57 lines
2.7 KiB
Text
57 lines
2.7 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_ops
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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_cond_congr {c₁ c₂ : Prop} {H₁ : decidable c₁} {H₂ : decidable c₂} (Heq : c₁ ↔ c₂) {A : Type} (t e : A)
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: (if c₁ then t else e) = (if c₂ then t else e) :=
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rec_on H₁
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(assume Hc₁ : c₁, rec_on H₂
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(assume Hc₂ : c₂, (if_pos Hc₁ t e) ⬝ (if_pos Hc₂ t e)⁻¹)
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(assume Hnc₂ : ¬c₂, absurd_elim _ (iff_elim_left Heq Hc₁) Hnc₂))
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(assume Hnc₁ : ¬c₁, rec_on H₂
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(assume Hc₂ : c₂, absurd_elim _ (iff_elim_right Heq Hc₂) Hnc₁)
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(assume Hnc₂ : ¬c₂, (if_neg Hnc₁ t e) ⬝ (if_neg Hnc₂ t e)⁻¹))
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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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Ht ▸ He ▸ (if_cond_congr Hc t₁ e₁)
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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_iff_equiv H₁ Hc) A t₂ e₂) :=
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have H2 [fact] : decidable c₂, from (decidable_iff_equiv H₁ Hc),
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if_congr_aux Hc Ht He
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