Datasets:

phanerozoic

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Coq-HoTT

like 2

{n A} {x y : A} : Tr n (x = y) -> (tr x = tr y :> Tr n.

Definition

Require Import Basics Types. Require Import TruncType. Require Import Truncations.Core Modalities.Modality Modalities.Descent.

Truncations\SeparatedTrunc.v

path_Tr

`{Univalence} {n} {A : Type} (x y : A) : Tr n (x = y) <~> (tr x = tr y :> Tr n.

Definition

Require Import Basics Types. Require Import TruncType. Require Import Truncations.Core Modalities.Modality Modalities.Descent.

Truncations\SeparatedTrunc.v

equiv_path_Tr

arrow@{u u0} (A : Type@{u}) (B : Type@{u0}) := A -> B.

Definition

Require Import Basics.Overture Basics.PathGroupoids Basics.Decidable Require Import Types.Forall.

Types\Arrow.v

arrow@

{A B : Type} (f g : A -> B) : (f == g) -> (f = g) := path_forall f g.

Definition

Require Import Basics.Overture Basics.PathGroupoids Basics.Decidable Require Import Types.Forall.

Types\Arrow.v

path_arrow

{A B : Type} (f g : A -> B) (h : f == g) : ap10 (path_arrow f g h) == h := apD10_path_forall f g h.

Definition

Require Import Basics.Overture Basics.PathGroupoids Basics.Decidable Require Import Types.Forall.

Types\Arrow.v

ap10_path_arrow

{A B : Type} (f g : A -> B) (h : f == g) : apD10 (path_arrow f g h) == h := apD10_path_forall f g h.

Definition

Require Import Basics.Overture Basics.PathGroupoids Basics.Decidable Require Import Types.Forall.

Types\Arrow.v

apD10_path_arrow

{A B : Type} (f g : A -> B) (h : f == g) : ap10 (path_forall f g h) == h := apD10_path_forall f g h.

Definition

Require Import Basics.Overture Basics.PathGroupoids Basics.Decidable Require Import Types.Forall.

Types\Arrow.v

ap10_path_forall

{A B : Type} (f g : A -> B) (p : f = g) : path_arrow f g (ap10 p) = p := eta_path_forall f g p.

Definition

Require Import Basics.Overture Basics.PathGroupoids Basics.Decidable Require Import Types.Forall.

Types\Arrow.v

eta_path_arrow

{A B : Type} (f : A -> B) : (path_arrow f f (fun x => 1)) = 1 := eta_path_arrow f f 1.

Definition

Require Import Basics.Overture Basics.PathGroupoids Basics.Decidable Require Import Types.Forall.

Types\Arrow.v

path_arrow_1

{A B : Type} f g : (f = g) <~> (f == g) := Build_Equiv _ _ (@ap10 A B f g) _.

Definition

Require Import Basics.Overture Basics.PathGroupoids Basics.Decidable Require Import Types.Forall.

Types\Arrow.v

equiv_ap10

{A B : Type} (f g : A -> B) : (f == g) <~> (f = g) := equiv_path_forall f g.

Definition

Require Import Basics.Overture Basics.PathGroupoids Basics.Decidable Require Import Types.Forall.

Types\Arrow.v

equiv_path_arrow

{X Y Z: Type} (f g : X -> Y -> Z) : (forall x y, f x y = g x y) <~> f = g. Proof. refine (equiv_path_arrow _ _ oE _). apply equiv_functor_forall_id; intro x. apply equiv_path_arrow. Defined.

Definition

Require Import Basics.Overture Basics.PathGroupoids Basics.Decidable Require Import Types.Forall.

Types\Arrow.v

equiv_path_arrow2

{X Y Z : Type} {f g : X -> Y -> Z} (h : forall x y, f x y = g x y) (x : X) (y : Y) : ap100 (equiv_path_arrow2 f g h) x y = h x y. Proof. unfold ap100. refine (ap (fun p => ap10 p y) _ @ _). 1: apply apD10_path_arrow. cbn. apply apD10_path_arrow. Defined.

Definition

Require Import Basics.Overture Basics.PathGroupoids Basics.Decidable Require Import Types.Forall.

Types\Arrow.v

ap100_path_arrow2

{A B : Type} (f g h : A -> B) (p : f == g) (q : g == h) : path_arrow f h (fun x => p x @ q x) = path_arrow f g p @ path_arrow g h q := path_forall_pp f g h p q.

Definition

Require Import Basics.Overture Basics.PathGroupoids Basics.Decidable Require Import Types.Forall.

Types\Arrow.v

path_arrow_pp

{A : Type} {B C : A -> Type} {x1 x2 : A} (p : x1 = x2) (f : B x1 -> C x1) (y : B x2) : (transport (fun x => B x -> C x) p f) y = p # (f (p^ # y)). Proof. destruct p; simpl; auto. Defined.

Definition

Require Import Basics.Overture Basics.PathGroupoids Basics.Decidable Require Import Types.Forall.

Types\Arrow.v

transport_arrow

transport_arrow' {A : Type} {B C : A -> Type} {x1 x2 : A} (p : x1 = x2) (f : B x1 -> C x1) : transport (fun x => B x -> C x) p f = transport _ p o f o transport _ p^. Proof. destruct p; auto. Defined.

Definition

Require Import Basics.Overture Basics.PathGroupoids Basics.Decidable Require Import Types.Forall.

Types\Arrow.v

transport_arrow'

{A : Type} {B : A -> Type} {C : Type} {x1 x2 : A} (p : x1 = x2) (f : B x1 -> C) (y : B x2) : (transport (fun x => B x -> C) p f) y = f (p^ # y). Proof. destruct p; simpl; auto. Defined.

Definition

Require Import Basics.Overture Basics.PathGroupoids Basics.Decidable Require Import Types.Forall.

Types\Arrow.v

transport_arrow_toconst

transport_arrow_toconst' {A : Type} {B : A -> Type} {C : Type} {x1 x2 : A} (p : x1 = x2) (f : B x1 -> C) : transport (fun x => B x -> C) p f = f o transport B p^. Proof. destruct p; auto. Defined.

Definition

Require Import Basics.Overture Basics.PathGroupoids Basics.Decidable Require Import Types.Forall.

Types\Arrow.v

transport_arrow_toconst'

{A B : Type} {C : A -> Type} {x1 x2 : A} (p : x1 = x2) (f : B -> C x1) (y : B) : (transport (fun x => B -> C x) p f) y = p # (f y). Proof. destruct p; simpl; auto. Defined.

Definition

Require Import Basics.Overture Basics.PathGroupoids Basics.Decidable Require Import Types.Forall.

Types\Arrow.v

transport_arrow_fromconst

{A : Type} {B : A -> Type} {C : Type} {x1 x2 : A} (p : x1 = x2) (f : B x1 -> C) {y1 y2 : B x2} (q : y1 = y2) : ap (transport (fun x => B x -> C) p f) q @ transport_arrow_toconst p f y2 = transport_arrow_toconst p f y1 @ ap (fun y => f (p^ # y)) q. Proof. destruct p, q; reflexivity. Defined.

Definition

Require Import Basics.Overture Basics.PathGroupoids Basics.Decidable Require Import Types.Forall.

Types\Arrow.v

ap_transport_arrow_toconst

{A:Type} (B C : A -> Type) {x1 x2:A} (p:x1=x2) (f : B x1 -> C x1) (g : B x2 -> C x2) : (forall (y1:B x1), transport C p (f y1) = g (transport B p y1)) <~> (transport (fun x => B x -> C x) p f = g). Proof. destruct p. apply equiv_path_arrow. Defined.

Definition

Require Import Basics.Overture Basics.PathGroupoids Basics.Decidable Require Import Types.Forall.

Types\Arrow.v

dpath_arrow

{A : Type} (B C : A -> Type) {x1 x2 : A} (p : x1 = x2) (f : B x1 -> C x1) (g : B x2 -> C x2) (h : forall (y1 : B x1), transport C p (f y1) = g (transport B p y1)) (u : B x1) : ap10 (dpath_arrow B C p f g h) (p # u) = transport_arrow p f (p # u) @ ap (fun x => p # (f x)) (transport_Vp B p u) @ h u. Proof. destruct p; simpl; unfold ap10. exact (apD10_path_forall f g h u @ (concat_1p _)^). Defined.

Definition

Require Import Basics.Overture Basics.PathGroupoids Basics.Decidable Require Import Types.Forall.

Types\Arrow.v

ap10_dpath_arrow

{A B : Type} {x y : A -> B} (p : x = y) (z : A) : ap (fun f => f z) p = ap10 p z := 1.

Definition

Require Import Basics.Overture Basics.PathGroupoids Basics.Decidable Require Import Types.Forall.

Types\Arrow.v

ap_apply_l

{A B C : Type} {x y : A} (p : x = y) (M : A -> B -> C) (z : B) : ap (fun a => (M a) z) p = ap10 (ap M p) z := match p with 1 => 1 end.

Definition

Require Import Basics.Overture Basics.PathGroupoids Basics.Decidable Require Import Types.Forall.

Types\Arrow.v

ap_apply_Fl

{A B C : Type} {x y : A} (p : x = y) (z : B -> C) (N : A -> B) : ap (fun a => z (N a)) p = ap01 z (ap N p) := (ap_compose N _ _).

Definition

Require Import Basics.Overture Basics.PathGroupoids Basics.Decidable Require Import Types.Forall.

Types\Arrow.v

ap_apply_Fr

{A B C : Type} {x y : A} (p : x = y) (M : A -> B -> C) (N : A -> B) : ap (fun a => (M a) (N a)) p = ap11 (ap M p) (ap N p) := match p with 1 => 1 end.

Definition

Require Import Basics.Overture Basics.PathGroupoids Basics.Decidable Require Import Types.Forall.

Types\Arrow.v

ap_apply_FlFr

{A B C : Type} {x y : A} (p : x = y) (M : A -> B -> C) : ap (fun a b => M a b) p = path_arrow _ _ (fun b => ap (fun a => M a b) p). Proof. destruct p; symmetry; simpl; apply path_arrow_1. Defined.

Definition

Require Import Basics.Overture Basics.PathGroupoids Basics.Decidable Require Import Types.Forall.

Types\Arrow.v

ap_lambda

`(f : B -> A) `(g : C -> D) : (A -> C) -> (B -> D) := @functor_forall A (fun _ => C) B (fun _ => D) f (fun _ => g).

Definition

Require Import Basics.Overture Basics.PathGroupoids Basics.Decidable Require Import Types.Forall.

Types\Arrow.v

functor_arrow

`(f : B -> A) : not A -> not B := functor_arrow f idmap.

Definition

Require Import Basics.Overture Basics.PathGroupoids Basics.Decidable Require Import Types.Forall.

Types\Arrow.v

not_contrapositive

iff_not@{u v k | u <= k, v <= k} (A : Type@{u}) (B : Type@{v}) : A <-> B -> iff@{u v k} (~A) (~B). Proof. intros e; split; apply not_contrapositive@{_ k}, e. Defined.

Definition

Require Import Basics.Overture Basics.PathGroupoids Basics.Decidable Require Import Types.Forall.

Types\Arrow.v

iff_not@

`(f : B -> A) `(g : C -> D) (h h' : A -> C) (p : h == h') : ap (functor_arrow f g) (path_arrow _ _ p) = path_arrow _ _ (fun b => ap g (p (f b))) := @ap_functor_forall _ A (fun _ => C) B (fun _ => D) f (fun _ => g) h h' p.

Definition

Require Import Basics.Overture Basics.PathGroupoids Basics.Decidable Require Import Types.Forall.

Types\Arrow.v

ap_functor_arrow

(A B : Type) `{Contr A} : (A -> B) <~> B. Proof. srapply (equiv_adjointify (fun f => f (center A)) const). - reflexivity. - intro f; funext a; unfold const; simpl. apply (ap f), contr. Defined.

Definition

Require Import Basics.Overture Basics.PathGroupoids Basics.Decidable Require Import Types.Forall.

Types\Arrow.v

equiv_arrow_from_contr

`{IsEquiv B A f} `{IsEquiv C D g} : (A -> C) <~> (B -> D) := @equiv_functor_forall _ A (fun _ => C) B (fun _ => D) f _ (fun _ => g) _.

Definition

Require Import Basics.Overture Basics.PathGroupoids Basics.Decidable Require Import Types.Forall.

Types\Arrow.v

equiv_functor_arrow

equiv_functor_arrow' `(f : B <~> A) `(g : C <~> D) : (A -> C) <~> (B -> D) := @equiv_functor_forall' _ A (fun _ => C) B (fun _ => D) f (fun _ => g). End AssumeFunext. Global Instance decidable_arrow {A B : Type} `{Decidable A} `{Decidable B} : Decidable (A -> B). Proof. destruct (dec B) as [x2|y2]. - exact (inl (fun _ => x2)). - destruct (dec A) as [x1|y1]. + apply inr; intros f. exact (y2 (f x1)). + apply inl; intros x1. elim (y1 x1). Defined.

Definition

Require Import Basics.Overture Basics.PathGroupoids Basics.Decidable Require Import Types.Forall.

Types\Arrow.v

equiv_functor_arrow'

Type0 := | true : | false : .

Inductive

Require Import HoTT.Basics. Require Import Types.Equiv.

Types\Bool.v

Bool

Bool_ind.

Definition

Require Import HoTT.Basics. Require Import Types.Equiv.

Types\Bool.v

Bool_rect

(b1 b2 : Bool) : Bool := if b1 then b2 else false.

Definition

Require Import HoTT.Basics. Require Import Types.Equiv.

Types\Bool.v

andb

(b1 b2 : Bool) : Bool := if b1 then true else b2.

Definition

Require Import HoTT.Basics. Require Import Types.Equiv.

Types\Bool.v

orb

(b : Bool) := if b then false else true.

Definition

Require Import HoTT.Basics. Require Import Types.Equiv.

Types\Bool.v

negb

(b1 b2 : Bool) : Bool := if b1 then b2 else true.

Definition

Require Import HoTT.Basics. Require Import Types.Equiv.

Types\Bool.v

implb

{b} : implb b true = true := if b as b return implb b true = true then idpath else idpath.

Definition

Require Import HoTT.Basics. Require Import Types.Equiv.

Types\Bool.v

implb_true

{a b} : (a -> b)%Bool = true <-> (a = true -> b = true). Proof. destruct a; simpl; split; trivial using idpath with nocore; destruct b; simpl; auto using idpath with nocore. Defined.

Definition

Require Import HoTT.Basics. Require Import Types.Equiv.

Types\Bool.v

implb_impl

~ (false = true) := fun H => match H in (_ = y) return (if y return Set then Empty else Bool) with | 1%path => true end.

Definition

Require Import HoTT.Basics. Require Import Types.Equiv.

Types\Bool.v

false_ne_true

~ (true = false) := fun H => false_ne_true (symmetry _ _ H).

Definition

Require Import HoTT.Basics. Require Import Types.Equiv.

Types\Bool.v

true_ne_false

IsHSet Bool := _.

Definition

Require Import HoTT.Basics. Require Import Types.Equiv.

Types\Bool.v

hset_bool

(b : Bool) : negb b <> b := match b return negb b <> b with | true => false_ne_true | false => true_ne_false end.

Definition

Require Import HoTT.Basics. Require Import Types.Equiv.

Types\Bool.v

not_fixed_negb

{b1 b2 : Bool} : (b1 <> b2) -> (b1 = negb b2). Proof. destruct b1, b2. - intros oops; case (oops idpath). - reflexivity. - reflexivity. - intros oops; case (oops idpath). Defined.

Definition

Require Import HoTT.Basics. Require Import Types.Equiv.

Types\Bool.v

negb_ne

negb_ne' {b1 b2 : Bool} : (b1 <> b2) -> (negb b1 = b2). Proof. intros oops. symmetry. apply negb_ne. exact (symmetric_neq oops). Defined.

Definition

Require Import HoTT.Basics. Require Import Types.Equiv.

Types\Bool.v

negb_ne'

`{Funext} : (forall b, P b) <~> P false * P true. Proof. apply (equiv_adjointify f g); repeat (reflexivity || intros [] || intro || apply path_forall). Defined.

Definition

Require Import HoTT.Basics. Require Import Types.Equiv.

Types\Bool.v

equiv_bool_forall_prod

`{Funext} (P : Type) : P * P <~> (Bool -> P) := (equiv_bool_forall_prod (fun _ => P))^-1%equiv. Global Instance isequiv_negb : IsEquiv negb. Proof. refine (@Build_IsEquiv _ _ negb negb (fun b => if b as b return negb (negb b) = b then idpath else idpath) (fun b => if b as b return negb (negb b) = b then idpath else idpath) _). intros []; simpl; exact idpath. Defined.

Definition

Require Import HoTT.Basics. Require Import Types.Equiv.

Types\Bool.v

equiv_bool_rec_uncurried

Bool <~> Bool := Build_Equiv Bool Bool negb _.

Definition

Require Import HoTT.Basics. Require Import Types.Equiv.

Types\Bool.v

equiv_negb

(f : Bool -> Bool) `{IsEquiv Bool Bool f} : f false = negb (f true). Proof. pose proof (eissect f true). pose proof (eissect f false). simpl in *. destruct (f true), (f false). - etransitivity; try (eassumption || (symmetry; eassumption)). - simpl. reflexivity. - simpl. reflexivity. - etransitivity; try (eassumption || (symmetry; eassumption)). Defined.

Lemma

Require Import HoTT.Basics. Require Import Types.Equiv.

Types\Bool.v

eval_bool_isequiv

(e : Bool <~> Bool) : e == g (f e). Proof. unfold f, g; clear f g; intros []; simpl. - destruct (e true); reflexivity. - refine (eval_bool_isequiv e @ _). destruct (e true); reflexivity. Defined.

Definition

Require Import HoTT.Basics. Require Import Types.Equiv.

Types\Bool.v

aut_bool_canonical

`{Funext} : Bool <~> (Bool <~> Bool). Proof. refine (equiv_adjointify g f _ _). - intro e. apply path_equiv, path_forall. intros b; symmetry; apply aut_bool_canonical. - intros []; reflexivity. Defined.

Lemma

Require Import HoTT.Basics. Require Import Types.Equiv.

Types\Bool.v

equiv_bool_aut_bool

`{Funext} (e : Bool <~> Bool) : (e = equiv_idmap Bool) + (e = equiv_negb). Proof. revert e. equiv_intro equiv_bool_aut_bool e. destruct e; simpl. - exact (inl idpath). - exact (inr idpath). Defined.

Definition

Require Import HoTT.Basics. Require Import Types.Equiv.

Types\Bool.v

aut_bool_idmap_or_negb

equiv_idmap Bool <> equiv_negb. Proof. intros oops. exact (true_ne_false (ap10_equiv oops true)). Defined.

Definition

Require Import HoTT.Basics. Require Import Types.Equiv.

Types\Bool.v

idmap_bool_ne_negb

(e1 e2 : Bool <~> Bool) : e1 o e2 == e2 o e1. Proof. intro b. refine (ap e1 (aut_bool_canonical e2 b) @ _). refine (aut_bool_canonical e1 _ @ _). refine (_ @ ap e2 (aut_bool_canonical e1 b)^). refine (_ @ (aut_bool_canonical e2 _)^). unfold f, g. destruct (e1 true), (e2 true), b; reflexivity. Defined.

Definition

Require Import HoTT.Basics. Require Import Types.Equiv.

Types\Bool.v

abelian_aut_bool

{T : Type} (falso: Empty) : T. Proof. case falso. Defined.

Lemma

Require Import Basics.Overture Basics.Equivalences Basics.Trunc.

Types\Empty.v

Empty_rec

equiv_empty_rec@{u} `{Funext} (A : Type@{u}) : Unit <~> ((Empty -> A) : Type@{u}) := (Build_Equiv@{Set u} _ _ (fun (_ : Unit) => @Empty_rec A) _). Global Instance istrunc_Empty@{} (n : trunc_index) : IsTrunc n.+1 Empty. Proof. refine (@istrunc_leq (-1) n.+1 tt _ _). apply istrunc_S. intros []. Defined.

Definition

Require Import Basics.Overture Basics.Equivalences Basics.Trunc.

Types\Empty.v

equiv_empty_rec@

{T : Type} (f : T -> Empty) : T <~> Empty := Build_Equiv T Empty f _.

Definition

Require Import Basics.Overture Basics.Equivalences Basics.Trunc.

Types\Empty.v

equiv_to_empty

{A B} (f : A -> B) `{IsTruncMap (-2) A B f} : IsEquiv f. Proof. srapply Build_IsEquiv. - intros b; exact (center {a : A & f a = b}).1. - intros b. exact (center {a : A & f a = b}).2. - intros a. exact (@contr {x : A & f x = f a} _ (a;1))..1. - intros a; cbn. apply moveL_M1. lhs_V nrapply transport_paths_l. lhs_V nrapply transport_compose. exact ((@contr {x : A & f x = f a} _ (a;1))..2). Defined.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths.

Types\Equiv.v

isequiv_contr_map

{A B} (f : A -> B) (feq : B -> IsEquiv f) : IsEquiv f. Proof. apply isequiv_contr_map. intros b. pose (feq b); exact _. Defined.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths.

Types\Equiv.v

isequiv_inhab_codomain

{A} (P Q : A -> Type) (g : forall a, P a -> Q a) `{!IsEquiv (functor_sigma idmap g)} : forall (a : A), IsEquiv (g a). Proof. intros a; apply isequiv_contr_map. apply istruncmap_from_functor_sigma. exact _. Defined.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths.

Types\Equiv.v

isequiv_from_functor_sigma

{A} (P Q : A -> Type) (g : forall a, P a -> Q a) : IsEquiv (functor_sigma idmap g) <-> forall a, IsEquiv (g a). Proof. split. - apply isequiv_from_functor_sigma. - intro K. apply isequiv_functor_sigma. Defined.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths.

Types\Equiv.v

equiv_total_iff_equiv_fiberwise

{A B} (f : A -> B) : IsTruncMap (-2) f <~> IsEquiv f. Proof. rapply equiv_iff_hprop. Defined.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths.

Types\Equiv.v

equiv_contr_map_isequiv

{A B : Type} (e1 e2 : A <~> B) : (e1 = e2 :> (A -> B)) <~> (e1 = e2 :> (A <~> B)). Proof. equiv_via ((issig_equiv A B) ^-1 e1 = (issig_equiv A B) ^-1 e2). 2: symmetry; apply equiv_ap; refine _. exact (equiv_path_sigma_hprop ((issig_equiv A B)^-1 e1) ((issig_equiv A B)^-1 e2)). Defined.

Lemma

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths.

Types\Equiv.v

equiv_path_equiv

{A B : Type} {e1 e2 : A <~> B} : (e1 = e2 :> (A -> B)) -> (e1 = e2 :> (A <~> B)) := equiv_path_equiv e1 e2. Global Instance isequiv_path_equiv {A B : Type} {e1 e2 : A <~> B} : IsEquiv (@ _ _ e1 e2) := equiv_isequiv (equiv_path_equiv e1 e2). Global Instance isequiv_ap_equiv_fun {A B : Type} (e1 e2 : A <~> B) : IsEquiv (ap (x:=e1) (y:=e2) (@equiv_fun A B)). Proof. snrapply isequiv_homotopic. - exact (equiv_path_equiv e1 e2)^-1%equiv. - exact _. - intro p. exact (ap_compose (fun v => (equiv_fun v; equiv_isequiv v)) pr1 p)^. Defined.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths.

Types\Equiv.v

path_equiv

{A B C D} (h : A <~> C) (k : B <~> D) : (A <~> B) -> (C <~> D) := fun f => ((k oE f) oE h^-1). Global Instance isequiv_functor_equiv {A B C D} (h : A <~> C) (k : B <~> D) : IsEquiv ( h k). Proof. refine (isequiv_adjointify _ ( (equiv_inverse h) (equiv_inverse k)) _ _). - intros f; apply path_equiv, path_arrow; intros x; simpl. exact (eisretr k _ @ ap f (eisretr h x)). - intros g; apply path_equiv, path_arrow; intros x; simpl. exact (eissect k _ @ ap g (eissect h x)). Defined.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths.

Types\Equiv.v

functor_equiv

{A B C D} (h : A <~> C) (k : B <~> D) : (A <~> B) <~> (C <~> D) := Build_Equiv _ _ (functor_equiv h k) _.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths.

Types\Equiv.v

equiv_functor_equiv

equiv_functor_precompose_equiv@{i j k u v | i <= u, j <= v, k <= u, k <= v} {X : Type@{i}} {Y : Type@{j}} (Z : Type@{k}) (e : X <~> Y) : Equiv@{v u} (Y <~> Z) (X <~> Z) := equiv_functor_equiv e^-1%equiv 1%equiv.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths.

Types\Equiv.v

equiv_functor_precompose_equiv@

equiv_functor_postcompose_equiv@{i j k u v | i <= u, j <= v, k <= u, k <= v} {X : Type@{i}} {Y : Type@{j}} (Z : Type@{k}) (e : X <~> Y) : Equiv@{u v} (Z <~> X) (Z <~> Y) := equiv_functor_equiv 1%equiv e. Global Instance isequiv_equiv_inverse {A B} : IsEquiv (@equiv_inverse A B). Proof. refine (isequiv_adjointify _ equiv_inverse _ _); intros e; apply path_equiv; reflexivity. Defined.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths.

Types\Equiv.v

equiv_functor_postcompose_equiv@

A B : (A <~> B) <~> (B <~> A) := Build_Equiv _ _ (@equiv_inverse A B) _.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths.

Types\Equiv.v

equiv_equiv_inverse

{A : Type} {B C : A -> Type} {x1 x2 : A} (p : x1 = x2) (f : B x1 <~> C x1) (y : B x2) : (transport (fun x => B x <~> C x) p f) y = p # (f (p^ # y)). Proof. destruct p; auto. Defined.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths.

Types\Equiv.v

transport_equiv

transport_equiv' {A : Type} {B C : A -> Type} {x1 x2 : A} (p : x1 = x2) (f : B x1 <~> C x1) : transport (fun x => B x <~> C x) p f = (equiv_transport _ p) oE f oE (equiv_transport _ p^) :> (B x2 -> C x2). Proof. destruct p; auto. Defined.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths.

Types\Equiv.v

transport_equiv'

transport_equiv'' `{Funext} {A : Type} {B C : A -> Type} {x1 x2 : A} (p : x1 = x2) (f : B x1 <~> C x1) : transport (fun x => B x <~> C x) p f = (equiv_transport _ p) oE f oE (equiv_transport _ p^). Proof. apply path_equiv. destruct p; auto. Defined.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths.

Types\Equiv.v

transport_equiv''

`{P : A -> Type} (f g : forall x, P x) (h : f == g) : apD10 (path_forall _ _ h) == h := apD10 (eisretr apD10 h).

Definition

Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids

Types\Forall.v

apD10_path_forall

`{P : A -> Type} (f g : forall x, P x) (p : f = g) : path_forall _ _ (apD10 p) = p := eissect apD10 p.

Definition

Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids

Types\Forall.v

eta_path_forall

`{P : A -> Type} (f : forall x, P x) : (path_forall f f (fun x => 1)) = 1 := eta_path_forall f f 1.

Definition

Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids

Types\Forall.v

path_forall_1

{A : Type} (P : A -> Type) f g : (f = g) <~> (f == g) := Build_Equiv _ _ (@apD10 A P f g) _.

Definition

Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids

Types\Forall.v

equiv_apD10

`{P : A -> Type} (f g : forall x, P x) : (f == g) <~> (f = g) := Build_Equiv _ _ (path_forall f g) _.

Definition

Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids

Types\Forall.v

equiv_path_forall

`{P : A -> Type} (f g h : forall x, P x) (p : f == g) (q : g == h) : path_forall f h (fun x => p x @ q x) = path_forall f g p @ path_forall g h q. Proof. revert p q. equiv_intro (@apD10 A P f g) p. equiv_intro (@apD10 A P g h) q. transitivity (path_forall f h (apD10 (p @ q))). - apply ap, path_forall; intros x. symmetry; apply apD10_pp. - refine (eta_path_forall _ _ _ @ _). apply concat2; symmetry; apply eta_path_forall. Defined.

Definition

Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids

Types\Forall.v

path_forall_pp

`{P : A -> Type} (f g : forall x, P x) (p : f == g) : path_forall _ _ (fun x => (p x)^) = (path_forall _ _ p)^. Proof. transitivity (path_forall _ _ (fun x => (apD10 (path_forall _ _ p) x)^)). - f_ap. symmetry. apply (@ap _ _ (fun h x => (h x)^)). apply eisretr. - transitivity (path_forall _ _ (apD10 (path_forall _ _ p)^)). + apply ap, inverse. apply path_forall; intros x. apply apD10_V. + apply eissect. Defined.

Definition

Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids

Types\Forall.v

path_forall_V

{A : Type} {P : A -> Type} {C : forall x, P x -> Type} {x1 x2 : A} (p : x1 = x2) (f : forall y : P x1, C x1 y) : (transport (fun x => forall y : P x, C x y) p f) == (fun y => transport (C x2) (transport_pV _ _ _) (transportD _ _ p _ (f (p^ # y)))) := match p with idpath => fun _ => 1 end.

Definition

Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids

Types\Forall.v

transport_forall

{A B : Type} {C : A -> B -> Type} {x1 x2 : A} (p : x1 = x2) (f : forall y : B, C x1 y) : (transport (fun x => forall y : B, C x y) p f) == (fun y => transport (fun x => C x y) p (f y)) := match p with idpath => fun _ => 1 end.

Definition

Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids

Types\Forall.v

transport_forall_constant

{A B : Type} (C : A -> B -> Type) {x1 x2 : A} (p : x1 = x2) (f : forall y : B, C x1 y) {y1 y2 : B} (q : y1 = y2) : apD (transport (fun x => forall y : B, C x y) p f) q = ap (transport (C x2) q) (transport_forall_constant p f y1) @ transport_transport C p q (f y1) @ ap (transport (fun x : A => C x y2) p) (apD f q) @ (transport_forall_constant p f y2)^. Proof. destruct p, q; reflexivity. Defined.

Definition

Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids

Types\Forall.v

apD_transport_forall_constant

{A} {B : A -> Type} {f g : forall x, B x} (p : f = g) (z : A) : ap (fun f => f z) p = apD10 p z := 1.

Definition

Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids

Types\Forall.v

ap_apply_lD

{A} {B : A -> Type} { C : forall x, B x -> Type} {f g : forall x y, C x y} (p : f = g) (z1 : A) (z2 : B z1) : ap (fun f => f z1 z2) p = apD10 (apD10 p z1) z2. Proof. by path_induction. Defined.

Definition

Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids

Types\Forall.v

ap_apply_lD2

{A B : Type} {C : B -> Type} {x y : A} (p : x = y) (M : forall a b, C b) : ap (fun a b => M a b) p = path_forall _ _ (fun b => ap (fun a => M a b) p). Proof. destruct p; symmetry; simpl; apply path_forall_1. Defined.

Definition

Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids

Types\Forall.v

ap_lambdaD

{A:Type} (B:A -> Type) (C:forall a, B a -> Type) (x1 x2:A) (p:x1=x2) (f:forall y1:B x1, C x1 y1) (g:forall (y2:B x2), C x2 y2) : (forall (y1:B x1), transportD B C p y1 (f y1) = g (transport B p y1)) <~> (transport (fun x => forall y:B x, C x y) p f = g). Proof. destruct p. apply equiv_path_forall. Defined.

Definition

Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids

Types\Forall.v

dpath_forall

{A B:Type} (C : A -> B -> Type) (x1 x2:A) (p:x1=x2) (f:forall (y1:B), C x1 y1) (g:forall (y2:B), C x2 y2) : (forall (y1:B), transport (fun x => C x y1) p (f y1) = g y1) <~> (transport (fun x => forall y:B, C x y) p f = g). Proof. destruct p. apply equiv_path_forall. Defined.

Definition

Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids

Types\Forall.v

dpath_forall_constant

`{P : A -> Type} `{Q : B -> Type} (f0 : B -> A) (f1 : forall b:B, P (f0 b) -> Q b) : (forall a:A, P a) -> (forall b:B, Q b) := (fun g b => f1 _ (g (f0 b))).

Definition

Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids

Types\Forall.v

functor_forall

`{P : A -> Type} `{Q : B -> Type} (f0 : B -> A) (f1 : forall b:B, P (f0 b) -> Q b) (g g' : forall a:A, P a) (h : g == g') : ap (functor_forall f0 f1) (path_forall _ _ h) = path_forall _ _ (fun b:B => (ap (f1 b) (h (f0 b)))). Proof. revert h. equiv_intro (@apD10 A P g g') h. destruct h. simpl. transitivity (idpath (functor_forall f0 f1 g)). - exact (ap (ap (functor_forall f0 f1)) (path_forall_1 g)). - symmetry. apply path_forall_1. Defined.

Definition

Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids

Types\Forall.v

ap_functor_forall

`{P : A -> Type} `{Q : B -> Type} `{R : C -> Type} (f0 : B -> A) (f1 : forall b:B, P (f0 b) -> Q b) (g0 : C -> B) (g1 : forall c:C, Q (g0 c) -> R c) (k : forall a, P a) : functor_forall g0 g1 (functor_forall f0 f1 k) == functor_forall (f0 o g0) (fun c => g1 c o f1 (g0 c)) k := fun a => 1.

Definition

Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids

Types\Forall.v

functor_forall_compose

`{P : A -> Type} `{Q : A -> Type} (f1 : forall a:A, P a -> Q a) : (forall a:A, P a) -> (forall a:A, Q a) := functor_forall idmap f1.

Definition

Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids

Types\Forall.v

functor_forall_id

{A B : Type} `{P : A -> Type} (f0 : B -> A) : (forall a:A, P a) -> (forall b:B, P (f0 b)) := functor_forall f0 (fun _ => idmap).

Definition

Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids

Types\Forall.v

functor_forall_pb

`{P : A -> Type} `{Q : B -> Type} (f0 : A -> B) `{!IsEquiv f0} (f1 : forall a:A, P a -> Q (f0 a)) : (forall a:A, P a) -> (forall b:B, Q b). Proof. nrapply (functor_forall f0^-1). intros b u. refine ((eisretr f0 b) # _). exact (f1 _ u). Defined.

Definition

Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids

Types\Forall.v

functor_forall_equiv

{A B : Type} `{Q : B -> Type} (f0 : A -> B) `{!IsEquiv f0} : (forall a:A, Q (f0 a)) -> (forall b:B, Q b) := functor_forall_equiv f0 (fun _ => idmap).

Definition

Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids

Types\Forall.v

functor_forall_equiv_pb

{A B : Type} {P : B -> Type} (f : A -> B) `{!IsEquiv f} (h : forall a, P (f a)) : forall a, functor_forall_equiv_pb f h (f a) = h a. Proof. intro a; srapply (_ @ apD h (eissect f a)); srapply (_ @ (transport_compose _ _ _ _)^). srapply ap10; apply ap; apply eisadj. Defined.

Definition

Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids

Types\Forall.v

functor_forall_equiv_pb_beta

`{P : A -> Type} `{Q : B -> Type} (f : B -> A) `{IsEquiv B A f} (g : forall b, P (f b) -> Q b) `{forall b, @IsEquiv (P (f b)) (Q b) (g b)} : (forall a, P a) <~> (forall b, Q b) := Build_Equiv _ _ (functor_forall f g) _.

Definition

Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids

Types\Forall.v

equiv_functor_forall

equiv_functor_forall' `{P : A -> Type} `{Q : B -> Type} (f : B <~> A) (g : forall b, P (f b) <~> Q b) : (forall a, P a) <~> (forall b, Q b) := equiv_functor_forall f g.

Definition

Require Import Basics.Overture Basics.Equivalences Basics.PathGroupoids

Types\Forall.v

equiv_functor_forall'