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{n} {A} `{IsHSet A} : IsTrunc n. | Definition | Require Import | Basics\Trunc.v | istrunc_hset | 1,800 |
A {B} (f : A -> B) `{IsTrunc n A} `{IsEquiv A B f} : IsTrunc n B. Proof. generalize dependent B; generalize dependent A. simple_induction n n IH; simpl; intros A ? B f ?. - exact (contr_equiv _ f). - apply istrunc_S. intros x y. refine (IH _ _ _ (ap (f^-1))^-1 _). Defined. | Definition | Require Import | Basics\Trunc.v | istrunc_isequiv_istrunc | 1,801 |
A {B} (f : A <~> B) `{IsTrunc n A} : IsTrunc n B := istrunc_isequiv_istrunc A f. | Definition | Require Import | Basics\Trunc.v | istrunc_equiv_istrunc | 1,802 |
(n : trunc_index) := { trunctype_type : Type ; trunctype_istrunc : IsTrunc n trunctype_type }. | Record | Require Import | Basics\Trunc.v | TruncType | 1,803 |
(A : Type) `{H : IsHProp A} (x : A) : Contr A. Proof. apply (Build_Contr _ x). intro y. rapply center. Defined. | Lemma | Require Import | Basics\Trunc.v | contr_inhabited_hprop | 1,804 |
`{H : IsHProp A} : forall x y : A, x = y. Proof. intros x y. rapply center. Defined. | Theorem | Require Import | Basics\Trunc.v | path_ishprop | 1,805 |
(A : Type) : (forall (x y : A), x = y) -> IsHProp A. Proof. intros H; apply istrunc_S; intros x y. nrapply contr_paths_contr. exact (Build_Contr _ x (H x)). Defined. | Theorem | Require Import | Basics\Trunc.v | hprop_allpath | 1,806 |
`{IsHProp A} `{IsHProp B} (f : A -> B) (g : B -> A) : IsEquiv f. Proof. apply (isequiv_adjointify f g); intros ?; apply path_ishprop. Defined. | Definition | Require Import | Basics\Trunc.v | isequiv_iff_hprop | 1,807 |
`{IsHProp A} `{IsHProp B} : (A <-> B) -> (A <~> B). Proof. intro fg. apply (equiv_adjointify (fst fg) (snd fg)); intros ?; apply path_ishprop. Defined. | Definition | Require Import | Basics\Trunc.v | equiv_iff_hprop_uncurried | 1,808 |
`{IsHProp A} `{IsHProp B} : (A -> B) -> (B -> A) -> (A <~> B) := fun f g => equiv_iff_hprop_uncurried (f, g). | Definition | Require Import | Basics\Trunc.v | equiv_iff_hprop | 1,809 |
(A : Type) `{IsHProp A} : Contr A <-> A. Proof. split. - apply center. - rapply contr_inhabited_hprop. Defined. | Corollary | Require Import | Basics\Trunc.v | iff_contr_hprop | 1,810 |
`{Funext} `{P : A -> Type} `{forall a, Contr (P a)} : Contr (forall a, P a). Proof. apply (Build_Contr _ (fun a => center (P a))). intro f. apply path_forall. intro a. apply contr. Defined. | Definition | Require Import | Basics\Trunc.v | contr_forall | 1,811 |
(A : Type) `{Funext} `{IsHProp A} : Contr A <~> A. Proof. exact (equiv_iff_hprop_uncurried (iff_contr_hprop A)). Defined. | Corollary | Require Import | Basics\Trunc.v | equiv_contr_hprop | 1,812 |
{n : trunc_index} {A : Type} (H : A -> IsTrunc n. | Definition | Require Import | Basics\Trunc.v | istrunc_inhabited_istrunc | 1,813 |
Type0 := | Nil | D0 (_:) | D1 (_:) | D2 (_:) | D3 (_:) | D4 (_:) | D5 (_:) | D6 (_:) | D7 (_:) | D8 (_:) | D9 (_:). | Inductive | Require Import Basics.Overture. | Basics\Numerals\Decimal.v | uint | 1,814 |
d := match d with | Nil => O | D0 d | D1 d | D2 d | D3 d | D4 d | D5 d | D6 d | D7 d | D8 d | D9 d => S ( d) end. | Fixpoint | Require Import Basics.Overture. | Basics\Numerals\Decimal.v | nb_digits | 1,815 |
d := match d with | D0 d => d | _ => d end. | Fixpoint | Require Import Basics.Overture. | Basics\Numerals\Decimal.v | nzhead | 1,816 |
d := match nzhead d with | Nil => zero | d => d end. | Definition | Require Import Basics.Overture. | Basics\Numerals\Decimal.v | unorm | 1,817 |
d := match d with | Pos d => Pos (unorm d) | Neg d => match nzhead d with | Nil => Pos zero | d => Neg d end end. | Definition | Require Import Basics.Overture. | Basics\Numerals\Decimal.v | norm | 1,818 |
(d:int) := match d with | Pos d => Neg d | Neg d => Pos d end. | Definition | Require Import Basics.Overture. | Basics\Numerals\Decimal.v | opp | 1,819 |
(d d' : uint) := match d with | Nil => d' | D0 d => d (D0 d') | D1 d => d (D1 d') | D2 d => d (D2 d') | D3 d => d (D3 d') | D4 d => d (D4 d') | D5 d => d (D5 d') | D6 d => d (D6 d') | D7 d => d (D7 d') | D8 d => d (D8 d') | D9 d => d (D9 d') end. | Fixpoint | Require Import Basics.Overture. | Basics\Numerals\Decimal.v | revapp | 1,820 |
d := revapp d Nil. | Definition | Require Import Basics.Overture. | Basics\Numerals\Decimal.v | rev | 1,821 |
d d' := revapp (rev d) d'. | Definition | Require Import Basics.Overture. | Basics\Numerals\Decimal.v | app | 1,822 |
d1 d2 := match d1 with Pos d1 => Pos (app d1 d2) | Neg d1 => Neg (app d1 d2) end. | Definition | Require Import Basics.Overture. | Basics\Numerals\Decimal.v | app_int | 1,823 |
d := let fix aux d_rev := match d_rev with | D0 d_rev => let (r, n) := aux d_rev in pair r (S n) | _ => pair d_rev O end in let (r, n) := aux (rev d) in pair (rev r) n. | Definition | Require Import Basics.Overture. | Basics\Numerals\Decimal.v | nztail | 1,824 |
d := match d with | Pos d => let (r, n) := nztail d in pair (Pos r) n | Neg d => let (r, n) := nztail d in pair (Neg r) n end. | Definition | Require Import Basics.Overture. | Basics\Numerals\Decimal.v | nztail_int | 1,825 |
d := match d with | Nil => D1 Nil | D0 d => D1 d | D1 d => D2 d | D2 d => D3 d | D3 d => D4 d | D4 d => D5 d | D5 d => D6 d | D6 d => D7 d | D7 d => D8 d | D8 d => D9 d | D9 d => D0 ( d) end. | Fixpoint | Require Import Basics.Overture. | Basics\Numerals\Decimal.v | succ | 1,826 |
d := match d with | Nil => Nil | D0 d => D0 ( d) | D1 d => D2 ( d) | D2 d => D4 ( d) | D3 d => D6 ( d) | D4 d => D8 ( d) | D5 d => D0 (succ_double d) | D6 d => D2 (succ_double d) | D7 d => D4 (succ_double d) | D8 d => D6 (succ_double d) | D9 d => D8 (succ_double d) end with succ_double d := match d with | Nil => D1 Nil | D0 d => D1 ( d) | D1 d => D3 ( d) | D2 d => D5 ( d) | D3 d => D7 ( d) | D4 d => D9 ( d) | D5 d => D1 (succ_double d) | D6 d => D3 (succ_double d) | D7 d => D5 (succ_double d) | D8 d => D7 (succ_double d) | D9 d => D9 (succ_double d) end. | Fixpoint | Require Import Basics.Overture. | Basics\Numerals\Decimal.v | double | 1,827 |
(i:uint) := i. | Definition | Require Import Basics.Overture. | Basics\Numerals\Decimal.v | uint_of_uint | 1,828 |
(i:int) := i. | Definition | Require Import Basics.Overture. | Basics\Numerals\Decimal.v | int_of_int | 1,829 |
Type0 := | Nil | D0 (_:) | D1 (_:) | D2 (_:) | D3 (_:) | D4 (_:) | D5 (_:) | D6 (_:) | D7 (_:) | D8 (_:) | D9 (_:) | Da (_:) | Db (_:) | Dc (_:) | Dd (_:) | De (_:) | Df (_:). | Inductive | Require Import Basics.Overture Basics.Numerals.Decimal. | Basics\Numerals\Hexadecimal.v | uint | 1,830 |
d := match d with | Nil => O | D0 d | D1 d | D2 d | D3 d | D4 d | D5 d | D6 d | D7 d | D8 d | D9 d | Da d | Db d | Dc d | Dd d | De d | Df d => S ( d) end. | Fixpoint | Require Import Basics.Overture Basics.Numerals.Decimal. | Basics\Numerals\Hexadecimal.v | nb_digits | 1,831 |
d := match d with | D0 d => d | _ => d end. | Fixpoint | Require Import Basics.Overture Basics.Numerals.Decimal. | Basics\Numerals\Hexadecimal.v | nzhead | 1,832 |
d := match nzhead d with | Nil => zero | d => d end. | Definition | Require Import Basics.Overture Basics.Numerals.Decimal. | Basics\Numerals\Hexadecimal.v | unorm | 1,833 |
d := match d with | Pos d => Pos (unorm d) | Neg d => match nzhead d with | Nil => Pos zero | d => Neg d end end. | Definition | Require Import Basics.Overture Basics.Numerals.Decimal. | Basics\Numerals\Hexadecimal.v | norm | 1,834 |
(d:int) := match d with | Pos d => Neg d | Neg d => Pos d end. | Definition | Require Import Basics.Overture Basics.Numerals.Decimal. | Basics\Numerals\Hexadecimal.v | opp | 1,835 |
(d d' : uint) := match d with | Nil => d' | D0 d => d (D0 d') | D1 d => d (D1 d') | D2 d => d (D2 d') | D3 d => d (D3 d') | D4 d => d (D4 d') | D5 d => d (D5 d') | D6 d => d (D6 d') | D7 d => d (D7 d') | D8 d => d (D8 d') | D9 d => d (D9 d') | Da d => d (Da d') | Db d => d (Db d') | Dc d => d (Dc d') | Dd d => d (Dd d') | De d => d (De d') | Df d => d (Df d') end. | Fixpoint | Require Import Basics.Overture Basics.Numerals.Decimal. | Basics\Numerals\Hexadecimal.v | revapp | 1,836 |
d := revapp d Nil. | Definition | Require Import Basics.Overture Basics.Numerals.Decimal. | Basics\Numerals\Hexadecimal.v | rev | 1,837 |
d d' := revapp (rev d) d'. | Definition | Require Import Basics.Overture Basics.Numerals.Decimal. | Basics\Numerals\Hexadecimal.v | app | 1,838 |
d1 d2 := match d1 with Pos d1 => Pos (app d1 d2) | Neg d1 => Neg (app d1 d2) end. | Definition | Require Import Basics.Overture Basics.Numerals.Decimal. | Basics\Numerals\Hexadecimal.v | app_int | 1,839 |
d := let fix aux d_rev := match d_rev with | D0 d_rev => let (r, n) := aux d_rev in pair r (S n) | _ => pair d_rev O end in let (r, n) := aux (rev d) in pair (rev r) n. | Definition | Require Import Basics.Overture Basics.Numerals.Decimal. | Basics\Numerals\Hexadecimal.v | nztail | 1,840 |
d := match d with | Pos d => let (r, n) := nztail d in pair (Pos r) n | Neg d => let (r, n) := nztail d in pair (Neg r) n end. | Definition | Require Import Basics.Overture Basics.Numerals.Decimal. | Basics\Numerals\Hexadecimal.v | nztail_int | 1,841 |
d := match d with | Nil => D1 Nil | D0 d => D1 d | D1 d => D2 d | D2 d => D3 d | D3 d => D4 d | D4 d => D5 d | D5 d => D6 d | D6 d => D7 d | D7 d => D8 d | D8 d => D9 d | D9 d => Da d | Da d => Db d | Db d => Dc d | Dc d => Dd d | Dd d => De d | De d => Df d | Df d => D0 ( d) end. | Fixpoint | Require Import Basics.Overture Basics.Numerals.Decimal. | Basics\Numerals\Hexadecimal.v | succ | 1,842 |
d := match d with | Nil => Nil | D0 d => D0 ( d) | D1 d => D2 ( d) | D2 d => D4 ( d) | D3 d => D6 ( d) | D4 d => D8 ( d) | D5 d => Da ( d) | D6 d => Dc ( d) | D7 d => De ( d) | D8 d => D0 (succ_double d) | D9 d => D2 (succ_double d) | Da d => D4 (succ_double d) | Db d => D6 (succ_double d) | Dc d => D8 (succ_double d) | Dd d => Da (succ_double d) | De d => Dc (succ_double d) | Df d => De (succ_double d) end with succ_double d := match d with | Nil => D1 Nil | D0 d => D1 ( d) | D1 d => D3 ( d) | D2 d => D5 ( d) | D3 d => D7 ( d) | D4 d => D9 ( d) | D5 d => Db ( d) | D6 d => Dd ( d) | D7 d => Df ( d) | D8 d => D1 (succ_double d) | D9 d => D3 (succ_double d) | Da d => D5 (succ_double d) | Db d => D7 (succ_double d) | Dc d => D9 (succ_double d) | Dd d => Db (succ_double d) | De d => Dd (succ_double d) | Df d => Df (succ_double d) end. | Fixpoint | Require Import Basics.Overture Basics.Numerals.Decimal. | Basics\Numerals\Hexadecimal.v | double | 1,843 |
PreCategory := @sub_pre_cat _ P HF. | Definition | Require Import Functor.Core Category.Strict. Require Import Cat.Core. Require Import GroupoidCategory.Core. Require Import Functor.Paths. | Categories\CategoryOfGroupoids.v | groupoid_cat | 1,844 |
PreCategory := @Build_PreCategory nat leq leq_refl (fun x y z p q => leq_trans q p) (fun _ _ _ _ _ _ _ => path_ishprop _ _) (fun _ _ _ => path_ishprop _ _) (fun _ _ _ => path_ishprop _ _) _. | Definition | Require Import Category.Subcategory.Full. Require Import Category.Sigma.Univalent. Require Import Category.Morphisms Category.Univalent Category.Strict. Require Import HoTT.Basics HoTT.Types HoTT.Spaces.Nat.Core. | Categories\ChainCategory.v | omega | 1,845 |
(n : nat) : PreCategory := { m : omega | m <= n }%category. | Definition | Require Import Category.Subcategory.Full. Require Import Category.Sigma.Univalent. Require Import Category.Morphisms Category.Univalent Category.Strict. Require Import HoTT.Basics HoTT.Types HoTT.Spaces.Nat.Core. | Categories\ChainCategory.v | chain | 1,846 |
IsStrictCategory omega. Proof. exact _. Defined. | Definition | Require Import Category.Subcategory.Full. Require Import Category.Sigma.Univalent. Require Import Category.Morphisms Category.Univalent Category.Strict. Require Import HoTT.Basics HoTT.Types HoTT.Spaces.Nat.Core. | Categories\ChainCategory.v | isstrict_omega | 1,847 |
{n} : IsStrictCategory [n]. Proof. exact _. Defined. | Definition | Require Import Category.Subcategory.Full. Require Import Category.Sigma.Univalent. Require Import Category.Morphisms Category.Univalent Category.Strict. Require Import HoTT.Basics HoTT.Types HoTT.Spaces.Nat.Core. | Categories\ChainCategory.v | isstrict_chain | 1,848 |
{n} : IsCategory [n]. Proof. exact _. Defined. | Definition | Require Import Category.Subcategory.Full. Require Import Category.Sigma.Univalent. Require Import Category.Morphisms Category.Univalent Category.Strict. Require Import HoTT.Basics HoTT.Types HoTT.Spaces.Nat.Core. | Categories\ChainCategory.v | iscategory_chain | 1,849 |
PreCategory := category_of_sections (Grothendieck. | Definition | Require Import Category.Core Functor.Core. Require Import Cat.Core. Require Import CategoryOfSections.Core. | Categories\DependentProduct.v | dependent_product | 1,850 |
X `{IsHSet X} := groupoid_category X. | Definition | Require Import HoTT.Basics GroupoidCategory.Core. | Categories\DiscreteCategory.v | discrete_category | 1,851 |
Functor cat cat := Build_Functor cat cat (fun C => (C. | Definition | Require Import Category.Core Functor.Core. Require Import Category.Dual Functor.Dual. Require Import Functor.Composition.Core Functor.Identity. Require Import Cat.Core Functor.Paths. Require Import Basics.Trunc Types.Sigma HoTT.Tactics Types.Forall. | Categories\DualFunctor.v | opposite_functor | 1,852 |
opposite_functor o opposite_functor = 1. Proof. path_functor. refine (path_forall _ _ opposite_functor_involutive_helper; _). repeat (apply path_forall; intro). rewrite !transport_forall_constant. transport_path_forall_hammer. unfold opposite_functor_involutive_helper. rewrite !transport_pr1_path_sigma_uncurried. simpl in *. repeat progress change (fun x => ?f x) with f in *. match goal with | [ |- context[transport (fun x' => ?f x'.1 ?y) (@path_sigma_uncurried ?A ?P ?u ?v ?pq)] ] => rewrite (@transport_pr1_path_sigma_uncurried A P u v pq (fun x => f x y)) end. simpl in *. hnf in *. subst_body. destruct_head @sig. destruct_head @Functor. destruct_head @PreCategory. reflexivity. Qed. | Definition | Require Import Category.Core Functor.Core. Require Import Category.Dual Functor.Dual. Require Import Functor.Composition.Core Functor.Identity. Require Import Cat.Core Functor.Paths. Require Import Basics.Trunc Types.Sigma HoTT.Tactics Types.Forall. | Categories\DualFunctor.v | opposite_functor_involutive | 1,853 |
s d d' (m : morphism d d') (m' : morphism s d) : morphism s d'. Proof. revert m'; apply Trunc_rec; intro m'. revert m; apply Trunc_rec; intro m. apply tr. exact (m' @ m). Defined. | Definition | Require Import Category.Core. Require Import HoTT.Truncations.Core. Require Import HoTT.Basics. | Categories\FundamentalPreGroupoidCategory.v | compose | 1,854 |
x : morphism x x := tr (reflexivity _). | Definition | Require Import Category.Core. Require Import HoTT.Truncations.Core. Require Import HoTT.Basics. | Categories\FundamentalPreGroupoidCategory.v | identity | 1,855 |
(X : Type) : PreCategory. Proof. refine (@Build_PreCategory X _ (@FundamentalPreGroupoidCategoryInternals.identity X) (@FundamentalPreGroupoidCategoryInternals.compose X) _ _ _ _); simpl; intros; abstract ( repeat match goal with | [ m : Trunc _ _ |- _ ] => revert m; apply Trunc_ind; [ intro; match goal with | [ |- IsHSet (?a = ?b :> ?T) ] => generalize a b; intros; let H := fresh in assert (H : forall x y : T, IsHProp (x = y)) end; typeclasses eauto | intro ] end; simpl; apply ap; first [ apply concat_p_pp | apply concat_1p | apply concat_p1 ] ). Defined. | Definition | Require Import Category.Core. Require Import HoTT.Truncations.Core. Require Import HoTT.Basics. | Categories\FundamentalPreGroupoidCategory.v | fundamental_pregroupoid_category | 1,856 |
Functor (C^op * C) set_cat. | Definition | Require Import Category.Core Functor.Core SetCategory.Core Category.Dual. Import Category.Prod.CategoryProdNotations Functor.Prod.Core.FunctorProdCoreNotations. Require Import Basics.Trunc. | Categories\HomFunctor.v | hom_functor | 1,857 |
(A : object C^op) := Eval simpl in Functor. | Definition | Require Import Category.Core Functor.Core SetCategory.Core Category.Dual. Import Category.Prod.CategoryProdNotations Functor.Prod.Core.FunctorProdCoreNotations. Require Import Basics.Trunc. | Categories\HomFunctor.v | covariant_hom_functor | 1,858 |
(A : C) := Eval simpl in Functor. | Definition | Require Import Category.Core Functor.Core SetCategory.Core Category.Dual. Import Category.Prod.CategoryProdNotations Functor.Prod.Core.FunctorProdCoreNotations. Require Import Basics.Trunc. | Categories\HomFunctor.v | contravariant_hom_functor | 1,859 |
s d d' (m : morphism d d') (m' : morphism s d) : morphism s d'. Proof. revert m'; apply Trunc_rec; intro m'. revert m; apply Trunc_rec; intro m. apply tr. exact (m o m')%core. Defined. | Definition | Require Import Category.Core. Require Import HoTT.Basics HoTT.Truncations.Core. | Categories\HomotopyPreCategory.v | compose | 1,860 |
x : morphism x x := tr idmap. | Definition | Require Import Category.Core. Require Import HoTT.Basics HoTT.Truncations.Core. | Categories\HomotopyPreCategory.v | identity | 1,861 |
PreCategory. Proof. refine (@Build_PreCategory Type _ (@HomotopyPreCategoryInternals.identity) (@HomotopyPreCategoryInternals.compose) _ _ _ _); simpl; intros; repeat match goal with | [ m : Trunc _ _ |- _ ] => revert m; apply Trunc_ind; [ intro; match goal with | [ |- IsHSet (?a = ?b :> ?T) ] => generalize a b; intros; let H := fresh in assert (H : forall x y : T, IsHProp (x = y)) end; typeclasses eauto | intro ] end; simpl; apply ap; exact idpath. Defined. | Definition | Require Import Category.Core. Require Import HoTT.Basics HoTT.Truncations.Core. | Categories\HomotopyPreCategory.v | homotopy_precategory | 1,862 |
PreCategory := @Build_PreCategory' X (fun _ _ => Unit) (fun _ => tt) (fun _ _ _ _ _ => tt) (fun _ _ _ _ _ _ _ => idpath) (fun _ _ _ _ _ _ _ => idpath) (fun _ _ f => match f with tt => idpath end) (fun _ _ f => match f with tt => idpath end) (fun _ => idpath) _. | Definition | Require Import Functor.Core Category.Strict Category.Univalent Category.Morphisms. Require Import Types.Unit Trunc HoTT.Tactics Equivalences. | Categories\IndiscreteCategory.v | indiscrete_category | 1,863 |
`{H : IsHSet X} : IsStrictCategory (indiscrete_category X) := H. Global Instance iscategory_indiscrete_category `{H : IsHProp X} : IsCategory (indiscrete_category X). Proof. intros. eapply (isequiv_adjointify (idtoiso (indiscrete_category X) (x := s) (y := d)) (fun _ => center _)); abstract ( repeat intro; destruct_head_hnf @Isomorphic; destruct_head_hnf @IsIsomorphism; destruct_head_hnf @Unit; path_induction_hammer ). Defined. | Definition | Require Import Functor.Core Category.Strict Category.Univalent Category.Morphisms. Require Import Types.Unit Trunc HoTT.Tactics Equivalences. | Categories\IndiscreteCategory.v | isstrict_indiscrete_category | 1,864 |
Functor C (indiscrete_category X) := Build_Functor C (indiscrete_category X) objOf (fun _ _ _ => tt) (fun _ _ _ _ _ => idpath) (fun _ => idpath). | Definition | Require Import Functor.Core Category.Strict Category.Univalent Category.Morphisms. Require Import Types.Unit Trunc HoTT.Tactics Equivalences. | Categories\IndiscreteCategory.v | to | 1,865 |
(n : nat) : Type0 := match n with | 0 => Empty | 1 => Unit | S n' => n' + Unit end. Coercion : nat >-> Sortclass. Global Instance trunc_cardinality_representative (n : nat) : IsHSet ( n). Proof. induction n; [ typeclasses eauto |]. induction n; [ typeclasses eauto |]. apply istrunc_S. intros [x|x] [y|y]; typeclasses eauto. Qed. | Fixpoint | Require Import Category.Core DiscreteCategory IndiscreteCategory. Require Import Types.Unit Trunc Types.Sum Types.Empty. Require Import Basics.Nat. | Categories\NatCategory.v | CardinalityRepresentative | 1,866 |
(n : nat) := match n with | 0 => indiscrete_category 0 | 1 => indiscrete_category 1 | S (S n') => discrete_category (S (S n')) end. | Definition | Require Import Category.Core DiscreteCategory IndiscreteCategory. Require Import Types.Unit Trunc Types.Sum Types.Empty. Require Import Basics.Nat. | Categories\NatCategory.v | nat_category | 1,867 |
(C D : PreCategory) : Functor (C * D) (D * C) := Build_Functor (C * D) (D * C) (fun cd => (snd_type cd, fst_type cd)%core) (fun _ _ m => (snd_type m, fst_type m)%core) (fun _ _ _ _ _ => idpath) (fun _ => idpath). | Definition | Require Import HoTT.Basics HoTT.Types. Require Import Category.Core Functor.Core InitialTerminalCategory.Core InitialTerminalCategory.Functors Category.Prod Functor.Prod Functor.Composition.Core Functor.Identity Functor.Composition.Laws. | Categories\ProductLaws.v | functor | 1,868 |
(C D : PreCategory) : functor C D o functor D C = 1 := idpath. | Definition | Require Import HoTT.Basics HoTT.Types. Require Import Category.Core Functor.Core InitialTerminalCategory.Core InitialTerminalCategory.Functors Category.Prod Functor.Prod Functor.Composition.Core Functor.Identity Functor.Composition.Laws. | Categories\ProductLaws.v | law | 1,869 |
Functor (A * (B * C)) ((A * B) * C) := (fst * (fst o snd)) * (snd o snd). | Definition | Require Import HoTT.Basics HoTT.Types. Require Import Category.Core Functor.Core InitialTerminalCategory.Core InitialTerminalCategory.Functors Category.Prod Functor.Prod Functor.Composition.Core Functor.Identity Functor.Composition.Laws. | Categories\ProductLaws.v | functor | 1,870 |
Functor ((A * B) * C) (A * (B * C)) := (fst o fst) * ((snd o fst) * snd). | Definition | Require Import HoTT.Basics HoTT.Types. Require Import Category.Core Functor.Core InitialTerminalCategory.Core InitialTerminalCategory.Functors Category.Prod Functor.Prod Functor.Composition.Core Functor.Identity Functor.Composition.Laws. | Categories\ProductLaws.v | inverse | 1,871 |
functor o inverse = 1 /\ inverse o functor = 1 := (idpath, idpath)%core. | Definition | Require Import HoTT.Basics HoTT.Types. Require Import Category.Core Functor.Core InitialTerminalCategory.Core InitialTerminalCategory.Functors Category.Prod Functor.Prod Functor.Composition.Core Functor.Identity Functor.Composition.Laws. | Categories\ProductLaws.v | law | 1,872 |
Functor (C * 0) 0 := Functors. | Definition | Require Import HoTT.Basics HoTT.Types. Require Import Category.Core Functor.Core InitialTerminalCategory.Core InitialTerminalCategory.Functors Category.Prod Functor.Prod Functor.Composition.Core Functor.Identity Functor.Composition.Laws. | Categories\ProductLaws.v | functor | 1,873 |
functor' : Functor (0 * C) 0 := Functors. | Definition | Require Import HoTT.Basics HoTT.Types. Require Import Category.Core Functor.Core InitialTerminalCategory.Core InitialTerminalCategory.Functors Category.Prod Functor.Prod Functor.Composition.Core Functor.Identity Functor.Composition.Laws. | Categories\ProductLaws.v | functor' | 1,874 |
Functor 0 (C * 0) := Functors. | Definition | Require Import HoTT.Basics HoTT.Types. Require Import Category.Core Functor.Core InitialTerminalCategory.Core InitialTerminalCategory.Functors Category.Prod Functor.Prod Functor.Composition.Core Functor.Identity Functor.Composition.Laws. | Categories\ProductLaws.v | inverse | 1,875 |
inverse' : Functor 0 (0 * C) := Functors. | Definition | Require Import HoTT.Basics HoTT.Types. Require Import Category.Core Functor.Core InitialTerminalCategory.Core InitialTerminalCategory.Functors Category.Prod Functor.Prod Functor.Composition.Core Functor.Identity Functor.Composition.Laws. | Categories\ProductLaws.v | inverse' | 1,876 |
functor o inverse = 1 /\ inverse o functor = 1 := center _. | Definition | Require Import HoTT.Basics HoTT.Types. Require Import Category.Core Functor.Core InitialTerminalCategory.Core InitialTerminalCategory.Functors Category.Prod Functor.Prod Functor.Composition.Core Functor.Identity Functor.Composition.Laws. | Categories\ProductLaws.v | law | 1,877 |
law' : functor' o inverse' = 1 /\ inverse' o functor' = 1 := center _. | Definition | Require Import HoTT.Basics HoTT.Types. Require Import Category.Core Functor.Core InitialTerminalCategory.Core InitialTerminalCategory.Functors Category.Prod Functor.Prod Functor.Composition.Core Functor.Identity Functor.Composition.Laws. | Categories\ProductLaws.v | law' | 1,878 |
Functor (C * 1) C := fst. | Definition | Require Import HoTT.Basics HoTT.Types. Require Import Category.Core Functor.Core InitialTerminalCategory.Core InitialTerminalCategory.Functors Category.Prod Functor.Prod Functor.Composition.Core Functor.Identity Functor.Composition.Laws. | Categories\ProductLaws.v | functor | 1,879 |
functor' : Functor (1 * C) C := snd. | Definition | Require Import HoTT.Basics HoTT.Types. Require Import Category.Core Functor.Core InitialTerminalCategory.Core InitialTerminalCategory.Functors Category.Prod Functor.Prod Functor.Composition.Core Functor.Identity Functor.Composition.Laws. | Categories\ProductLaws.v | functor' | 1,880 |
Functor C (C * 1) := 1 * Functors. | Definition | Require Import HoTT.Basics HoTT.Types. Require Import Category.Core Functor.Core InitialTerminalCategory.Core InitialTerminalCategory.Functors Category.Prod Functor.Prod Functor.Composition.Core Functor.Identity Functor.Composition.Laws. | Categories\ProductLaws.v | inverse | 1,881 |
inverse' : Functor C (1 * C) := Functors. | Definition | Require Import HoTT.Basics HoTT.Types. Require Import Category.Core Functor.Core InitialTerminalCategory.Core InitialTerminalCategory.Functors Category.Prod Functor.Prod Functor.Composition.Core Functor.Identity Functor.Composition.Laws. | Categories\ProductLaws.v | inverse' | 1,882 |
functor o inverse = 1 /\ inverse o functor = 1. Proof. unfold functor, inverse. t_prod. Qed. | Lemma | Require Import HoTT.Basics HoTT.Types. Require Import Category.Core Functor.Core InitialTerminalCategory.Core InitialTerminalCategory.Functors Category.Prod Functor.Prod Functor.Composition.Core Functor.Identity Functor.Composition.Laws. | Categories\ProductLaws.v | law1 | 1,883 |
law1' : functor' o inverse' = 1 /\ inverse' o functor' = 1. Proof. unfold functor', inverse'. t_prod. Qed. | Lemma | Require Import HoTT.Basics HoTT.Types. Require Import Category.Core Functor.Core InitialTerminalCategory.Core InitialTerminalCategory.Functors Category.Prod Functor.Prod Functor.Composition.Core Functor.Identity Functor.Composition.Laws. | Categories\ProductLaws.v | law1' | 1,884 |
PreCategory := wide simplex_category (@IsMonomorphism _) _ _ _. | Definition | Require Import Types Basics.Trunc. Require Import Category.Core Functor.Core. Require Import Category.Morphisms. Require Import Category.Dual FunctorCategory.Core. Require Import SetCategory.Core. Require Import SimplicialSets. Require Import Category.Sigma.OnMorphisms Category.Subcategory.Wide. | Categories\SemiSimplicialSets.v | semisimplex_category | 1,885 |
semisimplex_category -> simplex_category := pr1_mor. | Definition | Require Import Types Basics.Trunc. Require Import Category.Core Functor.Core. Require Import Category.Morphisms. Require Import Category.Dual FunctorCategory.Core. Require Import SetCategory.Core. Require Import SimplicialSets. Require Import Category.Sigma.OnMorphisms Category.Subcategory.Wide. | Categories\SemiSimplicialSets.v | semisimplicial_inclusion_functor | 1,886 |
(C : PreCategory) : PreCategory := semisimplex_category^op -> C. | Definition | Require Import Types Basics.Trunc. Require Import Category.Core Functor.Core. Require Import Category.Morphisms. Require Import Category.Dual FunctorCategory.Core. Require Import SetCategory.Core. Require Import SimplicialSets. Require Import Category.Sigma.OnMorphisms Category.Subcategory.Wide. | Categories\SemiSimplicialSets.v | semisimplicial_category | 1,887 |
semisimplicial_category set_cat. | Definition | Require Import Types Basics.Trunc. Require Import Category.Core Functor.Core. Require Import Category.Morphisms. Require Import Category.Dual FunctorCategory.Core. Require Import SetCategory.Core. Require Import SimplicialSets. Require Import Category.Sigma.OnMorphisms Category.Subcategory.Wide. | Categories\SemiSimplicialSets.v | semisimplicial_set | 1,888 |
semisimplicial_category prop_cat. | Definition | Require Import Types Basics.Trunc. Require Import Category.Core Functor.Core. Require Import Category.Morphisms. Require Import Category.Dual FunctorCategory.Core. Require Import SetCategory.Core. Require Import SimplicialSets. Require Import Category.Sigma.OnMorphisms Category.Subcategory.Wide. | Categories\SemiSimplicialSets.v | semisimplicial_prop | 1,889 |
@Build_PreCategory nat Functor identity compose associativity left_identity right_identity _. | Definition | Require Import Basics Types Spaces.Nat.Core. Require Import Category.Core Functor.Core Functor.Paths. Require Import SetCategory.Core. Require Import ChainCategory FunctorCategory.Core. Require Import Category.Dual. Require Import Functor.Identity Functor.Composition.Core Functor.Composition.Laws. | Categories\SimplicialSets.v | simplex_category | 1,890 |
(C : PreCategory) : PreCategory := simplex_category^op -> C. | Definition | Require Import Basics Types Spaces.Nat.Core. Require Import Category.Core Functor.Core Functor.Paths. Require Import SetCategory.Core. Require Import ChainCategory FunctorCategory.Core. Require Import Category.Dual. Require Import Functor.Identity Functor.Composition.Core Functor.Composition.Laws. | Categories\SimplicialSets.v | simplicial_category | 1,891 |
simplicial_category set_cat. | Definition | Require Import Basics Types Spaces.Nat.Core. Require Import Category.Core Functor.Core Functor.Paths. Require Import SetCategory.Core. Require Import ChainCategory FunctorCategory.Core. Require Import Category.Dual. Require Import Functor.Identity Functor.Composition.Core Functor.Composition.Laws. | Categories\SimplicialSets.v | simplicial_set | 1,892 |
simplicial_category prop_cat. | Definition | Require Import Basics Types Spaces.Nat.Core. Require Import Category.Core Functor.Core Functor.Paths. Require Import SetCategory.Core. Require Import ChainCategory FunctorCategory.Core. Require Import Category.Dual. Require Import Functor.Identity Functor.Composition.Core Functor.Composition.Laws. | Categories\SimplicialSets.v | simplicial_prop | 1,893 |
(Ap : object (X / U)) := IsInitialObject (X / U) Ap. | Definition | Require Import Category.Core Functor.Core. Require Import Category.Dual Functor.Dual. Require Import Category.Objects. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Import Comma.Core. Require Import Trunc Types.Sigma HoTT.Tactics. Require Import Basics.Tactics. | Categories\UniversalProperties.v | IsInitialMorphism | 1,894 |
(A : D) (p : morphism C X (U A)) (Ap := CommaCategory.Build_object !X U tt A p) (UniversalProperty : forall (A' : D) (p' : morphism C X (U A')), Contr { m : morphism D A A' | U _1 m o p = p' }) : IsInitialMorphism Ap. Proof. intro x. specialize (UniversalProperty (CommaCategory.b x) (CommaCategory.f x)). eapply istrunc_equiv_istrunc. - apply CommaCategory.issig_morphism. - apply contr_inhabited_hprop. + abstract univ_hprop_t UniversalProperty. + (exists tt). (exists (@center _ UniversalProperty).1). abstract (progress rewrite ?right_identity, ?left_identity; exact (@center _ UniversalProperty).2). Defined. | Definition | Require Import Category.Core Functor.Core. Require Import Category.Dual Functor.Dual. Require Import Category.Objects. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Import Comma.Core. Require Import Trunc Types.Sigma HoTT.Tactics. Require Import Basics.Tactics. | Categories\UniversalProperties.v | Build_IsInitialMorphism | 1,895 |
(A : D) (p : morphism C X (U A)) (Ap := CommaCategory. | Definition | Require Import Category.Core Functor.Core. Require Import Category.Dual Functor.Dual. Require Import Category.Objects. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Import Comma.Core. Require Import Trunc Types.Sigma HoTT.Tactics. Require Import Basics.Tactics. | Categories\UniversalProperties.v | Build_IsInitialMorphism_curried | 1,896 |
(univ : { A : D | { p : morphism C X (U A) | let Ap := CommaCategory. | Definition | Require Import Category.Core Functor.Core. Require Import Category.Dual Functor.Dual. Require Import Category.Objects. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Import Comma.Core. Require Import Trunc Types.Sigma HoTT.Tactics. Require Import Basics.Tactics. | Categories\UniversalProperties.v | Build_IsInitialMorphism_uncurried | 1,897 |
forall (univ : { A : D | { p : morphism C X (U A) | let Ap := CommaCategory. | Definition | Require Import Category.Core Functor.Core. Require Import Category.Dual Functor.Dual. Require Import Category.Objects. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Import Comma.Core. Require Import Trunc Types.Sigma HoTT.Tactics. Require Import Basics.Tactics. | Categories\UniversalProperties.v | Build_IsInitialMorphism_uncurried | 1,898 |
(M : IsInitialMorphism Ap) : D := CommaCategory. | Definition | Require Import Category.Core Functor.Core. Require Import Category.Dual Functor.Dual. Require Import Category.Objects. Require Import InitialTerminalCategory.Core InitialTerminalCategory.Functors. Import Comma.Core. Require Import Trunc Types.Sigma HoTT.Tactics. Require Import Basics.Tactics. | Categories\UniversalProperties.v | IsInitialMorphism_object | 1,899 |