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stringlengths 5
62.8k
| template
stringlengths 7
63.6k
| symbols
sequencelengths 0
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sequencelengths 0
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| input
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131k
| output
stringlengths 22
62.8k
|
|---|---|---|---|---|---|---|---|---|
Automatic_Refinement/Parametricity/Relators
|
Relators.in_br_conv
|
lemma in_br_conv: "(c,a)\<in>br \<alpha> I \<longleftrightarrow> a=\<alpha> c \<and> I c"
|
((?c, ?a) \<in> br ?\<alpha> ?I) = (?a = ?\<alpha> ?c \<and> ?I ?c)
|
((x_1, x_2) \<in> ?H1 x_3 x_4) = (x_2 = x_3 x_1 \<and> x_4 x_1)
|
[
"Relators.br"
] |
[] |
###template
((x_1, x_2) \<in> ?H1 x_3 x_4) = (x_2 = x_3 x_1 \<and> x_4 x_1)
###symbols
Relators.br
###defs
|
###output
((?c, ?a) \<in> br ?\<alpha> ?I) = (?a = ?\<alpha> ?c \<and> ?I ?c)
###end
|
Collections/GenCF/Gen/Gen_Set
|
Gen_Set.gen_diff
|
lemma gen_diff[autoref_rules_raw]:
assumes PRIO_TAG_GEN_ALGO
assumes DEL:
"GEN_OP del1 op_set_delete (Rk \<rightarrow> \<langle>Rk\<rangle>Rs1 \<rightarrow> \<langle>Rk\<rangle>Rs1)"
assumes IT: "SIDE_GEN_ALGO (is_set_to_list Rk Rs2 it2)"
shows "(gen_diff del1 (\<lambda>x. foldli (it2 x)),(-))
\<in> (\<langle>Rk\<rangle>Rs1) \<rightarrow> (\<langle>Rk\<rangle>Rs2) \<rightarrow> (\<langle>Rk\<rangle>Rs1)"
|
PRIO_TAG_GEN_ALGO \<Longrightarrow> GEN_OP ?del1.0 op_set_delete (?Rk \<rightarrow> \<langle>?Rk\<rangle>?Rs1.0 \<rightarrow> \<langle>?Rk\<rangle>?Rs1.0) \<Longrightarrow> SIDE_GEN_ALGO (is_set_to_list ?Rk ?Rs2.0 ?it2.0) \<Longrightarrow> (gen_diff ?del1.0 (\<lambda>x. foldli (?it2.0 x)), (-)) \<in> \<langle>?Rk\<rangle>?Rs1.0 \<rightarrow> \<langle>?Rk\<rangle>?Rs2.0 \<rightarrow> \<langle>?Rk\<rangle>?Rs1.0
|
\<lbrakk>?H1; ?H2 x_1 ?H3 (?H4 x_2 (?H5 (?H6 x_3 x_2) (?H6 x_3 x_2))); ?H7 (?H8 x_2 x_4 x_5)\<rbrakk> \<Longrightarrow> (?H9 x_1 (\<lambda>y_0. ?H10 (x_5 y_0)), ?H11) \<in> ?H12 (?H6 x_3 x_2) (?H13 (?H14 x_4 x_2) (?H6 x_3 x_2))
|
[
"Groups.minus_class.minus",
"Foldi.foldli",
"Gen_Set.gen_diff",
"Gen_Iterator.is_set_to_list",
"Autoref_Gen_Algo.SIDE_GEN_ALGO",
"Relators.relAPP",
"Relators.fun_rel_syn",
"Intf_Set.op_set_delete",
"Autoref_Fix_Rel.GEN_OP",
"Autoref_Fix_Rel.PRIO_TAG_GEN_ALGO"
] |
[
"class minus =\n fixes minus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"-\" 65)"
] |
###template
\<lbrakk>?H1; ?H2 x_1 ?H3 (?H4 x_2 (?H5 (?H6 x_3 x_2) (?H6 x_3 x_2))); ?H7 (?H8 x_2 x_4 x_5)\<rbrakk> \<Longrightarrow> (?H9 x_1 (\<lambda>y_0. ?H10 (x_5 y_0)), ?H11) \<in> ?H12 (?H6 x_3 x_2) (?H13 (?H14 x_4 x_2) (?H6 x_3 x_2))
###symbols
Groups.minus_class.minus
Foldi.foldli
Gen_Set.gen_diff
Gen_Iterator.is_set_to_list
Autoref_Gen_Algo.SIDE_GEN_ALGO
Relators.relAPP
Relators.fun_rel_syn
Intf_Set.op_set_delete
Autoref_Fix_Rel.GEN_OP
Autoref_Fix_Rel.PRIO_TAG_GEN_ALGO
###defs
class minus =
fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "-" 65)
|
###output
PRIO_TAG_GEN_ALGO \<Longrightarrow> GEN_OP ?del1.0 op_set_delete (?Rk \<rightarrow> \<langle>?Rk\<rangle>?Rs1.0 \<rightarrow> \<langle>?Rk\<rangle>?Rs1.0) \<Longrightarrow> SIDE_GEN_ALGO (is_set_to_list ?Rk ?Rs2.0 ?it2.0) \<Longrightarrow> (gen_diff ?del1.0 (\<lambda>x. foldli (?it2.0 x)), (-)) \<in> \<langle>?Rk\<rangle>?Rs1.0 \<rightarrow> \<langle>?Rk\<rangle>?Rs2.0 \<rightarrow> \<langle>?Rk\<rangle>?Rs1.0
###end
|
AODV/variants/b_fwdrreps/B_Fresher
|
B_Fresher.dhops_le_hops_imp_update_strictly_fresher
|
lemma dhops_le_hops_imp_update_strictly_fresher:
assumes "dip \<in> vD(rt2 nhip)"
and sqn: "sqn (rt2 nhip) dip = osn"
and hop: "the (dhops (rt2 nhip) dip) \<le> hops"
and **: "rt \<noteq> update rt dip (osn, kno, val, Suc hops, nhip, {})"
shows "update rt dip (osn, kno, val, Suc hops, nhip, {}) \<sqsubset>\<^bsub>dip\<^esub> rt2 nhip"
|
?dip \<in> vD (?rt2.0 ?nhip) \<Longrightarrow> sqn (?rt2.0 ?nhip) ?dip = ?osn \<Longrightarrow> the (dhops (?rt2.0 ?nhip) ?dip) \<le> ?hops \<Longrightarrow> ?rt \<noteq> update ?rt ?dip (?osn, kno, val, Suc ?hops, ?nhip, {}) \<Longrightarrow> update ?rt ?dip (?osn, kno, val, Suc ?hops, ?nhip, {}) \<sqsubset>\<^bsub>?dip\<^esub> ?rt2.0 ?nhip
|
\<lbrakk>x_1 \<in> ?H1 (x_2 x_3); ?H2 (x_2 x_3) x_1 = x_4; ?H3 (?H4 (x_2 x_3) x_1) \<le> x_5; x_6 \<noteq> ?H5 x_6 x_1 (x_4, ?H6, ?H7, ?H8 x_5, x_3, ?H9)\<rbrakk> \<Longrightarrow> ?H10 (?H5 x_6 x_1 (x_4, ?H6, ?H7, ?H8 x_5, x_3, ?H9)) x_1 (x_2 x_3)
|
[
"B_Fresher.rt_strictly_fresher_syn",
"Set.empty",
"Nat.Suc",
"Aodv_Basic.val",
"Aodv_Basic.kno",
"B_Aodv_Data.update",
"B_Aodv_Data.dhops",
"Option.option.the",
"B_Aodv_Data.sqn",
"B_Aodv_Data.vD"
] |
[
"abbreviation empty :: \"'a set\" (\"{}\")\n where \"{} \\<equiv> bot\"",
"definition Suc :: \"nat \\<Rightarrow> nat\"\n where \"Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))\"",
"abbreviation val where \"val \\<equiv> Valid\"",
"abbreviation kno where \"kno \\<equiv> Known\"",
"datatype 'a option =\n None\n | Some (the: 'a)"
] |
###template
\<lbrakk>x_1 \<in> ?H1 (x_2 x_3); ?H2 (x_2 x_3) x_1 = x_4; ?H3 (?H4 (x_2 x_3) x_1) \<le> x_5; x_6 \<noteq> ?H5 x_6 x_1 (x_4, ?H6, ?H7, ?H8 x_5, x_3, ?H9)\<rbrakk> \<Longrightarrow> ?H10 (?H5 x_6 x_1 (x_4, ?H6, ?H7, ?H8 x_5, x_3, ?H9)) x_1 (x_2 x_3)
###symbols
B_Fresher.rt_strictly_fresher_syn
Set.empty
Nat.Suc
Aodv_Basic.val
Aodv_Basic.kno
B_Aodv_Data.update
B_Aodv_Data.dhops
Option.option.the
B_Aodv_Data.sqn
B_Aodv_Data.vD
###defs
abbreviation empty :: "'a set" ("{}")
where "{} \<equiv> bot"
definition Suc :: "nat \<Rightarrow> nat"
where "Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))"
abbreviation val where "val \<equiv> Valid"
abbreviation kno where "kno \<equiv> Known"
datatype 'a option =
None
| Some (the: 'a)
|
###output
?dip \<in> vD (?rt2.0 ?nhip) \<Longrightarrow> sqn (?rt2.0 ?nhip) ?dip = ?osn \<Longrightarrow> the (dhops (?rt2.0 ?nhip) ?dip) \<le> ?hops \<Longrightarrow> ?rt \<noteq> update ?rt ?dip (?osn, kno, val, Suc ?hops, ?nhip, {}) \<Longrightarrow> update ?rt ?dip (?osn, kno, val, Suc ?hops, ?nhip, {}) \<sqsubset>\<^bsub>?dip\<^esub> ?rt2.0 ?nhip
###end
|
Multirelations_Heterogeneous/Multirelations
|
Multirelations_Basics.d_llp1
| null |
Dom ?R \<subseteq> Dom ?S \<Longrightarrow> ?R \<subseteq> Dom ?S * ?R
|
?H1 (?H2 x_1) (?H3 x_2) \<Longrightarrow> ?H4 x_1 (?H5 (?H3 x_2) x_1)
|
[
"Multirelations_Basics.s_prod",
"Multirelations_Basics.Dom",
"Set.subset_eq"
] |
[
"definition s_prod :: \"('a,'b) mrel \\<Rightarrow> ('b,'c) mrel \\<Rightarrow> ('a,'c) mrel\" (infixl \"\\<cdot>\" 75) where\n \"R \\<cdot> S = {(a,A). (\\<exists>B. (a,B) \\<in> R \\<and> (\\<exists>f. (\\<forall>b \\<in> B. (b,f b) \\<in> S) \\<and> A = \\<Union>(f ` B)))}\"",
"definition Dom :: \"('a,'b) mrel \\<Rightarrow> ('a,'a) mrel\" where\n \"Dom R = {(a,{a}) |a. \\<exists>B. (a,B) \\<in> R}\"",
"abbreviation subset_eq :: \"'a set \\<Rightarrow> 'a set \\<Rightarrow> bool\"\n where \"subset_eq \\<equiv> less_eq\""
] |
###template
?H1 (?H2 x_1) (?H3 x_2) \<Longrightarrow> ?H4 x_1 (?H5 (?H3 x_2) x_1)
###symbols
Multirelations_Basics.s_prod
Multirelations_Basics.Dom
Set.subset_eq
###defs
definition s_prod :: "('a,'b) mrel \<Rightarrow> ('b,'c) mrel \<Rightarrow> ('a,'c) mrel" (infixl "\<cdot>" 75) where
"R \<cdot> S = {(a,A). (\<exists>B. (a,B) \<in> R \<and> (\<exists>f. (\<forall>b \<in> B. (b,f b) \<in> S) \<and> A = \<Union>(f ` B)))}"
definition Dom :: "('a,'b) mrel \<Rightarrow> ('a,'a) mrel" where
"Dom R = {(a,{a}) |a. \<exists>B. (a,B) \<in> R}"
abbreviation subset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"
where "subset_eq \<equiv> less_eq"
|
###output
Dom ?R \<subseteq> Dom ?S \<Longrightarrow> ?R \<subseteq> Dom ?S * ?R
###end
|
Jordan_Normal_Form/Determinant_Impl
|
Determinant_Impl.sub1_det
|
lemma sub1_det:
assumes A: "A \<in> carrier_mat n n"
and sub1: "sub1 q k l (r,A) = (r'',A'')"
and r0: "r \<noteq> 0"
and All0: "q \<noteq> 0"
and l: "l + k < n"
shows "r * det A'' = r'' * det A"
|
mute_fun ?mf \<Longrightarrow> det_selection_fun ?sel_fun \<Longrightarrow> ?A \<in> carrier_mat ?n ?n \<Longrightarrow> ??.Determinant_Impl.sub1 ?mf ?q ?k ?l (?r, ?A) = (?r'', ?A'') \<Longrightarrow> ?r \<noteq> (0::?'a) \<Longrightarrow> ?q \<noteq> (0::?'a) \<Longrightarrow> ?l + ?k < ?n \<Longrightarrow> ?r * det ?A'' = ?r'' * det ?A
|
\<lbrakk>?H1 x_1; ?H2 x_2; x_3 \<in> ?H3 x_4 x_4; ?H4 x_1 x_5 x_6 x_7 (x_8, x_3) = (x_9, x_10); x_8 \<noteq> ?H5; x_5 \<noteq> ?H5; ?H6 x_7 x_6 < x_4\<rbrakk> \<Longrightarrow> ?H7 x_8 (?H8 x_10) = ?H7 x_9 (?H8 x_3)
|
[
"Determinant.det",
"Groups.times_class.times",
"Groups.plus_class.plus",
"Groups.zero_class.zero",
"Determinant_Impl.sub1",
"Matrix.carrier_mat",
"Determinant_Impl.det_selection_fun",
"Determinant_Impl.mute_fun"
] |
[
"definition det:: \"'a mat \\<Rightarrow> 'a :: comm_ring_1\" where\n \"det A = (if dim_row A = dim_col A then (\\<Sum> p \\<in> {p. p permutes {0 ..< dim_row A}}. \n signof p * (\\<Prod> i = 0 ..< dim_row A. A $$ (i, p i))) else 0)\"",
"class times =\n fixes times :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"*\" 70)",
"class plus =\n fixes plus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"+\" 65)",
"class zero =\n fixes zero :: 'a (\"0\")",
"fun sub1 :: \"'a \\<Rightarrow> nat \\<Rightarrow> nat \\<Rightarrow> 'a \\<times> 'a mat \\<Rightarrow> 'a \\<times> 'a mat\"\nwhere \"sub1 q 0 l rA = rA\"\n | \"sub1 q (Suc k) l rA = mute q (l + Suc k) l (sub1 q k l rA)\"",
"definition carrier_mat :: \"nat \\<Rightarrow> nat \\<Rightarrow> 'a mat set\"\n where \"carrier_mat nr nc = { m . dim_row m = nr \\<and> dim_col m = nc}\"",
"definition det_selection_fun :: \"'a det_selection_fun \\<Rightarrow> bool\" where \n \"det_selection_fun f = (\\<forall> xs. xs \\<noteq> [] \\<longrightarrow> f xs \\<in> fst ` set xs)\"",
"definition mute_fun :: \"('a :: comm_ring_1 \\<Rightarrow> 'a \\<Rightarrow> 'a \\<times> 'a \\<times> 'a) \\<Rightarrow> bool\" where\n \"mute_fun f = (\\<forall> x y x' y' g. f x y = (x',y',g) \\<longrightarrow> y \\<noteq> 0 \n \\<longrightarrow> x = x' * g \\<and> y * x' = x * y')\""
] |
###template
\<lbrakk>?H1 x_1; ?H2 x_2; x_3 \<in> ?H3 x_4 x_4; ?H4 x_1 x_5 x_6 x_7 (x_8, x_3) = (x_9, x_10); x_8 \<noteq> ?H5; x_5 \<noteq> ?H5; ?H6 x_7 x_6 < x_4\<rbrakk> \<Longrightarrow> ?H7 x_8 (?H8 x_10) = ?H7 x_9 (?H8 x_3)
###symbols
Determinant.det
Groups.times_class.times
Groups.plus_class.plus
Groups.zero_class.zero
Determinant_Impl.sub1
Matrix.carrier_mat
Determinant_Impl.det_selection_fun
Determinant_Impl.mute_fun
###defs
definition det:: "'a mat \<Rightarrow> 'a :: comm_ring_1" where
"det A = (if dim_row A = dim_col A then (\<Sum> p \<in> {p. p permutes {0 ..< dim_row A}}.
signof p * (\<Prod> i = 0 ..< dim_row A. A $$ (i, p i))) else 0)"
class times =
fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "*" 70)
class plus =
fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "+" 65)
class zero =
fixes zero :: 'a ("0")
fun sub1 :: "'a \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a \<times> 'a mat \<Rightarrow> 'a \<times> 'a mat"
where "sub1 q 0 l rA = rA"
| "sub1 q (Suc k) l rA = mute q (l + Suc k) l (sub1 q k l rA)"
definition carrier_mat :: "nat \<Rightarrow> nat \<Rightarrow> 'a mat set"
where "carrier_mat nr nc = { m . dim_row m = nr \<and> dim_col m = nc}"
definition det_selection_fun :: "'a det_selection_fun \<Rightarrow> bool" where
"det_selection_fun f = (\<forall> xs. xs \<noteq> [] \<longrightarrow> f xs \<in> fst ` set xs)"
definition mute_fun :: "('a :: comm_ring_1 \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a) \<Rightarrow> bool" where
"mute_fun f = (\<forall> x y x' y' g. f x y = (x',y',g) \<longrightarrow> y \<noteq> 0
\<longrightarrow> x = x' * g \<and> y * x' = x * y')"
|
###output
mute_fun ?mf \<Longrightarrow> det_selection_fun ?sel_fun \<Longrightarrow> ?A \<in> carrier_mat ?n ?n \<Longrightarrow> ??.Determinant_Impl.sub1 ?mf ?q ?k ?l (?r, ?A) = (?r'', ?A'') \<Longrightarrow> ?r \<noteq> (0::?'a) \<Longrightarrow> ?q \<noteq> (0::?'a) \<Longrightarrow> ?l + ?k < ?n \<Longrightarrow> ?r * det ?A'' = ?r'' * det ?A
###end
|
Regular_Tree_Relations/Util/Ground_Terms
|
Ground_Terms.parallel_replace_gterm_commute
|
lemma parallel_replace_gterm_commute [ac_simps]:
"p \<bottom> q \<Longrightarrow> s[p \<leftarrow> t]\<^sub>G[q \<leftarrow> u]\<^sub>G = s[q \<leftarrow> u]\<^sub>G[p \<leftarrow> t]\<^sub>G"
|
?p \<bottom> ?q \<Longrightarrow> ?s[?p \<leftarrow> ?t]\<^sub>G[?q \<leftarrow> ?u]\<^sub>G = ?s[?q \<leftarrow> ?u]\<^sub>G[?p \<leftarrow> ?t]\<^sub>G
|
?H1 x_1 x_2 \<Longrightarrow> ?H2 (?H2 x_3 x_1 x_4) x_2 x_5 = ?H2 (?H2 x_3 x_2 x_5) x_1 x_4
|
[
"Ground_Terms.replace_gterm_at",
"Term_Context.position_par"
] |
[] |
###template
?H1 x_1 x_2 \<Longrightarrow> ?H2 (?H2 x_3 x_1 x_4) x_2 x_5 = ?H2 (?H2 x_3 x_2 x_5) x_1 x_4
###symbols
Ground_Terms.replace_gterm_at
Term_Context.position_par
###defs
|
###output
?p \<bottom> ?q \<Longrightarrow> ?s[?p \<leftarrow> ?t]\<^sub>G[?q \<leftarrow> ?u]\<^sub>G = ?s[?q \<leftarrow> ?u]\<^sub>G[?p \<leftarrow> ?t]\<^sub>G
###end
|
Containers/DList_Set
|
DList_Set.member_filter
|
lemma member_filter [simp]: "member (filter P xs) = (\<lambda>x :: 'a. member xs x \<and> P x)"
|
ID CEQ(?'a) \<noteq> None \<Longrightarrow> DList_Set.member (DList_Set.filter ?P ?xs) = (\<lambda>x. DList_Set.member ?xs x \<and> ?P x)
|
?H1 ?H2 \<noteq> ?H3 \<Longrightarrow> ?H4 (?H5 x_1 x_2) = (\<lambda>y_0. ?H4 x_2 y_0 \<and> x_1 y_0)
|
[
"DList_Set.filter",
"DList_Set.member",
"Option.option.None",
"Collection_Eq.ceq_class.ceq",
"Containers_Auxiliary.ID"
] |
[
"datatype 'a option =\n None\n | Some (the: 'a)",
"class ceq =\n fixes ceq :: \"('a \\<Rightarrow> 'a \\<Rightarrow> bool) option\"\n assumes ceq: \"ceq = Some eq \\<Longrightarrow> eq = (=)\"\nbegin",
"definition ID :: \"'a \\<Rightarrow> 'a\" where \"ID = id\""
] |
###template
?H1 ?H2 \<noteq> ?H3 \<Longrightarrow> ?H4 (?H5 x_1 x_2) = (\<lambda>y_0. ?H4 x_2 y_0 \<and> x_1 y_0)
###symbols
DList_Set.filter
DList_Set.member
Option.option.None
Collection_Eq.ceq_class.ceq
Containers_Auxiliary.ID
###defs
datatype 'a option =
None
| Some (the: 'a)
class ceq =
fixes ceq :: "('a \<Rightarrow> 'a \<Rightarrow> bool) option"
assumes ceq: "ceq = Some eq \<Longrightarrow> eq = (=)"
begin
definition ID :: "'a \<Rightarrow> 'a" where "ID = id"
|
###output
ID CEQ(?'a) \<noteq> None \<Longrightarrow> DList_Set.member (DList_Set.filter ?P ?xs) = (\<lambda>x. DList_Set.member ?xs x \<and> ?P x)
###end
|
Akra_Bazzi/Akra_Bazzi_Library
|
Akra_Bazzi_Library.DERIV_nonneg_imp_mono
|
lemma DERIV_nonneg_imp_mono:
assumes "\<And>t. t\<in>{x..y} \<Longrightarrow> (f has_field_derivative f' t) (at t)"
assumes "\<And>t. t\<in>{x..y} \<Longrightarrow> f' t \<ge> 0"
assumes "(x::real) \<le> y"
shows "(f x :: real) \<le> f y"
|
(\<And>t. t \<in> {?x..?y} \<Longrightarrow> (?f has_real_derivative ?f' t) (at t)) \<Longrightarrow> (\<And>t. t \<in> {?x..?y} \<Longrightarrow> 0 \<le> ?f' t) \<Longrightarrow> ?x \<le> ?y \<Longrightarrow> ?f ?x \<le> ?f ?y
|
\<lbrakk>\<And>y_0. y_0 \<in> ?H1 x_1 x_2 \<Longrightarrow> ?H2 x_3 (x_4 y_0) (?H3 y_0); \<And>y_1. y_1 \<in> ?H1 x_1 x_2 \<Longrightarrow> ?H4 \<le> x_4 y_1; x_1 \<le> x_2\<rbrakk> \<Longrightarrow> x_3 x_1 \<le> x_3 x_2
|
[
"Groups.zero_class.zero",
"Topological_Spaces.topological_space_class.at",
"Deriv.has_real_derivative",
"Set_Interval.ord_class.atLeastAtMost"
] |
[
"class zero =\n fixes zero :: 'a (\"0\")",
"class topological_space = \"open\" +\n assumes open_UNIV [simp, intro]: \"open UNIV\"\n assumes open_Int [intro]: \"open S \\<Longrightarrow> open T \\<Longrightarrow> open (S \\<inter> T)\"\n assumes open_Union [intro]: \"\\<forall>S\\<in>K. open S \\<Longrightarrow> open (\\<Union>K)\"\nbegin",
"abbreviation has_real_derivative :: \"(real \\<Rightarrow> real) \\<Rightarrow> real \\<Rightarrow> real filter \\<Rightarrow> bool\"\n (infix \"(has'_real'_derivative)\" 50)\n where \"(f has_real_derivative D) F \\<equiv> (f has_field_derivative D) F\""
] |
###template
\<lbrakk>\<And>y_0. y_0 \<in> ?H1 x_1 x_2 \<Longrightarrow> ?H2 x_3 (x_4 y_0) (?H3 y_0); \<And>y_1. y_1 \<in> ?H1 x_1 x_2 \<Longrightarrow> ?H4 \<le> x_4 y_1; x_1 \<le> x_2\<rbrakk> \<Longrightarrow> x_3 x_1 \<le> x_3 x_2
###symbols
Groups.zero_class.zero
Topological_Spaces.topological_space_class.at
Deriv.has_real_derivative
Set_Interval.ord_class.atLeastAtMost
###defs
class zero =
fixes zero :: 'a ("0")
class topological_space = "open" +
assumes open_UNIV [simp, intro]: "open UNIV"
assumes open_Int [intro]: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<inter> T)"
assumes open_Union [intro]: "\<forall>S\<in>K. open S \<Longrightarrow> open (\<Union>K)"
begin
abbreviation has_real_derivative :: "(real \<Rightarrow> real) \<Rightarrow> real \<Rightarrow> real filter \<Rightarrow> bool"
(infix "(has'_real'_derivative)" 50)
where "(f has_real_derivative D) F \<equiv> (f has_field_derivative D) F"
|
###output
(\<And>t. t \<in> {?x..?y} \<Longrightarrow> (?f has_real_derivative ?f' t) (at t)) \<Longrightarrow> (\<And>t. t \<in> {?x..?y} \<Longrightarrow> 0 \<le> ?f' t) \<Longrightarrow> ?x \<le> ?y \<Longrightarrow> ?f ?x \<le> ?f ?y
###end
|
AOT/AOT_PossibleWorlds
|
AOT_PossibleWorlds.actual-s:3
| null |
\<^bold>\<turnstile>\<^sub>\<box> \<exists>p \<forall>s (Actual(s) \<rightarrow> \<not>s \<Turnstile> p)
|
H1 x_1 (H2 (\<lambda>y_0. H3 (\<lambda>y_1. H4 (H5 y_1) (H4 (H6 y_1) (H7 (H8 y_1 y_0))))))
|
[
"AOT_PossibleWorlds.TruthInSituation",
"AOT_syntax.AOT_not",
"AOT_PossibleWorlds.actual",
"AOT_PossibleWorlds.Situation",
"AOT_syntax.AOT_imp",
"AOT_syntax.AOT_forall",
"AOT_syntax.AOT_exists",
"AOT_model.AOT_model_valid_in"
] |
[
"definition AOT_model_valid_in :: \\<open>w\\<Rightarrow>\\<o>\\<Rightarrow>bool\\<close> where\n \\<open>AOT_model_valid_in w \\<phi> \\<equiv> AOT_model_d\\<o> \\<phi> w\\<close>"
] |
###template
H1 x_1 (H2 (\<lambda>y_0. H3 (\<lambda>y_1. H4 (H5 y_1) (H4 (H6 y_1) (H7 (H8 y_1 y_0))))))
###symbols
AOT_PossibleWorlds.TruthInSituation
AOT_syntax.AOT_not
AOT_PossibleWorlds.actual
AOT_PossibleWorlds.Situation
AOT_syntax.AOT_imp
AOT_syntax.AOT_forall
AOT_syntax.AOT_exists
AOT_model.AOT_model_valid_in
###defs
definition AOT_model_valid_in :: \<open>w\<Rightarrow>\<o>\<Rightarrow>bool\<close> where
\<open>AOT_model_valid_in w \<phi> \<equiv> AOT_model_d\<o> \<phi> w\<close>
|
###output
\<^bold>\<turnstile>\<^sub>\<box> \<exists>p \<forall>s (Actual(s) \<rightarrow> \<not>s \<Turnstile> p)
###end
|
Ordinal/Ordinal
|
OrdinalInverse.ordinal_div_self
| null |
0 < ?x \<Longrightarrow> ?x div ?x = 1
|
?H1 < x_1 \<Longrightarrow> ?H2 x_1 x_1 = ?H3
|
[
"Groups.one_class.one",
"Rings.divide_class.divide",
"Groups.zero_class.zero"
] |
[
"class one =\n fixes one :: 'a (\"1\")",
"class divide =\n fixes divide :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"div\" 70)",
"class zero =\n fixes zero :: 'a (\"0\")"
] |
###template
?H1 < x_1 \<Longrightarrow> ?H2 x_1 x_1 = ?H3
###symbols
Groups.one_class.one
Rings.divide_class.divide
Groups.zero_class.zero
###defs
class one =
fixes one :: 'a ("1")
class divide =
fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "div" 70)
class zero =
fixes zero :: 'a ("0")
|
###output
0 < ?x \<Longrightarrow> ?x div ?x = 1
###end
|
Tycon/Writer_Transformer
|
Writer_Transformer.invar'_bindWT
|
lemma invar'_bindWT: "\<lbrakk>invar' m; \<And>x. invar' (k\<cdot>x)\<rbrakk> \<Longrightarrow> invar' (bindWT\<cdot>m\<cdot>k)"
|
invar' ?m \<Longrightarrow> (\<And>x. invar' (?k\<cdot>x)) \<Longrightarrow> invar' (bindWT\<cdot>?m\<cdot>?k)
|
\<lbrakk>?H1 x_1; \<And>y_0. ?H2 (?H3 x_2 y_0)\<rbrakk> \<Longrightarrow> ?H2 (?H4 (?H5 ?H6 x_1) x_2)
|
[
"Writer_Transformer.bindWT",
"Cfun.cfun.Rep_cfun",
"Writer_Transformer.invar'"
] |
[
"definition bindWT :: \"'a\\<cdot>('m::monad,'w::monoid) writerT \\<rightarrow> ('a \\<rightarrow> 'b\\<cdot>('m,'w) writerT) \\<rightarrow> 'b\\<cdot>('m,'w) writerT\"\n where \"bindWT = (\\<Lambda> m k. WriterT\\<cdot>(bind\\<cdot>(runWriterT\\<cdot>m)\\<cdot>\n (\\<Lambda>(Writer\\<cdot>w\\<cdot>x). bind\\<cdot>(runWriterT\\<cdot>(k\\<cdot>x))\\<cdot>(\\<Lambda>(Writer\\<cdot>w'\\<cdot>y).\n return\\<cdot>(Writer\\<cdot>(mappend\\<cdot>w\\<cdot>w')\\<cdot>y)))))\"",
"definition invar' :: \"'a\\<cdot>('m::monad, 'w::monoid) writerT \\<Rightarrow> bool\"\n where \"invar' m \\<longleftrightarrow> fmapWT\\<cdot>ID\\<cdot>m = m\""
] |
###template
\<lbrakk>?H1 x_1; \<And>y_0. ?H2 (?H3 x_2 y_0)\<rbrakk> \<Longrightarrow> ?H2 (?H4 (?H5 ?H6 x_1) x_2)
###symbols
Writer_Transformer.bindWT
Cfun.cfun.Rep_cfun
Writer_Transformer.invar'
###defs
definition bindWT :: "'a\<cdot>('m::monad,'w::monoid) writerT \<rightarrow> ('a \<rightarrow> 'b\<cdot>('m,'w) writerT) \<rightarrow> 'b\<cdot>('m,'w) writerT"
where "bindWT = (\<Lambda> m k. WriterT\<cdot>(bind\<cdot>(runWriterT\<cdot>m)\<cdot>
(\<Lambda>(Writer\<cdot>w\<cdot>x). bind\<cdot>(runWriterT\<cdot>(k\<cdot>x))\<cdot>(\<Lambda>(Writer\<cdot>w'\<cdot>y).
return\<cdot>(Writer\<cdot>(mappend\<cdot>w\<cdot>w')\<cdot>y)))))"
definition invar' :: "'a\<cdot>('m::monad, 'w::monoid) writerT \<Rightarrow> bool"
where "invar' m \<longleftrightarrow> fmapWT\<cdot>ID\<cdot>m = m"
|
###output
invar' ?m \<Longrightarrow> (\<And>x. invar' (?k\<cdot>x)) \<Longrightarrow> invar' (bindWT\<cdot>?m\<cdot>?k)
###end
|
FSM_Tests/EquivalenceTesting/Simple_Convergence_Graph_Trie
|
Simple_Convergence_Graph_Trie.can_merge_by_suffix_code
| null |
can_merge_by_suffix ?x ?x1.0 ?x2.0 = ts.bex ?x (\<lambda>ys. ts.bex ?x1.0 (\<lambda>ys1. is_prefix ys ys1 \<and> (let ys'' = drop (length ys) ys1 in ts.bex ?x (\<lambda>ys'. ts.memb (ys' @ ys'') ?x2.0))))
|
?H1 x_1 x_2 x_3 = ?H2 x_1 (\<lambda>y_0. ?H2 x_2 (\<lambda>y_1. ?H3 y_0 y_1 \<and> (let y_2 = ?H4 (?H5 y_0) y_1 in ?H2 x_1 (\<lambda>y_3. ?H6 (?H7 y_3 y_2) x_3))))
|
[
"List.append",
"TrieSetImpl.ts.memb",
"List.length",
"List.drop",
"Util.is_prefix",
"TrieSetImpl.ts.bex",
"Simple_Convergence_Graph_Trie.can_merge_by_suffix"
] |
[
"primrec append :: \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" (infixr \"@\" 65) where\nappend_Nil: \"[] @ ys = ys\" |\nappend_Cons: \"(x#xs) @ ys = x # xs @ ys\"",
"abbreviation length :: \"'a list \\<Rightarrow> nat\" where\n\"length \\<equiv> size\"",
"primrec drop:: \"nat \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" where\ndrop_Nil: \"drop n [] = []\" |\ndrop_Cons: \"drop n (x # xs) = (case n of 0 \\<Rightarrow> x # xs | Suc m \\<Rightarrow> drop m xs)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>",
"fun is_prefix :: \"'a list \\<Rightarrow> 'a list \\<Rightarrow> bool\" where\n \"is_prefix [] _ = True\" |\n \"is_prefix (x#xs) [] = False\" |\n \"is_prefix (x#xs) (y#ys) = (x = y \\<and> is_prefix xs ys)\""
] |
###template
?H1 x_1 x_2 x_3 = ?H2 x_1 (\<lambda>y_0. ?H2 x_2 (\<lambda>y_1. ?H3 y_0 y_1 \<and> (let y_2 = ?H4 (?H5 y_0) y_1 in ?H2 x_1 (\<lambda>y_3. ?H6 (?H7 y_3 y_2) x_3))))
###symbols
List.append
TrieSetImpl.ts.memb
List.length
List.drop
Util.is_prefix
TrieSetImpl.ts.bex
Simple_Convergence_Graph_Trie.can_merge_by_suffix
###defs
primrec append :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" (infixr "@" 65) where
append_Nil: "[] @ ys = ys" |
append_Cons: "(x#xs) @ ys = x # xs @ ys"
abbreviation length :: "'a list \<Rightarrow> nat" where
"length \<equiv> size"
primrec drop:: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
drop_Nil: "drop n [] = []" |
drop_Cons: "drop n (x # xs) = (case n of 0 \<Rightarrow> x # xs | Suc m \<Rightarrow> drop m xs)"
\<comment> \<open>Warning: simpset does not contain this definition, but separate
theorems for \<open>n = 0\<close> and \<open>n = Suc k\<close>\<close>
fun is_prefix :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" where
"is_prefix [] _ = True" |
"is_prefix (x#xs) [] = False" |
"is_prefix (x#xs) (y#ys) = (x = y \<and> is_prefix xs ys)"
|
###output
can_merge_by_suffix ?x ?x1.0 ?x2.0 = ts.bex ?x (\<lambda>ys. ts.bex ?x1.0 (\<lambda>ys1. is_prefix ys ys1 \<and> (let ys'' = drop (length ys) ys1 in ts.bex ?x (\<lambda>ys'. ts.memb (ys' @ ys'') ?x2.0))))
###end
|
HOL-CSP_OpSem/NewLaws
|
NewLaws.Hiding_Mprefix_Sliding_non_disjoint
|
lemma Hiding_Mprefix_Sliding_non_disjoint:
\<open>((\<box>a \<in> A \<rightarrow> P a) \<rhd> Q) \ S = (\<box>a \<in> A - S \<rightarrow> (P a \ S)) \<rhd>
(Q \ S) \<sqinter> (\<sqinter>a \<in> A \<inter> S. (P a \ S))\<close>
if non_disjoint: \<open>A \<inter> S \<noteq> {}\<close>
|
?A \<inter> ?S \<noteq> {} \<Longrightarrow> Mprefix ?A ?P \<rhd> ?Q \ ?S = (\<box>a\<in>?A - ?S \<rightarrow> (?P a \ ?S)) \<rhd> (?Q \ ?S) \<sqinter> (\<sqinter> a\<in>?A \<inter> ?S. ?P a \ ?S)
|
?H1 x_1 x_2 \<noteq> ?H2 \<Longrightarrow> ?H3 (?H4 (?H5 x_1 x_3) x_4) x_2 = ?H4 (?H5 (?H6 x_1 x_2) (\<lambda>y_1. ?H3 (x_3 y_1) x_2)) (?H7 (?H3 x_4 x_2) (?H8 (?H1 x_1 x_2) (\<lambda>y_2. ?H3 (x_3 y_2) x_2)))
|
[
"GlobalNdet.GlobalNdet",
"Ndet.Ndet",
"Groups.minus_class.minus",
"Mprefix.Mprefix",
"Sliding.Sliding",
"Hiding.Hiding",
"Set.empty",
"Set.inter"
] |
[
"class minus =\n fixes minus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"-\" 65)",
"definition Sliding :: \\<open>'\\<alpha> process \\<Rightarrow> '\\<alpha> process \\<Rightarrow> '\\<alpha> process\\<close> (infixl \\<open>\\<rhd>\\<close> 78)\n where \\<open>P \\<rhd> Q \\<equiv> (P \\<box> Q) \\<sqinter> Q\\<close>\n\n\\<comment> \\<open>See if we want to define a MultiSliding operator like MultiSeq.\\<close>",
"abbreviation empty :: \"'a set\" (\"{}\")\n where \"{} \\<equiv> bot\"",
"abbreviation inter :: \"'a set \\<Rightarrow> 'a set \\<Rightarrow> 'a set\" (infixl \"\\<inter>\" 70)\n where \"(\\<inter>) \\<equiv> inf\""
] |
###template
?H1 x_1 x_2 \<noteq> ?H2 \<Longrightarrow> ?H3 (?H4 (?H5 x_1 x_3) x_4) x_2 = ?H4 (?H5 (?H6 x_1 x_2) (\<lambda>y_1. ?H3 (x_3 y_1) x_2)) (?H7 (?H3 x_4 x_2) (?H8 (?H1 x_1 x_2) (\<lambda>y_2. ?H3 (x_3 y_2) x_2)))
###symbols
GlobalNdet.GlobalNdet
Ndet.Ndet
Groups.minus_class.minus
Mprefix.Mprefix
Sliding.Sliding
Hiding.Hiding
Set.empty
Set.inter
###defs
class minus =
fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "-" 65)
definition Sliding :: \<open>'\<alpha> process \<Rightarrow> '\<alpha> process \<Rightarrow> '\<alpha> process\<close> (infixl \<open>\<rhd>\<close> 78)
where \<open>P \<rhd> Q \<equiv> (P \<box> Q) \<sqinter> Q\<close>
\<comment> \<open>See if we want to define a MultiSliding operator like MultiSeq.\<close>
abbreviation empty :: "'a set" ("{}")
where "{} \<equiv> bot"
abbreviation inter :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "\<inter>" 70)
where "(\<inter>) \<equiv> inf"
|
###output
?A \<inter> ?S \<noteq> {} \<Longrightarrow> Mprefix ?A ?P \<rhd> ?Q \ ?S = (\<box>a\<in>?A - ?S \<rightarrow> (?P a \ ?S)) \<rhd> (?Q \ ?S) \<sqinter> (\<sqinter> a\<in>?A \<inter> ?S. ?P a \ ?S)
###end
|
CZH_Elementary_Categories/czh_ecategories/CZH_SMC_FUNCT
|
CZH_SMC_FUNCT.smc_FUNCT_cs_intros(2)
| null |
?\<FF> : ?\<AA> \<mapsto>\<mapsto>\<^sub>C\<^bsub>?\<alpha>\<^esub> ?\<BB> \<Longrightarrow> cf_map ?\<FF> \<in>\<^sub>\<circ> cf_maps ?\<alpha> ?\<AA> ?\<BB>
|
?H1 x_1 x_2 x_3 x_4 \<Longrightarrow> ?H2 (?H3 x_4) (?H4 x_1 x_2 x_3)
|
[
"CZH_DG_FUNCT.cf_maps",
"CZH_DG_FUNCT.cf_map",
"CZH_Sets_Sets.vmember",
"CZH_ECAT_Functor.is_functor"
] |
[] |
###template
?H1 x_1 x_2 x_3 x_4 \<Longrightarrow> ?H2 (?H3 x_4) (?H4 x_1 x_2 x_3)
###symbols
CZH_DG_FUNCT.cf_maps
CZH_DG_FUNCT.cf_map
CZH_Sets_Sets.vmember
CZH_ECAT_Functor.is_functor
###defs
|
###output
?\<FF> : ?\<AA> \<mapsto>\<mapsto>\<^sub>C\<^bsub>?\<alpha>\<^esub> ?\<BB> \<Longrightarrow> cf_map ?\<FF> \<in>\<^sub>\<circ> cf_maps ?\<alpha> ?\<AA> ?\<BB>
###end
|
Jinja/BV/SemiType
|
SemiType.order_widen
|
lemma order_widen [intro,simp]:
"wf_prog m P \<Longrightarrow> order (subtype P) (types P)"
|
wf_prog ?m ?P \<Longrightarrow> order (subtype ?P) (types ?P)
|
?H1 x_1 x_2 \<Longrightarrow> ?H2 (?H3 x_2) (?H4 x_2)
|
[
"Decl.types",
"SemiType.subtype",
"Semilat.order",
"WellForm.wf_prog"
] |
[
"abbreviation\n \"types P == Collect (CONST is_type P)\"",
"definition\n wf_prog :: \"prog \\<Rightarrow> bool\" where\n \"wf_prog G = (let is = ifaces G; cs = classes G in\n ObjectC \\<in> set cs \\<and> \n (\\<forall> m\\<in>set Object_mdecls. accmodi m \\<noteq> Package) \\<and>\n (\\<forall>xn. SXcptC xn \\<in> set cs) \\<and>\n (\\<forall>i\\<in>set is. wf_idecl G i) \\<and> unique is \\<and>\n (\\<forall>c\\<in>set cs. wf_cdecl G c) \\<and> unique cs)\""
] |
###template
?H1 x_1 x_2 \<Longrightarrow> ?H2 (?H3 x_2) (?H4 x_2)
###symbols
Decl.types
SemiType.subtype
Semilat.order
WellForm.wf_prog
###defs
abbreviation
"types P == Collect (CONST is_type P)"
definition
wf_prog :: "prog \<Rightarrow> bool" where
"wf_prog G = (let is = ifaces G; cs = classes G in
ObjectC \<in> set cs \<and>
(\<forall> m\<in>set Object_mdecls. accmodi m \<noteq> Package) \<and>
(\<forall>xn. SXcptC xn \<in> set cs) \<and>
(\<forall>i\<in>set is. wf_idecl G i) \<and> unique is \<and>
(\<forall>c\<in>set cs. wf_cdecl G c) \<and> unique cs)"
|
###output
wf_prog ?m ?P \<Longrightarrow> order (subtype ?P) (types ?P)
###end
|
HOL-CSPM/CSPM
|
CSPM.prefix_MultiNdet_is_MultiDet_prefix
|
lemma prefix_MultiNdet_is_MultiDet_prefix:
\<open>A \<noteq> {} \<Longrightarrow> finite A \<Longrightarrow> (a \<rightarrow> (\<Sqinter> p \<in> A. P p) = \<^bold>\<box> p \<in> A. (a \<rightarrow> P p))\<close>
|
?A \<noteq> {} \<Longrightarrow> finite ?A \<Longrightarrow> ?a \<rightarrow> MultiNdet ?A ?P = \<^bold>\<box>p\<in>?A. ?a \<rightarrow> ?P p
|
\<lbrakk>x_1 \<noteq> ?H1; ?H2 x_1\<rbrakk> \<Longrightarrow> ?H3 x_2 (?H4 x_1 x_3) = ?H5 x_1 (\<lambda>y_1. ?H3 x_2 (x_3 y_1))
|
[
"MultiDet.MultiDet",
"MultiNdet.MultiNdet",
"Mprefix.write0",
"Finite_Set.finite",
"Set.empty"
] |
[
"definition MultiDet :: \\<open>['a set, 'a \\<Rightarrow> 'b process] \\<Rightarrow> 'b process\\<close>\n where \\<open>MultiDet A P = Finite_Set.fold (\\<lambda>a r. r \\<box> P a) STOP A\\<close>",
"definition MultiNdet :: \\<open>['a set, 'a \\<Rightarrow> 'b process] \\<Rightarrow> 'b process\\<close>\n where \\<open>MultiNdet A P = MultiNdet_list (SOME L. set L = A) P\\<close>",
"definition write0 :: \"['a, 'a process] \\<Rightarrow> 'a process\"\nwhere \"write0 a P \\<equiv> Mprefix {a} (\\<lambda> x. P)\"",
"class finite =\n assumes finite_UNIV: \"finite (UNIV :: 'a set)\"\nbegin",
"abbreviation empty :: \"'a set\" (\"{}\")\n where \"{} \\<equiv> bot\""
] |
###template
\<lbrakk>x_1 \<noteq> ?H1; ?H2 x_1\<rbrakk> \<Longrightarrow> ?H3 x_2 (?H4 x_1 x_3) = ?H5 x_1 (\<lambda>y_1. ?H3 x_2 (x_3 y_1))
###symbols
MultiDet.MultiDet
MultiNdet.MultiNdet
Mprefix.write0
Finite_Set.finite
Set.empty
###defs
definition MultiDet :: \<open>['a set, 'a \<Rightarrow> 'b process] \<Rightarrow> 'b process\<close>
where \<open>MultiDet A P = Finite_Set.fold (\<lambda>a r. r \<box> P a) STOP A\<close>
definition MultiNdet :: \<open>['a set, 'a \<Rightarrow> 'b process] \<Rightarrow> 'b process\<close>
where \<open>MultiNdet A P = MultiNdet_list (SOME L. set L = A) P\<close>
definition write0 :: "['a, 'a process] \<Rightarrow> 'a process"
where "write0 a P \<equiv> Mprefix {a} (\<lambda> x. P)"
class finite =
assumes finite_UNIV: "finite (UNIV :: 'a set)"
begin
abbreviation empty :: "'a set" ("{}")
where "{} \<equiv> bot"
|
###output
?A \<noteq> {} \<Longrightarrow> finite ?A \<Longrightarrow> ?a \<rightarrow> MultiNdet ?A ?P = \<^bold>\<box>p\<in>?A. ?a \<rightarrow> ?P p
###end
|
Consensus_Refined/MRU/CT_Proofs
|
CT_Proofs.step0_ref
|
lemma step0_ref:
"{ct_ref_rel}
(\<Union>r C. majorities.opt_mru_step0 r C),
CT_trans_step HOs HOs crds next0 send0 0 {> ct_ref_rel}"
|
\<forall>r. CT_commPerRd r (?HOs r) (?crds r) \<Longrightarrow> {ct_ref_rel} \<Union>r. \<Union> (range (majorities.opt_mru_step0 r)), CT_trans_step ?HOs ?HOs ?crds next0 send0 0 {> ct_ref_rel}
|
\<forall>y_0. ?H1 y_0 (x_1 y_0) (x_2 y_0) \<Longrightarrow> ?H2 ?H3 (?H4 (?H5 (\<lambda>y_1. ?H4 (?H6 (?H7 y_1))))) (?H8 x_1 x_1 x_2 ?H9 ?H10 ?H11) ?H3
|
[
"Groups.zero_class.zero",
"CT_Defs.send0",
"CT_Defs.next0",
"CT_Proofs.CT_trans_step",
"Three_Step_MRU.majorities.opt_mru_step0",
"Set.range",
"Complete_Lattices.Union",
"CT_Proofs.ct_ref_rel",
"Refinement.PO_rhoare",
"CT_Defs.CT_commPerRd"
] |
[
"class zero =\n fixes zero :: 'a (\"0\")",
"abbreviation range :: \"('a \\<Rightarrow> 'b) \\<Rightarrow> 'b set\" \\<comment> \\<open>of function\\<close>\n where \"range f \\<equiv> f ` UNIV\"",
"abbreviation Union :: \"'a set set \\<Rightarrow> 'a set\" (\"\\<Union>\")\n where \"\\<Union>S \\<equiv> \\<Squnion>S\"",
"definition\n PO_rhoare :: \n \"[('s \\<times> 't) set, ('s \\<times> 's) set, ('t \\<times> 't) set, ('s \\<times> 't) set] \\<Rightarrow> bool\"\n (\"(4{_} _, _ {> _})\" [0, 0, 0] 90)\nwhere\n \"{pre} Ra, Rc {> post} \\<equiv> pre O Rc \\<subseteq> Ra O post\""
] |
###template
\<forall>y_0. ?H1 y_0 (x_1 y_0) (x_2 y_0) \<Longrightarrow> ?H2 ?H3 (?H4 (?H5 (\<lambda>y_1. ?H4 (?H6 (?H7 y_1))))) (?H8 x_1 x_1 x_2 ?H9 ?H10 ?H11) ?H3
###symbols
Groups.zero_class.zero
CT_Defs.send0
CT_Defs.next0
CT_Proofs.CT_trans_step
Three_Step_MRU.majorities.opt_mru_step0
Set.range
Complete_Lattices.Union
CT_Proofs.ct_ref_rel
Refinement.PO_rhoare
CT_Defs.CT_commPerRd
###defs
class zero =
fixes zero :: 'a ("0")
abbreviation range :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b set" \<comment> \<open>of function\<close>
where "range f \<equiv> f ` UNIV"
abbreviation Union :: "'a set set \<Rightarrow> 'a set" ("\<Union>")
where "\<Union>S \<equiv> \<Squnion>S"
definition
PO_rhoare ::
"[('s \<times> 't) set, ('s \<times> 's) set, ('t \<times> 't) set, ('s \<times> 't) set] \<Rightarrow> bool"
("(4{_} _, _ {> _})" [0, 0, 0] 90)
where
"{pre} Ra, Rc {> post} \<equiv> pre O Rc \<subseteq> Ra O post"
|
###output
\<forall>r. CT_commPerRd r (?HOs r) (?crds r) \<Longrightarrow> {ct_ref_rel} \<Union>r. \<Union> (range (majorities.opt_mru_step0 r)), CT_trans_step ?HOs ?HOs ?crds next0 send0 0 {> ct_ref_rel}
###end
|
Chandy_Lamport/Example
|
Example.trace_snoc
| null |
trace ?c ?t ?c' \<Longrightarrow> next ?c' ?ev ?c'' \<Longrightarrow> trace ?c (?t @ [?ev]) ?c''
|
\<lbrakk>?H1 x_1 x_2 x_3; ?H2 x_3 x_4 x_5\<rbrakk> \<Longrightarrow> ?H1 x_1 (?H3 x_2 (?H4 x_4 ?H5)) x_5
|
[
"List.list.Nil",
"List.list.Cons",
"List.append",
"Example.next",
"Example.trace"
] |
[
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"primrec append :: \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" (infixr \"@\" 65) where\nappend_Nil: \"[] @ ys = ys\" |\nappend_Cons: \"(x#xs) @ ys = x # xs @ ys\""
] |
###template
\<lbrakk>?H1 x_1 x_2 x_3; ?H2 x_3 x_4 x_5\<rbrakk> \<Longrightarrow> ?H1 x_1 (?H3 x_2 (?H4 x_4 ?H5)) x_5
###symbols
List.list.Nil
List.list.Cons
List.append
Example.next
Example.trace
###defs
datatype (set: 'a) list =
Nil ("[]")
| Cons (hd: 'a) (tl: "'a list") (infixr "#" 65)
for
map: map
rel: list_all2
pred: list_all
where
"tl [] = []"
datatype (set: 'a) list =
Nil ("[]")
| Cons (hd: 'a) (tl: "'a list") (infixr "#" 65)
for
map: map
rel: list_all2
pred: list_all
where
"tl [] = []"
primrec append :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" (infixr "@" 65) where
append_Nil: "[] @ ys = ys" |
append_Cons: "(x#xs) @ ys = x # xs @ ys"
|
###output
trace ?c ?t ?c' \<Longrightarrow> next ?c' ?ev ?c'' \<Longrightarrow> trace ?c (?t @ [?ev]) ?c''
###end
|
Multirelations_Heterogeneous/Multirelations
|
Multirelations.ii_isotone
|
lemma ii_isotone:
"R \<subseteq> S \<Longrightarrow> P \<subseteq> Q \<Longrightarrow> R \<inter>\<inter> P \<subseteq> S \<inter>\<inter> Q"
|
?R \<subseteq> ?S \<Longrightarrow> ?P \<subseteq> ?Q \<Longrightarrow> ?R \<inter>\<inter> ?P \<subseteq> ?S \<inter>\<inter> ?Q
|
\<lbrakk>?H1 x_1 x_2; ?H1 x_3 x_4\<rbrakk> \<Longrightarrow> ?H1 (?H2 x_1 x_3) (?H2 x_2 x_4)
|
[
"Multirelations.inner_intersection",
"Set.subset_eq"
] |
[
"definition inner_intersection :: \"('a,'b) mrel \\<Rightarrow> ('a,'b) mrel \\<Rightarrow> ('a,'b) mrel\" (infixl \"\\<inter>\\<inter>\" 65) where\n \"R \\<inter>\\<inter> S \\<equiv> { (a,B) . \\<exists>C D . B = C \\<inter> D \\<and> (a,C) \\<in> R \\<and> (a,D) \\<in> S }\"",
"abbreviation subset_eq :: \"'a set \\<Rightarrow> 'a set \\<Rightarrow> bool\"\n where \"subset_eq \\<equiv> less_eq\""
] |
###template
\<lbrakk>?H1 x_1 x_2; ?H1 x_3 x_4\<rbrakk> \<Longrightarrow> ?H1 (?H2 x_1 x_3) (?H2 x_2 x_4)
###symbols
Multirelations.inner_intersection
Set.subset_eq
###defs
definition inner_intersection :: "('a,'b) mrel \<Rightarrow> ('a,'b) mrel \<Rightarrow> ('a,'b) mrel" (infixl "\<inter>\<inter>" 65) where
"R \<inter>\<inter> S \<equiv> { (a,B) . \<exists>C D . B = C \<inter> D \<and> (a,C) \<in> R \<and> (a,D) \<in> S }"
abbreviation subset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"
where "subset_eq \<equiv> less_eq"
|
###output
?R \<subseteq> ?S \<Longrightarrow> ?P \<subseteq> ?Q \<Longrightarrow> ?R \<inter>\<inter> ?P \<subseteq> ?S \<inter>\<inter> ?Q
###end
|
FO_Theory_Rewriting/Util/Multihole_Context
|
Multihole_Context.less_eq_mctxt_MVarE1
|
lemma less_eq_mctxt_MVarE1:
assumes "MVar v \<le> D"
obtains (MVar) "D = MVar v"
|
MVar ?v \<le> ?D \<Longrightarrow> (?D = MVar ?v \<Longrightarrow> ?thesis) \<Longrightarrow> ?thesis
|
\<lbrakk>?H1 x_1 \<le> x_2; x_2 = ?H1 x_1 \<Longrightarrow> x_3\<rbrakk> \<Longrightarrow> x_3
|
[
"Multihole_Context.mctxt.MVar"
] |
[] |
###template
\<lbrakk>?H1 x_1 \<le> x_2; x_2 = ?H1 x_1 \<Longrightarrow> x_3\<rbrakk> \<Longrightarrow> x_3
###symbols
Multihole_Context.mctxt.MVar
###defs
|
###output
MVar ?v \<le> ?D \<Longrightarrow> (?D = MVar ?v \<Longrightarrow> ?thesis) \<Longrightarrow> ?thesis
###end
|
Polynomial_Interpolation/Ring_Hom_Poly
|
Ring_Hom_Poly.degree_map_poly_le
|
lemma degree_map_poly_le: "degree (map_poly f p) \<le> degree p"
|
degree (map_poly ?f ?p) \<le> degree ?p
|
?H1 (?H2 x_1 x_2) \<le> ?H3 x_2
|
[
"Polynomial.map_poly",
"Polynomial.degree"
] |
[
"definition map_poly :: \"('a :: zero \\<Rightarrow> 'b :: zero) \\<Rightarrow> 'a poly \\<Rightarrow> 'b poly\"\n where \"map_poly f p = Poly (map f (coeffs p))\"",
"definition degree :: \"'a::zero poly \\<Rightarrow> nat\"\n where \"degree p = (LEAST n. \\<forall>i>n. coeff p i = 0)\""
] |
###template
?H1 (?H2 x_1 x_2) \<le> ?H3 x_2
###symbols
Polynomial.map_poly
Polynomial.degree
###defs
definition map_poly :: "('a :: zero \<Rightarrow> 'b :: zero) \<Rightarrow> 'a poly \<Rightarrow> 'b poly"
where "map_poly f p = Poly (map f (coeffs p))"
definition degree :: "'a::zero poly \<Rightarrow> nat"
where "degree p = (LEAST n. \<forall>i>n. coeff p i = 0)"
|
###output
degree (map_poly ?f ?p) \<le> degree ?p
###end
|
Poincare_Bendixson/Analysis_Misc
|
Analysis_Misc.eucl_eq_iff
|
lemma eucl_eq_iff: "x = y \<longleftrightarrow> (\<forall>i<DIM('a). nth_eucl x i = nth_eucl y i)"
for x y::"'a::executable_euclidean_space"
|
(?x = ?y) = (\<forall>i<DIM(?'a). ?x $\<^sub>e i = ?y $\<^sub>e i)
|
(x_1 = x_2) = (\<forall>y_0<?H1 ?H2. ?H3 x_1 y_0 = ?H3 x_2 y_0)
|
[
"Analysis_Misc.nth_eucl",
"Euclidean_Space.euclidean_space_class.Basis",
"Finite_Set.card"
] |
[
"definition nth_eucl :: \"'a::executable_euclidean_space \\<Rightarrow> nat \\<Rightarrow> real\" where\n \"nth_eucl x i = x \\<bullet> (Basis_list ! i)\"\n \\<comment> \\<open>TODO: why is that and some sort of \\<open>lambda_eucl\\<close> nowhere available?\\<close>",
"class euclidean_space = real_inner +\n fixes Basis :: \"'a set\"\n assumes nonempty_Basis [simp]: \"Basis \\<noteq> {}\"\n assumes finite_Basis [simp]: \"finite Basis\"\n assumes inner_Basis:\n \"\\<lbrakk>u \\<in> Basis; v \\<in> Basis\\<rbrakk> \\<Longrightarrow> inner u v = (if u = v then 1 else 0)\"\n assumes euclidean_all_zero_iff:\n \"(\\<forall>u\\<in>Basis. inner x u = 0) \\<longleftrightarrow> (x = 0)\""
] |
###template
(x_1 = x_2) = (\<forall>y_0<?H1 ?H2. ?H3 x_1 y_0 = ?H3 x_2 y_0)
###symbols
Analysis_Misc.nth_eucl
Euclidean_Space.euclidean_space_class.Basis
Finite_Set.card
###defs
definition nth_eucl :: "'a::executable_euclidean_space \<Rightarrow> nat \<Rightarrow> real" where
"nth_eucl x i = x \<bullet> (Basis_list ! i)"
\<comment> \<open>TODO: why is that and some sort of \<open>lambda_eucl\<close> nowhere available?\<close>
class euclidean_space = real_inner +
fixes Basis :: "'a set"
assumes nonempty_Basis [simp]: "Basis \<noteq> {}"
assumes finite_Basis [simp]: "finite Basis"
assumes inner_Basis:
"\<lbrakk>u \<in> Basis; v \<in> Basis\<rbrakk> \<Longrightarrow> inner u v = (if u = v then 1 else 0)"
assumes euclidean_all_zero_iff:
"(\<forall>u\<in>Basis. inner x u = 0) \<longleftrightarrow> (x = 0)"
|
###output
(?x = ?y) = (\<forall>i<DIM(?'a). ?x $\<^sub>e i = ?y $\<^sub>e i)
###end
|
Refine_Imperative_HOL/Sepref_Translate
|
Sepref_Translate.id_WHILEIT
|
lemma id_WHILEIT[id_rules]:
"PR_CONST (WHILEIT I) ::\<^sub>i TYPE(('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a nres) \<Rightarrow> 'a \<Rightarrow> 'a nres)"
|
PR_CONST WHILE\<^sub>T\<^bsup>?I\<^esup> ::\<^sub>i TYPE((?'a \<Rightarrow> bool) \<Rightarrow> (?'a \<Rightarrow> ?'a nres) \<Rightarrow> ?'a \<Rightarrow> ?'a nres)
|
?H1 (?H2 (?H3 x_1)) TYPE((?'a \<Rightarrow> bool) \<Rightarrow> (?'a \<Rightarrow> ?'a nres) \<Rightarrow> ?'a \<Rightarrow> ?'a nres)
|
[
"Refine_While.WHILEIT",
"Sepref_Id_Op.PR_CONST",
"Sepref_Id_Op.intf_type"
] |
[
"definition WHILEIT (\"WHILE\\<^sub>T\\<^bsup>_\\<^esup>\") where \n \"WHILEIT \\<equiv> iWHILEIT bind RETURN\"",
"definition intf_type :: \"'a \\<Rightarrow> 'b itself \\<Rightarrow> bool\" (infix \"::\\<^sub>i\" 10) where\n [simp]: \"c::\\<^sub>iI \\<equiv> True\""
] |
###template
?H1 (?H2 (?H3 x_1)) TYPE((?'a \<Rightarrow> bool) \<Rightarrow> (?'a \<Rightarrow> ?'a nres) \<Rightarrow> ?'a \<Rightarrow> ?'a nres)
###symbols
Refine_While.WHILEIT
Sepref_Id_Op.PR_CONST
Sepref_Id_Op.intf_type
###defs
definition WHILEIT ("WHILE\<^sub>T\<^bsup>_\<^esup>") where
"WHILEIT \<equiv> iWHILEIT bind RETURN"
definition intf_type :: "'a \<Rightarrow> 'b itself \<Rightarrow> bool" (infix "::\<^sub>i" 10) where
[simp]: "c::\<^sub>iI \<equiv> True"
|
###output
PR_CONST WHILE\<^sub>T\<^bsup>?I\<^esup> ::\<^sub>i TYPE((?'a \<Rightarrow> bool) \<Rightarrow> (?'a \<Rightarrow> ?'a nres) \<Rightarrow> ?'a \<Rightarrow> ?'a nres)
###end
|
IMP2/lib/IMP2_Aux_Lemmas
|
IMP2_Aux_Lemmas.mset_ran_eq_single_conv
|
lemma mset_ran_eq_single_conv: "mset_ran a r = {#x#} \<longleftrightarrow> (\<exists>i. r={i} \<and> x= a i)"
|
(mset_ran ?a ?r = {#?x#}) = (\<exists>i. ?r = {i} \<and> ?x = ?a i)
|
(?H1 x_1 x_2 = ?H2 x_3 ?H3) = (\<exists>y_0. x_2 = ?H4 y_0 ?H5 \<and> x_3 = x_1 y_0)
|
[
"Set.empty",
"Set.insert",
"Multiset.empty_mset",
"Multiset.add_mset",
"IMP2_Aux_Lemmas.mset_ran"
] |
[
"abbreviation empty :: \"'a set\" (\"{}\")\n where \"{} \\<equiv> bot\"",
"definition insert :: \"'a \\<Rightarrow> 'a set \\<Rightarrow> 'a set\"\n where insert_compr: \"insert a B = {x. x = a \\<or> x \\<in> B}\"",
"abbreviation empty_mset :: \\<open>'a multiset\\<close> (\\<open>{#}\\<close>)\n where \\<open>empty_mset \\<equiv> 0\\<close>"
] |
###template
(?H1 x_1 x_2 = ?H2 x_3 ?H3) = (\<exists>y_0. x_2 = ?H4 y_0 ?H5 \<and> x_3 = x_1 y_0)
###symbols
Set.empty
Set.insert
Multiset.empty_mset
Multiset.add_mset
IMP2_Aux_Lemmas.mset_ran
###defs
abbreviation empty :: "'a set" ("{}")
where "{} \<equiv> bot"
definition insert :: "'a \<Rightarrow> 'a set \<Rightarrow> 'a set"
where insert_compr: "insert a B = {x. x = a \<or> x \<in> B}"
abbreviation empty_mset :: \<open>'a multiset\<close> (\<open>{#}\<close>)
where \<open>empty_mset \<equiv> 0\<close>
|
###output
(mset_ran ?a ?r = {#?x#}) = (\<exists>i. ?r = {i} \<and> ?x = ?a i)
###end
|
CZH_Universal_Constructions/czh_ucategories/CZH_UCAT_Limit_IT
|
CZH_UCAT_Limit_IT.is_cat_obj_empty_initialE
| null |
?\<ZZ> : 0\<^sub>C\<^sub>F >\<^sub>C\<^sub>F\<^sub>.\<^sub>0 ?z : 0\<^sub>C \<mapsto>\<mapsto>\<^sub>C\<^bsub>?\<alpha>\<^esub> ?\<CC> \<Longrightarrow> (?\<ZZ> : cf_0 ?\<CC> >\<^sub>C\<^sub>F\<^sub>.\<^sub>c\<^sub>o\<^sub>l\<^sub>i\<^sub>m ?z : cat_0 \<mapsto>\<mapsto>\<^sub>C\<^bsub>?\<alpha>\<^esub> ?\<CC> \<Longrightarrow> ?W) \<Longrightarrow> ?W
|
\<lbrakk>?H1 x_1 x_2 x_3 x_4; ?H2 x_1 ?H3 x_2 (?H4 x_2) x_3 x_4 \<Longrightarrow> x_5\<rbrakk> \<Longrightarrow> x_5
|
[
"CZH_ECAT_Simple.cf_0",
"CZH_ECAT_Simple.cat_0",
"CZH_UCAT_Limit.is_cat_colimit",
"CZH_UCAT_Limit_IT.is_cat_obj_empty_initial"
] |
[] |
###template
\<lbrakk>?H1 x_1 x_2 x_3 x_4; ?H2 x_1 ?H3 x_2 (?H4 x_2) x_3 x_4 \<Longrightarrow> x_5\<rbrakk> \<Longrightarrow> x_5
###symbols
CZH_ECAT_Simple.cf_0
CZH_ECAT_Simple.cat_0
CZH_UCAT_Limit.is_cat_colimit
CZH_UCAT_Limit_IT.is_cat_obj_empty_initial
###defs
|
###output
?\<ZZ> : 0\<^sub>C\<^sub>F >\<^sub>C\<^sub>F\<^sub>.\<^sub>0 ?z : 0\<^sub>C \<mapsto>\<mapsto>\<^sub>C\<^bsub>?\<alpha>\<^esub> ?\<CC> \<Longrightarrow> (?\<ZZ> : cf_0 ?\<CC> >\<^sub>C\<^sub>F\<^sub>.\<^sub>c\<^sub>o\<^sub>l\<^sub>i\<^sub>m ?z : cat_0 \<mapsto>\<mapsto>\<^sub>C\<^bsub>?\<alpha>\<^esub> ?\<CC> \<Longrightarrow> ?W) \<Longrightarrow> ?W
###end
|
Universal_Turing_Machine/UF
|
UF.godel_code_eq_1
|
lemma godel_code_eq_1: "(godel_code nl = 1) = (nl = [])"
|
(godel_code ?nl = 1) = (?nl = [])
|
(?H1 x_1 = ?H2) = (x_1 = ?H3)
|
[
"List.list.Nil",
"Groups.one_class.one",
"UF.godel_code"
] |
[
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"class one =\n fixes one :: 'a (\"1\")",
"fun godel_code :: \"nat list \\<Rightarrow> nat\"\n where\n \"godel_code xs = (let lh = length xs in \n 2^lh * (godel_code' xs (Suc 0)))\""
] |
###template
(?H1 x_1 = ?H2) = (x_1 = ?H3)
###symbols
List.list.Nil
Groups.one_class.one
UF.godel_code
###defs
datatype (set: 'a) list =
Nil ("[]")
| Cons (hd: 'a) (tl: "'a list") (infixr "#" 65)
for
map: map
rel: list_all2
pred: list_all
where
"tl [] = []"
class one =
fixes one :: 'a ("1")
fun godel_code :: "nat list \<Rightarrow> nat"
where
"godel_code xs = (let lh = length xs in
2^lh * (godel_code' xs (Suc 0)))"
|
###output
(godel_code ?nl = 1) = (?nl = [])
###end
|
Deriving/Comparator_Generator/Compare_Instances
|
Compare_Instances.comparator_option_simps(1)
| null |
comparator_option ?comp\<^sub>'\<^sub>a None None = Eq
|
?H1 x_1 ?H2 ?H2 = ?H3
|
[
"Comparator.order.Eq",
"Option.option.None",
"Compare_Instances.comparator_option"
] |
[
"datatype 'a option =\n None\n | Some (the: 'a)"
] |
###template
?H1 x_1 ?H2 ?H2 = ?H3
###symbols
Comparator.order.Eq
Option.option.None
Compare_Instances.comparator_option
###defs
datatype 'a option =
None
| Some (the: 'a)
|
###output
comparator_option ?comp\<^sub>'\<^sub>a None None = Eq
###end
|
Decl_Sem_Fun_PL/ValuesFSetProps
|
ValuesFSetProps.le_any_fun_inv
| null |
?v \<sqsubseteq> VFun ?t \<Longrightarrow> (\<And>t1. ?v = VFun t1 \<Longrightarrow> fset t1 \<subseteq> fset ?t \<Longrightarrow> ?P) \<Longrightarrow> ?P
|
\<lbrakk>?H1 x_1 (?H2 x_2); \<And>y_0. \<lbrakk>x_1 = ?H2 y_0; ?H3 (?H4 y_0) (?H4 x_2)\<rbrakk> \<Longrightarrow> x_3\<rbrakk> \<Longrightarrow> x_3
|
[
"FSet.fset.fset",
"Set.subset_eq",
"ValuesFSet.val.VFun",
"ValuesFSet.val_le"
] |
[
"abbreviation subset_eq :: \"'a set \\<Rightarrow> 'a set \\<Rightarrow> bool\"\n where \"subset_eq \\<equiv> less_eq\"",
"datatype val = VNat nat | VFun \"(val \\<times> val) fset\"",
"inductive val_le :: \"val \\<Rightarrow> val \\<Rightarrow> bool\" (infix \"\\<sqsubseteq>\" 52) where\n vnat_le[intro!]: \"(VNat n) \\<sqsubseteq> (VNat n)\" |\n vfun_le[intro!]: \"fset t1 \\<subseteq> fset t2 \\<Longrightarrow> (VFun t1) \\<sqsubseteq> (VFun t2)\""
] |
###template
\<lbrakk>?H1 x_1 (?H2 x_2); \<And>y_0. \<lbrakk>x_1 = ?H2 y_0; ?H3 (?H4 y_0) (?H4 x_2)\<rbrakk> \<Longrightarrow> x_3\<rbrakk> \<Longrightarrow> x_3
###symbols
FSet.fset.fset
Set.subset_eq
ValuesFSet.val.VFun
ValuesFSet.val_le
###defs
abbreviation subset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"
where "subset_eq \<equiv> less_eq"
datatype val = VNat nat | VFun "(val \<times> val) fset"
inductive val_le :: "val \<Rightarrow> val \<Rightarrow> bool" (infix "\<sqsubseteq>" 52) where
vnat_le[intro!]: "(VNat n) \<sqsubseteq> (VNat n)" |
vfun_le[intro!]: "fset t1 \<subseteq> fset t2 \<Longrightarrow> (VFun t1) \<sqsubseteq> (VFun t2)"
|
###output
?v \<sqsubseteq> VFun ?t \<Longrightarrow> (\<And>t1. ?v = VFun t1 \<Longrightarrow> fset t1 \<subseteq> fset ?t \<Longrightarrow> ?P) \<Longrightarrow> ?P
###end
|
MSO_Regex_Equivalence/M2L_Equivalence_Checking
|
M2L_Equivalence_Checking.wf_pnPlus
| null |
M2L_Equivalence_Checking.wf ?\<Sigma> ?n ?r \<Longrightarrow> M2L_Equivalence_Checking.wf ?\<Sigma> ?n ?s \<Longrightarrow> M2L_Equivalence_Checking.wf ?\<Sigma> ?n (pnPlus ?r ?s)
|
\<lbrakk>?H1 x_1 x_2 x_3; ?H1 x_1 x_2 x_4\<rbrakk> \<Longrightarrow> ?H1 x_1 x_2 (?H2 x_3 x_4)
|
[
"PNormalization.pnPlus",
"M2L_Equivalence_Checking.wf"
] |
[
"fun pnPlus :: \"'a::linorder rexp \\<Rightarrow> 'a rexp \\<Rightarrow> 'a rexp\" where\n \"pnPlus Zero r = r\"\n| \"pnPlus r Zero = r\"\n(*<*)\n(*\n| \"pnPlus Full r = Full\"\n| \"pnPlus r Full = Full\"\n*)\n(*>*)\n| \"pnPlus (Plus r s) t = pnPlus r (pnPlus s t)\"\n| \"pnPlus r (Plus s t) =\n (if r = s then (Plus s t)\n else if r \\<le> s then Plus r (Plus s t)\n else Plus s (pnPlus r t))\"\n| \"pnPlus r s =\n (if r = s then r\n else if r \\<le> s then Plus r s\n else Plus s r)\""
] |
###template
\<lbrakk>?H1 x_1 x_2 x_3; ?H1 x_1 x_2 x_4\<rbrakk> \<Longrightarrow> ?H1 x_1 x_2 (?H2 x_3 x_4)
###symbols
PNormalization.pnPlus
M2L_Equivalence_Checking.wf
###defs
fun pnPlus :: "'a::linorder rexp \<Rightarrow> 'a rexp \<Rightarrow> 'a rexp" where
"pnPlus Zero r = r"
| "pnPlus r Zero = r"
(*<*)
(*
| "pnPlus Full r = Full"
| "pnPlus r Full = Full"
*)
(*>*)
| "pnPlus (Plus r s) t = pnPlus r (pnPlus s t)"
| "pnPlus r (Plus s t) =
(if r = s then (Plus s t)
else if r \<le> s then Plus r (Plus s t)
else Plus s (pnPlus r t))"
| "pnPlus r s =
(if r = s then r
else if r \<le> s then Plus r s
else Plus s r)"
|
###output
M2L_Equivalence_Checking.wf ?\<Sigma> ?n ?r \<Longrightarrow> M2L_Equivalence_Checking.wf ?\<Sigma> ?n ?s \<Longrightarrow> M2L_Equivalence_Checking.wf ?\<Sigma> ?n (pnPlus ?r ?s)
###end
|
Myhill-Nerode/Myhill
|
Myhill_1.Arden_removes_cl
| null |
rhss (Arden ?Y ?yrhs) = rhss ?yrhs - {?Y}
|
?H1 (?H2 x_1 x_2) = ?H3 (?H1 x_2) (?H4 x_1 ?H5)
|
[
"Set.empty",
"Set.insert",
"Groups.minus_class.minus",
"Myhill_1.Arden",
"Myhill_1.rhss"
] |
[
"abbreviation empty :: \"'a set\" (\"{}\")\n where \"{} \\<equiv> bot\"",
"definition insert :: \"'a \\<Rightarrow> 'a set \\<Rightarrow> 'a set\"\n where insert_compr: \"insert a B = {x. x = a \\<or> x \\<in> B}\"",
"class minus =\n fixes minus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"-\" 65)",
"definition \n \"Arden X rhs \\<equiv> \n Append_rexp_rhs (rhs - {Trn X r | r. Trn X r \\<in> rhs}) (Star (\\<Uplus> {r. Trn X r \\<in> rhs}))\"",
"definition \n \"rhss rhs \\<equiv> {X | X r. Trn X r \\<in> rhs}\""
] |
###template
?H1 (?H2 x_1 x_2) = ?H3 (?H1 x_2) (?H4 x_1 ?H5)
###symbols
Set.empty
Set.insert
Groups.minus_class.minus
Myhill_1.Arden
Myhill_1.rhss
###defs
abbreviation empty :: "'a set" ("{}")
where "{} \<equiv> bot"
definition insert :: "'a \<Rightarrow> 'a set \<Rightarrow> 'a set"
where insert_compr: "insert a B = {x. x = a \<or> x \<in> B}"
class minus =
fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "-" 65)
definition
"Arden X rhs \<equiv>
Append_rexp_rhs (rhs - {Trn X r | r. Trn X r \<in> rhs}) (Star (\<Uplus> {r. Trn X r \<in> rhs}))"
definition
"rhss rhs \<equiv> {X | X r. Trn X r \<in> rhs}"
|
###output
rhss (Arden ?Y ?yrhs) = rhss ?yrhs - {?Y}
###end
|
JiveDataStoreModel/Isabelle_Store/StoreProperties
|
StoreProperties.disj4'
|
lemma disj4': "\<lbrakk>disj (arrV T a) y s \<rbrakk>
\<Longrightarrow> disj (s@@(arrV T a).[i]) y s"
|
StoreProperties.disj (arrV ?T ?a) ?y ?s \<Longrightarrow> StoreProperties.disj (?s@@arrV ?T ?a.[?i]) ?y ?s
|
?H1 (?H2 x_1 x_2) x_3 x_4 \<Longrightarrow> ?H1 (?H3 x_4 (?H4 (?H2 x_1 x_2) x_5)) x_3 x_4
|
[
"Location.arr_loc",
"Store.access",
"Value.Value.arrV",
"StoreProperties.disj"
] |
[] |
###template
?H1 (?H2 x_1 x_2) x_3 x_4 \<Longrightarrow> ?H1 (?H3 x_4 (?H4 (?H2 x_1 x_2) x_5)) x_3 x_4
###symbols
Location.arr_loc
Store.access
Value.Value.arrV
StoreProperties.disj
###defs
|
###output
StoreProperties.disj (arrV ?T ?a) ?y ?s \<Longrightarrow> StoreProperties.disj (?s@@arrV ?T ?a.[?i]) ?y ?s
###end
|
Monomorphic_Monad/Monad_Overloading
|
Monad_Overloading.run_tell_envT
|
lemma run_tell_envT [simp]: "run_env (tell s m) r = tell s (run_env m r)"
|
run_env (tell ?s ?m) ?r = tell ?s (run_env ?m ?r)
|
?H1 (?H2 x_1 x_2) x_3 = ?H3 x_1 (?H1 x_2 x_3)
|
[
"Monad_Overloading.tell",
"Monomorphic_Monad.envT.run_env"
] |
[
"consts tell :: \"('w, 'm) tell\"",
"datatype ('r, 'm) envT = EnvT (run_env: \"'r \\<Rightarrow> 'm\")"
] |
###template
?H1 (?H2 x_1 x_2) x_3 = ?H3 x_1 (?H1 x_2 x_3)
###symbols
Monad_Overloading.tell
Monomorphic_Monad.envT.run_env
###defs
consts tell :: "('w, 'm) tell"
datatype ('r, 'm) envT = EnvT (run_env: "'r \<Rightarrow> 'm")
|
###output
run_env (tell ?s ?m) ?r = tell ?s (run_env ?m ?r)
###end
|
Robinson_Arithmetic/Robinson_Arithmetic
|
Robinson_Arithmetic.dsj_EH(4)
| null |
insert ?A (insert ?A1.0 (insert ?A2.0 (insert ?A3.0 ?H))) \<turnstile> ?B \<Longrightarrow> insert ?Ba (insert ?A1.0 (insert ?A2.0 (insert ?A3.0 ?H))) \<turnstile> ?B \<Longrightarrow> insert ?A1.0 (insert ?A2.0 (insert ?A3.0 (insert (?A OR ?Ba) ?H))) \<turnstile> ?B
|
\<lbrakk>?H1 (?H2 x_1 (?H2 x_2 (?H2 x_3 (?H2 x_4 x_5)))) x_6; ?H1 (?H2 x_7 (?H2 x_2 (?H2 x_3 (?H2 x_4 x_5)))) x_6\<rbrakk> \<Longrightarrow> ?H1 (?H2 x_2 (?H2 x_3 (?H2 x_4 (?H2 (?H3 x_1 x_7) x_5)))) x_6
|
[
"Robinson_Arithmetic.dsj",
"Set.insert",
"Robinson_Arithmetic.nprv"
] |
[
"definition insert :: \"'a \\<Rightarrow> 'a set \\<Rightarrow> 'a set\"\n where insert_compr: \"insert a B = {x. x = a \\<or> x \\<in> B}\"",
"inductive nprv :: \"fmla set \\<Rightarrow> fmla \\<Rightarrow> bool\" (infixl \"\\<turnstile>\" 55)\n where\n Hyp: \"A \\<in> H \\<Longrightarrow> H \\<turnstile> A\"\n | Q: \"A \\<in> Q_axioms \\<Longrightarrow> H \\<turnstile> A\"\n | Bool: \"A \\<in> boolean_axioms \\<Longrightarrow> H \\<turnstile> A\"\n | eql: \"A \\<in> equality_axioms \\<Longrightarrow> H \\<turnstile> A\"\n | Spec: \"A \\<in> special_axioms \\<Longrightarrow> H \\<turnstile> A\"\n | MP: \"H \\<turnstile> A IMP B \\<Longrightarrow> H' \\<turnstile> A \\<Longrightarrow> H \\<union> H' \\<turnstile> B\"\n | exiists: \"H \\<turnstile> A IMP B \\<Longrightarrow> atom i \\<sharp> B \\<Longrightarrow> \\<forall>C \\<in> H. atom i \\<sharp> C \\<Longrightarrow> H \\<turnstile> (exi i A) IMP B\""
] |
###template
\<lbrakk>?H1 (?H2 x_1 (?H2 x_2 (?H2 x_3 (?H2 x_4 x_5)))) x_6; ?H1 (?H2 x_7 (?H2 x_2 (?H2 x_3 (?H2 x_4 x_5)))) x_6\<rbrakk> \<Longrightarrow> ?H1 (?H2 x_2 (?H2 x_3 (?H2 x_4 (?H2 (?H3 x_1 x_7) x_5)))) x_6
###symbols
Robinson_Arithmetic.dsj
Set.insert
Robinson_Arithmetic.nprv
###defs
definition insert :: "'a \<Rightarrow> 'a set \<Rightarrow> 'a set"
where insert_compr: "insert a B = {x. x = a \<or> x \<in> B}"
inductive nprv :: "fmla set \<Rightarrow> fmla \<Rightarrow> bool" (infixl "\<turnstile>" 55)
where
Hyp: "A \<in> H \<Longrightarrow> H \<turnstile> A"
| Q: "A \<in> Q_axioms \<Longrightarrow> H \<turnstile> A"
| Bool: "A \<in> boolean_axioms \<Longrightarrow> H \<turnstile> A"
| eql: "A \<in> equality_axioms \<Longrightarrow> H \<turnstile> A"
| Spec: "A \<in> special_axioms \<Longrightarrow> H \<turnstile> A"
| MP: "H \<turnstile> A IMP B \<Longrightarrow> H' \<turnstile> A \<Longrightarrow> H \<union> H' \<turnstile> B"
| exiists: "H \<turnstile> A IMP B \<Longrightarrow> atom i \<sharp> B \<Longrightarrow> \<forall>C \<in> H. atom i \<sharp> C \<Longrightarrow> H \<turnstile> (exi i A) IMP B"
|
###output
insert ?A (insert ?A1.0 (insert ?A2.0 (insert ?A3.0 ?H))) \<turnstile> ?B \<Longrightarrow> insert ?Ba (insert ?A1.0 (insert ?A2.0 (insert ?A3.0 ?H))) \<turnstile> ?B \<Longrightarrow> insert ?A1.0 (insert ?A2.0 (insert ?A3.0 (insert (?A OR ?Ba) ?H))) \<turnstile> ?B
###end
|
Zeta_3_Irrational/Zeta_3_Irrational
|
Zeta_3_Irrational.beukers_integral1_different
|
lemma beukers_integral1_different:
assumes "r > s"
shows "beukers_integral1 r s = (\<Sum>k\<in>{s<..r}. 1 / k ^ 2) / (r - s)"
|
?s \<le> ?r \<Longrightarrow> ?s < ?r \<Longrightarrow> beukers_integral1 ?r ?s = (\<Sum>k\<in>{?s<..?r}. 1 / real (k\<^sup>2)) / real (?r - ?s)
|
\<lbrakk>x_1 \<le> x_2; x_1 < x_2\<rbrakk> \<Longrightarrow> ?H1 x_2 x_1 = ?H2 (?H3 (\<lambda>y_0. ?H2 ?H4 (?H5 (?H6 y_0))) (?H7 x_1 x_2)) (?H5 (?H8 x_2 x_1))
|
[
"Groups.minus_class.minus",
"Set_Interval.ord_class.greaterThanAtMost",
"Power.power_class.power2",
"Real.real",
"Groups.one_class.one",
"Groups_Big.comm_monoid_add_class.sum",
"Fields.inverse_class.inverse_divide",
"Zeta_3_Irrational.beukers_integral1"
] |
[
"class minus =\n fixes minus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"-\" 65)",
"primrec power :: \"'a \\<Rightarrow> nat \\<Rightarrow> 'a\" (infixr \"^\" 80)\n where\n power_0: \"a ^ 0 = 1\"\n | power_Suc: \"a ^ Suc n = a * a ^ n\"",
"abbreviation real :: \"nat \\<Rightarrow> real\"\n where \"real \\<equiv> of_nat\"",
"class one =\n fixes one :: 'a (\"1\")",
"class inverse = divide +\n fixes inverse :: \"'a \\<Rightarrow> 'a\"\nbegin",
"definition beukers_integral1 :: \"nat \\<Rightarrow> nat \\<Rightarrow> real\" where\n \"beukers_integral1 r s = (\\<integral>(x,y)\\<in>{0<..<1}\\<times>{0<..<1}. (-ln (x*y) / (1 - x*y) * x^r * y^s) \\<partial>lborel)\""
] |
###template
\<lbrakk>x_1 \<le> x_2; x_1 < x_2\<rbrakk> \<Longrightarrow> ?H1 x_2 x_1 = ?H2 (?H3 (\<lambda>y_0. ?H2 ?H4 (?H5 (?H6 y_0))) (?H7 x_1 x_2)) (?H5 (?H8 x_2 x_1))
###symbols
Groups.minus_class.minus
Set_Interval.ord_class.greaterThanAtMost
Power.power_class.power2
Real.real
Groups.one_class.one
Groups_Big.comm_monoid_add_class.sum
Fields.inverse_class.inverse_divide
Zeta_3_Irrational.beukers_integral1
###defs
class minus =
fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "-" 65)
primrec power :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixr "^" 80)
where
power_0: "a ^ 0 = 1"
| power_Suc: "a ^ Suc n = a * a ^ n"
abbreviation real :: "nat \<Rightarrow> real"
where "real \<equiv> of_nat"
class one =
fixes one :: 'a ("1")
class inverse = divide +
fixes inverse :: "'a \<Rightarrow> 'a"
begin
definition beukers_integral1 :: "nat \<Rightarrow> nat \<Rightarrow> real" where
"beukers_integral1 r s = (\<integral>(x,y)\<in>{0<..<1}\<times>{0<..<1}. (-ln (x*y) / (1 - x*y) * x^r * y^s) \<partial>lborel)"
|
###output
?s \<le> ?r \<Longrightarrow> ?s < ?r \<Longrightarrow> beukers_integral1 ?r ?s = (\<Sum>k\<in>{?s<..?r}. 1 / real (k\<^sup>2)) / real (?r - ?s)
###end
|
Grothendieck_Schemes/Topological_Space
|
Topological_Spaces.continuous_intros(94)
| null |
continuous_on ?s ?f \<Longrightarrow> \<forall>x\<in>?s. ?f x \<noteq> (0::?'b) \<Longrightarrow> continuous_on ?s (\<lambda>x. sgn (?f x))
|
\<lbrakk>?H1 x_1 x_2; \<forall>y_0\<in>x_1. x_2 y_0 \<noteq> ?H2\<rbrakk> \<Longrightarrow> ?H1 x_1 (\<lambda>y_1. ?H3 (x_2 y_1))
|
[
"Groups.sgn_class.sgn",
"Groups.zero_class.zero",
"Topological_Spaces.continuous_on"
] |
[
"class sgn =\n fixes sgn :: \"'a \\<Rightarrow> 'a\"",
"class zero =\n fixes zero :: 'a (\"0\")",
"definition continuous_on :: \"'a set \\<Rightarrow> ('a::topological_space \\<Rightarrow> 'b::topological_space) \\<Rightarrow> bool\"\n where \"continuous_on s f \\<longleftrightarrow> (\\<forall>x\\<in>s. (f \\<longlongrightarrow> f x) (at x within s))\""
] |
###template
\<lbrakk>?H1 x_1 x_2; \<forall>y_0\<in>x_1. x_2 y_0 \<noteq> ?H2\<rbrakk> \<Longrightarrow> ?H1 x_1 (\<lambda>y_1. ?H3 (x_2 y_1))
###symbols
Groups.sgn_class.sgn
Groups.zero_class.zero
Topological_Spaces.continuous_on
###defs
class sgn =
fixes sgn :: "'a \<Rightarrow> 'a"
class zero =
fixes zero :: 'a ("0")
definition continuous_on :: "'a set \<Rightarrow> ('a::topological_space \<Rightarrow> 'b::topological_space) \<Rightarrow> bool"
where "continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. (f \<longlongrightarrow> f x) (at x within s))"
|
###output
continuous_on ?s ?f \<Longrightarrow> \<forall>x\<in>?s. ?f x \<noteq> (0::?'b) \<Longrightarrow> continuous_on ?s (\<lambda>x. sgn (?f x))
###end
|
Coinductive/Coinductive
|
Coinductive_List.ltl_lappend
| null |
ltl (lappend ?xs ?ys) = (if lnull ?xs then ltl ?ys else lappend (ltl ?xs) ?ys)
|
?H1 (?H2 x_1 x_2) = (if ?H3 x_1 then ?H1 x_2 else ?H2 (?H1 x_1) x_2)
|
[
"Coinductive_List.llist.lnull",
"Coinductive_List.lappend",
"Coinductive_List.llist.ltl"
] |
[
"codatatype (lset: 'a) llist =\n lnull: LNil\n | LCons (lhd: 'a) (ltl: \"'a llist\")\nfor\n map: lmap\n rel: llist_all2\nwhere\n \"lhd LNil = undefined\"\n| \"ltl LNil = LNil\"",
"codatatype (lset: 'a) llist =\n lnull: LNil\n | LCons (lhd: 'a) (ltl: \"'a llist\")\nfor\n map: lmap\n rel: llist_all2\nwhere\n \"lhd LNil = undefined\"\n| \"ltl LNil = LNil\""
] |
###template
?H1 (?H2 x_1 x_2) = (if ?H3 x_1 then ?H1 x_2 else ?H2 (?H1 x_1) x_2)
###symbols
Coinductive_List.llist.lnull
Coinductive_List.lappend
Coinductive_List.llist.ltl
###defs
codatatype (lset: 'a) llist =
lnull: LNil
| LCons (lhd: 'a) (ltl: "'a llist")
for
map: lmap
rel: llist_all2
where
"lhd LNil = undefined"
| "ltl LNil = LNil"
codatatype (lset: 'a) llist =
lnull: LNil
| LCons (lhd: 'a) (ltl: "'a llist")
for
map: lmap
rel: llist_all2
where
"lhd LNil = undefined"
| "ltl LNil = LNil"
|
###output
ltl (lappend ?xs ?ys) = (if lnull ?xs then ltl ?ys else lappend (ltl ?xs) ?ys)
###end
|
Pi_Calculus/Strong_Late_Sim
|
Strong_Late_Sim.simCases
|
lemma simCases[case_names Bound Free]:
fixes P :: pi
and Q :: pi
and Rel :: "(pi \<times> pi) set"
assumes Bound: "\<And>a y Q'. \<lbrakk>Q \<longmapsto> a\<guillemotleft>y\<guillemotright> \<prec> Q'; y \<sharp> P\<rbrakk> \<Longrightarrow> \<exists>P'. P \<longmapsto> a\<guillemotleft>y\<guillemotright> \<prec> P' \<and> derivative P' Q' a y Rel"
and Free: "\<And>\<alpha> Q'. Q \<longmapsto> \<alpha> \<prec> Q' \<Longrightarrow> \<exists>P'. P \<longmapsto> \<alpha> \<prec> P' \<and> (P', Q') \<in> Rel"
shows "P \<leadsto>[Rel] Q"
|
(\<And>a y Q'. ?Q \<longmapsto> a\<guillemotleft>y\<guillemotright> \<prec> Q' \<Longrightarrow> y \<sharp> ?P \<Longrightarrow> \<exists>P'. ?P \<longmapsto> a\<guillemotleft>y\<guillemotright> \<prec> P' \<and> derivative P' Q' a y ?Rel) \<Longrightarrow> (\<And>\<alpha> Q'. ?Q \<longmapsto> \<alpha> \<prec> Q' \<Longrightarrow> \<exists>P'. ?P \<longmapsto> \<alpha> \<prec> P' \<and> (P', Q') \<in> ?Rel) \<Longrightarrow> ?P \<leadsto>[?Rel] ?Q
|
\<lbrakk>\<And>y_0 y_1 y_2. \<lbrakk>?H1 x_1 (?H2 y_0 y_1 y_2); ?H3 y_1 x_2\<rbrakk> \<Longrightarrow> \<exists>y_3. ?H1 x_2 (?H2 y_0 y_1 y_3) \<and> ?H4 y_3 y_2 y_0 y_1 x_3; \<And>y_4 y_5. ?H1 x_1 (?H5 y_4 y_5) \<Longrightarrow> \<exists>y_6. ?H1 x_2 (?H5 y_4 y_6) \<and> (y_6, y_5) \<in> x_3\<rbrakk> \<Longrightarrow> ?H6 x_2 x_3 x_1
|
[
"Strong_Late_Sim.simulation",
"Late_Semantics.residual.FreeR",
"Strong_Late_Sim.derivative",
"Nominal.fresh",
"Late_Semantics.residual.BoundR",
"Late_Semantics.transitions"
] |
[
"definition simulation :: \"pi \\<Rightarrow> (pi \\<times> pi) set \\<Rightarrow> pi \\<Rightarrow> bool\" (\"_ \\<leadsto>[_] _\" [80, 80, 80] 80) where\n \"P \\<leadsto>[Rel] Q \\<equiv> (\\<forall>a x Q'. Q \\<longmapsto>a\\<guillemotleft>x\\<guillemotright> \\<prec> Q' \\<and> x \\<sharp> P \\<longrightarrow> (\\<exists>P'. P \\<longmapsto>a\\<guillemotleft>x\\<guillemotright> \\<prec> P' \\<and> derivative P' Q' a x Rel)) \\<and>\n (\\<forall>\\<alpha> Q'. Q \\<longmapsto>\\<alpha> \\<prec> Q' \\<longrightarrow> (\\<exists>P'. P \\<longmapsto>\\<alpha> \\<prec> P' \\<and> (P', Q') \\<in> Rel))\"",
"definition derivative :: \"pi \\<Rightarrow> pi \\<Rightarrow> subject \\<Rightarrow> name \\<Rightarrow> (pi \\<times> pi) set \\<Rightarrow> bool\" where\n \"derivative P Q a x Rel \\<equiv> case a of InputS b \\<Rightarrow> (\\<forall>u. (P[x::=u], Q[x::=u]) \\<in> Rel)\n | BoundOutputS b \\<Rightarrow> (P, Q) \\<in> Rel\"",
"definition fresh :: \"'x \\<Rightarrow> 'a \\<Rightarrow> bool\" (\\<open>_ \\<sharp> _\\<close> [80,80] 80) where\n \"a \\<sharp> x \\<longleftrightarrow> a \\<notin> supp x\""
] |
###template
\<lbrakk>\<And>y_0 y_1 y_2. \<lbrakk>?H1 x_1 (?H2 y_0 y_1 y_2); ?H3 y_1 x_2\<rbrakk> \<Longrightarrow> \<exists>y_3. ?H1 x_2 (?H2 y_0 y_1 y_3) \<and> ?H4 y_3 y_2 y_0 y_1 x_3; \<And>y_4 y_5. ?H1 x_1 (?H5 y_4 y_5) \<Longrightarrow> \<exists>y_6. ?H1 x_2 (?H5 y_4 y_6) \<and> (y_6, y_5) \<in> x_3\<rbrakk> \<Longrightarrow> ?H6 x_2 x_3 x_1
###symbols
Strong_Late_Sim.simulation
Late_Semantics.residual.FreeR
Strong_Late_Sim.derivative
Nominal.fresh
Late_Semantics.residual.BoundR
Late_Semantics.transitions
###defs
definition simulation :: "pi \<Rightarrow> (pi \<times> pi) set \<Rightarrow> pi \<Rightarrow> bool" ("_ \<leadsto>[_] _" [80, 80, 80] 80) where
"P \<leadsto>[Rel] Q \<equiv> (\<forall>a x Q'. Q \<longmapsto>a\<guillemotleft>x\<guillemotright> \<prec> Q' \<and> x \<sharp> P \<longrightarrow> (\<exists>P'. P \<longmapsto>a\<guillemotleft>x\<guillemotright> \<prec> P' \<and> derivative P' Q' a x Rel)) \<and>
(\<forall>\<alpha> Q'. Q \<longmapsto>\<alpha> \<prec> Q' \<longrightarrow> (\<exists>P'. P \<longmapsto>\<alpha> \<prec> P' \<and> (P', Q') \<in> Rel))"
definition derivative :: "pi \<Rightarrow> pi \<Rightarrow> subject \<Rightarrow> name \<Rightarrow> (pi \<times> pi) set \<Rightarrow> bool" where
"derivative P Q a x Rel \<equiv> case a of InputS b \<Rightarrow> (\<forall>u. (P[x::=u], Q[x::=u]) \<in> Rel)
| BoundOutputS b \<Rightarrow> (P, Q) \<in> Rel"
definition fresh :: "'x \<Rightarrow> 'a \<Rightarrow> bool" (\<open>_ \<sharp> _\<close> [80,80] 80) where
"a \<sharp> x \<longleftrightarrow> a \<notin> supp x"
|
###output
(\<And>a y Q'. ?Q \<longmapsto> a\<guillemotleft>y\<guillemotright> \<prec> Q' \<Longrightarrow> y \<sharp> ?P \<Longrightarrow> \<exists>P'. ?P \<longmapsto> a\<guillemotleft>y\<guillemotright> \<prec> P' \<and> derivative P' Q' a y ?Rel) \<Longrightarrow> (\<And>\<alpha> Q'. ?Q \<longmapsto> \<alpha> \<prec> Q' \<Longrightarrow> \<exists>P'. ?P \<longmapsto> \<alpha> \<prec> P' \<and> (P', Q') \<in> ?Rel) \<Longrightarrow> ?P \<leadsto>[?Rel] ?Q
###end
|
CakeML/generated/LemExtraDefs
|
LemExtraDefs.is_digit_char_intro
|
lemma is_digit_char_intro[simp]:
"is_digit (char_to_digit c) = is_digit_char c"
|
is_digit (char_to_digit ?c) = is_digit_char ?c
|
?H1 (?H2 x_1) = ?H3 x_1
|
[
"LemExtraDefs.is_digit_char",
"LemExtraDefs.char_to_digit",
"LemExtraDefs.is_digit"
] |
[] |
###template
?H1 (?H2 x_1) = ?H3 x_1
###symbols
LemExtraDefs.is_digit_char
LemExtraDefs.char_to_digit
LemExtraDefs.is_digit
###defs
|
###output
is_digit (char_to_digit ?c) = is_digit_char ?c
###end
|
Jinja/J/WellTypeRT
|
WellTypeRT.WTs_implies_WTrts
| null |
?P,?E \<turnstile> ?es [::] ?Ts \<Longrightarrow> ?P,?E,?h \<turnstile> ?es [:] ?Ts
|
?H1 x_1 x_2 x_3 x_4 \<Longrightarrow> ?H2 x_1 x_2 x_5 x_3 x_4
|
[
"WellTypeRT.WTrts2",
"WellType.WTs"
] |
[] |
###template
?H1 x_1 x_2 x_3 x_4 \<Longrightarrow> ?H2 x_1 x_2 x_5 x_3 x_4
###symbols
WellTypeRT.WTrts2
WellType.WTs
###defs
|
###output
?P,?E \<turnstile> ?es [::] ?Ts \<Longrightarrow> ?P,?E,?h \<turnstile> ?es [:] ?Ts
###end
|
Gale_Shapley/Gale_Shapley2
|
Gale_Shapley2.list_array
|
lemma list_array: "list (array x n) = replicate n x"
|
list (array ?x ?n) = replicate ?n ?x
|
?H1 (?H2 x_1 x_2) = ?H3 x_2 x_1
|
[
"List.replicate",
"Gale_Shapley2.array",
"Gale_Shapley2.list"
] |
[
"primrec replicate :: \"nat \\<Rightarrow> 'a \\<Rightarrow> 'a list\" where\nreplicate_0: \"replicate 0 x = []\" |\nreplicate_Suc: \"replicate (Suc n) x = x # replicate n x\"",
"abbreviation \"array \\<equiv> new_array\"",
"abbreviation \"list \\<equiv> list_of_array\""
] |
###template
?H1 (?H2 x_1 x_2) = ?H3 x_2 x_1
###symbols
List.replicate
Gale_Shapley2.array
Gale_Shapley2.list
###defs
primrec replicate :: "nat \<Rightarrow> 'a \<Rightarrow> 'a list" where
replicate_0: "replicate 0 x = []" |
replicate_Suc: "replicate (Suc n) x = x # replicate n x"
abbreviation "array \<equiv> new_array"
abbreviation "list \<equiv> list_of_array"
|
###output
list (array ?x ?n) = replicate ?n ?x
###end
|
UTP/toolkit/List_Extra
|
List_Extra.nth_el_appendl
|
lemma nth_el_appendl[simp]: "i < length xs \<Longrightarrow> (xs @ ys)\<langle>i\<rangle>\<^sub>l = xs\<langle>i\<rangle>\<^sub>l"
|
?i < length ?xs \<Longrightarrow> (?xs @ ?ys)\<langle>?i\<rangle>\<^sub>l = ?xs\<langle>?i\<rangle>\<^sub>l
|
x_1 < ?H1 x_2 \<Longrightarrow> ?H2 (?H3 x_2 x_3) x_1 = ?H2 x_2 x_1
|
[
"List.append",
"List_Extra.nth_el",
"List.length"
] |
[
"primrec append :: \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" (infixr \"@\" 65) where\nappend_Nil: \"[] @ ys = ys\" |\nappend_Cons: \"(x#xs) @ ys = x # xs @ ys\"",
"abbreviation length :: \"'a list \\<Rightarrow> nat\" where\n\"length \\<equiv> size\""
] |
###template
x_1 < ?H1 x_2 \<Longrightarrow> ?H2 (?H3 x_2 x_3) x_1 = ?H2 x_2 x_1
###symbols
List.append
List_Extra.nth_el
List.length
###defs
primrec append :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" (infixr "@" 65) where
append_Nil: "[] @ ys = ys" |
append_Cons: "(x#xs) @ ys = x # xs @ ys"
abbreviation length :: "'a list \<Rightarrow> nat" where
"length \<equiv> size"
|
###output
?i < length ?xs \<Longrightarrow> (?xs @ ?ys)\<langle>?i\<rangle>\<^sub>l = ?xs\<langle>?i\<rangle>\<^sub>l
###end
|
FSM_Tests/Prime_Transformation
|
Prime_Transformation.uint64_nat_bij
|
lemma uint64_nat_bij : "(x :: nat) < 2^64 \<Longrightarrow> nat_of_uint64 (uint64_of_nat x) = x"
|
?x < 2 ^ 64 \<Longrightarrow> nat_of_uint64 (uint64_of_nat ?x) = ?x
|
x_1 < ?H1 (?H2 (?H3 ?H4)) (?H2 (?H3 (?H3 (?H3 (?H3 (?H3 (?H3 ?H4))))))) \<Longrightarrow> ?H5 (?H6 x_1) = x_1
|
[
"Uint64.uint64_of_nat",
"Uint64.nat_of_uint64",
"Num.num.One",
"Num.num.Bit0",
"Num.numeral_class.numeral",
"Power.power_class.power"
] |
[
"datatype num = One | Bit0 num | Bit1 num",
"datatype num = One | Bit0 num | Bit1 num",
"primrec numeral :: \"num \\<Rightarrow> 'a\"\n where\n numeral_One: \"numeral One = 1\"\n | numeral_Bit0: \"numeral (Bit0 n) = numeral n + numeral n\"\n | numeral_Bit1: \"numeral (Bit1 n) = numeral n + numeral n + 1\"",
"primrec power :: \"'a \\<Rightarrow> nat \\<Rightarrow> 'a\" (infixr \"^\" 80)\n where\n power_0: \"a ^ 0 = 1\"\n | power_Suc: \"a ^ Suc n = a * a ^ n\""
] |
###template
x_1 < ?H1 (?H2 (?H3 ?H4)) (?H2 (?H3 (?H3 (?H3 (?H3 (?H3 (?H3 ?H4))))))) \<Longrightarrow> ?H5 (?H6 x_1) = x_1
###symbols
Uint64.uint64_of_nat
Uint64.nat_of_uint64
Num.num.One
Num.num.Bit0
Num.numeral_class.numeral
Power.power_class.power
###defs
datatype num = One | Bit0 num | Bit1 num
datatype num = One | Bit0 num | Bit1 num
primrec numeral :: "num \<Rightarrow> 'a"
where
numeral_One: "numeral One = 1"
| numeral_Bit0: "numeral (Bit0 n) = numeral n + numeral n"
| numeral_Bit1: "numeral (Bit1 n) = numeral n + numeral n + 1"
primrec power :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixr "^" 80)
where
power_0: "a ^ 0 = 1"
| power_Suc: "a ^ Suc n = a * a ^ n"
|
###output
?x < 2 ^ 64 \<Longrightarrow> nat_of_uint64 (uint64_of_nat ?x) = ?x
###end
|
Stateful_Protocol_Composition_and_Typing/Messages
|
Messages.nonvar_term_has_composed_shallow_term
|
lemma nonvar_term_has_composed_shallow_term:
fixes t::"('f,'v) term"
assumes "\<not>(\<exists>x. t = Var x)"
shows "\<exists>f T. Fun f T \<sqsubseteq> t \<and> (\<forall>s \<in> set T. (\<exists>c. s = Fun c []) \<or> (\<exists>x. s = Var x))"
|
\<nexists>x. ?t = Var x \<Longrightarrow> \<exists>f T. Fun f T \<sqsubseteq> ?t \<and> (\<forall>s\<in>set T. (\<exists>c. s = Fun c []) \<or> (\<exists>x. s = Var x))
|
\<nexists>y_0. x_1 = ?H1 y_0 \<Longrightarrow> \<exists>y_1 y_2. ?H2 (?H3 y_1 y_2) x_1 \<and> (\<forall>y_3\<in>?H4 y_2. (\<exists>y_4. y_3 = ?H3 y_4 ?H5) \<or> (\<exists>y_5. y_3 = ?H1 y_5))
|
[
"List.list.Nil",
"List.list.set",
"Term.term.Fun",
"Messages.subtermeq",
"Term.term.Var"
] |
[
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"type_synonym \"term\" = \"(expr+stmt,var,expr list) sum3\"",
"abbreviation subtermeq (infix \"\\<sqsubseteq>\" 50) where \"t' \\<sqsubseteq> t \\<equiv> (t' \\<in> subterms t)\"",
"type_synonym \"term\" = \"(expr+stmt,var,expr list) sum3\""
] |
###template
\<nexists>y_0. x_1 = ?H1 y_0 \<Longrightarrow> \<exists>y_1 y_2. ?H2 (?H3 y_1 y_2) x_1 \<and> (\<forall>y_3\<in>?H4 y_2. (\<exists>y_4. y_3 = ?H3 y_4 ?H5) \<or> (\<exists>y_5. y_3 = ?H1 y_5))
###symbols
List.list.Nil
List.list.set
Term.term.Fun
Messages.subtermeq
Term.term.Var
###defs
datatype (set: 'a) list =
Nil ("[]")
| Cons (hd: 'a) (tl: "'a list") (infixr "#" 65)
for
map: map
rel: list_all2
pred: list_all
where
"tl [] = []"
datatype (set: 'a) list =
Nil ("[]")
| Cons (hd: 'a) (tl: "'a list") (infixr "#" 65)
for
map: map
rel: list_all2
pred: list_all
where
"tl [] = []"
type_synonym "term" = "(expr+stmt,var,expr list) sum3"
abbreviation subtermeq (infix "\<sqsubseteq>" 50) where "t' \<sqsubseteq> t \<equiv> (t' \<in> subterms t)"
type_synonym "term" = "(expr+stmt,var,expr list) sum3"
|
###output
\<nexists>x. ?t = Var x \<Longrightarrow> \<exists>f T. Fun f T \<sqsubseteq> ?t \<and> (\<forall>s\<in>set T. (\<exists>c. s = Fun c []) \<or> (\<exists>x. s = Var x))
###end
|
Planarity_Certificates/Planarity/Permutations_2
|
Permutations_2.orbit_perm_swap
|
lemma orbit_perm_swap: "orbit (perm_swap a b f) x = (a \<rightleftharpoons>\<^sub>F b) ` orbit f ((a \<rightleftharpoons>\<^sub>F b) x)"
|
orbit (perm_swap ?a ?b ?f) ?x = (?a \<rightleftharpoons>\<^sub>F ?b) ` orbit ?f ((?a \<rightleftharpoons>\<^sub>F ?b) ?x)
|
?H1 (?H2 x_1 x_2 x_3) x_4 = ?H3 (?H4 x_1 x_2) (?H1 x_3 (?H4 x_1 x_2 x_4))
|
[
"Permutations_2.funswapid",
"Set.image",
"Permutations_2.perm_swap",
"Orbits.orbit"
] |
[
"definition image :: \"('a \\<Rightarrow> 'b) \\<Rightarrow> 'a set \\<Rightarrow> 'b set\" (infixr \"`\" 90)\n where \"f ` A = {y. \\<exists>x\\<in>A. y = f x}\"",
"inductive_set orbit :: \"('a \\<Rightarrow> 'a) \\<Rightarrow> 'a \\<Rightarrow> 'a set\" for f x where\n base: \"f x \\<in> orbit f x\" |\n step: \"y \\<in> orbit f x \\<Longrightarrow> f y \\<in> orbit f x\""
] |
###template
?H1 (?H2 x_1 x_2 x_3) x_4 = ?H3 (?H4 x_1 x_2) (?H1 x_3 (?H4 x_1 x_2 x_4))
###symbols
Permutations_2.funswapid
Set.image
Permutations_2.perm_swap
Orbits.orbit
###defs
definition image :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'b set" (infixr "`" 90)
where "f ` A = {y. \<exists>x\<in>A. y = f x}"
inductive_set orbit :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a set" for f x where
base: "f x \<in> orbit f x" |
step: "y \<in> orbit f x \<Longrightarrow> f y \<in> orbit f x"
|
###output
orbit (perm_swap ?a ?b ?f) ?x = (?a \<rightleftharpoons>\<^sub>F ?b) ` orbit ?f ((?a \<rightleftharpoons>\<^sub>F ?b) ?x)
###end
|
Attack_Trees/AT
|
AT.length_last
|
lemma length_last[rule_format]: "(l @ [x]) ! (length (l @ [x]) - 1) = x"
|
(?l @ [?x]) ! (length (?l @ [?x]) - 1) = ?x
|
?H1 (?H2 x_1 (?H3 x_2 ?H4)) (?H5 (?H6 (?H2 x_1 (?H3 x_2 ?H4))) ?H7) = x_2
|
[
"Groups.one_class.one",
"List.length",
"Groups.minus_class.minus",
"List.list.Nil",
"List.list.Cons",
"List.append",
"List.nth"
] |
[
"class one =\n fixes one :: 'a (\"1\")",
"abbreviation length :: \"'a list \\<Rightarrow> nat\" where\n\"length \\<equiv> size\"",
"class minus =\n fixes minus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"-\" 65)",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"primrec append :: \"'a list \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" (infixr \"@\" 65) where\nappend_Nil: \"[] @ ys = ys\" |\nappend_Cons: \"(x#xs) @ ys = x # xs @ ys\"",
"primrec (nonexhaustive) nth :: \"'a list => nat => 'a\" (infixl \"!\" 100) where\nnth_Cons: \"(x # xs) ! n = (case n of 0 \\<Rightarrow> x | Suc k \\<Rightarrow> xs ! k)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>"
] |
###template
?H1 (?H2 x_1 (?H3 x_2 ?H4)) (?H5 (?H6 (?H2 x_1 (?H3 x_2 ?H4))) ?H7) = x_2
###symbols
Groups.one_class.one
List.length
Groups.minus_class.minus
List.list.Nil
List.list.Cons
List.append
List.nth
###defs
class one =
fixes one :: 'a ("1")
abbreviation length :: "'a list \<Rightarrow> nat" where
"length \<equiv> size"
class minus =
fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "-" 65)
datatype (set: 'a) list =
Nil ("[]")
| Cons (hd: 'a) (tl: "'a list") (infixr "#" 65)
for
map: map
rel: list_all2
pred: list_all
where
"tl [] = []"
datatype (set: 'a) list =
Nil ("[]")
| Cons (hd: 'a) (tl: "'a list") (infixr "#" 65)
for
map: map
rel: list_all2
pred: list_all
where
"tl [] = []"
primrec append :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" (infixr "@" 65) where
append_Nil: "[] @ ys = ys" |
append_Cons: "(x#xs) @ ys = x # xs @ ys"
primrec (nonexhaustive) nth :: "'a list => nat => 'a" (infixl "!" 100) where
nth_Cons: "(x # xs) ! n = (case n of 0 \<Rightarrow> x | Suc k \<Rightarrow> xs ! k)"
\<comment> \<open>Warning: simpset does not contain this definition, but separate
theorems for \<open>n = 0\<close> and \<open>n = Suc k\<close>\<close>
|
###output
(?l @ [?x]) ! (length (?l @ [?x]) - 1) = ?x
###end
|
JinjaThreads/Compiler/J1
|
J1State.call1_callE
| null |
call1 (?obj\<bullet>?M(?pns)) = \<lfloor>(?a, ?M', ?vs)\<rfloor> \<Longrightarrow> (call1 ?obj = \<lfloor>(?a, ?M', ?vs)\<rfloor> \<Longrightarrow> ?thesis) \<Longrightarrow> (\<And>v. ?obj = Val v \<Longrightarrow> calls1 ?pns = \<lfloor>(?a, ?M', ?vs)\<rfloor> \<Longrightarrow> ?thesis) \<Longrightarrow> (?obj = addr ?a \<Longrightarrow> ?pns = map Val ?vs \<Longrightarrow> ?M = ?M' \<Longrightarrow> ?thesis) \<Longrightarrow> ?thesis
|
\<lbrakk>?H1 (?H2 x_1 x_2 x_3) = ?H3 (x_4, x_5, x_6); ?H1 x_1 = ?H3 (x_4, x_5, x_6) \<Longrightarrow> x_7; \<And>y_0. \<lbrakk>x_1 = ?H4 y_0; ?H5 x_3 = ?H3 (x_4, x_5, x_6)\<rbrakk> \<Longrightarrow> x_7; \<lbrakk>x_1 = ?H6 x_4; x_3 = ?H7 ?H4 x_6; x_2 = x_5\<rbrakk> \<Longrightarrow> x_7\<rbrakk> \<Longrightarrow> x_7
|
[
"List.list.map",
"Expr.addr",
"J1State.calls1",
"Expr.exp.Val",
"Option.option.Some",
"Expr.exp.Call",
"J1State.call1"
] |
[
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"datatype 'a option =\n None\n | Some (the: 'a)"
] |
###template
\<lbrakk>?H1 (?H2 x_1 x_2 x_3) = ?H3 (x_4, x_5, x_6); ?H1 x_1 = ?H3 (x_4, x_5, x_6) \<Longrightarrow> x_7; \<And>y_0. \<lbrakk>x_1 = ?H4 y_0; ?H5 x_3 = ?H3 (x_4, x_5, x_6)\<rbrakk> \<Longrightarrow> x_7; \<lbrakk>x_1 = ?H6 x_4; x_3 = ?H7 ?H4 x_6; x_2 = x_5\<rbrakk> \<Longrightarrow> x_7\<rbrakk> \<Longrightarrow> x_7
###symbols
List.list.map
Expr.addr
J1State.calls1
Expr.exp.Val
Option.option.Some
Expr.exp.Call
J1State.call1
###defs
datatype (set: 'a) list =
Nil ("[]")
| Cons (hd: 'a) (tl: "'a list") (infixr "#" 65)
for
map: map
rel: list_all2
pred: list_all
where
"tl [] = []"
datatype 'a option =
None
| Some (the: 'a)
|
###output
call1 (?obj\<bullet>?M(?pns)) = \<lfloor>(?a, ?M', ?vs)\<rfloor> \<Longrightarrow> (call1 ?obj = \<lfloor>(?a, ?M', ?vs)\<rfloor> \<Longrightarrow> ?thesis) \<Longrightarrow> (\<And>v. ?obj = Val v \<Longrightarrow> calls1 ?pns = \<lfloor>(?a, ?M', ?vs)\<rfloor> \<Longrightarrow> ?thesis) \<Longrightarrow> (?obj = addr ?a \<Longrightarrow> ?pns = map Val ?vs \<Longrightarrow> ?M = ?M' \<Longrightarrow> ?thesis) \<Longrightarrow> ?thesis
###end
|
UTP/utp/utp_pred
|
utp_pred.subst_not
|
lemma subst_not [usubst]: "\<sigma> \<dagger> (\<not> P) = (\<not> \<sigma> \<dagger> P)"
|
?\<sigma> \<dagger> (\<not> ?P) = (\<not> ?\<sigma> \<dagger> ?P)
|
?H1 x_1 (?H2 x_2) = ?H3 (?H1 x_1 x_2)
|
[
"utp_pred.not_upred",
"utp_subst.subst"
] |
[] |
###template
?H1 x_1 (?H2 x_2) = ?H3 (?H1 x_1 x_2)
###symbols
utp_pred.not_upred
utp_subst.subst
###defs
|
###output
?\<sigma> \<dagger> (\<not> ?P) = (\<not> ?\<sigma> \<dagger> ?P)
###end
|
Abstract-Hoare-Logics/Procs/PsHoareTotal
|
PsHoareTotal.MGT_implies_complete
|
lemma MGT_implies_complete:
"{} |\<turnstile>\<^sub>t {MGT\<^sub>t c} \<Longrightarrow> {} \<Turnstile>\<^sub>t {P}c{Q} \<Longrightarrow> {} \<turnstile>\<^sub>t {P}c{Q::state assn}"
|
{} |\<turnstile>\<^sub>t {MGT\<^sub>t ?c} \<Longrightarrow> {} \<Turnstile>\<^sub>t {?P} ?c {?Q} \<Longrightarrow> {} \<turnstile>\<^sub>t {?P} ?c {?Q}
|
\<lbrakk>?H1 ?H2 (?H3 (?H4 x_1) ?H2); ?H5 ?H2 x_2 x_1 x_3\<rbrakk> \<Longrightarrow> ?H6 ?H2 x_2 x_1 x_3
|
[
"PsHoareTotal.thoare'",
"PsHoareTotal.ctvalid",
"PsHoareTotal.MGT\\<^sub>t",
"Set.insert",
"Set.empty",
"PsHoareTotal.thoare"
] |
[
"definition insert :: \"'a \\<Rightarrow> 'a set \\<Rightarrow> 'a set\"\n where insert_compr: \"insert a B = {x. x = a \\<or> x \\<in> B}\"",
"abbreviation empty :: \"'a set\" (\"{}\")\n where \"{} \\<equiv> bot\""
] |
###template
\<lbrakk>?H1 ?H2 (?H3 (?H4 x_1) ?H2); ?H5 ?H2 x_2 x_1 x_3\<rbrakk> \<Longrightarrow> ?H6 ?H2 x_2 x_1 x_3
###symbols
PsHoareTotal.thoare'
PsHoareTotal.ctvalid
PsHoareTotal.MGT\<^sub>t
Set.insert
Set.empty
PsHoareTotal.thoare
###defs
definition insert :: "'a \<Rightarrow> 'a set \<Rightarrow> 'a set"
where insert_compr: "insert a B = {x. x = a \<or> x \<in> B}"
abbreviation empty :: "'a set" ("{}")
where "{} \<equiv> bot"
|
###output
{} |\<turnstile>\<^sub>t {MGT\<^sub>t ?c} \<Longrightarrow> {} \<Turnstile>\<^sub>t {?P} ?c {?Q} \<Longrightarrow> {} \<turnstile>\<^sub>t {?P} ?c {?Q}
###end
|
DiskPaxos/DiskPaxos_Inv3
|
DiskPaxos_Inv3.InitPhase_HInv3_q
|
lemma InitPhase_HInv3_q:
"\<lbrakk> InitializePhase s s' q ; HInv3_L s' p q d \<rbrakk> \<Longrightarrow> HInv3_R s' p q d"
|
InitializePhase ?s ?s' ?q \<Longrightarrow> HInv3_L ?s' ?p ?q ?d \<Longrightarrow> HInv3_R ?s' ?p ?q ?d
|
\<lbrakk>?H1 x_1 x_2 x_3; ?H2 x_2 x_4 x_3 x_5\<rbrakk> \<Longrightarrow> ?H3 x_2 x_4 x_3 x_5
|
[
"DiskPaxos_Inv3.HInv3_R",
"DiskPaxos_Inv3.HInv3_L",
"DiskPaxos_Model.InitializePhase"
] |
[
"definition HInv3_R :: \"state \\<Rightarrow> Proc \\<Rightarrow> Proc \\<Rightarrow> Disk \\<Rightarrow> bool\"\nwhere\n \"HInv3_R s p q d = (\\<lparr>block= dblock s q, proc= q\\<rparr> \\<in> blocksRead s p d\n \\<or> \\<lparr>block= dblock s p, proc= p\\<rparr> \\<in> blocksRead s q d)\"",
"definition HInv3_L :: \"state \\<Rightarrow> Proc \\<Rightarrow> Proc \\<Rightarrow> Disk \\<Rightarrow> bool\"\nwhere\n \"HInv3_L s p q d = (phase s p \\<in> {1,2}\n \\<and> phase s q \\<in> {1,2} \n \\<and> hasRead s p d q\n \\<and> hasRead s q d p)\"",
"definition InitializePhase :: \"state \\<Rightarrow> state \\<Rightarrow> Proc \\<Rightarrow> bool\"\n where\n \"InitializePhase s s' p =\n (disksWritten s' = (disksWritten s)(p := {})\n & blocksRead s' = (blocksRead s)(p := (\\<lambda>d. {})))\""
] |
###template
\<lbrakk>?H1 x_1 x_2 x_3; ?H2 x_2 x_4 x_3 x_5\<rbrakk> \<Longrightarrow> ?H3 x_2 x_4 x_3 x_5
###symbols
DiskPaxos_Inv3.HInv3_R
DiskPaxos_Inv3.HInv3_L
DiskPaxos_Model.InitializePhase
###defs
definition HInv3_R :: "state \<Rightarrow> Proc \<Rightarrow> Proc \<Rightarrow> Disk \<Rightarrow> bool"
where
"HInv3_R s p q d = (\<lparr>block= dblock s q, proc= q\<rparr> \<in> blocksRead s p d
\<or> \<lparr>block= dblock s p, proc= p\<rparr> \<in> blocksRead s q d)"
definition HInv3_L :: "state \<Rightarrow> Proc \<Rightarrow> Proc \<Rightarrow> Disk \<Rightarrow> bool"
where
"HInv3_L s p q d = (phase s p \<in> {1,2}
\<and> phase s q \<in> {1,2}
\<and> hasRead s p d q
\<and> hasRead s q d p)"
definition InitializePhase :: "state \<Rightarrow> state \<Rightarrow> Proc \<Rightarrow> bool"
where
"InitializePhase s s' p =
(disksWritten s' = (disksWritten s)(p := {})
& blocksRead s' = (blocksRead s)(p := (\<lambda>d. {})))"
|
###output
InitializePhase ?s ?s' ?q \<Longrightarrow> HInv3_L ?s' ?p ?q ?d \<Longrightarrow> HInv3_R ?s' ?p ?q ?d
###end
|
Q0_Metatheory/Propositional_Wff
|
Propositional_Wff.\<V>\<^sub>B_neg
| null |
?A \<in> pwffs \<Longrightarrow> is_tv_assignment ?\<phi> \<Longrightarrow> \<V>\<^sub>B ?\<phi> (\<sim>\<^sup>\<Q> ?A) = \<sim> \<V>\<^sub>B ?\<phi> ?A
|
\<lbrakk>x_1 \<in> ?H1; ?H2 x_2\<rbrakk> \<Longrightarrow> ?H3 x_2 (?H4 x_1) = ?H5 (?H3 x_2 x_1)
|
[
"Boolean_Algebra.negation",
"Syntax.neg",
"Propositional_Wff.\\<V>\\<^sub>B",
"Propositional_Wff.is_tv_assignment",
"Propositional_Wff.pwffs"
] |
[
"definition negation :: \"V \\<Rightarrow> V\" (\"\\<sim> _\" [141] 141) where\n [simp]: \"\\<sim> p = bool_to_V (\\<not> bool_from_V p)\"",
"definition neg :: \"'fmla \\<Rightarrow> 'fmla\" where\n \"neg \\<phi> = imp \\<phi> fls\"",
"definition \\<V>\\<^sub>B :: \"(nat \\<Rightarrow> V) \\<Rightarrow> form \\<Rightarrow> V\" where\n [simp]: \"\\<V>\\<^sub>B \\<phi> A = (THE b. \\<V>\\<^sub>B_graph \\<phi> A b)\"",
"definition is_tv_assignment :: \"(nat \\<Rightarrow> V) \\<Rightarrow> bool\" where\n [iff]: \"is_tv_assignment \\<phi> \\<longleftrightarrow> (\\<forall>p. \\<phi> p \\<in> elts \\<bool>)\"",
"inductive_set pwffs :: \"form set\" where\n T_pwff: \"T\\<^bsub>o\\<^esub> \\<in> pwffs\"\n| F_pwff: \"F\\<^bsub>o\\<^esub> \\<in> pwffs\"\n| var_pwff: \"p\\<^bsub>o\\<^esub> \\<in> pwffs\"\n| neg_pwff: \"\\<sim>\\<^sup>\\<Q> A \\<in> pwffs\" if \"A \\<in> pwffs\"\n| conj_pwff: \"A \\<and>\\<^sup>\\<Q> B \\<in> pwffs\" if \"A \\<in> pwffs\" and \"B \\<in> pwffs\"\n| disj_pwff: \"A \\<or>\\<^sup>\\<Q> B \\<in> pwffs\" if \"A \\<in> pwffs\" and \"B \\<in> pwffs\"\n| imp_pwff: \"A \\<supset>\\<^sup>\\<Q> B \\<in> pwffs\" if \"A \\<in> pwffs\" and \"B \\<in> pwffs\"\n| eqv_pwff: \"A \\<equiv>\\<^sup>\\<Q> B \\<in> pwffs\" if \"A \\<in> pwffs\" and \"B \\<in> pwffs\""
] |
###template
\<lbrakk>x_1 \<in> ?H1; ?H2 x_2\<rbrakk> \<Longrightarrow> ?H3 x_2 (?H4 x_1) = ?H5 (?H3 x_2 x_1)
###symbols
Boolean_Algebra.negation
Syntax.neg
Propositional_Wff.\<V>\<^sub>B
Propositional_Wff.is_tv_assignment
Propositional_Wff.pwffs
###defs
definition negation :: "V \<Rightarrow> V" ("\<sim> _" [141] 141) where
[simp]: "\<sim> p = bool_to_V (\<not> bool_from_V p)"
definition neg :: "'fmla \<Rightarrow> 'fmla" where
"neg \<phi> = imp \<phi> fls"
definition \<V>\<^sub>B :: "(nat \<Rightarrow> V) \<Rightarrow> form \<Rightarrow> V" where
[simp]: "\<V>\<^sub>B \<phi> A = (THE b. \<V>\<^sub>B_graph \<phi> A b)"
definition is_tv_assignment :: "(nat \<Rightarrow> V) \<Rightarrow> bool" where
[iff]: "is_tv_assignment \<phi> \<longleftrightarrow> (\<forall>p. \<phi> p \<in> elts \<bool>)"
inductive_set pwffs :: "form set" where
T_pwff: "T\<^bsub>o\<^esub> \<in> pwffs"
| F_pwff: "F\<^bsub>o\<^esub> \<in> pwffs"
| var_pwff: "p\<^bsub>o\<^esub> \<in> pwffs"
| neg_pwff: "\<sim>\<^sup>\<Q> A \<in> pwffs" if "A \<in> pwffs"
| conj_pwff: "A \<and>\<^sup>\<Q> B \<in> pwffs" if "A \<in> pwffs" and "B \<in> pwffs"
| disj_pwff: "A \<or>\<^sup>\<Q> B \<in> pwffs" if "A \<in> pwffs" and "B \<in> pwffs"
| imp_pwff: "A \<supset>\<^sup>\<Q> B \<in> pwffs" if "A \<in> pwffs" and "B \<in> pwffs"
| eqv_pwff: "A \<equiv>\<^sup>\<Q> B \<in> pwffs" if "A \<in> pwffs" and "B \<in> pwffs"
|
###output
?A \<in> pwffs \<Longrightarrow> is_tv_assignment ?\<phi> \<Longrightarrow> \<V>\<^sub>B ?\<phi> (\<sim>\<^sup>\<Q> ?A) = \<sim> \<V>\<^sub>B ?\<phi> ?A
###end
|
LTL_to_DRA/LTL_Rabin_Unfold_Opt
|
LTL_Rabin_Unfold_Opt.ltl_to_generalized_rabin_af\<^sub>\<UU>_correct
| null |
finite ?\<Sigma> \<Longrightarrow> range ?w \<subseteq> ?\<Sigma> \<Longrightarrow> ?w \<Turnstile> ?\<phi> = accept\<^sub>G\<^sub>R (ltl_to_generalized_rabin_af\<^sub>\<UU> ?\<Sigma> ?\<phi>) ?w
|
\<lbrakk>?H1 x_1; ?H2 (?H3 x_2) x_1\<rbrakk> \<Longrightarrow> ?H4 x_2 x_3 = ?H5 (?H6 x_1 x_3) x_2
|
[
"LTL_Rabin_Unfold_Opt.ltl_to_generalized_rabin_af\\<^sub>\\<UU>",
"Rabin.accept\\<^sub>G\\<^sub>R",
"LTL_FGXU.ltl_semantics",
"Set.range",
"Set.subset_eq",
"Finite_Set.finite"
] |
[
"fun ltl_to_generalized_rabin_af\\<^sub>\\<UU>\nwhere\n \"ltl_to_generalized_rabin_af\\<^sub>\\<UU> \\<Sigma> \\<phi> = ltl_to_rabin_base_def.ltl_to_generalized_rabin \\<up>af\\<^sub>\\<UU> \\<up>af\\<^sub>G\\<^sub>\\<UU> (Abs \\<circ> Unf) (Abs \\<circ> Unf\\<^sub>G) M\\<^sub>\\<UU>_fin \\<Sigma> \\<phi>\"",
"definition accept\\<^sub>G\\<^sub>R :: \"('a, 'b) generalized_rabin_automaton \\<Rightarrow> 'b word \\<Rightarrow> bool\"\nwhere \n \"accept\\<^sub>G\\<^sub>R R w \\<equiv> (\\<exists>(Fin, Inf) \\<in> (snd (snd R)). accepting_pair\\<^sub>G\\<^sub>R (fst R) (fst (snd R)) (Fin, Inf) w)\"",
"fun ltl_semantics :: \"['a set word, 'a ltl] \\<Rightarrow> bool\" (infix \"\\<Turnstile>\" 80)\nwhere\n \"w \\<Turnstile> True = True\"\n| \"w \\<Turnstile> False = False\"\n| \"w \\<Turnstile> p(q) = (q \\<in> w 0)\"\n| \"w \\<Turnstile> np(q) = (q \\<notin> w 0)\"\n| \"w \\<Turnstile> \\<phi> and \\<psi> = (w \\<Turnstile> \\<phi> \\<and> w \\<Turnstile> \\<psi>)\"\n| \"w \\<Turnstile> \\<phi> or \\<psi> = (w \\<Turnstile> \\<phi> \\<or> w \\<Turnstile> \\<psi>)\"\n| \"w \\<Turnstile> X \\<phi> = (suffix 1 w \\<Turnstile> \\<phi>)\"\n| \"w \\<Turnstile> G \\<phi> = (\\<forall>k. suffix k w \\<Turnstile> \\<phi>)\"\n| \"w \\<Turnstile> F \\<phi> = (\\<exists>k. suffix k w \\<Turnstile> \\<phi>)\"\n| \"w \\<Turnstile> \\<phi> U \\<psi> = (\\<exists>k. suffix k w \\<Turnstile> \\<psi> \\<and> (\\<forall>j < k. suffix j w \\<Turnstile> \\<phi>))\"",
"abbreviation range :: \"('a \\<Rightarrow> 'b) \\<Rightarrow> 'b set\" \\<comment> \\<open>of function\\<close>\n where \"range f \\<equiv> f ` UNIV\"",
"abbreviation subset_eq :: \"'a set \\<Rightarrow> 'a set \\<Rightarrow> bool\"\n where \"subset_eq \\<equiv> less_eq\"",
"class finite =\n assumes finite_UNIV: \"finite (UNIV :: 'a set)\"\nbegin"
] |
###template
\<lbrakk>?H1 x_1; ?H2 (?H3 x_2) x_1\<rbrakk> \<Longrightarrow> ?H4 x_2 x_3 = ?H5 (?H6 x_1 x_3) x_2
###symbols
LTL_Rabin_Unfold_Opt.ltl_to_generalized_rabin_af\<^sub>\<UU>
Rabin.accept\<^sub>G\<^sub>R
LTL_FGXU.ltl_semantics
Set.range
Set.subset_eq
Finite_Set.finite
###defs
fun ltl_to_generalized_rabin_af\<^sub>\<UU>
where
"ltl_to_generalized_rabin_af\<^sub>\<UU> \<Sigma> \<phi> = ltl_to_rabin_base_def.ltl_to_generalized_rabin \<up>af\<^sub>\<UU> \<up>af\<^sub>G\<^sub>\<UU> (Abs \<circ> Unf) (Abs \<circ> Unf\<^sub>G) M\<^sub>\<UU>_fin \<Sigma> \<phi>"
definition accept\<^sub>G\<^sub>R :: "('a, 'b) generalized_rabin_automaton \<Rightarrow> 'b word \<Rightarrow> bool"
where
"accept\<^sub>G\<^sub>R R w \<equiv> (\<exists>(Fin, Inf) \<in> (snd (snd R)). accepting_pair\<^sub>G\<^sub>R (fst R) (fst (snd R)) (Fin, Inf) w)"
fun ltl_semantics :: "['a set word, 'a ltl] \<Rightarrow> bool" (infix "\<Turnstile>" 80)
where
"w \<Turnstile> True = True"
| "w \<Turnstile> False = False"
| "w \<Turnstile> p(q) = (q \<in> w 0)"
| "w \<Turnstile> np(q) = (q \<notin> w 0)"
| "w \<Turnstile> \<phi> and \<psi> = (w \<Turnstile> \<phi> \<and> w \<Turnstile> \<psi>)"
| "w \<Turnstile> \<phi> or \<psi> = (w \<Turnstile> \<phi> \<or> w \<Turnstile> \<psi>)"
| "w \<Turnstile> X \<phi> = (suffix 1 w \<Turnstile> \<phi>)"
| "w \<Turnstile> G \<phi> = (\<forall>k. suffix k w \<Turnstile> \<phi>)"
| "w \<Turnstile> F \<phi> = (\<exists>k. suffix k w \<Turnstile> \<phi>)"
| "w \<Turnstile> \<phi> U \<psi> = (\<exists>k. suffix k w \<Turnstile> \<psi> \<and> (\<forall>j < k. suffix j w \<Turnstile> \<phi>))"
abbreviation range :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b set" \<comment> \<open>of function\<close>
where "range f \<equiv> f ` UNIV"
abbreviation subset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"
where "subset_eq \<equiv> less_eq"
class finite =
assumes finite_UNIV: "finite (UNIV :: 'a set)"
begin
|
###output
finite ?\<Sigma> \<Longrightarrow> range ?w \<subseteq> ?\<Sigma> \<Longrightarrow> ?w \<Turnstile> ?\<phi> = accept\<^sub>G\<^sub>R (ltl_to_generalized_rabin_af\<^sub>\<UU> ?\<Sigma> ?\<phi>) ?w
###end
|
Transformer_Semantics/Isotone_Transformers
|
Isotone_Transformers.fiter_qiter1
|
lemma fiter_qiter1: "Abs_iso (fiter_fun (Rep_iso f) (Rep_iso g) (Rep_iso h)) = qiter_fun f g h"
|
Abs_iso (fiter_fun (Rep_iso ?f) (Rep_iso ?g) (Rep_iso ?h)) = qiter_fun ?f ?g ?h
|
?H1 (?H2 (?H3 x_1) (?H3 x_2) (?H3 x_3)) = ?H4 x_1 x_2 x_3
|
[
"Quantale_Star.unital_near_quantale_class.qiter_fun",
"Order_Lattice_Props.iso.Rep_iso",
"Isotone_Transformers.fiter_fun",
"Order_Lattice_Props.iso.Abs_iso"
] |
[
"definition fiter_fun :: \"('a \\<Rightarrow> 'c::semilattice_inf) \\<Rightarrow> ('b \\<Rightarrow> 'c) \\<Rightarrow> ('a \\<Rightarrow> 'b) \\<Rightarrow> 'a \\<Rightarrow> 'c\" where \n \"fiter_fun f g = (\\<sqinter>) f \\<circ> (\\<circ>) g\""
] |
###template
?H1 (?H2 (?H3 x_1) (?H3 x_2) (?H3 x_3)) = ?H4 x_1 x_2 x_3
###symbols
Quantale_Star.unital_near_quantale_class.qiter_fun
Order_Lattice_Props.iso.Rep_iso
Isotone_Transformers.fiter_fun
Order_Lattice_Props.iso.Abs_iso
###defs
definition fiter_fun :: "('a \<Rightarrow> 'c::semilattice_inf) \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'c" where
"fiter_fun f g = (\<sqinter>) f \<circ> (\<circ>) g"
|
###output
Abs_iso (fiter_fun (Rep_iso ?f) (Rep_iso ?g) (Rep_iso ?h)) = qiter_fun ?f ?g ?h
###end
|
ConcurrentIMP/Infinite_Sequences
|
Infinite_Sequences.stake_nth
|
lemma stake_nth[simp]:
assumes "i < j"
shows "stake j s ! i = s i"
|
?i < ?j \<Longrightarrow> stake ?j ?s ! ?i = ?s ?i
|
x_1 < x_2 \<Longrightarrow> ?H1 (?H2 x_2 x_3) x_1 = x_3 x_1
|
[
"Infinite_Sequences.stake",
"List.nth"
] |
[
"primrec stake :: \"nat \\<Rightarrow> 'a seq \\<Rightarrow> 'a list\" where\n \"stake 0 \\<sigma> = []\"\n| \"stake (Suc n) \\<sigma> = \\<sigma> 0 # stake n (\\<sigma> |\\<^sub>s 1)\"",
"primrec (nonexhaustive) nth :: \"'a list => nat => 'a\" (infixl \"!\" 100) where\nnth_Cons: \"(x # xs) ! n = (case n of 0 \\<Rightarrow> x | Suc k \\<Rightarrow> xs ! k)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>"
] |
###template
x_1 < x_2 \<Longrightarrow> ?H1 (?H2 x_2 x_3) x_1 = x_3 x_1
###symbols
Infinite_Sequences.stake
List.nth
###defs
primrec stake :: "nat \<Rightarrow> 'a seq \<Rightarrow> 'a list" where
"stake 0 \<sigma> = []"
| "stake (Suc n) \<sigma> = \<sigma> 0 # stake n (\<sigma> |\<^sub>s 1)"
primrec (nonexhaustive) nth :: "'a list => nat => 'a" (infixl "!" 100) where
nth_Cons: "(x # xs) ! n = (case n of 0 \<Rightarrow> x | Suc k \<Rightarrow> xs ! k)"
\<comment> \<open>Warning: simpset does not contain this definition, but separate
theorems for \<open>n = 0\<close> and \<open>n = Suc k\<close>\<close>
|
###output
?i < ?j \<Longrightarrow> stake ?j ?s ! ?i = ?s ?i
###end
|
CryptHOL/Computational_Model
|
Computational_Model.plus_intercept_parametric
|
lemma plus_intercept_parametric [transfer_rule]:
includes lifting_syntax shows
"((S ===> X1 ===> rel_gpv (rel_prod Y1 S) C)
===> (S ===> X2 ===> rel_gpv (rel_prod Y2 S) C)
===> S ===> rel_sum X1 X2 ===> rel_gpv (rel_prod (rel_sum Y1 Y2) S) C)
plus_intercept plus_intercept"
|
rel_fun (rel_fun ?S (rel_fun ?X1.0 (rel_gpv (rel_prod ?Y1.0 ?S) ?C))) (rel_fun (rel_fun ?S (rel_fun ?X2.0 (rel_gpv (rel_prod ?Y2.0 ?S) ?C))) (rel_fun ?S (rel_fun (rel_sum ?X1.0 ?X2.0) (rel_gpv (rel_prod (rel_sum ?Y1.0 ?Y2.0) ?S) ?C)))) plus_intercept plus_intercept
|
?H1 (?H2 x_1 (?H3 x_2 (?H4 (?H5 x_3 x_1) x_4))) (?H6 (?H7 x_1 (?H8 x_5 (?H9 (?H10 x_6 x_1) x_4))) (?H11 x_1 (?H12 (?H13 x_2 x_5) (?H14 (?H15 (?H16 x_3 x_6) x_1) x_4)))) ?H17 ?H18
|
[
"Computational_Model.plus_intercept",
"BNF_Def.rel_sum",
"Basic_BNFs.rel_prod",
"Generative_Probabilistic_Value.gpv.rel_gpv",
"BNF_Def.rel_fun"
] |
[
"primrec plus_intercept :: \"'s \\<Rightarrow> 'x1 + 'x2 \\<Rightarrow> (('y1 + 'y2) \\<times> 's, 'call, 'ret) gpv\"\nwhere\n \"plus_intercept s (Inl x) = map_gpv (apfst Inl) id (left s x)\"\n| \"plus_intercept s (Inr x) = map_gpv (apfst Inr) id (right s x)\"",
"inductive\n rel_sum :: \"('a \\<Rightarrow> 'c \\<Rightarrow> bool) \\<Rightarrow> ('b \\<Rightarrow> 'd \\<Rightarrow> bool) \\<Rightarrow> 'a + 'b \\<Rightarrow> 'c + 'd \\<Rightarrow> bool\" for R1 R2\nwhere\n \"R1 a c \\<Longrightarrow> rel_sum R1 R2 (Inl a) (Inl c)\"\n| \"R2 b d \\<Longrightarrow> rel_sum R1 R2 (Inr b) (Inr d)\"",
"inductive\n rel_prod :: \"('a \\<Rightarrow> 'b \\<Rightarrow> bool) \\<Rightarrow> ('c \\<Rightarrow> 'd \\<Rightarrow> bool) \\<Rightarrow> 'a \\<times> 'c \\<Rightarrow> 'b \\<times> 'd \\<Rightarrow> bool\" for R1 R2\nwhere\n \"\\<lbrakk>R1 a b; R2 c d\\<rbrakk> \\<Longrightarrow> rel_prod R1 R2 (a, c) (b, d)\"",
"codatatype (results'_gpv: 'a, outs'_gpv: 'out, 'in) gpv\n = GPV (the_gpv: \"('a, 'out, 'in \\<Rightarrow> ('a, 'out, 'in) gpv) generat spmf\")",
"definition\n rel_fun :: \"('a \\<Rightarrow> 'c \\<Rightarrow> bool) \\<Rightarrow> ('b \\<Rightarrow> 'd \\<Rightarrow> bool) \\<Rightarrow> ('a \\<Rightarrow> 'b) \\<Rightarrow> ('c \\<Rightarrow> 'd) \\<Rightarrow> bool\"\nwhere\n \"rel_fun A B = (\\<lambda>f g. \\<forall>x y. A x y \\<longrightarrow> B (f x) (g y))\""
] |
###template
?H1 (?H2 x_1 (?H3 x_2 (?H4 (?H5 x_3 x_1) x_4))) (?H6 (?H7 x_1 (?H8 x_5 (?H9 (?H10 x_6 x_1) x_4))) (?H11 x_1 (?H12 (?H13 x_2 x_5) (?H14 (?H15 (?H16 x_3 x_6) x_1) x_4)))) ?H17 ?H18
###symbols
Computational_Model.plus_intercept
BNF_Def.rel_sum
Basic_BNFs.rel_prod
Generative_Probabilistic_Value.gpv.rel_gpv
BNF_Def.rel_fun
###defs
primrec plus_intercept :: "'s \<Rightarrow> 'x1 + 'x2 \<Rightarrow> (('y1 + 'y2) \<times> 's, 'call, 'ret) gpv"
where
"plus_intercept s (Inl x) = map_gpv (apfst Inl) id (left s x)"
| "plus_intercept s (Inr x) = map_gpv (apfst Inr) id (right s x)"
inductive
rel_sum :: "('a \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'd \<Rightarrow> bool) \<Rightarrow> 'a + 'b \<Rightarrow> 'c + 'd \<Rightarrow> bool" for R1 R2
where
"R1 a c \<Longrightarrow> rel_sum R1 R2 (Inl a) (Inl c)"
| "R2 b d \<Longrightarrow> rel_sum R1 R2 (Inr b) (Inr d)"
inductive
rel_prod :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('c \<Rightarrow> 'd \<Rightarrow> bool) \<Rightarrow> 'a \<times> 'c \<Rightarrow> 'b \<times> 'd \<Rightarrow> bool" for R1 R2
where
"\<lbrakk>R1 a b; R2 c d\<rbrakk> \<Longrightarrow> rel_prod R1 R2 (a, c) (b, d)"
codatatype (results'_gpv: 'a, outs'_gpv: 'out, 'in) gpv
= GPV (the_gpv: "('a, 'out, 'in \<Rightarrow> ('a, 'out, 'in) gpv) generat spmf")
definition
rel_fun :: "('a \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'd \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool"
where
"rel_fun A B = (\<lambda>f g. \<forall>x y. A x y \<longrightarrow> B (f x) (g y))"
|
###output
rel_fun (rel_fun ?S (rel_fun ?X1.0 (rel_gpv (rel_prod ?Y1.0 ?S) ?C))) (rel_fun (rel_fun ?S (rel_fun ?X2.0 (rel_gpv (rel_prod ?Y2.0 ?S) ?C))) (rel_fun ?S (rel_fun (rel_sum ?X1.0 ?X2.0) (rel_gpv (rel_prod (rel_sum ?Y1.0 ?Y2.0) ?S) ?C)))) plus_intercept plus_intercept
###end
|
Lp/Functional_Spaces
|
Functional_Spaces.quasinorm_equivalent_trans
|
lemma quasinorm_equivalent_trans [trans]:
assumes "N1 =\<^sub>N N2" "N2 =\<^sub>N N3"
shows "N1 =\<^sub>N N3"
|
?N1.0 =\<^sub>N ?N2.0 \<Longrightarrow> ?N2.0 =\<^sub>N ?N3.0 \<Longrightarrow> ?N1.0 =\<^sub>N ?N3.0
|
\<lbrakk>?H1 x_1 x_2; ?H1 x_2 x_3\<rbrakk> \<Longrightarrow> ?H1 x_1 x_3
|
[
"Functional_Spaces.quasinorm_equivalent"
] |
[
"definition quasinorm_equivalent::\"('a::real_vector) quasinorm \\<Rightarrow> 'a quasinorm \\<Rightarrow> bool\" (infix \"=\\<^sub>N\" 60)\n where \"quasinorm_equivalent N1 N2 = ((N1 \\<subseteq>\\<^sub>N N2) \\<and> (N2 \\<subseteq>\\<^sub>N N1))\""
] |
###template
\<lbrakk>?H1 x_1 x_2; ?H1 x_2 x_3\<rbrakk> \<Longrightarrow> ?H1 x_1 x_3
###symbols
Functional_Spaces.quasinorm_equivalent
###defs
definition quasinorm_equivalent::"('a::real_vector) quasinorm \<Rightarrow> 'a quasinorm \<Rightarrow> bool" (infix "=\<^sub>N" 60)
where "quasinorm_equivalent N1 N2 = ((N1 \<subseteq>\<^sub>N N2) \<and> (N2 \<subseteq>\<^sub>N N1))"
|
###output
?N1.0 =\<^sub>N ?N2.0 \<Longrightarrow> ?N2.0 =\<^sub>N ?N3.0 \<Longrightarrow> ?N1.0 =\<^sub>N ?N3.0
###end
|
S_Finite_Measure_Monad/Measure_QuasiBorel_Adjunction
|
Measure_QuasiBorel_Adjunction.qbs_Mx_subset_of_measurable
|
lemma qbs_Mx_subset_of_measurable: "qbs_Mx X \<subseteq> borel \<rightarrow>\<^sub>M qbs_to_measure X"
|
qbs_Mx ?X \<subseteq> borel \<rightarrow>\<^sub>M qbs_to_measure ?X
|
?H1 (?H2 x_1) (?H3 ?H4 (?H5 x_1))
|
[
"Measure_QuasiBorel_Adjunction.qbs_to_measure",
"Borel_Space.topological_space_class.borel",
"Sigma_Algebra.measurable",
"QuasiBorel.qbs_Mx",
"Set.subset_eq"
] |
[
"definition qbs_to_measure :: \"'a quasi_borel \\<Rightarrow> 'a measure\" where\n\"qbs_to_measure X \\<equiv> Abs_measure (qbs_space X, sigma_Mx X, \\<lambda>A. (if A = {} then 0 else if A \\<in> - sigma_Mx X then 0 else \\<infinity>))\"",
"definition qbs_Mx :: \"'a quasi_borel \\<Rightarrow> (real \\<Rightarrow> 'a) set\" where\n \"qbs_Mx X \\<equiv> snd (Rep_quasi_borel X)\"",
"abbreviation subset_eq :: \"'a set \\<Rightarrow> 'a set \\<Rightarrow> bool\"\n where \"subset_eq \\<equiv> less_eq\""
] |
###template
?H1 (?H2 x_1) (?H3 ?H4 (?H5 x_1))
###symbols
Measure_QuasiBorel_Adjunction.qbs_to_measure
Borel_Space.topological_space_class.borel
Sigma_Algebra.measurable
QuasiBorel.qbs_Mx
Set.subset_eq
###defs
definition qbs_to_measure :: "'a quasi_borel \<Rightarrow> 'a measure" where
"qbs_to_measure X \<equiv> Abs_measure (qbs_space X, sigma_Mx X, \<lambda>A. (if A = {} then 0 else if A \<in> - sigma_Mx X then 0 else \<infinity>))"
definition qbs_Mx :: "'a quasi_borel \<Rightarrow> (real \<Rightarrow> 'a) set" where
"qbs_Mx X \<equiv> snd (Rep_quasi_borel X)"
abbreviation subset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"
where "subset_eq \<equiv> less_eq"
|
###output
qbs_Mx ?X \<subseteq> borel \<rightarrow>\<^sub>M qbs_to_measure ?X
###end
|
Security_Protocol_Refinement/Refinement/Message
|
Message.parts_insert_Number
|
lemma parts_insert_Number [simp]:
"parts (insert (Number N) H) = insert (Number N) (parts H)"
|
parts (insert (Number ?N) ?H) = insert (Number ?N) (parts ?H)
|
?H1 (?H2 (?H3 x_1) x_2) = ?H2 (?H3 x_1) (?H1 x_2)
|
[
"Message.msg.Number",
"Set.insert",
"Message.parts"
] |
[
"datatype\n msg = Agent agent \\<comment> \\<open>Agent names\\<close>\n | Number nat \\<comment> \\<open>Ordinary integers, timestamps, ...\\<close>\n | Nonce nat \\<comment> \\<open>Unguessable nonces\\<close>\n | Key key \\<comment> \\<open>Crypto keys\\<close>\n | Hash msg \\<comment> \\<open>Hashing\\<close>\n | MPair msg msg \\<comment> \\<open>Compound messages\\<close>\n | Crypt key msg \\<comment> \\<open>Encryption, public- or shared-key\\<close>",
"definition insert :: \"'a \\<Rightarrow> 'a set \\<Rightarrow> 'a set\"\n where insert_compr: \"insert a B = {x. x = a \\<or> x \\<in> B}\"",
"inductive_set\n parts :: \"msg set \\<Rightarrow> msg set\"\n for H :: \"msg set\"\n where\n Inj [intro]: \"X \\<in> H \\<Longrightarrow> X \\<in> parts H\"\n | Fst: \"\\<lbrace>X,Y\\<rbrace> \\<in> parts H \\<Longrightarrow> X \\<in> parts H\"\n | Snd: \"\\<lbrace>X,Y\\<rbrace> \\<in> parts H \\<Longrightarrow> Y \\<in> parts H\"\n | Body: \"Crypt K X \\<in> parts H \\<Longrightarrow> X \\<in> parts H\""
] |
###template
?H1 (?H2 (?H3 x_1) x_2) = ?H2 (?H3 x_1) (?H1 x_2)
###symbols
Message.msg.Number
Set.insert
Message.parts
###defs
datatype
msg = Agent agent \<comment> \<open>Agent names\<close>
| Number nat \<comment> \<open>Ordinary integers, timestamps, ...\<close>
| Nonce nat \<comment> \<open>Unguessable nonces\<close>
| Key key \<comment> \<open>Crypto keys\<close>
| Hash msg \<comment> \<open>Hashing\<close>
| MPair msg msg \<comment> \<open>Compound messages\<close>
| Crypt key msg \<comment> \<open>Encryption, public- or shared-key\<close>
definition insert :: "'a \<Rightarrow> 'a set \<Rightarrow> 'a set"
where insert_compr: "insert a B = {x. x = a \<or> x \<in> B}"
inductive_set
parts :: "msg set \<Rightarrow> msg set"
for H :: "msg set"
where
Inj [intro]: "X \<in> H \<Longrightarrow> X \<in> parts H"
| Fst: "\<lbrace>X,Y\<rbrace> \<in> parts H \<Longrightarrow> X \<in> parts H"
| Snd: "\<lbrace>X,Y\<rbrace> \<in> parts H \<Longrightarrow> Y \<in> parts H"
| Body: "Crypt K X \<in> parts H \<Longrightarrow> X \<in> parts H"
|
###output
parts (insert (Number ?N) ?H) = insert (Number ?N) (parts ?H)
###end
|
MiniSail/WellformedL
|
WellformedL.freshers(21)
| null |
?a \<sharp> (?c1.0 AND ?c2.0 ) = (?a \<sharp> ?c1.0 \<and> ?a \<sharp> ?c2.0)
|
?H1 x_1 (?H2 x_2 x_3) = (?H1 x_1 x_2 \<and> ?H1 x_1 x_3)
|
[
"Syntax.C_conj",
"Nominal2_Base.pt_class.fresh"
] |
[
"class pt =\n fixes permute :: \"perm \\<Rightarrow> 'a \\<Rightarrow> 'a\" (\"_ \\<bullet> _\" [76, 75] 75)\n assumes permute_zero [simp]: \"0 \\<bullet> x = x\"\n assumes permute_plus [simp]: \"(p + q) \\<bullet> x = p \\<bullet> (q \\<bullet> x)\"\nbegin"
] |
###template
?H1 x_1 (?H2 x_2 x_3) = (?H1 x_1 x_2 \<and> ?H1 x_1 x_3)
###symbols
Syntax.C_conj
Nominal2_Base.pt_class.fresh
###defs
class pt =
fixes permute :: "perm \<Rightarrow> 'a \<Rightarrow> 'a" ("_ \<bullet> _" [76, 75] 75)
assumes permute_zero [simp]: "0 \<bullet> x = x"
assumes permute_plus [simp]: "(p + q) \<bullet> x = p \<bullet> (q \<bullet> x)"
begin
|
###output
?a \<sharp> (?c1.0 AND ?c2.0 ) = (?a \<sharp> ?c1.0 \<and> ?a \<sharp> ?c2.0)
###end
|
CZH_Elementary_Categories/czh_ecategories/CZH_ECAT_Ordinal
|
CZH_ECAT_Ordinal.cat_ordinal_is_leD
|
lemma cat_ordinal_is_leD[dest]:
assumes "a \<le>\<^sub>O\<^bsub>cat_ordinal A\<^esub> b"
shows "[a, b]\<^sub>\<circ> : a \<mapsto>\<^bsub>cat_ordinal A\<^esub> b"
|
?a \<le>\<^sub>O\<^bsub>cat_ordinal ?A\<^esub> ?b \<Longrightarrow> [?a, ?b]\<^sub>\<circ> : ?a \<mapsto>\<^bsub>cat_ordinal ?A\<^esub> ?b
|
?H1 (?H2 x_1) x_2 x_3 \<Longrightarrow> ?H3 (?H2 x_1) x_2 x_3 (?H4 (?H4 ?H5 x_2) x_3)
|
[
"CZH_Sets_FSequences.vempty_vfsequence",
"CZH_Sets_FSequences.vcons",
"CZH_DG_Digraph.is_arr",
"CZH_ECAT_Ordinal.cat_ordinal",
"CZH_ECAT_Order.is_le"
] |
[] |
###template
?H1 (?H2 x_1) x_2 x_3 \<Longrightarrow> ?H3 (?H2 x_1) x_2 x_3 (?H4 (?H4 ?H5 x_2) x_3)
###symbols
CZH_Sets_FSequences.vempty_vfsequence
CZH_Sets_FSequences.vcons
CZH_DG_Digraph.is_arr
CZH_ECAT_Ordinal.cat_ordinal
CZH_ECAT_Order.is_le
###defs
|
###output
?a \<le>\<^sub>O\<^bsub>cat_ordinal ?A\<^esub> ?b \<Longrightarrow> [?a, ?b]\<^sub>\<circ> : ?a \<mapsto>\<^bsub>cat_ordinal ?A\<^esub> ?b
###end
|
Jordan_Normal_Form/Jordan_Normal_Form_Existence
|
Jordan_Normal_Form_Existence.swap_cols_rows_block_index
|
lemma swap_cols_rows_block_index[simp]:
"i < dim_row A \<Longrightarrow> i < dim_col A \<Longrightarrow> j < dim_row A \<Longrightarrow> j < dim_col A
\<Longrightarrow> low \<le> high \<Longrightarrow> high < dim_row A \<Longrightarrow> high < dim_col A
\<Longrightarrow> swap_cols_rows_block low high A $$ (i,j) = A $$
(if i = low then high else if i > low \<and> i \<le> high then i - 1 else i,
if j = low then high else if j > low \<and> j \<le> high then j - 1 else j)"
|
?i < dim_row ?A \<Longrightarrow> ?i < dim_col ?A \<Longrightarrow> ?j < dim_row ?A \<Longrightarrow> ?j < dim_col ?A \<Longrightarrow> ?low \<le> ?high \<Longrightarrow> ?high < dim_row ?A \<Longrightarrow> ?high < dim_col ?A \<Longrightarrow> ??.Jordan_Normal_Form_Existence.swap_cols_rows_block ?low ?high ?A $$ (?i, ?j) = ?A $$ (if ?i = ?low then ?high else if ?low < ?i \<and> ?i \<le> ?high then ?i - 1 else ?i, if ?j = ?low then ?high else if ?low < ?j \<and> ?j \<le> ?high then ?j - 1 else ?j)
|
\<lbrakk>x_1 < ?H1 x_2; x_1 < ?H2 x_2; x_3 < ?H1 x_2; x_3 < ?H2 x_2; x_4 \<le> x_5; x_5 < ?H1 x_2; x_5 < ?H2 x_2\<rbrakk> \<Longrightarrow> ?H3 (?H4 x_4 x_5 x_2) (x_1, x_3) = ?H3 x_2 (if x_1 = x_4 then x_5 else if x_4 < x_1 \<and> x_1 \<le> x_5 then ?H5 x_1 ?H6 else x_1, if x_3 = x_4 then x_5 else if x_4 < x_3 \<and> x_3 \<le> x_5 then ?H5 x_3 ?H6 else x_3)
|
[
"Groups.one_class.one",
"Groups.minus_class.minus",
"Jordan_Normal_Form_Existence.swap_cols_rows_block",
"Matrix.index_mat",
"Matrix.dim_col",
"Matrix.dim_row"
] |
[
"class one =\n fixes one :: 'a (\"1\")",
"class minus =\n fixes minus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"-\" 65)",
"function swap_cols_rows_block :: \"nat \\<Rightarrow> nat \\<Rightarrow> 'a mat \\<Rightarrow> 'a mat\" where\n \"swap_cols_rows_block i j A = (if i < j then\n swap_cols_rows_block (Suc i) j (swap_cols_rows i j A) else A)\""
] |
###template
\<lbrakk>x_1 < ?H1 x_2; x_1 < ?H2 x_2; x_3 < ?H1 x_2; x_3 < ?H2 x_2; x_4 \<le> x_5; x_5 < ?H1 x_2; x_5 < ?H2 x_2\<rbrakk> \<Longrightarrow> ?H3 (?H4 x_4 x_5 x_2) (x_1, x_3) = ?H3 x_2 (if x_1 = x_4 then x_5 else if x_4 < x_1 \<and> x_1 \<le> x_5 then ?H5 x_1 ?H6 else x_1, if x_3 = x_4 then x_5 else if x_4 < x_3 \<and> x_3 \<le> x_5 then ?H5 x_3 ?H6 else x_3)
###symbols
Groups.one_class.one
Groups.minus_class.minus
Jordan_Normal_Form_Existence.swap_cols_rows_block
Matrix.index_mat
Matrix.dim_col
Matrix.dim_row
###defs
class one =
fixes one :: 'a ("1")
class minus =
fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "-" 65)
function swap_cols_rows_block :: "nat \<Rightarrow> nat \<Rightarrow> 'a mat \<Rightarrow> 'a mat" where
"swap_cols_rows_block i j A = (if i < j then
swap_cols_rows_block (Suc i) j (swap_cols_rows i j A) else A)"
|
###output
?i < dim_row ?A \<Longrightarrow> ?i < dim_col ?A \<Longrightarrow> ?j < dim_row ?A \<Longrightarrow> ?j < dim_col ?A \<Longrightarrow> ?low \<le> ?high \<Longrightarrow> ?high < dim_row ?A \<Longrightarrow> ?high < dim_col ?A \<Longrightarrow> ??.Jordan_Normal_Form_Existence.swap_cols_rows_block ?low ?high ?A $$ (?i, ?j) = ?A $$ (if ?i = ?low then ?high else if ?low < ?i \<and> ?i \<le> ?high then ?i - 1 else ?i, if ?j = ?low then ?high else if ?low < ?j \<and> ?j \<le> ?high then ?j - 1 else ?j)
###end
|
Knot_Theory/Linkrel_Kauffman
|
Linkrel_Kauffman.straighten_topdown_inv
|
theorem straighten_topdown_inv:"straighten_topdown w1 w2 \<Longrightarrow> (kauff_mat w1) = (kauff_mat w2)"
|
straighten_topdown ?w1.0 ?w2.0 \<Longrightarrow> kauff_mat ?w1.0 = kauff_mat ?w2.0
|
?H1 x_1 x_2 \<Longrightarrow> ?H2 x_1 = ?H2 x_2
|
[
"Kauffman_Matrix.kauff_mat",
"Tangle_Moves.straighten_topdown"
] |
[
"primrec kauff_mat::\"wall \\<Rightarrow> rat_poly mat\"\nwhere \n\"kauff_mat (basic w) = (blockmat w)\"\n|\"kauff_mat (w*ws) = rat_poly.matrix_mult (blockmat w) (kauff_mat ws)\"",
"definition straighten_topdown::relation\nwhere\n\"straighten_topdown x y \\<equiv> ((x =((basic ((vert#cup#[])))\n \\<circ>(basic ((cap#vert#[])))))\n \\<and>(y = ((basic (vert#[]))\\<circ>(basic (vert#[])))))\""
] |
###template
?H1 x_1 x_2 \<Longrightarrow> ?H2 x_1 = ?H2 x_2
###symbols
Kauffman_Matrix.kauff_mat
Tangle_Moves.straighten_topdown
###defs
primrec kauff_mat::"wall \<Rightarrow> rat_poly mat"
where
"kauff_mat (basic w) = (blockmat w)"
|"kauff_mat (w*ws) = rat_poly.matrix_mult (blockmat w) (kauff_mat ws)"
definition straighten_topdown::relation
where
"straighten_topdown x y \<equiv> ((x =((basic ((vert#cup#[])))
\<circ>(basic ((cap#vert#[])))))
\<and>(y = ((basic (vert#[]))\<circ>(basic (vert#[])))))"
|
###output
straighten_topdown ?w1.0 ?w2.0 \<Longrightarrow> kauff_mat ?w1.0 = kauff_mat ?w2.0
###end
|
Winding_Number_Eval/Cauchy_Index_Theorem
|
Cauchy_Index_Theorem.jumpF_neg_has_sgnx
|
lemma jumpF_neg_has_sgnx:
assumes "jumpF f F < 0"
shows "(f has_sgnx -1) F"
|
jumpF ?f ?F < 0 \<Longrightarrow> (?f has_sgnx - 1) ?F
|
?H1 x_1 x_2 < ?H2 \<Longrightarrow> ?H3 x_1 (?H4 ?H5) x_2
|
[
"Groups.one_class.one",
"Groups.uminus_class.uminus",
"Cauchy_Index_Theorem.has_sgnx",
"Groups.zero_class.zero",
"Cauchy_Index_Theorem.jumpF"
] |
[
"class one =\n fixes one :: 'a (\"1\")",
"class uminus =\n fixes uminus :: \"'a \\<Rightarrow> 'a\" (\"- _\" [81] 80)",
"definition has_sgnx::\"(real \\<Rightarrow> real) \\<Rightarrow> real \\<Rightarrow> real filter \\<Rightarrow> bool\" \n (infixr \"has'_sgnx\" 55) where\n \"(f has_sgnx c) F= (eventually (\\<lambda>x. sgn(f x) = c) F)\"",
"class zero =\n fixes zero :: 'a (\"0\")",
"definition jumpF::\"(real \\<Rightarrow> real) \\<Rightarrow> real filter \\<Rightarrow> real\" where \n \"jumpF f F \\<equiv> (if filterlim f at_top F then 1/2 else \n if filterlim f at_bot F then -1/2 else (0::real))\""
] |
###template
?H1 x_1 x_2 < ?H2 \<Longrightarrow> ?H3 x_1 (?H4 ?H5) x_2
###symbols
Groups.one_class.one
Groups.uminus_class.uminus
Cauchy_Index_Theorem.has_sgnx
Groups.zero_class.zero
Cauchy_Index_Theorem.jumpF
###defs
class one =
fixes one :: 'a ("1")
class uminus =
fixes uminus :: "'a \<Rightarrow> 'a" ("- _" [81] 80)
definition has_sgnx::"(real \<Rightarrow> real) \<Rightarrow> real \<Rightarrow> real filter \<Rightarrow> bool"
(infixr "has'_sgnx" 55) where
"(f has_sgnx c) F= (eventually (\<lambda>x. sgn(f x) = c) F)"
class zero =
fixes zero :: 'a ("0")
definition jumpF::"(real \<Rightarrow> real) \<Rightarrow> real filter \<Rightarrow> real" where
"jumpF f F \<equiv> (if filterlim f at_top F then 1/2 else
if filterlim f at_bot F then -1/2 else (0::real))"
|
###output
jumpF ?f ?F < 0 \<Longrightarrow> (?f has_sgnx - 1) ?F
###end
|
Encodability_Process_Calculi/SimulationRelations
|
SimulationRelations.weak_reduction_contrasimulation_and_closures(2)
|
lemma weak_reduction_contrasimulation_and_closures:
fixes Rel :: "('proc \<times> 'proc) set"
and Cal :: "'proc processCalculus"
assumes contrasimulation: "weak_reduction_contrasimulation Rel Cal"
shows "weak_reduction_contrasimulation (Rel\<^sup>=) Cal"
and "weak_reduction_contrasimulation (Rel\<^sup>+) Cal"
and "weak_reduction_contrasimulation (Rel\<^sup>*) Cal"
|
weak_reduction_contrasimulation ?Rel ?Cal \<Longrightarrow> weak_reduction_contrasimulation (?Rel\<^sup>+) ?Cal
|
?H1 x_1 x_2 \<Longrightarrow> ?H1 (?H2 x_1) x_2
|
[
"Transitive_Closure.trancl",
"SimulationRelations.weak_reduction_contrasimulation"
] |
[
"inductive_set trancl :: \"('a \\<times> 'a) set \\<Rightarrow> ('a \\<times> 'a) set\" (\"(_\\<^sup>+)\" [1000] 999)\n for r :: \"('a \\<times> 'a) set\"\n where\n r_into_trancl [intro, Pure.intro]: \"(a, b) \\<in> r \\<Longrightarrow> (a, b) \\<in> r\\<^sup>+\"\n | trancl_into_trancl [Pure.intro]: \"(a, b) \\<in> r\\<^sup>+ \\<Longrightarrow> (b, c) \\<in> r \\<Longrightarrow> (a, c) \\<in> r\\<^sup>+\"",
"abbreviation weak_reduction_contrasimulation\n :: \"('proc \\<times> 'proc) set \\<Rightarrow> 'proc processCalculus \\<Rightarrow> bool\"\n where\n \"weak_reduction_contrasimulation Rel Cal \\<equiv>\n \\<forall>P Q P'. (P, Q) \\<in> Rel \\<and> P \\<longmapsto>Cal* P' \\<longrightarrow> (\\<exists>Q'. Q \\<longmapsto>Cal* Q' \\<and> (Q', P') \\<in> Rel)\""
] |
###template
?H1 x_1 x_2 \<Longrightarrow> ?H1 (?H2 x_1) x_2
###symbols
Transitive_Closure.trancl
SimulationRelations.weak_reduction_contrasimulation
###defs
inductive_set trancl :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" ("(_\<^sup>+)" [1000] 999)
for r :: "('a \<times> 'a) set"
where
r_into_trancl [intro, Pure.intro]: "(a, b) \<in> r \<Longrightarrow> (a, b) \<in> r\<^sup>+"
| trancl_into_trancl [Pure.intro]: "(a, b) \<in> r\<^sup>+ \<Longrightarrow> (b, c) \<in> r \<Longrightarrow> (a, c) \<in> r\<^sup>+"
abbreviation weak_reduction_contrasimulation
:: "('proc \<times> 'proc) set \<Rightarrow> 'proc processCalculus \<Rightarrow> bool"
where
"weak_reduction_contrasimulation Rel Cal \<equiv>
\<forall>P Q P'. (P, Q) \<in> Rel \<and> P \<longmapsto>Cal* P' \<longrightarrow> (\<exists>Q'. Q \<longmapsto>Cal* Q' \<and> (Q', P') \<in> Rel)"
|
###output
weak_reduction_contrasimulation ?Rel ?Cal \<Longrightarrow> weak_reduction_contrasimulation (?Rel\<^sup>+) ?Cal
###end
|
Crypto_Standards/FIPS180_4
|
FIPS180_4.SHA256_MessageSchedule_rec_valid
|
lemma SHA256_MessageSchedule_rec_valid:
assumes "word32s_valid W"
shows "word32s_valid (SHA256_MessageSchedule_rec n W)"
|
word32s_valid ?W \<Longrightarrow> word32s_valid (SHA256_MessageSchedule_rec ?n ?W)
|
?H1 x_1 \<Longrightarrow> ?H1 (?H2 x_2 x_1)
|
[
"FIPS180_4.SHA256_MessageSchedule_rec",
"Words.word32s_valid"
] |
[
"fun SHA256_MessageSchedule_rec :: \"nat \\<Rightarrow> words \\<Rightarrow> words\" where\n \"SHA256_MessageSchedule_rec n W = \n ( let t = length W in\n if t < 16 then W else (\n if n = 0 then W else\n SHA256_MessageSchedule_rec (n-1) (SHA256_MessageSchedule_1 W) )\n )\"",
"abbreviation word32s_valid :: \"words \\<Rightarrow> bool\" where\n \"word32s_valid bs \\<equiv> words_valid 32 bs\""
] |
###template
?H1 x_1 \<Longrightarrow> ?H1 (?H2 x_2 x_1)
###symbols
FIPS180_4.SHA256_MessageSchedule_rec
Words.word32s_valid
###defs
fun SHA256_MessageSchedule_rec :: "nat \<Rightarrow> words \<Rightarrow> words" where
"SHA256_MessageSchedule_rec n W =
( let t = length W in
if t < 16 then W else (
if n = 0 then W else
SHA256_MessageSchedule_rec (n-1) (SHA256_MessageSchedule_1 W) )
)"
abbreviation word32s_valid :: "words \<Rightarrow> bool" where
"word32s_valid bs \<equiv> words_valid 32 bs"
|
###output
word32s_valid ?W \<Longrightarrow> word32s_valid (SHA256_MessageSchedule_rec ?n ?W)
###end
|
Frequency_Moments/K_Smallest
|
K_Smallest.least_eq_iff
|
lemma least_eq_iff:
assumes "finite B"
assumes "A \<subseteq> B"
assumes "\<And>x. x \<in> B \<Longrightarrow> rank_of x B < k \<Longrightarrow> x \<in> A"
shows "least k A = least k B"
|
finite ?B \<Longrightarrow> ?A \<subseteq> ?B \<Longrightarrow> (\<And>x. x \<in> ?B \<Longrightarrow> rank_of x ?B < ?k \<Longrightarrow> x \<in> ?A) \<Longrightarrow> K_Smallest.least ?k ?A = K_Smallest.least ?k ?B
|
\<lbrakk>?H1 x_1; ?H2 x_2 x_1; \<And>y_0. \<lbrakk>y_0 \<in> x_1; ?H3 y_0 x_1 < x_3\<rbrakk> \<Longrightarrow> y_0 \<in> x_2\<rbrakk> \<Longrightarrow> ?H4 x_3 x_2 = ?H4 x_3 x_1
|
[
"K_Smallest.least",
"K_Smallest.rank_of",
"Set.subset_eq",
"Finite_Set.finite"
] |
[
"definition least where \"least k S = {y \\<in> S. rank_of y S < k}\"",
"definition rank_of :: \"'a :: linorder \\<Rightarrow> 'a set \\<Rightarrow> nat\" where \"rank_of x S = card {y \\<in> S. y < x}\"",
"abbreviation subset_eq :: \"'a set \\<Rightarrow> 'a set \\<Rightarrow> bool\"\n where \"subset_eq \\<equiv> less_eq\"",
"class finite =\n assumes finite_UNIV: \"finite (UNIV :: 'a set)\"\nbegin"
] |
###template
\<lbrakk>?H1 x_1; ?H2 x_2 x_1; \<And>y_0. \<lbrakk>y_0 \<in> x_1; ?H3 y_0 x_1 < x_3\<rbrakk> \<Longrightarrow> y_0 \<in> x_2\<rbrakk> \<Longrightarrow> ?H4 x_3 x_2 = ?H4 x_3 x_1
###symbols
K_Smallest.least
K_Smallest.rank_of
Set.subset_eq
Finite_Set.finite
###defs
definition least where "least k S = {y \<in> S. rank_of y S < k}"
definition rank_of :: "'a :: linorder \<Rightarrow> 'a set \<Rightarrow> nat" where "rank_of x S = card {y \<in> S. y < x}"
abbreviation subset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"
where "subset_eq \<equiv> less_eq"
class finite =
assumes finite_UNIV: "finite (UNIV :: 'a set)"
begin
|
###output
finite ?B \<Longrightarrow> ?A \<subseteq> ?B \<Longrightarrow> (\<And>x. x \<in> ?B \<Longrightarrow> rank_of x ?B < ?k \<Longrightarrow> x \<in> ?A) \<Longrightarrow> K_Smallest.least ?k ?A = K_Smallest.least ?k ?B
###end
|
Valuation/Valuation1
|
Valuation1.zle_less_trans
|
lemma zle_less_trans:"\<lbrakk>(i::int) \<le> j; j < k\<rbrakk> \<Longrightarrow> i < k"
|
?i \<le> ?j \<Longrightarrow> ?j < ?k \<Longrightarrow> ?i < ?k
|
\<lbrakk>x_1 \<le> x_2; x_2 < x_3\<rbrakk> \<Longrightarrow> x_1 < x_3
|
[] |
[] |
###template
\<lbrakk>x_1 \<le> x_2; x_2 < x_3\<rbrakk> \<Longrightarrow> x_1 < x_3
###symbols
###defs
|
###output
?i \<le> ?j \<Longrightarrow> ?j < ?k \<Longrightarrow> ?i < ?k
###end
|
S_Finite_Measure_Monad/Query
|
Query.Total_simp(1)
|
lemma Total_simp: "Total [] = 0" "Total (y#ls) = y + Total ls"
|
0 < ?\<sigma> \<Longrightarrow> Query.Total [] = 0
|
?H1 < x_1 \<Longrightarrow> ?H2 ?H3 = ?H1
|
[
"List.list.Nil",
"Query.Total",
"Groups.zero_class.zero"
] |
[
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"definition Total :: \"real list \\<Rightarrow> real\" where \"Total = (\\<lambda>l. foldr (+) l 0)\"",
"class zero =\n fixes zero :: 'a (\"0\")"
] |
###template
?H1 < x_1 \<Longrightarrow> ?H2 ?H3 = ?H1
###symbols
List.list.Nil
Query.Total
Groups.zero_class.zero
###defs
datatype (set: 'a) list =
Nil ("[]")
| Cons (hd: 'a) (tl: "'a list") (infixr "#" 65)
for
map: map
rel: list_all2
pred: list_all
where
"tl [] = []"
definition Total :: "real list \<Rightarrow> real" where "Total = (\<lambda>l. foldr (+) l 0)"
class zero =
fixes zero :: 'a ("0")
|
###output
0 < ?\<sigma> \<Longrightarrow> Query.Total [] = 0
###end
|
Robinson_Arithmetic/Instance
|
Instance.prv_eqv_imp_transi_rev
| null |
?\<phi>1.0 \<in> UNIV \<Longrightarrow> ?\<phi>2.0 \<in> UNIV \<Longrightarrow> ?\<chi> \<in> UNIV \<Longrightarrow> {} \<turnstile> eqv ?\<phi>1.0 ?\<phi>2.0 \<Longrightarrow> {} \<turnstile> ?\<phi>1.0 IMP ?\<chi> \<Longrightarrow> {} \<turnstile> ?\<phi>2.0 IMP ?\<chi>
|
\<lbrakk>x_1 \<in> ?H1; x_2 \<in> ?H1; x_3 \<in> ?H1; ?H2 ?H3 (?H4 x_1 x_2); ?H2 ?H3 (?H5 x_1 x_3)\<rbrakk> \<Longrightarrow> ?H2 ?H3 (?H5 x_2 x_3)
|
[
"Robinson_Arithmetic.imp",
"Instance.eqv",
"Set.empty",
"Robinson_Arithmetic.nprv",
"Set.UNIV"
] |
[
"abbreviation imp :: \"fmla \\<Rightarrow> fmla \\<Rightarrow> fmla\" (infixr \"IMP\" 125)\n where \"imp A B \\<equiv> dsj (neg A) B\"",
"abbreviation empty :: \"'a set\" (\"{}\")\n where \"{} \\<equiv> bot\"",
"inductive nprv :: \"fmla set \\<Rightarrow> fmla \\<Rightarrow> bool\" (infixl \"\\<turnstile>\" 55)\n where\n Hyp: \"A \\<in> H \\<Longrightarrow> H \\<turnstile> A\"\n | Q: \"A \\<in> Q_axioms \\<Longrightarrow> H \\<turnstile> A\"\n | Bool: \"A \\<in> boolean_axioms \\<Longrightarrow> H \\<turnstile> A\"\n | eql: \"A \\<in> equality_axioms \\<Longrightarrow> H \\<turnstile> A\"\n | Spec: \"A \\<in> special_axioms \\<Longrightarrow> H \\<turnstile> A\"\n | MP: \"H \\<turnstile> A IMP B \\<Longrightarrow> H' \\<turnstile> A \\<Longrightarrow> H \\<union> H' \\<turnstile> B\"\n | exiists: \"H \\<turnstile> A IMP B \\<Longrightarrow> atom i \\<sharp> B \\<Longrightarrow> \\<forall>C \\<in> H. atom i \\<sharp> C \\<Longrightarrow> H \\<turnstile> (exi i A) IMP B\"",
"abbreviation UNIV :: \"'a set\"\n where \"UNIV \\<equiv> top\""
] |
###template
\<lbrakk>x_1 \<in> ?H1; x_2 \<in> ?H1; x_3 \<in> ?H1; ?H2 ?H3 (?H4 x_1 x_2); ?H2 ?H3 (?H5 x_1 x_3)\<rbrakk> \<Longrightarrow> ?H2 ?H3 (?H5 x_2 x_3)
###symbols
Robinson_Arithmetic.imp
Instance.eqv
Set.empty
Robinson_Arithmetic.nprv
Set.UNIV
###defs
abbreviation imp :: "fmla \<Rightarrow> fmla \<Rightarrow> fmla" (infixr "IMP" 125)
where "imp A B \<equiv> dsj (neg A) B"
abbreviation empty :: "'a set" ("{}")
where "{} \<equiv> bot"
inductive nprv :: "fmla set \<Rightarrow> fmla \<Rightarrow> bool" (infixl "\<turnstile>" 55)
where
Hyp: "A \<in> H \<Longrightarrow> H \<turnstile> A"
| Q: "A \<in> Q_axioms \<Longrightarrow> H \<turnstile> A"
| Bool: "A \<in> boolean_axioms \<Longrightarrow> H \<turnstile> A"
| eql: "A \<in> equality_axioms \<Longrightarrow> H \<turnstile> A"
| Spec: "A \<in> special_axioms \<Longrightarrow> H \<turnstile> A"
| MP: "H \<turnstile> A IMP B \<Longrightarrow> H' \<turnstile> A \<Longrightarrow> H \<union> H' \<turnstile> B"
| exiists: "H \<turnstile> A IMP B \<Longrightarrow> atom i \<sharp> B \<Longrightarrow> \<forall>C \<in> H. atom i \<sharp> C \<Longrightarrow> H \<turnstile> (exi i A) IMP B"
abbreviation UNIV :: "'a set"
where "UNIV \<equiv> top"
|
###output
?\<phi>1.0 \<in> UNIV \<Longrightarrow> ?\<phi>2.0 \<in> UNIV \<Longrightarrow> ?\<chi> \<in> UNIV \<Longrightarrow> {} \<turnstile> eqv ?\<phi>1.0 ?\<phi>2.0 \<Longrightarrow> {} \<turnstile> ?\<phi>1.0 IMP ?\<chi> \<Longrightarrow> {} \<turnstile> ?\<phi>2.0 IMP ?\<chi>
###end
|
UpDown_Scheme/Triangular_Function
|
Triangular_Function.\<phi>_right_support'
| null |
?x \<in> {real_of_int ?i / 2 ^ (?l + 1)..real_of_int (?i + 1) / 2 ^ (?l + 1)} \<Longrightarrow> \<phi> (?l, ?i) ?x = 1 - ?x * 2 ^ (?l + 1) + real_of_int ?i
|
x_1 \<in> ?H1 (?H2 (?H3 x_2) (?H4 (?H5 (?H6 ?H7)) (?H8 x_3 ?H9))) (?H2 (?H3 (?H10 x_2 ?H11)) (?H4 (?H5 (?H6 ?H7)) (?H8 x_3 ?H9))) \<Longrightarrow> ?H12 (x_3, x_2) x_1 = ?H13 (?H14 ?H15 (?H16 x_1 (?H4 (?H5 (?H6 ?H7)) (?H8 x_3 ?H9)))) (?H3 x_2)
|
[
"Groups.times_class.times",
"Groups.minus_class.minus",
"Triangular_Function.\\<phi>",
"Groups.one_class.one",
"Groups.plus_class.plus",
"Num.num.One",
"Num.num.Bit0",
"Num.numeral_class.numeral",
"Power.power_class.power",
"Real.real_of_int",
"Fields.inverse_class.inverse_divide",
"Set_Interval.ord_class.atLeastAtMost"
] |
[
"class times =\n fixes times :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"*\" 70)",
"class minus =\n fixes minus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"-\" 65)",
"definition \\<phi> :: \"(nat \\<times> int) \\<Rightarrow> real \\<Rightarrow> real\" where\n \"\\<phi> \\<equiv> (\\<lambda>(l,i) x. max 0 (1 - \\<bar> x * 2^(l + 1) - real_of_int i \\<bar>))\"",
"class one =\n fixes one :: 'a (\"1\")",
"class plus =\n fixes plus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"+\" 65)",
"datatype num = One | Bit0 num | Bit1 num",
"datatype num = One | Bit0 num | Bit1 num",
"primrec numeral :: \"num \\<Rightarrow> 'a\"\n where\n numeral_One: \"numeral One = 1\"\n | numeral_Bit0: \"numeral (Bit0 n) = numeral n + numeral n\"\n | numeral_Bit1: \"numeral (Bit1 n) = numeral n + numeral n + 1\"",
"primrec power :: \"'a \\<Rightarrow> nat \\<Rightarrow> 'a\" (infixr \"^\" 80)\n where\n power_0: \"a ^ 0 = 1\"\n | power_Suc: \"a ^ Suc n = a * a ^ n\"",
"abbreviation real_of_int :: \"int \\<Rightarrow> real\"\n where \"real_of_int \\<equiv> of_int\"",
"class inverse = divide +\n fixes inverse :: \"'a \\<Rightarrow> 'a\"\nbegin"
] |
###template
x_1 \<in> ?H1 (?H2 (?H3 x_2) (?H4 (?H5 (?H6 ?H7)) (?H8 x_3 ?H9))) (?H2 (?H3 (?H10 x_2 ?H11)) (?H4 (?H5 (?H6 ?H7)) (?H8 x_3 ?H9))) \<Longrightarrow> ?H12 (x_3, x_2) x_1 = ?H13 (?H14 ?H15 (?H16 x_1 (?H4 (?H5 (?H6 ?H7)) (?H8 x_3 ?H9)))) (?H3 x_2)
###symbols
Groups.times_class.times
Groups.minus_class.minus
Triangular_Function.\<phi>
Groups.one_class.one
Groups.plus_class.plus
Num.num.One
Num.num.Bit0
Num.numeral_class.numeral
Power.power_class.power
Real.real_of_int
Fields.inverse_class.inverse_divide
Set_Interval.ord_class.atLeastAtMost
###defs
class times =
fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "*" 70)
class minus =
fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "-" 65)
definition \<phi> :: "(nat \<times> int) \<Rightarrow> real \<Rightarrow> real" where
"\<phi> \<equiv> (\<lambda>(l,i) x. max 0 (1 - \<bar> x * 2^(l + 1) - real_of_int i \<bar>))"
class one =
fixes one :: 'a ("1")
class plus =
fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "+" 65)
datatype num = One | Bit0 num | Bit1 num
datatype num = One | Bit0 num | Bit1 num
primrec numeral :: "num \<Rightarrow> 'a"
where
numeral_One: "numeral One = 1"
| numeral_Bit0: "numeral (Bit0 n) = numeral n + numeral n"
| numeral_Bit1: "numeral (Bit1 n) = numeral n + numeral n + 1"
primrec power :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixr "^" 80)
where
power_0: "a ^ 0 = 1"
| power_Suc: "a ^ Suc n = a * a ^ n"
abbreviation real_of_int :: "int \<Rightarrow> real"
where "real_of_int \<equiv> of_int"
class inverse = divide +
fixes inverse :: "'a \<Rightarrow> 'a"
begin
|
###output
?x \<in> {real_of_int ?i / 2 ^ (?l + 1)..real_of_int (?i + 1) / 2 ^ (?l + 1)} \<Longrightarrow> \<phi> (?l, ?i) ?x = 1 - ?x * 2 ^ (?l + 1) + real_of_int ?i
###end
|
SenSocialChoice/FSext
|
FSext.has_0
|
lemma has_0: "has 0 S"
|
has 0 ?S
|
?H1 ?H2 x_1
|
[
"Groups.zero_class.zero",
"FSext.has"
] |
[
"class zero =\n fixes zero :: 'a (\"0\")",
"definition has :: \"nat \\<Rightarrow> 'a set \\<Rightarrow> bool\" where\n \"has n S \\<equiv> \\<exists>xs. hasw xs S \\<and> length xs = n\""
] |
###template
?H1 ?H2 x_1
###symbols
Groups.zero_class.zero
FSext.has
###defs
class zero =
fixes zero :: 'a ("0")
definition has :: "nat \<Rightarrow> 'a set \<Rightarrow> bool" where
"has n S \<equiv> \<exists>xs. hasw xs S \<and> length xs = n"
|
###output
has 0 ?S
###end
|
CZH_Elementary_Categories/czh_ecategories/CZH_ECAT_PCategory
|
CZH_ECAT_PCategory.cat_prod_Comp_app_component
| null |
?g : ?b \<mapsto>\<^bsub>cat_prod ?I ?\<AA>\<^esub> ?c \<Longrightarrow> ?f : ?a \<mapsto>\<^bsub>cat_prod ?I ?\<AA>\<^esub> ?b \<Longrightarrow> ?i \<in>\<^sub>\<circ> ?I \<Longrightarrow> (?g \<circ>\<^sub>A\<^bsub>cat_prod ?I ?\<AA>\<^esub> ?f)\<lparr>?i\<rparr> = ?g\<lparr>?i\<rparr> \<circ>\<^sub>A\<^bsub>?\<AA> ?i\<^esub> ?f\<lparr>?i\<rparr>
|
\<lbrakk>?H1 (?H2 x_1 x_2) x_3 x_4 x_5; ?H1 (?H2 x_1 x_2) x_6 x_3 x_7; ?H3 x_8 x_1\<rbrakk> \<Longrightarrow> ?H4 (?H5 (?H2 x_1 x_2) x_5 x_7) x_8 = ?H5 (x_2 x_8) (?H4 x_5 x_8) (?H4 x_7 x_8)
|
[
"CZH_SMC_Semicategory.Comp_app",
"ZFC_Cardinals.app",
"CZH_Sets_Sets.vmember",
"CZH_ECAT_PCategory.cat_prod",
"CZH_DG_Digraph.is_arr"
] |
[
"definition app :: \"[V,V] \\<Rightarrow> V\"\n where \"app f x \\<equiv> THE y. \\<langle>x,y\\<rangle> \\<in> elts f\""
] |
###template
\<lbrakk>?H1 (?H2 x_1 x_2) x_3 x_4 x_5; ?H1 (?H2 x_1 x_2) x_6 x_3 x_7; ?H3 x_8 x_1\<rbrakk> \<Longrightarrow> ?H4 (?H5 (?H2 x_1 x_2) x_5 x_7) x_8 = ?H5 (x_2 x_8) (?H4 x_5 x_8) (?H4 x_7 x_8)
###symbols
CZH_SMC_Semicategory.Comp_app
ZFC_Cardinals.app
CZH_Sets_Sets.vmember
CZH_ECAT_PCategory.cat_prod
CZH_DG_Digraph.is_arr
###defs
definition app :: "[V,V] \<Rightarrow> V"
where "app f x \<equiv> THE y. \<langle>x,y\<rangle> \<in> elts f"
|
###output
?g : ?b \<mapsto>\<^bsub>cat_prod ?I ?\<AA>\<^esub> ?c \<Longrightarrow> ?f : ?a \<mapsto>\<^bsub>cat_prod ?I ?\<AA>\<^esub> ?b \<Longrightarrow> ?i \<in>\<^sub>\<circ> ?I \<Longrightarrow> (?g \<circ>\<^sub>A\<^bsub>cat_prod ?I ?\<AA>\<^esub> ?f)\<lparr>?i\<rparr> = ?g\<lparr>?i\<rparr> \<circ>\<^sub>A\<^bsub>?\<AA> ?i\<^esub> ?f\<lparr>?i\<rparr>
###end
|
Goedel_HFSet_Semanticless/Instance
|
Instance.imp
| null |
?\<phi>1.0 \<in> UNIV \<Longrightarrow> ?\<phi>2.0 \<in> UNIV \<Longrightarrow> ?\<phi>1.0 IMP ?\<phi>2.0 \<in> UNIV
|
\<lbrakk>x_1 \<in> ?H1; x_2 \<in> ?H1\<rbrakk> \<Longrightarrow> ?H2 x_1 x_2 \<in> ?H1
|
[
"SyntaxN.Imp",
"Set.UNIV"
] |
[
"abbreviation Imp :: \"fm \\<Rightarrow> fm \\<Rightarrow> fm\" (infixr \"IMP\" 125)\n where \"Imp A B \\<equiv> Disj (Neg A) B\"",
"abbreviation UNIV :: \"'a set\"\n where \"UNIV \\<equiv> top\""
] |
###template
\<lbrakk>x_1 \<in> ?H1; x_2 \<in> ?H1\<rbrakk> \<Longrightarrow> ?H2 x_1 x_2 \<in> ?H1
###symbols
SyntaxN.Imp
Set.UNIV
###defs
abbreviation Imp :: "fm \<Rightarrow> fm \<Rightarrow> fm" (infixr "IMP" 125)
where "Imp A B \<equiv> Disj (Neg A) B"
abbreviation UNIV :: "'a set"
where "UNIV \<equiv> top"
|
###output
?\<phi>1.0 \<in> UNIV \<Longrightarrow> ?\<phi>2.0 \<in> UNIV \<Longrightarrow> ?\<phi>1.0 IMP ?\<phi>2.0 \<in> UNIV
###end
|
FSM_Tests/EquivalenceTesting/SPYH_Method_Implementations
|
SPYH_Method_Implementations.spyh_method_via_h_framework_lists_completeness
|
lemma spyh_method_via_h_framework_lists_completeness :
fixes M1 :: "('a::linorder,'b::linorder,'c::linorder) fsm"
fixes M2 :: "('d,'b,'c) fsm"
assumes "observable M1"
and "observable M2"
and "minimal M1"
and "minimal M2"
and "size_r M1 \<le> m"
and "size M2 \<le> m"
and "inputs M2 = inputs M1"
and "outputs M2 = outputs M1"
shows "(L M1 = L M2) \<longleftrightarrow> list_all (passes_test_case M2 (initial M2)) (spyh_method_via_h_framework_lists M1 m completeInputTraces useInputHeuristic)"
|
observable ?M1.0 \<Longrightarrow> observable ?M2.0 \<Longrightarrow> minimal ?M1.0 \<Longrightarrow> minimal ?M2.0 \<Longrightarrow> size_r ?M1.0 \<le> ?m \<Longrightarrow> FSM.size ?M2.0 \<le> ?m \<Longrightarrow> FSM.inputs ?M2.0 = FSM.inputs ?M1.0 \<Longrightarrow> FSM.outputs ?M2.0 = FSM.outputs ?M1.0 \<Longrightarrow> (L ?M1.0 = L ?M2.0) = list_all (passes_test_case ?M2.0 (FSM.initial ?M2.0)) (spyh_method_via_h_framework_lists ?M1.0 ?m ?completeInputTraces ?useInputHeuristic)
|
\<lbrakk>?H1 x_1; ?H2 x_2; ?H3 x_1; ?H4 x_2; ?H5 x_1 \<le> x_3; ?H6 x_2 \<le> x_3; ?H7 x_2 = ?H8 x_1; ?H9 x_2 = ?H10 x_1\<rbrakk> \<Longrightarrow> (?H11 x_1 = ?H12 x_2) = ?H13 (?H14 x_2 (?H15 x_2)) (?H16 x_1 x_3 x_4 x_5)
|
[
"SPYH_Method_Implementations.spyh_method_via_h_framework_lists",
"FSM.initial",
"Test_Suite_Representations.passes_test_case",
"List.list.list_all",
"FSM.L",
"FSM.outputs",
"FSM.inputs",
"FSM.size",
"FSM.size_r",
"FSM.minimal",
"FSM.observable"
] |
[
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"abbreviation \"L M \\<equiv> LS M (initial M)\"",
"definition size where [simp, code]: \"size (m::('a, 'b, 'c) fsm) = card (states m)\"",
"abbreviation \"size_r M \\<equiv> card (reachable_states M)\"",
"fun minimal :: \"('a,'b,'c) fsm \\<Rightarrow> bool\" where\n \"minimal M = (\\<forall> q \\<in> states M . \\<forall> q' \\<in> states M . q \\<noteq> q' \\<longrightarrow> LS M q \\<noteq> LS M q')\"",
"fun observable :: \"('in, 'out, 'state) FSM \\<Rightarrow> bool\" where\n \"observable M = (\\<forall> t . \\<forall> s1 . ((succ M) t s1 = {}) \n \\<or> (\\<exists> s2 . (succ M) t s1 = {s2}))\""
] |
###template
\<lbrakk>?H1 x_1; ?H2 x_2; ?H3 x_1; ?H4 x_2; ?H5 x_1 \<le> x_3; ?H6 x_2 \<le> x_3; ?H7 x_2 = ?H8 x_1; ?H9 x_2 = ?H10 x_1\<rbrakk> \<Longrightarrow> (?H11 x_1 = ?H12 x_2) = ?H13 (?H14 x_2 (?H15 x_2)) (?H16 x_1 x_3 x_4 x_5)
###symbols
SPYH_Method_Implementations.spyh_method_via_h_framework_lists
FSM.initial
Test_Suite_Representations.passes_test_case
List.list.list_all
FSM.L
FSM.outputs
FSM.inputs
FSM.size
FSM.size_r
FSM.minimal
FSM.observable
###defs
datatype (set: 'a) list =
Nil ("[]")
| Cons (hd: 'a) (tl: "'a list") (infixr "#" 65)
for
map: map
rel: list_all2
pred: list_all
where
"tl [] = []"
abbreviation "L M \<equiv> LS M (initial M)"
definition size where [simp, code]: "size (m::('a, 'b, 'c) fsm) = card (states m)"
abbreviation "size_r M \<equiv> card (reachable_states M)"
fun minimal :: "('a,'b,'c) fsm \<Rightarrow> bool" where
"minimal M = (\<forall> q \<in> states M . \<forall> q' \<in> states M . q \<noteq> q' \<longrightarrow> LS M q \<noteq> LS M q')"
fun observable :: "('in, 'out, 'state) FSM \<Rightarrow> bool" where
"observable M = (\<forall> t . \<forall> s1 . ((succ M) t s1 = {})
\<or> (\<exists> s2 . (succ M) t s1 = {s2}))"
|
###output
observable ?M1.0 \<Longrightarrow> observable ?M2.0 \<Longrightarrow> minimal ?M1.0 \<Longrightarrow> minimal ?M2.0 \<Longrightarrow> size_r ?M1.0 \<le> ?m \<Longrightarrow> FSM.size ?M2.0 \<le> ?m \<Longrightarrow> FSM.inputs ?M2.0 = FSM.inputs ?M1.0 \<Longrightarrow> FSM.outputs ?M2.0 = FSM.outputs ?M1.0 \<Longrightarrow> (L ?M1.0 = L ?M2.0) = list_all (passes_test_case ?M2.0 (FSM.initial ?M2.0)) (spyh_method_via_h_framework_lists ?M1.0 ?m ?completeInputTraces ?useInputHeuristic)
###end
|
Incompleteness/Pseudo_Coding
|
Pseudo_Coding.perm_self_inverseI
|
lemma perm_self_inverseI: "\<lbrakk>-p = q; a \<sharp> p; a' \<sharp> p\<rbrakk> \<Longrightarrow> - ((a \<rightleftharpoons> a') + p) = (a \<rightleftharpoons> a') + q"
|
- ?p = ?q \<Longrightarrow> ?a \<sharp> ?p \<Longrightarrow> ?a' \<sharp> ?p \<Longrightarrow> - ((?a \<rightleftharpoons> ?a') + ?p) = (?a \<rightleftharpoons> ?a') + ?q
|
\<lbrakk>?H1 x_1 = x_2; ?H2 x_3 x_1; ?H2 x_4 x_1\<rbrakk> \<Longrightarrow> ?H1 (?H3 (?H4 x_3 x_4) x_1) = ?H3 (?H4 x_3 x_4) x_2
|
[
"Nominal2_Base.swap",
"Groups.plus_class.plus",
"Nominal2_Base.pt_class.fresh",
"Groups.uminus_class.uminus"
] |
[
"definition\n swap :: \"atom \\<Rightarrow> atom \\<Rightarrow> perm\" (\"'(_ \\<rightleftharpoons> _')\")\nwhere\n \"(a \\<rightleftharpoons> b) =\n Abs_perm (if sort_of a = sort_of b\n then (\\<lambda>c. if a = c then b else if b = c then a else c)\n else id)\"",
"class plus =\n fixes plus :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"+\" 65)",
"class pt =\n fixes permute :: \"perm \\<Rightarrow> 'a \\<Rightarrow> 'a\" (\"_ \\<bullet> _\" [76, 75] 75)\n assumes permute_zero [simp]: \"0 \\<bullet> x = x\"\n assumes permute_plus [simp]: \"(p + q) \\<bullet> x = p \\<bullet> (q \\<bullet> x)\"\nbegin",
"class uminus =\n fixes uminus :: \"'a \\<Rightarrow> 'a\" (\"- _\" [81] 80)"
] |
###template
\<lbrakk>?H1 x_1 = x_2; ?H2 x_3 x_1; ?H2 x_4 x_1\<rbrakk> \<Longrightarrow> ?H1 (?H3 (?H4 x_3 x_4) x_1) = ?H3 (?H4 x_3 x_4) x_2
###symbols
Nominal2_Base.swap
Groups.plus_class.plus
Nominal2_Base.pt_class.fresh
Groups.uminus_class.uminus
###defs
definition
swap :: "atom \<Rightarrow> atom \<Rightarrow> perm" ("'(_ \<rightleftharpoons> _')")
where
"(a \<rightleftharpoons> b) =
Abs_perm (if sort_of a = sort_of b
then (\<lambda>c. if a = c then b else if b = c then a else c)
else id)"
class plus =
fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "+" 65)
class pt =
fixes permute :: "perm \<Rightarrow> 'a \<Rightarrow> 'a" ("_ \<bullet> _" [76, 75] 75)
assumes permute_zero [simp]: "0 \<bullet> x = x"
assumes permute_plus [simp]: "(p + q) \<bullet> x = p \<bullet> (q \<bullet> x)"
begin
class uminus =
fixes uminus :: "'a \<Rightarrow> 'a" ("- _" [81] 80)
|
###output
- ?p = ?q \<Longrightarrow> ?a \<sharp> ?p \<Longrightarrow> ?a' \<sharp> ?p \<Longrightarrow> - ((?a \<rightleftharpoons> ?a') + ?p) = (?a \<rightleftharpoons> ?a') + ?q
###end
|
Refine_Imperative_HOL/Sepref
|
Sepref_HOL_Bindings.invalid_list_merge(2)
| null |
?A \<or>\<^sub>A hn_ctxt (list_assn (invalid_assn ?Aa)) ?l ?l' \<Longrightarrow>\<^sub>t ?C \<Longrightarrow> ?A \<or>\<^sub>A hn_invalid (list_assn ?Aa) ?l ?l' \<Longrightarrow>\<^sub>t ?C
|
?H1 (?H2 x_1 (?H3 (?H4 (?H5 x_2)) x_3 x_4)) x_5 \<Longrightarrow> ?H1 (?H2 x_1 (?H6 (?H4 x_2) x_3 x_4)) x_5
|
[
"Sepref_Basic.hn_invalid",
"Sepref_Basic.invalid_assn",
"Sepref_HOL_Bindings.list_assn",
"Sepref_Basic.hn_ctxt",
"Assertions.sup_assn",
"Assertions.entailst"
] |
[
"abbreviation \"hn_invalid R \\<equiv> hn_ctxt (invalid_assn R)\"",
"definition \"invalid_assn R x y \\<equiv> \\<up>(\\<exists>h. h\\<Turnstile>R x y) * True\"",
"fun list_assn :: \"('a \\<Rightarrow> 'c \\<Rightarrow> assn) \\<Rightarrow> 'a list \\<Rightarrow> 'c list \\<Rightarrow> assn\" where\n \"list_assn P [] [] = emp\"\n| \"list_assn P (a#as) (c#cs) = P a c * list_assn P as cs\"\n| \"list_assn _ _ _ = False\"",
"definition hn_ctxt :: \"('a\\<Rightarrow>'c\\<Rightarrow>assn) \\<Rightarrow> 'a \\<Rightarrow> 'c \\<Rightarrow> assn\" \n \\<comment> \\<open>Tag for refinement assertion\\<close>\n where\n \"hn_ctxt P a c \\<equiv> P a c\"",
"abbreviation sup_assn::\"assn\\<Rightarrow>assn\\<Rightarrow>assn\" (infixr \"\\<or>\\<^sub>A\" 61) \n where \"sup_assn \\<equiv> sup\"",
"definition entailst :: \"assn \\<Rightarrow> assn \\<Rightarrow> bool\" (infix \"\\<Longrightarrow>\\<^sub>t\" 10)\n where \"entailst A B \\<equiv> A \\<Longrightarrow>\\<^sub>A B * True\""
] |
###template
?H1 (?H2 x_1 (?H3 (?H4 (?H5 x_2)) x_3 x_4)) x_5 \<Longrightarrow> ?H1 (?H2 x_1 (?H6 (?H4 x_2) x_3 x_4)) x_5
###symbols
Sepref_Basic.hn_invalid
Sepref_Basic.invalid_assn
Sepref_HOL_Bindings.list_assn
Sepref_Basic.hn_ctxt
Assertions.sup_assn
Assertions.entailst
###defs
abbreviation "hn_invalid R \<equiv> hn_ctxt (invalid_assn R)"
definition "invalid_assn R x y \<equiv> \<up>(\<exists>h. h\<Turnstile>R x y) * True"
fun list_assn :: "('a \<Rightarrow> 'c \<Rightarrow> assn) \<Rightarrow> 'a list \<Rightarrow> 'c list \<Rightarrow> assn" where
"list_assn P [] [] = emp"
| "list_assn P (a#as) (c#cs) = P a c * list_assn P as cs"
| "list_assn _ _ _ = False"
definition hn_ctxt :: "('a\<Rightarrow>'c\<Rightarrow>assn) \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> assn"
\<comment> \<open>Tag for refinement assertion\<close>
where
"hn_ctxt P a c \<equiv> P a c"
abbreviation sup_assn::"assn\<Rightarrow>assn\<Rightarrow>assn" (infixr "\<or>\<^sub>A" 61)
where "sup_assn \<equiv> sup"
definition entailst :: "assn \<Rightarrow> assn \<Rightarrow> bool" (infix "\<Longrightarrow>\<^sub>t" 10)
where "entailst A B \<equiv> A \<Longrightarrow>\<^sub>A B * True"
|
###output
?A \<or>\<^sub>A hn_ctxt (list_assn (invalid_assn ?Aa)) ?l ?l' \<Longrightarrow>\<^sub>t ?C \<Longrightarrow> ?A \<or>\<^sub>A hn_invalid (list_assn ?Aa) ?l ?l' \<Longrightarrow>\<^sub>t ?C
###end
|
Real_Time_Deque/States_Proof
|
States_Proof.list_small_first_pop_small
|
lemma list_small_first_pop_small [simp]: "\<lbrakk>
invar (States dir big small);
0 < size small;
Small.pop small = (x, small')\<rbrakk>
\<Longrightarrow> x # list_small_first (States dir big small') = list_small_first (States dir big small)"
|
invar (States ?dir ?big ?small) \<Longrightarrow> 0 < size ?small \<Longrightarrow> Small.pop ?small = (?x, ?small') \<Longrightarrow> ?x # list_small_first (States ?dir ?big ?small') = list_small_first (States ?dir ?big ?small)
|
\<lbrakk>?H1 (?H2 x_1 x_2 x_3); ?H3 < ?H4 x_3; ?H5 x_3 = (x_4, x_5)\<rbrakk> \<Longrightarrow> ?H6 x_4 (?H7 (?H2 x_1 x_2 x_5)) = ?H7 (?H2 x_1 x_2 x_3)
|
[
"States_Aux.list_small_first",
"List.list.Cons",
"Small.pop",
"Nat.size_class.size",
"Groups.zero_class.zero",
"States.states.States",
"Type_Classes.invar_class.invar"
] |
[
"fun list_small_first :: \"'a states \\<Rightarrow> 'a list\" where\n \"list_small_first states = (let (big, small) = lists states in small @ (rev big))\"",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"fun pop :: \"'a small_state \\<Rightarrow> 'a * 'a small_state\" where\n \"pop (Small3 state) = (\n let (x, state) = Common.pop state \n in (x, Small3 state)\n )\"\n| \"pop (Small1 current small auxS) = \n (first current, Small1 (drop_first current) small auxS)\"\n| \"pop (Small2 current auxS big newS count) = \n (first current, Small2 (drop_first current) auxS big newS count)\"",
"class size =\n fixes size :: \"'a \\<Rightarrow> nat\" \\<comment> \\<open>see further theory \\<open>Wellfounded\\<close>\\<close>",
"class zero =\n fixes zero :: 'a (\"0\")",
"datatype 'a states = States direction \"'a big_state\" \"'a small_state\"",
"class invar =\n fixes invar :: \"'a \\<Rightarrow> bool\""
] |
###template
\<lbrakk>?H1 (?H2 x_1 x_2 x_3); ?H3 < ?H4 x_3; ?H5 x_3 = (x_4, x_5)\<rbrakk> \<Longrightarrow> ?H6 x_4 (?H7 (?H2 x_1 x_2 x_5)) = ?H7 (?H2 x_1 x_2 x_3)
###symbols
States_Aux.list_small_first
List.list.Cons
Small.pop
Nat.size_class.size
Groups.zero_class.zero
States.states.States
Type_Classes.invar_class.invar
###defs
fun list_small_first :: "'a states \<Rightarrow> 'a list" where
"list_small_first states = (let (big, small) = lists states in small @ (rev big))"
datatype (set: 'a) list =
Nil ("[]")
| Cons (hd: 'a) (tl: "'a list") (infixr "#" 65)
for
map: map
rel: list_all2
pred: list_all
where
"tl [] = []"
fun pop :: "'a small_state \<Rightarrow> 'a * 'a small_state" where
"pop (Small3 state) = (
let (x, state) = Common.pop state
in (x, Small3 state)
)"
| "pop (Small1 current small auxS) =
(first current, Small1 (drop_first current) small auxS)"
| "pop (Small2 current auxS big newS count) =
(first current, Small2 (drop_first current) auxS big newS count)"
class size =
fixes size :: "'a \<Rightarrow> nat" \<comment> \<open>see further theory \<open>Wellfounded\<close>\<close>
class zero =
fixes zero :: 'a ("0")
datatype 'a states = States direction "'a big_state" "'a small_state"
class invar =
fixes invar :: "'a \<Rightarrow> bool"
|
###output
invar (States ?dir ?big ?small) \<Longrightarrow> 0 < size ?small \<Longrightarrow> Small.pop ?small = (?x, ?small') \<Longrightarrow> ?x # list_small_first (States ?dir ?big ?small') = list_small_first (States ?dir ?big ?small)
###end
|
HOL-CSP/Process
|
Process.F_T
|
lemma F_T: "(s, X) \<in> \<F> P \<Longrightarrow> s \<in> \<T> P"
|
(?s, ?X) \<in> \<F> ?P \<Longrightarrow> ?s \<in> \<T> ?P
|
(x_1, x_2) \<in> ?H1 x_3 \<Longrightarrow> x_1 \<in> ?H2 x_3
|
[
"Process.Traces",
"Process.Failures"
] |
[] |
###template
(x_1, x_2) \<in> ?H1 x_3 \<Longrightarrow> x_1 \<in> ?H2 x_3
###symbols
Process.Traces
Process.Failures
###defs
|
###output
(?s, ?X) \<in> \<F> ?P \<Longrightarrow> ?s \<in> \<T> ?P
###end
|
Graph_Saturation/LabeledGraphs
|
LabeledGraphs.subgraph_restrict
|
lemma subgraph_restrict[simp]:
"subgraph G (restrict G) = graph G"
|
subgraph ?G (restrict ?G) = graph ?G
|
?H1 x_1 (?H2 x_1) = ?H3 x_1
|
[
"LabeledGraphs.graph",
"LabeledGraphs.restrict",
"LabeledGraphs.subgraph"
] |
[
"abbreviation graph where\n \"graph X \\<equiv> X = restrict X\"",
"definition restrict where\n \"restrict G = LG {(l,v1,v2) \\<in> edges G. v1 \\<in> vertices G \\<and> v2 \\<in> vertices G } (vertices G)\"",
"abbreviation subgraph\n where \"subgraph G\\<^sub>1 G\\<^sub>2 \n \\<equiv> graph_homomorphism G\\<^sub>1 G\\<^sub>2 (Id_on (vertices G\\<^sub>1))\""
] |
###template
?H1 x_1 (?H2 x_1) = ?H3 x_1
###symbols
LabeledGraphs.graph
LabeledGraphs.restrict
LabeledGraphs.subgraph
###defs
abbreviation graph where
"graph X \<equiv> X = restrict X"
definition restrict where
"restrict G = LG {(l,v1,v2) \<in> edges G. v1 \<in> vertices G \<and> v2 \<in> vertices G } (vertices G)"
abbreviation subgraph
where "subgraph G\<^sub>1 G\<^sub>2
\<equiv> graph_homomorphism G\<^sub>1 G\<^sub>2 (Id_on (vertices G\<^sub>1))"
|
###output
subgraph ?G (restrict ?G) = graph ?G
###end
|
ROBDD/Middle_Impl
|
Middle_Impl.True_rep
|
lemma True_rep[simp]: "bdd_sane s \<Longrightarrow> (ni,Trueif)\<in>Rmi s \<longleftrightarrow> ni=Suc 0"
|
bdd_sane ?s \<Longrightarrow> ((?ni, Trueif) \<in> Rmi ?s) = (?ni = Suc 0)
|
?H1 x_1 \<Longrightarrow> ((x_2, ?H2) \<in> ?H3 x_1) = (x_2 = ?H4 ?H5)
|
[
"Groups.zero_class.zero",
"Nat.Suc",
"Middle_Impl.Rmi",
"BDT.ifex.Trueif",
"Middle_Impl.bdd_sane"
] |
[
"class zero =\n fixes zero :: 'a (\"0\")",
"definition Suc :: \"nat \\<Rightarrow> nat\"\n where \"Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))\"",
"definition \"Rmi s \\<equiv> {(a,b)|a b. Rmi_g a b s}\"",
"datatype 'a ifex = Trueif | Falseif | IF 'a \"'a ifex\" \"'a ifex\"",
"definition \"bdd_sane bdd \\<equiv> pointermap_sane (dpm bdd) \\<and> mi_pre.map_invar_impl (dcl bdd) bdd\""
] |
###template
?H1 x_1 \<Longrightarrow> ((x_2, ?H2) \<in> ?H3 x_1) = (x_2 = ?H4 ?H5)
###symbols
Groups.zero_class.zero
Nat.Suc
Middle_Impl.Rmi
BDT.ifex.Trueif
Middle_Impl.bdd_sane
###defs
class zero =
fixes zero :: 'a ("0")
definition Suc :: "nat \<Rightarrow> nat"
where "Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))"
definition "Rmi s \<equiv> {(a,b)|a b. Rmi_g a b s}"
datatype 'a ifex = Trueif | Falseif | IF 'a "'a ifex" "'a ifex"
definition "bdd_sane bdd \<equiv> pointermap_sane (dpm bdd) \<and> mi_pre.map_invar_impl (dcl bdd) bdd"
|
###output
bdd_sane ?s \<Longrightarrow> ((?ni, Trueif) \<in> Rmi ?s) = (?ni = Suc 0)
###end
|
Differential_Dynamic_Logic/Denotational_Semantics
|
Denotational_Semantics.stimes_case
|
lemma stimes_case:
"sterm_sem I (Times t1 t2) = (\<lambda>v. (sterm_sem I t1 v) * (sterm_sem I t2 v))"
|
sterm_sem ?I (Times ?t1.0 ?t2.0) = (\<lambda>v. sterm_sem ?I ?t1.0 v * sterm_sem ?I ?t2.0 v)
|
?H1 x_1 (?H2 x_2 x_3) = (\<lambda>y_0. ?H3 (?H1 x_1 x_2 y_0) (?H1 x_1 x_3 y_0))
|
[
"Groups.times_class.times",
"Syntax.trm.Times",
"Denotational_Semantics.sterm_sem"
] |
[
"class times =\n fixes times :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"*\" 70)",
"datatype ('a, 'c) trm =\n\\<comment> \\<open>Real-valued variables given meaning by the state and modified by programs.\\<close>\n Var 'c\n\\<comment> \\<open>N.B. This is technically more expressive than True dL since most reals\\<close>\n\\<comment> \\<open>can't be written down.\\<close>\n| Const real\n\\<comment> \\<open>A function (applied to its arguments) consists of an identifier for the function\\<close>\n\\<comment> \\<open>and a function \\<open>'c \\<Rightarrow> ('a, 'c) trm\\<close> (where \\<open>'c\\<close> is a finite type) which specifies one\\<close>\n\\<comment> \\<open>argument of the function for each element of type \\<open>'c\\<close>. To simulate a function with\\<close>\n\\<comment> \\<open>less than \\<open>'c\\<close> arguments, set the remaining arguments to a constant, such as \\<open>Const 0\\<close>\\<close>\n| Function 'a \"'c \\<Rightarrow> ('a, 'c) trm\" (\"$f\")\n| Plus \"('a, 'c) trm\" \"('a, 'c) trm\"\n| Times \"('a, 'c) trm\" \"('a, 'c) trm\"\n\\<comment> \\<open>A (real-valued) variable standing for a differential, such as \\<open>x'\\<close>, given meaning by the state\\<close>\n\\<comment> \\<open>and modified by programs.\\<close>\n| DiffVar 'c (\"$''\")\n\\<comment> \\<open>The differential of an arbitrary term \\<open>(\\<theta>)'\\<close>\\<close>\n| Differential \"('a, 'c) trm\"",
"primrec sterm_sem :: \"('a::finite, 'b::finite, 'c::finite) interp \\<Rightarrow> ('a, 'c) trm \\<Rightarrow> 'c simple_state \\<Rightarrow> real\"\nwhere\n \"sterm_sem I (Var x) v = v $ x\"\n| \"sterm_sem I (Function f args) v = Functions I f (\\<chi> i. sterm_sem I (args i) v)\"\n| \"sterm_sem I (Plus t1 t2) v = sterm_sem I t1 v + sterm_sem I t2 v\"\n| \"sterm_sem I (Times t1 t2) v = sterm_sem I t1 v * sterm_sem I t2 v\"\n| \"sterm_sem I (Const r) v = r\"\n| \"sterm_sem I ($' c) v = undefined\"\n| \"sterm_sem I (Differential d) v = undefined\"\n \n\\<comment> \\<open>\\<open>frechet I \\<theta> \\<nu>\\<close> syntactically computes the frechet derivative of the term \\<open>\\<theta>\\<close> in the interpretation\\<close>\n\\<comment> \\<open>\\<open>I\\<close> at state \\<open>\\<nu>\\<close> (containing only the unprimed variables). The frechet derivative is a\\<close>\n\\<comment> \\<open>linear map from the differential state \\<open>\\<nu>\\<close> to reals.\\<close>"
] |
###template
?H1 x_1 (?H2 x_2 x_3) = (\<lambda>y_0. ?H3 (?H1 x_1 x_2 y_0) (?H1 x_1 x_3 y_0))
###symbols
Groups.times_class.times
Syntax.trm.Times
Denotational_Semantics.sterm_sem
###defs
class times =
fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "*" 70)
datatype ('a, 'c) trm =
\<comment> \<open>Real-valued variables given meaning by the state and modified by programs.\<close>
Var 'c
\<comment> \<open>N.B. This is technically more expressive than True dL since most reals\<close>
\<comment> \<open>can't be written down.\<close>
| Const real
\<comment> \<open>A function (applied to its arguments) consists of an identifier for the function\<close>
\<comment> \<open>and a function \<open>'c \<Rightarrow> ('a, 'c) trm\<close> (where \<open>'c\<close> is a finite type) which specifies one\<close>
\<comment> \<open>argument of the function for each element of type \<open>'c\<close>. To simulate a function with\<close>
\<comment> \<open>less than \<open>'c\<close> arguments, set the remaining arguments to a constant, such as \<open>Const 0\<close>\<close>
| Function 'a "'c \<Rightarrow> ('a, 'c) trm" ("$f")
| Plus "('a, 'c) trm" "('a, 'c) trm"
| Times "('a, 'c) trm" "('a, 'c) trm"
\<comment> \<open>A (real-valued) variable standing for a differential, such as \<open>x'\<close>, given meaning by the state\<close>
\<comment> \<open>and modified by programs.\<close>
| DiffVar 'c ("$''")
\<comment> \<open>The differential of an arbitrary term \<open>(\<theta>)'\<close>\<close>
| Differential "('a, 'c) trm"
primrec sterm_sem :: "('a::finite, 'b::finite, 'c::finite) interp \<Rightarrow> ('a, 'c) trm \<Rightarrow> 'c simple_state \<Rightarrow> real"
where
"sterm_sem I (Var x) v = v $ x"
| "sterm_sem I (Function f args) v = Functions I f (\<chi> i. sterm_sem I (args i) v)"
| "sterm_sem I (Plus t1 t2) v = sterm_sem I t1 v + sterm_sem I t2 v"
| "sterm_sem I (Times t1 t2) v = sterm_sem I t1 v * sterm_sem I t2 v"
| "sterm_sem I (Const r) v = r"
| "sterm_sem I ($' c) v = undefined"
| "sterm_sem I (Differential d) v = undefined"
\<comment> \<open>\<open>frechet I \<theta> \<nu>\<close> syntactically computes the frechet derivative of the term \<open>\<theta>\<close> in the interpretation\<close>
\<comment> \<open>\<open>I\<close> at state \<open>\<nu>\<close> (containing only the unprimed variables). The frechet derivative is a\<close>
\<comment> \<open>linear map from the differential state \<open>\<nu>\<close> to reals.\<close>
|
###output
sterm_sem ?I (Times ?t1.0 ?t2.0) = (\<lambda>v. sterm_sem ?I ?t1.0 v * sterm_sem ?I ?t2.0 v)
###end
|
DiskPaxos/DiskPaxos_Inv3
|
DiskPaxos_Inv3.EndPhase0_HInv3_q
|
lemma EndPhase0_HInv3_q:
"\<lbrakk> EndPhase0 s s' q; HInv3_L s' p q d \<rbrakk>
\<Longrightarrow> HInv3_R s' p q d"
|
EndPhase0 ?s ?s' ?q \<Longrightarrow> HInv3_L ?s' ?p ?q ?d \<Longrightarrow> HInv3_R ?s' ?p ?q ?d
|
\<lbrakk>?H1 x_1 x_2 x_3; ?H2 x_2 x_4 x_3 x_5\<rbrakk> \<Longrightarrow> ?H3 x_2 x_4 x_3 x_5
|
[
"DiskPaxos_Inv3.HInv3_R",
"DiskPaxos_Inv3.HInv3_L",
"DiskPaxos_Model.EndPhase0"
] |
[
"definition HInv3_R :: \"state \\<Rightarrow> Proc \\<Rightarrow> Proc \\<Rightarrow> Disk \\<Rightarrow> bool\"\nwhere\n \"HInv3_R s p q d = (\\<lparr>block= dblock s q, proc= q\\<rparr> \\<in> blocksRead s p d\n \\<or> \\<lparr>block= dblock s p, proc= p\\<rparr> \\<in> blocksRead s q d)\"",
"definition HInv3_L :: \"state \\<Rightarrow> Proc \\<Rightarrow> Proc \\<Rightarrow> Disk \\<Rightarrow> bool\"\nwhere\n \"HInv3_L s p q d = (phase s p \\<in> {1,2}\n \\<and> phase s q \\<in> {1,2} \n \\<and> hasRead s p d q\n \\<and> hasRead s q d p)\"",
"definition EndPhase0 :: \"state \\<Rightarrow> state \\<Rightarrow> Proc \\<Rightarrow> bool\"\nwhere\n \"EndPhase0 s s' p =\n (phase s p = 0\n \\<and> IsMajority ({d. hasRead s p d p})\n \\<and> (\\<exists>b \\<in> Ballot p. \n (\\<forall>r \\<in> allBlocksRead s p. mbal r < b)\n \\<and> dblock s' = (dblock s) ( p:= \n (SOME r. r \\<in> allBlocksRead s p \n \\<and> (\\<forall>s \\<in> allBlocksRead s p. bal s \\<le> bal r)) \\<lparr> mbal := b \\<rparr> ))\n \\<and> InitializePhase s s' p\n \\<and> phase s' = (phase s) (p:= 1)\n \\<and> inpt s' = inpt s \\<and> outpt s' = outpt s \\<and> disk s' = disk s)\""
] |
###template
\<lbrakk>?H1 x_1 x_2 x_3; ?H2 x_2 x_4 x_3 x_5\<rbrakk> \<Longrightarrow> ?H3 x_2 x_4 x_3 x_5
###symbols
DiskPaxos_Inv3.HInv3_R
DiskPaxos_Inv3.HInv3_L
DiskPaxos_Model.EndPhase0
###defs
definition HInv3_R :: "state \<Rightarrow> Proc \<Rightarrow> Proc \<Rightarrow> Disk \<Rightarrow> bool"
where
"HInv3_R s p q d = (\<lparr>block= dblock s q, proc= q\<rparr> \<in> blocksRead s p d
\<or> \<lparr>block= dblock s p, proc= p\<rparr> \<in> blocksRead s q d)"
definition HInv3_L :: "state \<Rightarrow> Proc \<Rightarrow> Proc \<Rightarrow> Disk \<Rightarrow> bool"
where
"HInv3_L s p q d = (phase s p \<in> {1,2}
\<and> phase s q \<in> {1,2}
\<and> hasRead s p d q
\<and> hasRead s q d p)"
definition EndPhase0 :: "state \<Rightarrow> state \<Rightarrow> Proc \<Rightarrow> bool"
where
"EndPhase0 s s' p =
(phase s p = 0
\<and> IsMajority ({d. hasRead s p d p})
\<and> (\<exists>b \<in> Ballot p.
(\<forall>r \<in> allBlocksRead s p. mbal r < b)
\<and> dblock s' = (dblock s) ( p:=
(SOME r. r \<in> allBlocksRead s p
\<and> (\<forall>s \<in> allBlocksRead s p. bal s \<le> bal r)) \<lparr> mbal := b \<rparr> ))
\<and> InitializePhase s s' p
\<and> phase s' = (phase s) (p:= 1)
\<and> inpt s' = inpt s \<and> outpt s' = outpt s \<and> disk s' = disk s)"
|
###output
EndPhase0 ?s ?s' ?q \<Longrightarrow> HInv3_L ?s' ?p ?q ?d \<Longrightarrow> HInv3_R ?s' ?p ?q ?d
###end
|
Count_Complex_Roots/Extended_Sturm
|
Extended_Sturm.cindex_polyE_mod
|
lemma cindex_polyE_mod:
fixes p q::"real poly"
shows "cindex_polyE a b q p = cindex_polyE a b (q mod p) p"
|
cindex_polyE ?a ?b ?q ?p = cindex_polyE ?a ?b (?q mod ?p) ?p
|
?H1 x_1 x_2 x_3 x_4 = ?H1 x_1 x_2 (?H2 x_3 x_4) x_4
|
[
"Rings.modulo_class.modulo",
"Extended_Sturm.cindex_polyE"
] |
[
"class modulo = dvd + divide +\n fixes modulo :: \"'a \\<Rightarrow> 'a \\<Rightarrow> 'a\" (infixl \"mod\" 70)",
"definition cindex_polyE:: \"real \\<Rightarrow> real \\<Rightarrow> real poly \\<Rightarrow> real poly \\<Rightarrow> real\" where\n \"cindex_polyE a b q p = jumpF_polyR q p a + cindex_poly a b q p - jumpF_polyL q p b\""
] |
###template
?H1 x_1 x_2 x_3 x_4 = ?H1 x_1 x_2 (?H2 x_3 x_4) x_4
###symbols
Rings.modulo_class.modulo
Extended_Sturm.cindex_polyE
###defs
class modulo = dvd + divide +
fixes modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "mod" 70)
definition cindex_polyE:: "real \<Rightarrow> real \<Rightarrow> real poly \<Rightarrow> real poly \<Rightarrow> real" where
"cindex_polyE a b q p = jumpF_polyR q p a + cindex_poly a b q p - jumpF_polyL q p b"
|
###output
cindex_polyE ?a ?b ?q ?p = cindex_polyE ?a ?b (?q mod ?p) ?p
###end
|
Given_Clause_Loops/Given_Clause_Loops_Util
|
Given_Clause_Loops_Util.set_drop_fold_removeAll
|
lemma set_drop_fold_removeAll: "set (drop k (fold removeAll ys xs)) \<subseteq> set (drop k xs)"
|
set (drop ?k (fold removeAll ?ys ?xs)) \<subseteq> set (drop ?k ?xs)
|
?H1 (?H2 (?H3 x_1 (?H4 ?H5 x_2 x_3))) (?H2 (?H3 x_1 x_3))
|
[
"List.removeAll",
"List.fold",
"List.drop",
"List.list.set",
"Set.subset_eq"
] |
[
"primrec removeAll :: \"'a \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" where\n\"removeAll x [] = []\" |\n\"removeAll x (y # xs) = (if x = y then removeAll x xs else y # removeAll x xs)\"",
"primrec fold :: \"('a \\<Rightarrow> 'b \\<Rightarrow> 'b) \\<Rightarrow> 'a list \\<Rightarrow> 'b \\<Rightarrow> 'b\" where\nfold_Nil: \"fold f [] = id\" |\nfold_Cons: \"fold f (x # xs) = fold f xs \\<circ> f x\"",
"primrec drop:: \"nat \\<Rightarrow> 'a list \\<Rightarrow> 'a list\" where\ndrop_Nil: \"drop n [] = []\" |\ndrop_Cons: \"drop n (x # xs) = (case n of 0 \\<Rightarrow> x # xs | Suc m \\<Rightarrow> drop m xs)\"\n \\<comment> \\<open>Warning: simpset does not contain this definition, but separate\n theorems for \\<open>n = 0\\<close> and \\<open>n = Suc k\\<close>\\<close>",
"datatype (set: 'a) list =\n Nil (\"[]\")\n | Cons (hd: 'a) (tl: \"'a list\") (infixr \"#\" 65)\nfor\n map: map\n rel: list_all2\n pred: list_all\nwhere\n \"tl [] = []\"",
"abbreviation subset_eq :: \"'a set \\<Rightarrow> 'a set \\<Rightarrow> bool\"\n where \"subset_eq \\<equiv> less_eq\""
] |
###template
?H1 (?H2 (?H3 x_1 (?H4 ?H5 x_2 x_3))) (?H2 (?H3 x_1 x_3))
###symbols
List.removeAll
List.fold
List.drop
List.list.set
Set.subset_eq
###defs
primrec removeAll :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where
"removeAll x [] = []" |
"removeAll x (y # xs) = (if x = y then removeAll x xs else y # removeAll x xs)"
primrec fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b \<Rightarrow> 'b" where
fold_Nil: "fold f [] = id" |
fold_Cons: "fold f (x # xs) = fold f xs \<circ> f x"
primrec drop:: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
drop_Nil: "drop n [] = []" |
drop_Cons: "drop n (x # xs) = (case n of 0 \<Rightarrow> x # xs | Suc m \<Rightarrow> drop m xs)"
\<comment> \<open>Warning: simpset does not contain this definition, but separate
theorems for \<open>n = 0\<close> and \<open>n = Suc k\<close>\<close>
datatype (set: 'a) list =
Nil ("[]")
| Cons (hd: 'a) (tl: "'a list") (infixr "#" 65)
for
map: map
rel: list_all2
pred: list_all
where
"tl [] = []"
abbreviation subset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"
where "subset_eq \<equiv> less_eq"
|
###output
set (drop ?k (fold removeAll ?ys ?xs)) \<subseteq> set (drop ?k ?xs)
###end
|
HOL-CSP/Assertions
|
Assertions.CHAOS\<^sub>S\<^sub>K\<^sub>I\<^sub>P_CHAOS_refine_F
| null |
CHAOS\<^sub>S\<^sub>K\<^sub>I\<^sub>P ?A \<sqsubseteq>\<^sub>F CHAOS ?A
|
?H1 (?H2 x_1) (?H3 x_1)
|
[
"Assertions.CHAOS",
"Assertions.CHAOS\\<^sub>S\\<^sub>K\\<^sub>I\\<^sub>P",
"Process_Order.failure_refine"
] |
[
"definition CHAOS :: \"'a set \\<Rightarrow> 'a process\" \n where \"CHAOS A \\<equiv> \\<mu> X. (STOP \\<sqinter> (\\<box> x \\<in> A \\<rightarrow> X))\"",
"definition CHAOS\\<^sub>S\\<^sub>K\\<^sub>I\\<^sub>P :: \"'a set \\<Rightarrow> 'a process\" \n where \"CHAOS\\<^sub>S\\<^sub>K\\<^sub>I\\<^sub>P A \\<equiv> \\<mu> X. (SKIP \\<sqinter> STOP \\<sqinter> (\\<box> x \\<in> A \\<rightarrow> X))\"",
"definition failure_refine :: \\<open>'a process \\<Rightarrow> 'a process \\<Rightarrow> bool\\<close> (infix \\<open>\\<sqsubseteq>\\<^sub>F\\<close> 60)\n where \\<open>P \\<sqsubseteq>\\<^sub>F Q \\<equiv> \\<F> Q \\<subseteq> \\<F> P\\<close>"
] |
###template
?H1 (?H2 x_1) (?H3 x_1)
###symbols
Assertions.CHAOS
Assertions.CHAOS\<^sub>S\<^sub>K\<^sub>I\<^sub>P
Process_Order.failure_refine
###defs
definition CHAOS :: "'a set \<Rightarrow> 'a process"
where "CHAOS A \<equiv> \<mu> X. (STOP \<sqinter> (\<box> x \<in> A \<rightarrow> X))"
definition CHAOS\<^sub>S\<^sub>K\<^sub>I\<^sub>P :: "'a set \<Rightarrow> 'a process"
where "CHAOS\<^sub>S\<^sub>K\<^sub>I\<^sub>P A \<equiv> \<mu> X. (SKIP \<sqinter> STOP \<sqinter> (\<box> x \<in> A \<rightarrow> X))"
definition failure_refine :: \<open>'a process \<Rightarrow> 'a process \<Rightarrow> bool\<close> (infix \<open>\<sqsubseteq>\<^sub>F\<close> 60)
where \<open>P \<sqsubseteq>\<^sub>F Q \<equiv> \<F> Q \<subseteq> \<F> P\<close>
|
###output
CHAOS\<^sub>S\<^sub>K\<^sub>I\<^sub>P ?A \<sqsubseteq>\<^sub>F CHAOS ?A
###end
|
Green/Derivs
|
Derivs.has_vector_derivative_transform_at
|
lemma has_vector_derivative_transform_at:
assumes "0 < d"
and "\<forall>x'. dist x' x < d \<longrightarrow> f x' = g x'"
and "(f has_vector_derivative f') (at x)"
shows "(g has_vector_derivative f') (at x)"
|
0 < ?d \<Longrightarrow> \<forall>x'. dist x' ?x < ?d \<longrightarrow> ?f x' = ?g x' \<Longrightarrow> (?f has_vector_derivative ?f') (at ?x) \<Longrightarrow> (?g has_vector_derivative ?f') (at ?x)
|
\<lbrakk>?H1 < x_1; \<forall>y_0. ?H2 y_0 x_2 < x_1 \<longrightarrow> x_3 y_0 = x_4 y_0; ?H3 x_3 x_5 (?H4 x_2)\<rbrakk> \<Longrightarrow> ?H3 x_4 x_5 (?H4 x_2)
|
[
"Topological_Spaces.topological_space_class.at",
"Deriv.has_vector_derivative",
"Real_Vector_Spaces.dist_class.dist",
"Groups.zero_class.zero"
] |
[
"class topological_space = \"open\" +\n assumes open_UNIV [simp, intro]: \"open UNIV\"\n assumes open_Int [intro]: \"open S \\<Longrightarrow> open T \\<Longrightarrow> open (S \\<inter> T)\"\n assumes open_Union [intro]: \"\\<forall>S\\<in>K. open S \\<Longrightarrow> open (\\<Union>K)\"\nbegin",
"definition has_vector_derivative :: \"(real \\<Rightarrow> 'b::real_normed_vector) \\<Rightarrow> 'b \\<Rightarrow> real filter \\<Rightarrow> bool\"\n (infix \"has'_vector'_derivative\" 50)\n where \"(f has_vector_derivative f') net \\<longleftrightarrow> (f has_derivative (\\<lambda>x. x *\\<^sub>R f')) net\"",
"class dist =\n fixes dist :: \"'a \\<Rightarrow> 'a \\<Rightarrow> real\"",
"class zero =\n fixes zero :: 'a (\"0\")"
] |
###template
\<lbrakk>?H1 < x_1; \<forall>y_0. ?H2 y_0 x_2 < x_1 \<longrightarrow> x_3 y_0 = x_4 y_0; ?H3 x_3 x_5 (?H4 x_2)\<rbrakk> \<Longrightarrow> ?H3 x_4 x_5 (?H4 x_2)
###symbols
Topological_Spaces.topological_space_class.at
Deriv.has_vector_derivative
Real_Vector_Spaces.dist_class.dist
Groups.zero_class.zero
###defs
class topological_space = "open" +
assumes open_UNIV [simp, intro]: "open UNIV"
assumes open_Int [intro]: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<inter> T)"
assumes open_Union [intro]: "\<forall>S\<in>K. open S \<Longrightarrow> open (\<Union>K)"
begin
definition has_vector_derivative :: "(real \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'b \<Rightarrow> real filter \<Rightarrow> bool"
(infix "has'_vector'_derivative" 50)
where "(f has_vector_derivative f') net \<longleftrightarrow> (f has_derivative (\<lambda>x. x *\<^sub>R f')) net"
class dist =
fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"
class zero =
fixes zero :: 'a ("0")
|
###output
0 < ?d \<Longrightarrow> \<forall>x'. dist x' ?x < ?d \<longrightarrow> ?f x' = ?g x' \<Longrightarrow> (?f has_vector_derivative ?f') (at ?x) \<Longrightarrow> (?g has_vector_derivative ?f') (at ?x)
###end
|
ConcurrentGC/Global_Invariants_Lemmas
|
Global_Invariants_Lemmas.valid_refs_invD(7)
|
lemma valid_refs_invD[elim]:
"\<lbrakk> x \<in> mut_m.mut_roots m s; (x reaches y) s; valid_refs_inv s \<rbrakk> \<Longrightarrow> valid_ref y s"
"\<lbrakk> x \<in> mut_m.mut_roots m s; (x reaches y) s; valid_refs_inv s \<rbrakk> \<Longrightarrow> \<exists>obj. sys_heap s y = Some obj"
"\<lbrakk> x \<in> mut_m.tso_store_refs m s; (x reaches y) s; valid_refs_inv s \<rbrakk> \<Longrightarrow> valid_ref y s"
"\<lbrakk> x \<in> mut_m.tso_store_refs m s; (x reaches y) s; valid_refs_inv s \<rbrakk> \<Longrightarrow> \<exists>obj. sys_heap s y = Some obj"
"\<lbrakk> w \<in> set (sys_mem_store_buffers (mutator m) s); x \<in> store_refs w; (x reaches y) s; valid_refs_inv s \<rbrakk> \<Longrightarrow> valid_ref y s"
"\<lbrakk> w \<in> set (sys_mem_store_buffers (mutator m) s); x \<in> store_refs w; (x reaches y) s; valid_refs_inv s \<rbrakk> \<Longrightarrow> \<exists>obj. sys_heap s y = Some obj"
"\<lbrakk> grey x s; (x reaches y) s; valid_refs_inv s \<rbrakk> \<Longrightarrow> valid_ref y s"
"\<lbrakk> mut_m.reachable m x s; valid_refs_inv s \<rbrakk> \<Longrightarrow> valid_ref x s"
"\<lbrakk> mut_m.reachable m x s; valid_refs_inv s \<rbrakk> \<Longrightarrow> \<exists>obj. sys_heap s x = Some obj"
"\<lbrakk> x \<in> mut_m.mut_ghost_honorary_root m s; (x reaches y) s; valid_refs_inv s \<rbrakk> \<Longrightarrow> valid_ref y s"
"\<lbrakk> x \<in> mut_m.mut_ghost_honorary_root m s; (x reaches y) s; valid_refs_inv s \<rbrakk> \<Longrightarrow> \<exists>obj. sys_heap s y = Some obj"
|
grey ?x ?s \<Longrightarrow> (?x reaches ?y) ?s \<Longrightarrow> valid_refs_inv ?s \<Longrightarrow> obj_at (\<lambda>s. True) ?y ?s
|
\<lbrakk>?H1 x_1 x_2; ?H2 x_1 x_3 x_2; ?H3 x_2\<rbrakk> \<Longrightarrow> ?H4 (\<lambda>y_0. True) x_3 x_2
|
[
"Proofs_Basis.obj_at",
"Global_Invariants.valid_refs_inv",
"Proofs_Basis.reaches",
"Proofs_Basis.grey"
] |
[
"definition obj_at :: \"(('field, 'payload, 'ref) object \\<Rightarrow> bool) \\<Rightarrow> 'ref \\<Rightarrow> ('field, 'mut, 'payload, 'ref) lsts_pred\" where\n \"obj_at P r \\<equiv> \\<lambda>s. case sys_heap s r of None \\<Rightarrow> False | Some obj \\<Rightarrow> P obj\"",
"definition valid_refs_inv :: \"('field, 'mut, 'payload, 'ref) lsts_pred\" where\n \"valid_refs_inv = (\\<^bold>\\<forall>m x. mut_m.reachable m x \\<^bold>\\<or> grey_reachable x \\<^bold>\\<longrightarrow> valid_ref x)\"",
"definition reaches :: \"'ref \\<Rightarrow> 'ref \\<Rightarrow> ('field, 'mut, 'payload, 'ref) lsts_pred\" (infix \"reaches\" 51) where\n \"x reaches y = (\\<lambda>s. (\\<lambda>x y. (x points_to y) s)\\<^sup>*\\<^sup>* x y)\"",
"definition grey :: \"'ref \\<Rightarrow> ('field, 'mut, 'payload, 'ref) lsts_pred\" where\n \"grey r = (\\<^bold>\\<exists>p. \\<langle>r\\<rangle> \\<^bold>\\<in> WL p)\""
] |
###template
\<lbrakk>?H1 x_1 x_2; ?H2 x_1 x_3 x_2; ?H3 x_2\<rbrakk> \<Longrightarrow> ?H4 (\<lambda>y_0. True) x_3 x_2
###symbols
Proofs_Basis.obj_at
Global_Invariants.valid_refs_inv
Proofs_Basis.reaches
Proofs_Basis.grey
###defs
definition obj_at :: "(('field, 'payload, 'ref) object \<Rightarrow> bool) \<Rightarrow> 'ref \<Rightarrow> ('field, 'mut, 'payload, 'ref) lsts_pred" where
"obj_at P r \<equiv> \<lambda>s. case sys_heap s r of None \<Rightarrow> False | Some obj \<Rightarrow> P obj"
definition valid_refs_inv :: "('field, 'mut, 'payload, 'ref) lsts_pred" where
"valid_refs_inv = (\<^bold>\<forall>m x. mut_m.reachable m x \<^bold>\<or> grey_reachable x \<^bold>\<longrightarrow> valid_ref x)"
definition reaches :: "'ref \<Rightarrow> 'ref \<Rightarrow> ('field, 'mut, 'payload, 'ref) lsts_pred" (infix "reaches" 51) where
"x reaches y = (\<lambda>s. (\<lambda>x y. (x points_to y) s)\<^sup>*\<^sup>* x y)"
definition grey :: "'ref \<Rightarrow> ('field, 'mut, 'payload, 'ref) lsts_pred" where
"grey r = (\<^bold>\<exists>p. \<langle>r\<rangle> \<^bold>\<in> WL p)"
|
###output
grey ?x ?s \<Longrightarrow> (?x reaches ?y) ?s \<Longrightarrow> valid_refs_inv ?s \<Longrightarrow> obj_at (\<lambda>s. True) ?y ?s
###end
|
Lazy-Lists-II/LList2
|
LList2.lappend_is_LNil_conv
|
lemma lappend_is_LNil_conv [iff]:
"(s @@ t = LNil) = (s = LNil \<and> t = LNil)"
|
(?s @@ ?t = LNil) = (?s = LNil \<and> ?t = LNil)
|
(?H1 x_1 x_2 = ?H2) = (x_1 = ?H2 \<and> x_2 = ?H2)
|
[
"Coinductive_List.llist.LNil",
"Coinductive_List.lappend"
] |
[
"codatatype (lset: 'a) llist =\n lnull: LNil\n | LCons (lhd: 'a) (ltl: \"'a llist\")\nfor\n map: lmap\n rel: llist_all2\nwhere\n \"lhd LNil = undefined\"\n| \"ltl LNil = LNil\""
] |
###template
(?H1 x_1 x_2 = ?H2) = (x_1 = ?H2 \<and> x_2 = ?H2)
###symbols
Coinductive_List.llist.LNil
Coinductive_List.lappend
###defs
codatatype (lset: 'a) llist =
lnull: LNil
| LCons (lhd: 'a) (ltl: "'a llist")
for
map: lmap
rel: llist_all2
where
"lhd LNil = undefined"
| "ltl LNil = LNil"
|
###output
(?s @@ ?t = LNil) = (?s = LNil \<and> ?t = LNil)
###end
|
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