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Elevate the real numbers to actually existing (#65)

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Patrick Stevens
2019-11-04 07:34:59 +00:00
committed by GitHub
parent d95f510cdd
commit 82b81b4bab
5 changed files with 55 additions and 49 deletions
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19
Fields/FieldOfFractions/Field.agda
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@@ -24,15 +24,16 @@ open import Fields.FieldOfFractions.Ring I
fieldOfFractions : Field fieldOfFractionsRing
Field.allInvertible fieldOfFractions (fst ,, (b , _)) prA = (b ,, (fst , ans)) , need
where
open Setoid S
open Equivalence eq
need : ((b * fst) * Ring.1R R) ∼ ((fst * b) * Ring.1R R)
need = Ring.*WellDefined R (Ring.*Commutative R) reflexive
ans : fst ∼ Ring.0R R → False
ans pr = prA need'
where
need' : (fst * Ring.1R R) ∼ (b * Ring.0R R)
need' = transitive (Ring.*WellDefined R pr reflexive) (transitive (transitive (Ring.*Commutative R) (Ring.timesZero R)) (symmetric (Ring.timesZero R)))
abstract
open Setoid S
open Equivalence eq
need : ((b * fst) * Ring.1R R) ∼ ((fst * b) * Ring.1R R)
need = Ring.*WellDefined R (Ring.*Commutative R) reflexive
ans : fst ∼ Ring.0R R → False
ans pr = prA need'
where
need' : (fst * Ring.1R R) ∼ (b * Ring.0R R)
need' = transitive (Ring.*WellDefined R pr reflexive) (transitive (transitive (Ring.*Commutative R) (Ring.timesZero R)) (symmetric (Ring.timesZero R)))
Field.nontrivial fieldOfFractions pr = IntegralDomain.nontrivial I (symmetric (transitive (symmetric (Ring.timesZero R)) (transitive (Ring.*Commutative R) (transitive pr (Ring.identIsIdent R)))))
where
open Setoid S