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Split out Integers and remove use of K (#39)

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Patrick Stevens
2019-08-22 08:37:41 +01:00
committed by GitHub
parent 26f20b2025
commit 9c14e7c439
15 changed files with 462 additions and 1583 deletions
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11
Groups/LectureNotes/Lecture1.agda
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@@ -5,13 +5,14 @@ open import Setoids.Setoids
open import Functions
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
open import Numbers.Naturals.Naturals
open import Numbers.Integers
open import Numbers.Integers.Integers
open import Numbers.Rationals
open import Sets.FinSet
open import Groups.Definition
open import Groups.Groups
open import Rings.Definition
open import IntegersModN
open import Semirings.Definition
module Groups.LectureNotes.Lecture1 where
@@ -27,13 +28,19 @@ module Groups.LectureNotes.Lecture1 where
integersMinusNotGroup record { wellDefined = wellDefined ; identity = identity ; inverse = inverse ; multAssoc = multAssoc ; multIdentRight = multIdentRight ; multIdentLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } with multAssoc {nonneg 3} {nonneg 2} {nonneg 1}
integersMinusNotGroup record { wellDefined = wellDefined ; identity = identity ; inverse = inverse ; multAssoc = multAssoc ; multIdentRight = multIdentRight ; multIdentLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } | ()
negSuccInjective : {a b : ℕ} → (negSucc a ≡ negSucc b) → a ≡ b
negSuccInjective {a} {.a} refl = refl
nonnegInjective : {a b : ℕ} → (nonneg a ≡ nonneg b) → a ≡ b
nonnegInjective {a} {.a} refl = refl
integersTimesNotGroup : Group (reflSetoid ℤ) (_*Z_) → False
integersTimesNotGroup record { wellDefined = wellDefined ; identity = (nonneg zero) ; inverse = inverse ; multAssoc = multAssoc ; multIdentRight = multIdentRight ; multIdentLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } with multIdentLeft {negSucc 1}
... | ()
integersTimesNotGroup record { wellDefined = wellDefined ; identity = (nonneg (succ zero)) ; inverse = inverse ; multAssoc = multAssoc ; multIdentRight = multIdentRight ; multIdentLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } with invLeft {nonneg zero}
... | bl with inverse (nonneg zero)
integersTimesNotGroup record { wellDefined = wellDefined ; identity = (nonneg (succ zero)) ; inverse = inverse ; multAssoc = multAssoc ; multIdentRight = multIdentRight ; multIdentLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } | () | nonneg zero
integersTimesNotGroup record { wellDefined = wellDefined ; identity = (nonneg (succ zero)) ; inverse = inverse ; multAssoc = multAssoc ; multIdentRight = multIdentRight ; multIdentLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } | () | nonneg (succ x)
integersTimesNotGroup record { wellDefined = wellDefined ; identity = (nonneg (succ zero)) ; inverse = inverse ; multAssoc = multAssoc ; multIdentRight = multIdentRight ; multIdentLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } | p | nonneg (succ x) = naughtE (nonnegInjective (transitivity (applyEquality nonneg (equalityCommutative (Semiring.productZeroRight ℕSemiring x))) p))
integersTimesNotGroup record { wellDefined = wellDefined ; identity = (nonneg (succ zero)) ; inverse = inverse ; multAssoc = multAssoc ; multIdentRight = multIdentRight ; multIdentLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } | () | negSucc x
integersTimesNotGroup record { wellDefined = wellDefined ; identity = (nonneg (succ (succ x))) ; inverse = inverse ; multAssoc = multAssoc ; multIdentRight = multIdentRight ; multIdentLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } with succInjective (negSuccInjective (multIdentLeft {negSucc 1}))
... | ()