78 lines
2 KiB
Text
78 lines
2 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.to_string init.prod init.sum
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inductive ordering
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| lt | eq | gt
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open ordering
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attribute [instance]
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definition ordering.has_to_string : 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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structure [class] has_ordering (A : Type) :=
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(cmp : A → A → ordering)
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definition 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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attribute [instance]
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definition nat_has_ordering : 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 {A B : Type} [has_ordering A] [has_ordering B]
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definition prod.cmp : A × B → A × B → 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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attribute [instance]
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definition prod_has_ordering {A B : Type} [has_ordering A] [has_ordering B] : has_ordering (A × B) :=
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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 {A B : Type} [has_ordering A] [has_ordering B]
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definition sum.cmp : A ⊕ B → A ⊕ B → 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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attribute [instance]
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definition sum_has_ordering {A B : Type} [has_ordering A] [has_ordering B] : has_ordering (A ⊕ B) :=
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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 {A : Type} [has_ordering A]
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definition option.cmp : option A → option A → 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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attribute [instance]
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definition option_has_ordering {A : Type} [has_ordering A] : has_ordering (option A) :=
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⟨option.cmp⟩
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end
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