lean2/library/logic/cast.lean

-- 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 .eq .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

inductive heq {A : Type} (a : A) : Π {B : Type}, B → Prop :=
refl : heq a a
infixl `==`:50 := heq

namespace heq
  universe variable u
  variables {A B C : Type.{u}} {a a' : A} {b b' : B} {c : C}
  theorem drec_on {C : Π {B : Type} (b : B), a == b → Type} (H₁ : a == b) (H₂ : C a (refl a)) : C b H₁ :=
  rec (λ H₁ : a == a, show C a H₁, from H₂) H₁ H₁

  theorem subst {P : ∀T : Type, T → Prop} (H₁ : a == b) (H₂ : P A a) : P B b :=
  rec_on H₁ H₂

  theorem symm (H : a == b) : b == a :=
  subst H (refl a)

  theorem type_eq (H : a == b) : A = B :=
  subst H (eq.refl A)

  theorem from_eq (H : a = a') : a == a' :=
  eq.subst H (refl a)

  theorem trans (H₁ : a == b) (H₂ : b == c) : a == c :=
  subst H₂ H₁

  theorem trans_left (H₁ : a == b) (H₂ : b = b') : a == b' :=
  trans H₁ (from_eq H₂)

  theorem trans_right (H₁ : a = a') (H₂ : a' == b) : a == b :=
  trans (from_eq H₁) H₂

  theorem to_cast_eq (H : a == b) : cast (type_eq H) a = b :=
  drec_on H !cast_eq

  theorem to_eq (H : a == a') : a = a' :=
  calc
    a = cast (eq.refl A) a : cast_eq
  ... = a'                 : to_cast_eq H

  theorem elim {D : Type} (H₁ : a == b) (H₂ : ∀ (Hab : A = B), cast Hab a = b → D) : D :=
  H₂ (type_eq H₁) (to_cast_eq H₁)

end heq

calc_trans heq.trans
calc_trans heq.trans_left
calc_trans heq.trans_right
calc_symm  heq.symm

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)

  theorem heq.true_elim {a : Prop} (H : a == true) : a :=
  eq_true_elim (heq.to_eq H)

  --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