64 lines
2.1 KiB
Text
64 lines
2.1 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 logic.inhabited logic.eq logic.decidable
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-- data.prod
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-- =========
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open inhabited decidable eq.ops
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-- The cartesian product.
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inductive prod (A B : Type) : Type :=
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mk : A → B → prod A B
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definition pair := @prod.mk
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namespace prod
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infixr `×` := prod
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-- notation for n-ary tuples
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notation `(` h `,` t:(foldl `,` (e r, prod.mk r e) h) `)` := t
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section
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parameters {A B : Type}
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protected theorem destruct {P : A × B → Prop} (p : A × B) (H : ∀a b, P (a, b)) : P p :=
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rec H p
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definition pr1 (p : prod A B) := rec (λ x y, x) p
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definition pr2 (p : prod A B) := rec (λ x y, y) p
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notation `pr₁`:max := pr1
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notation `pr₂`:max := pr2
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variables (a : A) (b : B)
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theorem pr1.pair : pr₁ (a, b) = a :=
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rfl
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theorem pr2.pair : pr₂ (a, b) = b :=
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rfl
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theorem prod_ext (p : prod A B) : pair (pr₁ p) (pr₂ p) = p :=
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destruct p (λx y, eq.refl (x, y))
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variables {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
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end prod
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