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38 lines
900 B
Idris
38 lines
900 B
Idris
||| Implementing `Decidable.Order.Strict` for `Data.Nat.LT`
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module Data.Nat.Order.Strict
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import Data.Nat
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import Decidable.Order.Strict
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import Decidable.Equality
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import Data.Nat.Order
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%default total
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public export
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Irreflexive Nat LT where
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irreflexive {x = 0} _ impossible
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irreflexive {x = S _} (LTESucc prf) =
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irreflexive {rel = Nat.LT} prf
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public export
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Transitive Nat LT where
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transitive {x} {y} xy yz =
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transitive {rel = LTE} (lteSuccRight xy) yz
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public export
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StrictPreorder Nat LT where
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public export
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decLT : (a, b : Nat) -> DecOrdering {lt = LT} a b
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decLT 0 0 = DecEQ Refl
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decLT 0 (S b) = DecLT (LTESucc LTEZero)
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decLT (S a) 0 = DecGT (LTESucc LTEZero)
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decLT (S a) (S b) = case decLT a b of
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DecLT a_lt_b => DecLT (LTESucc a_lt_b)
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DecEQ Refl => DecEQ Refl
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DecGT b_lt_a => DecGT (LTESucc b_lt_a)
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public export
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StrictOrdered Nat LT where
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order = decLT
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