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.
Module: data.sum
Authors: Leonardo de Moura, Jeremy Avigad
The sum type, aka disjoint union.
-/
import logic.connectives
open inhabited eq.ops
notation A ⊎ B := sum A B
namespace sum
notation A + B := sum A B
namespace low_precedence_plus
reserve infixr `+`:25 -- conflicts with notation for addition
infixr `+` := sum
end low_precedence_plus
variables {A B : Type}
definition inl_ne_inr (a : A) (b : B) : inl a ≠ inr b :=
by contradiction
definition inr_ne_inl (b : B) (a : A) : inr b ≠ inl a :=
by contradiction
definition inl_inj {a₁ a₂ : A} : intro_left B a₁ = intro_left B a₂ → a₁ = a₂ :=
assume H, sum.no_confusion H (λe, e)
definition inr_inj {b₁ b₂ : B} : intro_right A b₁ = intro_right A b₂ → b₁ = b₂ :=
assume H, sum.no_confusion H (λe, e)
protected definition is_inhabited_left [instance] [h : inhabited A] : inhabited (A + B) :=
inhabited.mk (inl (default A))
protected definition is_inhabited_right [instance] [h : inhabited B] : inhabited (A + B) :=
inhabited.mk (inr (default B))
protected definition has_decidable_eq [instance] [h₁ : decidable_eq A] [h₂ : decidable_eq B] : ∀ s₁ s₂ : A + B, decidable (s₁ = s₂)
| has_decidable_eq (inl a₁) (inl a₂) :=
match h₁ a₁ a₂ with
| decidable.inl hp := decidable.inl (hp ▸ rfl)
| decidable.inr hn := decidable.inr (λ he, absurd (inl_inj he) hn)
end
| has_decidable_eq (inl a₁) (inr b₂) := decidable.inr (by contradiction)
| has_decidable_eq (inr b₁) (inl a₂) := decidable.inr (by contradiction)
| has_decidable_eq (inr b₁) (inr b₂) :=
match h₂ b₁ b₂ with
| decidable.inl hp := decidable.inl (hp ▸ rfl)
| decidable.inr hn := decidable.inr (λ he, absurd (inr_inj he) hn)
end
end sum