59 lines
1.4 KiB
Idris
59 lines
1.4 KiB
Idris
module Misc
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import Data.Nat
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%default total
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public export
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Index : Type
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Index = Nat
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public export
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Name : Type
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Name = String
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public export
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PI : Type -> Type
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PI = Either String
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public export
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oops : String -> PI a
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oops = Left
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public export
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lteTransp : LTE a b -> a = c -> b = d -> LTE c d
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lteTransp p Refl Refl = p
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public export
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lteS : {n : _} -> LTE n (S n)
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lteS {n = Z} = LTEZero
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lteS {n = S n} = LTESucc lteS
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public export
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lteSplit : {m : _} -> LTE n m -> Either (n = m) (LT n m)
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lteSplit {m = Z} LTEZero = Left Refl
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lteSplit {m = S m} LTEZero = Right (LTESucc LTEZero)
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lteSplit {m = S m} (LTESucc p) = case lteSplit p of
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Left p2 => Left (cong S p2)
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Right p2 => Right (LTESucc p2)
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public export
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minusLte : {m,n : _} -> LTE (minus n (S m)) n
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minusLte {n = Z} = LTEZero
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minusLte {n = S n} {m = Z} = rewrite minusZeroRight n in lteSuccRight reflexive
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minusLte {n = S n} {m = S m} = lteSuccRight (minusLte {n = n} {m = m})
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public export
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prevEq : (i, j : Nat) -> S i = S j -> i = j
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prevEq Z Z Refl = Refl
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prevEq (S i) (S _) Refl = Refl
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public export
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natEqDecid : (n, m : Nat) -> Either (Not (n = m)) (n = m)
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natEqDecid Z Z = Right Refl
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natEqDecid (S _) Z = Left absurd
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natEqDecid Z (S _) = Left absurd
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natEqDecid (S n) (S m) = case natEqDecid n m of
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Right p => Right (cong S p)
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Left p => Left (p . prevEq n m)
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