Datasets:

phanerozoic

/

Coq-HoTT

like 2

(A : Type) := @const A Unit tt.

Definition

Require Import Basics.Overture Basics.Equivalences.

Types\Unit.v

const_tt

(A B : Type@{u}) (p : A = B) : A <~> B := equiv_transport (fun X => X) p.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

equiv_path

`{Funext} (A B : Type) (p : A = B) : equiv_path B A (p^) = (equiv_path A B p)^-1%equiv. Proof. apply path_equiv. reflexivity. Defined.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

equiv_path_V

forall `{Univalence} (A B : Type@{u}), IsEquiv (equiv_path A B).

Axiom

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

isequiv_equiv_path

{A B : Type} (f : A <~> B) : A = B := (equiv_path A B)^-1 f.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

path_universe_uncurried

{A B : Type} (f : A -> B) {feq : IsEquiv f} : (A = B) := path_universe_uncurried (Build_Equiv _ _ f feq).

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

path_universe

{A B : Type} (p : A = B) : path_universe (equiv_path A B p) = p := eissect (equiv_path A B) p.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

eta_path_universe

{A B : Type} (p : A = B) : path_universe_uncurried (equiv_path A B p) = p := eissect (equiv_path A B) p.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

eta_path_universe_uncurried

{A B : Type} : IsEquiv (@path_universe_uncurried A B) := _.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

isequiv_path_universe

(A B : Type) : (A <~> B) <~> (A = B) := Build_Equiv _ _ (@path_universe_uncurried A B) isequiv_path_universe.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

equiv_path_universe

(A B : Type) : (A = B) <~> (A <~> B) := (equiv_path_universe A B)^-1%equiv.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

equiv_equiv_path

{A B : Type} (p : A = B) : path_universe (equiv_path A B p) = p := eissect (equiv_path A B) p.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

path_universe_equiv_path

{A B : Type} (p : A = B) : path_universe_uncurried (equiv_path A B p) = p := eissect (equiv_path A B) p.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

path_universe_uncurried_equiv_path

{A B : Type} (p : A = B) : path_universe (transport idmap p) = p := eissect (equiv_path A B) p.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

path_universe_transport_idmap

{A B : Type} (p : A = B) : path_universe_uncurried (equiv_transport idmap p) = p := eissect (equiv_path A B) p.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

path_universe_uncurried_transport_idmap

{A B : Type} (f : A <~> B) : equiv_path A B (path_universe f) = f := eisretr (equiv_path A B) f.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

equiv_path_path_universe

{A B : Type} (f : A <~> B) : equiv_path A B (path_universe_uncurried f) = f := eisretr (equiv_path A B) f.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

equiv_path_path_universe_uncurried

{A B : Type} (f : A <~> B) : transport idmap (path_universe f) = f := ap equiv_fun (eisretr (equiv_path A B) f).

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

transport_idmap_path_universe

{A B : Type} (f : A <~> B) : transport idmap (path_universe_uncurried f) = f := ap equiv_fun (eisretr (equiv_path A B) f).

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

transport_idmap_path_universe_uncurried

`{Funext} {A B C : Type} (p : A = B) (q : B = C) : equiv_path A C (p @ q) = equiv_path B C q oE equiv_path A B p. Proof. apply path_equiv, path_arrow. nrapply transport_pp. Defined.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

equiv_path_pp

{A B C : Type} (f : A <~> B) (g : B <~> C) : path_universe_uncurried (equiv_compose g f) = path_universe_uncurried f @ path_universe_uncurried g. Proof. revert f. equiv_intro (equiv_path A B) f. revert g. equiv_intro (equiv_path B C) g. refine ((ap path_universe_uncurried (equiv_path_pp f g))^ @ _). refine (eta_path_universe (f @ g) @ _). apply concat2; symmetry; apply eta_path_universe. Defined.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

path_universe_compose_uncurried

{A B C : Type} (f : A <~> B) (g : B <~> C) : path_universe (g o f) = path_universe f @ path_universe g := path_universe_compose_uncurried f g.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

path_universe_compose

{A : Type} : path_universe (equiv_idmap A) = 1 := eta_path_universe 1.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

path_universe_1

{A B : Type} (f : A <~> B) : path_universe_uncurried f^-1 = (path_universe_uncurried f)^. Proof. revert f. equiv_intro ((equiv_path_universe A B)^-1) p. simpl. transitivity (p^). 2: exact (inverse2 (eisretr (equiv_path_universe A B) p)^). transitivity (path_universe_uncurried (equiv_path B A p^)). - by refine (ap _ (equiv_path_V A B p)^). - by refine (eissect (equiv_path B A) p^). Defined.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

path_universe_V_uncurried

`(f : A -> B) `{IsEquiv A B f} : path_universe (f^-1) = (path_universe f)^ := path_universe_V_uncurried (Build_Equiv A B f _).

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

path_universe_V

A {B C} (f : B <~> C) : equiv_path (A <~> B) (A <~> C) (ap (Equiv A) (path_universe f)) = equiv_functor_equiv (equiv_idmap A) f. Proof. revert f. equiv_intro (equiv_path B C) f. rewrite (eissect (equiv_path B C) f : path_universe (equiv_path B C f) = f). destruct f; simpl. apply path_equiv, path_forall; intros g. apply path_equiv, path_forall; intros a. reflexivity. Defined.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

ap_equiv_path_universe

A {B C} (f : B <~> C) : equiv_path (A * B) (A * C) (ap (prod A) (path_universe f)) = equiv_functor_prod_l f. Proof. revert f. equiv_intro (equiv_path B C) f. rewrite (eissect (equiv_path B C) f : path_universe (equiv_path B C f) = f). destruct f. apply path_equiv, path_arrow; intros x; reflexivity. Defined.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

ap_prod_l_path_universe

A {B C} (f : B <~> C) : equiv_path (B * A) (C * A) (ap (fun Z => Z * A) (path_universe f)) = equiv_functor_prod_r f. Proof. revert f. equiv_intro (equiv_path B C) f. rewrite (eissect (equiv_path B C) f : path_universe (equiv_path B C f) = f). destruct f. apply path_equiv, path_arrow; intros x; reflexivity. Defined.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

ap_prod_r_path_universe

{A B : Type} (f : A <~> B) (z : A) : transport (fun X:Type => X) (path_universe_uncurried f) z = f z. Proof. revert f. equiv_intro (equiv_path A B) p. exact (ap (fun s => transport idmap s z) (eissect _ p)). Defined.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

transport_path_universe_uncurried

{A B : Type} (f : A -> B) {feq : IsEquiv f} (z : A) : transport (fun X:Type => X) (path_universe f) z = f z := transport_path_universe_uncurried (Build_Equiv A B f feq) z.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

transport_path_universe

{A B : Type} (p : A = B) (z : A) : transport_path_universe (equiv_path A B p) z = (ap (fun s => transport idmap s z) (eissect _ p)) := equiv_ind_comp _ _ _.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

transport_path_universe_equiv_path

transport_path_universe' {A : Type} (P : A -> Type) {x y : A} (p : x = y) (f : P x <~> P y) (q : ap P p = path_universe f) (u : P x) : transport P p u = f u := transport_compose idmap P p u @ ap10 (ap (transport idmap) q) u @ transport_path_universe f u.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

transport_path_universe'

{A B : Type} (f : A <~> B) (z : B) : transport (fun X:Type => X) (path_universe_uncurried f)^ z = f^-1 z. Proof. revert f. equiv_intro (equiv_path A B) p. exact (ap (fun s => transport idmap s z) (inverse2 (eissect _ p))). Defined.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

transport_path_universe_V_uncurried

{A B : Type} (f : A -> B) {feq : IsEquiv f} (z : B) : transport (fun X:Type => X) (path_universe f)^ z = f^-1 z := transport_path_universe_V_uncurried (Build_Equiv _ _ f feq) z.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

transport_path_universe_V

{A B : Type} (p : A = B) (z : B) : transport_path_universe_V (equiv_path A B p) z = ap (fun s => transport idmap s z) (inverse2 (eissect _ p)) := equiv_ind_comp _ _ _.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

transport_path_universe_V_equiv_path

{A B : Type} (f : A <~> B) (z : A) : ap (transport idmap (path_universe f)^) (transport_path_universe f z) @ transport_path_universe_V f (f z) @ eissect f z = transport_Vp idmap (path_universe f) z. Proof. pattern f. refine (equiv_ind (equiv_path A B) _ _ _); intros p. refine (_ @ ap_transport_Vp_idmap p (path_universe (equiv_path A B p)) (eissect (equiv_path A B) p) z). apply whiskerR. apply concat2. - apply ap. apply transport_path_universe_equiv_path. - apply transport_path_universe_V_equiv_path. Defined.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

transport_path_universe_Vp_uncurried

{A B : Type} (f : A -> B) {feq : IsEquiv f} (z : A) : ap (transport idmap (path_universe f)^) (transport_path_universe f z) @ transport_path_universe_V f (f z) @ eissect f z = transport_Vp idmap (path_universe f) z := transport_path_universe_Vp_uncurried (Build_Equiv A B f feq) z.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

transport_path_universe_Vp

{A U V : Type} (w : U <~> V) : forall f : U -> A, transport (fun E : Type => E -> A) (path_universe w) f = (f o w^-1). Proof. intros f. funext y. refine (transport_arrow_toconst _ _ _ @ _). apply ap. apply transport_path_universe_V. Defined.

Theorem

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

transport_arrow_toconst_path_universe

{A B : Type} (f g : A <~> B) : (f == g) <~> (path_universe f = path_universe g). Proof. refine (_ oE equiv_path_arrow f g). refine (_ oE equiv_path_equiv f g). exact (equiv_ap (equiv_path A B)^-1 _ _). Defined.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

equiv_path2_universe

{A B : Type} {f g : A <~> B} : (f == g) -> (path_universe f = path_universe g) := equiv_path2_universe f g.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

path2_universe

{A B : Type} (f : A <~> B) : equiv_path2_universe f f (fun x => 1) = 1. Proof. simpl. rewrite concat_1p, concat_p1, path_forall_1, path_sigma_hprop_1. reflexivity. Qed.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

equiv_path2_universe_1

{A B : Type} (f : A <~> B) : @path2_universe _ _ f f (fun x => 1) = 1 := equiv_path2_universe_1 f.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

path2_universe_1

{A B C : Type} {f1 f2 : A <~> B} (p : f1 == f2) (g : B <~> C) : equiv_path2_universe (g o f1) (g o f2) (fun a => ap g (p a)) = path_universe_compose f1 g @ whiskerR (path2_universe p) (path_universe g) @ (path_universe_compose f2 g)^.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

path2_universe_postcompose

{A B C : Type} {f1 f2 : B <~> C} (p : f1 == f2) (g : A <~> B) : equiv_path2_universe (f1 o g) (f2 o g) (fun a => (p (g a))) = path_universe_compose g f1 @ whiskerL (path_universe g) (path2_universe p) @ (path_universe_compose g f2)^.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

path2_universe_precompose

{A C : Type} (p : forall a:A, a = a) (g : A <~> C) : equiv_path2_universe g g (fun a => ap g (p a)) = (concat_1p _)^ @ whiskerR (path_universe_1)^ (path_universe g) @ whiskerR (equiv_path2_universe (equiv_idmap A) (equiv_idmap A) p) (path_universe g) @ whiskerR path_universe_1 (path_universe g) @ concat_1p _. Proof. transitivity ((eta_path_universe (path_universe g))^ @ equiv_path2_universe (equiv_path A C (path_universe g)) (equiv_path A C (path_universe g)) (fun a => ap (equiv_path A C (path_universe g)) (p a)) @ eta_path_universe (path_universe g)). - refine ((apD (fun g' => equiv_path2_universe g' g' (fun a => ap g' (p a))) (eisretr (equiv_path A C) g))^ @ _). refine (transport_paths_FlFr (eisretr (equiv_path A C) g) _ @ _). apply concat2. + apply whiskerR. apply inverse2, symmetry. refine (eisadj (equiv_path A C)^-1 g). + symmetry; refine (eisadj (equiv_path A C)^-1 g). - generalize (path_universe g). intros h. destruct h. cbn. rewrite !concat_1p, !concat_p1. refine (_ @ whiskerR (whiskerR_pp 1 path_universe_1^ _) _). refine (_ @ whiskerR_pp 1 _ path_universe_1). refine (_ @ (whiskerR_p1_1 _)^). apply whiskerR, whiskerL, ap, ap, ap. apply path_forall; intros x; apply ap_idmap. Defined.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

path2_universe_postcompose_idmap

{A B : Type} (p : forall b:B, b = b) (g : A <~> B) : equiv_path2_universe g g (fun a => (p (g a))) = (concat_p1 _)^ @ whiskerL (path_universe g) (path_universe_1)^ @ whiskerL (path_universe g) (equiv_path2_universe (equiv_idmap B) (equiv_idmap B) p) @ whiskerL (path_universe g) (path_universe_1) @ concat_p1 _. Proof. transitivity ((eta_path_universe (path_universe g))^ @ equiv_path2_universe (equiv_path A B (path_universe g)) (equiv_path A B (path_universe g)) (fun a => p (equiv_path A B (path_universe g) a)) @ eta_path_universe (path_universe g)). - refine ((apD (fun g' => equiv_path2_universe g' g' (fun a => p (g' a))) (eisretr (equiv_path A B) g))^ @ _). refine (transport_paths_FlFr (eisretr (equiv_path A B) g) _ @ _). apply concat2. + apply whiskerR. apply inverse2, symmetry. refine (eisadj (equiv_path A B)^-1 g). + symmetry; refine (eisadj (equiv_path A B)^-1 g). - generalize (path_universe g). intros h. destruct h. cbn. rewrite !concat_p1. refine (_ @ (((concat_1p (whiskerL 1 path_universe_1^))^ @@ 1) @@ 1)). refine (_ @ whiskerR (whiskerL_pp 1 path_universe_1^ _) _). refine (_ @ whiskerL_pp 1 _ path_universe_1). exact ((whiskerL_1p_1 _)^). Defined.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

path2_universe_precompose_idmap

{A B : Type} {f g : A <~> B} (p q : f == g) : (p == q) <~> (path2_universe p = path2_universe q). Proof. refine (_ oE equiv_path_forall p q). refine (_ oE equiv_ap (equiv_path_arrow f g) p q). refine (_ oE equiv_ap (equiv_path_equiv f g) _ _). unfold path2_universe, equiv_path2_universe. simpl. refine (equiv_ap (ap (equiv_path A B)^-1) _ _). Defined.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

equiv_path3_universe

{A B : Type} {f g : A <~> B} {p q : f == g} : (p == q) -> (path2_universe p = path2_universe q) := equiv_path3_universe p q.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

path3_universe

{A B : Type} (f : A <~> B) (z : B) : transport_path_universe f (transport idmap (path_universe f)^ z) @ ap f (transport_path_universe_V f z) @ eisretr f z = transport_pV idmap (path_universe f) z. Proof. pattern f. refine (equiv_ind (equiv_path A B) _ _ _); intros p. refine (_ @ ap_transport_pV_idmap p (path_universe (equiv_path A B p)) (eissect (equiv_path A B) p) z). apply whiskerR. refine ((concat_Ap _ _)^ @ _). apply concat2. - apply ap. refine (transport_path_universe_V_equiv_path _ _ @ _). unfold inverse2. symmetry; apply (ap_compose inverse (fun s => transport idmap s z)). - apply transport_path_universe_equiv_path. Defined.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

transport_path_universe_pV_uncurried

{A B : Type} (f : A -> B) {feq : IsEquiv f } (z : B) : transport_path_universe f (transport idmap (path_universe f)^ z) @ ap f (transport_path_universe_V f z) @ eisretr f z = transport_pV idmap (path_universe f) z := transport_path_universe_pV_uncurried (Build_Equiv A B f feq) z.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

transport_path_universe_pV

{U : Type} (P : forall V, U <~> V -> Type) : (P U (equiv_idmap U)) -> (forall V (w : U <~> V), P V w). Proof. intros H0 V. apply (equiv_ind (equiv_path U V)). intro p; induction p; exact H0. Defined.

Theorem

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

equiv_induction

{U : Type} (P : forall V, U <~> V -> Type) (didmap : P U (equiv_idmap U)) : equiv_induction P didmap U (equiv_idmap U) = didmap := (equiv_ind_comp (P U) _ 1).

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

equiv_induction_comp

equiv_induction' (P : forall U V, U <~> V -> Type) : (forall T, P T T (equiv_idmap T)) -> (forall U V (w : U <~> V), P U V w). Proof. intros H0 U V w. apply (equiv_ind (equiv_path U V)). intro p; induction p; apply H0. Defined.

Theorem

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

equiv_induction'

(P : forall U V, U <~> V -> Type) (didmap : forall T, P T T (equiv_idmap T)) (U : Type) : equiv_induction' P didmap U U (equiv_idmap U) = didmap U := (equiv_ind_comp (P U U) _ 1).

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

equiv_induction'_comp

{U : Type} (P : forall V, V <~> U -> Type) : (P U (equiv_idmap U)) -> (forall V (w : V <~> U), P V w). Proof. intros H0 V. apply (equiv_ind (equiv_path V U)). revert V; rapply paths_ind_r; apply H0. Defined.

Theorem

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

equiv_induction_inv

{U : Type} (P : forall V, V <~> U -> Type) (didmap : P U (equiv_idmap U)) : equiv_induction_inv P didmap U (equiv_idmap U) = didmap := (equiv_ind_comp (P U) _ 1). Global Instance contr_basedequiv@{u +} {X : Type@{u}} : Contr {Y : Type@{u} & X <~> Y}. Proof. apply (Build_Contr _ (X; equiv_idmap)). intros [Y f]; revert Y f. exact (equiv_induction _ idpath). Defined.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

equiv_induction_inv_comp

~ (IsHSet Type). Proof. intro HT. apply true_ne_false. pose (r := path_ishprop (path_universe equiv_negb) 1). refine (_ @ (ap (fun q => transport idmap q false) r)). symmetry; apply transport_path_universe. Defined.

Definition

Require Import HoTT.Basics. Require Import Types.Sigma Types.Forall Types.Arrow Types.Paths Types.Equiv Types.Bool Types.Prod.

Types\Universe.v

not_hset_Type

(A : Type) (B : A -> Type) : Type := w_sup (x : A) : (B x -> A B) -> A B.

Inductive

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

Types\WType.v

W

{A B} (w : W A B) : A := match w with | w_sup x y => x end.

Definition

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

Types\WType.v

w_label

{A B} (w : W A B) : B (w_label w) -> W A B := match w with | w_sup x y => y end.

Definition

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

Types\WType.v

w_arg

(A : Type) (B : A -> Type) : _ <~> W A B := ltac:(issig).

Definition

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

Types\WType.v

issig_W

{A B} (z z' : W A B) : (w_label z;w_arg z) = (w_label z';w_arg z') :> {a : _ & B a -> W A B} <~> z = z' := (equiv_ap' (issig_W A B)^-1%equiv z z')^-1%equiv.

Definition

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

Types\WType.v

equiv_path_wtype

equiv_path_wtype' {A B} (z z' : W A B) : {p : w_label z = w_label z' & w_arg z = w_arg z' o transport B p} <~> z = z'. Proof. refine (equiv_path_wtype _ _ oE equiv_path_sigma _ _ _ oE _). apply equiv_functor_sigma_id. destruct z as [z1 z2], z' as [z1' z2']. cbn; intros p. destruct p. reflexivity. Defined.

Definition

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

Types\WType.v

equiv_path_wtype'

Type <~> Type. Proof. refine (((equiv_decidable_sum (fun X:Type => merely (X=A)))^-1) oE _ oE (equiv_decidable_sum (fun X:Type => merely (X=A)))). refine ((equiv_functor_sum_l (equiv_decidable_sum (fun X => merely (X.1=B)))^-1) oE _ oE (equiv_functor_sum_l (equiv_decidable_sum (fun X => merely (X.1=B))))). refine ((equiv_sum_assoc _ _ _) oE _ oE (equiv_sum_assoc _ _ _)^-1). apply equiv_functor_sum_r. assert (q : BAut B <~> {x : {x : Type & ~ merely (x = A)} & merely (x.1 = B)}). { refine (equiv_sigma_assoc _ _ oE _). apply equiv_functor_sigma_id; intros X. apply equiv_iff_hprop. - intros p. refine (fun q => _ ; p). strip_truncations. destruct q. exact (ne (equiv_path X B p)). - exact pr2. } refine (_ oE equiv_sum_symm _ _). apply equiv_functor_sum'. - exact (e^-1 oE q^-1). - exact (q oE e). Defined.

Definition

Require Import Basics Types. Require Import HoTT.Truncations. Require Import Universes.BAut Universes.Rigid. Require Import ExcludedMiddle.

Universes\Automorphisms.v

equiv_swap_types

merely (equiv_swap_types A = B). Proof. assert (ea := (e (point _)).2). cbn in ea. strip_truncations; apply tr. unfold equiv_swap_types. apply moveR_equiv_V. rewrite (equiv_decidable_sum_l (fun X => merely (X=A)) A (tr 1)). assert (ne' : ~ merely (B=A)) by (intros p; strip_truncations; exact (ne (equiv_path A B p^))). rewrite (equiv_decidable_sum_r (fun X => merely (X=A)) B ne'). cbn. apply ap, path_sigma_hprop; cbn. exact ea. Defined.

Definition

Require Import Basics Types. Require Import HoTT.Truncations. Require Import Universes.BAut Universes.Rigid. Require Import ExcludedMiddle.

Universes\Automorphisms.v

equiv_swap_types_swaps

equiv_swap_types <> equiv_idmap. Proof. intros p. assert (q := equiv_swap_types_swaps). strip_truncations. apply ne. apply equiv_path. rewrite p in q; exact q. Qed.

Definition

Require Import Basics Types. Require Import HoTT.Truncations. Require Import Universes.BAut Universes.Rigid. Require Import ExcludedMiddle.

Universes\Automorphisms.v

equiv_swap_types_not_id

`{Univalence} `{ExcludedMiddle} (A B : Type) `{IsRigid A} `{IsRigid B} (ne : ~(A <~> B)) : Type <~> Type. Proof. refine (equiv_swap_types A B ne _). apply equiv_contr_contr. Defined.

Definition

Require Import Basics Types. Require Import HoTT.Truncations. Require Import Universes.BAut Universes.Rigid. Require Import ExcludedMiddle.

Universes\Automorphisms.v

equiv_swap_rigid

`{Univalence} `{ExcludedMiddle} : Type <~> Type := equiv_swap_rigid Empty Unit (fun e => e^-1 tt).

Definition

Require Import Basics Types. Require Import HoTT.Truncations. Require Import Universes.BAut Universes.Rigid. Require Import ExcludedMiddle.

Universes\Automorphisms.v

equiv_swap_empty_unit

`{Univalence} `{ExcludedMiddle} (A B : Type) `{IsRigid A} `{IsRigid B} (ne : ~(A <~> B)) : equiv_swap_rigid A B ne A = B. Proof. unfold equiv_swap_rigid, equiv_swap_types. apply moveR_equiv_V. rewrite (equiv_decidable_sum_l (fun X => merely (X=A)) A (tr 1)). assert (ne' : ~ merely (B=A)) by (intros p; strip_truncations; exact (ne (equiv_path A B p^))). rewrite (equiv_decidable_sum_r (fun X => merely (X=A)) B ne'). cbn. apply ap, path_sigma_hprop; cbn. exact ((path_contr (center (BAut B)) (point (BAut B)))..1). Defined.

Definition

Require Import Basics Types. Require Import HoTT.Truncations. Require Import Universes.BAut Universes.Rigid. Require Import ExcludedMiddle.

Universes\Automorphisms.v

equiv_swap_rigid_swaps

`{Univalence} `{ExcludedMiddle} (X A B : Type) (n : trunc_index) (ne : ~(X*A <~> X*B)) `{IsRigid A} `{IsConnected n.+1 A} `{IsRigid B} `{IsConnected n.+1 B} `{IsTrunc n.+1 X} : Type <~> Type. Proof. refine (equiv_swap_types (X*A) (X*B) ne _). transitivity (BAut X). - symmetry; exact (baut_prod_rigid_equiv X A n). - exact (baut_prod_rigid_equiv X B n). Defined.

Definition

Require Import Basics Types. Require Import HoTT.Truncations. Require Import Universes.BAut Universes.Rigid. Require Import ExcludedMiddle.

Universes\Automorphisms.v

equiv_swap_prod_rigid

`{Univalence} (f : Type <~> Type) (eu : f Unit = Empty) : ExcludedMiddle_type. Proof. apply DNE_to_LEM, DNE_from_allneg; intros P ?. exists (f P); split. - intros p. assert (Contr P) by (apply contr_inhabited_hprop; assumption). assert (q : Unit = P) by (apply path_universe_uncurried, equiv_contr_contr). destruct q. rewrite eu. auto. - intros nfp. assert (q : f P = Empty) by (apply path_universe_uncurried, equiv_to_empty, nfp). rewrite <- eu in q. apply ((ap f)^-1) in q. rewrite q; exact tt. Defined.

Definition

Require Import Basics Types. Require Import HoTT.Truncations. Require Import Universes.BAut Universes.Rigid. Require Import ExcludedMiddle.

Universes\Automorphisms.v

lem_from_aut_type_unit_empty

`{Univalence} (A : Type) (P : Type) (a : merely A) `{IsHProp P} : P <-> (P * A = A). Proof. split. - intros p; apply path_universe with snd. apply isequiv_adjointify with (fun a => (p,a)). + intros x; reflexivity. + intros [p' x]. apply path_prod; [ apply path_ishprop | reflexivity ]. - intros q. strip_truncations. apply equiv_path in q. exact (fst (q^-1 a)). Defined.

Lemma

Require Import Basics Types. Require Import HoTT.Truncations. Require Import Universes.BAut Universes.Rigid. Require Import ExcludedMiddle.

Universes\Automorphisms.v

equiv_hprop_idprod

`{Univalence} (f : Type <~> Type) (A : Type) (a : merely A) (eu : f A = Empty) : ExcludedMiddle_type. Proof. apply DNE_to_LEM, DNE_from_allneg; intros P ?. exists (f (P * A)); split. - intros p. assert (q := fst (equiv_hprop_idprod A P a) p). apply (ap f) in q. rewrite eu in q. rewrite q; auto. - intros q. apply equiv_to_empty in q. apply path_universe_uncurried in q. rewrite <- eu in q. apply ((ap f)^-1) in q. exact (snd (equiv_hprop_idprod A P a) q). Defined.

Definition

Require Import Basics Types. Require Import HoTT.Truncations. Require Import Universes.BAut Universes.Rigid. Require Import ExcludedMiddle.

Universes\Automorphisms.v

lem_from_aut_type_inhabited_empty

`{Univalence} (g : Type <~> Type) (ge : g Empty <> Empty) : ~~ExcludedMiddle_type. Proof. pose (f := equiv_inverse g). intros nlem. apply ge. apply path_universe_uncurried, equiv_to_empty; intros gz. apply nlem. apply (lem_from_aut_type_inhabited_empty f (g Empty) (tr gz)). unfold f; apply eissect. Defined.

Definition

Require Import Basics Types. Require Import HoTT.Truncations. Require Import Universes.BAut Universes.Rigid. Require Import ExcludedMiddle.

Universes\Automorphisms.v

zero_beers

`{Univalence} (g : Type <~> Type) (ge : g ExcludedMiddle_type <> ExcludedMiddle_type) : ~~ExcludedMiddle_type. Proof. intros nlem. pose (nlem' := equiv_to_empty nlem). apply path_universe_uncurried in nlem'. rewrite nlem' in ge. apply (zero_beers g) in ge. exact (ge nlem). Defined.

Definition

Require Import Basics Types. Require Import HoTT.Truncations. Require Import Universes.BAut Universes.Rigid. Require Import ExcludedMiddle.

Universes\Automorphisms.v

lem_beers

(X : Type@{u}) := { Z : Type@{u} & merely (Z = X) }.

Definition

Require Import HoTT.Basics HoTT.Types. Require Import Constant. Require Import HoTT.Truncations. Require Import ObjectClassifier Homotopy.ExactSequence Pointed.

Universes\BAut.v

BAut

(X : Type) : pType := [BAut X, _].

Definition

Require Import HoTT.Basics HoTT.Types. Require Import Constant. Require Import HoTT.Truncations. Require Import ObjectClassifier Homotopy.ExactSequence Pointed.

Universes\BAut.v

pBAut

`{Univalence} {X} (Z Z' : BAut X) : (Z <~> Z') <~> (Z = Z' :> BAut X) := equiv_path_sigma_hprop _ _ oE equiv_path_universe _ _.

Definition

Require Import HoTT.Basics HoTT.Types. Require Import Constant. Require Import HoTT.Truncations. Require Import ObjectClassifier Homotopy.ExactSequence Pointed.

Universes\BAut.v

path_baut

`{Univalence} {X} {Z Z' : BAut X} (f : Z <~> Z') : ap (BAut_pr1 X) (path_baut Z Z' f) = path_universe f. Proof. unfold path_baut, BAut_pr1; simpl. apply ap_pr1_path_sigma_hprop. Defined.

Definition

Require Import HoTT.Basics HoTT.Types. Require Import Constant. Require Import HoTT.Truncations. Require Import ObjectClassifier Homotopy.ExactSequence Pointed.

Universes\BAut.v

ap_pr1_path_baut

`{Univalence} {X} {Z Z' : BAut X} (f : Z <~> Z') (z : Z) : transport (fun (W:BAut X) => W) (path_baut Z Z' f) z = f z. Proof. refine (transport_compose idmap (BAut_pr1 X) _ _ @ _). refine (_ @ transport_path_universe f z). apply ap10, ap, ap_pr1_path_baut. Defined.

Definition

Require Import HoTT.Basics HoTT.Types. Require Import Constant. Require Import HoTT.Truncations. Require Import ObjectClassifier Homotopy.ExactSequence Pointed.

Universes\BAut.v

transport_path_baut

(X A : Type) : BAut X -> BAut (X * A) := fun Z:BAut X => (Z * A ; Trunc_functor (-1) (ap (fun W => W * A)) (pr2 Z)) : BAut (X * A).

Definition

Require Import HoTT.Basics HoTT.Types. Require Import Constant. Require Import HoTT.Truncations. Require Import ObjectClassifier Homotopy.ExactSequence Pointed.

Universes\BAut.v

baut_prod_r

`{Univalence} (X A : Type) {Z W : BAut X} (e : Z <~> W) : ap (baut_prod_r X A) (path_baut Z W e) = path_baut (baut_prod_r X A Z) (baut_prod_r X A W) (equiv_functor_prod_r e). Proof. cbn. apply moveL_equiv_M; cbn; unfold pr1_path. rewrite <- (ap_compose (baut_prod_r X A) pr1 (path_sigma_hprop Z W _)). rewrite <- ((ap_compose pr1 (fun Z => Z * A) (path_sigma_hprop Z W _))^). rewrite ap_pr1_path_sigma_hprop. apply moveL_equiv_M; cbn. apply ap_prod_r_path_universe. Qed.

Definition

Require Import HoTT.Basics HoTT.Types. Require Import Constant. Require Import HoTT.Truncations. Require Import ObjectClassifier Homotopy.ExactSequence Pointed.

Universes\BAut.v

ap_baut_prod_r

`{Univalence} X (P : Type -> Type) `{forall (Z : BAut X), IsHSet (P Z)} : { e : P (point (BAut X)) & forall g : X <~> X, transport P (path_universe g) e = e } <~> (forall (Z:BAut X), P Z). Proof. refine (equiv_sig_ind _ oE _). refine ((equiv_functor_forall' (P := fun Z => { f : (Z=X) -> P Z & WeaklyConstant f }) 1 (fun Z => equiv_merely_rec_hset_if_domain _ _)) oE _); simpl. { intros p. change (IsHSet (P (BAut_pr1 X (Z ; tr p)))). exact _. } unfold WeaklyConstant. refine (equiv_sig_coind _ _ oE _). srapply equiv_functor_sigma'. 1:apply (equiv_paths_ind_r X (fun x _ => P x)). intros p; cbn. refine (equiv_paths_ind_r X _ oE _). srapply equiv_functor_forall'. 1:apply equiv_equiv_path. intros e; cbn. refine (_ oE equiv_moveL_transport_V _ _ _ _). apply equiv_concat_r. rewrite path_universe_transport_idmap, paths_ind_r_transport. reflexivity. Defined.

Lemma

Require Import HoTT.Basics HoTT.Types. Require Import Constant. Require Import HoTT.Truncations. Require Import ObjectClassifier Homotopy.ExactSequence Pointed.

Universes\BAut.v

baut_ind_hset

`{Univalence} X `{IsHSet X} : { f : X <~> X & forall g:X<~>X, g o f == f o g } <~> (forall Z:BAut X, Z = Z). Proof. refine (equiv_functor_forall_id (fun Z => equiv_path_sigma_hprop Z Z) oE _). refine (baut_ind_hset X (fun Z => Z = Z) oE _). simpl. refine (equiv_functor_sigma' (equiv_path_universe X X) _); intros f. apply equiv_functor_forall_id; intros g; simpl. refine (_ oE equiv_path_arrow _ _). refine (_ oE equiv_path_equiv (g oE f) (f oE g)). revert g. equiv_intro (equiv_path X X) g. revert f. equiv_intro (equiv_path X X) f. refine (_ oE equiv_concat_l (equiv_path_pp _ _) _). refine (_ oE equiv_concat_r (equiv_path_pp _ _)^ _). refine (_ oE (equiv_ap (equiv_path X X) _ _)^-1). refine (equiv_concat_l (transport_paths_lr _ _) _ oE _). refine (equiv_concat_l (concat_pp_p _ _ _) _ oE _). refine (equiv_moveR_Vp _ _ _ oE _). refine (equiv_concat_l _ _ oE equiv_concat_r _ _). - apply concat2; apply eissect. - symmetry; apply concat2; apply eissect. Defined.

Definition

Require Import HoTT.Basics HoTT.Types. Require Import Constant. Require Import HoTT.Truncations. Require Import ObjectClassifier Homotopy.ExactSequence Pointed.

Universes\BAut.v

center_baut

`{Univalence} X `{IsHSet X} : center_baut X (exist (fun (f:X<~>X) => forall (g:X<~>X), g o f == f o g) (equiv_idmap X) (fun (g:X<~>X) (x:X) => idpath (g x))) = fun Z => idpath Z. Proof. apply path_forall; intros Z. assert (IsHSet (Z.1 = Z.1)) by exact _. baut_reduce. exact (ap (path_sigma_hprop _ _) path_universe_1 @ path_sigma_hprop_1 _). Defined.

Definition

Require Import HoTT.Basics HoTT.Types. Require Import Constant. Require Import HoTT.Truncations. Require Import ObjectClassifier Homotopy.ExactSequence Pointed.

Universes\BAut.v

id_center_baut

`{Univalence} X `{IsTrunc 1 X} : { f : forall x:X, x=x & forall (g:X<~>X) (x:X), ap g (f x) = f (g x) } <~> (forall Z:BAut X, (idpath Z) = (idpath Z)). Proof. refine ((equiv_functor_forall_id (fun Z => (equiv_concat_lr _ _) oE (equiv_ap (equiv_path_sigma_hprop Z Z) 1%path 1%path))) oE _). { symmetry; apply path_sigma_hprop_1. } { apply path_sigma_hprop_1. } assert (forall Z:BAut X, IsHSet (idpath Z.1 = idpath Z.1)) by exact _. refine (baut_ind_hset X (fun Z => idpath Z = idpath Z) oE _). simple refine (equiv_functor_sigma' _ _). { refine (_ oE equiv_path2_universe 1 1). apply equiv_concat_lr. - symmetry; apply path_universe_1. - apply path_universe_1. } intros f. apply equiv_functor_forall_id; intros g. refine (_ oE equiv_path3_universe _ _). refine (dpath_paths2 (path_universe g) _ _ oE _). cbn. change (equiv_idmap X == equiv_idmap X) in f. refine (equiv_concat_lr _ _). - refine (_ @ (path2_universe_postcompose_idmap f g)^). abstract (rewrite !whiskerR_pp, !concat_pp_p; reflexivity). - refine (path2_universe_precompose_idmap f g @ _). abstract (rewrite !whiskerL_pp, !concat_pp_p; reflexivity). Defined.

Definition

Require Import HoTT.Basics HoTT.Types. Require Import Constant. Require Import HoTT.Truncations. Require Import ObjectClassifier Homotopy.ExactSequence Pointed.

Universes\BAut.v

center2_baut

`{Univalence} X `{IsTrunc 1 X} : center2_baut X (exist (fun (f:forall x:X, x=x) => forall (g:X<~>X) (x:X), ap g (f x) = f (g x)) (fun x => idpath x) (fun (g:X<~>X) (x:X) => idpath (idpath (g x)))) = fun Z => idpath (idpath Z). Proof. apply path_forall; intros Z. assert (IsHSet (idpath Z.1 = idpath Z.1)) by exact _. baut_reduce. cbn. unfold functor_forall, sig_rect, merely_rec_hset. cbn. rewrite equiv_path2_universe_1. rewrite !concat_p1, !concat_Vp. simpl. rewrite !concat_p1, !concat_Vp. reflexivity. Defined.

Definition

Require Import HoTT.Basics HoTT.Types. Require Import Constant. Require Import HoTT.Truncations. Require Import ObjectClassifier Homotopy.ExactSequence Pointed.

Universes\BAut.v

id_center2_baut

(F : Type) : Subuniverse. Proof. rapply (Build_Subuniverse (fun E => merely (E <~> F))). intros T U mere_eq f iseq_f. strip_truncations. pose (feq:=Build_Equiv _ _ f iseq_f). exact (tr (mere_eq oE feq^-1)). Defined.

Definition

Require Import HoTT.Basics HoTT.Types. Require Import Constant. Require Import HoTT.Truncations. Require Import ObjectClassifier Homotopy.ExactSequence Pointed.

Universes\BAut.v

subuniverse_merely_equiv

`{Univalence} {F : Type} : BAut F <~> Type_ (subuniverse_merely_equiv F). Proof. srapply equiv_functor_sigma_id; intro X; cbn. rapply Trunc_functor_equiv. exact (equiv_path_universe _ _)^-1%equiv. Defined.

Proposition

Require Import HoTT.Basics HoTT.Types. Require Import Constant. Require Import HoTT.Truncations. Require Import ObjectClassifier Homotopy.ExactSequence Pointed.

Universes\BAut.v

equiv_baut_typeO

`{Univalence} {Y : pType} {F : Type} : (Y -> BAut F) <~> { p : Slice Y & forall y:Y, merely (hfiber p.2 y <~> F) }. Proof. refine (_ oE equiv_postcompose' equiv_baut_typeO). refine (_ oE equiv_sigma_fibration_O). snrapply equiv_functor_sigma_id; intro p. rapply equiv_functor_forall_id; intro y. by apply Trunc_functor_equiv. Defined.

Corollary

Require Import HoTT.Basics HoTT.Types. Require Import Constant. Require Import HoTT.Truncations. Require Import ObjectClassifier Homotopy.ExactSequence Pointed.

Universes\BAut.v

equiv_map_baut_fibration

pequiv_pbaut_typeOp@{u v +} `{Univalence} {F : Type@{u}} : pBAut@{u v} F <~>* [Type_ (subuniverse_merely_equiv F), (F; tr equiv_idmap)]. Proof. snrapply Build_pEquiv'; cbn. 1: exact equiv_baut_typeO. by apply path_sigma_hprop. Defined.

Proposition

Require Import HoTT.Basics HoTT.Types. Require Import Constant. Require Import HoTT.Truncations. Require Import ObjectClassifier Homotopy.ExactSequence Pointed.

Universes\BAut.v

pequiv_pbaut_typeOp@

`{Univalence} {Y F : pType@{u}} : (Y ->* pBAut@{u v} F) <~> { p : { q : pSlice Y & forall y:Y, merely (hfiber q.

Definition

Require Import HoTT.Basics HoTT.Types. Require Import Constant. Require Import HoTT.Truncations. Require Import ObjectClassifier Homotopy.ExactSequence Pointed.

Universes\BAut.v

equiv_pmap_pbaut_pfibration

`{Univalence} {Y : pType@{u}} {F : Type@{u}} `{IsConnected 0 Y} : (Y ->* pBAut F) <~> (Y ->* [Type@{u}, F]). Proof. refine (_ oE pequiv_pequiv_postcompose pequiv_pbaut_typeOp). rapply equiv_pmap_typeO_type_connected. Defined.

Proposition

Require Import HoTT.Basics HoTT.Types. Require Import Constant. Require Import HoTT.Truncations. Require Import ObjectClassifier Homotopy.ExactSequence Pointed.

Universes\BAut.v

equiv_pmap_pbaut_type_p

`{Univalence} {Y F : pType} `{IsConnected 0 Y} : (Y ->* pBAut F) <~> { X : pType & FiberSeq F X Y } := classify_fiberseq oE equiv_pmap_pbaut_type_p.

Definition

Require Import HoTT.Basics HoTT.Types. Require Import Constant. Require Import HoTT.Truncations. Require Import ObjectClassifier Homotopy.ExactSequence Pointed.

Universes\BAut.v

equiv_pmap_pbaut_pfibration_connected

{ dprop_type : Type ; ishprop_dprop : Funext -> IsHProp dprop_type ; dec_dprop : Decidable dprop_type }.

Record

Require Import HoTT.Basics HoTT.Types. Require Import TruncType HProp. Require Import Truncations.Core Modalities.ReflectiveSubuniverse.

Universes\DProp.v

DProp

Record

Require Import HoTT.Basics HoTT.Types. Require Import TruncType HProp. Require Import Truncations.Core Modalities.ReflectiveSubuniverse.

Universes\DProp.v

DProp

Record

Require Import HoTT.Basics HoTT.Types. Require Import TruncType HProp. Require Import Truncations.Core Modalities.ReflectiveSubuniverse.

Universes\DProp.v

DHProp

DHProp -> DProp := fun P => Build_DProp P (fun _ => _) _.

Definition

Require Import HoTT.Basics HoTT.Types. Require Import TruncType HProp. Require Import Truncations.Core Modalities.ReflectiveSubuniverse.

Universes\DProp.v

dhprop_to_dprop

DHProp := Build_DHProp Unit_hp (inl tt).

Definition

Require Import HoTT.Basics HoTT.Types. Require Import TruncType HProp. Require Import Truncations.Core Modalities.ReflectiveSubuniverse.

Universes\DProp.v

True

DHProp := Build_DHProp False_hp (inr idmap).

Definition

Require Import HoTT.Basics HoTT.Types. Require Import TruncType HProp. Require Import Truncations.Core Modalities.ReflectiveSubuniverse.

Universes\DProp.v

False