/- Copyright (c) 2014 Microsoft Corporation. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Jeremy Avigad, Jakob von Raumer, Floris van Doorn Ported from Coq HoTT -/ prelude import .function .tactic open function eq /- Path equality -/ namespace eq variables {A B C : Type} {P : A → Type} {a a' x y z t : A} {b b' : B} --notation a = b := eq a b notation x = y `:>`:50 A:49 := @eq A x y definition idp [reducible] [constructor] {a : A} := refl a definition idpath [reducible] [constructor] (a : A) := refl a -- unbased path induction definition rec' [reducible] [unfold 6] {P : Π (a b : A), (a = b) → Type} (H : Π (a : A), P a a idp) {a b : A} (p : a = b) : P a b p := eq.rec (H a) p definition rec_on' [reducible] [unfold 5] {P : Π (a b : A), (a = b) → Type} {a b : A} (p : a = b) (H : Π (a : A), P a a idp) : P a b p := eq.rec (H a) p /- Concatenation and inverse -/ definition concat [trans] [unfold 6] (p : x = y) (q : y = z) : x = z := by induction q; exact p definition inverse [symm] [unfold 4] (p : x = y) : y = x := by induction p; reflexivity infix ⬝ := concat postfix ⁻¹ := inverse --a second notation for the inverse, which is not overloaded postfix [parsing_only] `⁻¹ᵖ`:std.prec.max_plus := inverse /- The 1-dimensional groupoid structure -/ -- The identity path is a right unit. definition con_idp [unfold_full] (p : x = y) : p ⬝ idp = p := idp -- The identity path is a right unit. definition idp_con [unfold 4] (p : x = y) : idp ⬝ p = p := by induction p; reflexivity -- Concatenation is associative. definition con.assoc' (p : x = y) (q : y = z) (r : z = t) : p ⬝ (q ⬝ r) = (p ⬝ q) ⬝ r := by induction r; reflexivity definition con.assoc (p : x = y) (q : y = z) (r : z = t) : (p ⬝ q) ⬝ r = p ⬝ (q ⬝ r) := by induction r; reflexivity -- The left inverse law. definition con.right_inv [unfold 4] (p : x = y) : p ⬝ p⁻¹ = idp := by induction p; reflexivity -- The right inverse law. definition con.left_inv [unfold 4] (p : x = y) : p⁻¹ ⬝ p = idp := by induction p; reflexivity /- Several auxiliary theorems about canceling inverses across associativity. These are somewhat redundant, following from earlier theorems. -/ definition inv_con_cancel_left (p : x = y) (q : y = z) : p⁻¹ ⬝ (p ⬝ q) = q := by induction q; induction p; reflexivity definition con_inv_cancel_left (p : x = y) (q : x = z) : p ⬝ (p⁻¹ ⬝ q) = q := by induction q; induction p; reflexivity definition con_inv_cancel_right (p : x = y) (q : y = z) : (p ⬝ q) ⬝ q⁻¹ = p := by induction q; reflexivity definition inv_con_cancel_right (p : x = z) (q : y = z) : (p ⬝ q⁻¹) ⬝ q = p := by induction q; reflexivity -- Inverse distributes over concatenation definition con_inv (p : x = y) (q : y = z) : (p ⬝ q)⁻¹ = q⁻¹ ⬝ p⁻¹ := by induction q; induction p; reflexivity definition inv_con_inv_left (p : y = x) (q : y = z) : (p⁻¹ ⬝ q)⁻¹ = q⁻¹ ⬝ p := by induction q; induction p; reflexivity -- universe metavariables definition inv_con_inv_right (p : x = y) (q : z = y) : (p ⬝ q⁻¹)⁻¹ = q ⬝ p⁻¹ := by induction q; induction p; reflexivity definition inv_con_inv_inv (p : y = x) (q : z = y) : (p⁻¹ ⬝ q⁻¹)⁻¹ = q ⬝ p := by induction q; induction p; reflexivity -- Inverse is an involution. definition inv_inv (p : x = y) : p⁻¹⁻¹ = p := by induction p; reflexivity -- auxiliary definition used by 'cases' tactic definition elim_inv_inv {A : Type} {a b : A} {C : a = b → Type} (H₁ : a = b) (H₂ : C (H₁⁻¹⁻¹)) : C H₁ := eq.rec_on (inv_inv H₁) H₂ /- Theorems for moving things around in equations -/ definition con_eq_of_eq_inv_con {p : x = z} {q : y = z} {r : y = x} : p = r⁻¹ ⬝ q → r ⬝ p = q := begin induction r, intro h, exact !idp_con ⬝ h ⬝ !idp_con end definition con_eq_of_eq_con_inv [unfold 5] {p : x = z} {q : y = z} {r : y = x} : r = q ⬝ p⁻¹ → r ⬝ p = q := by induction p; exact id definition inv_con_eq_of_eq_con {p : x = z} {q : y = z} {r : x = y} : p = r ⬝ q → r⁻¹ ⬝ p = q := by induction r; intro h; exact !idp_con ⬝ h ⬝ !idp_con definition con_inv_eq_of_eq_con [unfold 5] {p : z = x} {q : y = z} {r : y = x} : r = q ⬝ p → r ⬝ p⁻¹ = q := by induction p; exact id definition eq_con_of_inv_con_eq {p : x = z} {q : y = z} {r : y = x} : r⁻¹ ⬝ q = p → q = r ⬝ p := by induction r; intro h; exact !idp_con⁻¹ ⬝ h ⬝ !idp_con⁻¹ definition eq_con_of_con_inv_eq [unfold 5] {p : x = z} {q : y = z} {r : y = x} : q ⬝ p⁻¹ = r → q = r ⬝ p := by induction p; exact id definition eq_inv_con_of_con_eq {p : x = z} {q : y = z} {r : x = y} : r ⬝ q = p → q = r⁻¹ ⬝ p := by induction r; intro h; exact !idp_con⁻¹ ⬝ h ⬝ !idp_con⁻¹ definition eq_con_inv_of_con_eq [unfold 5] {p : z = x} {q : y = z} {r : y = x} : q ⬝ p = r → q = r ⬝ p⁻¹ := by induction p; exact id definition eq_of_con_inv_eq_idp [unfold 5] {p q : x = y} : p ⬝ q⁻¹ = idp → p = q := by induction q; exact id definition eq_of_inv_con_eq_idp {p q : x = y} : q⁻¹ ⬝ p = idp → p = q := by induction q; intro h; exact !idp_con⁻¹ ⬝ h definition eq_inv_of_con_eq_idp' [unfold 5] {p : x = y} {q : y = x} : p ⬝ q = idp → p = q⁻¹ := by induction q; exact id definition eq_inv_of_con_eq_idp {p : x = y} {q : y = x} : q ⬝ p = idp → p = q⁻¹ := by induction q; intro h; exact !idp_con⁻¹ ⬝ h definition eq_of_idp_eq_inv_con {p q : x = y} : idp = p⁻¹ ⬝ q → p = q := by induction p; intro h; exact h ⬝ !idp_con definition eq_of_idp_eq_con_inv [unfold 4] {p q : x = y} : idp = q ⬝ p⁻¹ → p = q := by induction p; exact id definition inv_eq_of_idp_eq_con [unfold 4] {p : x = y} {q : y = x} : idp = q ⬝ p → p⁻¹ = q := by induction p; exact id definition inv_eq_of_idp_eq_con' {p : x = y} {q : y = x} : idp = p ⬝ q → p⁻¹ = q := by induction p; intro h; exact h ⬝ !idp_con definition con_inv_eq_idp [unfold 6] {p q : x = y} (r : p = q) : p ⬝ q⁻¹ = idp := by cases r; apply con.right_inv definition inv_con_eq_idp [unfold 6] {p q : x = y} (r : p = q) : q⁻¹ ⬝ p = idp := by cases r; apply con.left_inv definition con_eq_idp {p : x = y} {q : y = x} (r : p = q⁻¹) : p ⬝ q = idp := by cases q; exact r definition idp_eq_inv_con {p q : x = y} (r : p = q) : idp = p⁻¹ ⬝ q := by cases r; exact !con.left_inv⁻¹ definition idp_eq_con_inv {p q : x = y} (r : p = q) : idp = q ⬝ p⁻¹ := by cases r; exact !con.right_inv⁻¹ definition idp_eq_con {p : x = y} {q : y = x} (r : p⁻¹ = q) : idp = q ⬝ p := by cases p; exact r definition eq_idp_of_con_right {p : x = x} {q : x = y} (r : p ⬝ q = q) : p = idp := by cases q; exact r definition eq_idp_of_con_left {p : x = x} {q : y = x} (r : q ⬝ p = q) : p = idp := by cases q; exact (idp_con p)⁻¹ ⬝ r definition idp_eq_of_con_right {p : x = x} {q : x = y} (r : q = p ⬝ q) : idp = p := by cases q; exact r definition idp_eq_of_con_left {p : x = x} {q : y = x} (r : q = q ⬝ p) : idp = p := by cases q; exact r ⬝ idp_con p /- Transport -/ definition transport [subst] [reducible] [unfold 5] (P : A → Type) {x y : A} (p : x = y) (u : P x) : P y := by induction p; exact u -- This idiom makes the operation right associative. infixr ` ▸ ` := transport _ definition cast [reducible] [unfold 3] {A B : Type} (p : A = B) (a : A) : B := p ▸ a definition cast_def [reducible] [unfold_full] {A B : Type} (p : A = B) (a : A) : cast p a = p ▸ a := idp definition tr_rev [reducible] [unfold 6] (P : A → Type) {x y : A} (p : x = y) (u : P y) : P x := p⁻¹ ▸ u definition ap [unfold 6] ⦃A B : Type⦄ (f : A → B) {x y:A} (p : x = y) : f x = f y := by induction p; reflexivity abbreviation ap01 [parsing_only] := ap definition homotopy [reducible] (f g : Πx, P x) : Type := Πx : A, f x = g x infix ~ := homotopy protected definition homotopy.refl [refl] [reducible] [unfold_full] (f : Πx, P x) : f ~ f := λ x, idp protected definition homotopy.symm [symm] [reducible] [unfold_full] {f g : Πx, P x} (H : f ~ g) : g ~ f := λ x, (H x)⁻¹ protected definition homotopy.trans [trans] [reducible] [unfold_full] {f g h : Πx, P x} (H1 : f ~ g) (H2 : g ~ h) : f ~ h := λ x, H1 x ⬝ H2 x definition homotopy_of_eq {f g : Πx, P x} (H1 : f = g) : f ~ g := H1 ▸ homotopy.refl f definition apd10 [unfold 5] {f g : Πx, P x} (H : f = g) : f ~ g := λx, by induction H; reflexivity --the next theorem is useful if you want to write "apply (apd10' a)" definition apd10' [unfold 6] {f g : Πx, P x} (a : A) (H : f = g) : f a = g a := by induction H; reflexivity --apd10 is also ap evaluation definition apd10_eq_ap_eval {f g : Πx, P x} (H : f = g) (a : A) : apd10 H a = ap (λs : Πx, P x, s a) H := by induction H; reflexivity definition ap10 [reducible] [unfold 5] {f g : A → B} (H : f = g) : f ~ g := apd10 H definition ap11 {f g : A → B} (H : f = g) {x y : A} (p : x = y) : f x = g y := by induction H; exact ap f p -- [apd] is defined in init.pathover using pathover instead of an equality with transport. definition apdt [unfold 6] (f : Πa, P a) {x y : A} (p : x = y) : p ▸ f x = f y := by induction p; reflexivity definition ap011 [unfold 9] (f : A → B → C) (Ha : a = a') (Hb : b = b') : f a b = f a' b' := by cases Ha; exact ap (f a) Hb /- More theorems for moving things around in equations -/ definition tr_eq_of_eq_inv_tr {P : A → Type} {x y : A} {p : x = y} {u : P x} {v : P y} : u = p⁻¹ ▸ v → p ▸ u = v := by induction p; exact id definition inv_tr_eq_of_eq_tr {P : A → Type} {x y : A} {p : y = x} {u : P x} {v : P y} : u = p ▸ v → p⁻¹ ▸ u = v := by induction p; exact id definition eq_inv_tr_of_tr_eq {P : A → Type} {x y : A} {p : x = y} {u : P x} {v : P y} : p ▸ u = v → u = p⁻¹ ▸ v := by induction p; exact id definition eq_tr_of_inv_tr_eq {P : A → Type} {x y : A} {p : y = x} {u : P x} {v : P y} : p⁻¹ ▸ u = v → u = p ▸ v := by induction p; exact id /- Functoriality of functions -/ -- Here we prove that functions behave like functors between groupoids, and that [ap] itself is -- functorial. -- Functions take identity paths to identity paths definition ap_idp [unfold_full] (x : A) (f : A → B) : ap f idp = idp :> (f x = f x) := idp -- Functions commute with concatenation. definition ap_con [unfold 8] (f : A → B) {x y z : A} (p : x = y) (q : y = z) : ap f (p ⬝ q) = ap f p ⬝ ap f q := by induction q; reflexivity definition con_ap_con_eq_con_ap_con_ap (f : A → B) {w x y z : A} (r : f w = f x) (p : x = y) (q : y = z) : r ⬝ ap f (p ⬝ q) = (r ⬝ ap f p) ⬝ ap f q := by induction q; induction p; reflexivity definition ap_con_con_eq_ap_con_ap_con (f : A → B) {w x y z : A} (p : x = y) (q : y = z) (r : f z = f w) : ap f (p ⬝ q) ⬝ r = ap f p ⬝ (ap f q ⬝ r) := by induction q; induction p; apply con.assoc -- Functions commute with path inverses. definition ap_inv' [unfold 6] (f : A → B) {x y : A} (p : x = y) : (ap f p)⁻¹ = ap f p⁻¹ := by induction p; reflexivity definition ap_inv [unfold 6] (f : A → B) {x y : A} (p : x = y) : ap f p⁻¹ = (ap f p)⁻¹ := by induction p; reflexivity -- [ap] itself is functorial in the first argument. definition ap_id [unfold 4] (p : x = y) : ap id p = p := by induction p; reflexivity definition ap_compose [unfold 8] (g : B → C) (f : A → B) {x y : A} (p : x = y) : ap (g ∘ f) p = ap g (ap f p) := by induction p; reflexivity -- Sometimes we don't have the actual function [compose]. definition ap_compose' [unfold 8] (g : B → C) (f : A → B) {x y : A} (p : x = y) : ap (λa, g (f a)) p = ap g (ap f p) := by induction p; reflexivity -- The action of constant maps. definition ap_constant [unfold 5] (p : x = y) (z : B) : ap (λu, z) p = idp := by induction p; reflexivity -- Naturality of [ap]. -- see also natural_square in cubical.square definition ap_con_eq_con_ap {f g : A → B} (p : f ~ g) {x y : A} (q : x = y) : ap f q ⬝ p y = p x ⬝ ap g q := by induction q; apply idp_con -- Naturality of [ap] at identity. definition ap_con_eq_con {f : A → A} (p : Πx, f x = x) {x y : A} (q : x = y) : ap f q ⬝ p y = p x ⬝ q := by induction q; apply idp_con definition con_ap_eq_con {f : A → A} (p : Πx, x = f x) {x y : A} (q : x = y) : p x ⬝ ap f q = q ⬝ p y := by induction q; exact !idp_con⁻¹ -- Naturality of [ap] with constant function definition ap_con_eq {f : A → B} {b : B} (p : Πx, f x = b) {x y : A} (q : x = y) : ap f q ⬝ p y = p x := by induction q; apply idp_con -- Naturality with other paths hanging around. definition con_ap_con_con_eq_con_con_ap_con {f g : A → B} (p : f ~ g) {x y : A} (q : x = y) {w z : B} (r : w = f x) (s : g y = z) : (r ⬝ ap f q) ⬝ (p y ⬝ s) = (r ⬝ p x) ⬝ (ap g q ⬝ s) := by induction s; induction q; reflexivity definition con_ap_con_eq_con_con_ap {f g : A → B} (p : f ~ g) {x y : A} (q : x = y) {w : B} (r : w = f x) : (r ⬝ ap f q) ⬝ p y = (r ⬝ p x) ⬝ ap g q := by induction q; reflexivity -- TODO: try this using the simplifier, and compare proofs definition ap_con_con_eq_con_ap_con {f g : A → B} (p : f ~ g) {x y : A} (q : x = y) {z : B} (s : g y = z) : ap f q ⬝ (p y ⬝ s) = p x ⬝ (ap g q ⬝ s) := begin induction s, induction q, apply idp_con end definition con_ap_con_con_eq_con_con_con {f : A → A} (p : f ~ id) {x y : A} (q : x = y) {w z : A} (r : w = f x) (s : y = z) : (r ⬝ ap f q) ⬝ (p y ⬝ s) = (r ⬝ p x) ⬝ (q ⬝ s) := by induction s; induction q; reflexivity definition con_con_ap_con_eq_con_con_con {g : A → A} (p : id ~ g) {x y : A} (q : x = y) {w z : A} (r : w = x) (s : g y = z) : (r ⬝ p x) ⬝ (ap g q ⬝ s) = (r ⬝ q) ⬝ (p y ⬝ s) := by induction s; induction q; reflexivity definition con_ap_con_eq_con_con {f : A → A} (p : f ~ id) {x y : A} (q : x = y) {w : A} (r : w = f x) : (r ⬝ ap f q) ⬝ p y = (r ⬝ p x) ⬝ q := by induction q; reflexivity definition ap_con_con_eq_con_con {f : A → A} (p : f ~ id) {x y : A} (q : x = y) {z : A} (s : y = z) : ap f q ⬝ (p y ⬝ s) = p x ⬝ (q ⬝ s) := by induction s; induction q; apply idp_con definition con_con_ap_eq_con_con {g : A → A} (p : id ~ g) {x y : A} (q : x = y) {w : A} (r : w = x) : (r ⬝ p x) ⬝ ap g q = (r ⬝ q) ⬝ p y := begin cases q, exact idp end definition con_ap_con_eq_con_con' {g : A → A} (p : id ~ g) {x y : A} (q : x = y) {z : A} (s : g y = z) : p x ⬝ (ap g q ⬝ s) = q ⬝ (p y ⬝ s) := by induction s; induction q; exact !idp_con⁻¹ /- Action of [apd10] and [ap10] on paths -/ -- Application of paths between functions preserves the groupoid structure definition apd10_idp (f : Πx, P x) (x : A) : apd10 (refl f) x = idp := idp definition apd10_con {f f' f'' : Πx, P x} (h : f = f') (h' : f' = f'') (x : A) : apd10 (h ⬝ h') x = apd10 h x ⬝ apd10 h' x := by induction h; induction h'; reflexivity definition apd10_inv {f g : Πx : A, P x} (h : f = g) (x : A) : apd10 h⁻¹ x = (apd10 h x)⁻¹ := by induction h; reflexivity definition ap10_idp {f : A → B} (x : A) : ap10 (refl f) x = idp := idp definition ap10_con {f f' f'' : A → B} (h : f = f') (h' : f' = f'') (x : A) : ap10 (h ⬝ h') x = ap10 h x ⬝ ap10 h' x := apd10_con h h' x definition ap10_inv {f g : A → B} (h : f = g) (x : A) : ap10 h⁻¹ x = (ap10 h x)⁻¹ := apd10_inv h x -- [ap10] also behaves nicely on paths produced by [ap] definition ap_ap10 (f g : A → B) (h : B → C) (p : f = g) (a : A) : ap h (ap10 p a) = ap10 (ap (λ f', h ∘ f') p) a:= by induction p; reflexivity /- Transport and the groupoid structure of paths -/ definition idp_tr {P : A → Type} {x : A} (u : P x) : idp ▸ u = u := idp definition con_tr [unfold 7] {P : A → Type} {x y z : A} (p : x = y) (q : y = z) (u : P x) : p ⬝ q ▸ u = q ▸ p ▸ u := by induction q; reflexivity definition tr_inv_tr {P : A → Type} {x y : A} (p : x = y) (z : P y) : p ▸ p⁻¹ ▸ z = z := (con_tr p⁻¹ p z)⁻¹ ⬝ ap (λr, transport P r z) (con.left_inv p) definition inv_tr_tr {P : A → Type} {x y : A} (p : x = y) (z : P x) : p⁻¹ ▸ p ▸ z = z := (con_tr p p⁻¹ z)⁻¹ ⬝ ap (λr, transport P r z) (con.right_inv p) definition con_con_tr {P : A → Type} {x y z w : A} (p : x = y) (q : y = z) (r : z = w) (u : P x) : ap (λe, e ▸ u) (con.assoc' p q r) ⬝ (con_tr (p ⬝ q) r u) ⬝ ap (transport P r) (con_tr p q u) = (con_tr p (q ⬝ r) u) ⬝ (con_tr q r (p ▸ u)) :> ((p ⬝ (q ⬝ r)) ▸ u = r ▸ q ▸ p ▸ u) := by induction r; induction q; induction p; reflexivity -- Here is another coherence lemma for transport. definition tr_inv_tr_lemma {P : A → Type} {x y : A} (p : x = y) (z : P x) : tr_inv_tr p (transport P p z) = ap (transport P p) (inv_tr_tr p z) := by induction p; reflexivity /- some properties for apdt -/ definition apdt_idp (x : A) (f : Πx, P x) : apdt f idp = idp :> (f x = f x) := idp definition apdt_con (f : Πx, P x) {x y z : A} (p : x = y) (q : y = z) : apdt f (p ⬝ q) = con_tr p q (f x) ⬝ ap (transport P q) (apdt f p) ⬝ apdt f q := by cases p; cases q; apply idp definition apdt_inv (f : Πx, P x) {x y : A} (p : x = y) : apdt f p⁻¹ = (eq_inv_tr_of_tr_eq (apdt f p))⁻¹ := by cases p; apply idp -- Dependent transport in a doubly dependent type. definition transportD [unfold 6] {P : A → Type} (Q : Πa, P a → Type) {a a' : A} (p : a = a') (b : P a) (z : Q a b) : Q a' (p ▸ b) := by induction p; exact z -- In Coq the variables P, Q and b are explicit, but in Lean we can probably have them implicit -- using the following notation notation p ` ▸D `:65 x:64 := transportD _ p _ x -- Transporting along higher-dimensional paths definition transport2 [unfold 7] (P : A → Type) {x y : A} {p q : x = y} (r : p = q) (z : P x) : p ▸ z = q ▸ z := ap (λp', p' ▸ z) r notation p ` ▸2 `:65 x:64 := transport2 _ p _ x -- An alternative definition. definition tr2_eq_ap10 (Q : A → Type) {x y : A} {p q : x = y} (r : p = q) (z : Q x) : transport2 Q r z = ap10 (ap (transport Q) r) z := by induction r; reflexivity definition tr2_con {P : A → Type} {x y : A} {p1 p2 p3 : x = y} (r1 : p1 = p2) (r2 : p2 = p3) (z : P x) : transport2 P (r1 ⬝ r2) z = transport2 P r1 z ⬝ transport2 P r2 z := by induction r1; induction r2; reflexivity definition tr2_inv (Q : A → Type) {x y : A} {p q : x = y} (r : p = q) (z : Q x) : transport2 Q r⁻¹ z = (transport2 Q r z)⁻¹ := by induction r; reflexivity definition transportD2 [unfold 7] {B C : A → Type} (D : Π(a:A), B a → C a → Type) {x1 x2 : A} (p : x1 = x2) (y : B x1) (z : C x1) (w : D x1 y z) : D x2 (p ▸ y) (p ▸ z) := by induction p; exact w notation p ` ▸D2 `:65 x:64 := transportD2 _ p _ _ x definition ap_tr_con_tr2 (P : A → Type) {x y : A} {p q : x = y} {z w : P x} (r : p = q) (s : z = w) : ap (transport P p) s ⬝ transport2 P r w = transport2 P r z ⬝ ap (transport P q) s := by induction r; exact !idp_con⁻¹ definition fn_tr_eq_tr_fn {P Q : A → Type} {x y : A} (p : x = y) (f : Πx, P x → Q x) (z : P x) : f y (p ▸ z) = p ▸ f x z := by induction p; reflexivity /- Transporting in particular fibrations -/ /- From the Coq HoTT library: One frequently needs lemmas showing that transport in a certain dependent type is equal to some more explicitly defined operation, defined according to the structure of that dependent type. For most dependent types, we prove these lemmas in the appropriate file in the types/ subdirectory. Here we consider only the most basic cases. -/ -- Transporting in a constant fibration. definition tr_constant (p : x = y) (z : B) : transport (λx, B) p z = z := by induction p; reflexivity definition tr2_constant {p q : x = y} (r : p = q) (z : B) : tr_constant p z = transport2 (λu, B) r z ⬝ tr_constant q z := by induction r; exact !idp_con⁻¹ -- Transporting in a pulled back fibration. definition tr_compose (P : B → Type) (f : A → B) (p : x = y) (z : P (f x)) : transport (P ∘ f) p z = transport P (ap f p) z := by induction p; reflexivity definition ap_precompose (f : A → B) (g g' : B → C) (p : g = g') : ap (λh, h ∘ f) p = transport (λh : B → C, g ∘ f = h ∘ f) p idp := by induction p; reflexivity definition apd10_ap_precompose (f : A → B) (g g' : B → C) (p : g = g') : apd10 (ap (λh : B → C, h ∘ f) p) = λa, apd10 p (f a) := by induction p; reflexivity definition apd10_ap_precompose_dependent {C : B → Type} (f : A → B) {g g' : Πb : B, C b} (p : g = g') : apd10 (ap (λ(h : (Πb : B, C b))(a : A), h (f a)) p) = λa, apd10 p (f a) := by induction p; reflexivity definition apd10_ap_postcompose (f : B → C) (g g' : A → B) (p : g = g') : apd10 (ap (λh : A → B, f ∘ h) p) = λa, ap f (apd10 p a) := by induction p; reflexivity -- A special case of [tr_compose] which seems to come up a lot. definition tr_eq_cast_ap {P : A → Type} {x y} (p : x = y) (u : P x) : p ▸ u = cast (ap P p) u := by induction p; reflexivity definition tr_eq_cast_ap_fn {P : A → Type} {x y} (p : x = y) : transport P p = cast (ap P p) := by induction p; reflexivity /- The behavior of [ap] and [apdt] -/ -- In a constant fibration, [apdt] reduces to [ap], modulo [transport_const]. definition apdt_eq_tr_constant_con_ap (f : A → B) (p : x = y) : apdt f p = tr_constant p (f x) ⬝ ap f p := by induction p; reflexivity /- The 2-dimensional groupoid structure -/ -- Horizontal composition of 2-dimensional paths. definition concat2 [unfold 9 10] {p p' : x = y} {q q' : y = z} (h : p = p') (h' : q = q') : p ⬝ q = p' ⬝ q' := ap011 concat h h' -- 2-dimensional path inversion definition inverse2 [unfold 6] {p q : x = y} (h : p = q) : p⁻¹ = q⁻¹ := ap inverse h infixl ` ◾ `:75 := concat2 postfix [parsing_only] `⁻²`:(max+10) := inverse2 --this notation is abusive, should we use it? /- Whiskering -/ definition whisker_left [unfold 8] (p : x = y) {q r : y = z} (h : q = r) : p ⬝ q = p ⬝ r := idp ◾ h definition whisker_right [unfold 7] {p q : x = y} (h : p = q) (r : y = z) : p ⬝ r = q ⬝ r := h ◾ idp -- Unwhiskering, a.k.a. cancelling definition cancel_left {x y z : A} (p : x = y) {q r : y = z} : (p ⬝ q = p ⬝ r) → (q = r) := λs, !inv_con_cancel_left⁻¹ ⬝ whisker_left p⁻¹ s ⬝ !inv_con_cancel_left definition cancel_right {x y z : A} {p q : x = y} (r : y = z) : (p ⬝ r = q ⬝ r) → (p = q) := λs, !con_inv_cancel_right⁻¹ ⬝ whisker_right s r⁻¹ ⬝ !con_inv_cancel_right -- Whiskering and identity paths. definition whisker_right_idp {p q : x = y} (h : p = q) : whisker_right h idp = h := by induction h; induction p; reflexivity definition whisker_right_idp_left [unfold_full] (p : x = y) (q : y = z) : whisker_right idp q = idp :> (p ⬝ q = p ⬝ q) := idp definition whisker_left_idp_right [unfold_full] (p : x = y) (q : y = z) : whisker_left p idp = idp :> (p ⬝ q = p ⬝ q) := idp definition whisker_left_idp {p q : x = y} (h : p = q) : (idp_con p)⁻¹ ⬝ whisker_left idp h ⬝ idp_con q = h := by induction h; induction p; reflexivity definition whisker_left_idp2 {A : Type} {a : A} (p : idp = idp :> a = a) : whisker_left idp p = p := begin refine _ ⬝ whisker_left_idp p, exact !idp_con⁻¹ end definition con2_idp [unfold_full] {p q : x = y} (h : p = q) : h ◾ idp = whisker_right h idp :> (p ⬝ idp = q ⬝ idp) := idp definition idp_con2 [unfold_full] {p q : x = y} (h : p = q) : idp ◾ h = whisker_left idp h :> (idp ⬝ p = idp ⬝ q) := idp definition inv2_con2 {p p' : x = y} (h : p = p') : h⁻² ◾ h = con.left_inv p ⬝ (con.left_inv p')⁻¹ := by induction h; induction p; reflexivity -- The interchange law for concatenation. definition con2_con_con2 {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) := by induction d; induction c; induction b;induction a; reflexivity definition con2_eq_rl {A : Type} {x y z : A} {p p' : x = y} {q q' : y = z} (a : p = p') (b : q = q') : a ◾ b = whisker_right a q ⬝ whisker_left p' b := by induction b; induction a; reflexivity definition con2_eq_lf {A : Type} {x y z : A} {p p' : x = y} {q q' : y = z} (a : p = p') (b : q = q') : a ◾ b = whisker_left p b ⬝ whisker_right a q' := by induction b; induction a; reflexivity definition whisker_right_con_whisker_left {x y z : A} {p p' : x = y} {q q' : y = z} (a : p = p') (b : q = q') : (whisker_right a q) ⬝ (whisker_left p' b) = (whisker_left p b) ⬝ (whisker_right a q') := by induction b; induction a; reflexivity -- Structure corresponding to the coherence equations of a bicategory. -- The "pentagonator": the 3-cell witnessing the associativity pentagon. definition pentagon {v w x y z : A} (p : v = w) (q : w = x) (r : x = y) (s : y = z) : whisker_left p (con.assoc' q r s) ⬝ con.assoc' p (q ⬝ r) s ⬝ whisker_right (con.assoc' p q r) s = con.assoc' p q (r ⬝ s) ⬝ con.assoc' (p ⬝ q) r s := by induction s;induction r;induction q;induction p;reflexivity -- The 3-cell witnessing the left unit triangle. definition triangulator (p : x = y) (q : y = z) : con.assoc' p idp q ⬝ whisker_right (con_idp p) q = whisker_left p (idp_con q) := by induction q; induction p; reflexivity definition eckmann_hilton (p q : idp = idp :> a = a) : p ⬝ q = q ⬝ p := begin refine (whisker_right_idp p ◾ whisker_left_idp2 q)⁻¹ ⬝ _, refine !whisker_right_con_whisker_left ⬝ _, refine !whisker_left_idp2 ◾ !whisker_right_idp end definition con_eq_con2 (p q : idp = idp :> a = a) : p ⬝ q = p ◾ q := begin refine (whisker_right_idp p ◾ whisker_left_idp2 q)⁻¹ ⬝ _, exact !con2_eq_rl⁻¹ end definition inv_eq_inv2 (p : idp = idp :> a = a) : p⁻¹ = p⁻² := begin apply eq.cancel_right p, refine !con.left_inv ⬝ _, refine _ ⬝ !con_eq_con2⁻¹, exact !inv2_con2⁻¹, end -- The action of functions on 2-dimensional paths definition ap02 [unfold 8] [reducible] (f : A → B) {x y : A} {p q : x = y} (r : p = q) : ap f p = ap f q := ap (ap f) r definition ap02_con (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' := by induction r; induction r'; reflexivity definition ap02_con2 (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_con f p q ⬝ (ap02 f r ◾ ap02 f s) ⬝ (ap_con f p' q')⁻¹ := by induction r; induction s; induction q; induction p; reflexivity definition apdt02 [unfold 8] {p q : x = y} (f : Π x, P x) (r : p = q) : apdt f p = transport2 P r (f x) ⬝ apdt f q := by induction r; exact !idp_con⁻¹ -- And now for a lemma whose statement is much longer than its proof. definition apdt02_con {P : A → Type} (f : Π x:A, P x) {x y : A} {p1 p2 p3 : x = y} (r1 : p1 = p2) (r2 : p2 = p3) : apdt02 f (r1 ⬝ r2) = apdt02 f r1 ⬝ whisker_left (transport2 P r1 (f x)) (apdt02 f r2) ⬝ con.assoc' _ _ _ ⬝ (whisker_right (tr2_con r1 r2 (f x))⁻¹ (apdt f p3)) := by induction r2; induction r1; induction p1; reflexivity end eq