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