46149d0d50
prod is needed for some automatically generated constructions. So, it is important it is loaded in the environment as early as possible.
28 lines
1.3 KiB
Text
28 lines
1.3 KiB
Text
-- 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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import data.prod.decl logic.inhabited logic.eq logic.decidable
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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}
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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 equal {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] : inhabited A → inhabited B → inhabited (prod A B) :=
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take (H₁ : inhabited A) (H₂ : inhabited B),
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inhabited.destruct H₁ (λa, inhabited.destruct H₂ (λb, inhabited.mk (pair a b)))
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protected definition has_decidable_eq [instance] : decidable_eq A → decidable_eq B → decidable_eq (A × B) :=
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take (H₁ : decidable_eq A) (H₂ : decidable_eq B) (u v : A × B),
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have H₃ : u = v ↔ (pr₁ u = pr₁ v) ∧ (pr₂ u = pr₂ v), from
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iff.intro
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(assume H, H ▸ and.intro rfl rfl)
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(assume H, and.elim H (assume H₄ H₅, equal H₄ H₅)),
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decidable_iff_equiv _ (iff.symm H₃)
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end prod
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