/- Copyright (c) 2014 Floris van Doorn. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Author: Floris van Doorn Partially ported from Coq HoTT Theorems about path types (identity types) -/ open eq sigma sigma.ops equiv is_equiv equiv.ops -- TODO: Rename transport_eq_... and pathover_eq_... to eq_transport_... and eq_pathover_... namespace eq /- Path spaces -/ variables {A B : Type} {a a1 a2 a3 a4 : A} {b b1 b2 : B} {f g : A → B} {h : B → A} /- The path spaces of a path space are not, of course, determined; they are just the higher-dimensional structure of the original space. -/ /- some lemmas about whiskering -/ definition whisker_left_con_right (p : a1 = a2) {q q' q'' : a2 = a3} (r : q = q') (s : q' = q'') : whisker_left p (r ⬝ s) = whisker_left p r ⬝ whisker_left p s := begin cases p, cases r, cases s, exact idp end definition whisker_right_con_right {p p' p'' : a1 = a2} (q : a2 = a3) (r : p = p') (s : p' = p'') : whisker_right (r ⬝ s) q = whisker_right r q ⬝ whisker_right s q := begin cases q, cases r, cases s, exact idp end definition whisker_left_con_left (p : a1 = a2) (p' : a2 = a3) {q q' : a3 = a4} (r : q = q') : whisker_left (p ⬝ p') r = !con.assoc ⬝ whisker_left p (whisker_left p' r) ⬝ !con.assoc' := begin cases p', cases p, cases r, cases q, exact idp end definition whisker_right_con_left {p p' : a1 = a2} (q : a2 = a3) (q' : a3 = a4) (r : p = p') : whisker_right r (q ⬝ q') = !con.assoc' ⬝ whisker_right (whisker_right r q) q' ⬝ !con.assoc := begin cases q', cases q, cases r, cases p, exact idp end definition whisker_left_inv_left (p : a2 = a1) {q q' : a2 = a3} (r : q = q') : !con_inv_cancel_left⁻¹ ⬝ whisker_left p (whisker_left p⁻¹ r) ⬝ !con_inv_cancel_left = r := begin cases p, cases r, cases q, exact idp end /- Transporting in path spaces. There are potentially a lot of these lemmas, so we adopt a uniform naming scheme: - `l` means the left endpoint varies - `r` means the right endpoint varies - `F` means application of a function to that (varying) endpoint. -/ definition transport_eq_l (p : a1 = a2) (q : a1 = a3) : transport (λx, x = a3) p q = p⁻¹ ⬝ q := by cases p; cases q; reflexivity definition transport_eq_r (p : a2 = a3) (q : a1 = a2) : transport (λx, a1 = x) p q = q ⬝ p := by cases p; cases q; reflexivity definition transport_eq_lr (p : a1 = a2) (q : a1 = a1) : transport (λx, x = x) p q = p⁻¹ ⬝ q ⬝ p := by cases p; rewrite [▸*,idp_con] definition transport_eq_Fl (p : a1 = a2) (q : f a1 = b) : transport (λx, f x = b) p q = (ap f p)⁻¹ ⬝ q := by cases p; cases q; reflexivity definition transport_eq_Fr (p : a1 = a2) (q : b = f a1) : transport (λx, b = f x) p q = q ⬝ (ap f p) := by cases p; reflexivity definition transport_eq_FlFr (p : a1 = a2) (q : f a1 = g a1) : transport (λx, f x = g x) p q = (ap f p)⁻¹ ⬝ q ⬝ (ap g p) := by cases p; rewrite [▸*,idp_con] definition transport_eq_FlFr_D {B : A → Type} {f g : Πa, B a} (p : a1 = a2) (q : f a1 = g a1) : transport (λx, f x = g x) p q = (apd f p)⁻¹ ⬝ ap (transport B p) q ⬝ (apd g p) := by cases p; rewrite [▸*,idp_con,ap_id] definition transport_eq_FFlr (p : a1 = a2) (q : h (f a1) = a1) : transport (λx, h (f x) = x) p q = (ap h (ap f p))⁻¹ ⬝ q ⬝ p := by cases p; rewrite [▸*,idp_con] definition transport_eq_lFFr (p : a1 = a2) (q : a1 = h (f a1)) : transport (λx, x = h (f x)) p q = p⁻¹ ⬝ q ⬝ (ap h (ap f p)) := by cases p; rewrite [▸*,idp_con] /- Pathovers -/ -- In the comment we give the fibration of the pathover definition pathover_eq_l (p : a1 = a2) (q : a1 = a3) : q =[p] p⁻¹ ⬝ q := /-(λx, x = a3)-/ by cases p; cases q; exact idpo definition pathover_eq_r (p : a2 = a3) (q : a1 = a2) : q =[p] q ⬝ p := /-(λx, a1 = x)-/ by cases p; cases q; exact idpo definition pathover_eq_lr (p : a1 = a2) (q : a1 = a1) : q =[p] p⁻¹ ⬝ q ⬝ p := /-(λx, x = x)-/ by cases p; rewrite [▸*,idp_con]; exact idpo definition pathover_eq_Fl (p : a1 = a2) (q : f a1 = b) : q =[p] (ap f p)⁻¹ ⬝ q := /-(λx, f x = b)-/ by cases p; cases q; exact idpo definition pathover_eq_Fr (p : a1 = a2) (q : b = f a1) : q =[p] q ⬝ (ap f p) := /-(λx, b = f x)-/ by cases p; exact idpo definition pathover_eq_FlFr (p : a1 = a2) (q : f a1 = g a1) : q =[p] (ap f p)⁻¹ ⬝ q ⬝ (ap g p) := /-(λx, f x = g x)-/ by cases p; rewrite [▸*,idp_con]; exact idpo definition pathover_eq_FlFr_D {B : A → Type} {f g : Πa, B a} (p : a1 = a2) (q : f a1 = g a1) : q =[p] (apd f p)⁻¹ ⬝ ap (transport B p) q ⬝ (apd g p) := /-(λx, f x = g x)-/ by cases p; rewrite [▸*,idp_con,ap_id];exact idpo definition pathover_eq_FFlr (p : a1 = a2) (q : h (f a1) = a1) : q =[p] (ap h (ap f p))⁻¹ ⬝ q ⬝ p := /-(λx, h (f x) = x)-/ by cases p; rewrite [▸*,idp_con];exact idpo definition pathover_eq_lFFr (p : a1 = a2) (q : a1 = h (f a1)) : q =[p] p⁻¹ ⬝ q ⬝ (ap h (ap f p)) := /-(λx, x = h (f x))-/ by cases p; rewrite [▸*,idp_con];exact idpo definition pathover_eq_r_idp (p : a1 = a2) : idp =[p] p := /-(λx, a1 = x)-/ by cases p; exact idpo definition pathover_eq_l_idp (p : a1 = a2) : idp =[p] p⁻¹ := /-(λx, x = a1)-/ by cases p; exact idpo definition pathover_eq_l_idp' (p : a1 = a2) : idp =[p⁻¹] p := /-(λx, x = a2)-/ by cases p; exact idpo -- The Functorial action of paths is [ap]. /- Equivalences between path spaces -/ /- [ap_closed] is in init.equiv -/ definition equiv_ap (f : A → B) [H : is_equiv f] (a1 a2 : A) : (a1 = a2) ≃ (f a1 = f a2) := equiv.mk (ap f) _ /- Path operations are equivalences -/ definition is_equiv_eq_inverse (a1 a2 : A) : is_equiv (@inverse A a1 a2) := is_equiv.mk inverse inverse inv_inv inv_inv (λp, eq.rec_on p idp) local attribute is_equiv_eq_inverse [instance] definition eq_equiv_eq_symm (a1 a2 : A) : (a1 = a2) ≃ (a2 = a1) := equiv.mk inverse _ definition is_equiv_concat_left [instance] (p : a1 = a2) (a3 : A) : is_equiv (concat p : a2 = a3 → a1 = a3) := is_equiv.mk (concat p) (concat p⁻¹) (con_inv_cancel_left p) (inv_con_cancel_left p) (λq, by cases p;cases q;exact idp) local attribute is_equiv_concat_left [instance] definition equiv_eq_closed_left (a3 : A) (p : a1 = a2) : (a1 = a3) ≃ (a2 = a3) := equiv.mk (concat p⁻¹) _ definition is_equiv_concat_right [instance] (p : a2 = a3) (a1 : A) : is_equiv (λq : a1 = a2, q ⬝ p) := is_equiv.mk (λq, q ⬝ p) (λq, q ⬝ p⁻¹) (λq, inv_con_cancel_right q p) (λq, con_inv_cancel_right q p) (λq, by cases p;cases q;exact idp) local attribute is_equiv_concat_right [instance] definition equiv_eq_closed_right (a1 : A) (p : a2 = a3) : (a1 = a2) ≃ (a1 = a3) := equiv.mk (λq, q ⬝ p) _ definition eq_equiv_eq_closed (p : a1 = a2) (q : a3 = a4) : (a1 = a3) ≃ (a2 = a4) := equiv.trans (equiv_eq_closed_left a3 p) (equiv_eq_closed_right a2 q) definition is_equiv_whisker_left (p : a1 = a2) (q r : a2 = a3) : is_equiv (@whisker_left A a1 a2 a3 p q r) := begin fapply adjointify, {intro s, apply (!cancel_left s)}, {intro s, apply concat, {apply whisker_left_con_right}, apply concat, rotate_left 1, apply (whisker_left_inv_left p s), apply concat2, {apply concat, {apply whisker_left_con_right}, apply concat2, {cases p, cases q, exact idp}, {exact idp}}, {cases p, cases r, exact idp}}, {intro s, cases s, cases q, cases p, exact idp} end definition eq_equiv_con_eq_con_left (p : a1 = a2) (q r : a2 = a3) : (q = r) ≃ (p ⬝ q = p ⬝ r) := equiv.mk _ !is_equiv_whisker_left definition is_equiv_whisker_right {p q : a1 = a2} (r : a2 = a3) : is_equiv (λs, @whisker_right A a1 a2 a3 p q s r) := begin fapply adjointify, {intro s, apply (!cancel_right s)}, {intro s, cases r, cases s, cases q, reflexivity}, {intro s, cases s, cases r, cases p, reflexivity} end definition eq_equiv_con_eq_con_right (p q : a1 = a2) (r : a2 = a3) : (p = q) ≃ (p ⬝ r = q ⬝ r) := equiv.mk _ !is_equiv_whisker_right definition is_equiv_con_eq_of_eq_inv_con (p : a1 = a3) (q : a2 = a3) (r : a2 = a1) : is_equiv (con_eq_of_eq_inv_con : p = r⁻¹ ⬝ q → r ⬝ p = q) := begin cases r, apply (@is_equiv_compose _ _ _ _ _ !is_equiv_concat_left !is_equiv_concat_right), end definition eq_inv_con_equiv_con_eq (p : a1 = a3) (q : a2 = a3) (r : a2 = a1) : (p = r⁻¹ ⬝ q) ≃ (r ⬝ p = q) := equiv.mk _ !is_equiv_con_eq_of_eq_inv_con definition is_equiv_con_eq_of_eq_con_inv (p : a1 = a3) (q : a2 = a3) (r : a2 = a1) : is_equiv (con_eq_of_eq_con_inv : r = q ⬝ p⁻¹ → r ⬝ p = q) := begin cases p, apply (@is_equiv_compose _ _ _ _ _ !is_equiv_concat_left !is_equiv_concat_right) end definition eq_con_inv_equiv_con_eq (p : a1 = a3) (q : a2 = a3) (r : a2 = a1) : (r = q ⬝ p⁻¹) ≃ (r ⬝ p = q) := equiv.mk _ !is_equiv_con_eq_of_eq_con_inv definition is_equiv_inv_con_eq_of_eq_con (p : a1 = a3) (q : a2 = a3) (r : a1 = a2) : is_equiv (inv_con_eq_of_eq_con : p = r ⬝ q → r⁻¹ ⬝ p = q) := begin cases r, apply (@is_equiv_compose _ _ _ _ _ !is_equiv_concat_left !is_equiv_concat_right) end definition eq_con_equiv_inv_con_eq (p : a1 = a3) (q : a2 = a3) (r : a1 = a2) : (p = r ⬝ q) ≃ (r⁻¹ ⬝ p = q) := equiv.mk _ !is_equiv_inv_con_eq_of_eq_con definition is_equiv_con_inv_eq_of_eq_con (p : a3 = a1) (q : a2 = a3) (r : a2 = a1) : is_equiv (con_inv_eq_of_eq_con : r = q ⬝ p → r ⬝ p⁻¹ = q) := begin cases p, apply (@is_equiv_compose _ _ _ _ _ !is_equiv_concat_left !is_equiv_concat_right) end definition eq_con_equiv_con_inv_eq (p : a3 = a1) (q : a2 = a3) (r : a2 = a1) : (r = q ⬝ p) ≃ (r ⬝ p⁻¹ = q) := equiv.mk _ !is_equiv_con_inv_eq_of_eq_con definition is_equiv_eq_con_of_inv_con_eq (p : a1 = a3) (q : a2 = a3) (r : a2 = a1) : is_equiv (eq_con_of_inv_con_eq : r⁻¹ ⬝ q = p → q = r ⬝ p) := begin cases r, apply (@is_equiv_compose _ _ _ _ _ !is_equiv_concat_left !is_equiv_concat_right) end definition inv_con_eq_equiv_eq_con (p : a1 = a3) (q : a2 = a3) (r : a2 = a1) : (r⁻¹ ⬝ q = p) ≃ (q = r ⬝ p) := equiv.mk _ !is_equiv_eq_con_of_inv_con_eq definition is_equiv_eq_con_of_con_inv_eq (p : a1 = a3) (q : a2 = a3) (r : a2 = a1) : is_equiv (eq_con_of_con_inv_eq : q ⬝ p⁻¹ = r → q = r ⬝ p) := begin cases p, apply (@is_equiv_compose _ _ _ _ _ !is_equiv_concat_left !is_equiv_concat_right) end definition con_inv_eq_equiv_eq_con (p : a1 = a3) (q : a2 = a3) (r : a2 = a1) : (q ⬝ p⁻¹ = r) ≃ (q = r ⬝ p) := equiv.mk _ !is_equiv_eq_con_of_con_inv_eq /- Pathover Equivalences -/ definition pathover_eq_equiv_l (p : a1 = a2) (q : a1 = a3) (r : a2 = a3) : q =[p] r ≃ q = p ⬝ r := /-(λx, x = a3)-/ by cases p; exact !pathover_idp ⬝e !equiv_eq_closed_right !idp_con⁻¹ definition pathover_eq_equiv_r (p : a2 = a3) (q : a1 = a2) (r : a1 = a3) : q =[p] r ≃ q ⬝ p = r := /-(λx, a1 = x)-/ by cases p; apply pathover_idp definition pathover_eq_equiv_lr (p : a1 = a2) (q : a1 = a1) (r : a2 = a2) : q =[p] r ≃ q ⬝ p = p ⬝ r := /-(λx, x = x)-/ by cases p; exact !pathover_idp ⬝e !equiv_eq_closed_right !idp_con⁻¹ definition pathover_eq_equiv_Fl (p : a1 = a2) (q : f a1 = b) (r : f a2 = b) : q =[p] r ≃ q = ap f p ⬝ r := /-(λx, f x = b)-/ by cases p; exact !pathover_idp ⬝e !equiv_eq_closed_right !idp_con⁻¹ definition pathover_eq_equiv_Fr (p : a1 = a2) (q : b = f a1) (r : b = f a2) : q =[p] r ≃ q ⬝ ap f p = r := /-(λx, b = f x)-/ by cases p; apply pathover_idp definition pathover_eq_equiv_FlFr (p : a1 = a2) (q : f a1 = g a1) (r : f a2 = g a2) : q =[p] r ≃ q ⬝ ap g p = ap f p ⬝ r := /-(λx, f x = g x)-/ by cases p; exact !pathover_idp ⬝e !equiv_eq_closed_right !idp_con⁻¹ definition pathover_eq_equiv_FFlr (p : a1 = a2) (q : h (f a1) = a1) (r : h (f a2) = a2) : q =[p] r ≃ q ⬝ p = ap h (ap f p) ⬝ r := /-(λx, h (f x) = x)-/ by cases p; exact !pathover_idp ⬝e !equiv_eq_closed_right !idp_con⁻¹ definition pathover_eq_equiv_lFFr (p : a1 = a2) (q : a1 = h (f a1)) (r : a2 = h (f a2)) : q =[p] r ≃ q ⬝ ap h (ap f p) = p ⬝ r := /-(λx, x = h (f x))-/ by cases p; exact !pathover_idp ⬝e !equiv_eq_closed_right !idp_con⁻¹ -- a lot of this library still needs to be ported from Coq HoTT end eq