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Patrick Stevens
2020-04-19 07:54:37 +01:00
committed by GitHub
parent 3afdc6d45b
commit e660eceb43
11 changed files with 348 additions and 346 deletions
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14
Numbers/BinaryNaturals/Subtraction.agda
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@@ -15,6 +15,7 @@ open import Maybe
module Numbers.BinaryNaturals.Subtraction where
private
aMinusAGo : (a : BinNat) → mapMaybe canonical (go zero a a) ≡ yes []
aMinusAGo [] = refl
aMinusAGo (zero :: a) with aMinusAGo a
@@ -397,13 +398,14 @@ module Numbers.BinaryNaturals.Subtraction where
bad' : one :: a :: as ≡ one :: [] → False
bad' ()
doublingLemma : (y : BinNat) → NToBinNat (2 *N binNatToN y) ≡ canonical (zero :: y)
doublingLemma y with inspect (canonical y)
doublingLemma y | [] with≡ pr rewrite binNatToNZero' y pr | pr = refl
doublingLemma y | (a :: as) with≡ pr with inspect (binNatToN y)
doublingLemma y | (a :: as) with≡ pr | zero with≡ pr2 rewrite binNatToNZero y pr2 = exFalso (nonEmptyNotEmpty (equalityCommutative pr))
doublingLemma y | (a :: as) with≡ pr | succ bl with≡ pr2 rewrite pr | pr2 | doubleIsBitShift' bl | equalityCommutative pr = applyEquality (zero ::_) (equalityCommutative (transitivity (equalityCommutative (binToBin y)) (applyEquality NToBinNat pr2)))
doublingLemma : (y : BinNat) → NToBinNat (2 *N binNatToN y) ≡ canonical (zero :: y)
doublingLemma y with inspect (canonical y)
doublingLemma y | [] with≡ pr rewrite binNatToNZero' y pr | pr = refl
doublingLemma y | (a :: as) with≡ pr with inspect (binNatToN y)
doublingLemma y | (a :: as) with≡ pr | zero with≡ pr2 rewrite binNatToNZero y pr2 = exFalso (nonEmptyNotEmpty (equalityCommutative pr))
doublingLemma y | (a :: as) with≡ pr | succ bl with≡ pr2 rewrite pr | pr2 | doubleIsBitShift' bl | equalityCommutative pr = applyEquality (zero ::_) (equalityCommutative (transitivity (equalityCommutative (binToBin y)) (applyEquality NToBinNat pr2)))
private
doubling : (a : ℕ) {y : BinNat} → (NToBinNat a ≡ zero :: y) → binNatToN y +N (binNatToN y +N 0) ≡ a
doubling a {y} pr = NToBinNatInj (binNatToN y +N (binNatToN y +N zero)) a (transitivity (transitivity (equalityCommutative (NToBinNatIsCanonical (binNatToN y +N (binNatToN y +N zero)))) (doublingLemma y)) (applyEquality canonical (equalityCommutative pr)))