# Theory Group_ZF_2

theory Group_ZF_2
imports AbelianGroup_ZF EquivClass1
(*     This file is a part of IsarMathLib -     a library of formalized mathematics for Isabelle/Isar (ZF logic).    Copyright (C) 2005, 2006  Slawomir Kolodynski    This program is free software; Redistribution and use in source and binary forms,     with or without modification, are permitted provided that the following conditions are met:   1. Redistributions of source code must retain the above copyright notice,    this list of conditions and the following disclaimer.   2. Redistributions in binary form must reproduce the above copyright notice,    this list of conditions and the following disclaimer in the documentation and/or    other materials provided with the distribution.   3. The name of the author may not be used to endorse or promote products    derived from this software without specific prior written permission.THIS SOFTWARE IS PROVIDED BY THE AUTHOR AS IS'' AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.*)header{*\isaheader{Group\_ZF\_2.thy}*}theory Group_ZF_2 imports AbelianGroup_ZF func_ZF EquivClass1begin;text{*This theory continues Group\_ZF.thy and considers lifting the group   structure to function spaces and projecting the group structure to   quotient spaces, in particular the quotient qroup.*}section{*Lifting groups to function spaces*}text{*If we have a monoid (group) $G$ than we get a monoid (group)   structure on a space of functions valued in   in $G$ by defining $(f\cdot g)(x) := f(x)\cdot g(x)$.   We call this process ''lifting the monoid (group) to function space''.  This section formalizes this lifting.*}text{*The lifted operation is an operation on the function space.*}lemma (in monoid0) Group_ZF_2_1_L0A:  assumes A1: "F = f {lifted to function space over} X"  shows "F : (X->G)×(X->G)->(X->G)"proof -  from monoidAsssum have "f : G×G->G"    using IsAmonoid_def IsAssociative_def by simp;  with A1 show ?thesis    using func_ZF_1_L3 group0_1_L3B by auto;qed;text{*The result of the lifted operation is in the function space.*}lemma (in monoid0) Group_ZF_2_1_L0:   assumes A1:"F = f {lifted to function space over} X"  and A2:"s:X->G" "r:X->G"  shows "F⟨ s,r⟩ : X->G"proof -  from A1 have "F : (X->G)×(X->G)->(X->G)"    using Group_ZF_2_1_L0A    by simp;  with A2 show ?thesis using apply_funtype    by simp;qed;    text{*The lifted monoid operation has a neutral element, namely  the constant function with the neutral element as the value. *}lemma (in monoid0) Group_ZF_2_1_L1:   assumes A1: "F = f {lifted to function space over} X"  and A2: "E = ConstantFunction(X,TheNeutralElement(G,f))"  shows "E : X->G ∧ (∀s∈X->G. F⟨ E,s⟩ = s ∧ F⟨ s,E⟩ = s)"proof  from A2 show T1:"E : X->G"    using unit_is_neutral func1_3_L1 by simp;  show "∀s∈X->G. F⟨ E,s⟩ = s ∧ F⟨ s,E⟩ = s"  proof;    fix s assume A3:"s:X->G"    from monoidAsssum have T2:"f : G×G->G"      using IsAmonoid_def IsAssociative_def by simp;    from A3 A1 T1 have       "F⟨ E,s⟩ : X->G" "F⟨ s,E⟩ : X->G" "s : X->G"      using Group_ZF_2_1_L0 by auto;    moreover from T2 A1 T1 A2 A3 have      "∀x∈X. (F⟨ E,s⟩)(x) = s(x)"      "∀x∈X. (F⟨ s,E⟩)(x) = s(x)"      using func_ZF_1_L4 group0_1_L3B func1_3_L2 	apply_type unit_is_neutral by auto;    ultimately show       "F⟨ E,s⟩ = s ∧ F⟨ s,E⟩ = s"      using fun_extension_iff by auto;  qed;qed;text{*Monoids can be lifted to a function space.*}lemma (in monoid0) Group_ZF_2_1_T1:   assumes A1: "F = f {lifted to function space over} X"  shows "IsAmonoid(X->G,F)"proof -;  from monoidAsssum A1 have     "F {is associative on} (X->G)"    using IsAmonoid_def func_ZF_2_L4 group0_1_L3B    by auto;  moreover from A1 have     "∃ E ∈ X->G. ∀s ∈ X->G. F⟨ E,s⟩ = s ∧ F⟨ s,E⟩ = s"    using Group_ZF_2_1_L1 by blast;  ultimately show ?thesis using IsAmonoid_def    by simpqed;text{*The constant function with the neutral element as the value is the  neutral element of the lifted monoid.*}lemma Group_ZF_2_1_L2:  assumes A1: "IsAmonoid(G,f)"  and A2: "F = f {lifted to function space over} X"  and A3: "E = ConstantFunction(X,TheNeutralElement(G,f))"  shows "E = TheNeutralElement(X->G,F)"proof -   from A1 A2 have      T1:"monoid0(G,f)" and T2:"monoid0(X->G,F)"    using monoid0_def monoid0.Group_ZF_2_1_T1    by auto;  from T1 A2 A3 have     "E : X->G ∧ (∀s∈X->G. F⟨ E,s⟩ = s ∧ F⟨ s,E⟩ = s)"    using monoid0.Group_ZF_2_1_L1 by simp;  with T2 show ?thesis    using monoid0.group0_1_L4 by auto;qed;text{*The lifted operation acts on the functions in a natural way defined  by the monoid operation.*}lemma (in monoid0) lifted_val:  assumes "F = f {lifted to function space over} X"  and "s:X->G"  "r:X->G"  and "x∈X"  shows "(F⟨s,r⟩)(x) = s(x) ⊕ r(x)"  using monoidAsssum assms IsAmonoid_def IsAssociative_def      group0_1_L3B func_ZF_1_L4  by auto;text{*The lifted operation acts on the functions in a natural way defined  by the group operation. This is the same as @{text "lifted_val"}, but  in the @{text "group0"} context.*}lemma (in group0) Group_ZF_2_1_L3:  assumes "F = P {lifted to function space over} X"  and "s:X->G" "r:X->G"  and "x∈X"  shows "(F⟨s,r⟩)(x) = s(x)·r(x)"  using assms group0_2_L1 monoid0.lifted_val by simp;    text{*In the group0 context we can apply theorems proven in monoid0 context  to the lifted monoid.*}lemma (in group0) Group_ZF_2_1_L4:  assumes A1: "F = P {lifted to function space over} X"  shows "monoid0(X->G,F)"proof -;  from A1 show ?thesis    using group0_2_L1 monoid0.Group_ZF_2_1_T1 monoid0_def    by simp;qed;text{*The compostion of a function $f:X\rightarrow G$ with the group inverse  is a right inverse for the lifted group. *}lemma (in group0) Group_ZF_2_1_L5:   assumes A1: "F = P {lifted to function space over} X"  and A2: "s : X->G"  and A3: "i = GroupInv(G,P) O s"  shows "i: X->G" and "F⟨ s,i⟩ = TheNeutralElement(X->G,F)"proof -;  let ?E = "ConstantFunction(X,\<one>)"  have "?E : X->G"     using group0_2_L2 func1_3_L1 by simp;  moreover from groupAssum A2 A3 A1 have    "F⟨ s,i⟩ :  X->G" using group0_2_T2 comp_fun       Group_ZF_2_1_L4 monoid0.group0_1_L1    by simp;  moreover from groupAssum A2 A3 A1 have     "∀x∈X. (F⟨ s,i⟩)(x) = ?E(x)"    using group0_2_T2 comp_fun Group_ZF_2_1_L3       comp_fun_apply apply_funtype group0_2_L6 func1_3_L2    by simp;  moreover from groupAssum A1 have    "?E = TheNeutralElement(X->G,F)"    using IsAgroup_def Group_ZF_2_1_L2 by simp;  ultimately show "F⟨ s,i⟩ = TheNeutralElement(X->G,F)"    using fun_extension_iff IsAgroup_def Group_ZF_2_1_L2    by simp  from groupAssum A2 A3 show "i: X->G"     using group0_2_T2 comp_fun by simpqed;text{*Groups can be lifted to the function space.*}theorem (in group0) Group_ZF_2_1_T2:  assumes A1: "F = P {lifted to function space over} X"  shows "IsAgroup(X->G,F)"proof -;  from A1 have "IsAmonoid(X->G,F)"    using group0_2_L1 monoid0.Group_ZF_2_1_T1    by simp;  moreover have     "∀s∈X->G. ∃i∈X->G. F⟨ s,i⟩ = TheNeutralElement(X->G,F)"  proof;    fix s assume A2: "s : X->G"    let ?i = "GroupInv(G,P) O s"    from groupAssum A2 have "?i:X->G"      using group0_2_T2 comp_fun by simp;    moreover from A1 A2 have       "F⟨ s,?i⟩ = TheNeutralElement(X->G,F)"      using Group_ZF_2_1_L5 by fast;   ultimately show "∃i∈X->G. F⟨ s,i⟩ = TheNeutralElement(X->G,F)"       by auto;  qed;  ultimately show ?thesis using IsAgroup_def    by simp;qed;text{*What is the group inverse for the lifted group?*}lemma (in group0) Group_ZF_2_1_L6:   assumes A1: "F = P {lifted to function space over} X"  shows "∀s∈(X->G). GroupInv(X->G,F)(s) = GroupInv(G,P) O s"proof -;  from A1 have  "group0(X->G,F)"    using group0_def Group_ZF_2_1_T2     by simp;  moreover from A1 have "∀s∈X->G. GroupInv(G,P) O s : X->G ∧     F⟨ s,GroupInv(G,P) O s⟩ = TheNeutralElement(X->G,F)"    using Group_ZF_2_1_L5 by simp;  ultimately have     "∀s∈(X->G).  GroupInv(G,P) O s = GroupInv(X->G,F)(s)"    by (rule group0.group0_2_L9A);  thus ?thesis by simp;qed;text{*What is the value of the group inverse for the lifted group?*}corollary (in group0) lift_gr_inv_val:    assumes "F = P {lifted to function space over} X" and  "s : X->G" and "x∈X"  shows  "(GroupInv(X->G,F)(s))(x) = (s(x))¯"  using groupAssum assms Group_ZF_2_1_L6 group0_2_T2 comp_fun_apply  by simp;text{*What is the group inverse in a subgroup of the lifted group?*}lemma (in group0) Group_ZF_2_1_L6A:  assumes A1: "F = P {lifted to function space over} X"  and A2: "IsAsubgroup(H,F)"  and A3: "g = restrict(F,H×H)"  and A4: "s∈H"  shows "GroupInv(H,g)(s) = GroupInv(G,P) O s"proof -;  from A1 have T1: "group0(X->G,F)"    using group0_def Group_ZF_2_1_T2     by simp;  with A2 A3 A4 have "GroupInv(H,g)(s) = GroupInv(X->G,F)(s)"    using group0.group0_3_T1 restrict by simp;  moreover from T1 A1 A2 A4 have    "GroupInv(X->G,F)(s) = GroupInv(G,P) O s"    using group0.group0_3_L2 Group_ZF_2_1_L6 by blast;  ultimately show ?thesis by simp;qed;text{*If a group is abelian, then its lift to a function space is also   abelian.*}lemma (in group0) Group_ZF_2_1_L7:   assumes A1: "F = P {lifted to function space over} X"  and A2: "P {is commutative on} G"  shows "F {is commutative on} (X->G)"proof-  from A1 A2  have    "F {is commutative on} (X->range(P))"    using group_oper_assocA func_ZF_2_L2    by simp;  moreover from groupAssum have "range(P) = G"    using group0_2_L1 monoid0.group0_1_L3B    by simp;  ultimately show ?thesis by simp;qed;    section{*Equivalence relations on groups*}text{*The goal of this section is to establish that (under some conditions)   given an equivalence  relation on a group or (monoid )we can project the group (monoid)  structure on the quotient and obtain another group.*}text{*The neutral element class is neutral in the projection.*}lemma (in monoid0) Group_ZF_2_2_L1:   assumes A1: "equiv(G,r)" and A2:"Congruent2(r,f)"  and A3: "F = ProjFun2(G,r,f)"   and A4: "e = TheNeutralElement(G,f)"  shows "r{e} ∈ G//r ∧   (∀c ∈ G//r. F⟨ r{e},c⟩ = c ∧  F⟨ c,r{e}⟩ = c)"proof;  from A4 show T1:"r{e} ∈ G//r"    using unit_is_neutral quotientI    by simp;  show     "∀c ∈ G//r. F⟨ r{e},c⟩ = c ∧  F⟨ c,r{e}⟩ = c"  proof;    fix c assume A5:"c ∈ G//r"    then obtain g where D1:"g∈G" "c = r{g}"      using quotient_def by auto    with A1 A2 A3 A4 D1 show       "F⟨ r{e},c⟩ = c ∧  F⟨ c,r{e}⟩ = c"      using unit_is_neutral EquivClass_1_L10 (*group0_1_L3*)      by simp;  qed;qed;text{*The projected structure is a monoid.*}theorem (in monoid0) Group_ZF_2_2_T1:  assumes A1: "equiv(G,r)" and A2: "Congruent2(r,f)"  and A3: "F = ProjFun2(G,r,f)"  shows "IsAmonoid(G//r,F)"proof -  let ?E = "r{TheNeutralElement(G,f)}"  from A1 A2 A3 have     "?E ∈ G//r ∧ (∀c∈G//r. F⟨ ?E,c⟩ = c ∧ F⟨ c,?E⟩ = c)"    using Group_ZF_2_2_L1 by simp;  hence    "∃E∈G//r. ∀ c∈G//r. F⟨ E,c⟩ = c ∧ F⟨ c,E⟩ = c"    by auto;  with monoidAsssum A1 A2 A3 show ?thesis    using IsAmonoid_def EquivClass_2_T2    by simp;qed;text{*The class of the neutral element is the neutral element of the  projected monoid.*}lemma Group_ZF_2_2_L1:  assumes A1: "IsAmonoid(G,f)"  and A2: "equiv(G,r)" and A3: "Congruent2(r,f)"  and A4: "F = ProjFun2(G,r,f)"  and A5: "e = TheNeutralElement(G,f)"  shows " r{e} = TheNeutralElement(G//r,F)"proof -;  from A1 A2 A3 A4 have     T1:"monoid0(G,f)" and T2:"monoid0(G//r,F)"    using monoid0_def monoid0.Group_ZF_2_2_T1 by auto;  from T1 A2 A3 A4 A5 have "r{e} ∈ G//r ∧     (∀c ∈ G//r. F⟨ r{e},c⟩ = c ∧  F⟨ c,r{e}⟩ = c)"    using monoid0.Group_ZF_2_2_L1 by simp;  with T2 show ?thesis using monoid0.group0_1_L4    by auto;qed;text{*The projected operation can be defined in terms of the group operation  on representants in a natural way.*}lemma (in group0) Group_ZF_2_2_L2:  assumes A1: "equiv(G,r)" and A2: "Congruent2(r,P)"  and A3: "F = ProjFun2(G,r,P)"  and A4: "a∈G" "b∈G"  shows "F⟨ r{a},r{b}⟩ = r{a·b}"proof -;  from A1 A2 A3 A4 show ?thesis    using EquivClass_1_L10 by simp;qed;text{*The class of the inverse is a right inverse of the class.*}lemma (in group0) Group_ZF_2_2_L3:  assumes A1: "equiv(G,r)" and A2: "Congruent2(r,P)"  and A3: "F = ProjFun2(G,r,P)"  and A4: "a∈G"  shows "F⟨r{a},r{a¯}⟩ = TheNeutralElement(G//r,F)"proof -;  from A1 A2 A3 A4 have    "F⟨r{a},r{a¯}⟩ = r{\<one>}"    using inverse_in_group Group_ZF_2_2_L2 group0_2_L6     by simp;  with groupAssum A1 A2 A3 show ?thesis    using IsAgroup_def Group_ZF_2_2_L1 by simp;qed;text{*The group structure can be projected to the quotient space.*}theorem (in group0) Group_ZF_3_T2:  assumes A1: "equiv(G,r)" and A2: "Congruent2(r,P)"  shows "IsAgroup(G//r,ProjFun2(G,r,P))"proof -;  let ?F = "ProjFun2(G,r,P)"  let ?E = "TheNeutralElement(G//r,?F)"  from groupAssum A1 A2 have "IsAmonoid(G//r,?F)"    using IsAgroup_def monoid0_def monoid0.Group_ZF_2_2_T1    by simp;  moreover have    "∀c∈G//r. ∃b∈G//r. ?F⟨ c,b⟩ = ?E"  proof;    fix c assume A3: "c ∈ G//r"    then obtain g where D1: "g∈G"  "c = r{g}"      using quotient_def by auto;    let ?b = "r{g¯}"    from D1 have "?b ∈ G//r"      using inverse_in_group quotientI      by simp;    moreover from A1 A2 D1 have       "?F⟨ c,?b⟩ = ?E"      using Group_ZF_2_2_L3 by simp;    ultimately show "∃b∈G//r. ?F⟨ c,b⟩ = ?E"      by auto;  qed;  ultimately show ?thesis    using IsAgroup_def by simp;qed;text{*The group inverse (in the projected group) of a class is the class  of the inverse.*}lemma (in group0) Group_ZF_2_2_L4:  assumes A1: "equiv(G,r)" and   A2: "Congruent2(r,P)" and   A3: "F = ProjFun2(G,r,P)" and  A4: "a∈G"  shows "r{a¯} = GroupInv(G//r,F)(r{a})"proof -  from A1 A2 A3 have "group0(G//r,F)"    using Group_ZF_3_T2 group0_def by simp;  moreover from A4 have     "r{a} ∈ G//r"  "r{a¯} ∈ G//r"    using inverse_in_group quotientI by auto;  moreover from A1 A2 A3 A4 have     "F⟨r{a},r{a¯}⟩ = TheNeutralElement(G//r,F)"    using Group_ZF_2_2_L3 by simp;  ultimately show ?thesis    by (rule group0.group0_2_L9);qed;section{*Normal subgroups and quotient groups*}text{*If $H$ is a subgroup of $G$, then for every $a\in G$  we can cosider the sets $\{a\cdot h. h \in H\}$   and $\{ h\cdot a. h \in H\}$ (called a left and right ''coset of H'', resp.)  These sets sometimes form a group, called the ''quotient group''.  This section discusses the notion of quotient groups.*}text{*A normal subgorup $N$ of a group $G$ is such that $aba^{-1}$ belongs to   $N$ if $a\in G, b\in N$. *}definition  "IsAnormalSubgroup(G,P,N) ≡ IsAsubgroup(N,P) ∧   (∀n∈N.∀g∈G. P⟨  P⟨  g,n ⟩,GroupInv(G,P)(g) ⟩ ∈ N)"text{*Having a group and a normal subgroup $N$   we can create another group  consisting of eqivalence classes of the relation   $a\sim b \equiv a\cdot b^{-1} \in N$.  We will refer to this relation  as the quotient group relation. The classes of this relation are in   fact cosets of subgroup $H$.*}definition  "QuotientGroupRel(G,P,H) ≡   {⟨ a,b⟩ ∈ G×G. P⟨ a, GroupInv(G,P)(b)⟩ ∈ H}"text{*Next we define the operation in the quotient group as the  projection of the group operation on the classses of the  quotient group relation.*}definition  "QuotientGroupOp(G,P,H) ≡ ProjFun2(G,QuotientGroupRel(G,P,H ),P)";text{*Definition of a normal subgroup in a more readable notation.*}lemma (in group0) Group_ZF_2_4_L0:   assumes "IsAnormalSubgroup(G,P,H)"  and "g∈G" "n∈H"  shows "g·n·g¯ ∈ H"  using assms IsAnormalSubgroup_def by simp;text{*The quotient group relation is reflexive.*}lemma (in group0) Group_ZF_2_4_L1:   assumes "IsAsubgroup(H,P)"  shows "refl(G,QuotientGroupRel(G,P,H))"  using assms  group0_2_L6 group0_3_L5     QuotientGroupRel_def refl_def by simp;text{*The quotient group relation is symmetric.*}lemma (in group0) Group_ZF_2_4_L2:  assumes A1:"IsAsubgroup(H,P)"  shows "sym(QuotientGroupRel(G,P,H))"proof -;  {      fix a b assume A2: "⟨ a,b⟩ ∈ QuotientGroupRel(G,P,H)"    with A1 have "(a·b¯)¯ ∈ H"       using QuotientGroupRel_def group0_3_T3A      by simp;    moreover from A2 have "(a·b¯)¯ =  b·a¯"      using QuotientGroupRel_def group0_2_L12      by simp;    ultimately have "b·a¯ ∈ H" by simp;    with A2 have "⟨ b,a⟩ ∈ QuotientGroupRel(G,P,H)"      using QuotientGroupRel_def by simp   }  then show ?thesis using symI by simp;qed;text{*The quotient group relation is transistive.*}lemma (in group0) Group_ZF_2_4_L3A:  assumes A1: "IsAsubgroup(H,P)" and   A2: "⟨ a,b⟩ ∈ QuotientGroupRel(G,P,H)" and   A3: "⟨ b,c⟩ ∈ QuotientGroupRel(G,P,H)"  shows "⟨ a,c⟩ ∈ QuotientGroupRel(G,P,H)"proof -;  let ?r = "QuotientGroupRel(G,P,H)"  from A2 A3 have T1:"a∈G" "b∈G" "c∈G"    using QuotientGroupRel_def by auto  from A1 A2 A3 have "(a·b¯)·(b·c¯) ∈ H"    using  QuotientGroupRel_def group0_3_L6    by simp;  moreover from T1 have     "a·c¯ = (a·b¯)·(b·c¯)"    using group0_2_L14A by blast;  ultimately have "a·c¯ ∈ H"     by simp;  with T1 show ?thesis using QuotientGroupRel_def    by simp;qed;text{*The quotient group relation is an equivalence relation. Note  we do not need the subgroup to be normal for this to be true.*}lemma (in group0) Group_ZF_2_4_L3: assumes A1:"IsAsubgroup(H,P)"  shows "equiv(G,QuotientGroupRel(G,P,H))"proof -;  let ?r = "QuotientGroupRel(G,P,H)"  from A1 have      "∀a b c. (⟨a, b⟩ ∈ ?r  ∧  ⟨b, c⟩ ∈ ?r --> ⟨a, c⟩ ∈ ?r)"    using Group_ZF_2_4_L3A by blast;  then have "trans(?r)"    using Fol1_L2 by blast;  with A1 show ?thesis     using Group_ZF_2_4_L1 Group_ZF_2_4_L2       QuotientGroupRel_def equiv_def    by auto;qed;text{*The next lemma states the essential condition for congruency of   the group operation with respect to the quotient group relation.*}lemma (in group0) Group_ZF_2_4_L4:   assumes A1: "IsAnormalSubgroup(G,P,H)"  and A2: "⟨a1,a2⟩ ∈ QuotientGroupRel(G,P,H)"  and A3: "⟨b1,b2⟩ ∈ QuotientGroupRel(G,P,H)"  shows "⟨a1·b1, a2·b2⟩ ∈ QuotientGroupRel(G,P,H)"proof -;  from A2 A3 have T1:    "a1∈G"  "a2∈G"  "b1∈G"  "b2∈G"    "a1·b1 ∈ G"  "a2·b2 ∈ G"    "b1·b2¯ ∈ H"  "a1·a2¯ ∈ H"    using QuotientGroupRel_def group0_2_L1 monoid0.group0_1_L1    by auto;  with A1 show ?thesis using    IsAnormalSubgroup_def group0_3_L6 group0_2_L15    QuotientGroupRel_def by simp;qed;text{*If the subgroup is normal, the group operation is congruent   with respect to the quotient group relation.*}lemma Group_ZF_2_4_L5A:  assumes "IsAgroup(G,P)"  and "IsAnormalSubgroup(G,P,H)"   shows "Congruent2(QuotientGroupRel(G,P,H),P)"  using assms group0_def group0.Group_ZF_2_4_L4 Congruent2_def  by simp;text{*The quotient group is indeed a group.*}theorem Group_ZF_2_4_T1:  assumes "IsAgroup(G,P)" and "IsAnormalSubgroup(G,P,H)"  shows     "IsAgroup(G//QuotientGroupRel(G,P,H),QuotientGroupOp(G,P,H))"  using assms group0_def group0.Group_ZF_2_4_L3 IsAnormalSubgroup_def    Group_ZF_2_4_L5A group0.Group_ZF_3_T2 QuotientGroupOp_def  by simp;text{*The class (coset) of the neutral element is the neutral  element of the quotient group.*}lemma Group_ZF_2_4_L5B:   assumes "IsAgroup(G,P)" and "IsAnormalSubgroup(G,P,H)"  and "r = QuotientGroupRel(G,P,H)"  and "e = TheNeutralElement(G,P)"  shows " r{e} = TheNeutralElement(G//r,QuotientGroupOp(G,P,H))"  using assms IsAnormalSubgroup_def group0_def    IsAgroup_def group0.Group_ZF_2_4_L3 Group_ZF_2_4_L5A    QuotientGroupOp_def Group_ZF_2_2_L1  by simp;text{*A group element is equivalent to the neutral element iff it is in the  subgroup we divide the group by.*}lemma (in group0) Group_ZF_2_4_L5C: assumes "a∈G"  shows "⟨a,\<one>⟩ ∈ QuotientGroupRel(G,P,H) <-> a∈H"  using assms QuotientGroupRel_def group_inv_of_one group0_2_L2  by auto;text{*A group element is in $H$ iff its class is the neutral element of  $G/H$.*}lemma (in group0) Group_ZF_2_4_L5D:  assumes A1: "IsAnormalSubgroup(G,P,H)" and   A2: "a∈G" and   A3: "r = QuotientGroupRel(G,P,H)" and  A4: "TheNeutralElement(G//r,QuotientGroupOp(G,P,H)) = e"  shows "r{a} = e <-> ⟨a,\<one>⟩ ∈ r"proof  assume "r{a} = e"  with groupAssum assms have     "r{\<one>} = r{a}" and I: "equiv(G,r)"    using Group_ZF_2_4_L5B IsAnormalSubgroup_def Group_ZF_2_4_L3    by auto;  with A2 have "⟨\<one>,a⟩ ∈ r" using eq_equiv_class     by simp;  with I show "⟨a,\<one>⟩ ∈ r" by (rule equiv_is_sym);next assume "⟨a,\<one>⟩ ∈ r"  moreover from A1 A3 have "equiv(G,r)"    using IsAnormalSubgroup_def Group_ZF_2_4_L3    by simp;  ultimately have "r{a} = r{\<one>}"    using equiv_class_eq by simp;  with groupAssum A1 A3 A4 show "r{a} = e"    using Group_ZF_2_4_L5B by simp;qed;  text{*The class of $a\in G$ is the neutral   element of the quotient $G/H$ iff $a\in H$.*}lemma (in group0) Group_ZF_2_4_L5E:   assumes "IsAnormalSubgroup(G,P,H)" and   "a∈G" and "r = QuotientGroupRel(G,P,H)" and   "TheNeutralElement(G//r,QuotientGroupOp(G,P,H)) = e"  shows "r{a} = e <-> a∈H"  using assms Group_ZF_2_4_L5C  Group_ZF_2_4_L5D  by simp;text{*Essential condition to show that every subgroup of an abelian group   is normal.*}lemma (in group0) Group_ZF_2_4_L5:  assumes A1: "P {is commutative on} G"   and A2: "IsAsubgroup(H,P)"  and A3: "g∈G"  "h∈H"   shows "g·h·g¯ ∈ H"proof -;  from A2 A3 have T1:"h∈G" "g¯ ∈ G"     using group0_3_L2 inverse_in_group by auto;  with A3 A1 have "g·h·g¯ = g¯·g·h"    using group0_4_L4A by simp;  with A3 T1 show ?thesis using    group0_2_L6 group0_2_L2    by simp;qed;text{*Every subgroup of an abelian group is normal. Moreover, the quotient  group is also abelian.*}lemma Group_ZF_2_4_L6:  assumes A1: "IsAgroup(G,P)"  and A2: "P {is commutative on} G"   and A3: "IsAsubgroup(H,P)"  shows  "IsAnormalSubgroup(G,P,H)"  "QuotientGroupOp(G,P,H) {is commutative on} (G//QuotientGroupRel(G,P,H))"proof -;  from A1 A2 A3 show T1: "IsAnormalSubgroup(G,P,H)" using    group0_def IsAnormalSubgroup_def group0.Group_ZF_2_4_L5     by simp;  let ?r = "QuotientGroupRel(G,P,H)"  from A1 A3 T1 have "equiv(G,?r)" "Congruent2(?r,P)"    using group0_def group0.Group_ZF_2_4_L3 Group_ZF_2_4_L5A    by auto;  with A2 show     "QuotientGroupOp(G,P,H) {is commutative on} (G//QuotientGroupRel(G,P,H))"    using EquivClass_2_T1 QuotientGroupOp_def    by simp;qed;text{*The group inverse (in the quotient group) of a class (coset) is the class  of the inverse.*}lemma (in group0) Group_ZF_2_4_L7:   assumes "IsAnormalSubgroup(G,P,H)"   and "a∈G" and "r = QuotientGroupRel(G,P,H)"   and "F = QuotientGroupOp(G,P,H)"  shows "r{a¯} = GroupInv(G//r,F)(r{a})"  using groupAssum assms IsAnormalSubgroup_def Group_ZF_2_4_L3     Group_ZF_2_4_L5A QuotientGroupOp_def Group_ZF_2_2_L4  by simp;section{*Function spaces as monoids*}text{*On every space of functions $\{f : X\rightarrow X\}$   we can define a natural   monoid structure with composition as the operation. This section explores   this fact.*}text{*The next lemma states that composition has a neutral element,   namely the identity function on $X$   (the one that maps $x\in X$ into itself).*}  lemma Group_ZF_2_5_L1: assumes A1: "F = Composition(X)"  shows "∃I∈(X->X). ∀f∈(X->X). F⟨ I,f⟩ = f ∧ F⟨ f,I⟩ = f"proof-;  let ?I = "id(X)"  from A1 have     "?I ∈ X->X ∧ (∀f∈(X->X). F⟨ ?I,f⟩ = f ∧ F⟨ f,?I⟩ = f)"     using id_type func_ZF_6_L1A by simp;  thus ?thesis by auto;qedtext{*The space of functions that map a set $X$ into   itsef is a monoid with composition as operation and the identity function  as the neutral element.*}lemma Group_ZF_2_5_L2: shows  "IsAmonoid(X->X,Composition(X))"  "id(X) = TheNeutralElement(X->X,Composition(X))"proof -;  let ?I = "id(X)"  let ?F = "Composition(X)"  show "IsAmonoid(X->X,Composition(X))"     using func_ZF_5_L5 Group_ZF_2_5_L1 IsAmonoid_def    by auto;    then have "monoid0(X->X,?F)"    using monoid0_def by simp;  moreover have    "?I ∈ X->X ∧ (∀f∈(X->X). ?F⟨ ?I,f⟩ = f ∧ ?F⟨ f,?I⟩ = f)"    using id_type func_ZF_6_L1A by simp;  ultimately show "?I = TheNeutralElement(X->X,?F)"    using monoid0.group0_1_L4 by auto;qed;end;