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Reshuffle in preparation to break the dependency on N's implementation (#75)

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Patrick Stevens
2019-11-17 10:01:39 +00:00
committed by GitHub
parent ff6ef4f1a1
commit c55dd5f63e
53 changed files with 2493 additions and 2388 deletions
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42
Numbers/BinaryNaturals/Order.agda
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@@ -4,7 +4,9 @@ open import WellFoundedInduction
open import LogicalFormulae
open import Functions
open import Lists.Lists
open import Numbers.Naturals.Naturals
open import Numbers.Naturals.Order
open import Numbers.Naturals.Order.Lemmas
open import Numbers.Naturals.Semiring
open import Groups.Definition
open import Numbers.BinaryNaturals.Definition
open import Orders
@@ -27,7 +29,7 @@ module Numbers.BinaryNaturals.Order where
badCompare'' ()
_<BInherited_ : BinNat → BinNat → Compare
a <BInherited b with orderIsTotal (binNatToN a) (binNatToN b)
a <BInherited b with TotalOrder.totality ℕTotalOrder (binNatToN a) (binNatToN b)
(a <BInherited b) | inl (inl x) = FirstLess
(a <BInherited b) | inl (inr x) = FirstGreater
(a <BInherited b) | inr x = Equal
@@ -283,24 +285,24 @@ module Numbers.BinaryNaturals.Order where
chopFirstBit m n {one} FirstGreater = refl
chopDouble : (a b : BinNat) (i : Bit) → (i :: a) <BInherited (i :: b) ≡ a <BInherited b
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 i with TotalOrder.totality ℕTotalOrder (binNatToN a) (binNatToN b)
chopDouble a b zero | inl (inl a<b) with TotalOrder.totality ℕTotalOrder (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) | 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) with TotalOrder.totality ℕTotalOrder (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) | 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) with TotalOrder.totality ℕTotalOrder (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 (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) with TotalOrder.totality ℕTotalOrder (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 (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))
chopDouble a b i | inr x with TotalOrder.totality ℕTotalOrder (binNatToN (i :: a)) (binNatToN (i :: b))
chopDouble a b zero | inr a=b | inl (inl a<b) rewrite a=b = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) a<b)
chopDouble a b one | inr a=b | inl (inl a<b) rewrite a=b = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) a<b)
chopDouble a b zero | inr a=b | inl (inr b<a) rewrite a=b = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) b<a)
@@ -311,23 +313,23 @@ module Numbers.BinaryNaturals.Order where
succNotLess {succ n} (le x proof) = succNotLess {n} (le x (succInjective (transitivity (applyEquality succ (transitivity (Semiring.commutative ℕSemiring (succ x) (succ n)) (transitivity (applyEquality succ (transitivity (Semiring.commutative ℕSemiring n (succ x)) (applyEquality succ (Semiring.commutative ℕSemiring x n)))) (Semiring.commutative ℕSemiring (succ (succ n)) x)))) proof)))
<BIsInherited : (a b : BinNat) → a <BInherited b ≡ a <B b
<BIsInherited [] b with orderIsTotal 0 (binNatToN b)
<BIsInherited [] b with TotalOrder.totality ℕTotalOrder 0 (binNatToN b)
<BIsInherited [] b | inl (inl x) with inspect (binNatToN b)
<BIsInherited [] b | inl (inl x) | 0 with≡ pr rewrite binNatToNZero b pr | pr = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) x)
<BIsInherited [] b | inl (inl x) | (succ bl) with≡ pr rewrite pr = equalityCommutative (zeroLess b λ p → zeroNotSucc bl b p pr)
<BIsInherited [] b | inr 0=b rewrite canonicalSecond [] b Equal | binNatToNZero b (equalityCommutative 0=b) = refl
<BIsInherited (a :: as) [] with orderIsTotal (binNatToN (a :: as)) 0
<BIsInherited (a :: as) [] with TotalOrder.totality ℕTotalOrder (binNatToN (a :: as)) 0
<BIsInherited (a :: as) [] | inl (inr x) with inspect (binNatToN (a :: as))
<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) = 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) with TotalOrder.totality ℕTotalOrder (binNatToN (zero :: a)) (binNatToN (one :: b))
<BIsInherited (zero :: a) (one :: b) | inl (inl 2a<2b+1) with TotalOrder.totality ℕTotalOrder (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))
where
t : a <BInherited b ≡ FirstLess
t with orderIsTotal (binNatToN a) (binNatToN b)
t with TotalOrder.totality ℕTotalOrder (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 | inr x rewrite x = exFalso (PartialOrder.irreflexive (TotalOrder.order ℕTotalOrder) a<b)
@@ -335,12 +337,12 @@ module Numbers.BinaryNaturals.Order where
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) with TotalOrder.totality ℕTotalOrder (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 (inr b<a) = equalityCommutative (equalToFirstGreater FirstLess a b (equalityCommutative indHyp))
where
t : a <BInherited b ≡ FirstGreater
t with orderIsTotal (binNatToN a) (binNatToN b)
t with TotalOrder.totality ℕTotalOrder (binNatToN a) (binNatToN b)
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)
@@ -348,12 +350,12 @@ module Numbers.BinaryNaturals.Order where
indHyp = transitivity (equalityCommutative t) (<BIsInherited a b)
<BIsInherited (zero :: a) (one :: b) | inl (inr 2b+1<2a) | inr a=b rewrite a=b = exFalso (succNotLess 2b+1<2a)
<BIsInherited (zero :: a) (one :: b) | inr 2a=2b+1 = exFalso (parity (binNatToN b) (binNatToN a) (equalityCommutative 2a=2b+1))
<BIsInherited (one :: a) (zero :: b) with orderIsTotal (binNatToN (one :: a)) (binNatToN (zero :: b))
<BIsInherited (one :: a) (zero :: b) | inl (inl 2a+1<2b) with orderIsTotal (binNatToN a) (binNatToN b)
<BIsInherited (one :: a) (zero :: b) with TotalOrder.totality ℕTotalOrder (binNatToN (one :: a)) (binNatToN (zero :: b))
<BIsInherited (one :: a) (zero :: b) | inl (inl 2a+1<2b) with TotalOrder.totality ℕTotalOrder (binNatToN a) (binNatToN b)
<BIsInherited (one :: a) (zero :: b) | inl (inl 2a+1<2b) | inl (inl a<b) = equalityCommutative (equalToFirstLess FirstGreater a b (equalityCommutative indHyp))
where
t : a <BInherited b ≡ FirstLess
t with orderIsTotal (binNatToN a) (binNatToN b)
t with TotalOrder.totality ℕTotalOrder (binNatToN a) (binNatToN 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)
@@ -361,12 +363,12 @@ module Numbers.BinaryNaturals.Order where
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) | 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) with TotalOrder.totality ℕTotalOrder (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)
<BIsInherited (one :: a) (zero :: b) | inl (inr 2b<2a+1) | inl (inr b<a) = equalityCommutative (equalToFirstGreater FirstGreater a b (equalityCommutative indHyp))
where
t : a <BInherited b ≡ FirstGreater
t with orderIsTotal (binNatToN a) (binNatToN b)
t with TotalOrder.totality ℕTotalOrder (binNatToN a) (binNatToN b)
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)