/- Copyright (c) 2015 Floris van Doorn. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Author: Floris van Doorn Basic theorems about pathovers -/ prelude import .path .equiv open equiv is_equiv function variables {A A' : Type} {B B' : A → Type} {B'' : A' → Type} {C : Π⦃a⦄, B a → Type} {a a₂ a₃ a₄ : A} {p p' : a = a₂} {p₂ : a₂ = a₃} {p₃ : a₃ = a₄} {p₁₃ : a = a₃} {b b' : B a} {b₂ b₂' : B a₂} {b₃ : B a₃} {b₄ : B a₄} {c : C b} {c₂ : C b₂} namespace eq inductive pathover.{l} (B : A → Type.{l}) (b : B a) : Π{a₂ : A}, a = a₂ → B a₂ → Type.{l} := idpatho : pathover B b (refl a) b notation b ` =[`:50 p:0 `] `:0 b₂:50 := pathover _ b p b₂ definition idpo [reducible] [constructor] : b =[refl a] b := pathover.idpatho b /- equivalences with equality using transport -/ definition pathover_of_tr_eq [unfold 5 8] (r : p ▸ b = b₂) : b =[p] b₂ := by cases p; cases r; constructor definition pathover_of_eq_tr [unfold 5 8] (r : b = p⁻¹ ▸ b₂) : b =[p] b₂ := by cases p; cases r; constructor definition tr_eq_of_pathover [unfold 8] (r : b =[p] b₂) : p ▸ b = b₂ := by cases r; reflexivity definition eq_tr_of_pathover [unfold 8] (r : b =[p] b₂) : b = p⁻¹ ▸ b₂ := by cases r; reflexivity definition pathover_equiv_tr_eq [constructor] (p : a = a₂) (b : B a) (b₂ : B a₂) : (b =[p] b₂) ≃ (p ▸ b = b₂) := begin fapply equiv.MK, { exact tr_eq_of_pathover}, { exact pathover_of_tr_eq}, { intro r, cases p, cases r, apply idp}, { intro r, cases r, apply idp}, end definition pathover_equiv_eq_tr [constructor] (p : a = a₂) (b : B a) (b₂ : B a₂) : (b =[p] b₂) ≃ (b = p⁻¹ ▸ b₂) := begin fapply equiv.MK, { exact eq_tr_of_pathover}, { exact pathover_of_eq_tr}, { intro r, cases p, cases r, apply idp}, { intro r, cases r, apply idp}, end definition pathover_tr [unfold 5] (p : a = a₂) (b : B a) : b =[p] p ▸ b := by cases p;constructor definition tr_pathover [unfold 5] (p : a = a₂) (b : B a₂) : p⁻¹ ▸ b =[p] b := by cases p;constructor definition concato [unfold 12] (r : b =[p] b₂) (r₂ : b₂ =[p₂] b₃) : b =[p ⬝ p₂] b₃ := pathover.rec_on r₂ r definition inverseo [unfold 8] (r : b =[p] b₂) : b₂ =[p⁻¹] b := pathover.rec_on r idpo definition apdo [unfold 6] (f : Πa, B a) (p : a = a₂) : f a =[p] f a₂ := eq.rec_on p idpo definition concato_eq [unfold 10] (r : b =[p] b₂) (q : b₂ = b₂') : b =[p] b₂' := eq.rec_on q r definition eq_concato [unfold 9] (q : b = b') (r : b' =[p] b₂) : b =[p] b₂ := by induction q;exact r definition change_path [unfold 9] (q : p = p') (r : b =[p] b₂) : b =[p'] b₂ := q ▸ r -- infix ` ⬝ ` := concato infix ` ⬝o `:72 := concato infix ` ⬝op `:73 := concato_eq infix ` ⬝po `:73 := eq_concato -- postfix `⁻¹` := inverseo postfix `⁻¹ᵒ`:(max+10) := inverseo definition pathover_cancel_right (q : b =[p ⬝ p₂] b₃) (r : b₃ =[p₂⁻¹] b₂) : b =[p] b₂ := change_path !con_inv_cancel_right (q ⬝o r) definition pathover_cancel_right' (q : b =[p₁₃ ⬝ p₂⁻¹] b₂) (r : b₂ =[p₂] b₃) : b =[p₁₃] b₃ := change_path !inv_con_cancel_right (q ⬝o r) definition pathover_cancel_left (q : b₂ =[p⁻¹] b) (r : b =[p ⬝ p₂] b₃) : b₂ =[p₂] b₃ := change_path !inv_con_cancel_left (q ⬝o r) definition pathover_cancel_left' (q : b =[p] b₂) (r : b₂ =[p⁻¹ ⬝ p₁₃] b₃) : b =[p₁₃] b₃ := change_path !con_inv_cancel_left (q ⬝o r) /- Some of the theorems analogous to theorems for = in init.path -/ definition cono_idpo (r : b =[p] b₂) : r ⬝o idpo =[con_idp p] r := pathover.rec_on r idpo definition idpo_cono (r : b =[p] b₂) : idpo ⬝o r =[idp_con p] r := pathover.rec_on r idpo definition cono.assoc' (r : b =[p] b₂) (r₂ : b₂ =[p₂] b₃) (r₃ : b₃ =[p₃] b₄) : r ⬝o (r₂ ⬝o r₃) =[!con.assoc'] (r ⬝o r₂) ⬝o r₃ := pathover.rec_on r₃ (pathover.rec_on r₂ (pathover.rec_on r idpo)) definition cono.assoc (r : b =[p] b₂) (r₂ : b₂ =[p₂] b₃) (r₃ : b₃ =[p₃] b₄) : (r ⬝o r₂) ⬝o r₃ =[!con.assoc] r ⬝o (r₂ ⬝o r₃) := pathover.rec_on r₃ (pathover.rec_on r₂ (pathover.rec_on r idpo)) definition cono.right_inv (r : b =[p] b₂) : r ⬝o r⁻¹ᵒ =[!con.right_inv] idpo := pathover.rec_on r idpo definition cono.left_inv (r : b =[p] b₂) : r⁻¹ᵒ ⬝o r =[!con.left_inv] idpo := pathover.rec_on r idpo definition eq_of_pathover {a' a₂' : A'} (q : a' =[p] a₂') : a' = a₂' := by cases q;reflexivity definition pathover_of_eq [unfold 5 8] {a' a₂' : A'} (q : a' = a₂') : a' =[p] a₂' := by cases p;cases q;constructor definition pathover_constant [constructor] (p : a = a₂) (a' a₂' : A') : a' =[p] a₂' ≃ a' = a₂' := begin fapply equiv.MK, { exact eq_of_pathover}, { exact pathover_of_eq}, { intro r, cases p, cases r, reflexivity}, { intro r, cases r, reflexivity}, end definition pathover_of_eq_tr_constant_inv (p : a = a₂) (a' : A') : pathover_of_eq (tr_constant p a')⁻¹ = pathover_tr p a' := by cases p; constructor definition eq_of_pathover_idp [unfold 6] {b' : B a} (q : b =[idpath a] b') : b = b' := tr_eq_of_pathover q --should B be explicit in the next two definitions? definition pathover_idp_of_eq [unfold 6] {b' : B a} (q : b = b') : b =[idpath a] b' := pathover_of_tr_eq q definition pathover_idp [constructor] (b : B a) (b' : B a) : b =[idpath a] b' ≃ b = b' := equiv.MK eq_of_pathover_idp (pathover_idp_of_eq) (to_right_inv !pathover_equiv_tr_eq) (to_left_inv !pathover_equiv_tr_eq) -- definition pathover_idp (b : B a) (b' : B a) : b =[idpath a] b' ≃ b = b' := -- pathover_equiv_tr_eq idp b b' -- definition eq_of_pathover_idp [reducible] {b' : B a} (q : b =[idpath a] b') : b = b' := -- to_fun !pathover_idp q -- definition pathover_idp_of_eq [reducible] {b' : B a} (q : b = b') : b =[idpath a] b' := -- to_inv !pathover_idp q definition idp_rec_on [recursor] {P : Π⦃b₂ : B a⦄, b =[idpath a] b₂ → Type} {b₂ : B a} (r : b =[idpath a] b₂) (H : P idpo) : P r := have H2 : P (pathover_idp_of_eq (eq_of_pathover_idp r)), from eq.rec_on (eq_of_pathover_idp r) H, proof left_inv !pathover_idp r ▸ H2 qed definition rec_on_right [recursor] {P : Π⦃b₂ : B a₂⦄, b =[p] b₂ → Type} {b₂ : B a₂} (r : b =[p] b₂) (H : P !pathover_tr) : P r := by cases r; exact H definition rec_on_left [recursor] {P : Π⦃b : B a⦄, b =[p] b₂ → Type} {b : B a} (r : b =[p] b₂) (H : P !tr_pathover) : P r := by cases r; exact H --pathover with fibration B' ∘ f definition pathover_ap [unfold 10] (B' : A' → Type) (f : A → A') {p : a = a₂} {b : B' (f a)} {b₂ : B' (f a₂)} (q : b =[p] b₂) : b =[ap f p] b₂ := by cases q; constructor definition pathover_of_pathover_ap (B' : A' → Type) (f : A → A') {p : a = a₂} {b : B' (f a)} {b₂ : B' (f a₂)} (q : b =[ap f p] b₂) : b =[p] b₂ := by cases p; apply (idp_rec_on q); apply idpo definition pathover_compose [constructor] (B' : A' → Type) (f : A → A') (p : a = a₂) (b : B' (f a)) (b₂ : B' (f a₂)) : b =[p] b₂ ≃ b =[ap f p] b₂ := begin fapply equiv.MK, { exact pathover_ap B' f}, { exact pathover_of_pathover_ap B' f}, { intro q, cases p, esimp, apply (idp_rec_on q), apply idp}, { intro q, cases q, reflexivity}, end definition apdo_con (f : Πa, B a) (p : a = a₂) (q : a₂ = a₃) : apdo f (p ⬝ q) = apdo f p ⬝o apdo f q := by cases p; cases q; reflexivity definition apdo_inv (f : Πa, B a) (p : a = a₂) : apdo f p⁻¹ = (apdo f p)⁻¹ᵒ := by cases p; reflexivity definition apdo_eq_pathover_of_eq_ap (f : A → A') (p : a = a₂) : apdo f p = pathover_of_eq (ap f p) := eq.rec_on p idp definition pathover_of_pathover_tr (q : b =[p ⬝ p₂] p₂ ▸ b₂) : b =[p] b₂ := pathover_cancel_right q !pathover_tr⁻¹ᵒ definition pathover_tr_of_pathover (q : b =[p₁₃ ⬝ p₂⁻¹] b₂) : b =[p₁₃] p₂ ▸ b₂ := pathover_cancel_right' q !pathover_tr definition pathover_of_tr_pathover (q : p ▸ b =[p⁻¹ ⬝ p₁₃] b₃) : b =[p₁₃] b₃ := pathover_cancel_left' !pathover_tr q definition tr_pathover_of_pathover (q : b =[p ⬝ p₂] b₃) : p ▸ b =[p₂] b₃ := pathover_cancel_left !pathover_tr⁻¹ᵒ q definition pathover_tr_of_eq (q : b = b') : b =[p] p ▸ b' := by cases q;apply pathover_tr definition tr_pathover_of_eq (q : b₂ = b₂') : p⁻¹ ▸ b₂ =[p] b₂' := by cases q;apply tr_pathover variable (C) definition transporto (r : b =[p] b₂) (c : C b) : C b₂ := by induction r;exact c infix ` ▸o `:75 := transporto _ definition fn_tro_eq_tro_fn (C' : Π ⦃a : A⦄, B a → Type) (q : b =[p] b₂) (f : Π(b : B a), C b → C' b) (c : C b) : f b (q ▸o c) = (q ▸o (f b c)) := by induction q;reflexivity variable {C} definition apo {f : A → A'} (g : Πa, B a → B'' (f a)) (q : b =[p] b₂) : g a b =[p] g a₂ b₂ := by induction q; constructor definition apo011 [unfold 10] (f : Πa, B a → A') (Ha : a = a₂) (Hb : b =[Ha] b₂) : f a b = f a₂ b₂ := by cases Hb; reflexivity definition apo0111 (f : Πa b, C b → A') (Ha : a = a₂) (Hb : b =[Ha] b₂) (Hc : c =[apo011 C Ha Hb] c₂) : f a b c = f a₂ b₂ c₂ := by cases Hb; apply (idp_rec_on Hc); apply idp definition apod11 {f : Πb, C b} {g : Πb₂, C b₂} (r : f =[p] g) {b : B a} {b₂ : B a₂} (q : b =[p] b₂) : f b =[apo011 C p q] g b₂ := by cases r; apply (idp_rec_on q); constructor definition apdo10 {f : Πb, C b} {g : Πb₂, C b₂} (r : f =[p] g) (b : B a) : f b =[apo011 C p !pathover_tr] g (p ▸ b) := by cases r; constructor definition apo10 [unfold 9] {f : B a → B' a} {g : B a₂ → B' a₂} (r : f =[p] g) (b : B a) : f b =[p] g (p ▸ b) := by cases r; constructor definition apo10_constant_right [unfold 9] {f : B a → A'} {g : B a₂ → A'} (r : f =[p] g) (b : B a) : f b = g (p ▸ b) := by cases r; constructor definition apo10_constant_left [unfold 9] {f : A' → B a} {g : A' → B a₂} (r : f =[p] g) (a' : A') : f a' =[p] g a' := by cases r; constructor definition apo11 {f : B a → B' a} {g : B a₂ → B' a₂} (r : f =[p] g) (q : b =[p] b₂) : f b =[p] g b₂ := by induction q; exact apo10 r b definition apdo_compose1 (g : Πa, B a → B' a) (f : Πa, B a) (p : a = a₂) : apdo (g ∘' f) p = apo g (apdo f p) := by induction p; reflexivity definition apdo_compose2 (g : Πa', B'' a') (f : A → A') (p : a = a₂) : apdo (λa, g (f a)) p = pathover_of_pathover_ap B'' f (apdo g (ap f p)) := by induction p; reflexivity definition cono.right_inv_eq (q : b = b') : concato_eq (pathover_idp_of_eq q) q⁻¹ = (idpo : b =[refl a] b) := by induction q;constructor definition cono.right_inv_eq' (q : b = b') : eq_concato q (pathover_idp_of_eq q⁻¹) = (idpo : b =[refl a] b) := by induction q;constructor definition cono.left_inv_eq (q : b = b') : concato_eq (pathover_idp_of_eq q⁻¹) q = (idpo : b' =[refl a] b') := by induction q;constructor definition cono.left_inv_eq' (q : b = b') : eq_concato q⁻¹ (pathover_idp_of_eq q) = (idpo : b' =[refl a] b') := by induction q;constructor definition pathover_of_fn_pathover_fn (f : Π{a}, B a ≃ B' a) (r : f b =[p] f b₂) : b =[p] b₂ := (left_inv f b)⁻¹ ⬝po apo (λa, f⁻¹ᵉ) r ⬝op left_inv f b₂ definition change_path_of_pathover (s : p = p') (r : b =[p] b₂) (r' : b =[p'] b₂) (q : r =[s] r') : change_path s r = r' := by induction s; eapply idp_rec_on q; reflexivity definition pathover_of_change_path (s : p = p') (r : b =[p] b₂) (r' : b =[p'] b₂) (q : change_path s r = r') : r =[s] r' := by induction s; induction q; constructor definition pathover_pathover_path [constructor] (s : p = p') (r : b =[p] b₂) (r' : b =[p'] b₂) : (r =[s] r') ≃ change_path s r = r' := begin fapply equiv.MK, { apply change_path_of_pathover}, { apply pathover_of_change_path}, { intro q, induction s, induction q, reflexivity}, { intro q, induction s, eapply idp_rec_on q, reflexivity}, end definition inverseo2 [unfold 10] {r r' : b =[p] b₂} (s : r = r') : r⁻¹ᵒ = r'⁻¹ᵒ := by induction s; reflexivity definition concato2 [unfold 15 16] {r r' : b =[p] b₂} {r₂ r₂' : b₂ =[p₂] b₃} (s : r = r') (s₂ : r₂ = r₂') : r ⬝o r₂ = r' ⬝o r₂' := by induction s; induction s₂; reflexivity infixl ` ◾o `:75 := concato2 postfix [parsing_only] `⁻²ᵒ`:(max+10) := inverseo2 --this notation is abusive, should we use it? end eq