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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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Fields/CauchyCompletion/Addition.agda
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@@ -47,7 +47,7 @@ CauchyCompletion.converges (record { elts = a ; converges = convA } +C record {
t {m} {n} <m <n with prA {m} {n} (inequalityShrinkLeft <m) (inequalityShrinkLeft <n)
... | am-an<e/2 with prB {m} {n} (inequalityShrinkRight <m) (inequalityShrinkRight <n)
... | bm-bn<e/2 with triangleInequality (index a m + inverse (index a n)) (index b m + inverse (index b n))
... | inl tri rewrite lemm m a b | lemm n a b = SetoidPartialOrder.<WellDefined pOrder (Equivalence.reflexive eq) e/2Pr (SetoidPartialOrder.transitive pOrder {_} {(abs ((index a m) + (inverse (index a n)))) + (abs ((index b m) + (inverse (index b n))))} (<WellDefined (absWellDefined _ _ (Equivalence.transitive eq (Equivalence.symmetric eq (+Associative {index a m})) (Equivalence.transitive eq (+WellDefined (Equivalence.reflexive eq {index a m}) (Equivalence.transitive eq groupIsAbelian (Equivalence.transitive eq (Equivalence.symmetric eq (+Associative {index b m})) (+WellDefined (Equivalence.reflexive eq {index b m}) (Equivalence.symmetric eq (invContravariant additiveGroup)))))) (+Associative {index a m})))) (Equivalence.reflexive eq) tri) (ringAddInequalities am-an<e/2 bm-bn<e/2))
... | inl tri rewrite lemm m a b | lemm n a b = SetoidPartialOrder.<WellDefined pOrder (Equivalence.reflexive eq) e/2Pr (SetoidPartialOrder.<Transitive pOrder {_} {(abs ((index a m) + (inverse (index a n)))) + (abs ((index b m) + (inverse (index b n))))} (<WellDefined (absWellDefined _ _ (Equivalence.transitive eq (Equivalence.symmetric eq (+Associative {index a m})) (Equivalence.transitive eq (+WellDefined (Equivalence.reflexive eq {index a m}) (Equivalence.transitive eq groupIsAbelian (Equivalence.transitive eq (Equivalence.symmetric eq (+Associative {index b m})) (+WellDefined (Equivalence.reflexive eq {index b m}) (Equivalence.symmetric eq (invContravariant additiveGroup)))))) (+Associative {index a m})))) (Equivalence.reflexive eq) tri) (ringAddInequalities am-an<e/2 bm-bn<e/2))
... | inr tri rewrite lemm m a b | lemm n a b = SetoidPartialOrder.<WellDefined pOrder (Equivalence.reflexive eq) e/2Pr (<WellDefined (Equivalence.transitive eq (Equivalence.symmetric eq tri) (absWellDefined _ _ (Equivalence.transitive eq (Equivalence.symmetric eq (+Associative {index a m})) (Equivalence.transitive eq (+WellDefined (Equivalence.reflexive eq {index a m}) (Equivalence.transitive eq groupIsAbelian (Equivalence.transitive eq (Equivalence.symmetric eq (+Associative {index b m})) (+WellDefined (Equivalence.reflexive eq {index b m}) (Equivalence.symmetric eq (invContravariant additiveGroup)))))) (+Associative {index a m}))))) (Equivalence.reflexive eq) (ringAddInequalities am-an<e/2 bm-bn<e/2))
inverseDistributes : {r s : A} → inverse (r + s) ∼ inverse r + inverse s