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Move some more things over to Without K (#40)

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Patrick Stevens
2019-08-22 20:32:34 +01:00
committed by GitHub
parent 9c14e7c439
commit 61f3dc7306
8 changed files with 48 additions and 42 deletions
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6
Numbers/BinaryNaturals/Multiplication.agda
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@@ -1,4 +1,4 @@
{-# OPTIONS --warning=error --safe #-}
{-# OPTIONS --warning=error --safe --without-K #-}
open import LogicalFormulae
open import Functions
@@ -144,7 +144,9 @@ module Numbers.BinaryNaturals.Multiplication where
NToBinNatDistributesPlus : (a b : ℕ) → NToBinNat (a +N b) ≡ NToBinNat a +B NToBinNat b
NToBinNatDistributesPlus zero b = refl
NToBinNatDistributesPlus (succ a) b rewrite equalityCommutative (NToBinNatSucc (a +N b)) | NToBinNatDistributesPlus a b = incrPullsOut (NToBinNat a) (NToBinNat b)
NToBinNatDistributesPlus (succ a) b with inspect (NToBinNat a)
... | bl with≡ prA with inspect (NToBinNat (a +N b))
... | q with≡ prAB = transitivity (applyEquality incr (NToBinNatDistributesPlus a b)) (incrPullsOut (NToBinNat a) (NToBinNat b))
timesCommutative : (a b : BinNat) → canonical (a *B b) ≡ canonical (b *B a)
timesCommLemma : (a b : BinNat) → canonical (zero :: (b *B a)) ≡ canonical (b *B (zero :: a))

6
Numbers/BinaryNaturals/Order.agda
View File

@@ -1,4 +1,4 @@
{-# OPTIONS --warning=error --safe #-}
{-# OPTIONS --warning=error --safe --without-K #-}
open import LogicalFormulae
open import Functions
@@ -292,7 +292,7 @@ module Numbers.BinaryNaturals.Order where
<BIsInherited (a :: as) [] | inl (inr x) | zero with≡ pr rewrite binNatToNZero (a :: as) pr | pr = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) x)
<BIsInherited (a :: as) [] | inl (inr x) | succ y with≡ pr rewrite pr = equalityCommutative (zeroLess' (a :: as) λ i → zeroNotSucc y (a :: as) i pr)
<BIsInherited (a :: as) [] | inr x rewrite canonicalFirst (a :: as) [] Equal | binNatToNZero (a :: as) x = refl
<BIsInherited (zero :: a) (zero :: b) rewrite chopFirstBit a b {zero} Equal = transitivity (chopDouble a b zero) (<BIsInherited a b)
<BIsInherited (zero :: a) (zero :: b) = transitivity (chopDouble a b zero) (<BIsInherited a b)
<BIsInherited (zero :: a) (one :: b) with orderIsTotal (binNatToN (zero :: a)) (binNatToN (one :: b))
<BIsInherited (zero :: a) (one :: b) | inl (inl 2a<2b+1) with orderIsTotal (binNatToN a) (binNatToN b)
<BIsInherited (zero :: a) (one :: b) | inl (inl 2a<2b+1) | inl (inl a<b) = equalityCommutative (equalToFirstLess FirstLess a b (equalityCommutative indHyp))
@@ -345,4 +345,4 @@ module Numbers.BinaryNaturals.Order where
indHyp = transitivity (equalityCommutative t) (<BIsInherited a b)
<BIsInherited (one :: a) (zero :: b) | inl (inr 2b<2a+1) | inr a=b rewrite a=b | canonicalFirst a b FirstGreater | canonicalSecond (canonical a) b FirstGreater | transitivity (equalityCommutative (binToBin a)) (transitivity (applyEquality NToBinNat a=b) (binToBin b)) = equalityCommutative (lemma1 (canonical b))
<BIsInherited (one :: a) (zero :: b) | inr x = exFalso (parity (binNatToN a) (binNatToN b) x)
<BIsInherited (one :: a) (one :: b) rewrite chopFirstBit a b {one} Equal = transitivity (chopDouble a b one) (<BIsInherited a b)
<BIsInherited (one :: a) (one :: b) = transitivity (chopDouble a b one) (<BIsInherited a b)