lean2/library/logic/cast.lean
2014-11-30 13:52:34 -08:00

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-- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
-- Released under Apache 2.0 license as described in the file LICENSE.
-- Author: Leonardo de Moura
import logic.eq logic.heq logic.quantifiers
open eq.ops
-- cast.lean
-- =========
section
universe variable u
variables {A B : Type.{u}}
definition cast (H : A = B) (a : A) : B :=
eq.rec a H
theorem cast_refl (a : A) : cast (eq.refl A) a = a :=
rfl
theorem cast_proof_irrel (H₁ H₂ : A = B) (a : A) : cast H₁ a = cast H₂ a :=
rfl
theorem cast_eq (H : A = A) (a : A) : cast H a = a :=
rfl
end
namespace heq
universe variable u
variables {A B C : Type.{u}} {a a' : A} {b b' : B} {c : C}
theorem to_cast_eq (H : a == b) : cast (type_eq H) a = b :=
drec_on H !cast_eq
end heq
section
universe variables u v
variables {A A' B C : Type.{u}} {P P' : A → Type.{v}} {a a' : A} {b : B}
theorem eq_rec_heq (H : a = a') (p : P a) : eq.rec_on H p == p :=
eq.drec_on H !heq.refl
-- should H₁ be explicit (useful in e.g. hproof_irrel)
theorem eq_rec_to_heq {H₁ : a = a'} {p : P a} {p' : P a'} (H₂ : eq.rec_on H₁ p = p') : p == p' :=
calc
p == eq.rec_on H₁ p : heq.symm (eq_rec_heq H₁ p)
... = p' : H₂
theorem cast_to_heq {H₁ : A = B} (H₂ : cast H₁ a = b) : a == b :=
eq_rec_to_heq H₂
theorem hproof_irrel {a b : Prop} (H : a = b) (H₁ : a) (H₂ : b) : H₁ == H₂ :=
eq_rec_to_heq (proof_irrel (cast H H₁) H₂)
--TODO: generalize to eq.rec. This is a special case of rec_on_compose in eq.lean
theorem cast_trans (Hab : A = B) (Hbc : B = C) (a : A) :
cast Hbc (cast Hab a) = cast (Hab ⬝ Hbc) a :=
heq.to_eq (calc
cast Hbc (cast Hab a) == cast Hab a : eq_rec_heq Hbc (cast Hab a)
... == a : eq_rec_heq Hab a
... == cast (Hab ⬝ Hbc) a : heq.symm (eq_rec_heq (Hab ⬝ Hbc) a))
theorem pi_eq (H : P = P') : (Π x, P x) = (Π x, P' x) :=
H ▸ (eq.refl (Π x, P x))
theorem hcongr_arg (f : Πx, P x) {a b : A} (H : a = b) : f a == f b :=
H ▸ (heq.refl (f a))
theorem rec_on_app (H : P = P') (f : Π x, P x) (a : A) : eq.rec_on H f a == f a :=
have aux : ∀ H : P = P, eq.rec_on H f a == f a, from
take H : P = P, heq.refl (eq.rec_on H f a),
(H ▸ aux) H
theorem hcongr_fun {f : Π x, P x} {f' : Π x, P' x} (a : A) (H₁ : f == f') (H₂ : P = P') : f a == f' a :=
have aux : ∀ (f : Π x, P x) (f' : Π x, P x), f == f' → f a == f' a, from
take f f' H, heq.to_eq H ▸ heq.refl (f a),
(H₂ ▸ aux) f f' H₁
theorem rec_on_pull (H : P = P') (f : Π x, P x) (a : A) : eq.rec_on H f a = eq.rec_on (congr_fun H a) (f a) :=
heq.to_eq (calc
eq.rec_on H f a == f a : rec_on_app H f a
... == eq.rec_on (congr_fun H a) (f a) : heq.symm (eq_rec_heq (congr_fun H a) (f a)))
theorem cast_app (H : P = P') (f : Π x, P x) (a : A) : cast (pi_eq H) f a == f a :=
have H₁ : ∀ (H : (Π x, P x) = (Π x, P x)), cast H f a == f a, from
assume H, heq.from_eq (congr_fun (cast_eq H f) a),
have H₂ : ∀ (H : (Π x, P x) = (Π x, P' x)), cast H f a == f a, from
H ▸ H₁,
H₂ (pi_eq H)
theorem hcongr {P' : A' → Type} {f : Π a, P a} {f' : Π a', P' a'} {a : A} {a' : A'}
(Hf : f == f') (HP : P == P') (Ha : a == a') : f a == f' a' :=
have H1 : ∀ (B P' : A → Type) (f : Π x, P x) (f' : Π x, P' x), f == f' → (λx, P x) == (λx, P' x)
→ f a == f' a, from
take P P' f f' Hf HB, hcongr_fun a Hf (heq.to_eq HB),
have H2 : ∀ (B : A → Type) (P' : A' → Type) (f : Π x, P x) (f' : Π x, P' x),
f == f' → (λx, P x) == (λx, P' x) → f a == f' a', from heq.subst Ha H1,
H2 P P' f f' Hf HP
end
section
variables {A : 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 : Type}
variables {a a' : A}
{b : B a} {b' : B a'}
{c : C a b} {c' : C a' b'}
{d : D a b c} {d' : D a' b' c'}
theorem hcongr_arg2 (f : Πa b, C a b) (Ha : a = a') (Hb : b == b') : f a b == f a' b' :=
hcongr (hcongr_arg f Ha) (hcongr_arg C Ha) Hb
theorem hcongr_arg3 (f : Πa b c, D a b c) (Ha : a = a') (Hb : b == b') (Hc : c == c')
: f a b c == f a' b' c' :=
hcongr (hcongr_arg2 f Ha Hb) (hcongr_arg2 D Ha Hb) Hc
theorem hcongr_arg4 (f : Πa b c d, E a b c d)
(Ha : a = a') (Hb : b == b') (Hc : c == c') (Hd : d == d') : f a b c d == f a' b' c' d' :=
hcongr (hcongr_arg3 f Ha Hb Hc) (hcongr_arg3 E Ha Hb Hc) Hd
theorem dcongr_arg2 (f : Πa, B a → F) (Ha : a = a') (Hb : eq.rec_on Ha b = b')
: f a b = f a' b' :=
heq.to_eq (hcongr_arg2 f Ha (eq_rec_to_heq Hb))
theorem dcongr_arg3 (f : Πa b, C a b → F) (Ha : a = a') (Hb : eq.rec_on Ha b = b')
(Hc : cast (dcongr_arg2 C Ha Hb) c = c') : f a b c = f a' b' c' :=
heq.to_eq (hcongr_arg3 f Ha (eq_rec_to_heq Hb) (eq_rec_to_heq Hc))
theorem dcongr_arg4 (f : Πa b c, D a b c → F) (Ha : a = a') (Hb : eq.rec_on Ha b = b')
(Hc : cast (dcongr_arg2 C Ha Hb) c = c')
(Hd : cast (dcongr_arg3 D Ha Hb Hc) d = d') : f a b c d = f a' b' c' d' :=
heq.to_eq (hcongr_arg4 f Ha (eq_rec_to_heq Hb) (eq_rec_to_heq Hc) (eq_rec_to_heq Hd))
--mixed versions (we want them for example if C a' b' is a subsingleton, like a proposition. Then proving eq is easier than proving heq)
theorem hdcongr_arg3 (f : Πa b, C a b → F) (Ha : a = a') (Hb : b == b')
(Hc : cast (heq.to_eq (hcongr_arg2 C Ha Hb)) c = c')
: f a b c = f a' b' c' :=
heq.to_eq (hcongr_arg3 f Ha Hb (eq_rec_to_heq Hc))
theorem hhdcongr_arg4 (f : Πa b c, D a b c → F) (Ha : a = a') (Hb : b == b')
(Hc : c == c')
(Hd : cast (dcongr_arg3 D Ha (!eq.rec_on_irrel_arg ⬝ heq.to_cast_eq Hb)
(!eq.rec_on_irrel_arg ⬝ heq.to_cast_eq Hc)) d = d')
: f a b c d = f a' b' c' d' :=
heq.to_eq (hcongr_arg4 f Ha Hb Hc (eq_rec_to_heq Hd))
theorem hddcongr_arg4 (f : Πa b c, D a b c → F) (Ha : a = a') (Hb : b == b')
(Hc : cast (heq.to_eq (hcongr_arg2 C Ha Hb)) c = c')
(Hd : cast (hdcongr_arg3 D Ha Hb Hc) d = d')
: f a b c d = f a' b' c' d' :=
heq.to_eq (hcongr_arg4 f Ha Hb (eq_rec_to_heq Hc) (eq_rec_to_heq Hd))
--Is a reasonable version of "hcongr2" provable without pi_ext and funext?
--It looks like you need some ugly extra conditions
-- theorem hcongr2' {A A' : Type} {B : A → Type} {B' : A' → Type} {C : Πa, B a → Type} {C' : Πa, B' a → Type}
-- {f : Π a b, C a b} {f' : Π a' b', C' a' b'} {a : A} {a' : A'} {b : B a} {b' : B' a'}
-- (HBB' : B == B') (HCC' : C == C')
-- (Hff' : f == f') (Haa' : a == a') (Hbb' : b == b') : f a b == f' a' b' :=
-- hcongr (hcongr Hff' (sorry) Haa') (hcongr HCC' (sorry) Haa') Hbb'
end