73 lines
3.5 KiB
Text
73 lines
3.5 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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-- Author: Leonardo de Moura
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----------------------------------------------------------------------------------------------------
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import logic.classes.decidable tools.tactic
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using decidable tactic eq_proofs
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definition ite (c : Prop) {H : decidable c} {A : Type} (t e : A) : A :=
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rec_on H (assume Hc, t) (assume Hnc, e)
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notation `if` c `then` t `else` e:45 := ite c t e
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theorem if_pos {c : Prop} {H : decidable c} (Hc : c) {A : Type} (t e : A) : (if c then t else e) = t :=
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decidable_rec
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(assume Hc : c, refl (@ite c (inl Hc) A t e))
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(assume Hnc : ¬c, absurd_elim (@ite c (inr Hnc) A t e = t) Hc Hnc)
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H
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theorem if_neg {c : Prop} {H : decidable c} (Hnc : ¬c) {A : Type} (t e : A) : (if c then t else e) = e :=
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decidable_rec
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(assume Hc : c, absurd_elim (@ite c (inl Hc) A t e = e) Hc Hnc)
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(assume Hnc : ¬c, refl (@ite c (inr Hnc) A t e))
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H
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theorem if_t_t (c : Prop) {H : decidable c} {A : Type} (t : A) : (if c then t else t) = t :=
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decidable_rec
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(assume Hc : c, refl (@ite c (inl Hc) A t t))
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(assume Hnc : ¬c, refl (@ite c (inr Hnc) A t t))
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H
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theorem if_true {A : Type} (t e : A) : (if true then t else e) = t :=
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if_pos trivial t e
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theorem if_false {A : Type} (t e : A) : (if false then t else e) = e :=
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if_neg not_false_trivial t e
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theorem if_congr_aux {c₁ c₂ : Prop} {H₁ : decidable c₁} {H₂ : decidable c₂} {A : Type} {t₁ t₂ e₁ e₂ : A}
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(Hc : c₁ = c₂) (Ht : t₁ = t₂) (He : e₁ = e₂) :
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(if c₁ then t₁ else e₁) = (if c₂ then t₂ else e₂) :=
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have H1 : ∀ (H : decidable c₁), (@ite c₁ H₁ A t₁ e₁) = (@ite c₁ H A t₁ e₁), from
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take H : decidable c₁,
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have Hd : H₁ = H, from irrelevant H₁ H,
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Hd ▸ refl (@ite c₁ H₁ A t₁ e₁),
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have H2 : (@ite c₁ H₁ A t₁ e₁) = (@ite c₂ H₂ A t₁ e₁), from
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(Hc ▸ H1) H₂,
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Ht ▸ He ▸ H2
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theorem if_congr {c₁ c₂ : Prop} {H₁ : decidable c₁} {A : Type} {t₁ t₂ e₁ e₂ : A} (Hc : c₁ = c₂) (Ht : t₁ = t₂) (He : e₁ = e₂) :
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(if c₁ then t₁ else e₁) = (@ite c₂ (decidable_eq_equiv H₁ Hc) A t₂ e₂) :=
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have H2 [fact] : decidable c₂, from (decidable_eq_equiv H₁ Hc),
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if_congr_aux Hc Ht He
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exit
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theorem app_if_distribute {A B : Type} (c : Prop) {H : decidable c} (f : A → B) (a b : A) : f (if c then a else b) = if c then f a else f b :=
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or_elim (decidable.em H)
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(assume Hc : c, calc
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f (if c then a else b) = f (if true then a else b) : { eqt_intro Hc }
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(assume Hnc : ¬c, sorry)
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exit
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:= or_elim (em c)
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(λ Hc : c , calc
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f (if c then a else b) = f (if true then a else b) : { eqt_intro Hc }
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... = f a : { if_true a b }
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... = if true then f a else f b : symm (if_true (f a) (f b))
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... = if c then f a else f b : { symm (eqt_intro Hc) })
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(λ Hnc : ¬ c, calc
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f (if c then a else b) = f (if false then a else b) : { eqf_intro Hnc }
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... = f b : { if_false a b }
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... = if false then f a else f b : symm (if_false (f a) (f b))
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... = if c then f a else f b : { symm (eqf_intro Hnc) })
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