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https://github.com/Smaug123/agdaproofs
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Move Bool to live on its own (#118)
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Patrick Stevens
GitHub
@@ -107,42 +107,14 @@ private
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ans : (m : ℕ) → (index (CauchyCompletion.elts (injection b +C a)) m + index (map inverse (CauchyCompletion.elts a)) m) ∼ b
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ans m rewrite indexAndApply (CauchyCompletion.elts (injection b)) (CauchyCompletion.elts a) _+_ {m} | indexAndConst b m | equalityCommutative (mapAndIndex (CauchyCompletion.elts a) inverse m) = transitive (symmetric +Associative) (transitive (+WellDefined reflexive invRight) identRight)
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{-
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minAddLemma : (a b : A) → (0R < a) → (min a b) < (a + b)
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minAddLemma a b 0<a with totality a b
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... | inl (inl x) = <WellDefined identLeft groupIsAbelian (orderRespectsAddition (<Transitive 0<a x) a)
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... | inl (inr x) = <WellDefined identLeft reflexive (orderRespectsAddition 0<a b)
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... | inr x = <WellDefined identLeft groupIsAbelian (orderRespectsAddition (<WellDefined reflexive x 0<a) a)
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addInequalities : {a b : CauchyCompletion} → 0R r<C a → 0R r<C b → 0R r<C (a +C b)
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addInequalities {a} {b} record { e = eA ; 0<e = 0<eA ; N = Na ; pr = prA } record { e = eB ; 0<e = 0<eB ; N = Nb ; pr = prB } = record { e = min eA eB ; 0<e = minInequalitiesR 0<eA 0<eB ; N = N ; pr = λ m N<m → <Transitive (<WellDefined (symmetric identLeft) reflexive (minAddLemma eA eB 0<eA)) (<WellDefined (+WellDefined identLeft identLeft) (identityOfIndiscernablesLeft _∼_ reflexive (indexAndApply (CauchyCompletion.elts a) (CauchyCompletion.elts b) _+_ {m})) (ringAddInequalities (prA m (p2 m N<m)) (prB m (p1 m N<m)))) }
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where
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N = succ (TotalOrder.max ℕTotalOrder Na Nb)
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Nb<N : Nb <N succ (TotalOrder.max ℕTotalOrder Na Nb)
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Nb<N with TotalOrder.totality ℕTotalOrder Na Nb
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... | inl (inl x) = le 0 refl
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... | inl (inr x) = lessTransitive x (le 0 refl)
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... | inr x = le 0 refl
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Na<N : Na <N succ (TotalOrder.max ℕTotalOrder Na Nb)
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Na<N with TotalOrder.totality ℕTotalOrder Na Nb
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... | inl (inl x) = lessTransitive x (le 0 refl)
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... | inl (inr x) = le 0 refl
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... | inr x = le 0 (applyEquality succ x)
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p1 : (m : ℕ) → (N <N m) → Nb <N m
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p1 m N<m = lessTransitive Nb<N N<m
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p2 : (m : ℕ) → (N <N m) → Na <N m
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p2 m N<m = lessTransitive Na<N N<m
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-}
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invInj : (i : A) → Setoid._∼_ cauchyCompletionSetoid (injection (inverse i) +C injection i) (injection 0R)
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invInj i = Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (injection (inverse i) +C injection i) }} {record { converges = CauchyCompletion.converges ((-C (injection i)) +C injection i) }} {record { converges = CauchyCompletion.converges (injection 0R) }} (Group.+WellDefined CGroup {record { converges = CauchyCompletion.converges (injection (inverse i)) }} {record { converges = CauchyCompletion.converges (injection i)}} {record { converges = CauchyCompletion.converges (-C (injection i)) }} {record { converges = CauchyCompletion.converges (injection i) }} homRespectsInverse (Equivalence.reflexive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (injection i) }})) (Group.invLeft CGroup {injection i})
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invInj i = Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (injection (inverse i) +C injection i) }} {record { converges = CauchyCompletion.converges ((-C (injection i)) +C injection i) }} {record { converges = CauchyCompletion.converges (injection 0R) }} (Group.+WellDefinedLeft CGroup {record { converges = CauchyCompletion.converges (injection (inverse i)) }} {record { converges = CauchyCompletion.converges (injection i)}} {record { converges = CauchyCompletion.converges (-C (injection i)) }} homRespectsInverse) (Group.invLeft CGroup {injection i})
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cOrderMove : (a b : A) (c : CauchyCompletion) → (injection a +C c) <Cr b → a r<C (injection b +C (-C c))
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cOrderMove a b c record { e = e ; 0<e = 0<e ; N = N ; property = property } = record { e = e ; 0<e = 0<e ; N = N ; pr = λ m N<m → <WellDefined (transitive (symmetric +Associative) (transitive (+WellDefined reflexive (transitive (+WellDefined (identityOfIndiscernablesRight _∼_ reflexive (indexAndApply (CauchyCompletion.elts (injection a)) (CauchyCompletion.elts c) _+_ {m})) reflexive) (transitive (symmetric +Associative) (transitive (+WellDefined (identityOfIndiscernablesRight _∼_ reflexive (indexAndConst a m)) invRight) identRight)))) groupIsAbelian)) (transitive (+WellDefined (identityOfIndiscernablesLeft _∼_ reflexive (indexAndConst b m)) (identityOfIndiscernablesRight _∼_ reflexive (mapAndIndex (CauchyCompletion.elts c) inverse m))) (identityOfIndiscernablesLeft _∼_ reflexive (indexAndApply (CauchyCompletion.elts (injection b)) (CauchyCompletion.elts (-C c)) _+_ {m}))) (orderRespectsAddition (property m N<m) (inverse (index (CauchyCompletion.elts c) m))) }
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cOrderMove' : (a b : A) (c : CauchyCompletion) → (injection a +C (-C c)) <Cr b → a r<C (injection b +C c)
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cOrderMove' a b c pr = r<CWellDefinedRight _ _ _ (Group.+WellDefined CGroup {record { converges = CauchyCompletion.converges (injection b) }} {record { converges = CauchyCompletion.converges (-C (-C c)) }} {record { converges = CauchyCompletion.converges (injection b) }} {record {converges = CauchyCompletion.converges c }} (Equivalence.reflexive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (injection b) }}) (invTwice CGroup record { converges = CauchyCompletion.converges c })) (cOrderMove a b (-C c) pr)
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cOrderMove' a b c pr = r<CWellDefinedRight _ _ _ (Group.+WellDefinedRight CGroup {record { converges = CauchyCompletion.converges (injection b) }} {record { converges = CauchyCompletion.converges (-C (-C c)) }} {record {converges = CauchyCompletion.converges c }} (invTwice CGroup record { converges = CauchyCompletion.converges c })) (cOrderMove a b (-C c) pr)
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cOrderMove'' : (a : CauchyCompletion) (b c : A) → (a +C (-C injection b)) <Cr c → a <Cr (b + c)
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cOrderMove'' a b c record { e = e ; 0<e = 0<e ; N = N ; property = property } = record { e = e ; 0<e = 0<e ; N = N ; property = λ m N<m → <WellDefined (transitive (symmetric +Associative) (+WellDefined reflexive (transitive (+WellDefined (identityOfIndiscernablesRight _∼_ reflexive (indexAndApply (CauchyCompletion.elts a) _ _+_ {m})) reflexive) (transitive (transitive (symmetric +Associative) (+WellDefined reflexive (transitive (+WellDefined (transitive (identityOfIndiscernablesLeft _∼_ reflexive (mapAndIndex _ inverse m)) (inverseWellDefined additiveGroup (identityOfIndiscernablesRight _∼_ reflexive (indexAndConst b m)))) reflexive) invLeft))) identRight)))) groupIsAbelian (orderRespectsAddition (property m N<m) b) }
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@@ -157,7 +129,7 @@ private
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g' : ((injection aboveC +C (-C c)) +C injection a) <C (injection (b-a/2 + a))
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g' = <CWellDefined (Equivalence.reflexive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges ((injection aboveC +C (-C c)) +C injection a) }}) (Equivalence.symmetric (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (injection (b-a/2 + a)) }} {record { converges = CauchyCompletion.converges (injection b-a/2 +C injection a) }} (GroupHom.groupHom CInjectionGroupHom)) g
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t : (injection (a + aboveC) +C (-C c)) <Cr (b-a/2 + a)
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t = <CRelaxR' (<CWellDefined (Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (((injection aboveC) +C (-C c)) +C injection a) }} {record { converges = CauchyCompletion.converges (injection a +C ((injection aboveC) +C (-C c))) }} {record { converges = CauchyCompletion.converges (injection (a + aboveC) +C (-C c)) }} (Ring.groupIsAbelian CRing {record { converges = CauchyCompletion.converges (injection aboveC +C (-C c)) }} {record { converges = CauchyCompletion.converges (injection a) }}) (Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (injection a +C ((injection aboveC) +C (-C c))) }} {record { converges = CauchyCompletion.converges (((injection a) +C (injection aboveC)) +C (-C c)) }} {record { converges = CauchyCompletion.converges ((injection (a + aboveC)) +C (-C c)) }} (Group.+Associative CGroup {record { converges = CauchyCompletion.converges (injection a) }} {record { converges = CauchyCompletion.converges (injection aboveC) }} {record { converges = CauchyCompletion.converges (-C c) }}) (Group.+WellDefined CGroup {record { converges = CauchyCompletion.converges (injection a +C injection aboveC) }} {record { converges = CauchyCompletion.converges (-C c) }} {record { converges = CauchyCompletion.converges (injection (a + aboveC)) }} {record { converges = CauchyCompletion.converges (-C c) }} (Equivalence.symmetric (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (injection (a + aboveC)) }} {record { converges = CauchyCompletion.converges (injection a +C injection aboveC) }} (GroupHom.groupHom CInjectionGroupHom)) (Equivalence.reflexive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (-C c)}})))) (Equivalence.reflexive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (injection (b-a/2 + a)) }}) g')
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t = <CRelaxR' (<CWellDefined (Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (((injection aboveC) +C (-C c)) +C injection a) }} {record { converges = CauchyCompletion.converges (injection a +C ((injection aboveC) +C (-C c))) }} {record { converges = CauchyCompletion.converges (injection (a + aboveC) +C (-C c)) }} (Ring.groupIsAbelian CRing {record { converges = CauchyCompletion.converges (injection aboveC +C (-C c)) }} {record { converges = CauchyCompletion.converges (injection a) }}) (Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (injection a +C ((injection aboveC) +C (-C c))) }} {record { converges = CauchyCompletion.converges (((injection a) +C (injection aboveC)) +C (-C c)) }} {record { converges = CauchyCompletion.converges ((injection (a + aboveC)) +C (-C c)) }} (Group.+Associative CGroup {record { converges = CauchyCompletion.converges (injection a) }} {record { converges = CauchyCompletion.converges (injection aboveC) }} {record { converges = CauchyCompletion.converges (-C c) }}) (Group.+WellDefinedLeft CGroup {record { converges = CauchyCompletion.converges (injection a +C injection aboveC) }} {record { converges = CauchyCompletion.converges (-C c) }} {record { converges = CauchyCompletion.converges (injection (a + aboveC)) }} (Equivalence.symmetric (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (injection (a + aboveC)) }} {record { converges = CauchyCompletion.converges (injection a +C injection aboveC) }} (GroupHom.groupHom CInjectionGroupHom))))) (Equivalence.reflexive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (injection (b-a/2 + a)) }}) g')
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lemm : 0R < b-a/2
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lemm = halvePositive' prDiff (moveInequality a<b)
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u : (b-a/2 + a) < b
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@@ -167,16 +139,7 @@ private
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cOrderRespectsAdditionLeft''Flip a b c a<b = <CWellDefined ((Ring.groupIsAbelian CRing {record { converges = CauchyCompletion.converges (injection a) }} {record { converges = CauchyCompletion.converges c }})) (Ring.groupIsAbelian CRing {record { converges = CauchyCompletion.converges (injection b) }} {record { converges = CauchyCompletion.converges c }}) (cOrderRespectsAdditionLeft'' a b c a<b)
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cOrderRespectsAdditionLeft''' : (a b : CauchyCompletion) (c : A) → (a <C b) → (a +C injection c) <C (b +C injection c)
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cOrderRespectsAdditionLeft''' a b c record { i = i ; a<i = a<i ; i<b = i<b } = <CTransitive (cOrderRespectsAdditionLeft' a i c a<i) (<CWellDefined (Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (injection (c + i)) }} {record { converges = CauchyCompletion.converges (injection c +C injection i) }} {record { converges = CauchyCompletion.converges (injection i +C injection c) }} (GroupHom.groupHom CInjectionGroupHom) (Ring.groupIsAbelian CRing {record { converges = CauchyCompletion.converges (injection c) }} {record { converges = CauchyCompletion.converges (injection i) }})) (Ring.groupIsAbelian CRing {record { converges = CauchyCompletion.converges (injection c) }} {record { converges = CauchyCompletion.converges b }}) (flip<C' {record { converges = CauchyCompletion.converges (injection (c + i)) }} {record { converges = CauchyCompletion.converges (injection c +C b) }} (<CWellDefined (Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges ((-C b) +C injection (inverse c)) }} {record { converges = CauchyCompletion.converges ((-C b) +C (-C (injection c))) }} {record { converges = CauchyCompletion.converges (-C (injection c +C b)) }} (Group.+WellDefined CGroup {record { converges = CauchyCompletion.converges (-C b) }} {record { converges = CauchyCompletion.converges (injection (inverse c)) }} {record { converges = CauchyCompletion.converges (-C b) }} {record { converges = CauchyCompletion.converges (-C (injection c)) }} (Equivalence.reflexive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (-C b) }}) homRespectsInverse) (Equivalence.symmetric (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (-C (injection c +C b)) }} {record { converges = CauchyCompletion.converges ((-C b) +C (-C (injection c))) }} (invContravariant CGroup {record { converges = CauchyCompletion.converges (injection c) }} {record { converges = CauchyCompletion.converges b }}))) homRespectsInverse' (cOrderRespectsAdditionLeft' (-C b) (inverse i) (inverse c) (flipR<C i<b)))))
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{-
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Have 0<a, so 0 < a- < a < a+
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Have c < a- + c, by cOrderRespectsAdditionLeft''
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Have a- + c < a+ + c by cOrderRespectsAdditionLeft''
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so a- + c < K < a+ + c
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-}
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cOrderRespectsAdditionLeft''' a b c record { i = i ; a<i = a<i ; i<b = i<b } = <CTransitive (cOrderRespectsAdditionLeft' a i c a<i) (<CWellDefined (Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (injection (c + i)) }} {record { converges = CauchyCompletion.converges (injection c +C injection i) }} {record { converges = CauchyCompletion.converges (injection i +C injection c) }} (GroupHom.groupHom CInjectionGroupHom) (Ring.groupIsAbelian CRing {record { converges = CauchyCompletion.converges (injection c) }} {record { converges = CauchyCompletion.converges (injection i) }})) (Ring.groupIsAbelian CRing {record { converges = CauchyCompletion.converges (injection c) }} {record { converges = CauchyCompletion.converges b }}) (flip<C' {record { converges = CauchyCompletion.converges (injection (c + i)) }} {record { converges = CauchyCompletion.converges (injection c +C b) }} (<CWellDefined (Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges ((-C b) +C injection (inverse c)) }} {record { converges = CauchyCompletion.converges ((-C b) +C (-C (injection c))) }} {record { converges = CauchyCompletion.converges (-C (injection c +C b)) }} (Group.+WellDefinedRight CGroup {record { converges = CauchyCompletion.converges (-C b) }} {record { converges = CauchyCompletion.converges (injection (inverse c)) }} {record { converges = CauchyCompletion.converges (-C (injection c)) }} homRespectsInverse) (Equivalence.symmetric (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (-C (injection c +C b)) }} {record { converges = CauchyCompletion.converges ((-C b) +C (-C (injection c))) }} (invContravariant CGroup {record { converges = CauchyCompletion.converges (injection c) }} {record { converges = CauchyCompletion.converges b }}))) homRespectsInverse' (cOrderRespectsAdditionLeft' (-C b) (inverse i) (inverse c) (flipR<C i<b)))))
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cOrderRespectsAdditionRightZero : (a : CauchyCompletion) → (0R r<C a) → (c : CauchyCompletion) → c <C (a +C c)
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cOrderRespectsAdditionRightZero a record { e = e ; 0<e = 0<e ; N = N1 ; pr = pr } c with halve charNot2 e
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@@ -211,7 +174,7 @@ Have a- + c < a+ + c by cOrderRespectsAdditionLeft''
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open Equivalence (Setoid.eq cauchyCompletionSetoid) renaming (transitive to tr)
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cOrderRespectsAdditionRight : (a : A) (b : CauchyCompletion) (c : CauchyCompletion) → (a r<C b) → (injection a +C c) <C (b +C c)
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cOrderRespectsAdditionRight a b c a<b = <CWellDefined (Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (Group.inverse CGroup (injection (inverse a) +C (-C c))) }} {record { converges = CauchyCompletion.converges (Group.inverse CGroup ((-C injection a) +C (-C c))) }} {record { converges = CauchyCompletion.converges (injection a +C c) }} (inverseWellDefined CGroup {record { converges = CauchyCompletion.converges (injection (inverse a) +C (-C c)) }} {record { converges = CauchyCompletion.converges ((-C (injection a)) +C (-C c)) }} (Group.+WellDefined CGroup {record { converges = CauchyCompletion.converges (injection (inverse a)) }} {record { converges = CauchyCompletion.converges (-C c) }} {record { converges = CauchyCompletion.converges (-C (injection a)) }} {record { converges = CauchyCompletion.converges (-C c) }} homRespectsInverse (Equivalence.reflexive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (-C c) }}))) (lemma2 (injection a) c)) (lemma2 b c) (flip<C (cOrderRespectsAdditionLeft _ _ (Group.inverse CGroup c) (flipR<C a<b)))
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cOrderRespectsAdditionRight a b c a<b = <CWellDefined (Equivalence.transitive (Setoid.eq cauchyCompletionSetoid) {record { converges = CauchyCompletion.converges (Group.inverse CGroup (injection (inverse a) +C (-C c))) }} {record { converges = CauchyCompletion.converges (Group.inverse CGroup ((-C injection a) +C (-C c))) }} {record { converges = CauchyCompletion.converges (injection a +C c) }} (inverseWellDefined CGroup {record { converges = CauchyCompletion.converges (injection (inverse a) +C (-C c)) }} {record { converges = CauchyCompletion.converges ((-C (injection a)) +C (-C c)) }} (Group.+WellDefinedLeft CGroup {record { converges = CauchyCompletion.converges (injection (inverse a)) }} {record { converges = CauchyCompletion.converges (-C c) }} {record { converges = CauchyCompletion.converges (-C (injection a)) }} homRespectsInverse)) (lemma2 (injection a) c)) (lemma2 b c) (flip<C (cOrderRespectsAdditionLeft _ _ (Group.inverse CGroup c) (flipR<C a<b)))
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cOrderRespectsAddition : (a b : CauchyCompletion) → (a <C b) → (c : CauchyCompletion) → (a +C c) <C (b +C c)
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cOrderRespectsAddition a b record { i = i ; a<i = a<i ; i<b = i<b } c = SetoidPartialOrder.<Transitive <COrder (cOrderRespectsAdditionLeft a i c a<i) (cOrderRespectsAdditionRight i b c i<b)
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