45 lines
1.8 KiB
Text
45 lines
1.8 KiB
Text
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/-
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Copyright (c) 2015 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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PropF has decidable equality
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-/
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import .soundness
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open bool decidable nat
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namespace PropF
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-- Show that PropF has decidable equality
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definition equal : PropF → PropF → bool
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| (Var x) (Var y) := if x = y then tt else ff
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| Bot Bot := tt
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| (Conj p₁ p₂) (Conj q₁ q₂) := equal p₁ q₁ && equal p₂ q₂
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| (Disj p₁ p₂) (Disj q₁ q₂) := equal p₁ q₁ && equal p₂ q₂
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| (Impl p₁ p₂) (Impl q₁ q₂) := equal p₁ q₁ && equal p₂ q₂
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| _ _ := ff
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lemma equal_refl : ∀ p, equal p p = tt
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| (Var x) := if_pos rfl
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| Bot := rfl
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| (Conj p₁ p₂) := begin change (equal p₁ p₁ && equal p₂ p₂ = tt), rewrite *equal_refl end
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| (Disj p₁ p₂) := begin change (equal p₁ p₁ && equal p₂ p₂ = tt), rewrite *equal_refl end
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| (Impl p₁ p₂) := begin change (equal p₁ p₁ && equal p₂ p₂ = tt), rewrite *equal_refl end
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lemma equal_to_eq : ∀ ⦃p q⦄, equal p q = tt → p = q
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| (Var x) (Var y) H :=
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if H₁ : x = y then congr_arg Var H₁
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else by rewrite [▸ (if x = y then tt else ff) = tt at H, if_neg H₁ at H]; exact (absurd H ff_ne_tt)
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| Bot Bot H := rfl
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| (Conj p₁ p₂) (Conj q₁ q₂) H :=
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by rewrite [equal_to_eq (band_elim_left H), equal_to_eq (band_elim_right H)]
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| (Disj p₁ p₂) (Disj q₁ q₂) H :=
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by rewrite [equal_to_eq (band_elim_left H), equal_to_eq (band_elim_right H)]
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| (Impl p₁ p₂) (Impl q₁ q₂) H :=
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by rewrite [equal_to_eq (band_elim_left H), equal_to_eq (band_elim_right H)]
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lemma has_decidable_eq [instance] : decidable_eq PropF :=
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decidable_eq_of_bool_pred equal_to_eq equal_refl
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end PropF
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