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https://github.com/Smaug123/agdaproofs
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Restructure towards ideals
This commit is contained in:
Smaug123
@@ -16,12 +16,14 @@ open import Groups.Groups
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open import Groups.Abelian.Lemmas
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open import Groups.DirectSum.Definition
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open import Groups.QuotientGroup.Definition
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open import Groups.QuotientGroup.Lemmas
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open import Groups.FiniteGroups.Definition
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open import Groups.Homomorphisms.Lemmas
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open import Groups.Homomorphisms.Examples
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open import Groups.Isomorphisms.Lemmas
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open import Groups.FinitePermutations
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open import Groups.Lemmas
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open import Groups.Groups2
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open import Groups.FirstIsomorphismTheorem
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open import Groups.SymmetricGroups.Definition
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open import Groups.Actions.Stabiliser
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open import Groups.Actions.Orbit
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@@ -30,6 +32,7 @@ open import Groups.ActionIsSymmetry
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open import Groups.Cyclic.Definition
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open import Groups.Cyclic.DefinitionLemmas
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open import Groups.Polynomials.Examples
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open import Groups.Cosets
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open import Fields.Fields
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open import Fields.Orders.Partial.Definition
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@@ -50,6 +53,7 @@ open import Rings.Polynomial.Ring
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open import Rings.Polynomial.Evaluation
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open import Rings.Ideals.Definition
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open import Rings.Isomorphisms.Definition
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open import Rings.Quotients.Definition
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open import Setoids.Setoids
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open import Setoids.DirectSum
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@@ -11,7 +11,6 @@ open import Groups.Lemmas
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open import Groups.Homomorphisms.Definition
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open import Groups.Groups
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open import Groups.SymmetricGroups.Definition
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open import Groups.Groups2
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open import Groups.Actions.Definition
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open import Sets.EquivalenceRelations
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@@ -9,7 +9,6 @@ open import Sets.FinSet
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open import Groups.Definition
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open import Groups.Lemmas
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open import Groups.Groups
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open import Groups.Groups2
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open import Sets.EquivalenceRelations
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module Groups.Actions.Definition where
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@@ -10,7 +10,6 @@ open import Groups.Definition
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open import Groups.Lemmas
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open import Groups.Groups
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open import Groups.Actions.Definition
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open import Groups.Groups2
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open import Sets.EquivalenceRelations
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module Groups.Actions.Orbit where
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@@ -12,7 +12,6 @@ open import Groups.Groups
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open import Groups.Subgroups.Definition
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open import Groups.Homomorphisms.Definition
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open import Groups.Actions.Definition
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open import Groups.Groups2
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open import Sets.EquivalenceRelations
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open import Groups.Actions.Definition
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@@ -29,12 +28,13 @@ stabiliserWellDefined x {g} {h} g=h gx=x = transitive (actionWellDefined1 (Equiv
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where
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open Equivalence eq
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stabiliserSubgroup : (x : B) → subgroup G (stabiliserWellDefined x)
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_&_&_.one (stabiliserSubgroup x) gx=x hx=x = transitive associativeAction (transitive (actionWellDefined2 hx=x) gx=x)
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open Setoid T
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open Equivalence (Setoid.eq T)
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stabiliserSubgroup : (x : B) → subgroup G (stabiliserPred x)
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_&&_.fst (stabiliserSubgroup x) = stabiliserWellDefined x
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_&_&_.one (_&&_.snd (stabiliserSubgroup x)) gx=x hx=x = transitive associativeAction (transitive (actionWellDefined2 hx=x) gx=x)
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_&_&_.two (_&&_.snd (stabiliserSubgroup x)) = identityAction
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_&_&_.three (_&&_.snd (stabiliserSubgroup x)) {g} gx=x = transitive (transitive (transitive (actionWellDefined2 (symmetric gx=x)) (symmetric associativeAction)) (actionWellDefined1 (invLeft {g}))) identityAction
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where
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open Equivalence eq
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_&_&_.two (stabiliserSubgroup x) = identityAction
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_&_&_.three (stabiliserSubgroup x) {g = g} gx=x = transitive (transitive (transitive (actionWellDefined2 (symmetric gx=x)) (symmetric associativeAction)) (actionWellDefined1 (invLeft {g}))) identityAction
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where
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open Equivalence eq
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open Group G
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@@ -1,51 +1,50 @@
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{-# OPTIONS --safe --warning=error #-}
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{-# OPTIONS --safe --warning=error --without-K #-}
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open import LogicalFormulae
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open import Sets.EquivalenceRelations
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open import Setoids.Setoids
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open import Setoids.Subset
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open import Functions
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open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
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open import Numbers.Naturals
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open import Numbers.Integers
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open import Sets.FinSet
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open import Groups.GroupDefinition
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open import Groups.Groups
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open import Groups.Definition
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open import Groups.Subgroups.Definition
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open import Groups.Subgroups.Normal.Definition
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module Groups.GroupCosets where
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data GroupCoset {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+A_ : A → A → A} {_+B_ : B → B → B} {G : Group S _+A_} {H : Group T _+B_} {f : B → A} {fHom : GroupHom H G f} (subgrp : Subgroup G H fHom) : Set (a ⊔ b ⊔ c ⊔ d) where
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cosetOfElt : (x : A) → GroupCoset subgrp
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module Groups.Cosets {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} (G : Group S _+_) {c : _} {pred : A → Set c} (subgrp : subgroup G pred) where
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groupCosetSetoid : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+A_ : A → A → A} {_+B_ : B → B → B} {G : Group S _+A_} {H : Group T _+B_} {f : B → A} {fHom : GroupHom H G f} (subgrp : Subgroup G H fHom) → Setoid (GroupCoset subgrp)
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Setoid._∼_ (groupCosetSetoid {A = A} {S = S} {_+A_ = _+A_} {G = G} subgrp) (cosetOfElt x) (cosetOfElt y) = Sg (A && A) (λ p → x +A (_&&_.fst p) ∼ y +A (_&&_.snd p))
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where
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open Setoid S
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Reflexive.reflexive (Equivalence.reflexiveEq (Setoid.eq (groupCosetSetoid {S = S} {G = G} subgrp))) {cosetOfElt x} = (identity ,, identity) , transitive (multIdentRight) (symmetric multIdentRight)
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where
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open Group G
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open Setoid S
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open Transitive (Equivalence.transitiveEq eq)
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open Symmetric (Equivalence.symmetricEq eq)
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Symmetric.symmetric (Equivalence.symmetricEq (Setoid.eq (groupCosetSetoid {S = S} subgrp))) {cosetOfElt x} {cosetOfElt y} ((xMul ,, yMul) , pr) = (yMul ,, xMul) , Symmetric.symmetric (Equivalence.symmetricEq (Setoid.eq S)) pr
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Transitive.transitive (Equivalence.transitiveEq (Setoid.eq (groupCosetSetoid {S = S} {_+A_ = _+A_} {G = G} subgrp))) {cosetOfElt x} {cosetOfElt y} {cosetOfElt z} ((xMul ,, yMul) , pr) ((yMul' ,, zMul) , pr2) = (((xMul +A (inverse yMul)) +A yMul') ,, zMul) , transitive (transitive multAssoc (wellDefined (transitive multAssoc (transitive (wellDefined pr reflexive) (transitive (symmetric multAssoc) (transitive (wellDefined reflexive invRight) multIdentRight)))) reflexive)) pr2
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where
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open Group G
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open Setoid S
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open Transitive (Equivalence.transitiveEq eq)
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open Reflexive (Equivalence.reflexiveEq eq)
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open Symmetric (Equivalence.symmetricEq eq)
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open Equivalence (Setoid.eq S)
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open import Groups.Lemmas G
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open _&_&_ (_&&_.snd subgrp)
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open Group G
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groupCosetBijection : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+A_ : A → A → A} {_+B_ : B → B → B} {G : Group S _+A_} {H : Group T _+B_} {f : B → A} {fHom : GroupHom H G f} (subgrp : Subgroup G H fHom) → SetoidsBiject T (groupCosetSetoid subgrp)
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SetoidsBiject.bij (groupCosetBijection {f = f} subgrp) x = cosetOfElt (f x)
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SetoidInjection.wellDefined (SetoidBijection.inj (SetoidsBiject.bijIsBijective (groupCosetBijection {S = S} {G = G} {fHom = fHom} subgrp))) x~y = (identity ,, identity) , transitive multIdentRight (transitive (GroupHom.wellDefined fHom x~y) (symmetric multIdentRight))
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where
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open Group G
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open Setoid S
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open Transitive (Equivalence.transitiveEq eq)
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open Symmetric (Equivalence.symmetricEq eq)
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SetoidInjection.injective (SetoidBijection.inj (SetoidsBiject.bijIsBijective (groupCosetBijection {S = S} {_+A_ = _+A_} {G = G} {f = f} subgrp))) {x} {y} ((xH ,, yH) , b) = {!!}
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SetoidSurjection.wellDefined (SetoidBijection.surj (SetoidsBiject.bijIsBijective (groupCosetBijection {S = S} {G = G} {fHom = fHom} subgrp))) x~y = (identity ,, identity) , transitive multIdentRight (transitive (GroupHom.wellDefined fHom x~y) (symmetric multIdentRight))
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where
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open Group G
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open Setoid S
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open Transitive (Equivalence.transitiveEq eq)
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open Symmetric (Equivalence.symmetricEq eq)
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SetoidSurjection.surjective (SetoidBijection.surj (SetoidsBiject.bijIsBijective (groupCosetBijection subgrp))) {cosetOfElt x} = {!!}
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private
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subs = _&&_.fst subgrp
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cosetSetoid : Setoid A
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Setoid._∼_ cosetSetoid g h = pred ((Group.inverse G h) + g)
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Equivalence.reflexive (Setoid.eq cosetSetoid) = subs (symmetric (Group.invLeft G)) two
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Equivalence.symmetric (Setoid.eq cosetSetoid) yx = subs (transitive invContravariant (+WellDefined reflexive invInv)) (three yx)
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Equivalence.transitive (Setoid.eq cosetSetoid) yx zy = subs (transitive +Associative (+WellDefined (transitive (symmetric +Associative) (transitive (+WellDefined reflexive invRight) identRight)) reflexive)) (one zy yx)
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cosetGroup : normalSubgroup G subgrp → Group cosetSetoid _+_
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Group.+WellDefined (cosetGroup norm) {m} {n} {x} {y} m=x n=y = ans
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where
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t : pred (inverse y + n)
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t = n=y
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u : pred (inverse x + m)
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u = m=x
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v : pred (m + inverse x)
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v = subs (+WellDefined reflexive (transitive (symmetric +Associative) (transitive (+WellDefined reflexive invRight) identRight))) (norm u)
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ans' : pred ((inverse y) + ((inverse x + m) + inverse (inverse y)))
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ans' = norm u
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ans'' : pred ((inverse y) + ((inverse x + m) + y))
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ans'' = subs (+WellDefined reflexive (+WellDefined reflexive (invTwice y))) ans'
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ans : pred (inverse (x + y) + (m + n))
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ans = subs (transitive (transitive +Associative (transitive (+WellDefined (transitive (symmetric +Associative) (transitive (+WellDefined reflexive (transitive (symmetric +Associative) (transitive (+WellDefined reflexive invRight) identRight))) +Associative)) reflexive) (symmetric +Associative))) (symmetric (+WellDefined invContravariant reflexive))) (one ans'' t)
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Group.0G (cosetGroup norm) = 0G
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Group.inverse (cosetGroup norm) = inverse
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Group.+Associative (cosetGroup norm) {a} {b} {c} = subs (symmetric (transitive (+WellDefined (inverseWellDefined (symmetric +Associative)) reflexive) (invLeft {a + (b + c)}))) two
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Group.identRight (cosetGroup norm) = subs (symmetric (transitive +Associative (transitive (+WellDefined invLeft reflexive) identRight))) two
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Group.identLeft (cosetGroup norm) = subs (symmetric (transitive (+WellDefined reflexive identLeft) invLeft)) two
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Group.invLeft (cosetGroup norm) = subs (symmetric (transitive (+WellDefined reflexive invLeft) invLeft)) two
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Group.invRight (cosetGroup norm) = subs (symmetric (transitive (+WellDefined reflexive invRight) invLeft)) two
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49
Groups/FirstIsomorphismTheorem.agda
Normal file
49
Groups/FirstIsomorphismTheorem.agda
Normal file
@@ -0,0 +1,49 @@
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{-# OPTIONS --safe --warning=error --without-K #-}
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open import Groups.Groups
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open import Groups.Definition
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open import Orders
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open import Numbers.Integers.Integers
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open import Setoids.Setoids
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open import LogicalFormulae
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open import Sets.FinSet
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open import Functions
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open import Sets.EquivalenceRelations
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open import Numbers.Naturals.Naturals
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open import Groups.Homomorphisms.Definition
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open import Groups.Homomorphisms.Lemmas
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open import Groups.Isomorphisms.Definition
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open import Groups.Subgroups.Definition
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open import Groups.Subgroups.Normal.Definition
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open import Groups.Subgroups.Normal.Lemmas
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open import Groups.Lemmas
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open import Groups.Abelian.Definition
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open import Groups.QuotientGroup.Definition
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open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
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module Groups.FirstIsomorphismTheorem {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+G_ : A → A → A} {_+H_ : B → B → B} {G : Group S _+G_} {H : Group T _+H_} {f : A → B} (fHom : GroupHom G H f) where
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open import Groups.Homomorphisms.Image fHom
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open import Groups.Homomorphisms.Kernel fHom
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open import Groups.Cosets G groupKernelIsSubgroup
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open Setoid T
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open Equivalence eq
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open import Setoids.Subset T
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groupFirstIsomorphismTheoremWellDefined : {x y : A} → groupKernelPred (Group.inverse G y +G x) → Setoid._∼_ (subsetSetoid imageGroupSubset) (f x , (x , reflexive)) (f y , (y , reflexive))
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groupFirstIsomorphismTheoremWellDefined {x} {y} pr = transitive (symmetric (invTwice H _)) (transitive (symmetric u) (invTwice H _))
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where
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t : Setoid._∼_ T (Group.inverse H (f y) +H (f x)) (Group.0G H)
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t = transitive (transitive (Group.+WellDefined H (symmetric (homRespectsInverse fHom)) reflexive) (symmetric (GroupHom.groupHom fHom))) pr
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u : Setoid._∼_ T (Group.inverse H (Group.inverse H (f y))) (Group.inverse H (Group.inverse H (f x)))
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u = inverseWellDefined H (transferToRight' H t)
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groupFirstIsomorphismTheorem : GroupsIsomorphic (cosetGroup groupKernelIsNormalSubgroup) (subgroupIsGroup H imageGroupSubset imageGroupSubgroup)
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GroupsIsomorphic.isomorphism groupFirstIsomorphismTheorem x = f x , (x , reflexive)
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GroupHom.groupHom (GroupIso.groupHom (GroupsIsomorphic.proof groupFirstIsomorphismTheorem)) {x} {y} = GroupHom.groupHom fHom
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GroupHom.wellDefined (GroupIso.groupHom (GroupsIsomorphic.proof groupFirstIsomorphismTheorem)) {x} {y} = groupFirstIsomorphismTheoremWellDefined {x} {y}
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SetoidInjection.wellDefined (SetoidBijection.inj (GroupIso.bij (GroupsIsomorphic.proof groupFirstIsomorphismTheorem))) = groupFirstIsomorphismTheoremWellDefined
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SetoidInjection.injective (SetoidBijection.inj (GroupIso.bij (GroupsIsomorphic.proof groupFirstIsomorphismTheorem))) fx=fy = transitive (GroupHom.groupHom fHom) (transitive (Group.+WellDefined H reflexive fx=fy) (transitive (symmetric (GroupHom.groupHom fHom)) (transitive (GroupHom.wellDefined fHom (Group.invLeft G)) (imageOfIdentityIsIdentity fHom))))
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SetoidSurjection.wellDefined (SetoidBijection.surj (GroupIso.bij (GroupsIsomorphic.proof groupFirstIsomorphismTheorem))) = groupFirstIsomorphismTheoremWellDefined
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SetoidSurjection.surjective (SetoidBijection.surj (GroupIso.bij (GroupsIsomorphic.proof groupFirstIsomorphismTheorem))) {b , (a , fa=b)} = a , fa=b
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@@ -73,6 +73,20 @@ quotientGroupSetoid {A = A} {S = S} {T = T} {_·A_ = _·A_} {_·B_ = _·B_} G {H
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k : f (m ·A ((Group.0G G) ·A Group.inverse G o)) ∼ 0G
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k = transitive (GroupHom.wellDefined fHom (Group.+WellDefined G (Equivalence.reflexive (Setoid.eq S)) (Group.+WellDefined G (Equivalence.symmetric (Setoid.eq S) (Group.invLeft G)) (Equivalence.reflexive (Setoid.eq S))))) j
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quotientGroupLemma : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} (G : Group S _·A_) {H : Group T _·B_} → {underf : A → B} → (f : GroupHom G H underf) → {x y : A} → Setoid._∼_ T (underf x) (underf y) → Setoid._∼_ (quotientGroupSetoid G f) x y
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quotientGroupLemma {S = S} {T = T} G {H = H} fHom {x} {y} fx=fy = transitive (GroupHom.groupHom fHom) (transitive (Group.+WellDefined H (Equivalence.reflexive (Setoid.eq T)) (homRespectsInverse fHom)) (transferToRight'' H fx=fy))
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where
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open Group G
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open Setoid T
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open Equivalence eq
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quotientGroupLemma' : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} (G : Group S _·A_) {H : Group T _·B_} → {underf : A → B} → (f : GroupHom G H underf) → {x y : A} → Setoid._∼_ (quotientGroupSetoid G f) x y → Setoid._∼_ T (underf x) (underf y)
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quotientGroupLemma' {S = S} {T = T} G {H = H} f fx=fy = transferToRight H (transitive (transitive (Group.+WellDefined H (Equivalence.reflexive (Setoid.eq T)) (symmetric (homRespectsInverse f))) (symmetric (GroupHom.groupHom f))) fx=fy)
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where
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open Group G
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open Setoid T
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open Equivalence eq
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{-
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quotientHom : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} (G : Group S _·A_) {H : Group T _·B_} → {f : A → B} → (fHom : GroupHom G H f) → A → A
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quotientHom {S = S} {T = T} {_·A_ = _·A_} {_·B_ = _·B_} G {f = f} fHom a = {!!}
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@@ -1,54 +0,0 @@
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{-# OPTIONS --safe --warning=error --without-K #-}
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open import Groups.Groups
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open import Groups.Definition
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open import Orders
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open import Numbers.Integers.Integers
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open import Setoids.Setoids
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open import LogicalFormulae
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open import Sets.FinSet
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open import Functions
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open import Sets.EquivalenceRelations
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open import Numbers.Naturals.Naturals
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open import Groups.Homomorphisms.Definition
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open import Groups.Homomorphisms.Lemmas
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open import Groups.Homomorphisms.Image
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open import Groups.Isomorphisms.Definition
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open import Groups.Subgroups.Definition
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open import Groups.Subgroups.Normal.Definition
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open import Groups.Subgroups.Normal.Lemmas
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open import Groups.Lemmas
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open import Groups.Abelian.Definition
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open import Groups.QuotientGroup.Definition
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open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
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module Groups.Groups2 where
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groupFirstIsomorphismIso : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+G_ : A → A → A} {_+H_ : B → B → B} {G : Group S _+G_} {H : Group T _+H_} {f : A → B} (fHom : GroupHom G H f) → GroupHomImageElement fHom → A
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groupFirstIsomorphismIso fHom (ofElt a) = a
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groupFirstIsomorphismIsoHom : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+G_ : A → A → A} {_+H_ : B → B → B} {G : Group S _+G_} {H : Group T _+H_} {f : A → B} (fHom : GroupHom G H f) → GroupHom (imageGroup fHom) (quotientGroup G fHom) (groupFirstIsomorphismIso fHom)
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GroupHom.groupHom (groupFirstIsomorphismIsoHom {G = G} fHom) {ofElt a} {ofElt b} = Equivalence.reflexive (Setoid.eq (quotientGroupSetoid G fHom))
|
||||
GroupHom.wellDefined (groupFirstIsomorphismIsoHom {T = T} {_+G_ = _+G_} {G = G} {H = H} {f = f} fHom) {ofElt a} {ofElt b} pr = ans
|
||||
where
|
||||
open Setoid T
|
||||
open Equivalence (Setoid.eq T)
|
||||
ans : f (a +G Group.inverse G b) ∼ Group.0G H
|
||||
ans = transitive (GroupHom.groupHom fHom) (transitive (Group.+WellDefined H pr reflexive) (transitive (symmetric (GroupHom.groupHom fHom)) (transitive (GroupHom.wellDefined fHom (Group.invRight G)) (imageOfIdentityIsIdentity fHom))))
|
||||
|
||||
groupFirstIsomorphismTheorem' : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+G_ : A → A → A} {_+H_ : B → B → B} {G : Group S _+G_} {H : Group T _+H_} {f : A → B} (fHom : GroupHom G H f) → GroupIso (imageGroup fHom) (quotientGroup G fHom) (groupFirstIsomorphismIso fHom)
|
||||
GroupIso.groupHom (groupFirstIsomorphismTheorem' fHom) = groupFirstIsomorphismIsoHom fHom
|
||||
SetoidInjection.wellDefined (SetoidBijection.inj (GroupIso.bij (groupFirstIsomorphismTheorem' fHom))) {x} {y} x~y = GroupHom.wellDefined (groupFirstIsomorphismIsoHom fHom) {x} {y} x~y
|
||||
SetoidInjection.injective (SetoidBijection.inj (GroupIso.bij (groupFirstIsomorphismTheorem' {T = T} {H = H} {f = f} fHom))) {ofElt a} {ofElt b} pr = need
|
||||
where
|
||||
open Setoid T
|
||||
open Equivalence eq
|
||||
need : f a ∼ f b
|
||||
need = transferToRight H (transitive (transitive (Group.+WellDefined H reflexive (symmetric (homRespectsInverse fHom))) (symmetric (GroupHom.groupHom fHom))) pr)
|
||||
SetoidSurjection.wellDefined (SetoidBijection.surj (GroupIso.bij (groupFirstIsomorphismTheorem' fHom))) {x} {y} x~y = GroupHom.wellDefined (groupFirstIsomorphismIsoHom fHom) {x} {y} x~y
|
||||
SetoidSurjection.surjective (SetoidBijection.surj (GroupIso.bij (groupFirstIsomorphismTheorem' {G = G} fHom))) {a} = ofElt a , Equivalence.reflexive (Setoid.eq (quotientGroupSetoid G fHom))
|
||||
|
||||
groupFirstIsomorphismTheorem : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+G_ : A → A → A} {_+H_ : B → B → B} {G : Group S _+G_} {H : Group T _+H_} {f : A → B} (fHom : GroupHom G H f) → GroupsIsomorphic (imageGroup fHom) (quotientGroup G fHom)
|
||||
groupFirstIsomorphismTheorem fHom = record { isomorphism = groupFirstIsomorphismIso fHom ; proof = groupFirstIsomorphismTheorem' fHom }
|
||||
|
||||
18
Groups/Homomorphisms/Examples.agda
Normal file
18
Groups/Homomorphisms/Examples.agda
Normal file
@@ -0,0 +1,18 @@
|
||||
{-# OPTIONS --safe --warning=error --without-K #-}
|
||||
|
||||
open import LogicalFormulae
|
||||
open import Setoids.Setoids
|
||||
open import Functions
|
||||
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
|
||||
open import Numbers.Naturals.Naturals
|
||||
open import Sets.FinSet
|
||||
open import Groups.Definition
|
||||
open import Sets.EquivalenceRelations
|
||||
open import Groups.Homomorphisms.Definition
|
||||
open import Groups.Lemmas
|
||||
|
||||
module Groups.Homomorphisms.Examples where
|
||||
|
||||
identityHom : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+A_ : A → A → A} (G : Group S _+A_) → GroupHom G G id
|
||||
GroupHom.groupHom (identityHom {S = S} G) = Equivalence.reflexive (Setoid.eq S)
|
||||
GroupHom.wellDefined (identityHom G) = id
|
||||
@@ -5,6 +5,7 @@ open import Groups.Definition
|
||||
open import Orders
|
||||
open import Numbers.Integers.Integers
|
||||
open import Setoids.Setoids
|
||||
open import Setoids.Subset
|
||||
open import LogicalFormulae
|
||||
open import Sets.FinSet
|
||||
open import Functions
|
||||
@@ -20,45 +21,25 @@ open import Groups.QuotientGroup.Definition
|
||||
|
||||
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
|
||||
|
||||
module Groups.Homomorphisms.Image where
|
||||
module Groups.Homomorphisms.Image {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+A_ : A → A → A} {_+B_ : B → B → B} {G : Group S _+A_} {H : Group T _+B_} {f : A → B} (fHom : GroupHom G H f) where
|
||||
|
||||
data GroupHomImageElement {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+A_ : A → A → A} {_+B_ : B → B → B} {G : Group S _+A_} {H : Group T _+B_} {f : A → B} (fHom : GroupHom G H f) : Set (a ⊔ b ⊔ c ⊔ d) where
|
||||
ofElt : (x : A) → GroupHomImageElement fHom
|
||||
imageGroupPred : B → Set (a ⊔ d)
|
||||
imageGroupPred b = Sg A (λ a → Setoid._∼_ T (f a) b)
|
||||
|
||||
imageGroupSetoid : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+A_ : A → A → A} {_+B_ : B → B → B} {G : Group S _+A_} {H : Group T _+B_} {f : A → B} (fHom : GroupHom G H f) → Setoid (GroupHomImageElement fHom)
|
||||
(imageGroupSetoid {T = T} {f = f} fHom Setoid.∼ ofElt b1) (ofElt b2) = Setoid._∼_ T (f b1) (f b2)
|
||||
Equivalence.reflexive (Setoid.eq (imageGroupSetoid {T = T} fHom)) {ofElt b1} = Equivalence.reflexive (Setoid.eq T)
|
||||
Equivalence.symmetric (Setoid.eq (imageGroupSetoid {T = T} fHom)) {ofElt b1} {ofElt b2} = Equivalence.symmetric (Setoid.eq T)
|
||||
Equivalence.transitive (Setoid.eq (imageGroupSetoid {T = T} fHom)) {ofElt b1} {ofElt b2} {ofElt b3} = Equivalence.transitive (Setoid.eq T)
|
||||
|
||||
imageGroupOp : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+A_ : A → A → A} {_+B_ : B → B → B} {G : Group S _+A_} {H : Group T _+B_} {f : A → B} (fHom : GroupHom G H f) → GroupHomImageElement fHom → GroupHomImageElement fHom → GroupHomImageElement fHom
|
||||
imageGroupOp {_+A_ = _+A_} fHom (ofElt a) (ofElt b) = ofElt (a +A b)
|
||||
|
||||
imageGroup : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+A_ : A → A → A} {_+B_ : B → B → B} {G : Group S _+A_} {H : Group T _+B_} {f : A → B} (fHom : GroupHom G H f) → Group (imageGroupSetoid fHom) (imageGroupOp fHom)
|
||||
Group.+WellDefined (imageGroup {T = T} {_+A_ = _+A_} {H = H} {f = f} fHom) {ofElt x} {ofElt y} {ofElt a} {ofElt b} x~a y~b = ans
|
||||
imageGroupSubset : subset T imageGroupPred
|
||||
imageGroupSubset {x} {y} x=y (a , fa=x) = a , transitive fa=x x=y
|
||||
where
|
||||
open Setoid T
|
||||
open Equivalence eq
|
||||
ans : f (x +A y) ∼ f (a +A b)
|
||||
ans = transitive (GroupHom.groupHom fHom) (transitive (Group.+WellDefined H x~a y~b) (symmetric (GroupHom.groupHom fHom)))
|
||||
Group.0G (imageGroup {G = G} fHom) = ofElt (Group.0G G)
|
||||
Group.inverse (imageGroup {G = G} fHom) (ofElt a) = ofElt (Group.inverse G a)
|
||||
Group.+Associative (imageGroup {G = G} fHom) {ofElt a} {ofElt b} {ofElt c} = GroupHom.wellDefined fHom (Group.+Associative G)
|
||||
Group.identRight (imageGroup {G = G} fHom) {ofElt a} = GroupHom.wellDefined fHom (Group.identRight G)
|
||||
Group.identLeft (imageGroup {G = G} fHom) {ofElt a} = GroupHom.wellDefined fHom (Group.identLeft G)
|
||||
Group.invLeft (imageGroup {G = G} fHom) {ofElt a} = GroupHom.wellDefined fHom (Group.invLeft G)
|
||||
Group.invRight (imageGroup {G = G} fHom) {ofElt a} = GroupHom.wellDefined fHom (Group.invRight G)
|
||||
|
||||
groupHomImageInclusion : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+G_ : A → A → A} {_+H_ : B → B → B} {G : Group S _+G_} {H : Group T _+H_} {f : A → B} (fHom : GroupHom G H f) → GroupHomImageElement fHom → B
|
||||
groupHomImageInclusion {f = f} fHom (ofElt x) = f x
|
||||
|
||||
groupHomImageIncludes : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+G_ : A → A → A} {_+H_ : B → B → B} {G : Group S _+G_} {H : Group T _+H_} {f : A → B} (fHom : GroupHom G H f) → GroupHom (imageGroup fHom) H (groupHomImageInclusion fHom)
|
||||
GroupHom.groupHom (groupHomImageIncludes fHom) {ofElt x} {ofElt y} = GroupHom.groupHom fHom
|
||||
GroupHom.wellDefined (groupHomImageIncludes fHom) {ofElt x} {ofElt y} x~y = x~y
|
||||
|
||||
groupHomImageInjects : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+G_ : A → A → A} {_+H_ : B → B → B} {G : Group S _+G_} {H : Group T _+H_} {f : A → B} (fHom : GroupHom G H f) → SetoidInjection (imageGroupSetoid fHom) T (groupHomImageInclusion fHom)
|
||||
groupHomImageInjects {S = S} {T} {_+G_} {_+H_} {G} {H} {f} fHom = record { wellDefined = λ {x} {y} → GroupHom.wellDefined (groupHomImageIncludes fHom) {x} {y} ; injective = λ {x} {y} → inj {x} {y} }
|
||||
imageGroupSubgroup : subgroup H imageGroupPred
|
||||
_&&_.fst imageGroupSubgroup = imageGroupSubset
|
||||
_&_&_.one (_&&_.snd imageGroupSubgroup) {x} {y} (a , fa=x) (b , fb=y) = (a +A b) , transitive (GroupHom.groupHom fHom) (Group.+WellDefined H fa=x fb=y)
|
||||
where
|
||||
inj : {x y : GroupHomImageElement fHom} → (Setoid._∼_ T (groupHomImageInclusion fHom x) (groupHomImageInclusion fHom y)) → Setoid._∼_ (imageGroupSetoid fHom) x y
|
||||
inj {ofElt x} {ofElt y} x~y = x~y
|
||||
|
||||
open Setoid T
|
||||
open Equivalence eq
|
||||
_&_&_.two (_&&_.snd imageGroupSubgroup) = Group.0G G , imageOfIdentityIsIdentity fHom
|
||||
_&_&_.three (_&&_.snd imageGroupSubgroup) {x} (a , fa=x) = Group.inverse G a , transitive (homRespectsInverse fHom) (inverseWellDefined H fa=x)
|
||||
where
|
||||
open Setoid T
|
||||
open Equivalence eq
|
||||
|
||||
44
Groups/Homomorphisms/Kernel.agda
Normal file
44
Groups/Homomorphisms/Kernel.agda
Normal file
@@ -0,0 +1,44 @@
|
||||
{-# OPTIONS --safe --warning=error --without-K #-}
|
||||
|
||||
open import Groups.Groups
|
||||
open import Groups.Definition
|
||||
open import Orders
|
||||
open import Numbers.Integers.Integers
|
||||
open import Setoids.Setoids
|
||||
open import LogicalFormulae
|
||||
open import Sets.FinSet
|
||||
open import Functions
|
||||
open import Sets.EquivalenceRelations
|
||||
open import Numbers.Naturals.Naturals
|
||||
open import Groups.Homomorphisms.Definition
|
||||
open import Groups.Homomorphisms.Lemmas
|
||||
open import Groups.Isomorphisms.Definition
|
||||
open import Groups.Subgroups.Definition
|
||||
open import Groups.Subgroups.Normal.Definition
|
||||
open import Groups.Subgroups.Normal.Lemmas
|
||||
open import Groups.Lemmas
|
||||
open import Groups.Abelian.Definition
|
||||
open import Groups.QuotientGroup.Definition
|
||||
open import Groups.Cosets
|
||||
|
||||
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
|
||||
|
||||
module Groups.Homomorphisms.Kernel {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+G_ : A → A → A} {_+H_ : B → B → B} {G : Group S _+G_} {H : Group T _+H_} {f : A → B} (fHom : GroupHom G H f) where
|
||||
|
||||
open Setoid T
|
||||
open Equivalence eq
|
||||
|
||||
groupKernelPred : A → Set d
|
||||
groupKernelPred a = Setoid._∼_ T (f a) (Group.0G H)
|
||||
|
||||
groupKernelPredWd : {x y : A} → (Setoid._∼_ S x y) → groupKernelPred x → groupKernelPred y
|
||||
groupKernelPredWd x=y fx=0 = transitive (GroupHom.wellDefined fHom (Equivalence.symmetric (Setoid.eq S) x=y)) fx=0
|
||||
|
||||
groupKernelIsSubgroup : subgroup G groupKernelPred
|
||||
_&_&_.one (_&&_.snd groupKernelIsSubgroup) fg=0 fh=0 = transitive (transitive (GroupHom.groupHom fHom) (Group.+WellDefined H fg=0 fh=0)) (Group.identLeft H)
|
||||
_&_&_.two (_&&_.snd groupKernelIsSubgroup) = imageOfIdentityIsIdentity fHom
|
||||
_&_&_.three (_&&_.snd groupKernelIsSubgroup) fg=0 = transitive (homRespectsInverse fHom) (transitive (inverseWellDefined H fg=0) (invIdent H))
|
||||
_&&_.fst groupKernelIsSubgroup = groupKernelPredWd
|
||||
|
||||
groupKernelIsNormalSubgroup : normalSubgroup G groupKernelIsSubgroup
|
||||
groupKernelIsNormalSubgroup {g} fk=0 = transitive (transitive (transitive (GroupHom.groupHom fHom) (transitive (Group.+WellDefined H reflexive (transitive (GroupHom.groupHom fHom) (transitive (Group.+WellDefined H fk=0 reflexive) (Group.identLeft H)))) (symmetric (GroupHom.groupHom fHom)))) (GroupHom.wellDefined fHom (Group.invRight G {g}))) (imageOfIdentityIsIdentity fHom)
|
||||
@@ -11,36 +11,38 @@ open import Sets.EquivalenceRelations
|
||||
open import Groups.Homomorphisms.Definition
|
||||
open import Groups.Lemmas
|
||||
|
||||
module Groups.Homomorphisms.Lemmas where
|
||||
module Groups.Homomorphisms.Lemmas {a b c d : _} {A : Set a} {S : Setoid {a} {c} A} {B : Set b} {T : Setoid {b} {d} B} {_+A_ : A → A → A} {_+B_ : B → B → B} {G : Group S _+A_} {H : Group T _+B_} {f : A → B} (hom : GroupHom G H f) where
|
||||
|
||||
imageOfIdentityIsIdentity : {m n o p : _} {A : Set m} {S : Setoid {m} {o} A} {B : Set n} {T : Setoid {n} {p} B} {_·A_ : A → A → A} {_·B_ : B → B → B} {G : Group S _·A_} {H : Group T _·B_} {f : A → B} → (hom : GroupHom G H f) → Setoid._∼_ T (f (Group.0G G)) (Group.0G H)
|
||||
imageOfIdentityIsIdentity {S = S} {T = T} {_·A_ = _·A_} {_·B_ = _·B_} {G = G} {H = H} {f = f} hom = Equivalence.symmetric (Setoid.eq T) t
|
||||
imageOfIdentityIsIdentity : Setoid._∼_ T (f (Group.0G G)) (Group.0G H)
|
||||
imageOfIdentityIsIdentity = Equivalence.symmetric (Setoid.eq T) t
|
||||
where
|
||||
open Group H
|
||||
open Setoid T
|
||||
id2 : Setoid._∼_ S (Group.0G G) ((Group.0G G) ·A (Group.0G G))
|
||||
id2 : Setoid._∼_ S (Group.0G G) ((Group.0G G) +A (Group.0G G))
|
||||
id2 = Equivalence.symmetric (Setoid.eq S) (Group.identRight G)
|
||||
r : f (Group.0G G) ∼ f (Group.0G G) ·B f (Group.0G G)
|
||||
s : 0G ·B f (Group.0G G) ∼ f (Group.0G G) ·B f (Group.0G G)
|
||||
r : f (Group.0G G) ∼ f (Group.0G G) +B f (Group.0G G)
|
||||
s : 0G +B f (Group.0G G) ∼ f (Group.0G G) +B f (Group.0G G)
|
||||
t : 0G ∼ f (Group.0G G)
|
||||
t = groupsHaveRightCancellation H (f (Group.0G G)) 0G (f (Group.0G G)) s
|
||||
s = Equivalence.transitive (Setoid.eq T) identLeft r
|
||||
r = Equivalence.transitive (Setoid.eq T) (GroupHom.wellDefined hom id2) (GroupHom.groupHom hom)
|
||||
|
||||
groupHomsCompose : {m n o r s t : _} {A : Set m} {S : Setoid {m} {r} A} {_+A_ : A → A → A} {B : Set n} {T : Setoid {n} {s} B} {_+B_ : B → B → B} {C : Set o} {U : Setoid {o} {t} C} {_+C_ : C → C → C} {G : Group S _+A_} {H : Group T _+B_} {I : Group U _+C_} {f : A → B} {g : B → C} (fHom : GroupHom G H f) (gHom : GroupHom H I g) → GroupHom G I (g ∘ f)
|
||||
GroupHom.wellDefined (groupHomsCompose {G = G} {H} {I} {f} {g} fHom gHom) {x} {y} pr = GroupHom.wellDefined gHom (GroupHom.wellDefined fHom pr)
|
||||
GroupHom.groupHom (groupHomsCompose {S = S} {_+A_ = _·A_} {T = T} {U = U} {_+C_ = _·C_} {G = G} {H} {I} {f} {g} fHom gHom) {x} {y} = answer
|
||||
groupHomsCompose : {o t : _} {C : Set o} {U : Setoid {o} {t} C} {_+C_ : C → C → C} {I : Group U _+C_} {g : B → C} (gHom : GroupHom H I g) → GroupHom G I (g ∘ f)
|
||||
GroupHom.wellDefined (groupHomsCompose {I} {f} gHom) {x} {y} pr = GroupHom.wellDefined gHom (GroupHom.wellDefined hom pr)
|
||||
GroupHom.groupHom (groupHomsCompose {U = U} {_+C_ = _·C_} {I} {g} gHom) {x} {y} = answer
|
||||
where
|
||||
open Group I
|
||||
answer : (Setoid._∼_ U) ((g ∘ f) (x ·A y)) ((g ∘ f) x ·C (g ∘ f) y)
|
||||
answer = (Equivalence.transitive (Setoid.eq U)) (GroupHom.wellDefined gHom (GroupHom.groupHom fHom {x} {y}) ) (GroupHom.groupHom gHom {f x} {f y})
|
||||
answer : (Setoid._∼_ U) ((g ∘ f) (x +A y)) ((g ∘ f) x ·C (g ∘ f) y)
|
||||
answer = (Equivalence.transitive (Setoid.eq U)) (GroupHom.wellDefined gHom (GroupHom.groupHom hom {x} {y}) ) (GroupHom.groupHom gHom {f x} {f y})
|
||||
|
||||
homRespectsInverse : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} {G : Group S _·A_} {H : Group T _·B_} {underF : A → B} → (f : GroupHom G H underF) → {x : A} → Setoid._∼_ T (underF (Group.inverse G x)) (Group.inverse H (underF x))
|
||||
homRespectsInverse {T = T} {_·A_ = _·A_} {_·B_ = _·B_} {G = G} {H = H} {underF = f} fHom {x} = rightInversesAreUnique H (f x) (f (Group.inverse G x)) (transitive (symmetric (GroupHom.groupHom fHom)) (transitive (GroupHom.wellDefined fHom (Group.invLeft G)) (imageOfIdentityIsIdentity fHom)))
|
||||
homRespectsInverse : {x : A} → Setoid._∼_ T (f (Group.inverse G x)) (Group.inverse H (f x))
|
||||
homRespectsInverse {x} = rightInversesAreUnique H (f x) (f (Group.inverse G x)) (transitive (symmetric (GroupHom.groupHom hom)) (transitive (GroupHom.wellDefined hom (Group.invLeft G)) imageOfIdentityIsIdentity))
|
||||
where
|
||||
open Setoid T
|
||||
open Equivalence eq
|
||||
|
||||
identityHom : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+A_ : A → A → A} (G : Group S _+A_) → GroupHom G G id
|
||||
GroupHom.groupHom (identityHom {S = S} G) = Equivalence.reflexive (Setoid.eq S)
|
||||
GroupHom.wellDefined (identityHom G) = id
|
||||
zeroImpliesInverseZero : {x : A} → Setoid._∼_ T (f x) (Group.0G H) → Setoid._∼_ T (f (Group.inverse G x)) (Group.0G H)
|
||||
zeroImpliesInverseZero {x} fx=0 = transitive homRespectsInverse (transitive (inverseWellDefined H fx=0) (invIdent H))
|
||||
where
|
||||
open Setoid T
|
||||
open Equivalence eq
|
||||
|
||||
@@ -14,11 +14,13 @@ open import Fields.FieldOfFractions.Setoid
|
||||
open import Sets.EquivalenceRelations
|
||||
open import Groups.Lemmas
|
||||
open import Groups.Homomorphisms.Lemmas
|
||||
open import Groups.Subgroups.Definition
|
||||
open import Groups.Subgroups.Normal.Definition
|
||||
|
||||
module Groups.QuotientGroup.Definition where
|
||||
|
||||
quotientGroup : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} (G : Group S _·A_) {H : Group T _·B_} → {underf : A → B} → (f : GroupHom G H underf) → Group (quotientGroupSetoid G f) _·A_
|
||||
Group.+WellDefined (quotientGroup {S = S} {T = T} {_·A_ = _·A_} {_·B_ = _·B_} G {H = H} {underf = f} fHom) {x} {y} {m} {n} x~m y~n = ans
|
||||
quotientGroupByHom : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} (G : Group S _·A_) {H : Group T _·B_} → {underf : A → B} → (f : GroupHom G H underf) → Group (quotientGroupSetoid G f) _·A_
|
||||
Group.+WellDefined (quotientGroupByHom {S = S} {T = T} {_·A_ = _·A_} {_·B_ = _·B_} G {H = H} {underf = f} fHom) {x} {y} {m} {n} x~m y~n = ans
|
||||
where
|
||||
open Setoid T
|
||||
open Equivalence (Setoid.eq T)
|
||||
@@ -40,15 +42,15 @@ Group.+WellDefined (quotientGroup {S = S} {T = T} {_·A_ = _·A_} {_·B_ = _·B_
|
||||
p8 = x~m
|
||||
ans : f ((x ·A y) ·A (Group.inverse G (m ·A n))) ∼ Group.0G H
|
||||
ans = transitive p1 (transitive p2 (transitive p3 (transitive p4 (transitive p5 (transitive p6 (transitive p7 p8))))))
|
||||
Group.0G (quotientGroup G fHom) = Group.0G G
|
||||
Group.inverse (quotientGroup G fHom) = Group.inverse G
|
||||
Group.+Associative (quotientGroup {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {a} {b} {c} = ans
|
||||
Group.0G (quotientGroupByHom G fHom) = Group.0G G
|
||||
Group.inverse (quotientGroupByHom G fHom) = Group.inverse G
|
||||
Group.+Associative (quotientGroupByHom {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {a} {b} {c} = ans
|
||||
where
|
||||
open Setoid T
|
||||
open Equivalence (Setoid.eq T)
|
||||
ans : f ((a ·A (b ·A c)) ·A (Group.inverse G ((a ·A b) ·A c))) ∼ Group.0G H
|
||||
ans = transitive (GroupHom.wellDefined fHom (transferToRight'' G (Group.+Associative G))) (imageOfIdentityIsIdentity fHom)
|
||||
Group.identRight (quotientGroup {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {a} = ans
|
||||
Group.identRight (quotientGroupByHom {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {a} = ans
|
||||
where
|
||||
open Group G
|
||||
open Setoid T
|
||||
@@ -56,7 +58,7 @@ Group.identRight (quotientGroup {S = S} {T = T} {_·A_ = _·A_} G {H = H} {under
|
||||
transitiveG = Equivalence.transitive (Setoid.eq S)
|
||||
ans : f ((a ·A 0G) ·A inverse a) ∼ Group.0G H
|
||||
ans = transitive (GroupHom.wellDefined fHom (transitiveG (Group.+WellDefined G (Group.identRight G) (Equivalence.reflexive (Setoid.eq S))) (Group.invRight G))) (imageOfIdentityIsIdentity fHom)
|
||||
Group.identLeft (quotientGroup {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {a} = ans
|
||||
Group.identLeft (quotientGroupByHom {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {a} = ans
|
||||
where
|
||||
open Group G
|
||||
open Setoid T
|
||||
@@ -64,14 +66,14 @@ Group.identLeft (quotientGroup {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf
|
||||
transitiveG = Equivalence.transitive (Setoid.eq S)
|
||||
ans : f ((0G ·A a) ·A (inverse a)) ∼ Group.0G H
|
||||
ans = transitive (GroupHom.wellDefined fHom (transitiveG (Group.+WellDefined G (Group.identLeft G) (Equivalence.reflexive (Setoid.eq S))) (Group.invRight G))) (imageOfIdentityIsIdentity fHom)
|
||||
Group.invLeft (quotientGroup {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {x} = ans
|
||||
Group.invLeft (quotientGroupByHom {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {x} = ans
|
||||
where
|
||||
open Group G
|
||||
open Setoid T
|
||||
open Equivalence eq
|
||||
ans : f ((inverse x ·A x) ·A (inverse 0G)) ∼ (Group.0G H)
|
||||
ans = transitive (GroupHom.wellDefined fHom (Equivalence.transitive (Setoid.eq S) (replaceGroupOp G (Equivalence.symmetric (Setoid.eq S) (Group.invLeft G)) (Equivalence.symmetric (Setoid.eq S) (invIdent G)) (Equivalence.reflexive (Setoid.eq S)) ((Equivalence.reflexive (Setoid.eq S))) ((Equivalence.reflexive (Setoid.eq S)))) (identRight {0G}))) (imageOfIdentityIsIdentity fHom)
|
||||
Group.invRight (quotientGroup {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {x} = ans
|
||||
Group.invRight (quotientGroupByHom {S = S} {T = T} {_·A_ = _·A_} G {H = H} {underf = f} fHom) {x} = ans
|
||||
where
|
||||
open Group G
|
||||
open Setoid T
|
||||
|
||||
21
Groups/QuotientGroup/Lemmas.agda
Normal file
21
Groups/QuotientGroup/Lemmas.agda
Normal file
@@ -0,0 +1,21 @@
|
||||
{-# OPTIONS --warning=error --safe --without-K #-}
|
||||
|
||||
open import Functions
|
||||
open import Sets.FinSet
|
||||
open import LogicalFormulae
|
||||
open import Groups.Definition
|
||||
open import Groups.Groups
|
||||
open import Groups.FiniteGroups.Definition
|
||||
open import Groups.Homomorphisms.Definition
|
||||
open import Groups.Abelian.Definition
|
||||
open import Setoids.Setoids
|
||||
open import Fields.FieldOfFractions.Setoid
|
||||
open import Sets.EquivalenceRelations
|
||||
open import Groups.Lemmas
|
||||
open import Groups.QuotientGroup.Definition
|
||||
|
||||
module Groups.QuotientGroup.Lemmas {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+A_ : A → A → A} {_+B_ : B → B → B} (G : Group S _+A_) (H : Group T _+B_) {f : A → B} (fHom : GroupHom G H f) where
|
||||
|
||||
projectionMapIsGroupHom : GroupHom G (quotientGroupByHom G fHom) id
|
||||
GroupHom.groupHom projectionMapIsGroupHom {x} {y} = quotientGroupLemma G fHom (Equivalence.reflexive (Setoid.eq T))
|
||||
GroupHom.wellDefined projectionMapIsGroupHom x=y = quotientGroupLemma G fHom (GroupHom.wellDefined fHom x=y)
|
||||
@@ -11,7 +11,21 @@ open import Groups.Homomorphisms.Definition
|
||||
|
||||
module Groups.Subgroups.Definition {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} (G : Group S _+_) where
|
||||
|
||||
open import Setoids.Subset S
|
||||
open Group G
|
||||
|
||||
subgroup : {c : _} {pred : A → Set c} → (wd : {x y : A} → (Setoid._∼_ S x y) → (pred x → pred y)) → Set (a ⊔ c)
|
||||
subgroup {pred = pred} wd = ({g h : A} → (pred g) → (pred h) → pred (g + h)) & pred 0G & ({g : A} → (pred g) → (pred (inverse g)))
|
||||
subgroup : {c : _} (pred : A → Set c) → Set (a ⊔ b ⊔ c)
|
||||
subgroup pred = subset pred && (({g h : A} → (pred g) → (pred h) → pred (g + h)) & pred 0G & ({g : A} → (pred g) → (pred (inverse g))))
|
||||
|
||||
subgroupOp : {c : _} {pred : A → Set c} → (s : subgroup pred) → Sg A pred → Sg A pred → Sg A pred
|
||||
subgroupOp {pred = pred} (_ ,, record { one = one ; two = two ; three = three }) (a , prA) (b , prB) = (a + b) , one prA prB
|
||||
|
||||
subgroupIsGroup : {c : _} {pred : A → Set c} → (subs : subset pred) → (s : subgroup pred) → Group (subsetSetoid subs) (subgroupOp s)
|
||||
Group.+WellDefined (subgroupIsGroup _ s) {m , prM} {n , prN} {x , prX} {y , prY} m=x n=y = +WellDefined m=x n=y
|
||||
Group.0G (subgroupIsGroup _ (_ ,, record { two = two })) = 0G , two
|
||||
Group.inverse (subgroupIsGroup _ (_ ,, record { three = three })) (a , prA) = (inverse a) , three prA
|
||||
Group.+Associative (subgroupIsGroup _ s) {a , prA} {b , prB} {c , prC} = +Associative
|
||||
Group.identRight (subgroupIsGroup _ s) {a , prA} = identRight
|
||||
Group.identLeft (subgroupIsGroup _ s) {a , prA} = identLeft
|
||||
Group.invLeft (subgroupIsGroup _ s) {a , prA} = invLeft
|
||||
Group.invRight (subgroupIsGroup _ s) {a , prA} = invRight
|
||||
|
||||
@@ -16,11 +16,10 @@ open import Groups.Isomorphisms.Definition
|
||||
open import Groups.Subgroups.Definition
|
||||
open import Groups.Lemmas
|
||||
open import Groups.Abelian.Definition
|
||||
open import Groups.QuotientGroup.Definition
|
||||
|
||||
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
|
||||
|
||||
module Groups.Subgroups.Normal.Definition {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} (G : Group S _+_) where
|
||||
|
||||
normalSubgroup : {c : _} {pred : A → Set c} (wd : {x y : A} → (Setoid._∼_ S x y) → (pred x → pred y)) (sub : subgroup G wd) → Set (a ⊔ c)
|
||||
normalSubgroup {pred = pred} wd sub = {g k : A} → pred k → pred (g + (k + Group.inverse G g))
|
||||
normalSubgroup : {c : _} {pred : A → Set c} (sub : subgroup G pred) → Set (a ⊔ c)
|
||||
normalSubgroup {pred = pred} sub = {g k : A} → pred k → pred (g + (k + Group.inverse G g))
|
||||
|
||||
@@ -64,12 +64,13 @@ groupKernelGroupPred {T = T} G {H = H} {f = f} hom a = Setoid._∼_ T (f a) (Gro
|
||||
groupKernelGroupPredWd : {a b c d : _} {A : Set a} {B : Set c} {S : Setoid {a} {b} A} {T : Setoid {c} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} (G : Group S _·A_) {H : Group T _·B_} {f : A → B} (hom : GroupHom G H f) → {x y : A} → (Setoid._∼_ S x y) → (groupKernelGroupPred G hom x → groupKernelGroupPred G hom y)
|
||||
groupKernelGroupPredWd {S = S} {T = T} G hom {x} {y} x=y fx=0 = Equivalence.transitive (Setoid.eq T) (GroupHom.wellDefined hom (Equivalence.symmetric (Setoid.eq S) x=y)) fx=0
|
||||
|
||||
groupKernelGroupIsSubgroup : {a b c d : _} {A : Set a} {B : Set c} {S : Setoid {a} {b} A} {T : Setoid {c} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} (G : Group S _·A_) {H : Group T _·B_} {f : A → B} (hom : GroupHom G H f) → subgroup G (groupKernelGroupPredWd G hom)
|
||||
_&_&_.one (groupKernelGroupIsSubgroup {S = S} {T = T} G {H = H} hom) g=0 h=0 = Equivalence.transitive (Setoid.eq T) (GroupHom.groupHom hom) (Equivalence.transitive (Setoid.eq T) (Group.+WellDefined H g=0 h=0) (Group.identLeft H))
|
||||
_&_&_.two (groupKernelGroupIsSubgroup G hom) = imageOfIdentityIsIdentity hom
|
||||
_&_&_.three (groupKernelGroupIsSubgroup {S = S} {T = T} G {H = H} hom) g=0 = Equivalence.transitive (Setoid.eq T) (homRespectsInverse hom) (Equivalence.transitive (Setoid.eq T) (inverseWellDefined H g=0) (invIdent H))
|
||||
groupKernelGroupIsSubgroup : {a b c d : _} {A : Set a} {B : Set c} {S : Setoid {a} {b} A} {T : Setoid {c} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} (G : Group S _·A_) {H : Group T _·B_} {f : A → B} (hom : GroupHom G H f) → subgroup G (groupKernelGroupPred G hom)
|
||||
_&_&_.one (_&&_.snd (groupKernelGroupIsSubgroup {S = S} {T = T} G {H = H} hom)) g=0 h=0 = Equivalence.transitive (Setoid.eq T) (GroupHom.groupHom hom) (Equivalence.transitive (Setoid.eq T) (Group.+WellDefined H g=0 h=0) (Group.identLeft H))
|
||||
_&_&_.two (_&&_.snd (groupKernelGroupIsSubgroup G hom)) = imageOfIdentityIsIdentity hom
|
||||
_&_&_.three (_&&_.snd (groupKernelGroupIsSubgroup {S = S} {T = T} G {H = H} hom)) g=0 = Equivalence.transitive (Setoid.eq T) (homRespectsInverse hom) (Equivalence.transitive (Setoid.eq T) (inverseWellDefined H g=0) (invIdent H))
|
||||
_&&_.fst (groupKernelGroupIsSubgroup G hom) = groupKernelGroupPredWd G hom
|
||||
|
||||
groupKernelGroupIsNormalSubgroup : {a b c d : _} {A : Set a} {B : Set c} {S : Setoid {a} {b} A} {T : Setoid {c} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} (G : Group S _·A_) {H : Group T _·B_} {f : A → B} (hom : GroupHom G H f) → normalSubgroup G (groupKernelGroupPredWd G hom) (groupKernelGroupIsSubgroup G hom)
|
||||
groupKernelGroupIsNormalSubgroup : {a b c d : _} {A : Set a} {B : Set c} {S : Setoid {a} {b} A} {T : Setoid {c} {d} B} {_·A_ : A → A → A} {_·B_ : B → B → B} (G : Group S _·A_) {H : Group T _·B_} {f : A → B} (hom : GroupHom G H f) → normalSubgroup G (groupKernelGroupIsSubgroup G hom)
|
||||
groupKernelGroupIsNormalSubgroup {T = T} G {H = H} hom k=0 = transitive groupHom (transitive (+WellDefined reflexive groupHom) (transitive (+WellDefined reflexive (transitive (+WellDefined k=0 reflexive) identLeft)) (transitive (symmetric groupHom) (transitive (wellDefined (Group.invRight G)) (imageOfIdentityIsIdentity hom)))))
|
||||
where
|
||||
open Setoid T
|
||||
@@ -77,8 +78,8 @@ groupKernelGroupIsNormalSubgroup {T = T} G {H = H} hom k=0 = transitive groupHom
|
||||
open Equivalence eq
|
||||
open GroupHom hom
|
||||
|
||||
abelianGroupSubgroupIsNormal : {a b c : _} {A : Set a} {B : Set b} {S : Setoid {a} {b} A} {_+_ : A → A → A} {G : Group S _+_} {pred : A → Set c} (predWd : {x y : A} → (Setoid._∼_ S x y) → (pred x → pred y)) → (s : subgroup G predWd) → AbelianGroup G → normalSubgroup G predWd s
|
||||
abelianGroupSubgroupIsNormal {S = S} {_+_ = _+_} {G = G} predWd record { one = respectsPlus ; two = respectsId ; three = respectsInv } abelian {k} {l} prK = predWd (transitive (transitive (transitive (symmetric identLeft) (+WellDefined (symmetric invRight) reflexive)) (symmetric +Associative)) (+WellDefined reflexive commutative)) prK
|
||||
abelianGroupSubgroupIsNormal : {a b c : _} {A : Set a} {B : Set b} {S : Setoid {a} {b} A} {_+_ : A → A → A} {G : Group S _+_} {pred : A → Set c} → (s : subgroup G pred) → AbelianGroup G → normalSubgroup G s
|
||||
abelianGroupSubgroupIsNormal {S = S} {_+_ = _+_} {G = G} (predWd ,, record { one = respectsPlus ; two = respectsId ; three = respectsInv }) abelian {k} {l} prK = predWd (transitive (transitive (transitive (symmetric identLeft) (+WellDefined (symmetric invRight) reflexive)) (symmetric +Associative)) (+WellDefined reflexive commutative)) prK
|
||||
where
|
||||
open Group G
|
||||
open AbelianGroup abelian
|
||||
|
||||
@@ -13,7 +13,6 @@ open import Groups.Subgroups.Definition
|
||||
open import Groups.Homomorphisms.Definition
|
||||
open import Groups.Homomorphisms.Lemmas
|
||||
open import Groups.Actions.Definition
|
||||
open import Groups.Groups2
|
||||
open import Sets.EquivalenceRelations
|
||||
|
||||
module Groups.SymmetricGroups.Lemmas where
|
||||
|
||||
@@ -17,11 +17,13 @@ open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
|
||||
|
||||
module Rings.Ideals.Definition {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ _*_ : A → A → A} (R : Ring S _+_ _*_) where
|
||||
|
||||
open import Groups.Subgroups.Definition (Ring.additiveGroup R)
|
||||
|
||||
ringKernel : {c d : _} {C : Set c} {T : Setoid {c} {d} C} {_+2_ _*2_ : C → C → C} (R2 : Ring T _+2_ _*2_) {f : A → C} (fHom : RingHom R R2 f) → Set (a ⊔ d)
|
||||
ringKernel {T = T} R2 {f} fHom = Sg A (λ a → Setoid._∼_ T (f a) (Ring.0R R2))
|
||||
|
||||
ideal : {c : _} {pred : A → Set c} → (wd : {x y : A} → (Setoid._∼_ S x y) → (pred x → pred y)) → Set (a ⊔ c)
|
||||
ideal {pred = pred} wd = (pred (Ring.0R R)) & ({x y : A} → pred x → pred y → pred (x + y)) & ({x : A} → {y : A} → pred x → pred (x * y))
|
||||
ideal : {c : _} (pred : A → Set c) → Set (a ⊔ b ⊔ c)
|
||||
ideal pred = subgroup pred && ({x : A} → {y : A} → pred x → pred (x * y))
|
||||
|
||||
idealPredForKernel : {c d : _} {C : Set c} {T : Setoid {c} {d} C} {_+2_ _*2_ : C → C → C} (R2 : Ring T _+2_ _*2_) {f : A → C} (fHom : RingHom R R2 f) → A → Set d
|
||||
idealPredForKernel {T = T} R2 {f} fHom a = Setoid._∼_ T (f a) (Ring.0R R2)
|
||||
@@ -29,17 +31,24 @@ idealPredForKernel {T = T} R2 {f} fHom a = Setoid._∼_ T (f a) (Ring.0R R2)
|
||||
idealPredForKernelWellDefined : {c d : _} {C : Set c} {T : Setoid {c} {d} C} {_+2_ _*2_ : C → C → C} (R2 : Ring T _+2_ _*2_) {f : A → C} (fHom : RingHom R R2 f) → {x y : A} → (Setoid._∼_ S x y) → (idealPredForKernel R2 fHom x → idealPredForKernel R2 fHom y)
|
||||
idealPredForKernelWellDefined {T = T} R2 {f} fHom a x=0 = Equivalence.transitive (Setoid.eq T) (GroupHom.wellDefined (RingHom.groupHom fHom) (Equivalence.symmetric (Setoid.eq S) a)) x=0
|
||||
|
||||
kernelIdealIsIdeal : {c d : _} {C : Set c} {T : Setoid {c} {d} C} {_+2_ _*2_ : C → C → C} {R2 : Ring T _+2_ _*2_} {f : A → C} (fHom : RingHom R R2 f) → ideal {pred = idealPredForKernel R2 fHom} (idealPredForKernelWellDefined R2 fHom)
|
||||
_&_&_.one (kernelIdealIsIdeal fHom) = imageOfIdentityIsIdentity (RingHom.groupHom fHom)
|
||||
_&_&_.two (kernelIdealIsIdeal {T = T} {R2 = R2} fHom) fx=0 fy=0 = transitive (transitive (GroupHom.groupHom (RingHom.groupHom fHom)) (+WellDefined fx=0 fy=0)) identLeft
|
||||
kernelIdealIsIdeal : {c d : _} {C : Set c} {T : Setoid {c} {d} C} {_+2_ _*2_ : C → C → C} {R2 : Ring T _+2_ _*2_} {f : A → C} (fHom : RingHom R R2 f) → ideal (idealPredForKernel R2 fHom)
|
||||
_&&_.fst (_&&_.fst (kernelIdealIsIdeal {R2 = R2} fHom)) = idealPredForKernelWellDefined R2 fHom
|
||||
_&_&_.one (_&&_.snd (_&&_.fst (kernelIdealIsIdeal {T = T} {R2 = R2} fHom))) {x} {y} fx=0 fy=0 = transitive (transitive (GroupHom.groupHom (RingHom.groupHom fHom)) (+WellDefined fx=0 fy=0)) identLeft
|
||||
where
|
||||
open Ring R2
|
||||
open Group (Ring.additiveGroup R2)
|
||||
open Setoid T
|
||||
open Equivalence eq
|
||||
_&_&_.three (kernelIdealIsIdeal {T = T} {R2 = R2} fHom) fx=0 = transitive (RingHom.ringHom fHom) (transitive (Ring.*WellDefined R2 fx=0 reflexive) (transitive (Ring.*Commutative R2) (Ring.timesZero R2)))
|
||||
_&_&_.two (_&&_.snd (_&&_.fst (kernelIdealIsIdeal fHom))) = imageOfIdentityIsIdentity (RingHom.groupHom fHom)
|
||||
_&_&_.three (_&&_.snd (_&&_.fst (kernelIdealIsIdeal {T = T} {R2 = R2} fHom))) {x} fx=0 = zeroImpliesInverseZero (RingHom.groupHom fHom) fx=0
|
||||
where
|
||||
open Ring R2
|
||||
open Group (Ring.additiveGroup R2)
|
||||
open Setoid T
|
||||
open Equivalence eq
|
||||
_&&_.snd (kernelIdealIsIdeal {T = T} {R2 = R2} {f = f} fHom) {x} {y} fx=0 = transitive (RingHom.ringHom fHom) (transitive (Ring.*WellDefined R2 fx=0 reflexive) (transitive (Ring.*Commutative R2) (Ring.timesZero R2 {f y})))
|
||||
where
|
||||
open Setoid T
|
||||
open Equivalence eq
|
||||
|
||||
-- TODO : define the quotient by an ideal; note that the result is a ring
|
||||
|
||||
@@ -6,115 +6,116 @@ open import LogicalFormulae
|
||||
open import Sets.EquivalenceRelations
|
||||
|
||||
module Setoids.Setoids where
|
||||
record Setoid {a} {b} (A : Set a) : Set (a ⊔ lsuc b) where
|
||||
infix 1 _∼_
|
||||
field
|
||||
_∼_ : A → A → Set b
|
||||
eq : Equivalence _∼_
|
||||
|
||||
open Equivalence eq
|
||||
record Setoid {a} {b} (A : Set a) : Set (a ⊔ lsuc b) where
|
||||
infix 1 _∼_
|
||||
field
|
||||
_∼_ : A → A → Set b
|
||||
eq : Equivalence _∼_
|
||||
|
||||
~refl : {r : A} → (r ∼ r)
|
||||
~refl {r} = reflexive {r}
|
||||
open Equivalence eq
|
||||
|
||||
record Quotient {a} {b} {c} {A : Set a} {image : Set b} (S : Setoid {a} {c} A) : Set (a ⊔ b ⊔ c) where
|
||||
~refl : {r : A} → (r ∼ r)
|
||||
~refl {r} = reflexive {r}
|
||||
|
||||
record Quotient {a} {b} {c} {A : Set a} {image : Set b} (S : Setoid {a} {c} A) : Set (a ⊔ b ⊔ c) where
|
||||
open Setoid S
|
||||
field
|
||||
map : A → image
|
||||
mapWellDefined : {x y : A} → x ∼ y → map x ≡ map y
|
||||
mapSurjective : Surjection map
|
||||
mapInjective : {x y : A} → map x ≡ map y → x ∼ y
|
||||
|
||||
record SetoidToSet {a b c : _} {A : Set a} (S : Setoid {a} {c} A) (quotient : Set b) : Set (a ⊔ b ⊔ c) where
|
||||
open Setoid S
|
||||
field
|
||||
project : A → quotient
|
||||
wellDefined : (x y : A) → (x ∼ y) → project x ≡ project y
|
||||
surj : Surjection project
|
||||
inj : (x y : A) → project x ≡ project y → x ∼ y
|
||||
|
||||
open Setoid
|
||||
|
||||
reflSetoid : {a : _} (A : Set a) → Setoid A
|
||||
_∼_ (reflSetoid A) a b = a ≡ b
|
||||
Equivalence.reflexive (eq (reflSetoid A)) = refl
|
||||
Equivalence.symmetric (eq (reflSetoid A)) {b} {.b} refl = refl
|
||||
Equivalence.transitive (eq (reflSetoid A)) {b} {.b} {.b} refl refl = refl
|
||||
|
||||
record SetoidInjection {a b c d : _} {A : Set a} {B : Set b} (S : Setoid {a} {c} A) (T : Setoid {b} {d} B) (f : A → B) : Set (a ⊔ b ⊔ c ⊔ d) where
|
||||
open Setoid S renaming (_∼_ to _∼A_)
|
||||
open Setoid T renaming (_∼_ to _∼B_)
|
||||
field
|
||||
wellDefined : {x y : A} → x ∼A y → (f x) ∼B (f y)
|
||||
injective : {x y : A} → (f x) ∼B (f y) → x ∼A y
|
||||
|
||||
record SetoidSurjection {a b c d : _} {A : Set a} {B : Set b} (S : Setoid {a} {c} A) (T : Setoid {b} {d} B) (f : A → B) : Set (a ⊔ b ⊔ c ⊔ d) where
|
||||
open Setoid S renaming (_∼_ to _∼A_)
|
||||
open Setoid T renaming (_∼_ to _∼B_)
|
||||
field
|
||||
wellDefined : {x y : A} → x ∼A y → (f x) ∼B (f y)
|
||||
surjective : {x : B} → Sg A (λ a → f a ∼B x)
|
||||
|
||||
record SetoidBijection {a b c d : _} {A : Set a} {B : Set b} (S : Setoid {a} {c} A) (T : Setoid {b} {d} B) (f : A → B) : Set (a ⊔ b ⊔ c ⊔ d) where
|
||||
field
|
||||
inj : SetoidInjection S T f
|
||||
surj : SetoidSurjection S T f
|
||||
wellDefined : {x y : A} → Setoid._∼_ S x y → Setoid._∼_ T (f x) (f y)
|
||||
wellDefined = SetoidInjection.wellDefined inj
|
||||
|
||||
record SetoidsBiject {a b c d : _} {A : Set a} {B : Set b} (S : Setoid {a} {c} A) (T : Setoid {b} {d} B) : Set (a ⊔ b ⊔ c ⊔ d) where
|
||||
field
|
||||
bij : A → B
|
||||
bijIsBijective : SetoidBijection S T bij
|
||||
|
||||
setoidInjComp : {a b c d e f : _} {A : Set a} {B : Set b} {C : Set c} {S : Setoid {a} {d} A} {T : Setoid {b} {e} B} {U : Setoid {c} {f} C} {g : A → B} {h : B → C} → (gB : SetoidInjection S T g) (hB : SetoidInjection T U h) → SetoidInjection S U (h ∘ g)
|
||||
SetoidInjection.wellDefined (setoidInjComp gI hI) x~y = SetoidInjection.wellDefined hI (SetoidInjection.wellDefined gI x~y)
|
||||
SetoidInjection.injective (setoidInjComp gI hI) hgx~hgy = SetoidInjection.injective gI (SetoidInjection.injective hI hgx~hgy)
|
||||
|
||||
setoidSurjComp : {a b c d e f : _} {A : Set a} {B : Set b} {C : Set c} {S : Setoid {a} {d} A} {T : Setoid {b} {e} B} {U : Setoid {c} {f} C} {g : A → B} {h : B → C} → (gB : SetoidSurjection S T g) (hB : SetoidSurjection T U h) → SetoidSurjection S U (h ∘ g)
|
||||
SetoidSurjection.wellDefined (setoidSurjComp gI hI) x~y = SetoidSurjection.wellDefined hI (SetoidSurjection.wellDefined gI x~y)
|
||||
SetoidSurjection.surjective (setoidSurjComp gI hI) {x} with SetoidSurjection.surjective hI {x}
|
||||
SetoidSurjection.surjective (setoidSurjComp gI hI) {x} | a , prA with SetoidSurjection.surjective gI {a}
|
||||
SetoidSurjection.surjective (setoidSurjComp {U = U} gI hI) {x} | a , prA | b , prB = b , Equivalence.transitive (Setoid.eq U) (SetoidSurjection.wellDefined hI prB) prA
|
||||
where
|
||||
open Setoid U
|
||||
|
||||
setoidBijComp : {a b c d e f : _} {A : Set a} {B : Set b} {C : Set c} {S : Setoid {a} {d} A} {T : Setoid {b} {e} B} {U : Setoid {c} {f} C} {g : A → B} {h : B → C} → (gB : SetoidBijection S T g) (hB : SetoidBijection T U h) → SetoidBijection S U (h ∘ g)
|
||||
SetoidBijection.inj (setoidBijComp gB hB) = setoidInjComp (SetoidBijection.inj gB) (SetoidBijection.inj hB)
|
||||
SetoidBijection.surj (setoidBijComp gB hB) = setoidSurjComp (SetoidBijection.surj gB) (SetoidBijection.surj hB)
|
||||
|
||||
setoidIdIsBijective : {a b : _} {A : Set a} {S : Setoid {a} {b} A} → SetoidBijection S S (λ i → i)
|
||||
SetoidInjection.wellDefined (SetoidBijection.inj (setoidIdIsBijective {A = A})) = id
|
||||
SetoidInjection.injective (SetoidBijection.inj (setoidIdIsBijective {A = A})) = id
|
||||
SetoidSurjection.wellDefined (SetoidBijection.surj (setoidIdIsBijective {A = A})) = id
|
||||
SetoidSurjection.surjective (SetoidBijection.surj (setoidIdIsBijective {S = S})) {x} = x , Equivalence.reflexive (Setoid.eq S)
|
||||
|
||||
record SetoidInvertible {a b c d : _} {A : Set a} {B : Set b} (S : Setoid {a} {c} A) (T : Setoid {b} {d} B) (f : A → B) : Set (a ⊔ b ⊔ c ⊔ d) where
|
||||
field
|
||||
fWellDefined : {x y : A} → Setoid._∼_ S x y → Setoid._∼_ T (f x) (f y)
|
||||
inverse : B → A
|
||||
inverseWellDefined : {x y : B} → Setoid._∼_ T x y → Setoid._∼_ S (inverse x) (inverse y)
|
||||
isLeft : (b : B) → Setoid._∼_ T (f (inverse b)) b
|
||||
isRight : (a : A) → Setoid._∼_ S (inverse (f a)) a
|
||||
|
||||
setoidBijectiveImpliesInvertible : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {f : A → B} (bij : SetoidBijection S T f) → SetoidInvertible S T f
|
||||
SetoidInvertible.fWellDefined (setoidBijectiveImpliesInvertible bij) x~y = SetoidInjection.wellDefined (SetoidBijection.inj bij) x~y
|
||||
SetoidInvertible.inverse (setoidBijectiveImpliesInvertible bij) x with SetoidSurjection.surjective (SetoidBijection.surj bij) {x}
|
||||
SetoidInvertible.inverse (setoidBijectiveImpliesInvertible bij) x | a , b = a
|
||||
SetoidInvertible.inverseWellDefined (setoidBijectiveImpliesInvertible bij) {x} {y} x~y with SetoidSurjection.surjective (SetoidBijection.surj bij) {x}
|
||||
SetoidInvertible.inverseWellDefined (setoidBijectiveImpliesInvertible {T = T} bij) {x} {y} x~y | a , prA with SetoidSurjection.surjective (SetoidBijection.surj bij) {y}
|
||||
... | b , prB = SetoidInjection.injective (SetoidBijection.inj bij) (Equivalence.transitive (Setoid.eq T) prA (Equivalence.transitive (Setoid.eq T) x~y (Equivalence.symmetric (Setoid.eq T) prB)))
|
||||
where
|
||||
open Setoid T
|
||||
SetoidInvertible.isLeft (setoidBijectiveImpliesInvertible bij) b with SetoidSurjection.surjective (SetoidBijection.surj bij) {b}
|
||||
... | a , prA = prA
|
||||
SetoidInvertible.isRight (setoidBijectiveImpliesInvertible {f = f} bij) b with SetoidSurjection.surjective (SetoidBijection.surj bij) {f b}
|
||||
... | fb , prFb = SetoidInjection.injective (SetoidBijection.inj bij) prFb
|
||||
|
||||
setoidInvertibleImpliesBijective : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {f : A → B} (inv : SetoidInvertible S T f) → SetoidBijection S T f
|
||||
SetoidInjection.wellDefined (SetoidBijection.inj (setoidInvertibleImpliesBijective inv)) x~y = SetoidInvertible.fWellDefined inv x~y
|
||||
SetoidInjection.injective (SetoidBijection.inj (setoidInvertibleImpliesBijective {S = S} {f = f} inv)) {x} {y} pr = Equivalence.transitive (Setoid.eq S) (Equivalence.symmetric (Setoid.eq S) (SetoidInvertible.isRight inv x)) (Equivalence.transitive (Setoid.eq S) (SetoidInvertible.inverseWellDefined inv pr) (SetoidInvertible.isRight inv y))
|
||||
where
|
||||
open Setoid S
|
||||
field
|
||||
map : A → image
|
||||
mapWellDefined : {x y : A} → x ∼ y → map x ≡ map y
|
||||
mapSurjective : Surjection map
|
||||
mapInjective : {x y : A} → map x ≡ map y → x ∼ y
|
||||
|
||||
record SetoidToSet {a b c : _} {A : Set a} (S : Setoid {a} {c} A) (quotient : Set b) : Set (a ⊔ b ⊔ c) where
|
||||
open Setoid S
|
||||
field
|
||||
project : A → quotient
|
||||
wellDefined : (x y : A) → (x ∼ y) → project x ≡ project y
|
||||
surj : Surjection project
|
||||
inj : (x y : A) → project x ≡ project y → x ∼ y
|
||||
|
||||
open Setoid
|
||||
|
||||
reflSetoid : {a : _} (A : Set a) → Setoid A
|
||||
_∼_ (reflSetoid A) a b = a ≡ b
|
||||
Equivalence.reflexive (eq (reflSetoid A)) = refl
|
||||
Equivalence.symmetric (eq (reflSetoid A)) {b} {.b} refl = refl
|
||||
Equivalence.transitive (eq (reflSetoid A)) {b} {.b} {.b} refl refl = refl
|
||||
|
||||
record SetoidInjection {a b c d : _} {A : Set a} {B : Set b} (S : Setoid {a} {c} A) (T : Setoid {b} {d} B) (f : A → B) : Set (a ⊔ b ⊔ c ⊔ d) where
|
||||
open Setoid S renaming (_∼_ to _∼A_)
|
||||
open Setoid T renaming (_∼_ to _∼B_)
|
||||
field
|
||||
wellDefined : {x y : A} → x ∼A y → (f x) ∼B (f y)
|
||||
injective : {x y : A} → (f x) ∼B (f y) → x ∼A y
|
||||
|
||||
record SetoidSurjection {a b c d : _} {A : Set a} {B : Set b} (S : Setoid {a} {c} A) (T : Setoid {b} {d} B) (f : A → B) : Set (a ⊔ b ⊔ c ⊔ d) where
|
||||
open Setoid S renaming (_∼_ to _∼A_)
|
||||
open Setoid T renaming (_∼_ to _∼B_)
|
||||
field
|
||||
wellDefined : {x y : A} → x ∼A y → (f x) ∼B (f y)
|
||||
surjective : {x : B} → Sg A (λ a → f a ∼B x)
|
||||
|
||||
record SetoidBijection {a b c d : _} {A : Set a} {B : Set b} (S : Setoid {a} {c} A) (T : Setoid {b} {d} B) (f : A → B) : Set (a ⊔ b ⊔ c ⊔ d) where
|
||||
field
|
||||
inj : SetoidInjection S T f
|
||||
surj : SetoidSurjection S T f
|
||||
wellDefined : {x y : A} → Setoid._∼_ S x y → Setoid._∼_ T (f x) (f y)
|
||||
wellDefined = SetoidInjection.wellDefined inj
|
||||
|
||||
record SetoidsBiject {a b c d : _} {A : Set a} {B : Set b} (S : Setoid {a} {c} A) (T : Setoid {b} {d} B) : Set (a ⊔ b ⊔ c ⊔ d) where
|
||||
field
|
||||
bij : A → B
|
||||
bijIsBijective : SetoidBijection S T bij
|
||||
|
||||
setoidInjComp : {a b c d e f : _} {A : Set a} {B : Set b} {C : Set c} {S : Setoid {a} {d} A} {T : Setoid {b} {e} B} {U : Setoid {c} {f} C} {g : A → B} {h : B → C} → (gB : SetoidInjection S T g) (hB : SetoidInjection T U h) → SetoidInjection S U (h ∘ g)
|
||||
SetoidInjection.wellDefined (setoidInjComp gI hI) x~y = SetoidInjection.wellDefined hI (SetoidInjection.wellDefined gI x~y)
|
||||
SetoidInjection.injective (setoidInjComp gI hI) hgx~hgy = SetoidInjection.injective gI (SetoidInjection.injective hI hgx~hgy)
|
||||
|
||||
setoidSurjComp : {a b c d e f : _} {A : Set a} {B : Set b} {C : Set c} {S : Setoid {a} {d} A} {T : Setoid {b} {e} B} {U : Setoid {c} {f} C} {g : A → B} {h : B → C} → (gB : SetoidSurjection S T g) (hB : SetoidSurjection T U h) → SetoidSurjection S U (h ∘ g)
|
||||
SetoidSurjection.wellDefined (setoidSurjComp gI hI) x~y = SetoidSurjection.wellDefined hI (SetoidSurjection.wellDefined gI x~y)
|
||||
SetoidSurjection.surjective (setoidSurjComp gI hI) {x} with SetoidSurjection.surjective hI {x}
|
||||
SetoidSurjection.surjective (setoidSurjComp gI hI) {x} | a , prA with SetoidSurjection.surjective gI {a}
|
||||
SetoidSurjection.surjective (setoidSurjComp {U = U} gI hI) {x} | a , prA | b , prB = b , Equivalence.transitive (Setoid.eq U) (SetoidSurjection.wellDefined hI prB) prA
|
||||
where
|
||||
open Setoid U
|
||||
|
||||
setoidBijComp : {a b c d e f : _} {A : Set a} {B : Set b} {C : Set c} {S : Setoid {a} {d} A} {T : Setoid {b} {e} B} {U : Setoid {c} {f} C} {g : A → B} {h : B → C} → (gB : SetoidBijection S T g) (hB : SetoidBijection T U h) → SetoidBijection S U (h ∘ g)
|
||||
SetoidBijection.inj (setoidBijComp gB hB) = setoidInjComp (SetoidBijection.inj gB) (SetoidBijection.inj hB)
|
||||
SetoidBijection.surj (setoidBijComp gB hB) = setoidSurjComp (SetoidBijection.surj gB) (SetoidBijection.surj hB)
|
||||
|
||||
setoidIdIsBijective : {a b : _} {A : Set a} {S : Setoid {a} {b} A} → SetoidBijection S S (λ i → i)
|
||||
SetoidInjection.wellDefined (SetoidBijection.inj (setoidIdIsBijective {A = A})) = id
|
||||
SetoidInjection.injective (SetoidBijection.inj (setoidIdIsBijective {A = A})) = id
|
||||
SetoidSurjection.wellDefined (SetoidBijection.surj (setoidIdIsBijective {A = A})) = id
|
||||
SetoidSurjection.surjective (SetoidBijection.surj (setoidIdIsBijective {S = S})) {x} = x , Equivalence.reflexive (Setoid.eq S)
|
||||
|
||||
record SetoidInvertible {a b c d : _} {A : Set a} {B : Set b} (S : Setoid {a} {c} A) (T : Setoid {b} {d} B) (f : A → B) : Set (a ⊔ b ⊔ c ⊔ d) where
|
||||
field
|
||||
fWellDefined : {x y : A} → Setoid._∼_ S x y → Setoid._∼_ T (f x) (f y)
|
||||
inverse : B → A
|
||||
inverseWellDefined : {x y : B} → Setoid._∼_ T x y → Setoid._∼_ S (inverse x) (inverse y)
|
||||
isLeft : (b : B) → Setoid._∼_ T (f (inverse b)) b
|
||||
isRight : (a : A) → Setoid._∼_ S (inverse (f a)) a
|
||||
|
||||
setoidBijectiveImpliesInvertible : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {f : A → B} (bij : SetoidBijection S T f) → SetoidInvertible S T f
|
||||
SetoidInvertible.fWellDefined (setoidBijectiveImpliesInvertible bij) x~y = SetoidInjection.wellDefined (SetoidBijection.inj bij) x~y
|
||||
SetoidInvertible.inverse (setoidBijectiveImpliesInvertible bij) x with SetoidSurjection.surjective (SetoidBijection.surj bij) {x}
|
||||
SetoidInvertible.inverse (setoidBijectiveImpliesInvertible bij) x | a , b = a
|
||||
SetoidInvertible.inverseWellDefined (setoidBijectiveImpliesInvertible bij) {x} {y} x~y with SetoidSurjection.surjective (SetoidBijection.surj bij) {x}
|
||||
SetoidInvertible.inverseWellDefined (setoidBijectiveImpliesInvertible {T = T} bij) {x} {y} x~y | a , prA with SetoidSurjection.surjective (SetoidBijection.surj bij) {y}
|
||||
... | b , prB = SetoidInjection.injective (SetoidBijection.inj bij) (Equivalence.transitive (Setoid.eq T) prA (Equivalence.transitive (Setoid.eq T) x~y (Equivalence.symmetric (Setoid.eq T) prB)))
|
||||
where
|
||||
open Setoid T
|
||||
SetoidInvertible.isLeft (setoidBijectiveImpliesInvertible bij) b with SetoidSurjection.surjective (SetoidBijection.surj bij) {b}
|
||||
... | a , prA = prA
|
||||
SetoidInvertible.isRight (setoidBijectiveImpliesInvertible {f = f} bij) b with SetoidSurjection.surjective (SetoidBijection.surj bij) {f b}
|
||||
... | fb , prFb = SetoidInjection.injective (SetoidBijection.inj bij) prFb
|
||||
|
||||
setoidInvertibleImpliesBijective : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {f : A → B} (inv : SetoidInvertible S T f) → SetoidBijection S T f
|
||||
SetoidInjection.wellDefined (SetoidBijection.inj (setoidInvertibleImpliesBijective inv)) x~y = SetoidInvertible.fWellDefined inv x~y
|
||||
SetoidInjection.injective (SetoidBijection.inj (setoidInvertibleImpliesBijective {S = S} {f = f} inv)) {x} {y} pr = Equivalence.transitive (Setoid.eq S) (Equivalence.symmetric (Setoid.eq S) (SetoidInvertible.isRight inv x)) (Equivalence.transitive (Setoid.eq S) (SetoidInvertible.inverseWellDefined inv pr) (SetoidInvertible.isRight inv y))
|
||||
where
|
||||
open Setoid S
|
||||
SetoidSurjection.wellDefined (SetoidBijection.surj (setoidInvertibleImpliesBijective inv)) x~y = SetoidInvertible.fWellDefined inv x~y
|
||||
SetoidSurjection.surjective (SetoidBijection.surj (setoidInvertibleImpliesBijective inv)) {x} = SetoidInvertible.inverse inv x , SetoidInvertible.isLeft inv x
|
||||
SetoidSurjection.wellDefined (SetoidBijection.surj (setoidInvertibleImpliesBijective inv)) x~y = SetoidInvertible.fWellDefined inv x~y
|
||||
SetoidSurjection.surjective (SetoidBijection.surj (setoidInvertibleImpliesBijective inv)) {x} = SetoidInvertible.inverse inv x , SetoidInvertible.isLeft inv x
|
||||
|
||||
21
Setoids/Subset.agda
Normal file
21
Setoids/Subset.agda
Normal file
@@ -0,0 +1,21 @@
|
||||
{-# OPTIONS --safe --warning=error --without-K #-}
|
||||
|
||||
open import Functions
|
||||
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
|
||||
open import LogicalFormulae
|
||||
open import Sets.EquivalenceRelations
|
||||
open import Setoids.Setoids
|
||||
|
||||
module Setoids.Subset {a b : _} {A : Set a} (S : Setoid {a} {b} A) where
|
||||
|
||||
open Setoid S
|
||||
open Equivalence eq
|
||||
|
||||
subset : {c : _} (pred : A → Set c) → Set (a ⊔ b ⊔ c)
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subset pred = ({x y : A} → x ∼ y → pred x → pred y)
|
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|
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subsetSetoid : {c : _} {pred : A → Set c} → (subs : subset pred) → Setoid (Sg A pred)
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Setoid._∼_ (subsetSetoid subs) (x , predX) (y , predY) = Setoid._∼_ S x y
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Equivalence.reflexive (Setoid.eq (subsetSetoid subs)) {a , b} = reflexive
|
||||
Equivalence.symmetric (Setoid.eq (subsetSetoid subs)) {a , prA} {b , prB} x = symmetric x
|
||||
Equivalence.transitive (Setoid.eq (subsetSetoid subs)) {a , prA} {b , prB} {c , prC} x y = transitive x y
|
||||
Reference in New Issue
Block a user