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Functor B A := Build_Functor B A (fun xym__Fx_to_Gy => fun x => xym__Fx_to_Gy x x 1) (fun x__s x__d x__m => fun x => x__m x x 1) (fun _ _ _ _ _ => idpath) (fun _ => idpath). | Definition | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Pseudofunctor.Core. Require Import Category.Morphisms FunctorCategory.Morphisms. Require Import Functor.Composition.Core. Require Import NaturalTransformation.Composition.Core NaturalTransformation.Composition.Laws. Require Import NaturalTransformation.Identity. | Categories\PseudonaturalTransformation\Core.v | B_to_A | 2,500 |
(a : a_part) (b : b_part a) := Eval simpl in forall x, (((left_identity_natural_transformation_1 _) o (p_identity_of G _ oR _) o (B_to_A _1 b x) o (_ oL (p_identity_of F _)^-1) o (left_identity_natural_transformation_2 _)) = 1)%natural_transformation. | Definition | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Pseudofunctor.Core. Require Import Category.Morphisms FunctorCategory.Morphisms. Require Import Functor.Composition.Core. Require Import NaturalTransformation.Composition.Core NaturalTransformation.Composition.Laws. Require Import NaturalTransformation.Identity. | Categories\PseudonaturalTransformation\Core.v | b_id_part | 2,501 |
Functor B C. Proof. refine (Build_Functor B C (fun xym__Fx_to_Gy => fun x y z m1 m2 => xym__Fx_to_Gy y z m1 o p_morphism_of F m2)%functor (fun xym__s xym__d xym__m => fun x y z m1 m2 => xym__m y z m1 oR p_morphism_of F m2) _ _); simpl; repeat (intro || apply path_forall); [ apply composition_of_whisker_r | apply whisker_r_left_identity ]. Defined. | Definition | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Pseudofunctor.Core. Require Import Category.Morphisms FunctorCategory.Morphisms. Require Import Functor.Composition.Core. Require Import NaturalTransformation.Composition.Core NaturalTransformation.Composition.Laws. Require Import NaturalTransformation.Identity. | Categories\PseudonaturalTransformation\Core.v | B_to_C_1 | 2,502 |
Functor B C. Proof. refine (Build_Functor B C (fun xym__Fx_to_Gy => fun x y z m1 m2 => p_morphism_of G m1 o xym__Fx_to_Gy x y m2)%functor (fun xym__s xym__d xym__m => fun x y z m1 m2 => p_morphism_of G m1 oL xym__m x y m2) _ _); simpl; repeat (intro || apply path_forall); [ apply composition_of_whisker_l | apply whisker_l_right_identity ]. Defined. | Definition | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Pseudofunctor.Core. Require Import Category.Morphisms FunctorCategory.Morphisms. Require Import Functor.Composition.Core. Require Import NaturalTransformation.Composition.Core NaturalTransformation.Composition.Laws. Require Import NaturalTransformation.Identity. | Categories\PseudonaturalTransformation\Core.v | B_to_C_2 | 2,503 |
Functor B C := Build_Functor B C (fun xym__Fx_to_Gy => fun x y z m1 m2 => xym__Fx_to_Gy x z (m1 o m2)) (fun xym__s xym__d xym__m => fun x y z m1 m2 => xym__m x z (m1 o m2)) (fun _ _ _ _ _ => idpath) (fun _ => idpath). | Definition | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Pseudofunctor.Core. Require Import Category.Morphisms FunctorCategory.Morphisms. Require Import Functor.Composition.Core. Require Import NaturalTransformation.Composition.Core NaturalTransformation.Composition.Laws. Require Import NaturalTransformation.Identity. | Categories\PseudonaturalTransformation\Core.v | B_to_C_3 | 2,504 |
c_part' (a : a_part) (b : b_part a) : forall (x y z : X) (m1 : morphism X y z) (m2 : morphism X x y), Type. Proof. hnf in a, b. pose (fun x y m => (b x y m : morphism _ _ _)) as bB; simpl in *. intros x y z m1 m2. exact (((associator_2 _ _ _) o (B_to_C_2 _1 bB x y z m1 m2) o (associator_1 _ _ _) o (B_to_C_1 _1 bB x y z m1 m2) o (associator_2 _ _ _)) = ((p_composition_of G _ _ _ m1 m2 oR _) o (B_to_C_3 _1 bB x y z m1 m2) o (_ oL (p_composition_of F _ _ _ m1 m2)^-1)))%natural_transformation. Defined. | Definition | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Pseudofunctor.Core. Require Import Category.Morphisms FunctorCategory.Morphisms. Require Import Functor.Composition.Core. Require Import NaturalTransformation.Composition.Core NaturalTransformation.Composition.Laws. Require Import NaturalTransformation.Identity. | Categories\PseudonaturalTransformation\Core.v | c_part' | 2,505 |
(a : a_part) (b : b_part a) := Eval simpl in forall x y z m1 m2, @' a b x y z m1 m2. | Definition | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Pseudofunctor.Core. Require Import Category.Morphisms FunctorCategory.Morphisms. Require Import Functor.Composition.Core. Require Import NaturalTransformation.Composition.Core NaturalTransformation.Composition.Laws. Require Import NaturalTransformation.Identity. | Categories\PseudonaturalTransformation\Core.v | c_part | 2,506 |
`{Funext} (X : PreCategory) | Record | Require Import Category.Core Functor.Core NaturalTransformation.Core. Require Import Pseudofunctor.Core. Require Import Category.Morphisms FunctorCategory.Morphisms. Require Import Functor.Composition.Core. Require Import NaturalTransformation.Composition.Core NaturalTransformation.Composition.Laws. Require Import NaturalTransformation.Identity. | Categories\PseudonaturalTransformation\Core.v | PseudonaturalTransformation | 2,507 |
`{Funext} : PreCategory := cat_of HProp. | Definition | Require Import Category.Strict. Require Import HoTT.Basics HoTT.Types TruncType. | Categories\SetCategory\Core.v | prop_cat | 2,508 |
`{Funext} : PreCategory := cat_of HSet. | Definition | Require Import Category.Strict. Require Import HoTT.Basics HoTT.Types TruncType. | Categories\SetCategory\Core.v | set_cat | 2,509 |
`{fs : Funext} (X : set_cat) C D F G (T1 T2 : morphism set_cat X (Build_HSet (@NaturalTransformation C D F G))) (H : forall x y, T1 x y = T2 x y) `{@IsIsomorphism set_cat _ _ T1} : @IsIsomorphism set_cat _ _ T2. Proof. exists (T1^-1)%morphism; abstract ( first [ apply @iso_moveR_Vp | apply @iso_moveR_pV ]; repeat first [ intro | solve [ auto | symmetry; auto ] | apply @path_forall | path_natural_transformation ] ). Defined. | Lemma | Require Import Category.Core NaturalTransformation.Core. Require Import Category.Morphisms NaturalTransformation.Paths. Require Import Category.Univalent. Require Import SetCategory.Core. Require Import HoTT.Basics HoTT.Types TruncType. | Categories\SetCategory\Morphisms.v | isisomorphism_set_cat_natural_transformation_paths | 2,510 |
s d (m : morphism set_cat s d) `{IsEquiv _ _ m} : IsIsomorphism m := Build_IsIsomorphism set_cat s d m m^-1%function (path_forall _ _ (eissect m)) (path_forall _ _ (eisretr m)). | Definition | Require Import Category.Core NaturalTransformation.Core. Require Import Category.Morphisms NaturalTransformation.Paths. Require Import Category.Univalent. Require Import SetCategory.Core. Require Import HoTT.Basics HoTT.Types TruncType. | Categories\SetCategory\Morphisms.v | isiso_isequiv | 2,511 |
s d (m : morphism set_cat s d) `{IsIsomorphism _ _ _ m} : IsEquiv m := Build_IsEquiv _ _ m m^-1%morphism (ap10 right_inverse) (ap10 left_inverse) (fun _ => path_ishprop _ _). | Definition | Require Import Category.Core NaturalTransformation.Core. Require Import Category.Morphisms NaturalTransformation.Paths. Require Import Category.Univalent. Require Import SetCategory.Core. Require Import HoTT.Basics HoTT.Types TruncType. | Categories\SetCategory\Morphisms.v | isequiv_isiso | 2,512 |
(s d : set_cat) (m : s <~> d) : s <~=~> d := Build_Isomorphic (@isiso_isequiv s d m _). Global Instance isequiv_isiso_isequiv s d : IsEquiv (@ s d) | 0. Proof. refine (isequiv_adjointify (@ s d) (fun m => Build_Equiv _ _ _ (@isequiv_isiso s d m m)) _ _); simpl in *; clear; abstract ( intros [? ?]; simpl; unfold ; simpl; apply ap; apply path_ishprop ). Defined. | Definition | Require Import Category.Core NaturalTransformation.Core. Require Import Category.Morphisms NaturalTransformation.Paths. Require Import Category.Univalent. Require Import SetCategory.Core. Require Import HoTT.Basics HoTT.Types TruncType. | Categories\SetCategory\Morphisms.v | iso_equiv | 2,513 |
(s d : set_cat) (p : s = d) : iso_equiv s d (equiv_path _ _ (ap trunctype_type p)) = idtoiso set_cat p. Proof. apply path_isomorphic. case p. reflexivity. Defined. | Lemma | Require Import Category.Core NaturalTransformation.Core. Require Import Category.Morphisms NaturalTransformation.Paths. Require Import Category.Univalent. Require Import SetCategory.Core. Require Import HoTT.Basics HoTT.Types TruncType. | Categories\SetCategory\Morphisms.v | path_idtoequiv_idtoiso | 2,514 |
s d (m : morphism prop_cat s d) `{IsEquiv _ _ m} : IsIsomorphism m := Build_IsIsomorphism prop_cat s d m m^-1%function (path_forall _ _ (eissect m)) (path_forall _ _ (eisretr m)). | Definition | Require Import Category.Core NaturalTransformation.Core. Require Import Category.Morphisms NaturalTransformation.Paths. Require Import Category.Univalent. Require Import SetCategory.Core. Require Import HoTT.Basics HoTT.Types TruncType. | Categories\SetCategory\Morphisms.v | isiso_isequiv_prop | 2,515 |
s d (m : morphism prop_cat s d) `{IsIsomorphism _ _ _ m} : IsEquiv m := Build_IsEquiv _ _ m m^-1%morphism (ap10 right_inverse) (ap10 left_inverse) (fun _ => path_ishprop _ _). | Definition | Require Import Category.Core NaturalTransformation.Core. Require Import Category.Morphisms NaturalTransformation.Paths. Require Import Category.Univalent. Require Import SetCategory.Core. Require Import HoTT.Basics HoTT.Types TruncType. | Categories\SetCategory\Morphisms.v | isequiv_isiso_prop | 2,516 |
(s d : prop_cat) (m : s <~> d) : s <~=~> d := Build_Isomorphic (@isiso_isequiv_prop s d m _). Global Instance isequiv_isiso_isequiv_prop s d : IsEquiv (@ s d) | 0. Proof. refine (isequiv_adjointify (@ s d) (fun m => Build_Equiv _ _ _ (@isequiv_isiso_prop s d m _)) _ _); simpl in *; clear; abstract ( intros [? ?]; simpl; unfold ; simpl; apply ap; apply path_ishprop ). Defined. | Definition | Require Import Category.Core NaturalTransformation.Core. Require Import Category.Morphisms NaturalTransformation.Paths. Require Import Category.Univalent. Require Import SetCategory.Core. Require Import HoTT.Basics HoTT.Types TruncType. | Categories\SetCategory\Morphisms.v | iso_equiv_prop | 2,517 |
(s d : prop_cat) (p : s = d) : iso_equiv_prop s d (equiv_path _ _ (ap trunctype_type p)) = idtoiso prop_cat p. Proof. apply path_isomorphic. case p. reflexivity. Defined. | Lemma | Require Import Category.Core NaturalTransformation.Core. Require Import Category.Morphisms NaturalTransformation.Paths. Require Import Category.Univalent. Require Import SetCategory.Core. Require Import HoTT.Basics HoTT.Types TruncType. | Categories\SetCategory\Morphisms.v | path_idtoequiv_idtoiso_prop | 2,518 |
Functor C prop_cat. | Definition | Require Import Category.Core Functor.Core SetCategory.Core. Require Import Basics.Trunc. | Categories\SetCategory\Functors\SetProp.v | to_prop | 2,519 |
Functor C set_cat. | Definition | Require Import Category.Core Functor.Core SetCategory.Core. Require Import Basics.Trunc. | Categories\SetCategory\Functors\SetProp.v | to_set | 2,520 |
Functor prop_cat C. | Definition | Require Import Category.Core Functor.Core SetCategory.Core. Require Import Basics.Trunc. | Categories\SetCategory\Functors\SetProp.v | from_prop | 2,521 |
Functor set_cat C. | Definition | Require Import Category.Core Functor.Core SetCategory.Core. Require Import Basics.Trunc. | Categories\SetCategory\Functors\SetProp.v | from_set | 2,522 |
(F : to_prop C) : to_set C := Build_Functor C set_cat (fun x => Build_HSet (F x)) (fun s d m => (F _1 m)%morphism) (fun s d d' m m' => composition_of F s d d' m m') (fun x => identity_of F x). | Definition | Require Import Category.Core Functor.Core SetCategory.Core. Require Import Basics.Trunc. | Categories\SetCategory\Functors\SetProp.v | to_prop2set | 2,523 |
(F : from_set C) : from_prop C := Build_Functor prop_cat C (fun x => F (Build_HSet x)) (fun s d m => (F _1 (m : morphism set_cat (Build_HSet s) (Build_HSet d)))%morphism) (fun s d d' m m' => composition_of F (Build_HSet s) (Build_HSet d) (Build_HSet d') m m') (fun x => identity_of F (Build_HSet x)). | Definition | Require Import Category.Core Functor.Core SetCategory.Core. Require Import Basics.Trunc. | Categories\SetCategory\Functors\SetProp.v | from_set2prop | 2,524 |
(X : PreCategory) := | Record | Require Import Category.Core. Require Import HoTT.Basics HoTT.Types HSet. Import PreCategoryOfStructures. | Categories\Structure\Core.v | NotionOfStructure | 2,525 |
forall xa yb (H : x xa = x yb), transport P H (a xa) = a yb -> xa = yb. Proof. intros [? ?] [? ?] H H'; simpl in *; path_induction; reflexivity. Defined. | Lemma | Require Import Category.Core. Require Import HoTT.Basics HoTT.Types HSet. Import PreCategoryOfStructures. | Categories\Structure\Core.v | path_object | 2,526 |
(xa yb : object) := | Record | Require Import Category.Core. Require Import HoTT.Basics HoTT.Types HSet. Import PreCategoryOfStructures. | Categories\Structure\Core.v | morphism | 2,527 |
(xa yb : object) : { f : Category.Core.morphism X xa.1 yb.1 | is_structure_homomorphism _ _ _ f xa.2 yb.2 } <~> morphism xa yb. Proof. issig. Defined. | Lemma | Require Import Category.Core. Require Import HoTT.Basics HoTT.Types HSet. Import PreCategoryOfStructures. | Categories\Structure\Core.v | issig_morphism | 2,528 |
forall xa yb (fh gi : morphism xa yb), f fh = f gi -> fh = gi. Proof. intros ? ? [? ?] [? ?] H; simpl in *; path_induction; apply ap. apply path_ishprop. Defined. | Lemma | Require Import Category.Core. Require Import HoTT.Basics HoTT.Types HSet. Import PreCategoryOfStructures. | Categories\Structure\Core.v | path_morphism | 2,529 |
X (P : NotionOfStructure X) : PreCategory. Proof. refine (@Build_PreCategory _ (@morphism _ P) (fun xa => {| f := identity xa.1; h := is_structure_homomorphism_identity _ _ xa.2 |}) (fun xa yb zc gi fh => {| f := (f gi) o (f fh); h := is_structure_homomorphism_composition _ _ _ _ _ _ _ _ _ (h fh) (h gi) |}) _ _ _ (fun s d => istrunc_equiv_istrunc _ (issig_morphism P s d))); simpl; abstract ( repeat match goal with | |- @morphism _ P _ _ -> _ => intros [? ?]; simpl in * | |- _ -> _ => intro end; first [ apply path_morphism; exact (associativity _ _ _ _ _ _ _ _) | apply path_morphism; exact (left_identity _ _ _ _) | apply path_morphism; exact (right_identity _ _ _ _) ] ). Defined. | Definition | Require Import Category.Core. Require Import HoTT.Basics HoTT.Types HSet. Import PreCategoryOfStructures. | Categories\Structure\Core.v | precategory_of_structures | 2,530 |
(xa yb : StrX) (f : xa <~=~> yb) : xa.1 <~=~> yb.1. Proof. exists (PreCategoryOfStructures.f (f : morphism _ _ _)). exists (PreCategoryOfStructures.f f^-1). - exact (ap (@PreCategoryOfStructures.f _ _ _ _) (@left_inverse _ _ _ _ f)). - exact (ap (@PreCategoryOfStructures.f _ _ _ _) (@right_inverse _ _ _ _ f)). Defined. | Definition | Require Import Category.Core Category.Univalent Category.Morphisms. Require Import Structure.Core. Require Import Types.Sigma Trunc Equivalences. Require Import Basics.Iff Basics.Tactics. | Categories\Structure\IdentityPrinciple.v | sip_isotoid_helper | 2,531 |
(xa : StrX) : @sip_isotoid_helper xa xa (reflexivity _) = reflexivity _. Proof. unfold sip_isotoid_helper, reflexivity, isomorphic_refl. apply ap. apply path_ishprop. Defined. | Lemma | Require Import Category.Core Category.Univalent Category.Morphisms. Require Import Structure.Core. Require Import Types.Sigma Trunc Equivalences. Require Import Basics.Iff Basics.Tactics. | Categories\Structure\IdentityPrinciple.v | sip_isotoid_helper_refl | 2,532 |
x y (p : x = y) (a : P x) (b : P y) : transport P p a = b <-> is_structure_homomorphism P _ _ (idtoiso X p) a b * is_structure_homomorphism P _ _ (idtoiso X p)^-1 b a. Proof. split. - intros; path_induction; split; apply reflexivity. - intros [H0 H1]; path_induction; simpl in *. apply antisymmetry_structure; assumption. Defined. | Lemma | Require Import Category.Core Category.Univalent Category.Morphisms. Require Import Structure.Core. Require Import Types.Sigma Trunc Equivalences. Require Import Basics.Iff Basics.Tactics. | Categories\Structure\IdentityPrinciple.v | sip_helper | 2,533 |
(xa yb : StrX) (f : xa <~=~> yb) : xa = yb. Proof. refine (path_sigma_uncurried _ _ _ (isotoid X xa.1 yb.1 (sip_isotoid_helper f); _)). apply sip_helper; simpl. split; lazymatch goal with | [ |- context[idtoiso ?X ((isotoid ?X ?x ?y) ?m)] ] => pose proof (eisretr (@idtoiso X x y) m) as H'; pattern (idtoiso X ((isotoid X x y) m)) end; refine (transport _ H'^ _); clear H'; simpl; apply PreCategoryOfStructures.h. Defined. | Definition | Require Import Category.Core Category.Univalent Category.Morphisms. Require Import Structure.Core. Require Import Types.Sigma Trunc Equivalences. Require Import Basics.Iff Basics.Tactics. | Categories\Structure\IdentityPrinciple.v | sip_isotoid | 2,534 |
xa : @sip_isotoid xa xa (reflexivity _) = reflexivity _. Proof. refine (_ @ eta_path_sigma_uncurried _). refine (ap (path_sigma_uncurried _ _ _) _). apply equiv_path_sigma_hprop. simpl. refine (_ @ eisretr (isotoid X xa.1 xa.1) 1%path). apply ap. apply sip_isotoid_helper_refl. Defined. | Lemma | Require Import Category.Core Category.Univalent Category.Morphisms. Require Import Structure.Core. Require Import Types.Sigma Trunc Equivalences. Require Import Basics.Iff Basics.Tactics. | Categories\Structure\IdentityPrinciple.v | sip_isotoid_refl | 2,535 |
xa yb (p : xa = yb) : PreCategoryOfStructures.f (idtoiso (precategory_of_structures P) p : morphism _ _ _) = idtoiso X p..1. Proof. induction p; reflexivity. Defined. | Lemma | Require Import Category.Core Category.Univalent Category.Morphisms. Require Import Structure.Core. Require Import Types.Sigma Trunc Equivalences. Require Import Basics.Iff Basics.Tactics. | Categories\Structure\IdentityPrinciple.v | path_f_idtoiso_precategory_of_structures | 2,536 |
(xa yb : StrX) (x : xa <~=~> yb) : PreCategoryOfStructures.f (idtoiso (precategory_of_structures P) (sip_isotoid x) : morphism _ _ _) = PreCategoryOfStructures.f (x : morphism _ _ _). Proof. refine (path_f_idtoiso_precategory_of_structures _ @ _). refine ((ap _ (ap _ _)) @ (ap (@morphism_isomorphic _ _ _) (eisretr (@idtoiso X xa.1 yb.1) (sip_isotoid_helper _)))). exact (pr1_path_sigma_uncurried _). Defined. | Lemma | Require Import Category.Core Category.Univalent Category.Morphisms. Require Import Structure.Core. Require Import Types.Sigma Trunc Equivalences. Require Import Basics.Iff Basics.Tactics. | Categories\Structure\IdentityPrinciple.v | structure_identity_principle_helper | 2,537 |
`{Funext} (Ο : Signature) : PreCategory. Proof. apply (@Build_PreCategory (SetAlgebra Ο) Homomorphism hom_id (@hom_compose Ο)); [intros; by apply path_hset_homomorphism .. | exact _]. Defined. | Lemma | Require Import Import Morphisms.CategoryMorphismsNotations isomorphic_notations. | Classes\categories\ua_category.v | precategory_algebra | 2,538 |
`{Funext} {Ο} {A B : object (precategory_algebra Ο)} : A <~=~> B β A β
B. Proof. intros [f [a b c]]. unshelve eapply (@BuildIsomorphic _ _ _ f). intros s. refine (isequiv_adjointify (f s) (a s) _ _). - exact (apD10_homomorphism c s). - exact (apD10_homomorphism b s). Defined. | Lemma | Require Import Import Morphisms.CategoryMorphismsNotations isomorphic_notations. | Classes\categories\ua_category.v | catiso_to_uaiso | 2,539 |
`{Funext} {Ο} {A B : object (precategory_algebra Ο)} : A β
B β A <~=~> B. Proof. intros [f F G]. set (h := BuildHomomorphism f). apply (@Morphisms.Build_Isomorphic _ A B h). apply (@Morphisms.Build_IsIsomorphism _ A B h (hom_inv h)). - apply path_hset_homomorphism. funext s x. apply eissect. - apply path_hset_homomorphism. funext s x. apply eisretr. Defined. | Lemma | Require Import Import Morphisms.CategoryMorphismsNotations isomorphic_notations. | Classes\categories\ua_category.v | uaiso_to_catiso | 2,540 |
`{Funext} {Ο : Signature} (A B : object (precategory_algebra Ο)) : @Morphisms.idtoiso (precategory_algebra Ο) A B = catiso_to_uaiso^-1 o isomorphic_id o (path_setalgebra A B)^-1. Proof. funext p. destruct p. by apply Morphisms.path_isomorphic. Defined. | Lemma | Require Import Import Morphisms.CategoryMorphismsNotations isomorphic_notations. | Classes\categories\ua_category.v | path_idtoiso_isomorphic_id | 2,541 |
`{Univalence} (Ο : Signature) : IsCategory (precategory_algebra Ο). Proof. intros A B. rewrite path_idtoiso_isomorphic_id. apply @isequiv_compose. - apply isequiv_compose. - apply isequiv_inverse. Qed. | Lemma | Require Import Import Morphisms.CategoryMorphismsNotations isomorphic_notations. | Classes\categories\ua_category.v | iscategory_algebra | 2,542 |
`{Univalence} (Ο : Signature) : Category := Build_Category (iscategory_algebra Ο). | Definition | Require Import Import Morphisms.CategoryMorphismsNotations isomorphic_notations. | Classes\categories\ua_category.v | category_algebra | 2,543 |
Type0 := | bzero : | double1 : -> | double2 : -> . | Inductive | Require Import Require Import Require Import | Classes\implementations\binary_naturals.v | binnat | 2,544 |
(n : binnat) : binnat := match n with | bzero => double1 bzero | double1 n' => double2 n' | double2 n' => double1 ( n') end. | Fixpoint | Require Import Require Import Require Import | Classes\implementations\binary_naturals.v | Succ | 2,545 |
(n : binnat) : binnat := match n with | bzero => bzero | double1 n' => double2 ( n') | double2 n' => double2 (Succ ( n')) end. | Fixpoint | Require Import Require Import Require Import | Classes\implementations\binary_naturals.v | double | 2,546 |
(n : nat) : nat := match n with | O => O | S n' => S (S ( n')) end. | Fixpoint | Require Import Require Import Require Import | Classes\implementations\binary_naturals.v | Double | 2,547 |
(n : nat) : nat := S (Double n). | Definition | Require Import Require Import Require Import | Classes\implementations\binary_naturals.v | Double1 | 2,548 |
(n : nat) : nat := S (S (Double n)). | Definition | Require Import Require Import Require Import | Classes\implementations\binary_naturals.v | Double2 | 2,549 |
(n : nat) : binnat := match n with | O => bzero | S n' => Succ ( n') end. End basics. Section binary_equiv. Local unary' (n : binnat) : nat := match n with | bzero => O | double1 n' => Double1 (unary' n') | double2 n' => Double2 (unary' n') end. Local Definition succunary (n : binnat) : unary' (Succ n) = S (unary' n). Proof. induction n. - reflexivity. - reflexivity. - simpl. rewrite IHn. reflexivity. Qed. | Fixpoint | Require Import Require Import Require Import | Classes\implementations\binary_naturals.v | binary | 2,550 |
(n : nat) : binary (Double1 n) = double1 (binary n). Proof. induction n. - reflexivity. - change (binary (Double1 n.+1)) with (Succ (Succ (binary (Double n).+1))). rewrite IHn. reflexivity. Qed. | Definition | Require Import Require Import Require Import | Classes\implementations\binary_naturals.v | double1binary | 2,551 |
(n : nat) : binary (Double2 n) = double2 (binary n). Proof. induction n. - reflexivity. - change (binary (Double2 n.+1)) with (Succ (Succ (binary (Double n).+2))). rewrite IHn. reflexivity. Qed. | Definition | Require Import Require Import Require Import | Classes\implementations\binary_naturals.v | double2binary | 2,552 |
nat <~> binnat := Build_Equiv _ _ binary isequiv_binary. | Definition | Require Import Require Import Require Import | Classes\implementations\binary_naturals.v | equiv_binary | 2,553 |
(n : nat) : binary n.+1 = Succ (binary n). Proof. reflexivity. Qed. | Definition | Require Import Require Import Require Import | Classes\implementations\binary_naturals.v | binarysucc | 2,554 |
forall m, unary (Succ m) = S (unary m). Proof. equiv_intros binary n. rewrite <- binarysucc. rewrite eissect, eissect. reflexivity. Qed. | Definition | Require Import Require Import Require Import | Classes\implementations\binary_naturals.v | unarysucc | 2,555 |
forall (m n : binnat), (Succ m) + n = Succ (m + n). Proof. induction m; induction n; try reflexivity; simpl; rewrite <- IHm; done. Qed. | Definition | Require Import Require Import Require Import | Classes\implementations\binary_naturals.v | binnatplussucc | 2,556 |
(m n : nat) : binary m + binary n = binary (m + n). Proof. induction m; induction n; try reflexivity. - simpl. rewrite binnatplussucc. apply ap. done. - simpl. rewrite <- IHm. rewrite binnatplussucc. done. Qed. | Definition | Require Import Require Import Require Import | Classes\implementations\binary_naturals.v | binaryplus | 2,557 |
(m n : binnat) : unary m + unary n = unary (m + n). Proof. etransitivity (unary (binary (_^-1 m + _^-1 n))). - apply ((eissect binary (unary m + unary n)) ^). - rewrite <- binaryplus. rewrite (eisretr binary m), (eisretr binary n). reflexivity. Qed. | Definition | Require Import Require Import Require Import | Classes\implementations\binary_naturals.v | unaryplus | 2,558 |
forall (m n : binnat), (Succ m) * n = n + (m * n). Proof. induction m. - intros n. change (bzero + n = n + bzero). apply commutativity. - intros n. simpl. change (double2 m * n) with ((m * n) + (m * n) + n + n). apply commutativity. - intros n. simpl. change (double1 (Succ m) * n) with ((Succ m) * n + (Succ m) * n + n). rewrite IHm. rewrite (commutativity n (double2 m * n)). rewrite (commutativity n (m * n)). rewrite <- (associativity (m * n) n (m * n + n)). rewrite (commutativity n (m * n + n)). rewrite (associativity (m * n) _ _). rewrite (associativity (m * n) (m * n) n). done. Qed. | Definition | Require Import Require Import Require Import | Classes\implementations\binary_naturals.v | binnatmultsucc | 2,559 |
(m n : nat) : binary m * binary n = binary (m * n). Proof. induction m; induction n; try reflexivity; rewrite binnatmultsucc, IHm, binaryplus; done. Qed. | Definition | Require Import Require Import Require Import | Classes\implementations\binary_naturals.v | binarymult | 2,560 |
(m n : binnat) : unary m * unary n = unary (m * n). Proof. etransitivity (unary (binary (_^-1 m * _^-1 n))). - apply ((eissect binary (unary m * unary n)) ^). - rewrite <- binarymult. rewrite (eisretr binary m), (eisretr binary n). reflexivity. Qed. | Definition | Require Import Require Import Require Import | Classes\implementations\binary_naturals.v | unarymult | 2,561 |
NaturalsToSemiRing nat := fun _ _ _ _ _ _ => fix f (n: nat) := match n with | 0%nat => 0 | S n' => 1 + f n' end. | Definition | Require Import Require Import Require Import | Classes\implementations\binary_naturals.v | nat_to_semiring_helper | 2,562 |
(m : binnat) : toR (Succ m) = (toR m)+1. Proof. induction m. - change (2 * 0 + 1 = 0 + 1). rewrite mult_comm. rewrite mult_0_l. done. - change (2 * (toR m) + 2 = 2 * (toR m) + 1 + 1). apply plus_assoc. - induction m as [|m _|m _]. + change (2 * (2 * 0 + 1) + 1 = 2 * 0 + 2 + 1). rewrite plus_mult_distr_l. rewrite (@mult_1_r _ Aplus Amult Azero Aone H _). rewrite mult_0_r, mult_0_r. reflexivity. + change (2 * (2 * (toR m) + 2) + 1 = 2 * (2 * (toR m) + 1 ) + 2 + 1). apply (ap (fun z => z + 1)). assert (L : 2 * toR m + 2 = 2 * toR m + 1 + 1) by by rewrite plus_assoc. rewrite L; clear L. rewrite plus_mult_distr_l. rewrite mult_1_r. reflexivity. + simpl in IHm. change ((2 * (toR (double1 (Succ m))) + 1 = 2 * (toR (double2 m)) + 2 + 1)). rewrite IHm; clear IHm. rewrite plus_mult_distr_l. rewrite mult_1_r. reflexivity. Qed. | Definition | Require Import Require Import Require Import | Classes\implementations\binary_naturals.v | f_suc | 2,563 |
forall m : binnat, toR m = toR_vianat m. Proof. equiv_intro binary n. induction n as [|n IHn]. - reflexivity. - induction n as [|n _]. + change ((1 + 1) * 0 + 1 = 1). rewrite mult_0_r. apply plus_0_l. + rewrite f_suc. rewrite IHn. assert (L : (toR_fromnat β binary^-1) (binary n.+1) + 1 = toR_fromnat ((binary^-1 (binary n.+1)).+1)%nat). { simpl rewrite (plus_comm _ 1). simpl rewrite unarysucc. reflexivity. } rewrite L; clear L. rewrite <- unarysucc. rewrite <- binarysucc. reflexivity. Qed. | Definition | Require Import Require Import Require Import | Classes\implementations\binary_naturals.v | f_nat | 2,564 |
(h : binnat -> R) `{!IsSemiRingPreserving h} : forall x, toR x = h x. Proof. equiv_intro binary n. rewrite f_nat; unfold Compose. rewrite eissect. refine (toR_unique (h β binary) n). Qed. | Lemma | Require Import Require Import Require Import | Classes\implementations\binary_naturals.v | binnat_toR_unique | 2,565 |
(z:binnat) : LeftCancellation plus z. Proof. intros x y p. apply (equiv_inj unary). apply (ap unary) in p. rewrite <- unaryplus, <- unaryplus in p. exact (left_cancellation _ _ _ _ p). Qed. | Instance | Require Import Require Import Require Import | Classes\implementations\binary_naturals.v | binnat_plus_cancel_l | 2,566 |
(z : binnat): PropHolds (z <> 0) -> LeftCancellation (.*.) z. Proof. intros E. hnf in E. assert (H : unary z <> unary 0). { intros q. apply (equiv_inj unary) in q. exact (E q). } intros x y p. apply (ap unary) in p. rewrite <- unarymult, <- unarymult in p. exact (equiv_inj unary (nat_mult_cancel_l (unary z) H _ _ p)). Qed. | Instance | Require Import Require Import Require Import | Classes\implementations\binary_naturals.v | binnat_mult_cancel_l | 2,567 |
Trichotomy (lt:Lt binnat). Proof. intros x y. pose (T := nat_trichotomy (unary x) (unary y)). destruct T as [l|[c|r]]. - left; assumption. - right; left. apply (equiv_inj unary); assumption. - right; right; assumption. Defined. | Instance | Require Import Require Import Require Import | Classes\implementations\binary_naturals.v | binnat_trichotomy | 2,568 |
(F : I β Type) : list I β Type := fold_right (Ξ» (i:I) (A:Type), F i * A) Unit. | Definition | Require Import | Classes\implementations\family_prod.v | FamilyProd | 2,569 |
{F G : I β Type} {β : list I} (f : β i, F i β G i) : FamilyProd F β β FamilyProd G β := match β with | nil => const_tt _ | i :: β' => Ξ» '(x,s), (f i x, f s) end. | Fixpoint | Require Import | Classes\implementations\family_prod.v | map_family_prod | 2,570 |
(F : I β Type) {β : list I} (P : β i, F i -> Type) : FamilyProd F β β Type := match β with | nil => Ξ» _, Unit | i :: _ => Ξ» '(x,s), P i x β§ F P s end. | Fixpoint | Require Import | Classes\implementations\family_prod.v | for_all_family_prod | 2,571 |
(F G : I β Type) {β : list I} (R : β i, F i -> G i -> Type) : FamilyProd F β β FamilyProd G β β Type := match β with | nil => Ξ» _ _, Unit | i :: _ => Ξ» '(x,s) '(y,t), R i x y β§ F G R s t end. | Fixpoint | Require Import | Classes\implementations\family_prod.v | for_all_2_family_prod | 2,572 |
(F : I β Type) (R : β i, Relation (F i)) `{!β i, Reflexive (R i)} {β : list I} (s : FamilyProd F β) : for_all_2_family_prod F F R s s. | Lemma | Require Import | Classes\implementations\family_prod.v | reflexive_for_all_2_family_prod | 2,573 |
Frac@{} : Type | Record | Require Import HoTT.HIT.quotient. Require Import Import Frac. | Classes\implementations\field_of_fractions.v | Frac@ | 2,574 |
Frac_ishset' : IsHSet Frac. Proof. assert (E : sig (fun n : R => sig (fun d : R => d <> 0 )) <~> Frac). - issig. - apply (istrunc_equiv_istrunc _ E). Qed. | Lemma | Require Import HoTT.HIT.quotient. Require Import Import Frac. | Classes\implementations\field_of_fractions.v | Frac_ishset' | 2,575 |
pl@{} : Plus Frac. Proof. intros q r; refine (frac (num q * den r + num r * den q) (den q * den r) _). Defined. | Instance | Require Import HoTT.HIT.quotient. Require Import Import Frac. | Classes\implementations\field_of_fractions.v | pl@ | 2,576 |
equiv@{} := fun x y => num x * den y = num y * den x. Global Instance equiv_equiv_rel@{} : EquivRel equiv. Proof. split. - intros x. hnf. reflexivity. - intros x y. unfold equiv. apply symmetry. - intros x y z. unfold equiv. intros E1 E2. apply (mult_left_cancel (den y)). + solve_propholds. + rewrite !mult_assoc, !(mult_comm (den y)). rewrite E1, <-E2. rewrite <-!mult_assoc. rewrite (mult_comm (den x)). reflexivity. Qed. | Definition | Require Import HoTT.HIT.quotient. Require Import Import Frac. | Classes\implementations\field_of_fractions.v | equiv@ | 2,577 |
pl_respect@{} : forall q1 q2, equiv q1 q2 -> forall r1 r2, equiv r1 r2 -> equiv (q1 + r1) (q2 + r2). Proof. unfold equiv;intros q1 q2 Eq r1 r2 Er. simpl. rewrite plus_mult_distr_r. rewrite <-(associativity (num q1) (den r1)). rewrite (associativity (den r1)), (mult_comm (den r1)), <-(associativity (den q2)). rewrite (associativity (num q1)), Eq. rewrite (mult_comm (den q2)), <-(associativity (num r1)), (associativity (den q1)). rewrite (mult_comm (den q1)), <-(associativity (den r2)), (associativity (num r1)). rewrite Er. rewrite (mult_comm (den r1)), <-(associativity (num q2)), (associativity (den q1)). rewrite (mult_comm (den q1)), <-(associativity (den r2)), (associativity (num q2)). rewrite <-(associativity (num r2)), (associativity (den r1)), (mult_comm _ (den q2)). rewrite (mult_comm (den r1)), (associativity (num r2)). apply symmetry;apply plus_mult_distr_r. Qed. | Lemma | Require Import HoTT.HIT.quotient. Require Import Import Frac. | Classes\implementations\field_of_fractions.v | pl_respect@ | 2,578 |
pl_comm@{} : forall q r, equiv (pl q r) (pl r q). Proof. intros q r;unfold equiv;simpl. rewrite (mult_comm (den r)), plus_comm. reflexivity. Qed. | Lemma | Require Import HoTT.HIT.quotient. Require Import Import Frac. | Classes\implementations\field_of_fractions.v | pl_comm@ | 2,579 |
pl_assoc@{} : forall q r t, equiv (pl q (pl r t)) (pl (pl q r) t). Proof. intros;unfold equiv;simpl. apply ap011;[|apply symmetry,associativity]. rewrite plus_mult_distr_r. rewrite (plus_mult_distr_r _ _ (den t)). rewrite plus_assoc. apply ap011;[apply ap011|]. - apply associativity. - rewrite <-(associativity (num r)), <-(associativity (num r) (den q)). rewrite (mult_comm (den t)). reflexivity. - rewrite (mult_comm (den q));apply symmetry,associativity. Qed. | Lemma | Require Import HoTT.HIT.quotient. Require Import Import Frac. | Classes\implementations\field_of_fractions.v | pl_assoc@ | 2,580 |
ml@{} : Mult Frac. Proof. intros q r; refine (frac (num q * num r) (den q * den r) _). Defined. | Instance | Require Import HoTT.HIT.quotient. Require Import Import Frac. | Classes\implementations\field_of_fractions.v | ml@ | 2,581 |
ml_respect@{} : forall q1 q2, equiv q1 q2 -> forall r1 r2, equiv r1 r2 -> equiv (q1 * r1) (q2 * r2). Proof. unfold equiv;intros q1 q2 Eq r1 r2 Er. simpl. rewrite <-(associativity (num q1)), (associativity (num r1)). rewrite (mult_comm (num r1)), <-(associativity (den q2)), (associativity (num q1)). rewrite Eq, Er. rewrite <-(associativity (num q2)), (associativity (den q1)), (mult_comm (den q1)). rewrite <-(simple_associativity (num r2)), <-(simple_associativity (num q2)). reflexivity. Qed. | Lemma | Require Import HoTT.HIT.quotient. Require Import Import Frac. | Classes\implementations\field_of_fractions.v | ml_respect@ | 2,582 |
neg@{} : Negate Frac. Proof. intros q;refine (frac (- num q) (den q) _). Defined. | Instance | Require Import HoTT.HIT.quotient. Require Import Import Frac. | Classes\implementations\field_of_fractions.v | neg@ | 2,583 |
neg_respect@{} : forall q r, equiv q r -> equiv (- q) (- r). Proof. unfold equiv;simpl;intros q r E. rewrite <-2!negate_mult_distr_l. rewrite E;reflexivity. Qed. | Lemma | Require Import HoTT.HIT.quotient. Require Import Import Frac. | Classes\implementations\field_of_fractions.v | neg_respect@ | 2,584 |
nonzero_num@{} x : ~ (equiv x 0) <-> num x <> 0. Proof. split; intros E F; apply E. - red. rewrite F. simpl. rewrite 2!mult_0_l. reflexivity. - red in F;simpl in F. rewrite mult_1_r, mult_0_l in F. trivial. Qed. | Lemma | Require Import HoTT.HIT.quotient. Require Import Import Frac. | Classes\implementations\field_of_fractions.v | nonzero_num@ | 2,585 |
pl_0_l@{} x : equiv (0 + x) x. Proof. red;simpl. rewrite mult_1_r, mult_0_l, mult_1_l, plus_0_l. reflexivity. Qed. | Lemma | Require Import HoTT.HIT.quotient. Require Import Import Frac. | Classes\implementations\field_of_fractions.v | pl_0_l@ | 2,586 |
pl_0_r@{} x : equiv (x + 0) x. Proof. red;simpl. rewrite 2!mult_1_r, mult_0_l, plus_0_r. reflexivity. Qed. | Lemma | Require Import HoTT.HIT.quotient. Require Import Import Frac. | Classes\implementations\field_of_fractions.v | pl_0_r@ | 2,587 |
pl_neg_l@{} x : equiv (- x + x) 0. Proof. red;simpl. rewrite mult_1_r, mult_0_l. rewrite <-plus_mult_distr_r. rewrite plus_negate_l. apply mult_0_l. Qed. | Lemma | Require Import HoTT.HIT.quotient. Require Import Import Frac. | Classes\implementations\field_of_fractions.v | pl_neg_l@ | 2,588 |
ml_assoc@{} q r t : equiv (ml q (ml r t)) (ml (ml q r) t). Proof. red;simpl. rewrite (associativity (num q)), (associativity (den q)). reflexivity. Qed. | Lemma | Require Import HoTT.HIT.quotient. Require Import Import Frac. | Classes\implementations\field_of_fractions.v | ml_assoc@ | 2,589 |
dec_rec@{} : DecRecip Frac := fun x => match decide_rel (=) (num x) 0 with | inl _ => 0 | inr P => frac (den x) (num x) P end. | Instance | Require Import HoTT.HIT.quotient. Require Import Import Frac. | Classes\implementations\field_of_fractions.v | dec_rec@ | 2,590 |
dec_recip_respect@{} : forall q r, equiv q r -> equiv (/ q) (/ r). Proof. unfold equiv,dec_recip,dec_rec;intros q r E;simpl. destruct (decide_rel paths (num q) 0) as [E1|E1], (decide_rel paths (num r) 0) as [E2|E2];simpl. - trivial. - rewrite E1 in E;rewrite mult_0_l in E. destruct E2. apply (right_cancellation_ne_0 mult (den q));try solve_propholds. rewrite mult_0_l;apply symmetry,E. - rewrite E2 in E;rewrite mult_0_l in E. destruct E1. apply (right_cancellation_ne_0 mult (den r));try solve_propholds. rewrite mult_0_l;trivial. - rewrite (mult_comm (den q)), (mult_comm (den r)). apply symmetry, E. Qed. | Lemma | Require Import HoTT.HIT.quotient. Require Import Import Frac. | Classes\implementations\field_of_fractions.v | dec_recip_respect@ | 2,591 |
(x : Frac R1) : Frac R2. Proof. apply (frac (f (num x)) (f (den x))). apply isinjective_ne_0. apply (den_ne_0 x). Defined. | Definition | Require Import HoTT.HIT.quotient. Require Import Import Frac. | Classes\implementations\field_of_fractions.v | lift | 2,592 |
forall q r, equiv q r -> equiv (lift q) (lift r). Proof. unfold equiv;simpl;intros q r E. rewrite <-2!preserves_mult. apply ap,E. Qed. | Lemma | Require Import HoTT.HIT.quotient. Require Import Import Frac. | Classes\implementations\field_of_fractions.v | lift_respects | 2,593 |
F@{} := quotient equiv. | Definition | Require Import HoTT.HIT.quotient. Require Import Import Frac. | Classes\implementations\field_of_fractions.v | F@ | 2,594 |
path@{} {x y} : equiv x y -> ' x = ' y := related_classes_eq _. | Definition | Require Import HoTT.HIT.quotient. Require Import Import Frac. | Classes\implementations\field_of_fractions.v | path@ | 2,595 |
F_rect@{i} (P : F -> Type@{i}) {sP : forall x, IsHSet (P x)} (dclass : forall x : Frac R, P (' x)) (dequiv : forall x y E, (path E) # (dclass x) = (dclass y)) : forall q, P q := quotient_ind equiv P dclass dequiv. | Definition | Require Import HoTT.HIT.quotient. Require Import Import Frac. | Classes\implementations\field_of_fractions.v | F_rect@ | 2,596 |
P {sP} dclass dequiv x : @F_rect P sP dclass dequiv (' x) = dclass x := 1. | Definition | Require Import HoTT.HIT.quotient. Require Import Import Frac. | Classes\implementations\field_of_fractions.v | F_compute | 2,597 |
P {sP} dclass dequiv q r (E : equiv q r) : apD (@F_rect P sP dclass dequiv) (path E) = dequiv q r E := quotient_ind_compute_path _ _ _ _ _ _ _ _. | Definition | Require Import HoTT.HIT.quotient. Require Import Import Frac. | Classes\implementations\field_of_fractions.v | F_compute_path | 2,598 |
F_ind@{i} (P : F -> Type@{i}) {sP : forall x, IsHProp (P x)} (dclass : forall x : Frac R, P (' x)) : forall x, P x. Proof. apply (@F_rect P (fun _ => istrunc_hprop) dclass). intros;apply path_ishprop. Qed. | Definition | Require Import HoTT.HIT.quotient. Require Import Import Frac. | Classes\implementations\field_of_fractions.v | F_ind@ | 2,599 |