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Move equiv rels (#46)
This commit is contained in:
Patrick Stevens
GitHub
@@ -12,6 +12,7 @@ open import Groups.Groups
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open import Rings.Definition
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open import Rings.Lemmas
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open import Fields.Fields
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open import Sets.EquivalenceRelations
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module Groups.Examples.ExampleSheet1 where
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@@ -23,9 +24,7 @@ module Groups.Examples.ExampleSheet1 where
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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 (Setoid.eq S))
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open Reflexive (Equivalence.reflexiveEq (Setoid.eq S))
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open Symmetric (Equivalence.symmetricEq (Setoid.eq S))
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open Equivalence eq
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question1' : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} → (G : Group S _+_) → Setoid._∼_ S ((Group.identity G) + (Group.identity G)) (Group.identity G)
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question1' G = Group.multIdentRight G
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@@ -42,32 +41,26 @@ module Groups.Examples.ExampleSheet1 where
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subgroupIntersectionOp {S = S} {_+_ = _+_} {_+H1_ = _+H1_} {_+H2_ = _+H2_} G {h1Hom = h1Hom} {h2Hom = h2Hom} H1 H2 (ofElt (b , prB) (c , prC)) (ofElt (b2 , prB2) (c2 , prC2)) = ofElt ((b +H1 b2) , GroupHom.groupHom h1Hom) ((c +H2 c2) , transitive (GroupHom.groupHom h2Hom) (transitive (Group.wellDefined G prC prC2) (Group.wellDefined G (symmetric prB) (symmetric prB2))))
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where
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open Setoid S
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open Transitive (Equivalence.transitiveEq (Setoid.eq S))
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open Symmetric (Equivalence.symmetricEq (Setoid.eq S))
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open Equivalence eq
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subgroupIntersectionSetoid : {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} {_+_ : A → A → A} {_+H1_ : B → B → B} {_+H2_ : C → C → C} (G : Group S _+_) {H1grp : Group T _+H1_} {H2grp : Group U _+H2_} {h1Inj : B → A} {h2Inj : C → A} {h1Hom : GroupHom H1grp G h1Inj} {h2Hom : GroupHom H2grp G h2Inj} (H1 : Subgroup G H1grp h1Hom) (H2 : Subgroup G H2grp h2Hom) → Setoid (SubgroupIntersectionElement G H1 H2)
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Setoid._∼_ (subgroupIntersectionSetoid {T = T} {U = U} G {h1Inj = h1} {h2Inj = h2} H1 H2) (ofElt (xH1 , prxH1) (xH2 , prxH2)) (ofElt (yH1 , pryH1) (yH2 , pryH2)) = (Setoid._∼_ T xH1 yH1) && (Setoid._∼_ U xH2 yH2)
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Reflexive.reflexive (Equivalence.reflexiveEq (Setoid.eq (subgroupIntersectionSetoid {T = T} {U = U} G H1 H2))) {ofElt (a , prA) (b , prB)} = (Reflexive.reflexive (Equivalence.reflexiveEq (Setoid.eq T))) ,, (Reflexive.reflexive (Equivalence.reflexiveEq (Setoid.eq U)))
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Symmetric.symmetric (Equivalence.symmetricEq (Setoid.eq (subgroupIntersectionSetoid {T = T} {U = U} G H1 H2))) {ofElt (a , prA) (b , prB)} {ofElt (c , prC) (d , prD)} (fst ,, snd) = Symmetric.symmetric (Equivalence.symmetricEq (Setoid.eq T)) fst ,, Symmetric.symmetric (Equivalence.symmetricEq (Setoid.eq U)) snd
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Transitive.transitive (Equivalence.transitiveEq (Setoid.eq (subgroupIntersectionSetoid {T = T} {U = U} G H1 H2))) {ofElt (_ , _) (_ , _)} {ofElt (_ , _) (_ , _)} {ofElt (_ , _) (_ , _)} (fst1 ,, snd1) (fst2 ,, snd2) = Transitive.transitive (Equivalence.transitiveEq (Setoid.eq T)) fst1 fst2 ,, Transitive.transitive (Equivalence.transitiveEq (Setoid.eq U)) snd1 snd2
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Equivalence.reflexive (Setoid.eq (subgroupIntersectionSetoid {T = T} {U = U} G H1 H2)) {ofElt (a , prA) (b , prB)} = (Equivalence.reflexive (Setoid.eq T)) ,, (Equivalence.reflexive (Setoid.eq U))
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Equivalence.symmetric (Setoid.eq (subgroupIntersectionSetoid {T = T} {U = U} G H1 H2)) {ofElt (a , prA) (b , prB)} {ofElt (c , prC) (d , prD)} (fst ,, snd) = Equivalence.symmetric (Setoid.eq T) fst ,, Equivalence.symmetric (Setoid.eq U) snd
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Equivalence.transitive (Setoid.eq (subgroupIntersectionSetoid {T = T} {U = U} G H1 H2)) {ofElt (_ , _) (_ , _)} {ofElt (_ , _) (_ , _)} {ofElt (_ , _) (_ , _)} (fst1 ,, snd1) (fst2 ,, snd2) = Equivalence.transitive (Setoid.eq T) fst1 fst2 ,, Equivalence.transitive (Setoid.eq U) snd1 snd2
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subgroupIntersectionGroup : {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} {_+_ : A → A → A} {_+H1_ : B → B → B} {_+H2_ : C → C → C} (G : Group S _+_) {H1grp : Group T _+H1_} {H2grp : Group U _+H2_} {h1Inj : B → A} {h2Inj : C → A} {h1Hom : GroupHom H1grp G h1Inj} {h2Hom : GroupHom H2grp G h2Inj} (H1 : Subgroup G H1grp h1Hom) (H2 : Subgroup G H2grp h2Hom) → Group (subgroupIntersectionSetoid G H1 H2) (subgroupIntersectionOp G H1 H2)
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Group.wellDefined (subgroupIntersectionGroup {S = S} {T = T} {U = U} G {H1grp = h1} {H2grp = h2} H1 H2) {ofElt (_ , _) (_ , _)} {ofElt (_ , _ ) (_ , _)} {ofElt (_ , _) (_ , _)} {ofElt (_ , _) (_ , _)} (pr1 ,, pr2) (pr3 ,, pr4) = transitiveT (Group.wellDefined h1 pr1 reflexiveT) (Group.wellDefined h1 reflexiveT pr3) ,, transitiveU (Group.wellDefined h2 pr2 reflexiveU) ((Group.wellDefined h2 reflexiveU pr4))
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where
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open Group G
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open Setoid T
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open Transitive (Equivalence.transitiveEq (Setoid.eq T)) renaming (transitive to transitiveT)
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open Symmetric (Equivalence.symmetricEq (Setoid.eq T)) renaming (symmetric to symmetricT)
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open Reflexive (Equivalence.reflexiveEq (Setoid.eq T)) renaming (reflexive to reflexiveT)
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open Transitive (Equivalence.transitiveEq (Setoid.eq U)) renaming (transitive to transitiveU)
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open Symmetric (Equivalence.symmetricEq (Setoid.eq U)) renaming (symmetric to symmetricU)
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open Reflexive (Equivalence.reflexiveEq (Setoid.eq U)) renaming (reflexive to reflexiveU)
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open Equivalence (Setoid.eq T) renaming (transitive to transitiveT ; symmetric to symmetricT ; reflexive to reflexiveT)
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open Equivalence (Setoid.eq U) renaming (transitive to transitiveU ; symmetric to symmetricU ; reflexive to reflexiveU)
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Group.identity (subgroupIntersectionGroup G {H1grp = H1grp} {H2grp = H2grp} {h1Hom = h1Hom} {h2Hom = h2Hom} H1 H2) = ofElt {x = Group.identity G} (Group.identity H1grp , imageOfIdentityIsIdentity h1Hom) (Group.identity H2grp , imageOfIdentityIsIdentity h2Hom)
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Group.inverse (subgroupIntersectionGroup {S = S} G {H1grp = h1} {H2grp = h2} {h1Hom = h1hom} {h2Hom = h2hom} H1 H2) (ofElt (a , prA) (b , prB)) = ofElt (Group.inverse h1 a , homRespectsInverse h1hom) (Group.inverse h2 b , transitive (homRespectsInverse h2hom) (inverseWellDefined G (transitive prB (symmetric prA))))
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where
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open Setoid S
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open Transitive (Equivalence.transitiveEq (Setoid.eq S))
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open Symmetric (Equivalence.symmetricEq (Setoid.eq S))
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open Equivalence eq
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Group.multAssoc (subgroupIntersectionGroup G {H1grp = h1} {H2grp = h2} H1 H2) {ofElt (a , prA) (b , prB)} {ofElt (c , prC) (d , prD)} {ofElt (e , prE) (f , prF)} = Group.multAssoc h1 ,, Group.multAssoc h2
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Group.multIdentRight (subgroupIntersectionGroup G {H1grp = h1} {H2grp = h2} H1 H2) {ofElt (_ , _) (_ , _)} = Group.multIdentRight h1 ,, Group.multIdentRight h2
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Group.multIdentLeft (subgroupIntersectionGroup G {H1grp = h1} {H2grp = h2} H1 H2) {ofElt (_ , _) (_ , _)} = Group.multIdentLeft h1 ,, Group.multIdentLeft h2
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@@ -86,8 +79,7 @@ module Groups.Examples.ExampleSheet1 where
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SetoidInjection.injective (Subgroup.fInj (subgroupIntersectionIsSubgroup {S = S} G H1 H2)) {ofElt (a , prA) (b , prB)} {ofElt (c , prC) (d , prD)} x~y = SetoidInjection.injective (Subgroup.fInj H1) x~y ,, SetoidInjection.injective (Subgroup.fInj H2) (transitive prB (transitive (transitive (symmetric prA) (transitive x~y prC) ) (symmetric prD)))
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where
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open Setoid S
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open Transitive (Equivalence.transitiveEq (Setoid.eq S))
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open Symmetric (Equivalence.symmetricEq (Setoid.eq S))
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open Equivalence eq
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{-
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To make sure we haven't defined something stupid, check that the intersection doesn't care which order the two subgroups came in, and check that the subgroup intersection is isomorphic to the original group in the case that the two were the same, and check that the intersection injects into the first subgroup.
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@@ -95,25 +87,24 @@ module Groups.Examples.ExampleSheet1 where
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subgroupIntersectionIsomorphic : {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} {_+_ : A → A → A} {_+H1_ : B → B → B} {_+H2_ : C → C → C} (G : Group S _+_) {H1grp : Group T _+H1_} {H2grp : Group U _+H2_} {h1Inj : B → A} {h2Inj : C → A} {h1Hom : GroupHom H1grp G h1Inj} {h2Hom : GroupHom H2grp G h2Inj} (H1 : Subgroup G H1grp h1Hom) (H2 : Subgroup G H2grp h2Hom) → GroupsIsomorphic (subgroupIntersectionGroup G H1 H2) (subgroupIntersectionGroup G H2 H1)
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GroupsIsomorphic.isomorphism (subgroupIntersectionIsomorphic G H1 H2) (ofElt (a , prA) (b , prB)) = ofElt (b , prB) (a , prA)
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GroupHom.groupHom (GroupIso.groupHom (GroupsIsomorphic.proof (subgroupIntersectionIsomorphic {T = T} {U = U} G H1 H2))) {ofElt (_ , _) (_ , _)} {ofElt (_ , _) (_ , _)} = Reflexive.reflexive (Equivalence.reflexiveEq (Setoid.eq U)) ,, Reflexive.reflexive (Equivalence.reflexiveEq (Setoid.eq T))
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GroupHom.groupHom (GroupIso.groupHom (GroupsIsomorphic.proof (subgroupIntersectionIsomorphic {T = T} {U = U} G H1 H2))) {ofElt (_ , _) (_ , _)} {ofElt (_ , _) (_ , _)} = Equivalence.reflexive (Setoid.eq U) ,, Equivalence.reflexive (Setoid.eq T)
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GroupHom.wellDefined (GroupIso.groupHom (GroupsIsomorphic.proof (subgroupIntersectionIsomorphic G H1 H2))) {ofElt (a , prA) (b , prB)} {ofElt (_ , _) (_ , _)} (fst ,, snd) = snd ,, fst
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SetoidInjection.wellDefined (SetoidBijection.inj (GroupIso.bij (GroupsIsomorphic.proof (subgroupIntersectionIsomorphic G H1 H2)))) {ofElt (_ , _) (_ , _)} {ofElt (_ , _) (_ , _)} (fst ,, snd) = snd ,, fst
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SetoidInjection.injective (SetoidBijection.inj (GroupIso.bij (GroupsIsomorphic.proof (subgroupIntersectionIsomorphic G H1 H2)))) {ofElt (_ , _) (_ , _)} {ofElt (_ , _) (_ , _)} (fst ,, snd) = snd ,, fst
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SetoidSurjection.wellDefined (SetoidBijection.surj (GroupIso.bij (GroupsIsomorphic.proof (subgroupIntersectionIsomorphic G H1 H2)))) {ofElt (_ , _) (_ , _)} {ofElt (_ , _) (_ , _)} (fst ,, snd) = snd ,, fst
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SetoidSurjection.surjective (SetoidBijection.surj (GroupIso.bij (GroupsIsomorphic.proof (subgroupIntersectionIsomorphic {T = T} {U = U} G H1 H2)))) {ofElt (a , prA) (b , prB)} = ofElt (b , prB) (a , prA) , (Reflexive.reflexive (Equivalence.reflexiveEq (Setoid.eq U)) ,, Reflexive.reflexive (Equivalence.reflexiveEq (Setoid.eq T)))
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SetoidSurjection.surjective (SetoidBijection.surj (GroupIso.bij (GroupsIsomorphic.proof (subgroupIntersectionIsomorphic {T = T} {U = U} G H1 H2)))) {ofElt (a , prA) (b , prB)} = ofElt (b , prB) (a , prA) , (Equivalence.reflexive (Setoid.eq U) ,, Equivalence.reflexive (Setoid.eq T))
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subgroupIntersectionOfSame : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+_ : A → A → A} {_+H1_ : B → B → B} (G : Group S _+_) {H1grp : Group T _+H1_} {h1Inj : B → A} {h1Hom : GroupHom H1grp G h1Inj} (H1 : Subgroup G H1grp h1Hom) → GroupsIsomorphic (subgroupIntersectionGroup G H1 H1) H1grp
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GroupsIsomorphic.isomorphism (subgroupIntersectionOfSame G H1) (ofElt (a , prA) (b , prB)) = a
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GroupHom.groupHom (GroupIso.groupHom (GroupsIsomorphic.proof (subgroupIntersectionOfSame {T = T} G H1))) {ofElt (_ , _) (_ , _)} {ofElt (_ , _) (_ , _)} = Reflexive.reflexive (Equivalence.reflexiveEq (Setoid.eq T))
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GroupHom.groupHom (GroupIso.groupHom (GroupsIsomorphic.proof (subgroupIntersectionOfSame {T = T} G H1))) {ofElt (_ , _) (_ , _)} {ofElt (_ , _) (_ , _)} = Equivalence.reflexive (Setoid.eq T)
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GroupHom.wellDefined (GroupIso.groupHom (GroupsIsomorphic.proof (subgroupIntersectionOfSame G H1))) {ofElt (_ , _) (_ , _)} {ofElt (_ , _) (_ , _)} (fst ,, _) = fst
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SetoidInjection.wellDefined (SetoidBijection.inj (GroupIso.bij (GroupsIsomorphic.proof (subgroupIntersectionOfSame G H1)))) {ofElt (_ , _) (_ , _)} {ofElt (_ , _) (_ , _)} (fst ,, _) = fst
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SetoidInjection.injective (SetoidBijection.inj (GroupIso.bij (GroupsIsomorphic.proof (subgroupIntersectionOfSame {S = S} {T = T} G {h1Hom = h1Hom} H1)))) {ofElt (a , prA) (b , prB)} {ofElt (c , prC) (d , prD)} a~b = a~b ,, SetoidInjection.injective (Subgroup.fInj H1) (transitive prB (transitive (transitive (symmetric prA) (transitive (GroupHom.wellDefined h1Hom a~b) prC)) (symmetric prD)))
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where
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open Setoid S
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open Transitive (Equivalence.transitiveEq (Setoid.eq S))
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open Symmetric (Equivalence.symmetricEq (Setoid.eq S))
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open Equivalence eq
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SetoidSurjection.wellDefined (SetoidBijection.surj (GroupIso.bij (GroupsIsomorphic.proof (subgroupIntersectionOfSame G H1)))) {ofElt (_ , _) (_ , _)} {ofElt (_ , _) (_ , _)} (fst ,, _) = fst
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SetoidSurjection.surjective (SetoidBijection.surj (GroupIso.bij (GroupsIsomorphic.proof (subgroupIntersectionOfSame {S = S} {T = T} G H1)))) {b} = ofElt (b , Reflexive.reflexive (Equivalence.reflexiveEq (Setoid.eq S))) (b , Reflexive.reflexive (Equivalence.reflexiveEq (Setoid.eq S))) , (Reflexive.reflexive (Equivalence.reflexiveEq (Setoid.eq T)))
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SetoidSurjection.surjective (SetoidBijection.surj (GroupIso.bij (GroupsIsomorphic.proof (subgroupIntersectionOfSame {S = S} {T = T} G H1)))) {b} = ofElt (b , Equivalence.reflexive (Setoid.eq S)) (b , Equivalence.reflexive (Setoid.eq S)) , (Equivalence.reflexive (Setoid.eq T))
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{- TODO: finish question 2 -}
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@@ -137,9 +128,7 @@ module Groups.Examples.ExampleSheet1 where
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where
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open Setoid S
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open Ring R
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open Transitive (Equivalence.transitiveEq (Setoid.eq S))
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open Reflexive (Equivalence.reflexiveEq (Setoid.eq S))
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open Symmetric (Equivalence.symmetricEq (Setoid.eq S))
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open Equivalence eq
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bad : Setoid._∼_ S xCoeff2 0R
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bad with Field.allInvertible F xCoeff1 xCoeffNonzero1
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... | xinv , pr' = transitive (symmetric multIdentIsIdent) (transitive (multWellDefined (symmetric pr') reflexive) (transitive (symmetric multAssoc) (transitive (multWellDefined reflexive pr) (ringTimesZero R))))
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@@ -150,9 +139,7 @@ module Groups.Examples.ExampleSheet1 where
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where
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open Setoid S
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open Ring R
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open Transitive (Equivalence.transitiveEq eq)
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open Symmetric (Equivalence.symmetricEq eq)
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open Reflexive (Equivalence.reflexiveEq eq)
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open Equivalence eq
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ans : (((xCoeff * xCoeff') * r) + ((xCoeff * constant') + constant)) ∼ (xCoeff * ((xCoeff' * r) + constant')) + constant
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ans = transitive (Group.multAssoc additiveGroup) (Group.wellDefined additiveGroup (transitive (Group.wellDefined additiveGroup (symmetric (Ring.multAssoc R)) reflexive) (symmetric (Ring.multDistributes R))) (reflexive {constant}))
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@@ -160,56 +147,47 @@ module Groups.Examples.ExampleSheet1 where
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Setoid._∼_ (linearFunctionsSetoid {S = S} I) f1 f2 = ((LinearFunction.xCoeff f1) ∼ (LinearFunction.xCoeff f2)) && ((LinearFunction.constant f1) ∼ (LinearFunction.constant f2))
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where
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open Setoid S
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Reflexive.reflexive (Equivalence.reflexiveEq (Setoid.eq (linearFunctionsSetoid {S = S} I))) = Reflexive.reflexive (Equivalence.reflexiveEq (Setoid.eq S)) ,, Reflexive.reflexive (Equivalence.reflexiveEq (Setoid.eq S))
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Symmetric.symmetric (Equivalence.symmetricEq (Setoid.eq (linearFunctionsSetoid {S = S} I))) (fst ,, snd) = Symmetric.symmetric (Equivalence.symmetricEq (Setoid.eq S)) fst ,, Symmetric.symmetric (Equivalence.symmetricEq (Setoid.eq S)) snd
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Transitive.transitive (Equivalence.transitiveEq (Setoid.eq (linearFunctionsSetoid {S = S} I))) (fst1 ,, snd1) (fst2 ,, snd2) = Transitive.transitive (Equivalence.transitiveEq (Setoid.eq S)) fst1 fst2 ,, Transitive.transitive (Equivalence.transitiveEq (Setoid.eq S)) snd1 snd2
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Equivalence.reflexive (Setoid.eq (linearFunctionsSetoid {S = S} I)) = Equivalence.reflexive (Setoid.eq S) ,, Equivalence.reflexive (Setoid.eq S)
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Equivalence.symmetric (Setoid.eq (linearFunctionsSetoid {S = S} I)) (fst ,, snd) = Equivalence.symmetric (Setoid.eq S) fst ,, Equivalence.symmetric (Setoid.eq S) snd
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Equivalence.transitive (Setoid.eq (linearFunctionsSetoid {S = S} I)) (fst1 ,, snd1) (fst2 ,, snd2) = Equivalence.transitive (Setoid.eq S) fst1 fst2 ,, Equivalence.transitive (Setoid.eq S) snd1 snd2
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linearFunctionsGroup : {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} (F : Field R) → Group (linearFunctionsSetoid F) (composeLinearFunctions)
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Group.wellDefined (linearFunctionsGroup {R = R} F) {record { xCoeff = xCoeffM ; xCoeffNonzero = xCoeffNonzeroM ; constant = constantM }} {record { xCoeff = xCoeffN ; xCoeffNonzero = xCoeffNonzeroN ; constant = constantN }} {record { xCoeff = xCoeffX ; xCoeffNonzero = xCoeffNonzeroX ; constant = constantX }} {record { xCoeff = xCoeff ; xCoeffNonzero = xCoeffNonzero ; constant = constant }} (fst1 ,, snd1) (fst2 ,, snd2) = multWellDefined fst1 fst2 ,, Group.wellDefined additiveGroup (multWellDefined fst1 snd2) snd1
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where
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open Ring R
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Group.identity (linearFunctionsGroup {S = S} {R = R} F) = record { xCoeff = Ring.1R R ; constant = Ring.0R R ; xCoeffNonzero = λ p → Field.nontrivial F (Symmetric.symmetric (Equivalence.symmetricEq (Setoid.eq S)) p) }
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Group.identity (linearFunctionsGroup {S = S} {R = R} F) = record { xCoeff = Ring.1R R ; constant = Ring.0R R ; xCoeffNonzero = λ p → Field.nontrivial F (Equivalence.symmetric (Setoid.eq S) p) }
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Group.inverse (linearFunctionsGroup {S = S} {_*_ = _*_} {R = R} F) record { xCoeff = xCoeff ; constant = c ; xCoeffNonzero = pr } with Field.allInvertible F xCoeff pr
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... | (inv , pr') = record { xCoeff = inv ; constant = inv * (Group.inverse (Ring.additiveGroup R) c) ; xCoeffNonzero = λ p → Field.nontrivial F (transitive (symmetric (transitive (Ring.multWellDefined R p reflexive) (transitive (Ring.multCommutative R) (ringTimesZero R)))) pr') }
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where
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open Setoid S
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open Transitive (Equivalence.transitiveEq (Setoid.eq S))
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open Reflexive (Equivalence.reflexiveEq (Setoid.eq S))
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open Symmetric (Equivalence.symmetricEq (Setoid.eq S))
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open Equivalence eq
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Group.multAssoc (linearFunctionsGroup {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} F) {record { xCoeff = xA ; xCoeffNonzero = xANonzero ; constant = cA }} {record { xCoeff = xB ; xCoeffNonzero = xBNonzero ; constant = cB }} {record { xCoeff = xC ; xCoeffNonzero = xCNonzero ; constant = cC }} = Ring.multAssoc R ,, transitive (Group.wellDefined additiveGroup (transitive multDistributes (Group.wellDefined additiveGroup multAssoc reflexive)) reflexive) (symmetric (Group.multAssoc additiveGroup))
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where
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open Transitive (Equivalence.transitiveEq (Setoid.eq S))
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open Reflexive (Equivalence.reflexiveEq (Setoid.eq S))
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open Symmetric (Equivalence.symmetricEq (Setoid.eq S))
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open Setoid S
|
||||
open Equivalence eq
|
||||
open Ring R
|
||||
Group.multIdentRight (linearFunctionsGroup {S = S} {_+_ = _+_} {_*_ = _*_} {R = R} F) {record { xCoeff = xCoeff ; xCoeffNonzero = xCoeffNonzero ; constant = constant }} = transitive (Ring.multCommutative R) (Ring.multIdentIsIdent R) ,, transitive (Group.wellDefined additiveGroup (ringTimesZero R) reflexive) (Group.multIdentLeft additiveGroup)
|
||||
where
|
||||
open Ring R
|
||||
open Setoid S
|
||||
open Transitive (Equivalence.transitiveEq (Setoid.eq S))
|
||||
open Reflexive (Equivalence.reflexiveEq (Setoid.eq S))
|
||||
open Equivalence eq
|
||||
Group.multIdentLeft (linearFunctionsGroup {S = S} {R = R} F) {record { xCoeff = xCoeff ; xCoeffNonzero = xCoeffNonzero ; constant = constant }} = multIdentIsIdent ,, transitive (Group.multIdentRight additiveGroup) multIdentIsIdent
|
||||
where
|
||||
open Setoid S
|
||||
open Ring R
|
||||
open Transitive (Equivalence.transitiveEq (Setoid.eq S))
|
||||
open Equivalence eq
|
||||
Group.invLeft (linearFunctionsGroup F) {record { xCoeff = xCoeff ; xCoeffNonzero = xCoeffNonzero ; constant = constant }} with Field.allInvertible F xCoeff xCoeffNonzero
|
||||
Group.invLeft (linearFunctionsGroup {S = S} {R = R} F) {record { xCoeff = xCoeff ; xCoeffNonzero = xCoeffNonzero ; constant = constant }} | inv , prInv = prInv ,, transitive (symmetric multDistributes) (transitive (multWellDefined reflexive (Group.invRight additiveGroup)) (ringTimesZero R))
|
||||
where
|
||||
open Setoid S
|
||||
open Ring R
|
||||
open Transitive (Equivalence.transitiveEq (Setoid.eq S))
|
||||
open Reflexive (Equivalence.reflexiveEq (Setoid.eq S))
|
||||
open Symmetric (Equivalence.symmetricEq (Setoid.eq S))
|
||||
open Equivalence eq
|
||||
Group.invRight (linearFunctionsGroup {S = S} {R = R} F) {record { xCoeff = xCoeff ; xCoeffNonzero = xCoeffNonzero ; constant = constant }} with Field.allInvertible F xCoeff xCoeffNonzero
|
||||
... | inv , pr = transitive multCommutative pr ,, transitive (Group.wellDefined additiveGroup multAssoc reflexive) (transitive (Group.wellDefined additiveGroup (multWellDefined (transitive multCommutative pr) reflexive) reflexive) (transitive (Group.wellDefined additiveGroup multIdentIsIdent reflexive) (Group.invLeft additiveGroup)))
|
||||
where
|
||||
open Setoid S
|
||||
open Ring R
|
||||
open Transitive (Equivalence.transitiveEq (Setoid.eq S))
|
||||
open Reflexive (Equivalence.reflexiveEq (Setoid.eq S))
|
||||
open Symmetric (Equivalence.symmetricEq (Setoid.eq S))
|
||||
open Equivalence eq
|
||||
|
||||
{-
|
||||
Question 3, part 2: prove that linearFunctionsGroup is not abelian
|
||||
@@ -221,9 +199,7 @@ module Groups.Examples.ExampleSheet1 where
|
||||
where
|
||||
open Ring R
|
||||
open Group additiveGroup
|
||||
open Symmetric (Equivalence.symmetricEq (Setoid.eq S)) renaming (symmetric to symmetricS)
|
||||
open Transitive (Equivalence.transitiveEq (Setoid.eq S)) renaming (transitive to transitiveS)
|
||||
open Reflexive (Equivalence.reflexiveEq (Setoid.eq S)) renaming (reflexive to reflexiveS)
|
||||
open Equivalence (Setoid.eq S) renaming (symmetric to symmetricS ; transitive to transitiveS ; reflexive to reflexiveS)
|
||||
|
||||
f : LinearFunction F
|
||||
f = record { xCoeff = 1R ; xCoeffNonzero = λ p → Field.nontrivial F (symmetricS p) ; constant = 1R }
|
||||
@@ -242,8 +218,7 @@ module Groups.Examples.ExampleSheet1 where
|
||||
otherWay : Setoid._∼_ (linearFunctionsSetoid F) fg (composeLinearFunctions f g)
|
||||
otherWay = symmetricS multIdentIsIdent ,, transitiveS (symmetricS (Group.multIdentLeft additiveGroup)) (Group.wellDefined additiveGroup (symmetricS multIdentIsIdent) (reflexiveS {1R}))
|
||||
|
||||
open Transitive (Equivalence.transitiveEq (Setoid.eq (linearFunctionsSetoid F)))
|
||||
open Symmetric (Equivalence.symmetricEq (Setoid.eq (linearFunctionsSetoid F)))
|
||||
open Equivalence (Setoid.eq (linearFunctionsSetoid F))
|
||||
bad : Setoid._∼_ (linearFunctionsSetoid F) gf fg
|
||||
bad = transitive {gf} {composeLinearFunctions g f} {fg} oneWay (transitive {composeLinearFunctions g f} {composeLinearFunctions f g} {fg} (commutative {g} {f}) (symmetric {fg} {composeLinearFunctions f g} otherWay))
|
||||
|
||||
|
||||
Reference in New Issue
Block a user