More lecture notes (#126)

2025-12-15 19:45:40 +00:00 · 2020-04-19 20:15:02 +01:00
parent 485b27e009
commit cab004f6d8
11 changed files with 163 additions and 36 deletions
--- a/Everything/Guardedness.agda
+++ b/Everything/Guardedness.agda
@@ -14,4 +14,7 @@ open import Sets.Cardinality.Infinite.Examples

 open import ProjectEuler.Problem2

+open import LectureNotes.NumbersAndSets.Examples1
+open import LectureNotes.Groups.Lecture1
+
 module Everything.Guardedness where
--- a/Everything/Safe.agda
+++ b/Everything/Safe.agda
@@ -150,7 +150,6 @@ open import Graphs.InducedSubgraph

 open import LectureNotes.MetAndTop.Chapter1
 open import LectureNotes.NumbersAndSets.Lecture1
-open import LectureNotes.Groups.Lecture1

 open import Modules.Examples
 open import Modules.PolynomialModule
--- a/Fields/Lemmas.agda
+++ b/Fields/Lemmas.agda
@@ -8,6 +8,7 @@ open import Sets.EquivalenceRelations
 open import LogicalFormulae
 open import Rings.IntegralDomains.Definition
 open import Rings.IntegralDomains.Lemmas
+open import Setoids.Subset

 module Fields.Lemmas {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} (F : Field R) where

@@ -30,13 +31,30 @@ abstract
  halfHalves : {x : A} (1/2 : A) (pr : 1/2 + 1/2 ∼ 1R) → (x + x) * 1/2 ∼ x
  halfHalves {x} 1/2 pr = transitive (transitive (transitive *Commutative (transitive (transitive *DistributesOver+ (transitive (+WellDefined *Commutative *Commutative) (symmetric *DistributesOver+))) *Commutative)) (*WellDefined pr (reflexive))) identIsIdent

-fieldIsIntDom : (Setoid._∼_ S (Ring.1R R) (Ring.0R R) → False) → IntegralDomain R
-IntegralDomain.intDom (fieldIsIntDom 1!=0) {a} {b} ab=0 a!=0 with Field.allInvertible F a a!=0
-IntegralDomain.intDom (fieldIsIntDom _) {a} {b} ab=0 a!=0 | 1/a , prA = transitive (symmetric identIsIdent) (transitive (*WellDefined (symmetric prA) reflexive) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive ab=0) (Ring.timesZero R))))
-IntegralDomain.nontrivial (fieldIsIntDom 1!=0) = 1!=0
+fieldIsIntDom : IntegralDomain R
+IntegralDomain.intDom fieldIsIntDom {a} {b} ab=0 a!=0 with Field.allInvertible F a a!=0
+IntegralDomain.intDom fieldIsIntDom {a} {b} ab=0 a!=0 | 1/a , prA = transitive (symmetric identIsIdent) (transitive (*WellDefined (symmetric prA) reflexive) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive ab=0) (Ring.timesZero R))))
+IntegralDomain.nontrivial fieldIsIntDom 1=0 = Field.nontrivial F (symmetric 1=0)

 allInvertibleWellDefined : {a b : A} {a!=0 : (a ∼ 0F) → False} {b!=0 : (b ∼ 0F) → False} → (a ∼ b) → underlying (allInvertible a a!=0) ∼ underlying (allInvertible b b!=0)
 allInvertibleWellDefined {a} {b} {a!=0} {b!=0} a=b with allInvertible a a!=0
 ... | x , prX with allInvertible b b!=0
 ... | y , prY with transitive (transitive prX (symmetric prY)) (*WellDefined reflexive (symmetric a=b))
-... | xa=ya = cancelIntDom (fieldIsIntDom λ p → a!=0 (oneZeroImpliesAllZero (symmetric p))) (transitive *Commutative (transitive xa=ya *Commutative)) a!=0
+... | xa=ya = cancelIntDom fieldIsIntDom (transitive *Commutative (transitive xa=ya *Commutative)) a!=0
+
+private
+  mulNonzeros : Sg A (λ m → (Setoid._∼_ S m (Ring.0R R)) → False) → Sg A (λ m → (Setoid._∼_ S m (Ring.0R R)) → False) → Sg A (λ m → (Setoid._∼_ S m (Ring.0R R)) → False)
+  mulNonzeros (a , a!=0) (b , b!=0) = (a * b) , λ ab=0 → b!=0 (IntegralDomain.intDom (fieldIsIntDom) ab=0 a!=0)
+
+fieldMultiplicativeGroup : Group (subsetSetoid S {pred = λ m → ((Setoid._∼_ S m (Ring.0R R)) → False)}(λ {x} {y} x=y x!=0 → λ y=0 → x!=0 (Equivalence.transitive (Setoid.eq S) x=y y=0))) (mulNonzeros)
+Group.+WellDefined (fieldMultiplicativeGroup) {x , prX} {y , prY} {z , prZ} {w , prW} = Ring.*WellDefined R
+Group.0G (fieldMultiplicativeGroup) = Ring.1R R , λ 1=0 → Field.nontrivial F (Equivalence.symmetric (Setoid.eq S) 1=0)
+Group.inverse (fieldMultiplicativeGroup) (x , pr) with Field.allInvertible F x pr
+... | 1/x , pr1/x = 1/x , λ 1/x=0 → Field.nontrivial F (Equivalence.transitive (Setoid.eq S) (Equivalence.symmetric (Setoid.eq S) (Equivalence.transitive (Setoid.eq S) (Ring.*WellDefined R 1/x=0 (Equivalence.reflexive (Setoid.eq S))) (Ring.timesZero' R))) pr1/x)
+Group.+Associative (fieldMultiplicativeGroup) {x , prX} {y , prY} {z , prZ} = Ring.*Associative R
+Group.identRight (fieldMultiplicativeGroup) {x , prX} = Ring.identIsIdent' R
+Group.identLeft (fieldMultiplicativeGroup) {x , prX} = Ring.identIsIdent R
+Group.invLeft (fieldMultiplicativeGroup) {x , prX} with Field.allInvertible F x prX
+... | 1/x , pr1/x = pr1/x
+Group.invRight (fieldMultiplicativeGroup) {x , prX} with Field.allInvertible F x prX
+... | 1/x , pr1/x = Equivalence.transitive (Setoid.eq S) (Ring.*Commutative R) pr1/x
--- a/LectureNotes/Groups/Lecture1.agda
+++ b/LectureNotes/Groups/Lecture1.agda
@@ -1,5 +1,6 @@
-{-# OPTIONS --warning=error --safe --without-K #-}
+{-# OPTIONS --warning=error --safe --without-K --guardedness #-}

+open import Sets.EquivalenceRelations
 open import Functions.Definition
 open import Functions.Lemmas
 open import Sets.FinSet.Definition
@@ -15,11 +16,17 @@ open import Numbers.Naturals.Semiring
 open import Numbers.Naturals.Order
 open import Numbers.Integers.Integers
 open import Numbers.Rationals.Definition
+open import Numbers.Reals.Definition
 open import Rings.Definition
 open import Fields.FieldOfFractions.Setoid
 open import Semirings.Definition
 open import Numbers.Modulo.Definition
 open import Numbers.Modulo.Group
+open import Fields.Fields
+open import Setoids.Subset
+open import Rings.IntegralDomains.Definition
+open import Fields.Lemmas
+open import Rings.Examples.Examples

 module LectureNotes.Groups.Lecture1 where

@@ -28,9 +35,12 @@ module LectureNotes.Groups.Lecture1 where
 groupExample1 : Group (reflSetoid ℤ) (_+Z_)
 groupExample1 = ℤGroup

-groupExample2 : Group (fieldOfFractionsSetoid ℤIntDom) (_+Q_)
+groupExample2 : Group ℚSetoid (_+Q_)
 groupExample2 = Ring.additiveGroup ℚRing

+groupExample2' : Group ℝSetoid (_+R_)
+groupExample2' = Ring.additiveGroup ℝRing
+
 groupExample3 : Group (reflSetoid ℤ) (_-Z_) → False
 groupExample3 record { +Associative = multAssoc } with multAssoc {nonneg 3} {nonneg 2} {nonneg 1}
 groupExample3 record { +WellDefined = wellDefined } | ()
@@ -42,20 +52,14 @@ 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 ; 0G = (nonneg zero) ; inverse = inverse ; +Associative = multAssoc ; identRight = multIdentRight ; identLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } with multIdentLeft {negSucc 1}
-... | ()
-integersTimesNotGroup record { +WellDefined = wellDefined ; 0G = (nonneg (succ zero)) ; inverse = inverse ; +Associative = multAssoc ; identRight = multIdentRight ; identLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } with invLeft {nonneg zero}
-... | bl with inverse (nonneg zero)
-integersTimesNotGroup record { +WellDefined = wellDefined ; 0G = (nonneg (succ zero)) ; inverse = inverse ; +Associative = multAssoc ; identRight = multIdentRight ; identLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } | () | nonneg zero
-integersTimesNotGroup record { +WellDefined = wellDefined ; 0G = (nonneg (succ zero)) ; inverse = inverse ; +Associative = multAssoc ; identRight = multIdentRight ; identLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } | p | nonneg (succ x) = naughtE (nonnegInjective (transitivity (applyEquality nonneg (equalityCommutative (Semiring.productZeroRight ℕSemiring x))) p))
-integersTimesNotGroup record { +WellDefined = wellDefined ; 0G = (nonneg (succ zero)) ; inverse = inverse ; +Associative = multAssoc ; identRight = multIdentRight ; identLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } | () | negSucc x
-integersTimesNotGroup record { +WellDefined = wellDefined ; 0G = (nonneg (succ (succ x))) ; inverse = inverse ; +Associative = multAssoc ; identRight = multIdentRight ; identLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } with succInjective (negSuccInjective (multIdentLeft {negSucc 1}))
-... | ()
-integersTimesNotGroup record { +WellDefined = wellDefined ; 0G = (negSucc x) ; inverse = inverse ; +Associative = multAssoc ; identRight = multIdentRight ; identLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } with multIdentLeft {nonneg 2}
-integersTimesNotGroup record { +WellDefined = wellDefined ; 0G = (negSucc x) ; inverse = inverse ; +Associative = multAssoc ; identRight = multIdentRight ; identLeft = multIdentLeft ; invLeft = invLeft ; invRight = invRight } | ()
+integersTimesNotGroup = multiplicationNotGroup ℤRing λ ()
+
+rationalsTimesNotGroup : Group ℚSetoid (_*Q_) → False
+rationalsTimesNotGroup = multiplicationNotGroup ℚRing λ ()
+
+QNonzeroGroup : Group _ _
+QNonzeroGroup = fieldMultiplicativeGroup ℚField

-- TODO: Q is not a group with *Q
-- TODO: Q without 0 is a group with *Q
 -- TODO: {1, -1} is a group with *

 ℤnIsGroup : (n : ℕ) → (0<n : 0 <N n) → _
--- a/LectureNotes/NumbersAndSets/Examples1.agda
+++ b/LectureNotes/NumbersAndSets/Examples1.agda
@@ -0,0 +1,97 @@
+{-# OPTIONS --warning=error --safe --without-K --guardedness #-}
+
+open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
+open import LogicalFormulae
+open import Numbers.Integers.Integers
+open import Numbers.Naturals.Semiring
+open import Numbers.Naturals.Subtraction
+open import Numbers.Naturals.Naturals
+open import Numbers.Naturals.Order
+open import Numbers.Naturals.Exponentiation
+open import Numbers.Primes.PrimeNumbers
+open import Maybe
+open import Semirings.Definition
+open import Numbers.Naturals.EuclideanAlgorithm
+open import Sequences
+open import Vectors
+
+module LectureNotes.NumbersAndSets.Examples1 where
+
+open Semiring ℕSemiring
+open import Semirings.Solver ℕSemiring multiplicationNIsCommutative
+
+private
+  abstract
+    q1Lemma1 : {q r : ℕ} → 3 *N succ q +N r ≡ succ (succ (succ (3 *N q +N r)))
+    q1Lemma1 {q} {r} rewrite sumZeroRight q | commutative q (succ q) | commutative q (succ (succ (q +N q))) | +Associative q q q = refl
+
+    q1Lemma : {n : ℕ} → (2 <N n) → ((3 ∣ n) || (3 ∣ (n +N 2))) || (3 ∣ (n +N 4))
+    q1Lemma {zero} ()
+    q1Lemma {succ zero} (le zero ())
+    q1Lemma {succ zero} (le (succ x) ())
+    q1Lemma {succ (succ zero)} (le (succ x) proof) rewrite Semiring.commutative ℕSemiring x 2 = exFalso (naughtE (equalityCommutative (succInjective (succInjective proof))))
+    q1Lemma {succ (succ (succ zero))} 2<n = inl (inl (aDivA 3))
+    q1Lemma {succ (succ (succ (succ zero)))} 2<n = inl (inr (divides (record { quot = 2 ; rem = zero ; pr = refl ; remIsSmall = inl (le 2 refl) ; quotSmall = inl (le 2 refl)}) refl))
+    q1Lemma {succ (succ (succ (succ (succ zero))))} 2<n = inr (divides (record { quot = 3 ; rem = zero ; pr = refl ; remIsSmall = inl (le 2 refl) ; quotSmall = inl (le 2 refl)}) refl)
+    q1Lemma {succ (succ (succ (succ (succ (succ n)))))} 2<n with q1Lemma {succ (succ (succ n))} (le n (applyEquality succ (Semiring.commutative ℕSemiring n 2)))
+    q1Lemma {succ (succ (succ (succ (succ (succ n)))))} 2<n | inl (inl (divides record { quot = quot ; rem = rem ; pr = pr ; remIsSmall = remIsSmall ; quotSmall = quotSmall } x)) = inl (inl (divides (record { quot = succ quot ; rem = rem ; pr = transitivity (q1Lemma1 {quot} {rem}) (applyEquality (λ x → succ (succ (succ x))) pr) ; remIsSmall = remIsSmall ; quotSmall = inl (le 2 refl) }) x))
+    q1Lemma {succ (succ (succ (succ (succ (succ n)))))} 2<n | inl (inr (divides record { quot = quot ; rem = rem ; pr = pr ; remIsSmall = remIsSmall ; quotSmall = quotSmall } x)) = inl (inr (divides record { quot = succ quot ; rem = rem ; pr = transitivity (q1Lemma1 {quot} {rem}) (applyEquality (λ x → succ (succ (succ x))) pr) ; remIsSmall = remIsSmall ; quotSmall = inl (le 2 refl) } x))
+    q1Lemma {succ (succ (succ (succ (succ (succ n)))))} 2<n | inr (divides record { quot = quot ; rem = rem ; pr = pr ; remIsSmall = remIsSmall ; quotSmall = quotSmall } x) = inr (divides record { quot = succ quot ; rem = rem ; pr = transitivity (q1Lemma1 {quot} {rem}) (applyEquality (λ x → succ (succ (succ x))) pr) ; remIsSmall = remIsSmall ; quotSmall = inl (le 2 refl) } x)
+
+q1 : {n : ℕ} → Prime n → Prime (n +N 2) → Prime (n +N 4) → n ≡ 3
+q1 {succ zero} record { p>1 = (le zero ()) ; pr = pr } pN+2 pN+4
+q1 {succ zero} record { p>1 = (le (succ x) ()) ; pr = pr } pN+2 pN+4
+q1 {succ (succ zero)} pN record { p>1 = p>1 ; pr = pr } pN+4 with pr {2} (divides (record { quot = 2 ; rem = zero ; pr = refl ; remIsSmall = inl (le 1 refl) ; quotSmall = inl (le 1 refl)}) refl) (le 1 refl) (le 1 refl)
+q1 {succ (succ zero)} pN record { p>1 = p>1 ; pr = pr } pN+4 | ()
+q1 {succ (succ (succ n))} pN pN+2 pN+4 with q1Lemma {succ (succ (succ n))} (le n (applyEquality succ (commutative n 2)))
+q1 {succ (succ (succ n))} pN pN+2 pN+4 | inl (inl 3|n) = equalityCommutative (primeDivPrimeImpliesEqual threeIsPrime pN 3|n)
+q1 {succ (succ (succ n))} pN pN+2 pN+4 | inl (inr 3|n+2) with primeDivPrimeImpliesEqual threeIsPrime pN+2 3|n+2
+... | bl rewrite commutative n 2 = exFalso (naughtE (succInjective (succInjective (succInjective bl))))
+q1 {succ (succ (succ n))} pN pN+2 pN+4 | inr 3|n+4 with primeDivPrimeImpliesEqual threeIsPrime pN+4 3|n+4
+... | bl rewrite commutative n 4 = exFalso (naughtE (succInjective (succInjective (succInjective bl))))
+
+q3Differences : ℕ → ℕ
+q3Differences a = 2 *N a
+q3Seq : (start : ℕ) → ℕ → ℕ
+q3Seq start zero = start
+q3Seq start (succ n) = q3Differences (succ n) +N q3Seq start n
+q3SeqFirst : take 5 (funcToSequence (q3Seq 41)) ≡ 41 ,- 43 ,- 47 ,- 53 ,- 61 ,- []
+q3SeqFirst = refl
+q3N : (start : ℕ) (a : ℕ) → q3Seq start a ≡ start +N (a +N (a *N a))
+q3N start zero = equalityCommutative (sumZeroRight start)
+q3N start (succ a) rewrite q3N start a | sumZeroRight a =
+  from (succ (plus (plus (const a) (succ (const a))) (plus (const start) (plus (const a) (times (const a) (const a))))))
+  to (plus (const start) (succ (plus (const a) (succ (plus (const a) (times (const a) (succ (const a))))))))
+  by applyEquality (λ i → succ (succ i)) (transitivity (+Associative (a +N a) start _) (transitivity (transitivity (applyEquality (_+N (a +N a *N a)) (commutative (a +N a) start)) (equalityCommutative (+Associative start _ _))) (applyEquality (start +N_) (equalityCommutative (+Associative a a (a +N a *N a))))))
+q3NDivision : (start : ℕ) → start ∣ (q3Seq start start)
+q3NDivision 0 = aDivA 0
+q3NDivision (succ start) rewrite q3N (succ start) (succ start) = dividesBothImpliesDividesSum {succ start} {succ start} (aDivA (succ start)) (dividesBothImpliesDividesSum {succ start} {succ start} (aDivA (succ start)) (divides (record { quot = succ start ; rem = 0 ; pr = sumZeroRight _ ; remIsSmall = inl (succIsPositive start) ; quotSmall = inl (succIsPositive start) }) refl))
+
+q3 : (start : ℕ) → ((n : ℕ) → Prime (q3Seq start n)) → False
+q3 zero a with a 2
+... | record { p>1 = p>1 ; pr = pr } with pr {2} (divides (record { quot = 3 ; rem = 0 ; pr = refl ; remIsSmall = inl (le 1 refl) ; quotSmall = inl (le 1 refl) }) refl) (le 3 refl) (succIsPositive 1)
+... | ()
+q3 (succ zero) a with a 0
+... | record { p>1 = le zero () ; pr = pr }
+... | record { p>1 = le (succ x) () ; pr = pr }
+q3 (succ (succ start)) a with q3NDivision (succ (succ start))
+... | r with a (succ (succ start))
+... | s = compositeImpliesNotPrime (succ (succ start)) _ (succPreservesInequality (succIsPositive start)) (le (succ (succ start) +N ((start +N succ start) +N q3Seq (succ (succ start)) start)) (applyEquality (λ i → succ (succ i)) (from (succ (plus (plus (const start) (plus (plus (const start) (succ (const start))) (const (q3Seq (succ (succ start)) start)))) (succ (succ (const start))))) to plus (plus (const start) (succ (succ (plus (const start) zero)))) (succ (plus (plus (const start) (succ (plus (const start) zero))) (const (q3Seq (succ (succ start)) start)))) by applyEquality (λ i → succ (succ (succ (succ i)))) (transitivity (commutative _ start) (+Associative start start _))))) r s
+
+q3' : ((n : ℕ) → Prime (q3Seq 41 n)) → False
+q3' = q3 41
+
+q6 : {n : ℕ} → 3 ∣ (n *N n) → 3 ∣ n
+q6 {n} 3|n^2 with primesArePrime {3} {n} {n} threeIsPrime 3|n^2
+q6 {n} 3|n^2 | inl x = x
+q6 {n} 3|n^2 | inr x = x
+
+_*q10_ : {a : ℕ} → {b : ℕ} → (0 <N a) → (0 <N b) → ℕ
+*q10Pos : {a : ℕ} → {b : ℕ} → (0<a : 0 <N a) → (0<b : 0 <N b) → 0 <N (_*q10_ 0<a 0<b)
+
+_*q10_ {succ zero} {succ b} 0<a 0<b = succ (succ b)
+_*q10_ {succ (succ a)} {succ zero} 0<a 0<b = _*q10_ {succ a} {2} (le a (applyEquality succ (sumZeroRight a))) (le 1 refl)
+_*q10_ {succ (succ a)} {succ (succ b)} 0<a 0<b = _*q10_ {succ a} {_*q10_ {succ (succ a)} {succ b} 0<a (le b (applyEquality succ (sumZeroRight b)))} (le a (applyEquality succ (sumZeroRight a))) (*q10Pos 0<a (le b (applyEquality succ (sumZeroRight b))))
+*q10Pos {succ zero} {succ b} 0<a 0<b = le (succ b) (applyEquality (λ i → succ (succ i)) (sumZeroRight b))
+*q10Pos {succ (succ a)} {succ zero} 0<a 0<b = *q10Pos (le a (applyEquality succ (sumZeroRight a))) (le 1 refl)
+*q10Pos {succ (succ a)} {succ (succ b)} 0<a 0<b = *q10Pos (le a (applyEquality succ (sumZeroRight a))) (*q10Pos 0<a (le b (applyEquality succ (sumZeroRight b))))
--- a/Numbers/Naturals/EuclideanAlgorithm.agda
+++ b/Numbers/Naturals/EuclideanAlgorithm.agda
@@ -250,7 +250,6 @@ reverseHCF : {a b : ℕ} → (ab : extendedHcf a b) → extendedHcf b a
 reverseHCF {a} {b} record { hcf = record { c = c ; c|a = c|a ; c|b = c|b ; hcf = hcf } ; extended1 = extended1 ; extended2 = extended2 ; extendedProof = (inl x) } = record { hcf = record { c = c ; c|a = c|b ; c|b = c|a ; hcf = λ x z z₁ → hcf x z₁ z } ; extended1 = extended2 ; extended2 = extended1 ; extendedProof = inr (equalityCommutative x) }
 reverseHCF {a} {b} record { hcf = record { c = c ; c|a = c|a ; c|b = c|b ; hcf = hcf } ; extended1 = extended1 ; extended2 = extended2 ; extendedProof = (inr x) } = record { hcf = record { c = c ; c|a = c|b ; c|b = c|a ; hcf = λ x z z₁ → hcf x z₁ z } ; extended1 = extended2 ; extended2 = extended1 ; extendedProof = inl (equalityCommutative x) }

-
 oneDivN : (a : ℕ) → 1 ∣ a
 oneDivN a = divides (record { quot = a ; rem = zero ; pr = pr ; remIsSmall = inl (succIsPositive zero) ; quotSmall = inl (le zero refl) }) refl
  where
--- a/Rings/Definition.agda
+++ b/Rings/Definition.agda
@@ -35,3 +35,5 @@ record Ring {n m} {A : Set n} (S : Setoid {n} {m} A) (_+_ : A → A → A) (_*_
  *DistributesOver+' = transitive *Commutative (transitive *DistributesOver+ (+WellDefined *Commutative *Commutative))
  *Associative' : {a b c : A} → ((a * b) * c) ∼ (a * (b * c))
  *Associative' = symmetric *Associative
+  identIsIdent' : {a : A} → a * 1R ∼ a
+  identIsIdent' = transitive *Commutative identIsIdent
--- a/Rings/EuclideanDomains/Examples.agda
+++ b/Rings/EuclideanDomains/Examples.agda
@@ -14,7 +14,7 @@ open import Fields.Lemmas
 module Rings.EuclideanDomains.Examples where

 polynomialField : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ _*_ : A → A → A} {R : Ring S _+_ _*_} (F : Field R) → (Setoid._∼_ S (Ring.1R R) (Ring.0R R) → False) → EuclideanDomain R
-EuclideanDomain.isIntegralDomain (polynomialField F 1!=0) = fieldIsIntDom F 1!=0
+EuclideanDomain.isIntegralDomain (polynomialField F 1!=0) = fieldIsIntDom F
 EuclideanDomain.norm (polynomialField F _) a!=0 = zero
 EuclideanDomain.normSize (polynomialField F _) a!=0 b!=0 c b=ac = inr refl
 DivisionAlgorithmResult.quotient (EuclideanDomain.divisionAlg (polynomialField {_*_ = _*_} F _) {a = a} {b} a!=0 b!=0) with Field.allInvertible F b b!=0
--- a/Rings/Examples/Examples.agda
+++ b/Rings/Examples/Examples.agda
@@ -10,17 +10,28 @@ open import Numbers.Modulo.Group
 open import Numbers.Modulo.Definition
 open import Rings.Examples.Proofs
 open import Numbers.Primes.PrimeNumbers
+open import Setoids.Setoids
+open import Rings.Definition
+open import Groups.Definition
+open import Groups.Lemmas
+open import Sets.EquivalenceRelations

 module Rings.Examples.Examples where

-  nToZn : (n : ℕ) (pr : 0 <N n) (x : ℕ) → ℤn n pr
-  nToZn n pr x = nToZn' n pr x
+multiplicationNotGroup : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ _*_ : A → A → A} (R : Ring S _+_ _*_) → (nontrivial : Setoid._∼_ S (Ring.1R R) (Ring.0R R) → False) → Group S _*_ → False
+multiplicationNotGroup {S = S} R 1!=0 gr = exFalso (1!=0 (groupsHaveLeftCancellation gr (Ring.0R R) (Ring.1R R) (Ring.0R R) (transitive (Ring.timesZero' R) (symmetric (Ring.timesZero' R)))))
+  where
+    open Setoid S
+    open Equivalence eq

-  mod : (n : ℕ) → (pr : 0 <N n) → ℤ → ℤn n pr
-  mod n pr a = mod' n pr a
+nToZn : (n : ℕ) (pr : 0 <N n) (x : ℕ) → ℤn n pr
+nToZn n pr x = nToZn' n pr x

-  modNExampleSurjective : (n : ℕ) → (pr : 0 <N n) → Surjection (mod n pr)
-  modNExampleSurjective n pr = modNExampleSurjective' n pr
+mod : (n : ℕ) → (pr : 0 <N n) → ℤ → ℤn n pr
+mod n pr a = mod' n pr a
+
+modNExampleSurjective : (n : ℕ) → (pr : 0 <N n) → Surjection (mod n pr)
+modNExampleSurjective n pr = modNExampleSurjective' n pr

 {-
  modNExampleGroupHom : (n : ℕ) → (pr : 0 <N n) → GroupHom ℤGroup (ℤnGroup n pr) (mod n pr)
--- a/Rings/Ideals/Maximal/Lemmas.agda
+++ b/Rings/Ideals/Maximal/Lemmas.agda
@@ -80,12 +80,7 @@ MaximalIdeal.isMaximal (quotientFieldImpliesIdealMaximal f) {bigger} biggerIdeal
    v = Ideal.isSubset biggerIdeal (invTwice additiveGroup 1R) (Ideal.closedUnderInverse biggerIdeal (Ideal.isSubset biggerIdeal (transitive (symmetric +Associative) (transitive (+WellDefined reflexive invRight) identRight)) (Ideal.closedUnderPlus biggerIdeal t u)))

 idealMaximalImpliesIdealPrime : ({d : _} → MaximalIdeal i {d}) → PrimeIdeal i
-idealMaximalImpliesIdealPrime max = quotientIntDomImpliesIdealPrime i f'
-  where
-    f : Field (cosetRing R i)
-    f = idealMaximalImpliesQuotientField max
-    f' : IntegralDomain (cosetRing R i)
-    f' = fieldIsIntDom f (λ p → Field.nontrivial f (Equivalence.symmetric (Setoid.eq (cosetSetoid additiveGroup (Ideal.isSubgroup i))) p))
+idealMaximalImpliesIdealPrime max = quotientIntDomImpliesIdealPrime i (fieldIsIntDom (idealMaximalImpliesQuotientField max))

 maximalIdealWellDefined : {d : _} {pred2 : A → Set d} (i2 : Ideal R pred2) → ({x : A} → pred x → pred2 x) → ({x : A} → pred2 x → pred x) → {e : _} → MaximalIdeal i {e} → MaximalIdeal i2 {e}
 MaximalIdeal.notContained (maximalIdealWellDefined i2 pToP2 p2ToP record { notContained = notContained ; notContainedIsNotContained = notContainedIsNotContained ; isMaximal = isMaximal }) = notContained
--- a/Rings/Lemmas.agda
+++ b/Rings/Lemmas.agda
@@ -58,4 +58,3 @@ abstract

 abelianUnderlyingGroup : AbelianGroup additiveGroup
 abelianUnderlyingGroup = record { commutative = groupIsAbelian }
-