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Rename order transitivity (#62)

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Patrick Stevens
2019-11-02 19:05:52 +00:00
committed by GitHub
parent 763ddb8dbb
commit 1325236359
20 changed files with 220 additions and 220 deletions
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20
Numbers/BinaryNaturals/Order.agda
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@@ -286,18 +286,18 @@ module Numbers.BinaryNaturals.Order where
chopDouble a b i with orderIsTotal (binNatToN a) (binNatToN b)
chopDouble a b zero | inl (inl a<b) with orderIsTotal (2 *N binNatToN a) (2 *N binNatToN b)
chopDouble a b zero | inl (inl a<b) | inl (inl x) = refl
chopDouble a b zero | inl (inl a<b) | inl (inr b<a) = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) (PartialOrder.transitive (TotalOrder.order ℕTotalOrder) b<a (lessRespectsMultiplicationLeft (binNatToN a) (binNatToN b) 2 a<b (le 1 refl))))
chopDouble a b zero | inl (inl a<b) | inl (inr b<a) = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) (PartialOrder.<Transitive (TotalOrder.order ℕTotalOrder) b<a (lessRespectsMultiplicationLeft (binNatToN a) (binNatToN b) 2 a<b (le 1 refl))))
chopDouble a b zero | inl (inl a<b) | inr a=b rewrite productCancelsLeft 2 (binNatToN a) (binNatToN b) (le 1 refl) a=b = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) a<b)
chopDouble a b one | inl (inl a<b) with orderIsTotal (2 *N binNatToN a) (2 *N binNatToN b)
chopDouble a b one | inl (inl a<b) | inl (inl 2a<2b) = refl
chopDouble a b one | inl (inl a<b) | inl (inr 2b<2a) = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) (PartialOrder.transitive (TotalOrder.order ℕTotalOrder) a<b (cancelInequalityLeft {2} 2b<2a)))
chopDouble a b one | inl (inl a<b) | inl (inr 2b<2a) = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) (PartialOrder.<Transitive (TotalOrder.order ℕTotalOrder) a<b (cancelInequalityLeft {2} 2b<2a)))
chopDouble a b one | inl (inl a<b) | inr 2a=2b rewrite productCancelsLeft 2 (binNatToN a) (binNatToN b) (le 1 refl) 2a=2b = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) a<b)
chopDouble a b zero | inl (inr b<a) with orderIsTotal (2 *N binNatToN a) (2 *N binNatToN b)
chopDouble a b zero | inl (inr b<a) | inl (inl 2a<2b) = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) (PartialOrder.transitive (TotalOrder.order ℕTotalOrder) b<a (cancelInequalityLeft {2} {binNatToN a} {binNatToN b} 2a<2b)))
chopDouble a b zero | inl (inr b<a) | inl (inl 2a<2b) = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) (PartialOrder.<Transitive (TotalOrder.order ℕTotalOrder) b<a (cancelInequalityLeft {2} {binNatToN a} {binNatToN b} 2a<2b)))
chopDouble a b zero | inl (inr b<a) | inl (inr 2b<2a) = refl
chopDouble a b zero | inl (inr b<a) | inr 2a=2b rewrite productCancelsLeft 2 (binNatToN a) (binNatToN b) (le 1 refl) 2a=2b = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) b<a)
chopDouble a b one | inl (inr b<a) with orderIsTotal (2 *N binNatToN a) (2 *N binNatToN b)
chopDouble a b one | inl (inr b<a) | inl (inl 2a<2b) = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) (PartialOrder.transitive (TotalOrder.order ℕTotalOrder) b<a (cancelInequalityLeft {2} 2a<2b)))
chopDouble a b one | inl (inr b<a) | inl (inl 2a<2b) = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) (PartialOrder.<Transitive (TotalOrder.order ℕTotalOrder) b<a (cancelInequalityLeft {2} 2a<2b)))
chopDouble a b one | inl (inr b<a) | inl (inr x) = refl
chopDouble a b one | inl (inr b<a) | inr 2a=2b rewrite productCancelsLeft 2 (binNatToN a) (binNatToN b) (le 1 refl) 2a=2b = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) b<a)
chopDouble a b i | inr x with orderIsTotal (binNatToN (i :: a)) (binNatToN (i :: b))
@@ -329,19 +329,19 @@ module Numbers.BinaryNaturals.Order where
t : a <BInherited b ≡ FirstLess
t with orderIsTotal (binNatToN a) (binNatToN b)
t | inl (inl x) = refl
t | inl (inr x) = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) (PartialOrder.transitive (TotalOrder.order ℕTotalOrder) x a<b))
t | inl (inr x) = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) (PartialOrder.<Transitive (TotalOrder.order ℕTotalOrder) x a<b))
t | inr x rewrite x = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) a<b)
indHyp : FirstLess ≡ go<B Equal a b
indHyp = transitivity (equalityCommutative t) (<BIsInherited a b)
<BIsInherited (zero :: a) (one :: b) | inl (inl 2a<2b+1) | inl (inr b<a) = exFalso (noIntegersBetweenXAndSuccX {2 *N binNatToN a} (2 *N binNatToN b) (lessRespectsMultiplicationLeft (binNatToN b) (binNatToN a) 2 b<a (le 1 refl)) 2a<2b+1)
<BIsInherited (zero :: a) (one :: b) | inl (inl 2a<2b+1) | inr a=b rewrite a=b | canonicalFirst a b FirstLess | canonicalSecond (canonical a) b FirstLess | transitivity (equalityCommutative (binToBin a)) (transitivity (applyEquality NToBinNat a=b) (binToBin b)) = equalityCommutative (lemma1 (canonical b))
<BIsInherited (zero :: a) (one :: b) | inl (inr 2b+1<2a) with orderIsTotal (binNatToN a) (binNatToN b)
<BIsInherited (zero :: a) (one :: b) | inl (inr 2b+1<2a) | inl (inl a<b) = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) (PartialOrder.transitive (TotalOrder.order ℕTotalOrder) 2b+1<2a (PartialOrder.transitive (TotalOrder.order ℕTotalOrder) (lessRespectsMultiplicationLeft (binNatToN a) (binNatToN b) 2 a<b (le 1 refl)) (le zero refl))))
<BIsInherited (zero :: a) (one :: b) | inl (inr 2b+1<2a) | inl (inl a<b) = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) (PartialOrder.<Transitive (TotalOrder.order ℕTotalOrder) 2b+1<2a (PartialOrder.<Transitive (TotalOrder.order ℕTotalOrder) (lessRespectsMultiplicationLeft (binNatToN a) (binNatToN b) 2 a<b (le 1 refl)) (le zero refl))))
<BIsInherited (zero :: a) (one :: b) | inl (inr 2b+1<2a) | inl (inr b<a) = equalityCommutative (equalToFirstGreater FirstLess a b (equalityCommutative indHyp))
where
t : a <BInherited b ≡ FirstGreater
t with orderIsTotal (binNatToN a) (binNatToN b)
t | inl (inl x) = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) (PartialOrder.transitive (TotalOrder.order ℕTotalOrder) x b<a))
t | inl (inl x) = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) (PartialOrder.<Transitive (TotalOrder.order ℕTotalOrder) x b<a))
t | inl (inr x) = refl
t | inr x rewrite x = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) b<a)
indHyp : FirstGreater ≡ go<B Equal a b
@@ -354,12 +354,12 @@ module Numbers.BinaryNaturals.Order where
where
t : a <BInherited b ≡ FirstLess
t with orderIsTotal (binNatToN a) (binNatToN b)
t | inl (inr x) = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) (PartialOrder.transitive (TotalOrder.order ℕTotalOrder) x a<b))
t | inl (inr x) = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) (PartialOrder.<Transitive (TotalOrder.order ℕTotalOrder) x a<b))
t | inl (inl x) = refl
t | inr x rewrite x = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) a<b)
indHyp : FirstLess ≡ go<B Equal a b
indHyp = transitivity (equalityCommutative t) (<BIsInherited a b)
<BIsInherited (one :: a) (zero :: b) | inl (inl 2a+1<2b) | inl (inr b<a) = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) (PartialOrder.transitive (TotalOrder.order ℕTotalOrder) 2a+1<2b (PartialOrder.transitive (TotalOrder.order ℕTotalOrder) (lessRespectsMultiplicationLeft (binNatToN b) (binNatToN a) 2 b<a (le 1 refl)) (le zero refl))))
<BIsInherited (one :: a) (zero :: b) | inl (inl 2a+1<2b) | inl (inr b<a) = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) (PartialOrder.<Transitive (TotalOrder.order ℕTotalOrder) 2a+1<2b (PartialOrder.<Transitive (TotalOrder.order ℕTotalOrder) (lessRespectsMultiplicationLeft (binNatToN b) (binNatToN a) 2 b<a (le 1 refl)) (le zero refl))))
<BIsInherited (one :: a) (zero :: b) | inl (inl 2a+1<2b) | inr a=b rewrite a=b = exFalso (succNotLess 2a+1<2b)
<BIsInherited (one :: a) (zero :: b) | inl (inr 2b<2a+1) with orderIsTotal (binNatToN a) (binNatToN b)
<BIsInherited (one :: a) (zero :: b) | inl (inr 2b<2a+1) | inl (inl a<b) = exFalso (noIntegersBetweenXAndSuccX {2 *N binNatToN b} (2 *N binNatToN a) (lessRespectsMultiplicationLeft (binNatToN a) (binNatToN b) 2 a<b (le 1 refl)) 2b<2a+1)
@@ -367,7 +367,7 @@ module Numbers.BinaryNaturals.Order where
where
t : a <BInherited b ≡ FirstGreater
t with orderIsTotal (binNatToN a) (binNatToN b)
t | inl (inl x) = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) (PartialOrder.transitive (TotalOrder.order ℕTotalOrder) x b<a))
t | inl (inl x) = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) (PartialOrder.<Transitive (TotalOrder.order ℕTotalOrder) x b<a))
t | inl (inr x) = refl
t | inr x rewrite x = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) b<a)
indHyp : FirstGreater ≡ go<B Equal a b

4
Numbers/BinaryNaturals/Subtraction.agda
View File

@@ -138,7 +138,7 @@ module Numbers.BinaryNaturals.Subtraction where
subLemma2 : (a b : ℕ) → a <N b → 2 *N a <N succ (2 *N b)
subLemma2 a b a<b with TotalOrder.totality ℕTotalOrder (2 *N a) (succ (2 *N b))
subLemma2 a b a<b | inl (inl x) = x
subLemma2 a b a<b | inl (inr x) = exFalso (TotalOrder.irreflexive ℕTotalOrder (TotalOrder.transitive ℕTotalOrder x (TotalOrder.transitive ℕTotalOrder (lessRespectsMultiplicationLeft a b 2 a<b (le 1 refl)) (le 0 refl))))
subLemma2 a b a<b | inl (inr x) = exFalso (TotalOrder.irreflexive ℕTotalOrder (TotalOrder.<Transitive ℕTotalOrder x (TotalOrder.<Transitive ℕTotalOrder (lessRespectsMultiplicationLeft a b 2 a<b (le 1 refl)) (le 0 refl))))
subLemma2 a b a<b | inr x = exFalso (parity b a (equalityCommutative x))
subtraction : (a b : BinNat) → a -B b ≡ no → binNatToN a <N binNatToN b
@@ -457,7 +457,7 @@ module Numbers.BinaryNaturals.Subtraction where
subtraction2'' a b pr with -N pr
subtraction2'' a b pr | record { result = result ; pr = subPr } with inspect (go zero (NToBinNat a) (NToBinNat b))
subtraction2'' a b (inl pr) | record { result = result ; pr = subPr } | no with≡ pr2 with subtraction (NToBinNat a) (NToBinNat b) pr2
... | bl rewrite nToN a | nToN b = exFalso (TotalOrder.irreflexive ℕTotalOrder (TotalOrder.transitive ℕTotalOrder pr bl))
... | bl rewrite nToN a | nToN b = exFalso (TotalOrder.irreflexive ℕTotalOrder (TotalOrder.<Transitive ℕTotalOrder pr bl))
subtraction2'' a b (inr pr) | record { result = result ; pr = subPr } | no with≡ pr2 with subtraction (NToBinNat a) (NToBinNat b) pr2
... | bl rewrite nToN a | nToN b | pr = exFalso (TotalOrder.irreflexive ℕTotalOrder bl)
subtraction2'' a b pr | record { result = result ; pr = subPr } | yes x with≡ pr2 with subtraction2 a b pr2