626 lines
24 KiB
Text
626 lines
24 KiB
Text
-- 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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-- Author: Jeremy Avigad
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-- Ported from Coq HoTT
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notation `assume` binders `,` r:(scoped f, f) := r
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notation `take` binders `,` r:(scoped f, f) := r
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abbreviation id {A : Type} (a : A) := a
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abbreviation compose {A : Type} {B : Type} {C : Type} (g : B → C) (f : A → B) := λ x, g (f x)
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infixl `∘`:60 := compose
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-- Path
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-- ----
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inductive path {A : Type} (a : A) : A → Type :=
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idpath : path a a
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infix `≈`:50 := path
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-- TODO: is this right?
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notation x `≈` y:50 `:>`:0 A:0 := @path A x y
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notation `idp`:max := idpath _ -- TODO: can we / should we use `1`?
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namespace path
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abbreviation induction_on {A : Type} {a b : A} (p : a ≈ b)
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{C : Π (b : A) (p : a ≈ b), Type} (H : C a (idpath a)) : C b p :=
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path_rec H p
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end path
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using path
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-- Concatenation and inverse
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-- -------------------------
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definition concat {A : Type} {x y z : A} (p : x ≈ y) (q : y ≈ z) : x ≈ z :=
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path_rec (λu, u) q p
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-- TODO: should this be an abbreviation?
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definition inverse {A : Type} {x y : A} (p : x ≈ y) : y ≈ x :=
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path_rec (idpath x) p
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infixl `@`:75 := concat
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postfix `^`:100 := inverse
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-- In Coq, these are not needed, because concat and inv are kept transparent
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definition inv_1 {A : Type} (x : A) : (idpath x)^ ≈ idpath x := idp
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definition concat_11 {A : Type} (x : A) : idpath x @ idpath x ≈ idpath x := idp
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-- The 1-dimensional groupoid structure
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-- ------------------------------------
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-- The identity path is a right unit.
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definition concat_p1 {A : Type} {x y : A} (p : x ≈ y) : p @ idp ≈ p :=
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induction_on p idp
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-- The identity path is a right unit.
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definition concat_1p {A : Type} {x y : A} (p : x ≈ y) : idp @ p ≈ p :=
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induction_on p idp
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-- Concatenation is associative.
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definition concat_p_pp {A : Type} {x y z t : A} (p : x ≈ y) (q : y ≈ z) (r : z ≈ t) :
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p @ (q @ r) ≈ (p @ q) @ r :=
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induction_on r (induction_on q idp)
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definition concat_pp_p {A : Type} {x y z t : A} (p : x ≈ y) (q : y ≈ z) (r : z ≈ t) :
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(p @ q) @ r ≈ p @ (q @ r) :=
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induction_on r (induction_on q idp)
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-- The left inverse law.
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definition concat_pV {A : Type} {x y : A} (p : x ≈ y) : p @ p^ ≈ idp :=
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induction_on p idp
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-- The right inverse law.
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definition concat_Vp {A : Type} {x y : A} (p : x ≈ y) : p^ @ p ≈ idp :=
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induction_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 concat_V_pp {A : Type} {x y z : A} (p : x ≈ y) (q : y ≈ z) : p^ @ (p @ q) ≈ q :=
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induction_on q (induction_on p idp)
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definition concat_p_Vp {A : Type} {x y z : A} (p : x ≈ y) (q : x ≈ z) : p @ (p^ @ q) ≈ q :=
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induction_on q (induction_on p idp)
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definition concat_pp_V {A : Type} {x y z : A} (p : x ≈ y) (q : y ≈ z) : (p @ q) @ q^ ≈ p :=
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induction_on q (induction_on p idp)
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definition concat_pV_p {A : Type} {x y z : A} (p : x ≈ z) (q : y ≈ z) : (p @ q^) @ q ≈ p :=
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induction_on q (take p, induction_on p idp) p
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-- Inverse distributes over concatenation
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definition inv_pp {A : Type} {x y z : A} (p : x ≈ y) (q : y ≈ z) : (p @ q)^ ≈ q^ @ p^ :=
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induction_on q (induction_on p idp)
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definition inv_Vp {A : Type} {x y z : A} (p : y ≈ x) (q : y ≈ z) : (p^ @ q)^ ≈ q^ @ p :=
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induction_on q (induction_on p idp)
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-- universe metavariables
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definition inv_pV {A : Type} {x y z : A} (p : x ≈ y) (q : z ≈ y) : (p @ q^)^ ≈ q @ p^ :=
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induction_on p (λq, induction_on q idp) q
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definition inv_VV {A : Type} {x y z : A} (p : y ≈ x) (q : z ≈ y) : (p^ @ q^)^ ≈ q @ p :=
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induction_on p (induction_on q idp)
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-- Inverse is an involution.
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definition inv_V {A : Type} {x y : A} (p : x ≈ y) : p^^ ≈ p :=
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induction_on p idp
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-- Theorems for moving things around in equations
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-- ----------------------------------------------
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definition moveR_Mp {A : Type} {x y z : A} (p : x ≈ z) (q : y ≈ z) (r : y ≈ x) :
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p ≈ (r^ @ q) → (r @ p) ≈ q :=
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have gen : Πp q, p ≈ (r^ @ q) → (r @ p) ≈ q, from
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induction_on r
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(take p q,
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assume h : p ≈ idp^ @ q,
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show idp @ p ≈ q, from concat_1p _ @ h @ concat_1p _),
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gen p q
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definition moveR_pM {A : Type} {x y z : A} (p : x ≈ z) (q : y ≈ z) (r : y ≈ x) :
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r ≈ q @ p^ → r @ p ≈ q :=
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induction_on p (take q r h, (concat_p1 _ @ h @ concat_p1 _)) q r
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definition moveR_Vp {A : Type} {x y z : A} (p : x ≈ z) (q : y ≈ z) (r : x ≈ y) :
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p ≈ r @ q → r^ @ p ≈ q :=
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induction_on r (take p q h, concat_1p _ @ h @ concat_1p _) p q
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definition moveR_pV {A : Type} {x y z : A} (p : z ≈ x) (q : y ≈ z) (r : y ≈ x) :
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r ≈ q @ p → r @ p^ ≈ q :=
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induction_on p (take q r h, concat_p1 _ @ h @ concat_p1 _) q r
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definition moveL_Mp {A : Type} {x y z : A} (p : x ≈ z) (q : y ≈ z) (r : y ≈ x) :
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r^ @ q ≈ p → q ≈ r @ p :=
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induction_on r (take p q h, (concat_1p _)^ @ h @ (concat_1p _)^) p q
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definition moveL_pM {A : Type} {x y z : A} (p : x ≈ z) (q : y ≈ z) (r : y ≈ x) :
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q @ p^ ≈ r → q ≈ r @ p :=
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induction_on p (take q r h, (concat_p1 _)^ @ h @ (concat_p1 _)^) q r
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definition moveL_Vp {A : Type} {x y z : A} (p : x ≈ z) (q : y ≈ z) (r : x ≈ y) :
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r @ q ≈ p → q ≈ r^ @ p :=
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induction_on r (take p q h, (concat_1p _)^ @ h @ (concat_1p _)^) p q
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definition moveL_pV {A : Type} {x y z : A} (p : z ≈ x) (q : y ≈ z) (r : y ≈ x) :
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q @ p ≈ r → q ≈ r @ p^ :=
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induction_on p (take q r h, (concat_p1 _)^ @ h @ (concat_p1 _)^) q r
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definition moveL_1M {A : Type} {x y : A} (p q : x ≈ y) :
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p @ q^ ≈ idp → p ≈ q :=
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induction_on q (take p h, (concat_p1 _)^ @ h) p
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definition moveL_M1 {A : Type} {x y : A} (p q : x ≈ y) :
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q^ @ p ≈ idp → p ≈ q :=
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induction_on q (take p h, (concat_1p _)^ @ h) p
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definition moveL_1V {A : Type} {x y : A} (p : x ≈ y) (q : y ≈ x) :
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p @ q ≈ idp → p ≈ q^ :=
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induction_on q (take p h, (concat_p1 _)^ @ h) p
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definition moveL_V1 {A : Type} {x y : A} (p : x ≈ y) (q : y ≈ x) :
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q @ p ≈ idp → p ≈ q^ :=
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induction_on q (take p h, (concat_1p _)^ @ h) p
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definition moveR_M1 {A : Type} {x y : A} (p q : x ≈ y) :
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idp ≈ p^ @ q → p ≈ q :=
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induction_on p (take q h, h @ (concat_1p _)) q
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definition moveR_1M {A : Type} {x y : A} (p q : x ≈ y) :
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idp ≈ q @ p^ → p ≈ q :=
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induction_on p (take q h, h @ (concat_p1 _)) q
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definition moveR_1V {A : Type} {x y : A} (p : x ≈ y) (q : y ≈ x) :
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idp ≈ q @ p → p^ ≈ q :=
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induction_on p (take q h, h @ (concat_p1 _)) q
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definition moveR_V1 {A : Type} {x y : A} (p : x ≈ y) (q : y ≈ x) :
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idp ≈ p @ q → p^ ≈ q :=
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induction_on p (take q h, h @ (concat_1p _)) q
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-- Transport
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-- ---------
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-- keep transparent, so transport _ idp p is definitionally equal to p
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abbreviation transport {A : Type} (P : A → Type) {x y : A} (p : x ≈ y) (u : P x) : P y :=
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path.induction_on p u
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definition transport_1 {A : Type} (P : A → Type) {x : A} (u : P x) : transport _ idp u ≈ u :=
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idp
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-- TODO: is the binding strength on x reasonable? (It is modeled on the notation for subst
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-- in the standard library.)
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-- This idiom makes the operation right associative.
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notation p `#`:65 x:64 := transport _ p x
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definition ap ⦃A B : Type⦄ (f : A → B) {x y:A} (p : x ≈ y) : f x ≈ f y :=
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path.induction_on p idp
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-- TODO: is this better than an alias? Note use of curly brackets
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abbreviation ap01 := ap
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abbreviation pointwise_paths {A : Type} {P : A → Type} (f g : Πx, P x) : Type :=
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Πx : A, f x ≈ g x
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infix `∼`:50 := pointwise_paths
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definition apD10 {A} {B : A → Type} {f g : Πx, B x} (H : f ≈ g) : f ∼ g :=
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λx, path.induction_on H idp
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definition ap10 {A B} {f g : A → B} (H : f ≈ g) : f ∼ g := apD10 H
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definition ap11 {A B} {f g : A → B} (H : f ≈ g) {x y : A} (p : x ≈ y) : f x ≈ g y :=
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induction_on H (induction_on p idp)
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-- TODO: Note that the next line breaks the proof!
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-- opaque_hint (hiding induction_on)
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-- set_option pp.implicit true
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definition apD {A:Type} {B : A → Type} (f : Πa:A, B a) {x y : A} (p : x ≈ y) : p # (f x) ≈ f y :=
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induction_on p idp
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-- More theorems for moving things around in equations
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-- ---------------------------------------------------
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definition moveR_transport_p {A : Type} (P : A → Type) {x y : A} (p : x ≈ y) (u : P x) (v : P y) :
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u ≈ p^ # v → p # u ≈ v :=
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induction_on p (take u v, id) u v
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definition moveR_transport_V {A : Type} (P : A → Type) {x y : A} (p : y ≈ x) (u : P x) (v : P y) :
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u ≈ p # v → p^ # u ≈ v :=
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induction_on p (take u v, id) u v
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definition moveL_transport_V {A : Type} (P : A → Type) {x y : A} (p : x ≈ y) (u : P x) (v : P y) :
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p # u ≈ v → u ≈ p^ # v :=
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induction_on p (take u v, id) u v
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definition moveL_transport_p {A : Type} (P : A → Type) {x y : A} (p : y ≈ x) (u : P x) (v : P y) :
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p^ # u ≈ v → u ≈ p # v :=
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induction_on p (take u v, id) u v
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-- Functoriality of functions
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-- --------------------------
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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_1 {A B : Type} (x : A) (f : A → B) : (ap f idp) ≈ idp :> (f x ≈ f x) := idp
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definition apD_1 {A B} (x : A) (f : forall x : A, B x) : apD f idp ≈ idp :> (f x ≈ f x) := idp
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-- Functions commute with concatenation.
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definition ap_pp {A B : Type} (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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induction_on q (induction_on p idp)
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definition ap_p_pp {A B : Type} (f : A → B) {w x y z : A} (r : f w ≈ f x) (p : x ≈ y) (q : y ≈ z) :
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r @ (ap f (p @ q)) ≈ (r @ ap f p) @ (ap f q) :=
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induction_on p (take r q, induction_on q (concat_p_pp r idp idp)) r q
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definition ap_pp_p {A B : Type} (f : A → B) {w x y z : A} (p : x ≈ y) (q : y ≈ z) (r : f z ≈ f w) :
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(ap f (p @ q)) @ r ≈ (ap f p) @ (ap f q @ r) :=
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induction_on p (take q, induction_on q (take r, concat_pp_p _ _ _)) q r
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-- Functions commute with path inverses.
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definition inverse_ap {A B : Type} (f : A → B) {x y : A} (p : x ≈ y) : (ap f p)^ ≈ ap f (p^) :=
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induction_on p idp
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definition ap_V {A B : Type} (f : A → B) {x y : A} (p : x ≈ y) : ap f (p^) ≈ (ap f p)^ :=
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induction_on p idp
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-- TODO: rename id to idmap?
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definition ap_idmap {A : Type} {x y : A} (p : x ≈ y) : ap id p ≈ p :=
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induction_on p idp
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definition ap_compose {A B C : Type} (f : A → B) (g : B → C) {x y : A} (p : x ≈ y) :
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ap (g ∘ f) p ≈ ap g (ap f p) :=
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induction_on p idp
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-- Sometimes we don't have the actual function [compose].
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definition ap_compose' {A B C : Type} (f : A → B) (g : B → C) {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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induction_on p idp
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-- The action of constant maps.
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definition ap_const {A B : Type} {x y : A} (p : x ≈ y) (z : B) :
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ap (λu, z) p ≈ idp :=
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induction_on p idp
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-- Naturality of [ap].
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definition concat_Ap {A B : Type} {f g : A → B} (p : forall x, f x ≈ g x) {x y : A} (q : x ≈ y) :
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(ap f q) @ (p y) ≈ (p x) @ (ap g q) :=
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induction_on q (concat_1p _ @ (concat_p1 _)^)
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-- Naturality of [ap] at identity.
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definition concat_A1p {A : Type} {f : A → A} (p : forall x, 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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induction_on q (concat_1p _ @ (concat_p1 _)^)
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definition concat_pA1 {A : Type} {f : A → A} (p : forall x, 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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induction_on q (concat_p1 _ @ (concat_1p _)^)
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--TODO: note that the Coq proof for the preceding is
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--
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-- match q as i in (_ ≈ y) return (p x @ ap f i ≈ i @ p y) with
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-- | idpath => concat_p1 _ @ (concat_1p _)^
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-- end.
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--
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-- It is nice that we don't have to give the predicate.
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-- Naturality with other paths hanging around.
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definition concat_pA_pp {A B : Type} {f g : A → B} (p : forall x, f x ≈ g x)
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{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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induction_on q (take s, induction_on s (take r, idp)) s r
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-- Action of [apD10] and [ap10] on paths
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-- -------------------------------------
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-- Application of paths between functions preserves the groupoid structure
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definition apD10_1 {A} {B : A → Type} (f : Πx, B x) (x : A) : apD10 (idpath f) x ≈ idp := idp
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definition apD10_pp {A} {B : A → Type} {f f' f'' : Πx, B 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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induction_on h (take h', induction_on h' idp) h'
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definition apD10_V {A : Type} {B : A → Type} {f g : Πx : A, B x} (h : f ≈ g) (x : A) :
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apD10 (h^) x ≈ (apD10 h x)^ :=
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induction_on h idp
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definition ap10_1 {A B} {f : A → B} (x : A) : ap10 (idpath f) x ≈ idp := idp
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definition ap10_pp {A B} {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_pp h h' x
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definition ap10_V {A B} {f g : A→B} (h : f ≈ g) (x:A) : ap10 (h^) x ≈ (ap10 h x)^ := apD10_V h x
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-- [ap10] also behaves nicely on paths produced by [ap]
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definition ap_ap10 {A B C} (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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induction_on p idp
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-- Transport and the groupoid structure of paths
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-- ---------------------------------------------
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-- TODO: move from above?
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-- definition transport_1 {A : Type} (P : A → Type) {x : A} (u : P x)
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-- : idp # u ≈ u := idp
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definition transport_pp {A : Type} (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 :=
|
||
induction_on q (induction_on p idp)
|
||
|
||
definition transport_pV {A : Type} (P : A → Type) {x y : A} (p : x ≈ y) (z : P y) :
|
||
p # p^ # z ≈ z :=
|
||
(transport_pp P (p^) p z)^ @ ap (λr, transport P r z) (concat_Vp p)
|
||
|
||
definition transport_Vp {A : Type} (P : A → Type) {x y : A} (p : x ≈ y) (z : P x) :
|
||
p^ # p # z ≈ z :=
|
||
(transport_pp P p (p^) z)^ @ ap (λr, transport P r z) (concat_pV p)
|
||
|
||
|
||
-----------------------------------------------
|
||
-- *** Examples of difficult induction problems
|
||
-----------------------------------------------
|
||
|
||
theorem double_induction
|
||
{A : Type} {x y z : A} (p : x ≈ y) (q : y ≈ z)
|
||
{C : Π(x y z : A), Π(p : x ≈ y), Π(q : y ≈ z), Type}
|
||
(H : C x x x (idpath x) (idpath x)) :
|
||
C x y z p q :=
|
||
induction_on p (take z q, induction_on q H) z q
|
||
|
||
theorem double_induction2
|
||
{A : Type} {x y z : A} (p : x ≈ y) (q : z ≈ y)
|
||
{C : Π(x y z : A), Π(p : x ≈ y), Π(q : z ≈ y), Type}
|
||
(H : C z z z (idpath z) (idpath z)) :
|
||
C x y z p q :=
|
||
induction_on p (take y q, induction_on q H) y q
|
||
|
||
theorem double_induction2'
|
||
{A : Type} {x y z : A} (p : x ≈ y) (q : z ≈ y)
|
||
{C : Π(x y z : A), Π(p : x ≈ y), Π(q : z ≈ y), Type}
|
||
(H : C z z z (idpath z) (idpath z)) : C x y z p q :=
|
||
induction_on p (take y q, induction_on q H) y q
|
||
|
||
theorem triple_induction
|
||
{A : Type} {x y z w : A} (p : x ≈ y) (q : y ≈ z) (r : z ≈ w)
|
||
{C : Π(x y z w : A), Π(p : x ≈ y), Π(q : y ≈ z), Π(r: z ≈ w), Type}
|
||
(H : C x x x x (idpath x) (idpath x) (idpath x)) :
|
||
C x y z w p q r :=
|
||
induction_on p (take z q, induction_on q (take w r, induction_on r H)) z q w r
|
||
|
||
-- try this again
|
||
definition concat_pV_p_new {A : Type} {x y z : A} (p : x ≈ z) (q : y ≈ z) : (p @ q^) @ q ≈ p :=
|
||
double_induction2 p q idp
|
||
|
||
definition transport_p_pp {A : Type} (P : A → Type)
|
||
{x y z w : A} (p : x ≈ y) (q : y ≈ z) (r : z ≈ w) (u : P x) :
|
||
ap (λe, e # u) (concat_p_pp p q r) @ (transport_pp P (p @ q) r u) @
|
||
ap (transport P r) (transport_pp P p q u)
|
||
≈ (transport_pp P p (q @ r) u) @ (transport_pp P q r (p # u))
|
||
:> ((p @ (q @ r)) # u ≈ r # q # p # u) :=
|
||
triple_induction p q r (take u, idp) u
|
||
|
||
-- Here is another coherence lemma for transport.
|
||
definition transport_pVp {A} (P : A → Type) {x y : A} (p : x ≈ y) (z : P x) :
|
||
transport_pV P p (transport P p z) ≈ ap (transport P p) (transport_Vp P p z)
|
||
:= induction_on p idp
|
||
|
||
-- Dependent transport in a doubly dependent type.
|
||
definition transportD {A : Type} (B : A → Type) (C : Π a : A, B a → Type)
|
||
{x1 x2 : A} (p : x1 ≈ x2) (y : B x1) (z : C x1 y) :
|
||
C x2 (p # y) :=
|
||
induction_on p z
|
||
|
||
-- Transporting along higher-dimensional paths
|
||
definition transport2 {A : Type} (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
|
||
|
||
-- An alternative definition.
|
||
definition transport2_is_ap10 {A : Type} (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 :=
|
||
induction_on r idp
|
||
|
||
definition transport2_p2p {A : Type} (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 :=
|
||
induction_on r1 (induction_on r2 idp)
|
||
|
||
-- TODO: another interesting case
|
||
definition transport2_V {A : Type} (Q : A → Type) {x y : A} {p q : x ≈ y} (r : p ≈ q) (z : Q x) :
|
||
transport2 Q (r^) z ≈ ((transport2 Q r z)^) :=
|
||
-- induction_on r idp -- doesn't work
|
||
induction_on r (idpath (inverse (transport2 Q (idpath p) z)))
|
||
|
||
definition concat_AT {A : Type} (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 :=
|
||
induction_on r (concat_p1 _ @ (concat_1p _)^)
|
||
|
||
-- TODO (from Coq library): What should this be called?
|
||
definition ap_transport {A} {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)) :=
|
||
induction_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 transport_const {A B : Type} {x1 x2 : A} (p : x1 ≈ x2) (y : B) :
|
||
transport (λx, B) p y ≈ y :=
|
||
induction_on p idp
|
||
|
||
definition transport2_const {A B : Type} {x1 x2 : A} {p q : x1 ≈ x2} (r : p ≈ q) (y : B) :
|
||
transport_const p y ≈ transport2 (λu, B) r y @ transport_const q y :=
|
||
induction_on r (concat_1p _)^
|
||
|
||
-- Transporting in a pulled back fibration.
|
||
definition transport_compose {A B} {x y : A} (P : B → Type) (f : A → B) (p : x ≈ y) (z : P (f x)) :
|
||
transport (λx, P (f x)) p z ≈ transport P (ap f p) z :=
|
||
induction_on p idp
|
||
|
||
definition transport_precompose {A B C} (f : A → B) (g g' : B → C) (p : g ≈ g') :
|
||
transport (λh : B → C, g ∘ f ≈ h ∘ f) p idp ≈ ap (λh, h ∘ f) p :=
|
||
induction_on p idp
|
||
|
||
definition apD10_ap_precompose {A B C} (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) :=
|
||
induction_on p idp
|
||
|
||
definition apD10_ap_postcompose {A B C} (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) :=
|
||
induction_on p idp
|
||
|
||
-- TODO: another example where a term has to be given explicitly
|
||
-- A special case of [transport_compose] which seems to come up a lot.
|
||
definition transport_idmap_ap A (P : A → Type) x y (p : x ≈ y) (u : P x) :
|
||
transport P p u ≈ transport (λz, z) (ap P p) u :=
|
||
induction_on p (idpath (transport (λ (z : Type), z) (ap P (idpath x)) u))
|
||
|
||
|
||
-- The behavior of [ap] and [apD]
|
||
-- ------------------------------
|
||
|
||
-- In a constant fibration, [apD] reduces to [ap], modulo [transport_const].
|
||
definition apD_const {A B} {x y : A} (f : A → B) (p: x ≈ y) :
|
||
apD f p ≈ transport_const p (f x) @ ap f p :=
|
||
induction_on p idp
|
||
|
||
|
||
-- The 2-dimensional groupoid structure
|
||
-- ------------------------------------
|
||
|
||
-- Horizontal composition of 2-dimensional paths.
|
||
definition concat2 {A} {x y z : A} {p p' : x ≈ y} {q q' : y ≈ z} (h : p ≈ p') (h' : q ≈ q') :
|
||
p @ q ≈ p' @ q' :=
|
||
induction_on h (induction_on h' idp)
|
||
|
||
infixl `@@`:75 := concat2
|
||
|
||
-- 2-dimensional path inversion
|
||
definition inverse2 {A : Type} {x y : A} {p q : x ≈ y} (h : p ≈ q) : p^ ≈ q^ :=
|
||
induction_on h idp
|
||
|
||
-- Whiskering
|
||
-- ----------
|
||
|
||
definition whiskerL {A : Type} {x y z : A} (p : x ≈ y) {q r : y ≈ z} (h : q ≈ r) : p @ q ≈ p @ r :=
|
||
idp @@ h
|
||
|
||
definition whiskerR {A : Type} {x y z : A} {p q : x ≈ y} (h : p ≈ q) (r : y ≈ z) : p @ r ≈ q @ r :=
|
||
h @@ idp
|
||
|
||
|
||
-- Unwhiskering, a.k.a. cancelling
|
||
-- -------------------------------
|
||
|
||
definition cancelL {A} {x y z : A} (p : x ≈ y) (q r : y ≈ z) : (p @ q ≈ p @ r) → (q ≈ r) :=
|
||
induction_on p (take r, induction_on r (take q a, (concat_1p q)^ @ a)) r q
|
||
|
||
definition cancelR {A} {x y z : A} (p q : x ≈ y) (r : y ≈ z) : (p @ r ≈ q @ r) → (p ≈ q) :=
|
||
induction_on r (take p, induction_on p (take q a, a @ concat_p1 q)) p q
|
||
|
||
-- Whiskering and identity paths.
|
||
|
||
definition whiskerR_p1 {A : Type} {x y : A} {p q : x ≈ y} (h : p ≈ q) :
|
||
(concat_p1 p)^ @ whiskerR h idp @ concat_p1 q ≈ h :=
|
||
induction_on h (induction_on p idp)
|
||
|
||
definition whiskerR_1p {A : Type} {x y z : A} (p : x ≈ y) (q : y ≈ z) :
|
||
whiskerR idp q ≈ idp :> (p @ q ≈ p @ q) :=
|
||
induction_on q idp
|
||
|
||
definition whiskerL_p1 {A : Type} {x y z : A} (p : x ≈ y) (q : y ≈ z) :
|
||
whiskerL p idp ≈ idp :> (p @ q ≈ p @ q) :=
|
||
induction_on q idp
|
||
|
||
definition whiskerL_1p {A : Type} {x y : A} {p q : x ≈ y} (h : p ≈ q) :
|
||
(concat_1p p) ^ @ whiskerL idp h @ concat_1p q ≈ h :=
|
||
induction_on h (induction_on p idp)
|
||
|
||
definition concat2_p1 {A : Type} {x y : A} {p q : x ≈ y} (h : p ≈ q) :
|
||
h @@ idp ≈ whiskerR h idp :> (p @ idp ≈ q @ idp) :=
|
||
induction_on h idp
|
||
|
||
definition concat2_1p {A : Type} {x y : A} {p q : x ≈ y} (h : p ≈ q) :
|
||
idp @@ h ≈ whiskerL idp h :> (idp @ p ≈ idp @ q) :=
|
||
induction_on h idp
|
||
|
||
-- TODO: note, 4 inductions
|
||
-- The interchange law for concatenation.
|
||
definition concat_concat2 {A : Type} {x y z : A} {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) :=
|
||
induction_on d (induction_on c (induction_on b (induction_on a idp)))
|
||
|
||
definition concat_whisker {A} {x y z : A} (p p' : x ≈ y) (q q' : y ≈ z) (a : p ≈ p') (b : q ≈ q') :
|
||
(whiskerR a q) @ (whiskerL p' b) ≈ (whiskerL p b) @ (whiskerR a q') :=
|
||
induction_on b (induction_on a (concat_1p _)^)
|
||
|
||
-- Structure corresponding to the coherence equations of a bicategory.
|
||
|
||
-- The "pentagonator": the 3-cell witnessing the associativity pentagon.
|
||
definition pentagon {A : Type} {v w x y z : A} (p : v ≈ w) (q : w ≈ x) (r : x ≈ y) (s : y ≈ z) :
|
||
whiskerL p (concat_p_pp q r s)
|
||
@ concat_p_pp p (q @ r) s
|
||
@ whiskerR (concat_p_pp p q r) s
|
||
≈ concat_p_pp p q (r @ s) @ concat_p_pp (p @ q) r s :=
|
||
induction_on p (take q, induction_on q (take r, induction_on r (take s, induction_on s idp))) q r s
|
||
|
||
-- The 3-cell witnessing the left unit triangle.
|
||
definition triangulator {A : Type} {x y z : A} (p : x ≈ y) (q : y ≈ z) :
|
||
concat_p_pp p idp q @ whiskerR (concat_p1 p) q ≈ whiskerL p (concat_1p q) :=
|
||
induction_on p (take q, induction_on q idp) q
|
||
|
||
definition eckmann_hilton {A : Type} {x:A} (p q : idp ≈ idp :> (x ≈ x)) : p @ q ≈ q @ p :=
|
||
(whiskerR_p1 p @@ whiskerL_1p q)^
|
||
@ (concat_p1 _ @@ concat_p1 _)
|
||
@ (concat_1p _ @@ concat_1p _)
|
||
@ (concat_whisker _ _ _ _ p q)
|
||
@ (concat_1p _ @@ concat_1p _)^
|
||
@ (concat_p1 _ @@ concat_p1 _)^
|
||
@ (whiskerL_1p q @@ whiskerR_p1 p)
|
||
|
||
|
||
-- The action of functions on 2-dimensional paths
|
||
definition ap02 {A B : Type} (f:A → B) {x y : A} {p q : x ≈ y} (r : p ≈ q) : ap f p ≈ ap f q :=
|
||
induction_on r idp
|
||
|
||
definition ap02_pp {A B} (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' :=
|
||
induction_on r (induction_on r' idp)
|
||
|
||
definition ap02_p2p {A B} (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_pp f p q
|
||
@ (ap02 f r @@ ap02 f s)
|
||
@ (ap_pp f p' q')^ :=
|
||
induction_on r (induction_on s (induction_on q (induction_on p idp)))
|
||
|
||
definition apD02 {A : Type} {B : A → Type} {x y : A} {p q : x ≈ y} (f : Π x, B x) (r : p ≈ q) :
|
||
apD f p ≈ transport2 B r (f x) @ apD f q :=
|
||
induction_on r (concat_1p _)^
|