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Move towards base-n expansions (#112)

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Patrick Stevens
2020-04-11 19:46:26 +01:00
committed by GitHub
parent e9aa1bcc05
commit 380548134d
22 changed files with 312 additions and 102 deletions
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28
Numbers/Naturals/Order.agda
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@@ -92,9 +92,23 @@ canRemoveSuccFrom<N {a} {b} (le x proof) rewrite commutative x (succ a) | commut
a<SuccA : (a : ℕ) → a <N succ a
a<SuccA a = le zero refl
<NWellDefined : {a b : ℕ} → (p1 : a <N b) → (p2 : a <N b) → _<N_.x p1 ≡ _<N_.x p2
<NWellDefined {a} {b} (le x proof) (le y proof1) = equalityCommutative r
where
q : y +N a ≡ x +N a
q = succInjective {y +N a} {x +N a} (transitivity proof1 (equalityCommutative proof))
r : y ≡ x
r = canSubtractFromEqualityRight q
canAddToOneSideOfInequality : {a b : ℕ} (c : ℕ) → a <N b → a <N (b +N c)
canAddToOneSideOfInequality {a} {b} c (le x proof) = le (x +N c) (transitivity (applyEquality succ (equalityCommutative (+Associative x c a))) (transitivity (applyEquality (λ i → (succ x) +N i) (commutative c a)) (transitivity (applyEquality succ (+Associative x a c)) (applyEquality (_+N c) proof))))
canAddToOneSideOfInequality' : {a b : ℕ} (c : ℕ) → a <N b → a <N (c +N b)
canAddToOneSideOfInequality' {a} {b} c (le x proof) = le (x +N c) ans
where
ans : succ ((x +N c) +N a) ≡ c +N b
ans rewrite (equalityCommutative (+Associative x c a)) | commutative c a | (+Associative x a c) = transitivity (applyEquality (_+N c) proof) (commutative b c)
addingIncreases : (a b : ℕ) → a <N a +N succ b
addingIncreases zero b = succIsPositive b
addingIncreases (succ a) b = succPreservesInequality (addingIncreases a b)
@@ -107,14 +121,6 @@ noIntegersBetweenXAndSuccX {zero} x x<a a<sx = zeroNeverGreater x<a
noIntegersBetweenXAndSuccX {succ a} x (le y proof) (le z proof1) with succInjective proof1
... | ah rewrite (equalityCommutative proof) | (succExtracts z (y +N x)) | +Associative (succ z) y x | commutative (succ (z +N y)) x = lessIrreflexive {x} (le (z +N y) (transitivity (commutative _ x) ah))
<NWellDefined : {a b : ℕ} → (p1 : a <N b) → (p2 : a <N b) → _<N_.x p1 ≡ _<N_.x p2
<NWellDefined {a} {b} (le x proof) (le y proof1) = equalityCommutative r
where
q : y +N a ≡ x +N a
q = succInjective {y +N a} {x +N a} (transitivity proof1 (equalityCommutative proof))
r : y ≡ x
r = canSubtractFromEqualityRight q
private
orderIsTotal : (a b : ℕ) → ((a <N b) || (b <N a)) || (a ≡ b)
orderIsTotal zero zero = inr refl
@@ -192,6 +198,12 @@ canFlipMultiplicationsInIneq {a} {b} {c} {d} pr = identityOfIndiscernablesRight
lessRespectsMultiplication : (a b c : ℕ) → (a <N b) → (zero <N c) → (a *N c <N b *N c)
lessRespectsMultiplication a b c prAB cPos = canFlipMultiplicationsInIneq {c} {a} {c} {b} (lessRespectsMultiplicationLeft a b c prAB cPos)
squash<N : {a b : ℕ} → .(a <N b) → a <N b
squash<N {a} {b} a<b with orderIsTotal a b
... | inl (inl x) = x
... | inl (inr x) = exFalso (lessIrreflexive (orderIsTransitive x a<b))
squash<N {a} {b} a<b | inr refl = exFalso (lessIrreflexive a<b)
<NTo<N' : {a b : ℕ} → a <N b → a <N' b
<NTo<N' (le x proof) = le' x proof