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phanerozoic

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Coq-HoTT

like 2

`(f : A -> B) `{IsEquiv A B f} (x y : B) : (f^-1 x = f^-1 y) <~> (x = y) := (@equiv_ap B A f^-1 _ x y)^-1%equiv.

Definition

Require Import

Basics\Equivalences.v

equiv_ap_inv

equiv_ap_inv' `(f : A <~> B) (x y : B) : (f^-1 x = f^-1 y) <~> (x = y) := (equiv_ap' f^-1%equiv x y)^-1%equiv.

Definition

Require Import

Basics\Equivalences.v

equiv_ap_inv'

{A B C} (f : A -> B) {g : B -> C} `{IsEquiv A B f} `{IsEquiv A C (g o f)} : IsEquiv g := isequiv_homotopic ((g o f) o f^-1) (fun b => ap g (eisretr f b)).

Definition

Require Import

Basics\Equivalences.v

cancelR_isequiv

{A B C} (f : A -> B) {g : B -> C} `{IsEquiv A B f} `{IsEquiv A C (g o f)} : B <~> C := Build_Equiv B C g (cancelR_isequiv f).

Definition

Require Import

Basics\Equivalences.v

cancelR_equiv

{A B C} (g : B -> C) {f : A -> B} `{IsEquiv B C g} `{IsEquiv A C (g o f)} : IsEquiv f := isequiv_homotopic (g^-1 o (g o f)) (fun a => eissect g (f a)).

Definition

Require Import

Basics\Equivalences.v

cancelL_isequiv

{A B C} (g : B -> C) {f : A -> B} `{IsEquiv B C g} `{IsEquiv A C (g o f)} : A <~> B := Build_Equiv _ _ f (cancelL_isequiv g).

Definition

Require Import

Basics\Equivalences.v

cancelL_equiv

{A B C D} (f : A -> B) (g : C -> D) (h : A -> C) (k : B -> D) (p : k o f == g o h) `{IsEquiv _ _ f} `{IsEquiv _ _ h} `{IsEquiv _ _ k} : IsEquiv g. Proof. refine (@cancelR_isequiv _ _ _ h g _ _). refine (isequiv_homotopic _ p). Defined.

Definition

Require Import

Basics\Equivalences.v

isequiv_commsq

isequiv_commsq' {A B C D} (f : A -> B) (g : C -> D) (h : A -> C) (k : B -> D) (p : g o h == k o f) `{IsEquiv _ _ g} `{IsEquiv _ _ h} `{IsEquiv _ _ k} : IsEquiv f. Proof. refine (@cancelL_isequiv _ _ _ k f _ _). refine (isequiv_homotopic _ p). Defined.

Definition

Require Import

Basics\Equivalences.v

isequiv_commsq'

`{IsEquiv A B f} (P : B -> Type) : (forall x:A, P (f x)) -> forall y:B, P y := fun g y => transport P (eisretr f y) (g (f^-1 y)).

Definition

Require Import

Basics\Equivalences.v

equiv_ind

`{IsEquiv A B f} (P : B -> Type) (df : forall x:A, P (f x)) (x : A) : equiv_ind f P df (f x) = df x. Proof. unfold equiv_ind. rewrite eisadj. rewrite <- transport_compose. exact (apD df (eissect f x)). Defined.

Definition

Require Import

Basics\Equivalences.v

equiv_ind_comp

{A} {a : A} {X : A -> Type} (e : forall (b : A), a = b <~> X b) (P : forall (b : A), X b -> Type) (r : P a (e a 1)) : forall (b : A) (x : X b), P b x. Proof. intro b. srapply (equiv_ind (e b)). intros []. exact r. Defined.

Definition

Require Import

Basics\Equivalences.v

equiv_path_ind

equiv_composeR' {A B C} (f : A <~> B) (g : B <~> C) := equiv_compose' g f. Ltac equiv_via mid := apply @equiv_composeR' with (B := mid). Ltac ev_equiv := repeat match goal with | [ |- context[equiv_fun (equiv_inverse (equiv_inverse ?f))] ] => change (equiv_fun (equiv_inverse (equiv_inverse f))) with (equiv_fun f) | [ |- context[(@equiv_inv ?B ?A (equiv_fun (equiv_inverse ?f)) ?iseq)] ] => change (@equiv_inv B A (equiv_fun (equiv_inverse f)) iseq) with (equiv_fun f) | [ |- context[((equiv_fun ?f)^-1)^-1] ] => change ((equiv_fun f)^-1)^-1 with (equiv_fun f) | [ |- context[equiv_fun (equiv_compose' ?g ?f) ?a] ] => change (equiv_fun (equiv_compose' g f) a) with (g (f a)) | [ |- context[equiv_fun (equiv_compose ?g ?f) ?a] ] => change (equiv_fun (equiv_compose g f) a) with (g (f a)) | [ |- context[equiv_fun (equiv_inverse ?f) ?a] ] => change (equiv_fun (equiv_inverse f) a) with (f^-1 a) | [ |- context[equiv_fun (equiv_compose' ?g ?f)] ] => change (equiv_fun (equiv_compose' g f)) with (g o f) | [ |- context[equiv_fun (equiv_compose ?g ?f)] ] => change (equiv_fun (equiv_compose g f)) with (g o f) | [ |- context[equiv_fun (equiv_inverse ?f)] ] => change (equiv_fun (equiv_inverse f)) with (f^-1) end. Ltac decomposing_intros := let x := fresh in intros x; hnf in x; cbn in x; try lazymatch type of x with | ?a = ?b => idtac | forall y:?A, ?B => idtac | Contr ?A => revert x; match goal with |- (forall y, ?P y) => snrefine (Contr_ind A P _) end | _ => elim x; clear x end; try decomposing_intros. Ltac multi_assumption := multimatch goal with [ H : ?A |- _ ] => exact H end. Ltac build_record := cbn; multi_assumption + (unshelve econstructor; build_record). Ltac make_equiv := snrefine (equiv_adjointify _ _ _ _); [ decomposing_intros; build_record | decomposing_intros; build_record | decomposing_intros; exact idpath | decomposing_intros; exact idpath ]. Ltac make_equiv_without_adjointification := snrefine (Build_Equiv _ _ _ _); [ decomposing_intros; build_record | snrefine (Build_IsEquiv _ _ _ _ _ _ _); [ decomposing_intros; build_record | decomposing_intros; exact idpath | decomposing_intros; exact idpath | decomposing_intros; exact idpath ] ]. snrefine (equiv_adjointify _ _ _ _). - intros x; cbn in x. elim x; clear x. intros x; cbn in x. elim x; clear x. intros a; cbn in a. intros b; cbn in b. intros x; cbn in x. elim x; clear x. intros c; cbn in c. intros d; cbn in d. cbn; unshelve econstructor. { cbn; exact a. } { cbn; unshelve econstructor. { cbn; unshelve econstructor. { cbn; exact b. } { cbn; exact c. } } { cbn; exact d. } } - intros x; cbn in x. elim x; clear x. intros a; cbn in a. intros x; cbn in x. elim x; clear x. intros x; cbn in x. elim x; clear x. intros b; cbn in b. intros c; cbn in c. intros d; cbn in d. cbn; unshelve econstructor. { cbn; unshelve econstructor. { cbn; exact a. } { cbn; exact b. } } { cbn; unshelve econstructor. { cbn; exact c. } { cbn; exact d. } } - intros x; cbn in x. elim x; clear x. intros a; cbn in a. intros x; cbn in x. elim x; clear x. intros x; cbn in x. elim x; clear x. intros b; cbn in b. intros c; cbn in c. intros d; cbn in d. cbn; exact idpath. - intros x; cbn in x. elim x; clear x. intros x; cbn in x. elim x; clear x. intros a; cbn in a. intros b; cbn in b. intros x; cbn in x. elim x; clear x. intros c; cbn in c. intros d; cbn in d. cbn; exact idpath. Defined.

Definition

Require Import

Basics\Equivalences.v

equiv_composeR'

(A B : Type) := prod (A -> B) (B -> A).

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\Iff.v

iff

{A} : A <-> A := (idmap , idmap).

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\Iff.v

iff_refl

{A B} : (A <-> B) -> (B <-> A) := fun f => (snd f , fst f).

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\Iff.v

iff_inverse

{A B C} (f : A <-> B) (g : B <-> C) : A <-> C := (fst g o fst f , snd f o snd g).

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\Iff.v

iff_compose

A B : ~ (A + B) <-> ~ A * ~ B. Proof. split. - intros ns. exact (ns o inl, ns o inr). - by intros []; snrapply sum_ind. Defined.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\Iff.v

iff_not_sum

A : A * ~A <-> Empty. Proof. split. - intros [a na]; exact (na a). - intros e; exact (Empty_rec _ e). Defined.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\Iff.v

iff_contradiction

n m := match n with | O => m | S n => n (S m) end.

Fixpoint

Require Import Basics.Overture Basics.Numeral.

Basics\Nat.v

tail_add

r n m := match n with | O => r | S n => (tail_add m r) n m end.

Fixpoint

Require Import Basics.Overture Basics.Numeral.

Basics\Nat.v

tail_addmul

n m := tail_addmul O n m.

Definition

Require Import Basics.Overture Basics.Numeral.

Basics\Nat.v

tail_mul

(d:Decimal.

Fixpoint

Require Import Basics.Overture Basics.Numeral.

Basics\Nat.v

of_uint_acc

(d:Decimal.

Definition

Require Import Basics.Overture Basics.Numeral.

Basics\Nat.v

of_uint

(d:Hexadecimal.

Fixpoint

Require Import Basics.Overture Basics.Numeral.

Basics\Nat.v

of_hex_uint_acc

(d:Hexadecimal.

Definition

Require Import Basics.Overture Basics.Numeral.

Basics\Nat.v

of_hex_uint

(d:Numeral.

Definition

Require Import Basics.Overture Basics.Numeral.

Basics\Nat.v

of_num_uint

n acc := match n with | O => acc | S n => n (Decimal.

Fixpoint

Require Import Basics.Overture Basics.Numeral.

Basics\Nat.v

to_little_uint

n := Decimal.

Definition

Require Import Basics.Overture Basics.Numeral.

Basics\Nat.v

to_uint

n := Numeral.

Definition

Require Import Basics.Overture Basics.Numeral.

Basics\Nat.v

to_num_uint

(d:Decimal.

Definition

Require Import Basics.Overture Basics.Numeral.

Basics\Nat.v

of_int

(d:Hexadecimal.

Definition

Require Import Basics.Overture Basics.Numeral.

Basics\Nat.v

of_hex_int

(d:Numeral.

Definition

Require Import Basics.Overture Basics.Numeral.

Basics\Nat.v

of_num_int

n := Decimal.

Definition

Require Import Basics.Overture Basics.Numeral.

Basics\Nat.v

to_int

n := Numeral.

Definition

Require Import Basics.Overture Basics.Numeral.

Basics\Nat.v

to_num_int

(i:uint) := i.

Definition

Require Import Basics.Overture Basics.Numerals.Decimal Basics.Numerals.Hexadecimal.

Basics\Numeral.v

uint_of_uint

(i:int) := i.

Definition

Require Import Basics.Overture Basics.Numerals.Decimal Basics.Numerals.Hexadecimal.

Basics\Numeral.v

int_of_int

(A : Type) : Type := | Some : A -> A | None : A.

Inductive

Basics\Overture.v

option

(A B : Type) : Type := | inl : A -> A B | inr : B -> A B.

Inductive

Basics\Overture.v

sum

(A B : Type) := pair { fst : A ; snd : B }.

Record

Basics\Overture.v

prod

(A : Type) := A -> A -> Type.

Definition

Basics\Overture.v

Relation

{A B} (b : B) := fun x : A => b.

Definition

Basics\Overture.v

const

{A} (P : A -> Type) := exist { proj1 : A ; proj2 : P proj1 ; }.

Record

Basics\Overture.v

sig

{A B C} (f : A -> B -> C) (p : A * B) : C := f (fst p) (snd p).

Definition

Basics\Overture.v

uncurry

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

Definition

Basics\Overture.v

Compose

{A B C} (g : forall b, C b) (f : A -> B) := fun x : A => g (f x).

Definition

Basics\Overture.v

composeD

{A : Type} (a : A) : A -> Type := idpath : a a.

Inductive

Basics\Overture.v

paths

paths_ind.

Definition

Basics\Overture.v

paths_rect

paths_ind' {A : Type} (P : forall (a b : A), (a = b) -> Type) : (forall (a : A), P a a idpath) -> forall (a b : A) (p : a = b), P a b p. Proof. intros H ? ? []. apply H. Defined.

Definition

Basics\Overture.v

paths_ind'

{A : Type} (a : A) (P : forall b : A, b = a -> Type) (u : P a idpath) : forall (y : A) (p : y = a), P y p. Proof. intros y p. destruct p. exact u. Defined.

Definition

Basics\Overture.v

paths_ind_r

{A : Type} {x y : A} (p : x = y) : y = x := match p with idpath => idpath end.

Definition

Basics\Overture.v

inverse

{A : Type} {x y : A} (p : x = y) : y = x := match p with idpath => idpath end.

Definition

Basics\Overture.v

inverse

{A : Type} {x y z : A} (p : x = y) (q : y = z) : x = z := match p, q with idpath, idpath => idpath end.

Definition

Basics\Overture.v

concat

{A : Type} (P : A -> Type) {x y : A} (p : x = y) (u : P x) : P y := match p with idpath => u end. Arguments {A}%_type_scope P%_function_scope {x y} p%_path_scope u : simpl nomatch. Notation "p # u" := ( _ p u) (only parsing) : path_scope. Local Lemma define_internal_paths_rew A x y P (u : P x) (H : x = y :> A) : P y. Proof. rewrite <- H. exact u. Defined.

Definition

Basics\Overture.v

transport

{A B : Type} (f : A -> B) {x y : A} (p : x = y) : f x = f y := match p with idpath => idpath end.

Definition

Basics\Overture.v

ap

{A:Type} {B:A->Type} (f:forall a:A, B a) {x y:A} (p:x=y): p # (f x) = f y := match p with idpath => idpath end.

Definition

Basics\Overture.v

apD

A (P : A -> Type) (f g : forall x, P x) := forall x, f x = g x.

Definition

Basics\Overture.v

pointwise_paths

{A} {P : A -> Type} {f g h : forall x, P x} : pointwise_paths A P f g -> pointwise_paths A P g h -> pointwise_paths A P f h := fun p q x => p x @ q x. Global Instance reflexive_pointwise_paths A P : Reflexive (pointwise_paths A P). Proof. intros ? ?; reflexivity. Defined.

Definition

Basics\Overture.v

pointwise_paths_concat

{A} {B : A -> Type} {f g : forall x, B x} (h : f = g) : f == g := fun x => match h with idpath => 1 end.

Definition

Basics\Overture.v

apD10

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

Definition

Basics\Overture.v

ap10

{A B} {f g : A -> B} (h : f = g) {x y : A} (p : x = y) : f x = g y. Proof. case h, p; reflexivity. Defined.

Definition

Basics\Overture.v

ap11

A B := { equiv_fun : A -> B ; equiv_isequiv : IsEquiv equiv_fun }.

Record

Basics\Overture.v

Equiv

{A B : Type} {f g : A <~> B} (h : f = g) : f == g := ap10 (ap equiv_fun h).

Definition

Basics\Overture.v

ap10_equiv

forall `{Funext} (A : Type) (P : A -> Type) f g, IsEquiv (@apD10 A P f g).

Axiom

Basics\Overture.v

isequiv_apD10

`{Funext} {A : Type} {P : A -> Type} (f g : forall x : A, P x) : f == g -> f = g := (@apD10 A P f g)^-1.

Definition

Basics\Overture.v

path_forall

Type0 := | minus_two : | trunc_S : -> .

Inductive

Basics\Overture.v

trunc_index

trunc_index_ind.

Definition

Basics\Overture.v

trunc_index_rect

(A : Type@{u}) : trunc_index -> Type@{u} := | Build_Contr : forall (center : A) (contr : forall y, center = y), A minus_two | istrunc_S : forall {n:trunc_index}, (forall x y:A, (x = y) n) -> A (trunc_S n).

Inductive

Basics\Overture.v

IsTrunc_internal

IsTrunc_internal_ind.

Definition

Basics\Overture.v

IsTrunc_internal_rect

(n : trunc_index) (A : Type) := match n with | minus_two => { center : A & forall y, center = y } | n.

Definition

Basics\Overture.v

IsTrunc_unfolded

(n : trunc_index) (A : Type) : IsTrunc n A -> IsTrunc_unfolded n A. Proof. intros [center contr|k istrunc]. - exact (center; contr). - exact istrunc. Defined.

Definition

Basics\Overture.v

istrunc_unfold

(n : trunc_index) (A : Type) : IsEquiv (istrunc_unfold n A). Proof. simple refine (Build_IsEquiv _ _ (istrunc_unfold n A) _ _ _ _). - destruct n. + intros [center contr]; exact (Build_Contr _ center contr). + intros H. exact (istrunc_S _ H). - destruct n; reflexivity. - intros [center contr|k istrunc]; reflexivity. - intros [center contr|k istrunc]; reflexivity. Defined.

Definition

Basics\Overture.v

isequiv_istrunc_unfold

(n : trunc_index) (A : Type) := Build_Equiv _ _ _ (isequiv_istrunc_unfold n A).

Definition

Basics\Overture.v

equiv_istrunc_unfold

(A : Type) {H : Contr A} : A := pr1 (istrunc_unfold _ _ H).

Definition

Basics\Overture.v

center

{A : Type} {H : Contr A} (y : A) : center A = y := pr2 (istrunc_unfold _ _ H) y.

Definition

Basics\Overture.v

contr

{n : trunc_index} {A : Type} (istrunc : IsTrunc n A) : A -> Type. Proof. intro y. destruct n. - exact (center A = y). - exact (forall x : A, IsTrunc n (y = x)). Defined.

Definition

Basics\Overture.v

istrunc_codomain_fam

{n : trunc_index} {A : Type} (istrunc : IsTrunc n A) : forall y : A, istrunc_codomain_fam istrunc y. Proof. destruct n. - exact (@contr A istrunc). - exact (istrunc_unfold _ _ istrunc). Defined.

Definition

Basics\Overture.v

istrunc_fun

Type0 := | O : | S : -> .

Inductive

Basics\Overture.v

nat

Type0 := .

Inductive

Basics\Overture.v

Empty

Empty_ind.

Definition

Basics\Overture.v

Empty_rect

(A : Type) := A -> Empty.

Definition

Basics\Overture.v

not

{A} {x y : A} : x <> y -> y <> x := fun np p => np (p^).

Definition

Basics\Overture.v

symmetric_neq

{A} (R : Relation A) : Relation A := fun x y => ~ (R x y).

Definition

Basics\Overture.v

complement

Type0 := tt : .

Inductive

Basics\Overture.v

Unit

Unit_ind.

Definition

Basics\Overture.v

Unit_rect

Record

Basics\Overture.v

pType

{A B : Type} (f : A -> B) (y : B) := { x : A & f x = y }.

Definition

Basics\Overture.v

hfiber

issmall_hprop@{i j | } : forall `{PropResizing} (X : Type@{j}) (T : IsHProp X), IsSmall@{i j} X.

Axiom

Basics\Overture.v

issmall_hprop@

{A : Type} {x y : A} (p : x = y) : p @ 1 = p := match p with idpath => 1 end.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

concat_p1

{A : Type} {x y : A} (p : x = y) : 1 @ p = p := match p with idpath => 1 end.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

concat_1p

{A : Type} {x y : A} (p : x = y) : p @ 1 = 1 @ p := concat_p1 p @ (concat_1p p)^.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

concat_p1_1p

{A : Type} {x y : A} (p : x = y) : 1 @ p = p @ 1 := concat_1p p @ (concat_p1 p)^.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

concat_1p_p1

{A : Type} {x y z t : A} (p : x = y) (q : y = z) (r : z = t) : p @ (q @ r) = (p @ q) @ r := match r with idpath => match q with idpath => match p with idpath => 1 end end end.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

concat_p_pp

{A : Type} {x y z t : A} (p : x = y) (q : y = z) (r : z = t) : (p @ q) @ r = p @ (q @ r) := match r with idpath => match q with idpath => match p with idpath => 1 end end end.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

concat_pp_p

{A : Type} {x y : A} (p : x = y) : p @ p^ = 1 := match p with idpath => 1 end.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

concat_pV

{A : Type} {x y : A} (p : x = y) : p^ @ p = 1 := match p with idpath => 1 end.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

concat_Vp

{A : Type} {x y z : A} (p : x = y) (q : y = z) : p^ @ (p @ q) = q := match q with idpath => match p with idpath => 1 end end.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

concat_V_pp

{A : Type} {x y z : A} (p : x = y) (q : x = z) : p @ (p^ @ q) = q := match q with idpath => match p with idpath => 1 end end.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

concat_p_Vp

{A : Type} {x y z : A} (p : x = y) (q : y = z) : (p @ q) @ q^ = p := match q with idpath => match p with idpath => 1 end end.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

concat_pp_V

{A : Type} {x y z : A} (p : x = z) (q : y = z) : (p @ q^) @ q = p := (match q as i return forall p, (p @ i^) @ i = p with idpath => fun p => match p with idpath => 1 end end) p.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

concat_pV_p

{A : Type} {x y z : A} (p : x = y) (q : y = z) : (p @ q)^ = q^ @ p^ := match q with idpath => match p with idpath => 1 end end.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

inv_pp