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Lots of without-K (#110)

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Patrick Stevens
2020-04-11 12:14:03 +01:00
committed by GitHub
parent 412edaf4c7
commit e9aa1bcc05
30 changed files with 424 additions and 98 deletions
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13
Fields/Orders/Limits/Definition.agda
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@@ -54,6 +54,19 @@ abstract
halfHalves : {a : A} → (1/2 * (a + a)) ∼ a
halfHalves {a} = transitive (*WellDefined reflexive (+WellDefined (symmetric identIsIdent) (symmetric identIsIdent))) (transitive (*WellDefined reflexive (symmetric *DistributesOver+')) (transitive *Associative (transitive (*WellDefined 1/2is1/2 reflexive) identIsIdent)))
abstract
bothNearImpliesNear : {t r s : A} (e : A) .(0<e : 0R < e) → (abs (r + inverse t) < e) → (abs (s + inverse t) < e) → abs (r + inverse s) < (e + e)
bothNearImpliesNear {t} {r} {s} e 0<e rNearT sNearT = u
where
pr : ((abs (r + inverse t)) + (abs (s + inverse t))) < (e + e)
pr = ringAddInequalities rNearT sNearT
t' : (abs (r + inverse t) + abs (t + inverse s)) < (e + e)
t' = <WellDefined (+WellDefined reflexive (transitive (absNegation _) (absWellDefined _ _ (transitive invContravariant (+WellDefined (invTwice _) reflexive))))) reflexive pr
u : abs (r + inverse s) < (e + e)
u with triangleInequality (r + inverse t) (t + inverse s)
... | inl t'' = <Transitive (<WellDefined (absWellDefined _ _ (transitive +Associative (+WellDefined (transitive (symmetric +Associative) (transitive (+WellDefined reflexive invLeft) identRight)) reflexive))) reflexive t'') t'
... | inr eq = <WellDefined (transitive (symmetric eq) (absWellDefined _ _ (transitive +Associative (+WellDefined (transitive (symmetric +Associative) (transitive (+WellDefined reflexive invLeft) identRight)) reflexive)))) reflexive t'
private
limitsUniqueLemma : (x : Sequence A) {r s : A} (xr : x ~> r) (xs : x ~> s) (r<s : r < s) → False
limitsUniqueLemma x {r} {s} xr xs r<s = irreflexive (<WellDefined reflexive (symmetric prE) u')