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Mostly show sqrt 2 is irrational (#53)

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Patrick Stevens
2019-10-22 20:56:58 +01:00
committed by GitHub
parent 6eaa181104
commit f5f4cf1376
8 changed files with 244 additions and 37 deletions
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35
Rings/Lemmas.agda
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@@ -10,28 +10,33 @@ open import Setoids.Setoids
open import Setoids.Orders
open import Sets.EquivalenceRelations
module Rings.Lemmas where
module Rings.Lemmas {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} (R : Ring S _+_ _*_) where
ringMinusExtracts : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} (R : Ring S _+_ _*_) → {x y : A} → Setoid._∼_ S (x * Group.inverse (Ring.additiveGroup R) y) (Group.inverse (Ring.additiveGroup R) (x * y))
ringMinusExtracts {S = S} {_+_ = _+_} {_*_ = _*_} R {x = x} {y} = transferToRight' additiveGroup (transitive (symmetric *DistributesOver+) (transitive (*WellDefined reflexive invLeft) (Ring.timesZero R)))
open Setoid S
open Ring R
open Group additiveGroup
ringMinusExtracts : {x y : A} → Setoid._∼_ S (x * Group.inverse (Ring.additiveGroup R) y) (Group.inverse (Ring.additiveGroup R) (x * y))
ringMinusExtracts {x = x} {y} = transferToRight' additiveGroup (transitive (symmetric *DistributesOver+) (transitive (*WellDefined reflexive invLeft) (Ring.timesZero R)))
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 = x} {y} = transitive *Commutative (transitive ringMinusExtracts (inverseWellDefined additiveGroup *Commutative))
where
open Equivalence eq
twoNegativesTimes : {a b : A} → (inverse a) * (inverse b) ∼ a * b
twoNegativesTimes {a} {b} = transitive (ringMinusExtracts) (transitive (inverseWellDefined additiveGroup ringMinusExtracts') (invTwice additiveGroup (a * b)))
where
open Setoid S
open Equivalence eq
open Ring R
open Group additiveGroup
groupLemmaMove0G : {a b : _} → {A : Set a} → {_·_ : A → A → A} → {S : Setoid {a} {b} A} → (G : Group S _·_) → {x : A} → (Setoid._∼_ S (Group.0G G) (Group.inverse G x)) → Setoid._∼_ S x (Group.0G G)
groupLemmaMove0G {S = S} G {x} pr = transitive (symmetric (invInv G)) (transitive (symmetric p) (invIdent G))
where
open Group G
open Setoid S
open Equivalence eq
p : inverse 0G ∼ inverse (inverse x)
open Equivalence (Setoid.eq S)
p : Setoid._∼_ S (Group.inverse G (Group.0G G)) (Group.inverse G (Group.inverse G x))
p = inverseWellDefined G pr
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 (transitive identLeft pr)
where
open Group G
open Setoid S
open Equivalence eq
groupLemmaMove0G' {S = S} G {x} pr = transferToRight' G (Equivalence.transitive (Setoid.eq S) (Group.identLeft G) pr)