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Implement subtraction on the binary naturals (#45)

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Patrick Stevens
2019-09-22 08:36:41 +01:00
committed by GitHub
parent d4ebcc04ce
commit b92e6b2dd8
8 changed files with 922 additions and 2 deletions
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29
Numbers/BinaryNaturals/Order.agda
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@@ -1,5 +1,6 @@
{-# OPTIONS --warning=error --safe --without-K #-}
open import WellFoundedInduction
open import LogicalFormulae
open import Functions
open import Lists.Lists
@@ -16,6 +17,15 @@ module Numbers.BinaryNaturals.Order where
FirstLess : Compare
FirstGreater : Compare
badCompare : Equal ≡ FirstLess → False
badCompare ()
badCompare' : Equal ≡ FirstGreater → False
badCompare' ()
badCompare'' : FirstLess ≡ FirstGreater → False
badCompare'' ()
_<BInherited_ : BinNat → BinNat → Compare
a <BInherited b with orderIsTotal (binNatToN a) (binNatToN b)
(a <BInherited b) | inl (inl x) = FirstLess
@@ -195,6 +205,25 @@ module Numbers.BinaryNaturals.Order where
equalContaminated' (one :: n) (zero :: m) pr = equalContaminated' n m pr
equalContaminated' (one :: n) (one :: m) pr = equalContaminated' n m pr
comparisonEqual : (a b : BinNat) → (a <B b ≡ Equal) → canonical a ≡ canonical b
comparisonEqual [] [] pr = refl
comparisonEqual [] (zero :: b) pr with inspect (canonical b)
comparisonEqual [] (zero :: b) pr | [] with≡ p rewrite p = refl
comparisonEqual [] (zero :: b) pr | (x :: r) with≡ p rewrite zeroLess b (λ i → nonEmptyNotEmpty (transitivity (equalityCommutative p) i)) = exFalso (badCompare (equalityCommutative pr))
comparisonEqual (zero :: a) [] pr with inspect (canonical a)
comparisonEqual (zero :: a) [] pr | [] with≡ x rewrite x = refl
comparisonEqual (zero :: a) [] pr | (x₁ :: y) with≡ x rewrite zeroLess' a (λ i → nonEmptyNotEmpty (transitivity (equalityCommutative x) i)) = exFalso (badCompare' (equalityCommutative pr))
comparisonEqual (zero :: a) (zero :: b) pr with inspect (canonical a)
comparisonEqual (zero :: a) (zero :: b) pr | [] with≡ x with inspect (canonical b)
comparisonEqual (zero :: a) (zero :: b) pr | [] with≡ pr2 | [] with≡ pr3 rewrite pr2 | pr3 = refl
comparisonEqual (zero :: a) (zero :: b) pr | [] with≡ pr2 | (x₂ :: y) with≡ pr3 rewrite pr2 | pr3 | comparisonEqual a b pr = exFalso (nonEmptyNotEmpty (transitivity (equalityCommutative pr3) pr2))
comparisonEqual (zero :: a) (zero :: b) pr | (c :: cs) with≡ pr2 with inspect (canonical b)
comparisonEqual (zero :: a) (zero :: b) pr | (c :: cs) with≡ pr2 | [] with≡ pr3 rewrite pr2 | pr3 | comparisonEqual a b pr = exFalso (nonEmptyNotEmpty (transitivity (equalityCommutative pr2) pr3))
comparisonEqual (zero :: a) (zero :: b) pr | (c :: cs) with≡ pr2 | (x₁ :: y) with≡ pr3 rewrite pr2 | pr3 | comparisonEqual a b pr = applyEquality (zero ::_) (transitivity (equalityCommutative pr2) pr3)
comparisonEqual (zero :: a) (one :: b) pr = exFalso (equalContaminated a b pr)
comparisonEqual (one :: a) (zero :: b) pr = exFalso (equalContaminated' a b pr)
comparisonEqual (one :: a) (one :: b) pr = applyEquality (one ::_) (comparisonEqual a b pr)
equalSymmetric : (n m : BinNat) → n <B m ≡ Equal → m <B n ≡ Equal
equalSymmetric [] [] n=m = refl
equalSymmetric [] (zero :: m) n=m rewrite equalSymmetric [] m n=m = refl