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Definition singleton {A : Type} (x : key) (v : A) : t A (Dom.singleton x) := exist (OK (Dom.singleton x)) (S.add x v (@S.empty A)) (singleton_OK x v). | Utf8 MSets FMaps Orders Coqlib DepMapInterface | coq-contribs-dep-map/DepMapImplementation | coq-contribs-dep-map | 836 |
Definition remove {A : Type} {dom : Dom.t} (x : key) (m : @t A dom) : @t A (Dom.remove x dom) := exist (OK (Dom.remove x dom)) (S.remove x (proj1_sig m)) (remove_OK x m). | Utf8 MSets FMaps Orders Coqlib DepMapInterface | coq-contribs-dep-map/DepMapImplementation | coq-contribs-dep-map | 837 |
Definition constant (A : Type) dom (v : A) : t A dom := exist (OK dom) (Dom.fold (fun x m => S.add x v m) dom (@S.empty A)) (constant_OK dom v). | Utf8 MSets FMaps Orders Coqlib DepMapInterface | coq-contribs-dep-map/DepMapImplementation | coq-contribs-dep-map | 838 |
Definition fold {A B : Type} (f : key -> A -> B -> B) dom (m : t A dom) (i : B) : B := S.fold f (proj1_sig m) i. | Utf8 MSets FMaps Orders Coqlib DepMapInterface | coq-contribs-dep-map/DepMapImplementation | coq-contribs-dep-map | 839 |
Definition map {A B : Type} (f : A -> B) dom (m : t A dom) : t B dom := exist (OK dom) (S.map f (proj1_sig m)) (map_OK f m). | Utf8 MSets FMaps Orders Coqlib DepMapInterface | coq-contribs-dep-map/DepMapImplementation | coq-contribs-dep-map | 840 |
Definition Scombine {A B C : Type} (f : A -> B -> C) (g : A -> C) (h : B -> C) (m₁ : S.t A) (m₂ : S.t B) : S.t C := Dom.fold (fun x acc => match S.find x m₁, S.find x m₂ with | Some v₁, Some v₂ => S.add x (f v₁ v₂) acc | Some v, None => S.add x (g v) acc | None, Some v => S.add x (h v) acc | None, None => acc end) (Dom.union (S.fold (fun x _ acc => Dom.add x acc) m₁ Dom.empty) (S.fold (fun x _ acc => Dom.add x acc) m₂ Dom.empty)) (S.empty C). | Utf8 MSets FMaps Orders Coqlib DepMapInterface | coq-contribs-dep-map/DepMapImplementation | coq-contribs-dep-map | 841 |
Definition combine A B C f g₁ g₂ dom₁ dom₂ (m₁ : t A dom₁) (m₂ : t B dom₂) : t C (Dom.union dom₁ dom₂) := exist (OK (Dom.union dom₁ dom₂)) (Scombine f g₁ g₂ (proj1_sig m₁) (proj1_sig m₂)) (combine_OK f g₁ g₂ m₁ m₂). | Utf8 MSets FMaps Orders Coqlib DepMapInterface | coq-contribs-dep-map/DepMapImplementation | coq-contribs-dep-map | 842 |
Definition cast {A : Type} dom₁ dom₂ (Heq : Dom.eq dom₁ dom₂) (m : t A dom₁) : t A dom₂ := exist (OK dom₂) (proj1_sig m) (cast_OK Heq (proj2_sig m)). | Utf8 MSets FMaps Orders Coqlib DepMapInterface | coq-contribs-dep-map/DepMapImplementation | coq-contribs-dep-map | 843 |
Definition elements A dom (m : t A dom) := S.elements (proj1_sig m). | Utf8 MSets FMaps Orders Coqlib DepMapInterface | coq-contribs-dep-map/DepMapImplementation | coq-contribs-dep-map | 844 |
Definition from_elements A (l : list (key * A)) : t A (List.fold_left (fun acc xv => Dom.add (fst xv) acc) l Dom.empty) := exist (OK (List.fold_left (fun acc xv => Dom.add (fst xv) acc) l Dom.empty)) (List.fold_left (fun acc xv => S.add (fst xv) (snd xv) acc) l (@S.empty A)) (from_elements_OK l). | Utf8 MSets FMaps Orders Coqlib DepMapInterface | coq-contribs-dep-map/DepMapImplementation | coq-contribs-dep-map | 845 |
Definition full_relation {A : Type} : relation A := fun x y : A => True. | SetoidList Orders | coq-contribs-dep-map/Coqlib | coq-contribs-dep-map | 846 |
Definition t := O.t. | SetoidList Orders | coq-contribs-dep-map/Coqlib | coq-contribs-dep-map | 847 |
Definition eq := O.eq. | SetoidList Orders | coq-contribs-dep-map/Coqlib | coq-contribs-dep-map | 848 |
Definition lt := O.lt. | SetoidList Orders | coq-contribs-dep-map/Coqlib | coq-contribs-dep-map | 849 |
Definition eq_refl : forall x, eq x x := reflexivity. | SetoidList Orders | coq-contribs-dep-map/Coqlib | coq-contribs-dep-map | 850 |
Definition eq_dec := O.eq_dec. | SetoidList Orders | coq-contribs-dep-map/Coqlib | coq-contribs-dep-map | 851 |
Definition eq {A : Type} {dom₁ dom₂} (m₁ : t A dom₁) (m₂ : t A dom₂) := forall x, find x m₁ = find x m₂. | Utf8 Orders Coqlib DepMapInterface | coq-contribs-dep-map/DepMapFactsInterface | coq-contribs-dep-map | 852 |
Definition incl {A : Type} {dom₁ dom₂} (m₁ : t A dom₁) (m₂ : t A dom₂) := forall x v, find x m₁ = Some v -> find x m₂ = Some v. | Utf8 Orders Coqlib DepMapInterface | coq-contribs-dep-map/DepMapFactsInterface | coq-contribs-dep-map | 853 |
Parameter eq_sym : forall A dom₁ dom₂ (m₁ : t A dom₁) (m₂ : t A dom₂), eq m₁ m₂ -> eq m₂ m₁. | Utf8 Orders Coqlib DepMapInterface | coq-contribs-dep-map/DepMapFactsInterface | coq-contribs-dep-map | 854 |
Parameter eq_trans : forall A dom₁ dom₂ dom3 (m₁ : t A dom₁) (m₂ : t A dom₂) (m3 : t A dom3), eq m₁ m₂ -> eq m₂ m3 -> eq m₁ m3. | Utf8 Orders Coqlib DepMapInterface | coq-contribs-dep-map/DepMapFactsInterface | coq-contribs-dep-map | 855 |
Parameter incl_trans : forall A dom₁ dom₂ dom3 (m₁ : t A dom₁) (m₂ : t A dom₂) (m3 : t A dom3), incl m₁ m₂ -> incl m₂ m3 -> incl m₁ m3. | Utf8 Orders Coqlib DepMapInterface | coq-contribs-dep-map/DepMapFactsInterface | coq-contribs-dep-map | 856 |
Parameter find_compat : forall A x y dom₁ dom₂ (m₁ : t A dom₁) (m₂ : t A dom₂), X.eq x y -> eq m₁ m₂ -> find x m₁ = find y m₂. | Utf8 Orders Coqlib DepMapInterface | coq-contribs-dep-map/DepMapFactsInterface | coq-contribs-dep-map | 857 |
Parameter set_compat : forall A x y v dom₁ dom₂ (m₁ : t A dom₁) (m₂ : t A dom₂) (Hin₁ : Dom.In x dom₁) (Hin₂ : Dom.In y dom₂), X.eq x y -> eq m₁ m₂ -> eq (set x v m₁ Hin₁) (set y v m₂ Hin₂). | Utf8 Orders Coqlib DepMapInterface | coq-contribs-dep-map/DepMapFactsInterface | coq-contribs-dep-map | 858 |
Parameter add_compat : forall A x y v dom₁ dom₂ (m₁ : t A dom₁) (m₂ : t A dom₂), X.eq x y -> eq m₁ m₂ -> eq (add x v m₁) (add y v m₂). | Utf8 Orders Coqlib DepMapInterface | coq-contribs-dep-map/DepMapFactsInterface | coq-contribs-dep-map | 859 |
Parameter remove_compat : forall A x y dom₁ dom₂ (m₁ : t A dom₁) (m₂ : t A dom₂), X.eq x y -> eq m₁ m₂ -> eq (remove x m₁) (remove y m₂). | Utf8 Orders Coqlib DepMapInterface | coq-contribs-dep-map/DepMapFactsInterface | coq-contribs-dep-map | 860 |
Parameter find_None : forall A dom x (m : t A dom), find x m = None <-> ¬Dom.In x dom. | Utf8 Orders Coqlib DepMapInterface | coq-contribs-dep-map/DepMapFactsInterface | coq-contribs-dep-map | 861 |
Parameter find_dom : forall A x v dom (m : t A dom), find x m = Some v -> Dom.In x dom. | Utf8 Orders Coqlib DepMapInterface | coq-contribs-dep-map/DepMapFactsInterface | coq-contribs-dep-map | 862 |
Parameter set_Some : forall A x y v u dom (m : t A dom) (Hin : Dom.In x dom), find y (set x v m Hin) = Some u <-> X.eq y x ∧ u = v ∨ ¬X.eq y x ∧ find y m = Some u. | Utf8 Orders Coqlib DepMapInterface | coq-contribs-dep-map/DepMapFactsInterface | coq-contribs-dep-map | 863 |
Parameter set_None : forall A x y v dom (m : t A dom) (Hin : Dom.In x dom), find y (set x v m Hin) = None <-> ¬X.eq y x ∧ find y m = None. | Utf8 Orders Coqlib DepMapInterface | coq-contribs-dep-map/DepMapFactsInterface | coq-contribs-dep-map | 864 |
Parameter add_Some : forall A x y v u dom (m : t A dom), find y (add x v m) = Some u <-> X.eq y x ∧ u = v ∨ ¬X.eq y x ∧ find y m = Some u. | Utf8 Orders Coqlib DepMapInterface | coq-contribs-dep-map/DepMapFactsInterface | coq-contribs-dep-map | 865 |
Parameter add_None : forall A x y v dom (m : t A dom), find y (add x v m) = None <-> ¬X.eq y x ∧ find y m = None. | Utf8 Orders Coqlib DepMapInterface | coq-contribs-dep-map/DepMapFactsInterface | coq-contribs-dep-map | 866 |
Parameter remove_Some : forall A x y u dom (m : t A dom), find y (remove x m) = Some u <-> ¬X.eq y x ∧ find y m = Some u. | Utf8 Orders Coqlib DepMapInterface | coq-contribs-dep-map/DepMapFactsInterface | coq-contribs-dep-map | 867 |
Parameter remove_None : forall A x y dom (m : t A dom), find y (remove x m) = None <-> X.eq y x ∨ find y m = None. | Utf8 Orders Coqlib DepMapInterface | coq-contribs-dep-map/DepMapFactsInterface | coq-contribs-dep-map | 868 |
Parameter add_cancel : forall A x v dom (m : t A dom), find x m = Some v -> eq (add x v m) m. | Utf8 Orders Coqlib DepMapInterface | coq-contribs-dep-map/DepMapFactsInterface | coq-contribs-dep-map | 869 |
Parameter remove_cancel : forall A x dom (m : t A dom), find x m = None -> eq (remove x m) m. | Utf8 Orders Coqlib DepMapInterface | coq-contribs-dep-map/DepMapFactsInterface | coq-contribs-dep-map | 870 |
Parameter add_merge : forall A x v₁ v₂ dom (m : t A dom), eq (add x v₂ (add x v₁ m)) (add x v₂ m). | Utf8 Orders Coqlib DepMapInterface | coq-contribs-dep-map/DepMapFactsInterface | coq-contribs-dep-map | 871 |
Parameter add_comm : forall A x y v₁ v₂ dom (m : t A dom), ¬X.eq x y -> eq (add y v₂ (add x v₁ m)) (add x v₁ (add y v₂ m)). | Utf8 Orders Coqlib DepMapInterface | coq-contribs-dep-map/DepMapFactsInterface | coq-contribs-dep-map | 872 |
Parameter remove_add_cancel : forall A s v dom (m : t A dom), eq (remove s (add s v m)) (remove s m). | Utf8 Orders Coqlib DepMapInterface | coq-contribs-dep-map/DepMapFactsInterface | coq-contribs-dep-map | 873 |
Parameter map_None : forall A B (f : A -> B) dom (m : t A dom) x, find x (map f m) = None <-> ¬Dom.In x dom. | Utf8 Orders Coqlib DepMapInterface | coq-contribs-dep-map/DepMapFactsInterface | coq-contribs-dep-map | 874 |
Parameter combine_None : forall A B C (f : A -> B -> C) g₁ g₂ dom₁ dom₂ (m₁ : t A dom₁) (m₂ : t B dom₂) x, find x (combine f g₁ g₂ m₁ m₂) = None <-> find x m₁ = None ∧ find x m₂ = None. | Utf8 Orders Coqlib DepMapInterface | coq-contribs-dep-map/DepMapFactsInterface | coq-contribs-dep-map | 875 |
Parameter add_incl : forall A x v dom (m : t A dom), ¬Dom.In x dom -> incl m (add x v m). | Utf8 Orders Coqlib DepMapInterface | coq-contribs-dep-map/DepMapFactsInterface | coq-contribs-dep-map | 876 |
Parameter remove_incl : forall A x dom (m : t A dom), incl (remove x m) m. | Utf8 Orders Coqlib DepMapInterface | coq-contribs-dep-map/DepMapFactsInterface | coq-contribs-dep-map | 877 |
Parameter cast_spec : forall A dom₁ dom₂ (Heq : Dom.eq dom₁ dom₂) (m : t A dom₁), eq (cast Heq m) m. | Utf8 Orders Coqlib DepMapInterface | coq-contribs-dep-map/DepMapFactsInterface | coq-contribs-dep-map | 878 |
Parameter eq_dom : forall A dom₁ dom₂ (m₁ : t A dom₁) (m₂ : t A dom₂), eq m₁ m₂ -> Dom.eq dom₁ dom₂. | Utf8 Orders Coqlib DepMapInterface | coq-contribs-dep-map/DepMapFactsInterface | coq-contribs-dep-map | 879 |
Parameter for_all : forall {A}, (key -> A -> bool) -> forall {dom}, t A dom -> bool. | Utf8 Orders Coqlib DepMapInterface | coq-contribs-dep-map/DepMapFactsInterface | coq-contribs-dep-map | 880 |
Parameter for_all_spec : forall A f, Proper (X.eq ==> Logic.eq ==> Logic.eq) f -> forall dom (m : t A dom), for_all f m = true <-> forall x v, find x m = Some v -> f x v = true. | Utf8 Orders Coqlib DepMapInterface | coq-contribs-dep-map/DepMapFactsInterface | coq-contribs-dep-map | 881 |
Parameter exists_ : forall {A}, (key -> A -> bool) -> forall {dom}, t A dom -> bool. | Utf8 Orders Coqlib DepMapInterface | coq-contribs-dep-map/DepMapFactsInterface | coq-contribs-dep-map | 882 |
Parameter exists_spec : forall A f, Proper (X.eq ==> Logic.eq ==> Logic.eq) f -> forall dom (m : t A dom), exists_ f m = true <-> exists x v, find x m = Some v ∧ f x v = true. | Utf8 Orders Coqlib DepMapInterface | coq-contribs-dep-map/DepMapFactsInterface | coq-contribs-dep-map | 883 |
Definition eq {A : Type} dom₁ dom₂ (m₁ : t A dom₁) (m₂ : t A dom₂) := forall x, find x m₁ = find x m₂. | Utf8 Bool SetoidList RelationPairs Orders Coqlib DepMapInterface DepMapFactsInterface | coq-contribs-dep-map/DepMapFactsImplementation | coq-contribs-dep-map | 884 |
Definition incl {A : Type} dom₁ dom₂ (m₁ : t A dom₁) (m₂ : t A dom₂) := forall x v, find x m₁ = Some v -> find x m₂ = Some v. | Utf8 Bool SetoidList RelationPairs Orders Coqlib DepMapInterface DepMapFactsInterface | coq-contribs-dep-map/DepMapFactsImplementation | coq-contribs-dep-map | 885 |
Definition for_all {A : Type} (f : key -> A -> bool) dom (m : t A dom) := fold (fun x v b => b && f x v) m true. | Utf8 Bool SetoidList RelationPairs Orders Coqlib DepMapInterface DepMapFactsInterface | coq-contribs-dep-map/DepMapFactsImplementation | coq-contribs-dep-map | 886 |
Definition exists_ {A : Type} (f : key -> A -> bool) dom (m : t A dom) := fold (fun x v b => b || f x v) m false. | Utf8 Bool SetoidList RelationPairs Orders Coqlib DepMapInterface DepMapFactsInterface | coq-contribs-dep-map/DepMapFactsImplementation | coq-contribs-dep-map | 887 |
Definition OpenTerm0 a := OpenTerm (ne_list.one a). | Coq.Classes.RelationClasses Coq.Relations.Relation_Definitions Coq.Lists.List Coq.Classes.Morphisms MathClasses.interfaces.universal_algebra MathClasses.interfaces.abstract_algebra MathClasses.interfaces.canonical_names MathClasses.theory.categories MathClasses.categories.varieties | coq-community-math-classes/varieties/open_terms | coq-community-math-classes | 888 |
Fixpoint app_tree {o}: OpenTerm o → op_type OpenTerm0 o := match o with | ne_list.one _ => id | ne_list.cons _ _ => λ x y, app_tree (App _ _ _ _ x y) end. | Coq.Classes.RelationClasses Coq.Relations.Relation_Definitions Coq.Lists.List Coq.Classes.Morphisms MathClasses.interfaces.universal_algebra MathClasses.interfaces.abstract_algebra MathClasses.interfaces.canonical_names MathClasses.theory.categories MathClasses.categories.varieties | coq-community-math-classes/varieties/open_terms | coq-community-math-classes | 889 |
Definition the_object: varieties.Object et := varieties.object et OpenTerm0. | Coq.Classes.RelationClasses Coq.Relations.Relation_Definitions Coq.Lists.List Coq.Classes.Morphisms MathClasses.interfaces.universal_algebra MathClasses.interfaces.abstract_algebra MathClasses.interfaces.canonical_names MathClasses.theory.categories MathClasses.categories.varieties | coq-community-math-classes/varieties/open_terms | coq-community-math-classes | 890 |
Definition sig: Signature := single_sorted_signature (λ o, match o with mult => 2%nat end). | MathClasses.interfaces.abstract_algebra MathClasses.interfaces.universal_algebra MathClasses.theory.ua_homomorphisms MathClasses.misc.workaround_tactics categories.categories | coq-community-math-classes/varieties/semigroups | coq-community-math-classes | 891 |
Definition theory: EquationalTheory := Build_EquationalTheory sig Laws. | MathClasses.interfaces.abstract_algebra MathClasses.interfaces.universal_algebra MathClasses.theory.ua_homomorphisms MathClasses.misc.workaround_tactics categories.categories | coq-community-math-classes/varieties/semigroups | coq-community-math-classes | 892 |
Definition Object := varieties.Object theory. | MathClasses.interfaces.abstract_algebra MathClasses.interfaces.universal_algebra MathClasses.theory.ua_homomorphisms MathClasses.misc.workaround_tactics categories.categories | coq-community-math-classes/varieties/semigroups | coq-community-math-classes | 893 |
Definition forget: Object → setoids.Object := @product.project unit (λ _, setoids.Object) (λ _, _: Arrows setoids.Object) _ (λ _, _: CatId setoids.Object) (λ _, _: CatComp setoids.Object) (λ _, _: Category setoids.Object) tt ∘ forget_algebra.object theory ∘ forget_variety.forget theory. | MathClasses.interfaces.abstract_algebra MathClasses.interfaces.universal_algebra MathClasses.theory.ua_homomorphisms MathClasses.misc.workaround_tactics categories.categories | coq-community-math-classes/varieties/semigroups | coq-community-math-classes | 894 |
Definition object: Object := varieties.object theory (λ _, A). | MathClasses.interfaces.abstract_algebra MathClasses.interfaces.universal_algebra MathClasses.theory.ua_homomorphisms MathClasses.misc.workaround_tactics categories.categories | coq-community-math-classes/varieties/semigroups | coq-community-math-classes | 895 |
Definition op := False. | MathClasses.interfaces.abstract_algebra MathClasses.interfaces.universal_algebra MathClasses.theory.ua_homomorphisms | coq-community-math-classes/varieties/setoids | coq-community-math-classes | 896 |
Definition sig: Signature := Build_Signature unit op (False_rect _). | MathClasses.interfaces.abstract_algebra MathClasses.interfaces.universal_algebra MathClasses.theory.ua_homomorphisms | coq-community-math-classes/varieties/setoids | coq-community-math-classes | 897 |
Definition Laws: EqEntailment sig → Prop := λ _, False. | MathClasses.interfaces.abstract_algebra MathClasses.interfaces.universal_algebra MathClasses.theory.ua_homomorphisms | coq-community-math-classes/varieties/setoids | coq-community-math-classes | 898 |
Definition object: varieties.Object theory := varieties.object theory (λ _, A). | MathClasses.interfaces.abstract_algebra MathClasses.interfaces.universal_algebra MathClasses.theory.ua_homomorphisms | coq-community-math-classes/varieties/setoids | coq-community-math-classes | 900 |
Definition sig: Signature := single_sorted_signature (λ o, match o with one => O | mult => 2%nat end). | MathClasses.interfaces.abstract_algebra MathClasses.interfaces.universal_algebra MathClasses.theory.ua_homomorphisms MathClasses.misc.workaround_tactics MathClasses.categories.categories | coq-community-math-classes/varieties/monoids | coq-community-math-classes | 901 |
Definition forget: Object → setoids.Object := @product.project unit (λ _, setoids.Object) (λ _, _) _ (λ _, _) (λ _, _) (λ _, _) tt ∘ forget_algebra.object theory ∘ forget_variety.forget theory. | MathClasses.interfaces.abstract_algebra MathClasses.interfaces.universal_algebra MathClasses.theory.ua_homomorphisms MathClasses.misc.workaround_tactics MathClasses.categories.categories | coq-community-math-classes/varieties/monoids | coq-community-math-classes | 904 |
Variable et: EquationalTheory. | Coq.Classes.RelationClasses Coq.Relations.Relation_Definitions Coq.Lists.List Coq.Classes.Morphisms MathClasses.interfaces.universal_algebra MathClasses.theory.ua_homomorphisms MathClasses.interfaces.abstract_algebra MathClasses.interfaces.canonical_names MathClasses.theory.categories MathClasses.categories.varieties | coq-community-math-classes/varieties/closed_terms | coq-community-math-classes | 906 |
Fixpoint app_tree {o}: ClosedTerm o → op_type ClosedTerm0 o := match o with | ne_list.one _ => id | ne_list.cons _ _ => λ x y, app_tree (App _ _ _ _ x y) end. | Coq.Classes.RelationClasses Coq.Relations.Relation_Definitions Coq.Lists.List Coq.Classes.Morphisms MathClasses.interfaces.universal_algebra MathClasses.theory.ua_homomorphisms MathClasses.interfaces.abstract_algebra MathClasses.interfaces.canonical_names MathClasses.theory.categories MathClasses.categories.varieties | coq-community-math-classes/varieties/closed_terms | coq-community-math-classes | 907 |
Definition the_object: varieties.Object et := varieties.object et ClosedTerm0. | Coq.Classes.RelationClasses Coq.Relations.Relation_Definitions Coq.Lists.List Coq.Classes.Morphisms MathClasses.interfaces.universal_algebra MathClasses.theory.ua_homomorphisms MathClasses.interfaces.abstract_algebra MathClasses.interfaces.canonical_names MathClasses.theory.categories MathClasses.categories.varieties | coq-community-math-classes/varieties/closed_terms | coq-community-math-classes | 908 |
Variable other: varieties.Object et. | Coq.Classes.RelationClasses Coq.Relations.Relation_Definitions Coq.Lists.List Coq.Classes.Morphisms MathClasses.interfaces.universal_algebra MathClasses.theory.ua_homomorphisms MathClasses.interfaces.abstract_algebra MathClasses.interfaces.canonical_names MathClasses.theory.categories MathClasses.categories.varieties | coq-community-math-classes/varieties/closed_terms | coq-community-math-classes | 909 |
Definition eval_in_other {o}: ClosedTerm o → op_type other o := @eval et other _ False o (no_vars _ other). | Coq.Classes.RelationClasses Coq.Relations.Relation_Definitions Coq.Lists.List Coq.Classes.Morphisms MathClasses.interfaces.universal_algebra MathClasses.theory.ua_homomorphisms MathClasses.interfaces.abstract_algebra MathClasses.interfaces.canonical_names MathClasses.theory.categories MathClasses.categories.varieties | coq-community-math-classes/varieties/closed_terms | coq-community-math-classes | 910 |
Definition morph a: the_object a → other a := eval_in_other. | Coq.Classes.RelationClasses Coq.Relations.Relation_Definitions Coq.Lists.List Coq.Classes.Morphisms MathClasses.interfaces.universal_algebra MathClasses.theory.ua_homomorphisms MathClasses.interfaces.abstract_algebra MathClasses.interfaces.canonical_names MathClasses.theory.categories MathClasses.categories.varieties | coq-community-math-classes/varieties/closed_terms | coq-community-math-classes | 911 |
Definition sig: Signature := single_sorted_signature (λ o, match o with zero | one => O | negate => 1%nat | plus | mult => 2%nat end). | Coq.setoid_ring.Ring MathClasses.interfaces.abstract_algebra MathClasses.interfaces.universal_algebra MathClasses.theory.ua_homomorphisms MathClasses.misc.workaround_tactics | coq-community-math-classes/varieties/rings | coq-community-math-classes | 912 |
Definition sig: Signature := single_sorted_signature (λ o, match o with one => O | inv => 1%nat | mult => 2%nat end). | MathClasses.interfaces.abstract_algebra MathClasses.interfaces.universal_algebra MathClasses.theory.ua_homomorphisms MathClasses.misc.workaround_tactics categories.categories | coq-community-math-classes/varieties/abgroup | coq-community-math-classes | 916 |
Definition sig: Signature := single_sorted_signature (λ o, match o with zero | one => O | plus | mult => 2%nat end). | MathClasses.interfaces.abstract_algebra MathClasses.interfaces.universal_algebra MathClasses.theory.ua_homomorphisms MathClasses.misc.workaround_tactics | coq-community-math-classes/varieties/semirings | coq-community-math-classes | 925 |
Definition sig: Signature := Build_Signature False False (False_rect _). | MathClasses.interfaces.abstract_algebra MathClasses.interfaces.universal_algebra | coq-community-math-classes/varieties/empty | coq-community-math-classes | 930 |
Definition Laws (_: EqEntailment sig): Prop := False. | MathClasses.interfaces.abstract_algebra MathClasses.interfaces.universal_algebra | coq-community-math-classes/varieties/empty | coq-community-math-classes | 931 |
Definition object: Object := varieties.object theory carriers. | MathClasses.interfaces.abstract_algebra MathClasses.interfaces.universal_algebra | coq-community-math-classes/varieties/empty | coq-community-math-classes | 934 |
Definition in_domain: Equiv A := λ x y, f x = f y. | Coq.Classes.Morphisms Coq.Classes.RelationClasses Coq.Relations.Relation_Definitions Coq.Lists.List MathClasses.interfaces.abstract_algebra MathClasses.interfaces.universal_algebra MathClasses.theory.ua_homomorphisms MathClasses.interfaces.canonical_names MathClasses.theory.ua_subalgebraT MathClasses.misc.util | coq-community-math-classes/theory/ua_congruence | coq-community-math-classes | 935 |
Definition image s (b: B s): Type := sigT (λ a, f s a = b). | Coq.Classes.Morphisms Coq.Classes.RelationClasses Coq.Relations.Relation_Definitions Coq.Lists.List MathClasses.interfaces.abstract_algebra MathClasses.interfaces.universal_algebra MathClasses.theory.ua_homomorphisms MathClasses.interfaces.canonical_names MathClasses.theory.ua_subalgebraT MathClasses.misc.util | coq-community-math-classes/theory/ua_congruence | coq-community-math-classes | 936 |
Definition quot_obj: algebras.Object σ := algebras.object σ A (algebra_equiv:=Φ). | Coq.Classes.Morphisms Coq.Classes.RelationClasses Coq.Relations.Relation_Definitions Coq.Lists.List MathClasses.interfaces.abstract_algebra MathClasses.interfaces.universal_algebra MathClasses.theory.ua_homomorphisms MathClasses.interfaces.canonical_names MathClasses.theory.ua_subalgebraT MathClasses.misc.util | coq-community-math-classes/theory/ua_congruence | coq-community-math-classes | 937 |
Definition subobject: algebras.Object σ := algebras.object σ (ua_subalgebraT.carrier image). | Coq.Classes.Morphisms Coq.Classes.RelationClasses Coq.Relations.Relation_Definitions Coq.Lists.List MathClasses.interfaces.abstract_algebra MathClasses.interfaces.universal_algebra MathClasses.theory.ua_homomorphisms MathClasses.interfaces.canonical_names MathClasses.theory.ua_subalgebraT MathClasses.misc.util | coq-community-math-classes/theory/ua_congruence | coq-community-math-classes | 938 |
Variable Sorts: Set. | MathClasses.interfaces.abstract_algebra MathClasses.interfaces.universal_algebra | coq-community-math-classes/theory/ua_mapped_operations | coq-community-math-classes | 939 |
Fixpoint map_op {o: OpType Sorts}: op_type A o → op_type B o := match o return op_type A o → op_type B o with | ne_list.one u => ab u | ne_list.cons _ _ => λ x y, map_op (x (ba _ y)) end. | MathClasses.interfaces.abstract_algebra MathClasses.interfaces.universal_algebra | coq-community-math-classes/theory/ua_mapped_operations | coq-community-math-classes | 940 |
Definition Pvars (vars: Vars et (carrier P) nat): Vars et A nat := λ s n, ` (vars s n). | Coq.Classes.RelationClasses Coq.Classes.Morphisms Coq.Program.Program MathClasses.interfaces.abstract_algebra MathClasses.interfaces.universal_algebra MathClasses.interfaces.canonical_names MathClasses.theory.ua_subalgebra | coq-community-math-classes/theory/ua_subvariety | coq-community-math-classes | 941 |
Definition homFrom (y: C): setoids.Object := setoids.object (x ⟶ y). | MathClasses.interfaces.abstract_algebra MathClasses.theory.setoids MathClasses.interfaces.functors MathClasses.theory.categories | coq-community-math-classes/theory/hom_functor | coq-community-math-classes | 942 |
Definition rationals_to_rationals Q1 Q2 `{Rationals Q1} `{Rationals Q2} : Q1 → Q2 := (rationals_to_frac Q2 (SRpair nat))⁻¹ ∘ rationals_to_frac Q1 (SRpair nat). | Coq.setoid_ring.Ring MathClasses.interfaces.abstract_algebra MathClasses.interfaces.integers MathClasses.interfaces.naturals MathClasses.interfaces.rationals MathClasses.implementations.field_of_fractions MathClasses.implementations.natpair_integers MathClasses.theory.rings MathClasses.theory.integers MathClasses.theory.dec_fields | coq-community-math-classes/theory/rationals | coq-community-math-classes | 943 |
Fixpoint op_closed {o: OpType (sorts sign)}: op_type A o → Prop := match o with | ne_list.one x => P x | ne_list.cons _ _ => λ d, ∀ z, P _ z → op_closed (d z) end. | Coq.Classes.RelationClasses MathClasses.interfaces.universal_algebra MathClasses.theory.ua_homomorphisms MathClasses.interfaces.canonical_names MathClasses.theory.categories MathClasses.interfaces.abstract_algebra | coq-community-math-classes/theory/ua_subalgebra | coq-community-math-classes | 944 |
Definition carrier s := sig (P s). | Coq.Classes.RelationClasses MathClasses.interfaces.universal_algebra MathClasses.theory.ua_homomorphisms MathClasses.interfaces.canonical_names MathClasses.theory.categories MathClasses.interfaces.abstract_algebra | coq-community-math-classes/theory/ua_subalgebra | coq-community-math-classes | 945 |
Definition close_op_proper d (o0 o1: op_type A d) (P': op_closed o0) (Q: op_closed o1): o0 = o1 → close_op o0 P' = close_op o1 Q. | Coq.Classes.RelationClasses MathClasses.interfaces.universal_algebra MathClasses.theory.ua_homomorphisms MathClasses.interfaces.canonical_names MathClasses.theory.categories MathClasses.interfaces.abstract_algebra | coq-community-math-classes/theory/ua_subalgebra | coq-community-math-classes | 946 |
Definition proj s := @proj1_sig (A s) (P s). | Coq.Classes.RelationClasses MathClasses.interfaces.universal_algebra MathClasses.theory.ua_homomorphisms MathClasses.interfaces.canonical_names MathClasses.theory.categories MathClasses.interfaces.abstract_algebra | coq-community-math-classes/theory/ua_subalgebra | coq-community-math-classes | 947 |
Definition forget (v: varieties.Object et) := algebras.object et v. | MathClasses.interfaces.canonical_names MathClasses.interfaces.abstract_algebra MathClasses.interfaces.universal_algebra MathClasses.interfaces.functors MathClasses.theory.categories MathClasses.categories.varieties MathClasses.categories.algebras | coq-community-math-classes/theory/forget_variety | coq-community-math-classes | 949 |
Definition prod_equiv `{Equiv A} `{Equiv B} : Equiv (A * B) := λ p q, fst p = fst q ∧ snd p = snd q. | MathClasses.interfaces.abstract_algebra | coq-community-math-classes/theory/products | coq-community-math-classes | 950 |
Definition prod_fst_equiv X A `{Equiv X} : relation (X * A) := λ x y, fst x = fst y. | MathClasses.interfaces.abstract_algebra | coq-community-math-classes/theory/products | coq-community-math-classes | 951 |
Definition fset_car_setoid A `{FSet A} : Setoid A := setoidmor_a singleton. | MathClasses.theory.lattices MathClasses.varieties.monoids MathClasses.implementations.bool MathClasses.implementations.list_finite_set MathClasses.orders.lattices MathClasses.interfaces.abstract_algebra MathClasses.interfaces.finite_sets MathClasses.interfaces.orders | coq-community-math-classes/theory/finite_sets | coq-community-math-classes | 952 |
Variable σ: Signature. | MathClasses.interfaces.abstract_algebra MathClasses.interfaces.universal_algebra | coq-community-math-classes/theory/ua_homomorphisms | coq-community-math-classes | 953 |
Fixpoint Preservation {n : OpType}: op_type A n → op_type B n → Prop := match n with | ne_list.one d => λ oA oB, f oA = oB | ne_list.cons x y => λ oA oB, ∀ x, Preservation (oA x) (oB (f x)) end. | MathClasses.interfaces.abstract_algebra MathClasses.interfaces.universal_algebra | coq-community-math-classes/theory/ua_homomorphisms | coq-community-math-classes | 954 |
CoInductive Stream_eq_coind (s1 s2: ∞A) : Prop := stream_eq_coind : hd s1 = hd s2 → Stream_eq_coind (tl s1) (tl s2) → Stream_eq_coind s1 s2. | MathClasses.theory.CoqStreams MathClasses.implementations.peano_naturals MathClasses.interfaces.abstract_algebra | coq-community-math-classes/theory/streams | coq-community-math-classes | 955 |
Definition EventuallyForAll (P : ∞A → Prop) := ForAll (λ s, P s → P (tl s)). | MathClasses.theory.CoqStreams MathClasses.implementations.peano_naturals MathClasses.interfaces.abstract_algebra | coq-community-math-classes/theory/streams | coq-community-math-classes | 956 |
CoFixpoint iterate (f:A → A) (x:A) : ∞A := x ::: iterate f (f x). | MathClasses.theory.CoqStreams MathClasses.implementations.peano_naturals MathClasses.interfaces.abstract_algebra | coq-community-math-classes/theory/streams | coq-community-math-classes | 957 |
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