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https://github.com/Smaug123/agdaproofs
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Lots of speedups (#116)
This commit is contained in:
Patrick Stevens
GitHub
@@ -10,7 +10,8 @@ open import Groups.Lemmas
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open import Fields.Fields
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open import Sets.EquivalenceRelations
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open import Sequences
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open import Setoids.Orders
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open import Setoids.Orders.Partial.Definition
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open import Setoids.Orders.Total.Definition
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open import Functions
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open import LogicalFormulae
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open import Numbers.Naturals.Semiring
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@@ -33,6 +34,8 @@ open import Fields.Orders.Total.Lemmas {F = F} record { oRing = order }
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open import Fields.Lemmas F
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open import Rings.Orders.Partial.Lemmas pRing
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open import Rings.Orders.Total.Lemmas order
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open import Rings.Orders.Total.Cauchy order
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open import Rings.Orders.Total.AbsoluteValue order
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open import Fields.CauchyCompletion.Definition order F
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open import Fields.CauchyCompletion.Setoid order F
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open import Fields.CauchyCompletion.Approximation order F
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@@ -44,112 +47,115 @@ private
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littleLemma : {a b c d : A} → ((a * b) + inverse (c * d)) ∼ ((a * (b + inverse d)) + (d * (a + inverse c)))
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littleLemma {a} {b} {c} {d} = Equivalence.transitive eq (Equivalence.transitive eq (+WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.symmetric eq identLeft) (+WellDefined (Equivalence.symmetric eq (Equivalence.transitive eq (+WellDefined (Equivalence.transitive eq (ringMinusExtracts R) (inverseWellDefined additiveGroup *Commutative)) (Equivalence.reflexive eq)) (invLeft {d * a}))) (Equivalence.transitive eq (Equivalence.symmetric eq (ringMinusExtracts' R)) *Commutative))) (Equivalence.symmetric eq +Associative))) (+Associative)) (Equivalence.symmetric eq (+WellDefined (*DistributesOver+) (*DistributesOver+)))
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timesConverges : (a b : CauchyCompletion) → cauchy (apply _*_ (CauchyCompletion.elts a) (CauchyCompletion.elts b))
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timesConverges record { elts = a ; converges = aConv } record { elts = b ; converges = bConv } e 0<e with boundModulus record { elts = a ; converges = aConv }
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... | aBound , (Na , prABound) with boundModulus record { elts = b ; converges = bConv }
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... | bBound , (Nb , prBBound) = N , ans
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where
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boundBoth : A
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boundBoth = aBound + bBound
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abstract
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ab<bb : aBound < boundBoth
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ab<bb = <WellDefined identLeft groupIsAbelian (orderRespectsAddition {0R} {bBound} (greaterThanAbsImpliesGreaterThan0 (prBBound (succ Nb) (le 0 refl))) aBound)
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bb<bb : bBound < boundBoth
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bb<bb = <WellDefined identLeft (Equivalence.reflexive eq) (orderRespectsAddition {0R} {aBound} (greaterThanAbsImpliesGreaterThan0 (prABound (succ Na) (le 0 refl))) bBound)
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0<boundBoth : 0R < boundBoth
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0<boundBoth = <WellDefined identLeft (Equivalence.reflexive eq) (ringAddInequalities (greaterThanAbsImpliesGreaterThan0 (prABound (succ Na) (le 0 refl))) (greaterThanAbsImpliesGreaterThan0 (prBBound (succ Nb) (le 0 refl))))
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abstract
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1/boundBoothAndPr : Sg A λ i → i * (aBound + bBound) ∼ 1R
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1/boundBoothAndPr = allInvertible boundBoth λ pr → irreflexive (<WellDefined (Equivalence.reflexive eq) pr 0<boundBoth)
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1/boundBooth : A
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1/boundBooth with 1/boundBoothAndPr
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... | a , _ = a
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1/boundBoothPr : 1/boundBooth * (aBound + bBound) ∼ 1R
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1/boundBoothPr with 1/boundBoothAndPr
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... | _ , pr = pr
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0<1/boundBooth : 0G < 1/boundBooth
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0<1/boundBooth = inversePositiveIsPositive 1/boundBoothPr 0<boundBoth
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abstract
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miniEAndPr : Sg A (λ i → (i + i) ∼ (e * 1/boundBooth))
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miniEAndPr = halve charNot2 (e * 1/boundBooth)
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miniE : A
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miniE with miniEAndPr
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... | a , _ = a
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miniEPr : (miniE + miniE) ∼ (e * 1/boundBooth)
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miniEPr with miniEAndPr
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... | _ , pr = pr
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0<miniE : 0R < miniE
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0<miniE = halvePositive miniE (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq miniEPr) (orderRespectsMultiplication 0<e 0<1/boundBooth))
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abstract
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reallyNAAndPr : Sg ℕ (λ N → {m n : ℕ} → N <N m → N <N n → abs (index a m + inverse (index a n)) < miniE)
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reallyNAAndPr = aConv miniE 0<miniE
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reallyNa : ℕ
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reallyNa with reallyNAAndPr
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... | a , _ = a
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reallyNaPr : {m n : ℕ} → reallyNa <N m → reallyNa <N n → abs (index a m + inverse (index a n)) < miniE
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reallyNaPr with reallyNAAndPr
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... | _ , pr = pr
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reallyNBAndPr : Sg ℕ (λ N → {m n : ℕ} → N <N m → N <N n → abs (index b m + inverse (index b n)) < miniE)
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reallyNBAndPr = bConv miniE 0<miniE
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reallyNb : ℕ
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reallyNb with reallyNBAndPr
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... | a , _ = a
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reallyNbPr : {m n : ℕ} → reallyNb <N m → reallyNb <N n → abs (index b m + inverse (index b n)) < miniE
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reallyNbPr with reallyNBAndPr
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... | _ , pr = pr
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N : ℕ
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N = (Na +N (Nb +N (reallyNa +N reallyNb)))
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ans : {m : ℕ} {n : ℕ} → N <N m → N <N n → abs (index (apply _*_ a b) m + inverse (index (apply _*_ a b) n)) < e
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ans {m} {n} N<m N<n rewrite indexAndApply a b _*_ {m} | indexAndApply a b _*_ {n} = ans'''
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where
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Na<m : Na <N m
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Na<m = inequalityShrinkLeft N<m
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Nb<n : Nb <N n
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Nb<n = inequalityShrinkLeft (inequalityShrinkRight {Na} N<n)
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abstract
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sum : {m n : ℕ} → (reallyNa +N reallyNb) <N m → (reallyNa +N reallyNb) <N n → (boundBoth * ((abs (index b m + inverse (index b n))) + (abs (index a m + inverse (index a n))))) < e
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sum {m} {n} <m <n = <WellDefined *Commutative (Equivalence.transitive eq (*WellDefined miniEPr (Equivalence.reflexive eq)) (Equivalence.transitive eq (Equivalence.symmetric eq *Associative) (Equivalence.transitive eq (Equivalence.transitive eq (*WellDefined (Equivalence.reflexive eq) 1/boundBoothPr) *Commutative) (identIsIdent)))) (ringCanMultiplyByPositive {c = boundBoth} 0<boundBoth (ringAddInequalities (reallyNbPr {m} {n} (inequalityShrinkRight <m) (inequalityShrinkRight <n)) (reallyNaPr {m} {n} (inequalityShrinkLeft <m) (inequalityShrinkLeft <n))))
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q : ((boundBoth * (abs (index b m + inverse (index b n)))) + (boundBoth * (abs (index a m + inverse (index a n))))) < e
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q = <WellDefined *DistributesOver+ (Equivalence.reflexive eq) (sum {m} {n} (inequalityShrinkRight {Nb} (inequalityShrinkRight {Na} N<m)) (inequalityShrinkRight {Nb} (inequalityShrinkRight {Na} N<n)))
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p : ((abs (index a m) * abs (index b m + inverse (index b n))) + (abs (index b n) * abs (index a m + inverse (index a n)))) < e
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p with totality 0R (index b m + inverse (index b n))
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p | inl (inl 0<bm-bn) with totality 0R (index a m + inverse (index a n))
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p | inl (inl 0<bm-bn) | inl (inl 0<am-an) = SetoidPartialOrder.<Transitive pOrder (ringAddInequalities (<WellDefined (Equivalence.reflexive eq) (*WellDefined (Equivalence.reflexive eq) (greaterZeroImpliesEqualAbs 0<bm-bn)) (ringCanMultiplyByPositive 0<bm-bn (SetoidPartialOrder.<Transitive pOrder (prABound m Na<m) ab<bb))) (<WellDefined (Equivalence.reflexive eq) (*WellDefined (Equivalence.reflexive eq) (greaterZeroImpliesEqualAbs 0<am-an)) (ringCanMultiplyByPositive 0<am-an (SetoidPartialOrder.<Transitive pOrder (prBBound n Nb<n) bb<bb)))) q
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p | inl (inl 0<bm-bn) | inl (inr am-an<0) = SetoidPartialOrder.<Transitive pOrder (ringAddInequalities (<WellDefined (*WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq (greaterZeroImpliesEqualAbs 0<bm-bn))) (Equivalence.reflexive eq) (ringCanMultiplyByPositive (<WellDefined (Equivalence.reflexive eq) (greaterZeroImpliesEqualAbs 0<bm-bn) 0<bm-bn) (SetoidPartialOrder.<Transitive pOrder (prABound m Na<m) ab<bb))) (<WellDefined (*WellDefined (Equivalence.reflexive eq) (lessZeroImpliesEqualNegAbs am-an<0)) (Equivalence.reflexive eq) (ringCanMultiplyByPositive (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq (lessZeroImpliesEqualNegAbs am-an<0)) (lemm2 _ am-an<0)) (SetoidPartialOrder.<Transitive pOrder (prBBound n Nb<n) bb<bb)))) q
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p | inl (inl 0<bm-bn) | inr 0=am-an = <WellDefined (+WellDefined (Equivalence.reflexive eq) (*WellDefined (Equivalence.reflexive eq) 0=am-an)) (Equivalence.reflexive eq) (<WellDefined (+WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq timesZero)) (Equivalence.reflexive eq) (<WellDefined (Equivalence.symmetric eq identRight) (Equivalence.reflexive eq) (SetoidPartialOrder.<Transitive pOrder (<WellDefined (Equivalence.reflexive eq) (+WellDefined (Equivalence.reflexive eq) (*WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq (Equivalence.symmetric eq absZeroIsZero) (absWellDefined _ _ 0=am-an)))) (<WellDefined (Equivalence.reflexive eq) (+WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq timesZero)) (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq identRight) (<WellDefined (*WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq (greaterZeroImpliesEqualAbs 0<bm-bn))) (Equivalence.reflexive eq) (ringCanMultiplyByPositive (<WellDefined (Equivalence.reflexive eq) (greaterZeroImpliesEqualAbs 0<bm-bn) 0<bm-bn) (SetoidPartialOrder.<Transitive pOrder (prABound m Na<m) ab<bb)))))) q)))
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p | inl (inr bm-bn<0) = SetoidPartialOrder.<Transitive pOrder ans'' q
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where
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bar : ((abs (index a m)) * (inverse (index b m + inverse (index b n)))) < (boundBoth * abs (index b m + inverse (index b n)))
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bar = <WellDefined (*WellDefined (Equivalence.reflexive eq) (lessZeroImpliesEqualNegAbs bm-bn<0)) (Equivalence.reflexive eq) (ringCanMultiplyByPositive (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq (lessZeroImpliesEqualNegAbs bm-bn<0)) (lemm2 _ bm-bn<0)) (SetoidPartialOrder.<Transitive pOrder (prABound m Na<m) (ab<bb)))
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foo : (((abs (index b n)) * (abs (index a m + inverse (index a n)))) < (boundBoth * abs (index a m + inverse (index a n)))) || (((abs (index b n)) * (abs (index a m + inverse (index a n)))) ∼ (boundBoth * abs (index a m + inverse (index a n))))
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foo with totality 0R (index a m + inverse (index a n))
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foo | inl (inl 0<am-an) = inl (ringCanMultiplyByPositive 0<am-an (SetoidPartialOrder.<Transitive pOrder (prBBound n Nb<n) bb<bb))
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foo | inl (inr am-an<0) = inl (ringCanMultiplyByPositive (lemm2 _ am-an<0) (SetoidPartialOrder.<Transitive pOrder (prBBound n Nb<n) bb<bb))
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foo | inr 0=am-an = inr (Equivalence.transitive eq (*WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq 0=am-an)) (Equivalence.transitive eq (Equivalence.transitive eq timesZero (Equivalence.symmetric eq timesZero)) (*WellDefined (Equivalence.reflexive eq) 0=am-an)))
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ans'' : _
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ans'' with foo
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... | inl pr = ringAddInequalities bar pr
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... | inr pr = <WellDefined (Equivalence.reflexive eq) (+WellDefined (Equivalence.reflexive eq) pr) (orderRespectsAddition bar ((abs (index b n)) * (abs (index a m + inverse (index a n)))))
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p | inr 0=bm-bn = ans''
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where
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bar : (boundBoth * abs (index b m + inverse (index b n))) ∼ 0R
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bar = Equivalence.transitive eq (*WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq (absWellDefined _ _ (Equivalence.symmetric eq 0=bm-bn)) (absZeroIsZero))) timesZero
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bar' : (abs (index a m) * (index b m + inverse (index b n))) ∼ 0R
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bar' = Equivalence.transitive eq (*WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq 0=bm-bn)) timesZero
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foo : (((abs (index b n)) * (abs (index a m + inverse (index a n)))) < (boundBoth * abs (index a m + inverse (index a n)))) || (0R ∼ (index a m + inverse (index a n)))
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foo with totality 0R (index a m + inverse (index a n))
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foo | inl (inl 0<am-an) = inl (SetoidPartialOrder.<Transitive pOrder (ringCanMultiplyByPositive 0<am-an (prBBound n Nb<n)) (ringCanMultiplyByPositive 0<am-an bb<bb))
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foo | inl (inr am-an<0) = inl (SetoidPartialOrder.<Transitive pOrder (ringCanMultiplyByPositive (lemm2 _ am-an<0) (prBBound n Nb<n)) (ringCanMultiplyByPositive (lemm2 _ am-an<0) bb<bb))
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foo | inr 0=am-an = inr 0=am-an
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ans'' : _
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ans'' with foo
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... | inl pr = SetoidPartialOrder.<Transitive pOrder (<WellDefined groupIsAbelian groupIsAbelian (<WellDefined (+WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq bar')) (+WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq bar)) (orderRespectsAddition pr 0R))) q
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... | inr pr = <WellDefined (Equivalence.transitive eq (Equivalence.symmetric eq identRight) (+WellDefined (Equivalence.symmetric eq bar') (Equivalence.symmetric eq (Equivalence.transitive eq (*WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq (absWellDefined _ _ (Equivalence.symmetric eq pr)) absZeroIsZero)) timesZero)))) (Equivalence.reflexive eq) 0<e
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ans''' : abs ((index a m * index b m) + inverse (index a n * index b n)) < e
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ans''' with triangleInequality (index a m * (index b m + inverse (index b n))) (index b n * (index a m + inverse (index a n)))
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... | inl less = <WellDefined (Equivalence.symmetric eq (absWellDefined ((index a m * index b m) + (inverse (index a n * index b n))) (((index a m) * (index b m + (inverse (index b n)))) + ((index b n) * (index a m + inverse (index a n)))) littleLemma)) (Equivalence.reflexive eq) (SetoidPartialOrder.<Transitive pOrder less (<WellDefined (+WellDefined (Equivalence.symmetric eq (absRespectsTimes (index a m) _)) (Equivalence.symmetric eq (absRespectsTimes (index b n) _))) (Equivalence.reflexive eq) p))
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... | inr equal = <WellDefined {_ + _} {abs _} {e} {e} (symmetric (transitive (transitive (absWellDefined ((index a m * index b m) + (inverse (index a n * index b n))) (((index a m) * (index b m + (inverse (index b n)))) + ((index b n) * (index a m + inverse (index a n)))) littleLemma) equal) (+WellDefined (absRespectsTimes (index a m) _) (absRespectsTimes (index b n) _)))) reflexive p
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_*C_ : CauchyCompletion → CauchyCompletion → CauchyCompletion
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CauchyCompletion.elts (record { elts = a ; converges = aConv } *C record { elts = b ; converges = bConv }) = apply _*_ a b
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CauchyCompletion.converges (record { elts = a ; converges = aConv } *C record { elts = b ; converges = bConv }) e 0<e with boundModulus record { elts = a ; converges = aConv }
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... | aBound , (Na , prABound) with boundModulus record { elts = b ; converges = bConv }
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... | bBound , (Nb , prBBound) = N , ans
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where
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boundBoth : A
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boundBoth = aBound + bBound
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abstract
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ab<bb : aBound < boundBoth
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ab<bb = <WellDefined identLeft groupIsAbelian (orderRespectsAddition {0R} {bBound} (greaterThanAbsImpliesGreaterThan0 (prBBound (succ Nb) (le 0 refl))) aBound)
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bb<bb : bBound < boundBoth
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bb<bb = <WellDefined identLeft (Equivalence.reflexive eq) (orderRespectsAddition {0R} {aBound} (greaterThanAbsImpliesGreaterThan0 (prABound (succ Na) (le 0 refl))) bBound)
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0<boundBoth : 0R < boundBoth
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0<boundBoth = <WellDefined identLeft (Equivalence.reflexive eq) (ringAddInequalities (greaterThanAbsImpliesGreaterThan0 (prABound (succ Na) (le 0 refl))) (greaterThanAbsImpliesGreaterThan0 (prBBound (succ Nb) (le 0 refl))))
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abstract
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1/boundBoothAndPr : Sg A λ i → i * (aBound + bBound) ∼ 1R
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1/boundBoothAndPr = allInvertible boundBoth λ pr → irreflexive (<WellDefined (Equivalence.reflexive eq) pr 0<boundBoth)
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1/boundBooth : A
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1/boundBooth with 1/boundBoothAndPr
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... | a , _ = a
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1/boundBoothPr : 1/boundBooth * (aBound + bBound) ∼ 1R
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1/boundBoothPr with 1/boundBoothAndPr
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... | _ , pr = pr
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0<1/boundBooth : 0G < 1/boundBooth
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0<1/boundBooth = inversePositiveIsPositive 1/boundBoothPr 0<boundBoth
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abstract
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miniEAndPr : Sg A (λ i → (i + i) ∼ (e * 1/boundBooth))
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miniEAndPr = halve charNot2 (e * 1/boundBooth)
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miniE : A
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miniE with miniEAndPr
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... | a , _ = a
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miniEPr : (miniE + miniE) ∼ (e * 1/boundBooth)
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miniEPr with miniEAndPr
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... | _ , pr = pr
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0<miniE : 0R < miniE
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0<miniE = halvePositive miniE (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq miniEPr) (orderRespectsMultiplication 0<e 0<1/boundBooth))
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abstract
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reallyNAAndPr : Sg ℕ (λ N → {m n : ℕ} → N <N m → N <N n → abs (index a m + inverse (index a n)) < miniE)
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reallyNAAndPr = aConv miniE 0<miniE
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reallyNa : ℕ
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reallyNa with reallyNAAndPr
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... | a , _ = a
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reallyNaPr : {m n : ℕ} → reallyNa <N m → reallyNa <N n → abs (index a m + inverse (index a n)) < miniE
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reallyNaPr with reallyNAAndPr
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... | _ , pr = pr
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reallyNBAndPr : Sg ℕ (λ N → {m n : ℕ} → N <N m → N <N n → abs (index b m + inverse (index b n)) < miniE)
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reallyNBAndPr = bConv miniE 0<miniE
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reallyNb : ℕ
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reallyNb with reallyNBAndPr
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... | a , _ = a
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reallyNbPr : {m n : ℕ} → reallyNb <N m → reallyNb <N n → abs (index b m + inverse (index b n)) < miniE
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reallyNbPr with reallyNBAndPr
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... | _ , pr = pr
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N : ℕ
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N = (Na +N (Nb +N (reallyNa +N reallyNb)))
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ans : {m : ℕ} {n : ℕ} → N <N m → N <N n → abs (index (apply _*_ a b) m + inverse (index (apply _*_ a b) n)) < e
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ans {m} {n} N<m N<n rewrite indexAndApply a b _*_ {m} | indexAndApply a b _*_ {n} = ans'''
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where
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Na<m : Na <N m
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Na<m = inequalityShrinkLeft N<m
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Nb<n : Nb <N n
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Nb<n = inequalityShrinkLeft (inequalityShrinkRight {Na} N<n)
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abstract
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sum : {m n : ℕ} → (reallyNa +N reallyNb) <N m → (reallyNa +N reallyNb) <N n → (boundBoth * ((abs (index b m + inverse (index b n))) + (abs (index a m + inverse (index a n))))) < e
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sum {m} {n} <m <n = <WellDefined *Commutative (Equivalence.transitive eq (*WellDefined miniEPr (Equivalence.reflexive eq)) (Equivalence.transitive eq (Equivalence.symmetric eq *Associative) (Equivalence.transitive eq (Equivalence.transitive eq (*WellDefined (Equivalence.reflexive eq) 1/boundBoothPr) *Commutative) (identIsIdent)))) (ringCanMultiplyByPositive {c = boundBoth} 0<boundBoth (ringAddInequalities (reallyNbPr {m} {n} (inequalityShrinkRight <m) (inequalityShrinkRight <n)) (reallyNaPr {m} {n} (inequalityShrinkLeft <m) (inequalityShrinkLeft <n))))
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q : ((boundBoth * (abs (index b m + inverse (index b n)))) + (boundBoth * (abs (index a m + inverse (index a n))))) < e
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q = <WellDefined *DistributesOver+ (Equivalence.reflexive eq) (sum {m} {n} (inequalityShrinkRight {Nb} (inequalityShrinkRight {Na} N<m)) (inequalityShrinkRight {Nb} (inequalityShrinkRight {Na} N<n)))
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p : ((abs (index a m) * abs (index b m + inverse (index b n))) + (abs (index b n) * abs (index a m + inverse (index a n)))) < e
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p with totality 0R (index b m + inverse (index b n))
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p | inl (inl 0<bm-bn) with totality 0R (index a m + inverse (index a n))
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p | inl (inl 0<bm-bn) | inl (inl 0<am-an) = SetoidPartialOrder.<Transitive pOrder (ringAddInequalities (<WellDefined (Equivalence.reflexive eq) (*WellDefined (Equivalence.reflexive eq) (greaterZeroImpliesEqualAbs 0<bm-bn)) (ringCanMultiplyByPositive 0<bm-bn (SetoidPartialOrder.<Transitive pOrder (prABound m Na<m) ab<bb))) (<WellDefined (Equivalence.reflexive eq) (*WellDefined (Equivalence.reflexive eq) (greaterZeroImpliesEqualAbs 0<am-an)) (ringCanMultiplyByPositive 0<am-an (SetoidPartialOrder.<Transitive pOrder (prBBound n Nb<n) bb<bb)))) q
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p | inl (inl 0<bm-bn) | inl (inr am-an<0) = SetoidPartialOrder.<Transitive pOrder (ringAddInequalities (<WellDefined (*WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq (greaterZeroImpliesEqualAbs 0<bm-bn))) (Equivalence.reflexive eq) (ringCanMultiplyByPositive (<WellDefined (Equivalence.reflexive eq) (greaterZeroImpliesEqualAbs 0<bm-bn) 0<bm-bn) (SetoidPartialOrder.<Transitive pOrder (prABound m Na<m) ab<bb))) (<WellDefined (*WellDefined (Equivalence.reflexive eq) (lessZeroImpliesEqualNegAbs am-an<0)) (Equivalence.reflexive eq) (ringCanMultiplyByPositive (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq (lessZeroImpliesEqualNegAbs am-an<0)) (lemm2 _ am-an<0)) (SetoidPartialOrder.<Transitive pOrder (prBBound n Nb<n) bb<bb)))) q
|
||||
p | inl (inl 0<bm-bn) | inr 0=am-an = <WellDefined (+WellDefined (Equivalence.reflexive eq) (*WellDefined (Equivalence.reflexive eq) 0=am-an)) (Equivalence.reflexive eq) (<WellDefined (+WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq timesZero)) (Equivalence.reflexive eq) (<WellDefined (Equivalence.symmetric eq identRight) (Equivalence.reflexive eq) (SetoidPartialOrder.<Transitive pOrder (<WellDefined (Equivalence.reflexive eq) (+WellDefined (Equivalence.reflexive eq) (*WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq (Equivalence.symmetric eq absZeroIsZero) (absWellDefined _ _ 0=am-an)))) (<WellDefined (Equivalence.reflexive eq) (+WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq timesZero)) (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq identRight) (<WellDefined (*WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq (greaterZeroImpliesEqualAbs 0<bm-bn))) (Equivalence.reflexive eq) (ringCanMultiplyByPositive (<WellDefined (Equivalence.reflexive eq) (greaterZeroImpliesEqualAbs 0<bm-bn) 0<bm-bn) (SetoidPartialOrder.<Transitive pOrder (prABound m Na<m) ab<bb)))))) q)))
|
||||
p | inl (inr bm-bn<0) = SetoidPartialOrder.<Transitive pOrder ans'' q
|
||||
where
|
||||
bar : ((abs (index a m)) * (inverse (index b m + inverse (index b n)))) < (boundBoth * abs (index b m + inverse (index b n)))
|
||||
bar = <WellDefined (*WellDefined (Equivalence.reflexive eq) (lessZeroImpliesEqualNegAbs bm-bn<0)) (Equivalence.reflexive eq) (ringCanMultiplyByPositive (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq (lessZeroImpliesEqualNegAbs bm-bn<0)) (lemm2 _ bm-bn<0)) (SetoidPartialOrder.<Transitive pOrder (prABound m Na<m) (ab<bb)))
|
||||
foo : (((abs (index b n)) * (abs (index a m + inverse (index a n)))) < (boundBoth * abs (index a m + inverse (index a n)))) || (((abs (index b n)) * (abs (index a m + inverse (index a n)))) ∼ (boundBoth * abs (index a m + inverse (index a n))))
|
||||
foo with totality 0R (index a m + inverse (index a n))
|
||||
foo | inl (inl 0<am-an) = inl (ringCanMultiplyByPositive 0<am-an (SetoidPartialOrder.<Transitive pOrder (prBBound n Nb<n) bb<bb))
|
||||
foo | inl (inr am-an<0) = inl (ringCanMultiplyByPositive (lemm2 _ am-an<0) (SetoidPartialOrder.<Transitive pOrder (prBBound n Nb<n) bb<bb))
|
||||
foo | inr 0=am-an = inr (Equivalence.transitive eq (*WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq 0=am-an)) (Equivalence.transitive eq (Equivalence.transitive eq timesZero (Equivalence.symmetric eq timesZero)) (*WellDefined (Equivalence.reflexive eq) 0=am-an)))
|
||||
ans'' : _
|
||||
ans'' with foo
|
||||
... | inl pr = ringAddInequalities bar pr
|
||||
... | inr pr = <WellDefined (Equivalence.reflexive eq) (+WellDefined (Equivalence.reflexive eq) pr) (orderRespectsAddition bar ((abs (index b n)) * (abs (index a m + inverse (index a n)))))
|
||||
p | inr 0=bm-bn = ans''
|
||||
where
|
||||
bar : (boundBoth * abs (index b m + inverse (index b n))) ∼ 0R
|
||||
bar = Equivalence.transitive eq (*WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq (absWellDefined _ _ (Equivalence.symmetric eq 0=bm-bn)) (absZeroIsZero))) timesZero
|
||||
bar' : (abs (index a m) * (index b m + inverse (index b n))) ∼ 0R
|
||||
bar' = Equivalence.transitive eq (*WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq 0=bm-bn)) timesZero
|
||||
foo : (((abs (index b n)) * (abs (index a m + inverse (index a n)))) < (boundBoth * abs (index a m + inverse (index a n)))) || (0R ∼ (index a m + inverse (index a n)))
|
||||
foo with totality 0R (index a m + inverse (index a n))
|
||||
foo | inl (inl 0<am-an) = inl (SetoidPartialOrder.<Transitive pOrder (ringCanMultiplyByPositive 0<am-an (prBBound n Nb<n)) (ringCanMultiplyByPositive 0<am-an bb<bb))
|
||||
foo | inl (inr am-an<0) = inl (SetoidPartialOrder.<Transitive pOrder (ringCanMultiplyByPositive (lemm2 _ am-an<0) (prBBound n Nb<n)) (ringCanMultiplyByPositive (lemm2 _ am-an<0) bb<bb))
|
||||
foo | inr 0=am-an = inr 0=am-an
|
||||
ans'' : _
|
||||
ans'' with foo
|
||||
... | inl pr = SetoidPartialOrder.<Transitive pOrder (<WellDefined groupIsAbelian groupIsAbelian (<WellDefined (+WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq bar')) (+WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq bar)) (orderRespectsAddition pr 0R))) q
|
||||
... | inr pr = <WellDefined (Equivalence.transitive eq (Equivalence.symmetric eq identRight) (+WellDefined (Equivalence.symmetric eq bar') (Equivalence.symmetric eq (Equivalence.transitive eq (*WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq (absWellDefined _ _ (Equivalence.symmetric eq pr)) absZeroIsZero)) timesZero)))) (Equivalence.reflexive eq) 0<e
|
||||
ans''' : abs ((index a m * index b m) + inverse (index a n * index b n)) < e
|
||||
ans''' with triangleInequality (index a m * (index b m + inverse (index b n))) (index b n * (index a m + inverse (index a n)))
|
||||
... | inl less = <WellDefined (Equivalence.symmetric eq (absWellDefined ((index a m * index b m) + (inverse (index a n * index b n))) (((index a m) * (index b m + (inverse (index b n)))) + ((index b n) * (index a m + inverse (index a n)))) littleLemma)) (Equivalence.reflexive eq) (SetoidPartialOrder.<Transitive pOrder less (<WellDefined (+WellDefined (Equivalence.symmetric eq (absRespectsTimes (index a m) _)) (Equivalence.symmetric eq (absRespectsTimes (index b n) _))) (Equivalence.reflexive eq) p))
|
||||
... | inr equal = <WellDefined {_ + _} {abs _} {e} {e} (symmetric (transitive (transitive (absWellDefined ((index a m * index b m) + (inverse (index a n * index b n))) (((index a m) * (index b m + (inverse (index b n)))) + ((index b n) * (index a m + inverse (index a n)))) littleLemma) equal) (+WellDefined (absRespectsTimes (index a m) _) (absRespectsTimes (index b n) _)))) reflexive p
|
||||
CauchyCompletion.converges (a *C b) = timesConverges a b
|
||||
|
||||
*CCommutative : {a b : CauchyCompletion} → Setoid._∼_ cauchyCompletionSetoid (a *C b) (b *C a)
|
||||
*CCommutative {a} {b} ε 0<e = 0 , ans
|
||||
@@ -159,7 +165,7 @@ CauchyCompletion.converges (record { elts = a ; converges = aConv } *C record {
|
||||
ans : {m : ℕ} → 0 <N m → abs (index (apply _+_ (CauchyCompletion.elts (a *C b)) (map inverse (CauchyCompletion.elts (b *C a)))) m) < ε
|
||||
ans {m} 0<m rewrite indexAndApply (apply _*_ (CauchyCompletion.elts a) (CauchyCompletion.elts b)) (map inverse (apply _*_ (CauchyCompletion.elts b) (CauchyCompletion.elts a))) _+_ {m} | indexAndApply (CauchyCompletion.elts a) (CauchyCompletion.elts b) _*_ {m} | equalityCommutative (mapAndIndex (apply _*_ (CauchyCompletion.elts b) (CauchyCompletion.elts a)) inverse m) | indexAndApply (CauchyCompletion.elts b) (CauchyCompletion.elts a) _*_ {m} = <WellDefined (Equivalence.symmetric eq (Equivalence.transitive eq (absWellDefined _ _ foo) (identityOfIndiscernablesRight _∼_ (Equivalence.reflexive eq) absZero))) (Equivalence.reflexive eq) 0<e
|
||||
|
||||
abstract
|
||||
private
|
||||
|
||||
multiplicationWellDefinedLeft' : (0!=1 : 0R ∼ 1R → False) (a b c : CauchyCompletion) → Setoid._∼_ cauchyCompletionSetoid a b → Setoid._∼_ cauchyCompletionSetoid (a *C c) (b *C c)
|
||||
multiplicationWellDefinedLeft' 0!=1 a b c a=b ε 0<e = N , ans
|
||||
@@ -229,11 +235,12 @@ abstract
|
||||
multiplicationPreservedRight : {a b : A} {c : CauchyCompletion} → (a ∼ b) → Setoid._∼_ cauchyCompletionSetoid (c *C injection a) (c *C injection b)
|
||||
multiplicationPreservedRight {a} {b} {c} a=b = Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {c *C injection a} {injection a *C c} {c *C injection b} (*CCommutative {c} {injection a}) (Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {injection a *C c} {injection b *C c} {c *C injection b} (multiplicationPreservedLeft {a} {b} {c} a=b) (*CCommutative {injection b} {c}))
|
||||
|
||||
multiplicationPreserved : {a b c d : A} → (a ∼ b) → (c ∼ d) → Setoid._∼_ cauchyCompletionSetoid (injection a *C injection c) (injection b *C injection d)
|
||||
multiplicationPreserved {a} {b} {c} {d} a=b c=d = Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {injection a *C injection c} {injection a *C injection d} {injection b *C injection d} (multiplicationPreservedRight {c} {d} {injection a} c=d) (multiplicationPreservedLeft {a} {b} {injection d} a=b)
|
||||
multiplicationPreserved : {a b c d : A} → (a ∼ b) → (c ∼ d) → Setoid._∼_ cauchyCompletionSetoid (injection a *C injection c) (injection b *C injection d)
|
||||
multiplicationPreserved {a} {b} {c} {d} a=b c=d = Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {injection a *C injection c} {injection a *C injection d} {injection b *C injection d} (multiplicationPreservedRight {c} {d} {injection a} c=d) (multiplicationPreservedLeft {a} {b} {injection d} a=b)
|
||||
|
||||
private
|
||||
multiplicationWellDefinedRight : (a b c : CauchyCompletion) → Setoid._∼_ cauchyCompletionSetoid b c → Setoid._∼_ cauchyCompletionSetoid (a *C b) (a *C c)
|
||||
multiplicationWellDefinedRight a b c b=c = Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {a *C b} {b *C a} {a *C c} (*CCommutative {a} {b}) (Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {b *C a} {c *C a} {a *C c} (multiplicationWellDefinedLeft b c a b=c) (*CCommutative {c} {a}))
|
||||
|
||||
multiplicationWellDefined : {a b c d : CauchyCompletion} → Setoid._∼_ cauchyCompletionSetoid a b → Setoid._∼_ cauchyCompletionSetoid c d → Setoid._∼_ cauchyCompletionSetoid (a *C c) (b *C d)
|
||||
multiplicationWellDefined {a} {b} {c} {d} a=b c=d = Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {a *C c} {a *C d} {b *C d} (multiplicationWellDefinedRight a c d c=d) (multiplicationWellDefinedLeft a b d a=b)
|
||||
multiplicationWellDefined : {a b c d : CauchyCompletion} → Setoid._∼_ cauchyCompletionSetoid a b → Setoid._∼_ cauchyCompletionSetoid c d → Setoid._∼_ cauchyCompletionSetoid (a *C c) (b *C d)
|
||||
multiplicationWellDefined {a} {b} {c} {d} a=b c=d = Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {a *C c} {a *C d} {b *C d} (multiplicationWellDefinedRight a c d c=d) (multiplicationWellDefinedLeft a b d a=b)
|
||||
|
||||
Reference in New Issue
Block a user