetI A B) : fset_scope. Notation "A `` B" := (fsetM A B) : fset_scope. Notation "A `\` B" := (fsetD A B) : fset_scope. Notation "A `\ a" := (A `\` [fset a]) : fset_scope. Notation "A `<=` B" := (fsubset A B) (at level 70, no associativity) : fset_scope. Notation "A `<` B" := (fproper A B) (at level 70, no associativity) : fset_scope. Notation "[ 'disjoint' A & B ]" := (fdisjoint A B) : fset_scope. (* Comprehensions ) Notation "[ 'fset' E \| x 'in' A & P ]" := [fset E \| x in [pred x in A \| P]] (at level 0, E, x at level 99, only parsing) : fset_scope. Notation "[ 'fset' E \| x : A & P ]" := [fset E \| x in {: A} & P] (at level 0, E, x at level 99, only parsing) : fset_scope. Notation "[ 'fset' E \| x : A , y 'in' B ]" := [fset E \| x in {: A}, y in B] (at level 0, E, x, y at level 99, only parsing) : fset_scope. Notation "[ 'fset' E \| x 'in' A , y : B ]" := [fset E \| x in A, y in {: B}] (at level 0, E, x, y at level 99, only parsing) : fset_scope. Notation "[ 'fset' E \| x : A , y : B ]" := [fset E \| x in {: A}, y in {: B}] (at level 0, E, x, y at level 99, only parsing) : fset_scope. Notation "[ 'fset' E \| x 'in' A , y 'in' B & P ]" := [fset E \| x in A, y in [pred y in B \| P]] (at level 0, E, x, y at level 99, only parsing) : fset_scope. Notation "[ 'fset' E \| x : A , y 'in' B & P ]" := [fset E \| x in {: A}, y in B & P] (at level 0, E, x, y at level 99, only parsing) : fset_scope. Notation "[ 'fset' E \| x 'in' A , y : B & P ]" := [fset E \| x in A, y in {: B} & P] (at level 0, E, x, y at level 99, only parsing) : fset_scope. Notation "[ 'fset' E \| x : A , y : B & P ]" := [fset E \| x in {: A}, y in {: B} & P] (at level 0, E, x, y at level 99, only parsing) : fset_scope. Notation "[ 'fsetval' x 'in' A ]" := [fset val x \| x in A] (at level 0, x at level 99, only parsing) : fset_scope. Notation "[ 'fsetval' x 'in' A \| P ]" := [fset val x \| x in A & P] (at level 0, x at level 99, only parsing) : fset_scope. Notation "[ 'fsetval' x 'in' A \| P & Q ]" := [fsetval x in A \| (P && Q)] (at level 0, x at level 99, only parsing) : fset_scope. Notation "[ 'fsetval' x : A ]" := [fset val x \| x in {: A}] (at level 0, x at level 99, only parsing) : fset_scope. Notation "[ 'fsetval' x : A \| P ]" := [fset val x \| x in {: A} & P] (at level 0, x at level 99, only parsing) : fset_scope. Notation "[ 'fsetval' x : A \| P & Q ]" := [fsetval x in {: A} \| (P && Q)] (at level 0, x at level 99, only parsing) : fset_scope. ( keyed parse only ) Notation "[ 'fset[' key ] E \| x 'in' A & P ]" := [fset[key] E \| x in [pred x in A \| P]] (at level 0, E, x at level 99, only parsing). Notation "[ 'fset[' key ] E \| x : A & P ]" := [fset[key] E \| x in {: A} & P] (at level 0, E, x at level 99, only parsing). Notation "[ 'fset[' key ] E \| x : A , y 'in' B ]" := [fset[key] E \| x in {: A}, y in B] (at level 0, E, x, y at level 99, only parsing). Notation "[ 'fset[' key ] E \| x 'in' A , y : B ]" := [fset[key] E \| x in A, y in {: B}] (at level 0, E, x, y at level 99, only parsing) : fset_scope. Notation "[ 'fset[' key ] E \| x : A , y : B ]" := [fset[key] E \| x in {: A}, y in {: B}] (at level 0, E, x, y at level 99, only parsing) : fset_scope. Notation "[ 'fset[' key ] E \| x 'in' A , y 'in' B & P ]" := [fset[key] E \| x in A, y in [pred y in B \| P]] (at level 0, E, x, y at level 99, only parsing) : fset_scope. Notation "[ 'fset[' key ] E \| x : A , y 'in' B & P ]" := [fset[key] E \| x in {: A}, y in B & P] (at level 0, E, x, y at level 99, only parsing) : fset_scope. Notation "[ 'fset[' key ] E \| x 'in' A , y : B & P ]" := [fset[key] E \| x in A, y in {: B} & P] (at level 0, E, x, y at level 99, only parsing) : fset_scope. Notation "[ 'fset[' key ] E \| x : A , y : B & P ]" := [fset[key] E \| x in {: A}, y in {: B} & P] (at level 0, E, x, y at level 99, only parsing) : fset_scope. Notation "[ 'fsetval[' key ] x 'in' A ]" := [fset[key] val x \| x in A] (at level 0, x at level 99, only parsing) : fset_scope. Notation "[ 'fsetval[' key ] x 'in' A \| P ]" := [fset[key] val x \| x in A & P] (at level 0, x at level 99, only parsing) : fset_scope. Notation "[ 'fsetval[' key ] x 'in' A \| P & Q ]" := [fsetval[key] x in A \| (P && Q)] (at level 0, x at level 99, only parsing) : fset_scope. Notation "[ 'fsetval[' key ] x : A ]" := [fset[key] val x \| x in {: A}] (at level 0, x at level 99, only parsing) : fset_scope. Notation "[ 'fsetval[' key ] x : A \| P ]" := [fset[key] val x \| x in {: A} & P] (at level 0, x at level 99, only parsing) : fset_scope. Notation "[ 'fsetval[' key ] x : A \| P & Q ]" := [fsetval[key] x in {: A} \| (P && Q)] (at level 0, x at level 99, only parsing) : fset_scope. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Print-only variants to work around the Coq pretty-printer K-term kink. ) Notation "[ 'f' 'set' E \| x 'in' A & P ]" := [fset[_] E \| x in A & P] (at level 0, E, x at level 99, format "[ '[hv' 'f' 'set' E '/ ' \| x 'in' A '/ ' & P ] ']'" ) : fset_scope. Notation "[ 'f' 'set' E \| x : A & P ]" := [fset[_] E \| x : A & P] (at level 0, E, x at level 99, format "[ '[hv' 'f' 'set' E '/ ' \| x : A '/ ' & P ] ']'") : fset_scope. Notation "[ 'f' 'set' E \| x : A , y 'in' B ]" := [fset[_] E \| x : A, y in B] (at level 0, E, x, y at level 99, format "[ '[hv' 'f' 'set' E '/ ' \| x : A , '/ ' y 'in' B ] ']'" ) : fset_scope. Notation "[ 'f' 'set' E \| x 'in' A , y : B ]" := [fset[_] E \| x in A, y : B] (at level 0, E, x, y at level 99, format "[ '[hv' 'f' 'set' E '/ ' \| x 'in' A , '/ ' y : B ] ']'" ) : fset_scope. Notation "[ 'f' 'set' E \| x : A , y : B ]" := [fset[_] E \| x : A, y : B] (at level 0, E, x, y at level 99, format "[ '[hv' 'f' 'set' E '/ ' \| x : A , '/ ' y : B ] ']'" ) : fset_scope. Notation "[ 'f' 'set' E \| x 'in' A , y 'in' B & P ]" := [fset[_] E \| x in A, y in B & P] (at level 0, E, x, y at level 99, format "[ '[hv' 'f' 'set' E '/ ' \| x 'in' A , '/ ' y 'in' B '/ ' & P ] ']'" ) : fset_scope. Notation "[ 'f' 'set' E \| x : A , y 'in' B & P ]" := [fset[_] E \| x : A, y in B & P] (at level 0, E, x, y at level 99, format "[ '[hv' 'f' 'set' E '/ ' \| x : A , '/ ' y 'in' B & P ] ']'" ) : fset_scope. Notation "[ 'f' 'set' E \| x 'in' A , y : B & P ]" := [fset[_] E \| x in A, y : B & P] (at level 0, E, x, y at level 99, format "[ '[hv' 'f' 'set' E '/ ' \| x 'in' A , '/ ' y : B & P ] ']'" ) : fset_scope. Notation "[ 'f' 'set' E \| x : A , y : B & P ]" := [fset[_] E \| x : A, y : B & P] (at level 0, E, x, y at level 99, format "[ '[hv' 'f' 'set' E '/ ' \| x : A , '/ ' y : B & P ] ']'" ) : fset_scope. Notation "[ 'f' 'setval' x 'in' A ]" := [fset[_] val x \| x in A] (at level 0, x at level 99, format "[ 'f' 'setval' x 'in' A ]") : fset_scope. Notation "[ 'f' 'setval' x 'in' A \| P ]" := [fset[_] val x \| x in A & P] (at level 0, x at level 99, format "[ 'f' 'setval' x 'in' A \| P ]") : fset_scope. Notation "[ 'f' 'setval' x 'in' A \| P & Q ]" := [fsetval[_] x in A \| (P && Q)] (at level 0, x at level 99, format "[ 'f' 'setval' x 'in' A \| P & Q ]") : fset_scope. Notation "[ 'f' 'setval' x : A ]" := [fsetval[_] x : A] (at level 0, x at level 99, format "[ 'f' 'setval' x : A ]") : fset_scope. Notation "[ 'f' 'setval' x : A \| P ]" := [fsetval[_] x : A \| P] (at level 0, x at level 99, format "[ 'f' 'setval' x : A \| P ]") : fset_scope. Notation "[ 'f' 'setval' x : A \| P & Q ]" := [fsetval[_] x : A \| (P && Q)] (at level 0, x at level 99, format "[ 'f' 'setval' x : A \| P & Q ]") : fset_scope. Section imfset. Variables (key : unit) (K : choiceType). Implicit Types (A B : {fset K}). ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma imfsetP (T : choiceType) (f : T -> K) (p : finmempred T) (k : K) : reflect (exists2 x : T, in_mem x p & k = f x) (k \in imfset key f p). Proof. rewrite unlock seq_fsetE /=; apply: (iffP mapP) => [] [x xp eqkf]; by exists x => //=; move: xp; rewrite enum_finmemE. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma in_imfset (T : choiceType) (f : T -> K) (p : finmempred T) (x : T) : ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) in_mem x p -> f x \in imfset key f p. Proof. by move=> px; apply/imfsetP; exists x. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma imfset_rec (T : choiceType) (f : T -> K) (p : finmempred T) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) (P : imfset key f p -> Prop) : ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) (forall (x : T) (px : in_mem x p), P [` in_imfset f px ]) -> forall k, P k. Proof. ( Goal: forall (_ : forall (x : Choice.sort T) (px : is_true (@in_mem (Choice.sort T) x (@pred_of_finmempred (Choice.eqType T) p))), P (@FSetSub K (@Imfset.imfset key T K f p (Phantom (mem_pred (Equality.sort (Choice.eqType T))) (@pred_of_finmempred (Choice.eqType T) p))) (f x) (@in_imfset T f p x px))) (k : @fset_sub_type K (@Imfset.imfset key T K f p (Phantom (mem_pred (Equality.sort (Choice.eqType T))) (@pred_of_finmempred (Choice.eqType T) p)))), P k ) move=> PP v; have /imfsetP [k pk vv_eq] := valP v. ( Goal: P v ) pose vP := in_imfset f pk; suff -> : P v = P [` vP] by apply: PP. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by congr P; apply/val_inj => /=; rewrite vv_eq. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma mem_imfset (T : choiceType) (f : T -> K) (p : finmempred T) : ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) injective f -> forall (x : T), (f x \in imfset key f p) = (in_mem x p). Proof. by move=> f_inj x; rewrite unlock seq_fsetE mem_map// enum_finmemE. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma imfset2P (T1 : choiceType) (T2 : T1 -> choiceType) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) (f : forall x, T2 x -> K) (p1 : finmempred T1) (p2 : forall x, finmempred (T2 x)) k : reflect (exists2 x : T1, in_mem x p1 & exists2 y : T2 x, in_mem y (p2 x) & k = f x y) (k \in imfset2 key f p1 p2). Proof. ( Goal: is_true (@in_mem (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f (@fsval K (@FinSupp.fs K (Choice.eqType K) (fun x : Choice.sort K => x) f) a)) (@mem (Choice.sort K) (finSetPredType K) (@FinSupp.fs K (Choice.eqType K) (fun x : Choice.sort K => x) f))) ) ( Goal: @ex2 (@finset_of K (Phant (Choice.sort K))) (fun S : @finset_of K (Phant (Choice.sort K)) => is_true (@fsubset K (@FinSupp.fs K (Choice.eqType K) (fun x : Choice.sort K => x) f) S)) (fun S : @finset_of K (Phant (Choice.sort K)) => and (@prop_in2 (Choice.sort K) (@mem (Choice.sort K) (finSetPredType K) S) (fun x1 x2 : Choice.sort K => forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f x1) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f x2), @eq (Choice.sort K) x1 x2) (inPhantom (@injective (Choice.sort K) (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f)))) (forall a : @fset_sub_type K S, is_true (@in_mem (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f (@fsval K S a)) (@mem (Choice.sort K) (finSetPredType K) S)))) ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) rewrite unlock !seq_fsetE; apply: (iffP (allsigsP _ _ _ _)). ( Goal: forall (_ : @eq (@finset_of K (Phant (Choice.sort K))) (@domf K V f) (@domf K V g)) (_ : forall (k : Choice.sort K) (kMf : is_true (@in_mem (Choice.sort K) k (@mem (Choice.sort K) (finMapPredType K V) f))) (kMg : is_true (@in_mem (Choice.sort K) k (@mem (Choice.sort K) (finMapPredType K V) g))), @eq V (@FunFinfun.fun_of_fin (@fset_sub_finType K (@domf K V f)) V (@ffun_of_fmap K V f) (@FSetSub K (@domf K V f) k kMf)) (@FunFinfun.fun_of_fin (@fset_sub_finType K (@domf K V g)) V (@ffun_of_fmap K V g) (@FSetSub K (@domf K V g) k kMg))), @eq (@finmap_of K V (Phant (forall _ : Choice.sort K, V))) f g ) move=> [[/=x y]]; rewrite !enum_finmemE => -[xp yp ->]. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by exists x => //; exists y. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by move=> [x xp [y yp ->]]; exists (Tagged T2 y); rewrite ?enum_finmemE. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma in_imfset2 (T1 : choiceType) (T2 : T1 -> choiceType) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) (f : forall x, T2 x -> K) (p1 : finmempred T1) (p2 : forall x, finmempred (T2 x)) (x : T1) (y : T2 x) : ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) in_mem x p1 -> in_mem y (p2 x) -> f x y \in imfset2 key f p1 p2. Proof. by move=> xD1 yD2; apply/imfset2P; exists x => //; exists y. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma mem_imfset2 (T1 : choiceType) (T2 : T1 -> choiceType) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) (f : forall x, T2 x -> K) (p1 : finmempred T1) (p2 : forall x, finmempred (T2 x)) (x : T1) (y : T2 x) : ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) injective (fun x : sigT T2 => f (tag x) (tagged x)) -> f x y \in imfset2 key f p1 p2 = (in_mem x p1) && (in_mem y (p2 x)). Proof. ( Goal: forall (_ : is_true (@fdisjoint K (@domf K V f) (@domf K V h))) (_ : is_true (@fdisjoint K (@domf K V g) (@domf K V h))) (_ : @eq (finMap K V) (catf h f) (catf h g)), @eq (@finmap_of K V (Phant (forall _ : Choice.sort K, V))) f g ) move=> f_inj; rewrite unlock seq_fsetE. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) apply/allsigsP/idP => [[t]\|]; last first. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by move=> /andP[xp1 xp2]; exists (Tagged T2 y); rewrite ?enum_finmemE. rewrite !enum_finmemE => -[pt1 pt2]; pose xy := Tagged T2 y. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite -[x]/(tag xy) -[y]/(tagged xy) => /f_inj ->; apply/andP. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma enum_imfset (T : choiceType) (f : T -> K) (p : finmempred T) : ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) {in p &, injective f} -> ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) perm_eq (imfset key f p) [seq f x \| x <- enum_finmem p]. Proof. ( Goal: forall (_ : is_true (@fdisjoint K (@domf K V f) (@domf K V h))) (_ : is_true (@fdisjoint K (@domf K V g) (@domf K V h))) (_ : @eq (finMap K V) (catf h f) (catf h g)), @eq (@finmap_of K V (Phant (forall _ : Choice.sort K, V))) f g ) move=> f_inj; rewrite unlock -[X in perm_eq _ X]undup_id ?seq_fset_perm//. ( Goal: @big_rel_class_of (@finset_of K (Phant (Choice.sort K))) (@fsubset K) ) rewrite map_inj_in_uniq ?enum_finmem_uniq // => ??. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite ?enum_finmemE; apply: f_inj. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma enum_imfset2 (T1 : choiceType) (T2 : T1 -> choiceType) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) (f : forall x, T2 x -> K) (p1 : finmempred T1) (p2 : forall x, finmempred (T2 x)) : {in [pred t \| p1 (tag t) & p2 _ (tagged t)] &, ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) injective (fun t : sigT T2 => f (tag t) (tagged t))} -> perm_eq (imfset2 key f p1 p2) (allsigs f (enum_finmem p1) (fun x => enum_finmem (p2 x))). Proof. ( Goal: forall (_ : is_true (@fdisjoint K (@domf K V f) (@domf K V h))) (_ : is_true (@fdisjoint K (@domf K V g) (@domf K V h))) (_ : @eq (finMap K V) (catf h f) (catf h g)), @eq (@finmap_of K V (Phant (forall _ : Choice.sort K, V))) f g ) move=> f_inj; rewrite unlock. ( Goal: is_true (@fsubset K (@FinSupp.fs K V (fun a : Choice.sort K => default a) (@fsfun_of_ffun key K V S (fun a : @fset_sub_type K S => h a) (fun a : Choice.sort K => default a))) S) ) apply: uniq_perm_eq => [\|\|i]; rewrite ?seq_fset_uniq ?seq_fsetE //. ( Goal: @eq (Equality.sort V) (@fun_of_fsfun K V default (of_ffun key) x) (@Option.default (Equality.sort V) (default x) (@Option.map (@fset_sub_type K S) (Equality.sort V) h (@insub (Choice.sort K) (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S) x))) ) rewrite allsigs_uniq ?enum_finmem_uniq//. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by move=> x; rewrite enum_finmem_uniq. ( Goal: forall (_ : @eq (@finset_of K (Phant (Choice.sort K))) (@domf K V f) (@domf K V g)) (_ : forall (k : Choice.sort K) (kMf : is_true (@in_mem (Choice.sort K) k (@mem (Choice.sort K) (finMapPredType K V) f))) (kMg : is_true (@in_mem (Choice.sort K) k (@mem (Choice.sort K) (finMapPredType K V) g))), @eq V (@FunFinfun.fun_of_fin (@fset_sub_finType K (@domf K V f)) V (@ffun_of_fmap K V f) (@FSetSub K (@domf K V f) k kMf)) (@FunFinfun.fun_of_fin (@fset_sub_finType K (@domf K V g)) V (@ffun_of_fmap K V g) (@FSetSub K (@domf K V g) k kMg))), @eq (@finmap_of K V (Phant (forall _ : Choice.sort K, V))) f g ) move=> t0 t0' /allsigsP[t]; rewrite !enum_finmemE => -[tp1 tp2 ->]. ( Goal: forall (_ : @eq (@finset_of K (Phant (Choice.sort K))) (@domf K V f) (@domf K V g)) (_ : forall (k : Choice.sort K) (kMf : is_true (@in_mem (Choice.sort K) k (@mem (Choice.sort K) (finMapPredType K V) f))) (kMg : is_true (@in_mem (Choice.sort K) k (@mem (Choice.sort K) (finMapPredType K V) g))), @eq V (@FunFinfun.fun_of_fin (@fset_sub_finType K (@domf K V f)) V (@ffun_of_fmap K V f) (@FSetSub K (@domf K V f) k kMf)) (@FunFinfun.fun_of_fin (@fset_sub_finType K (@domf K V g)) V (@ffun_of_fmap K V g) (@FSetSub K (@domf K V g) k kMg))), @eq (@finmap_of K V (Phant (forall _ : Choice.sort K, V))) f g ) move=> /allsigsP[t']; rewrite !enum_finmemE => -[t'p1 t'p2 ->]. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by apply: f_inj; rewrite inE; apply/andP. Qed. End imfset. Section in_imfset. Variable (key : unit) (K : choiceType). Implicit Types (A B : {fset K}) (a b : K). Lemma in_fset (p : finmempred K) (k : K) : (k \in imfset key id p) = in_mem k p. Proof. by rewrite mem_imfset; apply: inj_id. Qed. Lemma val_in_fset A (p : finmempred _) (k : A) : (val k \in imfset key val p) = (in_mem k p). Proof. by rewrite mem_imfset ?in_finmempred //; exact: val_inj. Qed. Lemma in_fset_val A (p : finmempred [eqType of A]) (k : K) : (k \in imfset key val p) = if insub k is Some a then in_mem a p else false. Proof. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) have [a _ <- /=\|kNA] := insubP; first by rewrite val_in_fset. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by apply/imfsetP => [] [a _ k_def]; move: kNA; rewrite k_def [_ \in _]valP. Qed. Lemma in_fset_valT A (p : finmempred _) (k : K) (kA : k \in A) : (k \in imfset key val p) = in_mem [` kA] p. Proof. by rewrite in_fset_val insubT /=. Qed. Lemma in_fset_valP A (p : finmempred _) (k : K) : reflect {kA : k \in A & in_mem [` kA] p} (k \in imfset key val p). Proof. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) apply: (iffP (imfsetP _ _ _ _)) => [\|[kA xkA]]; last by exists [`kA]. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by move=> /sig2_eqW[/= x Xx ->]; exists (valP x); rewrite fsetsubE. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma in_fset_valF A (p : finmempred [eqType of A]) (k : K) : k \notin A -> (k \in imfset key val p) = false. Proof. by apply: contraNF => /imfsetP[/= a Xa->]. Qed. Lemma in_fset_nil a : a \in [fset[key] x in [::]] = false. Proof. by rewrite !mem_imfset. Qed. Lemma in_fset_cons x (xs : seq K) a : (a \in [fset[key] x in x :: xs]) = ((a == x) \|\| (a \in [fset[key] x in xs])). Proof. by rewrite !mem_imfset. Qed. Lemma in_fset_cat (xs ys : seq K) a : (a \in [fset[key] x in xs ++ ys]) = ((a \in [fset[key] x in xs]) \|\| (a \in [fset[key] x in ys])). Proof. by rewrite !mem_imfset//= mem_cat. Qed. Definition in_fset_ (key : unit) := (in_fset_cons, in_fset_nil, in_fset_cat, in_fset). ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma card_in_imfset (T T' : choiceType) (f : T -> T') (p : finmempred T) : ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) {in p &, injective f} -> #\|` (imfset key f p)\| = (size (enum_finmem p)). Proof. ( Goal: forall (_ : is_true (@fdisjoint K (@domf K V f) (@domf K V h))) (_ : is_true (@fdisjoint K (@domf K V g) (@domf K V h))) (_ : @eq (finMap K V) (catf h f) (catf h g)), @eq (@finmap_of K V (Phant (forall _ : Choice.sort K, V))) f g ) move=> f_inj; rewrite unlock /= size_seq_fset undup_id ?size_map//. ( Goal: @big_rel_class_of (@finset_of K (Phant (Choice.sort K))) (@fsubset K) ) rewrite map_inj_in_uniq ?enum_finmem_uniq// => ??. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite !enum_finmemE; apply: f_inj. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma card_imfset (T T' : choiceType) (f : T -> T') (p : finmempred _) : ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) injective f -> #\|` (imfset key f p)\| = size (enum_finmem p). Proof. by move=> f_inj; rewrite card_in_imfset //= => x y ? ?; apply: f_inj. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma leq_imfset_card (T T' : choiceType) (f : T -> T') (p : finmempred _) : (#\|` imfset key f p\| <= size (enum_finmem p))%N. Proof. by rewrite unlock size_seq_fset (leq_trans (size_undup _)) ?size_map. Qed. End in_imfset. Section Theory. Variables (key : unit) (K K': choiceType). Implicit Types (a b x : K) (A B C D : {fset K}) (E : {fset K'}) (pA pB pC : pred K) (s : seq K). Lemma fsetP {A B} : A =i B <-> A = B. Proof. by split=> [eqAB\|-> //]; apply/val_inj/canonical_eq_keys => //= a. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) CoInductive in_fset_spec (A : {fset K}) (x : K) : K -> bool -> Prop := \| InFset (u : A) & x = val u : in_fset_spec A x (val u) true \| OutFset of x \notin A : in_fset_spec A x x false. Lemma in_fsetP A x : in_fset_spec A x x (x \in A). Proof. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) have [xA\|xNA] := boolP (x \in A); last by constructor. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by have {2}-> : x = val [` xA] by []; constructor. Qed. Lemma fset_eqP {A B} : reflect (A =i B) (A == B). Proof. exact: (equivP eqP (iff_sym fsetP)). Qed. Lemma in_fset0 x : x \in fset0 = false. Proof. by []. Qed. Lemma in_fset1 a' a : a \in [fset a'] = (a == a'). Proof. by rewrite !in_fset_ orbF. Qed. Lemma in_fsetU A B a : (a \in A `\|` B) = (a \in A) \|\| (a \in B). Proof. by rewrite !in_fset_. Qed. Lemma in_fset1U a' A a : (a \in a' \|` A) = (a == a') \|\| (a \in A). Proof. by rewrite in_fsetU in_fset1. Qed. Lemma in_fsetI A B a : (a \in A `&` B) = (a \in A) && (a \in B). Proof. by rewrite in_fset. Qed. Lemma in_fsetD A B a : (a \in A `\` B) = (a \notin B) && (a \in A). Proof. by rewrite in_fset andbC. Qed. Lemma in_fsetD1 A b a : (a \in A `\ b) = (a != b) && (a \in A). Proof. by rewrite in_fsetD in_fset1. Qed. Lemma in_fsetM A E (u : K K') : (u \in A `` E) = (u.1 \in A) && (u.2 \in E). Proof. by case: u => /= x y; rewrite mem_imfset2//= => -[??] [??] [-> ->]. Qed. Definition in_fsetE := (@in_fset_ imfset_key, val_in_fset, in_fset0, in_fset1, in_fsetU, in_fsetI, in_fsetD, in_fsetM, in_fset1U, in_fsetD1). Let inE := (inE, in_fsetE). Lemma fsetIC (A B : {fset K}) : A `&` B = B `&` A. Proof. by apply/fsetP => a; rewrite !inE andbC. Qed. Lemma fsetUC (A B : {fset K}) : A `\|` B = B `\|` A. Proof. by apply/fsetP => a; rewrite !inE orbC. Qed. Lemma fset0I A : fset0 `&` A = fset0. Proof. by apply/fsetP => x; rewrite !inE andFb. Qed. Lemma fsetI0 A : A `&` fset0 = fset0. Proof. by rewrite fsetIC fset0I. Qed. Lemma fsetIA A B C : A `&` (B `&` C) = A `&` B `&` C. Proof. by apply/fsetP=> x; rewrite !inE andbA. Qed. Lemma fsetICA A B C : A `&` (B `&` C) = B `&` (A `&` C). Proof. by rewrite !fsetIA (fsetIC A). Qed. Lemma fsetIAC A B C : A `&` B `&` C = A `&` C `&` B. Proof. by rewrite -!fsetIA (fsetIC B). Qed. Lemma fsetIACA A B C D : (A `&` B) `&` (C `&` D) = (A `&` C) `&` (B `&` D). Proof. by rewrite -!fsetIA (fsetICA B). Qed. Lemma fsetIid A : A `&` A = A. Proof. by apply/fsetP=> x; rewrite inE andbb. Qed. Lemma fsetIIl A B C : A `&` B `&` C = (A `&` C) `&` (B `&` C). Proof. by rewrite fsetIA !(fsetIAC _ C) -(fsetIA _ C) fsetIid. Qed. Lemma fsetIIr A B C : A `&` (B `&` C) = (A `&` B) `&` (A `&` C). Proof. by rewrite !(fsetIC A) fsetIIl. Qed. Lemma fsetUA A B C : A `\|` (B `\|` C) = A `\|` B `\|` C. Proof. by apply/fsetP => x; rewrite !inE orbA. Qed. Lemma fsetUCA A B C : A `\|` (B `\|` C) = B `\|` (A `\|` C). Proof. by rewrite !fsetUA (fsetUC B). Qed. Lemma fsetUAC A B C : A `\|` B `\|` C = A `\|` C `\|` B. Proof. by rewrite -!fsetUA (fsetUC B). Qed. Lemma fsetUACA A B C D : (A `\|` B) `\|` (C `\|` D) = (A `\|` C) `\|` (B `\|` D). Proof. by rewrite -!fsetUA (fsetUCA B). Qed. Lemma fsetUid A : A `\|` A = A. Proof. by apply/fsetP=> x; rewrite inE orbb. Qed. Lemma fsetUUl A B C : A `\|` B `\|` C = (A `\|` C) `\|` (B `\|` C). Proof. by rewrite fsetUA !(fsetUAC _ C) -(fsetUA _ C) fsetUid. Qed. Lemma fsetUUr A B C : A `\|` (B `\|` C) = (A `\|` B) `\|` (A `\|` C). Proof. by rewrite !(fsetUC A) fsetUUl. Qed. ( distribute /cancel ) Lemma fsetIUr A B C : A `&` (B `\|` C) = (A `&` B) `\|` (A `&` C). Proof. by apply/fsetP=> x; rewrite !inE andb_orr. Qed. Lemma fsetIUl A B C : (A `\|` B) `&` C = (A `&` C) `\|` (B `&` C). Proof. by apply/fsetP=> x; rewrite !inE andb_orl. Qed. Lemma fsetUIr A B C : A `\|` (B `&` C) = (A `\|` B) `&` (A `\|` C). Proof. by apply/fsetP=> x; rewrite !inE orb_andr. Qed. Lemma fsetUIl A B C : (A `&` B) `\|` C = (A `\|` C) `&` (B `\|` C). Proof. by apply/fsetP=> x; rewrite !inE orb_andl. Qed. Lemma fsetUKC A B : (A `\|` B) `&` A = A. Proof. by apply/fsetP=> x; rewrite !inE orbK. Qed. Lemma fsetUK A B : (B `\|` A) `&` A = A. Proof. by rewrite fsetUC fsetUKC. Qed. Lemma fsetKUC A B : A `&` (B `\|` A) = A. Proof. by rewrite fsetIC fsetUK. Qed. Lemma fsetKU A B : A `&` (A `\|` B) = A. Proof. by rewrite fsetIC fsetUKC. Qed. Lemma fsetIKC A B : (A `&` B) `\|` A = A. Proof. by apply/fsetP=> x; rewrite !inE andbK. Qed. Lemma fsetIK A B : (B `&` A) `\|` A = A. Proof. by rewrite fsetIC fsetIKC. Qed. Lemma fsetKIC A B : A `\|` (B `&` A) = A. Proof. by rewrite fsetUC fsetIK. Qed. Lemma fsetKI A B : A `\|` (A `&` B) = A. Proof. by rewrite fsetIC fsetKIC. Qed. Lemma fsetUKid A B : B `\|` A `\|` A = B `\|` A. Proof. by rewrite -fsetUA fsetUid. Qed. Lemma fsetUKidC A B : A `\|` B `\|` A = A `\|` B. Proof. by rewrite fsetUAC fsetUid. Qed. Lemma fsetKUid A B : A `\|` (A `\|` B) = A `\|` B. Proof. by rewrite fsetUA fsetUid. Qed. Lemma fsetKUidC A B : A `\|` (B `\|` A) = B `\|` A. Proof. by rewrite fsetUCA fsetUid. Qed. Lemma fsetIKid A B : B `&` A `&` A = B `&` A. Proof. by rewrite -fsetIA fsetIid. Qed. Lemma fsetIKidC A B : A `&` B `&` A = A `&` B. Proof. by rewrite fsetIAC fsetIid. Qed. Lemma fsetKIid A B : A `&` (A `&` B) = A `&` B. Proof. by rewrite fsetIA fsetIid. Qed. Lemma fsetKIidC A B : A `&` (B `&` A) = B `&` A. Proof. by rewrite fsetICA fsetIid. Qed. ( subset ) Lemma fsubsetP {A B} : reflect {subset A <= B} (A `<=` B). Proof. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) apply: (iffP fset_eqP) => AsubB a; first by rewrite -AsubB inE => /andP[]. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite inE; have [/AsubB\|] := boolP (a \in A). Qed. Lemma fset_sub_val A (p : finmempred [eqType of A]) : (imfset key val p) `<=` A. Proof. by apply/fsubsetP => k /in_fset_valP []. Qed. Lemma fset_sub A (P : pred K) : [fset x in A \| P x] `<=` A. Proof. by apply/fsubsetP => k; rewrite in_fset inE /= => /andP []. Qed. Lemma fsetD_eq0 (A B : {fset K}) : (A `\` B == fset0) = (A `<=` B). Proof. ( Goal: Bool.reflect (@ex (Choice.sort K) (fun x : Choice.sort K => @eq (option (Choice.sort V)) (@fnd K (Choice.sort V) (@domf K (Choice.sort V) f) (@ffun_of_fmap K (Choice.sort V) f) x) (@Some (Choice.sort V) v))) (@in_mem (Choice.sort V) v (@mem (Choice.sort V) (finSetPredType V) (codomf f))) ) apply/fset_eqP/fsubsetP => sAB a. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by move=> aA; have := sAB a; rewrite !inE aA andbT => /negPn. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite !inE andbC; apply/negP => /andP [/sAB ->]. Qed. Lemma fsubset_refl A : A `<=` A. Proof. exact/fsubsetP. Qed. Hint Resolve fsubset_refl. Definition fincl A B (AsubB : A `<=` B) (a : A) : B := [` (fsubsetP AsubB) _ (valP a)]. Definition fsub B A : {set B} := [set x : B \| val x \in A]. Lemma fsubE A B (AsubB : A `<=` B) : fsub B A = [set fincl AsubB x \| x : A]. Proof. ( Goal: @eq (@finset_of V (Phant (Choice.sort V))) (codomf (@fmap0 K (Choice.sort V))) (@fset0 V) ) apply/setP => x; rewrite in_set; apply/idP/imsetP => [\|[[a aA] aA' ->]] //. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by move=> xA; exists [` xA]=> //; apply: val_inj. Qed. Lemma fincl_fsub A B (AsubB : A `<=` B) (a : A) : fincl AsubB a \in fsub B A. Proof. by rewrite inE /= (valP a). Qed. Lemma in_fsub B A (b : B) : (b \in fsub B A) = (val b \in A). Proof. by rewrite inE. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma subset_fsubE C A B : A `<=` C -> B `<=` C -> (fsub C A \subset fsub C B) = (A `<=` B). Proof. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) move=> sAC sBC; apply/subsetP/fsubsetP => sAB a; last first. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite !inE => /sAB. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by move=> aA; have := sAB _ (fincl_fsub sAC [` aA]); rewrite inE. Qed. Lemma fsubset_trans : transitive (@fsubset K). Proof. by move=>??? s t ; apply/fsubsetP => a /(fsubsetP s) /(fsubsetP t). Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma subset_fsub A B C : A `<=` B -> B `<=` C -> fsub C A \subset fsub C B. Proof. by move=> sAB sBC; rewrite subset_fsubE // (fsubset_trans sAB). Qed. Lemma fsetIidPl {A B} : reflect (A `&` B = A) (A `<=` B). Proof. exact: eqP. Qed. Lemma fsetIidPr {A B} : reflect (A `&` B = B) (B `<=` A). Proof. by rewrite fsetIC; apply: fsetIidPl. Qed. Lemma fsubsetIidl A B : (A `<=` A `&` B) = (A `<=` B). Proof. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by apply/fsubsetP/fsubsetP=> sAB a aA; have := sAB _ aA; rewrite !inE ?aA. Qed. Lemma fsubsetIidr A B : (B `<=` A `&` B) = (B `<=` A). Proof. by rewrite fsetIC fsubsetIidl. Qed. Lemma fsetUidPr A B : reflect (A `\|` B = B) (A `<=` B). Proof. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) apply: (iffP fsubsetP) => sAB; last by move=> a aA; rewrite -sAB inE aA. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by apply/fsetP => b; rewrite inE; have [/sAB\|//] := boolP (_ \in _). Qed. Lemma fsetUidPl A B : reflect (A `\|` B = A) (B `<=` A). Proof. by rewrite fsetUC; apply/fsetUidPr. Qed. Lemma fsubsetUl A B : A `<=` A `\|` B. Proof. by apply/fsubsetP => a; rewrite inE => ->. Qed. Hint Resolve fsubsetUl. Lemma fsubsetUr A B : B `<=` A `\|` B. Proof. by rewrite fsetUC. Qed. Hint Resolve fsubsetUr. Lemma fsubsetU1 x A : A `<=` x \|` A. Proof. by rewrite fsubsetUr. Qed. Hint Resolve fsubsetU1. Lemma fsubsetU A B C : (A `<=` B) \|\| (A `<=` C) -> A `<=` B `\|` C. Proof. by move=> /orP [] /fsubset_trans ->. Qed. Lemma fincl_inj A B (AsubB : A `<=` B) : injective (fincl AsubB). Proof. by move=> a b [eq_ab]; apply: val_inj. Qed. Hint Resolve fincl_inj. Lemma fsub_inj B : {in [pred A \| A `<=` B] &, injective (fsub B)}. Proof. ( Goal: @prop_in2 (@finset_of K (Phant (Choice.sort K))) (@mem (@finset_of K (Phant (Choice.sort K))) (simplPredType (@finset_of K (Phant (Choice.sort K)))) (@SimplPred (@finset_of K (Phant (Choice.sort K))) (fun A : @finset_of K (Phant (Choice.sort K)) => @fsubset K A B))) (fun x1 x2 : @finset_of K (Phant (Choice.sort K)) => forall _ : @eq (@set_of (@fset_sub_finType K B) (Phant (@fset_sub_type K B))) (fsub B x1) (fsub B x2), @eq (@finset_of K (Phant (Choice.sort K))) x1 x2) (inPhantom (@injective (@set_of (@fset_sub_finType K B) (Phant (@fset_sub_type K B))) (@finset_of K (Phant (Choice.sort K))) (fsub B))) ) move=> A A'; rewrite -!topredE /= => sAB sA'B /setP eqAA'; apply/fsetP => a. ( Goal: Bool.reflect (@ex (Choice.sort K) (fun x : Choice.sort K => @eq (option (Choice.sort V)) (@fnd K (Choice.sort V) (@domf K (Choice.sort V) f) (@ffun_of_fmap K (Choice.sort V) f) x) (@Some (Choice.sort V) v))) (@in_mem (Choice.sort V) v (@mem (Choice.sort V) (finSetPredType V) (codomf f))) ) apply/idP/idP => mem_a. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by have := eqAA' (fincl sAB [` mem_a]); rewrite !inE // => <-. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by have := eqAA' (fincl sA'B [` mem_a]); rewrite !inE // => ->. Qed. Hint Resolve fsub_inj. Lemma eqEfsubset A B : (A == B) = (A `<=` B) && (B `<=` A). Proof. ( Goal: Bool.reflect (@ex (Choice.sort K) (fun x : Choice.sort K => @eq (option (Choice.sort V)) (@fnd K (Choice.sort V) (@domf K (Choice.sort V) f) (@ffun_of_fmap K (Choice.sort V) f) x) (@Some (Choice.sort V) v))) (@in_mem (Choice.sort V) v (@mem (Choice.sort V) (finSetPredType V) (codomf f))) ) apply/eqP/andP => [-> //\|[/fsubsetP AB /fsubsetP BA]]. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by apply/fsetP=> x; apply/idP/idP=> [/AB\|/BA]. Qed. Lemma subEfproper A B : A `<=` B = (A == B) \|\| (A `<` B). Proof. by rewrite eqEfsubset -andb_orr orbN andbT. Qed. Lemma fproper_sub A B : A `<` B -> A `<=` B. Proof. by rewrite subEfproper orbC => ->. Qed. Lemma eqVfproper A B : A `<=` B -> A = B \/ A `<` B. Proof. by rewrite subEfproper => /predU1P. Qed. Lemma fproperEneq A B : A `<` B = (A != B) && (A `<=` B). Proof. by rewrite andbC eqEfsubset negb_and andb_orr andbN. Qed. Lemma fproper_neq A B : A `<` B -> A != B. Proof. by rewrite fproperEneq; case/andP. Qed. Lemma fproper_irrefl A : ~~ (A `<` A). Proof. by rewrite fproperEneq eqxx. Qed. Lemma eqEfproper A B : (A == B) = (A `<=` B) && ~~ (A `<` B). Proof. by rewrite negb_and negbK andb_orr andbN eqEfsubset. Qed. Lemma card_fsub B A : A `<=` B -> #\|fsub B A\| = #\|` A\|. Proof. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by move=> sAB; rewrite cardfE fsubE card_imset //; apply: fincl_inj. Qed. Lemma eqEfcard A B : (A == B) = (A `<=` B) && (#\|` B\| <= #\|` A\|)%N. Proof. ( Goal: @eq (Equality.sort V) (@fun_of_fsfun K V default (of_ffun key) x) (@Option.default (Equality.sort V) (default x) (@Option.map (@fset_sub_type K S) (Equality.sort V) h (@insub (Choice.sort K) (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S) x))) ) rewrite -(inj_in_eq (@fsub_inj (A `\|` B))) -?topredE //=. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite eqEcard !(@subset_fsubE (A `\|` B)) ?(@card_fsub (A `\|` B)). Qed. Lemma fproperEcard A B : (A `<` B) = (A `<=` B) && (#\|` A\| < #\|` B\|)%N. Proof. by rewrite fproperEneq ltnNge andbC eqEfcard; case: (A `<=` B). Qed. Lemma fsubset_leqif_cards A B : A `<=` B -> (#\|` A\| <= #\|` B\| ?= iff (A == B))%N. Proof. ( Goal: @eq (Equality.sort V) (@fun_of_fsfun K V default (of_ffun key) x) (@Option.default (Equality.sort V) (default x) (@Option.map (@fset_sub_type K S) (Equality.sort V) h (@insub (Choice.sort K) (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S) x))) ) rewrite -!(@card_fsub (A `\|` B)) // -(@subset_fsubE (A `\|` B)) //. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by move=> /subset_leqif_cards; rewrite (inj_in_eq (@fsub_inj _)) -?topredE /=. Qed. Lemma fsub0set A : fset0 `<=` A. Proof. by apply/fsubsetP=> x; rewrite inE. Qed. Hint Resolve fsub0set. Lemma fsubset0 A : (A `<=` fset0) = (A == fset0). Proof. by rewrite eqEfsubset fsub0set andbT. Qed. Lemma fproper0 A : (fset0 `<` A) = (A != fset0). Proof. by rewrite /fproper fsub0set fsubset0. Qed. Lemma fproperE A B : (A `<` B) = (A `<=` B) && ~~ (B `<=` A). Proof. by []. Qed. Lemma fsubEproper A B : (A `<=` B) = (A == B) \|\| (A `<` B). Proof. by rewrite fproperEneq; case: eqP => //= ->; apply: fsubset_refl. Qed. Lemma fsubset_leq_card A B : A `<=` B -> (#\|` A\| <= #\|` B\|)%N. Proof. by move=> /fsubset_leqif_cards ->. Qed. Lemma fproper_ltn_card A B : A `<` B -> (#\|` A\| < #\|` B\|)%N. Proof. by rewrite fproperEcard => /andP []. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma fsubset_cardP A B : #\|` A\| = #\|` B\| -> reflect (A =i B) (A `<=` B). Proof. ( Goal: forall (_ : is_true (negb (@in_mem (Choice.sort K) k (@mem (Choice.sort K) (finMapPredType K V) f)))) (_ : is_true (negb (@in_mem (Choice.sort K) k (@mem (Choice.sort K) (finMapPredType K V) f')))) (_ : @eq (@finmap_of K V (Phant (forall _ : Choice.sort K, V))) (@setf K V f k v) (@setf K V f' k v)), @eq (@finmap_of K V (Phant (forall _ : Choice.sort K, V))) f f' ) move=> eq_cardAB; apply: (iffP idP) => [/eqVfproper [->//\|]\|/fsetP -> //]. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite fproperEcard eq_cardAB ltnn andbF. Qed. Lemma fproper_sub_trans B A C : A `<` B -> B `<=` C -> A `<` C. Proof. ( Goal: @big_rel_class_of (@finset_of K (Phant (Choice.sort K))) (@fsubset K) ) rewrite !fproperEcard => /andP [sAB lt_AB] sBC. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite (fsubset_trans sAB) //= (leq_trans lt_AB) // fsubset_leq_card. Qed. Lemma fsub_proper_trans B A C : ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) A `<=` B -> B `<` C -> A `<` C. Proof. ( Goal: @big_rel_class_of (@finset_of K (Phant (Choice.sort K))) (@fsubset K) ) rewrite !fproperEcard => sAB /andP [sBC lt_BC]. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite (fsubset_trans sAB) //= (leq_ltn_trans _ lt_BC) // fsubset_leq_card. Qed. Lemma fsubset_neq0 A B : A `<=` B -> A != fset0 -> B != fset0. Proof. by rewrite -!fproper0 => sAB /fproper_sub_trans->. Qed. ( fsub is a morphism ) Lemma fsub0 A : fsub A fset0 = set0 :> {set A}. Proof. by apply/setP => x; rewrite !inE. Qed. Lemma fsubT A : fsub A A = [set : A]. Proof. by apply/setP => x; rewrite !inE (valP x). Qed. Lemma fsub1 A a (aA : a \in A) : fsub A [fset a] = [set [` aA]] :> {set A}. Proof. by apply/setP=> x; rewrite !inE; congr eq_op. Qed. Lemma fsubU C A B : fsub C (A `\|` B) = fsub C A :\|: fsub C B. Proof. by apply/setP => x; rewrite !inE. Qed. Lemma fsubI C A B : fsub C (A `&` B) = fsub C A :&: fsub C B. Proof. by apply/setP => x; rewrite !inE. Qed. Lemma fsubD C A B : fsub C (A `\` B) = fsub C A :\: fsub C B. Proof. by apply/setP => x; rewrite !inE andbC. Qed. Lemma fsubD1 C A b (bC : b \in C) : fsub C (A `\ b) = fsub C A :\ [` bC]. Proof. by rewrite fsubD fsub1. Qed. Lemma fsub_eq0 A B : A `<=` B -> (fsub B A == set0) = (A == fset0). Proof. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by move=> sAB; rewrite -fsub0 (inj_in_eq (@fsub_inj _)) -?topredE /=. Qed. Lemma fset_0Vmem A : (A = fset0) + {x : K \| x \in A}. Proof. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) have [\|[x mem_x]] := set_0Vmem (fsub A A); last first. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by right; exists (val x); rewrite inE // in mem_x. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by move=> /eqP; rewrite fsub_eq0 // => /eqP; left. Qed. Lemma fset1P x a : reflect (x = a) (x \in [fset a]). Proof. by rewrite inE; exact: eqP. Qed. Lemma fset11 x : x \in [fset x]. Proof. by rewrite inE. Qed. Lemma fset1_inj : injective (@fset1 K). Proof. by move=> a b eqsab; apply/fset1P; rewrite -eqsab fset11. Qed. Lemma fset1UP x a B : reflect (x = a \/ x \in B) (x \in a \|` B). Proof. by rewrite !inE; exact: predU1P. Qed. Lemma fset_cons a s : [fset[key] x in a :: s] = a \|` [fset[key] x in s]. Proof. by apply/fsetP=> x; rewrite in_fset_cons !inE. Qed. Lemma fset_nil : [fset[key] x in [::] : seq K] = fset0. Proof. by apply/fsetP=> x; rewrite in_fset_nil. Qed. Lemma fset_cat s s' : [fset[key] x in s ++ s'] = [fset[key] x in s] `\|` [fset[key] x in s']. Proof. by apply/fsetP=> x; rewrite !inE !in_fset_cat. Qed. Lemma fset1U1 x B : x \in x \|` B. Proof. by rewrite !inE eqxx. Qed. Lemma fset1Ur x a B : x \in B -> x \in a \|` B. Proof. by move=> Bx; rewrite !inE predU1r. Qed. ( We need separate lemmas for the explicit enumerations since they ) ( associate on the left. ) Lemma fsetU1l x A b : x \in A -> x \in A `\|` [fset b]. Proof. by move=> Ax; rewrite !inE Ax. Qed. Lemma fsetU1r A b : b \in A `\|` [fset b]. Proof. by rewrite !inE eqxx orbT. Qed. Lemma fsetD1P x A b : reflect (x != b /\ x \in A) (x \in A `\ b). Proof. by rewrite !inE; exact: andP. Qed. Lemma fsetD11 b A : (b \in A `\ b) = false. Proof. by rewrite !inE eqxx. Qed. Lemma fsetD1K a A : a \in A -> a \|` (A `\ a) = A. Proof. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by move=> Aa; apply/fsetP=> x; rewrite !inE; case: eqP => // ->. Qed. Lemma fsetU1K a B : a \notin B -> (a \|` B) `\ a = B. Proof. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by move/negPf=> nBa; apply/fsetP=> x; rewrite !inE; case: eqP => // ->. Qed. Lemma fset2P x a b : reflect (x = a \/ x = b) (x \in [fset a; b]). Proof. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite !inE; apply: (iffP orP) => [] [] /eqP ->; [left\|right\|left\|right]. Qed. Lemma in_fset2 x a b : (x \in [fset a; b]) = (x == a) \|\| (x == b). Proof. by rewrite !inE. Qed. Lemma fset21 a b : a \in [fset a; b]. Proof. by rewrite fset1U1. Qed. Lemma fset22 a b : b \in [fset a; b]. Proof. by rewrite in_fset2 eqxx orbT. Qed. Lemma fsetUP x A B : reflect (x \in A \/ x \in B) (x \in A `\|` B). Proof. by rewrite !inE; exact: orP. Qed. Lemma fsetULVR x A B : x \in A `\|` B -> (x \in A) + (x \in B). Proof. by rewrite inE; case: (x \in A); [left\|right]. Qed. Lemma fsetUS A B C : A `<=` B -> C `\|` A `<=` C `\|` B. Proof. ( Goal: forall (_ : is_true (@fdisjoint K A B)) (C : @finset_of K (Phant (Choice.sort K))), is_true (@disjoint (@fset_sub_finType K C) (@mem (Finite.sort (@fset_sub_finType K C)) (predPredType (Finite.sort (@fset_sub_finType K C))) (@SetDef.pred_of_set (@fset_sub_finType K C) (fsub C A))) (@mem (Finite.sort (@fset_sub_finType K C)) (predPredType (Finite.sort (@fset_sub_finType K C))) (@SetDef.pred_of_set (@fset_sub_finType K C) (fsub C B)))) ) move=> sAB; apply/fsubsetP=> x; rewrite !inE. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by case: (x \in C) => //; exact: (fsubsetP sAB). Qed. Lemma fsetSU A B C : A `<=` B -> A `\|` C `<=` B `\|` C. Proof. by move=> sAB; rewrite -!(fsetUC C) fsetUS. Qed. Lemma fsetUSS A B C D : A `<=` C -> B `<=` D -> A `\|` B `<=` C `\|` D. Proof. by move=> /(fsetSU B) /fsubset_trans sAC /(fsetUS C)/sAC. Qed. Lemma fset0U A : fset0 `\|` A = A. Proof. by apply/fsetP => x; rewrite !inE orFb. Qed. Lemma fsetU0 A : A `\|` fset0 = A. Proof. by rewrite fsetUC fset0U. Qed. ( intersection ) Lemma fsetIP x A B : reflect (x \in A /\ x \in B) (x \in A `&` B). Proof. by rewrite inE; apply: andP. Qed. Lemma fsetIS A B C : A `<=` B -> C `&` A `<=` C `&` B. Proof. ( Goal: forall (_ : is_true (@fdisjoint K A B)) (C : @finset_of K (Phant (Choice.sort K))), is_true (@disjoint (@fset_sub_finType K C) (@mem (Finite.sort (@fset_sub_finType K C)) (predPredType (Finite.sort (@fset_sub_finType K C))) (@SetDef.pred_of_set (@fset_sub_finType K C) (fsub C A))) (@mem (Finite.sort (@fset_sub_finType K C)) (predPredType (Finite.sort (@fset_sub_finType K C))) (@SetDef.pred_of_set (@fset_sub_finType K C) (fsub C B)))) ) move=> sAB; apply/fsubsetP=> x; rewrite !inE. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by case: (x \in C) => //; exact: (fsubsetP sAB). Qed. Lemma fsetSI A B C : A `<=` B -> A `&` C `<=` B `&` C. Proof. by move=> sAB; rewrite -!(fsetIC C) fsetIS. Qed. Lemma fsetISS A B C D : A `<=` C -> B `<=` D -> A `&` B `<=` C `&` D. Proof. by move=> /(fsetSI B) /fsubset_trans sAC /(fsetIS C) /sAC. Qed. ( difference ) Lemma fsetDP A B x : reflect (x \in A /\ x \notin B) (x \in A `\` B). Proof. by rewrite inE andbC; apply: andP. Qed. Lemma fsetSD C A B : A `<=` B -> A `\` C `<=` B `\` C. Proof. ( Goal: forall (_ : is_true (@fdisjoint K A B)) (C : @finset_of K (Phant (Choice.sort K))), is_true (@disjoint (@fset_sub_finType K C) (@mem (Finite.sort (@fset_sub_finType K C)) (predPredType (Finite.sort (@fset_sub_finType K C))) (@SetDef.pred_of_set (@fset_sub_finType K C) (fsub C A))) (@mem (Finite.sort (@fset_sub_finType K C)) (predPredType (Finite.sort (@fset_sub_finType K C))) (@SetDef.pred_of_set (@fset_sub_finType K C) (fsub C B)))) ) move=> sAB; apply/fsubsetP=> x; rewrite !inE. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by case: (x \in C) => //; exact: (fsubsetP sAB). Qed. Lemma fsetDS C A B : A `<=` B -> C `\` B `<=` C `\` A. Proof. ( Goal: forall (_ : is_true (@fdisjoint K A B)) (C : @finset_of K (Phant (Choice.sort K))), is_true (@disjoint (@fset_sub_finType K C) (@mem (Finite.sort (@fset_sub_finType K C)) (predPredType (Finite.sort (@fset_sub_finType K C))) (@SetDef.pred_of_set (@fset_sub_finType K C) (fsub C A))) (@mem (Finite.sort (@fset_sub_finType K C)) (predPredType (Finite.sort (@fset_sub_finType K C))) (@SetDef.pred_of_set (@fset_sub_finType K C) (fsub C B)))) ) move=> sAB; apply/fsubsetP=> x; rewrite !inE ![_ && (_ \in _)]andbC. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by case: (x \in C) => //; apply: contra; exact: (fsubsetP sAB). Qed. Lemma fsetDSS A B C D : A `<=` C -> D `<=` B -> A `\` B `<=` C `\` D. Proof. by move=> /(fsetSD B) /fsubset_trans sAC /(fsetDS C) /sAC. Qed. Lemma fsetD0 A : A `\` fset0 = A. Proof. by apply/fsetP=> x; rewrite !inE. Qed. Lemma fset0D A : fset0 `\` A = fset0. Proof. by apply/fsetP=> x; rewrite !inE andbF. Qed. Lemma fsetDv A : A `\` A = fset0. Proof. by apply/fsetP=> x; rewrite !inE andNb. Qed. Lemma fsetID B A : A `&` B `\|` A `\` B = A. Proof. by apply/fsetP=> x; rewrite !inE; do ?case: (_ \in _). Qed. Lemma fsetDUl A B C : (A `\|` B) `\` C = (A `\` C) `\|` (B `\` C). Proof. by apply/fsetP=> x; rewrite !inE; do ?case: (_ \in _). Qed. Lemma fsetDUr A B C : A `\` (B `\|` C) = (A `\` B) `&` (A `\` C). Proof. by apply/fsetP=> x; rewrite !inE; do ?case: (_ \in _). Qed. Lemma fsetDIl A B C : (A `&` B) `\` C = (A `\` C) `&` (B `\` C). Proof. by apply/fsetP=> x; rewrite !inE; do ?case: (_ \in _). Qed. Lemma fsetIDA A B C : A `&` (B `\` C) = (A `&` B) `\` C. Proof. by apply/fsetP=> x; rewrite !inE; do ?case: (_ \in _). Qed. Lemma fsetIDAC A B C : (A `\` B) `&` C = (A `&` C) `\` B. Proof. by apply/fsetP=> x; rewrite !inE; do ?case: (_ \in _). Qed. Lemma fsetDIr A B C : A `\` (B `&` C) = (A `\` B) `\|` (A `\` C). Proof. by apply/fsetP=> x; rewrite !inE; do ?case: (_ \in _). Qed. Lemma fsetDDl A B C : (A `\` B) `\` C = A `\` (B `\|` C). Proof. by apply/fsetP=> x; rewrite !inE; do ?case: (_ \in _). Qed. Lemma fsetDDr A B C : A `\` (B `\` C) = (A `\` B) `\|` (A `&` C). Proof. by apply/fsetP=> x; rewrite !inE; do ?case: (_ \in _). Qed. Lemma fsetDK A B : B `<=` A -> A `\` (A `\` B) = B. Proof. by rewrite fsetDDr => /fsetIidPr->; rewrite fsetDv fset0U. Qed. Lemma fsetUDl (A B C : {fset K}) : A `\|` (B `\` C) = (A `\|` B) `\` (C `\` A). Proof. by apply/fsetP=> x; rewrite !inE; do ?case: (_ \in _). Qed. Lemma fsetUDr (A B C : {fset K}) : (A `\` B) `\|` C = (A `\|` C) `\` (B `\` C). Proof. by apply/fsetP=> x; rewrite !inE; do ?case: (_ \in _). Qed. ( other inclusions ) Lemma fsubsetIl A B : A `&` B `<=` A. Proof. by apply/fsubsetP=> x; rewrite inE => /andP []. Qed. Lemma fsubsetIr A B : A `&` B `<=` B. Proof. by apply/fsubsetP=> x; rewrite inE => /andP []. Qed. Lemma fsubsetDl A B : A `\` B `<=` A. Proof. by apply/fsubsetP=> x; rewrite inE => /andP []. Qed. Lemma fsubD1set A x : A `\ x `<=` A. Proof. by rewrite fsubsetDl. Qed. Lemma fsubsetD2l C A B : A `<=` C -> B `<=` C -> (C `\` B `<=` C `\` A) = (A `<=` B). Proof. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) move=> sAC sBC; apply/idP/idP; last exact: fsetDS. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by move=> /(@fsetDS C); rewrite !fsetDK //; apply; apply: fsubsetDl. Qed. Hint Resolve fsubsetIl fsubsetIr fsubsetDl fsubD1set. ( cardinal lemmas for fsets ) Lemma card_finset (T : finType) (P : pred T) : #\|` [fset x in P] \| = #\|P\|. Proof. by rewrite card_imfset //= -cardE. Qed. Lemma card_fset (T : choiceType) (A : {fset T}) : #\|` [fset x in A] \| = #\|` A\|. Proof. by rewrite card_imfset. Qed. Lemma card_fseq (T : choiceType) (s : seq T) : #\|` [fset x in s] \| = size (undup s). Proof. by rewrite card_imfset. Qed. Lemma cardfs0 : #\|` @fset0 K\| = 0. Proof. by rewrite -(@card_fsub fset0) // fsub0 cards0. Qed. Lemma cardfT0 : #\|{: @fset0 K}\| = 0. Proof. by rewrite -cardfE cardfs0. Qed. Lemma cardfs_eq0 A : (#\|` A\| == 0) = (A == fset0). Proof. by rewrite -(@card_fsub A) // cards_eq0 fsub_eq0. Qed. Lemma cardfs0_eq A : #\|` A\| = 0 -> A = fset0. Proof. by move=> /eqP; rewrite cardfs_eq0 => /eqP. Qed. Lemma fset0Pn A : reflect (exists x, x \in A) (A != fset0). Proof. ( Goal: is_true (@in_mem (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f (@fsval K (@FinSupp.fs K (Choice.eqType K) (fun x : Choice.sort K => x) f) a)) (@mem (Choice.sort K) (finSetPredType K) (@FinSupp.fs K (Choice.eqType K) (fun x : Choice.sort K => x) f))) ) ( Goal: @ex2 (@finset_of K (Phant (Choice.sort K))) (fun S : @finset_of K (Phant (Choice.sort K)) => is_true (@fsubset K (@FinSupp.fs K (Choice.eqType K) (fun x : Choice.sort K => x) f) S)) (fun S : @finset_of K (Phant (Choice.sort K)) => and (@prop_in2 (Choice.sort K) (@mem (Choice.sort K) (finSetPredType K) S) (fun x1 x2 : Choice.sort K => forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f x1) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f x2), @eq (Choice.sort K) x1 x2) (inPhantom (@injective (Choice.sort K) (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f)))) (forall a : @fset_sub_type K S, is_true (@in_mem (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f (@fsval K S a)) (@mem (Choice.sort K) (finSetPredType K) S)))) ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) rewrite -cardfs_eq0 cardfE; apply: (equivP existsP). ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by split=> [] [a aP]; [exists (val a); apply: valP\|exists [` aP]]. Qed. Lemma cardfs_gt0 A : (0 < #\|` A\|)%N = (A != fset0). Proof. by rewrite lt0n cardfs_eq0. Qed. Lemma cardfs1 x : #\|` [fset x]\| = 1. Proof. by rewrite card_imfset. Qed. Lemma cardfsUI A B : #\|` A `\|` B\| + #\|` A `&` B\| = #\|` A\| + #\|` B\|. Proof. ( Goal: @eq (Equality.sort V) (@fun_of_fsfun K V default (of_ffun key) x) (@Option.default (Equality.sort V) (default x) (@Option.map (@fset_sub_type K S) (Equality.sort V) h (@insub (Choice.sort K) (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S) x))) ) rewrite -!(@card_fsub (A `\|` B)) ?(fsubset_trans (fsubsetIl _ _)) //. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite fsubU fsubI cardsUI. Qed. Lemma cardfsU A B : #\|` A `\|` B\| = (#\|` A\| + #\|` B\| - #\|` A `&` B\|)%N. Proof. by rewrite -cardfsUI addnK. Qed. Lemma cardfsI A B : #\|` A `&` B\| = (#\|` A\| + #\|` B\| - #\|` A `\|` B\|)%N. Proof. by rewrite -cardfsUI addKn. Qed. Lemma cardfsID B A : #\|` A `&` B\| + #\|` A `\` B\| = #\|` A\|. Proof. by rewrite -!(@card_fsub A) // fsubI fsubD cardsID. Qed. Lemma cardfsD A B : #\|` A `\` B\| = (#\|` A\| - #\|` A `&` B\|)%N. Proof. by rewrite -(cardfsID B A) addKn. Qed. Lemma mem_fset1U a A : a \in A -> a \|` A = A. Proof. ( Goal: forall (_ : is_true (@fdisjoint K A B)) (C : @finset_of K (Phant (Choice.sort K))), is_true (@disjoint (@fset_sub_finType K C) (@mem (Finite.sort (@fset_sub_finType K C)) (predPredType (Finite.sort (@fset_sub_finType K C))) (@SetDef.pred_of_set (@fset_sub_finType K C) (fsub C A))) (@mem (Finite.sort (@fset_sub_finType K C)) (predPredType (Finite.sort (@fset_sub_finType K C))) (@SetDef.pred_of_set (@fset_sub_finType K C) (fsub C B)))) ) move=> aA; apply/fsetP => x; rewrite !inE orbC. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by have [//\|/=] := boolP (_ \in A); apply: contraNF => /eqP ->. Qed. Lemma mem_fsetD1 a A : a \notin A -> A `\ a = A. Proof. ( Goal: forall (_ : is_true (@fdisjoint K A B)) (C : @finset_of K (Phant (Choice.sort K))), is_true (@disjoint (@fset_sub_finType K C) (@mem (Finite.sort (@fset_sub_finType K C)) (predPredType (Finite.sort (@fset_sub_finType K C))) (@SetDef.pred_of_set (@fset_sub_finType K C) (fsub C A))) (@mem (Finite.sort (@fset_sub_finType K C)) (predPredType (Finite.sort (@fset_sub_finType K C))) (@SetDef.pred_of_set (@fset_sub_finType K C) (fsub C B)))) ) move=> aA; apply/fsetP => x; rewrite !inE andbC. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by have [/= xA\|//] := boolP (_ \in A); apply: contraNneq aA => <-. Qed. Lemma fsetI1 a A : A `&` [fset a] = if a \in A then [fset a] else fset0. Proof. ( Goal: is_true (@fsubset K (@FinSupp.fs K V (fun a : Choice.sort K => default a) (@fsfun_of_ffun key K V S (fun a : @fset_sub_type K S => h a) (fun a : Choice.sort K => default a))) S) ) apply/fsetP => x; rewrite (fun_if (fun X => _ \in X)) !inE. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by have [[->\|?] []] := (altP (x =P a), boolP (a \in A)); rewrite ?andbF. Qed. Lemma cardfsU1 a A : #\|` a \|` A\| = (a \notin A) + #\|` A\|. Proof. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) have [aA\|aNA] := boolP (a \in A); first by rewrite mem_fset1U. ( Goal: @eq (Equality.sort V) (@fun_of_fsfun K V default (of_ffun key) x) (@Option.default (Equality.sort V) (default x) (@Option.map (@fset_sub_type K S) (Equality.sort V) h (@insub (Choice.sort K) (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S) x))) ) rewrite cardfsU -addnBA ?fsubset_leq_card // fsetIC -cardfsD. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite mem_fsetD1 // cardfs1. Qed. Lemma cardfs2 a b : #\|` [fset a; b]\| = (a != b).+1. Proof. by rewrite !cardfsU1 cardfs1 inE addn1. Qed. Lemma cardfsD1 a A : #\|` A\| = (a \in A) + #\|` A `\ a\|. Proof. ( Goal: @eq (Equality.sort V) (@fun_of_fsfun K V default (of_ffun key) x) (@Option.default (Equality.sort V) (default x) (@Option.map (@fset_sub_type K S) (Equality.sort V) h (@insub (Choice.sort K) (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S) x))) ) rewrite -(cardfsID [fset a]) fsetI1 (fun_if (fun A => #\|` A\|)). ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite cardfs0 cardfs1; case: (_ \in _). Qed. ( other inclusions ) Lemma fsub1set A x : ([fset x] `<=` A) = (x \in A). Proof. ( Goal: @big_rel_class_of (@finset_of K (Phant (Choice.sort K))) (@fsubset K) ) rewrite -(@subset_fsubE (x \|` A)) // fsub1 ?fset1U1 // => xxA. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite sub1set inE. Qed. Lemma cardfs1P A : reflect (exists x, A = [fset x]) (#\|` A\| == 1). Proof. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) apply: (iffP idP) => [\|[x ->]]; last by rewrite cardfs1. ( Goal: @big_rel_class_of (@finset_of K (Phant (Choice.sort K))) (@fsubset K) ) rewrite eq_sym eqn_leq cardfs_gt0=> /andP[/fset0Pn[x Ax] leA1]. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by exists x; apply/eqP; rewrite eq_sym eqEfcard fsub1set cardfs1 leA1 Ax. Qed. Lemma fsubset1 A x : (A `<=` [fset x]) = (A == [fset x]) \|\| (A == fset0). Proof. ( Goal: @eq (Equality.sort V) (@fun_of_fsfun K V default (of_ffun key) x) (@Option.default (Equality.sort V) (default x) (@Option.map (@fset_sub_type K S) (Equality.sort V) h (@insub (Choice.sort K) (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S) x))) ) rewrite eqEfcard cardfs1 -cardfs_eq0 orbC andbC. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by case: posnP => // A0; rewrite (cardfs0_eq A0) fsub0set. Qed. Arguments fsetIidPl {A B}. Lemma cardfsDS A B : B `<=` A -> #\|` A `\` B\| = (#\|` A\| - #\|` B\|)%N. Proof. by rewrite cardfsD => /fsetIidPr->. Qed. Lemma fsubIset A B C : (B `<=` A) \|\| (C `<=` A) -> (B `&` C `<=` A). Proof. by case/orP; apply: fsubset_trans; rewrite (fsubsetIl, fsubsetIr). Qed. Lemma fsubsetI A B C : (A `<=` B `&` C) = (A `<=` B) && (A `<=` C). Proof. ( Goal: @eq bool (@fsubset K A (@fsetI K B C)) (andb (@fsubset K A B) (@fsubset K A C)) ) rewrite !(sameP fsetIidPl eqP) fsetIA; have [-> //\| ] := altP (A `&` B =P A). ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by apply: contraNF => /eqP <-; rewrite -fsetIA -fsetIIl fsetIAC. Qed. Lemma fsubsetIP A B C : reflect (A `<=` B /\ A `<=` C) (A `<=` B `&` C). Proof. by rewrite fsubsetI; exact: andP. Qed. Lemma fsubUset A B C : (B `\|` C `<=` A) = (B `<=` A) && (C `<=` A). Proof. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) apply/idP/idP => [subA\|/andP [AB CA]]; last by rewrite -[A]fsetUid fsetUSS. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite !(fsubset_trans _ subA). Qed. Lemma fsubUsetP A B C : reflect (A `<=` C /\ B `<=` C) (A `\|` B `<=` C). Proof. by rewrite fsubUset; exact: andP. Qed. Lemma fsubDset A B C : (A `\` B `<=` C) = (A `<=` B `\|` C). Proof. ( Goal: is_true (@fsubset K (@FinSupp.fs K V (fun a : Choice.sort K => default a) (@fsfun_of_ffun key K V S (fun a : @fset_sub_type K S => h a) (fun a : Choice.sort K => default a))) S) ) apply/fsubsetP/fsubsetP=> sABC x; rewrite !inE. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by case Bx: (x \in B) => // Ax; rewrite sABC ?in_fsetD ?Bx. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by case Bx: (x \in B) => //; move/sABC; rewrite inE Bx. Qed. Lemma fsetU_eq0 A B : (A `\|` B == fset0) = (A == fset0) && (B == fset0). Proof. by rewrite -!fsubset0 fsubUset. Qed. Lemma fsubsetD1 A B x : (A `<=` B `\ x) = (A `<=` B) && (x \notin A). Proof. ( Goal: @eq bool (@fsubset K A (@fsetD K B (@fset1 K x))) (andb (@fsubset K A B) (negb (@in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) A)))) ) do !rewrite -(@subset_fsubE (x \|` A `\|` B)) ?fsubDset ?fsetUA // 1?fsetUAC //. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) rewrite fsubD1 => [\|mem_x]; first by rewrite -fsetUA fset1U1. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite subsetD1 // inE. Qed. Lemma fsubsetD1P A B x : reflect (A `<=` B /\ x \notin A) (A `<=` B `\ x). Proof. by rewrite fsubsetD1; exact: andP. Qed. Lemma fsubsetPn A B : reflect (exists2 x, x \in A & x \notin B) (~~ (A `<=` B)). Proof. ( Goal: is_true (@in_mem (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f (@fsval K (@FinSupp.fs K (Choice.eqType K) (fun x : Choice.sort K => x) f) a)) (@mem (Choice.sort K) (finSetPredType K) (@FinSupp.fs K (Choice.eqType K) (fun x : Choice.sort K => x) f))) ) ( Goal: @ex2 (@finset_of K (Phant (Choice.sort K))) (fun S : @finset_of K (Phant (Choice.sort K)) => is_true (@fsubset K (@FinSupp.fs K (Choice.eqType K) (fun x : Choice.sort K => x) f) S)) (fun S : @finset_of K (Phant (Choice.sort K)) => and (@prop_in2 (Choice.sort K) (@mem (Choice.sort K) (finSetPredType K) S) (fun x1 x2 : Choice.sort K => forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f x1) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f x2), @eq (Choice.sort K) x1 x2) (inPhantom (@injective (Choice.sort K) (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f)))) (forall a : @fset_sub_type K S, is_true (@in_mem (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f (@fsval K S a)) (@mem (Choice.sort K) (finSetPredType K) S)))) ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) rewrite -fsetD_eq0; apply: (iffP (fset0Pn _)) => [[x]\|[x xA xNB]]. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite inE => /andP[]; exists x. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by exists x; rewrite inE xA xNB. Qed. Lemma fproperD1 A x : x \in A -> A `\ x `<` A. Proof. ( Goal: forall _ : is_true (@in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) A)), is_true (@fproper K (@fsetD K A (@fset1 K x)) A) ) move=> Ax; rewrite fproperE fsubsetDl; apply/fsubsetPn; exists x=> //. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite in_fsetD1 Ax eqxx. Qed. Lemma fproperIr A B : ~~ (B `<=` A) -> A `&` B `<` B. Proof. by move=> nsAB; rewrite fproperE fsubsetIr fsubsetI negb_and nsAB. Qed. Lemma fproperIl A B : ~~ (A `<=` B) -> A `&` B `<` A. Proof. by move=> nsBA; rewrite fproperE fsubsetIl fsubsetI negb_and nsBA orbT. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma fproperUr A B : ~~ (A `<=` B) -> B `<` A `\|` B. Proof. by rewrite fproperE fsubsetUr fsubUset fsubset_refl /= andbT. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma fproperUl A B : ~~ (B `<=` A) -> A `<` A `\|` B. Proof. by move=> not_sBA; rewrite fsetUC fproperUr. Qed. Lemma fproper1set A x : ([fset x] `<` A) -> (x \in A). Proof. by move/fproper_sub; rewrite fsub1set. Qed. Lemma fproperIset A B C : (B `<` A) \|\| (C `<` A) -> (B `&` C `<` A). Proof. by case/orP; apply: fsub_proper_trans; rewrite (fsubsetIl, fsubsetIr). Qed. Lemma fproperI A B C : (A `<` B `&` C) -> (A `<` B) && (A `<` C). Proof. ( Goal: forall (_ : is_true (negb (@in_mem (Choice.sort K) k (@mem (Choice.sort K) (finMapPredType K V) f)))) (_ : is_true (negb (@in_mem (Choice.sort K) k (@mem (Choice.sort K) (finMapPredType K V) f')))) (_ : @eq (@finmap_of K V (Phant (forall _ : Choice.sort K, V))) (@setf K V f k v) (@setf K V f' k v)), @eq (@finmap_of K V (Phant (forall _ : Choice.sort K, V))) f f' ) move=> pAI; apply/andP. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by split; apply: (fproper_sub_trans pAI); rewrite (fsubsetIl, fsubsetIr). Qed. Lemma fproperU A B C : (B `\|` C `<` A) -> (B `<` A) && (C `<` A). Proof. ( Goal: forall (_ : is_true (negb (@in_mem (Choice.sort K) k (@mem (Choice.sort K) (finMapPredType K V) f)))) (_ : is_true (negb (@in_mem (Choice.sort K) k (@mem (Choice.sort K) (finMapPredType K V) f')))) (_ : @eq (@finmap_of K V (Phant (forall _ : Choice.sort K, V))) (@setf K V f k v) (@setf K V f' k v)), @eq (@finmap_of K V (Phant (forall _ : Choice.sort K, V))) f f' ) move=> pUA; apply/andP. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by split; apply: fsub_proper_trans pUA; rewrite (fsubsetUr, fsubsetUl). Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma fsetDpS C A B : B `<=` C -> A `<` B -> C `\` B `<` C `\` A. Proof. ( Goal: forall (_ : is_true (@fdisjoint K (@domf K V f) (@domf K V h))) (_ : is_true (@fdisjoint K (@domf K V g) (@domf K V h))) (_ : @eq (finMap K V) (catf h f) (catf h g)), @eq (@finmap_of K V (Phant (forall _ : Choice.sort K, V))) f g ) move=> subBC subAB; rewrite fproperEneq fsetDS 1?fproper_sub// andbT. ( Goal: is_true (negb (@eq_op (fset_eqType K) (@fsetD K C B) (@fsetD K C A))) ) apply/negP => /eqP /(congr1 (fsetD C)); rewrite !fsetDK // => [eqAB//\|]. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite eqAB (negPf (fproper_irrefl _)) in subAB. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by apply: fsubset_trans subBC; apply: fproper_sub. Qed. Lemma fproperD2l C A B : A `<=` C -> B `<=` C -> (C `\` B `<` C `\` A) = (A `<` B). Proof. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) move=> sAC sBC; apply/idP/idP; last exact: fsetDpS. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by move=> /(@fsetDpS C); rewrite !fsetDK //; apply; apply: fsubsetDl. Qed. Lemma fsetI_eq0 A B : (A `&` B == fset0) = [disjoint A & B]. Proof. by []. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma fdisjoint_sub {A B} : [disjoint A & B]%fset -> forall C : {fset K}, [disjoint fsub C A & fsub C B]%bool. Proof. ( Goal: forall (_ : is_true (@fdisjoint K A B)) (C : @finset_of K (Phant (Choice.sort K))), is_true (@disjoint (@fset_sub_finType K C) (@mem (Finite.sort (@fset_sub_finType K C)) (predPredType (Finite.sort (@fset_sub_finType K C))) (@SetDef.pred_of_set (@fset_sub_finType K C) (fsub C A))) (@mem (Finite.sort (@fset_sub_finType K C)) (predPredType (Finite.sort (@fset_sub_finType K C))) (@SetDef.pred_of_set (@fset_sub_finType K C) (fsub C B)))) ) move=> disjointAB C; apply/pred0P => a /=; rewrite !inE. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by have /eqP /fsetP /(_ (val a)) := disjointAB; rewrite !inE. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma disjoint_fsub C A B : A `\|` B `<=` C -> [disjoint fsub C A & fsub C B]%bool = [disjoint A & B]. Proof. ( Goal: forall _ : is_true (@fsubset K (@fsetU K A B) C), @eq bool (@disjoint (@fset_sub_finType K C) (@mem (Finite.sort (@fset_sub_finType K C)) (predPredType (Finite.sort (@fset_sub_finType K C))) (@SetDef.pred_of_set (@fset_sub_finType K C) (fsub C A))) (@mem (Finite.sort (@fset_sub_finType K C)) (predPredType (Finite.sort (@fset_sub_finType K C))) (@SetDef.pred_of_set (@fset_sub_finType K C) (fsub C B)))) (@fdisjoint K A B) ) move=> ABsubC. ( Goal: @eq (@finset_of V (Phant (Choice.sort V))) (codomf (@restrictf K (Choice.sort V) f A)) (@Imfset.imfset imfset_key (Finite.choiceType (@fset_sub_finType K (@domf K (Choice.sort V) f))) V (fun k : Choice.sort (Finite.choiceType (@fset_sub_finType K (@domf K (Choice.sort V) f))) => @FunFinfun.fun_of_fin (@fset_sub_finType K (@domf K (Choice.sort V) f)) (Choice.sort V) (@ffun_of_fmap K (Choice.sort V) f) k) (@mem_fin (Choice.eqType (Finite.choiceType (@fset_sub_finType K (@domf K (Choice.sort V) f)))) (simplPredType (@fset_sub_type K (@domf K (Choice.sort V) f))) (@subfinset_finpred (Finite.choiceType (@fset_sub_finType K (@domf K (Choice.sort V) f))) (@mem_fin (Choice.eqType (Finite.choiceType (@fset_sub_finType K (@domf K (Choice.sort V) f)))) (predPredType (@fset_sub_type K (@domf K (Choice.sort V) f) : predArgType)) (@fin_finpred (Choice.eqType (Finite.choiceType (@fset_sub_finType K (@domf K (Choice.sort V) f)))) (pred_finpredType (@fset_sub_finType K (@domf K (Choice.sort V) f))) (@sort_of_simpl_pred (@fset_sub_type K (@domf K (Choice.sort V) f) : predArgType) (pred_of_argType (@fset_sub_type K (@domf K (Choice.sort V) f) : predArgType))))) (fun k : @fset_sub_type K (@domf K (Choice.sort V) f) => @in_mem (Choice.sort K) (@val (Choice.sort K) (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) (@domf K (Choice.sort V) f))) (@fset_sub_subType K (@domf K (Choice.sort V) f)) k) (@mem (Choice.sort K) (finSetPredType K) A)))) (Phantom (mem_pred (@fset_sub_type K (@domf K (Choice.sort V) f))) (@mem (@fset_sub_type K (@domf K (Choice.sort V) f)) (simplPredType (@fset_sub_type K (@domf K (Choice.sort V) f))) (@SimplPred (@fset_sub_type K (@domf K (Choice.sort V) f)) (fun k : @fset_sub_type K (@domf K (Choice.sort V) f) => andb (@in_mem (@fset_sub_type K (@domf K (Choice.sort V) f)) k (@mem (@fset_sub_type K (@domf K (Choice.sort V) f)) (predPredType (@fset_sub_type K (@domf K (Choice.sort V) f) : predArgType)) (@sort_of_simpl_pred (@fset_sub_type K (@domf K (Choice.sort V) f) : predArgType) (pred_of_argType (@fset_sub_type K (@domf K (Choice.sort V) f) : predArgType))))) (@in_mem (Choice.sort K) (@val (Choice.sort K) (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) (@domf K (Choice.sort V) f))) (@fset_sub_subType K (@domf K (Choice.sort V) f)) k) (@mem (Choice.sort K) (finSetPredType K) A))))))) ) apply/idP/idP=> [/pred0P DAB\|/fdisjoint_sub->//]; apply/eqP/fsetP=> a. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) rewrite !inE; have [aC\|] := boolP (a \in A `\|` B); last first. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite !inE => /norP [/negPf-> /negPf->]. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by have /= := DAB [` fsubsetP ABsubC _ aC]; rewrite !inE. Qed. Lemma fdisjointP {A B} : ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) reflect (forall a, a \in A -> a \notin B) [disjoint A & B]%fset. Proof. ( Goal: Bool.reflect (@ex (Choice.sort K) (fun x : Choice.sort K => @eq (option (Choice.sort V)) (@fnd K (Choice.sort V) (@domf K (Choice.sort V) f) (@ffun_of_fmap K (Choice.sort V) f) x) (@Some (Choice.sort V) v))) (@in_mem (Choice.sort V) v (@mem (Choice.sort V) (finSetPredType V) (codomf f))) ) apply: (iffP eqP) => [AIB_eq0 a aA\|neq_ab]. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by have /fsetP /(_ a) := AIB_eq0; rewrite !inE aA /= => ->. ( Goal: is_true (@fsubset K (@FinSupp.fs K V (fun a : Choice.sort K => default a) (@fsfun_of_ffun key K V S (fun a : @fset_sub_type K S => h a) (fun a : Choice.sort K => default a))) S) ) apply/fsetP => a; rewrite !inE. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by case: (boolP (a \in A)) => // /neq_ab /negPf ->. Qed. Lemma fsetDidPl A B : reflect (A `\` B = A) [disjoint A & B]%fset. Proof. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) apply: (iffP fdisjointP)=> [NB\|<- a]; last by rewrite inE => /andP[]. ( Goal: is_true (@fsubset K (@FinSupp.fs K V (fun a : Choice.sort K => default a) (@fsfun_of_ffun key K V S (fun a : @fset_sub_type K S => h a) (fun a : Choice.sort K => default a))) S) ) apply/fsetP => a; rewrite !inE andbC. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by case: (boolP (a \in A)) => //= /NB ->. Qed. Lemma disjoint_fsetI0 A B : [disjoint A & B] -> A `&` B = fset0. Proof. by rewrite -fsetI_eq0; move/eqP. Qed. Lemma fsubsetD A B C : (A `<=` (B `\` C)) = (A `<=` B) && [disjoint A & C]%fset. Proof. ( Goal: @eq bool (@fsubset K A (@fsetD K B C)) (andb (@fsubset K A B) (@fdisjoint K A C)) ) pose D := A `\|` B `\|` C. ( Goal: Bool.reflect (@ex2 (@finset_of K (Phant (Choice.sort K))) (fun S : @finset_of K (Phant (Choice.sort K)) => is_true (@fsubset K (@FinSupp.fs K (Choice.eqType K) (fun x : Choice.sort K => x) f) S)) (fun S : @finset_of K (Phant (Choice.sort K)) => and (@prop_in2 (Choice.sort K) (@mem (Choice.sort K) (finSetPredType K) S) (fun x1 x2 : Choice.sort K => forall _ : @eq (Equality.sort (Choice.eqType K)) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f x1) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f x2), @eq (Choice.sort K) x1 x2) (inPhantom (@injective (Equality.sort (Choice.eqType K)) (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f)))) (forall a : @fset_sub_type K S, is_true (@in_mem (Equality.sort (Choice.eqType K)) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f (@val (Choice.sort K) (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S) a)) (@mem (Choice.sort K) (finSetPredType K) S))))) (fsinjectiveb f) ) have AD : A `<=` D by rewrite /D -fsetUA fsubsetUl. ( Goal: Bool.reflect (@ex2 (@finset_of K (Phant (Choice.sort K))) (fun S : @finset_of K (Phant (Choice.sort K)) => is_true (@fsubset K (@FinSupp.fs K (Choice.eqType K) (fun x : Choice.sort K => x) f) S)) (fun S : @finset_of K (Phant (Choice.sort K)) => and (@prop_in2 (Choice.sort K) (@mem (Choice.sort K) (finSetPredType K) S) (fun x1 x2 : Choice.sort K => forall _ : @eq (Equality.sort (Choice.eqType K)) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f x1) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f x2), @eq (Choice.sort K) x1 x2) (inPhantom (@injective (Equality.sort (Choice.eqType K)) (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f)))) (forall a : @fset_sub_type K S, is_true (@in_mem (Equality.sort (Choice.eqType K)) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f (@val (Choice.sort K) (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S) a)) (@mem (Choice.sort K) (finSetPredType K) S))))) (fsinjectiveb f) ) have BD : B `<=` D by rewrite /D fsetUAC fsubsetUr. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) rewrite -(@subset_fsubE D) //; last first. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite fsubDset (fsubset_trans BD) // fsubsetUr. ( Goal: @eq (Equality.sort V) (@fun_of_fsfun K V default (of_ffun key) x) (@Option.default (Equality.sort V) (default x) (@Option.map (@fset_sub_type K S) (Equality.sort V) h (@insub (Choice.sort K) (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S) x))) ) rewrite fsubD subsetD !subset_fsubE // disjoint_fsub //. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite /D fsetUAC fsubsetUl. Qed. Lemma fsubsetDP A B C : reflect (A `<=` B /\ [disjoint A & C]%fset) (A `<=` (B `\` C)). Proof. by rewrite fsubsetD; apply: andP. Qed. Lemma fdisjoint_sym A B : [disjoint A & B] = [disjoint B & A]. Proof. by rewrite -!fsetI_eq0 fsetIC. Qed. Lemma fdisjointP_sym {A B} : ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) reflect (forall a, a \in A -> a \notin B) [disjoint B & A]%fset. Proof. by rewrite fdisjoint_sym; apply: fdisjointP. Qed. Lemma fdisjoint_trans A B C : ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) A `<=` B -> [disjoint B & C] -> [disjoint A & C]. Proof. ( Goal: forall (_ : is_true (@fdisjoint K (@domf K V f) (@domf K V h))) (_ : is_true (@fdisjoint K (@domf K V g) (@domf K V h))) (_ : @eq (finMap K V) (catf h f) (catf h g)), @eq (@finmap_of K V (Phant (forall _ : Choice.sort K, V))) f g ) move=> AsubB; rewrite -!(@disjoint_fsub (B `\|` C)) ?fsetSU //. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by apply: disjoint_trans; rewrite subset_fsub. Qed. Lemma fdisjoint0X A : [disjoint fset0 & A]. Proof. by rewrite -fsetI_eq0 fset0I. Qed. Lemma fdisjointX0 A : [disjoint A & fset0]. Proof. by rewrite -fsetI_eq0 fsetI0. Qed. Lemma fdisjoint1X x A : [disjoint [fset x] & A] = (x \notin A). Proof. ( Goal: @eq (Equality.sort V) (@fun_of_fsfun K V default (of_ffun key) x) (@Option.default (Equality.sort V) (default x) (@Option.map (@fset_sub_type K S) (Equality.sort V) h (@insub (Choice.sort K) (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S) x))) ) rewrite -(@disjoint_fsub (x \|` A)) //. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite (@eq_disjoint1 _ [` fset1U1 _ _]) ?inE =>// ?; rewrite !inE. Qed. Lemma fdisjointX1 x A : [disjoint A & [fset x]] = (x \notin A). Proof. by rewrite fdisjoint_sym fdisjoint1X. Qed. Lemma fdisjointUX A B C : [disjoint A `\|` B & C] = [disjoint A & C]%fset && [disjoint B & C]%fset. Proof. by rewrite -!fsetI_eq0 fsetIUl fsetU_eq0. Qed. Lemma fdisjointXU A B C : [disjoint A & B `\|` C] = [disjoint A & B]%fset && [disjoint A & C]%fset. Proof. by rewrite -!fsetI_eq0 fsetIUr fsetU_eq0. Qed. Lemma fdisjointU1X x A B : [disjoint x \|` A & B]%fset = (x \notin B) && [disjoint A & B]%fset. Proof. by rewrite fdisjointUX fdisjoint1X. Qed. Lemma fsubK A B : A `<=` B -> [fsetval k in fsub B A] = A. Proof. ( Goal: forall _ : is_true (@fsubset K A B), @eq (@finset_of K (Phant (Choice.sort K))) (@Imfset.imfset imfset_key (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) B)) (@subFin_sort K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) B)) (@fset_sub_subfinType K B))) K (fun k : Choice.sort (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) B)) (@subFin_sort K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) B)) (@fset_sub_subfinType K B))) => @val (Choice.sort K) (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) B)) (@subFin_sort K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) B)) (@fset_sub_subfinType K B)) k) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) B)) (@subFin_sort K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) B)) (@fset_sub_subfinType K B)))) (predPredType (Finite.sort (@fset_sub_finType K B))) (@fin_finpred (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) B)) (@subFin_sort K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) B)) (@fset_sub_subfinType K B)))) (pred_finpredType (@subFinType_finType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) B)) (@fset_sub_subfinType K B))) (@SetDef.pred_of_set (@fset_sub_finType K B) (fsub B A)))) (Phantom (mem_pred (Finite.sort (@fset_sub_finType K B))) (@mem (Finite.sort (@fset_sub_finType K B)) (predPredType (Finite.sort (@fset_sub_finType K B))) (@SetDef.pred_of_set (@fset_sub_finType K B) (fsub B A))))) A ) move=> AsubB; apply/fsetP => k /=; symmetry. ( Goal: Bool.reflect (@ex2 (@finset_of K (Phant (Choice.sort K))) (fun S : @finset_of K (Phant (Choice.sort K)) => is_true (@fsubset K (@FinSupp.fs K (Choice.eqType K) (fun x : Choice.sort K => x) f) S)) (fun S : @finset_of K (Phant (Choice.sort K)) => and (@prop_in2 (Choice.sort K) (@mem (Choice.sort K) (finSetPredType K) S) (fun x1 x2 : Choice.sort K => forall _ : @eq (Equality.sort (Choice.eqType K)) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f x1) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f x2), @eq (Choice.sort K) x1 x2) (inPhantom (@injective (Equality.sort (Choice.eqType K)) (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f)))) (forall a : @fset_sub_type K S, is_true (@in_mem (Equality.sort (Choice.eqType K)) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f (@val (Choice.sort K) (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S) a)) (@mem (Choice.sort K) (finSetPredType K) S))))) (fsinjectiveb f) ) have [kB\|kNB] := (boolP (k \in B)). ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite in_fset_valT /= inE. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite in_fset_valF //; apply: contraNF kNB; apply/fsubsetP. Qed. Lemma FSetK A (X : {set A}) : fsub A [fsetval k in X] = X. Proof. by apply/setP => x; rewrite !inE. Qed. End Theory. Hint Resolve fsubset_refl. Hint Resolve fsubset_trans. Hint Resolve fproper_irrefl. Hint Resolve fsub0set. Module Import FSetInE. Definition inE := (inE, in_fsetE). End FSetInE. Section Card. ( Lemma card_finset (T : finType) (P : pred T) : #\|` [fset x in P] \| = #\|P\|. ) ( Proof. ) ( rewrite cardfE cardE; apply/eqP. ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( rewrite -uniq_size_uniq ?fset_uniq ?enum_uniq // => x. ) ( by rewrite !inE mem_enum. ) ( Qed. ) End Card. Section Enum. Lemma enum_fset0 (T : choiceType) : enum [finType of fset0] = [::] :> seq (@fset0 T). Proof. by rewrite enumT unlock. Qed. Lemma enum_fset1 (T : choiceType) (x : T) : enum [finType of [fset x]] = [:: [`fset11 x]]. Proof. ( Goal: @eq (@finset_of V (Phant (Choice.sort V))) (codomf (@restrictf K (Choice.sort V) f A)) (@Imfset.imfset imfset_key (Finite.choiceType (@fset_sub_finType K (@domf K (Choice.sort V) f))) V (fun k : Choice.sort (Finite.choiceType (@fset_sub_finType K (@domf K (Choice.sort V) f))) => @FunFinfun.fun_of_fin (@fset_sub_finType K (@domf K (Choice.sort V) f)) (Choice.sort V) (@ffun_of_fmap K (Choice.sort V) f) k) (@mem_fin (Choice.eqType (Finite.choiceType (@fset_sub_finType K (@domf K (Choice.sort V) f)))) (simplPredType (@fset_sub_type K (@domf K (Choice.sort V) f))) (@subfinset_finpred (Finite.choiceType (@fset_sub_finType K (@domf K (Choice.sort V) f))) (@mem_fin (Choice.eqType (Finite.choiceType (@fset_sub_finType K (@domf K (Choice.sort V) f)))) (predPredType (@fset_sub_type K (@domf K (Choice.sort V) f) : predArgType)) (@fin_finpred (Choice.eqType (Finite.choiceType (@fset_sub_finType K (@domf K (Choice.sort V) f)))) (pred_finpredType (@fset_sub_finType K (@domf K (Choice.sort V) f))) (@sort_of_simpl_pred (@fset_sub_type K (@domf K (Choice.sort V) f) : predArgType) (pred_of_argType (@fset_sub_type K (@domf K (Choice.sort V) f) : predArgType))))) (fun k : @fset_sub_type K (@domf K (Choice.sort V) f) => @in_mem (Choice.sort K) (@val (Choice.sort K) (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) (@domf K (Choice.sort V) f))) (@fset_sub_subType K (@domf K (Choice.sort V) f)) k) (@mem (Choice.sort K) (finSetPredType K) A)))) (Phantom (mem_pred (@fset_sub_type K (@domf K (Choice.sort V) f))) (@mem (@fset_sub_type K (@domf K (Choice.sort V) f)) (simplPredType (@fset_sub_type K (@domf K (Choice.sort V) f))) (@SimplPred (@fset_sub_type K (@domf K (Choice.sort V) f)) (fun k : @fset_sub_type K (@domf K (Choice.sort V) f) => andb (@in_mem (@fset_sub_type K (@domf K (Choice.sort V) f)) k (@mem (@fset_sub_type K (@domf K (Choice.sort V) f)) (predPredType (@fset_sub_type K (@domf K (Choice.sort V) f) : predArgType)) (@sort_of_simpl_pred (@fset_sub_type K (@domf K (Choice.sort V) f) : predArgType) (pred_of_argType (@fset_sub_type K (@domf K (Choice.sort V) f) : predArgType))))) (@in_mem (Choice.sort K) (@val (Choice.sort K) (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) (@domf K (Choice.sort V) f))) (@fset_sub_subType K (@domf K (Choice.sort V) f)) k) (@mem (Choice.sort K) (finSetPredType K) A))))))) ) apply/perm_eq_small=> //; apply/uniq_perm_eq => //. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by apply/enum_uniq. ( Goal: @eq (Equality.sort V) (@Option.default (Equality.sort V) (default x) (@fnd K (Equality.sort V) (@Imfset.imfset imfset_key (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) K (fun a : @fset_sub_type K S => @fsval K S a) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (simplPredType (@fset_sub_type K S)) (@subfinset_finpred (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (predPredType (@fset_sub_type K S)) (@fin_finpred (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (pred_finpredType (@subFinType_finType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subfinType K S))) (@sort_of_simpl_pred (@fset_sub_type K S) (pred_of_argType (@fset_sub_type K S))))) (fun a : @fset_sub_type K S => negb (@eq_op V (h a) (default (@fsval K S a)))))) (Phantom (mem_pred (@fset_sub_type K S)) (@mem (@fset_sub_type K S) (simplPredType (@fset_sub_type K S)) (@SimplPred (@fset_sub_type K S) (fun a : @fset_sub_type K S => negb (@eq_op V (h a) (default (@fsval K S a)))))))) (@FunFinfun.finfun (@fset_sub_finType K (@Imfset.imfset imfset_key (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) K (fun a : @fset_sub_type K S => @fsval K S a) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (simplPredType (@fset_sub_type K S)) (@subfinset_finpred (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (predPredType (@fset_sub_type K S)) (@fin_finpred (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (pred_finpredType (@subFinType_finType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subfinType K S))) (@sort_of_simpl_pred (@fset_sub_type K S) (pred_of_argType (@fset_sub_type K S))))) (fun a : @fset_sub_type K S => negb (@eq_op V (h a) (default (@fsval K S a)))))) (Phantom (mem_pred (@fset_sub_type K S)) (@mem (@fset_sub_type K S) (simplPredType (@fset_sub_type K S)) (@SimplPred (@fset_sub_type K S) (fun a : @fset_sub_type K S => negb (@eq_op V (h a) (default (@fsval K S a))))))))) (Equality.sort V) (fun a : @fset_sub_type K (@Imfset.imfset imfset_key (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) K (fun a : @fset_sub_type K S => @fsval K S a) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (simplPredType (@fset_sub_type K S)) (@subfinset_finpred (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (predPredType (@fset_sub_type K S)) (@fin_finpred (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (pred_finpredType (@subFinType_finType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subfinType K S))) (@sort_of_simpl_pred (@fset_sub_type K S) (pred_of_argType (@fset_sub_type K S))))) (fun a : @fset_sub_type K S => negb (@eq_op V (h a) (default (@fsval K S a)))))) (Phantom (mem_pred (@fset_sub_type K S)) (@mem (@fset_sub_type K S) (simplPredType (@fset_sub_type K S)) (@SimplPred (@fset_sub_type K S) (fun a : @fset_sub_type K S => negb (@eq_op V (h a) (default (@fsval K S a)))))))) => h (@fincl K (@Imfset.imfset imfset_key (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) K (@fsval K S) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (simplPredType (@fset_sub_type K S)) (@subfinset_finpred (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (predPredType (@fset_sub_type K S)) (@fin_finpred (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (pred_finpredType (@subFinType_finType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subfinType K S))) (@sort_of_simpl_pred (@fset_sub_type K S) (pred_of_argType (@fset_sub_type K S))))) (fun a0 : @fset_sub_type K S => negb (@eq_op V (h a0) (default (@fsval K S a0)))))) (Phantom (mem_pred (@fset_sub_type K S)) (@mem (@fset_sub_type K S) (simplPredType (@fset_sub_type K S)) (@SimplPred (@fset_sub_type K S) (fun x : @fset_sub_type K S => negb (@eq_op V (h x) (default (@fsval K S x)))))))) S (@fset_sub_val imfset_key K S (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (simplPredType (@fset_sub_type K S)) (@subfinset_finpred (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (predPredType (@fset_sub_type K S)) (@fin_finpred (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (pred_finpredType (@subFinType_finType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subfinType K S))) (@sort_of_simpl_pred (@fset_sub_type K S) (pred_of_argType (@fset_sub_type K S))))) (fun a0 : @fset_sub_type K S => negb (@eq_op V (h a0) (default (@fsval K S a0))))))) a))) x)) (default x) ) ( Goal: @eq (Equality.sort V) (@Option.default (Equality.sort V) (default (@fsval K S u)) (@fnd K (Equality.sort V) (@Imfset.imfset imfset_key (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) K (fun a : @fset_sub_type K S => @fsval K S a) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (simplPredType (@fset_sub_type K S)) (@subfinset_finpred (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (predPredType (@fset_sub_type K S)) (@fin_finpred (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (pred_finpredType (@subFinType_finType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subfinType K S))) (@sort_of_simpl_pred (@fset_sub_type K S) (pred_of_argType (@fset_sub_type K S))))) (fun a : @fset_sub_type K S => negb (@eq_op V (h a) (default (@fsval K S a)))))) (Phantom (mem_pred (@fset_sub_type K S)) (@mem (@fset_sub_type K S) (simplPredType (@fset_sub_type K S)) (@SimplPred (@fset_sub_type K S) (fun a : @fset_sub_type K S => negb (@eq_op V (h a) (default (@fsval K S a)))))))) (@FunFinfun.finfun (@fset_sub_finType K (@Imfset.imfset imfset_key (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) K (fun a : @fset_sub_type K S => @fsval K S a) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (simplPredType (@fset_sub_type K S)) (@subfinset_finpred (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (predPredType (@fset_sub_type K S)) (@fin_finpred (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (pred_finpredType (@subFinType_finType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subfinType K S))) (@sort_of_simpl_pred (@fset_sub_type K S) (pred_of_argType (@fset_sub_type K S))))) (fun a : @fset_sub_type K S => negb (@eq_op V (h a) (default (@fsval K S a)))))) (Phantom (mem_pred (@fset_sub_type K S)) (@mem (@fset_sub_type K S) (simplPredType (@fset_sub_type K S)) (@SimplPred (@fset_sub_type K S) (fun a : @fset_sub_type K S => negb (@eq_op V (h a) (default (@fsval K S a))))))))) (Equality.sort V) (fun a : @fset_sub_type K (@Imfset.imfset imfset_key (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) K (fun a : @fset_sub_type K S => @fsval K S a) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (simplPredType (@fset_sub_type K S)) (@subfinset_finpred (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (predPredType (@fset_sub_type K S)) (@fin_finpred (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (pred_finpredType (@subFinType_finType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subfinType K S))) (@sort_of_simpl_pred (@fset_sub_type K S) (pred_of_argType (@fset_sub_type K S))))) (fun a : @fset_sub_type K S => negb (@eq_op V (h a) (default (@fsval K S a)))))) (Phantom (mem_pred (@fset_sub_type K S)) (@mem (@fset_sub_type K S) (simplPredType (@fset_sub_type K S)) (@SimplPred (@fset_sub_type K S) (fun a : @fset_sub_type K S => negb (@eq_op V (h a) (default (@fsval K S a)))))))) => h (@fincl K (@Imfset.imfset imfset_key (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) K (@fsval K S) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (simplPredType (@fset_sub_type K S)) (@subfinset_finpred (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (predPredType (@fset_sub_type K S)) (@fin_finpred (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (pred_finpredType (@subFinType_finType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subfinType K S))) (@sort_of_simpl_pred (@fset_sub_type K S) (pred_of_argType (@fset_sub_type K S))))) (fun a0 : @fset_sub_type K S => negb (@eq_op V (h a0) (default (@fsval K S a0)))))) (Phantom (mem_pred (@fset_sub_type K S)) (@mem (@fset_sub_type K S) (simplPredType (@fset_sub_type K S)) (@SimplPred (@fset_sub_type K S) (fun x : @fset_sub_type K S => negb (@eq_op V (h x) (default (@fsval K S x)))))))) S (@fset_sub_val imfset_key K S (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (simplPredType (@fset_sub_type K S)) (@subfinset_finpred (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (predPredType (@fset_sub_type K S)) (@fin_finpred (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (pred_finpredType (@subFinType_finType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subfinType K S))) (@sort_of_simpl_pred (@fset_sub_type K S) (pred_of_argType (@fset_sub_type K S))))) (fun a0 : @fset_sub_type K S => negb (@eq_op V (h a0) (default (@fsval K S a0))))))) a))) (@fsval K S u))) (h u) ) case=> [y hy]; rewrite mem_seq1 mem_enum /in_mem /=. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite eqE /=; rewrite in_fset1 in hy. Qed. End Enum. Section ImfsetTh. Variables (key : unit) (K V : choiceType). Implicit Types (f : K -> V) (g : V -> K) (A V : {fset K}). Lemma imfset_id (A : {fset K}) : id @` A = A. Proof. by apply/fsetP=> a; rewrite in_fset. Qed. Lemma imfset_comp f g (p : finmempred _) : imfset key (g \o f) p = g @` (imfset key f p). Proof. ( Goal: @eq (@finset_of V (Phant (Choice.sort V))) (codomf (@restrictf K (Choice.sort V) f A)) (@Imfset.imfset imfset_key (Finite.choiceType (@fset_sub_finType K (@domf K (Choice.sort V) f))) V (fun k : Choice.sort (Finite.choiceType (@fset_sub_finType K (@domf K (Choice.sort V) f))) => @FunFinfun.fun_of_fin (@fset_sub_finType K (@domf K (Choice.sort V) f)) (Choice.sort V) (@ffun_of_fmap K (Choice.sort V) f) k) (@mem_fin (Choice.eqType (Finite.choiceType (@fset_sub_finType K (@domf K (Choice.sort V) f)))) (simplPredType (@fset_sub_type K (@domf K (Choice.sort V) f))) (@subfinset_finpred (Finite.choiceType (@fset_sub_finType K (@domf K (Choice.sort V) f))) (@mem_fin (Choice.eqType (Finite.choiceType (@fset_sub_finType K (@domf K (Choice.sort V) f)))) (predPredType (@fset_sub_type K (@domf K (Choice.sort V) f) : predArgType)) (@fin_finpred (Choice.eqType (Finite.choiceType (@fset_sub_finType K (@domf K (Choice.sort V) f)))) (pred_finpredType (@fset_sub_finType K (@domf K (Choice.sort V) f))) (@sort_of_simpl_pred (@fset_sub_type K (@domf K (Choice.sort V) f) : predArgType) (pred_of_argType (@fset_sub_type K (@domf K (Choice.sort V) f) : predArgType))))) (fun k : @fset_sub_type K (@domf K (Choice.sort V) f) => @in_mem (Choice.sort K) (@val (Choice.sort K) (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) (@domf K (Choice.sort V) f))) (@fset_sub_subType K (@domf K (Choice.sort V) f)) k) (@mem (Choice.sort K) (finSetPredType K) A)))) (Phantom (mem_pred (@fset_sub_type K (@domf K (Choice.sort V) f))) (@mem (@fset_sub_type K (@domf K (Choice.sort V) f)) (simplPredType (@fset_sub_type K (@domf K (Choice.sort V) f))) (@SimplPred (@fset_sub_type K (@domf K (Choice.sort V) f)) (fun k : @fset_sub_type K (@domf K (Choice.sort V) f) => andb (@in_mem (@fset_sub_type K (@domf K (Choice.sort V) f)) k (@mem (@fset_sub_type K (@domf K (Choice.sort V) f)) (predPredType (@fset_sub_type K (@domf K (Choice.sort V) f) : predArgType)) (@sort_of_simpl_pred (@fset_sub_type K (@domf K (Choice.sort V) f) : predArgType) (pred_of_argType (@fset_sub_type K (@domf K (Choice.sort V) f) : predArgType))))) (@in_mem (Choice.sort K) (@val (Choice.sort K) (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) (@domf K (Choice.sort V) f))) (@fset_sub_subType K (@domf K (Choice.sort V) f)) k) (@mem (Choice.sort K) (finSetPredType K) A))))))) ) apply/fsetP=> a; apply/imfsetP/imfsetP=> [[/= x xA ->]\|]. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by exists (f x); rewrite // in_imfset. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by move=> [/= x /imfsetP [/= y yA ->] ->]; exists y. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma subset_imfset f (p q : finmempred _) : {subset p <= q} -> imfset key f p `<=` imfset key f q. Proof. ( Goal: forall (_ : is_true (negb (@in_mem (Choice.sort K) k (@mem (Choice.sort K) (finMapPredType K V) f)))) (_ : is_true (negb (@in_mem (Choice.sort K) k (@mem (Choice.sort K) (finMapPredType K V) f')))) (_ : @eq (@finmap_of K V (Phant (forall _ : Choice.sort K, V))) (@setf K V f k v) (@setf K V f' k v)), @eq (@finmap_of K V (Phant (forall _ : Choice.sort K, V))) f f' ) move=> subPQ; apply/fsubsetP=> x /imfsetP [y /= yA ->]. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite in_imfset //= [in_mem _ _]subPQ. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma eq_imfset (f f' : K -> V) (p q : finmempred _): ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) f =1 f' -> (forall x, in_mem x p = in_mem x q) -> imfset key f p = imfset key f' q. Proof. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) move=> eq_f eqP; apply/fsetP => x; apply/imfsetP/imfsetP => /= [] [k Pk ->]; by exists k => //=; rewrite ?eq_f ?eqP in Pk . Qed. End ImfsetTh. Section PowerSetTheory. Variable (K : choiceType). Fact fpowerset_key : unit. Proof. exact: tt. Qed. Definition fpowerset (A : {fset K}) : {fset {fset K}} := [fset[fpowerset_key] [fsetval y in Y : {set A}] \| Y in powerset [set: A]]. Lemma fpowersetE A B : (B \in fpowerset A) = (B `<=` A). Proof. (* Goal: Bool.reflect (@ex (Choice.sort K) (fun x : Choice.sort K => @eq (option (Choice.sort V)) (@fnd K (Choice.sort V) (@domf K (Choice.sort V) f) (@ffun_of_fmap K (Choice.sort V) f) x) (@Some (Choice.sort V) v))) (@in_mem (Choice.sort V) v (@mem (Choice.sort V) (finSetPredType V) (codomf f))) ) apply/imfsetP/fsubsetP => /= [[Z _ -> y /in_fset_valP [] //]\|/fsubsetP subYX]. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) exists (fsub _ B); last by rewrite fsubK. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite powersetE /= -fsubT subset_fsub ?fsubset_refl. Qed. Lemma fpowersetCE (X A B : {fset K}) : (A \in fpowerset (X `\` B)) = (A `<=` X) && [disjoint A & B]%fset. Proof. by rewrite fpowersetE fsubsetD. Qed. Lemma fpowersetS A B : (fpowerset A `<=` fpowerset B) = (A `<=` B). Proof. ( Goal: Bool.reflect (@ex (Choice.sort K) (fun x : Choice.sort K => @eq (option (Choice.sort V)) (@fnd K (Choice.sort V) (@domf K (Choice.sort V) f) (@ffun_of_fmap K (Choice.sort V) f) x) (@Some (Choice.sort V) v))) (@in_mem (Choice.sort V) v (@mem (Choice.sort V) (finSetPredType V) (codomf f))) ) apply/fsubsetP/fsubsetP => [sub_pA_pB a\|subAB X]. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by have := sub_pA_pB [fset a]; rewrite !fpowersetE !fsub1set. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite !fpowersetE => /fsubsetP XA; apply/fsubsetP => x /XA /subAB. Qed. Lemma fpowerset0 : fpowerset fset0 = [fset fset0]. Proof. by apply/fsetP=> X; rewrite inE fpowersetE fsubset0. Qed. Lemma fpowerset1 (x : K) : fpowerset [fset x] = [fset fset0; [fset x]]. Proof. by apply/fsetP => X; rewrite !inE fpowersetE fsubset1 orbC. Qed. Lemma fpowersetI A B : fpowerset (A `&` B) = fpowerset A `&` fpowerset B. Proof. by apply/fsetP=> X; rewrite inE !fpowersetE fsubsetI. Qed. Lemma card_fpowerset (A : {fset K}) : #\|` fpowerset A\| = 2 ^ #\|` A\|. Proof. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) rewrite !card_imfset; first by rewrite -cardE card_powerset cardsE cardfE. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by move=> X Y /fsetP eqXY; apply/setP => x; have := eqXY (val x); rewrite !inE. Qed. End PowerSetTheory. Section BigFSet. Variable (R : Type) (idx : R) (op : Monoid.law idx). Variable (I : choiceType). ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma big_seq_fsetE (X : {fset I}) (P : pred I) (F : I -> R) : ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) \big[op/idx]_(i <- X \| P i) F i = \big[op/idx]_(x : X \| P (val x)) F (val x). Proof. by rewrite enum_fsetE big_map enumT. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma big1_fset (X : {fset I}) (P : pred I) (F : I -> R) : ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) (forall i, i \in X -> P i -> F i = idx) -> ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) \big[op/idx]_(i <- X \| P i) F i = idx. Proof. by move=> Fid; rewrite big_seq_fsetE big1//= => -[]. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma big_fset0 (P : pred fset0) (F : @fset0 I -> R) : \big[op/idx]_(i : fset0 \| P i) F i = idx. Proof. by rewrite /index_enum -enumT /= enum_fset0 big_nil. Qed. Lemma big_seq_fset0 (F : I -> R): \big[op/idx]_(i <- fset0) F i = idx. Proof. by rewrite big_seq_fsetE big_fset0. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma big_fset1 (a : I) (F : [fset a] -> R) : \big[op/idx]_(i : [fset a]) F i = F [` fset11 a]. Proof. by rewrite /index_enum -enumT enum_fset1 big_seq1. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma big_seq_fset1 (a : I) (F : I -> R) : ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) \big[op/idx]_(i <- [fset a]) F i = F a. Proof. by rewrite big_seq_fsetE big_fset1. Qed. End BigFSet. Notation eq_big_imfset := (eq_big_perm _ (enum_imfset _ _)). Section BigComFSet. Variable (R : Type) (idx : R) (op : Monoid.com_law idx). Variable (I J : choiceType). ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma big_fset (X : finmempred _) (P : pred I) (F : I -> R) : ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) \big[op/idx]_(i <- [fset i in X \| P i]) F i = \big[op/idx]_(i <- enum_finmem X \| P i) F i. Proof. by rewrite !eq_big_imfset//= !big_map !big_filter_cond big_andbC. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma big_fset_condE (X : {fset I}) (P : pred I) (F : I -> R) : ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) \big[op/idx]_(i <- X \| P i) F i = \big[op/idx]_(i <- [fset i in X \| P i]) F i. Proof. by rewrite big_fset. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma eq_fbig_cond (A B : {fset I}) (P Q : pred I) (F G : I -> R) : ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) [fset x in A \| P x] =i [fset x in B \| Q x] -> ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) (forall x, x \in A -> P x -> F x = G x) -> ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) \big[op/idx]_(i <- A \| P i) F i = \big[op/idx]_(i <- B \| Q i) G i. Proof. ( Goal: forall (_ : is_true (@fdisjoint K (@domf K V f) (@domf K V h))) (_ : is_true (@fdisjoint K (@domf K V g) (@domf K V h))) (_ : @eq (finMap K V) (catf h f) (catf h g)), @eq (@finmap_of K V (Phant (forall _ : Choice.sort K, V))) f g ) move=> /fsetP eqABPQ FG; rewrite big_fset_condE [in RHS]big_fset_condE -eqABPQ. ( Goal: is_true (@in_mem (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f (@fsval K (@FinSupp.fs K (Choice.eqType K) (fun x : Choice.sort K => x) f) a)) (@mem (Choice.sort K) (finSetPredType K) (@FinSupp.fs K (Choice.eqType K) (fun x : Choice.sort K => x) f))) ) ( Goal: @ex2 (@finset_of K (Phant (Choice.sort K))) (fun S : @finset_of K (Phant (Choice.sort K)) => is_true (@fsubset K (@FinSupp.fs K (Choice.eqType K) (fun x : Choice.sort K => x) f) S)) (fun S : @finset_of K (Phant (Choice.sort K)) => and (@prop_in2 (Choice.sort K) (@mem (Choice.sort K) (finSetPredType K) S) (fun x1 x2 : Choice.sort K => forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f x1) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f x2), @eq (Choice.sort K) x1 x2) (inPhantom (@injective (Choice.sort K) (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f)))) (forall a : @fset_sub_type K S, is_true (@in_mem (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f (@fsval K S a)) (@mem (Choice.sort K) (finSetPredType K) S)))) ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) rewrite big_seq_cond [in RHS]big_seq_cond; apply: eq_bigr => i. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite in_fset !inE => /andP[/andP[??] _]; apply: FG. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma eq_fbig (A B : {fset I}) (F G : I -> R) : ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) A =i B -> (forall x, x \in A -> F x = G x) -> ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) \big[op/idx]_(i <- A) F i = \big[op/idx]_(i <- B) G i. Proof. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by move=> AB FG; apply: eq_fbig_cond => x; rewrite ?inE/= -?AB// => /FG. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma eq_fbigl_cond (A B : {fset I}) (P Q : pred I) (F : I -> R) : ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) [fset x in A \| P x] =i [fset x in B \| Q x] -> ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) \big[op/idx]_(i <- A \| P i) F i = \big[op/idx]_(i <- B \| Q i) F i. Proof. by move=> AB; apply: eq_fbig_cond. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma eq_fbigl (A B : {fset I}) (F : I -> R) : ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) A =i B -> \big[op/idx]_(i <- A) F i = \big[op/idx]_(i <- B) F i. Proof. by move=> AB; apply: eq_fbig. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma eq_fbigr (A : {fset I}) (P : pred I) (F G : I -> R) : ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) (forall x, x \in A -> P x -> F x = G x) -> ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) \big[op/idx]_(i <- A \| P i) F i = \big[op/idx]_(i <- A \| P i) G i. Proof. by apply: eq_fbig_cond. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma big_fsetID (B : pred I) (A : {fset I}) (F : I -> R) : ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) \big[op/idx]_(i <- A) F i = ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) op (\big[op/idx]_(i <- [fset x in A \| B x]) F i) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) (\big[op/idx]_(i <- [fset x in A \| ~~ B x]) F i). Proof. by rewrite !big_fset; apply: bigID. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma big_fsetIDcond (B : pred I) (A : {fset I}) (P : pred I) (F : I -> R) : ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) \big[op/idx]_(i <- A \| P i) F i = ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) op (\big[op/idx]_(i <- [fset x in A \| B x] \| P i) F i) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) (\big[op/idx]_(i <- [fset x in A \| ~~ B x] \| P i) F i). Proof. by rewrite big_mkcond (big_fsetID B) // -!big_mkcond. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma big_fsetD1 (a : I) (A : {fset I}) (F : I -> R) : a \in A -> ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) \big[op/idx]_(i <- A) F i = op (F a) (\big[op/idx]_(i <- A `\ a) F i). Proof. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) move=> aA; rewrite (big_fsetID (mem [fset a])); congr (op _ _); last first. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by apply: eq_fbigl=> i; rewrite !inE/= andbC. ( Goal: @eq (Equality.sort V) (@fun_of_fsfun K V default (of_ffun key) x) (@Option.default (Equality.sort V) (default x) (@Option.map (@fset_sub_type K S) (Equality.sort V) h (@insub (Choice.sort K) (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S) x))) ) rewrite (_ : [fset _ \| _ in _ & _] = [fset a]) ?big_seq_fset1//=. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by apply/fsetP=> i; rewrite !inE /= andbC; case: eqP => //->. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma big_setU1 (a : I) (A : {fset I}) (F : I -> R) : a \notin A -> ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) \big[op/idx]_(i <- (a \|` A)) F i = op (F a) (\big[op/idx]_(i <- A) F i). Proof. ( Goal: forall (_ : is_true (@fdisjoint K (@domf K V f) (@domf K V h))) (_ : is_true (@fdisjoint K (@domf K V g) (@domf K V h))) (_ : @eq (finMap K V) (catf h f) (catf h g)), @eq (@finmap_of K V (Phant (forall _ : Choice.sort K, V))) f g ) move=> aNA; rewrite eq_big_imfset//= big_map undup_id ?big_cat ?big_seq_fset1//. ( Goal: @eq (@finset_of V (Phant (Choice.sort V))) (codomf (@setf K (Choice.sort V) f k v)) (@fsetU V (@fset1 V v) (codomf (@restrictf K (Choice.sort V) f (@fsetD K (@domf K (Choice.sort V) f) (@fset1 K k))))) ) rewrite cat_uniq ?fset_uniq andbT//=; apply/hasPn=> /= x xA; rewrite !inE/=. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by apply: contraNneq aNA => <-. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma big_fset_incl (A : {fset I}) B F : A `<=` B -> ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) (forall x, x \in B -> x \notin A -> F x = idx) -> ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) \big[op/idx]_(x <- A) F x = \big[op/idx]_(x <- B) F x. Proof. ( Goal: forall (_ : is_true (@fdisjoint K (@domf K V f) (@domf K V h))) (_ : is_true (@fdisjoint K (@domf K V g) (@domf K V h))) (_ : @eq (finMap K V) (catf h f) (catf h g)), @eq (@finmap_of K V (Phant (forall _ : Choice.sort K, V))) f g ) move=> subAB Fid; rewrite [in RHS](big_fsetID (mem A)) /=. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) rewrite [X in op _ X]big1_fset ?Monoid.mulm1; last first. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by move=> i; rewrite !inE /= => /andP[iB iNA _]; apply: Fid. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by apply: eq_fbigl => i; rewrite !inE /= -(@in_fsetI _ B A) (fsetIidPr _). Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma big_imfset key (h : I -> J) (A : finmempred _) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) (G : J -> R) : {in A &, injective h} -> ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) \big[op/idx]_(j <- imfset key h A) G j = ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) \big[op/idx]_(i <- enum_finmem A) G (h i). Proof. by move=> h_inj; rewrite eq_big_imfset// big_map. Qed. End BigComFSet. Arguments big_fsetD1 {R idx op I} a [A F]. Notation eq_big_imfset2 := (eq_big_perm _ (enum_imfset2 _ _)). Section BigComImfset2. Variables (R : Type) (idx : R) (op : Monoid.com_law idx) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) (I : choiceType) (J : I -> choiceType) (K : choiceType). Lemma big_imfset2 key (A : finmempred I) (B : forall i, finmempred (J i)) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) (h : forall i : I, J i -> K) (F : K -> R) : {in [pred t : sigT J \| A (tag t) & B _ (tagged t)] &, ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) injective (fun t => h (tag t) (tagged t))} -> ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) \big[op/idx]_(k <- imfset2 key h A B) F k = ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) \big[op/idx]_(i <- enum_finmem A) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) \big[op/idx]_(j <- enum_finmem (B i)) F (h i j). Proof. ( Goal: forall (_ : is_true (@fdisjoint K (@domf K V f) (@domf K V h))) (_ : is_true (@fdisjoint K (@domf K V g) (@domf K V h))) (_ : @eq (finMap K V) (catf h f) (catf h g)), @eq (@finmap_of K V (Phant (forall _ : Choice.sort K, V))) f g ) move=> h_inj; rewrite eq_big_imfset2 //. ( Goal: @eq (Equality.sort V) (@fun_of_fsfun K V default (of_ffun key) x) (@Option.default (Equality.sort V) (default x) (@Option.map (@fset_sub_type K S) (Equality.sort V) h (@insub (Choice.sort K) (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S) x))) ) rewrite (allsigs_comp (fun _ j => Tagged _ j) (fun t => h (tag t) (tagged t))). ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite big_map big_allsigs /=. Qed. End BigComImfset2. Section BigFsetDep. Variables (R : Type) (idx : R) (op : Monoid.com_law idx) (I : choiceType) (J : choiceType) (K : choiceType). ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma pair_big_dep_cond (A : {fset I}) (B : I -> {fset J}) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) (P : pred I) (Q : I -> pred J) (F : I -> J -> R) : ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) \big[op/idx]_(i <- A \| P i) \big[op/idx]_(j <- B i \| Q i j) F i j = ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) \big[op/idx]_(p <- [fset ((i, j) : I J) \| i in [fset i in A \| P i], j in [fset j in B i \| Q i j]]) F p.1 p.2. Proof. (* Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) rewrite big_imfset2 //=; last by move=> [??] [??] _ _ /= [-> ->]. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite big_fset /=; apply: eq_bigr => i _; rewrite big_fset. Qed. End BigFsetDep. Section BigComImfset. Variables (R : Type) (idx : R) (op : Monoid.com_law idx) (I : choiceType) (J : choiceType) (K : choiceType). ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma partition_big_imfset (h : I -> J) (A : {fset I}) (F : I -> R) : ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) \big[op/idx]_(i <- A) F i = ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) \big[op/idx]_(j <- [fset h x \| x in A]) \big[op/idx]_(i <- A \| h i == j) F i. Proof. transitivity (\big[op/idx]_(i <- [fset (h i, i) \| i in A]) F i.2). ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite eq_big_imfset ?big_map// => i j ? ? []. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) transitivity (\big[op/idx]_(i <- [fset ij \| ij in ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) [seq (i1, i2) \| i1 <- [fset h x \| x in A], i2 <- A]]) if h i.2 == i.1 then F i.2 else idx). ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) rewrite -big_mkcond; apply: eq_fbigl_cond; move=> x; rewrite !inE/= andbT. ( Goal: Bool.reflect (@ex (Choice.sort K) (fun x : Choice.sort K => @eq (option (Choice.sort V)) (@fnd K (Choice.sort V) (@domf K (Choice.sort V) f) (@ffun_of_fmap K (Choice.sort V) f) x) (@Some (Choice.sort V) v))) (@in_mem (Choice.sort V) v (@mem (Choice.sort V) (finSetPredType V) (codomf f))) ) apply/imfsetP/idP => [[i /= iA -> /=]\|]. rewrite eqxx andbT; apply/allpairsP; exists (h i, i) => /=. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite in_imfset. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by move=> /andP[/allpairsP[[/= j i] [/imfsetP[/=a aA ->] iA ->/= /eqP<-]]]; exists i. ( Goal: @eq (Equality.sort V) (@fun_of_fsfun K V default (of_ffun key) x) (@Option.default (Equality.sort V) (default x) (@Option.map (@fset_sub_type K S) (Equality.sort V) h (@insub (Choice.sort K) (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S) x))) ) rewrite eq_big_imfset //= big_map undup_id. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite big_allpairs; apply: eq_bigr => i /= _; rewrite -big_mkcond. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite allpairs_uniq => //= -[j0 i0] [j1 i1] /=. Qed. End BigComImfset. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Notation "\bigcup_ ( i <- r \| P ) F" := ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) (\big[@fsetU _/fset0]_(i <- r \| P%fset) F%fset) : fset_scope. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Notation "\bigcup_ ( i <- r ) F" := ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) (\big[@fsetU _/fset0]_(i <- r) F%fset) : fset_scope. Notation "\bigcup_ ( i \| P ) F" := (\big[@fsetU _/fset0]_(i \| P) F%fset) : fset_scope. Notation "\bigcup_ ( i 'in' A \| P ) F" := (\big[@fsetU _/fset0]_(i in A \| P) F%fset) : fset_scope. Notation "\bigcup_ ( i 'in' A ) F" := (\big[@fsetU _/fset0]_(i in A) F%fset) : fset_scope. Section FSetMonoids. Import Monoid. Variable (T : choiceType). Canonical fsetU_monoid := Law (@fsetUA T) (@fset0U T) (@fsetU0 T). Canonical fsetU_comoid := ComLaw (@fsetUC T). End FSetMonoids. Section BigFOpsSeq. Variables (T : choiceType) (I : eqType) (r : seq I). ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Implicit Types (P : pred I) (F : I -> {fset T}). Lemma bigfcup_undup P F : ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) \bigcup_(i <- undup r \| P i) F i = \bigcup_(i <- r \| P i) F i. Proof. by rewrite big_undup => //= A; rewrite fsetUid. Qed. Lemma bigfcup_sup j P F : j \in r -> P j -> F j `<=` \bigcup_(i <- r \| P i) F i. Proof. ( Goal: forall (_ : is_true (@fdisjoint K (@domf K V f) (@domf K V h))) (_ : is_true (@fdisjoint K (@domf K V g) (@domf K V h))) (_ : @eq (finMap K V) (catf h f) (catf h g)), @eq (@finmap_of K V (Phant (forall _ : Choice.sort K, V))) f g ) move=> jr Pj; rewrite -bigfcup_undup big_mkcond. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite (bigD1_seq j) ?mem_undup ?undup_uniq ?Pj //= fsubsetUl. Qed. Lemma bigfcupP x F P : reflect (exists2 i : I, (i \in r) && P i & x \in F i) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) (x \in \bigcup_(i <- r \| P i) F i). Proof. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) apply: (iffP idP) => [\|[i /andP[ri Pi]]]; last first. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by apply: fsubsetP x; rewrite bigfcup_sup. rewrite big_seq_cond; elim/big_rec: _ => [\|i _ /andP[ri Pi] _ /fsetUP[\|//]]. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite in_fset0. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by exists i; rewrite ?ri. Qed. Lemma bigfcupsP (U : {fset T}) P F : ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) reflect (forall i : I, i \in r -> P i -> F i `<=` U) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) (\bigcup_(i <- r \| P i) F i `<=` U). Proof. ( Goal: Bool.reflect (@ex (Choice.sort K) (fun x : Choice.sort K => @eq (option (Choice.sort V)) (@fnd K (Choice.sort V) (@domf K (Choice.sort V) f) (@ffun_of_fmap K (Choice.sort V) f) x) (@Some (Choice.sort V) v))) (@in_mem (Choice.sort V) v (@mem (Choice.sort V) (finSetPredType V) (codomf f))) ) apply: (iffP idP) => [sFU i ri Pi\| sFU]. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by apply: fsubset_trans sFU; apply: bigfcup_sup. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by apply/fsubsetP=> x /bigfcupP[i /andP[ri Pi]]; apply/fsubsetP/sFU. Qed. End BigFOpsSeq. ( ** Induction Principles ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma finSet_rect (T : choiceType) (P : {fset T} -> Type) : ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) P fset0 -> (forall X, (forall Y, Y `<` X -> P Y) -> P X) -> forall X, P X. Proof. ( Goal: forall (_ : P (@fset0 T)) (_ : forall (X : @finset_of T (Phant (Choice.sort T))) (_ : forall (Y : @finset_of T (Phant (Choice.sort T))) (_ : is_true (@fproper T Y X)), P Y), P X) (X : @finset_of T (Phant (Choice.sort T))), P X ) move=> P0 Psub X; move: (leqnn #\|` X\|); move: (X in Y in _ <= Y) => Y. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) elim: #\|` _\| X => [\|n IHn] {Y} X; first by rewrite leqn0 cardfs_eq0 => /eqP->. ( Goal: forall _ : is_true (leq (@size (Choice.sort T) (@enum_fset T X)) (S n)), P X ) move=> Xleq; apply: Psub => Y XsubY; apply: IHn. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite -ltnS (leq_trans _ Xleq) // fproper_ltn_card. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma fset_bounded_coind (T : choiceType) (P : {fset T} -> Type) (U : {fset T}): ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) (forall X, (forall Y, Y `<=` U -> X `<` Y -> P Y) -> P X) -> ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) forall X, X `<=` U -> P X. Proof. ( Goal: forall (_ : is_true (@fdisjoint K (@domf K V f) (@domf K V h))) (_ : is_true (@fdisjoint K (@domf K V g) (@domf K V h))) (_ : @eq (finMap K V) (catf h f) (catf h g)), @eq (@finmap_of K V (Phant (forall _ : Choice.sort K, V))) f g ) move=> Psuper X XsubU; rewrite -[X](fsetDK XsubU)//. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) have {XsubU}: (U `\` X) `<=` U by rewrite fsubsetDl. ( Goal: forall _ : is_true (@fsubset T (@fsetD T U X) U), P (@fsetD T U (@fsetD T U X)) ) elim: (_ `\` X) => {X} [\|X IHX] XsubU. ( Goal: is_true (@in_mem (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f (@fsval K (@FinSupp.fs K (Choice.eqType K) (fun x : Choice.sort K => x) f) a)) (@mem (Choice.sort K) (finSetPredType K) (@FinSupp.fs K (Choice.eqType K) (fun x : Choice.sort K => x) f))) ) ( Goal: @ex2 (@finset_of K (Phant (Choice.sort K))) (fun S : @finset_of K (Phant (Choice.sort K)) => is_true (@fsubset K (@FinSupp.fs K (Choice.eqType K) (fun x : Choice.sort K => x) f) S)) (fun S : @finset_of K (Phant (Choice.sort K)) => and (@prop_in2 (Choice.sort K) (@mem (Choice.sort K) (finSetPredType K) S) (fun x1 x2 : Choice.sort K => forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f x1) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f x2), @eq (Choice.sort K) x1 x2) (inPhantom (@injective (Choice.sort K) (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f)))) (forall a : @fset_sub_type K S, is_true (@in_mem (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f (@fsval K S a)) (@mem (Choice.sort K) (finSetPredType K) S)))) ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) rewrite fsetD0; apply: Psuper => Y /fsub_proper_trans UY/UY. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite fproperEneq eqxx. ( Goal: is_true (@fsubset K (@FinSupp.fs K V (fun a : Choice.sort K => default a) (@fsfun_of_ffun key K V S (fun a : @fset_sub_type K S => h a) (fun a : Choice.sort K => default a))) S) ) apply: Psuper => Y /fsetDK<-; rewrite fproperD2l ?fsubsetDl //. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by move=> /IHX; apply; rewrite fsubsetDl. Qed. (* ** Fixpoints ) Lemma iter_fix T (f : T -> T) x n : f x = x -> iter n f x = x. Proof. by move=> fixf; elim: n => //= n ->. Qed. Section SetFixpoint. (* Least Fixpoints ) Section Least. Variables (T : finType) (F : {set T} -> {set T}). Hypothesis (F_mono : {homo F : X Y / X \subset Y}). Let n := #\|T\|. Notation iterF := (fun i => iter i F set0). Lemma set_iterF_sub i : iterF i \subset iterF i.+1. Proof. by elim: i => [\| i IHi]; rewrite /= ?sub0set ?F_mono. Qed. Lemma set_iterF_mono : {homo iterF : i j / i <= j >-> i \subset j}. Proof. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by apply: homo_leq => //[???\|]; [apply: subset_trans\|apply: set_iterF_sub]. Qed. Definition set_fix := iterF n. Lemma set_fixK : F set_fix = set_fix. Proof. suff /'exists_eqP[x /= e]: [exists k : 'I_n.+1, iterF k == iterF k.+1]. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite /set_fix -(subnK (leq_ord x)) iter_add iter_fix. ( Goal: is_true (negb (@eq_op (fset_eqType K) (@fsetD K C B) (@fsetD K C A))) ) apply: contraT; rewrite negb_exists => /forallP /(_ (Ordinal _)) /= neq_iter. suff iter_big k : k <= n.+1 -> k <= #\|iter k F set0\|. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by have := iter_big _ (leqnn _); rewrite ltnNge max_card. elim: k => [\|k IHk] k_lt //=; apply: (leq_ltn_trans (IHk (ltnW k_lt))). ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite proper_card// properEneq// set_iterF_sub neq_iter. Qed. Hint Resolve set_fixK. Lemma set_fixKn k : iter k F set_fix = set_fix. Proof. by rewrite iter_fix. Qed. Lemma iter_sub_fix k : iterF k \subset set_fix. Proof. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) have [/set_iterF_mono//\|/ltnW/subnK<-] := leqP k n; by rewrite iter_add set_fixKn. Qed. Lemma fix_order_proof x : x \in set_fix -> exists n, x \in iterF n. Proof. by move=> x_fix; exists n. Qed. Definition fix_order (x : T) := if (x \in set_fix) =P true isn't ReflectT x_fix then 0 else (ex_minn (fix_order_proof x_fix)). Lemma fix_order_le_max (x : T) : fix_order x <= n. Proof. ( Goal: @eq bool (@in_mem (Choice.sort V) x (@mem (Choice.sort V) (simplPredType (Choice.sort V)) (@SimplPred (Choice.sort V) (fun x : Choice.sort V => andb (@in_mem (Choice.sort V) x (@mem (Choice.sort V) (predPredType (Choice.sort V)) (fun x0 : Choice.sort V => @pred_of_eq_seq (Choice.eqType V) (@enum_fset V (codomf f)) x0))) (negb (@FiniteQuant.quant0b (@fset_sub_finType K (@domf K (Choice.sort V) f)) (fun k : @fset_sub_type K (@domf K (Choice.sort V) f) => @FiniteQuant.ex (@fset_sub_finType K (@domf K (Choice.sort V) f)) (FiniteQuant.Quantified (andb (@in_mem (Choice.sort K) (@fsval K (@domf K (Choice.sort V) f) k) (@mem (Choice.sort K) (finSetPredType K) (@fsetD K (@domf K (Choice.sort V) f) A))) (@eq_op (Choice.eqType V) (@FunFinfun.fun_of_fin (@fset_sub_finType K (@domf K (Choice.sort V) f)) (Choice.sort V) (@ffun_of_fmap K (Choice.sort V) f) k) x))) k))))))) (@in_mem (Choice.sort V) x (@mem (Choice.sort V) (simplPredType (Choice.sort V)) (@SimplPred (Choice.sort V) (fun x : Choice.sort V => andb (@in_mem (Choice.sort V) x (@mem (Choice.sort V) (predPredType (Choice.sort V)) (fun x0 : Choice.sort V => @pred_of_eq_seq (Choice.eqType V) (@enum_fset V (codomf f)) x0))) (negb (@FiniteQuant.quant0b (@fset_sub_finType K (@domf K (Choice.sort V) f)) (fun k : @fset_sub_type K (@domf K (Choice.sort V) f) => @FiniteQuant.ex (@fset_sub_finType K (@domf K (Choice.sort V) f)) (FiniteQuant.Quantified (andb (negb (@in_mem (Choice.sort K) (@fsval K (@domf K (Choice.sort V) f) k) (@mem (Choice.sort K) (finSetPredType K) A))) (@eq_op (Choice.eqType V) (@FunFinfun.fun_of_fin (@fset_sub_finType K (@domf K (Choice.sort V) f)) (Choice.sort V) (@ffun_of_fmap K (Choice.sort V) f) k) x))) k))))))) ) rewrite /fix_order; case: eqP => //= x_in. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by case: ex_minnP => //= ??; apply. Qed. Lemma in_iter_fix_orderE (x : T) : (x \in iterF (fix_order x)) = (x \in set_fix). Proof. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) rewrite /fix_order; case: eqP; last by move=>/negP/negPf->; rewrite inE. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by move=> x_in; case: ex_minnP => m ->; rewrite x_in. Qed. Lemma fix_order_gt0 (x : T) : (fix_order x > 0) = (x \in set_fix). Proof. ( Goal: @eq bool (@in_mem (Choice.sort V) x (@mem (Choice.sort V) (simplPredType (Choice.sort V)) (@SimplPred (Choice.sort V) (fun x : Choice.sort V => andb (@in_mem (Choice.sort V) x (@mem (Choice.sort V) (predPredType (Choice.sort V)) (fun x0 : Choice.sort V => @pred_of_eq_seq (Choice.eqType V) (@enum_fset V (codomf f)) x0))) (negb (@FiniteQuant.quant0b (@fset_sub_finType K (@domf K (Choice.sort V) f)) (fun k : @fset_sub_type K (@domf K (Choice.sort V) f) => @FiniteQuant.ex (@fset_sub_finType K (@domf K (Choice.sort V) f)) (FiniteQuant.Quantified (andb (@in_mem (Choice.sort K) (@fsval K (@domf K (Choice.sort V) f) k) (@mem (Choice.sort K) (finSetPredType K) (@fsetD K (@domf K (Choice.sort V) f) A))) (@eq_op (Choice.eqType V) (@FunFinfun.fun_of_fin (@fset_sub_finType K (@domf K (Choice.sort V) f)) (Choice.sort V) (@ffun_of_fmap K (Choice.sort V) f) k) x))) k))))))) (@in_mem (Choice.sort V) x (@mem (Choice.sort V) (simplPredType (Choice.sort V)) (@SimplPred (Choice.sort V) (fun x : Choice.sort V => andb (@in_mem (Choice.sort V) x (@mem (Choice.sort V) (predPredType (Choice.sort V)) (fun x0 : Choice.sort V => @pred_of_eq_seq (Choice.eqType V) (@enum_fset V (codomf f)) x0))) (negb (@FiniteQuant.quant0b (@fset_sub_finType K (@domf K (Choice.sort V) f)) (fun k : @fset_sub_type K (@domf K (Choice.sort V) f) => @FiniteQuant.ex (@fset_sub_finType K (@domf K (Choice.sort V) f)) (FiniteQuant.Quantified (andb (negb (@in_mem (Choice.sort K) (@fsval K (@domf K (Choice.sort V) f) k) (@mem (Choice.sort K) (finSetPredType K) A))) (@eq_op (Choice.eqType V) (@FunFinfun.fun_of_fin (@fset_sub_finType K (@domf K (Choice.sort V) f)) (Choice.sort V) (@ffun_of_fmap K (Choice.sort V) f) k) x))) k))))))) ) rewrite /fix_order; case: eqP => [x_in\|/negP/negPf->//]. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite x_in; case: ex_minnP => -[\|m]; rewrite ?inE//= => _; apply. Qed. Lemma fix_order_eq0 (x : T) : (fix_order x == 0) = (x \notin set_fix). Proof. by rewrite -fix_order_gt0 -ltnNge ltnS leqn0. Qed. Lemma in_iter_fixE (x : T) k : (x \in iterF k) = (0 < fix_order x <= k). Proof. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) rewrite /fix_order; case: eqP => //= [x_in\|/negP xNin]; last first. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by apply: contraNF xNin; apply/subsetP/iter_sub_fix. ( Goal: @eq bool (@in_mem (Finite.sort T) x (@mem (Finite.sort T) (predPredType (Finite.sort T)) (@SetDef.pred_of_set T (@iter (@set_of T (Phant (Finite.sort T))) k F (@set0 T))))) (andb (leq (S O) (@ex_minn (fun n : nat => @in_mem (Finite.sort T) x (@mem (Finite.sort T) (predPredType (Finite.sort T)) (@SetDef.pred_of_set T (@iter (@set_of T (Phant (Finite.sort T))) n F (@set0 T))))) (@fix_order_proof x x_in))) (leq (@ex_minn (fun n : nat => @in_mem (Finite.sort T) x (@mem (Finite.sort T) (predPredType (Finite.sort T)) (@SetDef.pred_of_set T (@iter (@set_of T (Phant (Finite.sort T))) n F (@set0 T))))) (@fix_order_proof x x_in)) k)) ) case: ex_minnP => -[\|m]; rewrite ?inE// => xm mP. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by apply/idP/idP=> [/mP//\|lt_mk]; apply: subsetP xm; apply: set_iterF_mono. Qed. Lemma in_iter (x : T) k : x \in set_fix -> fix_order x <= k -> x \in iterF k. Proof. by move=> x_in xk; rewrite in_iter_fixE fix_order_gt0 x_in xk. Qed. Lemma notin_iter (x : T) k : k < fix_order x -> x \notin iterF k. Proof. by move=> k_le; rewrite in_iter_fixE negb_and orbC -ltnNge k_le. Qed. Lemma fix_order_small x k : x \in iterF k -> fix_order x <= k. Proof. by rewrite in_iter_fixE => /andP[]. Qed. Lemma fix_order_big x k : x \in set_fix -> x \notin iterF k -> fix_order x > k. Proof. by move=> x_in; rewrite in_iter_fixE fix_order_gt0 x_in /= -ltnNge. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma le_fix_order (x y : T) : y \in iterF (fix_order x) -> fix_order y <= fix_order x. Proof. exact: fix_order_small. Qed. End Least. Section Greatest. Variables (T : finType) (F : {set T} -> {set T}). Hypothesis (F_mono : {homo F : X Y / X \subset Y}). Notation n := #\|T\|. Definition funsetC X := ~: (F (~: X)). Notation G := funsetC. Lemma funsetC_mono : {homo G : X Y / X \subset Y}. Proof. by move=> ; rewrite subCset setCK F_mono// subCset setCK. Qed. Hint Resolve funsetC_mono. Definition set_cofix := ~: set_fix G. Lemma set_cofixK : F set_cofix = set_cofix. Proof. by rewrite /set_cofix -[in RHS]set_fixK ?setCK. Qed. End Greatest. End SetFixpoint. Section Fixpoints. Variables (T : choiceType) (U : {fset T}). (* Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Definition sub_fun (F : {fset T} -> {fset T}) (X : {set U}) : {set U} := fsub U (F [fsetval x in X]). Lemma fset_fsub X : X `<=` U -> [fsetval x in fsub U X] = X. Proof. ( Goal: forall _ : is_true (leq (@size (Choice.sort T) (@enum_fset T X)) (S n)), P X ) move=> XU; apply/fsetP => x; apply/in_fset_valP/idP. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by move=> [xU/=]; rewrite in_fsub. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by move=> xX; exists (fsubsetP XU x xX); rewrite /= in_fsub. Qed. Variable (F : {fset T} -> {fset T}). Hypothesis (F_mono : {homo F : X Y / X `<=` Y}) (F_bound : {homo F : X / X `<=` U}). Notation Fsub := (sub_fun F). Notation iterF := (fun i => iter i F fset0). Lemma Fsub_mono : {homo Fsub : X Y / X \subset Y}. Proof. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) move=> X Y subXY; apply: subset_fsub; last by apply/F_bound/fset_sub_val. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by apply/F_mono/subset_imfset/subsetP. Qed. Hint Resolve Fsub_mono. Definition fset_fix := [fsetval x in set_fix Fsub]. Lemma fset_iterFE i : iterF i = [fsetval x in iter i Fsub set0]. Proof. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) elim: i => [\|i /= -> /=]; last by rewrite fset_fsub // F_bound//= fset_sub_val. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by apply/fsetP=> x; rewrite in_fset_val /=; case: insub=> //= ?; rewrite !inE. Qed. Lemma fset_iterF_sub i : iterF i `<=` U. Proof. by rewrite /= fset_iterFE fset_sub_val. Qed. Lemma fset_fixK : F fset_fix = fset_fix. Proof. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite /fset_fix -[in RHS]set_fixK// fset_fsub// F_bound//= fset_sub_val. Qed. Hint Resolve fset_fixK. Lemma fset_fixKn k : iter k F fset_fix = fset_fix. Proof. by rewrite iter_fix. Qed. Lemma iter_sub_ffix k : iterF k `<=` fset_fix. Proof. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite /fset_fix !fset_iterFE; apply/subset_imfset/subsetP/iter_sub_fix. Qed. Definition ffix_order (x : T) := if x \in U =P true is ReflectT xU then fix_order Fsub [` xU] else 0. Lemma ffix_order_le_max (x : T) : ffix_order x <= #\|` U\|. Proof. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite /ffix_order; case: eqP => //= x_in; rewrite cardfE fix_order_le_max. Qed. Lemma in_iter_ffix_orderE (x : T) : (x \in iterF (ffix_order x)) = (x \in fset_fix). Proof. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) rewrite /ffix_order; case: eqP => [xU\|/negP xNU]; last first. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite !inE /fset_fix in_fset_valF. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite fset_iterFE !in_fset_valT /= in_iter_fix_orderE. Qed. Lemma ffix_order_gt0 (x : T) : (ffix_order x > 0) = (x \in fset_fix). Proof. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) rewrite /ffix_order; case: eqP => [xU\|/negP xNU]; last by rewrite in_fset_valF. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite fix_order_gt0 in_fset_valT. Qed. Lemma ffix_order_eq0 (x : T) : (ffix_order x == 0) = (x \notin fset_fix). Proof. by rewrite -ffix_order_gt0 -ltnNge ltnS leqn0. Qed. Lemma in_iter_ffixE (x : T) k : (x \in iterF k) = (0 < ffix_order x <= k). Proof. rewrite /ffix_order fset_iterFE; case: eqP => [xU\|/negP xNU]; by [rewrite in_fset_valF\|rewrite in_fset_valT /= in_iter_fixE]. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma in_iter_ffix (x : T) k : x \in fset_fix -> ffix_order x <= k -> x \in iterF k. Proof. by move=> x_in xk; rewrite in_iter_ffixE ffix_order_gt0 x_in xk. Qed. Lemma notin_iter_ffix (x : T) k : k < ffix_order x -> x \notin iterF k. Proof. by move=> k_le; rewrite in_iter_ffixE negb_and orbC -ltnNge k_le. Qed. Lemma ffix_order_small x k : x \in iterF k -> ffix_order x <= k. Proof. by rewrite in_iter_ffixE => /andP[]. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma ffix_order_big x k : x \in fset_fix -> x \notin iterF k -> ffix_order x > k. Proof. by move=> x_in; rewrite in_iter_ffixE ffix_order_gt0 x_in -ltnNge. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma le_ffix_order (x y : T) : y \in iterF (ffix_order x) -> ffix_order y <= ffix_order x. Proof. exact: ffix_order_small. Qed. End Fixpoints. ( apply/apply/fsetP=> x /=. ) ( apply//in_fset_valP. ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( rewrite -IHi /=. ) ( rewrite in_imfset. ) Section DefMap. Variables (K : choiceType) (V : Type). Record finMap : Type := FinMap { domf : {fset K}; ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ffun_of_fmap :> {ffun domf -> V} }. Definition finmap_of (_ : phant (K -> V)) := finMap. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Let T_ (domf : {fset K}) := {ffun domf -> V}. Local Notation finMap' := {domf : _ & T_ domf}. End DefMap. Notation "{fmap T }" := (@finmap_of _ _ (Phant T)) : type_scope. Definition pred_of_finmap (K : choiceType) (V : Type) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) (f : {fmap K -> V}) : pred K := mem (domf f). Canonical finMapPredType (K : choiceType) (V : Type) := Eval hnf in mkPredType (@pred_of_finmap K V). Delimit Scope fmap_scope with fmap. Local Open Scope fmap_scope. Notation "f .[ kf ]" := (f [` kf]) : fmap_scope. Arguments ffun_of_fmap : simpl never. Notation "[ 'fmap' x : aT => F ]" := (FinMap [ffun x : aT => F]) (at level 0, x ident, only parsing) : fun_scope. Notation "[ 'fmap' : aT => F ]" := (FinMap [ffun : aT => F]) (at level 0, only parsing) : fun_scope. Notation "[ 'fmap' x => F ]" := [fmap x : _ => F] (at level 0, x ident, format "[ 'fmap' x => F ]") : fun_scope. Notation "[ 'fmap' => F ]" := [fmap: _ => F] (at level 0, format "[ 'fmap' => F ]") : fun_scope. Canonical finmap_of_finfun (K : choiceType) V (A : {fset K}) (f : {ffun A -> V}) := FinMap f. Arguments finmap_of_finfun /. Arguments ffun_of_fmap : simpl nomatch. Section OpsMap. Variables (K : choiceType). Definition fmap0 V : {fmap K -> V} := FinMap (ffun0 _ (cardfT0 K)). ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Definition fnd V (A : {fset K}) (f : {ffun A -> V}) (k : K) := omap f (insub k). ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Inductive fnd_spec V (A : {fset K}) (f : {ffun A -> V}) k : ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) bool -> option A -> option V -> Type := \| FndIn (kf : k \in A) : fnd_spec f k true (some [` kf]) (some (f.[kf])) \| FndOut (kNf : k \notin A) : fnd_spec f k false None None. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Definition setf V (f : {fmap K -> V}) (k0 : K) (v0 : V) : {fmap K -> V} := [fmap k : k0 \|` domf f => if val k == k0 then v0 else odflt v0 (fnd f (val k))]. End OpsMap. Prenex Implicits fnd setf. Arguments fmap0 {K V}. Arguments setf : simpl never. Arguments fnd : simpl never. Notation "[ 'fmap' 'of' T ]" := (fmap0 : {fmap T}) (only parsing) : fmap_scope. Notation "[fmap]" := fmap0 : fmap_scope. Notation "x .[ k <- v ]" := (setf x k v) : fmap_scope. Notation "f .[? k ]" := (fnd f k) : fmap_scope. Section FinMapCanonicals. Variable K : choiceType. Let finMap_on (V : Type) (d : {fset K}) := {ffun d -> V}. Local Notation finMap_ V := {d : _ & finMap_on V d}. Definition finMap_encode V (f : {fmap K -> V}) := Tagged (finMap_on V) (ffun_of_fmap f). Definition finMap_decode V (f : finMap_ V) := FinMap (tagged f). Lemma finMap_codeK V : cancel (@finMap_encode V) (@finMap_decode V). Proof. by case. Qed. Section FinMapEqType. Variable V : eqType. Definition finMap_eqMixin := CanEqMixin (@finMap_codeK V). Canonical finMap_eqType := EqType {fmap K -> V} finMap_eqMixin. End FinMapEqType. Section FinMapChoiceType. Variable V : choiceType. Definition finMap_choiceMixin := CanChoiceMixin (@finMap_codeK V). Canonical finMap_choiceType := ChoiceType {fmap K -> V} finMap_choiceMixin. End FinMapChoiceType. End FinMapCanonicals. Section FinMapTheory. Variables (K : choiceType). ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma fndP V (f : {fmap K -> V}) k : fnd_spec f k (k \in domf f) (insub k) (f.[? k]). Proof. ( Goal: @eq bool (@in_mem (Choice.sort V) x (@mem (Choice.sort V) (simplPredType (Choice.sort V)) (@SimplPred (Choice.sort V) (fun x : Choice.sort V => andb (@in_mem (Choice.sort V) x (@mem (Choice.sort V) (predPredType (Choice.sort V)) (fun x0 : Choice.sort V => @pred_of_eq_seq (Choice.eqType V) (@enum_fset V (codomf f)) x0))) (negb (@FiniteQuant.quant0b (@fset_sub_finType K (@domf K (Choice.sort V) f)) (fun k : @fset_sub_type K (@domf K (Choice.sort V) f) => @FiniteQuant.ex (@fset_sub_finType K (@domf K (Choice.sort V) f)) (FiniteQuant.Quantified (andb (@in_mem (Choice.sort K) (@fsval K (@domf K (Choice.sort V) f) k) (@mem (Choice.sort K) (finSetPredType K) (@fsetD K (@domf K (Choice.sort V) f) A))) (@eq_op (Choice.eqType V) (@FunFinfun.fun_of_fin (@fset_sub_finType K (@domf K (Choice.sort V) f)) (Choice.sort V) (@ffun_of_fmap K (Choice.sort V) f) k) x))) k))))))) (@in_mem (Choice.sort V) x (@mem (Choice.sort V) (simplPredType (Choice.sort V)) (@SimplPred (Choice.sort V) (fun x : Choice.sort V => andb (@in_mem (Choice.sort V) x (@mem (Choice.sort V) (predPredType (Choice.sort V)) (fun x0 : Choice.sort V => @pred_of_eq_seq (Choice.eqType V) (@enum_fset V (codomf f)) x0))) (negb (@FiniteQuant.quant0b (@fset_sub_finType K (@domf K (Choice.sort V) f)) (fun k : @fset_sub_type K (@domf K (Choice.sort V) f) => @FiniteQuant.ex (@fset_sub_finType K (@domf K (Choice.sort V) f)) (FiniteQuant.Quantified (andb (negb (@in_mem (Choice.sort K) (@fsval K (@domf K (Choice.sort V) f) k) (@mem (Choice.sort K) (finSetPredType K) A))) (@eq_op (Choice.eqType V) (@FunFinfun.fun_of_fin (@fset_sub_finType K (@domf K (Choice.sort V) f)) (Choice.sort V) (@ffun_of_fmap K (Choice.sort V) f) k) x))) k))))))) ) rewrite /fnd; case: insubP=> [[k' k'f] _ {k} <- /=\|kNf]. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite k'f; constructor. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite (negPf kNf); constructor. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma fndSome V (f : {fmap K -> V}) (k : K) : f.[? k] = (k \in f) :> bool. Proof. by case: fndP. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma not_fnd V (f : {fmap K -> V}) (k : K) : ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) k \notin f -> f.[? k] = None. Proof. by case: fndP. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma getfE V (f : {fmap K -> V}) (k : domf f) (kf : val k \in domf f) : f.[kf] = f k :> V. Proof. by congr (_ _); apply: val_inj. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma eq_getf V (f : {fmap K -> V}) k (kf kf' : k \in domf f) : f.[kf] = f.[kf'] :> V. Proof. by rewrite (@getfE _ _ [` kf']). Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma Some_fnd V (f : {fmap K -> V}) (k : domf f) : Some (f k) = f.[? val k]. Proof. by case: fndP (valP k) => // ? _; rewrite getfE. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma in_fnd V (f : {fmap K -> V}) (k : K) (kf : k \in domf f) : f.[? k] = Some f.[kf]. Proof. by rewrite Some_fnd. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma fnd_if V (cond : bool) (f g : {fmap K -> V}) (k : K) : ((if cond then f else g) : finMap _ _).[? k] = if cond then f.[? k] else g.[? k]. Proof. by case: cond. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma getfP V (f g : {fmap K -> V}) : domf f = domf g -> ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) (forall k (kMf : k \in f) (kMg : k \in g), f.[kMf] = g.[kMg]) -> f = g. Proof. ( Goal: forall (_ : @eq (@finset_of K (Phant (Choice.sort K))) (@domf K V f) (@domf K V g)) (_ : forall (k : Choice.sort K) (kMf : is_true (@in_mem (Choice.sort K) k (@mem (Choice.sort K) (finMapPredType K V) f))) (kMg : is_true (@in_mem (Choice.sort K) k (@mem (Choice.sort K) (finMapPredType K V) g))), @eq V (@FunFinfun.fun_of_fin (@fset_sub_finType K (@domf K V f)) V (@ffun_of_fmap K V f) (@FSetSub K (@domf K V f) k kMf)) (@FunFinfun.fun_of_fin (@fset_sub_finType K (@domf K V g)) V (@ffun_of_fmap K V g) (@FSetSub K (@domf K V g) k kMg))), @eq (@finmap_of K V (Phant (forall _ : Choice.sort K, V))) f g ) move: f g => [kf f] [kg g] /= eq_kfg; case: _ / eq_kfg in g => {kg}. (* Goal: forall _ : forall (k : Choice.sort K) (kMf : is_true (@in_mem (Choice.sort K) k (@mem (Choice.sort K) (finMapPredType K V) (@FinMap K V kf f)))) (kMg : is_true (@in_mem (Choice.sort K) k (@mem (Choice.sort K) (finMapPredType K V) (@FinMap K V kf g)))), @eq V (@FunFinfun.fun_of_fin (@fset_sub_finType K kf) V f (@FSetSub K kf k kMf)) (@FunFinfun.fun_of_fin (@fset_sub_finType K kf) V g (@FSetSub K kf k kMg)), @eq (@finmap_of K V (Phant (forall _ : Choice.sort K, V))) (@FinMap K V kf f) (@FinMap K V kf g) ) move=> eq_fg; congr FinMap; apply/ffunP => /= x. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by do [rewrite -!getfE; do ?exact: valP] => . Qed. (* Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma fmapP V (f g : {fmap K -> V}) : ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) (forall k, f.[? k] = g.[? k]) <-> f = g. Proof. ( Goal: iff (@eqfun (Equality.sort V) (Choice.sort K) (@fun_of_fsfun K V default f) (@fun_of_fsfun K V default f')) (@eq (@fsfun K V default) f f') ) split=> [fnd_fg\|-> //]; apply: getfP => [\|k kMf kMg]. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by apply/fsetP => x; rewrite -!fndSome fnd_fg. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by apply: Some_inj; rewrite !Some_fnd. Qed. Lemma fnd_fmap0 V k : ([fmap] : {fmap K -> V}).[? k] = None. Proof. by rewrite not_fnd // in_fset0. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma mem_setf V (f : {fmap K -> V}) (k0 : K) (v0 : V) : ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) f.[k0 <- v0] =i predU1 k0 (mem (domf f)). Proof. by move=> k; rewrite !inE. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma dom_setf V (f : {fmap K -> V}) (k0 : K) (v0 : V) : ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) domf (f.[k0 <- v0]) = k0 \|` domf f. Proof. by apply/fsetP=> k; rewrite mem_setf. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma fnd_set_in V (f : {fmap K -> V}) k0 v0 (x : domf f.[k0 <- v0]) : ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) val x != k0 -> val x \in f. Proof. by have := valP x; rewrite mem_setf inE; case: eqP. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma setfK V (f : {fmap K -> V}) k0 v0 (x : domf f.[k0 <- v0]): ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) f.[k0 <- v0] x = if eqVneq (val x) k0 is right xNk0 then f.[fnd_set_in xNk0] else v0. Proof. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) case: eqVneq => [\|xNk0]; rewrite ?ffunE /=; first by move->; rewrite eqxx. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite (negPf xNk0) in_fnd ?fnd_set_in //= => xf; apply: eq_getf. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma fnd_set V (f : {fmap K -> V}) k0 v0 k : ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) f.[k0 <- v0].[? k] = if k == k0 then Some v0 else f.[? k]. Proof. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) case: fndP => [ksf\|]; last first. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite mem_setf inE negb_or => /andP [/negPf ->]; case: fndP. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) rewrite setfK; case: eqVneq => //= [->\|kNk0]; first by rewrite eqxx. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite Some_fnd (negPf kNk0). Qed. Lemma fmap_nil V (f : {fmap K -> V}) : domf f = fset0 -> f = [fmap]. Proof. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by move=> kf0; apply: getfP => //= k ? kMg; have := kMg; rewrite !inE. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma getf_set V (f : {fmap K -> V}) (k : K) (v : V) (kf' : k \in _) : ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) f.[k <- v].[kf'] = v. Proof. by apply: Some_inj; rewrite Some_fnd fnd_set eqxx. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma setf_get V (f : {fmap K -> V}) (k : domf f) : ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) f.[val k <- f k] = f. Proof. by apply/fmapP=> k'; rewrite fnd_set Some_fnd; case: eqP => [->\|]. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma setfNK V (f : {fmap K -> V}) (k k' : K) (v : V) (k'f : k' \in _) (k'f' : k' \in _): ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) f.[k <- v].[k'f'] = if k' == k then v else f.[k'f]. Proof. by apply: Some_inj; rewrite Some_fnd !fnd_set in_fnd; case: ifP. Qed. Lemma domf0 V : domf [fmap of K -> V] = fset0. Proof. by apply/fsetP => x. Qed. End FinMapTheory. Section ReduceOp. Variable (K : choiceType) (V : Type). Implicit Types (f : {fmap K -> option V}). Lemma reducef_subproof f (x : [fsetval x : domf f \| f x]) : f (fincl (fset_sub_val _ _) x). Proof. ( Goal: is_true (@isSome V (@FunFinfun.fun_of_fin (@fset_sub_finType K (@domf K (option V) f)) (option V) (@ffun_of_fmap K (option V) f) (@fincl K (@Imfset.imfset imfset_key (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) (@domf K (option V) f))) (@fset_sub_subType K (@domf K (option V) f))) K (@val (Choice.sort K) (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) (@domf K (option V) f))) (@fset_sub_subType K (@domf K (option V) f))) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) (@domf K (option V) f))) (@fset_sub_subType K (@domf K (option V) f)))) (simplPredType (@fset_sub_type K (@domf K (option V) f))) (@subfinset_finpred (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) (@domf K (option V) f))) (@fset_sub_subType K (@domf K (option V) f))) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) (@domf K (option V) f))) (@fset_sub_subType K (@domf K (option V) f)))) (predPredType (@fset_sub_type K (@domf K (option V) f) : predArgType)) (@fin_finpred (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) (@domf K (option V) f))) (@fset_sub_subType K (@domf K (option V) f)))) (pred_finpredType (@subFinType_finType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) (@domf K (option V) f))) (@fset_sub_subfinType K (@domf K (option V) f)))) (@sort_of_simpl_pred (@fset_sub_type K (@domf K (option V) f) : predArgType) (pred_of_argType (@fset_sub_type K (@domf K (option V) f) : predArgType))))) (fun x : @fset_sub_type K (@domf K (option V) f) => @isSome V (@FunFinfun.fun_of_fin (@fset_sub_finType K (@domf K (option V) f)) (option V) (@ffun_of_fmap K (option V) f) x)))) (Phantom (mem_pred (Equality.sort (@Equality.clone (@fset_sub_type K (@domf K (option V) f)) (@fset_sub_eqType K (@domf K (option V) f)) (@fset_sub_eqMixin K (@domf K (option V) f)) (fun x : Equality.sort (@fset_sub_eqType K (@domf K (option V) f)) => x) (fun x : phantom Equality.type (@Equality.Pack (@fset_sub_type K (@domf K (option V) f)) (@fset_sub_eqMixin K (@domf K (option V) f))) => x)))) (@pred_of_finmempred (@Equality.clone (@fset_sub_type K (@domf K (option V) f)) (@fset_sub_eqType K (@domf K (option V) f)) (@fset_sub_eqMixin K (@domf K (option V) f)) (fun x : Equality.sort (@fset_sub_eqType K (@domf K (option V) f)) => x) (fun x : phantom Equality.type (@Equality.Pack (@fset_sub_type K (@domf K (option V) f)) (@fset_sub_eqMixin K (@domf K (option V) f))) => x)) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) (@domf K (option V) f))) (@fset_sub_subType K (@domf K (option V) f)))) (simplPredType (@fset_sub_type K (@domf K (option V) f))) (@subfinset_finpred (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) (@domf K (option V) f))) (@fset_sub_subType K (@domf K (option V) f))) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) (@domf K (option V) f))) (@fset_sub_subType K (@domf K (option V) f)))) (predPredType (@fset_sub_type K (@domf K (option V) f) : predArgType)) (@fin_finpred (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) (@domf K (option V) f))) (@fset_sub_subType K (@domf K (option V) f)))) (pred_finpredType (@subFinType_finType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) (@domf K (option V) f))) (@fset_sub_subfinType K (@domf K (option V) f)))) (@sort_of_simpl_pred (@fset_sub_type K (@domf K (option V) f) : predArgType) (pred_of_argType (@fset_sub_type K (@domf K (option V) f) : predArgType))))) (fun x : @fset_sub_type K (@domf K (option V) f) => @isSome V (@FunFinfun.fun_of_fin (@fset_sub_finType K (@domf K (option V) f)) (option V) (@ffun_of_fmap K (option V) f) x))))))) (@domf K (option V) f) (@fset_sub_val imfset_key K (@domf K (option V) f) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) (@domf K (option V) f))) (@fset_sub_subType K (@domf K (option V) f)))) (simplPredType (@fset_sub_type K (@domf K (option V) f))) (@subfinset_finpred (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) (@domf K (option V) f))) (@fset_sub_subType K (@domf K (option V) f))) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) (@domf K (option V) f))) (@fset_sub_subType K (@domf K (option V) f)))) (predPredType (@fset_sub_type K (@domf K (option V) f) : predArgType)) (@fin_finpred (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) (@domf K (option V) f))) (@fset_sub_subType K (@domf K (option V) f)))) (pred_finpredType (@subFinType_finType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) (@domf K (option V) f))) (@fset_sub_subfinType K (@domf K (option V) f)))) (@sort_of_simpl_pred (@fset_sub_type K (@domf K (option V) f) : predArgType) (pred_of_argType (@fset_sub_type K (@domf K (option V) f) : predArgType))))) (fun x : @fset_sub_type K (@domf K (option V) f) => @isSome V (@FunFinfun.fun_of_fin (@fset_sub_finType K (@domf K (option V) f)) (option V) (@ffun_of_fmap K (option V) f) x))))) x))) ) set y := (y in f y); suff : val y \in [fsetval x : domf f \| f x]. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite val_in_fset. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by suff -> : val y = val x by exact: valP. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Definition reducef f : {fmap K -> V} := [fmap x => oextract (@reducef_subproof f x)]. Lemma domf_reduce f : domf (reducef f) = [fsetval x : domf f \| f x]. Proof. by []. Qed. Lemma mem_reducef f k : k \in reducef f = ojoin f.[? k]. Proof. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) rewrite inE; case: fndP => [kf\|] /=; first by rewrite in_fset_valT. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by apply: contraNF; apply: (fsubsetP (fset_sub_val _ _)). Qed. Lemma fnd_reducef f k : (reducef f).[? k] = ojoin f.[? k]. Proof. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) case: fndP => /= [kf\|]; last by rewrite mem_reducef; case: ojoin. ( Goal: @eq (@finset_of V (Phant (Choice.sort V))) (codomf (@setf K (Choice.sort V) f k v)) (@fsetU V (@fset1 V v) (codomf (@restrictf K (Choice.sort V) f (@fsetD K (@domf K (Choice.sort V) f) (@fset1 K k))))) ) rewrite ffunE /= Some_oextract; apply: Some_inj; rewrite Some_fnd. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite Some_ojoin // ojoinT // -mem_reducef. Qed. Lemma get_reducef f k (krf : k \in reducef f) (kf : k \in f): Some (reducef f).[krf] = f.[kf]. Proof. by rewrite Some_fnd fnd_reducef in_fnd. Qed. End ReduceOp. Arguments reducef : simpl never. Section RestrictionOps. Variable (K : choiceType) (V : Type). Implicit Types (f g : {fmap K -> V}). ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Definition filterf f (P : pred K) : {fmap K -> V} := [fmap x : [fset x in domf f \| P x] => f (fincl (fset_sub _ _) x)]. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Definition restrictf f (A : {fset K}) : {fmap K -> V} := filterf f (mem A). Notation "x .[& A ]" := (restrictf x A) : fmap_scope. Notation "x .[\ A ]" := (x.[& domf x `\` A]) : fmap_scope. Notation "x .[~ k ]" := (x.[\ [fset k]]) : fmap_scope. Lemma domf_filterf f (P : pred K) : domf (filterf f P) = [fset k in domf f \| P k]. Proof. by []. Qed. Lemma mem_filterf f (P : pred K) (k : K) : (k \in domf (filterf f P)) = (k \in f) && (P k) :> bool. Proof. by rewrite !inE. Qed. Lemma mem_restrictf f (A : {fset K}) (k : K) : k \in f.[& A] = (k \in A) && (k \in f) :> bool. Proof. by rewrite mem_filterf andbC. Qed. Lemma mem_remf f (A : {fset K}) (k : K) : k \in f.[\ A] = (k \notin A) && (k \in f) :> bool. Proof. by rewrite mem_restrictf inE -andbA andbb. Qed. Lemma mem_remf1 f (k' k : K) : k \in f.[~ k'] = (k != k') && (k \in f) :> bool. Proof. by rewrite mem_remf inE. Qed. Lemma domf_restrict f A : domf f.[& A] = A `&` domf f. Proof. by apply/fsetP=> k'; rewrite mem_restrictf !inE. Qed. Lemma domf_rem f A : domf f.[\ A] = domf f `\` A. Proof. by rewrite domf_restrict fsetIDAC fsetIid. Qed. Lemma mem_remfF f (k : K) : k \in f.[~ k] = false. Proof. by rewrite mem_remf1 eqxx. Qed. Lemma fnd_filterf f P k : (filterf f P).[? k] = if P k then f.[? k] else None. Proof. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) case: fndP => [kff\|]; last first. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite in_fset => /nandP [/not_fnd->\|/negPf-> //]; rewrite if_same. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by have := kff; rewrite in_fset => /andP [kf ->]; rewrite ffunE Some_fnd. Qed. Lemma get_filterf f P k (kff : k \in filterf f P) (kf : k \in f) : (filterf f P).[kff] = f.[kf]. Proof. ( Goal: is_true (@fsubset K (@FinSupp.fs K V (fun a : Choice.sort K => default a) (@fsfun_of_ffun key K V S (fun a : @fset_sub_type K S => h a) (fun a : Choice.sort K => default a))) S) ) apply: Some_inj; rewrite !Some_fnd /= fnd_filterf. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by move: kff; rewrite !(in_fset, inE) => /andP [? ->]. Qed. Lemma fnd_restrict f A (k : K) : f.[& A].[? k] = if k \in A then f.[? k] else None. Proof. by rewrite fnd_filterf. Qed. Lemma fnd_rem f A (k : K) : f.[\ A].[? k] = if k \in A then None else f.[? k]. Proof. ( Goal: @eq (Equality.sort V) (@fun_of_fsfun K V default (of_ffun key) x) (@Option.default (Equality.sort V) (default x) (@Option.map (@fset_sub_type K S) (Equality.sort V) h (@insub (Choice.sort K) (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S) x))) ) rewrite fnd_restrict inE. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by case: fndP => ?; rewrite ?(andbT, andbF) //=; case: (_ \in _). Qed. Lemma restrictf_comp f A B : f.[& A].[& B] = f.[& A `&` B]. Proof. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by apply/fmapP=> k; rewrite !fnd_restrict !inE; do !case: (_ \in _). Qed. Lemma remf_comp f A B : f.[\ A].[\ B] = f.[\ A `\|` B]. Proof. by apply/fmapP=> k; rewrite !fnd_rem inE; do !case: (_ \in _). Qed. Lemma restrictfT f : f.[& domf f] = f. Proof. by apply/fmapP=> k; rewrite fnd_restrict; case: fndP. Qed. Lemma restrictf0 f : f.[& fset0] = [fmap]. Proof. by apply/fmapP => k; rewrite fnd_restrict !(inE, not_fnd). Qed. Lemma remf0 f : f.[\ fset0] = f. Proof. by rewrite fsetD0 restrictfT. Qed. Lemma fnd_rem1 f (k k' : K) : f.[~ k].[? k'] = if k' != k then f.[? k'] else None. Proof. by rewrite fnd_rem inE; case: eqP. Qed. Lemma getf_restrict f A (k : K) (kf : k \in f) (kfA : k \in f.[& A]) : f.[& A].[kfA] = f.[kf]. Proof. by rewrite get_filterf. Qed. Lemma setf_restrict f A (k : K) (v : V) : ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) f.[& A].[k <- v] = f.[k <- v].[& k \|` A]. Proof. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by apply/fmapP=> k'; rewrite !(fnd_set, fnd_restrict, inE); case: eqP. Qed. Lemma setf_rem f A (k : K) (v : V) : ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) f.[\ A].[k <- v] = f.[k <- v].[\ (A `\ k)]. Proof. by rewrite setf_restrict fsetUDl. Qed. Lemma setf_rem1 f (k : K) (v : V) : f.[~ k].[k <- v] = f.[k <- v]. Proof. by rewrite setf_rem fsetDv remf0. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma setfC f k1 k2 v1 v2 : f.[k1 <- v1].[k2 <- v2] = ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) if k2 == k1 then f.[k2 <- v2] else f.[k2 <- v2].[k1 <- v1]. Proof. ( Goal: Bool.reflect (@ex (Choice.sort K) (fun x : Choice.sort K => @eq (option (Choice.sort V)) (@fnd K (Choice.sort V) (@domf K (Choice.sort V) f) (@ffun_of_fmap K (Choice.sort V) f) x) (@Some (Choice.sort V) v))) (@in_mem (Choice.sort V) v (@mem (Choice.sort V) (finSetPredType V) (codomf f))) ) apply/fmapP => k. rewrite fnd_if !fnd_set. ( Goal: Bool.reflect (@ex2 (@finset_of K (Phant (Choice.sort K))) (fun S : @finset_of K (Phant (Choice.sort K)) => is_true (@fsubset K (@FinSupp.fs K (Choice.eqType K) (fun x : Choice.sort K => x) f) S)) (fun S : @finset_of K (Phant (Choice.sort K)) => and (@prop_in2 (Choice.sort K) (@mem (Choice.sort K) (finSetPredType K) S) (fun x1 x2 : Choice.sort K => forall _ : @eq (Equality.sort (Choice.eqType K)) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f x1) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f x2), @eq (Choice.sort K) x1 x2) (inPhantom (@injective (Equality.sort (Choice.eqType K)) (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f)))) (forall a : @fset_sub_type K S, is_true (@in_mem (Equality.sort (Choice.eqType K)) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f (@val (Choice.sort K) (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S) a)) (@mem (Choice.sort K) (finSetPredType K) S))))) (fsinjectiveb f) ) have [[->\|kNk2] [// <-\|k2Nk1]] // := (altP (k =P k2), altP (k2 =P k1)). ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite (negPf kNk2). Qed. Lemma restrictf_mkdom f A : f.[& A] = f.[& domf f `&` A]. Proof. ( Goal: is_true (@fsubset K (@FinSupp.fs K V (fun a : Choice.sort K => default a) (@fsfun_of_ffun key K V S (fun a : @fset_sub_type K S => h a) (fun a : Choice.sort K => default a))) S) ) apply/fmapP=> k; rewrite !fnd_restrict inE. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by case: fndP => ?; rewrite ?(andbT, andbF) //=; case: (_ \in _). Qed. Lemma restrictf_id f A : [disjoint domf f & A] -> f.[& A] = [fmap]. Proof. by move=> dAf; rewrite restrictf_mkdom (eqP dAf) restrictf0. Qed. Lemma remf_id f A : [disjoint domf f & A] -> f.[\ A] = f. Proof. by move=> /fsetDidPl ->; rewrite restrictfT. Qed. Lemma remf1_id f k : k \notin f -> f.[~ k] = f. Proof. by move=> kNf; rewrite remf_id //= fdisjointX1. Qed. Lemma restrictf_set f A (k : K) (v : V) : ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) f.[k <- v].[& A] = if k \in A then f.[& A].[k <- v] else f.[& A]. Proof. ( Goal: is_true (@fsubset K (@FinSupp.fs K V (fun a : Choice.sort K => default a) (@fsfun_of_ffun key K V S (fun a : @fset_sub_type K S => h a) (fun a : Choice.sort K => default a))) S) ) apply/fmapP => k' /=; rewrite !(fnd_if, fnd_set, fnd_restrict). ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by case: eqP => [->\|]; do !case: ifP. Qed. Lemma remf_set f A (k : K) (v : V) : ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) f.[k <- v].[\ A] = if k \in A then f.[\ A] else f.[\ A].[k <- v]. Proof. ( Goal: is_true (@fsubset K (@FinSupp.fs K V (fun a : Choice.sort K => default a) (@fsfun_of_ffun key K V S (fun a : @fset_sub_type K S => h a) (fun a : Choice.sort K => default a))) S) ) apply/fmapP => k' /=; rewrite !(fnd_if, fnd_rem, fnd_set, inE). ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by case: eqP => [->\|]; do !case: (_ \in _). Qed. Lemma remf1_set f (k k' : K) (v : V) : ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) f.[k' <- v].[~ k] = if k == k' then f.[~ k] else f.[~ k].[k' <- v]. Proof. by rewrite remf_set inE eq_sym. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma setf_inj f f' k v : k \notin f -> k \notin f' -> ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) f.[k <- v] = f'.[k <- v]-> f = f'. Proof. ( Goal: forall (_ : is_true (negb (@in_mem (Choice.sort K) k (@mem (Choice.sort K) (finMapPredType K V) f)))) (_ : is_true (negb (@in_mem (Choice.sort K) k (@mem (Choice.sort K) (finMapPredType K V) f')))) (_ : @eq (@finmap_of K V (Phant (forall _ : Choice.sort K, V))) (@setf K V f k v) (@setf K V f' k v)), @eq (@finmap_of K V (Phant (forall _ : Choice.sort K, V))) f f' ) move=> kf kf' eq_fkv; apply/fmapP => k'. ( Goal: Bool.reflect (@ex2 (@finset_of K (Phant (Choice.sort K))) (fun S : @finset_of K (Phant (Choice.sort K)) => is_true (@fsubset K (@FinSupp.fs K (Choice.eqType K) (fun x : Choice.sort K => x) f) S)) (fun S : @finset_of K (Phant (Choice.sort K)) => and (@prop_in2 (Choice.sort K) (@mem (Choice.sort K) (finSetPredType K) S) (fun x1 x2 : Choice.sort K => forall _ : @eq (Equality.sort (Choice.eqType K)) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f x1) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f x2), @eq (Choice.sort K) x1 x2) (inPhantom (@injective (Equality.sort (Choice.eqType K)) (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f)))) (forall a : @fset_sub_type K S, is_true (@in_mem (Equality.sort (Choice.eqType K)) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f (@val (Choice.sort K) (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S) a)) (@mem (Choice.sort K) (finSetPredType K) S))))) (fsinjectiveb f) ) have := congr1 (fun g => g.[? k']) eq_fkv. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite !fnd_set; case: eqP => // ->; rewrite !not_fnd. Qed. End RestrictionOps. Arguments filterf : simpl never. Arguments restrictf : simpl never. Notation "x .[& A ]" := (restrictf x A) : fmap_scope. Notation "x .[\ A ]" := (x.[& domf x `\` A]) : fmap_scope. Notation "x .[~ k ]" := (x.[\ [fset k]]) : fmap_scope. Section Cat. Variables (K : choiceType) (V : Type). Implicit Types (f g : {fmap K -> V}). ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Definition catf (f g : {fmap K -> V}) := [fmap k : (domf f `\` domf g) `\|` domf g=> match fsetULVR (valP k) with \| inl kfDg => f.[fsubsetP (fsubsetDl _ _) _ kfDg] \| inr kg => g.[kg] end]. Local Notation "f + g" := (catf f g) : fset_scope. Lemma domf_cat f g : domf (f + g) = domf f `\|` domf g. Proof. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by apply/fsetP=> x; rewrite !inE; case: (boolP (_ \in domf _)); rewrite ?orbT. Qed. Lemma mem_catf f g k : k \in domf (f + g) = (k \in f) \|\| (k \in g). Proof. by rewrite domf_cat inE. Qed. Lemma fnd_cat f g k : (f + g).[? k] = if k \in domf g then g.[? k] else f.[? k]. Proof. ( Goal: @eq (Equality.sort V) (@Option.default (Equality.sort V) (default x) (@fnd K (Equality.sort V) (@Imfset.imfset imfset_key (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) K (fun a : @fset_sub_type K S => @fsval K S a) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (simplPredType (@fset_sub_type K S)) (@subfinset_finpred (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (predPredType (@fset_sub_type K S)) (@fin_finpred (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (pred_finpredType (@subFinType_finType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subfinType K S))) (@sort_of_simpl_pred (@fset_sub_type K S) (pred_of_argType (@fset_sub_type K S))))) (fun a : @fset_sub_type K S => negb (@eq_op V (h a) (default (@fsval K S a)))))) (Phantom (mem_pred (@fset_sub_type K S)) (@mem (@fset_sub_type K S) (simplPredType (@fset_sub_type K S)) (@SimplPred (@fset_sub_type K S) (fun a : @fset_sub_type K S => negb (@eq_op V (h a) (default (@fsval K S a)))))))) (@FunFinfun.finfun (@fset_sub_finType K (@Imfset.imfset imfset_key (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) K (fun a : @fset_sub_type K S => @fsval K S a) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (simplPredType (@fset_sub_type K S)) (@subfinset_finpred (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (predPredType (@fset_sub_type K S)) (@fin_finpred (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (pred_finpredType (@subFinType_finType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subfinType K S))) (@sort_of_simpl_pred (@fset_sub_type K S) (pred_of_argType (@fset_sub_type K S))))) (fun a : @fset_sub_type K S => negb (@eq_op V (h a) (default (@fsval K S a)))))) (Phantom (mem_pred (@fset_sub_type K S)) (@mem (@fset_sub_type K S) (simplPredType (@fset_sub_type K S)) (@SimplPred (@fset_sub_type K S) (fun a : @fset_sub_type K S => negb (@eq_op V (h a) (default (@fsval K S a))))))))) (Equality.sort V) (fun a : @fset_sub_type K (@Imfset.imfset imfset_key (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) K (fun a : @fset_sub_type K S => @fsval K S a) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (simplPredType (@fset_sub_type K S)) (@subfinset_finpred (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (predPredType (@fset_sub_type K S)) (@fin_finpred (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (pred_finpredType (@subFinType_finType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subfinType K S))) (@sort_of_simpl_pred (@fset_sub_type K S) (pred_of_argType (@fset_sub_type K S))))) (fun a : @fset_sub_type K S => negb (@eq_op V (h a) (default (@fsval K S a)))))) (Phantom (mem_pred (@fset_sub_type K S)) (@mem (@fset_sub_type K S) (simplPredType (@fset_sub_type K S)) (@SimplPred (@fset_sub_type K S) (fun a : @fset_sub_type K S => negb (@eq_op V (h a) (default (@fsval K S a)))))))) => h (@fincl K (@Imfset.imfset imfset_key (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) K (@fsval K S) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (simplPredType (@fset_sub_type K S)) (@subfinset_finpred (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (predPredType (@fset_sub_type K S)) (@fin_finpred (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (pred_finpredType (@subFinType_finType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subfinType K S))) (@sort_of_simpl_pred (@fset_sub_type K S) (pred_of_argType (@fset_sub_type K S))))) (fun a0 : @fset_sub_type K S => negb (@eq_op V (h a0) (default (@fsval K S a0)))))) (Phantom (mem_pred (@fset_sub_type K S)) (@mem (@fset_sub_type K S) (simplPredType (@fset_sub_type K S)) (@SimplPred (@fset_sub_type K S) (fun x : @fset_sub_type K S => negb (@eq_op V (h x) (default (@fsval K S x)))))))) S (@fset_sub_val imfset_key K S (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (simplPredType (@fset_sub_type K S)) (@subfinset_finpred (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (predPredType (@fset_sub_type K S)) (@fin_finpred (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (pred_finpredType (@subFinType_finType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subfinType K S))) (@sort_of_simpl_pred (@fset_sub_type K S) (pred_of_argType (@fset_sub_type K S))))) (fun a0 : @fset_sub_type K S => negb (@eq_op V (h a0) (default (@fsval K S a0))))))) a))) x)) (default x) ) ( Goal: @eq (Equality.sort V) (@Option.default (Equality.sort V) (default (@fsval K S u)) (@fnd K (Equality.sort V) (@Imfset.imfset imfset_key (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) K (fun a : @fset_sub_type K S => @fsval K S a) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (simplPredType (@fset_sub_type K S)) (@subfinset_finpred (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (predPredType (@fset_sub_type K S)) (@fin_finpred (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (pred_finpredType (@subFinType_finType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subfinType K S))) (@sort_of_simpl_pred (@fset_sub_type K S) (pred_of_argType (@fset_sub_type K S))))) (fun a : @fset_sub_type K S => negb (@eq_op V (h a) (default (@fsval K S a)))))) (Phantom (mem_pred (@fset_sub_type K S)) (@mem (@fset_sub_type K S) (simplPredType (@fset_sub_type K S)) (@SimplPred (@fset_sub_type K S) (fun a : @fset_sub_type K S => negb (@eq_op V (h a) (default (@fsval K S a)))))))) (@FunFinfun.finfun (@fset_sub_finType K (@Imfset.imfset imfset_key (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) K (fun a : @fset_sub_type K S => @fsval K S a) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (simplPredType (@fset_sub_type K S)) (@subfinset_finpred (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (predPredType (@fset_sub_type K S)) (@fin_finpred (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (pred_finpredType (@subFinType_finType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subfinType K S))) (@sort_of_simpl_pred (@fset_sub_type K S) (pred_of_argType (@fset_sub_type K S))))) (fun a : @fset_sub_type K S => negb (@eq_op V (h a) (default (@fsval K S a)))))) (Phantom (mem_pred (@fset_sub_type K S)) (@mem (@fset_sub_type K S) (simplPredType (@fset_sub_type K S)) (@SimplPred (@fset_sub_type K S) (fun a : @fset_sub_type K S => negb (@eq_op V (h a) (default (@fsval K S a))))))))) (Equality.sort V) (fun a : @fset_sub_type K (@Imfset.imfset imfset_key (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) K (fun a : @fset_sub_type K S => @fsval K S a) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (simplPredType (@fset_sub_type K S)) (@subfinset_finpred (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (predPredType (@fset_sub_type K S)) (@fin_finpred (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (pred_finpredType (@subFinType_finType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subfinType K S))) (@sort_of_simpl_pred (@fset_sub_type K S) (pred_of_argType (@fset_sub_type K S))))) (fun a : @fset_sub_type K S => negb (@eq_op V (h a) (default (@fsval K S a)))))) (Phantom (mem_pred (@fset_sub_type K S)) (@mem (@fset_sub_type K S) (simplPredType (@fset_sub_type K S)) (@SimplPred (@fset_sub_type K S) (fun a : @fset_sub_type K S => negb (@eq_op V (h a) (default (@fsval K S a)))))))) => h (@fincl K (@Imfset.imfset imfset_key (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) K (@fsval K S) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (simplPredType (@fset_sub_type K S)) (@subfinset_finpred (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (predPredType (@fset_sub_type K S)) (@fin_finpred (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (pred_finpredType (@subFinType_finType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subfinType K S))) (@sort_of_simpl_pred (@fset_sub_type K S) (pred_of_argType (@fset_sub_type K S))))) (fun a0 : @fset_sub_type K S => negb (@eq_op V (h a0) (default (@fsval K S a0)))))) (Phantom (mem_pred (@fset_sub_type K S)) (@mem (@fset_sub_type K S) (simplPredType (@fset_sub_type K S)) (@SimplPred (@fset_sub_type K S) (fun x : @fset_sub_type K S => negb (@eq_op V (h x) (default (@fsval K S x)))))))) S (@fset_sub_val imfset_key K S (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (simplPredType (@fset_sub_type K S)) (@subfinset_finpred (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (predPredType (@fset_sub_type K S)) (@fin_finpred (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (pred_finpredType (@subFinType_finType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subfinType K S))) (@sort_of_simpl_pred (@fset_sub_type K S) (pred_of_argType (@fset_sub_type K S))))) (fun a0 : @fset_sub_type K S => negb (@eq_op V (h a0) (default (@fsval K S a0))))))) a))) (@fsval K S u))) (h u) ) case: fndP => //= [kfg\|]; rewrite /catf /=. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) rewrite ffunE /=; case: fsetULVR => [kf\|kg]; last by rewrite Some_fnd kg. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite -in_fnd; move: kf; rewrite inE => /andP[/negPf ->]. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite mem_catf => /norP [kNf kNg]; rewrite !not_fnd // if_same. Qed. Lemma catfE f g : f + g = f.[\ domf g] + g. Proof. by apply/fmapP=> k; rewrite !(fnd_cat, fnd_rem); case: ifP. Qed. Lemma getf_catl f g k (kfg : k \in domf (f + g)) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) (kf : k \in domf f) : k \notin domf g -> (f + g).[kfg] = f.[kf]. Proof. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by move=> kNg; apply: Some_inj; rewrite Some_fnd fnd_cat (negPf kNg) in_fnd. Qed. Lemma getf_catr f g k (kfg : k \in domf (f + g)) (kg : k \in domf g) : (f + g).[kfg] = g.[kg]. Proof. by apply: Some_inj; rewrite Some_fnd fnd_cat kg in_fnd. Qed. Lemma catf0 f : f + [fmap] = f. Proof. by apply/fmapP => k; rewrite fnd_cat in_fset0. Qed. Lemma cat0f f : [fmap] + f = f. Proof. ( Goal: @eq (finMap K V) (catf (@fmap0 K V) f) f ) apply/fmapP => k; rewrite fnd_cat; case: ifPn => //= kf. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite !not_fnd ?inE. Qed. Lemma catf_setl f g k (v : V) : ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) f.[k <- v] + g = if k \in g then f + g else (f + g).[k <- v]. Proof. ( Goal: is_true (@fsubset K (@FinSupp.fs K V (fun a : Choice.sort K => default a) (@fsfun_of_ffun key K V S (fun a : @fset_sub_type K S => h a) (fun a : Choice.sort K => default a))) S) ) apply/fmapP=> k'; rewrite !(fnd_if, fnd_cat, fnd_set). ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by have [->\|Nkk'] := altP eqP; do !case: (_ \in _). Qed. Lemma catf_setr f g k (v : V) : f + g.[k <- v] = (f + g).[k <- v]. Proof. ( Goal: is_true (@fsubset K (@FinSupp.fs K V (fun a : Choice.sort K => default a) (@fsfun_of_ffun key K V S (fun a : @fset_sub_type K S => h a) (fun a : Choice.sort K => default a))) S) ) apply/fmapP=> k'; rewrite !(fnd_cat, fnd_set, mem_setf, inE). ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by have [->\|Nkk'] := altP eqP; do !case: (_ \in _). Qed. Lemma restrictf_cat f g A : (f + g).[& A] = f.[& A] + g.[& A]. Proof. ( Goal: is_true (@fsubset K (@FinSupp.fs K V (fun a : Choice.sort K => default a) (@fsfun_of_ffun key K V S (fun a : @fset_sub_type K S => h a) (fun a : Choice.sort K => default a))) S) ) apply/fmapP => k'; rewrite !(fnd_cat, fnd_restrict) mem_restrictf. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by case: (_ \in _). Qed. Lemma restrictf_cat_domr f g : (f + g).[& domf g] = g. Proof. ( Goal: @eq (Equality.sort V) (@fun_of_fsfun K V default (of_ffun key) x) (@Option.default (Equality.sort V) (default x) (@Option.map (@fset_sub_type K S) (Equality.sort V) h (@insub (Choice.sort K) (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S) x))) ) rewrite catfE restrictf_cat restrictf_comp. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite fsetIDAC fsetDIl fsetDv fsetI0 restrictf0 restrictfT cat0f. Qed. Lemma remf_cat f g A : (f + g).[\ A] = f.[\ A] + g.[\ A]. Proof. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by apply/fmapP => k'; rewrite !(fnd_cat, fnd_rem) mem_remf; case: (_ \in _). Qed. Lemma catf_restrictl A f g : f.[& A] + g = (f + g).[& A `\|` domf g]. Proof. ( Goal: is_true (@fsubset K (@FinSupp.fs K V (fun a : Choice.sort K => default a) (@fsfun_of_ffun key K V S (fun a : @fset_sub_type K S => h a) (fun a : Choice.sort K => default a))) S) ) apply/fmapP=> k; rewrite !(fnd_cat, fnd_restrict) !inE. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by do !case: (_ \in _). Qed. Lemma catf_reml A f g : f.[\ A] + g = (f + g).[\ A `\` domf g]. Proof. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by apply/fmapP=> k; rewrite !(fnd_cat, fnd_rem) inE; case: (_ \in _). Qed. Lemma catf_rem1l k f g : f.[~ k] + g = if k \in g then f + g else (f + g).[~ k]. Proof. ( Goal: is_true (@fsubset K (@FinSupp.fs K V (fun a : Choice.sort K => default a) (@fsfun_of_ffun key K V S (fun a : @fset_sub_type K S => h a) (fun a : Choice.sort K => default a))) S) ) apply/fmapP => k'; rewrite !(fnd_if, fnd_cat, fnd_rem1). ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by have [->\|?] := altP eqP; do !case: (_ \in _). Qed. Lemma setf_catr f g k (v : V) : (f + g).[k <- v] = f + g.[k <- v]. Proof. by rewrite catf_setr. Qed. Lemma setf_catl f g k (v : V) : (f + g).[k <- v] = f.[k <- v] + g.[~ k]. Proof. by rewrite catf_setl mem_remf1 eqxx /= !setf_catr setf_rem1. Qed. Lemma catfA f g h : f + (g + h) = f + g + h. Proof. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by apply/fmapP => k; rewrite !fnd_cat !mem_catf; do !case: (_ \in _). Qed. Lemma catfC f g : f + g = g + f.[\ domf g]. Proof. ( Goal: is_true (@fsubset K (@FinSupp.fs K V (fun a : Choice.sort K => default a) (@fsfun_of_ffun key K V S (fun a : @fset_sub_type K S => h a) (fun a : Choice.sort K => default a))) S) ) apply/fmapP=> k; rewrite !fnd_cat fnd_rem domf_rem inE. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by have [\|kNg] //= := boolP (_ \in domf g); rewrite (not_fnd kNg); case: fndP. Qed. Lemma disjoint_catfC f g : [disjoint domf f & domf g] -> f + g = g + f. Proof. by move=> dfg; rewrite catfC remf_id. Qed. Lemma catfAC f g h : f + g + h = f + h + g.[\ domf h]. Proof. by rewrite -!catfA [X in _ + X]catfC. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma disjoint_catfAC f g h : [disjoint domf g & domf h]%fmap -> f + g + h = f + h + g. Proof. by move=> dgh; rewrite catfAC remf_id. Qed. Lemma catfCA f g h : f + (g + h) = g + (f.[\ domf g] + h). Proof. by rewrite !catfA [X in X + _]catfC. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma disjoint_catfCA f g h : [disjoint domf f & domf g]%fmap -> f + (g + h) = g + (f + h). Proof. by move=> dfg; rewrite catfCA remf_id. Qed. Lemma catfIs f g h : f + h = g + h -> f.[\ domf h] = g.[\ domf h]. Proof. ( Goal: forall _ : @eq (finMap K V) (catf f g) (catf f h), @eq (@finmap_of K V (Phant (forall _ : Choice.sort K, V))) (@restrictf K V g (@domf K V h)) (@restrictf K V h (@domf K V g)) ) move=> /fmapP eq_fg_fh; apply/fmapP => k; have := eq_fg_fh k. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite !fnd_cat !fnd_rem; case: ifP. Qed. Lemma disjoint_catfIs h f g : ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) [disjoint domf f & domf h] -> [disjoint domf g & domf h] -> ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) f + h = g + h -> f = g. Proof. by move=> dfg dgh /catfIs; rewrite !remf_id. Qed. Lemma restrict_catfsI f g h : f + g = f + h -> g.[& domf h] = h.[& domf g]. Proof. ( Goal: forall _ : @eq (finMap K V) (catf f g) (catf f h), @eq (@finmap_of K V (Phant (forall _ : Choice.sort K, V))) (@restrictf K V g (@domf K V h)) (@restrictf K V h (@domf K V g)) ) move=> /fmapP eq_fg_fh; apply/fmapP => k; have := eq_fg_fh k. ( Goal: @eq (Equality.sort V) (@fun_of_fsfun K V default (of_ffun key) x) (@Option.default (Equality.sort V) (default x) (@Option.map (@fset_sub_type K S) (Equality.sort V) h (@insub (Choice.sort K) (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S) x))) ) rewrite !fnd_cat !fnd_restrict. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by do ![case: (boolP (_ \in domf _)) => ? //=] => _; rewrite not_fnd. Qed. Lemma disjoint_catfsI h f g : ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) [disjoint domf f & domf h] -> [disjoint domf g & domf h] -> ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) h + f = h + g -> f = g. Proof. ( Goal: forall (_ : is_true (@fdisjoint K (@domf K V f) (@domf K V h))) (_ : is_true (@fdisjoint K (@domf K V g) (@domf K V h))) (_ : @eq (finMap K V) (catf h f) (catf h g)), @eq (@finmap_of K V (Phant (forall _ : Choice.sort K, V))) f g ) move=> dfg dgh; rewrite -disjoint_catfC // -[RHS]disjoint_catfC //. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by apply: disjoint_catfIs. Qed. End Cat. Module Import FmapE. Definition fmapE := (fndSome, getfE, setfK, fnd_set, getf_set, setfNK, fnd_reducef, get_reducef, fnd_filterf, get_filterf, fnd_restrict, getf_restrict, fnd_rem, fnd_rem1, restrictfT, restrictf0, restrictf_id, remf_id, remf1_id, fnd_cat). End FmapE. Arguments catf : simpl never. Notation "f + g" := (catf f g) : fset_scope. Section FinMapKeyType. Variables (K V : choiceType). Implicit Types (f g : {fmap K -> V}). Definition codomf f : {fset V} := [fset f k \| k : domf f]. Lemma mem_codomf f v : (v \in codomf f) = [exists x : domf f, f x == v]. Proof. ( Goal: Bool.reflect (@ex (Choice.sort K) (fun x : Choice.sort K => @eq (option (Choice.sort V)) (@fnd K (Choice.sort V) (@domf K (Choice.sort V) f) (@ffun_of_fmap K (Choice.sort V) f) x) (@Some (Choice.sort V) v))) (@in_mem (Choice.sort V) v (@mem (Choice.sort V) (finSetPredType V) (codomf f))) ) apply: sameP existsP. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by apply: (iffP (imfsetP _ _ _ _)) => /= [[x _ ->]\|[x /eqP <-]]; exists x. Qed. Lemma codomfP f v : reflect (exists x, f.[? x] = Some v) (v \in codomf f). Proof. ( Goal: Bool.reflect (@ex (Choice.sort K) (fun x : Choice.sort K => @eq (option (Choice.sort V)) (@fnd K (Choice.sort V) (@domf K (Choice.sort V) f) (@ffun_of_fmap K (Choice.sort V) f) x) (@Some (Choice.sort V) v))) (@in_mem (Choice.sort V) v (@mem (Choice.sort V) (finSetPredType V) (codomf f))) ) apply: (iffP (imfsetP _ _ _ _)) => /= [[x _ ->]\|[k]]. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by exists (val x); rewrite Some_fnd. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by case: fndP => //= kf [<-]; exists [` kf]. Qed. Lemma codomfPn f v : reflect (forall x, f.[? x] != Some v) (v \notin codomf f). Proof. ( Goal: is_true (@in_mem (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f (@fsval K (@FinSupp.fs K (Choice.eqType K) (fun x : Choice.sort K => x) f) a)) (@mem (Choice.sort K) (finSetPredType K) (@FinSupp.fs K (Choice.eqType K) (fun x : Choice.sort K => x) f))) ) ( Goal: @ex2 (@finset_of K (Phant (Choice.sort K))) (fun S : @finset_of K (Phant (Choice.sort K)) => is_true (@fsubset K (@FinSupp.fs K (Choice.eqType K) (fun x : Choice.sort K => x) f) S)) (fun S : @finset_of K (Phant (Choice.sort K)) => and (@prop_in2 (Choice.sort K) (@mem (Choice.sort K) (finSetPredType K) S) (fun x1 x2 : Choice.sort K => forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f x1) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f x2), @eq (Choice.sort K) x1 x2) (inPhantom (@injective (Choice.sort K) (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f)))) (forall a : @fset_sub_type K S, is_true (@in_mem (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f (@fsval K S a)) (@mem (Choice.sort K) (finSetPredType K) S)))) ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) rewrite mem_codomf negb_exists; apply: (iffP forallP) => f_eq_v x /=. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by case: fndP => //= kf; rewrite f_eq_v. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by apply: contraNneq (f_eq_v (val x)) => <-; rewrite Some_fnd. Qed. Lemma codomf0 : codomf [fmap] = fset0. Proof. ( Goal: @eq (@finset_of V (Phant (Choice.sort V))) (codomf (@fmap0 K (Choice.sort V))) (@fset0 V) ) apply/fsetP=> k; rewrite inE; apply/negP => /codomfP [k']. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite not_fnd //= inE. Qed. Lemma in_codomf f (k : domf f) : f k \in codomf f. Proof. by rewrite in_imfset. Qed. Lemma fndSomeP f (k : K) (v : V): ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) (f.[? k] = Some v) <-> {kf : k \in f & f.[kf] = v}. Proof. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) split => [fk\|[kf fk]]; last by rewrite in_fnd fk. ( Goal: Bool.reflect (@ex2 (@finset_of K (Phant (Choice.sort K))) (fun S : @finset_of K (Phant (Choice.sort K)) => is_true (@fsubset K (@FinSupp.fs K (Choice.eqType K) (fun x : Choice.sort K => x) f) S)) (fun S : @finset_of K (Phant (Choice.sort K)) => and (@prop_in2 (Choice.sort K) (@mem (Choice.sort K) (finSetPredType K) S) (fun x1 x2 : Choice.sort K => forall _ : @eq (Equality.sort (Choice.eqType K)) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f x1) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f x2), @eq (Choice.sort K) x1 x2) (inPhantom (@injective (Equality.sort (Choice.eqType K)) (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f)))) (forall a : @fset_sub_type K S, is_true (@in_mem (Equality.sort (Choice.eqType K)) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f (@val (Choice.sort K) (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S) a)) (@mem (Choice.sort K) (finSetPredType K) S))))) (fsinjectiveb f) ) have kf : k \in f by rewrite -fndSome fk. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by exists kf; apply: Some_inj; rewrite Some_fnd. Qed. Lemma codomf_restrict f (A : {fset K}) : codomf f.[& A] = [fset f k \| k : domf f & val k \in A]. Proof. ( Goal: @eq (@finset_of V (Phant (Choice.sort V))) (codomf (@restrictf K (Choice.sort V) f A)) (@Imfset.imfset imfset_key (Finite.choiceType (@fset_sub_finType K (@domf K (Choice.sort V) f))) V (fun k : Choice.sort (Finite.choiceType (@fset_sub_finType K (@domf K (Choice.sort V) f))) => @FunFinfun.fun_of_fin (@fset_sub_finType K (@domf K (Choice.sort V) f)) (Choice.sort V) (@ffun_of_fmap K (Choice.sort V) f) k) (@mem_fin (Choice.eqType (Finite.choiceType (@fset_sub_finType K (@domf K (Choice.sort V) f)))) (simplPredType (@fset_sub_type K (@domf K (Choice.sort V) f))) (@subfinset_finpred (Finite.choiceType (@fset_sub_finType K (@domf K (Choice.sort V) f))) (@mem_fin (Choice.eqType (Finite.choiceType (@fset_sub_finType K (@domf K (Choice.sort V) f)))) (predPredType (@fset_sub_type K (@domf K (Choice.sort V) f) : predArgType)) (@fin_finpred (Choice.eqType (Finite.choiceType (@fset_sub_finType K (@domf K (Choice.sort V) f)))) (pred_finpredType (@fset_sub_finType K (@domf K (Choice.sort V) f))) (@sort_of_simpl_pred (@fset_sub_type K (@domf K (Choice.sort V) f) : predArgType) (pred_of_argType (@fset_sub_type K (@domf K (Choice.sort V) f) : predArgType))))) (fun k : @fset_sub_type K (@domf K (Choice.sort V) f) => @in_mem (Choice.sort K) (@val (Choice.sort K) (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) (@domf K (Choice.sort V) f))) (@fset_sub_subType K (@domf K (Choice.sort V) f)) k) (@mem (Choice.sort K) (finSetPredType K) A)))) (Phantom (mem_pred (@fset_sub_type K (@domf K (Choice.sort V) f))) (@mem (@fset_sub_type K (@domf K (Choice.sort V) f)) (simplPredType (@fset_sub_type K (@domf K (Choice.sort V) f))) (@SimplPred (@fset_sub_type K (@domf K (Choice.sort V) f)) (fun k : @fset_sub_type K (@domf K (Choice.sort V) f) => andb (@in_mem (@fset_sub_type K (@domf K (Choice.sort V) f)) k (@mem (@fset_sub_type K (@domf K (Choice.sort V) f)) (predPredType (@fset_sub_type K (@domf K (Choice.sort V) f) : predArgType)) (@sort_of_simpl_pred (@fset_sub_type K (@domf K (Choice.sort V) f) : predArgType) (pred_of_argType (@fset_sub_type K (@domf K (Choice.sort V) f) : predArgType))))) (@in_mem (Choice.sort K) (@val (Choice.sort K) (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) (@domf K (Choice.sort V) f))) (@fset_sub_subType K (@domf K (Choice.sort V) f)) k) (@mem (Choice.sort K) (finSetPredType K) A))))))) ) apply/fsetP=> v; apply/imfsetP/imfsetP => /= [] [k kP ->]. ( Goal: @ex2 (@fset_sub_type K (@domf K (Choice.sort V) f)) (fun x : @fset_sub_type K (@domf K (Choice.sort V) f) => is_true (@in_mem (@fset_sub_type K (@domf K (Choice.sort V) f)) x (@mem (@fset_sub_type K (@domf K (Choice.sort V) f)) (simplPredType (@fset_sub_type K (@domf K (Choice.sort V) f))) (@SimplPred (@fset_sub_type K (@domf K (Choice.sort V) f)) (fun x0 : @fset_sub_type K (@domf K (Choice.sort V) f) => @in_mem (Choice.sort K) (@fsval K (@domf K (Choice.sort V) f) x0) (@mem (Choice.sort K) (finSetPredType K) A)))))) (fun x : @fset_sub_type K (@domf K (Choice.sort V) f) => @eq (Choice.sort V) (@FunFinfun.fun_of_fin (@fset_sub_finType K (@Imfset.imfset imfset_key K K (fun x0 : Choice.sort K => x0) (@mem_fin (Choice.eqType K) (simplPredType (Choice.sort K)) (@subfinset_finpred K (@mem_fin (Choice.eqType K) (seq_predType (Choice.eqType K)) (@fset_finpred K (@domf K (Choice.sort V) f))) (@pred_of_simpl (Choice.sort K) (@pred_of_mem_pred (Choice.sort K) (@mem (Choice.sort K) (finSetPredType K) A))))) (Phantom (mem_pred (Choice.sort K)) (@mem (Choice.sort K) (simplPredType (Choice.sort K)) (@SimplPred (Choice.sort K) (fun x0 : Choice.sort K => andb (@in_mem (Choice.sort K) x0 (@mem (Choice.sort K) (finSetPredType K) (@domf K (Choice.sort V) f))) (@in_mem (Choice.sort K) x0 (@mem (Choice.sort K) (finSetPredType K) A)))))))) (Choice.sort V) (@FunFinfun.finfun (@fset_sub_finType K (@Imfset.imfset imfset_key K K (fun x0 : Choice.sort K => x0) (@mem_fin (Choice.eqType K) (simplPredType (Choice.sort K)) (@subfinset_finpred K (@mem_fin (Choice.eqType K) (seq_predType (Choice.eqType K)) (@fset_finpred K (@domf K (Choice.sort V) f))) (@pred_of_simpl (Choice.sort K) (@pred_of_mem_pred (Choice.sort K) (@mem (Choice.sort K) (finSetPredType K) A))))) (Phantom (mem_pred (Choice.sort K)) (@mem (Choice.sort K) (simplPredType (Choice.sort K)) (@SimplPred (Choice.sort K) (fun x0 : Choice.sort K => andb (@in_mem (Choice.sort K) x0 (@mem (Choice.sort K) (finSetPredType K) (@domf K (Choice.sort V) f))) (@in_mem (Choice.sort K) x0 (@mem (Choice.sort K) (finSetPredType K) A)))))))) (Choice.sort V) (fun x0 : @fset_sub_type K (@Imfset.imfset imfset_key K K (fun x0 : Choice.sort K => x0) (@mem_fin (Choice.eqType K) (simplPredType (Choice.sort K)) (@subfinset_finpred K (@mem_fin (Choice.eqType K) (seq_predType (Choice.eqType K)) (@fset_finpred K (@domf K (Choice.sort V) f))) (@pred_of_simpl (Choice.sort K) (@pred_of_mem_pred (Choice.sort K) (@mem (Choice.sort K) (finSetPredType K) A))))) (Phantom (mem_pred (Choice.sort K)) (@mem (Choice.sort K) (simplPredType (Choice.sort K)) (@SimplPred (Choice.sort K) (fun x0 : Choice.sort K => andb (@in_mem (Choice.sort K) x0 (@mem (Choice.sort K) (finSetPredType K) (@domf K (Choice.sort V) f))) (@in_mem (Choice.sort K) x0 (@mem (Choice.sort K) (finSetPredType K) A))))))) => @FunFinfun.fun_of_fin (@fset_sub_finType K (@domf K (Choice.sort V) f)) (Choice.sort V) (@ffun_of_fmap K (Choice.sort V) f) (@fincl K (@Imfset.imfset imfset_key K K (fun x1 : Choice.sort K => x1) (@mem_fin (Choice.eqType K) (simplPredType (Choice.sort K)) (@subfinset_finpred K (@mem_fin (Choice.eqType K) (seq_predType (Choice.eqType K)) (@fset_finpred K (@domf K (Choice.sort V) f))) (@pred_of_simpl (Choice.sort K) (@pred_of_mem_pred (Choice.sort K) (@mem (Choice.sort K) (finSetPredType K) A))))) (Phantom (mem_pred (Choice.sort K)) (@mem (Choice.sort K) (simplPredType (Choice.sort K)) (@SimplPred (Choice.sort K) (fun x1 : Choice.sort K => andb (@in_mem (Choice.sort K) x1 (@mem (Choice.sort K) (finSetPredType K) (@domf K (Choice.sort V) f))) (@in_mem (Choice.sort K) x1 (@mem (Choice.sort K) (finSetPredType K) A))))))) (@domf K (Choice.sort V) f) (@fset_sub K (@domf K (Choice.sort V) f) (@pred_of_simpl (Choice.sort K) (@pred_of_mem_pred (Choice.sort K) (@mem (Choice.sort K) (finSetPredType K) A)))) x0))) k) (@FunFinfun.fun_of_fin (@fset_sub_finType K (@domf K (Choice.sort V) f)) (Choice.sort V) (@ffun_of_fmap K (Choice.sort V) f) x)) ) ( Goal: @ex2 (@fset_sub_type K (@Imfset.imfset imfset_key K K (fun x : Choice.sort K => x) (@mem_fin (Choice.eqType K) (simplPredType (Choice.sort K)) (@subfinset_finpred K (@mem_fin (Choice.eqType K) (seq_predType (Choice.eqType K)) (@fset_finpred K (@domf K (Choice.sort V) f))) (@pred_of_simpl (Choice.sort K) (@pred_of_mem_pred (Choice.sort K) (@mem (Choice.sort K) (finSetPredType K) A))))) (Phantom (mem_pred (Choice.sort K)) (@mem (Choice.sort K) (simplPredType (Choice.sort K)) (@SimplPred (Choice.sort K) (fun x : Choice.sort K => andb (@in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) (@domf K (Choice.sort V) f))) (@in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) A)))))))) (fun x : @fset_sub_type K (@Imfset.imfset imfset_key K K (fun x : Choice.sort K => x) (@mem_fin (Choice.eqType K) (simplPredType (Choice.sort K)) (@subfinset_finpred K (@mem_fin (Choice.eqType K) (seq_predType (Choice.eqType K)) (@fset_finpred K (@domf K (Choice.sort V) f))) (@pred_of_simpl (Choice.sort K) (@pred_of_mem_pred (Choice.sort K) (@mem (Choice.sort K) (finSetPredType K) A))))) (Phantom (mem_pred (Choice.sort K)) (@mem (Choice.sort K) (simplPredType (Choice.sort K)) (@SimplPred (Choice.sort K) (fun x : Choice.sort K => andb (@in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) (@domf K (Choice.sort V) f))) (@in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) A))))))) => is_true (@in_mem (@fset_sub_type K (@Imfset.imfset imfset_key K K (fun x0 : Choice.sort K => x0) (@mem_fin (Choice.eqType K) (simplPredType (Choice.sort K)) (@subfinset_finpred K (@mem_fin (Choice.eqType K) (seq_predType (Choice.eqType K)) (@fset_finpred K (@domf K (Choice.sort V) f))) (@pred_of_simpl (Choice.sort K) (@pred_of_mem_pred (Choice.sort K) (@mem (Choice.sort K) (finSetPredType K) A))))) (Phantom (mem_pred (Choice.sort K)) (@mem (Choice.sort K) (simplPredType (Choice.sort K)) (@SimplPred (Choice.sort K) (fun x0 : Choice.sort K => andb (@in_mem (Choice.sort K) x0 (@mem (Choice.sort K) (finSetPredType K) (@domf K (Choice.sort V) f))) (@in_mem (Choice.sort K) x0 (@mem (Choice.sort K) (finSetPredType K) A)))))))) x (@mem (@fset_sub_type K (@Imfset.imfset imfset_key K K (fun x0 : Choice.sort K => x0) (@mem_fin (Choice.eqType K) (simplPredType (Choice.sort K)) (@subfinset_finpred K (@mem_fin (Choice.eqType K) (seq_predType (Choice.eqType K)) (@fset_finpred K (@domf K (Choice.sort V) f))) (@pred_of_simpl (Choice.sort K) (@pred_of_mem_pred (Choice.sort K) (@mem (Choice.sort K) (finSetPredType K) A))))) (Phantom (mem_pred (Choice.sort K)) (@mem (Choice.sort K) (simplPredType (Choice.sort K)) (@SimplPred (Choice.sort K) (fun x0 : Choice.sort K => andb (@in_mem (Choice.sort K) x0 (@mem (Choice.sort K) (finSetPredType K) (@domf K (Choice.sort V) f))) (@in_mem (Choice.sort K) x0 (@mem (Choice.sort K) (finSetPredType K) A)))))))) (predPredType (@fset_sub_type K (@Imfset.imfset imfset_key K K (fun x0 : Choice.sort K => x0) (@mem_fin (Choice.eqType K) (simplPredType (Choice.sort K)) (@subfinset_finpred K (@mem_fin (Choice.eqType K) (seq_predType (Choice.eqType K)) (@fset_finpred K (@domf K (Choice.sort V) f))) (@pred_of_simpl (Choice.sort K) (@pred_of_mem_pred (Choice.sort K) (@mem (Choice.sort K) (finSetPredType K) A))))) (Phantom (mem_pred (Choice.sort K)) (@mem (Choice.sort K) (simplPredType (Choice.sort K)) (@SimplPred (Choice.sort K) (fun x0 : Choice.sort K => andb (@in_mem (Choice.sort K) x0 (@mem (Choice.sort K) (finSetPredType K) (@domf K (Choice.sort V) f))) (@in_mem (Choice.sort K) x0 (@mem (Choice.sort K) (finSetPredType K) A))))))))) (@sort_of_simpl_pred (@fset_sub_type K (@Imfset.imfset imfset_key K K (fun x0 : Choice.sort K => x0) (@mem_fin (Choice.eqType K) (simplPredType (Choice.sort K)) (@subfinset_finpred K (@mem_fin (Choice.eqType K) (seq_predType (Choice.eqType K)) (@fset_finpred K (@domf K (Choice.sort V) f))) (@pred_of_simpl (Choice.sort K) (@pred_of_mem_pred (Choice.sort K) (@mem (Choice.sort K) (finSetPredType K) A))))) (Phantom (mem_pred (Choice.sort K)) (@mem (Choice.sort K) (simplPredType (Choice.sort K)) (@SimplPred (Choice.sort K) (fun x0 : Choice.sort K => andb (@in_mem (Choice.sort K) x0 (@mem (Choice.sort K) (finSetPredType K) (@domf K (Choice.sort V) f))) (@in_mem (Choice.sort K) x0 (@mem (Choice.sort K) (finSetPredType K) A)))))))) (pred_of_argType (@fset_sub_type K (@Imfset.imfset imfset_key K K (fun x0 : Choice.sort K => x0) (@mem_fin (Choice.eqType K) (simplPredType (Choice.sort K)) (@subfinset_finpred K (@mem_fin (Choice.eqType K) (seq_predType (Choice.eqType K)) (@fset_finpred K (@domf K (Choice.sort V) f))) (@pred_of_simpl (Choice.sort K) (@pred_of_mem_pred (Choice.sort K) (@mem (Choice.sort K) (finSetPredType K) A))))) (Phantom (mem_pred (Choice.sort K)) (@mem (Choice.sort K) (simplPredType (Choice.sort K)) (@SimplPred (Choice.sort K) (fun x0 : Choice.sort K => andb (@in_mem (Choice.sort K) x0 (@mem (Choice.sort K) (finSetPredType K) (@domf K (Choice.sort V) f))) (@in_mem (Choice.sort K) x0 (@mem (Choice.sort K) (finSetPredType K) A))))))))))))) (fun x : @fset_sub_type K (@Imfset.imfset imfset_key K K (fun x : Choice.sort K => x) (@mem_fin (Choice.eqType K) (simplPredType (Choice.sort K)) (@subfinset_finpred K (@mem_fin (Choice.eqType K) (seq_predType (Choice.eqType K)) (@fset_finpred K (@domf K (Choice.sort V) f))) (@pred_of_simpl (Choice.sort K) (@pred_of_mem_pred (Choice.sort K) (@mem (Choice.sort K) (finSetPredType K) A))))) (Phantom (mem_pred (Choice.sort K)) (@mem (Choice.sort K) (simplPredType (Choice.sort K)) (@SimplPred (Choice.sort K) (fun x : Choice.sort K => andb (@in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) (@domf K (Choice.sort V) f))) (@in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) A))))))) => @eq (Choice.sort V) (@FunFinfun.fun_of_fin (@fset_sub_finType K (@domf K (Choice.sort V) f)) (Choice.sort V) (@ffun_of_fmap K (Choice.sort V) f) k) (@FunFinfun.fun_of_fin (@fset_sub_finType K (@Imfset.imfset imfset_key K K (fun x0 : Choice.sort K => x0) (@mem_fin (Choice.eqType K) (simplPredType (Choice.sort K)) (@subfinset_finpred K (@mem_fin (Choice.eqType K) (seq_predType (Choice.eqType K)) (@fset_finpred K (@domf K (Choice.sort V) f))) (@pred_of_simpl (Choice.sort K) (@pred_of_mem_pred (Choice.sort K) (@mem (Choice.sort K) (finSetPredType K) A))))) (Phantom (mem_pred (Choice.sort K)) (@mem (Choice.sort K) (simplPredType (Choice.sort K)) (@SimplPred (Choice.sort K) (fun x0 : Choice.sort K => andb (@in_mem (Choice.sort K) x0 (@mem (Choice.sort K) (finSetPredType K) (@domf K (Choice.sort V) f))) (@in_mem (Choice.sort K) x0 (@mem (Choice.sort K) (finSetPredType K) A)))))))) (Choice.sort V) (@FunFinfun.finfun (@fset_sub_finType K (@Imfset.imfset imfset_key K K (fun x0 : Choice.sort K => x0) (@mem_fin (Choice.eqType K) (simplPredType (Choice.sort K)) (@subfinset_finpred K (@mem_fin (Choice.eqType K) (seq_predType (Choice.eqType K)) (@fset_finpred K (@domf K (Choice.sort V) f))) (@pred_of_simpl (Choice.sort K) (@pred_of_mem_pred (Choice.sort K) (@mem (Choice.sort K) (finSetPredType K) A))))) (Phantom (mem_pred (Choice.sort K)) (@mem (Choice.sort K) (simplPredType (Choice.sort K)) (@SimplPred (Choice.sort K) (fun x0 : Choice.sort K => andb (@in_mem (Choice.sort K) x0 (@mem (Choice.sort K) (finSetPredType K) (@domf K (Choice.sort V) f))) (@in_mem (Choice.sort K) x0 (@mem (Choice.sort K) (finSetPredType K) A)))))))) (Choice.sort V) (fun x0 : @fset_sub_type K (@Imfset.imfset imfset_key K K (fun x0 : Choice.sort K => x0) (@mem_fin (Choice.eqType K) (simplPredType (Choice.sort K)) (@subfinset_finpred K (@mem_fin (Choice.eqType K) (seq_predType (Choice.eqType K)) (@fset_finpred K (@domf K (Choice.sort V) f))) (@pred_of_simpl (Choice.sort K) (@pred_of_mem_pred (Choice.sort K) (@mem (Choice.sort K) (finSetPredType K) A))))) (Phantom (mem_pred (Choice.sort K)) (@mem (Choice.sort K) (simplPredType (Choice.sort K)) (@SimplPred (Choice.sort K) (fun x0 : Choice.sort K => andb (@in_mem (Choice.sort K) x0 (@mem (Choice.sort K) (finSetPredType K) (@domf K (Choice.sort V) f))) (@in_mem (Choice.sort K) x0 (@mem (Choice.sort K) (finSetPredType K) A))))))) => @FunFinfun.fun_of_fin (@fset_sub_finType K (@domf K (Choice.sort V) f)) (Choice.sort V) (@ffun_of_fmap K (Choice.sort V) f) (@fincl K (@Imfset.imfset imfset_key K K (fun x1 : Choice.sort K => x1) (@mem_fin (Choice.eqType K) (simplPredType (Choice.sort K)) (@subfinset_finpred K (@mem_fin (Choice.eqType K) (seq_predType (Choice.eqType K)) (@fset_finpred K (@domf K (Choice.sort V) f))) (@pred_of_simpl (Choice.sort K) (@pred_of_mem_pred (Choice.sort K) (@mem (Choice.sort K) (finSetPredType K) A))))) (Phantom (mem_pred (Choice.sort K)) (@mem (Choice.sort K) (simplPredType (Choice.sort K)) (@SimplPred (Choice.sort K) (fun x1 : Choice.sort K => andb (@in_mem (Choice.sort K) x1 (@mem (Choice.sort K) (finSetPredType K) (@domf K (Choice.sort V) f))) (@in_mem (Choice.sort K) x1 (@mem (Choice.sort K) (finSetPredType K) A))))))) (@domf K (Choice.sort V) f) (@fset_sub K (@domf K (Choice.sort V) f) (@pred_of_simpl (Choice.sort K) (@pred_of_mem_pred (Choice.sort K) (@mem (Choice.sort K) (finSetPredType K) A)))) x0))) x)) ) have := valP k; rewrite !inE => /andP [kf kA]; exists [` kf] => //. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite ffunE /= -getfE. ( Goal: Bool.reflect (@ex2 (@finset_of K (Phant (Choice.sort K))) (fun S : @finset_of K (Phant (Choice.sort K)) => is_true (@fsubset K (@FinSupp.fs K (Choice.eqType K) (fun x : Choice.sort K => x) f) S)) (fun S : @finset_of K (Phant (Choice.sort K)) => and (@prop_in2 (Choice.sort K) (@mem (Choice.sort K) (finSetPredType K) S) (fun x1 x2 : Choice.sort K => forall _ : @eq (Equality.sort (Choice.eqType K)) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f x1) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f x2), @eq (Choice.sort K) x1 x2) (inPhantom (@injective (Equality.sort (Choice.eqType K)) (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f)))) (forall a : @fset_sub_type K S, is_true (@in_mem (Equality.sort (Choice.eqType K)) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f (@val (Choice.sort K) (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S) a)) (@mem (Choice.sort K) (finSetPredType K) S))))) (fsinjectiveb f) ) have kA : val k \in [fset x \| x in domf f & x \in A] by rewrite !inE (valP k). ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by exists [` kA]; rewrite // ?ffunE !getfE. Qed. Lemma codomf_restrict_exists f (A : {fset K}) : codomf f.[& A] = [fset v in codomf f \| [exists k : domf f, (val k \in A) && (f k == v)]]. Proof. rewrite codomf_restrict; apply/fsetP => v; rewrite !(in_fset, inE) /=; apply/imfsetP/idP. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by move=> [k kA ->]; rewrite in_codomf /=; apply/existsP; exists k; apply/andP. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by move=> /andP[fkdom /existsP [k /andP[kA /eqP<-]]]; exists k. Qed. Lemma codomf_rem f (A : {fset K}) : codomf f.[\ A] = [fset f k \| k : domf f & val k \notin A]. Proof. ( Goal: @eq (Equality.sort V) (@fun_of_fsfun K V default (of_ffun key) x) (@Option.default (Equality.sort V) (default x) (@Option.map (@fset_sub_type K S) (Equality.sort V) h (@insub (Choice.sort K) (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S) x))) ) rewrite codomf_restrict. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by apply: eq_imfset => //= x /=; rewrite -!topredE /= !inE (valP x) andbT. Qed. Lemma codomf_rem_exists f (A : {fset K}) : codomf f.[\ A] = [fset v in codomf f \| [exists k : domf f, (val k \notin A) && (f k == v)]]. Proof. ( Goal: is_true (@in_mem (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f (@fsval K (@FinSupp.fs K (Choice.eqType K) (fun x : Choice.sort K => x) f) a)) (@mem (Choice.sort K) (finSetPredType K) (@FinSupp.fs K (Choice.eqType K) (fun x : Choice.sort K => x) f))) ) ( Goal: @ex2 (@finset_of K (Phant (Choice.sort K))) (fun S : @finset_of K (Phant (Choice.sort K)) => is_true (@fsubset K (@FinSupp.fs K (Choice.eqType K) (fun x : Choice.sort K => x) f) S)) (fun S : @finset_of K (Phant (Choice.sort K)) => and (@prop_in2 (Choice.sort K) (@mem (Choice.sort K) (finSetPredType K) S) (fun x1 x2 : Choice.sort K => forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f x1) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f x2), @eq (Choice.sort K) x1 x2) (inPhantom (@injective (Choice.sort K) (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f)))) (forall a : @fset_sub_type K S, is_true (@in_mem (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f (@fsval K S a)) (@mem (Choice.sort K) (finSetPredType K) S)))) ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) rewrite codomf_restrict_exists; apply: eq_imfset => x //=. ( Goal: @eq bool (@in_mem (Choice.sort V) x (@mem (Choice.sort V) (simplPredType (Choice.sort V)) (@SimplPred (Choice.sort V) (fun x : Choice.sort V => andb (@in_mem (Choice.sort V) x (@mem (Choice.sort V) (predPredType (Choice.sort V)) (fun x0 : Choice.sort V => @pred_of_eq_seq (Choice.eqType V) (@enum_fset V (codomf f)) x0))) (negb (@FiniteQuant.quant0b (@fset_sub_finType K (@domf K (Choice.sort V) f)) (fun k : @fset_sub_type K (@domf K (Choice.sort V) f) => @FiniteQuant.ex (@fset_sub_finType K (@domf K (Choice.sort V) f)) (FiniteQuant.Quantified (andb (@in_mem (Choice.sort K) (@fsval K (@domf K (Choice.sort V) f) k) (@mem (Choice.sort K) (finSetPredType K) (@fsetD K (@domf K (Choice.sort V) f) A))) (@eq_op (Choice.eqType V) (@FunFinfun.fun_of_fin (@fset_sub_finType K (@domf K (Choice.sort V) f)) (Choice.sort V) (@ffun_of_fmap K (Choice.sort V) f) k) x))) k))))))) (@in_mem (Choice.sort V) x (@mem (Choice.sort V) (simplPredType (Choice.sort V)) (@SimplPred (Choice.sort V) (fun x : Choice.sort V => andb (@in_mem (Choice.sort V) x (@mem (Choice.sort V) (predPredType (Choice.sort V)) (fun x0 : Choice.sort V => @pred_of_eq_seq (Choice.eqType V) (@enum_fset V (codomf f)) x0))) (negb (@FiniteQuant.quant0b (@fset_sub_finType K (@domf K (Choice.sort V) f)) (fun k : @fset_sub_type K (@domf K (Choice.sort V) f) => @FiniteQuant.ex (@fset_sub_finType K (@domf K (Choice.sort V) f)) (FiniteQuant.Quantified (andb (negb (@in_mem (Choice.sort K) (@fsval K (@domf K (Choice.sort V) f) k) (@mem (Choice.sort K) (finSetPredType K) A))) (@eq_op (Choice.eqType V) (@FunFinfun.fun_of_fin (@fset_sub_finType K (@domf K (Choice.sort V) f)) (Choice.sort V) (@ffun_of_fmap K (Choice.sort V) f) k) x))) k))))))) ) rewrite !inE; case: (_ \in _) => //=. apply/existsP/existsP => [] /= [k]; rewrite ?inE; by do 2?[move=>/andP []]; exists k; rewrite ?inE; do 2?[apply/andP; split]. Qed. Lemma in_codomf_rem1 f (k : K) (kf : k \in domf f) : codomf f.[~ k] = if [exists k' : domf f, (val k' != k) && (f k' == f.[kf])] then codomf f else codomf f `\ f.[kf]. Proof. ( Goal: is_true (@fsubset K (@FinSupp.fs K V (fun a : Choice.sort K => default a) (@fsfun_of_ffun key K V S (fun a : @fset_sub_type K S => h a) (fun a : Choice.sort K => default a))) S) ) apply/fsetP => v; rewrite codomf_rem_exists (fun_if (fun x => v \in x)) !(in_fset, inE). ( Goal: @eq (option (Equality.sort V)) (@fnd K (Equality.sort V) (@domf K (Equality.sort V) (@val (@finmap_of K (Equality.sort V) (Phant (forall _ : Choice.sort K, Equality.sort V))) (fun x : @finmap_of K (Equality.sort V) (Phant (forall _ : Choice.sort K, Equality.sort V)) => @FiniteQuant.quant0b (@fset_sub_finType K (@domf K (Equality.sort V) x)) (fun k : @fset_sub_type K (@domf K (Equality.sort V) x) => @FiniteQuant.all (@fset_sub_finType K (@domf K (Equality.sort V) x)) (FiniteQuant.Quantified (negb (@eq_op V (@FunFinfun.fun_of_fin (@fset_sub_finType K (@domf K (Equality.sort V) x)) (Equality.sort V) (@ffun_of_fmap K (Equality.sort V) x) k) (default (@val (Choice.sort K) (fun x0 : Choice.sort K => @in_mem (Choice.sort K) x0 (@mem (Choice.sort K) (finSetPredType K) (@domf K (Equality.sort V) x))) (@fset_sub_subType K (@domf K (Equality.sort V) x)) k))))) k)) (@fsfun_subType K V default) f)) (@ffun_of_fmap K (Equality.sort V) (@val (@finmap_of K (Equality.sort V) (Phant (forall _ : Choice.sort K, Equality.sort V))) (fun x : @finmap_of K (Equality.sort V) (Phant (forall _ : Choice.sort K, Equality.sort V)) => @FiniteQuant.quant0b (@fset_sub_finType K (@domf K (Equality.sort V) x)) (fun k : @fset_sub_type K (@domf K (Equality.sort V) x) => @FiniteQuant.all (@fset_sub_finType K (@domf K (Equality.sort V) x)) (FiniteQuant.Quantified (negb (@eq_op V (@FunFinfun.fun_of_fin (@fset_sub_finType K (@domf K (Equality.sort V) x)) (Equality.sort V) (@ffun_of_fmap K (Equality.sort V) x) k) (default (@val (Choice.sort K) (fun x0 : Choice.sort K => @in_mem (Choice.sort K) x0 (@mem (Choice.sort K) (finSetPredType K) (@domf K (Equality.sort V) x))) (@fset_sub_subType K (@domf K (Equality.sort V) x)) k))))) k)) (@fsfun_subType K V default) f)) k) (@fnd K (Equality.sort V) (@domf K (Equality.sort V) (@val (@finmap_of K (Equality.sort V) (Phant (forall _ : Choice.sort K, Equality.sort V))) (fun x : @finmap_of K (Equality.sort V) (Phant (forall _ : Choice.sort K, Equality.sort V)) => @FiniteQuant.quant0b (@fset_sub_finType K (@domf K (Equality.sort V) x)) (fun k : @fset_sub_type K (@domf K (Equality.sort V) x) => @FiniteQuant.all (@fset_sub_finType K (@domf K (Equality.sort V) x)) (FiniteQuant.Quantified (negb (@eq_op V (@FunFinfun.fun_of_fin (@fset_sub_finType K (@domf K (Equality.sort V) x)) (Equality.sort V) (@ffun_of_fmap K (Equality.sort V) x) k) (default (@val (Choice.sort K) (fun x0 : Choice.sort K => @in_mem (Choice.sort K) x0 (@mem (Choice.sort K) (finSetPredType K) (@domf K (Equality.sort V) x))) (@fset_sub_subType K (@domf K (Equality.sort V) x)) k))))) k)) (@fsfun_subType K V default) f')) (@ffun_of_fmap K (Equality.sort V) (@val (@finmap_of K (Equality.sort V) (Phant (forall _ : Choice.sort K, Equality.sort V))) (fun x : @finmap_of K (Equality.sort V) (Phant (forall _ : Choice.sort K, Equality.sort V)) => @FiniteQuant.quant0b (@fset_sub_finType K (@domf K (Equality.sort V) x)) (fun k : @fset_sub_type K (@domf K (Equality.sort V) x) => @FiniteQuant.all (@fset_sub_finType K (@domf K (Equality.sort V) x)) (FiniteQuant.Quantified (negb (@eq_op V (@FunFinfun.fun_of_fin (@fset_sub_finType K (@domf K (Equality.sort V) x)) (Equality.sort V) (@ffun_of_fmap K (Equality.sort V) x) k) (default (@val (Choice.sort K) (fun x0 : Choice.sort K => @in_mem (Choice.sort K) x0 (@mem (Choice.sort K) (finSetPredType K) (@domf K (Equality.sort V) x))) (@fset_sub_subType K (@domf K (Equality.sort V) x)) k))))) k)) (@fsfun_subType K V default) f')) k) ) have [vf\|vNf] := boolP (_ \in codomf f); rewrite /= ?(andbF,andbT) ?if_same //. ( Goal: is_true (@in_mem (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f (@fsval K (@FinSupp.fs K (Choice.eqType K) (fun x : Choice.sort K => x) f) a)) (@mem (Choice.sort K) (finSetPredType K) (@FinSupp.fs K (Choice.eqType K) (fun x : Choice.sort K => x) f))) ) ( Goal: @ex2 (@finset_of K (Phant (Choice.sort K))) (fun S : @finset_of K (Phant (Choice.sort K)) => is_true (@fsubset K (@FinSupp.fs K (Choice.eqType K) (fun x : Choice.sort K => x) f) S)) (fun S : @finset_of K (Phant (Choice.sort K)) => and (@prop_in2 (Choice.sort K) (@mem (Choice.sort K) (finSetPredType K) S) (fun x1 x2 : Choice.sort K => forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f x1) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f x2), @eq (Choice.sort K) x1 x2) (inPhantom (@injective (Choice.sort K) (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f)))) (forall a : @fset_sub_type K S, is_true (@in_mem (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f (@fsval K S a)) (@mem (Choice.sort K) (finSetPredType K) S)))) ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) rewrite -/(_ \|\| _); apply/existsP/idP => /= [[k' /andP[xk /eqP <-]]\|]. ( Goal: is_true (@in_mem (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f (@fsval K (@FinSupp.fs K (Choice.eqType K) (fun x : Choice.sort K => x) f) a)) (@mem (Choice.sort K) (finSetPredType K) (@FinSupp.fs K (Choice.eqType K) (fun x : Choice.sort K => x) f))) ) ( Goal: @ex2 (@finset_of K (Phant (Choice.sort K))) (fun S : @finset_of K (Phant (Choice.sort K)) => is_true (@fsubset K (@FinSupp.fs K (Choice.eqType K) (fun x : Choice.sort K => x) f) S)) (fun S : @finset_of K (Phant (Choice.sort K)) => and (@prop_in2 (Choice.sort K) (@mem (Choice.sort K) (finSetPredType K) S) (fun x1 x2 : Choice.sort K => forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f x1) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f x2), @eq (Choice.sort K) x1 x2) (inPhantom (@injective (Choice.sort K) (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f)))) (forall a : @fset_sub_type K S, is_true (@in_mem (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f (@fsval K S a)) (@mem (Choice.sort K) (finSetPredType K) S)))) ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) rewrite orbC -implybE; apply/implyP => eq_fk. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite inE /= in xk; apply/existsP; exists k'; rewrite // xk eq_fk. ( Goal: @eq (option (Equality.sort V)) (@fnd K (Equality.sort V) (@domf K (Equality.sort V) (@val (@finmap_of K (Equality.sort V) (Phant (forall _ : Choice.sort K, Equality.sort V))) (fun x : @finmap_of K (Equality.sort V) (Phant (forall _ : Choice.sort K, Equality.sort V)) => @FiniteQuant.quant0b (@fset_sub_finType K (@domf K (Equality.sort V) x)) (fun k : @fset_sub_type K (@domf K (Equality.sort V) x) => @FiniteQuant.all (@fset_sub_finType K (@domf K (Equality.sort V) x)) (FiniteQuant.Quantified (negb (@eq_op V (@FunFinfun.fun_of_fin (@fset_sub_finType K (@domf K (Equality.sort V) x)) (Equality.sort V) (@ffun_of_fmap K (Equality.sort V) x) k) (default (@val (Choice.sort K) (fun x0 : Choice.sort K => @in_mem (Choice.sort K) x0 (@mem (Choice.sort K) (finSetPredType K) (@domf K (Equality.sort V) x))) (@fset_sub_subType K (@domf K (Equality.sort V) x)) k))))) k)) (@fsfun_subType K V default) f)) (@ffun_of_fmap K (Equality.sort V) (@val (@finmap_of K (Equality.sort V) (Phant (forall _ : Choice.sort K, Equality.sort V))) (fun x : @finmap_of K (Equality.sort V) (Phant (forall _ : Choice.sort K, Equality.sort V)) => @FiniteQuant.quant0b (@fset_sub_finType K (@domf K (Equality.sort V) x)) (fun k : @fset_sub_type K (@domf K (Equality.sort V) x) => @FiniteQuant.all (@fset_sub_finType K (@domf K (Equality.sort V) x)) (FiniteQuant.Quantified (negb (@eq_op V (@FunFinfun.fun_of_fin (@fset_sub_finType K (@domf K (Equality.sort V) x)) (Equality.sort V) (@ffun_of_fmap K (Equality.sort V) x) k) (default (@val (Choice.sort K) (fun x0 : Choice.sort K => @in_mem (Choice.sort K) x0 (@mem (Choice.sort K) (finSetPredType K) (@domf K (Equality.sort V) x))) (@fset_sub_subType K (@domf K (Equality.sort V) x)) k))))) k)) (@fsfun_subType K V default) f)) k) (@fnd K (Equality.sort V) (@domf K (Equality.sort V) (@val (@finmap_of K (Equality.sort V) (Phant (forall _ : Choice.sort K, Equality.sort V))) (fun x : @finmap_of K (Equality.sort V) (Phant (forall _ : Choice.sort K, Equality.sort V)) => @FiniteQuant.quant0b (@fset_sub_finType K (@domf K (Equality.sort V) x)) (fun k : @fset_sub_type K (@domf K (Equality.sort V) x) => @FiniteQuant.all (@fset_sub_finType K (@domf K (Equality.sort V) x)) (FiniteQuant.Quantified (negb (@eq_op V (@FunFinfun.fun_of_fin (@fset_sub_finType K (@domf K (Equality.sort V) x)) (Equality.sort V) (@ffun_of_fmap K (Equality.sort V) x) k) (default (@val (Choice.sort K) (fun x0 : Choice.sort K => @in_mem (Choice.sort K) x0 (@mem (Choice.sort K) (finSetPredType K) (@domf K (Equality.sort V) x))) (@fset_sub_subType K (@domf K (Equality.sort V) x)) k))))) k)) (@fsfun_subType K V default) f')) (@ffun_of_fmap K (Equality.sort V) (@val (@finmap_of K (Equality.sort V) (Phant (forall _ : Choice.sort K, Equality.sort V))) (fun x : @finmap_of K (Equality.sort V) (Phant (forall _ : Choice.sort K, Equality.sort V)) => @FiniteQuant.quant0b (@fset_sub_finType K (@domf K (Equality.sort V) x)) (fun k : @fset_sub_type K (@domf K (Equality.sort V) x) => @FiniteQuant.all (@fset_sub_finType K (@domf K (Equality.sort V) x)) (FiniteQuant.Quantified (negb (@eq_op V (@FunFinfun.fun_of_fin (@fset_sub_finType K (@domf K (Equality.sort V) x)) (Equality.sort V) (@ffun_of_fmap K (Equality.sort V) x) k) (default (@val (Choice.sort K) (fun x0 : Choice.sort K => @in_mem (Choice.sort K) x0 (@mem (Choice.sort K) (finSetPredType K) (@domf K (Equality.sort V) x))) (@fset_sub_subType K (@domf K (Equality.sort V) x)) k))))) k)) (@fsfun_subType K V default) f')) k) ) have /imfsetP /= [[l lf] _ ->] := vf; rewrite orbC. ( Goal: Bool.reflect (@ex2 (@finset_of K (Phant (Choice.sort K))) (fun S : @finset_of K (Phant (Choice.sort K)) => is_true (@fsubset K (@FinSupp.fs K (Choice.eqType K) (fun x : Choice.sort K => x) f) S)) (fun S : @finset_of K (Phant (Choice.sort K)) => and (@prop_in2 (Choice.sort K) (@mem (Choice.sort K) (finSetPredType K) S) (fun x1 x2 : Choice.sort K => forall _ : @eq (Equality.sort (Choice.eqType K)) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f x1) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f x2), @eq (Choice.sort K) x1 x2) (inPhantom (@injective (Equality.sort (Choice.eqType K)) (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f)))) (forall a : @fset_sub_type K S, is_true (@in_mem (Equality.sort (Choice.eqType K)) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f (@val (Choice.sort K) (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S) a)) (@mem (Choice.sort K) (finSetPredType K) S))))) (fsinjectiveb f) ) have [->\|neq_f _] := altP (f.[lf] =P _). ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by move=> /existsP [m /andP[Nmk /eqP <-]]; exists m; rewrite eqxx inE Nmk. ( Goal: @ex (@fset_sub_type K (@domf K (Choice.sort V) f)) (fun x : @fset_sub_type K (@domf K (Choice.sort V) f) => is_true (andb (negb (@in_mem (Choice.sort K) (@fsval K (@domf K (Choice.sort V) f) x) (@mem (Choice.sort K) (finSetPredType K) (@fset1 K k)))) (@eq_op (Choice.eqType V) (@FunFinfun.fun_of_fin (@fset_sub_finType K (@domf K (Choice.sort V) f)) (Choice.sort V) (@ffun_of_fmap K (Choice.sort V) f) x) (@FunFinfun.fun_of_fin (@fset_sub_finType K (@domf K (Choice.sort V) f)) (Choice.sort V) (@ffun_of_fmap K (Choice.sort V) f) (@FSetSub K (@domf K (Choice.sort V) f) l lf))))) ) exists [` lf]; rewrite eqxx andbT inE /=. ( Goal: is_true (@fsubset K (@FinSupp.fs K V (fun a : Choice.sort K => default a) (@fsfun_of_ffun key K V S (fun a : @fset_sub_type K S => h a) (fun a : Choice.sort K => default a))) S) ) apply: contra neq_f => /eqP eq_lk; rewrite eq_lk in lf . (* Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by apply/eqP; congr f.[_]; apply: bool_irrelevance. Qed. Lemma codomf_set f (k : K) (v : V) (kf : k \in domf f) : ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) codomf f.[k <- v] = v \|` codomf f.[~ k]. Proof. ( Goal: @eq (@finset_of V (Phant (Choice.sort V))) (codomf (@setf K (Choice.sort V) f k v)) (@fsetU V (@fset1 V v) (codomf (@restrictf K (Choice.sort V) f (@fsetD K (@domf K (Choice.sort V) f) (@fset1 K k))))) ) rewrite -setf_rem1; apply/fsetP=> v'; rewrite !inE. ( Goal: Bool.reflect (@ex2 (@finset_of K (Phant (Choice.sort K))) (fun S : @finset_of K (Phant (Choice.sort K)) => is_true (@fsubset K (@FinSupp.fs K (Choice.eqType K) (fun x : Choice.sort K => x) f) S)) (fun S : @finset_of K (Phant (Choice.sort K)) => and (@prop_in2 (Choice.sort K) (@mem (Choice.sort K) (finSetPredType K) S) (fun x1 x2 : Choice.sort K => forall _ : @eq (Equality.sort (Choice.eqType K)) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f x1) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f x2), @eq (Choice.sort K) x1 x2) (inPhantom (@injective (Equality.sort (Choice.eqType K)) (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f)))) (forall a : @fset_sub_type K S, is_true (@in_mem (Equality.sort (Choice.eqType K)) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f (@val (Choice.sort K) (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S) a)) (@mem (Choice.sort K) (finSetPredType K) S))))) (fsinjectiveb f) ) have [->\|neq_v'v] /= := altP eqP. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by apply/codomfP; exists k; rewrite fnd_set eqxx. apply/codomfP/codomfP => [] [k' fk'_eq]; exists k'; move: fk'_eq; rewrite fnd_set. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by have [_ [eq_vv']\|//] := altP eqP; rewrite eq_vv' eqxx in neq_v'v . (* Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by have [->\|//] := altP eqP; rewrite fnd_rem inE eqxx. Qed. End FinMapKeyType. Module Import FinmapInE. Definition inE := (inE, mem_codomf, mem_catf, mem_remfF, mem_filterf, mem_reducef, mem_restrictf, mem_remf, mem_remf1, mem_setf). End FinmapInE. Section FsfunDef. Variables (K : choiceType) (V : eqType) (default : K -> V). Record fsfun := Fsfun { ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) fmap_of_fsfun : {fmap K -> V}; _ : [forall k : domf fmap_of_fsfun, fmap_of_fsfun k != default (val k)] }. Canonical fsfun_subType := Eval hnf in [subType for fmap_of_fsfun]. Definition fsfun_eqMixin := [eqMixin of fsfun by <:]. Canonical fsfun_eqType := EqType fsfun fsfun_eqMixin. Fact fsfun_subproof (f : fsfun) : forall (k : K) (kf : k \in fmap_of_fsfun f), (fmap_of_fsfun f).[kf]%fmap != default k. Proof. ( Goal: @eq (Equality.sort V) (@Option.default (Equality.sort V) (default (@fsval K S u)) (@fnd K (Equality.sort V) (@Imfset.imfset imfset_key (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) K (fun a : @fset_sub_type K S => @fsval K S a) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (simplPredType (@fset_sub_type K S)) (@subfinset_finpred (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (predPredType (@fset_sub_type K S)) (@fin_finpred (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (pred_finpredType (@subFinType_finType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subfinType K S))) (@sort_of_simpl_pred (@fset_sub_type K S) (pred_of_argType (@fset_sub_type K S))))) (fun a : @fset_sub_type K S => negb (@eq_op V (h a) (default (@fsval K S a)))))) (Phantom (mem_pred (@fset_sub_type K S)) (@mem (@fset_sub_type K S) (simplPredType (@fset_sub_type K S)) (@SimplPred (@fset_sub_type K S) (fun a : @fset_sub_type K S => negb (@eq_op V (h a) (default (@fsval K S a)))))))) (@FunFinfun.finfun (@fset_sub_finType K (@Imfset.imfset imfset_key (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) K (fun a : @fset_sub_type K S => @fsval K S a) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (simplPredType (@fset_sub_type K S)) (@subfinset_finpred (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (predPredType (@fset_sub_type K S)) (@fin_finpred (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (pred_finpredType (@subFinType_finType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subfinType K S))) (@sort_of_simpl_pred (@fset_sub_type K S) (pred_of_argType (@fset_sub_type K S))))) (fun a : @fset_sub_type K S => negb (@eq_op V (h a) (default (@fsval K S a)))))) (Phantom (mem_pred (@fset_sub_type K S)) (@mem (@fset_sub_type K S) (simplPredType (@fset_sub_type K S)) (@SimplPred (@fset_sub_type K S) (fun a : @fset_sub_type K S => negb (@eq_op V (h a) (default (@fsval K S a))))))))) (Equality.sort V) (fun a : @fset_sub_type K (@Imfset.imfset imfset_key (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) K (fun a : @fset_sub_type K S => @fsval K S a) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (simplPredType (@fset_sub_type K S)) (@subfinset_finpred (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (predPredType (@fset_sub_type K S)) (@fin_finpred (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (pred_finpredType (@subFinType_finType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subfinType K S))) (@sort_of_simpl_pred (@fset_sub_type K S) (pred_of_argType (@fset_sub_type K S))))) (fun a : @fset_sub_type K S => negb (@eq_op V (h a) (default (@fsval K S a)))))) (Phantom (mem_pred (@fset_sub_type K S)) (@mem (@fset_sub_type K S) (simplPredType (@fset_sub_type K S)) (@SimplPred (@fset_sub_type K S) (fun a : @fset_sub_type K S => negb (@eq_op V (h a) (default (@fsval K S a)))))))) => h (@fincl K (@Imfset.imfset imfset_key (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) K (@fsval K S) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (simplPredType (@fset_sub_type K S)) (@subfinset_finpred (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (predPredType (@fset_sub_type K S)) (@fin_finpred (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (pred_finpredType (@subFinType_finType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subfinType K S))) (@sort_of_simpl_pred (@fset_sub_type K S) (pred_of_argType (@fset_sub_type K S))))) (fun a0 : @fset_sub_type K S => negb (@eq_op V (h a0) (default (@fsval K S a0)))))) (Phantom (mem_pred (@fset_sub_type K S)) (@mem (@fset_sub_type K S) (simplPredType (@fset_sub_type K S)) (@SimplPred (@fset_sub_type K S) (fun x : @fset_sub_type K S => negb (@eq_op V (h x) (default (@fsval K S x)))))))) S (@fset_sub_val imfset_key K S (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (simplPredType (@fset_sub_type K S)) (@subfinset_finpred (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (predPredType (@fset_sub_type K S)) (@fin_finpred (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (pred_finpredType (@subFinType_finType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subfinType K S))) (@sort_of_simpl_pred (@fset_sub_type K S) (pred_of_argType (@fset_sub_type K S))))) (fun a0 : @fset_sub_type K S => negb (@eq_op V (h a0) (default (@fsval K S a0))))))) a))) (@fsval K S u))) (h u) ) case:f => f f_stable k kf /=. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) exact: (forallP f_stable [` kf]). Qed. Definition fun_of_fsfun (f : fsfun) (k : K) := odflt (default k) (fmap_of_fsfun f).[? k]%fmap. End FsfunDef. Coercion fun_of_fsfun : fsfun >-> Funclass. Module Type FinSuppSig. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Axiom fs : forall (K : choiceType) (V : eqType) (default : K -> V), ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) fsfun default -> {fset K}. Axiom E : fs = (fun K V d f => domf (@fmap_of_fsfun K V d f)). End FinSuppSig. Module FinSupp : FinSuppSig. Definition fs := (fun K V d f => domf (@fmap_of_fsfun K V d f)). Definition E := erefl fs. End FinSupp. Notation finsupp := FinSupp.fs. Canonical unlockable_finsupp := Unlockable FinSupp.E. Section FSfunBasics. Variables (K : choiceType) (V : eqType) (default : K -> V). Implicit Types (f : fsfun default) (k : K) (v : V). Lemma mem_finsupp f k : (k \in finsupp f) = (f k != default k). Proof. ( Goal: @eq bool (@in_mem (Choice.sort K) k (@mem (Choice.sort K) (finSetPredType K) (@FinSupp.fs K V default f))) (negb (@eq_op V (@fun_of_fsfun K V default f k) (default k))) ) rewrite /fun_of_fsfun [finsupp]unlock; case: fndP; rewrite ?eqxx //=. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by move=> kf; rewrite fsfun_subproof. Qed. Lemma memNfinsupp f k : (k \notin finsupp f) = (f k == default k). Proof. by rewrite mem_finsupp negbK. Qed. Lemma fsfun_dflt f k : k \notin finsupp f -> f k = default k. Proof. by rewrite mem_finsupp negbK => /eqP. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) CoInductive fsfun_spec f k : V -> bool -> Type := ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) \| FsfunOut : k \notin finsupp f -> fsfun_spec f k (default k) false \| FsfunIn (kf : k \in finsupp f) : fsfun_spec f k (f k) true. Lemma finsuppP f k : fsfun_spec f k (f k) (k \in finsupp f). Proof. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) have [kf\|kNf] := boolP (_ \in _); first by constructor. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite fsfun_dflt // ; constructor. Qed. Lemma fsfunP f f' : f =1 f' <-> f = f'. Proof. ( Goal: iff (@eqfun (Equality.sort V) (Choice.sort K) (@fun_of_fsfun K V default f) (@fun_of_fsfun K V default f')) (@eq (@fsfun K V default) f f') ) split=> [eq_fg\|->//]; apply/val_inj/fmapP => k. ( Goal: @eq (option (Equality.sort V)) (@fnd K (Equality.sort V) (@domf K (Equality.sort V) (@val (@finmap_of K (Equality.sort V) (Phant (forall _ : Choice.sort K, Equality.sort V))) (fun x : @finmap_of K (Equality.sort V) (Phant (forall _ : Choice.sort K, Equality.sort V)) => @FiniteQuant.quant0b (@fset_sub_finType K (@domf K (Equality.sort V) x)) (fun k : @fset_sub_type K (@domf K (Equality.sort V) x) => @FiniteQuant.all (@fset_sub_finType K (@domf K (Equality.sort V) x)) (FiniteQuant.Quantified (negb (@eq_op V (@FunFinfun.fun_of_fin (@fset_sub_finType K (@domf K (Equality.sort V) x)) (Equality.sort V) (@ffun_of_fmap K (Equality.sort V) x) k) (default (@val (Choice.sort K) (fun x0 : Choice.sort K => @in_mem (Choice.sort K) x0 (@mem (Choice.sort K) (finSetPredType K) (@domf K (Equality.sort V) x))) (@fset_sub_subType K (@domf K (Equality.sort V) x)) k))))) k)) (@fsfun_subType K V default) f)) (@ffun_of_fmap K (Equality.sort V) (@val (@finmap_of K (Equality.sort V) (Phant (forall _ : Choice.sort K, Equality.sort V))) (fun x : @finmap_of K (Equality.sort V) (Phant (forall _ : Choice.sort K, Equality.sort V)) => @FiniteQuant.quant0b (@fset_sub_finType K (@domf K (Equality.sort V) x)) (fun k : @fset_sub_type K (@domf K (Equality.sort V) x) => @FiniteQuant.all (@fset_sub_finType K (@domf K (Equality.sort V) x)) (FiniteQuant.Quantified (negb (@eq_op V (@FunFinfun.fun_of_fin (@fset_sub_finType K (@domf K (Equality.sort V) x)) (Equality.sort V) (@ffun_of_fmap K (Equality.sort V) x) k) (default (@val (Choice.sort K) (fun x0 : Choice.sort K => @in_mem (Choice.sort K) x0 (@mem (Choice.sort K) (finSetPredType K) (@domf K (Equality.sort V) x))) (@fset_sub_subType K (@domf K (Equality.sort V) x)) k))))) k)) (@fsfun_subType K V default) f)) k) (@fnd K (Equality.sort V) (@domf K (Equality.sort V) (@val (@finmap_of K (Equality.sort V) (Phant (forall _ : Choice.sort K, Equality.sort V))) (fun x : @finmap_of K (Equality.sort V) (Phant (forall _ : Choice.sort K, Equality.sort V)) => @FiniteQuant.quant0b (@fset_sub_finType K (@domf K (Equality.sort V) x)) (fun k : @fset_sub_type K (@domf K (Equality.sort V) x) => @FiniteQuant.all (@fset_sub_finType K (@domf K (Equality.sort V) x)) (FiniteQuant.Quantified (negb (@eq_op V (@FunFinfun.fun_of_fin (@fset_sub_finType K (@domf K (Equality.sort V) x)) (Equality.sort V) (@ffun_of_fmap K (Equality.sort V) x) k) (default (@val (Choice.sort K) (fun x0 : Choice.sort K => @in_mem (Choice.sort K) x0 (@mem (Choice.sort K) (finSetPredType K) (@domf K (Equality.sort V) x))) (@fset_sub_subType K (@domf K (Equality.sort V) x)) k))))) k)) (@fsfun_subType K V default) f')) (@ffun_of_fmap K (Equality.sort V) (@val (@finmap_of K (Equality.sort V) (Phant (forall _ : Choice.sort K, Equality.sort V))) (fun x : @finmap_of K (Equality.sort V) (Phant (forall _ : Choice.sort K, Equality.sort V)) => @FiniteQuant.quant0b (@fset_sub_finType K (@domf K (Equality.sort V) x)) (fun k : @fset_sub_type K (@domf K (Equality.sort V) x) => @FiniteQuant.all (@fset_sub_finType K (@domf K (Equality.sort V) x)) (FiniteQuant.Quantified (negb (@eq_op V (@FunFinfun.fun_of_fin (@fset_sub_finType K (@domf K (Equality.sort V) x)) (Equality.sort V) (@ffun_of_fmap K (Equality.sort V) x) k) (default (@val (Choice.sort K) (fun x0 : Choice.sort K => @in_mem (Choice.sort K) x0 (@mem (Choice.sort K) (finSetPredType K) (@domf K (Equality.sort V) x))) (@fset_sub_subType K (@domf K (Equality.sort V) x)) k))))) k)) (@fsfun_subType K V default) f')) k) ) have := eq_fg k; rewrite /(f _) /(f' _) /=. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) case: fndP; case: fndP => //= kf kf'; first by move->. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by move/eqP/negPn; rewrite fsfun_subproof. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by move/eqP/negPn; rewrite eq_sym fsfun_subproof. Qed. Lemma fsfun_injective_inP f (T : {fset K}) : reflect {in T &, injective f} (injectiveb [ffun x : T => f (val x)]). Proof. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) apply: (iffP (@injectiveP _ _ _)) => f_inj a b; last first. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite !ffunE => ; apply: val_inj; apply: f_inj => //; apply: valP. (* Goal: forall (_ : is_true (@in_mem (Choice.sort K) a (@mem (Choice.sort K) (finSetPredType K) T))) (_ : is_true (@in_mem (Choice.sort K) b (@mem (Choice.sort K) (finSetPredType K) T))) (_ : @eq (Equality.sort V) (@fun_of_fsfun K V default f a) (@fun_of_fsfun K V default f b)), @eq (Choice.sort K) a b ) move=> aT bT eq_ga_gb; have := f_inj.[aT].[bT]; rewrite !ffunE /=. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by move=> /(_ eq_ga_gb) /(congr1 val). Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Definition fsfun_of_can_ffun (T : {fset K}) (g : {ffun T -> V}) (can_g : forall k : T, g k != default (val k)) := @Fsfun K V default (FinMap g) (appP forallP idP can_g). ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma fsfun_of_can_ffunE (T : {fset K}) (g : {ffun T -> V}) (can_g : forall k : T , g k != default (val k)) k (kg : k \in T) : (fsfun_of_can_ffun can_g) k = g.[kg]. Proof. by rewrite/fun_of_fsfun in_fnd. Qed. End FSfunBasics. Notation "{ 'fsfun' ty 'for' dflt }" := (fsfun (dflt : ty)) (at level 0, format "{ 'fsfun' ty 'for' dflt }") : type_scope. Notation "{ 'fsfun' ty 'of' x => dflt }" := {fsfun ty for fun x => dflt} (at level 0, x at level 99, format "{ 'fsfun' ty 'of' x => dflt }") : type_scope. Notation "{ 'fsfun' ty 'with' dflt }" := {fsfun ty of _ => dflt} (at level 0, format "{ 'fsfun' ty 'with' dflt }") : type_scope. Notation "{ 'fsfun' ty }" := {fsfun ty of x => x} (at level 0, format "{ 'fsfun' ty }") : type_scope. Notation "{ 'fsfun' 'for' dflt }" := {fsfun _ for dflt} (at level 0, only parsing) : type_scope. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Notation "{ 'fsfun' 'of' x : T => dflt }" := {fsfun T -> _ of x => dflt} (at level 0, x at level 99, only parsing) : type_scope. Notation "{ 'fsfun' 'of' x => dflt }" := {fsfun of x : _ => dflt} (at level 0, x at level 99, only parsing) : type_scope. Notation "{ 'fsfun' 'with' dflt }" := {fsfun of _ => dflt} (at level 0, only parsing) : type_scope. Module Type FsfunSig. Section FsfunSig. Variables (K : choiceType) (V : eqType) (default : K -> V). Parameter of_ffun : forall (S : {fset K}), (S -> V) -> unit -> fsfun default. Variables (S : {fset K}) (h : S -> V). Axiom of_ffunE :forall key x, of_ffun h key x = odflt (default x) (omap h (insub x)). End FsfunSig. End FsfunSig. Module Fsfun : FsfunSig. Section FsfunOfFinfun. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Variables (K : choiceType) (V : eqType) (default : K -> V) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) (S : {fset K}) (h : S -> V). Let fmap := [fmap a : [fsetval a in {: S} \| h a != default (val a)] => h (fincl (fset_sub_val _ _) a)]. Fact fmapP a : fmap a != default (val a). Proof. ( Goal: @eq bool (@fsubset K A (@fsetI K B C)) (andb (@fsubset K A B) (@fsubset K A C)) ) rewrite ffunE; have /in_fset_valP [a_in_S] := valP a. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by have -> : [` a_in_S] = fincl (fset_sub_val _ _) a by exact/val_inj. Qed. Definition of_ffun (k : unit) := fsfun_of_can_ffun fmapP. Lemma of_ffunE key x : of_ffun key x = odflt (default x) (omap h (insub x)). Proof. ( Goal: @eq (Equality.sort V) (@fun_of_fsfun K V default (of_ffun key) x) (@Option.default (Equality.sort V) (default x) (@Option.map (@fset_sub_type K S) (Equality.sort V) h (@insub (Choice.sort K) (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S) x))) ) rewrite /fun_of_fsfun /=. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) case: insubP => /= [u _ <-\|xNS]; last first. ( Goal: @eq (Equality.sort V) (@Option.default (Equality.sort V) (default x) (@fnd K (Equality.sort V) (@Imfset.imfset imfset_key (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) K (fun a : @fset_sub_type K S => @fsval K S a) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (simplPredType (@fset_sub_type K S)) (@subfinset_finpred (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (predPredType (@fset_sub_type K S)) (@fin_finpred (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (pred_finpredType (@subFinType_finType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subfinType K S))) (@sort_of_simpl_pred (@fset_sub_type K S) (pred_of_argType (@fset_sub_type K S))))) (fun a : @fset_sub_type K S => negb (@eq_op V (h a) (default (@fsval K S a)))))) (Phantom (mem_pred (@fset_sub_type K S)) (@mem (@fset_sub_type K S) (simplPredType (@fset_sub_type K S)) (@SimplPred (@fset_sub_type K S) (fun a : @fset_sub_type K S => negb (@eq_op V (h a) (default (@fsval K S a)))))))) (@FunFinfun.finfun (@fset_sub_finType K (@Imfset.imfset imfset_key (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) K (fun a : @fset_sub_type K S => @fsval K S a) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (simplPredType (@fset_sub_type K S)) (@subfinset_finpred (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (predPredType (@fset_sub_type K S)) (@fin_finpred (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (pred_finpredType (@subFinType_finType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subfinType K S))) (@sort_of_simpl_pred (@fset_sub_type K S) (pred_of_argType (@fset_sub_type K S))))) (fun a : @fset_sub_type K S => negb (@eq_op V (h a) (default (@fsval K S a)))))) (Phantom (mem_pred (@fset_sub_type K S)) (@mem (@fset_sub_type K S) (simplPredType (@fset_sub_type K S)) (@SimplPred (@fset_sub_type K S) (fun a : @fset_sub_type K S => negb (@eq_op V (h a) (default (@fsval K S a))))))))) (Equality.sort V) (fun a : @fset_sub_type K (@Imfset.imfset imfset_key (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) K (fun a : @fset_sub_type K S => @fsval K S a) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (simplPredType (@fset_sub_type K S)) (@subfinset_finpred (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (predPredType (@fset_sub_type K S)) (@fin_finpred (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (pred_finpredType (@subFinType_finType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subfinType K S))) (@sort_of_simpl_pred (@fset_sub_type K S) (pred_of_argType (@fset_sub_type K S))))) (fun a : @fset_sub_type K S => negb (@eq_op V (h a) (default (@fsval K S a)))))) (Phantom (mem_pred (@fset_sub_type K S)) (@mem (@fset_sub_type K S) (simplPredType (@fset_sub_type K S)) (@SimplPred (@fset_sub_type K S) (fun a : @fset_sub_type K S => negb (@eq_op V (h a) (default (@fsval K S a)))))))) => h (@fincl K (@Imfset.imfset imfset_key (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) K (@fsval K S) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (simplPredType (@fset_sub_type K S)) (@subfinset_finpred (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (predPredType (@fset_sub_type K S)) (@fin_finpred (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (pred_finpredType (@subFinType_finType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subfinType K S))) (@sort_of_simpl_pred (@fset_sub_type K S) (pred_of_argType (@fset_sub_type K S))))) (fun a0 : @fset_sub_type K S => negb (@eq_op V (h a0) (default (@fsval K S a0)))))) (Phantom (mem_pred (@fset_sub_type K S)) (@mem (@fset_sub_type K S) (simplPredType (@fset_sub_type K S)) (@SimplPred (@fset_sub_type K S) (fun x : @fset_sub_type K S => negb (@eq_op V (h x) (default (@fsval K S x)))))))) S (@fset_sub_val imfset_key K S (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (simplPredType (@fset_sub_type K S)) (@subfinset_finpred (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (predPredType (@fset_sub_type K S)) (@fin_finpred (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (pred_finpredType (@subFinType_finType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subfinType K S))) (@sort_of_simpl_pred (@fset_sub_type K S) (pred_of_argType (@fset_sub_type K S))))) (fun a0 : @fset_sub_type K S => negb (@eq_op V (h a0) (default (@fsval K S a0))))))) a))) x)) (default x) ) ( Goal: @eq (Equality.sort V) (@Option.default (Equality.sort V) (default (@fsval K S u)) (@fnd K (Equality.sort V) (@Imfset.imfset imfset_key (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) K (fun a : @fset_sub_type K S => @fsval K S a) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (simplPredType (@fset_sub_type K S)) (@subfinset_finpred (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (predPredType (@fset_sub_type K S)) (@fin_finpred (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (pred_finpredType (@subFinType_finType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subfinType K S))) (@sort_of_simpl_pred (@fset_sub_type K S) (pred_of_argType (@fset_sub_type K S))))) (fun a : @fset_sub_type K S => negb (@eq_op V (h a) (default (@fsval K S a)))))) (Phantom (mem_pred (@fset_sub_type K S)) (@mem (@fset_sub_type K S) (simplPredType (@fset_sub_type K S)) (@SimplPred (@fset_sub_type K S) (fun a : @fset_sub_type K S => negb (@eq_op V (h a) (default (@fsval K S a)))))))) (@FunFinfun.finfun (@fset_sub_finType K (@Imfset.imfset imfset_key (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) K (fun a : @fset_sub_type K S => @fsval K S a) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (simplPredType (@fset_sub_type K S)) (@subfinset_finpred (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (predPredType (@fset_sub_type K S)) (@fin_finpred (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (pred_finpredType (@subFinType_finType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subfinType K S))) (@sort_of_simpl_pred (@fset_sub_type K S) (pred_of_argType (@fset_sub_type K S))))) (fun a : @fset_sub_type K S => negb (@eq_op V (h a) (default (@fsval K S a)))))) (Phantom (mem_pred (@fset_sub_type K S)) (@mem (@fset_sub_type K S) (simplPredType (@fset_sub_type K S)) (@SimplPred (@fset_sub_type K S) (fun a : @fset_sub_type K S => negb (@eq_op V (h a) (default (@fsval K S a))))))))) (Equality.sort V) (fun a : @fset_sub_type K (@Imfset.imfset imfset_key (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) K (fun a : @fset_sub_type K S => @fsval K S a) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (simplPredType (@fset_sub_type K S)) (@subfinset_finpred (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (predPredType (@fset_sub_type K S)) (@fin_finpred (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (pred_finpredType (@subFinType_finType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subfinType K S))) (@sort_of_simpl_pred (@fset_sub_type K S) (pred_of_argType (@fset_sub_type K S))))) (fun a : @fset_sub_type K S => negb (@eq_op V (h a) (default (@fsval K S a)))))) (Phantom (mem_pred (@fset_sub_type K S)) (@mem (@fset_sub_type K S) (simplPredType (@fset_sub_type K S)) (@SimplPred (@fset_sub_type K S) (fun a : @fset_sub_type K S => negb (@eq_op V (h a) (default (@fsval K S a)))))))) => h (@fincl K (@Imfset.imfset imfset_key (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) K (@fsval K S) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (simplPredType (@fset_sub_type K S)) (@subfinset_finpred (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (predPredType (@fset_sub_type K S)) (@fin_finpred (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (pred_finpredType (@subFinType_finType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subfinType K S))) (@sort_of_simpl_pred (@fset_sub_type K S) (pred_of_argType (@fset_sub_type K S))))) (fun a0 : @fset_sub_type K S => negb (@eq_op V (h a0) (default (@fsval K S a0)))))) (Phantom (mem_pred (@fset_sub_type K S)) (@mem (@fset_sub_type K S) (simplPredType (@fset_sub_type K S)) (@SimplPred (@fset_sub_type K S) (fun x : @fset_sub_type K S => negb (@eq_op V (h x) (default (@fsval K S x)))))))) S (@fset_sub_val imfset_key K S (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (simplPredType (@fset_sub_type K S)) (@subfinset_finpred (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (predPredType (@fset_sub_type K S)) (@fin_finpred (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (pred_finpredType (@subFinType_finType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subfinType K S))) (@sort_of_simpl_pred (@fset_sub_type K S) (pred_of_argType (@fset_sub_type K S))))) (fun a0 : @fset_sub_type K S => negb (@eq_op V (h a0) (default (@fsval K S a0))))))) a))) (@fsval K S u))) (h u) ) case: fndP => //= kf; rewrite !ffunE /=. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by set y := (X in h X); rewrite (valP y) in xNS. ( Goal: @eq (Equality.sort V) (@Option.default (Equality.sort V) (default (@fsval K S u)) (@fnd K (Equality.sort V) (@Imfset.imfset imfset_key (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) K (fun a : @fset_sub_type K S => @fsval K S a) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (simplPredType (@fset_sub_type K S)) (@subfinset_finpred (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (predPredType (@fset_sub_type K S)) (@fin_finpred (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (pred_finpredType (@subFinType_finType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subfinType K S))) (@sort_of_simpl_pred (@fset_sub_type K S) (pred_of_argType (@fset_sub_type K S))))) (fun a : @fset_sub_type K S => negb (@eq_op V (h a) (default (@fsval K S a)))))) (Phantom (mem_pred (@fset_sub_type K S)) (@mem (@fset_sub_type K S) (simplPredType (@fset_sub_type K S)) (@SimplPred (@fset_sub_type K S) (fun a : @fset_sub_type K S => negb (@eq_op V (h a) (default (@fsval K S a)))))))) (@FunFinfun.finfun (@fset_sub_finType K (@Imfset.imfset imfset_key (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) K (fun a : @fset_sub_type K S => @fsval K S a) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (simplPredType (@fset_sub_type K S)) (@subfinset_finpred (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (predPredType (@fset_sub_type K S)) (@fin_finpred (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (pred_finpredType (@subFinType_finType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subfinType K S))) (@sort_of_simpl_pred (@fset_sub_type K S) (pred_of_argType (@fset_sub_type K S))))) (fun a : @fset_sub_type K S => negb (@eq_op V (h a) (default (@fsval K S a)))))) (Phantom (mem_pred (@fset_sub_type K S)) (@mem (@fset_sub_type K S) (simplPredType (@fset_sub_type K S)) (@SimplPred (@fset_sub_type K S) (fun a : @fset_sub_type K S => negb (@eq_op V (h a) (default (@fsval K S a))))))))) (Equality.sort V) (fun a : @fset_sub_type K (@Imfset.imfset imfset_key (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) K (fun a : @fset_sub_type K S => @fsval K S a) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (simplPredType (@fset_sub_type K S)) (@subfinset_finpred (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (predPredType (@fset_sub_type K S)) (@fin_finpred (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (pred_finpredType (@subFinType_finType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subfinType K S))) (@sort_of_simpl_pred (@fset_sub_type K S) (pred_of_argType (@fset_sub_type K S))))) (fun a : @fset_sub_type K S => negb (@eq_op V (h a) (default (@fsval K S a)))))) (Phantom (mem_pred (@fset_sub_type K S)) (@mem (@fset_sub_type K S) (simplPredType (@fset_sub_type K S)) (@SimplPred (@fset_sub_type K S) (fun a : @fset_sub_type K S => negb (@eq_op V (h a) (default (@fsval K S a)))))))) => h (@fincl K (@Imfset.imfset imfset_key (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) K (@fsval K S) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (simplPredType (@fset_sub_type K S)) (@subfinset_finpred (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (predPredType (@fset_sub_type K S)) (@fin_finpred (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (pred_finpredType (@subFinType_finType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subfinType K S))) (@sort_of_simpl_pred (@fset_sub_type K S) (pred_of_argType (@fset_sub_type K S))))) (fun a0 : @fset_sub_type K S => negb (@eq_op V (h a0) (default (@fsval K S a0)))))) (Phantom (mem_pred (@fset_sub_type K S)) (@mem (@fset_sub_type K S) (simplPredType (@fset_sub_type K S)) (@SimplPred (@fset_sub_type K S) (fun x : @fset_sub_type K S => negb (@eq_op V (h x) (default (@fsval K S x)))))))) S (@fset_sub_val imfset_key K S (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (simplPredType (@fset_sub_type K S)) (@subfinset_finpred (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S)) (@mem_fin (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (predPredType (@fset_sub_type K S)) (@fin_finpred (Choice.eqType (@sub_choiceType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S))) (pred_finpredType (@subFinType_finType K (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subfinType K S))) (@sort_of_simpl_pred (@fset_sub_type K S) (pred_of_argType (@fset_sub_type K S))))) (fun a0 : @fset_sub_type K S => negb (@eq_op V (h a0) (default (@fsval K S a0))))))) a))) (@fsval K S u))) (h u) ) case: fndP => /= [kf\|]. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite ffunE; congr (h _); apply/val_inj => //. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite inE /= -topredE /= negbK => /eqP ->. Qed. End FsfunOfFinfun. End Fsfun. Canonical fsfun_of_funE K V default S h key x := Unlockable (@Fsfun.of_ffunE K V default S h key x). Fact fsfun_key : unit. Proof. exact: tt. Qed. Definition fsfun_of_ffun key (K : choiceType) (V : eqType) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) (S : {fset K}) (h : S -> V) (default : K -> V) := (Fsfun.of_ffun default h key). ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Definition fsfun_choiceMixin (K V : choiceType) (d : K -> V) := [choiceMixin of fsfun d by <:]. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Canonical fsfun_choiceType (K V : choiceType) (d : K -> V) := ChoiceType (fsfun d) (fsfun_choiceMixin d). Delimit Scope fsfun_scope with fsfun. Notation "[ 'fsfun[' key ] x : aT => F \| default ]" := (fsfun_of_ffun key (fun x : aT => F) (fun x => default)) (at level 0, x at level 99, only parsing) : fun_scope. Notation "[ 'fsfun' x : aT => F \| default ]" := [fsfun[fsfun_key] x : aT => F \| default] (at level 0, x at level 99, only parsing) : fun_scope. Notation "[ 'fsfun[' key ] x : aT => F ]" := [fsfun[key] x : aT => F \| _] (at level 0, x at level 99, only parsing) : fun_scope. Notation "[ 'fsfun' x : aT => F ]" := [fsfun x : aT => F \| _] (at level 0, x at level 99, only parsing) : fun_scope. Notation "[ 'fsfun[' key ] x => F \| default ]" := [fsfun[key] x : _ => F \| default ] (at level 0, x at level 99, only parsing) : fun_scope. Notation "[ 'fsfun' x => F \| default ]" := [fsfun x : _ => F \| default] (at level 0, x at level 99, only parsing) : fun_scope. Notation "[ 'fsfun[' key ] x => F ]" := [fsfun[key] x => F \| _] (at level 0, x at level 99, only parsing) : fun_scope. Notation "[ 'fsfun' x => F ]" := [fsfun x => F \| _] (at level 0, x at level 99, only parsing) : fun_scope. Notation "[ 'fsfun[' key ]=> F \| default ]" := [fsfun[key] _ => F \| default ] (at level 0, only parsing) : fun_scope. Notation "[ 'fsfun=>' F \| default ]" := [fsfun _ => F \| default] (at level 0, only parsing) : fun_scope. Notation "[ 'fsfun[' key ]=> F ]" := [fsfun[key]=> F \| _] (at level 0, only parsing) : fun_scope. Notation "[ 'fsfun=>' F ]" := [fsfun=> F \| _] (at level 0, only parsing) : fun_scope. Definition fsfun_of_fun key (K : choiceType) (V : eqType) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) (S : {fset K}) (h : K -> V) default := [fsfun[key] x : S => h (val x) \| default x]. Notation "[ 'fsfun[' key ] x 'in' S => F \| default ]" := (fsfun_of_fun key S (fun x => F) (fun x => default)) (at level 0, x at level 99, only parsing) : fun_scope. Notation "[ 'fsfun' x 'in' S => F \| default ]" := [fsfun[fsfun_key] x in S => F \| default] (at level 0, x at level 99, only parsing) : fun_scope. Notation "[ 'fsfun[' key ] x 'in' S => F ]" := [fsfun[key] x in S => F \| _] (at level 0, x at level 99, only parsing) : fun_scope. Notation "[ 'fsfun' x 'in' S => F ]" := [fsfun[fsfun_key] x in S => F \| _] (at level 0, x at level 99, only parsing) : fun_scope. Notation "[ 'fsfun[' key ] 'in' S => F \| default ]" := [fsfun[key] _ in S => F \| default] (at level 0, only parsing) : fun_scope. Notation "[ 'fsfun' 'in' S => F \| default ]" := [fsfun[fsfun_key] in S => F \| default] (at level 0, only parsing) : fun_scope. Notation "[ 'fsfun[' key ] 'in' S => F ]" := [fsfun[key] in S => F \| _] (at level 0, only parsing) : fun_scope. Notation "[ 'fsfun' 'in' S => F ]" := [fsfun[fsfun_key] in S => F \| _] (at level 0, only parsing) : fun_scope. ( only printing ) Notation "[ 'fs' 'fun' x : aT => F ]" := [fsfun[_] x : aT => F] (at level 0, x at level 99, format "[ 'fs' 'fun' x : aT => F ]") : fun_scope. Notation "[ 'fs' 'fun' x 'in' aT => F ]" := [fsfun[_] x in aT => F] (at level 0, x at level 99, format "[ 'fs' 'fun' x 'in' aT => F ]") : fun_scope. Fact fsfun0_key : unit. Proof. exact: tt. Qed. Notation "[ 'fsfun' 'for' default ]" := (fsfun_of_ffun fsfun0_key [fmap] default) (at level 0, only parsing) : fun_scope. Notation "[ 'fsfun' 'of' x => default ]" := [fsfun for fun x => default] (at level 0, x at level 99, only parsing) : fun_scope. Notation "[ 'fsfun' 'with' default ]" := [fsfun of _ => default] (at level 0, only parsing) : fun_scope. Notation "[ 'fsfun' ]" := [fsfun for _] (at level 0, format "[ 'fsfun' ]") : fun_scope. Section FsfunTheory. Variables (key : unit) (K : choiceType) (V : eqType) (default : K -> V). ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma fsfun_ffun (S : {fset K}) (h : S -> V) (x : K) : [fsfun[key] a : S => h a \| default a] x = odflt (default x) (omap h (insub x)). Proof. by rewrite unlock. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma fsfun_fun (S : {fset K}) (h : K -> V) (x : K) : [fsfun[key] a in S => h a \| default a] x = if x \in S then h x else (default x). Proof. by rewrite fsfun_ffun; case: insubP => //= [u -> ->\|/negPf ->]. Qed. Lemma fsfun0E : [fsfun for default] =1 default. Proof. by move=> x; rewrite unlock insubF ?inE. Qed. Definition fsfunE := (fsfun_fun, fsfun0E). ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Lemma finsupp_sub (S : {fset K}) (h : S -> V) : finsupp [fsfun[key] a : S => h a \| default a] `<=` S. Proof. ( Goal: is_true (@fsubset K (@FinSupp.fs K V (fun a : Choice.sort K => default a) (@fsfun_of_ffun key K V S (fun a : @fset_sub_type K S => h a) (fun a : Choice.sort K => default a))) S) ) apply/fsubsetP => a; rewrite mem_finsupp unlock /=. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by case: insubP => //=; rewrite eqxx. Qed. Lemma finsupp0 : finsupp [fsfun for default] = fset0. Proof. by apply/fsetP => x; rewrite !inE mem_finsupp fsfunE eqxx. Qed. Lemma fsfun0_inj : injective default -> injective [fsfun for default]. Proof. by move=> def_inj x y; rewrite !fsfunE => /def_inj. Qed. Lemma in_finsupp0 x : x \in finsupp [fsfun for default] = false. Proof. by rewrite finsupp0 inE. Qed. End FsfunTheory. Module Import FsfunInE2. Definition inE := (inE, in_finsupp0). End FsfunInE2. Section FsfunIdTheory. Variables (K : choiceType). Implicit Types (f g : {fsfun K -> K}). Fact fsfun_comp_key : unit. Proof. exact: tt. Qed. Definition fsfun_comp g f : {fsfun _} := [fsfun[fsfun_comp_key] k in finsupp f `\|` finsupp g => g (f k)]. Notation "g \o f" := (fsfun_comp g f) : fsfun_scope. Lemma fscompE g f : (g \o f)%fsfun =1 g \o f. Proof. ( Goal: @eqfun (Equality.sort (Choice.eqType K)) (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) (fsfun_comp g f)) (@funcomp (Equality.sort (Choice.eqType K)) (Choice.sort K) (Choice.sort K) tt (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) g) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f)) ) move=> k; rewrite fsfun_ffun; case: insubP => //= [u _ <- //\|]. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite inE => /norP [kNf kNg]; rewrite !fsfun_dflt. Qed. Lemma fscomp0f : left_id [fsfun] fsfun_comp. Proof. by move=> f; apply/fsfunP=> k; rewrite fscompE /= fsfun0E. Qed. Lemma fscompA : associative fsfun_comp. Proof. by move=> f g h; apply/fsfunP => k; rewrite !fscompE /= !fscompE. Qed. Lemma fscomp_inj g f : injective f -> injective g -> injective (g \o f)%fsfun. Proof. by move=> f_inj g_inj k k'; rewrite !fscompE => /= /g_inj /f_inj. Qed. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) Definition fsinjectiveb : pred {fsfun K -> K} := [pred f \| (injectiveb [ffun a : finsupp f => f (val a)]) && [forall a : finsupp f, f (val a) \in finsupp f]]. Let equivalent (Ps : seq Prop) := if Ps is P0 :: Ps then let fix aux (P : Prop) (Qs : seq Prop) := ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) if Qs is Q :: Qs then (P -> Q) /\ (aux Q Qs) else P -> P0 in aux P0 Ps else True. Lemma fsinjective_subproof f : equivalent [:: injective f ; let S := finsupp f in {in S &, injective f} /\ forall a : S, f (val a) \in S ; exists2 S : {fset K}, (finsupp f `<=` S) & {in S &, injective f} /\ forall a : S, f (val a) \in S]. Proof. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) split => /= [f_inj\|]; last split=> [[f_inj f_st]\|[S fS [f_inj f_st]] a b]. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) - split=> [a b ? ?\|a]; first exact: f_inj. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) rewrite mem_finsupp (inj_eq f_inj) -mem_finsupp; apply/valP. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) - by exists (finsupp f)=> //; apply: fsubset_refl. ( Goal: Bool.reflect (@ex2 (@finset_of K (Phant (Choice.sort K))) (fun S : @finset_of K (Phant (Choice.sort K)) => is_true (@fsubset K (@FinSupp.fs K (Choice.eqType K) (fun x : Choice.sort K => x) f) S)) (fun S : @finset_of K (Phant (Choice.sort K)) => and (@prop_in2 (Choice.sort K) (@mem (Choice.sort K) (finSetPredType K) S) (fun x1 x2 : Choice.sort K => forall _ : @eq (Equality.sort (Choice.eqType K)) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f x1) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f x2), @eq (Choice.sort K) x1 x2) (inPhantom (@injective (Equality.sort (Choice.eqType K)) (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f)))) (forall a : @fset_sub_type K S, is_true (@in_mem (Equality.sort (Choice.eqType K)) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f (@val (Choice.sort K) (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S) a)) (@mem (Choice.sort K) (finSetPredType K) S))))) (fsinjectiveb f) ) have Nfinsupp := contra (fsubsetP fS _). ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) wlog /andP [aS bNS] : a b / (a \in S) && (b \notin S) => [hwlog\|]; last first. ( Goal: @big_rel_class_of (@finset_of K (Phant (Choice.sort K))) (@fsubset K) ) rewrite (fsfun_dflt (Nfinsupp _ bNS)) => fb_eq_a. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) by move: bNS; rewrite -fb_eq_a (f_st.[aS]). ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) have [[aS\|aNS] [bS\|bNS]] := (boolP (a \in S), boolP (b \in S)); first 3 last. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) - by rewrite !fsfun_dflt // ?Nfinsupp. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) - exact: f_inj. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) - by apply: hwlog; rewrite aS. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by move=> fab; symmetry; apply: hwlog; rewrite // bS. Qed. Lemma fsinjectiveP f : reflect (injective f) (fsinjectiveb f). Proof. ( Goal: Bool.reflect (@ex2 (@finset_of K (Phant (Choice.sort K))) (fun S : @finset_of K (Phant (Choice.sort K)) => is_true (@fsubset K (@FinSupp.fs K (Choice.eqType K) (fun x : Choice.sort K => x) f) S)) (fun S : @finset_of K (Phant (Choice.sort K)) => and (@prop_in2 (Choice.sort K) (@mem (Choice.sort K) (finSetPredType K) S) (fun x1 x2 : Choice.sort K => forall _ : @eq (Equality.sort (Choice.eqType K)) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f x1) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f x2), @eq (Choice.sort K) x1 x2) (inPhantom (@injective (Equality.sort (Choice.eqType K)) (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f)))) (forall a : @fset_sub_type K S, is_true (@in_mem (Equality.sort (Choice.eqType K)) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f (@val (Choice.sort K) (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S) a)) (@mem (Choice.sort K) (finSetPredType K) S))))) (fsinjectiveb f) ) have [H1 [H2 H3]]:= fsinjective_subproof f. ( Goal: is_true (@in_mem (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f (@fsval K (@FinSupp.fs K (Choice.eqType K) (fun x : Choice.sort K => x) f) a)) (@mem (Choice.sort K) (finSetPredType K) (@FinSupp.fs K (Choice.eqType K) (fun x : Choice.sort K => x) f))) ) ( Goal: @ex2 (@finset_of K (Phant (Choice.sort K))) (fun S : @finset_of K (Phant (Choice.sort K)) => is_true (@fsubset K (@FinSupp.fs K (Choice.eqType K) (fun x : Choice.sort K => x) f) S)) (fun S : @finset_of K (Phant (Choice.sort K)) => and (@prop_in2 (Choice.sort K) (@mem (Choice.sort K) (finSetPredType K) S) (fun x1 x2 : Choice.sort K => forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f x1) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f x2), @eq (Choice.sort K) x1 x2) (inPhantom (@injective (Choice.sort K) (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f)))) (forall a : @fset_sub_type K S, is_true (@in_mem (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f (@fsval K S a)) (@mem (Choice.sort K) (finSetPredType K) S)))) ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) rewrite /fsinjectiveb; apply: (iffP idP)=> [\|]. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by move=> /andP [/fsfun_injective_inP ? /forallP ?]; apply/H3/H2. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by move=> /H1 [/fsfun_injective_inP ? /forallP ?]; apply/andP. Qed. Lemma fsinjectivebP f : reflect (exists2 S : {fset K}, (finsupp f `<=` S) & {in S &, injective f} /\ forall a : S, f (val a) \in S) (fsinjectiveb f). Proof. ( Goal: Bool.reflect (@ex2 (@finset_of K (Phant (Choice.sort K))) (fun S : @finset_of K (Phant (Choice.sort K)) => is_true (@fsubset K (@FinSupp.fs K (Choice.eqType K) (fun x : Choice.sort K => x) f) S)) (fun S : @finset_of K (Phant (Choice.sort K)) => and (@prop_in2 (Choice.sort K) (@mem (Choice.sort K) (finSetPredType K) S) (fun x1 x2 : Choice.sort K => forall _ : @eq (Equality.sort (Choice.eqType K)) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f x1) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f x2), @eq (Choice.sort K) x1 x2) (inPhantom (@injective (Equality.sort (Choice.eqType K)) (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f)))) (forall a : @fset_sub_type K S, is_true (@in_mem (Equality.sort (Choice.eqType K)) (@fun_of_fsfun K (Choice.eqType K) ((fun x : Choice.sort K => x) : forall _ : Choice.sort K, Choice.sort K) f (@val (Choice.sort K) (fun x : Choice.sort K => @in_mem (Choice.sort K) x (@mem (Choice.sort K) (finSetPredType K) S)) (@fset_sub_subType K S) a)) (@mem (Choice.sort K) (finSetPredType K) S))))) (fsinjectiveb f) ) have [H1 [H2 H3]]:= fsinjective_subproof f. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by apply: (iffP (fsinjectiveP _)) => //; by move=> /H1 /H2. Qed. End FsfunIdTheory. Definition inE := inE. Export BigEnough. Module BigEnoughFSet. Definition big_rel_fsubset_class K : big_rel_class_of (@fsubset K). Proof. ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) exists fsubset (fun G => \bigcup_(g <- G) g) => [\|g s\|g1 g2 j] //. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite big_cons fsubsetUl. ( Goal: forall _ : is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) j (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))), is_true (@fsubset K g1 (@BigOp.bigop (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) (@fset0 K) (@cons (@finset_of K (Phant (Choice.sort K))) g2 j) (fun g : @finset_of K (Phant (Choice.sort K)) => @BigBody (@finset_of K (Phant (Choice.sort K))) (@finset_of K (Phant (Choice.sort K))) g (@fsetU K) true g))) ) by rewrite big_cons => h; rewrite fsubsetU // h orbT. Qed. Canonical big_enough_fset K := BigRelOf (big_rel_fsubset_class K). Ltac fset_big_enough_trans := match goal with ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b ) ( Goal: forall _ : @eq (Choice.sort K) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f a) (@fun_of_fsfun K (Choice.eqType K) (fun x : Choice.sort K => x) f b), @eq (Choice.sort K) a b *) \| [leq : is_true (?A `<=` ?B) \|- is_true (?X `<=` ?B)] => apply: fsubset_trans leq; big_enough; olddone end. Ltac done := do [fset_big_enough_trans\|BigEnough.done]. Ltac pose_big_fset K i := evar (i : {fset K}); suff : closed i; first do [move=> _; instantiate (1 := bigger_than (@fsubset K) _) in (Value of i)]. End BigEnoughFSet.
From mathcomp Require Import ssreflect ssrbool eqtype ssrfun ssrnat choice seq. From mathcomp Require Import fintype tuple bigop path. (**********************************************************************) ( Experimental library of generic sets ) ( ==================================== ) ( Contains two structures: ) ( semisetType == families of sets, without total set (e.g. {fset T}) ) ( setType == families of sets, with total set ) ( (e.g. {set T} or {SAset R^n}) ) (*********************************************************************) From mathcomp Require Import order. Set Implicit Arguments. Unset Strict Implicit. Unset Printing Implicit Defensive. Reserved Notation "x \subset y" (at level 70, y at next level). Reserved Notation "x \contains y" (at level 70, y at next level, only parsing). Reserved Notation "x \proper y" (at level 70, y at next level). Reserved Notation "x \containsproper y" (at level 70, y at next level, only parsing). Reserved Notation "x \subset y :> T" (at level 70, y at next level). Reserved Notation "x \contains y :> T" (at level 70, y at next level, only parsing). Reserved Notation "x \proper y :> T" (at level 70, y at next level). Reserved Notation "x \containsproper y :> T" (at level 70, y at next level, only parsing). Reserved Notation "\subsets y" (at level 35). Reserved Notation "\supersets y" (at level 35). Reserved Notation "\propersets y" (at level 35). Reserved Notation "\superpropersets y" (at level 35). Reserved Notation "\subsets y :> T" (at level 35, y at next level). Reserved Notation "\supersets y :> T" (at level 35, y at next level). Reserved Notation "\propersets y :> T" (at level 35, y at next level). Reserved Notation "\superpropersets y :> T" (at level 35, y at next level). Reserved Notation "x \subset y \subset z" (at level 70, y, z at next level). Reserved Notation "x \proper y \subset z" (at level 70, y, z at next level). Reserved Notation "x \subset y \proper z" (at level 70, y, z at next level). Reserved Notation "x \proper y \proper z" (at level 70, y, z at next level). Reserved Notation "x \subset y ?= 'iff' c" (at level 70, y, c at next level, format "x '[hv' \subset y '/' ?= 'iff' c ']'"). Reserved Notation "x \subset y ?= 'iff' c :> T" (at level 70, y, c at next level, format "x '[hv' \subset y '/' ?= 'iff' c :> T ']'"). Reserved Notation "~: A" (at level 35, right associativity). Reserved Notation "[ 'set' ~ a ]" (at level 0, format "[ 'set' ~ a ]"). Reserved Notation "[ 'set' a1 ; a2 ; .. ; an ]" (at level 0, a1 at level 99, format "[ 'set' a1 ; a2 ; .. ; an ]"). Delimit Scope abstract_set_scope with set. Local Open Scope abstract_set_scope. Module SET. Import Order.Theory Order.Syntax Order.Def. Fact display_set : unit -> unit. Proof. exact. Qed. Module Import SetSyntax. Notation "\sub%set" := (@le (display_set _) _) : abstract_set_scope. Notation "\super%set" := (@ge (display_set _) _) : abstract_set_scope. Notation "\proper%set" := (@lt (display_set _) _) : abstract_set_scope. Notation "\superproper%set" := (@gt (display_set _) _) : abstract_set_scope. Notation "\sub?%set" := (@leif (display_set _) _) : abstract_set_scope. Notation "\subsets y" := (\super%set y) : abstract_set_scope. Notation "\subsets y :> T" := (\subsets (y : T)) : abstract_set_scope. Notation "\supersets y" := (\sub%set y) : abstract_set_scope. Notation "\supersets y :> T" := (\supersets (y : T)) : abstract_set_scope. Notation "\propersets y" := (\superproper%set y) : abstract_set_scope. Notation "\propersets y :> T" := (\propersets (y : T)) : abstract_set_scope. Notation "\superpropersets y" := (\proper%set y) : abstract_set_scope. Notation "\superpropersets y :> T" := (\superpropersets (y : T)) : abstract_set_scope. Notation "x \subset y" := (\sub%set x y) : abstract_set_scope. Notation "x \subset y :> T" := ((x : T) \subset (y : T)) : abstract_set_scope. Notation "x \proper y" := (\proper%set x y) : abstract_set_scope. Notation "x \proper y :> T" := ((x : T) \proper (y : T)) : abstract_set_scope. Notation "x \subset y \subset z" := ((x \subset y)%set && (y \subset z)%set) : abstract_set_scope. Notation "x \proper y \subset z" := ((x \proper y)%set && (y \subset z)%set) : abstract_set_scope. Notation "x \subset y \proper z" := ((x \subset y)%set && (y \proper z)%set) : abstract_set_scope. Notation "x \proper y \proper z" := ((x \proper y)%set && (y \proper z)%set) : abstract_set_scope. Notation "x \subset y ?= 'iff' C" := (\sub?%set x y C) : abstract_set_scope. Notation "x \subset y ?= 'iff' C :> R" := ((x : R) \subset (y : R) ?= iff C) (only parsing) : abstract_set_scope. Notation set0 := (@bottom (display_set _) _). Notation setT := (@top (display_set _) _). Notation setU := (@join (display_set _) _). Notation setI := (@meet (display_set _) _). Notation setD := (@sub (display_set _) _). Notation setC := (@compl (display_set _) _). Notation "x :&: y" := (setI x y). Notation "x :\|: y" := (setU x y). Notation "x :\: y" := (setD x y). Notation "~: x" := (setC x). Notation "x \subset y" := (\sub%set x y) : bool_scope. Notation "x \proper y" := (\proper%set x y) : bool_scope. End SetSyntax. Ltac EqualityPack cT xclass xT := match type of Equality.Pack with \| forall sort : Type, Equality.mixin_of sort -> eqType => ( mathcomp.dev ) exact (@Equality.Pack cT xclass) \| _ => ( mathcomp <= 1.7 ) exact (@Equality.Pack cT xclass xT) end. Ltac ChoicePack cT xclass xT := match type of Choice.Pack with \| forall sort : Type, Choice.class_of sort -> choiceType => ( mathcomp.dev ) exact (@Choice.Pack cT xclass) \| _ => ( mathcomp <= 1.7 ) exact (@Choice.Pack cT xclass xT) end. Module Semiset. Section ClassDef. Variable elementType : Type. ( Universe type ) Variable eqType_of_elementType : elementType -> eqType. Coercion eqType_of_elementType : elementType >-> eqType. Implicit Types (X Y : elementType). Structure mixin_of d (set : elementType -> (cblatticeType (display_set d))) := Mixin { memset : forall X, set X -> X -> bool; set1 : forall X, X -> set X; _ : forall X (x : X), ~~ memset set0 x; ( set0 is empty instead ) _ : forall X (x y : X), memset (set1 y) x = (x == y); _ : forall X (x : X) A, (set1 x \subset A) = (memset A x); _ : forall X (A : set X), (set0 \proper A) -> {x \| memset A x} ; ( exists or sig ?? ) _ : forall X (A B : set X), {subset memset A <= memset B} -> A \subset B; _ : forall X (x : X) A B, (memset (A :\|: B) x) = (memset A x) \|\| (memset B x); ( there is no closure in a set ) funsort : elementType -> elementType -> Type; fun_of_funsort : forall X Y, funsort X Y -> X -> Y; imset : forall X Y, funsort X Y -> set X -> set Y; _ : forall X Y (f : funsort X Y) (A : set X) (y : Y), reflect (exists2 x : X, memset A x & y = fun_of_funsort f x) (memset (imset f A) y) }. Record class_of d (set : elementType -> Type) := Class { base : forall X, @Order.CBLattice.class_of (display_set d) (set X); mixin : mixin_of (fun X => Order.CBLattice.Pack (base X) (set X)) }. Local Coercion base : class_of >-> Funclass. Structure type d := Pack { sort ; _ : class_of d sort; _ : elementType -> Type }. Local Coercion sort : type >-> Funclass. Variables (set : elementType -> Type) (disp : unit) (cT : type disp). Definition class := let: Pack _ c _ as cT' := cT return class_of _ cT' in c. Definition clone disp' c of (disp = disp') & phant_id class c := @Pack disp' set c set. Let xset := let: Pack set _ _ := cT in set. Notation xclass := (class : class_of _ xset). Definition pack b0 (m0 : mixin_of (fun X=> @Order.CBLattice.Pack (display_set disp) (set X) (b0 X) (set X))) := fun bT b & (forall X, phant_id (@Order.CBLattice.class (display_set disp) (bT X)) (b X)) => fun m & phant_id m0 m => Pack (@Class disp set b m) set. End ClassDef. Section CanonicalDef. Variable elementType : Type. Variable eqType_of_elementType : elementType -> eqType. Coercion eqType_of_elementType : elementType >-> eqType. Notation type := (type eqType_of_elementType). Local Coercion base : class_of >-> Funclass. Local Coercion sort : type >-> Funclass. Variables (set : elementType -> Type) (X : elementType). Variables (disp : unit) (cT : type disp). Local Notation ddisp := (display_set disp). Let xset := let: Pack set _ _ := cT in set. Notation xclass := (@class _ eqType_of_elementType _ cT : class_of eqType_of_elementType _ xset). Definition eqType := ltac:(EqualityPack (cT X) ((@class _ eqType_of_elementType _ cT : class_of eqType_of_elementType _ xset) X) (xset X)). Definition choiceType := ltac:(ChoicePack (cT X) ((@class _ eqType_of_elementType _ cT : class_of eqType_of_elementType _ xset) X) (xset X)). Definition porderType := @Order.POrder.Pack ddisp (cT X) (xclass X) (xset X). Definition latticeType := @Order.Lattice.Pack ddisp (cT X) (xclass X) (xset X). Definition blatticeType := @Order.BLattice.Pack ddisp (cT X) (xclass X) (xset X). Definition cblatticeType := @Order.CBLattice.Pack ddisp (cT X) (xclass X) (xset X). End CanonicalDef. Module Import Exports. Coercion mixin : class_of >-> mixin_of. Coercion base : class_of >-> Funclass. Coercion sort : type >-> Funclass. Coercion eqType : type >-> Equality.type. Coercion choiceType : type >-> Choice.type. Coercion porderType : type >-> Order.POrder.type. Coercion latticeType : type >-> Order.Lattice.type. Coercion blatticeType : type >-> Order.BLattice.type. Coercion cblatticeType : type >-> Order.CBLattice.type. Canonical eqType. Canonical choiceType. Canonical porderType. Canonical latticeType. Canonical blatticeType. Canonical cblatticeType. Notation semisetType := type. Notation semisetMixin := mixin_of. Notation SemisetMixin := Mixin. Notation SemisetType set m := (@pack _ _ set _ _ m _ _ (fun=> id) _ id). Notation "[ 'semisetType' 'of' set 'for' cset ]" := (@clone _ _ set _ cset _ _ erefl id) (at level 0, format "[ 'semisetType' 'of' set 'for' cset ]") : form_scope. Notation "[ 'semisetType' 'of' set 'for' cset 'with' disp ]" := (@clone _ _ set _ cset disp _ (unit_irrelevance _ _) id) (at level 0, format "[ 'semisetType' 'of' set 'for' cset 'with' disp ]") : form_scope. Notation "[ 'semisetType' 'of' set ]" := [semisetType of set for _] (at level 0, format "[ 'semisetType' 'of' set ]") : form_scope. Notation "[ 'semisetType' 'of' set 'with' disp ]" := [semisetType of set for _ with disp] (at level 0, format "[ 'semisetType' 'of' set 'with' disp ]") : form_scope. End Exports. End Semiset. Import Semiset.Exports. Section SemisetOperations. Context {elementType : Type} {eqType_of_elementType : elementType -> eqType}. Coercion eqType_of_elementType : elementType >-> eqType. Context {disp : unit}. Section setfun. Variable (set : semisetType eqType_of_elementType disp). Definition setfun := Semiset.funsort (Semiset.class set). Definition fun_of_setfun X Y (f : setfun X Y) : X -> Y := @Semiset.fun_of_funsort _ _ _ _ (Semiset.class set) _ _ f. Coercion fun_of_setfun : setfun >-> Funclass. End setfun. Context {set : semisetType eqType_of_elementType disp}. Variable X Y : elementType. Definition memset : set X -> X -> bool := @Semiset.memset _ _ _ _ (Semiset.class set) _. Definition set1 : X -> set X := @Semiset.set1 _ _ _ _ (Semiset.class set) _. Definition imset : setfun set X Y -> set X -> set Y:= @Semiset.imset _ _ _ _ (Semiset.class set) _ _. Canonical set_predType := Eval hnf in mkPredType memset. Structure setpredType := SetPredType { setpred_sort :> Type; tosetpred : setpred_sort -> pred X; _ : {mem : setpred_sort -> mem_pred X \| isMem tosetpred mem}; _ : {pred_fset : setpred_sort -> set X \| forall p x, x \in pred_fset p = tosetpred p x} }. Canonical setpredType_predType (fpX : setpredType) := @PredType X (setpred_sort fpX) (@tosetpred fpX) (let: SetPredType _ _ mem _ := fpX in mem). Definition predset (fpX : setpredType) : fpX -> set X := let: SetPredType _ _ _ (exist pred_fset _) := fpX in pred_fset. End SemisetOperations. Module Import SemisetSyntax. Notation "[ 'set' x : T \| P ]" := (predset (fun x : T => P%B)) (at level 0, x at level 99, only parsing) : abstract_set_scope. Notation "[ 'set' x \| P ]" := [set x : _ \| P] (at level 0, x, P at level 99, format "[ 'set' x \| P ]") : abstract_set_scope. Notation "[ 'set' x 'in' A ]" := [set x \| x \in A] (at level 0, x at level 99, format "[ 'set' x 'in' A ]") : abstract_set_scope. Notation "[ 'set' x : T 'in' A ]" := [set x : T \| x \in A] (at level 0, x at level 99, only parsing) : abstract_set_scope. Notation "[ 'set' x : T \| P & Q ]" := [set x : T \| P && Q] (at level 0, x at level 99, only parsing) : abstract_set_scope. Notation "[ 'set' x \| P & Q ]" := [set x \| P && Q ] (at level 0, x, P at level 99, format "[ 'set' x \| P & Q ]") : abstract_set_scope. Notation "[ 'set' x : T 'in' A \| P ]" := [set x : T \| x \in A & P] (at level 0, x at level 99, only parsing) : abstract_set_scope. Notation "[ 'set' x 'in' A \| P ]" := [set x \| x \in A & P] (at level 0, x at level 99, format "[ 'set' x 'in' A \| P ]") : abstract_set_scope. Notation "[ 'set' x 'in' A \| P & Q ]" := [set x in A \| P && Q] (at level 0, x at level 99, format "[ 'set' x 'in' A \| P & Q ]") : abstract_set_scope. Notation "[ 'set' x : T 'in' A \| P & Q ]" := [set x : T in A \| P && Q] (at level 0, x at level 99, only parsing) : abstract_set_scope. Notation "[ 'set' a ]" := (set1 a) (at level 0, a at level 99, format "[ 'set' a ]") : abstract_set_scope. Notation "[ 'set' a : T ]" := [set (a : T)] (at level 0, a at level 99, format "[ 'set' a : T ]") : abstract_set_scope. Notation "a \|: y" := ([set a] :\|: y) : abstract_set_scope. Notation "x :\ a" := (x :\: [set a]) : abstract_set_scope. Notation "[ 'set' a1 ; a2 ; .. ; an ]" := (setU .. (a1 \|: [set a2]) .. [set an]). Notation "f @: A" := (imset f A) (at level 24) : abstract_set_scope. End SemisetSyntax. Module Import SemisetTheory. Section SemisetTheory. Variable elementType : Type. Variable eqType_of_elementType : elementType -> eqType. Coercion eqType_of_elementType : elementType >-> eqType. Variable disp : unit. Variable set : semisetType eqType_of_elementType disp. Section setX. Variables X : elementType. Implicit Types (x y : X) (A B C : set X). Lemma notin_set0 (x : X) : x \notin (set0 : set X). Proof. ( Goal: @eq bool (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) (@join (display_set disp) (@Semiset.latticeType elementType eqType_of_elementType X disp set) A B))) (orb (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) A)) (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) B))) ) rewrite /set1 /in_mem /= /memset. ( Goal: forall (A : @Semiset.sort elementType eqType_of_elementType disp set X) (f : @Semiset.funsort elementType eqType_of_elementType disp (fun X : elementType => @Order.CBLattice.Pack (display_set disp) (@Semiset.sort elementType eqType_of_elementType disp set X) (@Semiset.base elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set) X) (@Semiset.sort elementType eqType_of_elementType disp set X)) (@Semiset.mixin elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set)) X Y), Bool.reflect (@ex2 (Equality.sort (eqType_of_elementType X)) (fun x : Equality.sort (eqType_of_elementType X) => is_true (@Semiset.memset elementType eqType_of_elementType disp (fun X : elementType => @Order.CBLattice.Pack (display_set disp) (@Semiset.sort elementType eqType_of_elementType disp set X) (@Semiset.base elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set) X) (@Semiset.sort elementType eqType_of_elementType disp set X)) (@Semiset.mixin elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set)) X A x)) (fun x : Equality.sort (eqType_of_elementType X) => @eq (Equality.sort (eqType_of_elementType Y)) y (@fun_of_setfun elementType eqType_of_elementType disp set X Y f x))) (@Semiset.memset elementType eqType_of_elementType disp (fun X : elementType => @Order.CBLattice.Pack (display_set disp) (@Semiset.sort elementType eqType_of_elementType disp set X) (@Semiset.base elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set) X) (@Semiset.sort elementType eqType_of_elementType disp set X)) (@Semiset.mixin elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set)) Y (@Semiset.imset elementType eqType_of_elementType disp (fun X : elementType => @Order.CBLattice.Pack (display_set disp) (@Semiset.sort elementType eqType_of_elementType disp set X) (@Semiset.base elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set) X) (@Semiset.sort elementType eqType_of_elementType disp set X)) (@Semiset.mixin elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set)) X Y f A) y) ) case: set => [S [base [memset set1 /= H ? ? ? ? ? ? ? ? ?]] ?] /=. ( Goal: Bool.reflect (@ex2 (Equality.sort (eqType_of_elementType X)) (fun x : Equality.sort (eqType_of_elementType X) => is_true (memset X A x)) (fun x : Equality.sort (eqType_of_elementType X) => @eq (Equality.sort (eqType_of_elementType Y)) y (@fun_of_setfun elementType eqType_of_elementType disp (@Semiset.Pack elementType eqType_of_elementType disp S (@Semiset.Class elementType eqType_of_elementType disp S base (@Semiset.Mixin elementType eqType_of_elementType disp (fun X : elementType => @Order.CBLattice.Pack (display_set disp) (S X) (base X) (S X)) memset set1 _i_ _e_ _e1_ _s_ _i1_ _e2_ _funsort_ _fun_of_funsort_ _imset_ H)) _T_) X Y f x))) (memset Y (_imset_ X Y f A) y) ) exact: H. Qed. Lemma in_set1 x y : x \in ([set y] : set X) = (x == y). Proof. ( Goal: @eq bool (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) (@join (display_set disp) (@Semiset.latticeType elementType eqType_of_elementType X disp set) A B))) (orb (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) A)) (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) B))) ) rewrite /set1 /in_mem /= /memset. ( Goal: forall (A : @Semiset.sort elementType eqType_of_elementType disp set X) (f : @Semiset.funsort elementType eqType_of_elementType disp (fun X : elementType => @Order.CBLattice.Pack (display_set disp) (@Semiset.sort elementType eqType_of_elementType disp set X) (@Semiset.base elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set) X) (@Semiset.sort elementType eqType_of_elementType disp set X)) (@Semiset.mixin elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set)) X Y), Bool.reflect (@ex2 (Equality.sort (eqType_of_elementType X)) (fun x : Equality.sort (eqType_of_elementType X) => is_true (@Semiset.memset elementType eqType_of_elementType disp (fun X : elementType => @Order.CBLattice.Pack (display_set disp) (@Semiset.sort elementType eqType_of_elementType disp set X) (@Semiset.base elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set) X) (@Semiset.sort elementType eqType_of_elementType disp set X)) (@Semiset.mixin elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set)) X A x)) (fun x : Equality.sort (eqType_of_elementType X) => @eq (Equality.sort (eqType_of_elementType Y)) y (@fun_of_setfun elementType eqType_of_elementType disp set X Y f x))) (@Semiset.memset elementType eqType_of_elementType disp (fun X : elementType => @Order.CBLattice.Pack (display_set disp) (@Semiset.sort elementType eqType_of_elementType disp set X) (@Semiset.base elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set) X) (@Semiset.sort elementType eqType_of_elementType disp set X)) (@Semiset.mixin elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set)) Y (@Semiset.imset elementType eqType_of_elementType disp (fun X : elementType => @Order.CBLattice.Pack (display_set disp) (@Semiset.sort elementType eqType_of_elementType disp set X) (@Semiset.base elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set) X) (@Semiset.sort elementType eqType_of_elementType disp set X)) (@Semiset.mixin elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set)) X Y f A) y) ) case: set => [S [base [memset set1 /= ? H ? ? ? ? ? ? ? ?]] ?] /=. ( Goal: Bool.reflect (@ex2 (Equality.sort (eqType_of_elementType X)) (fun x : Equality.sort (eqType_of_elementType X) => is_true (memset X A x)) (fun x : Equality.sort (eqType_of_elementType X) => @eq (Equality.sort (eqType_of_elementType Y)) y (@fun_of_setfun elementType eqType_of_elementType disp (@Semiset.Pack elementType eqType_of_elementType disp S (@Semiset.Class elementType eqType_of_elementType disp S base (@Semiset.Mixin elementType eqType_of_elementType disp (fun X : elementType => @Order.CBLattice.Pack (display_set disp) (S X) (base X) (S X)) memset set1 _i_ _e_ _e1_ _s_ _i1_ _e2_ _funsort_ _fun_of_funsort_ _imset_ H)) _T_) X Y f x))) (memset Y (_imset_ X Y f A) y) ) exact: H. Qed. Lemma sub1set x A : ([set x] \subset A) = (x \in A). Proof. ( Goal: @eq bool (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) (@join (display_set disp) (@Semiset.latticeType elementType eqType_of_elementType X disp set) A B))) (orb (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) A)) (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) B))) ) rewrite /set1 /in_mem /= /memset. ( Goal: forall (A : @Semiset.sort elementType eqType_of_elementType disp set X) (f : @Semiset.funsort elementType eqType_of_elementType disp (fun X : elementType => @Order.CBLattice.Pack (display_set disp) (@Semiset.sort elementType eqType_of_elementType disp set X) (@Semiset.base elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set) X) (@Semiset.sort elementType eqType_of_elementType disp set X)) (@Semiset.mixin elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set)) X Y), Bool.reflect (@ex2 (Equality.sort (eqType_of_elementType X)) (fun x : Equality.sort (eqType_of_elementType X) => is_true (@Semiset.memset elementType eqType_of_elementType disp (fun X : elementType => @Order.CBLattice.Pack (display_set disp) (@Semiset.sort elementType eqType_of_elementType disp set X) (@Semiset.base elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set) X) (@Semiset.sort elementType eqType_of_elementType disp set X)) (@Semiset.mixin elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set)) X A x)) (fun x : Equality.sort (eqType_of_elementType X) => @eq (Equality.sort (eqType_of_elementType Y)) y (@fun_of_setfun elementType eqType_of_elementType disp set X Y f x))) (@Semiset.memset elementType eqType_of_elementType disp (fun X : elementType => @Order.CBLattice.Pack (display_set disp) (@Semiset.sort elementType eqType_of_elementType disp set X) (@Semiset.base elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set) X) (@Semiset.sort elementType eqType_of_elementType disp set X)) (@Semiset.mixin elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set)) Y (@Semiset.imset elementType eqType_of_elementType disp (fun X : elementType => @Order.CBLattice.Pack (display_set disp) (@Semiset.sort elementType eqType_of_elementType disp set X) (@Semiset.base elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set) X) (@Semiset.sort elementType eqType_of_elementType disp set X)) (@Semiset.mixin elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set)) X Y f A) y) ) case: set A => [S [base [memset set1 /= ? ? H ? ? ? ? ? ? ?]] ?] A /=. ( Goal: Bool.reflect (@ex2 (Equality.sort (eqType_of_elementType X)) (fun x : Equality.sort (eqType_of_elementType X) => is_true (memset X A x)) (fun x : Equality.sort (eqType_of_elementType X) => @eq (Equality.sort (eqType_of_elementType Y)) y (@fun_of_setfun elementType eqType_of_elementType disp (@Semiset.Pack elementType eqType_of_elementType disp S (@Semiset.Class elementType eqType_of_elementType disp S base (@Semiset.Mixin elementType eqType_of_elementType disp (fun X : elementType => @Order.CBLattice.Pack (display_set disp) (S X) (base X) (S X)) memset set1 _i_ _e_ _e1_ _s_ _i1_ _e2_ _funsort_ _fun_of_funsort_ _imset_ H)) _T_) X Y f x))) (memset Y (_imset_ X Y f A) y) ) exact: H. Qed. Lemma set_gt0_ex A : set0 \proper A -> {x \| x \in A}. Proof. ( Goal: @eq bool (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) (@join (display_set disp) (@Semiset.latticeType elementType eqType_of_elementType X disp set) A B))) (orb (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) A)) (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) B))) ) rewrite /set1 /in_mem /= /memset. ( Goal: forall (A : @Semiset.sort elementType eqType_of_elementType disp set X) (f : @Semiset.funsort elementType eqType_of_elementType disp (fun X : elementType => @Order.CBLattice.Pack (display_set disp) (@Semiset.sort elementType eqType_of_elementType disp set X) (@Semiset.base elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set) X) (@Semiset.sort elementType eqType_of_elementType disp set X)) (@Semiset.mixin elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set)) X Y), Bool.reflect (@ex2 (Equality.sort (eqType_of_elementType X)) (fun x : Equality.sort (eqType_of_elementType X) => is_true (@Semiset.memset elementType eqType_of_elementType disp (fun X : elementType => @Order.CBLattice.Pack (display_set disp) (@Semiset.sort elementType eqType_of_elementType disp set X) (@Semiset.base elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set) X) (@Semiset.sort elementType eqType_of_elementType disp set X)) (@Semiset.mixin elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set)) X A x)) (fun x : Equality.sort (eqType_of_elementType X) => @eq (Equality.sort (eqType_of_elementType Y)) y (@fun_of_setfun elementType eqType_of_elementType disp set X Y f x))) (@Semiset.memset elementType eqType_of_elementType disp (fun X : elementType => @Order.CBLattice.Pack (display_set disp) (@Semiset.sort elementType eqType_of_elementType disp set X) (@Semiset.base elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set) X) (@Semiset.sort elementType eqType_of_elementType disp set X)) (@Semiset.mixin elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set)) Y (@Semiset.imset elementType eqType_of_elementType disp (fun X : elementType => @Order.CBLattice.Pack (display_set disp) (@Semiset.sort elementType eqType_of_elementType disp set X) (@Semiset.base elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set) X) (@Semiset.sort elementType eqType_of_elementType disp set X)) (@Semiset.mixin elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set)) X Y f A) y) ) case: set A => [S [base [memset set1 /= ? ? ? H ? ? ? ? ? ?]] ?] A /=. ( Goal: Bool.reflect (@ex2 (Equality.sort (eqType_of_elementType X)) (fun x : Equality.sort (eqType_of_elementType X) => is_true (memset X A x)) (fun x : Equality.sort (eqType_of_elementType X) => @eq (Equality.sort (eqType_of_elementType Y)) y (@fun_of_setfun elementType eqType_of_elementType disp (@Semiset.Pack elementType eqType_of_elementType disp S (@Semiset.Class elementType eqType_of_elementType disp S base (@Semiset.Mixin elementType eqType_of_elementType disp (fun X : elementType => @Order.CBLattice.Pack (display_set disp) (S X) (base X) (S X)) memset set1 _i_ _e_ _e1_ _s_ _i1_ _e2_ _funsort_ _fun_of_funsort_ _imset_ H)) _T_) X Y f x))) (memset Y (_imset_ X Y f A) y) ) exact: H. Qed. Lemma subsetP_subproof A B : {subset A <= B} -> A \subset B. Proof. ( Goal: @eq bool (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) (@join (display_set disp) (@Semiset.latticeType elementType eqType_of_elementType X disp set) A B))) (orb (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) A)) (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) B))) ) rewrite /set1 /in_mem /= /memset. ( Goal: forall (A : @Semiset.sort elementType eqType_of_elementType disp set X) (f : @Semiset.funsort elementType eqType_of_elementType disp (fun X : elementType => @Order.CBLattice.Pack (display_set disp) (@Semiset.sort elementType eqType_of_elementType disp set X) (@Semiset.base elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set) X) (@Semiset.sort elementType eqType_of_elementType disp set X)) (@Semiset.mixin elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set)) X Y), Bool.reflect (@ex2 (Equality.sort (eqType_of_elementType X)) (fun x : Equality.sort (eqType_of_elementType X) => is_true (@Semiset.memset elementType eqType_of_elementType disp (fun X : elementType => @Order.CBLattice.Pack (display_set disp) (@Semiset.sort elementType eqType_of_elementType disp set X) (@Semiset.base elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set) X) (@Semiset.sort elementType eqType_of_elementType disp set X)) (@Semiset.mixin elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set)) X A x)) (fun x : Equality.sort (eqType_of_elementType X) => @eq (Equality.sort (eqType_of_elementType Y)) y (@fun_of_setfun elementType eqType_of_elementType disp set X Y f x))) (@Semiset.memset elementType eqType_of_elementType disp (fun X : elementType => @Order.CBLattice.Pack (display_set disp) (@Semiset.sort elementType eqType_of_elementType disp set X) (@Semiset.base elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set) X) (@Semiset.sort elementType eqType_of_elementType disp set X)) (@Semiset.mixin elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set)) Y (@Semiset.imset elementType eqType_of_elementType disp (fun X : elementType => @Order.CBLattice.Pack (display_set disp) (@Semiset.sort elementType eqType_of_elementType disp set X) (@Semiset.base elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set) X) (@Semiset.sort elementType eqType_of_elementType disp set X)) (@Semiset.mixin elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set)) X Y f A) y) ) case: set A B => [S [base [memset set1 /= ? ? ? ? H ? ? ? ? ?]] ?] /=. ( Goal: Bool.reflect (@ex2 (Equality.sort (eqType_of_elementType X)) (fun x : Equality.sort (eqType_of_elementType X) => is_true (memset X A x)) (fun x : Equality.sort (eqType_of_elementType X) => @eq (Equality.sort (eqType_of_elementType Y)) y (@fun_of_setfun elementType eqType_of_elementType disp (@Semiset.Pack elementType eqType_of_elementType disp S (@Semiset.Class elementType eqType_of_elementType disp S base (@Semiset.Mixin elementType eqType_of_elementType disp (fun X : elementType => @Order.CBLattice.Pack (display_set disp) (S X) (base X) (S X)) memset set1 _i_ _e_ _e1_ _s_ _i1_ _e2_ _funsort_ _fun_of_funsort_ _imset_ H)) _T_) X Y f x))) (memset Y (_imset_ X Y f A) y) ) exact: H. Qed. Lemma in_setU (x : X) A B : (x \in A :\|: B) = (x \in A) \|\| (x \in B). Proof. ( Goal: @eq bool (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) (@join (display_set disp) (@Semiset.latticeType elementType eqType_of_elementType X disp set) A B))) (orb (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) A)) (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) B))) ) rewrite /set1 /in_mem /= /memset. ( Goal: forall (A : @Semiset.sort elementType eqType_of_elementType disp set X) (f : @Semiset.funsort elementType eqType_of_elementType disp (fun X : elementType => @Order.CBLattice.Pack (display_set disp) (@Semiset.sort elementType eqType_of_elementType disp set X) (@Semiset.base elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set) X) (@Semiset.sort elementType eqType_of_elementType disp set X)) (@Semiset.mixin elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set)) X Y), Bool.reflect (@ex2 (Equality.sort (eqType_of_elementType X)) (fun x : Equality.sort (eqType_of_elementType X) => is_true (@Semiset.memset elementType eqType_of_elementType disp (fun X : elementType => @Order.CBLattice.Pack (display_set disp) (@Semiset.sort elementType eqType_of_elementType disp set X) (@Semiset.base elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set) X) (@Semiset.sort elementType eqType_of_elementType disp set X)) (@Semiset.mixin elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set)) X A x)) (fun x : Equality.sort (eqType_of_elementType X) => @eq (Equality.sort (eqType_of_elementType Y)) y (@fun_of_setfun elementType eqType_of_elementType disp set X Y f x))) (@Semiset.memset elementType eqType_of_elementType disp (fun X : elementType => @Order.CBLattice.Pack (display_set disp) (@Semiset.sort elementType eqType_of_elementType disp set X) (@Semiset.base elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set) X) (@Semiset.sort elementType eqType_of_elementType disp set X)) (@Semiset.mixin elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set)) Y (@Semiset.imset elementType eqType_of_elementType disp (fun X : elementType => @Order.CBLattice.Pack (display_set disp) (@Semiset.sort elementType eqType_of_elementType disp set X) (@Semiset.base elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set) X) (@Semiset.sort elementType eqType_of_elementType disp set X)) (@Semiset.mixin elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set)) X Y f A) y) ) case: set A B => [S [base [memset set1 /= ? ? ? ? ? H ? ? ? ?]] ?] /=. ( Goal: Bool.reflect (@ex2 (Equality.sort (eqType_of_elementType X)) (fun x : Equality.sort (eqType_of_elementType X) => is_true (memset X A x)) (fun x : Equality.sort (eqType_of_elementType X) => @eq (Equality.sort (eqType_of_elementType Y)) y (@fun_of_setfun elementType eqType_of_elementType disp (@Semiset.Pack elementType eqType_of_elementType disp S (@Semiset.Class elementType eqType_of_elementType disp S base (@Semiset.Mixin elementType eqType_of_elementType disp (fun X : elementType => @Order.CBLattice.Pack (display_set disp) (S X) (base X) (S X)) memset set1 _i_ _e_ _e1_ _s_ _i1_ _e2_ _funsort_ _fun_of_funsort_ _imset_ H)) _T_) X Y f x))) (memset Y (_imset_ X Y f A) y) ) exact: H. Qed. Lemma in_set0 x : x \in (set0 : set X) = false. Proof. by rewrite (negPf (notin_set0 _)). Qed. Lemma subsetP {A B} : reflect {subset A <= B} (A <= B)%O. Proof. ( Goal: is_true (negb (@le (display_set disp) (@Semiset.porderType elementType eqType_of_elementType Y disp set) (@imset elementType eqType_of_elementType disp set X Y f A') (@imset elementType eqType_of_elementType disp set X Y f A))) ) apply: (iffP idP) => [sAB x xA\|/subsetP_subproof//]. ( Goal: forall _ : is_true (@in_mem (Equality.sort (eqType_of_elementType Y)) (@fun_of_setfun elementType eqType_of_elementType disp set X Y f x) (@mem (Equality.sort (eqType_of_elementType Y)) (@set_predType elementType eqType_of_elementType disp set Y) (@imset elementType eqType_of_elementType disp set X Y f A))), is_true (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) A)) ) by rewrite -sub1set (le_trans _ sAB) // sub1set. Qed. Lemma setP A B : A =i B <-> A = B. Proof. ( Goal: iff (@eq_mem (Equality.sort (eqType_of_elementType X)) (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) A) (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) B)) (@eq (@Semiset.sort elementType eqType_of_elementType disp set X) A B) ) split=> [eqAB\|->//]; apply/eqP; rewrite eq_le. ( Goal: @eq bool (@eq_op (@Semiset.eqType elementType eqType_of_elementType Y disp set) (@imset elementType eqType_of_elementType disp set X Y f A) (@bottom (display_set disp) (@Semiset.blatticeType elementType eqType_of_elementType Y disp set))) (@eq_op (@Semiset.eqType elementType eqType_of_elementType X disp set) A (@bottom (display_set disp) (@Semiset.blatticeType elementType eqType_of_elementType X disp set))) ) gen have leAB : A B eqAB / A \subset B; last by rewrite !leAB. ( Goal: forall _ : is_true (@in_mem (Equality.sort (eqType_of_elementType Y)) (@fun_of_setfun elementType eqType_of_elementType disp set X Y f x) (@mem (Equality.sort (eqType_of_elementType Y)) (@set_predType elementType eqType_of_elementType disp set Y) (@imset elementType eqType_of_elementType disp set X Y f A))), is_true (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) A)) ) by apply/subsetP => x; rewrite eqAB. Qed. Lemma set1_neq0 (x : X) : [set x] != set0 :> set X. Proof. by apply/negP=> /eqP /setP /(_ x); rewrite in_set0 in_set1 eqxx. Qed. Lemma set1_eq0 x : ([set x] == set0 :> set X) = false. Proof. by rewrite (negPf (set1_neq0 _)). Qed. Lemma set11 x : x \in ([set x] : set X). Proof. by rewrite -sub1set. Qed. Hint Resolve set11. Lemma set1_inj : injective (@set1 _ _ _ set X). Proof. ( Goal: @injective (@Semiset.sort elementType eqType_of_elementType disp set X) (Equality.sort (eqType_of_elementType X)) (@set1 elementType eqType_of_elementType disp set X) ) move=> x y /eqP; rewrite eq_le sub1set => /andP []. ( Goal: forall _ : is_true (@in_mem (Equality.sort (eqType_of_elementType Y)) (@fun_of_setfun elementType eqType_of_elementType disp set X Y f x) (@mem (Equality.sort (eqType_of_elementType Y)) (@set_predType elementType eqType_of_elementType disp set Y) (@imset elementType eqType_of_elementType disp set X Y f A))), is_true (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) A)) ) by rewrite in_set1 => /eqP. Qed. Lemma set_0Vmem A : (A = set0) + {x : X \| x \in A}. Proof. ( Goal: sum (@eq (@Semiset.sort elementType eqType_of_elementType disp set X) A (@bottom (display_set disp) (@Semiset.blatticeType elementType eqType_of_elementType X disp set))) (@sig (Equality.sort (eqType_of_elementType X)) (fun x : Equality.sort (eqType_of_elementType X) => is_true (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) A)))) ) have [\|AN0] := eqVneq A set0; [left\|right] => //. ( Goal: forall _ : is_true (@in_mem (Equality.sort (eqType_of_elementType Y)) (@fun_of_setfun elementType eqType_of_elementType disp set X Y f x) (@mem (Equality.sort (eqType_of_elementType Y)) (@set_predType elementType eqType_of_elementType disp set Y) (@imset elementType eqType_of_elementType disp set X Y f A))), is_true (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) A)) ) by move: AN0; rewrite -lt0x => /set_gt0_ex. Qed. Lemma set0Pn A : reflect (exists x, x \in A) (A != set0). Proof. ( Goal: forall _ : is_true (@le (display_set disp) (@Semiset.porderType elementType eqType_of_elementType X disp set) A (@set1 elementType eqType_of_elementType disp set X x)), @eq bool (orb (@le (display_set disp) (@Semiset.porderType elementType eqType_of_elementType X disp set) (@set1 elementType eqType_of_elementType disp set X x) A) (@eq_op (@Semiset.eqType elementType eqType_of_elementType X disp set) A (@bottom (display_set disp) (@Semiset.blatticeType elementType eqType_of_elementType X disp set)))) true ) have [->\|[x xA]] := set_0Vmem A; rewrite ?eqxx -?lt0x. ( Goal: forall _ : is_true (@in_mem (Equality.sort (eqType_of_elementType Y)) (@fun_of_setfun elementType eqType_of_elementType disp set X Y f x) (@mem (Equality.sort (eqType_of_elementType Y)) (@set_predType elementType eqType_of_elementType disp set Y) (@imset elementType eqType_of_elementType disp set X Y f A))), is_true (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) A)) ) by constructor=> [[x]]; rewrite in_set0. ( Goal: Bool.reflect (@ex (Equality.sort (eqType_of_elementType X)) (fun x : Equality.sort (eqType_of_elementType X) => is_true (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) A)))) (@lt (display_set disp) (@Order.BLattice.porderType (display_set disp) (@Semiset.blatticeType elementType eqType_of_elementType X disp set)) (@bottom (display_set disp) (@Semiset.blatticeType elementType eqType_of_elementType X disp set)) A) ) suff -> : set0 \proper A by constructor; exists x. ( Goal: forall _ : is_true (@in_mem (Equality.sort (eqType_of_elementType Y)) (@fun_of_setfun elementType eqType_of_elementType disp set X Y f x) (@mem (Equality.sort (eqType_of_elementType Y)) (@set_predType elementType eqType_of_elementType disp set Y) (@imset elementType eqType_of_elementType disp set X Y f A))), is_true (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) A)) ) by move: xA; rewrite -sub1set => /(lt_le_trans _)->; rewrite ?lt0x ?set1_eq0. Qed. Lemma subset1 A x : (A \subset [set x]) = (A == [set x]) \|\| (A == set0). Proof. ( Goal: @eq bool (@eq_op (@Semiset.eqType elementType eqType_of_elementType Y disp set) (@imset elementType eqType_of_elementType disp set X Y f A) (@bottom (display_set disp) (@Semiset.blatticeType elementType eqType_of_elementType Y disp set))) (@eq_op (@Semiset.eqType elementType eqType_of_elementType X disp set) A (@bottom (display_set disp) (@Semiset.blatticeType elementType eqType_of_elementType X disp set))) ) symmetry; rewrite eq_le; have [] /= := boolP (A \subset [set x]); last first. ( Goal: forall _ : is_true (@in_mem (Equality.sort (eqType_of_elementType Y)) (@fun_of_setfun elementType eqType_of_elementType disp set X Y f x) (@mem (Equality.sort (eqType_of_elementType Y)) (@set_predType elementType eqType_of_elementType disp set Y) (@imset elementType eqType_of_elementType disp set X Y f A))), is_true (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) A)) ) by apply: contraNF => /eqP ->; rewrite ?le0x. ( Goal: forall _ : is_true (@le (display_set disp) (@Semiset.porderType elementType eqType_of_elementType X disp set) A (@set1 elementType eqType_of_elementType disp set X x)), @eq bool (orb (@le (display_set disp) (@Semiset.porderType elementType eqType_of_elementType X disp set) (@set1 elementType eqType_of_elementType disp set X x) A) (@eq_op (@Semiset.eqType elementType eqType_of_elementType X disp set) A (@bottom (display_set disp) (@Semiset.blatticeType elementType eqType_of_elementType X disp set)))) true ) have [/eqP->\|[y yA]] := set_0Vmem A; rewrite ?orbT // ?sub1set. ( Goal: forall _ : is_true (@in_mem (Equality.sort (eqType_of_elementType Y)) (@fun_of_setfun elementType eqType_of_elementType disp set X Y f x) (@mem (Equality.sort (eqType_of_elementType Y)) (@set_predType elementType eqType_of_elementType disp set Y) (@imset elementType eqType_of_elementType disp set X Y f A))), is_true (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) A)) ) by move=> /subsetP /(_ _ yA); rewrite in_set1 => /eqP<-; rewrite yA. Qed. Lemma eq_set1 (x : X) A : (A == [set x]) = (set0 \proper A \subset [set x]). Proof. ( Goal: forall _ : is_true (@in_mem (Equality.sort (eqType_of_elementType Y)) (@fun_of_setfun elementType eqType_of_elementType disp set X Y f x) (@mem (Equality.sort (eqType_of_elementType Y)) (@set_predType elementType eqType_of_elementType disp set Y) (@imset elementType eqType_of_elementType disp set X Y f A))), is_true (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) A)) ) by rewrite subset1; have [->\|?] := posxP A; rewrite 1?eq_sym ?set1_eq0 ?orbF. Qed. Lemma in_setI A B (x : X) : (x \in A :&: B) = (x \in A) && (x \in B). Proof. ( Goal: @eq bool (@eq_op (@Semiset.eqType elementType eqType_of_elementType Y disp set) (@imset elementType eqType_of_elementType disp set X Y f A) (@bottom (display_set disp) (@Semiset.blatticeType elementType eqType_of_elementType Y disp set))) (@eq_op (@Semiset.eqType elementType eqType_of_elementType X disp set) A (@bottom (display_set disp) (@Semiset.blatticeType elementType eqType_of_elementType X disp set))) ) apply/idP/idP => [xAB\|?]; last by rewrite -sub1set lexI !sub1set. ( Goal: forall _ : is_true (@in_mem (Equality.sort (eqType_of_elementType Y)) (@fun_of_setfun elementType eqType_of_elementType disp set X Y f x) (@mem (Equality.sort (eqType_of_elementType Y)) (@set_predType elementType eqType_of_elementType disp set Y) (@imset elementType eqType_of_elementType disp set X Y f A))), is_true (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) A)) ) by rewrite (subsetP (leIr _ _) _ xAB) (subsetP (leIl _ _) _ xAB). Qed. Lemma set1U A x : [set x] :&: A = if x \in A then [set x] else set0. Proof. ( Goal: @eq (@Semiset.sort elementType eqType_of_elementType disp set Y) (@imset elementType eqType_of_elementType disp set X Y f (@bottom (display_set disp) (@Semiset.blatticeType elementType eqType_of_elementType X disp set))) (@bottom (display_set disp) (@Semiset.blatticeType elementType eqType_of_elementType Y disp set)) ) apply/setP => y; rewrite (fun_if (fun E => y \in E)) in_setI in_set1 in_set0. ( Goal: forall _ : is_true (@in_mem (Equality.sort (eqType_of_elementType Y)) (@fun_of_setfun elementType eqType_of_elementType disp set X Y f x) (@mem (Equality.sort (eqType_of_elementType Y)) (@set_predType elementType eqType_of_elementType disp set Y) (@imset elementType eqType_of_elementType disp set X Y f A))), is_true (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) A)) ) by have [->\|] := altP (y =P x); rewrite ?if_same //; case: (_ \in A). Qed. Lemma set1U_eq0 A x : ([set x] :&: A == set0) = (x \notin A). Proof. by rewrite set1U; case: (x \in A); rewrite ?set1_eq0 ?eqxx. Qed. Lemma in_setD A B x : (x \in A :\: B) = (x \notin B) && (x \in A). Proof. ( Goal: @eq bool (@eq_op (@Semiset.eqType elementType eqType_of_elementType Y disp set) (@imset elementType eqType_of_elementType disp set X Y f A) (@bottom (display_set disp) (@Semiset.blatticeType elementType eqType_of_elementType Y disp set))) (@eq_op (@Semiset.eqType elementType eqType_of_elementType X disp set) A (@bottom (display_set disp) (@Semiset.blatticeType elementType eqType_of_elementType X disp set))) ) apply/idP/idP => [\|/andP[xNB xA]]; last first. ( Goal: forall _ : is_true (@in_mem (Equality.sort (eqType_of_elementType Y)) (@fun_of_setfun elementType eqType_of_elementType disp set X Y f x) (@mem (Equality.sort (eqType_of_elementType Y)) (@set_predType elementType eqType_of_elementType disp set Y) (@imset elementType eqType_of_elementType disp set X Y f A))), is_true (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) A)) ) by rewrite -sub1set leBRL sub1set xA set1U_eq0. ( Goal: sum (@eq (@Semiset.sort elementType eqType_of_elementType disp set X) A (@bottom (display_set disp) (@Semiset.blatticeType elementType eqType_of_elementType X disp set))) (@sig (Equality.sort (eqType_of_elementType X)) (fun x : Equality.sort (eqType_of_elementType X) => is_true (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) A)))) ) rewrite -sub1set leBRL sub1set => /andP [-> dxB]. ( Goal: forall _ : is_true (@in_mem (Equality.sort (eqType_of_elementType Y)) (@fun_of_setfun elementType eqType_of_elementType disp set X Y f x) (@mem (Equality.sort (eqType_of_elementType Y)) (@set_predType elementType eqType_of_elementType disp set Y) (@imset elementType eqType_of_elementType disp set X Y f A))), is_true (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) A)) ) by rewrite -sub1set disj_le ?set1_eq0. Qed. Definition inE := ((in_set0, in_set1, in_setU, in_setI, in_setD), inE). Definition subset_trans B A C := (@le_trans _ _ B A C). Definition proper_trans B A C := (@lt_trans _ _ B A C). Definition sub_proper_trans B A C := (@le_lt_trans _ _ B A C). Definition proper_sub_trans B A C := (@lt_le_trans _ _ B A C). Definition proper_sub A B := (@ltW _ _ A B). Lemma properP A B : reflect (A \subset B /\ (exists2 x, x \in B & x \notin A)) (A \proper B). Proof. ( Goal: is_true (negb (@le (display_set disp) (@Semiset.porderType elementType eqType_of_elementType Y disp set) (@imset elementType eqType_of_elementType disp set X Y f A') (@imset elementType eqType_of_elementType disp set X Y f A))) ) apply: (iffP idP)=> [ltAB\|[leAB [x xB xNA]]]. ( Goal: and (is_true (@le (display_set disp) (@Semiset.porderType elementType eqType_of_elementType X disp set) A B)) (@ex2 (Equality.sort (eqType_of_elementType X)) (fun x : Equality.sort (eqType_of_elementType X) => is_true (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) B))) (fun x : Equality.sort (eqType_of_elementType X) => is_true (negb (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) A))))) ) ( Goal: is_true (@lt (display_set disp) (@Semiset.porderType elementType eqType_of_elementType X disp set) A B) ) rewrite ltW //; split => //; have := lt0B ltAB; rewrite lt0x. ( Goal: forall _ : is_true (@in_mem (Equality.sort (eqType_of_elementType Y)) (@fun_of_setfun elementType eqType_of_elementType disp set X Y f x) (@mem (Equality.sort (eqType_of_elementType Y)) (@set_predType elementType eqType_of_elementType disp set Y) (@imset elementType eqType_of_elementType disp set X Y f A))), is_true (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) A)) ) by move => /set0Pn [x]; rewrite in_setD => /andP [xNA xB]; exists x. ( Goal: is_true (@lt (display_set disp) (@Semiset.porderType elementType eqType_of_elementType X disp set) A B) ) rewrite lt_neqAle leAB andbT; apply: contraTneq xNA. ( Goal: forall _ : is_true (@in_mem (Equality.sort (eqType_of_elementType Y)) (@fun_of_setfun elementType eqType_of_elementType disp set X Y f x) (@mem (Equality.sort (eqType_of_elementType Y)) (@set_predType elementType eqType_of_elementType disp set Y) (@imset elementType eqType_of_elementType disp set X Y f A))), is_true (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) A)) ) by move=> /setP /(_ x) ->; rewrite xB. Qed. Lemma set1P x y : reflect (x = y) (x \in ([set y] : set X)). Proof. by rewrite in_set1; apply/eqP. Qed. Lemma subset_eqP A B : reflect (A =i B) (A \subset B \subset A)%set. Proof. ( Goal: @eq bool (@eq_op (@Semiset.eqType elementType eqType_of_elementType Y disp set) (@imset elementType eqType_of_elementType disp set X Y f A) (@bottom (display_set disp) (@Semiset.blatticeType elementType eqType_of_elementType Y disp set))) (@eq_op (@Semiset.eqType elementType eqType_of_elementType X disp set) A (@bottom (display_set disp) (@Semiset.blatticeType elementType eqType_of_elementType X disp set))) ) apply: (iffP andP) => [[AB BA] x\|eqAB]; first by apply/idP/idP; apply: subsetP. ( Goal: forall _ : is_true (@in_mem (Equality.sort (eqType_of_elementType Y)) (@fun_of_setfun elementType eqType_of_elementType disp set X Y f x) (@mem (Equality.sort (eqType_of_elementType Y)) (@set_predType elementType eqType_of_elementType disp set Y) (@imset elementType eqType_of_elementType disp set X Y f A))), is_true (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) A)) ) by split; apply/subsetP=> x; rewrite !eqAB. Qed. Lemma eqEsubset A B : (A == B) = (A \subset B) && (B \subset A). Proof. exact: eq_le. Qed. Lemma properE A B : A \proper B = (A \subset B) && ~~ (B \subset A). Proof. by case: comparableP. Qed. Lemma subEproper A B : A \subset B = (A == B) \|\| (A \proper B). Proof. exact: le_eqVlt. Qed. Lemma eqVproper A B : A \subset B -> A = B \/ A \proper B. Proof. by rewrite subEproper => /predU1P. Qed. Lemma properEneq A B : A \proper B = (A != B) && (A \subset B). Proof. exact: lt_neqAle. Qed. Lemma proper_neq A B : A \proper B -> A != B. Proof. by rewrite properEneq; case/andP. Qed. Lemma eqEproper A B : (A == B) = (A \subset B) && ~~ (A \proper B). Proof. by case: comparableP. Qed. Lemma sub0set A : set0 \subset A. Proof. by apply/subsetP=> x; rewrite inE. Qed. Lemma subset0 A : (A \subset set0) = (A == set0). Proof. by rewrite eqEsubset sub0set andbT. Qed. Lemma proper0 A : (set0 \proper A) = (A != set0). Proof. by rewrite properE sub0set subset0. Qed. Lemma subset_neq0 A B : A \subset B -> A != set0 -> B != set0. Proof. by rewrite -!proper0 => sAB /proper_sub_trans->. Qed. Lemma setU1r x a B : x \in B -> x \in a \|: B. Proof. by move=> Bx; rewrite !inE predU1r. Qed. Lemma setU1P x a B : reflect (x = a \/ x \in B) (x \in a \|: B). Proof. by rewrite !inE; apply: predU1P. Qed. ( We need separate lemmas for the explicit enumerations since they ) ( associate on the left. ) Lemma set1Ul x A b : x \in A -> x \in A :\|: [set b]. Proof. by move=> Ax; rewrite !inE Ax. Qed. Lemma set1Ur A b : b \in A :\|: [set b]. Proof. by rewrite !inE eqxx orbT. Qed. Lemma setD1P x A b : reflect (x != b /\ x \in A) (x \in A :\ b). Proof. by rewrite !inE; apply: andP. Qed. Lemma in_setD1 x A b : (x \in A :\ b) = (x != b) && (x \in A) . Proof. by rewrite !inE. Qed. Lemma setD11 b A : (b \in A :\ b) = false. Proof. by rewrite !inE eqxx. Qed. Lemma setD1K a A : a \in A -> a \|: (A :\ a) = A. Proof. by move=> Aa; apply/setP=> x; rewrite !inE; case: eqP => // ->. Qed. Lemma setU1K a B : a \notin B -> (a \|: B) :\ a = B. Proof. ( Goal: forall _ : is_true (@in_mem (Equality.sort (eqType_of_elementType Y)) (@fun_of_setfun elementType eqType_of_elementType disp set X Y f x) (@mem (Equality.sort (eqType_of_elementType Y)) (@set_predType elementType eqType_of_elementType disp set Y) (@imset elementType eqType_of_elementType disp set X Y f A))), is_true (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) A)) ) by move/negPf=> nBa; apply/setP=> x; rewrite !inE; case: eqP => // ->. Qed. Lemma set2P x a b : reflect (x = a \/ x = b) (x \in ([set a; b] : set X)). Proof. by rewrite !inE; apply: pred2P. Qed. Lemma in_set2 x a b : (x \in ([set a; b] : set X)) = (x == a) \|\| (x == b). Proof. by rewrite !inE. Qed. Lemma set21 a b : a \in ([set a; b] : set X). Proof. by rewrite !inE eqxx. Qed. Lemma set22 a b : b \in ([set a; b] : set X). Proof. by rewrite !inE eqxx orbT. Qed. Lemma setUP x A B : reflect (x \in A \/ x \in B) (x \in A :\|: B). Proof. by rewrite !inE; apply: orP. Qed. Lemma setUC A B : A :\|: B = B :\|: A. Proof. by apply/setP => x; rewrite !inE orbC. Qed. Lemma setUS A B C : A \subset B -> C :\|: A \subset C :\|: B. Proof. ( Goal: forall _ : is_true (@le (display_set disp) (@Semiset.porderType elementType eqType_of_elementType X disp set) A B), is_true (@le (display_set disp) (@Order.Lattice.porderType (display_set disp) (@Semiset.latticeType elementType eqType_of_elementType X disp set)) (@meet (display_set disp) (@Semiset.latticeType elementType eqType_of_elementType X disp set) C A) (@meet (display_set disp) (@Semiset.latticeType elementType eqType_of_elementType X disp set) C B)) ) move=> sAB; apply/subsetP=> x; rewrite !inE. ( Goal: forall _ : is_true (@in_mem (Equality.sort (eqType_of_elementType Y)) (@fun_of_setfun elementType eqType_of_elementType disp set X Y f x) (@mem (Equality.sort (eqType_of_elementType Y)) (@set_predType elementType eqType_of_elementType disp set Y) (@imset elementType eqType_of_elementType disp set X Y f A))), is_true (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) A)) ) by case: (x \in C) => //; apply: (subsetP sAB). Qed. Lemma setSU A B C : A \subset B -> A :\|: C \subset B :\|: C. Proof. by move=> sAB; rewrite -!(setUC C) setUS. Qed. Lemma setUSS A B C D : A \subset C -> B \subset D -> A :\|: B \subset C :\|: D. Proof. by move=> /(setSU B) /subset_trans sAC /(setUS C)/sAC. Qed. Lemma set0U A : set0 :\|: A = A. Proof. by apply/setP => x; rewrite !inE orFb. Qed. Lemma setU0 A : A :\|: set0 = A. Proof. by rewrite setUC set0U. Qed. Lemma setUA A B C : A :\|: (B :\|: C) = A :\|: B :\|: C. Proof. by apply/setP => x; rewrite !inE orbA. Qed. Lemma setUCA A B C : A :\|: (B :\|: C) = B :\|: (A :\|: C). Proof. by rewrite !setUA (setUC B). Qed. Lemma setUAC A B C : A :\|: B :\|: C = A :\|: C :\|: B. Proof. by rewrite -!setUA (setUC B). Qed. Lemma setUACA A B C D : (A :\|: B) :\|: (C :\|: D) = (A :\|: C) :\|: (B :\|: D). Proof. by rewrite -!setUA (setUCA B). Qed. Lemma setUid A : A :\|: A = A. Proof. by apply/setP=> x; rewrite inE orbb. Qed. Lemma setUUl A B C : A :\|: B :\|: C = (A :\|: C) :\|: (B :\|: C). Proof. by rewrite setUA !(setUAC _ C) -(setUA _ C) setUid. Qed. Lemma setUUr A B C : A :\|: (B :\|: C) = (A :\|: B) :\|: (A :\|: C). Proof. by rewrite !(setUC A) setUUl. Qed. ( intersection ) Lemma setIP x A B : reflect (x \in A /\ x \in B) (x \in A :&: B). Proof. by rewrite !inE; apply: andP. Qed. Lemma setIC A B : A :&: B = B :&: A. Proof. by apply/setP => x; rewrite !inE andbC. Qed. Lemma setIS A B C : A \subset B -> C :&: A \subset C :&: B. Proof. ( Goal: forall _ : is_true (@le (display_set disp) (@Semiset.porderType elementType eqType_of_elementType X disp set) A B), is_true (@le (display_set disp) (@Order.Lattice.porderType (display_set disp) (@Semiset.latticeType elementType eqType_of_elementType X disp set)) (@meet (display_set disp) (@Semiset.latticeType elementType eqType_of_elementType X disp set) C A) (@meet (display_set disp) (@Semiset.latticeType elementType eqType_of_elementType X disp set) C B)) ) move=> sAB; apply/subsetP=> x; rewrite !inE. ( Goal: forall _ : is_true (@in_mem (Equality.sort (eqType_of_elementType Y)) (@fun_of_setfun elementType eqType_of_elementType disp set X Y f x) (@mem (Equality.sort (eqType_of_elementType Y)) (@set_predType elementType eqType_of_elementType disp set Y) (@imset elementType eqType_of_elementType disp set X Y f A))), is_true (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) A)) ) by case: (x \in C) => //; apply: (subsetP sAB). Qed. Lemma setSI A B C : A \subset B -> A :&: C \subset B :&: C. Proof. by move=> sAB; rewrite -!(setIC C) setIS. Qed. Lemma setISS A B C D : A \subset C -> B \subset D -> A :&: B \subset C :&: D. Proof. by move=> /(setSI B) /subset_trans sAC /(setIS C) /sAC. Qed. Lemma set0I A : set0 :&: A = set0. Proof. by apply/setP => x; rewrite !inE andFb. Qed. Lemma setI0 A : A :&: set0 = set0. Proof. by rewrite setIC set0I. Qed. Lemma setIA A B C : A :&: (B :&: C) = A :&: B :&: C. Proof. by apply/setP=> x; rewrite !inE andbA. Qed. Lemma setICA A B C : A :&: (B :&: C) = B :&: (A :&: C). Proof. by rewrite !setIA (setIC A). Qed. Lemma setIAC A B C : A :&: B :&: C = A :&: C :&: B. Proof. by rewrite -!setIA (setIC B). Qed. Lemma setIACA A B C D : (A :&: B) :&: (C :&: D) = (A :&: C) :&: (B :&: D). Proof. by rewrite -!setIA (setICA B). Qed. Lemma setIid A : A :&: A = A. Proof. by apply/setP=> x; rewrite inE andbb. Qed. Lemma setIIl A B C : A :&: B :&: C = (A :&: C) :&: (B :&: C). Proof. by rewrite setIA !(setIAC _ C) -(setIA _ C) setIid. Qed. Lemma setIIr A B C : A :&: (B :&: C) = (A :&: B) :&: (A :&: C). Proof. by rewrite !(setIC A) setIIl. Qed. ( distribute /cancel ) Lemma setIUr A B C : A :&: (B :\|: C) = (A :&: B) :\|: (A :&: C). Proof. by apply/setP=> x; rewrite !inE andb_orr. Qed. Lemma setIUl A B C : (A :\|: B) :&: C = (A :&: C) :\|: (B :&: C). Proof. by apply/setP=> x; rewrite !inE andb_orl. Qed. Lemma setUIr A B C : A :\|: (B :&: C) = (A :\|: B) :&: (A :\|: C). Proof. by apply/setP=> x; rewrite !inE orb_andr. Qed. Lemma setUIl A B C : (A :&: B) :\|: C = (A :\|: C) :&: (B :\|: C). Proof. by apply/setP=> x; rewrite !inE orb_andl. Qed. Lemma setUK A B : (A :\|: B) :&: A = A. Proof. by apply/setP=> x; rewrite !inE orbK. Qed. Lemma setKU A B : A :&: (B :\|: A) = A. Proof. by apply/setP=> x; rewrite !inE orKb. Qed. Lemma setIK A B : (A :&: B) :\|: A = A. Proof. by apply/setP=> x; rewrite !inE andbK. Qed. Lemma setKI A B : A :\|: (B :&: A) = A. Proof. by apply/setP=> x; rewrite !inE andKb. Qed. ( difference ) Lemma setDP A B x : reflect (x \in A /\ x \notin B) (x \in A :\: B). Proof. by rewrite inE andbC; apply: andP. Qed. Lemma setSD A B C : A \subset B -> A :\: C \subset B :\: C. Proof. ( Goal: forall _ : is_true (@in_mem (Equality.sort (eqType_of_elementType Y)) (@fun_of_setfun elementType eqType_of_elementType disp set X Y f x) (@mem (Equality.sort (eqType_of_elementType Y)) (@set_predType elementType eqType_of_elementType disp set Y) (@imset elementType eqType_of_elementType disp set X Y f A))), is_true (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) A)) ) by move=> /subsetP AB; apply/subsetP => x; rewrite !inE => /andP[-> /AB]. Qed. Lemma setDS A B C : A \subset B -> C :\: B \subset C :\: A. Proof. ( Goal: forall _ : is_true (@le (display_set disp) (@Semiset.porderType elementType eqType_of_elementType X disp set) A B), is_true (@le (display_set disp) (@Order.CBLattice.porderType (display_set disp) (@Semiset.cblatticeType elementType eqType_of_elementType X disp set)) (@sub (display_set disp) (@Semiset.cblatticeType elementType eqType_of_elementType X disp set) C B) (@sub (display_set disp) (@Semiset.cblatticeType elementType eqType_of_elementType X disp set) C A)) ) move=> /subsetP AB; apply/subsetP => x; rewrite !inE => /andP[]. ( Goal: forall _ : is_true (@in_mem (Equality.sort (eqType_of_elementType Y)) (@fun_of_setfun elementType eqType_of_elementType disp set X Y f x) (@mem (Equality.sort (eqType_of_elementType Y)) (@set_predType elementType eqType_of_elementType disp set Y) (@imset elementType eqType_of_elementType disp set X Y f A))), is_true (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) A)) ) by move=> /(contra (AB _)) ->. Qed. Lemma setDSS A B C D : A \subset C -> D \subset B -> A :\: B \subset C :\: D. Proof. by move=> /(setSD B) /subset_trans sAC /(setDS C) /sAC. Qed. Lemma setD0 A : A :\: set0 = A. Proof. exact: subx0. Qed. Lemma set0D A : set0 :\: A = set0. Proof. exact: sub0x. Qed. Lemma setDv A : A :\: A = set0. Proof. exact: subxx. Qed. Lemma setID A B : A :&: B :\|: A :\: B = A. Proof. exact: joinIB. Qed. Lemma setDUl A B C : (A :\|: B) :\: C = (A :\: C) :\|: (B :\: C). Proof. exact: subUx. Qed. Lemma setDUr A B C : A :\: (B :\|: C) = (A :\: B) :&: (A :\: C). Proof. exact: subxU. Qed. Lemma setDIl A B C : (A :&: B) :\: C = (A :\: C) :&: (B :\: C). Proof. exact: subIx. Qed. Lemma setIDA A B C : A :&: (B :\: C) = (A :&: B) :\: C. Proof. exact: meetxB. Qed. Lemma setIDAC A B C : (A :\: B) :&: C = (A :&: C) :\: B. Proof. exact: meetBx. Qed. Lemma setDIr A B C : A :\: (B :&: C) = (A :\: B) :\|: (A :\: C). Proof. exact: subxI. Qed. Lemma setDDl A B C : (A :\: B) :\: C = A :\: (B :\|: C). Proof. exact: subBx. Qed. Lemma setDDr A B C : A :\: (B :\: C) = (A :\: B) :\|: (A :&: C). Proof. exact: subxB. Qed. ( other inclusions ) Lemma subsetIl A B : A :&: B \subset A. Proof. by apply/subsetP=> x; rewrite inE; case/andP. Qed. Lemma subsetIr A B : A :&: B \subset B. Proof. by apply/subsetP=> x; rewrite inE; case/andP. Qed. Lemma subsetUl A B : A \subset A :\|: B. Proof. by apply/subsetP=> x; rewrite inE => ->. Qed. Lemma subsetUr A B : B \subset A :\|: B. Proof. by apply/subsetP=> x; rewrite inE orbC => ->. Qed. Lemma subsetU1 x A : A \subset x \|: A. Proof. exact: subsetUr. Qed. Lemma subsetDl A B : A :\: B \subset A. Proof. exact: leBx. Qed. Lemma subD1set A x : A :\ x \subset A. Proof. by rewrite subsetDl. Qed. Lemma setIidPl A B : reflect (A :&: B = A) (A \subset B). Proof. ( Goal: is_true (negb (@le (display_set disp) (@Semiset.porderType elementType eqType_of_elementType Y disp set) (@imset elementType eqType_of_elementType disp set X Y f A') (@imset elementType eqType_of_elementType disp set X Y f A))) ) apply: (iffP subsetP) => [sAB \| <- x /setIP[] //]. ( Goal: forall _ : is_true (@in_mem (Equality.sort (eqType_of_elementType Y)) (@fun_of_setfun elementType eqType_of_elementType disp set X Y f x) (@mem (Equality.sort (eqType_of_elementType Y)) (@set_predType elementType eqType_of_elementType disp set Y) (@imset elementType eqType_of_elementType disp set X Y f A))), is_true (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) A)) ) by apply/setP=> x; rewrite inE; apply/andb_idr/sAB. Qed. Arguments setIidPl {A B}. Lemma setIidPr A B : reflect (A :&: B = B) (B \subset A). Proof. by rewrite setIC; apply: setIidPl. Qed. Lemma setUidPl A B : reflect (A :\|: B = A) (B \subset A). Proof. exact: join_idPr. Qed. Lemma setUidPr A B : reflect (A :\|: B = B) (A \subset B). Proof. by rewrite setUC; apply: setUidPl. Qed. Lemma subIset A B C : (B \subset A) \|\| (C \subset A) -> (B :&: C \subset A). Proof. by case/orP; apply: subset_trans; rewrite (subsetIl, subsetIr). Qed. Lemma subsetI A B C : (A \subset B :&: C) = (A \subset B) && (A \subset C). Proof. ( Goal: @eq bool (@le (display_set disp) (@Semiset.porderType elementType eqType_of_elementType X disp set) A (@sub (display_set disp) (@Semiset.cblatticeType elementType eqType_of_elementType X disp set) B (@set1 elementType eqType_of_elementType disp set X x))) (andb (@le (display_set disp) (@Semiset.porderType elementType eqType_of_elementType X disp set) A B) (negb (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) A)))) ) rewrite !(sameP setIidPl eqP) setIA; have [-> //\| ] := altP (A :&: B =P A). ( Goal: forall _ : is_true (@in_mem (Equality.sort (eqType_of_elementType Y)) (@fun_of_setfun elementType eqType_of_elementType disp set X Y f x) (@mem (Equality.sort (eqType_of_elementType Y)) (@set_predType elementType eqType_of_elementType disp set Y) (@imset elementType eqType_of_elementType disp set X Y f A))), is_true (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) A)) ) by apply: contraNF => /eqP <-; rewrite -setIA -setIIl setIAC. Qed. Lemma subsetIP A B C : reflect (A \subset B /\ A \subset C) (A \subset B :&: C). Proof. by rewrite subsetI; apply: andP. Qed. Lemma subsetIidl A B : (A \subset A :&: B) = (A \subset B). Proof. by rewrite subsetI lexx. Qed. Lemma subsetIidr A B : (B \subset A :&: B) = (B \subset A). Proof. by rewrite setIC subsetIidl. Qed. Lemma subUset A B C : (B :\|: C \subset A) = (B \subset A) && (C \subset A). Proof. exact: leUx. Qed. Lemma subsetU A B C : (A \subset B) \|\| (A \subset C) -> A \subset B :\|: C. Proof. exact: lexU. Qed. Lemma subUsetP A B C : reflect (A \subset C /\ B \subset C) (A :\|: B \subset C). Proof. by rewrite subUset; apply: andP. Qed. Lemma subDset A B C : (A :\: B \subset C) = (A \subset B :\|: C). Proof. exact: leBLR. Qed. Lemma setU_eq0 A B : (A :\|: B == set0) = (A == set0) && (B == set0). Proof. by rewrite -!subset0 subUset. Qed. Lemma setD_eq0 A B : (A :\: B == set0) = (A \subset B). Proof. by rewrite -subset0 subDset setU0. Qed. Lemma subsetD1 A B x : (A \subset B :\ x) = (A \subset B) && (x \notin A). Proof. ( Goal: @eq bool (@le (display_set disp) (@Semiset.porderType elementType eqType_of_elementType X disp set) A (@sub (display_set disp) (@Semiset.cblatticeType elementType eqType_of_elementType X disp set) B (@set1 elementType eqType_of_elementType disp set X x))) (andb (@le (display_set disp) (@Semiset.porderType elementType eqType_of_elementType X disp set) A B) (negb (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) A)))) ) rewrite andbC; have [xA\|] //= := boolP (x \in A). ( Goal: forall _ : is_true (@in_mem (Equality.sort (eqType_of_elementType Y)) (@fun_of_setfun elementType eqType_of_elementType disp set X Y f x) (@mem (Equality.sort (eqType_of_elementType Y)) (@set_predType elementType eqType_of_elementType disp set Y) (@imset elementType eqType_of_elementType disp set X Y f A))), is_true (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) A)) ) by apply: contraTF isT => /subsetP /(_ x xA); rewrite !inE eqxx. ( Goal: forall _ : is_true (@le (display_set disp) (@Semiset.porderType elementType eqType_of_elementType X disp set) A A'), is_true (@le (display_set disp) (@Semiset.porderType elementType eqType_of_elementType Y disp set) (@imset elementType eqType_of_elementType disp set X Y f A) (@imset elementType eqType_of_elementType disp set X Y f A')) ) move=> xNA; apply/subsetP/subsetP => sAB y yA. ( Goal: forall _ : is_true (@in_mem (Equality.sort (eqType_of_elementType Y)) (@fun_of_setfun elementType eqType_of_elementType disp set X Y f x) (@mem (Equality.sort (eqType_of_elementType Y)) (@set_predType elementType eqType_of_elementType disp set Y) (@imset elementType eqType_of_elementType disp set X Y f A))), is_true (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) A)) ) by have:= sAB y yA; rewrite !inE => /andP[]. ( Goal: forall _ : is_true (@in_mem (Equality.sort (eqType_of_elementType Y)) (@fun_of_setfun elementType eqType_of_elementType disp set X Y f x) (@mem (Equality.sort (eqType_of_elementType Y)) (@set_predType elementType eqType_of_elementType disp set Y) (@imset elementType eqType_of_elementType disp set X Y f A))), is_true (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) A)) ) by rewrite !inE sAB // andbT; apply: contraNneq xNA => <-. Qed. Lemma subsetD1P A B x : reflect (A \subset B /\ x \notin A) (A \subset B :\ x). Proof. by rewrite subsetD1; apply: andP. Qed. Lemma properD1 A x : x \in A -> A :\ x \proper A. Proof. by move=> Ax; rewrite properE subsetDl /= subsetD1 Ax andbF. Qed. Lemma properIr A B : ~~ (B \subset A) -> A :&: B \proper B. Proof. by move=> nsAB; rewrite properE subsetIr subsetI negb_and nsAB. Qed. Lemma properIl A B : ~~ (A \subset B) -> A :&: B \proper A. Proof. by move=> nsBA; rewrite properE subsetIl subsetI negb_and nsBA orbT. Qed. Lemma properUr A B : ~~ (A \subset B) -> B \proper A :\|: B. Proof. by rewrite properE subsetUr subUset lexx /= andbT. Qed. Lemma properUl A B : ~~ (B \subset A) -> A \proper A :\|: B. Proof. by move=> not_sBA; rewrite setUC properUr. Qed. Lemma proper1set A x : ([set x] \proper A) -> (x \in A). Proof. by move/proper_sub; rewrite sub1set. Qed. Lemma properIset A B C : (B \proper A) \|\| (C \proper A) -> (B :&: C \proper A). Proof. by case/orP; apply: sub_proper_trans; rewrite (subsetIl, subsetIr). Qed. Lemma properI A B C : (A \proper B :&: C) -> (A \proper B) && (A \proper C). Proof. ( Goal: forall _ : is_true (@le (display_set disp) (@Semiset.porderType elementType eqType_of_elementType X disp set) A A'), is_true (@le (display_set disp) (@Semiset.porderType elementType eqType_of_elementType Y disp set) (@imset elementType eqType_of_elementType disp set X Y f A) (@imset elementType eqType_of_elementType disp set X Y f A')) ) move=> pAI; apply/andP. ( Goal: forall _ : is_true (@in_mem (Equality.sort (eqType_of_elementType Y)) (@fun_of_setfun elementType eqType_of_elementType disp set X Y f x) (@mem (Equality.sort (eqType_of_elementType Y)) (@set_predType elementType eqType_of_elementType disp set Y) (@imset elementType eqType_of_elementType disp set X Y f A))), is_true (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) A)) ) by split; apply: (proper_sub_trans pAI); rewrite (subsetIl, subsetIr). Qed. Lemma properU A B C : (B :\|: C \proper A) -> (B \proper A) && (C \proper A). Proof. ( Goal: forall _ : is_true (@le (display_set disp) (@Semiset.porderType elementType eqType_of_elementType X disp set) A A'), is_true (@le (display_set disp) (@Semiset.porderType elementType eqType_of_elementType Y disp set) (@imset elementType eqType_of_elementType disp set X Y f A) (@imset elementType eqType_of_elementType disp set X Y f A')) ) move=> pUA; apply/andP. ( Goal: forall _ : is_true (@in_mem (Equality.sort (eqType_of_elementType Y)) (@fun_of_setfun elementType eqType_of_elementType disp set X Y f x) (@mem (Equality.sort (eqType_of_elementType Y)) (@set_predType elementType eqType_of_elementType disp set Y) (@imset elementType eqType_of_elementType disp set X Y f A))), is_true (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) A)) ) by split; apply: sub_proper_trans pUA; rewrite (subsetUr, subsetUl). Qed. End setX. Section setXY. Variables X Y : elementType. Implicit Types (x : X) (y : Y) (A : set X) (B : set Y) (f : setfun set X Y). Lemma imsetP (f : setfun set X Y) A y : reflect (exists2 x : X, x \in A & y = f x) (y \in imset f A). Proof. ( Goal: forall (_ : @prop_in2 (Equality.sort (eqType_of_elementType X)) (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) A') (fun x1 x2 : Equality.sort (eqType_of_elementType X) => forall _ : @eq (Equality.sort (eqType_of_elementType Y)) (@fun_of_setfun elementType eqType_of_elementType disp set X Y f x1) (@fun_of_setfun elementType eqType_of_elementType disp set X Y f x2), @eq (Equality.sort (eqType_of_elementType X)) x1 x2) (inPhantom (@injective (Equality.sort (eqType_of_elementType Y)) (Equality.sort (eqType_of_elementType X)) (@fun_of_setfun elementType eqType_of_elementType disp set X Y f)))) (_ : is_true (@lt (display_set disp) (@Semiset.porderType elementType eqType_of_elementType X disp set) A A')), is_true (@lt (display_set disp) (@Semiset.porderType elementType eqType_of_elementType Y disp set) (@imset elementType eqType_of_elementType disp set X Y f A) (@imset elementType eqType_of_elementType disp set X Y f A')) ) move: A f; rewrite /set1 /in_mem /= /memset /imset /setfun. ( Goal: forall (A : @Semiset.sort elementType eqType_of_elementType disp set X) (f : @Semiset.funsort elementType eqType_of_elementType disp (fun X : elementType => @Order.CBLattice.Pack (display_set disp) (@Semiset.sort elementType eqType_of_elementType disp set X) (@Semiset.base elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set) X) (@Semiset.sort elementType eqType_of_elementType disp set X)) (@Semiset.mixin elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set)) X Y), Bool.reflect (@ex2 (Equality.sort (eqType_of_elementType X)) (fun x : Equality.sort (eqType_of_elementType X) => is_true (@Semiset.memset elementType eqType_of_elementType disp (fun X : elementType => @Order.CBLattice.Pack (display_set disp) (@Semiset.sort elementType eqType_of_elementType disp set X) (@Semiset.base elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set) X) (@Semiset.sort elementType eqType_of_elementType disp set X)) (@Semiset.mixin elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set)) X A x)) (fun x : Equality.sort (eqType_of_elementType X) => @eq (Equality.sort (eqType_of_elementType Y)) y (@fun_of_setfun elementType eqType_of_elementType disp set X Y f x))) (@Semiset.memset elementType eqType_of_elementType disp (fun X : elementType => @Order.CBLattice.Pack (display_set disp) (@Semiset.sort elementType eqType_of_elementType disp set X) (@Semiset.base elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set) X) (@Semiset.sort elementType eqType_of_elementType disp set X)) (@Semiset.mixin elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set)) Y (@Semiset.imset elementType eqType_of_elementType disp (fun X : elementType => @Order.CBLattice.Pack (display_set disp) (@Semiset.sort elementType eqType_of_elementType disp set X) (@Semiset.base elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set) X) (@Semiset.sort elementType eqType_of_elementType disp set X)) (@Semiset.mixin elementType eqType_of_elementType disp (@Semiset.sort elementType eqType_of_elementType disp set) (@Semiset.class elementType eqType_of_elementType disp set)) X Y f A) y) ) case: set => [S [base [memset set1 /= ? ? ? ? ? ? ? ? ? H]]] ? /= A f. ( Goal: Bool.reflect (@ex2 (Equality.sort (eqType_of_elementType X)) (fun x : Equality.sort (eqType_of_elementType X) => is_true (memset X A x)) (fun x : Equality.sort (eqType_of_elementType X) => @eq (Equality.sort (eqType_of_elementType Y)) y (@fun_of_setfun elementType eqType_of_elementType disp (@Semiset.Pack elementType eqType_of_elementType disp S (@Semiset.Class elementType eqType_of_elementType disp S base (@Semiset.Mixin elementType eqType_of_elementType disp (fun X : elementType => @Order.CBLattice.Pack (display_set disp) (S X) (base X) (S X)) memset set1 _i_ _e_ _e1_ _s_ _i1_ _e2_ _funsort_ _fun_of_funsort_ _imset_ H)) _T_) X Y f x))) (memset Y (_imset_ X Y f A) y) ) exact: H. Qed. Lemma mem_imset f A x : x \in A -> f x \in imset f A. Proof. by move=> Dx; apply/imsetP; exists x. Qed. Lemma imset0 f : imset f set0 = set0. Proof. ( Goal: @eq (@Semiset.sort elementType eqType_of_elementType disp set Y) (@imset elementType eqType_of_elementType disp set X Y f (@bottom (display_set disp) (@Semiset.blatticeType elementType eqType_of_elementType X disp set))) (@bottom (display_set disp) (@Semiset.blatticeType elementType eqType_of_elementType Y disp set)) ) apply/setP => y; rewrite in_set0. ( Goal: forall _ : is_true (@in_mem (Equality.sort (eqType_of_elementType Y)) (@fun_of_setfun elementType eqType_of_elementType disp set X Y f x) (@mem (Equality.sort (eqType_of_elementType Y)) (@set_predType elementType eqType_of_elementType disp set Y) (@imset elementType eqType_of_elementType disp set X Y f A))), is_true (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) A)) ) by apply/imsetP => [[x]]; rewrite in_set0. Qed. Lemma imset_eq0 f A : (imset f A == set0) = (A == set0). Proof. ( Goal: @eq bool (@eq_op (@Semiset.eqType elementType eqType_of_elementType Y disp set) (@imset elementType eqType_of_elementType disp set X Y f A) (@bottom (display_set disp) (@Semiset.blatticeType elementType eqType_of_elementType Y disp set))) (@eq_op (@Semiset.eqType elementType eqType_of_elementType X disp set) A (@bottom (display_set disp) (@Semiset.blatticeType elementType eqType_of_elementType X disp set))) ) have [->\|/set_gt0_ex [x xA]] := posxP A; first by rewrite imset0 eqxx. ( Goal: forall _ : is_true (@in_mem (Equality.sort (eqType_of_elementType Y)) (@fun_of_setfun elementType eqType_of_elementType disp set X Y f x) (@mem (Equality.sort (eqType_of_elementType Y)) (@set_predType elementType eqType_of_elementType disp set Y) (@imset elementType eqType_of_elementType disp set X Y f A))), is_true (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) A)) ) by apply/set0Pn; exists (f x); rewrite mem_imset. Qed. Lemma imset_set1 f x : imset f [set x] = [set f x]. Proof. ( Goal: is_true (negb (@le (display_set disp) (@Semiset.porderType elementType eqType_of_elementType Y disp set) (@imset elementType eqType_of_elementType disp set X Y f A') (@imset elementType eqType_of_elementType disp set X Y f A))) ) apply/setP => y. ( Goal: forall _ : is_true (@in_mem (Equality.sort (eqType_of_elementType Y)) (@fun_of_setfun elementType eqType_of_elementType disp set X Y f x) (@mem (Equality.sort (eqType_of_elementType Y)) (@set_predType elementType eqType_of_elementType disp set Y) (@imset elementType eqType_of_elementType disp set X Y f A))), is_true (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) A)) ) by apply/imsetP/set1P=> [[x' /set1P-> //]\| ->]; exists x; rewrite ?set11. Qed. Lemma imsetS f A A' : A \subset A' -> imset f A \subset imset f A'. Proof. ( Goal: forall _ : is_true (@le (display_set disp) (@Semiset.porderType elementType eqType_of_elementType X disp set) A A'), is_true (@le (display_set disp) (@Semiset.porderType elementType eqType_of_elementType Y disp set) (@imset elementType eqType_of_elementType disp set X Y f A) (@imset elementType eqType_of_elementType disp set X Y f A')) ) move=> leAB; apply/subsetP => y /imsetP [x xA ->]. ( Goal: forall _ : is_true (@in_mem (Equality.sort (eqType_of_elementType Y)) (@fun_of_setfun elementType eqType_of_elementType disp set X Y f x) (@mem (Equality.sort (eqType_of_elementType Y)) (@set_predType elementType eqType_of_elementType disp set Y) (@imset elementType eqType_of_elementType disp set X Y f A))), is_true (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) A)) ) by rewrite mem_imset // (subsetP leAB). Qed. Lemma imset_proper f A A' : {in A' &, injective f} -> A \proper A' -> imset f A \proper imset f A'. Proof. ( Goal: forall (_ : @prop_in2 (Equality.sort (eqType_of_elementType X)) (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) A') (fun x1 x2 : Equality.sort (eqType_of_elementType X) => forall _ : @eq (Equality.sort (eqType_of_elementType Y)) (@fun_of_setfun elementType eqType_of_elementType disp set X Y f x1) (@fun_of_setfun elementType eqType_of_elementType disp set X Y f x2), @eq (Equality.sort (eqType_of_elementType X)) x1 x2) (inPhantom (@injective (Equality.sort (eqType_of_elementType Y)) (Equality.sort (eqType_of_elementType X)) (@fun_of_setfun elementType eqType_of_elementType disp set X Y f)))) (_ : is_true (@lt (display_set disp) (@Semiset.porderType elementType eqType_of_elementType X disp set) A A')), is_true (@lt (display_set disp) (@Semiset.porderType elementType eqType_of_elementType Y disp set) (@imset elementType eqType_of_elementType disp set X Y f A) (@imset elementType eqType_of_elementType disp set X Y f A')) ) move=> injf /properP[sAB [x Bx nAx]]; rewrite lt_leAnge imsetS //=. ( Goal: is_true (negb (@le (display_set disp) (@Semiset.porderType elementType eqType_of_elementType Y disp set) (@imset elementType eqType_of_elementType disp set X Y f A') (@imset elementType eqType_of_elementType disp set X Y f A))) ) apply: contra nAx => sfBA. ( Goal: is_true (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) A)) ) have: f x \in imset f A by rewrite (subsetP sfBA) ?mem_imset. ( Goal: forall _ : is_true (@in_mem (Equality.sort (eqType_of_elementType Y)) (@fun_of_setfun elementType eqType_of_elementType disp set X Y f x) (@mem (Equality.sort (eqType_of_elementType Y)) (@set_predType elementType eqType_of_elementType disp set Y) (@imset elementType eqType_of_elementType disp set X Y f A))), is_true (@in_mem (Equality.sort (eqType_of_elementType X)) x (@mem (Equality.sort (eqType_of_elementType X)) (@set_predType elementType eqType_of_elementType disp set X) A)) ) by case/imsetP=> y Ay /injf-> //; apply: subsetP sAB y Ay. Qed. End setXY. End SemisetTheory. End SemisetTheory. Module set. Section ClassDef. Variable elementType : Type. Variable eqType_of_elementType : elementType -> eqType. Coercion eqType_of_elementType : elementType >-> eqType. Implicit Types (X Y : elementType). Record class_of d (set : elementType -> Type) := Class { base : forall X, Order.CTBLattice.class_of (display_set d) (set X); mixin : Semiset.mixin_of eqType_of_elementType (fun X => Order.CBLattice.Pack (base X) (set X)) }. Local Coercion base : class_of >-> Funclass. Definition base2 d (set : elementType -> Type) (c : class_of d set) := Semiset.Class (@mixin _ set c). Local Coercion base2 : class_of >-> Semiset.class_of. Structure type d := Pack { sort ; _ : class_of d sort; _ : elementType -> Type }. Local Coercion sort : type >-> Funclass. Variables (set : elementType -> Type) (disp : unit) (cT : type disp). Definition class := let: Pack _ c _ as cT' := cT return class_of _ cT' in c. ( Definition clone c of phant_id class c := @Pack set c set. *) Let xset := let: Pack set _ _ := cT in set. Notation xclass := (class : class_of xset). Definition pack := fun bT (b : forall X, Order.CTBLattice.class_of _ _) & (forall X, phant_id (@Order.CTBLattice.class disp (bT X)) (b X)) => fun mT m & phant_id (@Semiset.class _ eqType_of_elementType mT) (@Semiset.Class _ _ disp set b m) => Pack (@Class _ set (fun x => b x) m) set. End ClassDef. Section CanonicalDef. Variable elementType : Type. Variable eqType_of_elementType : elementType -> eqType. Coercion eqType_of_elementType : elementType >-> eqType. Notation type := (type eqType_of_elementType). Local Coercion sort : type >-> Funclass. Local Coercion base : class_of >-> Funclass. Local Coercion base2 : class_of >-> Semiset.class_of. Variables (set : elementType -> Type) (X : elementType). Variable (disp : unit) (cT : type disp). Local Notation ddisp := (display_set disp). Let xset := let: Pack set _ _ := cT in set. Notation xclass := (@class _ eqType_of_elementType _ cT : class_of eqType_of_elementType _ xset). Definition eqType := ltac:(EqualityPack (cT X) ((@class _ eqType_of_elementType _ cT : class_of eqType_of_elementType _ xset) X) (xset X)). Definition choiceType := ltac:(ChoicePack (cT X) ((@class _ eqType_of_elementType _ cT : class_of eqType_of_elementType _ xset) X) (xset X)). Definition porderType := @Order.POrder.Pack ddisp (cT X) (xclass X) (xset X). Definition latticeType := @Order.Lattice.Pack ddisp (cT X) (xclass X) (xset X). Definition blatticeType := @Order.BLattice.Pack ddisp (cT X) (xclass X) (xset X). Definition cblatticeType := @Order.CBLattice.Pack ddisp (cT X) (xclass X) (xset X). Definition ctblatticeType := @Order.CTBLattice.Pack ddisp (cT X) (xclass X) (xset X). Definition semisetType := @Semiset.Pack _ _ disp cT xclass xset. Definition semiset_ctblatticeType := @Order.CTBLattice.Pack ddisp (semisetType X) (xclass X) (xset X). End CanonicalDef. Module Import Exports. Coercion base : class_of >-> Funclass. Coercion base2 : class_of >-> Semiset.class_of. Coercion sort : type >-> Funclass. Coercion eqType : type >-> Equality.type. Coercion choiceType : type >-> Choice.type. Coercion porderType : type >-> Order.POrder.type. Coercion latticeType : type >-> Order.Lattice.type. Coercion blatticeType : type >-> Order.BLattice.type. Coercion cblatticeType : type >-> Order.CBLattice.type. Coercion ctblatticeType : type >-> Order.CTBLattice.type. Coercion semisetType : type >-> Semiset.type. Canonical eqType. Canonical choiceType. Canonical porderType. Canonical latticeType. Canonical blatticeType. Canonical cblatticeType. Canonical ctblatticeType. Canonical semisetType. Notation setType := type. Notation "[ 'setType' 'of' set ]" := (@pack _ _ set _ _ _ (fun=> id) _ _ id) (at level 0, format "[ 'setType' 'of' set ]") : form_scope. End Exports. End set. Import set.Exports. Module Import setTheory. Section setTheory. Variable elementType : Type. Variable eqType_of_elementType : elementType -> eqType. Coercion eqType_of_elementType : elementType >-> eqType. Variable disp : unit. Variable set : setType eqType_of_elementType disp. Section setX. Variables X : elementType. Implicit Types (x y : X) (A B : set X). End setX. End setTheory. End setTheory. Module Theory. Export Semiset.Exports. Export set.Exports. Export SetSyntax. Export SemisetSyntax. Export SemisetTheory. Export setTheory. End Theory. End SET.