Datasets:

phanerozoic

/

Coq-HoTT

like 2

{A : Type} (P : A -> Type) {a b : A} (p : a = b) (x : P a) : transport_pp P p 1 x = transport2 P (concat_p1 p) x. Proof. by induction p. Defined.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

transport_pp_1

{A} {P Q : A -> Type} {x y : A} (p : x = y) (f : forall x, P x -> Q x) (z : P x) : f y (p # z) = (p # (f x z)). Proof. by induction p. Defined.

Lemma

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

ap_transport

{A : Type} (B : A -> Type) (C1 C2 : forall a : A, B a -> Type) (f : forall a b, C1 a b -> C2 a b) {x1 x2 : A} (p : x1 = x2) (y : B x1) (z : C1 x1 y) : f x2 (p # y) (transportD B C1 p y z) = transportD B C2 p y (f x1 y z). Proof. by induction p. Defined.

Lemma

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

ap_transportD

{A : Type} (B C : A -> Type) (D1 D2 : forall a, B a -> C a -> Type) (f : forall a b c, D1 a b c -> D2 a b c) {x1 x2 : A} (p : x1 = x2) (y : B x1) (z : C x1) (w : D1 x1 y z) : f x2 (p # y) (p # z) (transportD2 B C D1 p y z w) = transportD2 B C D2 p y z (f x1 y z w). Proof. by induction p. Defined.

Lemma

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

ap_transportD2

{X} (Y : X -> Type) {x1 x2 : X} (p : x1 = x2) {y1 y2 : Y x2} (q : y1 = y2) : ap (transport Y p) (ap (transport Y p^) q) = transport_pV Y p y1 @ q @ (transport_pV Y p y2)^. Proof. destruct p, q; reflexivity. Defined.

Lemma

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

ap_transport_pV

{X} (P : X -> Type) (f : forall x, P x) {x1 x2 : X} (p : x1 = x2) : ap (transport P p) (apD f p^) @ apD f p = transport_pV P p (f x2). Proof. destruct p; reflexivity. Defined.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

transport_pV_ap

{A} {P : A -> Type} (f : forall x, P x) {x y z : A} (p : x = y) (q : y = z) : apD f (p @ q) = transport_pp P p q (f x) @ ap (transport P q) (apD f p) @ apD f q. Proof. destruct p, q; reflexivity. Defined.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

apD_pp

{A} {P : A -> Type} (f : forall x, P x) {x y : A} (p : x = y) : apD f p^ = moveR_transport_V _ _ _ _ (apD f p)^. Proof. destruct p; reflexivity. Defined.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

apD_V

{A B : Type} {x1 x2 : A} (p : x1 = x2) (y : B) : transport (fun x => B) p y = y. Proof. destruct p. exact 1. Defined.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

transport_const

{A B : Type} {x1 x2 : A} {p q : x1 = x2} (r : p = q) (y : B) : transport_const p y = transport2 (fun _ => B) r y @ transport_const q y := match r with idpath => (concat_1p _)^ end.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

transport2_const

{A B} {x y : A} (P : B -> Type) (f : A -> B) (p : x = y) (z : P (f x)) : transport (fun x => P (f x)) p z = transport P (ap f p) z. Proof. destruct p; reflexivity. Defined.

Lemma

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

transport_compose

{A A'} B {x x' : A} (C : forall x : A', B x -> Type) (f : A -> A') (p : x = x') y (z : C (f x) y) : transportD (B o f) (C oD f) p y z = transport (C (f x')) (transport_compose B f p y)^ (transportD B C (ap f p) y z). Proof. destruct p; reflexivity. Defined.

Lemma

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

transportD_compose

{A} B (f : forall x : A, B x) (C : forall x, B x -> Type) {x1 x2 : A} (p : x1 = x2) (z : C x1 (f x1)) : apD f p # transportD B C p _ z = transport (fun x => C x (f x)) p z. Proof. destruct p; reflexivity. Defined.

Lemma

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

transport_apD_transportD

{A B C} (f : A -> B) (g g' : B -> C) (p : g = g') : transport (fun h : B -> C => g o f = h o f) p 1 = ap (fun h => h o f) p. Proof. destruct p; reflexivity. Defined.

Lemma

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

transport_precompose

{A} (P : A -> Type) {x y : A} (p : x = y) (u : P x) : transport P p u = transport idmap (ap P p) u := match p with idpath => idpath end.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

transport_idmap_ap

{A B} (C : A -> B -> Type) {x1 x2 : A} (p : x1 = x2) {y1 y2 : B} (q : y1 = y2) (c : C x1 y1) : transport (C x2) q (transport (fun x => C x y1) p c) = transport (fun x => C x y2) p (transport (C x1) q c). Proof. destruct p, q; reflexivity. Defined.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

transport_transport

{A B} {x y : A} (f : A -> B) (p: x = y) : apD f p = transport_const p (f x) @ ap f p. Proof. destruct p; reflexivity. Defined.

Lemma

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

apD_const

{A A' : Type} {B : A' -> Type} (g : A -> A') (f : forall a, B a) {x y : A} (p : x = y) : apD (f o g) p = (transport_compose _ _ _ _) @ apD f (ap g p). Proof. by destruct p. Defined.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

apD_compose

apD_compose' {A A' : Type} {B : A' -> Type} (g : A -> A') (f : forall a, B a) {x y : A} (p : x = y) : apD f (ap g p) = (transport_compose B _ _ _)^ @ apD (f o g) p. Proof. by destruct p. Defined.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

apD_compose'

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

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

concat2

{A B : Type} {x' y' z' : B} (f : A -> (x' = y')) (g : A -> (y' = z')) {x y : A} (p : x = y) : (ap f p) @@ (ap g p) = ap (fun u => f u @ g u) p. Proof. by path_induction. Defined.

Lemma

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

concat2_ap_ap

{A : Type} {x y : A} {p q : x = y} (h : p = q) : p^ = q^ := ap inverse h.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

inverse2

{A B} (f : A -> B) {a b : A} (p : a = b) : ap_pp f p 1 @ concat_p1 (ap f p) = ap (ap f) (concat_p1 p). Proof. destruct p; reflexivity. Defined.

Lemma

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

ap_pp_concat_p1

{A B} (f : A -> B) {a b : A} (p : a = b) : ap_pp f 1 p @ concat_1p (ap f p) = ap (ap f) (concat_1p p). Proof. destruct p; reflexivity. Defined.

Lemma

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

ap_pp_concat_1p

{A B} (f : A -> B) {x y : A} (p : x = y) : ap_pp f p p^ @ ((1 @@ ap_V f p) @ concat_pV (ap f p)) = ap (ap f) (concat_pV p). Proof. destruct p; reflexivity. Defined.

Lemma

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

ap_pp_concat_pV

{A B} (f : A -> B) {x y : A} (p : x = y) : ap_pp f p^ p @ ((ap_V f p @@ 1) @ concat_Vp (ap f p)) = ap (ap f) (concat_Vp p). Proof. destruct p; reflexivity. Defined.

Lemma

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

ap_pp_concat_Vp

{A} {x y : A} (p q : x = y) (r : p = q) : (r @@ inverse2 r) @ concat_pV q = concat_pV p. Proof. destruct r, p; reflexivity. Defined.

Lemma

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

concat_pV_inverse2

{A} {x y : A} (p q : x = y) (r : p = q) : (inverse2 r @@ r) @ concat_Vp q = concat_Vp p. Proof. destruct r, p; reflexivity. Defined.

Lemma

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

concat_Vp_inverse2

{A : Type} {x y z : A} (p : x = y) {q r : y = z} (h : q = r) : p @ q = p @ r := 1 @@ h.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

whiskerL

{A : Type} {x y z : A} {p q : x = y} (h : p = q) (r : y = z) : p @ r = q @ r := h @@ 1.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

whiskerR

{A} {x y z : A} (p : x = y) (q r : y = z) : (p @ q = p @ r) -> (q = r) := fun h => (concat_V_pp p q)^ @ whiskerL p^ h @ (concat_V_pp p r).

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

cancelL

{A} {x y z : A} (p q : x = y) (r : y = z) : (p @ r = q @ r) -> (p = q) := fun h => (concat_pp_V p r)^ @ whiskerR h r^ @ (concat_pp_V q r).

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

cancelR

{A : Type} {x y : A} {p q : x = y} (h : p = q) : (concat_p1 p)^ @ whiskerR h 1 @ concat_p1 q = h := match h with idpath => match p with idpath => 1 end end.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

whiskerR_p1

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

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

whiskerR_1p

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

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

whiskerL_p1

{A : Type} {x y : A} {p q : x = y} (h : p = q) : (concat_1p p) ^ @ whiskerL 1 h @ concat_1p q = h := match h with idpath => match p with idpath => 1 end end.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

whiskerL_1p

{A} {x : A} (h : idpath x = idpath x) : whiskerR h 1 = h. Proof. refine (_ @ whiskerR_p1 h); simpl. symmetry; refine (concat_p1 _ @ concat_1p _). Defined.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

whiskerR_p1_1

{A} {x : A} (h : idpath x = idpath x) : whiskerL 1 h = h. Proof. refine (_ @ whiskerL_1p h); simpl. symmetry; refine (concat_p1 _ @ concat_1p _). Defined.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

whiskerL_1p_1

{A : Type} {x y z : A} {p p' : x = y} {q q' : y = z} (g : p = p') (h k : q = q') : (g @@ h = g @@ k) -> (h = k). Proof. intro r. induction g, p, q. refine ((whiskerL_1p h)^ @ _). refine (_ @ (whiskerL_1p k)). refine (whiskerR _ _). refine (whiskerL _ _). apply r. Defined.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

cancel2L

{A : Type} {x y z : A} {p p' : x = y} {q q' : y = z} (g h : p = p') (k : q = q') : (g @@ k = h @@ k) -> (g = h). Proof. intro r. induction k, p, q. refine ((whiskerR_p1 g)^ @ _). refine (_ @ (whiskerR_p1 h)). refine (whiskerR _ _). refine (whiskerL _ _). apply r. Defined.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

cancel2R

{A} {x y z : A} (p : x = y) {q q' q'' : y = z} (r : q = q') (s : q' = q'') : whiskerL p (r @ s) = whiskerL p r @ whiskerL p s. Proof. destruct p, r, s; reflexivity. Defined.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

whiskerL_pp

{A} {x y z : A} {p p' p'' : x = y} (q : y = z) (r : p = p') (s : p' = p'') : whiskerR (r @ s) q = whiskerR r q @ whiskerR s q. Proof. destruct q, r, s; reflexivity. Defined.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

whiskerR_pp

{A} {x y z : A} (p : x = y) {q q' : y = z} (r : q = q') : (concat_V_pp p q)^ @ whiskerL p^ (whiskerL p r) @ concat_V_pp p q' = r. Proof. destruct p, r, q. reflexivity. Defined.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

whiskerL_VpL

{A} {x y z : A} (p : y = x) {q q' : y = z} (r : q = q') : (concat_p_Vp p q)^ @ whiskerL p (whiskerL p^ r) @ concat_p_Vp p q' = r. Proof. destruct p, r, q. reflexivity. Defined.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

whiskerL_pVL

{A} {x y z : A} {p p' : x = y} (r : p = p') (q : y = z) : (concat_pp_V p q)^ @ whiskerR (whiskerR r q) q^ @ concat_pp_V p' q = r. Proof. destruct p, r, q. reflexivity. Defined.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

whiskerR_pVR

{A} {x y z : A} {p p' : x = y} (r : p = p') (q : z = y) : (concat_pV_p p q)^ @ whiskerR (whiskerR r q^) q @ concat_pV_p p' q = r. Proof. destruct p, r, q. reflexivity. Defined.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

whiskerR_VpR

{A} {w x y z : A} {p p' : w = x} (h : p = p') (q : x = y) (r : y = z) : whiskerR h (q @ r) @ concat_p_pp p' q r = concat_p_pp p q r @ whiskerR (whiskerR h q) r. Proof. by destruct h, p, q, r. Defined.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

concat_p_pp_nat_l

{A} {w x y z : A} (p : w = x) {q q' : x = y} (h : q = q') (r : y = z) : whiskerL p (whiskerR h r) @ concat_p_pp p q' r = concat_p_pp p q r @ whiskerR (whiskerL p h) r. Proof. by destruct h, p, q, r. Defined.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

concat_p_pp_nat_m

{A} {w x y z : A} (p : w = x) (q : x = y) {r r' : y = z} (h : r = r') : whiskerL p (whiskerL q h) @ concat_p_pp p q r' = concat_p_pp p q r @ whiskerL (p @ q) h. Proof. by destruct h, p, q, r. Defined.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

concat_p_pp_nat_r

{A : Type} {x y z : A} {p p' p'' : x = y} {q q' q'' : y = z} (a : p = p') (b : p' = p'') (c : q = q') (d : q' = q'') : (a @@ c) @ (b @@ d) = (a @ b) @@ (c @ d). Proof. case d. case c. case b. case a. reflexivity. Defined.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

concat_concat2

{A} {x y z : A} (p p' : x = y) (q q' : y = z) (a : p = p') (b : q = q') : (whiskerR a q) @ (whiskerL p' b) = (whiskerL p b) @ (whiskerR a q') := match b with idpath => match a with idpath => (concat_1p _)^ end end.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

concat_whisker

{A : Type} {v w x y z : A} (p : v = w) (q : w = x) (r : x = y) (s : y = z) : whiskerL p (concat_p_pp q r s) @ concat_p_pp p (q@r) s @ whiskerR (concat_p_pp p q r) s = concat_p_pp p q (r@s) @ concat_p_pp (p@q) r s. Proof. case p, q, r, s. reflexivity. Defined.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

pentagon

{A : Type} {x y z : A} (p : x = y) (q : y = z) : concat_p_pp p 1 q @ whiskerR (concat_p1 p) q = whiskerL p (concat_1p q). Proof. case p, q. reflexivity. Defined.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

triangulator

{A : Type} {x:A} (p q : 1 = 1 :> (x = x)) : p @ q = q @ p := (whiskerR_p1 p @@ whiskerL_1p q)^ @ (concat_p1 _ @@ concat_p1 _) @ (concat_1p _ @@ concat_1p _) @ (concat_whisker _ _ _ _ p q) @ (concat_1p _ @@ concat_1p _)^ @ (concat_p1 _ @@ concat_p1 _)^ @ (whiskerL_1p q @@ whiskerR_p1 p).

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

eckmann_hilton

{A B : Type} (f:A->B) {x y:A} {p q:x=y} (r:p=q) : ap f p = ap f q := ap (ap f) r.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

ap02

{A B} (f:A->B) {x y:A} {p p' p'':x=y} (r:p=p') (r':p'=p'') : ap02 f (r @ r') = ap02 f r @ ap02 f r' := ap_pp (ap f) r r'.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

ap02_pp

{A B} (f:A->B) {x y z:A} {p p':x=y} {q q':y=z} (r:p=p') (s:q=q') : ap02 f (r @@ s) = ap_pp f p q @ (ap02 f r @@ ap02 f s) @ (ap_pp f p' q')^. Proof. case r, s, p, q. reflexivity. Defined.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

ap02_p2p

{A : Type} {B : A -> Type} {x y : A} {p q : x = y} (f : forall x, B x) (r : p = q) : apD f p = transport2 B r (f x) @ apD f q := match r with idpath => (concat_1p _)^ end.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

apD02

{A B : Type} (f : A -> B) {x y : A} {p q : x = y} (r : p = q) : apD02 f r = (apD_const f p) @ (transport2_const r (f x) @@ ap02 f r) @ (concat_p_pp _ _ _)^ @ (whiskerL (transport2 _ r (f x)) (apD_const f q)^) := match r with idpath => match p with idpath => 1 end end.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

apD02_const

{A} (B : A -> Type) (f : forall x:A, B x) {x y : A} {p1 p2 p3 : x = y} (r1 : p1 = p2) (r2 : p2 = p3) : apD02 f (r1 @ r2) = apD02 f r1 @ whiskerL (transport2 B r1 (f x)) (apD02 f r2) @ concat_p_pp _ _ _ @ (whiskerR (transport2_p2p B r1 r2 (f x))^ (apD f p3)). Proof. destruct r1, r2. destruct p1. reflexivity. Defined.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

apD02_pp

{A B C} (f : A -> B -> C) {x x' y y'} {p p' : x = x'} (r : p = p') {q q' : y = y'} (s : q = q') : ap011 f p q = ap011 f p' q'. Proof. destruct r, p. apply ap02. exact s. Defined.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

ap022

{A B} (p q : A = B) (r : q = p) (z : A) : ap (transport idmap q^) (ap (fun s => transport idmap s z) r) @ ap (fun s => transport idmap s (p # z)) (inverse2 r) @ transport_Vp idmap p z = transport_Vp idmap q z. Proof. by path_induction. Defined.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

ap_transport_Vp_idmap

{A B} (p q : A = B) (r : q = p) (z : B) : ap (transport idmap q) (ap (fun s => transport idmap s^ z) r) @ ap (fun s => transport idmap s (p^ # z)) r @ transport_pV idmap p z = transport_pV idmap q z. Proof. by path_induction. Defined.

Definition

Require Import Basics.Overture Basics.Tactics.

Basics\PathGroupoids.v

ap_transport_pV_idmap

{A : Type} (a : A) := a.

Definition

Require Import Basics.Overture.

Basics\Tactics.v

fix_proto

_ <~> MyRecord := ltac:(issig).

Definition

Require Import Basics.Overture.

Basics\Tactics.v

issig_myrecord

(A B : Type) : {f : A -> B & IsEquiv f} <~> Equiv A B. Proof. issig. Defined.

Definition

Require Import Basics.Overture.

Basics\Tactics.v

issig_equiv

{A B : Type} (f : A -> B) : {g : B -> A & {r : f o g == idmap & { s : g o f == idmap & forall x : A, r (f x) = ap f (s x)}}} <~> IsEquiv f. Proof. issig. Defined.

Definition

Require Import Basics.Overture.

Basics\Tactics.v

issig_isequiv

trunc_index_inc@{} (k : trunc_index) (n : nat) : trunc_index := match n with | O => k | S m => (trunc_index_inc k m).

Fixpoint

Require Import

Basics\Trunc.v

trunc_index_inc@

trunc_index_inc'@{} (k : trunc_index) (n : nat) : trunc_index := match n with | O => k | S m => (trunc_index_inc' k.

Fixpoint

Require Import

Basics\Trunc.v

trunc_index_inc'@

trunc_index_inc'_succ@{} (n : nat) (k : trunc_index) : trunc_index_inc' k.+1 n = (trunc_index_inc' k n).+1. Proof. revert k; simple_induction n n IHn; intro k. - reflexivity. - apply (IHn k.+1). Defined.

Definition

Require Import

Basics\Trunc.v

trunc_index_inc'_succ@

trunc_index_inc_agree@{} (k : trunc_index) (n : nat) : trunc_index_inc k n = trunc_index_inc' k n. Proof. simple_induction n n IHn. - reflexivity. - simpl. refine (ap _ IHn @ _). symmetry; apply trunc_index_inc'_succ. Defined.

Definition

Require Import

Basics\Trunc.v

trunc_index_inc_agree@

nat_to_trunc_index@{} (n : nat) : trunc_index := (trunc_index_inc minus_two n).

Definition

Require Import

Basics\Trunc.v

nat_to_trunc_index@

(n : nat) : trunc_index_inc' 0%nat n = n. Proof. induction n as [|n p]. 1: reflexivity. refine (trunc_index_inc'_succ _ _ @ _). exact (ap _ p). Defined.

Definition

Require Import

Basics\Trunc.v

trunc_index_inc'_0n

int_to_trunc_index@{} (v : Decimal.

Definition

Require Import

Basics\Trunc.v

int_to_trunc_index@

num_int_to_trunc_index@{} (v : Numeral.

Definition

Require Import

Basics\Trunc.v

num_int_to_trunc_index@

trunc_index_to_little_uint@{} n acc := match n with | minus_two => acc | minus_two.

Fixpoint

Require Import

Basics\Trunc.v

trunc_index_to_little_uint@

trunc_index_to_int@{} n := match n with | minus_two => Decimal.

Definition

Require Import

Basics\Trunc.v

trunc_index_to_int@

trunc_index_to_num_int@{} n := Numeral.

Definition

Require Import

Basics\Trunc.v

trunc_index_to_num_int@

(n : trunc_index) : nat := match n with | minus_two => 0%nat | n'.

Fixpoint

Require Import

Basics\Trunc.v

trunc_index_to_nat

trunc_index_add@{} (m n : trunc_index) : trunc_index := match m with | -2 => n | m'.

Fixpoint

Require Import

Basics\Trunc.v

trunc_index_add@

trunc_index_add_minus_two@{} m : m +2+ -2 = m. Proof. simple_induction m m IHm. 1: reflexivity. cbn; apply ap. assumption. Defined.

Definition

Require Import

Basics\Trunc.v

trunc_index_add_minus_two@

trunc_index_add_succ@{} m n : m +2+ n.+1 = (m +2+ n).+1. Proof. revert m; simple_induction n n IHn; intro m; simple_induction m m IHm. 1,3: reflexivity. all: cbn; apply ap. all: assumption. Defined.

Definition

Require Import

Basics\Trunc.v

trunc_index_add_succ@

trunc_index_add_comm@{} m n : m +2+ n = n +2+ m. Proof. simple_induction n n IHn. - apply trunc_index_add_minus_two. - exact (trunc_index_add_succ _ _ @ ap trunc_S IHn). Defined.

Definition

Require Import

Basics\Trunc.v

trunc_index_add_comm@

trunc_index_leq@{} (m n : trunc_index) : Type0 := match m, n with | -2, _ => Unit | m'.+1, -2 => Empty | m'.+1, n'.+1 => trunc_index_leq m' n' end. Existing Class trunc_index_leq. Notation "m <= n" := (trunc_index_leq m n) : trunc_scope. Global Instance trunc_index_leq_minus_two_n@{} n : -2 <= n := tt. Global Instance trunc_index_leq_succ@{} n : n <= n.+1. Proof. by induction n as [|n IHn] using trunc_index_ind. Defined.

Fixpoint

Require Import

Basics\Trunc.v

trunc_index_leq@

trunc_index_pred@{} : trunc_index -> trunc_index. Proof. intros [|m]. 1: exact (-2). exact m. Defined.

Definition

Require Import

Basics\Trunc.v

trunc_index_pred@

trunc_index_succ_pred@{} (n : nat) : (n.-1).+1 = n. Proof. simple_induction n n IHn. 1: reflexivity. unfold nat_to_trunc_index in *; cbn in *. refine (ap trunc_S IHn). Defined.

Definition

Require Import

Basics\Trunc.v

trunc_index_succ_pred@

trunc_index_leq_minus_two@{} {n} : n <= -2 -> n = -2. Proof. destruct n. 1: reflexivity. contradiction. Defined.

Definition

Require Import

Basics\Trunc.v

trunc_index_leq_minus_two@

trunc_index_leq_succ'@{} n m : n <= m -> n <= m.+1. Proof. revert m. induction n as [|n IHn] using trunc_index_ind. 1: exact _. intros m p; cbn. induction m as [|m IHm] using trunc_index_ind. 1: destruct p. apply IHn, p. Defined.

Definition

Require Import

Basics\Trunc.v

trunc_index_leq_succ'@

trunc_index_leq_add@{} n m : n <= m +2+ n. Proof. simple_induction m m IHm. - reflexivity. - rapply trunc_index_leq_transitive. Defined.

Definition

Require Import

Basics\Trunc.v

trunc_index_leq_add@

trunc_index_leq_add'@{} n m : n <= n +2+ m. Proof. rewrite trunc_index_add_comm. apply trunc_index_leq_add. Defined.

Definition

Require Import

Basics\Trunc.v

trunc_index_leq_add'@

trunc_index_min@{} (n m : trunc_index) : trunc_index. Proof. destruct n. 1: exact (-2). destruct m. 1: exact (-2). exact (trunc_index_min n m).+1. Defined.

Fixpoint

Require Import

Basics\Trunc.v

trunc_index_min@

trunc_index_min_minus_two@{} n : trunc_index_min n (-2) = -2. Proof. by destruct n. Defined.

Definition

Require Import

Basics\Trunc.v

trunc_index_min_minus_two@

trunc_index_min_swap@{} n m : trunc_index_min n m = trunc_index_min m n. Proof. revert m. simple_induction n n IHn; intro m. { symmetry. apply trunc_index_min_minus_two. } simple_induction m m IHm. 1: reflexivity. cbn; apply ap, IHn. Defined.

Definition

Require Import

Basics\Trunc.v

trunc_index_min_swap@

trunc_index_min_path@{} n m : (trunc_index_min n m = n) + (trunc_index_min n m = m). Proof. revert m; simple_induction n n IHn; intro m. 1: by apply inl. simple_induction m m IHm. 1: by apply inr. destruct (IHn m). 1: apply inl. 2: apply inr. 1,2: cbn; by apply ap. Defined.

Definition

Require Import

Basics\Trunc.v

trunc_index_min_path@

trunc_index_min_leq_left@{} (n m : trunc_index) : trunc_index_min n m <= n. Proof. revert n m. refine (trunc_index_ind _ _ _); [ | intros n IHn ]. all: refine (trunc_index_ind _ _ _); [ | intros m IHm ]. all: try exact tt. exact (IHn m). Defined.

Definition

Require Import

Basics\Trunc.v

trunc_index_min_leq_left@

trunc_index_min_leq_right@{} (n m : trunc_index) : trunc_index_min n m <= m. Proof. revert n m. refine (trunc_index_ind _ _ _); [ | intros n IHn ]. all: refine (trunc_index_ind _ _ _); [ | intros m IHm ]. all: try exact tt. exact (IHn m). Defined.

Definition

Require Import

Basics\Trunc.v

trunc_index_min_leq_right@

`{H : IsTrunc (-2) A} : Contr A := H. Global Instance istrunc_paths' {n : trunc_index} {A : Type} `{IsTrunc n A} : forall x y : A, IsTrunc n (x = y) | 1000. Proof. generalize dependent A. simple_induction n n IH; simpl; intros A H x y. - apply contr_paths_contr. - apply istrunc_S. rapply IH. Defined.

Definition

Require Import

Basics\Trunc.v

contr_istrunc_minus_two

{m n} (Hmn : m <= n) `{IsTrunc m A} : IsTrunc n A. Proof. generalize dependent A; generalize dependent m. simple_induction n n' IH; intros [ | m'] Hmn A ? . - assumption. - destruct Hmn. - apply @istrunc_succ, (IH (-2)); auto. - apply istrunc_S. intros x y; apply (IH m'); auto with typeclass_instances. Defined.

Definition

Require Import

Basics\Trunc.v

istrunc_leq

{n} {A} `{Contr A} : IsTrunc n.

Definition

Require Import

Basics\Trunc.v

istrunc_contr

{n} {A} `{IsHProp A} : IsTrunc n.

Definition

Require Import

Basics\Trunc.v

istrunc_hprop