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Diffstat (limited to 'test-suite/bugs/closed/5215.v')
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diff --git a/test-suite/bugs/closed/5215.v b/test-suite/bugs/closed/5215.v new file mode 100644 index 000000000..ecf529159 --- /dev/null +++ b/test-suite/bugs/closed/5215.v @@ -0,0 +1,286 @@ +Require Import Coq.Logic.FunctionalExtensionality. +Require Import Coq.Program.Tactics. + +Global Set Primitive Projections. + +Global Set Universe Polymorphism. + +Global Unset Universe Minimization ToSet. + +Class Category : Type := +{ + Obj : Type; + Hom : Obj -> Obj -> Type; + compose : forall {a b c : Obj}, (Hom a b) -> (Hom b c) -> (Hom a c); + id : forall {a : Obj}, Hom a a; +}. + +Arguments Obj {_}, _. +Arguments id {_ _}, {_} _, _ _. +Arguments Hom {_} _ _, _ _ _. +Arguments compose {_} {_ _ _} _ _, _ {_ _ _} _ _, _ _ _ _ _ _. + +Coercion Obj : Category >-> Sortclass. + +Definition Opposite (C : Category) : Category := +{| + + Obj := Obj C; + Hom := fun a b => Hom b a; + compose := + fun a b c (f : Hom b a) (g : Hom c b) => compose C c b a g f; + id := fun c => id C c; +|}. + +Record Functor (C C' : Category) : Type := +{ + FO : C -> C'; + FA : forall {a b}, Hom a b -> Hom (FO a) (FO b); +}. + +Arguments FO {_ _} _ _. +Arguments FA {_ _} _ {_ _} _, {_ _} _ _ _ _. + +Section Opposite_Functor. + Context {C D : Category} (F : Functor C D). + + Program Definition Opposite_Functor : (Functor (Opposite C) (Opposite D)) := + {| + FO := FO F; + FA := fun _ _ h => FA F h; + |}. + +End Opposite_Functor. + +Section Functor_Compose. + Context {C C' C'' : Category} (F : Functor C C') (F' : Functor C' C''). + + Program Definition Functor_compose : Functor C C'' := + {| + FO := fun c => FO F' (FO F c); + FA := fun c d f => FA F' (FA F f) + |}. + +End Functor_Compose. + +Section Algebras. + Context {C : Category} (T : Functor C C). + Record Algebra : Type := + { + Alg_Carrier : C; + Constructors : Hom (FO T Alg_Carrier) Alg_Carrier + }. + + Record Algebra_Hom (alg alg' : Algebra) : Type := + { + Alg_map : Hom (Alg_Carrier alg) (Alg_Carrier alg'); + + Alg_map_com : compose (FA T Alg_map) (Constructors alg') + = compose (Constructors alg) Alg_map + }. + + Arguments Alg_map {_ _} _. + Arguments Alg_map_com {_ _} _. + Program Definition Algebra_Hom_compose + {alg alg' alg'' : Algebra} + (h : Algebra_Hom alg alg') + (h' : Algebra_Hom alg' alg'') + : Algebra_Hom alg alg'' + := + {| + Alg_map := compose (Alg_map h) (Alg_map h') + |}. + + Next Obligation. Proof. Admitted. + + Lemma Algebra_Hom_eq_simplify (alg alg' : Algebra) + (ah ah' : Algebra_Hom alg alg') + : (Alg_map ah) = (Alg_map ah') -> ah = ah'. + Proof. Admitted. + + Program Definition Algebra_Hom_id (alg : Algebra) : Algebra_Hom alg alg := + {| + Alg_map := id + |}. + + Next Obligation. Admitted. + + Definition Algebra_Cat : Category := + {| + Obj := Algebra; + Hom := Algebra_Hom; + compose := @Algebra_Hom_compose; + id := Algebra_Hom_id; + |}. + +End Algebras. + +Arguments Alg_Carrier {_ _} _. +Arguments Constructors {_ _} _. +Arguments Algebra_Hom {_ _} _ _. +Arguments Alg_map {_ _ _ _} _. +Arguments Alg_map_com {_ _ _ _} _. +Arguments Algebra_Hom_id {_ _} _. + +Section CoAlgebras. + Context {C : Category}. + + Definition CoAlgebra (T : Functor C C) := + @Algebra (Opposite C) (Opposite_Functor T). + + Definition CoAlgebra_Hom {T : Functor C C} := + @Algebra_Hom (Opposite C) (Opposite_Functor T). + + Definition CoAlgebra_Hom_id {T : Functor C C} := + @Algebra_Hom_id (Opposite C) (Opposite_Functor T). + + Definition CoAlgebra_Cat (T : Functor C C) := + @Algebra_Cat (Opposite C) (Opposite_Functor T). + +End CoAlgebras. + +Program Definition Type_Cat : Category := +{| + Obj := Type; + Hom := (fun A B => A -> B); + compose := fun A B C (g : A -> B) (h : B -> C) => fun (x : A) => h (g x); + id := fun A => fun x => x +|}. + +Local Obligation Tactic := idtac. + +Program Definition Prod_Cat (C C' : Category) : Category := +{| + Obj := C * C'; + Hom := + fun a b => + ((Hom (fst a) (fst b)) * (Hom (snd a) (snd b)))%type; + compose := + fun a b c f g => + ((compose (fst f) (fst g)), (compose (snd f)(snd g))); + id := fun c => (id, id) +|}. + +Class Terminal (C : Category) : Type := +{ + terminal : C; + t_morph : forall (d : Obj), Hom d terminal; + t_morph_unique : forall (d : Obj) (f g : (Hom d terminal)), f = g +}. + +Arguments terminal {_} _. +Arguments t_morph {_} _ _. +Arguments t_morph_unique {_} _ _ _ _. + +Coercion terminal : Terminal >-> Obj. + +Definition Initial (C : Category) := Terminal (Opposite C). +Existing Class Initial. + +Record Product {C : Category} (c d : C) : Type := +{ + product : C; + Pi_1 : Hom product c; + Pi_2 : Hom product d; + Prod_morph_ex : forall (p' : Obj) (r1 : Hom p' c) (r2 : Hom p' d), (Hom p' product); +}. + +Arguments Product _ _ _, {_} _ _. + +Arguments Pi_1 {_ _ _ _}, {_ _ _} _. +Arguments Pi_2 {_ _ _ _}, {_ _ _} _. +Arguments Prod_morph_ex {_ _ _} _ _ _ _. + +Coercion product : Product >-> Obj. + +Definition Has_Products (C : Category) : Type := forall a b, Product a b. + +Existing Class Has_Products. + +Program Definition Prod_Func (C : Category) {HP : Has_Products C} + : Functor (Prod_Cat C C) C := +{| + FO := fun x => HP (fst x) (snd x); + FA := fun a b f => Prod_morph_ex _ _ (compose Pi_1 (fst f)) (compose Pi_2 (snd f)) +|}. + +Arguments Prod_Func _ _, _ {_}. + +Definition Sum (C : Category) := @Product (Opposite C). + +Arguments Sum _ _ _, {_} _ _. + +Definition Has_Sums (C : Category) : Type := forall (a b : C), (Sum a b). + +Existing Class Has_Sums. + +Program Definition sum_Sum (A B : Type) : (@Sum Type_Cat A B) := +{| + product := (A + B)%type; + Prod_morph_ex := + fun (p' : Type) + (r1 : A -> p') + (r2 : B -> p') + (X : A + B) => + match X return p' with + | inl a => r1 a + | inr b => r2 b + end +|}. +Next Obligation. simpl; auto. Defined. +Next Obligation. simpl; auto. Defined. + +Program Instance Type_Cat_Has_Sums : Has_Sums Type_Cat := sum_Sum. + +Definition Sum_Func {C : Category} {HS : Has_Sums C} : + Functor (Prod_Cat C C) C := Opposite_Functor (Prod_Func (Opposite C) HS). + +Arguments Sum_Func _ _, _ {_}. + +Program Instance unit_Type_term : Terminal Type_Cat := +{ + terminal := unit; + t_morph := fun _ _=> tt +}. + +Next Obligation. Proof. Admitted. + +Program Definition term_id : Functor Type_Cat (Prod_Cat Type_Cat Type_Cat) := +{| + FO := fun a => (@terminal Type_Cat _, a); + FA := fun a b f => (@id _ (@terminal Type_Cat _), f) +|}. + +Definition S_nat_func : Functor Type_Cat Type_Cat := + Functor_compose term_id (Sum_Func Type_Cat _). + +Definition S_nat_alg_cat := Algebra_Cat S_nat_func. + +CoInductive CoNat : Set := + | CoO : CoNat + | CoS : CoNat -> CoNat +. + +Definition S_nat_coalg_cat := @CoAlgebra_Cat Type_Cat S_nat_func. + +Set Printing Universes. +Program Definition CoNat_alg_term : Initial S_nat_coalg_cat := +{| + terminal := _; + t_morph := _ +|}. + +Next Obligation. Admitted. +Next Obligation. Admitted. + +Axiom Admit : False. + +Next Obligation. +Proof. + intros d f g. + assert(H1 := (@Alg_map_com _ _ _ _ f)). clear. + assert (inl tt = inr tt) by (exfalso; apply Admit). + discriminate. + all: exfalso; apply Admit. + Show Universes. +Qed. |