lean2/library/data/sum.lean

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-- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
-- Released under Apache 2.0 license as described in the file LICENSE.
-- Author: Leonardo de Moura, Jeremy Avigad
import logic.prop logic.inhabited logic.decidable
open inhabited decidable eq.ops
-- data.sum
-- ========
-- The sum type, aka disjoint union.
inductive sum (A B : Type) : Type :=
inl : A → sum A B,
inr : B → sum A B
namespace sum
notation A ⊎ B := sum A B
namespace extra_notation
reserve infixr `+`:25 -- conflicts with notation for addition
infixr `+` := sum
end extra_notation
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variables {A B : Type}
variables {a a₁ a₂ : A} {b b₁ b₂ : B}
-- Here is the trick for the theorems that follow:
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-- Fixing a₁, "f s" is a recursive description of "inl B a₁ = s".
-- When s is (inl B a₁), it reduces to a₁ = a₁.
-- When s is (inl B a₂), it reduces to a₁ = a₂.
-- When s is (inr A b), it reduces to false.
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theorem inl_inj : inl B a₁ = inl B a₂ → a₁ = a₂ :=
assume H,
let f := λs, rec_on s (λa, a₁ = a) (λb, false) in
have H₁ : f (inl B a₁), from rfl,
have H₂ : f (inl B a₂), from H ▸ H₁,
H₂
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theorem inl_neq_inr : inl B a ≠ inr A b :=
assume H,
let f := λs, rec_on s (λa', a = a') (λb, false) in
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have H₁ : f (inl B a), from rfl,
have H₂ : f (inr A b), from H ▸ H₁,
H₂
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theorem inr_inj : inr A b₁ = inr A b₂ → b₁ = b₂ :=
assume H,
let f := λs, rec_on s (λa, false) (λb, b₁ = b) in
have H₁ : f (inr A b₁), from rfl,
have H₂ : f (inr A b₂), from H ▸ H₁,
H₂
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protected definition is_inhabited_left [instance] : inhabited A → inhabited (A ⊎ B) :=
assume H : inhabited A, inhabited.mk (inl B (default A))
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protected definition is_inhabited_right [instance] : inhabited B → inhabited (A ⊎ B) :=
assume H : inhabited B, inhabited.mk (inr A (default B))
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protected definition has_eq_decidable [instance] : decidable_eq A → decidable_eq B → decidable_eq (A ⊎ B) :=
assume (H₁ : decidable_eq A) (H₂ : decidable_eq B),
take s₁ s₂ : A ⊎ B,
rec_on s₁
(take a₁, show decidable (inl B a₁ = s₂), from
rec_on s₂
(take a₂, show decidable (inl B a₁ = inl B a₂), from
decidable.rec_on (H₁ a₁ a₂)
(assume Heq : a₁ = a₂, decidable.inl (Heq ▸ rfl))
(assume Hne : a₁ ≠ a₂, decidable.inr (mt inl_inj Hne)))
(take b₂,
have H₃ : (inl B a₁ = inr A b₂) ↔ false,
from iff.intro inl_neq_inr (assume H₄, false_elim H₄),
show decidable (inl B a₁ = inr A b₂), from decidable_iff_equiv _ (iff.symm H₃)))
(take b₁, show decidable (inr A b₁ = s₂), from
rec_on s₂
(take a₂,
have H₃ : (inr A b₁ = inl B a₂) ↔ false,
from iff.intro (assume H₄, inl_neq_inr (H₄⁻¹)) (assume H₄, false_elim H₄),
show decidable (inr A b₁ = inl B a₂), from decidable_iff_equiv _ (iff.symm H₃))
(take b₂, show decidable (inr A b₁ = inr A b₂), from
decidable.rec_on (H₂ b₁ b₂)
(assume Heq : b₁ = b₂, decidable.inl (Heq ▸ rfl))
(assume Hne : b₁ ≠ b₂, decidable.inr (mt inr_inj Hne))))
end sum