669 lines
25 KiB
Text
669 lines
25 KiB
Text
/-
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Copyright (c) 2014 Microsoft Corporation. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Jeremy Avigad, Jakob von Raumer, Floris van Doorn
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Ported from Coq HoTT
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-/
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prelude
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import .function .tactic
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open function eq
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/- Path equality -/
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namespace eq
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variables {A B C : Type} {P : A → Type} {x y z t : A}
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--notation a = b := eq a b
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notation x = y `:>`:50 A:49 := @eq A x y
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definition idp [reducible] [constructor] {a : A} := refl a
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definition idpath [reducible] [constructor] (a : A) := refl a
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-- unbased path induction
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definition rec' [reducible] [unfold 6] {P : Π (a b : A), (a = b) → Type}
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(H : Π (a : A), P a a idp) {a b : A} (p : a = b) : P a b p :=
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eq.rec (H a) p
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definition rec_on' [reducible] [unfold 5] {P : Π (a b : A), (a = b) → Type}
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{a b : A} (p : a = b) (H : Π (a : A), P a a idp) : P a b p :=
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eq.rec (H a) p
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/- Concatenation and inverse -/
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definition concat [trans] [unfold 6] (p : x = y) (q : y = z) : x = z :=
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eq.rec (λp', p') q p
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definition inverse [symm] [unfold 4] (p : x = y) : y = x :=
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eq.rec (refl x) p
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infix ⬝ := concat
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postfix ⁻¹ := inverse
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--a second notation for the inverse, which is not overloaded
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postfix [parsing_only] `⁻¹ᵖ`:std.prec.max_plus := inverse
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/- The 1-dimensional groupoid structure -/
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-- The identity path is a right unit.
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definition con_idp [unfold_full] (p : x = y) : p ⬝ idp = p :=
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idp
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-- The identity path is a right unit.
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definition idp_con [unfold 4] (p : x = y) : idp ⬝ p = p :=
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eq.rec_on p idp
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-- Concatenation is associative.
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definition con.assoc' (p : x = y) (q : y = z) (r : z = t) :
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p ⬝ (q ⬝ r) = (p ⬝ q) ⬝ r :=
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eq.rec_on r (eq.rec_on q idp)
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definition con.assoc (p : x = y) (q : y = z) (r : z = t) :
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(p ⬝ q) ⬝ r = p ⬝ (q ⬝ r) :=
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eq.rec_on r (eq.rec_on q idp)
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-- The left inverse law.
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definition con.right_inv [unfold 4] (p : x = y) : p ⬝ p⁻¹ = idp :=
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eq.rec_on p idp
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-- The right inverse law.
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definition con.left_inv [unfold 4] (p : x = y) : p⁻¹ ⬝ p = idp :=
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eq.rec_on p idp
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/- Several auxiliary theorems about canceling inverses across associativity. These are somewhat
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redundant, following from earlier theorems. -/
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definition inv_con_cancel_left (p : x = y) (q : y = z) : p⁻¹ ⬝ (p ⬝ q) = q :=
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eq.rec_on q (eq.rec_on p idp)
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definition con_inv_cancel_left (p : x = y) (q : x = z) : p ⬝ (p⁻¹ ⬝ q) = q :=
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eq.rec_on q (eq.rec_on p idp)
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definition con_inv_cancel_right (p : x = y) (q : y = z) : (p ⬝ q) ⬝ q⁻¹ = p :=
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eq.rec_on q (eq.rec_on p idp)
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definition inv_con_cancel_right (p : x = z) (q : y = z) : (p ⬝ q⁻¹) ⬝ q = p :=
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eq.rec_on q (take p, eq.rec_on p idp) p
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-- Inverse distributes over concatenation
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definition con_inv (p : x = y) (q : y = z) : (p ⬝ q)⁻¹ = q⁻¹ ⬝ p⁻¹ :=
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eq.rec_on q (eq.rec_on p idp)
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definition inv_con_inv_left (p : y = x) (q : y = z) : (p⁻¹ ⬝ q)⁻¹ = q⁻¹ ⬝ p :=
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eq.rec_on q (eq.rec_on p idp)
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-- universe metavariables
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definition inv_con_inv_right (p : x = y) (q : z = y) : (p ⬝ q⁻¹)⁻¹ = q ⬝ p⁻¹ :=
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eq.rec_on p (take q, eq.rec_on q idp) q
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definition inv_con_inv_inv (p : y = x) (q : z = y) : (p⁻¹ ⬝ q⁻¹)⁻¹ = q ⬝ p :=
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eq.rec_on p (eq.rec_on q idp)
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-- Inverse is an involution.
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definition inv_inv (p : x = y) : p⁻¹⁻¹ = p :=
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eq.rec_on p idp
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-- auxiliary definition used by 'cases' tactic
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definition elim_inv_inv {A : Type} {a b : A} {C : a = b → Type} (H₁ : a = b) (H₂ : C (H₁⁻¹⁻¹)) : C H₁ :=
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eq.rec_on (inv_inv H₁) H₂
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/- Theorems for moving things around in equations -/
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definition con_eq_of_eq_inv_con {p : x = z} {q : y = z} {r : y = x} :
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p = r⁻¹ ⬝ q → r ⬝ p = q :=
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eq.rec_on r (take p h, !idp_con ⬝ h ⬝ !idp_con) p
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definition con_eq_of_eq_con_inv [unfold 5] {p : x = z} {q : y = z} {r : y = x} :
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r = q ⬝ p⁻¹ → r ⬝ p = q :=
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eq.rec_on p (take q h, h) q
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definition inv_con_eq_of_eq_con {p : x = z} {q : y = z} {r : x = y} :
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p = r ⬝ q → r⁻¹ ⬝ p = q :=
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eq.rec_on r (take q h, !idp_con ⬝ h ⬝ !idp_con) q
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definition con_inv_eq_of_eq_con [unfold 5] {p : z = x} {q : y = z} {r : y = x} :
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r = q ⬝ p → r ⬝ p⁻¹ = q :=
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eq.rec_on p (take r h, h) r
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definition eq_con_of_inv_con_eq {p : x = z} {q : y = z} {r : y = x} :
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r⁻¹ ⬝ q = p → q = r ⬝ p :=
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eq.rec_on r (take p h, !idp_con⁻¹ ⬝ h ⬝ !idp_con⁻¹) p
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definition eq_con_of_con_inv_eq [unfold 5] {p : x = z} {q : y = z} {r : y = x} :
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q ⬝ p⁻¹ = r → q = r ⬝ p :=
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eq.rec_on p (take q h, h) q
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definition eq_inv_con_of_con_eq {p : x = z} {q : y = z} {r : x = y} :
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r ⬝ q = p → q = r⁻¹ ⬝ p :=
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eq.rec_on r (take q h, !idp_con⁻¹ ⬝ h ⬝ !idp_con⁻¹) q
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definition eq_con_inv_of_con_eq [unfold 5] {p : z = x} {q : y = z} {r : y = x} :
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q ⬝ p = r → q = r ⬝ p⁻¹ :=
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eq.rec_on p (take r h, h) r
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definition eq_of_con_inv_eq_idp [unfold 5] {p q : x = y} : p ⬝ q⁻¹ = idp → p = q :=
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eq.rec_on q (take p h, h) p
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definition eq_of_inv_con_eq_idp {p q : x = y} : q⁻¹ ⬝ p = idp → p = q :=
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eq.rec_on q (take p h, !idp_con⁻¹ ⬝ h) p
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definition eq_inv_of_con_eq_idp' [unfold 5] {p : x = y} {q : y = x} : p ⬝ q = idp → p = q⁻¹ :=
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eq.rec_on q (take p h, h) p
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definition eq_inv_of_con_eq_idp {p : x = y} {q : y = x} : q ⬝ p = idp → p = q⁻¹ :=
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eq.rec_on q (take p h, !idp_con⁻¹ ⬝ h) p
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definition eq_of_idp_eq_inv_con {p q : x = y} : idp = p⁻¹ ⬝ q → p = q :=
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eq.rec_on p (take q h, h ⬝ !idp_con) q
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definition eq_of_idp_eq_con_inv [unfold 4] {p q : x = y} : idp = q ⬝ p⁻¹ → p = q :=
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eq.rec_on p (take q h, h) q
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definition inv_eq_of_idp_eq_con [unfold 4] {p : x = y} {q : y = x} : idp = q ⬝ p → p⁻¹ = q :=
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eq.rec_on p (take q h, h) q
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definition inv_eq_of_idp_eq_con' {p : x = y} {q : y = x} : idp = p ⬝ q → p⁻¹ = q :=
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eq.rec_on p (take q h, h ⬝ !idp_con) q
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definition con_inv_eq_idp [unfold 6] {p q : x = y} (r : p = q) : p ⬝ q⁻¹ = idp :=
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by cases r;apply con.right_inv
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definition inv_con_eq_idp [unfold 6] {p q : x = y} (r : p = q) : q⁻¹ ⬝ p = idp :=
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by cases r;apply con.left_inv
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definition con_eq_idp {p : x = y} {q : y = x} (r : p = q⁻¹) : p ⬝ q = idp :=
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by cases q;exact r
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definition idp_eq_inv_con {p q : x = y} (r : p = q) : idp = p⁻¹ ⬝ q :=
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by cases r;exact !con.left_inv⁻¹
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definition idp_eq_con_inv {p q : x = y} (r : p = q) : idp = q ⬝ p⁻¹ :=
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by cases r;exact !con.right_inv⁻¹
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definition idp_eq_con {p : x = y} {q : y = x} (r : p⁻¹ = q) : idp = q ⬝ p :=
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by cases p;exact r
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/- Transport -/
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definition transport [subst] [reducible] [unfold 5] (P : A → Type) {x y : A} (p : x = y)
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(u : P x) : P y :=
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eq.rec_on p u
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-- This idiom makes the operation right associative.
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infixr ` ▸ ` := transport _
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definition cast [reducible] [unfold 3] {A B : Type} (p : A = B) (a : A) : B :=
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p ▸ a
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definition cast_def [reducible] [unfold_full] {A B : Type} (p : A = B) (a : A)
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: cast p a = p ▸ a :=
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idp
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definition tr_rev [reducible] [unfold 6] (P : A → Type) {x y : A} (p : x = y) (u : P y) : P x :=
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p⁻¹ ▸ u
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definition ap [unfold 6] ⦃A B : Type⦄ (f : A → B) {x y:A} (p : x = y) : f x = f y :=
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eq.rec_on p idp
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abbreviation ap01 [parsing_only] := ap
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definition homotopy [reducible] (f g : Πx, P x) : Type :=
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Πx : A, f x = g x
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infix ~ := homotopy
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protected definition homotopy.refl [refl] [reducible] [unfold_full] (f : Πx, P x) : f ~ f :=
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λ x, idp
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protected definition homotopy.symm [symm] [reducible] [unfold_full] {f g : Πx, P x} (H : f ~ g)
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: g ~ f :=
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λ x, (H x)⁻¹
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protected definition homotopy.trans [trans] [reducible] [unfold_full] {f g h : Πx, P x}
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(H1 : f ~ g) (H2 : g ~ h) : f ~ h :=
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λ x, H1 x ⬝ H2 x
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definition homotopy_of_eq {f g : Πx, P x} (H1 : f = g) : f ~ g :=
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H1 ▸ homotopy.refl f
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definition apd10 [unfold 5] {f g : Πx, P x} (H : f = g) : f ~ g :=
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λx, eq.rec_on H idp
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--the next theorem is useful if you want to write "apply (apd10' a)"
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definition apd10' [unfold 6] {f g : Πx, P x} (a : A) (H : f = g) : f a = g a :=
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eq.rec_on H idp
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definition ap10 [reducible] [unfold 5] {f g : A → B} (H : f = g) : f ~ g := apd10 H
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definition ap11 {f g : A → B} (H : f = g) {x y : A} (p : x = y) : f x = g y :=
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eq.rec_on H (eq.rec_on p idp)
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definition apd [unfold 6] (f : Πa, P a) {x y : A} (p : x = y) : p ▸ f x = f y :=
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eq.rec_on p idp
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/- More theorems for moving things around in equations -/
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definition tr_eq_of_eq_inv_tr {P : A → Type} {x y : A} {p : x = y} {u : P x} {v : P y} :
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u = p⁻¹ ▸ v → p ▸ u = v :=
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eq.rec_on p (take v, id) v
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definition inv_tr_eq_of_eq_tr {P : A → Type} {x y : A} {p : y = x} {u : P x} {v : P y} :
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u = p ▸ v → p⁻¹ ▸ u = v :=
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eq.rec_on p (take u, id) u
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definition eq_inv_tr_of_tr_eq {P : A → Type} {x y : A} {p : x = y} {u : P x} {v : P y} :
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p ▸ u = v → u = p⁻¹ ▸ v :=
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eq.rec_on p (take v, id) v
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definition eq_tr_of_inv_tr_eq {P : A → Type} {x y : A} {p : y = x} {u : P x} {v : P y} :
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p⁻¹ ▸ u = v → u = p ▸ v :=
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eq.rec_on p (take u, id) u
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/- Functoriality of functions -/
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-- Here we prove that functions behave like functors between groupoids, and that [ap] itself is
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-- functorial.
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-- Functions take identity paths to identity paths
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definition ap_idp [unfold_full] (x : A) (f : A → B) : ap f idp = idp :> (f x = f x) := idp
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-- Functions commute with concatenation.
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definition ap_con [unfold 8] (f : A → B) {x y z : A} (p : x = y) (q : y = z) :
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ap f (p ⬝ q) = ap f p ⬝ ap f q :=
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eq.rec_on q idp
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definition con_ap_con_eq_con_ap_con_ap (f : A → B) {w x y z : A} (r : f w = f x)
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(p : x = y) (q : y = z) : r ⬝ ap f (p ⬝ q) = (r ⬝ ap f p) ⬝ ap f q :=
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eq.rec_on q (take p, eq.rec_on p idp) p
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definition ap_con_con_eq_ap_con_ap_con (f : A → B) {w x y z : A} (p : x = y) (q : y = z)
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(r : f z = f w) : ap f (p ⬝ q) ⬝ r = ap f p ⬝ (ap f q ⬝ r) :=
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eq.rec_on q (eq.rec_on p (take r, con.assoc _ _ _)) r
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-- Functions commute with path inverses.
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definition ap_inv' [unfold 6] (f : A → B) {x y : A} (p : x = y) : (ap f p)⁻¹ = ap f p⁻¹ :=
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eq.rec_on p idp
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definition ap_inv [unfold 6] (f : A → B) {x y : A} (p : x = y) : ap f p⁻¹ = (ap f p)⁻¹ :=
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eq.rec_on p idp
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-- [ap] itself is functorial in the first argument.
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definition ap_id [unfold 4] (p : x = y) : ap id p = p :=
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eq.rec_on p idp
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definition ap_compose [unfold 8] (g : B → C) (f : A → B) {x y : A} (p : x = y) :
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ap (g ∘ f) p = ap g (ap f p) :=
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eq.rec_on p idp
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-- Sometimes we don't have the actual function [compose].
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definition ap_compose' [unfold 8] (g : B → C) (f : A → B) {x y : A} (p : x = y) :
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ap (λa, g (f a)) p = ap g (ap f p) :=
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eq.rec_on p idp
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-- The action of constant maps.
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definition ap_constant [unfold 5] (p : x = y) (z : B) : ap (λu, z) p = idp :=
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eq.rec_on p idp
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-- Naturality of [ap].
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-- see also natural_square in cubical.square
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definition ap_con_eq_con_ap {f g : A → B} (p : f ~ g) {x y : A} (q : x = y) :
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ap f q ⬝ p y = p x ⬝ ap g q :=
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eq.rec_on q !idp_con
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-- Naturality of [ap] at identity.
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definition ap_con_eq_con {f : A → A} (p : Πx, f x = x) {x y : A} (q : x = y) :
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ap f q ⬝ p y = p x ⬝ q :=
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eq.rec_on q !idp_con
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definition con_ap_eq_con {f : A → A} (p : Πx, x = f x) {x y : A} (q : x = y) :
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p x ⬝ ap f q = q ⬝ p y :=
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eq.rec_on q !idp_con⁻¹
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-- Naturality of [ap] with constant function
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definition ap_con_eq {f : A → B} {b : B} (p : Πx, f x = b) {x y : A} (q : x = y) :
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ap f q ⬝ p y = p x :=
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eq.rec_on q !idp_con
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-- Naturality with other paths hanging around.
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definition con_ap_con_con_eq_con_con_ap_con {f g : A → B} (p : f ~ g) {x y : A} (q : x = y)
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{w z : B} (r : w = f x) (s : g y = z) :
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(r ⬝ ap f q) ⬝ (p y ⬝ s) = (r ⬝ p x) ⬝ (ap g q ⬝ s) :=
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eq.rec_on s (eq.rec_on q idp)
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definition con_ap_con_eq_con_con_ap {f g : A → B} (p : f ~ g) {x y : A} (q : x = y)
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{w : B} (r : w = f x) :
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(r ⬝ ap f q) ⬝ p y = (r ⬝ p x) ⬝ ap g q :=
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eq.rec_on q idp
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-- TODO: try this using the simplifier, and compare proofs
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definition ap_con_con_eq_con_ap_con {f g : A → B} (p : f ~ g) {x y : A} (q : x = y)
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{z : B} (s : g y = z) :
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ap f q ⬝ (p y ⬝ s) = p x ⬝ (ap g q ⬝ s) :=
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eq.rec_on s (eq.rec_on q
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(calc
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(ap f idp) ⬝ (p x ⬝ idp) = idp ⬝ p x : idp
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... = p x : !idp_con
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... = (p x) ⬝ (ap g idp ⬝ idp) : idp))
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-- This also works:
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-- eq.rec_on s (eq.rec_on q (!idp_con ▸ idp))
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definition con_ap_con_con_eq_con_con_con {f : A → A} (p : f ~ id) {x y : A} (q : x = y)
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{w z : A} (r : w = f x) (s : y = z) :
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(r ⬝ ap f q) ⬝ (p y ⬝ s) = (r ⬝ p x) ⬝ (q ⬝ s) :=
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eq.rec_on s (eq.rec_on q idp)
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definition con_con_ap_con_eq_con_con_con {g : A → A} (p : id ~ g) {x y : A} (q : x = y)
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{w z : A} (r : w = x) (s : g y = z) :
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(r ⬝ p x) ⬝ (ap g q ⬝ s) = (r ⬝ q) ⬝ (p y ⬝ s) :=
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eq.rec_on s (eq.rec_on q idp)
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definition con_ap_con_eq_con_con {f : A → A} (p : f ~ id) {x y : A} (q : x = y)
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{w : A} (r : w = f x) :
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(r ⬝ ap f q) ⬝ p y = (r ⬝ p x) ⬝ q :=
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eq.rec_on q idp
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definition ap_con_con_eq_con_con {f : A → A} (p : f ~ id) {x y : A} (q : x = y)
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{z : A} (s : y = z) :
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ap f q ⬝ (p y ⬝ s) = p x ⬝ (q ⬝ s) :=
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eq.rec_on s (eq.rec_on q (!idp_con ▸ idp))
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definition con_con_ap_eq_con_con {g : A → A} (p : id ~ g) {x y : A} (q : x = y)
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{w : A} (r : w = x) :
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(r ⬝ p x) ⬝ ap g q = (r ⬝ q) ⬝ p y :=
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begin cases q, exact idp end
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definition con_ap_con_eq_con_con' {g : A → A} (p : id ~ g) {x y : A} (q : x = y)
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{z : A} (s : g y = z) :
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p x ⬝ (ap g q ⬝ s) = q ⬝ (p y ⬝ s) :=
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begin
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apply (eq.rec_on s),
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apply (eq.rec_on q),
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apply (idp_con (p x) ▸ idp)
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end
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/- Action of [apd10] and [ap10] on paths -/
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-- Application of paths between functions preserves the groupoid structure
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definition apd10_idp (f : Πx, P x) (x : A) : apd10 (refl f) x = idp := idp
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definition apd10_con {f f' f'' : Πx, P x} (h : f = f') (h' : f' = f'') (x : A) :
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apd10 (h ⬝ h') x = apd10 h x ⬝ apd10 h' x :=
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eq.rec_on h (take h', eq.rec_on h' idp) h'
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definition apd10_inv {f g : Πx : A, P x} (h : f = g) (x : A) :
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apd10 h⁻¹ x = (apd10 h x)⁻¹ :=
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eq.rec_on h idp
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definition ap10_idp {f : A → B} (x : A) : ap10 (refl f) x = idp := idp
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definition ap10_con {f f' f'' : A → B} (h : f = f') (h' : f' = f'') (x : A) :
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ap10 (h ⬝ h') x = ap10 h x ⬝ ap10 h' x := apd10_con h h' x
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definition ap10_inv {f g : A → B} (h : f = g) (x : A) : ap10 h⁻¹ x = (ap10 h x)⁻¹ :=
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apd10_inv h x
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-- [ap10] also behaves nicely on paths produced by [ap]
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definition ap_ap10 (f g : A → B) (h : B → C) (p : f = g) (a : A) :
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ap h (ap10 p a) = ap10 (ap (λ f', h ∘ f') p) a:=
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eq.rec_on p idp
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/- Transport and the groupoid structure of paths -/
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definition idp_tr {P : A → Type} {x : A} (u : P x) : idp ▸ u = u := idp
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definition con_tr [unfold 7] {P : A → Type} {x y z : A} (p : x = y) (q : y = z) (u : P x) :
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p ⬝ q ▸ u = q ▸ p ▸ u :=
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eq.rec_on q idp
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definition tr_inv_tr {P : A → Type} {x y : A} (p : x = y) (z : P y) :
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p ▸ p⁻¹ ▸ z = z :=
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(con_tr p⁻¹ p z)⁻¹ ⬝ ap (λr, transport P r z) (con.left_inv p)
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definition inv_tr_tr {P : A → Type} {x y : A} (p : x = y) (z : P x) :
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p⁻¹ ▸ p ▸ z = z :=
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(con_tr p p⁻¹ z)⁻¹ ⬝ ap (λr, transport P r z) (con.right_inv p)
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definition con_tr_lemma {P : A → Type}
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{x y z w : A} (p : x = y) (q : y = z) (r : z = w) (u : P x) :
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ap (λe, e ▸ u) (con.assoc' p q r) ⬝ (con_tr (p ⬝ q) r u) ⬝
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ap (transport P r) (con_tr p q u)
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= (con_tr p (q ⬝ r) u) ⬝ (con_tr q r (p ▸ u))
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:> ((p ⬝ (q ⬝ r)) ▸ u = r ▸ q ▸ p ▸ u) :=
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eq.rec_on r (eq.rec_on q (eq.rec_on p idp))
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-- Here is another coherence lemma for transport.
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definition tr_inv_tr_lemma {P : A → Type} {x y : A} (p : x = y) (z : P x) :
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tr_inv_tr p (transport P p z) = ap (transport P p) (inv_tr_tr p z) :=
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eq.rec_on p idp
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/- some properties for apd -/
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definition apd_idp (x : A) (f : Πx, P x) : apd f idp = idp :> (f x = f x) := idp
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definition apd_con (f : Πx, P x) {x y z : A} (p : x = y) (q : y = z)
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: apd f (p ⬝ q) = con_tr p q (f x) ⬝ ap (transport P q) (apd f p) ⬝ apd f q :=
|
|
by cases p;cases q;apply idp
|
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definition apd_inv (f : Πx, P x) {x y : A} (p : x = y)
|
|
: apd f p⁻¹ = (eq_inv_tr_of_tr_eq (apd f p))⁻¹ :=
|
|
by cases p;apply idp
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-- Dependent transport in a doubly dependent type.
|
|
definition transportD [unfold 6] {P : A → Type} (Q : Πa, P a → Type)
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{a a' : A} (p : a = a') (b : P a) (z : Q a b) : Q a' (p ▸ b) :=
|
|
eq.rec_on p z
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|
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|
-- 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
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|
|
-- 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
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|
|
|
notation p ` ▸2 `:65 x:64 := transport2 _ p _ x
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|
|
-- 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 :=
|
|
eq.rec_on r idp
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|
|
|
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 :=
|
|
eq.rec_on r1 (eq.rec_on r2 idp)
|
|
|
|
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 :=
|
|
eq.rec_on r idp
|
|
|
|
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) :=
|
|
eq.rec_on p 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 :=
|
|
eq.rec_on r !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)) :=
|
|
eq.rec_on p idp
|
|
|
|
/- 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 :=
|
|
eq.rec_on p idp
|
|
|
|
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 :=
|
|
eq.rec_on r !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 :=
|
|
eq.rec_on p idp
|
|
|
|
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 :=
|
|
eq.rec_on p idp
|
|
|
|
definition apd10_ap_precompose (f : A → B) (g g' : B → C) (p : g = g') (a : A) :
|
|
apd10 (ap (λh : B → C, h ∘ f) p) a = apd10 p (f a) :=
|
|
eq.rec_on p idp
|
|
|
|
definition apd10_ap_postcompose (f : B → C) (g g' : A → B) (p : g = g') (a : A) :
|
|
apd10 (ap (λh : A → B, f ∘ h) p) a = ap f (apd10 p a) :=
|
|
eq.rec_on p idp
|
|
|
|
-- 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 :=
|
|
eq.rec_on p idp
|
|
|
|
definition tr_eq_cast_ap_fn {P : A → Type} {x y} (p : x = y) : transport P p = cast (ap P p) :=
|
|
eq.rec_on p idp
|
|
|
|
/- The behavior of [ap] and [apd] -/
|
|
|
|
-- In a constant fibration, [apd] reduces to [ap], modulo [transport_const].
|
|
definition apd_eq_tr_constant_con_ap (f : A → B) (p : x = y) :
|
|
apd f p = tr_constant p (f x) ⬝ ap f p :=
|
|
eq.rec_on p idp
|
|
|
|
|
|
/- 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' :=
|
|
eq.rec_on h (eq.rec_on h' idp)
|
|
|
|
-- 2-dimensional path inversion
|
|
definition inverse2 [unfold 6] {p q : x = y} (h : p = q) : p⁻¹ = q⁻¹ :=
|
|
eq.rec_on h idp
|
|
|
|
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 :=
|
|
eq.rec_on h (eq.rec_on p idp)
|
|
|
|
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 :=
|
|
eq.rec_on h (eq.rec_on p idp)
|
|
|
|
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
|
|
|
|
-- 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) :=
|
|
eq.rec_on d (eq.rec_on c (eq.rec_on b (eq.rec_on a idp)))
|
|
|
|
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') :=
|
|
eq.rec_on b (eq.rec_on a !idp_con⁻¹)
|
|
|
|
-- 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) :=
|
|
eq.rec_on q (eq.rec_on p idp)
|
|
|
|
definition eckmann_hilton {x:A} (p q : idp = idp :> x = x) : p ⬝ q = q ⬝ p :=
|
|
(!whisker_right_idp ◾ !whisker_left_idp)⁻¹
|
|
⬝ whisker_left _ !idp_con
|
|
⬝ !whisker_right_con_whisker_left
|
|
⬝ whisker_right !idp_con⁻¹ _
|
|
⬝ (!whisker_left_idp ◾ !whisker_right_idp)
|
|
|
|
-- 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' :=
|
|
eq.rec_on r (eq.rec_on r' idp)
|
|
|
|
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 :=
|
|
eq.rec_on r (eq.rec_on s (eq.rec_on q (eq.rec_on p idp)))
|
|
|
|
definition apd02 [unfold 8] {p q : x = y} (f : Π x, P x) (r : p = q) :
|
|
apd f p = transport2 P r (f x) ⬝ apd f q :=
|
|
eq.rec_on r !idp_con⁻¹
|
|
|
|
-- And now for a lemma whose statement is much longer than its proof.
|
|
definition apd02_con {P : A → Type} (f : Π x:A, P x) {x y : A}
|
|
{p1 p2 p3 : x = y} (r1 : p1 = p2) (r2 : p2 = p3) :
|
|
apd02 f (r1 ⬝ r2) = apd02 f r1
|
|
⬝ whisker_left (transport2 P r1 (f x)) (apd02 f r2)
|
|
⬝ con.assoc' _ _ _
|
|
⬝ (whisker_right (tr2_con r1 r2 (f x))⁻¹ (apd f p3)) :=
|
|
eq.rec_on r2 (eq.rec_on r1 (eq.rec_on p1 idp))
|
|
|
|
end eq
|