1+ <!DOCTYPE HTML>
2+ < html > < head > < meta charset ="utf-8 "> < title > Cubical.Algebra.AbGroup.FinitePresentation</ title > < link rel ="stylesheet " href ="Agda.css "> </ head > < body > < pre class ="Agda "> < a id ="1 " class ="Keyword "> module</ a > < a id ="8 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html " class ="Module "> Cubical.Algebra.AbGroup.FinitePresentation</ a > < a id ="51 " class ="Keyword "> where</ a >
3+
4+ < a id ="58 " class ="Keyword "> open</ a > < a id ="63 " class ="Keyword "> import</ a > < a id ="70 " href ="Cubical.Foundations.Prelude.html " class ="Module "> Cubical.Foundations.Prelude</ a >
5+
6+ < a id ="99 " class ="Keyword "> open</ a > < a id ="104 " class ="Keyword "> import</ a > < a id ="111 " href ="Cubical.Data.Nat.html " class ="Module "> Cubical.Data.Nat</ a >
7+ < a id ="128 " class ="Keyword "> open</ a > < a id ="133 " class ="Keyword "> import</ a > < a id ="140 " href ="Cubical.Data.Int.html " class ="Module "> Cubical.Data.Int</ a >
8+
9+ < a id ="158 " class ="Keyword "> open</ a > < a id ="163 " class ="Keyword "> import</ a > < a id ="170 " href ="Cubical.HITs.PropositionalTruncation.html " class ="Module "> Cubical.HITs.PropositionalTruncation</ a >
10+
11+ < a id ="208 " class ="Keyword "> open</ a > < a id ="213 " class ="Keyword "> import</ a > < a id ="220 " href ="Cubical.Algebra.Group.html " class ="Module "> Cubical.Algebra.Group</ a >
12+ < a id ="242 " class ="Keyword "> open</ a > < a id ="247 " class ="Keyword "> import</ a > < a id ="254 " href ="Cubical.Algebra.Group.Morphisms.html " class ="Module "> Cubical.Algebra.Group.Morphisms</ a >
13+ < a id ="286 " class ="Keyword "> open</ a > < a id ="291 " class ="Keyword "> import</ a > < a id ="298 " href ="Cubical.Algebra.Group.MorphismProperties.html " class ="Module "> Cubical.Algebra.Group.MorphismProperties</ a >
14+ < a id ="339 " class ="Keyword "> open</ a > < a id ="344 " class ="Keyword "> import</ a > < a id ="351 " href ="Cubical.Algebra.AbGroup.html " class ="Module "> Cubical.Algebra.AbGroup</ a >
15+ < a id ="375 " class ="Keyword "> open</ a > < a id ="380 " class ="Keyword "> import</ a > < a id ="387 " href ="Cubical.Algebra.AbGroup.Instances.FreeAbGroup.html " class ="Module "> Cubical.Algebra.AbGroup.Instances.FreeAbGroup</ a >
16+
17+ < a id ="434 " class ="Keyword "> open</ a > < a id ="439 " class ="Keyword "> import</ a > < a id ="446 " href ="Cubical.Algebra.Group.QuotientGroup.html " class ="Module "> Cubical.Algebra.Group.QuotientGroup</ a >
18+ < a id ="482 " class ="Keyword "> open</ a > < a id ="487 " class ="Keyword "> import</ a > < a id ="494 " href ="Cubical.Algebra.Group.Subgroup.html " class ="Module "> Cubical.Algebra.Group.Subgroup</ a >
19+
20+ < a id ="526 " class ="Keyword "> private</ a >
21+ < a id ="536 " class ="Keyword "> variable</ a >
22+ < a id ="549 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#549 " class ="Generalizable "> ℓ</ a > < a id ="551 " class ="Symbol "> :</ a > < a id ="553 " href ="Agda.Primitive.html#742 " class ="Postulate "> Level</ a >
23+
24+ < a id ="560 " class ="Keyword "> record</ a > < a id ="FinitePresentation "> </ a > < a id ="567 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#567 " class ="Record "> FinitePresentation</ a > < a id ="586 " class ="Symbol "> (</ a > < a id ="587 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#587 " class ="Bound "> A</ a > < a id ="589 " class ="Symbol "> :</ a > < a id ="591 " href ="Cubical.Algebra.AbGroup.Base.html#1758 " class ="Function "> AbGroup</ a > < a id ="599 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#549 " class ="Generalizable "> ℓ</ a > < a id ="600 " class ="Symbol "> )</ a > < a id ="602 " class ="Symbol "> :</ a > < a id ="604 " href ="Agda.Primitive.html#388 " class ="Primitive "> Type</ a > < a id ="609 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#599 " class ="Bound "> ℓ</ a > < a id ="611 " class ="Keyword "> where</ a >
25+ < a id ="619 " class ="Keyword "> field</ a >
26+ < a id ="FinitePresentation.nGens "> </ a > < a id ="629 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#629 " class ="Field "> nGens</ a > < a id ="635 " class ="Symbol "> :</ a > < a id ="637 " href ="Agda.Builtin.Nat.html#203 " class ="Datatype "> ℕ</ a >
27+ < a id ="FinitePresentation.nRels "> </ a > < a id ="643 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#643 " class ="Field "> nRels</ a > < a id ="649 " class ="Symbol "> :</ a > < a id ="651 " href ="Agda.Builtin.Nat.html#203 " class ="Datatype "> ℕ</ a >
28+ < a id ="FinitePresentation.rels "> </ a > < a id ="657 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#657 " class ="Field "> rels</ a > < a id ="662 " class ="Symbol "> :</ a > < a id ="664 " href ="Cubical.Algebra.AbGroup.Base.html#4235 " class ="Function "> AbGroupHom</ a > < a id ="675 " href ="Cubical.Algebra.AbGroup.Instances.FreeAbGroup.html#4192 " class ="Function Operator "> ℤ[Fin</ a > < a id ="681 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#643 " class ="Field "> nRels</ a > < a id ="687 " href ="Cubical.Algebra.AbGroup.Instances.FreeAbGroup.html#4192 " class ="Function Operator "> ]</ a > < a id ="689 " href ="Cubical.Algebra.AbGroup.Instances.FreeAbGroup.html#4192 " class ="Function Operator "> ℤ[Fin</ a > < a id ="695 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#629 " class ="Field "> nGens</ a > < a id ="701 " href ="Cubical.Algebra.AbGroup.Instances.FreeAbGroup.html#4192 " class ="Function Operator "> ]</ a >
29+ < a id ="FinitePresentation.fpiso "> </ a > < a id ="707 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#707 " class ="Field "> fpiso</ a > < a id ="713 " class ="Symbol "> :</ a > < a id ="715 " href ="Cubical.Algebra.AbGroup.Base.html#4364 " class ="Function "> AbGroupIso</ a > < a id ="726 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#587 " class ="Bound "> A</ a > < a id ="728 " class ="Symbol "> (</ a > < a id ="729 " href ="Cubical.Algebra.AbGroup.Instances.FreeAbGroup.html#4192 " class ="Function Operator "> ℤ[Fin</ a > < a id ="735 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#629 " class ="Field "> nGens</ a > < a id ="741 " href ="Cubical.Algebra.AbGroup.Instances.FreeAbGroup.html#4192 " class ="Function Operator "> ]</ a > < a id ="743 " href ="Cubical.Algebra.AbGroup.Properties.html#2532 " class ="Function Operator "> /Im</ a > < a id ="747 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#657 " class ="Field "> rels</ a > < a id ="751 " class ="Symbol "> )</ a >
30+
31+ < a id ="isFinitelyPresented "> </ a > < a id ="754 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#754 " class ="Function "> isFinitelyPresented</ a > < a id ="774 " class ="Symbol "> :</ a > < a id ="776 " href ="Cubical.Algebra.AbGroup.Base.html#1758 " class ="Function "> AbGroup</ a > < a id ="784 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#549 " class ="Generalizable "> ℓ</ a > < a id ="786 " class ="Symbol "> →</ a > < a id ="788 " href ="Agda.Primitive.html#388 " class ="Primitive "> Type</ a > < a id ="793 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#549 " class ="Generalizable "> ℓ</ a >
32+ < a id ="795 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#754 " class ="Function "> isFinitelyPresented</ a > < a id ="815 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#815 " class ="Bound "> G</ a > < a id ="817 " class ="Symbol "> =</ a > < a id ="819 " href ="Cubical.HITs.PropositionalTruncation.Base.html#226 " class ="Datatype Operator "> ∥</ a > < a id ="821 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#567 " class ="Record "> FinitePresentation</ a > < a id ="840 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#815 " class ="Bound "> G</ a > < a id ="842 " href ="Cubical.HITs.PropositionalTruncation.Base.html#226 " class ="Datatype Operator "> ∥₁</ a >
33+
34+ < a id ="846 " class ="Keyword "> open</ a > < a id ="851 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#567 " class ="Module "> FinitePresentation</ a >
35+ < a id ="GrIsoPresFinitePresentation "> </ a > < a id ="870 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#870 " class ="Function "> GrIsoPresFinitePresentation</ a > < a id ="898 " class ="Symbol "> :</ a > < a id ="900 " class ="Symbol "> ∀</ a > < a id ="902 " class ="Symbol "> {</ a > < a id ="903 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#903 " class ="Bound "> ℓ</ a > < a id ="905 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#905 " class ="Bound "> ℓ'</ a > < a id ="907 " class ="Symbol "> }</ a > < a id ="909 " class ="Symbol "> {</ a > < a id ="910 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#910 " class ="Bound "> A</ a > < a id ="912 " class ="Symbol "> :</ a > < a id ="914 " href ="Cubical.Algebra.AbGroup.Base.html#1758 " class ="Function "> AbGroup</ a > < a id ="922 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#903 " class ="Bound "> ℓ</ a > < a id ="923 " class ="Symbol "> }</ a > < a id ="925 " class ="Symbol "> {</ a > < a id ="926 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#926 " class ="Bound "> B</ a > < a id ="928 " class ="Symbol "> :</ a > < a id ="930 " href ="Cubical.Algebra.AbGroup.Base.html#1758 " class ="Function "> AbGroup</ a > < a id ="938 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#905 " class ="Bound "> ℓ'</ a > < a id ="940 " class ="Symbol "> }</ a >
36+ < a id ="944 " class ="Symbol "> →</ a > < a id ="946 " href ="Cubical.Algebra.AbGroup.Base.html#4364 " class ="Function "> AbGroupIso</ a > < a id ="957 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#910 " class ="Bound "> A</ a > < a id ="959 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#926 " class ="Bound "> B</ a > < a id ="961 " class ="Symbol "> →</ a > < a id ="963 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#567 " class ="Record "> FinitePresentation</ a > < a id ="982 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#910 " class ="Bound "> A</ a > < a id ="984 " class ="Symbol "> →</ a > < a id ="986 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#567 " class ="Record "> FinitePresentation</ a > < a id ="1005 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#926 " class ="Bound "> B</ a >
37+ < a id ="1007 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#629 " class ="Field "> nGens</ a > < a id ="1013 " class ="Symbol "> (</ a > < a id ="1014 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#870 " class ="Function "> GrIsoPresFinitePresentation</ a > < a id ="1042 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#1042 " class ="Bound "> abG</ a > < a id ="1046 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#1046 " class ="Bound "> fpA</ a > < a id ="1049 " class ="Symbol "> )</ a > < a id ="1051 " class ="Symbol "> =</ a > < a id ="1053 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#629 " class ="Field "> nGens</ a > < a id ="1059 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#1046 " class ="Bound "> fpA</ a >
38+ < a id ="1063 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#643 " class ="Field "> nRels</ a > < a id ="1069 " class ="Symbol "> (</ a > < a id ="1070 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#870 " class ="Function "> GrIsoPresFinitePresentation</ a > < a id ="1098 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#1098 " class ="Bound "> abG</ a > < a id ="1102 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#1102 " class ="Bound "> fpA</ a > < a id ="1105 " class ="Symbol "> )</ a > < a id ="1107 " class ="Symbol "> =</ a > < a id ="1109 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#643 " class ="Field "> nRels</ a > < a id ="1115 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#1102 " class ="Bound "> fpA</ a >
39+ < a id ="1119 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#657 " class ="Field "> rels</ a > < a id ="1124 " class ="Symbol "> (</ a > < a id ="1125 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#870 " class ="Function "> GrIsoPresFinitePresentation</ a > < a id ="1153 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#1153 " class ="Bound "> abG</ a > < a id ="1157 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#1157 " class ="Bound "> fpA</ a > < a id ="1160 " class ="Symbol "> )</ a > < a id ="1162 " class ="Symbol "> =</ a > < a id ="1164 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#657 " class ="Field "> rels</ a > < a id ="1169 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#1157 " class ="Bound "> fpA</ a >
40+ < a id ="1173 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#707 " class ="Field "> fpiso</ a > < a id ="1179 " class ="Symbol "> (</ a > < a id ="1180 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#870 " class ="Function "> GrIsoPresFinitePresentation</ a > < a id ="1208 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#1208 " class ="Bound "> abG</ a > < a id ="1212 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#1212 " class ="Bound "> fpA</ a > < a id ="1215 " class ="Symbol "> )</ a > < a id ="1217 " class ="Symbol "> =</ a >
41+ < a id ="1221 " href ="Cubical.Algebra.Group.MorphismProperties.html#9259 " class ="Function "> compGroupIso</ a > < a id ="1234 " class ="Symbol "> (</ a > < a id ="1235 " href ="Cubical.Algebra.Group.MorphismProperties.html#9455 " class ="Function "> invGroupIso</ a > < a id ="1247 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#1208 " class ="Bound "> abG</ a > < a id ="1250 " class ="Symbol "> )</ a > < a id ="1252 " class ="Symbol "> (</ a > < a id ="1253 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#707 " class ="Field "> fpiso</ a > < a id ="1259 " href ="Cubical.Algebra.AbGroup.FinitePresentation.html#1212 " class ="Bound "> fpA</ a > < a id ="1262 " class ="Symbol "> )</ a >
42+ </ pre > </ body > </ html >
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