228 lines
9.8 KiB
Text
228 lines
9.8 KiB
Text
/-
|
||
Copyright (c) 2014 Floris van Doorn. All rights reserved.
|
||
Released under Apache 2.0 license as described in the file LICENSE.
|
||
|
||
Module: init.path
|
||
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
|
||
|
||
notation f `∼2`:50 g := homotopy2 f g
|
||
notation f `∼3`:50 g := homotopy3 f g
|
||
|
||
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 {f g : Πa b, C a b} (p : f = g) : f ∼2 g :=
|
||
λa b, apd10 (apd10 p a) b
|
||
|
||
definition apd1000 {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
|
||
|
||
end eq
|
||
|
||
open eq 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, function.compose],
|
||
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, apply eq_of_homotopy, intro a,
|
||
apply concat,
|
||
{apply (ap (λx, @apd100 _ _ (λ(b : B a)(c : C a b), _) _ _ (x a))),
|
||
apply (right_inv apd10)},
|
||
--TODO: remove implicit argument after #469 is closed
|
||
apply (@right_inv _ _ apd100 !is_equiv_apd100) --is explicit argument needed here?
|
||
end
|
||
begin
|
||
intro p, cases p, apply eq_of_homotopy3_id
|
||
end
|
||
end funext
|
||
|
||
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 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
|