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Comparison on the reals (#55)

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Patrick Stevens
2019-10-26 10:36:24 +01:00
committed by GitHub
parent 4e56b68024
commit 0d68919127
18 changed files with 1343 additions and 691 deletions
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5
Rings/Lemmas.agda
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@@ -21,7 +21,7 @@ ringMinusExtracts {x = x} {y} = transferToRight' additiveGroup (transitive (symm
where
open Equivalence eq
ringMinusExtracts' : {x y : A} → Setoid._∼_ S ((Group.inverse (Ring.additiveGroup R) x) * y) (Group.inverse (Ring.additiveGroup R) (x * y))
ringMinusExtracts' : {x y : A} → ((inverse x) * y) ∼ inverse (x * y)
ringMinusExtracts' {x = x} {y} = transitive *Commutative (transitive ringMinusExtracts (inverseWellDefined additiveGroup *Commutative))
where
open Equivalence eq
@@ -40,3 +40,6 @@ groupLemmaMove0G {S = S} G {x} pr = transitive (symmetric (invInv G)) (transitiv
groupLemmaMove0G' : {a b : _} → {A : Set a} → {_·_ : A → A → A} → {S : Setoid {a} {b} A} → (G : Group S _·_) → {x : A} → Setoid._∼_ S x (Group.0G G) → (Setoid._∼_ S (Group.0G G) (Group.inverse G x))
groupLemmaMove0G' {S = S} G {x} pr = transferToRight' G (Equivalence.transitive (Setoid.eq S) (Group.identLeft G) pr)
oneZeroImpliesAllZero : 0R ∼ 1R → {x : A} → x ∼ 0R
oneZeroImpliesAllZero 0=1 = Equivalence.transitive eq (Equivalence.symmetric eq identIsIdent) (Equivalence.transitive eq (*WellDefined (Equivalence.symmetric eq 0=1) (Equivalence.reflexive eq)) (Equivalence.transitive eq *Commutative timesZero))