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| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -0,0 +1,102 @@ | ||
| module Haskell.Law.Ord.Integer where | ||
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| open import Haskell.Prim | ||
| open import Haskell.Prim.Bool | ||
| open import Haskell.Prim.Eq | ||
| open import Haskell.Prim.Ord | ||
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| open import Haskell.Law.Bool | ||
| open import Haskell.Law.Eq | ||
| open import Haskell.Law.Equality | ||
| open import Haskell.Law.Integer | ||
| open import Haskell.Law.Ord.Def | ||
| open import Haskell.Law.Ord.Nat | ||
| open import Haskell.Law.Nat | ||
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| instance | ||
| iLawfulOrdInteger : IsLawfulOrd Integer | ||
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| iLawfulOrdInteger .comparability (pos n) (pos m) = comparability n m | ||
| iLawfulOrdInteger .comparability (pos n) (negsuc m) = refl | ||
| iLawfulOrdInteger .comparability (negsuc n) (pos m) = refl | ||
| iLawfulOrdInteger .comparability (negsuc n) (negsuc m) | ||
| rewrite sym $ lte2gte m n | ||
| | sym $ lte2gte n m | ||
| = comparability m n | ||
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| iLawfulOrdInteger .transitivity (pos n) (pos m) (pos o) h₁ = transitivity n m o h₁ | ||
| iLawfulOrdInteger .transitivity (pos n) (pos m) (negsuc o) h₁ | ||
| rewrite &&-sym (n <= m) False | ||
| = h₁ | ||
| iLawfulOrdInteger .transitivity (negsuc n) y (pos o) h₁ = refl | ||
| iLawfulOrdInteger .transitivity (negsuc n) (negsuc m) (negsuc o) h₁ | ||
| rewrite eqSymmetry n o | ||
| = transitivity o m n (trans (reverseLte o m m n) h₁) | ||
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| iLawfulOrdInteger .reflexivity (pos n) = reflexivity n | ||
| iLawfulOrdInteger .reflexivity (negsuc n) = reflexivity n | ||
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| iLawfulOrdInteger .antisymmetry (pos n) (pos m) h₁ = antisymmetry n m h₁ | ||
| iLawfulOrdInteger .antisymmetry (negsuc n) (negsuc m) h₁ = antisymmetry n m | ||
| $ trans (reverseLte n m m n) h₁ | ||
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| iLawfulOrdInteger .lte2gte (pos n) (pos m) | ||
| rewrite eqSymmetry n m | ||
| = refl | ||
| iLawfulOrdInteger .lte2gte (pos n) (negsuc m) = refl | ||
| iLawfulOrdInteger .lte2gte (negsuc n) (pos m) = refl | ||
| iLawfulOrdInteger .lte2gte (negsuc n) (negsuc m) | ||
| rewrite eqSymmetry n m | ||
| = refl | ||
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| iLawfulOrdInteger .lt2LteNeq (pos n) (pos m) = lt2LteNeq n m | ||
| iLawfulOrdInteger .lt2LteNeq (pos n) (negsuc m) = refl | ||
| iLawfulOrdInteger .lt2LteNeq (negsuc n) (pos m) = refl | ||
| iLawfulOrdInteger .lt2LteNeq (negsuc n) (negsuc m) | ||
| rewrite eqSymmetry n m | ||
| = lt2LteNeq m n | ||
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| iLawfulOrdInteger .lt2gt x y = refl | ||
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| iLawfulOrdInteger .compareLt (pos n) (pos m) = compareLt n m | ||
| iLawfulOrdInteger .compareLt (pos n) (negsuc m) = refl | ||
| iLawfulOrdInteger .compareLt (negsuc n) (pos m) = refl | ||
| iLawfulOrdInteger .compareLt (negsuc n) (negsuc m) | ||
| rewrite eqSymmetry n m | ||
| = compareLt m n | ||
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| iLawfulOrdInteger .compareGt (pos n) (pos m) = compareGt n m | ||
| iLawfulOrdInteger .compareGt (pos n) (negsuc m) = refl | ||
| iLawfulOrdInteger .compareGt (negsuc n) (pos m) = refl | ||
| iLawfulOrdInteger .compareGt (negsuc n) (negsuc m) | ||
| rewrite eqSymmetry n m | ||
| = compareGt m n | ||
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| iLawfulOrdInteger .compareEq (pos n) (pos m) = compareEq n m | ||
| iLawfulOrdInteger .compareEq (pos n) (negsuc m) = refl | ||
| iLawfulOrdInteger .compareEq (negsuc n) (pos m) = refl | ||
| iLawfulOrdInteger .compareEq (negsuc n) (negsuc m) | ||
| rewrite eqSymmetry n m | ||
| = compareEq m n | ||
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| iLawfulOrdInteger .min2if (pos n) (pos m) | ||
| rewrite lte2ngt n m | ||
| | sym $ ifFlip (m < n) (pos m) (pos n) | ||
| = eqReflexivity (min (pos n) (pos m)) | ||
| iLawfulOrdInteger .min2if (pos n) (negsuc m) = eqReflexivity m | ||
| iLawfulOrdInteger .min2if (negsuc n) (pos m) = eqReflexivity n | ||
| iLawfulOrdInteger .min2if (negsuc n) (negsuc m) | ||
| rewrite gte2nlt n m | ||
| | sym $ ifFlip (n < m) (negsuc m) (negsuc n) | ||
| = eqReflexivity (min (negsuc n) (negsuc m)) | ||
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| iLawfulOrdInteger .max2if (pos n) (pos m) | ||
| rewrite gte2nlt n m | ||
| | sym (ifFlip (n < m) (pos m) (pos n)) | ||
| = eqReflexivity (max (pos n) (pos m)) | ||
| iLawfulOrdInteger .max2if (pos n) (negsuc m) = eqReflexivity n | ||
| iLawfulOrdInteger .max2if (negsuc n) (pos m) = eqReflexivity m | ||
| iLawfulOrdInteger .max2if (negsuc n) (negsuc m) | ||
| rewrite lte2ngt n m | ||
| | sym $ ifFlip (m < n) (negsuc m) (negsuc n) | ||
| = eqReflexivity (max (negsuc n) (negsuc m)) |