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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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46
Fields/CauchyCompletion/Approximation.agda
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@@ -42,7 +42,7 @@ open import Fields.CauchyCompletion.Comparison order F charNot2
abstract
chain : {a b : A} (c : CauchyCompletion) → (a r<C c) → (c <Cr b) → a < b
chain {a} {b} c (betweenAC , (0<betweenAC ,, (Nac , prAC))) (betweenCB , (0<betweenCB ,, (Nb , prBC))) = SetoidPartialOrder.transitive pOrder (SetoidPartialOrder.transitive pOrder (<WellDefined identLeft (Equivalence.reflexive eq) (orderRespectsAddition 0<betweenAC a)) (<WellDefined groupIsAbelian (Equivalence.reflexive eq) (SetoidPartialOrder.transitive pOrder (prAC (succ Nac +N Nb) (le Nb (applyEquality succ (Semiring.commutative ℕSemiring Nb Nac)))) (<WellDefined identLeft (Equivalence.reflexive eq) (orderRespectsAddition 0<betweenCB (index (Sequence.tail (CauchyCompletion.elts c)) (Nac +N Nb))))))) (prBC (succ Nac +N Nb) (le Nac refl))
chain {a} {b} c (betweenAC , (0<betweenAC ,, (Nac , prAC))) (betweenCB , (0<betweenCB ,, (Nb , prBC))) = SetoidPartialOrder.<Transitive pOrder (SetoidPartialOrder.<Transitive pOrder (<WellDefined identLeft (Equivalence.reflexive eq) (orderRespectsAddition 0<betweenAC a)) (<WellDefined groupIsAbelian (Equivalence.reflexive eq) (SetoidPartialOrder.<Transitive pOrder (prAC (succ Nac +N Nb) (le Nb (applyEquality succ (Semiring.commutative ℕSemiring Nb Nac)))) (<WellDefined identLeft (Equivalence.reflexive eq) (orderRespectsAddition 0<betweenCB (index (Sequence.tail (CauchyCompletion.elts c)) (Nac +N Nb))))))) (prBC (succ Nac +N Nb) (le Nac refl))
approxLemma : (a : CauchyCompletion) (e e/2 : A) → (0G < e) → (e/2 + e/2 ∼ e) → (m N : ℕ) → abs ((index (CauchyCompletion.elts a) m) + inverse (index (CauchyCompletion.elts a) N)) < e/2 → (e/2 + index (CauchyCompletion.elts a) m) < (index (CauchyCompletion.elts a) N + e)
approxLemma a e e/2 0<e prE/2 m N ans with totality 0R ((index (CauchyCompletion.elts a) m) + inverse (index (CauchyCompletion.elts a) N))
@@ -63,7 +63,7 @@ abstract
ans m N<m | bl | inl (inl 0<am-aN) with <WellDefined (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.symmetric eq +Associative) (+WellDefined (Equivalence.reflexive eq) (invLeft))) identRight) (Equivalence.reflexive eq) (orderRespectsAddition bl (index (CauchyCompletion.elts a) (succ N)))
... | am<1+an = <WellDefined (Equivalence.transitive eq (+WellDefined (Equivalence.reflexive eq) identLeft) groupIsAbelian) (Equivalence.transitive eq (+WellDefined groupIsAbelian (Equivalence.reflexive eq)) +Associative) (ringAddInequalities am<1+an (orderRespectsAddition (0<1 (charNot2ImpliesNontrivial R charNot2)) 1R))
ans m N<m | bl | inl (inr am-aN<0) with <WellDefined (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.symmetric eq +Associative) (+WellDefined (Equivalence.reflexive eq) invLeft)) identRight) identLeft (orderRespectsAddition am-aN<0 (index (CauchyCompletion.elts a) (succ N)))
... | am<aN = <WellDefined groupIsAbelian (Equivalence.reflexive eq) (orderRespectsAddition (SetoidPartialOrder.transitive pOrder am<aN (<WellDefined (Equivalence.reflexive eq) (+Associative) (<WellDefined identLeft groupIsAbelian (orderRespectsAddition (<WellDefined identLeft (Equivalence.reflexive eq) (ringAddInequalities (0<1 (charNot2ImpliesNontrivial R charNot2)) (0<1 (charNot2ImpliesNontrivial R charNot2)))) (index (CauchyCompletion.elts a) (succ N)))))) 1R)
... | am<aN = <WellDefined groupIsAbelian (Equivalence.reflexive eq) (orderRespectsAddition (SetoidPartialOrder.<Transitive pOrder am<aN (<WellDefined (Equivalence.reflexive eq) (+Associative) (<WellDefined identLeft groupIsAbelian (orderRespectsAddition (<WellDefined identLeft (Equivalence.reflexive eq) (ringAddInequalities (0<1 (charNot2ImpliesNontrivial R charNot2)) (0<1 (charNot2ImpliesNontrivial R charNot2)))) (index (CauchyCompletion.elts a) (succ N)))))) 1R)
ans m N<m | bl | inr 0=am-aN = <WellDefined (Equivalence.transitive eq (+WellDefined identLeft (Equivalence.reflexive eq)) identLeft) (Equivalence.transitive eq groupIsAbelian (Equivalence.transitive eq (+WellDefined (Equivalence.transitive eq groupIsAbelian (+WellDefined (transferToRight additiveGroup (Equivalence.symmetric eq 0=am-aN)) (Equivalence.reflexive eq))) (Equivalence.reflexive eq)) +Associative)) (orderRespectsAddition (ringAddInequalities (0<1 (charNot2ImpliesNontrivial R charNot2)) (0<1 (charNot2ImpliesNontrivial R charNot2))) (1R + (index (CauchyCompletion.elts a) m)))
rationalApproximatelyAbove : (a : CauchyCompletion) → (e : A) → (0G < e) → A
@@ -88,13 +88,13 @@ abstract
ans2 : (m : ℕ) → N +N N2 <N m → (e/8 + index (CauchyCompletion.elts a) m) < (index (CauchyCompletion.elts a) (succ N) + e/2)
ans2 m <m with cauchyBeyondN {m} {succ N} (inequalityShrinkLeft <m) (le 0 refl)
... | absam-aN<e/4 with totality 0R ((index (CauchyCompletion.elts a) m) + inverse (index (CauchyCompletion.elts a) (succ N)))
ans2 m <m | am-aN<e/4 | inl (inl 0<am-aN) = SetoidPartialOrder.transitive pOrder (<WellDefined groupIsAbelian (Equivalence.transitive eq groupIsAbelian +Associative) (orderRespectsAddition (<WellDefined (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.symmetric eq +Associative) (+WellDefined (Equivalence.reflexive eq) invLeft)) identRight) (Equivalence.reflexive eq) (orderRespectsAddition am-aN<e/4 (index (CauchyCompletion.elts a) (succ N)))) e/8)) (<WellDefined (Equivalence.reflexive eq) groupIsAbelian (orderRespectsAddition (<WellDefined (Equivalence.reflexive eq) prE/4 (orderRespectsAddition (halfLess e/8 e/4 0<e/4 prE/8) e/4)) (index (CauchyCompletion.elts a) (succ N))))
ans2 m <m | am-aN<e/4 | inl (inl 0<am-aN) = SetoidPartialOrder.<Transitive pOrder (<WellDefined groupIsAbelian (Equivalence.transitive eq groupIsAbelian +Associative) (orderRespectsAddition (<WellDefined (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.symmetric eq +Associative) (+WellDefined (Equivalence.reflexive eq) invLeft)) identRight) (Equivalence.reflexive eq) (orderRespectsAddition am-aN<e/4 (index (CauchyCompletion.elts a) (succ N)))) e/8)) (<WellDefined (Equivalence.reflexive eq) groupIsAbelian (orderRespectsAddition (<WellDefined (Equivalence.reflexive eq) prE/4 (orderRespectsAddition (halfLess e/8 e/4 0<e/4 prE/8) e/4)) (index (CauchyCompletion.elts a) (succ N))))
ans2 m <m | -[am-aN]<e/4 | inl (inr am-aN<0) with <WellDefined (Equivalence.transitive eq +Associative (Equivalence.transitive eq (+WellDefined invLeft (Equivalence.reflexive eq)) identLeft)) (Equivalence.reflexive eq) (orderRespectsAddition (<WellDefined (Equivalence.reflexive eq) (invTwice additiveGroup _) (ringSwapNegatives' -[am-aN]<e/4)) (e/4 + e/8))
... | e/8<am-aN+e/4+e/8 = SetoidPartialOrder.transitive pOrder (orderRespectsAddition e/8<am-aN+e/4+e/8 (index (CauchyCompletion.elts a) m)) (<WellDefined (Equivalence.reflexive eq) groupIsAbelian (ringAddInequalities (<WellDefined (Equivalence.reflexive eq) identLeft (ringAddInequalities am-aN<0 (<WellDefined groupIsAbelian prE/4 (orderRespectsAddition (halfLess e/8 e/4 0<e/4 prE/8) e/4)))) am<aN))
... | e/8<am-aN+e/4+e/8 = SetoidPartialOrder.<Transitive pOrder (orderRespectsAddition e/8<am-aN+e/4+e/8 (index (CauchyCompletion.elts a) m)) (<WellDefined (Equivalence.reflexive eq) groupIsAbelian (ringAddInequalities (<WellDefined (Equivalence.reflexive eq) identLeft (ringAddInequalities am-aN<0 (<WellDefined groupIsAbelian prE/4 (orderRespectsAddition (halfLess e/8 e/4 0<e/4 prE/8) e/4)))) am<aN))
where
am<aN : (index (CauchyCompletion.elts a) m) < (index (CauchyCompletion.elts a) (succ N))
am<aN = <WellDefined (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.symmetric eq +Associative) (+WellDefined (Equivalence.reflexive eq) groupIsAbelian)) (Equivalence.transitive eq (+WellDefined (Equivalence.reflexive eq) invRight) identRight)) identLeft (orderRespectsAddition am-aN<0 (index (CauchyCompletion.elts a) (succ N)))
ans2 m <m | absam-aN<e/4 | inr 0=am-aN = <WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq groupIsAbelian (+WellDefined (transferToRight additiveGroup (Equivalence.symmetric eq 0=am-aN)) (Equivalence.reflexive eq))) (orderRespectsAddition {b = e/2} (SetoidPartialOrder.transitive pOrder (halfLess e/8 e/4 0<e/4 prE/8) (halfLess e/4 e/2 0<e/2 prE/4)) (index (CauchyCompletion.elts a) m))
ans2 m <m | absam-aN<e/4 | inr 0=am-aN = <WellDefined (Equivalence.reflexive eq) (Equivalence.transitive eq groupIsAbelian (+WellDefined (transferToRight additiveGroup (Equivalence.symmetric eq 0=am-aN)) (Equivalence.reflexive eq))) (orderRespectsAddition {b = e/2} (SetoidPartialOrder.<Transitive pOrder (halfLess e/8 e/4 0<e/4 prE/8) (halfLess e/4 e/2 0<e/2 prE/4)) (index (CauchyCompletion.elts a) m))
rationalApproximatelyAboveIsNear : (a : CauchyCompletion) (e : A) → (0<e : 0G < e) → (injection (rationalApproximatelyAbove a e 0<e) +C (-C a)) <C (injection e)
rationalApproximatelyAboveIsNear a e 0<e with halve charNot2 e
@@ -121,7 +121,7 @@ abstract
r : ((inverse am) + aN) < e/4
r with t
... | f with totality 0G (am + inverse aN)
r | am-aN<e/4 | inl (inl 0<am-aN) = SetoidPartialOrder.transitive pOrder (<WellDefined (Equivalence.transitive eq (invContravariant additiveGroup) (Equivalence.transitive eq groupIsAbelian (+WellDefined (Equivalence.reflexive eq) (invTwice additiveGroup _)))) (Equivalence.reflexive eq) (lemm2' _ 0<am-aN)) 0<e/4
r | am-aN<e/4 | inl (inl 0<am-aN) = SetoidPartialOrder.<Transitive pOrder (<WellDefined (Equivalence.transitive eq (invContravariant additiveGroup) (Equivalence.transitive eq groupIsAbelian (+WellDefined (Equivalence.reflexive eq) (invTwice additiveGroup _)))) (Equivalence.reflexive eq) (lemm2' _ 0<am-aN)) 0<e/4
r | f | inl (inr x) = <WellDefined (Equivalence.transitive eq (invContravariant additiveGroup) (Equivalence.transitive eq groupIsAbelian (+WellDefined (Equivalence.reflexive eq) (invTwice additiveGroup _)))) (Equivalence.reflexive eq) f
r | am-aN<e/4 | inr 0=am-aN = <WellDefined (Equivalence.transitive eq (Equivalence.symmetric eq 0=am-aN) (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.symmetric eq (invIdentity additiveGroup)) (inverseWellDefined additiveGroup 0=am-aN)) (inverseWellDefined additiveGroup groupIsAbelian)) (invContravariant additiveGroup)) (+WellDefined (Equivalence.reflexive eq) (invTwice additiveGroup _)))) (Equivalence.reflexive eq) am-aN<e/4
q : ((inverse (index (CauchyCompletion.elts a) m)) + (index (CauchyCompletion.elts a) (succ N) + e/2)) < (e/4 + e/2)
@@ -145,56 +145,56 @@ abstract
... | below , (below<a ,, a-below<e) with approximateAbove a 1R (0<1 (charNot2ImpliesNontrivial R charNot2))
... | above , (a<above ,, above-a<e) with totality 0R below
boundModulus a | below , (below<a ,, a-below<e) | above , (a<above ,, above-a<e) | inl (inl 0<below) with totality 0R above
boundModulus a | below , ((belowBound , (0<belowBound ,, (Nbelow , prBelow))) ,, a-below<e) | above , ((bound , (0<bound ,, (N , ans))) ,, above-a<e) | inl (inl 0<below) | inl (inl 0<above) = above , ((N +N Nbelow) , λ m N<m → SetoidPartialOrder.transitive pOrder (res m N<m) (ans m (inequalityShrinkLeft N<m)))
boundModulus a | below , ((belowBound , (0<belowBound ,, (Nbelow , prBelow))) ,, a-below<e) | above , ((bound , (0<bound ,, (N , ans))) ,, above-a<e) | inl (inl 0<below) | inl (inl 0<above) = above , ((N +N Nbelow) , λ m N<m → SetoidPartialOrder.<Transitive pOrder (res m N<m) (ans m (inequalityShrinkLeft N<m)))
where
res : (m : ℕ) → ((N +N Nbelow) <N m) → (abs (index (CauchyCompletion.elts a) m)) < (bound + index (CauchyCompletion.elts a) m)
res m N<m with totality 0R (index (CauchyCompletion.elts a) m)
res m N<m | inl (inl _) = <WellDefined identLeft (Equivalence.reflexive eq) (orderRespectsAddition 0<bound (index (CauchyCompletion.elts a) m))
res m N<m | inl (inr am<0) = exFalso (irreflexive (SetoidPartialOrder.transitive pOrder 0<below (SetoidPartialOrder.transitive pOrder (SetoidPartialOrder.transitive pOrder (<WellDefined identLeft groupIsAbelian (orderRespectsAddition 0<belowBound below)) (prBelow m (inequalityShrinkRight N<m))) am<0)))
res m N<m | inl (inr am<0) = exFalso (irreflexive (SetoidPartialOrder.<Transitive pOrder 0<below (SetoidPartialOrder.<Transitive pOrder (SetoidPartialOrder.<Transitive pOrder (<WellDefined identLeft groupIsAbelian (orderRespectsAddition 0<belowBound below)) (prBelow m (inequalityShrinkRight N<m))) am<0)))
res m N<m | inr 0=am = <WellDefined 0=am (Equivalence.transitive eq (Equivalence.symmetric eq identRight) (+WellDefined (Equivalence.reflexive eq) 0=am)) 0<bound
boundModulus a | below , (below<a ,, a-below<e) | above , (a<above ,, above-a<e) | inl (inl 0<below) | inl (inr above<0) = exFalso (irreflexive (SetoidPartialOrder.transitive pOrder 0<below (SetoidPartialOrder.transitive pOrder (chain a below<a a<above) above<0)))
boundModulus a | below , (below<a ,, a-below<e) | above , (a<above ,, above-a<e) | inl (inl 0<below) | inr 0=above = exFalso (irreflexive (SetoidPartialOrder.transitive pOrder 0<below (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq 0=above) (chain a below<a a<above))))
boundModulus a | below , (below<a ,, a-below<e) | above , (a<above ,, above-a<e) | inl (inl 0<below) | inl (inr above<0) = exFalso (irreflexive (SetoidPartialOrder.<Transitive pOrder 0<below (SetoidPartialOrder.<Transitive pOrder (chain a below<a a<above) above<0)))
boundModulus a | below , (below<a ,, a-below<e) | above , (a<above ,, above-a<e) | inl (inl 0<below) | inr 0=above = exFalso (irreflexive (SetoidPartialOrder.<Transitive pOrder 0<below (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq 0=above) (chain a below<a a<above))))
boundModulus a | below , (below<a ,, a-below<e) | above , (a<above ,, above-a<e) | inl (inr below<0) with totality 0R above
boundModulus a | below , (below<a ,, a-below<e) | above , (a<above ,, above-a<e) | inl (inr below<0) | inl (inl 0<above) with totality (inverse below) above
boundModulus a | below , ((boundBelow , (0<boundBelow ,, (N , prBoundBelow))) ,, a-below<e) | above , ((boundAbove , (0<boundAbove ,, (Nabove , prBoundAbove))) ,, above-a<e) | inl (inr below<0) | inl (inl 0<above) | inl (inl -bel<ab) = above , ((N +N Nabove) , ans)
where
ans : (m : ℕ) → (N +N Nabove <N m) → abs (index (CauchyCompletion.elts a) m) < above
ans m N<m with totality 0G (index (CauchyCompletion.elts a) m)
ans m N<m | inl (inl 0<am) = SetoidPartialOrder.transitive pOrder (<WellDefined identLeft (Equivalence.reflexive eq) (orderRespectsAddition 0<boundAbove (index (CauchyCompletion.elts a) m))) (prBoundAbove m (inequalityShrinkRight N<m))
ans m N<m | inl (inr am<0) = SetoidPartialOrder.transitive pOrder (ringSwapNegatives' (prBoundBelow m (inequalityShrinkLeft N<m))) (SetoidPartialOrder.transitive pOrder (ringSwapNegatives' (<WellDefined identLeft groupIsAbelian (orderRespectsAddition 0<boundBelow below))) -bel<ab)
ans m N<m | inl (inl 0<am) = SetoidPartialOrder.<Transitive pOrder (<WellDefined identLeft (Equivalence.reflexive eq) (orderRespectsAddition 0<boundAbove (index (CauchyCompletion.elts a) m))) (prBoundAbove m (inequalityShrinkRight N<m))
ans m N<m | inl (inr am<0) = SetoidPartialOrder.<Transitive pOrder (ringSwapNegatives' (prBoundBelow m (inequalityShrinkLeft N<m))) (SetoidPartialOrder.<Transitive pOrder (ringSwapNegatives' (<WellDefined identLeft groupIsAbelian (orderRespectsAddition 0<boundBelow below))) -bel<ab)
ans m N<m | inr 0=am = <WellDefined 0=am (Equivalence.reflexive eq) 0<above
boundModulus a | below , ((boundBelow , (0<boundBelow ,, (N , prBoundBelow))) ,, a-below<e) | above , ((boundAbove , (0<boundAbove ,, (Nabove , prBoundAbove))) ,, above-a<e) | inl (inr below<0) | inl (inl 0<above) | inl (inr ab<-bel) = inverse below , ((N +N Nabove) , ans)
where
ans : (m : ℕ) → (N +N Nabove <N m) → abs (index (CauchyCompletion.elts a) m) < (inverse below)
ans m N<m with totality 0G (index (CauchyCompletion.elts a) m)
ans m N<m | inl (inl 0<am) = SetoidPartialOrder.transitive pOrder (SetoidPartialOrder.transitive pOrder (<WellDefined identLeft (Equivalence.reflexive eq) (orderRespectsAddition 0<boundAbove (index (CauchyCompletion.elts a) m))) (prBoundAbove m (inequalityShrinkRight N<m))) ab<-bel
ans m N<m | inl (inr am<0) = ringSwapNegatives' (SetoidPartialOrder.transitive pOrder (<WellDefined identLeft groupIsAbelian (orderRespectsAddition 0<boundBelow below)) (prBoundBelow m (inequalityShrinkLeft N<m)))
ans m N<m | inl (inl 0<am) = SetoidPartialOrder.<Transitive pOrder (SetoidPartialOrder.<Transitive pOrder (<WellDefined identLeft (Equivalence.reflexive eq) (orderRespectsAddition 0<boundAbove (index (CauchyCompletion.elts a) m))) (prBoundAbove m (inequalityShrinkRight N<m))) ab<-bel
ans m N<m | inl (inr am<0) = ringSwapNegatives' (SetoidPartialOrder.<Transitive pOrder (<WellDefined identLeft groupIsAbelian (orderRespectsAddition 0<boundBelow below)) (prBoundBelow m (inequalityShrinkLeft N<m)))
ans m N<m | inr 0=am = <WellDefined 0=am (Equivalence.reflexive eq) (lemm2 below below<0)
boundModulus a | below , ((boundBelow , (0<boundBelow ,, (N , prBoundBelow))) ,, a-below<e) | above , ((boundAbove , (0<boundAbove ,, (Nabove , prBoundAbove))) ,, above-a<e) | inl (inr below<0) | inl (inl 0<above) | inr -bel=ab = above , ((N +N Nabove) , ans)
where
ans : (m : ℕ) → (N +N Nabove <N m) → abs (index (CauchyCompletion.elts a) m) < above
ans m N<m with totality 0G (index (CauchyCompletion.elts a) m)
ans m N<m | inl (inl 0<am) = SetoidPartialOrder.transitive pOrder (<WellDefined identLeft (Equivalence.reflexive eq) (orderRespectsAddition 0<boundAbove (index (CauchyCompletion.elts a) m))) (prBoundAbove m (inequalityShrinkRight N<m))
ans m N<m | inl (inr am<0) = <WellDefined (Equivalence.reflexive eq) (-bel=ab) (ringSwapNegatives' (SetoidPartialOrder.transitive pOrder (<WellDefined identLeft groupIsAbelian (orderRespectsAddition 0<boundBelow below)) (prBoundBelow m (inequalityShrinkLeft N<m))))
ans m N<m | inl (inl 0<am) = SetoidPartialOrder.<Transitive pOrder (<WellDefined identLeft (Equivalence.reflexive eq) (orderRespectsAddition 0<boundAbove (index (CauchyCompletion.elts a) m))) (prBoundAbove m (inequalityShrinkRight N<m))
ans m N<m | inl (inr am<0) = <WellDefined (Equivalence.reflexive eq) (-bel=ab) (ringSwapNegatives' (SetoidPartialOrder.<Transitive pOrder (<WellDefined identLeft groupIsAbelian (orderRespectsAddition 0<boundBelow below)) (prBoundBelow m (inequalityShrinkLeft N<m))))
ans m N<m | inr 0=am = <WellDefined 0=am (Equivalence.reflexive eq) 0<above
boundModulus a | below , ((boundBelow , (0<boundBelow ,, (N , prBoundBelow))) ,, a-below<e) | above , ((boundAbove , (0<boundAbove ,, (Nabove , prBoundAbove))) ,, above-a<e) | inl (inr below<0) | inl (inr above<0) = inverse below , ((N +N Nabove) , ans)
where
ans : (m : ℕ) → ((N +N Nabove) <N m) → abs (index (CauchyCompletion.elts a) m) < inverse below
ans m N<m with totality 0R (index (CauchyCompletion.elts a) m)
ans m N<m | inl (inl 0<am) = exFalso (irreflexive {0G} (SetoidPartialOrder.transitive pOrder 0<am (SetoidPartialOrder.transitive pOrder (SetoidPartialOrder.transitive pOrder (<WellDefined identLeft (Equivalence.reflexive eq) (orderRespectsAddition 0<boundAbove (index (CauchyCompletion.elts a) m))) (prBoundAbove m (inequalityShrinkRight N<m))) above<0)))
ans m N<m | inl (inr am<0) = ringSwapNegatives' (SetoidPartialOrder.transitive pOrder (<WellDefined identLeft groupIsAbelian (orderRespectsAddition 0<boundBelow below)) (prBoundBelow m (inequalityShrinkLeft N<m)))
ans m N<m | inr 0=am = exFalso (irreflexive {0G} (<WellDefined (Equivalence.symmetric eq 0=am) (Equivalence.reflexive eq) (SetoidPartialOrder.transitive pOrder (SetoidPartialOrder.transitive pOrder (<WellDefined identLeft (Equivalence.reflexive eq) (orderRespectsAddition 0<boundAbove (index (CauchyCompletion.elts a) m))) (prBoundAbove m (inequalityShrinkRight N<m))) above<0)))
ans m N<m | inl (inl 0<am) = exFalso (irreflexive {0G} (SetoidPartialOrder.<Transitive pOrder 0<am (SetoidPartialOrder.<Transitive pOrder (SetoidPartialOrder.<Transitive pOrder (<WellDefined identLeft (Equivalence.reflexive eq) (orderRespectsAddition 0<boundAbove (index (CauchyCompletion.elts a) m))) (prBoundAbove m (inequalityShrinkRight N<m))) above<0)))
ans m N<m | inl (inr am<0) = ringSwapNegatives' (SetoidPartialOrder.<Transitive pOrder (<WellDefined identLeft groupIsAbelian (orderRespectsAddition 0<boundBelow below)) (prBoundBelow m (inequalityShrinkLeft N<m)))
ans m N<m | inr 0=am = exFalso (irreflexive {0G} (<WellDefined (Equivalence.symmetric eq 0=am) (Equivalence.reflexive eq) (SetoidPartialOrder.<Transitive pOrder (SetoidPartialOrder.<Transitive pOrder (<WellDefined identLeft (Equivalence.reflexive eq) (orderRespectsAddition 0<boundAbove (index (CauchyCompletion.elts a) m))) (prBoundAbove m (inequalityShrinkRight N<m))) above<0)))
boundModulus a | below , ((boundBelow , (0<boundBelow ,, (N , prBoundBelow))) ,, a-below<e) | above , ((boundAbove , (0<boundAbove ,, (Nabove , prBoundAbove))) ,, above-a<e) | inl (inr below<0) | inr 0=above = inverse below , ((N +N Nabove) , ans)
where
ans : (m : ℕ) → ((N +N Nabove) <N m) → abs (index (CauchyCompletion.elts a) m) < inverse below
ans m N<m with totality 0R (index (CauchyCompletion.elts a) m)
ans m N<m | inl (inl 0<am) = exFalso (irreflexive {0G} (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq 0=above) (SetoidPartialOrder.transitive pOrder 0<am (SetoidPartialOrder.transitive pOrder (<WellDefined identLeft (Equivalence.reflexive eq) (orderRespectsAddition 0<boundAbove (index (CauchyCompletion.elts a) m))) (prBoundAbove m (inequalityShrinkRight N<m))))))
ans m N<m | inl (inr am<0) = ringSwapNegatives' (SetoidPartialOrder.transitive pOrder (<WellDefined identLeft groupIsAbelian (orderRespectsAddition 0<boundBelow below)) (prBoundBelow m (inequalityShrinkLeft N<m)))
ans m N<m | inl (inl 0<am) = exFalso (irreflexive {0G} (<WellDefined (Equivalence.reflexive eq) (Equivalence.symmetric eq 0=above) (SetoidPartialOrder.<Transitive pOrder 0<am (SetoidPartialOrder.<Transitive pOrder (<WellDefined identLeft (Equivalence.reflexive eq) (orderRespectsAddition 0<boundAbove (index (CauchyCompletion.elts a) m))) (prBoundAbove m (inequalityShrinkRight N<m))))))
ans m N<m | inl (inr am<0) = ringSwapNegatives' (SetoidPartialOrder.<Transitive pOrder (<WellDefined identLeft groupIsAbelian (orderRespectsAddition 0<boundBelow below)) (prBoundBelow m (inequalityShrinkLeft N<m)))
ans m N<m | inr 0=am = <WellDefined 0=am (Equivalence.reflexive eq) (lemm2 _ below<0)
boundModulus a | below , ((boundBelow , ((boundBelowDiff ,, (Nb , ansBelow)))) ,, a-below<e) | above , ((bound , (0<bound ,, (N , ans))) ,, above-a<e) | inr 0=below = above , ((N +N Nb) , λ m N<m → SetoidPartialOrder.transitive pOrder (res m N<m) (ans m (inequalityShrinkLeft N<m)))
boundModulus a | below , ((boundBelow , ((boundBelowDiff ,, (Nb , ansBelow)))) ,, a-below<e) | above , ((bound , (0<bound ,, (N , ans))) ,, above-a<e) | inr 0=below = above , ((N +N Nb) , λ m N<m → SetoidPartialOrder.<Transitive pOrder (res m N<m) (ans m (inequalityShrinkLeft N<m)))
where
res : (m : ℕ) → (N +N Nb) <N m → (abs (index (CauchyCompletion.elts a) m)) < (bound + index (CauchyCompletion.elts a) m)
res m N<m with totality 0R (index (CauchyCompletion.elts a) m)
res m N<m | inl (inl 0<am) = <WellDefined identLeft (Equivalence.reflexive eq) (orderRespectsAddition 0<bound (index (CauchyCompletion.elts a) m))
res m N<m | inl (inr am<0) = exFalso (irreflexive (<WellDefined (Equivalence.symmetric eq 0=below) (Equivalence.reflexive eq) (SetoidPartialOrder.transitive pOrder (SetoidPartialOrder.transitive pOrder (<WellDefined identLeft groupIsAbelian (orderRespectsAddition boundBelowDiff below)) (ansBelow m (inequalityShrinkRight N<m))) am<0)))
res m N<m | inl (inr am<0) = exFalso (irreflexive (<WellDefined (Equivalence.symmetric eq 0=below) (Equivalence.reflexive eq) (SetoidPartialOrder.<Transitive pOrder (SetoidPartialOrder.<Transitive pOrder (<WellDefined identLeft groupIsAbelian (orderRespectsAddition boundBelowDiff below)) (ansBelow m (inequalityShrinkRight N<m))) am<0)))
res m N<m | inr 0=am = <WellDefined 0=am (Equivalence.transitive eq (Equivalence.symmetric eq identRight) (+WellDefined (Equivalence.reflexive eq) 0=am)) 0<bound