252 lines
11 KiB
Text
252 lines
11 KiB
Text
/-
|
|
Copyright (c) 2014 Floris van Doorn. All rights reserved.
|
|
Released under Apache 2.0 license as described in the file LICENSE.
|
|
Author: Floris van Doorn
|
|
|
|
Theorems about functions with multiple arguments
|
|
-/
|
|
|
|
variables {A U V W X Y Z : Type} {B : A → Type} {C : Πa, B a → Type} {D : Πa b, C a b → Type}
|
|
{E : Πa b c, D a b c → Type} {F : Πa b c d, E a b c d → Type}
|
|
{G : Πa b c d e, F a b c d e → Type} {H : Πa b c d e f, G a b c d e f → Type}
|
|
variables {a a' : A} {u u' : U} {v v' : V} {w w' : W} {x x' x'' : X} {y y' : Y} {z z' : Z}
|
|
{b : B a} {b' : B a'}
|
|
{c : C a b} {c' : C a' b'}
|
|
{d : D a b c} {d' : D a' b' c'}
|
|
{e : E a b c d} {e' : E a' b' c' d'}
|
|
{ff : F a b c d e} {f' : F a' b' c' d' e'}
|
|
{g : G a b c d e ff} {g' : G a' b' c' d' e' f'}
|
|
{h : H a b c d e ff g} {h' : H a' b' c' d' e' f' g'}
|
|
|
|
namespace eq
|
|
/-
|
|
Naming convention:
|
|
The theorem which states how to construct an path between two function applications is
|
|
api₀i₁...iₙ.
|
|
Here i₀, ... iₙ are digits, n is the arity of the function(s),
|
|
and iⱼ specifies the dimension of the path between the jᵗʰ argument
|
|
(i₀ specifies the dimension of the path between the functions).
|
|
A value iⱼ ≡ 0 means that the jᵗʰ arguments are definitionally equal
|
|
The functions are non-dependent, except when the theorem name contains trailing zeroes
|
|
(where the function is dependent only in the arguments where it doesn't result in any
|
|
transports in the theorem statement).
|
|
For the fully-dependent versions (except that the conclusion doesn't contain a transport)
|
|
we write
|
|
apdi₀i₁...iₙ.
|
|
|
|
For versions where only some arguments depend on some other arguments,
|
|
or for versions with transport in the conclusion (like apd), we don't have a
|
|
consistent naming scheme (yet).
|
|
|
|
We don't prove each theorem systematically, but prove only the ones which we actually need.
|
|
-/
|
|
|
|
definition homotopy2 [reducible] (f g : Πa b, C a b) : Type :=
|
|
Πa b, f a b = g a b
|
|
definition homotopy3 [reducible] (f g : Πa b c, D a b c) : Type :=
|
|
Πa b c, f a b c = g a b c
|
|
definition homotopy4 [reducible] (f g : Πa b c d, E a b c d) : Type :=
|
|
Πa b c d, f a b c d = g a b c d
|
|
|
|
infix ` ~2 `:50 := homotopy2
|
|
infix ` ~3 `:50 := homotopy3
|
|
|
|
definition ap011 (f : U → V → W) (Hu : u = u') (Hv : v = v') : f u v = f u' v' :=
|
|
by cases Hu; congruence; repeat assumption
|
|
|
|
definition ap0111 (f : U → V → W → X) (Hu : u = u') (Hv : v = v') (Hw : w = w')
|
|
: f u v w = f u' v' w' :=
|
|
by cases Hu; congruence; repeat assumption
|
|
|
|
definition ap01111 (f : U → V → W → X → Y)
|
|
(Hu : u = u') (Hv : v = v') (Hw : w = w') (Hx : x = x')
|
|
: f u v w x = f u' v' w' x' :=
|
|
by cases Hu; congruence; repeat assumption
|
|
|
|
definition ap011111 (f : U → V → W → X → Y → Z)
|
|
(Hu : u = u') (Hv : v = v') (Hw : w = w') (Hx : x = x') (Hy : y = y')
|
|
: f u v w x y = f u' v' w' x' y' :=
|
|
by cases Hu; congruence; repeat assumption
|
|
|
|
definition ap0111111 (f : U → V → W → X → Y → Z → A)
|
|
(Hu : u = u') (Hv : v = v') (Hw : w = w') (Hx : x = x') (Hy : y = y') (Hz : z = z')
|
|
: f u v w x y z = f u' v' w' x' y' z' :=
|
|
by cases Hu; congruence; repeat assumption
|
|
|
|
definition ap010 (f : X → Πa, B a) (Hx : x = x') : f x ~ f x' :=
|
|
by intros; cases Hx; reflexivity
|
|
|
|
definition ap0100 (f : X → Πa b, C a b) (Hx : x = x') : f x ~2 f x' :=
|
|
by intros; cases Hx; reflexivity
|
|
|
|
definition ap01000 (f : X → Πa b c, D a b c) (Hx : x = x') : f x ~3 f x' :=
|
|
by intros; cases Hx; reflexivity
|
|
|
|
definition apd011 (f : Πa, B a → Z) (Ha : a = a') (Hb : transport B Ha b = b')
|
|
: f a b = f a' b' :=
|
|
by cases Ha; cases Hb; reflexivity
|
|
|
|
definition apd0111 (f : Πa b, C a b → Z) (Ha : a = a') (Hb : transport B Ha b = b')
|
|
(Hc : cast (apd011 C Ha Hb) c = c')
|
|
: f a b c = f a' b' c' :=
|
|
by cases Ha; cases Hb; cases Hc; reflexivity
|
|
|
|
definition apd01111 (f : Πa b c, D a b c → Z) (Ha : a = a') (Hb : transport B Ha b = b')
|
|
(Hc : cast (apd011 C Ha Hb) c = c') (Hd : cast (apd0111 D Ha Hb Hc) d = d')
|
|
: f a b c d = f a' b' c' d' :=
|
|
by cases Ha; cases Hb; cases Hc; cases Hd; reflexivity
|
|
|
|
definition apd011111 (f : Πa b c d, E a b c d → Z) (Ha : a = a') (Hb : transport B Ha b = b')
|
|
(Hc : cast (apd011 C Ha Hb) c = c') (Hd : cast (apd0111 D Ha Hb Hc) d = d')
|
|
(He : cast (apd01111 E Ha Hb Hc Hd) e = e')
|
|
: f a b c d e = f a' b' c' d' e' :=
|
|
by cases Ha; cases Hb; cases Hc; cases Hd; cases He; reflexivity
|
|
|
|
definition apd0111111 (f : Πa b c d e, F a b c d e → Z) (Ha : a = a') (Hb : transport B Ha b = b')
|
|
(Hc : cast (apd011 C Ha Hb) c = c') (Hd : cast (apd0111 D Ha Hb Hc) d = d')
|
|
(He : cast (apd01111 E Ha Hb Hc Hd) e = e') (Hf : cast (apd011111 F Ha Hb Hc Hd He) ff = f')
|
|
: f a b c d e ff = f a' b' c' d' e' f' :=
|
|
begin cases Ha, cases Hb, cases Hc, cases Hd, cases He, cases Hf, reflexivity end
|
|
|
|
-- definition apd0111111 (f : Πa b c d e ff, G a b c d e ff → Z) (Ha : a = a') (Hb : transport B Ha b = b')
|
|
-- (Hc : cast (apd011 C Ha Hb) c = c') (Hd : cast (apd0111 D Ha Hb Hc) d = d')
|
|
-- (He : cast (apd01111 E Ha Hb Hc Hd) e = e') (Hf : cast (apd011111 F Ha Hb Hc Hd He) ff = f')
|
|
-- (Hg : cast (apd0111111 G Ha Hb Hc Hd He Hf) g = g')
|
|
-- : f a b c d e ff g = f a' b' c' d' e' f' g' :=
|
|
-- by cases Ha; cases Hb; cases Hc; cases Hd; cases He; cases Hf; cases Hg; reflexivity
|
|
|
|
-- definition apd01111111 (f : Πa b c d e ff g, G a b c d e ff g → Z) (Ha : a = a') (Hb : transport B Ha b = b')
|
|
-- (Hc : cast (apd011 C Ha Hb) c = c') (Hd : cast (apd0111 D Ha Hb Hc) d = d')
|
|
-- (He : cast (apd01111 E Ha Hb Hc Hd) e = e') (Hf : cast (apd011111 F Ha Hb Hc Hd He) ff = f')
|
|
-- (Hg : cast (apd0111111 G Ha Hb Hc Hd He Hf) g = g') (Hh : cast (apd01111111 H Ha Hb Hc Hd He Hf Hg) h = h')
|
|
-- : f a b c d e ff g h = f a' b' c' d' e' f' g' h' :=
|
|
-- by cases Ha; cases Hb; cases Hc; cases Hd; cases He; cases Hf; cases Hg; cases Hh; reflexivity
|
|
|
|
definition apd100 [unfold 6] {f g : Πa b, C a b} (p : f = g) : f ~2 g :=
|
|
λa b, apd10 (apd10 p a) b
|
|
|
|
definition apd1000 [unfold 7] {f g : Πa b c, D a b c} (p : f = g) : f ~3 g :=
|
|
λa b c, apd100 (apd10 p a) b c
|
|
|
|
/- some properties of these variants of ap -/
|
|
|
|
-- we only prove what we currently need
|
|
|
|
definition ap010_con (f : X → Πa, B a) (p : x = x') (q : x' = x'') :
|
|
ap010 f (p ⬝ q) a = ap010 f p a ⬝ ap010 f q a :=
|
|
eq.rec_on q (eq.rec_on p idp)
|
|
|
|
definition ap010_ap (f : X → Πa, B a) (g : Y → X) (p : y = y') :
|
|
ap010 f (ap g p) a = ap010 (λy, f (g y)) p a :=
|
|
eq.rec_on p idp
|
|
|
|
/- the following theorems are function extentionality for functions with multiple arguments -/
|
|
|
|
definition eq_of_homotopy2 {f g : Πa b, C a b} (H : f ~2 g) : f = g :=
|
|
eq_of_homotopy (λa, eq_of_homotopy (H a))
|
|
|
|
definition eq_of_homotopy3 {f g : Πa b c, D a b c} (H : f ~3 g) : f = g :=
|
|
eq_of_homotopy (λa, eq_of_homotopy2 (H a))
|
|
|
|
definition eq_of_homotopy2_id (f : Πa b, C a b)
|
|
: eq_of_homotopy2 (λa b, idpath (f a b)) = idpath f :=
|
|
begin
|
|
transitivity eq_of_homotopy (λ a, idpath (f a)),
|
|
{apply (ap eq_of_homotopy), apply eq_of_homotopy, intros, apply eq_of_homotopy_idp},
|
|
apply eq_of_homotopy_idp
|
|
end
|
|
|
|
definition eq_of_homotopy3_id (f : Πa b c, D a b c)
|
|
: eq_of_homotopy3 (λa b c, idpath (f a b c)) = idpath f :=
|
|
begin
|
|
transitivity _,
|
|
{apply (ap eq_of_homotopy), apply eq_of_homotopy, intros, apply eq_of_homotopy2_id},
|
|
apply eq_of_homotopy_idp
|
|
end
|
|
|
|
definition eq_of_homotopy2_inv {f g : Πa b, C a b} (H : f ~2 g)
|
|
: eq_of_homotopy2 (λa b, (H a b)⁻¹) = (eq_of_homotopy2 H)⁻¹ :=
|
|
ap eq_of_homotopy (eq_of_homotopy (λa, !eq_of_homotopy_inv)) ⬝ !eq_of_homotopy_inv
|
|
|
|
definition eq_of_homotopy3_inv {f g : Πa b c, D a b c} (H : f ~3 g)
|
|
: eq_of_homotopy3 (λa b c, (H a b c)⁻¹) = (eq_of_homotopy3 H)⁻¹ :=
|
|
ap eq_of_homotopy (eq_of_homotopy (λa, !eq_of_homotopy2_inv)) ⬝ !eq_of_homotopy_inv
|
|
|
|
definition eq_of_homotopy2_con {f g h : Πa b, C a b} (H1 : f ~2 g) (H2 : g ~2 h)
|
|
: eq_of_homotopy2 (λa b, H1 a b ⬝ H2 a b) = eq_of_homotopy2 H1 ⬝ eq_of_homotopy2 H2 :=
|
|
ap eq_of_homotopy (eq_of_homotopy (λa, !eq_of_homotopy_con)) ⬝ !eq_of_homotopy_con
|
|
|
|
definition eq_of_homotopy3_con {f g h : Πa b c, D a b c} (H1 : f ~3 g) (H2 : g ~3 h)
|
|
: eq_of_homotopy3 (λa b c, H1 a b c ⬝ H2 a b c) = eq_of_homotopy3 H1 ⬝ eq_of_homotopy3 H2 :=
|
|
ap eq_of_homotopy (eq_of_homotopy (λa, !eq_of_homotopy2_con)) ⬝ !eq_of_homotopy_con
|
|
|
|
end eq
|
|
|
|
open eq equiv is_equiv
|
|
namespace funext
|
|
definition is_equiv_apd100 [instance] (f g : Πa b, C a b)
|
|
: is_equiv (@apd100 A B C f g) :=
|
|
adjointify _
|
|
eq_of_homotopy2
|
|
begin
|
|
intro H, esimp [apd100, eq_of_homotopy2],
|
|
apply eq_of_homotopy, intro a,
|
|
apply concat, apply (ap (λx, apd10 (x a))), apply (right_inv apd10),
|
|
apply (right_inv apd10)
|
|
end
|
|
begin
|
|
intro p, cases p, apply eq_of_homotopy2_id
|
|
end
|
|
|
|
definition is_equiv_apd1000 [instance] (f g : Πa b c, D a b c)
|
|
: is_equiv (@apd1000 A B C D f g) :=
|
|
adjointify _
|
|
eq_of_homotopy3
|
|
begin
|
|
intro H, esimp,
|
|
apply eq_of_homotopy, intro a,
|
|
transitivity apd100 (eq_of_homotopy2 (H a)),
|
|
{apply ap (λx, apd100 (x a)),
|
|
apply right_inv apd10},
|
|
apply right_inv apd100
|
|
end
|
|
begin
|
|
intro p, cases p, apply eq_of_homotopy3_id
|
|
end
|
|
end funext
|
|
|
|
attribute funext.is_equiv_apd100 funext.is_equiv_apd1000 [constructor]
|
|
|
|
namespace eq
|
|
open funext
|
|
local attribute funext.is_equiv_apd100 [instance]
|
|
protected definition homotopy2.rec_on {f g : Πa b, C a b} {P : (f ~2 g) → Type}
|
|
(p : f ~2 g) (H : Π(q : f = g), P (apd100 q)) : P p :=
|
|
right_inv apd100 p ▸ H (eq_of_homotopy2 p)
|
|
|
|
protected definition homotopy3.rec_on {f g : Πa b c, D a b c} {P : (f ~3 g) → Type}
|
|
(p : f ~3 g) (H : Π(q : f = g), P (apd1000 q)) : P p :=
|
|
right_inv apd1000 p ▸ H (eq_of_homotopy3 p)
|
|
|
|
definition eq_equiv_homotopy2 [constructor] (f g : Πa b, C a b) : (f = g) ≃ (f ~2 g) :=
|
|
equiv.mk apd100 _
|
|
|
|
definition eq_equiv_homotopy3 [constructor] (f g : Πa b c, D a b c) : (f = g) ≃ (f ~3 g) :=
|
|
equiv.mk apd1000 _
|
|
|
|
definition apd10_ap (f : X → Πa, B a) (p : x = x')
|
|
: apd10 (ap f p) = ap010 f p :=
|
|
eq.rec_on p idp
|
|
|
|
definition eq_of_homotopy_ap010 (f : X → Πa, B a) (p : x = x')
|
|
: eq_of_homotopy (ap010 f p) = ap f p :=
|
|
inv_eq_of_eq !apd10_ap⁻¹
|
|
|
|
definition ap_eq_ap_of_homotopy {f : X → Πa, B a} {p q : x = x'} (H : ap010 f p ~ ap010 f q)
|
|
: ap f p = ap f q :=
|
|
calc
|
|
ap f p = eq_of_homotopy (ap010 f p) : eq_of_homotopy_ap010
|
|
... = eq_of_homotopy (ap010 f q) : eq_of_homotopy H
|
|
... = ap f q : eq_of_homotopy_ap010
|
|
|
|
end eq
|