42 lines
1.6 KiB
Text
42 lines
1.6 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, Jeremy Avigad
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-/
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import logic.eq
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open inhabited decidable eq.ops
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namespace prod
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variables {A B : Type} {a₁ a₂ : A} {b₁ b₂ : B} {u : A × B}
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theorem pair_eq : a₁ = a₂ → b₁ = b₂ → (a₁, b₁) = (a₂, b₂) :=
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assume H1 H2, H1 ▸ H2 ▸ rfl
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protected theorem eq {p₁ p₂ : prod A B} : pr₁ p₁ = pr₁ p₂ → pr₂ p₁ = pr₂ p₂ → p₁ = p₂ :=
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destruct p₁ (take a₁ b₁, destruct p₂ (take a₂ b₂ H₁ H₂, pair_eq H₁ H₂))
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protected definition is_inhabited [instance] [h₁ : inhabited A] [h₂ : inhabited B] : inhabited (prod A B) :=
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inhabited.mk (default A, default B)
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protected definition has_decidable_eq [instance] [h₁ : decidable_eq A] [h₂ : decidable_eq B] : ∀ p₁ p₂ : A × B, decidable (p₁ = p₂)
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| (a, b) (a', b') :=
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match h₁ a a' with
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| inl e₁ :=
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match h₂ b b' with
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| inl e₂ := by left; congruence; repeat assumption
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| inr n₂ := by right; intro h; injection h; contradiction
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end
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| inr n₁ := by right; intro h; injection h; contradiction
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end
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definition swap {A : Type} : A × A → A × A
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| (a, b) := (b, a)
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theorem swap_swap {A : Type} : ∀ p : A × A, swap (swap p) = p
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| (a, b) := rfl
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theorem eq_of_swap_eq {A : Type} : ∀ p₁ p₂ : A × A, swap p₁ = swap p₂ → p₁ = p₂ :=
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take p₁ p₂, assume seqs,
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have swap (swap p₁) = swap (swap p₂), from congr_arg swap seqs,
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by rewrite *swap_swap at this; exact this
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end prod
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