Datasets:

phanerozoic

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Coq-HoTT

like 2

{A} (f : A -> A) p a : pos_iter f (pos_succ p) a = f (pos_iter f p a). Proof. unfold pos_iter. by rewrite pos_peano_rec_beta_pos_succ. Qed.

Lemma

Require Import Basics.Overture Basics.Tactics. Require Import Pos.Core.

Spaces\Pos\Spec.v

pos_iter_succ_l

{A} (f : A -> A) p a : pos_iter f (pos_succ p) a = pos_iter f p (f a). Proof. revert p f a. srapply pos_peano_ind. 1: hnf; intros; trivial. hnf; intros p q f a. refine (_ @ _ @ _^). 1,3: unfold pos_iter; by rewrite pos_peano_rec_beta_pos_succ. apply ap. apply q. Qed.

Lemma

Require Import Basics.Overture Basics.Tactics. Require Import Pos.Core.

Spaces\Pos\Spec.v

pos_iter_succ_r

p q : p * q~0 = (p * q)~0. Proof. induction p; simpl; f_ap; f_ap; trivial. Qed.

Lemma

Require Import Basics.Overture Basics.Tactics. Require Import Pos.Core.

Spaces\Pos\Spec.v

mul_xO_r

p q : p * q~1 = p + (p * q)~0. Proof. induction p; simpl; trivial; f_ap. rewrite IHp. rewrite pos_add_assoc. rewrite (pos_add_comm q p). symmetry. apply pos_add_assoc. Qed.

Lemma

Require Import Basics.Overture Basics.Tactics. Require Import Pos.Core.

Spaces\Pos\Spec.v

mul_xI_r

p q : p * q = q * p. Proof. induction q; simpl. 1: apply pos_mul_1_r. + rewrite mul_xO_r. f_ap. + rewrite mul_xI_r. f_ap; f_ap. Qed.

Lemma

Require Import Basics.Overture Basics.Tactics. Require Import Pos.Core.

Spaces\Pos\Spec.v

pos_mul_comm

p q r : p * (q + r) = p * q + p * r. Proof. induction p; cbn; [reflexivity | f_ap | ]. rewrite IHp. set (m:=(p*q)~0). set (n:=(p*r)~0). change ((p*q+p*r)~0) with (m+n). rewrite 2 pos_add_assoc; f_ap. rewrite <- 2 pos_add_assoc; f_ap. apply pos_add_comm. Qed.

Theorem

Require Import Basics.Overture Basics.Tactics. Require Import Pos.Core.

Spaces\Pos\Spec.v

pos_mul_add_distr_l

p q r : (p + q) * r = p * r + q * r. Proof. rewrite 3 (pos_mul_comm _ r); apply pos_mul_add_distr_l. Qed.

Theorem

Require Import Basics.Overture Basics.Tactics. Require Import Pos.Core.

Spaces\Pos\Spec.v

pos_mul_add_distr_r

p q r : p * (q * r) = p * q * r. Proof. induction p; simpl; rewrite ?IHp; trivial. by rewrite pos_mul_add_distr_r. Qed.

Theorem

Require Import Basics.Overture Basics.Tactics. Require Import Pos.Core.

Spaces\Pos\Spec.v

pos_mul_assoc

p q : (pos_succ p) * q = p * q + q. Proof. by rewrite <- pos_add_1_r, pos_mul_add_distr_r, pos_mul_1_l. Qed.

Lemma

Require Import Basics.Overture Basics.Tactics. Require Import Pos.Core.

Spaces\Pos\Spec.v

pos_mul_succ_l

p q : p * (pos_succ q) = p + p * q. Proof. by rewrite <- pos_add_1_l, pos_mul_add_distr_l, pos_mul_1_r. Qed.

Lemma

Require Import Basics.Overture Basics.Tactics. Require Import Pos.Core.

Spaces\Pos\Spec.v

pos_mul_succ_r

tbase = tbase.

Axiom

Require Import Basics.Overture Basics.Equivalences Cubical.DPath

Spaces\Torus\Torus.v

loop_a

tbase = tbase.

Axiom

Require Import Basics.Overture Basics.Equivalences Cubical.DPath

Spaces\Torus\Torus.v

loop_b

PathSquare loop_a loop_a loop_b loop_b.

Axiom

Require Import Basics.Overture Basics.Equivalences Cubical.DPath

Spaces\Torus\Torus.v

surf

(P : Torus -> Type) (pb : P tbase) (pla : DPath P loop_a pb pb) (plb : DPath P loop_b pb pb) (ps : DPathSquare P surf pla pla plb plb) (x : Torus) : P x := (match x with tbase => fun _ _ _ => pb end) pla plb ps.

Definition

Require Import Basics.Overture Basics.Equivalences Cubical.DPath

Spaces\Torus\Torus.v

Torus_ind

forall (P : Torus -> Type) (pb : P tbase) (pla : DPath P loop_a pb pb) (plb : DPath P loop_b pb pb) (ps : DPathSquare P surf pla pla plb plb), DPathSquare P hr (apD (Torus_ind P pb pla plb ps) (loop_a)) pla 1%dpath 1%dpath.

Axiom

Require Import Basics.Overture Basics.Equivalences Cubical.DPath

Spaces\Torus\Torus.v

Torus_ind_beta_loop_a

forall (P : Torus -> Type) (pb : P tbase) (pla : DPath P loop_a pb pb) (plb : DPath P loop_b pb pb) (ps : DPathSquare P surf pla pla plb plb), DPathSquare P hr (apD (Torus_ind P pb pla plb ps) (loop_b)) plb 1%dpath 1%dpath.

Axiom

Require Import Basics.Overture Basics.Equivalences Cubical.DPath

Spaces\Torus\Torus.v

Torus_ind_beta_loop_b

forall (P : Torus -> Type) (pb : P tbase) (pla : DPath P loop_a pb pb) (plb : DPath P loop_b pb pb) (ps : DPathSquare P surf pla pla plb plb), DPathCube P (cu_refl_lr _) (ds_apD (Torus_ind P pb pla plb ps) surf) ps (Torus_ind_beta_loop_a _ _ _ _ _) (Torus_ind_beta_loop_a _ _ _ _ _) (Torus_ind_beta_loop_b _ _ _ _ _) (Torus_ind_beta_loop_b _ _ _ _ _).

Axiom

Require Import Basics.Overture Basics.Equivalences Cubical.DPath

Spaces\Torus\Torus.v

Torus_ind_beta_surf

(P : Type) (pb : P) (pla plb : pb = pb) (ps : PathSquare pla pla plb plb) : Torus -> P := Torus_ind _ pb (dp_const pla) (dp_const plb) (ds_const ps).

Definition

Require Import Basics.Overture Basics.Equivalences Cubical.DPath

Spaces\Torus\Torus.v

Torus_rec

(P : Type) (pb : P) (pla plb : pb = pb) (ps : PathSquare pla pla plb plb) : PathSquare (ap (Torus_rec P pb pla plb ps) loop_a) pla 1 1. Proof. refine (sq_GGcc _ (eissect _ _) (ds_const'^-1 (Torus_ind_beta_loop_a _ _ _ _ _))). apply moveR_equiv_V, dp_apD_const. Defined.

Lemma

Require Import Basics.Overture Basics.Equivalences Cubical.DPath

Spaces\Torus\Torus.v

Torus_rec_beta_loop_a

(P : Type) (pb : P) (pla plb : pb = pb) (ps : PathSquare pla pla plb plb) : PathSquare (ap (Torus_rec P pb pla plb ps) loop_b) plb 1 1. Proof. refine (sq_GGcc _ (eissect _ _) (ds_const'^-1 (Torus_ind_beta_loop_b _ _ _ _ _))). apply moveR_equiv_V, dp_apD_const. Defined.

Lemma

Require Import Basics.Overture Basics.Equivalences Cubical.DPath

Spaces\Torus\Torus.v

Torus_rec_beta_loop_b

(P : Type) (pb : P) (pla plb : pb = pb) (ps : PathSquare pla pla plb plb) : PathCube (sq_ap (Torus_rec P pb pla plb ps) surf) ps (Torus_rec_beta_loop_a P pb pla plb ps) (Torus_rec_beta_loop_a P pb pla plb ps) (Torus_rec_beta_loop_b P pb pla plb ps) (Torus_rec_beta_loop_b P pb pla plb ps). Proof. Admitted. Global Instance ispointed_torus : IsPointed Torus := tbase.

Definition

Require Import Basics.Overture Basics.Equivalences Cubical.DPath

Spaces\Torus\Torus.v

Torus_rec_beta_surf

loop_a @ loop_b = loop_b @ loop_a := equiv_sq_path^-1 surf.

Definition

Require Import Basics.Overture Basics.Equivalences Cubical.DPath

Spaces\Torus\Torus.v

loops_commute_torus

(P : Type) (pb : P) (pla plb : pb = pb) (ps : PathSquare pla pla plb plb) : { ba : PathSquare (ap (Torus_rec P pb pla plb ps) loop_a) pla 1 1 & { bb : PathSquare (ap (Torus_rec P pb pla plb ps) loop_b) plb 1 1 & PathCube (sq_ap (Torus_rec P pb pla plb ps) surf) ps ba ba bb bb}}. Proof. refine (_;_;_). set (cu_cGcccc (eissect ds_const' _) (dc_const'^-1 (Torus_ind_beta_surf (fun _ => P) pb (dp_const pla) (dp_const plb) (ds_const' (sq_GGGG (eissect _ _)^ (eissect _ _)^ (eissect _ _)^ (eissect _ _)^ ps))))). Admitted. *)

Definition

Require Import Basics.Overture Basics.Equivalences Cubical.DPath

Spaces\Torus\Torus.v

Torus_rec_beta_cube

{s : PathSquare loop_a loop_a (ap (Circle_rec _ tbase loop_b) loop) (ap (Circle_rec _ tbase loop_b) loop) & PathCube s surf hr hr (sq_G1 (Circle_rec_beta_loop _ _ _)) (sq_G1 (Circle_rec_beta_loop _ _ _))}. Proof. apply cu_fill_left. Defined.

Definition

Require Import Basics Types. Require Import Cubical.DPath Cubical.PathSquare Cubical.DPathSquare Require Import Spaces.Circle Spaces.Torus.Torus.

Spaces\Torus\TorusEquivCircles.v

c2t_square_and_cube

Torus -> Circle * Circle. Proof. snrapply Torus_rec. + exact (base, base). + exact (path_prod' loop 1). + exact (path_prod' 1 loop). + exact (sq_prod (hr, vr)). Defined.

Definition

Require Import Basics Types. Require Import Cubical.DPath Cubical.PathSquare Cubical.DPathSquare Require Import Spaces.Circle Spaces.Torus.Torus.

Spaces\Torus\TorusEquivCircles.v

t2c

c2t' `{Funext} : Circle -> Circle -> Torus. Proof. snrapply Circle_rec. + snrapply Circle_rec. - exact tbase. - exact loop_b. + apply path_forall. snrapply Circle_ind. - exact loop_a. - srapply sq_dp^-1. apply (pr1 c2t_square_and_cube). Defined.

Definition

Require Import Basics Types. Require Import Cubical.DPath Cubical.PathSquare Cubical.DPathSquare Require Import Spaces.Circle Spaces.Torus.Torus.

Spaces\Torus\TorusEquivCircles.v

c2t'

`{Funext} : Circle * Circle -> Torus. Proof. apply uncurry, '. Defined.

Definition

Require Import Basics Types. Require Import Cubical.DPath Cubical.PathSquare Cubical.DPathSquare Require Import Spaces.Circle Spaces.Torus.Torus.

Spaces\Torus\TorusEquivCircles.v

c2t

`{Funext} : {bl1 : PathSquare (ap (fun y => c2t' base y) loop) loop_b 1 1 & {bl2 : PathSquare (ap (fun x => c2t' x base) loop) loop_a 1 1 & PathCube (sq_ap011 c2t' loop loop) surf bl2 bl2 bl1 bl1}}. Proof. nrefine (_;_;_). unfold sq_ap011. nrefine (cu_concat_lr (cu_ds (dp_apD_nat (fun y => ap_compose _ (fun f => f y) _) _)) _ (sji0:=?[X1]) (sji1:=?X1) (sj0i:=?[Y1]) (sj1i:=?Y1) (pj11:=1)). nrefine (cu_concat_lr (cu_ds (dp_apD_nat (fun x => ap_apply_l _ _ @ apD10 (ap _(Circle_rec_beta_loop _ _ _)) x) _)) _ (sji0:=?[X2]) (sji1:=?X2) (sj0i:=?[Y2]) (sj1i:=?Y2) (pj11:=1)). nrefine (cu_concat_lr (cu_ds (dp_apD_nat (ap10_path_forall _ _ _) _)) _ (sji0:=?[X3]) (sji1:=?X3) (sj0i:=?[Y3]) (sj1i:=?Y3) (pj11:=1)). nrefine (cu_concat_lr (cu_G11 (ap _ (Circle_ind_beta_loop _ _ _))) _ (sji0:=?[X4]) (sji1:=?X4) (sj0i:=?[Y4]) (sj1i:=?Y4) (pj11:=1)). nrefine (cu_concat_lr (cu_G11 (eisretr _ _)) _ (sji0:=?[X5]) (sji1:=?X5) (sj0i:=?[Y5]) (sj1i:=?Y5) (pj11:=1)). apply c2t_square_and_cube.2. Defined.

Definition

Require Import Basics Types. Require Import Cubical.DPath Cubical.PathSquare Cubical.DPathSquare Require Import Spaces.Circle Spaces.Torus.Torus.

Spaces\Torus\TorusEquivCircles.v

c2t'_beta

`{Funext} : c2t o t2c == idmap. Proof. nrefine (Torus_ind _ 1 _ _ _). apply cu_ds^-1. refine (cu_GGGGcc (eisretr _ _)^ (eisretr _ _)^ (eisretr _ _)^ (eisretr _ _)^ _). apply cu_rot_tb_fb. refine (cu_ccGGGG (eisretr _ _)^ (eisretr _ _)^ (eisretr _ _)^ (eisretr _ _)^ _). nrefine ((sq_ap_compose t2c c2t surf) @lr (cu_ap c2t (Torus_rec_beta_surf _ _ _ _ _ )) @lr (sq_ap_uncurry _ _ _) @lr (pr2 (pr2 c2t'_beta)) @lr (cu_flip_lr (sq_ap_idmap _))). Defined.

Definition

Require Import Basics Types. Require Import Cubical.DPath Cubical.PathSquare Cubical.DPathSquare Require Import Spaces.Circle Spaces.Torus.Torus.

Spaces\Torus\TorusEquivCircles.v

t2c2t

{A B C D : Type} (f : A -> B -> C) (g : C -> D) {a a' : A} (p : a = a') {b b' : B} (q : b = b') : PathCube (sq_ap011 (fun x y => g (f x y)) p q) (sq_ap g (sq_ap011 f p q)) apcs apcs apcs apcs. Proof. by destruct p, q. Defined.

Definition

Require Import Basics Types. Require Import Cubical.DPath Cubical.PathSquare Cubical.DPathSquare Require Import Spaces.Circle Spaces.Torus.Torus.

Spaces\Torus\TorusEquivCircles.v

sq_ap011_compose

`{Funext} : t2c o c2t == idmap. Proof. nrapply prod_ind. snrefine (Circle_ind _ (Circle_ind _ 1 _) _). 1: apply sq_dp^-1, sq_tr^-1; shelve. apply dp_forall_domain. intro x; apply sq_dp^-1; revert x. snrefine (Circle_ind _ _ _). 1: apply sq_tr^-1; shelve. apply dp_cu. nrefine (cu_ccGGcc _ _ _). 1,2: nrefine (ap sq_dp (Circle_ind_beta_loop _ _ _) @ eisretr _ _)^. apply cu_rot_tb_fb. nrefine (cu_ccGGGG _ _ _ _ _). 1,2,3,4: exact (eisretr _ _)^. nrefine ((sq_ap011_compose c2t' t2c loop loop) @lr (cu_ap t2c (c2t'_beta.2.2)) @lr (Torus_rec_beta_surf _ _ _ _ _) @lr (cu_flip_lr (sq_ap_idmap _)) @lr (sq_ap_uncurry _ _ _)). Defined.

Definition

Require Import Basics Types. Require Import Cubical.DPath Cubical.PathSquare Cubical.DPathSquare Require Import Spaces.Circle Spaces.Torus.Torus.

Spaces\Torus\TorusEquivCircles.v

c2t2c

`{Funext} : Torus <~> Circle * Circle := equiv_adjointify t2c c2t c2t2c t2c2t.

Definition

Require Import Basics Types. Require Import Cubical.DPath Cubical.PathSquare Cubical.DPathSquare Require Import Spaces.Circle Spaces.Torus.Torus.

Spaces\Torus\TorusEquivCircles.v

equiv_torus_prod_Circle

`{Funext} : T <~>* S1 * S1. Proof. srapply Build_pEquiv'. 1: apply equiv_torus_prod_Circle. reflexivity. Defined.

Lemma

Require Import Basics Types. Require Import Pointed WildCat. Require Import Modalities.ReflectiveSubuniverse Truncations.Core. Require Import Algebra.AbGroups. Require Import Homotopy.HomotopyGroup. Require Import Homotopy.PinSn. Require Import Spaces.Int Spaces.Circle. Require Import Spaces.Torus.Torus. Require Import Spaces.Torus.TorusEquivCircles.

Spaces\Torus\TorusHomotopy.v

pequiv_torus_prod_circles

`{Univalence} : GroupIsomorphism (Pi 1 T) (grp_prod abgroup_Z abgroup_Z). Proof. etransitivity. 1: exact (emap (Pi 1) pequiv_torus_prod_circles). etransitivity. 1: apply grp_iso_pi_prod. apply grp_iso_prod. 1,2: apply pi1_circle. Defined.

Theorem

Require Import Basics Types. Require Import Pointed WildCat. Require Import Modalities.ReflectiveSubuniverse Truncations.Core. Require Import Algebra.AbGroups. Require Import Homotopy.HomotopyGroup. Require Import Homotopy.PinSn. Require Import Spaces.Int Spaces.Circle. Require Import Spaces.Torus.Torus. Require Import Spaces.Torus.TorusEquivCircles.

Spaces\Torus\TorusHomotopy.v

pi1_torus

`{Univalence} : loops T <~>* Int * Int. Proof. refine (_ o*E pequiv_ptr (n:=0)). nrapply pi1_torus. Defined.

Theorem

Require Import Basics Types. Require Import Pointed WildCat. Require Import Modalities.ReflectiveSubuniverse Truncations.Core. Require Import Algebra.AbGroups. Require Import Homotopy.HomotopyGroup. Require Import Homotopy.PinSn. Require Import Spaces.Int Spaces.Circle. Require Import Spaces.Torus.Torus. Require Import Spaces.Torus.TorusEquivCircles.

Spaces\Torus\TorusHomotopy.v

loops_torus

Record

Require Import HoTT.Basics HoTT.Types. Require Import Pointed.

Spectra\Spectrum.v

Prespectrum

(E : Prespectrum) `{IsSpectrum E} : forall n, E n <~>* loops (E n.

Definition

Require Import HoTT.Basics HoTT.Types. Require Import Pointed.

Spectra\Spectrum.v

equiv_glue

Record

Require Import HoTT.Basics HoTT.Types. Require Import Pointed.

Spectra\Spectrum.v

Spectrum

`{Univalence} (k : trunc_index) (E : Spectrum) : Spectrum. Proof. simple refine (Build_Spectrum (Build_Prespectrum (fun n => pTr (trunc_index_inc k n) (E n)) _) _). - intros n. exact ((ptr_loops _ (E n.+1)) o*E (pequiv_ptr_functor _ (equiv_glue E n))). - intros n. unfold glue. srapply isequiv_compose. Defined.

Definition

Require Import HoTT.Basics HoTT.Types. Require Import Pointed.

Spectra\Spectrum.v

strunc

get_lem' {KT VT} Key {lem} `{@respects_equivalence_db KT VT Key lem} : VT := lem. Notation get_lem key := ltac:(let res := constr:(get_lem' key) in let res' := (eval unfold get_lem' in res) in exact res') (only parsing). Section const. Context {A : Type} {T : Type}. Global Instance const_respects_equivalenceL : RespectsEquivalenceL A (fun _ _ => T). Proof. refine (fun _ _ => equiv_idmap T; fun _ => _). exact idpath. Defined.

Definition

Require Import Basics.Overture Basics.Equivalences Basics.Tactics. Require Import Types.Equiv Types.Prod Types.Forall Types.Sigma Types.Universe.

Tactics\EquivalenceInduction.v

get_lem'

RespectsEquivalenceL A (fun _ _ => Unit) := @const_respects_equivalenceL A Unit.

Definition

Require Import Basics.Overture Basics.Equivalences Basics.Tactics. Require Import Types.Equiv Types.Prod Types.Forall Types.Sigma Types.Universe.

Tactics\EquivalenceInduction.v

unit_respects_equivalenceL

RespectsEquivalenceR A (fun _ _ => Unit) := @const_respects_equivalenceR A Unit. End unit. Section prod. Global Instance prod_respects_equivalenceL {A} {P Q : forall B, (A <~> B) -> Type} `{RespectsEquivalenceL A P, RespectsEquivalenceL A Q} : RespectsEquivalenceL A (fun B e => P B e * Q B e). Proof. refine ((fun B e => equiv_functor_prod' (respects_equivalenceL.1 B e) (respects_equivalenceL.1 B e)); _). exact (fun fs => transport (fun e' => _ = equiv_functor_prod' e' _) (respects_equivalenceL.2 _) (transport (fun e' => _ = equiv_functor_prod' _ e') (respects_equivalenceL.2 _) idpath)). Defined.

Definition

Require Import Basics.Overture Basics.Equivalences Basics.Tactics. Require Import Types.Equiv Types.Prod Types.Forall Types.Sigma Types.Universe.

Tactics\EquivalenceInduction.v

unit_respects_equivalenceR

{A A'} {P : forall B, (A <~> B) -> Type} {P' : forall B, A' <~> B -> Type} (eA : A <~> A') (eP : forall B e, P B (equiv_compose' e eA) <~> P' B e) `{HP : RespectsEquivalenceL A P} : RespectsEquivalenceL A' P'. Proof. simple refine ((fun B e => _); _). { refine (equiv_compose' (eP _ _) (equiv_compose' (equiv_compose' (HP.1 _ _) (equiv_inverse (HP.1 _ _))) (equiv_inverse (eP _ _)))). } { t. } Defined.

Definition

Require Import Basics.Overture Basics.Equivalences Basics.Tactics. Require Import Types.Equiv Types.Prod Types.Forall Types.Sigma Types.Universe.

Tactics\EquivalenceInduction.v

respects_equivalenceL_equiv

{A A'} {P : forall B, (B <~> A) -> Type} {P' : forall B, B <~> A' -> Type} (eA : A' <~> A) (eP : forall B e, P B (equiv_compose' eA e) <~> P' B e) `{HP : RespectsEquivalenceR A P} : RespectsEquivalenceR A' P'. Proof. simple refine ((fun B e => _); _). { refine (equiv_compose' (eP _ _) (equiv_compose' (equiv_compose' (HP.1 _ _) (equiv_inverse (HP.1 _ _))) (equiv_inverse (eP _ _)))). } { t. } Defined.

Definition

Require Import Basics.Overture Basics.Equivalences Basics.Tactics. Require Import Types.Equiv Types.Prod Types.Forall Types.Sigma Types.Universe.

Tactics\EquivalenceInduction.v

respects_equivalenceR_equiv

respects_equivalenceL_equiv' {A} {P P' : forall B, (A <~> B) -> Type} (eP : forall B e, P B e <~> P' B e) `{HP : RespectsEquivalenceL A P} : RespectsEquivalenceL A P'. Proof. simple refine ((fun B e => _); _). { refine (equiv_compose' (eP _ _) (equiv_compose' (equiv_compose' (HP.1 _ _) (equiv_inverse (HP.1 _ _))) (equiv_inverse (eP _ _)))). } { t. } Defined.

Definition

Require Import Basics.Overture Basics.Equivalences Basics.Tactics. Require Import Types.Equiv Types.Prod Types.Forall Types.Sigma Types.Universe.

Tactics\EquivalenceInduction.v

respects_equivalenceL_equiv'

respects_equivalenceR_equiv' {A} {P P' : forall B, (B <~> A) -> Type} (eP : forall B e, P B e <~> P' B e) `{HP : RespectsEquivalenceR A P} : RespectsEquivalenceR A P'. Proof. simple refine ((fun B e => _); _). { refine (equiv_compose' (eP _ _) (equiv_compose' (equiv_compose' (HP.1 _ _) (equiv_inverse (HP.1 _ _))) (equiv_inverse (eP _ _)))). } { t. } Defined.

Definition

Require Import Basics.Overture Basics.Equivalences Basics.Tactics. Require Import Types.Equiv Types.Prod Types.Forall Types.Sigma Types.Universe.

Tactics\EquivalenceInduction.v

respects_equivalenceR_equiv'

(A:Type) := adummy : A. Ltac rewrite_refl H := match goal with | [ |- ?X ] => let dX' := eval_in ltac:(fun H' => rewrite H in H') (adummy X) in match type of dX' with | ?X' => change X' end end. Example rewrite_refl_example {A B : Type} (x : A) (f : A -> B) : ap f idpath = idpath :> (f x = f x). Proof. rewrite ap_1. reflexivity. Abort. Example rewrite_refl_example {A B : Type} (x : A) (f : A -> B) : ap f idpath = idpath :> (f x = f x). Proof. rewrite_refl @ap_1. reflexivity. Abort.

Inductive

Require Import Basics.Overture Basics.PathGroupoids.

Tactics\EvalIn.v

dummy

`{Funext} {m n : trunc_index} {A B : Type} (f : A -> B) `{IsConnMap n _ _ f} (P : B -> Type) {HP : forall b:B, IsTrunc (m +2+ n) (P b)} (d : forall a:A, P (f a)) : IsTrunc m (ExtensionAlong f P d). Proof. revert P HP d. simple_induction m m' IH; intros P HP d; simpl in *. - apply (Build_Contr _ (extension_conn_map_elim n f P d)). intros y. apply (allpath_extension_conn_map n); assumption. - apply istrunc_S. intros e e'. refine (istrunc_isequiv_istrunc _ (path_extension e e')). Defined.

Lemma

Require Import Basics. Require Import Types. Require Import Extensions. Require Import Factorization. Require Import Modalities.Descent. Require Import Truncations.Core Truncations.SeparatedTrunc.

Truncations\Connectedness.v

istrunc_extension_along_conn

`{Univalence} {n : trunc_index} {A : Type} (a0:A) `{IsConnected n.+1 A} : IsConnMap n (unit_name a0). Proof. rapply (OO_cancelL_conn_map (Tr n.+1) (Tr n) (unit_name a0) (const_tt A)). apply O_lex_leq_Tr. Defined.

Definition

Require Import Basics. Require Import Types. Require Import Extensions. Require Import Factorization. Require Import Modalities.Descent. Require Import Truncations.Core Truncations.SeparatedTrunc.

Truncations\Connectedness.v

conn_point_incl

`{Univalence} (n : trunc_index) {A : pType@{u}} `{IsConnected n.+1 A} (P : A -> Type@{u}) `{forall a, IsTrunc n (P a)} (p0 : P (point A)) : forall a, P a. Proof. intro a. pose proof (p := center (Tr n ((point A) = a))). strip_truncations. exact (p # p0). Defined.

Definition

Require Import Basics. Require Import Types. Require Import Extensions. Require Import Factorization. Require Import Modalities.Descent. Require Import Truncations.Core Truncations.SeparatedTrunc.

Truncations\Connectedness.v

conn_point_elim

n A `{IsConnected n.+1 A} : IsConnected n A. Proof. apply isconnected_from_elim; intros C ? f. refine (isconnected_elim n.+1 C f). Defined.

Definition

Require Import Basics. Require Import Types. Require Import Extensions. Require Import Factorization. Require Import Modalities.Descent. Require Import Truncations.Core Truncations.SeparatedTrunc.

Truncations\Connectedness.v

isconnected_pred

n m A `{H : IsConnected (n +2+ m) A} : IsConnected m A. Proof. induction n. 1: assumption. apply IHn. apply isconnected_pred. assumption. Defined.

Definition

Require Import Basics. Require Import Types. Require Import Extensions. Require Import Factorization. Require Import Modalities.Descent. Require Import Truncations.Core Truncations.SeparatedTrunc.

Truncations\Connectedness.v

isconnected_pred_add

isconnected_pred_add' n m A `{H : IsConnected (m +2+ n) A} : IsConnected m A. Proof. apply (isconnected_pred_add n m). destruct (trunc_index_add_comm m n); assumption. Defined.

Definition

Require Import Basics. Require Import Types. Require Import Extensions. Require Import Factorization. Require Import Modalities.Descent. Require Import Truncations.Core Truncations.SeparatedTrunc.

Truncations\Connectedness.v

isconnected_pred_add'

n A `{IsConnected n.

Definition

Require Import Basics. Require Import Types. Require Import Extensions. Require Import Factorization. Require Import Modalities.Descent. Require Import Truncations.Core Truncations.SeparatedTrunc.

Truncations\Connectedness.v

merely_isconnected

(n : trunc_index) A `{IsConnected n.

Definition

Require Import Basics. Require Import Types. Require Import Extensions. Require Import Factorization. Require Import Modalities.Descent. Require Import Truncations.Core Truncations.SeparatedTrunc.

Truncations\Connectedness.v

is0connected_isconnected

n m A B (f : A -> B) `{IsConnMap (n +2+ m) _ _ f} : IsConnMap m f. Proof. intro b. exact (isconnected_pred_add n m _). Defined.

Definition

Require Import Basics. Require Import Types. Require Import Extensions. Require Import Factorization. Require Import Modalities.Descent. Require Import Truncations.Core Truncations.SeparatedTrunc.

Truncations\Connectedness.v

isconnmap_pred_add

`{Univalence} (A : Type) `{IsConnected 0 A} (x y : A) : merely (x = y). Proof. rapply center. Defined.

Definition

Require Import Basics. Require Import Types. Require Import Extensions. Require Import Factorization. Require Import Modalities.Descent. Require Import Truncations.Core Truncations.SeparatedTrunc.

Truncations\Connectedness.v

merely_path_is0connected

`{Univalence} (A : Type) `{merely A} (p : forall (x y:A), merely (x = y)) : IsConnected 0 A. Proof. strip_truncations. apply contr_inhabited_hprop. - apply hprop_allpath; intros z w. strip_truncations. exact (equiv_path_Tr z w (p z w)). - apply tr; assumption. Defined.

Definition

Require Import Basics. Require Import Types. Require Import Extensions. Require Import Factorization. Require Import Modalities.Descent. Require Import Truncations.Core Truncations.SeparatedTrunc.

Truncations\Connectedness.v

is0connected_merely_allpath

{X : Type} {x : X} (z1 z2 : { z : X & merely (z = x) }) : merely (z1 = z2). Proof. destruct z1 as [z1 p1], z2 as [z2 p2]. strip_truncations. apply tr. apply path_sigma_hprop; cbn. exact (p1 @ p2^). Defined.

Definition

Require Import Basics. Require Import Types. Require Import Extensions. Require Import Factorization. Require Import Modalities.Descent. Require Import Truncations.Core Truncations.SeparatedTrunc.

Truncations\Connectedness.v

merely_path_component

{X : Type} (x : X) : { z : X & merely (z = x) } <~> image (Tr (-1)) (unit_name x). Proof. unfold image; simpl. apply equiv_functor_sigma_id; intros z; simpl. apply Trunc_functor_equiv; unfold hfiber. refine ((equiv_contr_sigma _)^-1 oE _). apply equiv_path_inverse. Defined.

Definition

Require Import Basics. Require Import Types. Require Import Extensions. Require Import Factorization. Require Import Modalities.Descent. Require Import Truncations.Core Truncations.SeparatedTrunc.

Truncations\Connectedness.v

equiv_component_image_unit

{ f : forall a b, P a b & { e_a0 : forall b, f a0 b = f_a0 b & { e_b0 : forall a, f a b0 = f_b0 a & e_b0 a0 = (e_a0 b0) @ f_a0b0 }}}. Proof. assert (goal_as_extension : ExtensionAlong (unit_name a0) (fun a => ExtensionAlong (unit_name b0) (P a) (unit_name (f_b0 a))) (unit_name (f_a0 ; (unit_name f_a0b0)))). - apply (extension_conn_map_elim m). + apply (conn_point_incl a0). + intros a. apply (istrunc_extension_along_conn (n := n)). * apply (conn_point_incl b0). * apply HP. - destruct goal_as_extension as [f_eb name_ea_eab]. assert (ea_eab := name_ea_eab tt); clear name_ea_eab. exists (fun a => pr1 (f_eb a)). exists (fun b => apD10 (ea_eab ..1) b). exists (fun a => pr2 (f_eb a) tt). apply moveL_Mp. apply (concatR (apD10 (ea_eab ..2) tt)). set (ea := ea_eab ..1). generalize ea; simpl. clear ea_eab ea. intros. rewrite transport_arrow. rewrite transport_const. rewrite transport_paths_Fl. exact 1%path. Qed.

Corollary

Require Import Basics. Require Import Types. Require Import Extensions. Require Import Factorization. Require Import Modalities.Descent. Require Import Truncations.Core Truncations.SeparatedTrunc.

Truncations\Connectedness.v

isconn_wedge_incl

forall a b, P a b := isconn_wedge_incl.

Definition

Require Import Basics. Require Import Types. Require Import Extensions. Require Import Factorization. Require Import Modalities.Descent. Require Import Truncations.Core Truncations.SeparatedTrunc.

Truncations\Connectedness.v

wedge_incl_elim

forall b, wedge_incl_elim a0 b = f_a0 b := isconn_wedge_incl.

Definition

Require Import Basics. Require Import Types. Require Import Extensions. Require Import Factorization. Require Import Modalities.Descent. Require Import Truncations.Core Truncations.SeparatedTrunc.

Truncations\Connectedness.v

wedge_incl_comp1

forall a, wedge_incl_elim a b0 = f_b0 a := isconn_wedge_incl.

Definition

Require Import Basics. Require Import Types. Require Import Extensions. Require Import Factorization. Require Import Modalities.Descent. Require Import Truncations.Core Truncations.SeparatedTrunc.

Truncations\Connectedness.v

wedge_incl_comp2

wedge_incl_comp2 a0 = (wedge_incl_comp1 b0) @ f_a0b0 := isconn_wedge_incl.

Definition

Require Import Basics. Require Import Types. Require Import Extensions. Require Import Factorization. Require Import Modalities.Descent. Require Import Truncations.Core Truncations.SeparatedTrunc.

Truncations\Connectedness.v

wedge_incl_comp3

`{Univalence} {m n : trunc_index} {A : Type} (a0 : A) `{IsConnected m.+1 A} {B : Type} (b0 : B) `{IsConnected n.+1 B} (P : A -> B -> Type) {HP : forall a b, IsTrunc (m +2+ n) (P a b)} (fs : {f_a0 : forall b:B, P a0 b & { f_b0 : forall a:A, P a b0 & f_a0 b0 = f_b0 a0 }}) : forall (a : A) (b : B), P a b. Proof. destruct fs as [f_a0 [f_b0 f_a0b0]]. refine (wedge_incl_elim _ _ _ _ _ f_a0b0). Defined.

Definition

Require Import Basics. Require Import Types. Require Import Extensions. Require Import Factorization. Require Import Modalities.Descent. Require Import Truncations.Core Truncations.SeparatedTrunc.

Truncations\Connectedness.v

wedge_incl_elim_uncurried

n (A : Type) : Type := | tr : A -> n A | istrunc_truncation : forall (f : Sphere n.

Inductive

Require Import Basics Types WildCat.Core WildCat.Universe HFiber. Require Import Modalities.Modality.

Truncations\Core.v

Trunc

{n A} (P : Trunc n A -> Type) {Pt : forall aa, IsTrunc n (P aa)} : (forall a, P (tr a)) -> (forall aa, P aa) := fun f aa => match aa with tr a => fun _ => f a end Pt.

Definition

Require Import Basics Types WildCat.Core WildCat.Universe HFiber. Require Import Modalities.Modality.

Truncations\Core.v

Trunc_ind

{n A X} `{IsTrunc n X} : (A -> X) -> (Trunc n A -> X) := Trunc_ind (fun _ => X).

Definition

Require Import Basics Types WildCat.Core WildCat.Universe HFiber. Require Import Modalities.Modality.

Truncations\Core.v

Trunc_rec

n {A : Type} : Trunc_rec (A:=A) (tr (n:=n)) == idmap := Trunc_ind _ (fun a => idpath).

Definition

Require Import Basics Types WildCat.Core WildCat.Universe HFiber. Require Import Modalities.Modality.

Truncations\Core.v

Trunc_rec_tr

(n : trunc_index) : Modality. Proof. srapply (Build_Modality (fun A => IsTrunc n A)); cbn. - intros A B ? f ?; rapply (istrunc_isequiv_istrunc A f). - exact (Trunc n). - intros; apply istrunc_truncation. - intros A; apply tr. - intros A B ? f oa; cbn in *. exact (Trunc_ind B f oa). - intros; reflexivity. - exact (@istrunc_paths' n). Defined.

Definition

Require Import Basics Types WildCat.Core WildCat.Universe HFiber. Require Import Modalities.Modality.

Truncations\Core.v

Tr

(A : Type) : IsTrunc n A <-> IsEquiv (@tr n A) := inO_iff_isequiv_to_O (Tr n) A.

Definition

Require Import Basics Types WildCat.Core WildCat.Universe HFiber. Require Import Modalities.Modality.

Truncations\Core.v

trunc_iff_isequiv_truncation

(A : Type) `{IsTrunc n A} : A <~> Tr n A := Build_Equiv _ _ (@tr n A) _.

Definition

Require Import Basics Types WildCat.Core WildCat.Universe HFiber. Require Import Modalities.Modality.

Truncations\Core.v

equiv_tr

{A : Type} `{IsTrunc n A} : Tr n A -> A := (@tr n A)^-1.

Definition

Require Import Basics Types WildCat.Core WildCat.Universe HFiber. Require Import Modalities.Modality.

Truncations\Core.v

untrunc_istrunc

Trunc_functor@{i j k | i <= k, j <= k} {X : Type@{i}} {Y : Type@{j}} (f : X -> Y) : Tr@{i} n X -> Tr@{j} n Y := O_functor@{k k k} (Tr n) f.

Definition

Require Import Basics Types WildCat.Core WildCat.Universe HFiber. Require Import Modalities.Modality.

Truncations\Core.v

Trunc_functor@

{X Y : Type} (f : X <~> Y) : Tr n X <~> Tr n Y := equiv_O_functor (Tr n) f.

Definition

Require Import Basics Types WildCat.Core WildCat.Universe HFiber. Require Import Modalities.Modality.

Truncations\Core.v

Trunc_functor_equiv

{X Y Z} (f : X -> Y) (g : Y -> Z) : Trunc_functor (g o f) == Trunc_functor g o Trunc_functor f := O_functor_compose (Tr n) f g.

Definition

Require Import Basics Types WildCat.Core WildCat.Universe HFiber. Require Import Modalities.Modality.

Truncations\Core.v

Trunc_functor_compose

(X : Type) : @Trunc_functor X X idmap == idmap := O_functor_idmap (Tr n) X.

Definition

Require Import Basics Types WildCat.Core WildCat.Universe HFiber. Require Import Modalities.Modality.

Truncations\Core.v

Trunc_functor_idmap

{X Y} : Tr n (X * Y) <~> Tr n X * Tr n Y := equiv_O_prod_cmp (Tr n) X Y. Global Instance is1functor_Tr : Is1Functor (Tr n). Proof. apply Build_Is1Functor. - apply @O_functor_homotopy. - apply @Trunc_functor_idmap. - apply @Trunc_functor_compose. Defined.

Definition

Require Import Basics Types WildCat.Core WildCat.Universe HFiber. Require Import Modalities.Modality.

Truncations\Core.v

equiv_Trunc_prod_cmp

{n : trunc_index} (A : Type) `{In (Tr n) A} : IsTrunc n A. Proof. assumption. Defined.

Definition

Require Import Basics Types WildCat.Core WildCat.Universe HFiber. Require Import Modalities.Modality.

Truncations\Core.v

istrunc_inO_tr

{n : trunc_index} {A B : Type} (f : A -> B) `{MapIn (Tr n) _ _ f} : IsTruncMap n f. Proof. assumption. Defined.

Definition

Require Import Basics Types WildCat.Core WildCat.Universe HFiber. Require Import Modalities.Modality.

Truncations\Core.v

istruncmap_mapinO_tr

(A : Type@{i}) : HProp@{i} := Build_HProp (Tr (-1) A).

Definition

Require Import Basics Types WildCat.Core WildCat.Universe HFiber. Require Import Modalities.Modality.

Truncations\Core.v

merely

{X} (P : X -> Type) : HProp := merely (sig P).

Definition

Require Import Basics Types WildCat.Core WildCat.Universe HFiber. Require Import Modalities.Modality.

Truncations\Core.v

hexists

(P Q : Type) : HProp := merely (P + Q).

Definition

Require Import Basics Types WildCat.Core WildCat.Universe HFiber. Require Import Modalities.Modality.

Truncations\Core.v

hor

{X Y} (f : X -> Y) := image (Tr (-1)) f.

Definition

Require Import Basics Types WildCat.Core WildCat.Universe HFiber. Require Import Modalities.Modality.

Truncations\Core.v

himage

{A} `{IsHProp A} (ma : merely A) : Contr A. Proof. refine (@contr_trunc_conn (Tr (-1)) A _ _); try assumption. refine (contr_inhabited_hprop _ ma). Defined.

Definition

Require Import Basics Types WildCat.Core WildCat.Universe HFiber. Require Import Modalities.Modality.

Truncations\Core.v

contr_inhab_prop

{A} {A_stable : Stable A} : Tr (-1) A <-> A. Proof. refine (_, tr). intro ma. apply stable; intro na. revert ma; rapply Trunc_ind; exact na. Defined.

Definition

Require Import Basics Types WildCat.Core WildCat.Universe HFiber. Require Import Modalities.Modality.

Truncations\Core.v

merely_inhabited_iff_inhabited_stable

{A B} (f : A -> B) : (forall b, merely (hfiber f b)) -> IsSurjection f. Proof. intros H b; refine (contr_inhabited_hprop _ _). apply H. Defined.

Definition

Require Import Basics Types WildCat.Core WildCat.Universe HFiber. Require Import Modalities.Modality.

Truncations\Core.v

BuildIsSurjection

{X : Type} (P : X -> Type) : (forall x, merely (P x)) <-> IsSurjection (pr1 : {x : X & P x} -> X). Proof. refine (iff_compose _ (iff_forall_inO_mapinO_pr1 (Conn _) P)). apply iff_functor_forall; intro a. symmetry; apply (iff_contr_hprop (Tr (-1) (P a))). Defined.

Lemma

Require Import Basics Types WildCat.Core WildCat.Universe HFiber. Require Import Modalities.Modality.

Truncations\Core.v

iff_merely_issurjection

`{Funext} {X : Type} (P : X -> Type) : (forall x, merely (P x)) <~> IsSurjection (pr1 : {x : X & P x} -> X). Proof. exact (equiv_iff_hprop_uncurried (iff_merely_issurjection P)). Defined.

Lemma

Require Import Basics Types WildCat.Core WildCat.Universe HFiber. Require Import Modalities.Modality.

Truncations\Core.v

equiv_merely_issurjection

{A B C : Type} (f : A -> B) (g : B -> C) (isconn : IsSurjection (g o f)) : IsSurjection g. Proof. intro c. rapply contr_inhabited_hprop. rapply (Trunc_functor _ (X:= (hfiber (g o f) c))). - intros [a p]. exact (f a; p). - apply center, isconn. Defined.

Lemma

Require Import Basics Types WildCat.Core WildCat.Universe HFiber. Require Import Modalities.Modality.

Truncations\Core.v

cancelR_issurjection

{X Y : Type} {r : X -> Y} (s : Y -> X) (h : forall y:Y, r (s y) = y) : IsSurjection r. Proof. intro y. rapply contr_inhabited_hprop. exact (tr (s y; h y)). Defined.

Definition

Require Import Basics Types WildCat.Core WildCat.Universe HFiber. Require Import Modalities.Modality.

Truncations\Core.v

issurj_retr

{A B} (f : A -> B) `{IsSurjection f} `{IsEmbedding f} : IsEquiv f. Proof. apply (@isequiv_conn_ino_map (Tr (-1))); assumption. Defined.

Definition

Require Import Basics Types WildCat.Core WildCat.Universe HFiber. Require Import Modalities.Modality.

Truncations\Core.v

isequiv_surj_emb

{X Y Z : Type} {f : X -> Y} (i : Y -> Z) `{IsEmbedding i} `{!IsEquiv (i o f)} : IsEquiv f. Proof. rapply (cancelL_isequiv i). refine (isequiv_surj_emb i). rapply (cancelR_issurjection f). Defined.

Definition

Require Import Basics Types WildCat.Core WildCat.Universe HFiber. Require Import Modalities.Modality.

Truncations\Core.v

isequiv_isequiv_compose_embedding

`{Funext} {X Y Z : Type} `{IsHSet X} (f : Y -> Z) `{IsSurjection f} : IsEmbedding (fun phi : Z -> X => phi o f). Proof. intros phi; apply istrunc_S. intros g0 g1; cbn. rapply contr_inhabited_hprop. apply path_sigma_hprop, equiv_path_arrow. rapply conn_map_elim; intro y. exact (ap10 (g0.2 @ g1.2^) y). Defined.

Definition

Require Import Basics Types WildCat.Core WildCat.Universe HFiber. Require Import Modalities.Modality.

Truncations\Core.v

isembedding_precompose_surjection_hset

`{Univalence} {X : Type} (x1 x2 : X) : paths x1 = paths x2 -> x1 = x2. Proof. refine (_ o @equiv_ap10 _ X Type (paths x1) (paths x2)). refine (_ o equiv_functor_forall_id (fun y => equiv_equiv_path (x1 = y) (x2 = y))). refine (_ o functor_forall_id (fun y => @equiv_fun (x1 = y) (x2 = y))). refine (_ o (equiv_paths_ind x1 (fun y p => x2 = y))^-1%equiv). exact (equiv_path_inverse x2 x1). Defined.

Definition

Require Import Basics Types WildCat.Core WildCat.Universe HFiber. Require Import Modalities.Modality.

Truncations\Core.v

ap_paths_inverse

`{Univalence} {X : Type@{u}} : IsEmbedding (@paths X). Proof. snrapply isembedding_isequiv_ap. intros x1 x2. snrapply (isequiv_isequiv_compose_embedding (ap_paths_inverse x1 x2)). - unfold ap_paths_inverse. nrefine (mapinO_compose (O:=Tr (-1)) _ (equiv_path_inverse x2 x1 oE _)). 2: exact _. nrefine (mapinO_compose _ (functor_forall_id _)). 1: exact _. rapply mapinO_functor_forall_id. intro y. apply isembedding_equiv_fun. - simpl. srapply (isequiv_homotopic idmap). intros []. reflexivity. Defined.

Definition

Require Import Basics Types WildCat.Core WildCat.Universe HFiber. Require Import Modalities.Modality.

Truncations\Core.v

isembedding_paths

{n m : trunc_index} (Hmn : m <= n) : O_leq (Tr m) (Tr n). Proof. intros A; rapply istrunc_leq. Defined.

Definition

Require Import Basics Types WildCat.Core WildCat.Universe HFiber. Require Import Modalities.Modality.

Truncations\Core.v

O_leq_Tr_leq

n m X : Tr (trunc_index_min n m) X <~> Tr n (Tr m X). Proof. destruct (trunc_index_min_path n m) as [p|q]. + assert (l := trunc_index_min_leq_right n m). destruct p^; clear p. snrapply (Build_Equiv _ _ (Trunc_functor _ tr)). nrapply O_inverts_conn_map. rapply (conn_map_O_leq _ (Tr m)). rapply O_leq_Tr_leq. + assert (l := trunc_index_min_leq_left n m). destruct q^; clear q. srapply equiv_tr. srapply istrunc_leq. Defined.

Definition

Require Import Basics Types WildCat.Core WildCat.Universe HFiber. Require Import Modalities.Modality.

Truncations\Core.v

Trunc_min

n m X : Tr n (Tr m X) <~> Tr m (Tr n X). Proof. refine (Trunc_min m n _ oE equiv_transport (fun k => Tr k _) _ oE (Trunc_min n m _)^-1). apply trunc_index_min_swap. Defined.

Definition

Require Import Basics Types WildCat.Core WildCat.Universe HFiber. Require Import Modalities.Modality.

Truncations\Core.v

Trunc_swap