73 lines
1.8 KiB
Text
73 lines
1.8 KiB
Text
/-
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Copyright (c) 2016 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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Authors: Leonardo de Moura
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-/
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prelude
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import init.data.to_string init.data.prod init.data.sum.basic
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inductive ordering
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| lt | eq | gt
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open ordering
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instance : has_to_string ordering :=
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has_to_string.mk (λ s, match s with | ordering.lt := "lt" | ordering.eq := "eq" | ordering.gt := "gt" end)
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class has_ordering (α : Type) :=
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(cmp : α → α → ordering)
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def nat.cmp (a b : nat) : ordering :=
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if a < b then ordering.lt
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else if a = b then ordering.eq
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else ordering.gt
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instance : has_ordering nat :=
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⟨nat.cmp⟩
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section
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open prod
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variables {α β : Type} [has_ordering α] [has_ordering β]
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def prod.cmp : α × β → α × β → ordering
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| (a₁, b₁) (a₂, b₂) :=
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match (has_ordering.cmp a₁ a₂) with
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| ordering.lt := lt
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| ordering.eq := has_ordering.cmp b₁ b₂
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| ordering.gt := gt
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end
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instance {α β : Type} [has_ordering α] [has_ordering β] : has_ordering (α × β) :=
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⟨prod.cmp⟩
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end
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section
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open sum
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variables {α β : Type} [has_ordering α] [has_ordering β]
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def sum.cmp : α ⊕ β → α ⊕ β → ordering
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| (inl a₁) (inl a₂) := has_ordering.cmp a₁ a₂
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| (inr b₁) (inr b₂) := has_ordering.cmp b₁ b₂
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| (inl a₁) (inr b₂) := lt
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| (inr b₁) (inl a₂) := gt
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instance {α β : Type} [has_ordering α] [has_ordering β] : has_ordering (α ⊕ β) :=
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⟨sum.cmp⟩
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end
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section
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open option
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variables {α : Type} [has_ordering α]
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def option.cmp : option α → option α → ordering
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| (some a₁) (some a₂) := has_ordering.cmp a₁ a₂
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| (some a₁) none := gt
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| none (some a₂) := lt
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| none none := eq
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instance {α : Type} [has_ordering α] : has_ordering (option α) :=
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⟨option.cmp⟩
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end
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