diff --git a/thys/Nominal_Myhill_Nerode/Nominal_Myhill_Nerode.thy b/thys/Nominal_Myhill_Nerode/Nominal_Myhill_Nerode.thy
--- a/thys/Nominal_Myhill_Nerode/Nominal_Myhill_Nerode.thy
+++ b/thys/Nominal_Myhill_Nerode/Nominal_Myhill_Nerode.thy
@@ -1,3732 +1,3681 @@
-(*
-  Formalization by Cárolos Laméris 
-  
-  Note that
-    bojanczyk2014automata ~
-    Bojańczyk, M., Klin, B., & Lasota, S. (2014).
-    Automata theory in nominal sets.
-    Logical Methods in Computer Science, 10.
-
-    ( https://lmcs.episciences.org/1157 )
-*)
-
-section \<open>
-Myhill-Nerode Theorem for $G$-automata
-\<close>
-text \<open>
-We prove the Myhill-Nerode Theorem for $G$-automata / nominal $G$-automata
-following the proofs from \cite{bojanczyk2014automata} (The standard Myhill-Nerode theorem is also
-proved, as a special case of the $G$-Myhill-Nerode theorem).
-Concretely, we formalize the following results from \cite{bojanczyk2014automata}:
-lemmas: 3.4, 3.5, 3.6, 3.7, 4.8, 4.9;
-proposition: 5.1;
-theorems: 3.8 (Myhill-Nerode for $G$-automata), 5.2 (Myhill-Nerode for nominal $G$-automata).
-
-Throughout this document, we maintain the following convention for isar proofs:
-If we \texttt{obtain} some term \texttt{t} for which some result holds, we name it \texttt{H\_t}.
-An assumption which is an induction hypothesis is named \texttt{A\_IH}
-Assumptions start with an "\texttt{A}" and intermediate results start with a "\texttt{H}".
-Typically we just name them via indexes, i.e. as \texttt{A\_i} and \texttt{H\_j}.
-When encountering nested isar proofs we add an index for how nested the assumption / intermediate
-result is.
-For example if we have an isar proof in an isar proof in an isar proof, we would name assumptions
-of the most nested proof \texttt{A3\_i}.
-\<close>
-
-theory Nominal_Myhill_Nerode
-  imports
-    Main 
-    HOL.Groups
-    HOL.Relation
-    HOL.Fun
-    "HOL-Algebra.Group_Action"
-    "HOL-Library.Adhoc_Overloading"
-    "HOL-Algebra.Elementary_Groups"
-
-begin
-
-text \<open>
-\texttt{GMN\_simps} will contain selection of lemmas / definitions is updated through out
-the document.
-\<close>
-
-named_theorems GMN_simps
-lemmas GMN_simps
-
-text \<open>
-We will use the $\star$-symbol for the set of words of elements of a set, $A^{\star}$, the induced
-group action on the set of words $\phi^{\star}$ and for the extended transition function
-$\delta^{\star}$, thus we introduce the map \texttt{star} and apply \texttt{adhoc\_overloading} to
-get the notation working in all three situations:
-\<close>
-
-consts star :: "'typ1 \<Rightarrow> 'typ2" ("_\<^sup>\<star>" [1000] 999)
-
-abbreviation
-  kleene_star_set :: "'alpha set \<Rightarrow> 'alpha list set"
-  where "kleene_star_set A \<equiv> lists A"
-
-adhoc_overloading
-  star kleene_star_set
-
-text \<open>
-We use $\odot$ to convert between the definition of group actions via group homomoprhisms
-and the more standard infix group action notation.
-We deviate from \cite{bojanczyk2014automata} in that we consider left group actions,
-rather than right group actions:
-\<close>
-
-definition
-  make_op :: "('grp \<Rightarrow> 'X \<Rightarrow> 'X) \<Rightarrow> 'grp \<Rightarrow> 'X \<Rightarrow> 'X" (infixl "(\<odot>\<index>)" 70)
-  where " (\<odot> \<^bsub>\<phi>\<^esub>) \<equiv> (\<lambda>g. (\<lambda>x. \<phi> g x))"
-
-lemmas make_op_def [simp, GMN_simps]
-
-subsection \<open>
-Extending Group Actions
-\<close>
-
-text \<open>
-The following lemma is used for a proof in the locale \texttt{alt\_grp\_act}:
-\<close>
-
-lemma pre_image_lemma:
-  "\<lbrakk>S \<subseteq> T; x \<in> T \<and> f \<in> Bij T; (restrict f S) ` S = S; f x \<in> S\<rbrakk> \<Longrightarrow> x \<in> S"
-  apply (clarsimp simp add: extensional_def subset_eq Bij_def bij_betw_def restrict_def inj_on_def)
-  by (metis imageE)
-
-text \<open>
-The locale \texttt{alt\_grp\_act} is just a renaming of the locale \texttt{group\_action}.
-This was done to obtain more easy to interpret type names and context variables closer
-to the notation of \cite{bojanczyk2014automata}:
-\<close>
-
-locale alt_grp_act = group_action G X \<phi>
-  for
-    G :: "('grp, 'b) monoid_scheme" and
-    X :: "'X set" (structure) and
-    \<phi>
-begin
-
-lemma alt_grp_act_is_left_grp_act:
-  shows "x \<in> X \<Longrightarrow> \<one>\<^bsub>G\<^esub> \<odot>\<^bsub>\<phi>\<^esub> x = x" and
-  "g \<in> carrier G \<Longrightarrow> h \<in> carrier G \<Longrightarrow> x \<in> X \<Longrightarrow> (g \<otimes>\<^bsub>G\<^esub> h) \<odot>\<^bsub>\<phi>\<^esub> x = g \<odot>\<^bsub>\<phi>\<^esub> (h \<odot>\<^bsub>\<phi>\<^esub> x)"
-proof-
-  assume
-    A_0: "x \<in> X"
-  show "\<one>\<^bsub>G\<^esub> \<odot>\<^bsub>\<phi>\<^esub> x = x"
-    using group_action_axioms
-    apply (simp add: group_action_def BijGroup_def)
-    by (metis A_0 id_eq_one restrict_apply')
-next
-  assume 
-    A_0: "g \<in> carrier G" and
-    A_1: "h \<in> carrier G" and
-    A_2: "x \<in> X"
-  show "g \<otimes>\<^bsub>G\<^esub> h \<odot>\<^bsub>\<phi>\<^esub> x = g \<odot>\<^bsub>\<phi>\<^esub> (h \<odot>\<^bsub>\<phi>\<^esub> x)"
-    using group_action_axioms
-    apply (simp add: group_action_def group_hom_def group_hom_axioms_def hom_def BijGroup_def)
-    using composition_rule A_0 A_1 A_2
-    by auto
-qed
-
-definition
-  induced_star_map :: "('grp \<Rightarrow> 'X \<Rightarrow>'X) \<Rightarrow> 'grp \<Rightarrow> 'X list \<Rightarrow> 'X list"
-  where "induced_star_map func = (\<lambda>g\<in>carrier G. (\<lambda>lst \<in> (lists X). map (func g) lst))"
-
-text \<open>
-Because the adhoc overloading is used within a locale, isues will be encountered later due to there
-being multiple instances of the locale \texttt{alt\_grp\_act} in a single context:
-\<close>
-
-adhoc_overloading
-  star induced_star_map
-
-definition 
-  induced_quot_map ::
-  "'Y set \<Rightarrow> ('grp \<Rightarrow> 'Y \<Rightarrow> 'Y) \<Rightarrow> ('Y \<times>'Y) set \<Rightarrow> 'grp \<Rightarrow> 'Y set \<Rightarrow> 'Y set" ("[_]\<^sub>_\<index>" 60)
-  where "([ func ]\<^sub>R \<^bsub>S\<^esub>) = (\<lambda>g\<in>carrier G. (\<lambda>x \<in> (S // R). R `` {(func g) (SOME z. z\<in>x)}))"
-
-lemmas induced_star_map_def [simp, GMN_simps]
-  induced_quot_map_def [simp, GMN_simps]
-
-lemma act_maps_n_distrib:
-  "\<forall>g\<in>carrier G. \<forall>w\<in>X\<^sup>\<star>. \<forall>v\<in>X\<^sup>\<star>. (\<phi>\<^sup>\<star>) g (w @ v) = ((\<phi>\<^sup>\<star>) g w) @ ((\<phi>\<^sup>\<star>) g v)"
-  by (auto simp add: group_action_def group_hom_def group_hom_axioms_def hom_def)
-
-lemma triv_act:
-  "a \<in> X \<Longrightarrow> (\<phi> \<one>\<^bsub>G\<^esub>) a = a"
-  using group_hom.hom_one[of "G" "BijGroup X" "\<phi>"] group_BijGroup[where S = "X"]
-  apply (clarsimp simp add: group_action_def group_hom_def group_hom_axioms_def BijGroup_def)
-  by (metis id_eq_one restrict_apply')
-
-lemma triv_act_map:
-  "\<forall>w\<in>X\<^sup>\<star>. ((\<phi>\<^sup>\<star>) \<one>\<^bsub>G\<^esub>) w = w"
-  using triv_act
-  apply clarsimp
-  apply (rule conjI; rule impI)
-   apply clarify
-  using map_idI
-   apply metis
-  using group.subgroup_self group_hom group_hom.axioms(1) subgroup.one_closed
-  by blast
-
-proposition lists_a_Gset:
-  "alt_grp_act G (X\<^sup>\<star>) (\<phi>\<^sup>\<star>)"
-proof-
-  have H_0: "\<And>g. g \<in> carrier G \<Longrightarrow>
-    restrict (map (\<phi> g)) (X\<^sup>\<star>) \<in> carrier (BijGroup (X\<^sup>\<star>))"
-  proof-
-    fix g
-    assume
-      A1_0: "g \<in> carrier G"
-    from A1_0 have H1_0: "inj_on (\<lambda>x. if x \<in> X\<^sup>\<star> then map (\<phi> g) x else undefined) (X\<^sup>\<star>)"
-      apply (clarsimp simp add: inj_on_def)
-      by (metis (mono_tags, lifting) inj_onD inj_prop list.inj_map_strong)
-    from A1_0 have H1_1: "\<And>y z. \<forall>x\<in>set y. x \<in> X \<Longrightarrow> z \<in> set y \<Longrightarrow> \<phi> g z \<in> X"
-      using element_image
-      by blast
-    have H1_2: "(inv \<^bsub>G\<^esub> g) \<in> carrier G"
-      by (meson A1_0 group.inv_closed group_hom group_hom.axioms(1))
-    have H1_3: "\<And>x. x \<in> X\<^sup>\<star> \<Longrightarrow>
-    map (comp (\<phi> g) (\<phi> (inv \<^bsub>G\<^esub> g))) x = map (\<phi> (g \<otimes>\<^bsub>G\<^esub> (inv \<^bsub>G\<^esub> g))) x"
-      using alt_grp_act_axioms
-      apply (simp add: alt_grp_act_def group_action_def group_hom_def group_hom_axioms_def hom_def
-          BijGroup_def)
-      apply (rule meta_mp[of "\<And>x. x \<in> carrier G \<Longrightarrow> \<phi> x \<in> Bij X"])
-       apply (metis A1_0 H1_2 composition_rule in_lists_conv_set)
-      by blast
-    from H1_2 have H1_4: "\<And>x. x \<in> X\<^sup>\<star> \<Longrightarrow> map (\<phi> (inv \<^bsub>G\<^esub> g)) x \<in> X\<^sup>\<star>"
-      using surj_prop
-      by fastforce
-    have H1_5: "\<And>y. \<forall>x\<in>set y. x \<in> X \<Longrightarrow> y \<in> map (\<phi> g) ` X\<^sup>\<star>"
-      apply (simp add: image_def)
-      using H1_3 H1_4 
-      by (metis A1_0 group.r_inv group_hom group_hom.axioms(1) in_lists_conv_set map_idI map_map
-          triv_act)
-    show "restrict (map (\<phi> g)) (X\<^sup>\<star>) \<in> carrier (BijGroup (X\<^sup>\<star>))"
-      apply (clarsimp simp add: restrict_def BijGroup_def Bij_def
-          extensional_def bij_betw_def) 
-      apply (rule conjI)
-      using H1_0
-       apply simp
-      using H1_1 H1_5
-      by (auto simp add: image_def) 
-  qed
-  have H_1: "\<And>x y. \<lbrakk>x \<in> carrier G; y \<in> carrier G; x \<otimes>\<^bsub>G\<^esub> y \<in> carrier G\<rbrakk> \<Longrightarrow>
-           restrict (map (\<phi> (x \<otimes>\<^bsub>G\<^esub> y))) (X\<^sup>\<star>) =
-           restrict (map (\<phi> x)) (X\<^sup>\<star>) \<otimes>\<^bsub>BijGroup (X\<^sup>\<star>)\<^esub>
-           restrict (map (\<phi> y)) (X\<^sup>\<star>)"
-  proof-
-    fix x y
-    assume
-      A1_0: "x \<in> carrier G" and
-      A1_1: " y \<in> carrier G" and
-      A1_2: "x \<otimes>\<^bsub>G\<^esub> y \<in> carrier G"
-    have H1_0: "\<And>z. z \<in> carrier G \<Longrightarrow>
-       bij_betw (\<lambda>x. if x \<in> X\<^sup>\<star> then map (\<phi> z) x else undefined) (X\<^sup>\<star>) (X\<^sup>\<star>)"
-      using H_0
-      by (auto simp add: BijGroup_def Bij_def bij_betw_def inj_on_def)
-    from A1_1 have H1_1: "\<And>lst. lst \<in> X\<^sup>\<star> \<Longrightarrow> (map (\<phi> y)) lst \<in> X\<^sup>\<star>"
-      by (metis group_action.surj_prop group_action_axioms lists_image rev_image_eqI)
-    have H1_2: "\<And>a. a \<in> X\<^sup>\<star> \<Longrightarrow> map (\<lambda>xb.
-    if xb \<in> X
-    then \<phi> x ((\<phi> y) xb)
-    else undefined) a = map (\<phi> x) (map (\<phi> y) a)"
-      by auto
-    have H1_3: "(\<lambda>xa. if xa \<in> X\<^sup>\<star> then map (\<phi> (x \<otimes>\<^bsub>G\<^esub> y)) xa else undefined) =
-    compose (X\<^sup>\<star>) (\<lambda>xa. if xa \<in> X\<^sup>\<star> then map (\<phi> x) xa else undefined)
-    (\<lambda>x. if x \<in> X\<^sup>\<star> then map (\<phi> y) x else undefined)"
-      using alt_grp_act_axioms
-      apply (clarsimp simp add: compose_def alt_grp_act_def group_action_def
-          group_hom_def group_hom_axioms_def hom_def BijGroup_def restrict_def)
-      using A1_0 A1_1 H1_2 H1_1 bij_prop0
-      by auto
-    show "restrict (map (\<phi> (x \<otimes>\<^bsub>G\<^esub> y))) (X\<^sup>\<star>) =
-  restrict (map (\<phi> x)) (X\<^sup>\<star>) \<otimes>\<^bsub>BijGroup (X\<^sup>\<star>)\<^esub>
-  restrict (map (\<phi> y)) (X\<^sup>\<star>)"
-      apply (clarsimp simp add: restrict_def BijGroup_def Bij_def extensional_def)
-      apply (simp add: H1_3)
-      using A1_0 A1_1 H1_0
-      by auto
-  qed
-  show "alt_grp_act G (X\<^sup>\<star>) (\<phi>\<^sup>\<star>)"
-    apply (clarsimp simp add: alt_grp_act_def group_action_def group_hom_def group_hom_axioms_def)
-    apply (intro conjI)
-    using group_hom group_hom_def
-      apply (auto)[1]
-     apply (simp add: group_BijGroup)
-    apply (clarsimp simp add: hom_def)
-    apply (intro conjI; clarify)
-     apply (rule H_0)
-     apply simp
-    apply (rule conjI; rule impI)
-     apply (rule H_1)
-       apply simp+
-    apply (rule meta_mp[of "\<And>x y. x \<in> carrier G \<Longrightarrow>
-    y \<in> carrier G \<Longrightarrow> x \<otimes>\<^bsub>G\<^esub> y \<in> carrier G"])
-     apply blast
-    by (meson group.subgroup_self group_hom group_hom.axioms(1) subgroup.m_closed)
-qed 
-end
-
-lemma alt_group_act_is_grp_act [simp, GMN_simps]:
-  "alt_grp_act = group_action"
-  using alt_grp_act_def
-  by blast
-
-lemma prod_group_act:
-  assumes
-    grp_act_A: "alt_grp_act G A \<phi>" and
-    grp_act_B: "alt_grp_act G B \<psi>"
-  shows "alt_grp_act G (A\<times>B) (\<lambda>g\<in>carrier G. \<lambda>(a, b) \<in> (A \<times> B). (\<phi> g a, \<psi> g b))"
-  apply (simp add: alt_grp_act_def group_action_def group_hom_def)
-  apply (intro conjI)
-  subgoal
-    using grp_act_A grp_act_B
-    by (auto simp add: alt_grp_act_def group_action_def group_hom_def)
-  subgoal
-    using grp_act_A grp_act_B
-    by (auto simp add: alt_grp_act_def group_action_def group_hom_def group_BijGroup)
-  apply (clarsimp simp add: group_hom_axioms_def hom_def BijGroup_def)
-  apply (intro conjI; clarify)
-  subgoal for g
-    apply (clarsimp simp add: Bij_def bij_betw_def inj_on_def restrict_def extensional_def)
-    apply (intro conjI)
-    using grp_act_A
-     apply (simp add: alt_grp_act_def group_action_def group_hom_def group_hom_axioms_def
-        BijGroup_def hom_def Pi_def compose_def Bij_def bij_betw_def inj_on_def)
-    using grp_act_B
-     apply (simp add: alt_grp_act_def group_action_def group_hom_def group_hom_axioms_def
-        BijGroup_def hom_def Pi_def compose_def Bij_def bij_betw_def inj_on_def)
-    apply (rule meta_mp[of "\<phi> g \<in> Bij A \<and> \<psi> g \<in> Bij B"])
-     apply (clarsimp simp add: Bij_def bij_betw_def)
-    using grp_act_A grp_act_B
-     apply (simp add: alt_grp_act_def group_action_def group_hom_def group_hom_axioms_def
-        BijGroup_def hom_def Pi_def Bij_def)
-    using grp_act_A grp_act_B
-     apply (clarsimp simp add: compose_def restrict_def image_def alt_grp_act_def
-        group_action_def group_hom_def group_hom_axioms_def BijGroup_def hom_def Pi_def Bij_def
-        bij_betw_def)
-     apply (rule subset_antisym)
-      apply blast+
-    by (metis alt_group_act_is_grp_act group_action.bij_prop0 grp_act_A grp_act_B)
-  apply (intro conjI; intro impI)
-   apply (clarify)
-   apply (intro conjI; intro impI)
-    apply (rule conjI)
-  subgoal for x y
-    apply unfold_locales
-    apply (clarsimp simp add: Bij_def compose_def restrict_def bij_betw_def)
-    apply (rule extensionalityI[where A = "A \<times> B"])
-      apply (clarsimp simp add: extensional_def)
-    using grp_act_A grp_act_B
-     apply (simp add: alt_grp_act_def group_action_def group_hom_def group_hom_axioms_def
-        BijGroup_def hom_def Pi_def Bij_def compose_def extensional_def)
-    apply (simp add: fun_eq_iff; rule conjI; rule impI)
-    using group_action.composition_rule[of G A \<phi>] group_action.composition_rule[of G B \<psi>] grp_act_A 
-      grp_act_B   
-     apply force
-    by blast
-    apply (simp add: \<open>\<And>g. g \<in> carrier G \<Longrightarrow> (\<lambda>(a, b)\<in>A \<times> B. (\<phi> g a, \<psi> g b)) \<in> Bij (A \<times> B)\<close>)
-   apply (simp add: \<open>Group.group G\<close> group.subgroup_self subgroup.m_closed)
-  by (simp add: \<open>\<And>g. g \<in> carrier G \<Longrightarrow> (\<lambda>(a, b)\<in>A \<times> B. (\<phi> g a, \<psi> g b)) \<in> Bij (A \<times> B)\<close>)+
-
-subsection \<open>
-Equivariance and Quotient Actions
-\<close>
-
-locale eq_var_subset = alt_grp_act G X \<phi>
-  for
-    G :: "('grp, 'b) monoid_scheme" and
-    X :: "'X set" (structure) and
-    \<phi> +
-  fixes
-    subset
-  assumes
-    is_subset: "subset \<subseteq> X" and
-    is_equivar: "\<forall>g\<in>carrier G. (\<phi> g) ` subset = subset" 
-
-text \<open>
-The following lemmas are used for proofs in the locale \texttt{eq\_var\_rel}:
-\<close>
-
-lemma some_equiv_class_id:
-  "\<lbrakk>equiv X R; w \<in> X // R; x \<in> w\<rbrakk> \<Longrightarrow> R `` {x} = R `` {SOME z. z \<in> w}"
-  by (smt (z3) Eps_cong equiv_Eps_in equiv_class_eq_iff quotient_eq_iff)
-
-lemma nested_somes:
-  "\<lbrakk>equiv X R; w \<in> X // R\<rbrakk> \<Longrightarrow> (SOME z. z \<in> w) = (SOME z. z \<in> R``{(SOME z'. z' \<in> w)})"
-  by (metis proj_Eps proj_def)
-
-locale eq_var_rel = alt_grp_act G X \<phi>
-  for
-    G :: "('grp, 'b) monoid_scheme" and
-    X :: "'X set" (structure) and
-    \<phi> +
-  fixes R 
-  assumes
-    is_subrel:
-    "R \<subseteq> X \<times> X" and
-    is_eq_var_rel:
-    "\<And>a b. (a, b) \<in> R \<Longrightarrow> \<forall>g \<in> carrier G. (g \<odot>\<^bsub>\<phi>\<^esub> a, g \<odot>\<^bsub>\<phi>\<^esub> b) \<in> R"
-begin
-
-lemma is_eq_var_rel' [simp, GMN_simps]:
-  "\<And>a b. (a, b) \<in> R \<Longrightarrow> \<forall>g \<in> carrier G. ((\<phi> g) a, (\<phi> g) b) \<in> R"
-  using is_eq_var_rel
-  by auto
-
-lemma is_eq_var_rel_rev:
-  "a \<in> X \<Longrightarrow> b \<in> X \<Longrightarrow> g \<in> carrier G \<Longrightarrow> (g \<odot>\<^bsub>\<phi>\<^esub> a, g \<odot>\<^bsub>\<phi>\<^esub> b) \<in> R \<Longrightarrow> (a, b) \<in> R"
-proof -
-  assume
-    A_0: "(g \<odot>\<^bsub>\<phi>\<^esub> a, g \<odot>\<^bsub>\<phi>\<^esub> b) \<in> R" and
-    A_1: "a \<in> X" and
-    A_2: "b \<in> X" and
-    A_3: "g \<in> carrier G" 
-  have H_0: " group_action G X \<phi>" and
-    H_1: "R \<subseteq> X \<times> X" and
-    H_2: "\<And>a b g. (a, b) \<in> R \<Longrightarrow> g \<in> carrier G \<Longrightarrow> (\<phi> g a, \<phi> g b) \<in> R" 
-    by (simp add: group_action_axioms is_subrel)+
-  from H_0 have H_3: "group G"
-    by (auto simp add: group_action_def group_hom_def)
-  have H_4: "(\<phi> (inv\<^bsub>G\<^esub> g) (\<phi> g a), \<phi> (inv\<^bsub>G\<^esub> g) (\<phi> g b)) \<in> R"
-    apply (rule H_2)
-    using A_0 apply simp
-    by (simp add: A_3 H_3)
-  from H_3 A_3 have H_5: "(inv\<^bsub>G\<^esub> g) \<in> carrier G"
-    by auto
-  hence H_6: "\<And>e. e \<in> X \<Longrightarrow> \<phi> (inv\<^bsub>G\<^esub> g) (\<phi> g e) = \<phi> ((inv\<^bsub>G\<^esub> g) \<otimes>\<^bsub>G\<^esub> g) e"
-    using H_0 A_3 group_action.composition_rule 
-    by fastforce
-  hence H_7: "\<And>e. e \<in> X \<Longrightarrow> \<phi> (inv\<^bsub>G\<^esub> g) (\<phi> g e) = \<phi> \<one>\<^bsub>G\<^esub> e"
-    using H_3 A_3 group.l_inv
-    by fastforce 
-  hence H_8: "\<And>e. e \<in> X \<Longrightarrow> \<phi> (inv\<^bsub>G\<^esub> g) (\<phi> g e) = e"
-    using H_0
-    by (simp add: A_3 group_action.orbit_sym_aux)
-  thus "(a, b) \<in> R"
-    using A_1 A_2 H_4
-    by simp
-qed
-
-lemma equiv_equivar_class_some_eq:
-  assumes
-    A_0: "equiv X R" and
-    A_1: "w \<in> X // R" and
-    A_2: "g \<in> carrier G"
-  shows "([ \<phi> ]\<^sub>R) g w = R `` {(SOME z'. z' \<in> \<phi> g ` w)}"
-proof-
-  obtain z where H_z: "w = R `` {z} \<and> z \<in> w"
-    by (metis A_0 A_1 equiv_class_self quotientE)
-  have H_0: "\<And>x. (\<phi> g z, x) \<in> R \<Longrightarrow> x \<in> \<phi> g ` {y. (z, y) \<in> R}"
-  proof-
-    fix y
-    assume
-      A1_0: "(\<phi> g z, y) \<in> R"
-    obtain y' where H2_y': "y' = \<phi> (inv\<^bsub>G\<^esub> g) y \<and> y' \<in> X" 
-      using eq_var_rel_axioms
-      apply (clarsimp simp add: eq_var_rel_def group_action_def group_hom_def) 
-      by (meson A_0 eq_var_rel_axioms A_2 A1_0 equiv_class_eq_iff eq_var_rel.is_eq_var_rel
-          group.inv_closed element_image) 
-    from A_1 A_2 H2_y' have H2_0: "\<phi> g y' = y"
-      apply (clarsimp simp add: eq_var_rel_def eq_var_rel_axioms_def)
-      using A_2 A1_0 group_action.bij_prop1[where G = "G" and E = "X" and \<phi> = "\<phi>"]
-      by (metis A_0 alt_group_act_is_grp_act alt_grp_act_axioms equiv_class_eq_iff orbit_sym_aux)
-    from A_1 A_2 A1_0 have H2_1: "(z, y') \<in> R"
-      by (metis H2_0 A_0 A_2 H2_y' H_z equiv_class_eq_iff is_eq_var_rel_rev
-          quotient_eq_iff make_op_def)
-    thus "y \<in> \<phi> g ` {v. (z, v) \<in> R}" 
-      using H2_0
-      by (auto simp add: image_def)
-  qed
-  have H_1: "\<phi> g ` (R `` {z}) = R `` {\<phi> g z}"
-    apply (clarsimp simp add: Relation.Image_def)
-    apply (rule subset_antisym; simp add: Set.subset_eq; rule allI; rule impI)
-    using eq_var_rel_axioms A_2 eq_var_rel.is_eq_var_rel
-     apply simp
-    by (simp add: H_0)
-  have H_2: "\<phi> g ` w \<in> X // R"
-    using eq_var_rel_axioms A_1 A_2 H_1 
-    by (metis A_0 H_z equiv_class_eq_iff quotientI quotient_eq_iff element_image)
-  thus "([\<phi>]\<^sub>R) g w = R `` {SOME z'. z' \<in> \<phi> g ` w}"
-    using A_0 A_1 A_2
-    apply (clarsimp simp add: Image_def)
-    apply (intro subset_antisym)
-     apply (clarify)
-    using A_0 H_z imageI insert_absorb insert_not_empty some_in_eq some_equiv_class_id 
-     apply (smt (z3) A_1 Eps_cong Image_singleton_iff equiv_Eps_in)
-    apply (clarify)
-    by (smt (z3) Eps_cong equiv_Eps_in image_iff in_quotient_imp_closed quotient_eq_iff)
-qed
-
-lemma ec_er_closed_under_action:
-  assumes
-    A_0: "equiv X R" and
-    A_1: "g \<in> carrier G" and
-    A_2: "w \<in> X//R"
-  shows "\<phi> g ` w \<in> X // R"
-proof-
-  obtain z where H_z: "R `` {z} = w \<and> z \<in> X"
-    by (metis A_2 quotientE)
-  have H_0: "equiv X R \<Longrightarrow> g \<in> carrier G \<Longrightarrow> w \<in> X // R \<Longrightarrow>
-      {y. (\<phi> g z, y) \<in> R} \<subseteq> {y. \<exists>x. (z, x) \<in> R \<and> y = \<phi> g x}"
-  proof (clarify)
-    fix x
-    assume
-      A1_0: "equiv X R" and
-      A1_1: "g \<in> carrier G" and
-      A1_2: "w \<in> X // R" and
-      A1_3: "(\<phi> g z, x) \<in> R"
-    obtain x' where H2_x': "x = \<phi> g x' \<and> x' \<in> X" 
-      using group_action_axioms
-      by (metis A1_1 is_subrel A1_3 SigmaD2 group_action.bij_prop1 subsetD)
-    thus "\<exists>y. (z, y) \<in> R \<and> x = \<phi> g y"
-      using is_eq_var_rel_rev[where g = "g" and a = "z" and b = "x'"] A1_3
-      by (auto simp add: eq_var_rel_def eq_var_rel_axioms_def A1_1 A1_2 group_action_axioms H_z
-          H2_x')
-  qed
-  have H_1: "\<phi> g ` R `` {z} = R `` {\<phi> g z}"
-    using A_0 A_1 A_2
-    apply (clarsimp simp add: eq_var_rel_axioms_def eq_var_rel_def
-        Image_def image_def)
-    apply (intro subset_antisym)
-     apply (auto)[1]
-    by (rule H_0)
-  thus "\<phi> g ` w \<in> X // R"
-    using H_1 H_z
-    by (metis A_1 quotientI element_image)
-qed
-
-text \<open>
-The following lemma corresponds to the first part of lemma 3.5 from \cite{bojanczyk2014automata}:
-\<close>
-
-lemma quot_act_wd:
-  "\<lbrakk>equiv X R; x \<in> X; g \<in> carrier G\<rbrakk> \<Longrightarrow> g \<odot>\<^bsub>[\<phi>]\<^sub>R\<^esub> (R `` {x}) = (R `` {g \<odot>\<^bsub>\<phi>\<^esub> x})"
-  apply (clarsimp simp add: eq_var_rel_def eq_var_rel_axioms_def)
-  apply (rule conjI; rule impI)
-   apply (smt (verit, best) Eps_cong Image_singleton_iff eq_var_rel.is_eq_var_rel'
-      eq_var_rel_axioms equiv_Eps_in equiv_class_eq)
-  by (simp add: quotientI)+
-
-text \<open>
-The following lemma corresponds to the second part of lemma 3.5 from \cite{bojanczyk2014automata}:
-\<close>
-
-lemma quot_act_is_grp_act:
-  "equiv X R \<Longrightarrow> alt_grp_act G (X // R) ([\<phi>]\<^sub>R)"
-proof-
-  assume A_0: "equiv X R"
-  have H_0: "\<And>x. Group.group G \<Longrightarrow>
-         Group.group (BijGroup X) \<Longrightarrow>
-         R \<subseteq> X \<times> X \<Longrightarrow>
-         \<phi> \<in> carrier G \<rightarrow> carrier (BijGroup X) \<Longrightarrow>
-         \<forall>x\<in>carrier G. \<forall>y\<in>carrier G. \<phi> (x \<otimes>\<^bsub>G\<^esub> y) = \<phi> x \<otimes>\<^bsub>BijGroup X\<^esub> \<phi> y \<Longrightarrow>
-         x \<in> carrier G \<Longrightarrow> (\<lambda>xa\<in>X // R. R `` {\<phi> x (SOME z. z \<in> xa)}) \<in> carrier (BijGroup (X // R))"
-  proof-
-    fix g
-    assume
-      A1_0: "Group.group G" and
-      A1_1: "Group.group (BijGroup X)" and
-      A1_2: "\<phi> \<in> carrier G \<rightarrow> carrier (BijGroup X)" and
-      A1_3: "\<forall>x\<in>carrier G. \<forall>y\<in>carrier G. \<phi> (x \<otimes>\<^bsub>G\<^esub> y) = \<phi> x \<otimes>\<^bsub>BijGroup X\<^esub> \<phi> y" and
-      A1_4: "g \<in> carrier G"
-    have H_0: "group_action G X \<phi>"
-      apply (clarsimp simp add: group_action_def group_hom_def group_hom_axioms_def)
-      apply (simp add: A1_0 A1_1)+
-      apply (simp add: hom_def)
-      apply (rule conjI)
-      using A1_2
-       apply blast
-      by (simp add: A1_3)
-    have H1_0: "\<And>x y. \<lbrakk>x \<in> X // R; y \<in> X // R; R `` {\<phi> g (SOME z. z \<in> x)} =
-                                                     R `` {\<phi> g (SOME z. z \<in> y)}\<rbrakk> \<Longrightarrow> x \<subseteq> y"
-    proof (clarify; rename_tac a)
-      fix x y a
-      assume
-        A2_0: "x \<in> X // R" and
-        A2_1: "y \<in> X // R" and
-        A2_2: "R `` {\<phi> g (SOME z. z \<in> x)} = R `` {\<phi> g (SOME z. z \<in> y)}" and
-        A2_3: "a \<in> x"
-      obtain b where H2_b: "R `` {b} = y \<and> b \<in> X" 
-        by (metis A2_1 quotientE)
-      obtain a' b' where H2_a'_b': "a'\<in> x \<and> b'\<in> y \<and> R `` {\<phi> g a'} = R `` {\<phi> g b'}"
-        by (metis A_0 A2_1 A2_2 A2_3 equiv_Eps_in some_eq_imp)
-      from H2_a'_b' have H2_2: "(\<phi> g a', \<phi> g b') \<in> R"
-        by (metis A_0 A1_4 A2_1 Image_singleton_iff eq_var_rel.is_eq_var_rel' eq_var_rel_axioms
-            quotient_eq_iff)
-      hence H2_0: "(\<phi> (inv\<^bsub>G\<^esub> g) (\<phi> g a'), \<phi> (inv\<^bsub>G\<^esub> g) (\<phi> g b')) \<in> R"
-        by (simp add: A1_0 is_eq_var_rel A1_4)
-      have H2_1: "a' \<in> X \<and> b' \<in> X" 
-        using A_0 A2_0 A2_1 H2_a'_b' in_quotient_imp_subset
-        by blast
-      hence H2_2: "(a', b') \<in> R"
-        using H2_0
-        by (metis A1_4 H_0 group_action.orbit_sym_aux)
-      have H2_3: "(a, a') \<in> R" 
-        by (meson A_0 A2_0 A2_3 H2_a'_b' quotient_eq_iff)
-      hence H2_4: "(b', a) \<in> R" 
-        using H2_2
-        by (metis A_0 A2_0 A2_1 A2_3 H2_a'_b' quotient_eqI quotient_eq_iff)
-      thus "a \<in> y"
-        by (metis A_0 A2_1 H2_a'_b' in_quotient_imp_closed)
-    qed
-    have H1_1: "\<And>x. x \<in> X // R \<Longrightarrow> \<exists>xa\<in>X // R. x = R `` {\<phi> g (SOME z. z \<in> xa)}"
-    proof -
-      fix x
-      assume
-        A2_0: "x \<in> X // R"
-      have H2_0: "\<And>e. R `` {e} \<in> X // R \<Longrightarrow> R `` {e} \<subseteq> R `` {\<phi> g (\<phi> (inv\<^bsub>G\<^esub> g) e)}"
-      proof (rule subsetI)
-        fix e y
-        assume
-          A3_0: "R `` {e} \<in> X // R" and
-          A3_1: "y \<in> R `` {e}"
-        have H3_0: "y \<in> X" 
-          using A3_1 is_subrel
-          by blast
-        from H_0 have H3_1: "\<phi> g (\<phi> (inv\<^bsub>G\<^esub> g) y) = y"
-          by (metis (no_types, lifting) A1_0 A1_4 H3_0 group.inv_closed group.inv_inv
-              group_action.orbit_sym_aux)
-        from A3_1 have H3_2: "(e, y) \<in> R"
-          by simp
-        hence H3_3: "((\<phi> (inv\<^bsub>G\<^esub> g) e), (\<phi> (inv\<^bsub>G\<^esub> g) y)) \<in> R"
-          using is_eq_var_rel A1_4 A1_0
-          by simp
-        hence H3_4: "(\<phi> g (\<phi> (inv\<^bsub>G\<^esub> g) e), \<phi> g (\<phi> (inv\<^bsub>G\<^esub> g) y)) \<in> R"
-          using is_eq_var_rel A1_4 A1_0
-          by simp
-        hence H3_5: "(\<phi> g (\<phi> (inv\<^bsub>G\<^esub> g) e), y) \<in> R"
-          using H3_1
-          by simp
-        thus "y \<in> R `` {\<phi> g (\<phi> (inv\<^bsub>G\<^esub> g) e)}"
-          by simp
-      qed
-      hence H2_1: "\<And>e. R `` {e} \<in> X // R \<Longrightarrow> R `` {e} = R `` {\<phi> g (\<phi> (inv\<^bsub>G\<^esub> g) e)}" 
-        by (metis A_0 proj_def proj_in_iff equiv_class_eq_iff subset_equiv_class)
-      have H2_2: "\<And>e f. R `` {e} \<in> X // R \<Longrightarrow> R `` {f} \<in> X // R \<Longrightarrow>
-        R `` {e} = R `` {f} \<Longrightarrow> \<forall>f'\<in> R `` {f}. R `` {e} = R `` {f'}"
-        by (metis A_0 Image_singleton_iff equiv_class_eq)
-      have H2_3: "x \<in> X // R \<Longrightarrow> \<exists>e\<in>X. x = R `` {e}" 
-        by (meson quotientE)
-      have H2_4: "\<And>e. R `` {e} \<in> X // R \<Longrightarrow> R `` {e} = R `` {\<phi> g (\<phi> (inv\<^bsub>G\<^esub> g) e)} \<and>
-        (\<phi> (inv\<^bsub>G\<^esub> g) e) \<in> R `` {\<phi> (inv\<^bsub>G\<^esub> g) e}"
-        by (smt (z3) A_0 A1_0 A1_4 H_0 H2_1 Image_singleton_iff equiv_class_eq_iff
-            group.inv_closed group_action.element_image in_quotient_imp_non_empty
-            subset_empty subset_emptyI)
-      have H2_5: "\<And>e. R `` {e} \<in> X // R \<Longrightarrow> \<forall>z\<in>R `` {\<phi> (inv\<^bsub>G\<^esub> g) e}. (\<phi> (inv\<^bsub>G\<^esub> g) e, z) \<in> R"
-        by simp
-      hence H2_6: "\<And>e. R `` {e} \<in> X // R \<Longrightarrow>
-        \<forall>z\<in>R `` {\<phi> (inv\<^bsub>G\<^esub> g) e}. (\<phi> g (\<phi> (inv\<^bsub>G\<^esub> g) e), \<phi> g z) \<in> R"
-        using is_eq_var_rel' A1_4 A1_0
-        by blast
-      hence H2_7: "\<And>e. R `` {e} \<in> X // R \<Longrightarrow> \<forall>z\<in>R `` {\<phi> (inv\<^bsub>G\<^esub> g) e}. (e, \<phi> g z) \<in> R"
-        using H2_1
-        by blast
-      hence H2_8: "\<And>e. R `` {e} \<in> X // R \<Longrightarrow> \<forall>z\<in>R `` {\<phi> (inv\<^bsub>G\<^esub> g) e}. R `` {e} = R `` {\<phi> g z}"
-        by (meson A_0 equiv_class_eq_iff)
-      have H2_9: "\<And>e. R `` {e} \<in> X // R \<Longrightarrow>
-      R `` {e} = R `` {\<phi> g (SOME z. z \<in> R `` {\<phi> (inv\<^bsub>G\<^esub> g) e})}"
-      proof-
-        fix e
-        assume
-          A3_0: "R `` {e} \<in> X // R"
-        show "R `` {e} = R `` {\<phi> g (SOME z. z \<in> R `` {\<phi> (inv\<^bsub>G\<^esub> g) e})}"
-          apply (rule someI2[where Q = "\<lambda>z. R `` {e} = R `` {\<phi> g z}" and
-                P = "\<lambda>z. z \<in> R `` {\<phi> (inv\<^bsub>G\<^esub> g) e}" and a = "\<phi> (inv\<^bsub>G\<^esub> g) e"])
-          using A3_0 H2_4
-           apply blast
-          using A3_0 H2_8
-          by auto
-      qed
-      have H2_10: "\<forall>e. (R `` {e} \<in> X // R \<longrightarrow>
-      (R `` {e} = R `` {\<phi> g (SOME z. z \<in> R `` {\<phi> (inv\<^bsub>G\<^esub> g) e})}))"
-        using H2_9
-        by auto
-      hence H2_11: "\<forall>e. (R `` {e} \<in> X // R \<longrightarrow>
-      (\<exists>xa\<in>X // R. R `` {e} = R `` {\<phi> g (SOME z. z \<in> xa)}))"
-        using H2_8
-        apply clarsimp
-        by (smt (verit, best) A_0 H2_3 H2_5 H2_4 equiv_Eps_in equiv_class_eq_iff quotientI)
-      have H2_12: "\<And>x. x \<in> X // R \<Longrightarrow> \<exists>e\<in>X. x = R `` {e} "
-        by (meson quotientE)
-      have H2_13: "\<And>x. x \<in> X // R \<Longrightarrow> \<exists>xa\<in>X // R. x = R `` {\<phi> g (SOME z. z \<in> xa)}"
-        using H2_11 H2_12
-        by blast
-      show "\<exists>xa\<in>X // R. x = R `` {\<phi> g (SOME z. z \<in> xa)}"
-        by (simp add: A2_0 H2_13)
-    qed
-    show "(\<lambda>x\<in>X // R. R `` {\<phi> g (SOME z. z \<in> x)}) \<in> carrier (BijGroup (X // R))"
-      apply (clarsimp simp add: BijGroup_def Bij_def bij_betw_def)
-      subgoal
-        apply (clarsimp simp add: inj_on_def)
-        apply (rule conjI)
-         apply (clarsimp)
-         apply (rule subset_antisym)
-          apply (simp add: H1_0)
-         apply (simp add: \<open>\<And>y x. \<lbrakk>x \<in> X // R;
-          y \<in> X // R; R `` {\<phi> g (SOME z. z \<in> x)} = R `` {\<phi> g (SOME z. z \<in> y)}\<rbrakk> \<Longrightarrow> x \<subseteq> y\<close>)
-        apply (rule subset_antisym; clarify)
-        subgoal for x y
-          by (metis A_0 is_eq_var_rel' A1_4 Eps_cong equiv_Eps_preserves equiv_class_eq_iff
-              quotientI) 
-        apply (clarsimp simp add: Set.image_def)
-        by (simp add: H1_1)
-      done
-  qed
-  have H_1: "\<And>x y. \<lbrakk>Group.group G; Group.group (BijGroup X); R \<subseteq> X \<times> X;
-            \<phi> \<in> carrier G \<rightarrow> carrier (BijGroup X);
-           \<forall>x\<in>carrier G. \<forall>y\<in>carrier G. \<phi> (x \<otimes>\<^bsub>G\<^esub> y) = \<phi> x \<otimes>\<^bsub>BijGroup X\<^esub> \<phi> y;
-           x \<in> carrier G; y \<in> carrier G; x \<otimes>\<^bsub>G\<^esub> y \<in> carrier G\<rbrakk> \<Longrightarrow>
-           (\<lambda>xa\<in>X // R. R `` {(\<phi> x \<otimes>\<^bsub>BijGroup X\<^esub> \<phi> y) (SOME z. z \<in> xa)}) =
-           (\<lambda>xa\<in>X // R. R `` {\<phi> x (SOME z. z \<in> xa)}) \<otimes>\<^bsub>BijGroup (X // R)\<^esub>
-           (\<lambda>x\<in>X // R. R `` {\<phi> y (SOME z. z \<in> x)})"
-  proof -
-    fix x y 
-    assume
-      A1_1: "Group.group G" and
-      A1_2: "Group.group (BijGroup X)" and
-      A1_3: "\<phi> \<in> carrier G \<rightarrow> carrier (BijGroup X)" and
-      A1_4: "\<forall>x\<in>carrier G. \<forall>y\<in>carrier G. \<phi> (x \<otimes>\<^bsub>G\<^esub> y) = \<phi> x \<otimes>\<^bsub>BijGroup X\<^esub> \<phi> y" and
-      A1_5: "x \<in> carrier G" and
-      A1_6: "y \<in> carrier G" and
-      A1_7: "x \<otimes>\<^bsub>G\<^esub> y \<in> carrier G"
-    have H1_0: "\<And>w::'X set. w \<in> X // R \<Longrightarrow>
-    R `` {(\<phi> x \<otimes>\<^bsub>BijGroup X\<^esub> \<phi> y) (SOME z. z \<in> w)} =
-    ((\<lambda>v\<in>X // R. R `` {\<phi> x (SOME z. z \<in> v)}) \<otimes>\<^bsub>BijGroup (X // R)\<^esub>
-    (\<lambda>x\<in>X // R. R `` {\<phi> y (SOME z. z \<in> x)})) w"
-    proof-
-      fix w
-      assume
-        A2_0: "w \<in> X // R" 
-      have H2_4: "\<phi> y ` w \<in> X // R"
-        using ec_er_closed_under_action[where w = "w" and g = "y"]
-        by (clarsimp simp add: group_hom_axioms_def hom_def A_0 A1_1 A1_2 is_eq_var_rel' A1_3 A1_4
-            A1_6 A2_0) 
-      hence H2_1: "R `` {(\<phi> x \<otimes>\<^bsub>BijGroup X\<^esub> \<phi> y) (SOME z. z \<in> w)} =
-      R `` {\<phi> (x \<otimes>\<^bsub>G\<^esub> y) (SOME z. z \<in> w)}"
-        using A1_4 A1_5 A1_6
-        by auto
-      also have H2_2: "... = R `` {SOME z. z \<in> \<phi> (x \<otimes>\<^bsub>G\<^esub> y) ` w}"
-        using A1_7 equiv_equivar_class_some_eq[where w = "w" and g = "x \<otimes>\<^bsub>G\<^esub> y"]
-        by (clarsimp simp add: A1_7 A_0 A2_0 group_action_def group_hom_def group_hom_axioms_def
-            hom_def)
-      also have H2_3: "... = R `` {SOME z. z \<in> \<phi> x ` \<phi> y ` w}"
-        apply (rule meta_mp[of "\<not>(\<exists>x. x \<in> w \<and> x \<notin> X)"])
-        using A1_1 is_eq_var_rel' A1_3 A1_4 A1_5 A1_6 A2_0
-         apply (clarsimp simp add: image_def BijGroup_def restrict_def compose_def Pi_def)
-         apply (smt (z3) Eps_cong)
-        apply (clarify) 
-        using A_0 A2_0 in_quotient_imp_subset
-        by auto
-      also have H2_5: "... = R `` {\<phi> x (SOME z. z \<in> \<phi> y ` w)}"
-        using equiv_equivar_class_some_eq[where w = "\<phi> y ` w" and g = "x"]
-        apply (clarsimp simp add: A_0 group_action_def group_hom_def group_hom_axioms_def hom_def)
-        by (simp add: A1_1 A1_2 is_eq_var_rel' A1_3 A1_4 A1_5 H2_4)
-      also have H2_6: "... = R `` {\<phi> x (SOME z. z \<in> R``{(SOME z'. z' \<in> \<phi> y ` w)})}"
-        using H2_4 nested_somes[where w = "\<phi> y ` w" and X = "X" and R = "R"] A_0
-        by presburger
-      also have H2_7: "... = R `` {\<phi> x (SOME z. z \<in> R `` {\<phi> y (SOME z'. z' \<in> w)})}"
-        using equiv_equivar_class_some_eq[where g = "y" and w = "w"] H2_6
-        by (simp add: A_0 group_action_def
-            group_hom_def group_hom_axioms_def hom_def A1_1 A1_2 is_eq_var_rel' A1_3 A1_4 A2_0 A1_6) 
-      also have H2_9: "... = ((\<lambda>v\<in>X // R. R `` {\<phi> x (SOME z. z \<in> v)}) \<otimes>\<^bsub>BijGroup (X // R)\<^esub>
-      (\<lambda>x\<in>X // R. R `` {\<phi> y (SOME z. z \<in> x)})) w"
-      proof-
-        have H3_0: "\<And>u. R `` {\<phi> y (SOME z. z \<in> w)} \<in> X // R \<Longrightarrow> u \<in> carrier G \<Longrightarrow>
-        (\<lambda>v\<in>X // R. R `` {\<phi> u (SOME z. z \<in> v)}) \<in> Bij (X // R)"
-        proof -
-          fix u
-          assume 
-            A4_0: "R `` {\<phi> y (SOME z. z \<in> w)} \<in> X // R" and
-            A4_1: "u \<in> carrier G"
-          have H4_0: "\<forall>g \<in> carrier G.
-          (\<lambda>x\<in>X // R. R `` {\<phi> g (SOME z. z \<in> x)}) \<in> carrier (BijGroup (X // R))"
-            by (simp add: A_0 A1_1 A1_2 A1_3 A1_4 H_0 is_subrel)
-          thus "(\<lambda>v\<in>X // R. R `` {\<phi> u (SOME z. z \<in> v)}) \<in> Bij (X // R)"
-            by (auto simp add: BijGroup_def A4_1)
-        qed
-        have H3_1: "R `` {\<phi> y (SOME z. z \<in> w)} \<in> X // R"
-        proof-
-          have H4_0: "\<phi> y ` w \<in> X // R"
-            using ec_er_closed_under_action
-            by (simp add: H2_4)
-          hence H4_1: "R `` {(SOME z. z \<in> \<phi> y ` w)} = \<phi> y ` w"
-            apply (clarsimp simp add: image_def) 
-            apply (rule subset_antisym)
-            using A_0 equiv_Eps_in in_quotient_imp_closed
-             apply fastforce
-            using A_0 equiv_Eps_in quotient_eq_iff
-            by fastforce
-          have H4_2: "R `` {\<phi> y (SOME z. z \<in> w)} = R `` {(SOME z. z \<in> \<phi> y ` w)}"
-            using equiv_equivar_class_some_eq[where g = "y" and w = "w"]
-            by (metis A_0 A2_0 H4_0 H4_1 equiv_Eps_in imageI some_equiv_class_id)
-          from H4_0 H4_1 H4_2 show "R `` {\<phi> y (SOME z. z \<in> w)} \<in> X // R"
-            by auto
-        qed
-        show ?thesis
-          apply (rule meta_mp[of "R `` {\<phi> y (SOME z. z \<in> w)} \<in> X // R"])
-           apply (rule meta_mp[of "\<forall>u \<in> carrier G.
-         (\<lambda>v\<in>X // R. R `` {\<phi> u (SOME z. z \<in> v)}) \<in> Bij (X // R)"])
-          using A2_0 A1_5 A1_6
-            apply ( simp add: BijGroup_def compose_def)
-           apply clarify
-          by (simp add: H3_0 H3_1)+
-      qed
-      finally show "R `` {(\<phi> x \<otimes>\<^bsub>BijGroup X\<^esub> \<phi> y) (SOME z. z \<in> w)} =
-      ((\<lambda>v\<in>X // R. R `` {\<phi> x (SOME z. z \<in> v)}) \<otimes>\<^bsub>BijGroup (X // R)\<^esub>
-      (\<lambda>x\<in>X // R. R `` {\<phi> y (SOME z. z \<in> x)})) w"
-        by simp
-    qed
-    have H1_1: "\<And>w::'X set. w \<notin> X // R \<Longrightarrow>
-    ((\<lambda>v\<in>X // R. R `` {\<phi> x (SOME z. z \<in> v)}) \<otimes>\<^bsub>BijGroup (X // R)\<^esub>
-    (\<lambda>x\<in>X // R. R `` {\<phi> y (SOME z. z \<in> x)})) w = undefined"
-    proof -
-      fix w
-      assume
-        A2_0: "w \<notin> X // R" 
-      have H2_0: "\<And>u. u\<in> carrier G \<Longrightarrow> (\<lambda>v\<in>X // R. R `` {\<phi> u (SOME z. z \<in> v)}) \<in> Bij (X // R)"
-        using H_0
-        apply (clarsimp simp add: A_0 A1_1 A1_2 is_eq_var_rel' A1_3 A1_4 is_subrel)
-        by (simp add: BijGroup_def)
-      hence H2_1: "(\<lambda>x'\<in>X // R. R `` {\<phi> y (SOME z. z \<in> x')}) \<in> Bij (X // R)"
-        using A1_6
-        by auto
-      from H2_0 have H2_2: "(\<lambda>x'\<in>X // R. R `` {\<phi> x (SOME z. z \<in> x')}) \<in> Bij (X // R)"
-        by (simp add: A1_5)
-      thus "((\<lambda>v\<in>X // R. R `` {\<phi> x (SOME z. z \<in> v)}) \<otimes>\<^bsub>BijGroup (X // R)\<^esub>
-      (\<lambda>x\<in>X // R. R `` {\<phi> y (SOME z. z \<in> x)})) w = undefined"
-        using H2_1 H2_2
-        by (auto simp add: BijGroup_def compose_def A2_0)
-    qed
-    from H1_0 H1_1 have "\<And>w. (\<lambda>xa\<in>X // R. R `` {(\<phi> x \<otimes>\<^bsub>BijGroup X\<^esub> \<phi> y) (SOME z. z \<in> xa)}) w =
-    ((\<lambda>xa\<in>X // R. R `` {\<phi> x (SOME z. z \<in> xa)}) \<otimes>\<^bsub>BijGroup (X // R)\<^esub>
-    (\<lambda>x'\<in>X // R. R `` {\<phi> y (SOME z. z \<in> x')})) w"
-      by auto
-    thus "(\<lambda>xa\<in>X // R. R `` {(\<phi> x \<otimes>\<^bsub>BijGroup X\<^esub> \<phi> y) (SOME z. z \<in> xa)}) =
-    (\<lambda>xa\<in>X // R. R `` {\<phi> x (SOME z. z \<in> xa)}) \<otimes>\<^bsub>BijGroup (X // R)\<^esub>
-    (\<lambda>x\<in>X // R. R `` {\<phi> y (SOME z. z \<in> x)})"
-      by (simp add: restrict_def)
-  qed
-  show ?thesis
-    apply (clarsimp simp add: group_action_def group_hom_def)
-    using eq_var_rel_axioms
-    apply (clarsimp simp add: eq_var_rel_def eq_var_rel_axioms_def
-        group_action_def group_hom_def)
-    apply (rule conjI)
-     apply (simp add: group_BijGroup)
-    apply (clarsimp simp add: group_hom_axioms_def hom_def)
-    apply (intro conjI)
-     apply (rule funcsetI; simp)
-     apply (simp add: H_0)
-    apply (clarify; rule conjI; intro impI)
-     apply (simp add: H_1)
-    by (auto simp add: group.is_monoid monoid.m_closed)
-qed
-end
-
-locale eq_var_func = GA_0: alt_grp_act G X \<phi> + GA_1: alt_grp_act G Y \<psi> 
-  for
-    G :: "('grp, 'b) monoid_scheme" and
-    X :: "'X set" and
-    \<phi> and
-    Y :: "'Y set" and
-    \<psi> +
-  fixes
-    f :: "'X \<Rightarrow> 'Y" 
-  assumes
-    is_ext_func_bet:
-    "f \<in> (X \<rightarrow>\<^sub>E Y)" and
-    is_eq_var_func:
-    "\<And>a g. a \<in> X \<Longrightarrow> g \<in> carrier G \<Longrightarrow> f (g \<odot>\<^bsub>\<phi>\<^esub> a) = g \<odot>\<^bsub>\<psi>\<^esub> (f a)"
-begin
-
-lemma is_eq_var_func' [simp]:
-  "a \<in> X \<Longrightarrow> g \<in> carrier G \<Longrightarrow> f (\<phi> g a) = \<psi> g (f a)"
-  using is_eq_var_func
-  by auto
-
-end
-
-lemma G_set_equiv:
-  "alt_grp_act G A \<phi> \<Longrightarrow> eq_var_subset G A \<phi> A"
-  by (auto simp add: eq_var_subset_def eq_var_subset_axioms_def group_action_def
-      group_hom_def group_hom_axioms_def hom_def BijGroup_def Bij_def bij_betw_def)
-
-subsection \<open>
-Basic ($G$)-Automata Theory
-\<close>
-
-
-locale language = 
-  fixes A :: "'alpha set" and
-    L
-  assumes
-    is_lang: "L \<subseteq> A\<^sup>\<star>"
-
-locale G_lang = alt_grp_act G A \<phi> + language A L 
-  for
-    G :: "('grp, 'b) monoid_scheme" and
-    A :: "'alpha set" (structure) and
-    \<phi> L +
-  assumes
-    L_is_equivar:
-    "eq_var_subset G (A\<^sup>\<star>) (induced_star_map \<phi>) L"
-begin
-lemma G_lang_is_lang[simp]: "language A L"
-  by (simp add: language_axioms)
-end
-
-sublocale G_lang \<subseteq> language
-  by simp
-
-fun give_input :: "('state \<Rightarrow> 'alpha \<Rightarrow> 'state) \<Rightarrow> 'state \<Rightarrow> 'alpha list \<Rightarrow> 'state"
-  where "give_input trans_func s Nil = s"
-  |   "give_input trans_func s (a#as) = give_input trans_func (trans_func s a) as"
-
-adhoc_overloading
-  star give_input
-
-locale det_aut =
-  fixes
-    labels :: "'alpha set" and
-    states :: "'state set" and
-    init_state :: "'state" and
-    fin_states :: "'state set" and
-    trans_func :: "'state \<Rightarrow> 'alpha \<Rightarrow> 'state" ("\<delta>")
-  assumes
-    init_state_is_a_state:
-    "init_state \<in> states" and
-    fin_states_are_states:
-    "fin_states \<subseteq> states" and
-    trans_func_ext:
-    "(\<lambda>(state, label). trans_func state label) \<in> (states \<times> labels) \<rightarrow>\<^sub>E states" 
-begin
-
-lemma trans_func_well_def:
-  "\<And>state label. state \<in> states \<Longrightarrow> label \<in> labels \<Longrightarrow> (\<delta> state label) \<in> states"
-  using trans_func_ext
-  by auto
-
-lemma give_input_closed:
-  "input \<in> (labels\<^sup>\<star>) \<Longrightarrow> s \<in> states \<Longrightarrow> (\<delta>\<^sup>\<star>) s input \<in> states"
-  apply (induction input arbitrary: s)
-  by (auto simp add: trans_func_well_def)
-
-lemma input_under_concat:
-  "w \<in> labels\<^sup>\<star> \<Longrightarrow> v \<in> labels\<^sup>\<star> \<Longrightarrow> (\<delta>\<^sup>\<star>) s (w @ v) = (\<delta>\<^sup>\<star>) ((\<delta>\<^sup>\<star>) s w) v"
-  apply (induction w arbitrary: s)
-  by auto
-
-lemma eq_pres_under_concat:
-  assumes
-    "w \<in> labels\<^sup>\<star>" and
-    "w' \<in> labels\<^sup>\<star>" and
-    "s \<in> states" and
-    "(\<delta>\<^sup>\<star>) s w = (\<delta>\<^sup>\<star>) s w'"
-  shows "\<forall>v \<in> labels\<^sup>\<star>. (\<delta>\<^sup>\<star>) s (w @ v) = (\<delta>\<^sup>\<star>) s (w' @ v)"
-  using input_under_concat[where w = w and s = s] input_under_concat[where w = w' and s = s] assms
-  by auto
-
-lemma trans_to_charact:
-  "\<And>s a w. \<lbrakk>s \<in> states; a \<in> labels; w \<in> labels\<^sup>\<star>; s = (\<delta>\<^sup>\<star>) i w\<rbrakk> \<Longrightarrow> (\<delta>\<^sup>\<star>) i (w @ [a]) = \<delta> s a"
-proof-
-  fix s a w
-  assume
-    A_0: "s \<in> states" and
-    A_1: "a \<in> labels" and
-    A_2: "w \<in> labels\<^sup>\<star>" and
-    A_3: "s = (\<delta>\<^sup>\<star>) i w"
-  have H_0: "trans_func s a = (\<delta>\<^sup>\<star>) s [a]"
-    by auto
-  from A_2 A_3 H_0 have H_1: "(\<delta>\<^sup>\<star>) s [a] = (\<delta>\<^sup>\<star>) ((\<delta>\<^sup>\<star>) i w) [a]"
-    by simp
-  from A_1 A_2 have H_2: "(\<delta>\<^sup>\<star>) ((\<delta>\<^sup>\<star>) i w) [a] = (\<delta>\<^sup>\<star>) i (w @ [a])"
-    using input_under_concat
-    by force
-  show "(\<delta>\<^sup>\<star>) i (w @ [a]) = \<delta> s a"
-    using A_1 H_0 A_3 H_1 H_2
-    by force
-qed
-
-end
-
-locale aut_hom = Aut0: det_aut A S\<^sub>0 i\<^sub>0 F\<^sub>0 \<delta>\<^sub>0 + Aut1: det_aut A S\<^sub>1 i\<^sub>1 F\<^sub>1 \<delta>\<^sub>1 for
-  A :: "'alpha set" and
-  S\<^sub>0 :: "'states_0 set" and
-  i\<^sub>0 and F\<^sub>0 and \<delta>\<^sub>0 and
-  S\<^sub>1 :: "'states_1 set" and
-  i\<^sub>1 and F\<^sub>1 and \<delta>\<^sub>1 +
-fixes f :: "'states_0 \<Rightarrow> 'states_1"
-assumes
-  hom_is_ext:
-  "f \<in> S\<^sub>0 \<rightarrow>\<^sub>E S\<^sub>1" and
-  pres_init:
-  "f i\<^sub>0 = i\<^sub>1" and
-  pres_final:
-  "s \<in> F\<^sub>0 \<longleftrightarrow> f s \<in> F\<^sub>1 \<and> s \<in> S\<^sub>0" and
-  pres_trans:
-  "s\<^sub>0 \<in> S\<^sub>0 \<Longrightarrow> a \<in> A \<Longrightarrow> f (\<delta>\<^sub>0 s\<^sub>0 a) = \<delta>\<^sub>1 (f s\<^sub>0) a"
-begin
-
-lemma hom_translation:
-  "input \<in> (A\<^sup>\<star>) \<Longrightarrow> s \<in> S\<^sub>0 \<Longrightarrow> (f ((\<delta>\<^sub>0\<^sup>\<star>) s input)) = ((\<delta>\<^sub>1\<^sup>\<star>) (f s) input)"
-  apply (induction input arbitrary: s)
-  by (auto simp add: Aut0.trans_func_well_def pres_trans)
-
-lemma recognise_same_lang:
-  "input \<in> A\<^sup>\<star> \<Longrightarrow> ((\<delta>\<^sub>0\<^sup>\<star>) i\<^sub>0 input) \<in> F\<^sub>0 \<longleftrightarrow> ((\<delta>\<^sub>1\<^sup>\<star>) i\<^sub>1 input) \<in> F\<^sub>1"
-  using hom_translation[where input = input and s = i\<^sub>0]
-  apply (clarsimp  simp add: Aut0.init_state_is_a_state pres_init pres_final)
-  apply (induction input)
-   apply (clarsimp simp add: Aut0.init_state_is_a_state)
-  using Aut0.give_input_closed Aut0.init_state_is_a_state
-  by blast
-
-end
-
-locale aut_epi = aut_hom + 
-  assumes
-    is_epi: "f ` S\<^sub>0 = S\<^sub>1"
-
-locale det_G_aut =
-  is_aut:       det_aut A S i F \<delta> +
-  labels_a_G_set:   alt_grp_act G A \<phi> + 
-  states_a_G_set:   alt_grp_act G S \<psi> +
-  accepting_is_eq_var: eq_var_subset G S \<psi> F +
-  init_is_eq_var:   eq_var_subset G S \<psi> "{i}" +
-  trans_is_eq_var:   eq_var_func G "S \<times> A"
-  "\<lambda>g\<in>carrier G. \<lambda>(s, a) \<in> (S \<times> A). (\<psi> g s, \<phi> g a)"
-  S "\<psi>" "(\<lambda>(s, a) \<in> (S \<times> A). \<delta> s a)"
-  for A :: "'alpha set" (structure) and
-    S :: "'states set" and
-    i F \<delta> and
-    G :: "('grp, 'b) monoid_scheme" and
-    \<phi> \<psi>
-begin
-
-adhoc_overloading
-  star labels_a_G_set.induced_star_map
-
-lemma give_input_eq_var:
-  "eq_var_func G
- (A\<^sup>\<star> \<times> S) (\<lambda>g\<in>carrier G. \<lambda>(w, s) \<in> (A\<^sup>\<star> \<times> S). ((\<phi>\<^sup>\<star>) g w, \<psi> g s))
- S \<psi>
- (\<lambda>(w, s) \<in> (A\<^sup>\<star> \<times> S). (\<delta>\<^sup>\<star>) s w)"
-proof-
-  have H_0: "\<And>a w s g.
-       (\<And>s. s \<in> S \<Longrightarrow> (\<phi>\<^sup>\<star>) g w \<in> A\<^sup>\<star> \<and> \<psi> g s \<in> S \<Longrightarrow>
-       (\<delta>\<^sup>\<star>) (\<psi> g s) ((\<phi>\<^sup>\<star>) g w) = \<psi> g ((\<delta>\<^sup>\<star>) s w)) \<Longrightarrow>
-       s \<in> S \<Longrightarrow>
-       g \<in> carrier G \<Longrightarrow>
-       a \<in> A \<Longrightarrow> \<forall>x\<in>set w. x \<in> A \<Longrightarrow> \<psi> g s \<in> S \<Longrightarrow> \<forall>x\<in>set ((\<phi>\<^sup>\<star>) g (a # w)). x \<in> A \<Longrightarrow>
-       (\<delta>\<^sup>\<star>) (\<psi> g s) ((\<phi>\<^sup>\<star>) g (a # w)) = \<psi> g ((\<delta>\<^sup>\<star>) (\<delta> s a) w)"
-  proof-
-    fix a w s g
-    assume
-      A_IH: "(\<And>s. s \<in> S \<Longrightarrow>
-             (\<phi>\<^sup>\<star>) g w \<in> A\<^sup>\<star> \<and> \<psi> g s \<in> S \<Longrightarrow>
-             (\<delta>\<^sup>\<star>) (\<psi> g s) ((\<phi>\<^sup>\<star>) g w) = \<psi> g ((\<delta>\<^sup>\<star>) s w))" and
-      A_0: "s \<in> S" and
-      A_1: "\<psi> g s \<in> S" and
-      A_2: "\<forall>x\<in>set ((\<phi>\<^sup>\<star>) g (a # w)). x \<in> A" and
-      A_3: "g \<in> carrier G" and
-      A_4: " a \<in> A" and
-      A_5: "\<forall>x\<in>set w. x \<in> A"
-    have H_0: "((\<phi>\<^sup>\<star>) g (a # w)) = (\<phi> g a) # (\<phi>\<^sup>\<star>) g w"
-      using A_4 A_5 A_3
-      by auto
-    hence H_1: "(\<delta>\<^sup>\<star>) (\<psi> g s) ((\<phi>\<^sup>\<star>) g (a # w))
-       = (\<delta>\<^sup>\<star>) (\<psi> g s) ((\<phi> g a) # (\<phi>\<^sup>\<star>) g w)"
-      by simp
-    have H_2: "...  = (\<delta>\<^sup>\<star>) ((\<delta>\<^sup>\<star>) (\<psi> g s) [\<phi> g a]) ((\<phi>\<^sup>\<star>) g w)"
-      using is_aut.input_under_concat
-      by simp
-    have H_3: "(\<delta>\<^sup>\<star>) (\<psi> g s) [\<phi> g a] = \<psi> g (\<delta> s a)"
-      using trans_is_eq_var.eq_var_func_axioms A_4 A_5 A_0 A_1 A_3 apply (clarsimp simp del:
-          GMN_simps simp add: eq_var_func_def eq_var_func_axioms_def make_op_def)
-      apply (rule meta_mp[of "\<psi> g s \<in> S \<and> \<phi> g a \<in> A \<and> s \<in> S \<and> a \<in> A"])
-       apply presburger
-      apply (clarify)
-      using labels_a_G_set.element_image
-      by presburger
-    have H_4: "(\<delta>\<^sup>\<star>) (\<psi> g (\<delta> s a)) ((\<phi>\<^sup>\<star>) g w) = \<psi> g ((\<delta>\<^sup>\<star>) (\<delta> s a) w)"
-      apply (rule A_IH[where s1 = "\<delta> s a"])
-      subgoal
-        using A_4 A_5 A_0
-        by (auto simp add: is_aut.trans_func_well_def)
-      using A_4 A_5 A_0 A_3 \<open>\<delta> s a \<in> S\<close> states_a_G_set.element_image
-      by (metis A_2 Cons_in_lists_iff H_0 in_listsI)
-    show "(\<delta>\<^sup>\<star>) (\<psi> g s) ((\<phi>\<^sup>\<star>) g (a # w)) = \<psi> g ((\<delta>\<^sup>\<star>) (\<delta> s a) w)"
-      using H_0 H_1 H_2 H_3 H_4
-      by presburger
-  qed
-  show ?thesis
-    apply (subst eq_var_func_def)
-    apply (subst eq_var_func_axioms_def)
-    apply (rule conjI)
-     apply (rule prod_group_act[where G = G and A = "A\<^sup>\<star>" and \<phi> = "(\<phi>\<^sup>\<star>)"
-          and B = S     and \<psi> = \<psi>])
-    using labels_a_G_set.lists_a_Gset
-      apply blast
-     apply (simp add: states_a_G_set.group_action_axioms)
-    apply (rule conjI)
-     apply (simp add: states_a_G_set.group_action_axioms)
-    apply (rule conjI)
-     apply (subst extensional_funcset_def)
-     apply (subst restrict_def)
-     apply (subst Pi_def)
-     apply (subst extensional_def)
-     apply (auto simp add: in_listsI is_aut.give_input_closed)[1]
-    apply (subst restrict_def)
-    apply (clarsimp simp del: GMN_simps simp add: make_op_def)
-    apply (rule conjI; intro impI)
-    subgoal for w s g
-      apply (induction w arbitrary: s) 
-       apply simp
-      apply (clarsimp simp del: GMN_simps)
-      by (simp add: H_0 del: GMN_simps)
-    apply clarsimp
-    by (metis (no_types, lifting) image_iff in_lists_conv_set labels_a_G_set.surj_prop list.set_map
-        states_a_G_set.element_image)
-qed
-
-definition
-  accepted_words :: "'alpha list set"
-  where "accepted_words = {w. w \<in> A\<^sup>\<star> \<and> ((\<delta>\<^sup>\<star>) i w) \<in> F}"
-
-lemma induced_g_lang:
-  "G_lang G A \<phi> accepted_words"
-proof-
-  have H_0: "\<And>g w. g \<in> carrier G \<Longrightarrow> w \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i w \<in> F \<Longrightarrow> map (\<phi> g) w \<in> A\<^sup>\<star>"
-    apply (clarsimp)
-    using labels_a_G_set.element_image
-    by blast
-  have H_1: "\<And>g w. g \<in> carrier G \<Longrightarrow> w \<in> A\<^sup>\<star> \<Longrightarrow> (\<delta>\<^sup>\<star>) i w \<in> F \<Longrightarrow> (\<delta>\<^sup>\<star>) i (map (\<phi> g) w) \<in> F"
-  proof -
-    fix g w
-    assume
-      A_0: "g \<in> carrier G" and
-      A_1: "w \<in> A\<^sup>\<star>" and
-      A_2: "(\<delta>\<^sup>\<star>) i w \<in> F"
-    have H1_0: "\<psi> g ((\<delta>\<^sup>\<star>) i w) \<in> F"
-      using accepting_is_eq_var.eq_var_subset_axioms
-        A_0 A_2 accepting_is_eq_var.is_equivar
-      by blast
-    have H1_1: "\<psi> g i = i"
-      using init_is_eq_var.eq_var_subset_axioms A_0
-        init_is_eq_var.is_equivar
-      by auto
-    have H1_2: "\<And>w g. \<lbrakk>g \<in> carrier G; w \<in> A\<^sup>\<star>; (\<delta>\<^sup>\<star>) i w \<in> F\<rbrakk> \<Longrightarrow> (\<phi>\<^sup>\<star>) g w \<in> A\<^sup>\<star>"
-      using H_0
-      by auto
-    from A_1 have H1_3: "w \<in> A\<^sup>\<star>"
-      by auto
-    show "(\<delta>\<^sup>\<star>) i (map (\<phi> g) w) \<in> F"
-      using give_input_eq_var A_0 A_1 H1_1 H1_3
-      apply (clarsimp simp del: GMN_simps simp add: eq_var_func_def eq_var_func_axioms_def
-          make_op_def)
-      using A_2 H1_0 is_aut.init_state_is_a_state H1_2
-      by (smt (verit, best) H1_3 labels_a_G_set.induced_star_map_def restrict_apply)
-  qed
-  have H_2: "\<And>g x. g \<in> carrier G \<Longrightarrow> x \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i x \<in> F \<Longrightarrow>
-            x \<in> map (\<phi> g) ` {w \<in> A\<^sup>\<star>. (\<delta>\<^sup>\<star>) i w \<in> F}"
-  proof-
-    fix g w
-    assume
-      A_0: "g \<in> carrier G" and
-      A_1: "w \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i w \<in> F" 
-    have H_0: "\<And>g xa. g \<in> carrier G \<Longrightarrow> xa \<in> A\<^sup>\<star> \<Longrightarrow> (\<delta>\<^sup>\<star>) i xa \<in> F \<Longrightarrow> (\<delta>\<^sup>\<star>) i ((\<phi>\<^sup>\<star>) g xa) \<in> F" 
-      using H_1
-      by auto
-    have H_1: "((\<phi>\<^sup>\<star>) (inv \<^bsub>G\<^esub> g) w) \<in> A\<^sup>\<star>"
-      by (smt (verit) A_0 A_1 in_listsI alt_grp_act_def group_action.bij_prop1
-          group_action.orbit_sym_aux labels_a_G_set.lists_a_Gset)
-    have H_2: "(\<delta>\<^sup>\<star>) i ((\<phi>\<^sup>\<star>) (inv \<^bsub>G\<^esub> g) w) \<in> F"
-      apply (rule H_0)
-      using A_0 A_1
-      by (meson group.inv_closed group_hom.axioms(1) labels_a_G_set.group_hom)+
-    have H_3: "((\<phi>\<^sup>\<star>) g ((\<phi>\<^sup>\<star>) (inv \<^bsub>G\<^esub> g) w))
-     \<in> (\<phi>\<^sup>\<star>) g ` {w \<in> A\<^sup>\<star>. (\<delta>\<^sup>\<star>) i w \<in> F}"
-      using H_1 H_2
-      by auto 
-    have H_4: "((\<phi>\<^sup>\<star>) g ((\<phi>\<^sup>\<star>) (inv \<^bsub>G\<^esub> g) w)) = w"
-      using A_0 A_1  
-      by (smt (verit) in_listsI alt_grp_act_def group_action.bij_prop1 group_action.orbit_sym_aux
-          labels_a_G_set.lists_a_Gset)
-    show "w \<in> map (\<phi> g) ` {w \<in> A\<^sup>\<star>. (\<delta>\<^sup>\<star>) i w \<in> F}"
-      using H_3 H_4 A_0
-      by auto
-  qed
-  show ?thesis
-    apply (clarsimp simp del: GMN_simps simp add: G_lang_def accepted_words_def G_lang_axioms_def)
-    apply (rule conjI)
-    using labels_a_G_set.alt_grp_act_axioms
-     apply (auto)[1]
-    apply (clarsimp simp del: GMN_simps simp add: eq_var_subset_def eq_var_subset_axioms_def)
-    apply (intro conjI)
-      apply (simp add: language.intro)
-    using labels_a_G_set.alt_grp_act_axioms labels_a_G_set.lists_a_Gset
-     apply blast
-    apply (clarsimp simp del: GMN_simps)
-    apply (rule subset_antisym; simp add: Set.subset_eq; rule allI; rule impI)
-     apply (rule conjI)
-      apply (simp add: H_0)
-     apply (simp add: H_1)
-    by (simp add: H_2)
-qed
-end
-
-locale reach_det_aut =
-  det_aut A S i F \<delta>  
-  for A :: "'alpha set" (structure) and
-    S :: "'states set" and
-    i F \<delta> +
-  assumes
-    is_reachable:
-    "s \<in> S \<Longrightarrow> \<exists>input \<in> A\<^sup>\<star>. (\<delta>\<^sup>\<star>) i input = s"
-
-locale reach_det_G_aut =
-  det_G_aut A S i F \<delta> G \<phi> \<psi> + reach_det_aut A S i F \<delta> 
-  for A :: "'alpha set" (structure) and
-    S :: "'states set" and
-    i and F and \<delta> and
-    G :: "('grp, 'b) monoid_scheme" and
-    \<phi> \<psi> 
-begin
-
-text \<open>
-To avoid duplicate variant of "star":
-\<close>
-
-no_adhoc_overloading
-  star labels_a_G_set.induced_star_map
-end
-
-sublocale reach_det_G_aut \<subseteq> reach_det_aut
-  using reach_det_aut_axioms
-  by simp
-
-locale G_aut_hom = Aut0: reach_det_G_aut A S\<^sub>0 i\<^sub>0 F\<^sub>0 \<delta>\<^sub>0 G \<phi> \<psi>\<^sub>0 +
-  Aut1: reach_det_G_aut A S\<^sub>1 i\<^sub>1 F\<^sub>1 \<delta>\<^sub>1 G \<phi> \<psi>\<^sub>1 +
-  hom_f: aut_hom A S\<^sub>0 i\<^sub>0 F\<^sub>0 \<delta>\<^sub>0 S\<^sub>1 i\<^sub>1 F\<^sub>1 \<delta>\<^sub>1 f +
-  eq_var_f: eq_var_func G S\<^sub>0 \<psi>\<^sub>0 S\<^sub>1 \<psi>\<^sub>1 f for
-  A :: "'alpha set" and
-  S\<^sub>0 :: "'states_0 set" and
-  i\<^sub>0 and F\<^sub>0 and \<delta>\<^sub>0 and
-  S\<^sub>1 :: "'states_1 set" and
-  i\<^sub>1 and F\<^sub>1 and \<delta>\<^sub>1 and
-  G :: "('grp, 'b) monoid_scheme" and
-  \<phi> \<psi>\<^sub>0 \<psi>\<^sub>1 f 
-
-locale G_aut_epi = G_aut_hom + 
-  assumes
-    is_epi: "f ` S\<^sub>0 = S\<^sub>1"
-
-locale det_aut_rec_lang = det_aut A S i F \<delta>  + language A L
-  for A :: "'alpha set" (structure) and
-    S :: "'states set" and
-    i F \<delta> L + 
-  assumes
-    is_recognised:
-    "w \<in> L \<longleftrightarrow> w \<in> A\<^sup>\<star> \<and> ((\<delta>\<^sup>\<star>) i w) \<in> F"
-
-locale det_G_aut_rec_lang = det_G_aut A S i F \<delta> G \<phi> \<psi> + det_aut_rec_lang A S i F \<delta> L 
-  for A :: "'alpha set" (structure) and
-    S :: "'states set" and
-    i F \<delta> and
-    G :: "('grp, 'b) monoid_scheme" and
-    \<phi> \<psi> L
-begin
-
-lemma lang_is_G_lang: "G_lang G A \<phi> L"
-proof-
-  have H0: "L = accepted_words"
-    apply (simp add: accepted_words_def)
-    apply (subst is_recognised [symmetric])
-    by simp
-  show "G_lang G A \<phi> L"
-    apply (subst H0)
-    apply (rule det_G_aut.induced_g_lang[of A S i F \<delta> G \<phi> \<psi>])
-    by (simp add: det_G_aut_axioms)
-qed  
-
-text \<open>
-To avoid ambiguous parse trees:
-\<close>
-
-no_notation trans_is_eq_var.GA_0.induced_quot_map ("[_]\<^sub>_\<index>" 60)
-no_notation states_a_G_set.induced_quot_map ("[_]\<^sub>_\<index>" 60)
-
-end
-
-locale reach_det_aut_rec_lang = reach_det_aut A S i F \<delta> + det_aut_rec_lang A S i F \<delta> L
-  for A :: "'alpha set" and
-    S :: "'states set" and
-    i F \<delta> and
-    L :: "'alpha list set"
-
-locale reach_det_G_aut_rec_lang = det_G_aut_rec_lang A S i F \<delta> G \<phi> \<psi> L +
-  reach_det_G_aut A S i F \<delta> G \<phi> \<psi>
-  for A :: "'alpha set" and
-    S :: "'states set" and
-    i F \<delta> and
-    G :: "('grp, 'b) monoid_scheme" and
-    \<phi> \<psi> and
-    L :: "'alpha list set"
-
-sublocale reach_det_G_aut_rec_lang \<subseteq> det_G_aut_rec_lang
-  apply (simp add: det_G_aut_rec_lang_def)
-  using reach_det_G_aut_rec_lang_axioms
-  by (simp add: det_G_aut_axioms det_aut_rec_lang_axioms) 
-
-locale det_G_aut_recog_G_lang = det_G_aut_rec_lang A S i F \<delta> G \<phi> \<psi> L + G_lang G A \<phi> L
-  for A :: "'alpha set" (structure) and
-    S :: "'states set" and
-    i F \<delta> and
-    G :: "('grp, 'b) monoid_scheme" and
-    \<phi> \<psi> and
-    L :: "'alpha list set"
-
-sublocale det_G_aut_rec_lang \<subseteq> det_G_aut_recog_G_lang
-  apply (simp add: det_G_aut_recog_G_lang_def)
-  apply (rule conjI)
-   apply (simp add: det_G_aut_rec_lang_axioms)
-  by (simp add: lang_is_G_lang)
-
-locale reach_det_G_aut_rec_G_lang = reach_det_G_aut_rec_lang A S i F \<delta> G \<phi> \<psi> L + G_lang G A \<phi> L
-  for A :: "'alpha set" (structure) and
-    S :: "'states set" and
-    i F \<delta> and
-    G :: "('grp, 'b) monoid_scheme" and
-    \<phi> \<psi> L 
-
-sublocale reach_det_G_aut_rec_lang \<subseteq> reach_det_G_aut_rec_G_lang
-  apply (simp add: reach_det_G_aut_rec_G_lang_def)
-  apply (rule conjI)
-   apply (simp add: reach_det_G_aut_rec_lang_axioms)
-  by (simp add: lang_is_G_lang)
-
-lemma (in reach_det_G_aut)
-  "reach_det_G_aut_rec_lang A S i F \<delta> G \<phi> \<psi> accepted_words"
-  apply (clarsimp simp del: simp add: reach_det_G_aut_rec_lang_def
-      det_G_aut_rec_lang_def det_aut_rec_lang_axioms_def)
-  apply (intro conjI)
-    apply (simp add: det_G_aut_axioms)
-   apply (clarsimp simp add: reach_det_G_aut_axioms accepted_words_def reach_det_aut_rec_lang_def)
-   apply (simp add: det_aut_rec_lang_def det_aut_rec_lang_axioms.intro is_aut.det_aut_axioms
-      language_def)
-  by (simp add: reach_det_G_aut_axioms)
-
-lemma (in det_G_aut) action_on_input:
-  "\<And> g w. g \<in> carrier G \<Longrightarrow> w \<in> A\<^sup>\<star> \<Longrightarrow> \<psi> g ((\<delta>\<^sup>\<star>) i w) = (\<delta>\<^sup>\<star>) i ((\<phi>\<^sup>\<star>) g w)"
-proof-
-  fix g w
-  assume
-    A_0: "g \<in> carrier G" and
-    A_1: "w \<in> A\<^sup>\<star>"
-  have H_0: "(\<delta>\<^sup>\<star>) (\<psi> g i) ((\<phi>\<^sup>\<star>) g w) = (\<delta>\<^sup>\<star>) i ((\<phi>\<^sup>\<star>) g w)"
-    using A_0 init_is_eq_var.is_equivar
-    by fastforce
-  have H_1: "\<psi> g ((\<delta>\<^sup>\<star>) i w) = (\<delta>\<^sup>\<star>) (\<psi> g i) ((\<phi>\<^sup>\<star>) g w)"
-    using A_0 A_1 give_input_eq_var
-    apply (clarsimp simp del: GMN_simps simp add: eq_var_func_axioms_def eq_var_func_def
-        make_op_def)
-    apply (rule meta_mp[of "((\<phi>\<^sup>\<star>) g w) \<in> A\<^sup>\<star> \<and> \<psi> g i \<in> S"]) 
-    using is_aut.init_state_is_a_state A_1
-     apply presburger
-    using det_G_aut_axioms
-    apply (clarsimp simp add: det_G_aut_def) 
-    apply (rule conjI; rule impI; rule conjI)
-    using labels_a_G_set.element_image
-       apply fastforce
-    using is_aut.init_state_is_a_state states_a_G_set.element_image
-    by blast+
-  show "\<psi> g ((\<delta>\<^sup>\<star>) i w) = (\<delta>\<^sup>\<star>) i ((\<phi>\<^sup>\<star>) g w)"
-    using H_0 H_1
-    by simp
-qed
-
-definition (in det_G_aut)
-  reachable_states :: "'states set" ("S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h")
-  where "S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h = {s . \<exists> w \<in> A\<^sup>\<star>. (\<delta>\<^sup>\<star>) i w = s}"
-
-definition (in det_G_aut)
-  reachable_trans :: "'states \<Rightarrow> 'alpha \<Rightarrow> 'states" ("\<delta>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h")
-  where "\<delta>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h s a = (\<lambda>(s', a') \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<times> A. \<delta> s' a') (s, a)"
-
-definition (in det_G_aut)
-  reachable_action :: "'grp \<Rightarrow> 'states \<Rightarrow> 'states" ("\<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h")
-  where "\<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g s = (\<lambda>(g', s') \<in> carrier G \<times> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h. \<psi> g' s') (g, s)"
-
-lemma (in det_G_aut) reachable_action_is_restict:
-  "\<And>g s. g \<in> carrier G \<Longrightarrow> s \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<Longrightarrow> \<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g s = \<psi> g s"
-  by (auto simp add: reachable_action_def reachable_states_def)
-
-lemma (in det_G_aut_rec_lang) reach_det_aut_is_det_aut_rec_L:
-  "reach_det_G_aut_rec_lang A S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h i (F \<inter> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h) \<delta>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h G \<phi> \<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h L"
-proof-
-  have H_0: "(\<lambda>(x, y). \<delta>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h x y) \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<times> A \<rightarrow>\<^sub>E S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h"
-  proof-
-    have H1_0: "(\<lambda>(x, y). \<delta> x y) \<in> extensional (S \<times> A)"
-      using is_aut.trans_func_ext
-      by (simp add: PiE_iff)
-    have H1_1: "(\<lambda>(s', a') \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<times> A. \<delta> s' a') \<in> extensional (S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<times> A)"
-      using H1_0
-      by simp
-    have H1_2: "(\<lambda>(s', a')\<in>S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<times> A. \<delta> s' a') = (\<lambda>(x, y). \<delta>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h x y)"
-      by (auto simp add: reachable_trans_def)
-    show "(\<lambda>(x, y). \<delta>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h x y) \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<times> A \<rightarrow>\<^sub>E S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h"
-      apply (clarsimp simp add: PiE_iff)
-      apply (rule conjI)
-       apply (clarify)
-      using reachable_trans_def
-       apply (simp add: reachable_states_def)[1]
-       apply (metis Cons_in_lists_iff append_Nil2 append_in_lists_conv is_aut.give_input_closed
-          is_aut.init_state_is_a_state is_aut.trans_to_charact)
-      using H1_1 H1_2
-      by simp
-  qed
-  have H_1: "\<And>g. g \<in> carrier G \<Longrightarrow>
-    (\<And>s. \<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g s = (if s \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h then case (g, s) of (x, xa) \<Rightarrow> \<psi> x xa else undefined)) \<Longrightarrow>
-    bij_betw (\<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g) S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h"
-  proof-
-    fix g
-    assume
-      A1_0: "g \<in> carrier G" and
-      A1_1: "(\<And>s. \<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g s =
-        (if s \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h
-        then case (g, s) of (x, xa) \<Rightarrow> \<psi> x xa
-        else undefined))" 
-    have H1_0: "\<And>r. r \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<Longrightarrow> (\<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g) r \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h"
-      using A1_0
-      apply (clarsimp simp add: reachable_states_def reachable_action_def)
-      apply (rule meta_mp[of "\<And>w. w \<in> A\<^sup>\<star> \<Longrightarrow> ((\<phi>\<^sup>\<star>) g w) \<in> A\<^sup>\<star>"])
-      using action_on_input[where g = g]
-       apply (metis in_listsI) 
-      by (metis alt_group_act_is_grp_act group_action.element_image labels_a_G_set.lists_a_Gset)
-    have H1_1: "\<And>f T U. bij_betw f T T \<Longrightarrow> f ` U = U \<Longrightarrow> U \<subseteq> T \<Longrightarrow> bij_betw (restrict f U) U U"
-      apply (clarsimp simp add: bij_betw_def inj_on_def image_def)
-      by (meson in_mono)
-    have H1_2: "\<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g = restrict (\<psi> g) S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h"
-      using reachable_action_def A1_0
-      by (auto simp add: restrict_def)
-    have H1_3: "bij_betw (\<psi> g) S S \<Longrightarrow> (\<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g) ` S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h = S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h
-      \<Longrightarrow> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<subseteq> S \<Longrightarrow> bij_betw (\<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g) S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h"
-      by (metis H1_2 bij_betw_imp_inj_on inj_on_imp_bij_betw inj_on_restrict_eq inj_on_subset)
-    have H1_4: "\<And>w s. s = (\<delta>\<^sup>\<star>) i w \<Longrightarrow>
-         \<forall>x\<in>set w. x \<in> A \<Longrightarrow>
-         \<exists>x. (\<exists>w\<in> A\<^sup>\<star>. (\<delta>\<^sup>\<star>) i w = x) \<and> (\<delta>\<^sup>\<star>) i w = \<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g x"
-    proof-
-      fix w s
-      assume
-        A2_0: "\<forall>x\<in>set w. x \<in> A" and
-        A2_1: "s = (\<delta>\<^sup>\<star>) i w"
-      have H2_0: "(inv \<^bsub>G\<^esub> g) \<in> carrier G"
-        apply (rule meta_mp[of "group G"])
-        using A1_0
-         apply simp
-        using det_G_aut_rec_lang_axioms
-        by (auto simp add: det_G_aut_rec_lang_def
-            det_aut_rec_lang_axioms_def det_G_aut_def group_action_def group_hom_def)
-      have H2_1: "\<psi> (inv \<^bsub>G\<^esub> g) s = (\<delta>\<^sup>\<star>) i ((\<phi>\<^sup>\<star>) (inv \<^bsub>G\<^esub> g) w)"
-        apply (simp del: GMN_simps add: A2_1)
-        apply (rule action_on_input[where g = "(inv \<^bsub>G\<^esub> g)" and w = w]) 
-        using H2_0 A2_0
-        by auto
-      have H2_2: "((\<phi>\<^sup>\<star>) (inv \<^bsub>G\<^esub> g) w) \<in> A\<^sup>\<star>"
-        using A2_0 H2_0 det_G_aut_rec_lang_axioms
-        apply (clarsimp)
-        using labels_a_G_set.surj_prop list.set_map
-        by fastforce
-      have H2_3: "\<exists>w\<in>A\<^sup>\<star>. (\<delta>\<^sup>\<star>) i w = \<psi> (inv \<^bsub>G\<^esub> g) s" 
-        by (metis H2_1 H2_2)
-      from H2_3 have H2_4: "\<psi> (inv \<^bsub>G\<^esub> g) s \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h"
-        by (simp add: reachable_states_def)
-      have H2_5: "\<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g (\<psi> (inv \<^bsub>G\<^esub> g) s) = \<psi> g (\<psi> (inv \<^bsub>G\<^esub> g) s)"
-        apply (rule reachable_action_is_restict)
-        using A1_0 H2_4
-        by simp+
-      have H2_6: "(\<delta>\<^sup>\<star>) i w = \<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g (\<psi> (inv \<^bsub>G\<^esub> g) s)"
-        apply (simp add: H2_5 A2_1) 
-        by (metis A1_0 A2_0 in_listsI A2_1 H2_5 is_aut.give_input_closed
-            is_aut.init_state_is_a_state states_a_G_set.bij_prop1 states_a_G_set.orbit_sym_aux)
-      show " \<exists>x. (\<exists>w\<in>A\<^sup>\<star>. (\<delta>\<^sup>\<star>) i w = x) \<and> (\<delta>\<^sup>\<star>) i w = \<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g x"
-        using H2_3 H2_6
-        by blast
-    qed
-    show "bij_betw (\<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g) S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h"
-      apply (rule H1_3)
-        apply (simp add: A1_0 bij_betw_def states_a_G_set.inj_prop states_a_G_set.surj_prop)
-       apply (clarsimp simp add: image_def H1_0)
-       apply (rule subset_antisym; simp add: Set.subset_eq; clarify)
-      using H1_0 
-        apply auto[1]
-      subgoal for s
-        apply (clarsimp simp add: reachable_states_def)
-        by (simp add: H1_4)
-      apply (simp add: reachable_states_def Set.subset_eq; rule allI; rule impI)
-      using is_aut.give_input_closed is_aut.init_state_is_a_state
-      by auto
-  qed
-  have H_2: "group G"
-    using det_G_aut_rec_lang_axioms
-    by (auto simp add: det_G_aut_rec_lang_def det_G_aut_def group_action_def
-        group_hom_def)
-  have H_3: "\<And>g. g \<in> carrier G \<Longrightarrow> \<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g \<in> carrier (BijGroup S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h)"
-    subgoal for g
-      using reachable_action_def[where g = g]
-      apply (simp add: BijGroup_def Bij_def extensional_def)
-      by (simp add: H_1)
-    done
-  have H_4: "\<And>x y. x \<in> carrier G \<Longrightarrow> y \<in> carrier G \<Longrightarrow> \<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h (x \<otimes>\<^bsub>G\<^esub> y) = \<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h x \<otimes>\<^bsub>BijGroup S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h\<^esub>
-                                                                           \<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h y"
-  proof -
-    fix g h
-    assume
-      A1_0: "g \<in> carrier G" and
-      A1_1: "h \<in> carrier G"
-    have H1_0: "\<And>g . g \<in> carrier G \<Longrightarrow> \<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g = restrict (\<psi> g) S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h"
-      using reachable_action_def
-      by (auto simp add: restrict_def)
-    from H1_0 have H1_1: "\<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h (g \<otimes>\<^bsub>G\<^esub> h) = restrict (\<psi> (g \<otimes>\<^bsub>G\<^esub> h)) S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h"
-      by (simp add: A1_0 A1_1 H_2 group.subgroup_self subgroup.m_closed)
-    have H1_2: "\<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g \<otimes>\<^bsub>BijGroup S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h\<^esub> \<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h h = 
-    (restrict (\<psi> g) S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h) \<otimes>\<^bsub>BijGroup S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h\<^esub>
-    (restrict (\<psi> h) S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h)"
-      using A1_0 A1_1 H1_0
-      by simp
-    have H1_3: "\<And>g. g \<in> carrier G \<Longrightarrow> \<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g \<in> carrier (BijGroup S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h)"
-      by (simp add: H_3)
-    have H1_4: "\<And>x y. x\<in>carrier G \<Longrightarrow> y\<in>carrier G \<Longrightarrow> \<psi> (x \<otimes>\<^bsub>G\<^esub> y) = \<psi> x \<otimes>\<^bsub>BijGroup S\<^esub> \<psi> y"
-      using det_G_aut_axioms
-      by (simp add: det_G_aut_def group_action_def group_hom_def group_hom_axioms_def hom_def)
-    hence H1_5: "\<psi> (g \<otimes>\<^bsub>G\<^esub> h) = \<psi> g \<otimes>\<^bsub>BijGroup S\<^esub> \<psi> h"
-      using A1_0 A1_1
-      by simp
-    have H1_6: "(\<lambda>x. if x \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h
-                   then if (if x \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h
-                            then \<psi> h x
-                            else undefined) \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h
-                        then \<psi> g (if x \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h
-                                  then \<psi> h x
-                                  else undefined)
-                        else undefined
-                   else undefined) =
-              (\<lambda>x. if x \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h
-                   then \<psi> g (\<psi> h x) 
-                   else undefined)" 
-      apply (rule meta_mp[of "\<And>x. x \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<Longrightarrow> (\<psi> h x) \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h"])
-      using H1_3[where g1 = h] A1_1 H1_0
-      by (auto simp add: A1_1 BijGroup_def Bij_def bij_betw_def) 
-    have H1_7: "... = (\<lambda>x. if x \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h
-                          then if x \<in> S
-                               then \<psi> g (\<psi> h x)
-                               else undefined
-                          else undefined)"
-      apply (clarsimp simp add: reachable_states_def) 
-      by (metis is_aut.give_input_closed is_aut.init_state_is_a_state)
-    have H1_8: "(restrict (\<psi> g) S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h) \<otimes>\<^bsub>BijGroup S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h\<^esub> (restrict (\<psi> h) S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h) =
-     restrict (\<psi> (g \<otimes>\<^bsub>G\<^esub> h)) S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h"
-      apply (rule meta_mp[of "\<And>g. g \<in> carrier G \<Longrightarrow> restrict (\<psi> g) S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<in> Bij S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<and>
-      \<psi> g \<in> Bij S"])
-       apply (clarsimp simp add: H1_5 BijGroup_def; intro conjI; intro impI)
-      subgoal
-        using A1_0 A1_1
-        apply (clarsimp simp add: compose_def restrict_def)
-        by (simp add: H1_6 H1_7) 
-       apply (simp add: A1_0 A1_1)+
-      subgoal for g
-        using H1_3[where g1 = g] H1_0[of g]
-        by (simp add: BijGroup_def states_a_G_set.bij_prop0)
-      done
-    show "\<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h (g \<otimes>\<^bsub>G\<^esub> h) =
-      \<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g \<otimes>\<^bsub>BijGroup S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h\<^esub> \<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h h"
-      by (simp add: H1_1 H1_2 H1_8)
-  qed
-  have H_5: "\<And>w' w g. g \<in> carrier G \<Longrightarrow>
-          (\<delta>\<^sup>\<star>) i w \<in> F \<Longrightarrow> \<forall>x\<in>set w. x \<in> A \<Longrightarrow> (\<delta>\<^sup>\<star>) i w' = (\<delta>\<^sup>\<star>) i w \<Longrightarrow> \<forall>x\<in>set w'. x \<in> A \<Longrightarrow>
-          \<exists>w'\<in> A\<^sup>\<star>. (\<delta>\<^sup>\<star>) i w' = \<psi> g ((\<delta>\<^sup>\<star>) i w)"
-  proof -
-    fix w' w g
-    assume
-      A1_0: "g \<in> carrier G" and
-      A1_1: "(\<delta>\<^sup>\<star>) i w \<in> F" and
-      A1_2: "\<forall>x\<in>set w. x \<in> A" and
-      A1_3: "(\<delta>\<^sup>\<star>) i w' = (\<delta>\<^sup>\<star>) i w" and
-      A1_4: "\<forall>x\<in>set w. x \<in> A"
-    from A1_1 A1_2 have H1_0: "((\<delta>\<^sup>\<star>) i w) \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h"
-      using reachable_states_def
-      by auto
-    have H1_1: "\<psi> g ((\<delta>\<^sup>\<star>) i w) = ((\<delta>\<^sup>\<star>) i ((\<phi>\<^sup>\<star>) g w))"
-      using give_input_eq_var
-      apply (clarsimp simp add: eq_var_func_def eq_var_func_axioms_def simp del: GMN_simps)
-      using A1_0 A1_2 action_on_input
-      by blast
-    have H1_2: "(\<phi>\<^sup>\<star>) g w \<in> A\<^sup>\<star>"
-      using A1_0 A1_2
-      by (metis in_listsI alt_group_act_is_grp_act group_action.element_image
-          labels_a_G_set.lists_a_Gset)
-    show "\<exists>wa\<in>A\<^sup>\<star>. (\<delta>\<^sup>\<star>) i wa = \<psi> g ((\<delta>\<^sup>\<star>) i w)"
-      by (metis H1_1 H1_2)
-  qed
-  have H_6: "alt_grp_act G S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h"
-    apply (clarsimp simp add: group_action_def group_hom_def group_hom_axioms_def hom_def)
-    apply (intro conjI)
-       apply (simp add: H_2)
-    subgoal 
-      by (simp add: group_BijGroup)
-     apply clarify
-     apply (simp add: H_3)
-    by (simp add: H_4)
-  have H_7: "\<And>g w. g \<in> carrier G \<Longrightarrow> (\<delta>\<^sup>\<star>) i w \<in> F \<Longrightarrow> \<forall>x\<in>set w. x \<in> A \<Longrightarrow>
-            \<exists>x. x \<in> F \<and> (\<exists>w\<in>A\<^sup>\<star>. (\<delta>\<^sup>\<star>) i w = x) \<and> (\<delta>\<^sup>\<star>) i w = \<psi> g x"
-  proof-
-    fix g w
-    assume
-      A1_0: "g \<in> carrier G" and
-      A1_1: "(\<delta>\<^sup>\<star>) i w \<in> F" and
-      A1_2: "\<forall>x\<in>set w. x \<in> A"
-    have H1_0: "(inv \<^bsub>G\<^esub> g) \<in> carrier G" 
-      by (meson A1_0 group.inv_closed group_hom.axioms(1) labels_a_G_set.group_hom)
-    have H1_1: "((\<delta>\<^sup>\<star>) i w) \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h"
-      using A1_1 A1_2 reachable_states_def
-      by auto
-    have H1_2: "\<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h (inv \<^bsub>G\<^esub> g) ((\<delta>\<^sup>\<star>) i w) = \<psi> (inv \<^bsub>G\<^esub> g) ((\<delta>\<^sup>\<star>) i w)"
-      apply (rule reachable_action_is_restict)
-      using H1_0 H1_1
-      by auto
-    have H1_3: "\<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g (\<psi> (inv \<^bsub>G\<^esub> g) ((\<delta>\<^sup>\<star>) i w)) = ((\<delta>\<^sup>\<star>) i w)" 
-      by (smt (verit) A1_0 H1_1 H_6 H1_2
-          alt_group_act_is_grp_act group_action.bij_prop1 group_action.orbit_sym_aux)
-    have H1_4: "\<psi> (inv \<^bsub>G\<^esub> g) ((\<delta>\<^sup>\<star>) i w) \<in> F"
-      using A1_1 H1_0 accepting_is_eq_var.is_equivar
-      by blast
-    have H1_5: "\<psi> (inv \<^bsub>G\<^esub> g) ((\<delta>\<^sup>\<star>) i w) \<in> F \<and> (\<delta>\<^sup>\<star>) i w = \<psi> g (\<psi> (inv \<^bsub>G\<^esub> g) ((\<delta>\<^sup>\<star>) i w))"
-      using H1_4 H1_3 A1_0 A1_1 H1_0 H1_1 reachable_action_is_restict
-      by (metis H_6 alt_group_act_is_grp_act
-          group_action.element_image)
-    have H1_6: "\<psi> (inv \<^bsub>G\<^esub> g) ((\<delta>\<^sup>\<star>) i w) = ((\<delta>\<^sup>\<star>) i ((\<phi>\<^sup>\<star>) (inv \<^bsub>G\<^esub> g) w))"
-      using give_input_eq_var
-      apply (clarsimp simp add: eq_var_func_def eq_var_func_axioms_def simp del: GMN_simps)
-      using A1_2 H1_0 action_on_input
-      by blast
-    have H1_7: "(\<phi>\<^sup>\<star>) (inv \<^bsub>G\<^esub> g) w \<in> A\<^sup>\<star>"
-      by (metis A1_2 in_listsI H1_0 alt_group_act_is_grp_act group_action.element_image
-          labels_a_G_set.lists_a_Gset)
-    thus "\<exists>x. x \<in> F \<and> (\<exists>w\<in>A\<^sup>\<star>. (\<delta>\<^sup>\<star>) i w = x) \<and> (\<delta>\<^sup>\<star>) i w = \<psi> g x"
-      using H1_5 H1_6 H1_7
-      by metis
-  qed
-  have H_8: "\<And>r a g . r \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<Longrightarrow> a \<in> A \<Longrightarrow> \<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g r \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<and> \<phi> g a \<in> A \<Longrightarrow> g \<in> carrier G \<Longrightarrow>
-            \<delta>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h (\<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g r) (\<phi> g a) = \<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g (\<delta>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h r a)"
-  proof-
-    fix r a g
-    assume
-      A1_0: "r \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h" and
-      A1_1: "a \<in> A" and
-      A1_2: "\<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g r \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<and> \<phi> g a \<in> A" and
-      A1_3: "g \<in> carrier G"
-    have H1_0: "r \<in> S \<and> \<psi> g r \<in> S"
-      apply (rule conjI)
-      subgoal
-        using A1_0
-        apply (clarsimp simp add: reachable_states_def) 
-        by (simp add: in_listsI is_aut.give_input_closed is_aut.init_state_is_a_state)
-      using \<open>r \<in> S\<close> A1_3 states_a_G_set.element_image
-      by blast
-    have H1_1: "\<And>a b g . a \<in> S \<and> b \<in> A \<Longrightarrow> g \<in> carrier G \<Longrightarrow>
-        (if \<psi> g a \<in> S \<and> \<phi> g b \<in> A then \<delta> (\<psi> g a) (\<phi> g b) else undefined) =
-        \<psi> g (\<delta> a b)"
-      using det_G_aut_axioms A1_0 A1_1 A1_3
-      apply (clarsimp simp add: det_G_aut_def eq_var_func_def eq_var_func_axioms_def) 
-      by presburger+
-    hence H1_2: "\<psi> g (\<delta> r a) = (\<delta> (\<psi> g r) (\<phi> g a))"
-      using H1_1[where a1 = r and b1 = a and g1 = g] H1_0 A1_1 A1_2 A1_3
-      by simp
-    have H1_3: "\<And>a w. a \<in> A \<Longrightarrow> w \<in> A\<^sup>\<star> \<Longrightarrow> \<exists>w'\<in> A\<^sup>\<star>. (\<delta>\<^sup>\<star>) i w' = \<delta> ((\<delta>\<^sup>\<star>) i w) a"
-    proof -
-      fix a w
-      assume
-        A2_0: "a \<in> A" and
-        A2_1: "w \<in> A\<^sup>\<star>" 
-      have H2_0: "(w @ [a]) \<in> A\<^sup>\<star> \<and> (w @ [a]) \<in> A\<^sup>\<star> \<Longrightarrow> (\<delta>\<^sup>\<star>) i (w @ [a]) = \<delta> ((\<delta>\<^sup>\<star>) i w) a"
-        by (simp add: is_aut.give_input_closed is_aut.trans_to_charact
-            is_aut.init_state_is_a_state)
-      show "\<exists>w'\<in>A\<^sup>\<star>. (\<delta>\<^sup>\<star>) i w' = \<delta> ((\<delta>\<^sup>\<star>) i w) a"
-        using H2_0
-        apply clarsimp 
-        by (metis A2_0 A2_1 append_in_lists_conv lists.Cons lists.Nil)
-    qed
-    have H1_4: "\<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g (\<delta>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h r a) = \<psi> g (\<delta> r a)"
-      apply (clarsimp simp add: reachable_action_def reachable_trans_def)
-      using A1_0 A1_1 A1_3 H1_0 H1_3 
-      using reachable_states_def by fastforce
-    have H1_5:"\<psi> g r = \<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g r"
-      using A1_0 A1_3
-      by (auto simp add: reachable_action_def)
-    hence H1_6: "\<psi> g r \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h"
-      using A1_2
-      by simp
-    have H1_7: "\<delta>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h (\<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g r) (\<phi> g a) = \<delta> (\<psi> g r) (\<phi> g a)"
-      using A1_0 A1_1 A1_2 A1_3
-      by (auto simp del: simp add:reachable_trans_def reachable_action_def )
-    show "\<delta>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h (\<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g r) (\<phi> g a) = \<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g (\<delta>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h r a)"
-      using H1_2 H1_4 H1_7
-      by auto
-  qed
-  have H_9: "\<And>a w s. \<lbrakk>(\<And>s. s \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<Longrightarrow> (\<delta>\<^sup>\<star>) s w = (\<delta>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h\<^sup>\<star>) s w);
-       a \<in> A \<and> (\<forall>x\<in>set w. x \<in> A); s \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h\<rbrakk> \<Longrightarrow> (\<delta>\<^sup>\<star>) (\<delta> s a) w = (\<delta>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h\<^sup>\<star>) (\<delta>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h s a) w"
-  proof-
-    fix a w s
-    assume 
-      A1_IH: "(\<And>s. s \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<Longrightarrow> (\<delta>\<^sup>\<star>) s w = (\<delta>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h\<^sup>\<star>) s w)" and
-      A1_0: "a \<in> A \<and> (\<forall>x\<in>set w. x \<in> A)" and
-      A1_1: "s \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h" 
-    have H1_0: "\<delta>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h s a = \<delta> s a"
-      using A1_1
-      apply (clarsimp simp add: reachable_trans_def)
-      apply (rule meta_mp[of "det_aut A S i F \<delta>"])
-      using det_aut.trans_func_ext[where labels = A and states = S and
-          init_state = i and fin_states = F and trans_func = \<delta>]
-       apply (simp add: extensional_def)
-      by (auto simp add: A1_0)
-    show "(\<delta>\<^sup>\<star>) (\<delta> s a) w = (\<delta>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h\<^sup>\<star>) (\<delta>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h s a) w"
-      apply (simp add: H1_0)
-      apply (rule A1_IH[where s1 = "\<delta> s a"])
-      using A1_0 A1_1
-      apply (simp add: reachable_states_def)
-      by (metis Cons_in_lists_iff append_Nil2 append_in_lists_conv is_aut.give_input_closed
-          is_aut.init_state_is_a_state is_aut.trans_to_charact)
-  qed
-  show ?thesis
-    apply (clarsimp simp del: GMN_simps simp add: reach_det_G_aut_rec_lang_def
-        det_G_aut_rec_lang_def det_G_aut_def det_aut_def)
-    apply (intro conjI)
-    subgoal
-      apply (simp add: reachable_states_def) 
-      by (meson give_input.simps(1) lists.Nil)
-           apply (simp add: H_0)
-    using labels_a_G_set.alt_grp_act_axioms
-          apply (auto)[1]
-         apply (rule H_6)
-    subgoal
-      apply (clarsimp simp add: eq_var_subset_def eq_var_subset_axioms_def)
-      apply (rule conjI)
-      using H_6
-       apply (auto)[1]
-      apply (simp del: add: reachable_states_def)[1] 
-      apply (clarify; rule subset_antisym; simp add: Set.subset_eq; clarify)
-       apply (rule conjI)
-      subgoal for g _ w
-        apply (clarsimp simp add: reachable_action_def reachable_states_def) 
-        using accepting_is_eq_var.is_equivar
-        by blast
-      subgoal for g _ w
-        apply (clarsimp simp add: reachable_action_def reachable_states_def)
-        apply (rule conjI; clarify)
-         apply (auto)[2]
-        by (simp add: H_5)
-      apply (clarsimp simp add: reachable_states_def Int_def reachable_action_def ) 
-      apply (clarsimp simp add: image_def)
-      by (simp add: H_7)
-    subgoal
-      apply (clarsimp simp add: eq_var_subset_def)
-      apply (rule conjI)
-      using H_6
-       apply (auto)[1]
-      apply (clarsimp simp add: eq_var_subset_axioms_def) 
-      apply (simp add: \<open>i \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h\<close>)
-      apply (simp add: reachable_action_def)
-      using \<open>i \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h\<close> init_is_eq_var.is_equivar
-      by fastforce
-    subgoal
-      apply (clarsimp simp add: eq_var_func_def eq_var_func_axioms_def) 
-      apply (intro conjI)
-      using H_6 alt_grp_act.axioms
-        labels_a_G_set.group_action_axioms prod_group_act labels_a_G_set.alt_grp_act_axioms
-         apply blast
-      using H_6
-        apply (auto)[1]
-       apply (rule funcsetI; clarsimp)
-      subgoal for s a
-        apply (clarsimp simp add: reachable_states_def reachable_trans_def)
-        by (metis Cons_in_lists_iff append_Nil2 append_in_lists_conv in_listsI
-            is_aut.give_input_closed is_aut.init_state_is_a_state is_aut.trans_to_charact)
-      apply (intro allI; clarify; rule conjI; intro impI)
-       apply (simp add: H_8)
-      using G_set_equiv H_6 eq_var_subset.is_equivar
-        labels_a_G_set.element_image
-      by fastforce
-     apply (rule meta_mp[of "\<And>w s. w \<in> A\<^sup>\<star> \<Longrightarrow> s \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<Longrightarrow> (\<delta>\<^sup>\<star>) s w = (\<delta>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h\<^sup>\<star>) s w"])
-    subgoal
-      using det_G_aut_rec_lang_axioms
-      apply (clarsimp simp add: det_aut_rec_lang_axioms_def det_aut_rec_lang_def
-          det_G_aut_rec_lang_def det_aut_def)
-      apply (intro conjI)
-      using \<open>i \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h\<close> 
-        apply blast 
-      using H_0
-       apply blast
-      by (metis (mono_tags, lifting) \<open>i \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h\<close> mem_Collect_eq reachable_states_def)
-    subgoal for w s
-      apply (induction w arbitrary: s)
-       apply (clarsimp)
-      apply (simp add: in_lists_conv_set)
-      by (simp add: H_9)
-    apply (clarsimp simp add: reach_det_G_aut_def det_G_aut_def det_aut_def)
-    apply (intro conjI)
-           apply (simp add: \<open>i \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h\<close>)
-          apply (simp add: H_0)
-         apply (simp add: labels_a_G_set.group_action_axioms)
-    using \<open>alt_grp_act G S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h\<close>
-        apply (auto)[1]
-       apply (simp add: \<open>eq_var_subset G S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h (F \<inter> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h)\<close>)
-      apply (simp add: \<open>eq_var_subset G S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h {i}\<close>)
-    using \<open>eq_var_func G (S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<times> A) (\<lambda>g\<in>carrier G. \<lambda>(s, a)\<in>S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<times> A. (\<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g s, \<phi> g a))
-  S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h (\<lambda>(x, y)\<in>S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<times> A. \<delta>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h x y)\<close>
-     apply blast
-    apply (simp add: reach_det_aut_axioms_def reach_det_aut_def reachable_states_def)
-    apply (rule meta_mp[of "\<And>s input. s \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<Longrightarrow> input \<in> A\<^sup>\<star> \<Longrightarrow>
-   (\<delta>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h\<^sup>\<star>) s input = (\<delta>\<^sup>\<star>) s input"])
-    using \<open>i \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h\<close>
-     apply (metis (no_types, lifting) \<open>(\<And>w s. \<lbrakk>w \<in> A\<^sup>\<star>; s \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h\<rbrakk> \<Longrightarrow>
-    (\<delta>\<^sup>\<star>) s w = (\<delta>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h\<^sup>\<star>) s w) \<Longrightarrow> det_aut_rec_lang A S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h i (F \<inter> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h) \<delta>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h L\<close> det_aut_rec_lang_def
-        reachable_states_def)
-    by (simp add: \<open>\<And>w s. \<lbrakk>w \<in> A\<^sup>\<star>; s \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h\<rbrakk> \<Longrightarrow> (\<delta>\<^sup>\<star>) s w = (\<delta>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h\<^sup>\<star>) s w\<close>)
-qed
-
-subsection \<open>
-Syntactic Automaton
-\<close>
-context language begin
-
-definition
-  rel_MN :: "('alpha list \<times> 'alpha list) set" ("\<equiv>\<^sub>M\<^sub>N")
-  where "rel_MN = {(w, w') \<in> (A\<^sup>\<star>)\<times>(A\<^sup>\<star>). (\<forall>v \<in> A\<^sup>\<star>. (w @ v) \<in> L \<longleftrightarrow> (w' @ v) \<in> L)}"
-
-lemma MN_rel_equival:
-  "equiv (A\<^sup>\<star>) rel_MN"
-  by (auto simp add: rel_MN_def equiv_def refl_on_def sym_def trans_def)
-
-abbreviation
-  MN_equiv
-  where "MN_equiv \<equiv> A\<^sup>\<star> // rel_MN" 
-
-definition
-  alt_natural_map_MN :: "'alpha list \<Rightarrow> 'alpha list set " ("[_]\<^sub>M\<^sub>N")
-  where "[w]\<^sub>M\<^sub>N = rel_MN `` {w}"
-
-definition
-  MN_trans_func :: "('alpha list set) \<Rightarrow> 'alpha \<Rightarrow> 'alpha list set" ("\<delta>\<^sub>M\<^sub>N")
-  where "MN_trans_func W' a' =
-     (\<lambda>(W,a) \<in> MN_equiv \<times> A. rel_MN `` {(SOME w. w \<in> W) @ [a]}) (W', a')"
-
-abbreviation
-  MN_init_state
-  where "MN_init_state \<equiv> [Nil::'alpha list]\<^sub>M\<^sub>N"
-
-abbreviation
-  MN_fin_states
-  where "MN_fin_states \<equiv> {v. \<exists>w\<in>L. v = [w]\<^sub>M\<^sub>N}"
-
-lemmas 
-  alt_natural_map_MN_def [simp, GMN_simps]
-  MN_trans_func_def [simp, GMN_simps]
-end
-
-context G_lang begin
-adhoc_overloading
-  star induced_star_map
-
-lemma MN_quot_act_wd:
-  "w' \<in> [w]\<^sub>M\<^sub>N \<Longrightarrow> \<forall>g \<in> carrier G. (g \<odot> \<^bsub>\<phi>\<^sup>\<star>\<^esub> w') \<in> [g \<odot> \<^bsub>\<phi>\<^sup>\<star>\<^esub> w]\<^sub>M\<^sub>N"
-proof-
-  assume A_0: "w' \<in> [w]\<^sub>M\<^sub>N"
-  have H_0: "\<And>g. \<lbrakk>(w, w') \<in> \<equiv>\<^sub>M\<^sub>N; g \<in> carrier G; group_hom G (BijGroup A) \<phi>;
-  group_hom G (BijGroup (A\<^sup>\<star>)) (\<lambda>g\<in>carrier G. restrict (map (\<phi> g)) (A\<^sup>\<star>)); L \<subseteq> A\<^sup>\<star>;
-  \<forall>g\<in>carrier G. map (\<phi> g) ` (L \<inter> A\<^sup>\<star>) \<union> (\<lambda>x. undefined) ` (L \<inter> {x. x \<notin> A\<^sup>\<star>}) = L;
-    \<forall>x\<in>set w. x \<in> A; w' \<in> A\<^sup>\<star>\<rbrakk> \<Longrightarrow> (map (\<phi> g) w, map (\<phi> g) w') \<in> \<equiv>\<^sub>M\<^sub>N"
-  proof-
-    fix g
-    assume
-      A1_0: "(w, w') \<in> \<equiv>\<^sub>M\<^sub>N" and
-      A1_1: "g \<in> carrier G" and
-      A1_2: "group_hom G (BijGroup A) \<phi>" and
-      A1_3: "group_hom G (BijGroup (lists A)) (\<lambda>g\<in>carrier G. restrict (map (\<phi> g)) (lists A))" and
-      A1_4: "L \<subseteq> lists A" and
-      A1_5: "\<forall>g\<in>carrier G.
-           map (\<phi> g) ` (L \<inter> lists A) \<union> (\<lambda>x. undefined) ` (L \<inter> {x. x \<notin> lists A}) = L" and
-      A1_6: "\<forall>x\<in>set w. x \<in> A" and
-      A1_7: "w' \<in> A\<^sup>\<star>"
-    have H1_0: "\<And>v w w'. \<lbrakk>g \<in> carrier G; group_hom G (BijGroup A) \<phi>;
-    group_hom G (BijGroup (lists A)) (\<lambda>g\<in>carrier G. restrict (map (\<phi> g)) (lists A));
-    L \<subseteq> lists A; \<forall>g\<in>carrier G.
-    {y. \<exists>x\<in>L \<inter> lists A. y = map (\<phi> g) x} \<union> {y. y = undefined \<and> (\<exists>x. x \<in> L \<and> x \<notin> lists A)} = L;
-    \<forall>x\<in>set w. x \<in> A; \<forall>v\<in>lists A. (w @ v \<in> L) = (w' @ v \<in> L); \<forall>x\<in>set w'. x \<in> A; \<forall>x\<in>set v. x \<in> A;
-    map (\<phi> g) w @ v \<in> L\<rbrakk> \<Longrightarrow> map (\<phi> g) w' @ v \<in> L"
-    proof -
-      fix v w w'
-      assume
-        A2_0: "g \<in> carrier G" and
-        A2_1: "L \<subseteq> A\<^sup>\<star>" and
-        A2_2: "group_hom G (BijGroup A) \<phi>" and
-        A2_3: "group_hom G (BijGroup (A\<^sup>\<star>)) (\<lambda>g\<in>carrier G. restrict (map (\<phi> g)) (A\<^sup>\<star>))" and
-        A2_4: "\<forall>g\<in>carrier G. {y. \<exists>x\<in>L \<inter> A\<^sup>\<star>. y = map (\<phi> g) x} \<union>
-        {y. y = undefined \<and> (\<exists>x. x \<in> L \<and> x \<notin> A\<^sup>\<star>)} = L" and
-        A2_5: "\<forall>x\<in>set w. x \<in> A" and
-        A2_6: "\<forall>x\<in>set w'. x \<in> A" and
-        A2_7: "\<forall>v\<in>A\<^sup>\<star>. (w @ v \<in> L) = (w' @ v \<in> L)" and
-        A2_8: "\<forall>x\<in>set v. x \<in> A" and
-        A2_9: "map (\<phi> g) w @ v \<in> L" 
-      have H2_0: "\<forall>g\<in>carrier G. {y. \<exists>x\<in>L \<inter> A\<^sup>\<star>. y = map (\<phi> g) x} = L"
-        using A2_1 A2_4 subset_eq
-        by (smt (verit, ccfv_SIG) Collect_mono_iff sup.orderE)
-      hence H2_1: "\<forall>g\<in>carrier G. {y. \<exists>x\<in>L. y = map (\<phi> g) x} = L"
-        using A2_1 inf.absorb_iff1
-        by (smt (verit, ccfv_SIG) Collect_cong)
-      hence H2_2: "\<forall>g\<in>carrier G.\<forall>x\<in>L. map (\<phi> g) x \<in> L"
-        by auto
-      from A2_2 have H2_3: "\<forall>h\<in>carrier G. \<forall>a\<in>A. (\<phi> h) a \<in> A"
-        by (auto simp add: group_hom_def BijGroup_def group_hom_axioms_def hom_def Bij_def
-            bij_betw_def)
-      from A2_8 have H2_4: "v \<in> lists A"
-        by (simp add: in_listsI)
-      hence H2_5: "\<forall>h\<in>carrier G. map (\<phi> h) v \<in> lists A"
-        using H2_3
-        by fastforce
-      hence H2_6: "\<forall>h\<in>carrier G. (w @ (map (\<phi> h) v) \<in> L) = (w' @ (map (\<phi> h) v) \<in> L)"
-        using A2_7
-        by force
-      hence H2_7: "(w @ (map (\<phi> (inv\<^bsub>G\<^esub> g)) v) \<in> L) = (w' @ (map (\<phi> (inv\<^bsub>G\<^esub> g)) v) \<in> L)"
-        using A2_0
-        by (meson A2_7 A2_1 append_in_lists_conv in_mono)
-      have "(map (\<phi> g) w) \<in> (A\<^sup>\<star>)"
-        using A2_0 A2_2 A2_5 H2_3
-        by (auto simp add: group_hom_def group_hom_axioms_def hom_def BijGroup_def Bij_def
-            bij_betw_def)
-      hence H2_8: "\<forall>w\<in>A\<^sup>\<star>. \<forall>g\<in>carrier G. map (\<phi> (inv\<^bsub>G\<^esub> g)) ((map (\<phi> g) w) @ v) =
-      w @ (map (\<phi> (inv\<^bsub>G\<^esub> g)) v)"
-        using act_maps_n_distrib triv_act_map A2_0 A2_2 A2_3 H2_4
-        apply (clarsimp)
-        by (smt (verit, del_insts) comp_apply group_action.intro group_action.orbit_sym_aux map_idI)
-      have H2_9: "map (\<phi> (inv\<^bsub>G\<^esub> g)) ((map (\<phi> g) w) @ v) \<in> L"
-        using A2_9 H2_1 H2_2 A2_1
-        apply clarsimp
-        by (metis A2_0 A2_2 group.inv_closed group_hom.axioms(1) list.map_comp map_append)
-      hence H2_10: "w @ (map (\<phi> (inv\<^bsub>G\<^esub> g)) v) \<in> L"
-        using H2_8 A2_0
-        by (auto simp add: A2_5 in_listsI)
-      hence H2_11: "w' @ (map (\<phi> (inv\<^bsub>G\<^esub> g)) v) \<in> L"
-        using H2_7
-        by simp
-      hence H2_12: "map (\<phi> (inv\<^bsub>G\<^esub> g)) ((map (\<phi> g) w') @ v) \<in> L"
-        using A2_0 H2_8 A2_1 subsetD
-        by (metis append_in_lists_conv)
-      have H2_13: "\<forall>g\<in>carrier G. restrict (map (\<phi> g)) (A\<^sup>\<star>) \<in> Bij (A\<^sup>\<star>)"
-        using alt_grp_act.lists_a_Gset[where G = "G" and X = "A" and \<phi> = "\<phi>"] A1_3 
-        by (auto simp add: group_action_def
-            group_hom_def group_hom_axioms_def Pi_def hom_def BijGroup_def)
-      have H2_14: "\<forall>g\<in>carrier G. restrict (map (\<phi> g)) L ` L = L"
-        using H2_2
-        apply (clarsimp simp add: Set.image_def)
-        using H2_1
-        by blast
-      have H2_15: "map (\<phi> g) w' \<in> lists A"
-        using A2_0 A2_1 H2_13 H2_2 
-        by (metis H2_11 append_in_lists_conv image_eqI lists_image subset_eq surj_prop)
-      have H2_16: "inv\<^bsub>G\<^esub> g \<in> carrier G" 
-        by (metis A2_0 A2_2 group.inv_closed group_hom.axioms(1))
-      thus "map (\<phi> g) w' @ v \<in> L" 
-        using A2_0 A2_1 A2_2 H2_4 H2_12 H2_13 H2_14 H2_15 H2_16 group.inv_closed group_hom.axioms(1)
-          alt_grp_act.lists_a_Gset[where G = "G" and X = "A" and \<phi> = "\<phi>"] 
-          pre_image_lemma[where S = "L" and T = "A\<^sup>\<star>" and f = "map (\<phi> (inv\<^bsub>G\<^esub> g))" and
-            x = "((map (\<phi> g) w') @ v)"]
-        apply (clarsimp simp add: group_action_def) 
-        by (smt (verit, best) A2_1 FuncSet.restrict_restrict H2_14 H2_15 H2_16 H2_4
-            append_in_lists_conv inf.absorb_iff2 map_append map_map pre_image_lemma restrict_apply'
-            restrict_apply') 
-    qed
-    show "(map (\<phi> g) w, map (\<phi> g) w') \<in> \<equiv>\<^sub>M\<^sub>N"
-      apply (clarsimp simp add: rel_MN_def Set.image_def)
-      apply (intro conjI)
-      using A1_1 A1_6 group_action.surj_prop group_action_axioms
-        apply fastforce
-      using A1_1 A1_7 image_iff surj_prop
-       apply fastforce
-      apply (clarify; rule iffI)
-      subgoal for v
-        apply (rule H1_0[where v1 ="v" and w1 = "w" and w'1 = "w'"])
-        using A1_0 A1_1 A1_2 A1_3 A1_4 A1_5 A1_6 A1_7
-        by (auto simp add: rel_MN_def Set.image_def)
-      apply (rule H1_0[where w1 = "w'" and w'1 = "w"])
-      using A1_0 A1_1 A1_2 A1_3 A1_4 A1_5 A1_6 A1_7
-      by (auto simp add: rel_MN_def Set.image_def)
-  qed
-  show ?thesis
-    using G_lang_axioms A_0
-    apply (clarsimp simp add: G_lang_def G_lang_axioms_def eq_var_subset_def
-        eq_var_subset_axioms_def alt_grp_act_def group_action_def)
-    apply (intro conjI; clarify)
-     apply (rule conjI; rule impI)
-      apply (simp add: H_0)
-    by (auto simp add: rel_MN_def)
-qed
-
-text \<open>
-The following lemma corresponds to lemma 3.4 from \cite{bojanczyk2014automata}:
-\<close>
-
-lemma MN_rel_eq_var:
-  "eq_var_rel G (A\<^sup>\<star>) (\<phi>\<^sup>\<star>) \<equiv>\<^sub>M\<^sub>N"
-  apply (clarsimp simp add: eq_var_rel_def alt_grp_act_def eq_var_rel_axioms_def)
-  apply (intro conjI)
-    apply (metis L_is_equivar alt_grp_act.axioms eq_var_subset.axioms(1) induced_star_map_def)
-  using L_is_equivar
-   apply (simp add: rel_MN_def eq_var_subset_def eq_var_subset_axioms_def)
-   apply fastforce
-  apply (clarify)
-  apply (intro conjI; rule impI; rule conjI; rule impI)
-     apply (simp add: in_lists_conv_set)
-     apply (clarsimp simp add: rel_MN_def)
-     apply (intro conjI)
-       apply (clarsimp simp add: rel_MN_def)
-  subgoal for w v g w'
-    using L_is_equivar
-    apply (clarsimp simp add: restrict_def eq_var_subset_def eq_var_subset_axioms_def)
-    by (meson element_image)
-      apply(metis image_mono in_listsI in_mono list.set_map lists_mono subset_code(1) surj_prop)
-     apply (clarify; rule iffI)
-  subgoal for w v g u
-    using G_lang_axioms MN_quot_act_wd[where w = "w" and w'= "v"]
-    by (auto simp add: rel_MN_def G_lang_def G_lang_axioms_def
-        eq_var_subset_def eq_var_subset_axioms_def Set.subset_eq element_image) 
-  subgoal for w v g u
-    using G_lang_axioms MN_quot_act_wd[where w = "w" and w'= "v"]
-    by (auto simp add: rel_MN_def G_lang_def G_lang_axioms_def
-        eq_var_subset_def eq_var_subset_axioms_def Set.subset_eq element_image) 
-  using G_lang_axioms MN_quot_act_wd
-  by (auto simp add: rel_MN_def G_lang_def G_lang_axioms_def
-      eq_var_subset_def eq_var_subset_axioms_def Set.subset_eq element_image)
-
-lemma quot_act_wd_alt_notation:
-  "w \<in> A\<^sup>\<star> \<Longrightarrow> g \<in> carrier G \<Longrightarrow> g \<odot>\<^bsub>[\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>\<^esub> ([w]\<^sub>M\<^sub>N) = ([g \<odot>\<^bsub>\<phi>\<^sup>\<star>\<^esub> w]\<^sub>M\<^sub>N)"
-  using eq_var_rel.quot_act_wd[where G = G and \<phi> = "\<phi>\<^sup>\<star>" and X = "A\<^sup>\<star>" and R = "\<equiv>\<^sub>M\<^sub>N" and x = w
-      and g = g]
-  by (simp del: GMN_simps add: alt_natural_map_MN_def MN_rel_eq_var MN_rel_equival)
-
-lemma MN_trans_func_characterization:
-  "v \<in> (A\<^sup>\<star>) \<Longrightarrow> a \<in> A \<Longrightarrow> \<delta>\<^sub>M\<^sub>N [v]\<^sub>M\<^sub>N a = [v @ [a]]\<^sub>M\<^sub>N"
-proof-
-  assume
-    A_0: "v \<in> (A\<^sup>\<star>)" and
-    A_1: "a \<in> A"
-  have H_0: "\<And>u. u \<in> [v]\<^sub>M\<^sub>N \<Longrightarrow> (u @ [a]) \<in> [v @ [a]]\<^sub>M\<^sub>N"
-    by (auto simp add: rel_MN_def A_1 A_0)
-  hence H_1: "(SOME w. (v, w) \<in> \<equiv>\<^sub>M\<^sub>N) \<in> [v]\<^sub>M\<^sub>N \<Longrightarrow> ((SOME w. (v, w) \<in> \<equiv>\<^sub>M\<^sub>N) @ [a]) \<in> [v @ [a]]\<^sub>M\<^sub>N"
-    by auto
-  from A_0 have "(v, v) \<in> \<equiv>\<^sub>M\<^sub>N \<and> v \<in> [v]\<^sub>M\<^sub>N"
-    by (auto simp add: rel_MN_def)
-  hence H_2: "(SOME w. (v, w) \<in> \<equiv>\<^sub>M\<^sub>N) \<in> [v]\<^sub>M\<^sub>N"
-    apply (clarsimp simp add: rel_MN_def)
-    apply (rule conjI)
-     apply (smt (verit, ccfv_SIG) A_0 in_listsD verit_sko_ex_indirect)
-    by (smt (verit, del_insts) A_0 in_listsI tfl_some)
-  hence H_3: " ((SOME w. (v, w) \<in> \<equiv>\<^sub>M\<^sub>N) @ [a]) \<in> [v @ [a]]\<^sub>M\<^sub>N"
-    using H_1
-    by simp
-  thus "\<delta>\<^sub>M\<^sub>N [v]\<^sub>M\<^sub>N a = [v @ [a]]\<^sub>M\<^sub>N"
-    using A_0 A_1 MN_rel_equival
-    apply (clarsimp simp add: equiv_def)
-    apply (rule conjI; rule impI)
-     apply (metis MN_rel_equival equiv_class_eq)
-    by (simp add: A_0 quotientI)
-qed
-
-lemma MN_trans_eq_var_func :
-  "eq_var_func G
- (MN_equiv \<times> A) (\<lambda>g\<in>carrier G. \<lambda>(W, a) \<in> (MN_equiv \<times> A). (([(\<phi>\<^sup>\<star>)]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>) g W, \<phi> g a))
-  MN_equiv ([(\<phi>\<^sup>\<star>)]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>)
-  (\<lambda>(w, a) \<in> MN_equiv \<times> A. \<delta>\<^sub>M\<^sub>N w a)"
-proof-
-  have H_0: "alt_grp_act G MN_equiv ([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>)"
-    using MN_rel_eq_var MN_rel_equival eq_var_rel.quot_act_is_grp_act 
-      alt_group_act_is_grp_act restrict_apply
-    by fastforce
-  have H_1: "\<And>a b g.
-       a \<in> MN_equiv \<Longrightarrow>
-       b \<in> A \<Longrightarrow>
-       (([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>) g a \<in> MN_equiv \<and> \<phi> g b \<in> A \<longrightarrow>
-       g \<in> carrier G \<longrightarrow> \<delta>\<^sub>M\<^sub>N (([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>) g a) (\<phi> g b) = ([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>) g (\<delta>\<^sub>M\<^sub>N a b)) \<and>
-       ((([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>) g a \<in> MN_equiv \<longrightarrow> \<phi> g b \<notin> A) \<longrightarrow>
-        g \<in> carrier G \<longrightarrow> undefined = ([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>) g (\<delta>\<^sub>M\<^sub>N a b))"
-  proof -
-    fix C a g
-    assume
-      A1_0: "C \<in> MN_equiv" and
-      A1_1: "a \<in> A"
-    have H1_0: "g \<in> carrier G \<Longrightarrow> \<phi> g a \<in> A" 
-      by (meson A1_1 element_image)
-    from A1_0 obtain c where H1_c: "[c]\<^sub>M\<^sub>N = C \<and> c \<in> A\<^sup>\<star>"
-      by (auto simp add: quotient_def) 
-    have H1_1: "g \<in> carrier G \<Longrightarrow> \<delta>\<^sub>M\<^sub>N (([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>) g C) (\<phi> g a) = ([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>) g (\<delta>\<^sub>M\<^sub>N [c]\<^sub>M\<^sub>N a)"
-    proof-
-      assume 
-        A2_0: "g \<in> carrier G" 
-      have H2_0: "\<phi> g a \<in> A"
-        using H1_0 A2_0
-        by simp
-      have H2_1: "(\<phi>\<^sup>\<star>) g \<in> Bij (A\<^sup>\<star>)" using G_lang_axioms lists_a_Gset A2_0
-        apply (clarsimp simp add: G_lang_def G_lang_axioms_def group_action_def
-            group_hom_def hom_def group_hom_axioms_def BijGroup_def image_def)  
-        by (meson Pi_iff restrict_Pi_cancel)
-      hence H2_2: "(\<phi>\<^sup>\<star>) g c \<in> (A\<^sup>\<star>)"
-        using H1_c
-        apply (clarsimp simp add: Bij_def bij_betw_def inj_on_def Image_def image_def)
-        apply (rule conjI; rule impI; clarify)
-        using surj_prop
-         apply fastforce
-        using A2_0
-        by blast
-      from H1_c have H2_1: "([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>) g (\<equiv>\<^sub>M\<^sub>N `` {c}) = ([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>) g C"
-        by auto
-      also have H2_2: "([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>) g C = [(\<phi>\<^sup>\<star>) g c]\<^sub>M\<^sub>N"
-        using eq_var_rel.quot_act_wd[where R = "\<equiv>\<^sub>M\<^sub>N" and G = G and X = "A\<^sup>\<star>" and \<phi> = "\<phi>\<^sup>\<star>" and g = g
-            and x = c]
-        by (clarsimp simp del: GMN_simps simp add: alt_natural_map_MN_def make_op_def MN_rel_eq_var
-            MN_rel_equival H1_c A2_0 H2_1) 
-      hence H2_3: "\<delta>\<^sub>M\<^sub>N (([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>) g C) (\<phi> g a) = \<delta>\<^sub>M\<^sub>N ([(\<phi>\<^sup>\<star>) g c]\<^sub>M\<^sub>N) (\<phi> g a)" 
-        using H2_2 
-        by simp
-      also have H2_4: "... = [((\<phi>\<^sup>\<star>) g c) @ [(\<phi> g a)]]\<^sub>M\<^sub>N"
-        using MN_trans_func_characterization[where v = "(\<phi>\<^sup>\<star>) g c" and a = "\<phi> g a"] H1_c  A2_0
-          G_set_equiv H2_0 eq_var_subset.is_equivar insert_iff lists_a_Gset
-        by blast
-      also have H2_5: "... = [(\<phi>\<^sup>\<star>) g (c @ [a])]\<^sub>M\<^sub>N"
-        using A2_0 H1_c A1_1
-        by auto
-      also have H2_6: "... = ([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>) g [(c @ [a])]\<^sub>M\<^sub>N"
-        apply (rule meta_mp[of "c @ [a] \<in> A\<^sup>\<star>"])
-        using eq_var_rel.quot_act_wd[where R = "\<equiv>\<^sub>M\<^sub>N" and G = G and X = "A\<^sup>\<star>" and \<phi> = "\<phi>\<^sup>\<star>" and g = g
-            and x = "c @ [a]"]
-         apply (clarsimp simp del: GMN_simps simp add: make_op_def MN_rel_eq_var MN_rel_equival H1_c
-            A2_0 H2_1) 
-        using H1_c A1_1
-        by auto 
-      also have H2_7: "... = ([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>) g (\<delta>\<^sub>M\<^sub>N [c]\<^sub>M\<^sub>N a)"
-        using MN_trans_func_characterization[where v = "c" and a = "a"] H1_c A1_1
-        by metis
-      finally show "\<delta>\<^sub>M\<^sub>N (([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>) g C) (\<phi> g a) = ([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>) g (\<delta>\<^sub>M\<^sub>N [c]\<^sub>M\<^sub>N a)"
-        using H2_1 
-        by metis
-    qed
-    show "(([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>) g C \<in> MN_equiv \<and> \<phi> g a \<in> A \<longrightarrow>
-    g \<in> carrier G \<longrightarrow>
-    \<delta>\<^sub>M\<^sub>N (([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>) g C) (\<phi> g a) =
-    ([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>) g (\<delta>\<^sub>M\<^sub>N C a)) \<and>
-    ((([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>) g C \<in> MN_equiv \<longrightarrow> \<phi> g a \<notin> A) \<longrightarrow>
-    g \<in> carrier G \<longrightarrow> undefined = ([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>) g (\<delta>\<^sub>M\<^sub>N C a))"
-      apply (rule conjI; clarify)
-      using H1_1 H1_c
-       apply blast
-      by (metis A1_0 H1_0 H_0 alt_group_act_is_grp_act
-          group_action.element_image)
-  qed
-  show ?thesis
-    apply (subst eq_var_func_def)
-    apply (subst eq_var_func_axioms_def)
-    apply (rule conjI)
-    subgoal 
-      apply (rule prod_group_act[where G = G and A = "MN_equiv" and \<phi> = "[(\<phi>\<^sup>\<star>)]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>"
-            and B = A     and \<psi> = \<phi>])
-       apply (rule H_0)
-      using G_lang_axioms
-      by (auto simp add: G_lang_def G_lang_axioms_def)
-    apply (rule conjI)
-    subgoal
-      using MN_rel_eq_var MN_rel_equival eq_var_rel.quot_act_is_grp_act
-      using alt_group_act_is_grp_act restrict_apply
-      by fastforce
-    apply (rule conjI)
-    subgoal
-      apply (subst extensional_funcset_def)
-      apply (subst restrict_def)
-      apply (subst Pi_def)
-      apply (subst extensional_def)
-      apply (clarsimp)
-      by (metis MN_rel_equival append_in_lists_conv equiv_Eps_preserves lists.Cons lists.Nil
-          quotientI)
-    apply (subst restrict_def)
-    apply (clarsimp simp del: GMN_simps simp add: make_op_def)
-    by (simp add: H_1 del: GMN_simps)
-qed
-
-lemma MN_quot_act_on_empty_str:
-  "\<And>g. \<lbrakk>g \<in> carrier G; ([], x) \<in> \<equiv>\<^sub>M\<^sub>N\<rbrakk> \<Longrightarrow> x \<in> map (\<phi> g) ` \<equiv>\<^sub>M\<^sub>N `` {[]}"
-proof-
-  fix g
-  assume
-    A_0: "g \<in> carrier G" and
-    A_1: "([], x) \<in> \<equiv>\<^sub>M\<^sub>N"
-  from A_1 have H_0: "x \<in> (A\<^sup>\<star>)"
-    by (auto simp add: rel_MN_def)
-  from A_0 H_0 have H_1: "x = (\<phi>\<^sup>\<star>) g ((\<phi>\<^sup>\<star>) (inv \<^bsub>G\<^esub> g) x)"
-    by (smt (verit) alt_grp_act_def group_action.bij_prop1 group_action.orbit_sym_aux lists_a_Gset) 
-  have H_2: "inv \<^bsub>G \<^esub> g \<in> carrier G"
-    using A_0 MN_rel_eq_var
-    by (auto simp add: eq_var_rel_def eq_var_rel_axioms_def group_action_def group_hom_def)
-  have H_3: "([], (\<phi>\<^sup>\<star>) (inv \<^bsub>G\<^esub> g) x) \<in> \<equiv>\<^sub>M\<^sub>N"
-    using A_0 A_1 H_0 MN_rel_eq_var
-    apply (clarsimp simp add: eq_var_rel_def eq_var_rel_axioms_def) 
-    apply (rule conjI; clarify)
-     apply (smt (verit, best) H_0 list.simps(8) lists.Nil)
-    using H_2
-    by simp
-  hence H_4: "\<exists>y\<in>\<equiv>\<^sub>M\<^sub>N `` {[]}. x = map (\<phi> g) y"
-    using A_0 H_0 H_1 H_2
-    apply clarsimp
-    by (metis H_0 Image_singleton_iff insert_iff insert_image lists_image surj_prop)
-  thus "x \<in> map (\<phi> g) ` \<equiv>\<^sub>M\<^sub>N `` {[]}"
-    by (auto simp add: image_def)
-qed
-
-lemma MN_init_state_equivar:
-  "eq_var_subset G (A\<^sup>\<star>) (\<phi>\<^sup>\<star>) MN_init_state"
-  apply (clarsimp simp add: eq_var_subset_def eq_var_subset_axioms_def)
-  apply (intro conjI)
-  using lists_a_Gset
-    apply (auto)[1]
-   apply (clarsimp)
-  subgoal for w a
-    by (auto simp add: rel_MN_def)
-  apply (clarify; rule subset_antisym; simp add: Set.subset_eq; clarify)
-   apply (clarsimp simp add: image_def Image_def Int_def) 
-   apply (erule disjE)
-  subgoal for g w
-    using MN_rel_eq_var
-    apply (clarsimp simp add: eq_var_rel_def eq_var_rel_axioms_def)
-    by (metis (full_types, opaque_lifting) in_listsI list.simps(8) lists.Nil)
-   apply (erule conjE)
-   apply (auto simp add: \<open>\<And>a w. \<lbrakk>([], w) \<in> \<equiv>\<^sub>M\<^sub>N; a \<in> set w\<rbrakk> \<Longrightarrow> a \<in> A\<close>)[1]  
-  apply (rule meta_mp[of "\<equiv>\<^sub>M\<^sub>N `` {[]} \<inter> lists A = \<equiv>\<^sub>M\<^sub>N `` {[]}"])
-  using MN_quot_act_on_empty_str
-   apply (clarsimp) 
-  apply (rule subset_antisym; simp add: Set.subset_eq)
-  by (simp add: \<open>\<And>w a. \<lbrakk>([], w) \<in> \<equiv>\<^sub>M\<^sub>N; a \<in> set w\<rbrakk> \<Longrightarrow> a \<in> A\<close> in_listsI)
-
-lemma MN_init_state_equivar_v2:
-  "eq_var_subset G (MN_equiv) ([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star> \<^esub>) {MN_init_state}"
-proof-
-  have H_0: "\<forall>g\<in>carrier G. (\<phi>\<^sup>\<star>) g ` MN_init_state = MN_init_state \<Longrightarrow>
-             \<forall>g\<in>carrier G. ([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>) g MN_init_state = MN_init_state"
-  proof (clarify)
-    fix g 
-    assume
-      A_0: "g \<in> carrier G" 
-    have H_0: "\<And>x. [x]\<^sub>M\<^sub>N = \<equiv>\<^sub>M\<^sub>N `` {x}"
-      by simp
-    have H_1: "([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>) g [[]]\<^sub>M\<^sub>N = [(\<phi>\<^sup>\<star>) g []]\<^sub>M\<^sub>N"
-      using eq_var_rel.quot_act_wd[where R = "\<equiv>\<^sub>M\<^sub>N" and G = G and X = "A\<^sup>\<star>" and \<phi> = "\<phi>\<^sup>\<star>" and g = g
-          and x = "[]"] MN_rel_eq_var MN_rel_equival
-      by (clarsimp simp del: GMN_simps simp add: H_0 make_op_def A_0) 
-    from A_0 H_1 show "([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>) g [[]]\<^sub>M\<^sub>N = [[]]\<^sub>M\<^sub>N"
-      by auto
-  qed
-  show ?thesis
-    using MN_init_state_equivar
-    apply (clarsimp simp add: eq_var_subset_def simp del: GMN_simps)
-    apply (rule conjI)
-    subgoal
-      by (metis MN_rel_eq_var MN_rel_equival eq_var_rel.quot_act_is_grp_act)
-    apply (clarsimp del: subset_antisym simp del: GMN_simps simp add: eq_var_subset_axioms_def)
-    apply (rule conjI)
-     apply (auto simp add: quotient_def)[1]
-    by (simp add: H_0 del: GMN_simps)
-qed
-
-lemma MN_final_state_equiv:
-  "eq_var_subset G (MN_equiv) ([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star> \<^esub>) MN_fin_states"
-proof-
-  have H_0: "\<And>g x w. g \<in> carrier G \<Longrightarrow> w \<in> L \<Longrightarrow> \<exists>wa\<in>L. ([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>) g [w]\<^sub>M\<^sub>N = [wa]\<^sub>M\<^sub>N"
-  proof-
-    fix g w
-    assume
-      A1_0: "g \<in> carrier G" and 
-      A1_1: "w \<in> L" 
-    have H1_0: "\<And>v. v \<in> L \<Longrightarrow> (\<phi>\<^sup>\<star>) g v \<in> L"
-      using A1_0 G_lang_axioms
-      apply (clarsimp simp add: G_lang_def G_lang_axioms_def eq_var_subset_def
-          eq_var_subset_axioms_def)
-      by blast
-    hence H1_1: "(\<phi>\<^sup>\<star>) g w \<in> L"
-      using A1_1
-      by simp
-    from A1_1 have H1_2: "\<And>v. v \<in> [w]\<^sub>M\<^sub>N \<Longrightarrow> v \<in> L"
-      apply (clarsimp simp add: rel_MN_def) 
-      by (metis lists.simps self_append_conv)
-    have H1_3: "([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>) g [w]\<^sub>M\<^sub>N = [(\<phi>\<^sup>\<star>) g w]\<^sub>M\<^sub>N"
-      using eq_var_rel.quot_act_wd[where R = "\<equiv>\<^sub>M\<^sub>N" and G = G and X = "A\<^sup>\<star>" and \<phi> = "\<phi>\<^sup>\<star>" and g = g
-          and x = "w"] MN_rel_eq_var MN_rel_equival G_lang_axioms
-      by (clarsimp simp add: A1_0 A1_1 G_lang_axioms_def G_lang_def eq_var_subset_def
-          eq_var_subset_axioms_def subset_eq) 
-    show "\<exists>wa\<in>L. ([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>) g [w]\<^sub>M\<^sub>N = [wa]\<^sub>M\<^sub>N"
-      using H1_1 H1_3
-      by blast
-  qed
-  have H_1: "\<And>g x w. g \<in> carrier G \<Longrightarrow> w \<in> L \<Longrightarrow> [w]\<^sub>M\<^sub>N \<in> ([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>) g ` MN_fin_states"
-  proof-
-    fix g x w
-    assume
-      A1_0: "g \<in> carrier G" and 
-      A1_1: "w \<in> L" 
-    have H1_0: "\<And>v h. v \<in> L \<Longrightarrow> h \<in> carrier G \<Longrightarrow> (\<phi>\<^sup>\<star>) h v \<in> L"
-      using G_lang_axioms
-      apply (clarsimp simp add: G_lang_def G_lang_axioms_def eq_var_subset_def
-          eq_var_subset_axioms_def)
-      by blast
-    have H1_1: "(\<phi>\<^sup>\<star>) (inv \<^bsub>G\<^esub> g) w \<in> L" 
-      apply (rule meta_mp[of "(inv \<^bsub>G\<^esub> g) \<in> carrier G"]) 
-      using A1_1 H1_0
-       apply blast using A1_0 
-      by (metis group.inv_closed group_hom group_hom.axioms(1))
-    have H1_2: "\<And>v. v \<in> [w]\<^sub>M\<^sub>N \<Longrightarrow> v \<in> L"
-      using A1_1 apply (clarsimp simp add: rel_MN_def) 
-      by (metis lists.simps self_append_conv)
-    have H1_3: "([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>) g [(\<phi>\<^sup>\<star>) (inv \<^bsub>G\<^esub> g) w]\<^sub>M\<^sub>N = [w]\<^sub>M\<^sub>N"
-      apply (rule meta_mp[of "(\<phi>\<^sup>\<star>) g ((\<phi>\<^sup>\<star>) (inv\<^bsub>G\<^esub> g) w) = w"])
-      using eq_var_rel.quot_act_wd[where R = "\<equiv>\<^sub>M\<^sub>N" and G = G and X = "A\<^sup>\<star>" and \<phi> = "\<phi>\<^sup>\<star>" and g = g
-          and x = "(\<phi>\<^sup>\<star>) (inv \<^bsub>G\<^esub> g) w"]
-        MN_rel_eq_var MN_rel_equival G_lang_axioms H1_1
-       apply (clarsimp simp del: GMN_simps simp add: make_op_def A1_0 A1_1 G_lang_axioms_def
-          G_lang_def eq_var_subset_def eq_var_subset_axioms_def subset_eq)
-      using A1_0 A1_1 H1_1 G_lang_axioms
-       apply (clarsimp simp del: GMN_simps simp add: alt_natural_map_MN_def make_op_def
-          G_lang_axioms_def G_lang_def eq_var_subset_def alt_grp_act_def)
-      using group_action.orbit_sym_aux[where G = G and E = "A\<^sup>\<star>" and g = "(inv\<^bsub>G\<^esub> g)" and x = w
-          and \<phi> = "\<phi>\<^sup>\<star>" and y = "((\<phi>\<^sup>\<star>) (inv\<^bsub>G\<^esub> g) w)"]
-      by (smt (verit) A1_0 A1_1 L_is_equivar alt_group_act_is_grp_act
-          eq_var_subset.is_subset group_action.bij_prop1 group_action.orbit_sym_aux insert_subset
-          lists_a_Gset mk_disjoint_insert)
-    show "[w]\<^sub>M\<^sub>N \<in> ([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>) g ` {v. \<exists>w\<in>L. v = [w]\<^sub>M\<^sub>N}"
-      apply (clarsimp simp del: GMN_simps simp add: image_def) 
-      using H1_1 H1_3
-      by blast
-  qed
-  show ?thesis
-    apply (clarsimp simp del: GMN_simps simp add: eq_var_subset_def) 
-    apply (rule conjI)
-    using MN_init_state_equivar_v2 eq_var_subset.axioms(1)
-     apply blast
-    apply (clarsimp simp del: GMN_simps simp add: eq_var_subset_axioms_def)
-    apply (rule conjI; clarsimp simp del: GMN_simps)
-    subgoal for w
-      using G_lang_axioms
-      by (auto simp add: quotient_def G_lang_axioms_def G_lang_def eq_var_subset_def
-          eq_var_subset_axioms_def)
-    apply (rule subset_antisym; simp add: Set.subset_eq del: GMN_simps; clarify)
-     apply (simp add: H_0 del: GMN_simps)
-    by (simp add: H_1 del: GMN_simps)
-qed
-
-interpretation syntac_aut :
-  det_aut "A" "MN_equiv" "MN_init_state" "MN_fin_states" "MN_trans_func"
-proof-
-  have H_0: "\<And>state label. state \<in> MN_equiv \<Longrightarrow> label \<in> A \<Longrightarrow> \<delta>\<^sub>M\<^sub>N state label \<in> MN_equiv"
-  proof -
-    fix state label
-    assume
-      A_0: "state \<in> MN_equiv" and
-      A_1: "label \<in> A" 
-    obtain w where H_w: "state = [w]\<^sub>M\<^sub>N \<and> w \<in> A\<^sup>\<star>"
-      by (metis A_0 alt_natural_map_MN_def quotientE)
-    have H_0: "\<delta>\<^sub>M\<^sub>N [w]\<^sub>M\<^sub>N label = [w @ [label]]\<^sub>M\<^sub>N"
-      using MN_trans_func_characterization[where v = w and a = label] H_w A_1
-      by simp
-    have H_1: "\<And>v. v \<in> A\<^sup>\<star> \<Longrightarrow> [v]\<^sub>M\<^sub>N \<in> MN_equiv"
-      by (simp add: in_listsI quotientI)
-    show "\<delta>\<^sub>M\<^sub>N state label \<in> MN_equiv"
-      using H_w H_0 H_1 
-      by (simp add: A_1)
-  qed
-  show "det_aut A MN_equiv MN_init_state MN_fin_states \<delta>\<^sub>M\<^sub>N"
-    apply (clarsimp simp del: GMN_simps simp add: det_aut_def alt_natural_map_MN_def)
-    apply (intro conjI)
-      apply (auto simp add: quotient_def)[1]
-    using G_lang_axioms
-     apply (auto simp add: quotient_def G_lang_axioms_def G_lang_def
-        eq_var_subset_def eq_var_subset_axioms_def)[1]
-    apply (auto simp add: extensional_def PiE_iff simp del: MN_trans_func_def)[1]
-        apply (simp add: H_0 del: GMN_simps)
-    by auto
-qed
-
-corollary syth_aut_is_det_aut:
-  "det_aut A MN_equiv MN_init_state MN_fin_states \<delta>\<^sub>M\<^sub>N"
-  using local.syntac_aut.det_aut_axioms
-  by simp
-
-lemma give_input_transition_func:
-  "w \<in> (A\<^sup>\<star>) \<Longrightarrow> \<forall>v \<in> (A\<^sup>\<star>). [v @ w]\<^sub>M\<^sub>N = (\<delta>\<^sub>M\<^sub>N\<^sup>\<star>) [v]\<^sub>M\<^sub>N w"
-proof-
-  assume 
-    A_0: "w \<in> A\<^sup>\<star>"
-  have H_0: "\<And> a w v. \<lbrakk>a \<in> A; w \<in> A\<^sup>\<star>; \<forall>v\<in>A\<^sup>\<star>. [v @ w]\<^sub>M\<^sub>N = (\<delta>\<^sub>M\<^sub>N\<^sup>\<star>) [v]\<^sub>M\<^sub>N w; v \<in> A\<^sup>\<star>\<rbrakk> \<Longrightarrow>
-            [v @ a # w]\<^sub>M\<^sub>N = (\<delta>\<^sub>M\<^sub>N\<^sup>\<star>) [v]\<^sub>M\<^sub>N (a # w)"
-  proof-
-    fix a w v
-    assume
-      A1_IH: "\<forall>v\<in> A\<^sup>\<star>. [v @ w]\<^sub>M\<^sub>N = (\<delta>\<^sub>M\<^sub>N\<^sup>\<star>) [v]\<^sub>M\<^sub>N w" and
-      A1_0: "a \<in> A" and
-      A1_1: "v \<in> A\<^sup>\<star>" and
-      A1_2: "w \<in> A\<^sup>\<star>"
-    from A1_IH A1_1 A1_2 have H1_1: "[v @ w]\<^sub>M\<^sub>N = (\<delta>\<^sub>M\<^sub>N\<^sup>\<star>) [v]\<^sub>M\<^sub>N w"
-      by auto
-    have H1_2: "[(v @ [a]) @ w]\<^sub>M\<^sub>N = (\<delta>\<^sub>M\<^sub>N\<^sup>\<star>) [v @ [a]]\<^sub>M\<^sub>N w"
-      apply (rule meta_mp[of "(v @ [a]) \<in> (A\<^sup>\<star>)"])
-      using A1_IH A1_2 H1_1
-       apply blast 
-      using A1_0 A1_1
-      by auto
-    have H1_3: "\<delta>\<^sub>M\<^sub>N [v]\<^sub>M\<^sub>N a = [v @ [a]]\<^sub>M\<^sub>N"
-      using MN_trans_func_characterization[where a = a] A1_0 A1_1
-      by auto
-    hence H1_4: "[v @ a # w]\<^sub>M\<^sub>N = (\<delta>\<^sub>M\<^sub>N\<^sup>\<star>) [v @ [a]]\<^sub>M\<^sub>N w"
-      using H1_2
-      by auto
-    also have H1_5: "... = (\<delta>\<^sub>M\<^sub>N\<^sup>\<star>) (\<delta>\<^sub>M\<^sub>N [v]\<^sub>M\<^sub>N a) w"
-      using H1_4 H1_3 A1_1
-      by auto 
-    thus "[v @ a # w]\<^sub>M\<^sub>N = (\<delta>\<^sub>M\<^sub>N\<^sup>\<star>) [v]\<^sub>M\<^sub>N (a # w)"
-      using calculation
-      by auto
-  qed
-  from A_0 show ?thesis
-    apply (induction w)
-     apply (auto)[1] 
-    by (simp add: H_0 del: GMN_simps)
-qed
-  
-
-lemma MN_unique_init_state:
-  "w \<in> (A\<^sup>\<star>) \<Longrightarrow> [w]\<^sub>M\<^sub>N = (\<delta>\<^sub>M\<^sub>N\<^sup>\<star>) [Nil]\<^sub>M\<^sub>N w"
-  using give_input_transition_func[where w = w] 
-  by (metis append_self_conv2 lists.Nil)
-
-lemma fin_states_rep_by_lang:
-  "w \<in> A\<^sup>\<star> \<Longrightarrow> [w]\<^sub>M\<^sub>N \<in> MN_fin_states \<Longrightarrow> w \<in> L"
-proof-
-  assume
-    A_0: "w \<in> A\<^sup>\<star>" and
-    A_1: "[w]\<^sub>M\<^sub>N \<in> MN_fin_states"
-  from A_1 have H_0: "\<exists>w'\<in>[w]\<^sub>M\<^sub>N. w' \<in> L"
-    apply (clarsimp) 
-    by (metis A_0 MN_rel_equival equiv_class_self proj_def proj_in_iff)
-  from H_0 obtain w' where H_w': "w'\<in>[w]\<^sub>M\<^sub>N \<and> w' \<in> L" 
-    by auto
-  have H_1: "\<And>v. v \<in> A\<^sup>\<star> \<Longrightarrow> w'@v \<in> L \<Longrightarrow> w@v \<in> L"
-    using H_w' A_1 A_0
-    by (auto simp add: rel_MN_def)
-  show "w \<in> L"
-    using H_1 H_w'
-    apply clarify 
-    by (metis append_Nil2 lists.Nil)
-qed
-
-text \<open>
-The following lemma corresponds to lemma 3.6 from \cite{bojanczyk2014automata}:
-\<close>
-
-lemma syntactic_aut_det_G_aut:
-  "det_G_aut A MN_equiv MN_init_state MN_fin_states MN_trans_func G \<phi> ([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>)"
-  apply (clarsimp simp add: det_G_aut_def simp del: GMN_simps)
-  apply (intro conjI)
-  using syth_aut_is_det_aut
-       apply (auto)[1]
-  using alt_grp_act_axioms
-      apply (auto)[1]
-  using MN_init_state_equivar_v2 eq_var_subset.axioms(1)
-     apply blast
-  using MN_final_state_equiv
-    apply presburger
-  using MN_init_state_equivar_v2
-   apply presburger
-  using MN_trans_eq_var_func
-  by linarith
-
-lemma syntactic_aut_det_G_aut_rec_L:
-  "det_G_aut_rec_lang A MN_equiv MN_init_state MN_fin_states MN_trans_func G \<phi>
- ([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>) L"
-  apply (clarsimp simp add: det_G_aut_rec_lang_def det_aut_rec_lang_axioms_def
-      det_aut_rec_lang_def simp del: GMN_simps)
-  apply (intro conjI)
-  using syntactic_aut_det_G_aut syth_aut_is_det_aut
-    apply (auto)[1]
-  using syntactic_aut_det_G_aut syth_aut_is_det_aut
-   apply (auto)[1]
-  apply (rule allI; rule iffI)
-   apply (rule conjI)
-  using L_is_equivar eq_var_subset.is_subset image_iff image_mono insert_image insert_subset
-    apply blast
-  using MN_unique_init_state L_is_equivar eq_var_subset.is_subset
-   apply blast
-  using MN_unique_init_state fin_states_rep_by_lang in_lists_conv_set 
-  by (smt (verit) mem_Collect_eq)
-
-lemma syntact_aut_is_reach_aut_rec_lang:
-  "reach_det_G_aut_rec_lang A MN_equiv MN_init_state MN_fin_states MN_trans_func G \<phi>
- ([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>) L"
-  apply (clarsimp simp del: GMN_simps simp add: reach_det_G_aut_rec_lang_def
-      det_G_aut_rec_lang_def det_aut_rec_lang_axioms_def reach_det_G_aut_def
-      reach_det_aut_def reach_det_aut_axioms_def det_G_aut_def det_aut_rec_lang_def)
-  apply (intro conjI)
-  using syth_aut_is_det_aut
-                 apply blast
-  using alt_grp_act_axioms
-                apply (auto)[1]
-  subgoal
-    using MN_init_state_equivar_v2 eq_var_subset.axioms(1)
-    by blast
-  using MN_final_state_equiv
-              apply presburger
-  using MN_init_state_equivar_v2
-  subgoal
-    by presburger
-  using MN_trans_eq_var_func
-            apply linarith
-  using syth_aut_is_det_aut
-           apply (auto)[1]
-          apply (metis (mono_tags, lifting) G_lang.MN_unique_init_state G_lang_axioms
-      det_G_aut_rec_lang_def det_aut_rec_lang.is_recognised syntactic_aut_det_G_aut_rec_L)
-  using syth_aut_is_det_aut
-         apply (auto)[1]
-  using alt_grp_act_axioms
-        apply (auto)[1]
-  using \<open>alt_grp_act G MN_equiv ([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>)\<close>
-       apply blast
-  using \<open>eq_var_subset G MN_equiv ([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>) MN_fin_states\<close>
-      apply blast
-  using \<open>eq_var_subset G MN_equiv ([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>) {MN_init_state}\<close>
-     apply blast
-  using MN_trans_eq_var_func
-    apply blast
-  using syth_aut_is_det_aut
-   apply auto[1]
-  by (metis MN_unique_init_state alt_natural_map_MN_def quotientE)
-end
-
-subsection \<open>
-Proving the Myhill-Nerode Theorem for $G$-Automata
-\<close>
-context det_G_aut begin
-no_adhoc_overloading
-  star labels_a_G_set.induced_star_map
-end
-
-context reach_det_G_aut_rec_lang begin
-adhoc_overloading
-  star labels_a_G_set.induced_star_map
-
-definition
-  states_to_words :: "'states \<Rightarrow> 'alpha list"
-  where "states_to_words = (\<lambda>s \<in> S. SOME w. w \<in> A\<^sup>\<star> \<and> ((\<delta>\<^sup>\<star>) i w = s))"
-
-definition
-  words_to_syth_states :: "'alpha list \<Rightarrow> 'alpha list set"
-  where "words_to_syth_states w = [w]\<^sub>M\<^sub>N"
-
-definition
-  induced_epi:: "'states \<Rightarrow> 'alpha list set"
-  where "induced_epi = compose S words_to_syth_states states_to_words"
-
-lemma induced_epi_wd1:
-  "s \<in> S \<Longrightarrow> \<exists>w. w \<in> A\<^sup>\<star> \<and> ((\<delta>\<^sup>\<star>) i w = s)"
-  using reach_det_G_aut_rec_lang_axioms is_reachable
-  by auto
-
-lemma induced_epi_wd2:
-  "w \<in> A\<^sup>\<star> \<Longrightarrow> w' \<in> A\<^sup>\<star> \<Longrightarrow> (\<delta>\<^sup>\<star>) i w = (\<delta>\<^sup>\<star>) i w' \<Longrightarrow> [w]\<^sub>M\<^sub>N = [w']\<^sub>M\<^sub>N"
-proof- 
-  assume
-    A_0: "w \<in> A\<^sup>\<star>" and
-    A_1: "w' \<in> A\<^sup>\<star>" and
-    A_2: "(\<delta>\<^sup>\<star>) i w = (\<delta>\<^sup>\<star>) i w'"
-  have H_0: "\<And>v. v \<in> A\<^sup>\<star> \<Longrightarrow> w @ v \<in> L \<longleftrightarrow> w' @ v \<in> L"
-    apply clarify 
-    by (smt (z3) A_0 A_1 A_2 append_in_lists_conv is_aut.eq_pres_under_concat
-        is_aut.init_state_is_a_state is_lang is_recognised subsetD)+
-  show "[w]\<^sub>M\<^sub>N = [w']\<^sub>M\<^sub>N "
-    apply (simp add: rel_MN_def)
-    using H_0 A_0 A_1
-    by auto
-qed
-
-lemma states_to_words_on_final:
-  "states_to_words \<in> (F \<rightarrow> L)"
-proof-
-  have H_0: "\<And>x. x \<in> F \<Longrightarrow> x \<in> S \<Longrightarrow> (SOME w. w \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i w = x) \<in> L"
-  proof-
-    fix s
-    assume
-      A1_0: "s \<in> F"
-    have H1_0: "\<exists>w. w \<in> lists A \<and> (\<delta>\<^sup>\<star>) i w = s"
-      using A1_0 is_reachable 
-      by (metis is_aut.fin_states_are_states subsetD)
-    have H1_1: "\<And>w. w \<in> lists A \<and> (\<delta>\<^sup>\<star>) i w = s \<Longrightarrow> w \<in> L"
-      using A1_0 is_recognised
-      by auto
-    show "(SOME w. w \<in> lists A \<and> (\<delta>\<^sup>\<star>) i w = s) \<in> L "
-      by (metis (mono_tags, lifting) H1_0 H1_1 someI_ex)
-  qed
-  show ?thesis
-    apply (clarsimp simp add: states_to_words_def)
-    apply (rule conjI; rule impI)
-     apply ( simp add: H_0)
-    using is_aut.fin_states_are_states
-    by blast
-qed
-
-
-lemma induced_epi_eq_var:
-  "eq_var_func G S \<psi> MN_equiv ([(\<phi>\<^sup>\<star>)]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>) induced_epi"
-proof-
-  have H_0: "\<And> s g. \<lbrakk>s \<in> S; g \<in> carrier G; \<psi> g s \<in> S\<rbrakk> \<Longrightarrow>
-    words_to_syth_states (states_to_words (\<psi> g s)) =
-    ([(\<phi>\<^sup>\<star>)]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>) g (words_to_syth_states (states_to_words s))"
-  proof-
-    fix s g 
-    assume
-      A1_0: "s \<in> S" and
-      A1_1: "g \<in> carrier G" and
-      A1_2: "\<psi> g s \<in> S"
-    have H1_0: "([(\<phi>\<^sup>\<star>)]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>) g (words_to_syth_states (states_to_words s)) =
-     [(\<phi>\<^sup>\<star>) g (SOME w. w \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i w = s)]\<^sub>M\<^sub>N"
-      apply (clarsimp simp del: GMN_simps simp add: words_to_syth_states_def
-          states_to_words_def A1_0)
-      apply (rule meta_mp[of "(SOME w. w \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i w = s) \<in> A\<^sup>\<star>"])
-      using quot_act_wd_alt_notation[where w = "(SOME w. w \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i w = s)" and g = g] A1_1
-       apply simp
-      using A1_0
-      by (metis (mono_tags, lifting) induced_epi_wd1 some_eq_imp)
-    have H1_1: "\<And>g s' w'. \<lbrakk>s'\<in> S; w'\<in> A\<^sup>\<star>; g \<in> carrier G; (\<phi>\<^sup>\<star>) g w' \<in> A\<^sup>\<star> \<and> \<psi> g s' \<in> S\<rbrakk>
-               \<Longrightarrow> (\<delta>\<^sup>\<star>) (\<psi> g s') ((\<phi>\<^sup>\<star>) g w') = \<psi> g ((\<delta>\<^sup>\<star>) s' w')"
-      using give_input_eq_var
-      apply (clarsimp simp del: GMN_simps simp add: eq_var_func_axioms_def eq_var_func_def
-          make_op_def)
-      by (meson in_listsI)
-    have H1_2: "{w. w \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i w = \<psi> g s} =
-     {w'. \<exists>w \<in> A\<^sup>\<star>. (\<phi>\<^sup>\<star>) g w = w' \<and> (\<delta>\<^sup>\<star>) i w = s}"
-    proof (rule subset_antisym; clarify)
-      fix w'
-      assume
-        A2_0: "(\<delta>\<^sup>\<star>) i w' = \<psi> g s" and
-        A2_1: "\<forall>x\<in>set w'. x \<in> A"
-      have H2_0: "(inv \<^bsub>G\<^esub> g) \<in> carrier G"
-        by (meson A1_1 group.inv_closed group_hom.axioms(1) states_a_G_set.group_hom)
-      have H2_1: "(\<phi>\<^sup>\<star>) g ((\<phi>\<^sup>\<star>) (inv \<^bsub>G\<^esub> g) w') = w'"
-        by (smt (verit) A1_1 A2_1 alt_group_act_is_grp_act group_action.bij_prop1
-            group_action.orbit_sym_aux in_listsI labels_a_G_set.lists_a_Gset)
-      have H2_2: "\<And>g w. g \<in> carrier G \<Longrightarrow> w \<in> A\<^sup>\<star> \<Longrightarrow> (\<delta>\<^sup>\<star>) i ((\<phi>\<^sup>\<star>) g w) = (\<delta>\<^sup>\<star>) (\<psi> g i) ((\<phi>\<^sup>\<star>) g w)"
-        using init_is_eq_var.eq_var_subset_axioms init_is_eq_var.is_equivar
-        by auto
-      have H2_3: "\<And>g w. g \<in> carrier G \<Longrightarrow> w \<in> A\<^sup>\<star> \<Longrightarrow> (\<delta>\<^sup>\<star>) (\<psi> g i) ((\<phi>\<^sup>\<star>) g w) = \<psi> g ((\<delta>\<^sup>\<star>) i w)"
-        apply (rule H1_1[where s'1 = i]) 
-           apply (simp add: A2_1 in_lists_conv_set H2_0 is_aut.init_state_is_a_state)+
-        using is_aut.init_state_is_a_state labels_a_G_set.element_image
-          states_a_G_set.element_image
-        by blast
-      have H2_4: "\<psi> (inv \<^bsub>G\<^esub> g) ((\<delta>\<^sup>\<star>) i w') = s"
-        using A2_0 H2_0
-        by (simp add: A1_0 A1_1 states_a_G_set.orbit_sym_aux)
-      have H2_5: "(\<delta>\<^sup>\<star>) i ((\<phi>\<^sup>\<star>) (inv \<^bsub>G\<^esub> g) w') = s"
-        apply (rule meta_mp[of "w'\<in> A\<^sup>\<star>"])
-        using H2_0 H2_1 H2_4 A2_1 H2_2 H2_3
-         apply presburger 
-        using A2_1
-        by auto
-      have H2_6: "(\<phi>\<^sup>\<star>) (inv\<^bsub>G\<^esub> g) w' \<in> lists A "
-        using H2_0 A2_1 
-        by (metis alt_group_act_is_grp_act group_action.element_image in_listsI
-            labels_a_G_set.lists_a_Gset)
-      thus "\<exists>w\<in>lists A. (\<phi>\<^sup>\<star>) g w = w' \<and> (\<delta>\<^sup>\<star>) i w = s"
-        using H2_1 H2_5 H2_6
-        by blast
-    next
-      fix x w
-      assume
-        A2_0: "\<forall>x\<in>set w. x \<in> A" and
-        A2_1: "s = (\<delta>\<^sup>\<star>) i w"
-      show "(\<phi>\<^sup>\<star>) g w \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i ((\<phi>\<^sup>\<star>) g w) = \<psi> g ((\<delta>\<^sup>\<star>) i w) "
-        apply (rule conjI)
-         apply (rule meta_mp[of "(inv \<^bsub>G\<^esub> g) \<in> carrier G"])
-        using alt_group_act_is_grp_act group_action.element_image in_listsI
-          labels_a_G_set.lists_a_Gset
-          apply (metis A1_1 A2_0)
-         apply (meson A1_1 group.inv_closed group_hom.axioms(1) states_a_G_set.group_hom)
-        apply (rule meta_mp[of "\<psi> g i = i"])
-        using H1_1[where s'1 = i and g1 = "g"]
-         apply (metis A1_1 A2_0 action_on_input in_listsI)
-        using init_is_eq_var.eq_var_subset_axioms init_is_eq_var.is_equivar
-        by (simp add: A1_1)
-    qed
-    have H1_3: "\<exists>w. w \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i w = s" 
-      using A1_0 is_reachable
-      by auto
-    have H1_4: "\<exists>w. w \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i w = \<psi> g s"
-      using A1_2 induced_epi_wd1
-      by auto
-    have H1_5: "[(\<phi>\<^sup>\<star>) g (SOME w. w \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i w = s)]\<^sub>M\<^sub>N = [SOME w. w \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i w = \<psi> g s]\<^sub>M\<^sub>N" 
-    proof (rule subset_antisym; clarify)
-      fix w'
-      assume
-        A2_0: "w' \<in> [(\<phi>\<^sup>\<star>) g (SOME w. w \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i w = s)]\<^sub>M\<^sub>N"
-      have H2_0: "\<And>w. w \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i w = s \<Longrightarrow> w' \<in> [(\<phi>\<^sup>\<star>) g w]\<^sub>M\<^sub>N"
-        using A2_0 H1_3 H1_2 H1_4 induced_epi_wd2 mem_Collect_eq tfl_some 
-        by (smt (verit, best))  
-      obtain w'' where H2_w'': "w' \<in> [(\<phi>\<^sup>\<star>) g w'']\<^sub>M\<^sub>N \<and> w'' \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i w'' = s"
-        using A2_0 H1_3 tfl_some
-        by (metis (mono_tags, lifting))
-      from H1_2 H2_w'' have H2_1: "(\<delta>\<^sup>\<star>) i ((\<phi>\<^sup>\<star>) g w'') = \<psi> g s"
-        by blast
-      have H2_2: "\<And>w. w \<in> A\<^sup>\<star> \<Longrightarrow> (\<delta>\<^sup>\<star>) i w = \<psi> g s \<Longrightarrow> w' \<in> [w]\<^sub>M\<^sub>N"
-      proof -
-        fix w''
-        assume
-          A3_0: "w'' \<in> A\<^sup>\<star>" and
-          A3_1: "(\<delta>\<^sup>\<star>) i w'' = \<psi> g s"
-        have H3_0: "(inv \<^bsub>G\<^esub>g) \<in> carrier G" 
-          by (metis A1_1 group.inv_closed group_hom.axioms(1) states_a_G_set.group_hom)
-        from A3_0 H3_0 have H3_1: "(\<phi>\<^sup>\<star>) (inv \<^bsub>G\<^esub> g) w'' \<in> A\<^sup>\<star>"
-          by (metis alt_grp_act.axioms group_action.element_image 
-              labels_a_G_set.lists_a_Gset)
-        have H3_2: "\<And>g w. g \<in> carrier G \<Longrightarrow> w \<in> A\<^sup>\<star> \<Longrightarrow> (\<delta>\<^sup>\<star>) i ((\<phi>\<^sup>\<star>) g w) = (\<delta>\<^sup>\<star>) (\<psi> g i) ((\<phi>\<^sup>\<star>) g w)"
-          using init_is_eq_var.eq_var_subset_axioms init_is_eq_var.is_equivar
-          by auto
-        have H3_3: "\<And>g w. g \<in> carrier G \<Longrightarrow> w \<in> A\<^sup>\<star> \<Longrightarrow> (\<delta>\<^sup>\<star>) (\<psi> g i) ((\<phi>\<^sup>\<star>) g w) = \<psi> g ((\<delta>\<^sup>\<star>) i w)"
-          apply (rule H1_1[where s'1 = i]) 
-             apply (simp add: A3_1 in_lists_conv_set H2_1 is_aut.init_state_is_a_state)+
-          using is_aut.init_state_is_a_state labels_a_G_set.element_image
-            states_a_G_set.element_image
-          by blast
-        have H3_4: "s = (\<delta>\<^sup>\<star>) i ((\<phi>\<^sup>\<star>) (inv \<^bsub>G\<^esub> g) w'')"
-          using A3_0 A3_1 H3_0 H3_2 H3_3 A1_0 A1_1 states_a_G_set.orbit_sym_aux
-          by auto
-        from H3_4 show " w' \<in> [w'']\<^sub>M\<^sub>N"
-          by (metis (mono_tags, lifting) A1_1 G_set_equiv H2_1 H2_w'' \<open>(\<delta>\<^sup>\<star>) i w'' = \<psi> g s\<close> A3_0
-              eq_var_subset.is_equivar image_eqI induced_epi_wd2
-              labels_a_G_set.lists_a_Gset)
-      qed
-      from H2_2 show "w' \<in> [SOME w. w \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i w = \<psi> g s]\<^sub>M\<^sub>N"
-        by (smt (verit) H1_4 some_eq_ex)
-    next
-      fix w'
-      assume
-        A2_0: "w' \<in> [SOME w. w \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i w = \<psi> g s]\<^sub>M\<^sub>N"
-      obtain w'' where H2_w'': "w' \<in> [(\<phi>\<^sup>\<star>) g w'']\<^sub>M\<^sub>N \<and> w'' \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i w'' = s"
-        using A2_0 H1_3 tfl_some 
-        by (smt (verit) H1_2 mem_Collect_eq)
-      from H1_2 H2_w'' have H2_0: "(\<delta>\<^sup>\<star>) i ((\<phi>\<^sup>\<star>) g w'') = \<psi> g s"
-        by blast
-      have H2_1: "\<And>w. w \<in> A\<^sup>\<star> \<Longrightarrow> (\<delta>\<^sup>\<star>) i w = s \<Longrightarrow> w' \<in> [(\<phi>\<^sup>\<star>) g w]\<^sub>M\<^sub>N"
-      proof -
-        fix w''
-        assume
-          A3_0: "w'' \<in> A\<^sup>\<star>" and
-          A3_1: "(\<delta>\<^sup>\<star>) i w'' = s"
-        have H3_0: "(inv \<^bsub>G\<^esub>g) \<in> carrier G" 
-          by (metis A1_1 group.inv_closed group_hom.axioms(1) states_a_G_set.group_hom)
-        have H3_1: "(\<phi>\<^sup>\<star>) (inv \<^bsub>G\<^esub> g) w'' \<in> A\<^sup>\<star>"
-          using A3_0 H3_0 
-          by (metis alt_group_act_is_grp_act group_action.element_image labels_a_G_set.lists_a_Gset)
-        have H3_2: "\<And>g w. g \<in> carrier G \<Longrightarrow> w \<in> A\<^sup>\<star> \<Longrightarrow> (\<delta>\<^sup>\<star>) i ((\<phi>\<^sup>\<star>) g w) =
-       (\<delta>\<^sup>\<star>) (\<psi> g i) ((\<phi>\<^sup>\<star>) g w)"
-          using init_is_eq_var.eq_var_subset_axioms init_is_eq_var.is_equivar
-          by auto
-        have H3_3: "\<And>g w. g \<in> carrier G \<Longrightarrow> w \<in> A\<^sup>\<star> \<Longrightarrow> (\<delta>\<^sup>\<star>) (\<psi> g i) ((\<phi>\<^sup>\<star>) g w) = 
-       \<psi> g ((\<delta>\<^sup>\<star>) i w)"
-          apply (rule H1_1[where s'1 = i]) 
-             apply (simp add: A3_1 in_lists_conv_set H2_0 is_aut.init_state_is_a_state)+
-          using is_aut.init_state_is_a_state labels_a_G_set.element_image
-            states_a_G_set.element_image
-          by blast
-        have H3_4: "\<psi> (inv \<^bsub>G\<^esub> g) s = (\<delta>\<^sup>\<star>) i ((\<phi>\<^sup>\<star>) (inv \<^bsub>G\<^esub> g) w'')"
-          using A3_0 A3_1 H3_0 H3_2 H3_3
-          by auto
-        show "w' \<in> [(\<phi>\<^sup>\<star>) g w'']\<^sub>M\<^sub>N "
-          using H3_4 H3_1 
-          by (smt (verit, del_insts) A1_1 A3_0 A3_1 in_listsI H3_2 H3_3
-              \<open>\<And>thesis. (\<And>w''. w' \<in> [(\<phi>\<^sup>\<star>) g w'']\<^sub>M\<^sub>N \<and> w'' \<in> A\<^sup>\<star> \<and>
-              (\<delta>\<^sup>\<star>) i w'' = s \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
-              alt_group_act_is_grp_act group_action.surj_prop image_eqI induced_epi_wd2
-              labels_a_G_set.lists_a_Gset)
-      qed
-      show "w' \<in> [(\<phi>\<^sup>\<star>) g (SOME w. w \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i w = s)]\<^sub>M\<^sub>N"
-        using H2_1 H1_3
-        by (metis (mono_tags, lifting) someI)
-    qed
-    show "words_to_syth_states (states_to_words (\<psi> g s)) =
-    ([(\<phi>\<^sup>\<star>)]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>) g (words_to_syth_states (states_to_words s))"
-      using H1_5
-      apply (clarsimp simp del: GMN_simps simp add: words_to_syth_states_def states_to_words_def)
-      apply (intro conjI; clarify; rule conjI)
-      using H1_0
-         apply (auto del: subset_antisym simp del: GMN_simps simp add: words_to_syth_states_def
-          states_to_words_def)[1]
-      using A1_2
-        apply blast 
-      using A1_0
-       apply blast
-      using A1_0
-      by blast
-  qed
-  show ?thesis
-    apply (clarsimp del: subset_antisym simp del: GMN_simps simp add: eq_var_func_def
-        eq_var_func_axioms_def)
-    apply (intro conjI)
-    subgoal
-      using states_a_G_set.alt_grp_act_axioms
-      by auto
-      apply (metis MN_rel_eq_var MN_rel_equival eq_var_rel.quot_act_is_grp_act)
-     apply (clarsimp simp add: FuncSet.extensional_funcset_def Pi_def)
-     apply (rule conjI)
-      apply (clarify)
-    subgoal for s
-      using is_reachable[where s = s]
-      apply (clarsimp simp add: induced_epi_def compose_def states_to_words_def
-          words_to_syth_states_def)
-      by (smt (verit) \<open>s \<in> S \<Longrightarrow> \<exists>input\<in>A\<^sup>\<star>. (\<delta>\<^sup>\<star>) i input = s\<close> alt_natural_map_MN_def
-          lists_eq_set quotientI rel_MN_def singleton_conv someI)
-     apply (clarsimp simp del: GMN_simps simp add: induced_epi_def make_op_def
-        compose_def)
-    apply (clarify)
-    apply (clarsimp simp del: GMN_simps simp add: induced_epi_def compose_def make_op_def)
-    apply (rule conjI; rule impI)
-     apply (simp add: H_0)
-    using states_a_G_set.element_image
-    by blast
-qed
-
-text \<open>
-The following lemma corresponds to lemma 3.7 from \cite{bojanczyk2014automata}:
-\<close>
-
-lemma reach_det_G_aut_rec_lang:
-  "G_aut_epi A S i F \<delta> MN_equiv MN_init_state MN_fin_states \<delta>\<^sub>M\<^sub>N G \<phi> \<psi> ([(\<phi>\<^sup>\<star>)]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>) induced_epi"
-proof-
-  have H_0: "\<And>s. s \<in> MN_equiv \<Longrightarrow> \<exists>input\<in> A\<^sup>\<star>. (\<delta>\<^sub>M\<^sub>N\<^sup>\<star>) MN_init_state input = s"
-  proof-
-    fix s
-    assume
-      A_0: "s \<in> MN_equiv"
-    from A_0 have H_0: "\<exists>w. w \<in> A\<^sup>\<star> \<and> s = [w]\<^sub>M\<^sub>N"
-      by (auto simp add: quotient_def)
-    show "\<exists>input\<in>A\<^sup>\<star>. (\<delta>\<^sub>M\<^sub>N\<^sup>\<star>) MN_init_state input = s"
-      using H_0 
-      by (metis MN_unique_init_state)
-  qed
-  have H_1: "\<And>s\<^sub>0 a. s\<^sub>0 \<in> S \<Longrightarrow> a \<in> A \<Longrightarrow> induced_epi (\<delta> s\<^sub>0 a) = \<delta>\<^sub>M\<^sub>N (induced_epi s\<^sub>0) a"
-  proof-
-    fix s\<^sub>0 a
-    assume
-      A1_0: "s\<^sub>0 \<in> S" and
-      A1_1: "a \<in> A" 
-    obtain w where H1_w: "w \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i w = s\<^sub>0" 
-      using A1_0 induced_epi_wd1
-      by auto
-    have H1_0: "[SOME w. w \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i w = s\<^sub>0]\<^sub>M\<^sub>N = [w]\<^sub>M\<^sub>N"
-      by (metis (mono_tags, lifting) H1_w induced_epi_wd2 some_eq_imp)
-    have H1_1: "(\<delta>\<^sup>\<star>) i (SOME w. w \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i w = \<delta> s\<^sub>0 a) = (\<delta>\<^sup>\<star>) i (w @ [a])"
-      using A1_0 A1_1 H1_w is_aut.trans_to_charact[where s = s\<^sub>0 and a = a and w = w] 
-      by (smt (verit, del_insts) induced_epi_wd1 is_aut.trans_func_well_def tfl_some)
-    have H1_2: "w @ [a] \<in> A\<^sup>\<star>" using H1_w A1_1 by auto
-    have H1_3: "[(SOME w. w \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i w = s\<^sub>0) @ [a]]\<^sub>M\<^sub>N = [w @ [a]]\<^sub>M\<^sub>N" 
-      by (metis (mono_tags, lifting) A1_1 H1_0 H1_w MN_trans_func_characterization someI)
-    have H1_4: "... = [SOME w. w \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i w = \<delta> s\<^sub>0 a]\<^sub>M\<^sub>N"
-      apply (rule sym)
-      apply (rule induced_epi_wd2[where w = "SOME w. w \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i w = \<delta> s\<^sub>0 a"
-            and w'= "w @ [a]"])
-        apply (metis (mono_tags, lifting) A1_0 A1_1 H1_w some_eq_imp H1_2 is_aut.trans_to_charact)
-       apply (rule H1_2)
-      using H1_1
-      by simp
-    show "induced_epi (\<delta> s\<^sub>0 a) = \<delta>\<^sub>M\<^sub>N (induced_epi s\<^sub>0) a"
-      apply (clarsimp del: subset_antisym simp del: GMN_simps simp add: induced_epi_def
-          words_to_syth_states_def states_to_words_def compose_def is_aut.trans_func_well_def)
-      using A1_1 H1_w H1_0 H1_3 H1_4 MN_trans_func_characterization A1_0
-        is_aut.trans_func_well_def
-      by presburger
-  qed
-  have H_2: "induced_epi ` S = MN_equiv"
-  proof-
-    have H1_0: "\<forall>s \<in> S. \<exists>v\<in> A\<^sup>\<star>. (\<delta>\<^sup>\<star>) i v = s \<and> [SOME w. w \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i w = s]\<^sub>M\<^sub>N = [v]\<^sub>M\<^sub>N"
-      by (smt (verit) is_reachable tfl_some)
-    have H1_1: "\<And>v. v\<in> A\<^sup>\<star> \<Longrightarrow> (\<delta>\<^sup>\<star>) i v \<in> S" 
-      using is_aut.give_input_closed
-      by (auto simp add: is_aut.init_state_is_a_state)
-    show ?thesis
-      apply (clarsimp simp del: GMN_simps simp add: induced_epi_def words_to_syth_states_def
-          states_to_words_def compose_def image_def)
-      using H1_0 H1_1
-      apply (clarsimp)
-      apply (rule subset_antisym; simp del: GMN_simps add: Set.subset_eq)
-       apply (metis (no_types, lifting) quotientI)
-      by (metis (no_types, lifting) alt_natural_map_MN_def induced_epi_wd2 quotientE)
-  qed
-  show ?thesis
-    apply (simp del: GMN_simps add: G_aut_epi_def G_aut_epi_axioms_def)
-    apply (rule conjI)
-    subgoal
-      apply (clarsimp simp del: GMN_simps simp add: G_aut_hom_def aut_hom_def reach_det_G_aut_def
-          is_reachable det_G_aut_def reach_det_aut_def reach_det_aut_axioms_def)
-      apply (intro conjI)
-                        apply (simp add: is_aut.det_aut_axioms)
-      using labels_a_G_set.alt_grp_act_axioms
-                       apply (auto)[1]
-      using states_a_G_set.alt_grp_act_axioms
-                      apply blast
-                     apply (simp add: accepting_is_eq_var.eq_var_subset_axioms)
-      using init_is_eq_var.eq_var_subset_axioms
-                    apply (auto)[1]
-                   apply (simp add: trans_is_eq_var.eq_var_func_axioms)
-                  apply (simp add: is_aut.det_aut_axioms)
-      using syth_aut_is_det_aut
-                 apply simp
-      using labels_a_G_set.alt_grp_act_axioms
-                apply (auto)[1]
-               apply (metis MN_rel_eq_var MN_rel_equival eq_var_rel.quot_act_is_grp_act)
-      using MN_final_state_equiv
-              apply presburger 
-      using MN_init_state_equivar_v2
-             apply presburger 
-      using MN_trans_eq_var_func
-            apply blast
-      using syth_aut_is_det_aut
-           apply auto[1]
-          apply (clarify)
-          apply (simp add: H_0 del: GMN_simps)
-         apply (simp add: is_aut.det_aut_axioms)
-      using syth_aut_is_det_aut
-        apply blast
-       apply (clarsimp del: subset_antisym simp del: GMN_simps simp add: aut_hom_axioms_def
-          FuncSet.extensional_funcset_def Pi_def extensional_def)[1]
-       apply (intro conjI)
-           apply (clarify)
-           apply (simp add: induced_epi_def)
-           apply (simp add: induced_epi_def words_to_syth_states_def states_to_words_def
-          compose_def)
-           apply (rule meta_mp[of "(\<delta>\<^sup>\<star>) i Nil = i"])
-      using induced_epi_wd2[where w = "Nil"]
-            apply (auto simp add: is_aut.init_state_is_a_state del: subset_antisym)[2] 
-      subgoal for x
-        apply (rule quotientI)
-        using is_reachable[where s = x] someI[where P = "\<lambda>w. w \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i w = x"]      
-        by blast
-          apply (auto simp add: induced_epi_def words_to_syth_states_def states_to_words_def
-          compose_def)[1]
-         apply (simp add: induced_epi_def states_to_words_def compose_def
-          is_aut.init_state_is_a_state)
-         apply (metis (mono_tags, lifting) \<open>\<And>w'. \<lbrakk>[] \<in> A\<^sup>\<star>; w' \<in> A\<^sup>\<star>;
-      (\<delta>\<^sup>\<star>) i [] = (\<delta>\<^sup>\<star>) i w'\<rbrakk> \<Longrightarrow> MN_init_state = [w']\<^sub>M\<^sub>N\<close>
-          alt_natural_map_MN_def give_input.simps(1) lists.Nil some_eq_imp
-          words_to_syth_states_def)
-        apply (clarify)
-      subgoal for s 
-        apply (rule iffI)
-         apply (smt (verit) Pi_iff compose_eq in_mono induced_epi_def is_aut.fin_states_are_states
-            states_to_words_on_final words_to_syth_states_def)
-        apply (clarsimp simp del: GMN_simps simp add: induced_epi_def words_to_syth_states_def
-            states_to_words_def compose_def)
-        apply (rule meta_mp[of "(SOME w. w \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i w = s) \<in> L"])
-         apply (smt (verit) induced_epi_wd1 is_recognised someI)
-        using fin_states_rep_by_lang is_reachable mem_Collect_eq
-        by (metis (mono_tags, lifting))
-       apply (clarsimp simp del: GMN_simps)
-       apply (simp add: H_1)
-      using induced_epi_eq_var
-      by blast
-    by (simp add: H_2)
-qed
-
-end
-
-lemma (in det_G_aut) finite_reachable:
-  "finite (orbits G S \<psi>) \<Longrightarrow> finite (orbits G S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h)"
-proof-
-  assume
-    A_0: "finite (orbits G S \<psi>)"
-  have H_0: "S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<subseteq> S" 
-    apply (clarsimp simp add: reachable_states_def) 
-    by (simp add: in_listsI is_aut.give_input_closed is_aut.init_state_is_a_state)
-  have H_1: "{{\<psi> g x |g. g \<in> carrier G} |x. x \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h} \<subseteq>
-    {{\<psi> g x |g. g \<in> carrier G} |x. x \<in> S}"
-    by (smt (verit, best) Collect_mono_iff H_0 subsetD)
-  have H_2: "\<And>x. x \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<Longrightarrow>
-     {\<psi> g x |g. g \<in> carrier G} = {\<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g x |g. g \<in> carrier G}"
-    using reachable_action_is_restict
-    by (metis)
-  hence H_3: "{{\<psi> g x |g. g \<in> carrier G} |x. x \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h} =
-    {{\<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g x |g. g \<in> carrier G} |x. x \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h}" 
-    by blast
-  show "finite (orbits G S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h)"
-    using A_0 apply (clarsimp simp add: orbits_def orbit_def)
-    using Finite_Set.finite_subset H_1 H_3
-    by auto
-qed
-
-lemma (in det_G_aut)
-  orbs_pos_card: "finite (orbits G S \<psi>) \<Longrightarrow> card (orbits G S \<psi>) > 0"
-  apply (clarsimp simp add: card_gt_0_iff orbits_def)
-  using is_aut.init_state_is_a_state
-  by auto
-
-lemma (in reach_det_G_aut_rec_lang) MN_B2T:
-  assumes
-    Fin: "finite (orbits G S \<psi>)"
-  shows
-    "finite (orbits G (language.MN_equiv A L) (([(\<phi>\<^sup>\<star>)]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>)))"
-proof- 
-  have H_0: "finite {{\<psi> g x |g. g \<in> carrier G} |x. x \<in> S}"
-    using Fin
-    by (auto simp add: orbits_def orbit_def)
-  have H_1: "induced_epi ` S = MN_equiv"
-    using reach_det_G_aut_rec_lang
-    by (auto simp del: GMN_simps simp add: G_aut_epi_def G_aut_epi_axioms_def)
-  have H_2: "\<And>B f. finite B \<Longrightarrow> finite {f b| b. b \<in> B} "
-    by auto
-  have H_3: "finite {{\<psi> g x |g. g \<in> carrier G} |x. x \<in> S} \<Longrightarrow>
-      finite {induced_epi ` b |b. b \<in> {{\<psi> g x |g. g \<in> carrier G} |x. x \<in> S}}"
-    using H_2[where f1 = "(\<lambda>x. induced_epi ` x)" and B1 = "{{\<psi> g x |g. g \<in> carrier G} |x. x \<in> S}"]
-    by auto
-  have H_4: "\<And>s. s \<in> S \<Longrightarrow> \<exists>b. {induced_epi (\<psi> g s) |g. g \<in> carrier G}
-              = {y. \<exists>x\<in>b. y = induced_epi x} \<and> (\<exists>x. b = {\<psi> g x |g. g \<in> carrier G} \<and> x \<in> S)"
-  proof-
-    fix s
-    assume
-      A2_0: "s \<in> S"
-    have H2_0: "{induced_epi (\<psi> g s) |g. g \<in> carrier G} = {y. \<exists>x \<in> {\<psi> g s |g. g \<in> carrier G}. y =
-    induced_epi x}"
-      by blast
-    have H2_1: "(\<exists>x. {\<psi> g s |g. g \<in> carrier G} = {\<psi> g x |g. g \<in> carrier G} \<and> x \<in> S)"
-      using A2_0 
-      by auto
-    show "\<exists>b. {induced_epi (\<psi> g s) |g. g \<in> carrier G} =
-    {y. \<exists>x\<in>b. y = induced_epi x} \<and> (\<exists>x. b = {\<psi> g x |g. g \<in> carrier G} \<and> x \<in> S)"
-      using A2_0 H2_0 H2_1 
-      by meson
-  qed
-  have H_5: "{induced_epi ` b |b. b \<in> {{\<psi> g x |g. g \<in> carrier G} |x. x \<in> S}} =
-  {{induced_epi (\<psi> g s) | g . g \<in> carrier G} |s. s \<in> S}"
-    apply (clarsimp simp add: image_def)
-    apply (rule subset_antisym; simp add: Set.subset_eq; clarify)
-     apply auto[1]
-    apply (simp)
-    by (simp add: H_4)
-  from H_3 H_5 have H_6: "finite {{\<psi> g x |g. g \<in> carrier G} |x. x \<in> S} \<Longrightarrow>
-      finite {{induced_epi (\<psi> g s) | g . g \<in> carrier G} |s. s \<in> S}"
-    by metis
-  have H_7: "finite {{induced_epi (\<psi> g x) |g. g \<in> carrier G} |x. x \<in> S}"
-    apply (rule H_6)
-    by (simp add: H_0)
-  have H_8: "\<And>x. x \<in> S \<Longrightarrow> {induced_epi (\<psi> g x) |g. g \<in> carrier G} =
-  {([(\<phi>\<^sup>\<star>)]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>) g (induced_epi x) |g. g \<in> carrier G}"
-    using induced_epi_eq_var
-    apply (simp del: GMN_simps add: eq_var_func_def eq_var_func_axioms_def make_op_def)
-    by blast
-  hence H_9: "{{induced_epi (\<psi> g x) |g. g \<in> carrier G} |x. x \<in> S} =
-  {{([(\<phi>\<^sup>\<star>)]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>) g (induced_epi x) |g. g \<in> carrier G} |x. x \<in> S}"
-    by blast
-  have H_10: "\<And>f g X B C. g ` B = C \<Longrightarrow>
-             {{f x (g b)|x. x\<in>X}|b. b \<in> B} = {{f x c|x. x \<in> X}|c. c \<in> C}"
-    by auto
-  have H_11: "{{([(\<phi>\<^sup>\<star>)]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>) g (induced_epi x) |g. g \<in> carrier G} |x. x \<in> S} =
-  {{([(\<phi>\<^sup>\<star>)]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>) g W |g. g \<in> carrier G} |W. W \<in> MN_equiv}"
-    apply (rule H_10[where f2 = "([(\<phi>\<^sup>\<star>)]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>)" and X2 = "carrier G" and g2 = induced_epi
-          and B2 = S and C2 = MN_equiv])
-    using H_1
-    by simp
-  have H_12: "{{([(\<phi>\<^sup>\<star>)]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>) g W |g. g \<in> carrier G} |W. W \<in> MN_equiv} =
-  orbits G (language.MN_equiv A L) (([(\<phi>\<^sup>\<star>)]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>))"
-    by (auto simp add: orbits_def orbit_def)
-  show "finite (orbits G (language.MN_equiv A L) (([(\<phi>\<^sup>\<star>)]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>)))"
-    using H_9 H_11 H_12 H_7
-    by presburger
-qed
-
-context det_G_aut_rec_lang begin
-
-text \<open>
-To avoid duplicate variant of "star":
-\<close>
-
-no_adhoc_overloading
-  star labels_a_G_set.induced_star_map
-
-end
-
-context det_G_aut_rec_lang begin
-adhoc_overloading
-  star labels_a_G_set.induced_star_map
-end
-
-
-lemma (in det_G_aut_rec_lang) MN_prep:
-  "\<exists>S'. \<exists>\<delta>'. \<exists>F'. \<exists>\<psi>'.
-  (reach_det_G_aut_rec_lang A S' i F' \<delta>' G \<phi> \<psi>' L \<and>
-  (finite (orbits G S \<psi>) \<longrightarrow> finite (orbits G S' \<psi>')))"
-  by (meson G_lang_axioms finite_reachable reach_det_G_aut_rec_lang.intro
-      reach_det_aut_is_det_aut_rec_L)
-
-lemma (in det_G_aut_rec_lang) MN_fin_orbs_imp_fin_states:
-  assumes
-    Fin: "finite (orbits G S \<psi>)"
-  shows
-    "finite (orbits G (language.MN_equiv A L) (([(\<phi>\<^sup>\<star>)]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>)))"
-  using MN_prep 
-  by (metis assms reach_det_G_aut_rec_lang.MN_B2T)
-
-text \<open>
-The following theorem corresponds to theorem 3.8 from \cite{bojanczyk2014automata}, i.e. the
-Myhill-Nerode theorem for G-automata.
-The left to right direction (see statement below) of the typical Myhill-Nerode theorem would
-qantify over types (if some condition holds, then there exists some automaton accepting the
-language). As it is not possible to qantify over types in this way, the equivalence is spit into
-two directions. In the left to right direction, the explicit type of the syntactic automaton is
-used. In the right to left direction some type, 's, is fixed.
-As the two directions are split, the opertunity was taken to strengthen the right to left direction:
-We do not assume the given automaton is reachable.
-
-This splitting of the directions will be present in all other Myhill-Nerode theorems that will be
-proved in this document.
-\<close>
-
-theorem (in G_lang) G_Myhill_Nerode :
-  assumes
-    "finite (orbits G A \<phi>)"
-  shows
-    G_Myhill_Nerode_LR: "finite (orbits G MN_equiv ([(\<phi>\<^sup>\<star>)]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>)) \<Longrightarrow>
-     (\<exists>S F :: 'alpha list set set. \<exists>i :: 'alpha list set. \<exists>\<delta>. \<exists>\<psi>.
-     reach_det_G_aut_rec_lang A S i F \<delta> G \<phi> \<psi> L \<and> finite (orbits G S \<psi>))" and
-    G_Myhill_Nerode_RL: "(\<exists>S F :: 's set. \<exists>i :: 's. \<exists>\<delta>. \<exists>\<psi>.
-     det_G_aut_rec_lang A S i F \<delta> G \<phi> \<psi> L \<and> finite (orbits G S \<psi>))
-     \<Longrightarrow> finite (orbits G MN_equiv ([(\<phi>\<^sup>\<star>)]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>))"
-  subgoal
-    using syntact_aut_is_reach_aut_rec_lang 
-    by blast
-  by (metis det_G_aut_rec_lang.MN_fin_orbs_imp_fin_states)
-
-subsection \<open>
-Proving the standard Myhill-Nerode Theorem
-\<close>
-
-text \<open>
-Any automaton is a $G$-automaton with respect to the trivial group and action,
-hence the standard Myhill-Nerode theorem is a special case of the $G$-Myhill-Nerode theorem.
-\<close>
-
-interpretation triv_act:
-  alt_grp_act "singleton_group (undefined)" X "(\<lambda>x \<in> {undefined}. one (BijGroup X))"
-  apply (simp add: group_action_def group_hom_def group_hom_axioms_def)
-  apply (intro conjI)
-   apply (simp add: group_BijGroup)
-  using trivial_hom 
-  by (smt (verit) carrier_singleton_group group.hom_restrict group_BijGroup restrict_apply
-      singleton_group)
-
-lemma (in det_aut) triv_G_aut:
-  fixes triv_G
-  assumes H_triv_G: "triv_G = (singleton_group (undefined))"
-  shows "det_G_aut labels states init_state fin_states \<delta> 
-  triv_G (\<lambda>x \<in> {undefined}. one (BijGroup labels)) (\<lambda>x \<in> {undefined}. one (BijGroup states))"
-  apply (simp add: det_G_aut_def group_hom_def group_hom_axioms_def
-      eq_var_subset_def eq_var_subset_axioms_def eq_var_func_def eq_var_func_axioms_def)
-  apply (intro conjI)
-             apply (rule det_aut_axioms)
-            apply (simp add: assms triv_act.group_action_axioms)+
-  using fin_states_are_states
-          apply (auto)[1]
-         apply (clarify; rule conjI; rule impI)
-          apply (simp add: H_triv_G BijGroup_def image_def)
-  using fin_states_are_states
-          apply auto[1]
-         apply (simp add: H_triv_G BijGroup_def image_def)
-        apply (simp add: assms triv_act.group_action_axioms)
-       apply (simp add: init_state_is_a_state)
-      apply (clarify; rule conjI; rule impI)
-       apply (simp add: H_triv_G BijGroup_def image_def init_state_is_a_state)+ 
-     apply (clarsimp simp add: group_action_def BijGroup_def hom_def group_hom_def
-      group_hom_axioms_def)
-     apply (rule conjI)
-      apply (smt (verit) BijGroup_def Bij_imp_funcset Id_compose SigmaE case_prod_conv
-      group_BijGroup id_Bij restrict_ext restrict_extensional)
-     apply (rule meta_mp[of "undefined \<otimes>\<^bsub>singleton_group undefined\<^esub> undefined = undefined"])
-      apply (auto)[1]
-     apply (metis carrier_singleton_group comm_groupE(1) singletonD singletonI
-      singleton_abelian_group)
-    apply (simp add: assms triv_act.group_action_axioms)
-   apply (auto simp add: trans_func_well_def)[1] 
-  by (clarsimp simp add: BijGroup_def trans_func_well_def H_triv_G) 
-
-lemma triv_orbits:
-  "orbits (singleton_group (undefined)) S (\<lambda>x \<in> {undefined}. one (BijGroup S)) =
-   {{s} |s. s \<in> S}"
-  apply (simp add: BijGroup_def singleton_group_def orbits_def orbit_def)
-  by auto
-
-lemma fin_triv_orbs:
-  "finite (orbits (singleton_group (undefined)) S (\<lambda>x \<in> {undefined}. one (BijGroup S))) = finite S"
-  apply (subst triv_orbits)
-  apply (rule meta_mp[of "bij_betw (\<lambda>s \<in> S. {s}) S {{s} |s. s \<in> S}"])
-  using bij_betw_finite
-   apply (auto)[1]
-  by (auto simp add: bij_betw_def image_def)
-
-context language begin
-
-interpretation triv_G_lang:
-  G_lang "singleton_group (undefined)" A "(\<lambda>x \<in> {undefined}. one (BijGroup A))" L
-  apply (simp add: G_lang_def G_lang_axioms_def eq_var_subset_def eq_var_subset_axioms_def)
-  apply (intro conjI)
-      apply (simp add: triv_act.group_action_axioms)
-     apply (simp add: language_axioms)
-  using triv_act.lists_a_Gset
-    apply fastforce
-   apply (rule is_lang)
-  apply (clarsimp simp add: BijGroup_def image_def)
-  apply (rule subset_antisym; simp add: Set.subset_eq; clarify)
-  using is_lang
-   apply (auto simp add: map_idI)[1] 
-  using is_lang map_idI 
-  by (metis in_listsD in_mono inf.absorb_iff1 restrict_apply)
-
-definition triv_G :: "'grp monoid"
-  where "triv_G = (singleton_group (undefined))"
-
-definition triv_act :: "'grp \<Rightarrow> 'alpha \<Rightarrow> 'alpha"
-  where "triv_act = (\<lambda>x \<in> {undefined}. \<one>\<^bsub>BijGroup A\<^esub>)"
-
-corollary standard_Myhill_Nerode:
-  assumes
-    H_fin_alph: "finite A"
-  shows
-    standard_Myhill_Nerode_LR: "finite MN_equiv \<Longrightarrow>
-     (\<exists>S F :: 'alpha list set set. \<exists>i :: 'alpha list set. \<exists>\<delta>.
-     reach_det_aut_rec_lang A S i F \<delta> L \<and> finite S)" and
-    standard_Myhill_Nerode_RL: "(\<exists>S F :: 's set. \<exists>i :: 's. \<exists>\<delta>.
-     det_aut_rec_lang A S i F \<delta> L \<and> finite S) \<Longrightarrow> finite MN_equiv"
-proof-
-  assume
-    A_0: "finite MN_equiv"
-  have H_0: "reach_det_aut_rec_lang A MN_equiv MN_init_state MN_fin_states \<delta>\<^sub>M\<^sub>N L"
-    using triv_G_lang.syntact_aut_is_reach_aut_rec_lang
-    apply (clarsimp simp add: reach_det_G_aut_rec_lang_def det_G_aut_rec_lang_def
-        reach_det_aut_rec_lang_def reach_det_aut_def reach_det_aut_axioms_def det_G_aut_def)
-    by (smt (z3) alt_natural_map_MN_def quotientE triv_G_lang.MN_unique_init_state)
-  show "\<exists>S F:: 'alpha list set set. \<exists>i :: 'alpha list set. \<exists>\<delta>.
-  reach_det_aut_rec_lang A S i F \<delta> L \<and> finite S"
-    using A_0 H_0 
-    by auto
-next
-  assume
-    A_0: "\<exists>S F:: 's set. \<exists>i :: 's. \<exists>\<delta>. det_aut_rec_lang A S i F \<delta> L \<and> finite S"
-  obtain S F :: "'s set" and i :: "'s"  and \<delta>
-    where H_MN: "det_aut_rec_lang A S i F \<delta> L \<and> finite S"
-    using A_0
-    by auto
-  have H_0: "det_G_aut A S i F \<delta> triv_G (\<lambda>x\<in>{undefined}. \<one>\<^bsub>BijGroup A\<^esub>)
-   (\<lambda>x\<in>{undefined}. \<one>\<^bsub>BijGroup S\<^esub>)"
-    apply (rule det_aut.triv_G_aut[of A S i F \<delta> triv_G])
-    using H_MN
-     apply (simp add: det_aut_rec_lang_def)
-    by (rule triv_G_def)
-  have H_1: "det_G_aut_rec_lang A S i F \<delta> triv_G (\<lambda>x\<in>{undefined}. \<one>\<^bsub>BijGroup A\<^esub>)
-   (\<lambda>x\<in>{undefined}. \<one>\<^bsub>BijGroup S\<^esub>) L"
-    by (auto simp add: det_G_aut_rec_lang_def H_0 H_MN)
-  have H_2: "(\<exists>S F:: 's set. \<exists>i :: 's. \<exists>\<delta> \<psi>.
-         det_G_aut_rec_lang A S i F \<delta> (singleton_group undefined) (\<lambda>x\<in>{undefined}. \<one>\<^bsub>BijGroup A\<^esub>)
-         \<psi> L \<and> finite (orbits (singleton_group undefined) S \<psi>))"
-    using H_1
-    by (metis H_MN fin_triv_orbs triv_G_def)
-  have H_3: "finite (orbits triv_G A triv_act)"
-    apply (subst triv_G_def; subst triv_act_def; subst fin_triv_orbs[of A])
-    by (rule H_fin_alph)
-  have H_4:"finite (orbits triv_G MN_equiv (triv_act.induced_quot_map (A\<^sup>\<star>)
-     (triv_act.induced_star_map A triv_act) \<equiv>\<^sub>M\<^sub>N))"
-    using H_3
-    apply (simp add: triv_G_def triv_act_def del: GMN_simps)
-    using triv_G_lang.G_Myhill_Nerode H_2
-    by blast    
-  have H_5:"triv_act.induced_star_map A triv_act = (\<lambda>x \<in> {undefined}. \<one>\<^bsub>BijGroup (A\<^sup>\<star>)\<^esub>)"
-    apply (simp add: BijGroup_def restrict_def fun_eq_iff triv_act_def)
-    by (clarsimp simp add: list.map_ident_strong)
-  have H_6: "(triv_act.induced_quot_map (A\<^sup>\<star>) (triv_act.induced_star_map A
-  triv_act) \<equiv>\<^sub>M\<^sub>N) = (\<lambda>x \<in> {undefined}. \<one>\<^bsub>BijGroup MN_equiv\<^esub>)"
-    apply (subst H_5)
-    apply (simp add: BijGroup_def fun_eq_iff Image_def)
-    apply (rule allI; rule conjI; intro impI)
-     apply (smt (verit) Collect_cong Collect_mem_eq Eps_cong MN_rel_equival equiv_Eps_in
-        in_quotient_imp_closed quotient_eq_iff)
-    using MN_rel_equival equiv_Eps_preserves
-    by auto
-  show "finite MN_equiv"
-    apply (subst fin_triv_orbs [symmetric]; subst H_6 [symmetric]; subst triv_G_def [symmetric])
-    by (rule H_4)
-qed
-end
-
-section \<open>
-Myhill-Nerode Theorem for Nominal $G$-Automata
-\<close>
-
-subsection \<open>
-Data Symmetries, Supports and Nominal Actions
-\<close>
-
-text \<open>
-The following locale corresponds to the definition 2.2 from \cite{bojanczyk2014automata}.
-Note that we let $G$ be an arbitrary group instead of a subgroup of \texttt{BijGroup} $D$,
-but assume there is a homomoprhism $\pi: G \to \texttt{BijGroup} D$.
-By \texttt{group\_hom.img\_is\_subgroup} this is an equivalent definition:
-\<close>
-
-locale data_symm = group_action G D \<pi>
-  for 
-    G :: "('grp, 'b) monoid_scheme" and
-    D :: "'D set" ("\<bbbD>") and
-    \<pi> 
-
-text \<open>
-The following locales corresponds to definition 4.3 from \cite{bojanczyk2014automata}:
-\<close>
-
-locale supports = data_symm G D \<pi> + alt_grp_act G X \<phi> 
-  for
-    G :: "('grp, 'b) monoid_scheme" and
-    D :: "'D set" ("\<bbbD>") and
-    \<pi> and
-    X :: "'X set" (structure) and
-    \<phi> +
-  fixes
-    C :: "'D set" and
-    x :: "'X"
-  assumes
-    is_in_set:
-    "x \<in> X" and
-    is_subset:
-    "C \<subseteq> \<bbbD>" and
-    supports:
-    "g \<in> carrier G \<Longrightarrow> (\<forall>c. c \<in> C \<longrightarrow> \<pi> g c = c) \<Longrightarrow> g \<odot>\<^bsub>\<phi>\<^esub> x = x"
-begin
-
-text \<open>
-The following lemma corresponds to lemma 4.9 from \cite{bojanczyk2014automata}:
-\<close>
-
-lemma image_supports:
-  "\<And>g. g \<in> carrier G \<Longrightarrow> supports G D \<pi> X \<phi> (\<pi> g ` C) (g \<odot>\<^bsub>\<phi>\<^esub> x)"
-proof-
-  fix g
-  assume 
-    A_0: "g \<in> carrier G"
-  have H_0: "\<And>h. data_symm G \<bbbD> \<pi> \<Longrightarrow>
-         group_action G X \<phi> \<Longrightarrow>
-         x \<in> X \<Longrightarrow>
-         C \<subseteq> \<bbbD> \<Longrightarrow>
-         \<forall>g. g \<in> carrier G \<longrightarrow> (\<forall>c. c \<in> C \<longrightarrow> \<pi> g c = c) \<longrightarrow> \<phi> g x = x \<Longrightarrow>
-         h \<in> carrier G \<Longrightarrow> \<forall>c. c \<in> \<pi> g ` C \<longrightarrow> \<pi> h c = c \<Longrightarrow>
-         \<phi> h (\<phi> g x) = \<phi> g x"
-  proof-
-    fix h
-    assume
-      A1_0: "data_symm G \<bbbD> \<pi>" and
-      A1_1: "group_action G X \<phi>" and
-      A1_2: "\<forall>g. g \<in> carrier G \<longrightarrow> (\<forall>c. c \<in> C \<longrightarrow> \<pi> g c = c) \<longrightarrow> \<phi> g x = x" and
-      A1_3: "h \<in> carrier G" and
-      A1_4: "\<forall>c. c \<in> \<pi> g ` C \<longrightarrow> \<pi> h c = c"
-    have H1_0: "\<And>g. g \<in> carrier G \<Longrightarrow> (\<forall>c. c \<in> C \<longrightarrow> \<pi> g c = c) \<Longrightarrow> \<phi> g x = x"
-      using A1_2
-      by auto 
-    have H1_1: "\<forall>c. c \<in> C \<longrightarrow> \<pi> ((inv\<^bsub>G\<^esub> g) \<otimes>\<^bsub>G\<^esub> h \<otimes>\<^bsub>G\<^esub> g) c = c \<Longrightarrow>
-    \<phi> ((inv\<^bsub>G\<^esub> g) \<otimes>\<^bsub>G\<^esub> h \<otimes>\<^bsub>G\<^esub> g) x = x"
-      apply (rule H1_0[of "((inv\<^bsub>G\<^esub> g) \<otimes>\<^bsub>G\<^esub> h \<otimes>\<^bsub>G\<^esub> g)"])
-       apply (meson A_0 A1_3 group.subgroupE(3) group.subgroup_self group_hom group_hom.axioms(1)
-          subgroup.m_closed)
-      by simp
-    have H2: "\<pi> (((inv\<^bsub>G\<^esub> g) \<otimes>\<^bsub>G\<^esub> h) \<otimes>\<^bsub>G\<^esub> g) = compose \<bbbD> (\<pi> ((inv\<^bsub>G\<^esub> g) \<otimes>\<^bsub>G\<^esub> h)) (\<pi> g)"
-      using A1_0
-      apply (clarsimp simp add: data_symm_def group_action_def BijGroup_def group_hom_def
-          group_hom_axioms_def hom_def restrict_def)
-      apply (rule meta_mp[of "\<pi> g \<in> Bij \<bbbD> \<and> \<pi> ((inv\<^bsub>G\<^esub> g) \<otimes>\<^bsub>G\<^esub> h) \<in> Bij \<bbbD>"])
-       apply (smt (verit) A_0 A1_3 data_symm.axioms data_symm_axioms group.inv_closed
-          group.surj_const_mult group_action.bij_prop0 image_eqI)
-      apply (rule conjI)
-      using A_0
-       apply blast
-      by (meson A_0 A1_3 data_symm.axioms data_symm_axioms group.subgroupE(3) group.subgroupE(4)
-          group.subgroup_self group_action.bij_prop0)
-    also have H1_3: "... = compose \<bbbD> (compose \<bbbD> (\<pi> (inv\<^bsub>G\<^esub> g) ) (\<pi> h)) (\<pi> g)"
-      using A1_0
-      apply (clarsimp simp add: data_symm_def group_action_def BijGroup_def comp_def
-          group_hom_def group_hom_axioms_def hom_def restrict_def)
-      apply (rule meta_mp[of "\<pi> (inv\<^bsub>G\<^esub> g) \<in> Bij \<bbbD> \<and> \<pi> h \<in> Bij \<bbbD>"])
-       apply (simp add: A_0 A1_3)
-      apply (rule conjI)
-       apply (simp add: A_0 Pi_iff)
-      using A1_3
-      by blast
-    also have H1_4: "... = compose \<bbbD> ((\<pi> (inv\<^bsub>G\<^esub> g)) \<circ> (\<pi> h)) (\<pi> g)"
-      using A1_0
-      apply (clarsimp simp add: data_symm_def group_action_def BijGroup_def comp_def group_hom_def
-          group_hom_axioms_def hom_def restrict_def compose_def)
-      using A_0 A1_3  
-      by (meson data_symm.axioms data_symm_axioms group.inv_closed group_action.element_image)
-    also have H1_5: "... = (\<lambda>d \<in> \<bbbD>. ((\<pi> (inv\<^bsub>G\<^esub> g)) \<circ> (\<pi> h) \<circ> (\<pi> g)) d)"
-      by (simp add: compose_def)
-    have H1_6: "\<And>c. c \<in> C \<Longrightarrow> ((\<pi> h) \<circ> (\<pi> g)) c = (\<pi> g) c"
-      using A1_4
-      by auto
-    have H1_7: "\<And>c. c \<in> C \<Longrightarrow> ((\<pi> (inv\<^bsub>G\<^esub> g)) \<circ> (\<pi> h) \<circ> (\<pi> g)) c = c"
-      using H1_6 A1_0
-      apply (simp add: data_symm_def group_action_def BijGroup_def compose_def group_hom_def
-          group_hom_axioms_def hom_def)
-      by (meson A_0 data_symm.axioms data_symm_axioms group_action.orbit_sym_aux is_subset subsetD)
-    have H1_8: "\<forall>c. c \<in> C \<longrightarrow> \<pi> ((inv\<^bsub>G\<^esub> g) \<otimes>\<^bsub>G\<^esub> h \<otimes>\<^bsub>G\<^esub> g) c = c"
-      using H1_7 H1_5
-      by (metis calculation is_subset restrict_apply' subsetD)
-    have H1_9: "\<phi> ((inv\<^bsub>G\<^esub> g) \<otimes>\<^bsub>G\<^esub> h \<otimes>\<^bsub>G\<^esub> g) x = x"
-      using H1_8
-      by (simp add: H1_1)
-    hence H1_10: "\<phi> ((inv\<^bsub>G\<^esub> g) \<otimes>\<^bsub>G\<^esub> h \<otimes>\<^bsub>G\<^esub> g) x = \<phi> ((inv\<^bsub>G\<^esub> g) \<otimes>\<^bsub>G\<^esub> (h \<otimes>\<^bsub>G\<^esub> g)) x"
-      by (smt (verit, ccfv_SIG) A_0 A1_3 group.inv_closed group.subgroupE(4) group.subgroup_self
-          group_action.composition_rule group_action.element_image group_action_axioms group_hom
-          group_hom.axioms(1) is_in_set)
-    have H1_11: "... = ((\<phi> (inv\<^bsub>G\<^esub> g)) \<circ> (\<phi> (h \<otimes>\<^bsub>G\<^esub> g))) x"
-      using A_0 A1_3 group.subgroupE(4) group.subgroup_self group_action.composition_rule
-        group_action_axioms group_hom group_hom.axioms(1) is_in_set
-      by fastforce
-    have H1_12: "... = ((the_inv_into X (\<phi> g)) \<circ> (\<phi> (h \<otimes>\<^bsub>G\<^esub> g))) x"
-      using A1_1
-      apply (simp add: group_action_def)
-      by (smt (verit) A_0 A1_3 group.inv_closed group.subgroupE(4) group.subgroup_self
-          group_action.element_image group_action.inj_prop group_action.orbit_sym_aux
-          group_action_axioms group_hom.axioms(1) is_in_set the_inv_into_f_f)
-    have H1_13: "((the_inv_into X (\<phi> g)) \<circ> (\<phi> (h \<otimes>\<^bsub>G\<^esub> g))) x = x" 
-      using H1_9 H1_10 H1_11 H1_12
-      by auto
-    hence H1_14: "(\<phi> (h \<otimes>\<^bsub>G\<^esub> g)) x = \<phi> g x"
-      using H1_13
-      by (metis A_0 A1_3 comp_apply composition_rule element_image f_the_inv_into_f inj_prop is_in_set
-          surj_prop)
-    show "\<phi> h (\<phi> g x) = \<phi> g x"
-      using A1_3 A1_2 A_0 H1_14 composition_rule
-      by (simp add: is_in_set)
-  qed
-  show "supports G D \<pi> X \<phi> (\<pi> g ` C) (g \<odot>\<^bsub>\<phi>\<^esub> x)"
-    using supports_axioms
-    apply (clarsimp simp add: supports_def supports_axioms_def)
-    apply (intro conjI)
-    using element_image is_in_set A_0
-      apply blast
-     apply (metis A_0 data_symm_def group_action.surj_prop image_mono is_subset)
-    apply (rule allI; intro impI)
-    apply (rename_tac h)
-    by (simp add: H_0)
-qed
-end
-
-locale nominal = data_symm G D \<pi> + alt_grp_act G X \<phi>
-  for
-    G :: "('grp, 'b) monoid_scheme" and
-    D :: "'D set" ("\<bbbD>") and
-    \<pi> and
-    X :: "'X set" (structure) and
-    \<phi> +
-  assumes
-    is_nominal:
-    "\<And>g x. g \<in> carrier G \<Longrightarrow> x \<in> X \<Longrightarrow> \<exists>C. C \<subseteq> \<bbbD> \<and> finite C \<and> supports G \<bbbD> \<pi> X \<phi> C x"
-
-locale nominal_det_G_aut = det_G_aut + 
-  nominal G D \<pi> A \<phi> + nominal G D \<pi> S \<psi>
-  for
-    D :: "'D set" ("\<bbbD>") and
-    \<pi> 
-
-text \<open>
-The following lemma corresponds to lemma 4.8 from \cite{bojanczyk2014automata}:
-\<close>
-
-lemma (in eq_var_func) supp_el_pres:
-  "supports G D \<pi> X \<phi> C x \<Longrightarrow> supports G D \<pi> Y \<psi> C (f x)"
-  apply (clarsimp simp add: supports_def supports_axioms_def)
-  apply (rule conjI)
-  using eq_var_func_axioms
-   apply (simp add: eq_var_func_def eq_var_func_axioms_def)
-  apply (intro conjI)
-  using is_ext_func_bet 
-   apply blast
-  apply clarify
-  by (metis is_eq_var_func')
-
-lemma (in nominal) support_union_lem:
-  fixes f sup_C col
-  assumes H_f: "f = (\<lambda>x. (SOME C. C \<subseteq> \<bbbD> \<and> finite C \<and> supports G \<bbbD> \<pi> X \<phi> C x))"
-    and H_col: "col \<subseteq> X \<and> finite col"
-    and H_sup_C: "sup_C = \<Union>{Cx. Cx \<in> f ` col}"
-  shows "\<And>x. x \<in> col \<Longrightarrow> sup_C \<subseteq> \<bbbD> \<and> finite sup_C \<and> supports G \<bbbD> \<pi> X \<phi> sup_C x"
-proof - 
-  fix x
-  assume A_0: "x \<in> col"
-  have H_0: "\<And>x. x \<in> X \<Longrightarrow> \<exists>C. C \<subseteq> \<bbbD> \<and> finite C \<and> supports G \<bbbD> \<pi> X \<phi> C x"
-    using nominal_axioms 
-    apply (clarsimp simp add: nominal_def nominal_axioms_def)
-    using stabilizer_one_closed stabilizer_subset 
-    by blast
-  have H_1: "\<And>x. x \<in> col \<Longrightarrow> f x \<subseteq> \<bbbD> \<and> finite (f x) \<and> supports G \<bbbD> \<pi> X \<phi> (f x) x"
-    apply (subst H_f)
-    using someI_ex H_col H_f H_0
-    by (metis (no_types, lifting) in_mono)
-  have H_2: "sup_C \<subseteq> \<bbbD>"
-    using H_1 
-    by (simp add: H_sup_C UN_least)
-  have H_3: "finite sup_C"
-    using H_1 H_col H_sup_C
-    by simp
-  have H_4: "f x \<subseteq> sup_C"
-    using H_1 H_sup_C A_0
-    by blast
-  have H_5: "\<And>g c. \<lbrakk>g \<in> carrier G; (c \<in> sup_C \<longrightarrow> \<pi> g c = c); c \<in> (f x)\<rbrakk> \<Longrightarrow> \<pi> g c = c"
-    using H_4 H_1 A_0
-    by (auto simp add: image_def supports_def supports_axioms_def) 
-  have H_6: "supports G \<bbbD> \<pi> X \<phi> sup_C x"
-    apply (simp add: supports_def supports_axioms_def)
-    apply (intro conjI)
-        apply (simp add: data_symm_axioms)
-    using A_0 H_1 supports.axioms(2)
-       apply fastforce
-    using H_col A_0
-      apply blast
-     apply (rule H_2)
-    apply (clarify)
-    using supports_axioms_def[of G D \<pi> X \<phi> sup_C]
-    apply (clarsimp)
-    using H_1 A_0
-    apply (clarsimp simp add: supports_def supports_axioms_def)
-    using A_0 H_5
-    by presburger
-  show "sup_C \<subseteq> \<bbbD> \<and> finite sup_C \<and> supports G \<bbbD> \<pi> X \<phi> sup_C x"
-    using H_2 H_3 H_6 by auto
-qed
-
-lemma (in nominal) set_of_list_nom:
-  "nominal G D \<pi> (X\<^sup>\<star>) (\<phi>\<^sup>\<star>)"
-proof-
-  have H_0: "\<And>g x. g \<in> carrier G \<Longrightarrow> \<forall>x\<in>set x. x \<in> X \<Longrightarrow>
-            \<exists>C\<subseteq>\<bbbD>. finite C \<and> supports G \<bbbD> \<pi> (X\<^sup>\<star>) (\<phi>\<^sup>\<star>) C x"
-  proof-
-    fix g w
-    assume
-      A1_0: "g \<in> carrier G" and
-      A1_1: "\<forall>x\<in>set w. x \<in> X"
-    have H1_0: "\<And>x. x \<in> X \<Longrightarrow> \<exists>C\<subseteq>\<bbbD>. finite C \<and> supports G \<bbbD> \<pi> X \<phi> C x"
-      using A1_0 is_nominal by force
-    define f where H1_f: "f = (\<lambda>x. (SOME C. C \<subseteq> \<bbbD> \<and>  finite C \<and> supports G \<bbbD> \<pi> X \<phi> C x))"
-    define sup_C :: "'D set " where H1_sup_C: "sup_C = \<Union>{Cx. Cx \<in> f ` set w}"
-    have H1_1: "\<And>x. x \<in> set w \<Longrightarrow> sup_C \<subseteq> \<bbbD> \<and> finite sup_C \<and> supports G \<bbbD> \<pi> X \<phi> sup_C x"
-      apply (rule support_union_lem[where f = f and col = "set w"])
-         apply (rule H1_f)
-      using A1_0
-        apply (simp add: A1_1 subset_code(1))
-       apply (rule H1_sup_C)
-      by simp
-    have H1_2: "supports G \<bbbD> \<pi> (X\<^sup>\<star>) (\<phi>\<^sup>\<star>) sup_C w"
-      apply (clarsimp simp add: supports_def supports_axioms_def simp del: GMN_simps)
-      apply (intro conjI) 
-          apply (simp add: data_symm_axioms)
-      using lists_a_Gset
-         apply (auto)[1]
-        apply (simp add: A1_1 in_listsI)
-      using H1_1 H1_sup_C
-       apply blast
-      apply (rule allI; intro impI)
-      apply clarsimp
-      apply (rule conjI)
-      using H1_1
-      by (auto simp add: supports_def supports_axioms_def map_idI)
-    show "\<exists>C\<subseteq>\<bbbD>. finite C \<and> supports G \<bbbD> \<pi> (X\<^sup>\<star>) (\<phi>\<^sup>\<star>) C w"
-      using nominal_axioms_def
-      apply (clarsimp simp add: nominal_def simp del: GMN_simps)
-      using H1_1 H1_2
-      by (metis Collect_empty_eq H1_sup_C Union_empty empty_iff image_empty infinite_imp_nonempty
-          subset_empty subset_emptyI supports.is_subset)
-  qed
-  show ?thesis
-    apply (clarsimp simp add: nominal_def nominal_axioms_def simp del: GMN_simps)
-    apply (intro conjI)
-    using group.subgroupE(2) group.subgroup_self group_hom group_hom.axioms(1)
-      apply (simp add: data_symm_axioms)
-     apply (rule lists_a_Gset)
-    apply (clarify)
-    by (simp add: H_0 del: GMN_simps)
-qed
-
-subsection \<open>
-Proving the Myhill-Nerode Theorem for Nominal $G$-Automata
-\<close>
-
-context det_G_aut begin
-adhoc_overloading
-  star labels_a_G_set.induced_star_map
-end
-
-lemma (in det_G_aut) input_to_init_eqvar:
-  "eq_var_func G (A\<^sup>\<star>) (\<phi>\<^sup>\<star>) S \<psi> (\<lambda>w\<in>A\<^sup>\<star>. (\<delta>\<^sup>\<star>) i w)"
-proof-
-  have H_0: "\<And>a g. \<lbrakk>\<forall>x\<in>set a. x \<in> A; map (\<phi> g) a \<in> A\<^sup>\<star>; g \<in> carrier G\<rbrakk> \<Longrightarrow>
-            (\<delta>\<^sup>\<star>) i (map (\<phi> g) a) = \<psi> g ((\<delta>\<^sup>\<star>) i a)"
-  proof-
-    fix w g
-    assume
-      A1_0: "\<forall>x\<in>set w. x \<in> A" and
-      A1_1: "map (\<phi> g) w \<in> A\<^sup>\<star>" and
-      A1_2: "g \<in> carrier G"
-    have H1_0: "(\<delta>\<^sup>\<star>) (\<psi> g i) (map (\<phi> g) w) = \<psi> g ((\<delta>\<^sup>\<star>) i w)"
-      using give_input_eq_var
-      apply (clarsimp simp add: eq_var_func_axioms_def eq_var_func_def)
-      using A1_0 A1_1 A1_2 in_listsI is_aut.init_state_is_a_state states_a_G_set.element_image 
-      by (smt (verit, del_insts))
-    have H1_1: "(\<psi> g i) = i"
-      using A1_2 is_aut.init_state_is_a_state init_is_eq_var.is_equivar
-      by force
-    show "(\<delta>\<^sup>\<star>) i (map (\<phi> g) w) = \<psi> g ((\<delta>\<^sup>\<star>) i w)"
-      using H1_0 H1_1
-      by simp
-  qed
-  show ?thesis
-    apply (clarsimp simp add: eq_var_func_def eq_var_func_axioms_def)
-    apply (intro conjI)
-    using labels_a_G_set.lists_a_Gset
-       apply fastforce
-      apply (simp add: states_a_G_set.group_action_axioms del: GMN_simps)
-     apply (simp add: in_listsI is_aut.give_input_closed is_aut.init_state_is_a_state)
-    apply clarify
-    apply (rule conjI; intro impI)
-     apply (simp add: H_0)
-    using labels_a_G_set.surj_prop
-    by fastforce
-qed
-
-lemma (in reach_det_G_aut) input_to_init_surj:
-  "(\<lambda>w\<in>A\<^sup>\<star>. (\<delta>\<^sup>\<star>) i w) ` (A\<^sup>\<star>) = S" 
-  using reach_det_G_aut_axioms
-  apply (clarsimp simp add: image_def reach_det_G_aut_def reach_det_aut_def
-      reach_det_aut_axioms_def)
-  using is_aut.give_input_closed is_aut.init_state_is_a_state
-  by blast
-
-context reach_det_G_aut begin
-adhoc_overloading
-  star labels_a_G_set.induced_star_map
-end
-
-text \<open>
-The following lemma corresponds to proposition 5.1 from \cite{bojanczyk2014automata}:
-\<close>
-
-proposition (in reach_det_G_aut) alpha_nom_imp_states_nom:
-  "nominal G D \<pi> A \<phi> \<Longrightarrow> nominal G D \<pi> S \<psi>"
-proof-
-  assume
-    A_0: "nominal G D \<pi> A \<phi>"
-  have H_0: "\<And>g x. \<lbrakk>g \<in> carrier G; data_symm G D \<pi>; group_action G A \<phi>;
-             \<forall>x. x \<in> A \<longrightarrow> (\<exists>C\<subseteq>D. finite C \<and> supports G D \<pi> A \<phi> C x); x \<in> S\<rbrakk>
-           \<Longrightarrow> \<exists>C\<subseteq>D. finite C \<and> supports G D \<pi> S \<psi> C x"
-  proof -
-    fix g s
-    assume
-      A1_0: "g \<in> carrier G" and
-      A1_1: "data_symm G D \<pi>" and
-      A1_2: "group_action G A \<phi>" and
-      A1_3: "\<forall>x. x \<in> A \<longrightarrow> (\<exists>C\<subseteq>D. finite C \<and> supports G D \<pi> A \<phi> C x)" and
-      A1_4: "s \<in> S"
-    have H1_0: "\<And>x. x \<in> (A\<^sup>\<star>) \<Longrightarrow> \<exists>C\<subseteq>D. finite C \<and> supports G D \<pi> (A\<^sup>\<star>) (\<phi>\<^sup>\<star>) C x"
-      using nominal.set_of_list_nom[of G D \<pi> A \<phi>] A1_2
-      apply (clarsimp simp add: nominal_def)
-      by (metis A1_0 A1_1 A1_3 in_listsI labels_a_G_set.induced_star_map_def nominal_axioms_def)   
-    define f where H1_f: "f = (\<lambda>w\<in>A\<^sup>\<star>. (\<delta>\<^sup>\<star>) i w)"      
-    obtain w where H1_w0: "s = f w" and H1_w1: "w \<in> (A\<^sup>\<star>)"
-      using input_to_init_surj A1_4
-      apply (simp add: H1_f image_def)
-      by (metis is_reachable)
-    obtain C where H1_C: "finite C \<and> supports G D \<pi> (A\<^sup>\<star>) (labels_a_G_set.induced_star_map \<phi>) C w"
-      by (meson H1_0 H1_w0 H1_w1)
-    have H1_2: "supports G D \<pi> S \<psi> C s"
-      apply (subst H1_w0)
-      apply (rule eq_var_func.supp_el_pres[where X = "A\<^sup>\<star>" and \<phi> = "\<phi>\<^sup>\<star>"])
-       apply (subst H1_f)
-       apply (rule det_G_aut.input_to_init_eqvar[of A S i F \<delta> G \<phi> \<psi>])
-      using reach_det_G_aut_axioms
-       apply (simp add: reach_det_G_aut_def) 
-      using H1_C
-      by simp
-    show "\<exists>C\<subseteq>D. finite C \<and> supports G D \<pi> S \<psi> C s"
-      using H1_2 H1_C 
-      by (meson supports.is_subset)
-  qed
-  show ?thesis
-    apply (rule meta_mp[of "(\<exists>g. g \<in> carrier G)"])
-    subgoal 
-      using A_0 apply (clarsimp simp add: nominal_def nominal_axioms_def)
-      apply (rule conjI)
-      subgoal for g
-        by (clarsimp simp add: states_a_G_set.group_action_axioms)
-      apply clarify
-      by (simp add: H_0)
-    by (metis bot.extremum_unique ex_in_conv is_aut.init_state_is_a_state
-        states_a_G_set.stabilizer_one_closed states_a_G_set.stabilizer_subset)
-qed
-
-text \<open>
-The following theorem corresponds to theorem 5.2 from \cite{bojanczyk2014automata}:
-\<close>
-
-theorem (in G_lang) Nom_G_Myhill_Nerode:
-  assumes
-    orb_fin: "finite (orbits G A \<phi>)" and
-    nom: "nominal G D \<pi> A \<phi>"
-  shows
-    Nom_G_Myhill_Nerode_LR: "finite (orbits G MN_equiv ([(\<phi>\<^sup>\<star>)]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>)) \<Longrightarrow>
-     (\<exists>S F :: 'alpha list set set. \<exists>i :: 'alpha list set. \<exists>\<delta>. \<exists>\<psi>.
-     reach_det_G_aut_rec_lang A S i F \<delta> G \<phi> \<psi> L \<and> finite (orbits G S \<psi>)
-     \<and> nominal_det_G_aut A S i F \<delta> G \<phi> \<psi> D \<pi>)" and
-    Nom_G_Myhill_Nerode_RL: "(\<exists>S F :: 's set. \<exists>i :: 's. \<exists>\<delta>. \<exists>\<psi>.
-     det_G_aut_rec_lang A S i F \<delta> G \<phi> \<psi> L \<and> finite (orbits G S \<psi>)
-     \<and> nominal_det_G_aut A S i F \<delta> G \<phi> \<psi> D \<pi>)
-    \<Longrightarrow> finite (orbits G MN_equiv ([(\<phi>\<^sup>\<star>)]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>))"
-proof-
-  assume
-    A_0: "finite (orbits G MN_equiv ([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>))"
-  obtain S F :: "'alpha list set set" and i :: "'alpha list set" and \<delta> \<psi>
-    where H_MN: "reach_det_G_aut_rec_lang A S i F \<delta> G \<phi> \<psi> L \<and> finite (orbits G S \<psi>)"
-    using A_0 orb_fin G_Myhill_Nerode_LR
-    by blast
-  have H_0: "nominal G D \<pi> S \<psi>"
-    using H_MN
-    apply (clarsimp simp del: GMN_simps simp add: reach_det_G_aut_rec_lang_def)
-    using nom reach_det_G_aut.alpha_nom_imp_states_nom 
-    by metis
-  show "\<exists>S F :: 'alpha list set set. \<exists>i :: 'alpha list set. \<exists>\<delta>. \<exists>\<psi>.
-       reach_det_G_aut_rec_lang A S i F \<delta> G \<phi> \<psi> L \<and>
-       finite (orbits G S \<psi>) \<and> nominal_det_G_aut A S i F \<delta> G \<phi> \<psi> D \<pi>"
-    apply (simp add: nominal_det_G_aut_def reach_det_G_aut_rec_lang_def)
-    using nom H_MN H_0
-    apply (clarsimp simp add: reach_det_G_aut_rec_lang_def reach_det_G_aut_def
-        reach_det_aut_axioms_def)
-    by blast
-next
-  assume A0: "\<exists>S F i \<delta> \<psi>. det_G_aut_rec_lang A S i F \<delta> G \<phi> \<psi> L \<and> finite (orbits G S \<psi>)
-                \<and> nominal_det_G_aut A S i F \<delta> G \<phi> \<psi> D \<pi>"
-  show "finite (orbits G MN_equiv ([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>))"
-    using A0 orb_fin
-    by (meson G_Myhill_Nerode_RL)
-qed
+(*
+  Formalization by Cárolos Laméris 
+  
+  Note that
+    bojanczyk2014automata ~
+    Bojańczyk, M., Klin, B., & Lasota, S. (2014).
+    Automata theory in nominal sets.
+    Logical Methods in Computer Science, 10.
+
+    ( https://lmcs.episciences.org/1157 )
+*)
+
+section \<open>
+Myhill-Nerode Theorem for $G$-automata
+\<close>
+text \<open>
+We prove the Myhill-Nerode Theorem for $G$-automata / nominal $G$-automata
+following the proofs from \cite{bojanczyk2014automata} (The standard Myhill-Nerode theorem is also
+proved, as a special case of the $G$-Myhill-Nerode theorem).
+Concretely, we formalize the following results from \cite{bojanczyk2014automata}:
+lemmas: 3.4, 3.5, 3.6, 3.7, 4.8, 4.9;
+proposition: 5.1;
+theorems: 3.8 (Myhill-Nerode for $G$-automata), 5.2 (Myhill-Nerode for nominal $G$-automata).
+
+Throughout this document, we maintain the following convention for isar proofs:
+If we \texttt{obtain} some term \texttt{t} for which some result holds, we name it \texttt{H\_t}.
+An assumption which is an induction hypothesis is named \texttt{A\_IH}
+Assumptions start with an "\texttt{A}" and intermediate results start with a "\texttt{H}".
+Typically we just name them via indexes, i.e. as \texttt{A\_i} and \texttt{H\_j}.
+When encountering nested isar proofs we add an index for how nested the assumption / intermediate
+result is.
+For example if we have an isar proof in an isar proof in an isar proof, we would name assumptions
+of the most nested proof \texttt{A3\_i}.
+\<close>
+
+theory Nominal_Myhill_Nerode
+  imports
+    Main 
+    HOL.Groups
+    HOL.Relation
+    HOL.Fun
+    "HOL-Algebra.Group_Action"
+    "HOL-Library.Adhoc_Overloading"
+    "HOL-Algebra.Elementary_Groups"
+
+begin
+
+text \<open>
+\texttt{GMN\_simps} will contain selection of lemmas / definitions is updated through out
+the document.
+\<close>
+
+named_theorems GMN_simps
+lemmas GMN_simps
+
+text \<open>
+We will use the $\star$-symbol for the set of words of elements of a set, $A^{\star}$, the induced
+group action on the set of words $\phi^{\star}$ and for the extended transition function
+$\delta^{\star}$, thus we introduce the map \texttt{star} and apply \texttt{adhoc\_overloading} to
+get the notation working in all three situations:
+\<close>
+
+consts star :: "'typ1 \<Rightarrow> 'typ2" ("_\<^sup>\<star>" [1000] 999)
+
+abbreviation
+  kleene_star_set :: "'alpha set \<Rightarrow> 'alpha list set"
+  where "kleene_star_set A \<equiv> lists A"
+
+adhoc_overloading
+  star kleene_star_set
+
+text \<open>
+We use $\odot$ to convert between the definition of group actions via group homomoprhisms
+and the more standard infix group action notation.
+We deviate from \cite{bojanczyk2014automata} in that we consider left group actions,
+rather than right group actions:
+\<close>
+
+definition
+  make_op :: "('grp \<Rightarrow> 'X \<Rightarrow> 'X) \<Rightarrow> 'grp \<Rightarrow> 'X \<Rightarrow> 'X" (infixl "(\<odot>\<index>)" 70)
+  where " (\<odot> \<^bsub>\<phi>\<^esub>) \<equiv> (\<lambda>g. (\<lambda>x. \<phi> g x))"
+
+lemmas make_op_def [simp, GMN_simps]
+
+subsection \<open>
+Extending Group Actions
+\<close>
+
+text \<open>
+The following lemma is used for a proof in the locale \texttt{alt\_grp\_act}:
+\<close>
+
+lemma pre_image_lemma:
+  "\<lbrakk>S \<subseteq> T; x \<in> T \<and> f \<in> Bij T; (restrict f S) ` S = S; f x \<in> S\<rbrakk> \<Longrightarrow> x \<in> S"
+  apply (clarsimp simp add: extensional_def subset_eq Bij_def bij_betw_def restrict_def inj_on_def)
+  by (metis imageE)
+
+text \<open>
+The locale \texttt{alt\_grp\_act} is just a renaming of the locale \texttt{group\_action}.
+This was done to obtain more easy to interpret type names and context variables closer
+to the notation of \cite{bojanczyk2014automata}:
+\<close>
+
+locale alt_grp_act = group_action G X \<phi>
+  for
+    G :: "('grp, 'b) monoid_scheme" and
+    X :: "'X set" (structure) and
+    \<phi>
+begin
+
+lemma alt_grp_act_is_left_grp_act:
+  shows "x \<in> X \<Longrightarrow> \<one>\<^bsub>G\<^esub> \<odot>\<^bsub>\<phi>\<^esub> x = x" and
+  "g \<in> carrier G \<Longrightarrow> h \<in> carrier G \<Longrightarrow> x \<in> X \<Longrightarrow> (g \<otimes>\<^bsub>G\<^esub> h) \<odot>\<^bsub>\<phi>\<^esub> x = g \<odot>\<^bsub>\<phi>\<^esub> (h \<odot>\<^bsub>\<phi>\<^esub> x)"
+proof-
+  assume
+    A_0: "x \<in> X"
+  show "\<one>\<^bsub>G\<^esub> \<odot>\<^bsub>\<phi>\<^esub> x = x"
+    using group_action_axioms
+    apply (simp add: group_action_def BijGroup_def)
+    by (metis A_0 id_eq_one restrict_apply')
+next
+  assume 
+    A_0: "g \<in> carrier G" and
+    A_1: "h \<in> carrier G" and
+    A_2: "x \<in> X"
+  show "g \<otimes>\<^bsub>G\<^esub> h \<odot>\<^bsub>\<phi>\<^esub> x = g \<odot>\<^bsub>\<phi>\<^esub> (h \<odot>\<^bsub>\<phi>\<^esub> x)"
+    using group_action_axioms
+    apply (simp add: group_action_def group_hom_def group_hom_axioms_def hom_def BijGroup_def)
+    using composition_rule A_0 A_1 A_2
+    by auto
+qed
+
+definition
+  induced_star_map :: "('grp \<Rightarrow> 'X \<Rightarrow>'X) \<Rightarrow> 'grp \<Rightarrow> 'X list \<Rightarrow> 'X list"
+  where "induced_star_map func = (\<lambda>g\<in>carrier G. (\<lambda>lst \<in> (lists X). map (func g) lst))"
+
+text \<open>
+Because the adhoc overloading is used within a locale, isues will be encountered later due to there
+being multiple instances of the locale \texttt{alt\_grp\_act} in a single context:
+\<close>
+
+adhoc_overloading
+  star induced_star_map
+
+definition 
+  induced_quot_map ::
+  "'Y set \<Rightarrow> ('grp \<Rightarrow> 'Y \<Rightarrow> 'Y) \<Rightarrow> ('Y \<times>'Y) set \<Rightarrow> 'grp \<Rightarrow> 'Y set \<Rightarrow> 'Y set" ("[_]\<^sub>_\<index>" 60)
+  where "([ func ]\<^sub>R \<^bsub>S\<^esub>) = (\<lambda>g\<in>carrier G. (\<lambda>x \<in> (S // R). R `` {(func g) (SOME z. z\<in>x)}))"
+
+lemmas induced_star_map_def [simp, GMN_simps]
+  induced_quot_map_def [simp, GMN_simps]
+
+lemma act_maps_n_distrib:
+  "\<forall>g\<in>carrier G. \<forall>w\<in>X\<^sup>\<star>. \<forall>v\<in>X\<^sup>\<star>. (\<phi>\<^sup>\<star>) g (w @ v) = ((\<phi>\<^sup>\<star>) g w) @ ((\<phi>\<^sup>\<star>) g v)"
+  by (auto simp add: group_action_def group_hom_def group_hom_axioms_def hom_def)
+
+lemma triv_act:
+  "a \<in> X \<Longrightarrow> (\<phi> \<one>\<^bsub>G\<^esub>) a = a"
+  using group_hom.hom_one[of "G" "BijGroup X" "\<phi>"] group_BijGroup[where S = "X"]
+  apply (clarsimp simp add: group_action_def group_hom_def group_hom_axioms_def BijGroup_def)
+  by (metis id_eq_one restrict_apply')
+
+lemma triv_act_map:
+  "\<forall>w\<in>X\<^sup>\<star>. ((\<phi>\<^sup>\<star>) \<one>\<^bsub>G\<^esub>) w = w"
+  using triv_act
+  apply clarsimp
+  apply (rule conjI; rule impI)
+   apply clarify
+  using map_idI
+   apply metis
+  using group.subgroup_self group_hom group_hom.axioms(1) subgroup.one_closed
+  by blast
+
+proposition lists_a_Gset:
+  "alt_grp_act G (X\<^sup>\<star>) (\<phi>\<^sup>\<star>)"
+proof-
+  have H_0: "\<And>g. g \<in> carrier G \<Longrightarrow>
+    restrict (map (\<phi> g)) (kleene_star_set X) \<in> carrier (BijGroup (kleene_star_set X))"
+  proof-
+    fix g
+    assume
+      A1_0: "g \<in> carrier G"
+    from A1_0 have H1_0: "inj_on (\<lambda>x. if x \<in> lists X then map (\<phi> g) x else undefined) (lists X)"
+      apply (clarsimp simp add: inj_on_def)
+      by (metis (mono_tags, lifting) inj_onD inj_prop list.inj_map_strong)
+    from A1_0 have H1_1: "\<And>y z. \<forall>x\<in>set y. x \<in> X \<Longrightarrow> z \<in> set y \<Longrightarrow> \<phi> g z \<in> X"
+      using element_image
+      by blast
+    have H1_2: "(inv \<^bsub>G\<^esub> g) \<in> carrier G"
+      by (meson A1_0 group.inv_closed group_hom group_hom.axioms(1))
+    have H1_3: "\<And>x. x \<in> lists X \<Longrightarrow>
+    map (comp (\<phi> g) (\<phi> (inv \<^bsub>G\<^esub> g))) x = map (\<phi> (g \<otimes>\<^bsub>G\<^esub> (inv \<^bsub>G\<^esub> g))) x"
+      using alt_grp_act_axioms
+      apply (simp add: alt_grp_act_def group_action_def group_hom_def group_hom_axioms_def hom_def
+          BijGroup_def)
+      apply (rule meta_mp[of "\<And>x. x \<in> carrier G \<Longrightarrow> \<phi> x \<in> Bij X"])
+       apply (metis A1_0 H1_2 composition_rule in_lists_conv_set)
+      by blast
+    from H1_2 have H1_4: "\<And>x. x \<in> lists X \<Longrightarrow> map (\<phi> (inv \<^bsub>G\<^esub> g)) x \<in> lists X"
+      using surj_prop
+      by fastforce
+    have H1_5: "\<And>y. \<forall>x\<in>set y. x \<in> X \<Longrightarrow> y \<in> map (\<phi> g) ` lists X"
+      apply (simp add: image_def)
+      using H1_3 H1_4 
+      by (metis A1_0 group.r_inv group_hom group_hom.axioms(1) in_lists_conv_set map_idI map_map
+          triv_act)
+    show "restrict (map (\<phi> g)) (lists X) \<in> carrier (BijGroup (lists X))"
+      apply (clarsimp simp add: restrict_def BijGroup_def Bij_def
+          extensional_def bij_betw_def) 
+      apply (rule conjI)
+      using H1_0
+       apply simp
+      using H1_1 H1_5
+      by (auto simp add: image_def) 
+  qed
+  have H_1: "\<And>x y. \<lbrakk>x \<in> carrier G; y \<in> carrier G; x \<otimes>\<^bsub>G\<^esub> y \<in> carrier G\<rbrakk> \<Longrightarrow>
+           restrict (map (\<phi> (x \<otimes>\<^bsub>G\<^esub> y))) (kleene_star_set X) =
+           restrict (map (\<phi> x)) (kleene_star_set X) \<otimes>\<^bsub>BijGroup (kleene_star_set X)\<^esub>
+           restrict (map (\<phi> y)) (kleene_star_set X)"
+  proof-
+    fix x y
+    assume
+      A1_0: "x \<in> carrier G" and
+      A1_1: " y \<in> carrier G" and
+      A1_2: "x \<otimes>\<^bsub>G\<^esub> y \<in> carrier G"
+    have H1_0: "\<And>z. z \<in> carrier G \<Longrightarrow>
+       bij_betw (\<lambda>x. if x \<in> lists X then map (\<phi> z) x else undefined) (lists X) (lists X)"
+      using \<open>\<And>g. g \<in> carrier G \<Longrightarrow> restrict (map (\<phi> g)) (lists X) \<in> carrier (BijGroup (lists X))\<close>
+      by (auto simp add: BijGroup_def Bij_def bij_betw_def inj_on_def)
+    from A1_1 have H1_1: "\<And>lst. lst \<in> lists X \<Longrightarrow> (map (\<phi> y)) lst \<in> lists X"
+      by (metis group_action.surj_prop group_action_axioms lists_image rev_image_eqI)
+    have H1_2: "\<And>a. a \<in> lists X \<Longrightarrow> map (\<lambda>xb.
+    if xb \<in> X
+    then \<phi> x ((\<phi> y) xb)
+    else undefined) a = map (\<phi> x) (map (\<phi> y) a)"
+      by auto
+    have H1_3: "(\<lambda>xa. if xa \<in> lists X then map (\<phi> (x \<otimes>\<^bsub>G\<^esub> y)) xa else undefined) =
+    compose (lists X) (\<lambda>xa. if xa \<in> lists X then map (\<phi> x) xa else undefined)
+    (\<lambda>x. if x \<in> lists X then map (\<phi> y) x else undefined)"
+      using alt_grp_act_axioms
+      apply (clarsimp simp add: compose_def alt_grp_act_def group_action_def
+          group_hom_def group_hom_axioms_def hom_def BijGroup_def restrict_def)
+      using A1_0 A1_1 H1_2 H1_1 bij_prop0
+      by auto
+    show "restrict (map (\<phi> (x \<otimes>\<^bsub>G\<^esub> y))) (lists X) =
+  restrict (map (\<phi> x)) (lists X) \<otimes>\<^bsub>BijGroup (lists X)\<^esub>
+  restrict (map (\<phi> y)) (lists X)"
+      apply (clarsimp simp add: restrict_def BijGroup_def Bij_def extensional_def)
+      apply (simp add: H1_3)
+      using A1_0 A1_1 H1_0
+      by auto
+  qed
+  show "alt_grp_act G (X\<^sup>\<star>) (\<phi>\<^sup>\<star>)"
+    apply (clarsimp simp add: alt_grp_act_def group_action_def group_hom_def group_hom_axioms_def)
+    apply (intro conjI)
+    using group_hom group_hom_def
+      apply (auto)[1]
+     apply (simp add: group_BijGroup)
+    apply (clarsimp simp add: hom_def)
+    apply (intro conjI; clarify)
+     apply (rule H_0)
+     apply simp
+    apply (rule conjI; rule impI)
+     apply (rule H_1)
+       apply simp+
+    apply (rule meta_mp[of "\<And>x y. x \<in> carrier G \<Longrightarrow>
+    y \<in> carrier G \<Longrightarrow> x \<otimes>\<^bsub>G\<^esub> y \<in> carrier G"])
+     apply blast
+    by (meson group.subgroup_self group_hom group_hom.axioms(1) subgroup.m_closed)
+qed
+end
+
+lemma alt_group_act_is_grp_act [simp, GMN_simps]:
+  "alt_grp_act = group_action"
+  using alt_grp_act_def
+  by blast
+
+lemma prod_group_act:
+  assumes
+    grp_act_A: "alt_grp_act G A \<phi>" and
+    grp_act_B: "alt_grp_act G B \<psi>"
+  shows "alt_grp_act G (A\<times>B) (\<lambda>g\<in>carrier G. \<lambda>(a, b) \<in> (A \<times> B). (\<phi> g a, \<psi> g b))"
+  apply (simp add: alt_grp_act_def group_action_def group_hom_def)
+  apply (intro conjI)
+  subgoal
+    using grp_act_A grp_act_B
+    by (auto simp add: alt_grp_act_def group_action_def group_hom_def)
+  subgoal
+    using grp_act_A grp_act_B
+    by (auto simp add: alt_grp_act_def group_action_def group_hom_def group_BijGroup)
+  apply (clarsimp simp add: group_hom_axioms_def hom_def BijGroup_def)
+  apply (intro conjI; clarify)
+  subgoal for g
+    apply (clarsimp simp add: Bij_def bij_betw_def inj_on_def restrict_def extensional_def)
+    apply (intro conjI)
+    using grp_act_A
+     apply (simp add: alt_grp_act_def group_action_def group_hom_def group_hom_axioms_def
+        BijGroup_def hom_def Pi_def compose_def Bij_def bij_betw_def inj_on_def)
+    using grp_act_B
+     apply (simp add: alt_grp_act_def group_action_def group_hom_def group_hom_axioms_def
+        BijGroup_def hom_def Pi_def compose_def Bij_def bij_betw_def inj_on_def)
+    apply (rule meta_mp[of "\<phi> g \<in> Bij A \<and> \<psi> g \<in> Bij B"])
+     apply (clarsimp simp add: Bij_def bij_betw_def)
+    using grp_act_A grp_act_B
+     apply (simp add: alt_grp_act_def group_action_def group_hom_def group_hom_axioms_def
+        BijGroup_def hom_def Pi_def Bij_def)
+    using grp_act_A grp_act_B
+     apply (clarsimp simp add: compose_def restrict_def image_def alt_grp_act_def
+        group_action_def group_hom_def group_hom_axioms_def BijGroup_def hom_def Pi_def Bij_def
+        bij_betw_def)
+     apply (rule subset_antisym)
+      apply blast+
+    by (metis alt_group_act_is_grp_act group_action.bij_prop0 grp_act_A grp_act_B)
+  apply (intro conjI; intro impI)
+   apply (clarify)
+   apply (intro conjI; intro impI)
+    apply (rule conjI)
+  subgoal for x y
+    apply unfold_locales
+    apply (clarsimp simp add: Bij_def compose_def restrict_def bij_betw_def)
+    apply (rule extensionalityI[where A = "A \<times> B"])
+      apply (clarsimp simp add: extensional_def)
+    using grp_act_A grp_act_B
+     apply (simp add: alt_grp_act_def group_action_def group_hom_def group_hom_axioms_def
+        BijGroup_def hom_def Pi_def Bij_def compose_def extensional_def)
+    apply (simp add: fun_eq_iff; rule conjI; rule impI)
+    using group_action.composition_rule[of G A \<phi>] group_action.composition_rule[of G B \<psi>] grp_act_A 
+      grp_act_B   
+     apply force
+    by blast
+    apply (simp add: \<open>\<And>g. g \<in> carrier G \<Longrightarrow> (\<lambda>(a, b)\<in>A \<times> B. (\<phi> g a, \<psi> g b)) \<in> Bij (A \<times> B)\<close>)
+   apply (simp add: \<open>Group.group G\<close> group.subgroup_self subgroup.m_closed)
+  by (simp add: \<open>\<And>g. g \<in> carrier G \<Longrightarrow> (\<lambda>(a, b)\<in>A \<times> B. (\<phi> g a, \<psi> g b)) \<in> Bij (A \<times> B)\<close>)+
+
+subsection \<open>
+Equivariance and Quotient Actions
+\<close>
+
+locale eq_var_subset = alt_grp_act G X \<phi>
+  for
+    G :: "('grp, 'b) monoid_scheme" and
+    X :: "'X set" (structure) and
+    \<phi> +
+  fixes
+    Y
+  assumes
+    is_subset: "Y \<subseteq> X" and
+    is_equivar: "\<forall>g\<in>carrier G. (\<phi> g) ` Y = Y" 
+
+lemma (in alt_grp_act) eq_var_one_direction:
+  "\<And>Y. Y \<subseteq> X \<Longrightarrow> \<forall>g\<in>carrier G. (\<phi> g) ` Y \<subseteq> Y \<Longrightarrow> eq_var_subset G X \<phi> Y"
+proof-
+  fix Y 
+  assume
+    A_0: "Y \<subseteq> X" and
+    A_1: "\<forall>g\<in>carrier G. (\<phi> g) ` Y \<subseteq> Y"
+  have H_0: "\<And>g. g \<in> carrier G \<Longrightarrow> (inv\<^bsub>G\<^esub> g) \<in> carrier G"
+    by (meson group.inv_closed group_hom group_hom.axioms(1))
+  hence H_1: "\<And>g y. g \<in> carrier G \<Longrightarrow> y \<in> Y \<Longrightarrow> (\<phi> (inv\<^bsub>G\<^esub> g)) y \<in> Y"
+    using A_1  
+    by (simp add: image_subset_iff)
+  have H_2: "\<And>g y. g \<in> carrier G \<Longrightarrow> y \<in> Y \<Longrightarrow> \<phi> g ((\<phi> (inv\<^bsub>G\<^esub> g)) y) = y"
+    by (metis A_0 bij_prop1 orbit_sym_aux subsetD)
+  show "eq_var_subset G X \<phi> Y"
+    apply (simp add: eq_var_subset_def eq_var_subset_axioms_def)
+    apply (intro conjI)
+      apply (simp add: group_action_axioms)
+     apply (rule A_0)
+    apply (clarify)
+    apply (rule subset_antisym)
+    using A_1
+     apply simp
+    apply (simp add: image_def)
+    apply (rule subsetI)
+    apply clarify
+    using H_1 H_2
+    by metis
+qed
+
+text \<open>
+The following lemmas are used for proofs in the locale \texttt{eq\_var\_rel}:
+\<close>
+
+lemma some_equiv_class_id:
+  "\<lbrakk>equiv X R; w \<in> X // R; x \<in> w\<rbrakk> \<Longrightarrow> R `` {x} = R `` {SOME z. z \<in> w}"
+  by (smt (z3) Eps_cong equiv_Eps_in equiv_class_eq_iff quotient_eq_iff)
+
+lemma nested_somes:
+  "\<lbrakk>equiv X R; w \<in> X // R\<rbrakk> \<Longrightarrow> (SOME z. z \<in> w) = (SOME z. z \<in> R``{(SOME z'. z' \<in> w)})"
+  by (metis proj_Eps proj_def)
+
+locale eq_var_rel = alt_grp_act G X \<phi>
+  for
+    G :: "('grp, 'b) monoid_scheme" and
+    X :: "'X set" (structure) and
+    \<phi> +
+  fixes R 
+  assumes
+    is_subrel:
+    "R \<subseteq> X \<times> X" and
+    is_eq_var_rel:
+    "\<And>a b. (a, b) \<in> R \<Longrightarrow> \<forall>g \<in> carrier G. (g \<odot>\<^bsub>\<phi>\<^esub> a, g \<odot>\<^bsub>\<phi>\<^esub> b) \<in> R"
+begin
+
+lemma is_eq_var_rel' [simp, GMN_simps]:
+  "\<And>a b. (a, b) \<in> R \<Longrightarrow> \<forall>g \<in> carrier G. ((\<phi> g) a, (\<phi> g) b) \<in> R"
+  using is_eq_var_rel
+  by auto
+
+lemma is_eq_var_rel_rev:
+  "a \<in> X \<Longrightarrow> b \<in> X \<Longrightarrow> g \<in> carrier G \<Longrightarrow> (g \<odot>\<^bsub>\<phi>\<^esub> a, g \<odot>\<^bsub>\<phi>\<^esub> b) \<in> R \<Longrightarrow> (a, b) \<in> R"
+proof -
+  assume
+    A_0: "(g \<odot>\<^bsub>\<phi>\<^esub> a, g \<odot>\<^bsub>\<phi>\<^esub> b) \<in> R" and
+    A_1: "a \<in> X" and
+    A_2: "b \<in> X" and
+    A_3: "g \<in> carrier G" 
+  have H_0: " group_action G X \<phi>" and
+    H_1: "R \<subseteq> X \<times> X" and
+    H_2: "\<And>a b g. (a, b) \<in> R \<Longrightarrow> g \<in> carrier G \<Longrightarrow> (\<phi> g a, \<phi> g b) \<in> R" 
+    by (simp add: group_action_axioms is_subrel)+
+  from H_0 have H_3: "group G"
+    by (auto simp add: group_action_def group_hom_def)
+  have H_4: "(\<phi> (inv\<^bsub>G\<^esub> g) (\<phi> g a), \<phi> (inv\<^bsub>G\<^esub> g) (\<phi> g b)) \<in> R"
+    apply (rule H_2)
+    using A_0 apply simp
+    by (simp add: A_3 H_3)
+  from H_3 A_3 have H_5: "(inv\<^bsub>G\<^esub> g) \<in> carrier G"
+    by auto
+  hence H_6: "\<And>e. e \<in> X \<Longrightarrow> \<phi> (inv\<^bsub>G\<^esub> g) (\<phi> g e) = \<phi> ((inv\<^bsub>G\<^esub> g) \<otimes>\<^bsub>G\<^esub> g) e"
+    using H_0 A_3 group_action.composition_rule 
+    by fastforce
+  hence H_7: "\<And>e. e \<in> X \<Longrightarrow> \<phi> (inv\<^bsub>G\<^esub> g) (\<phi> g e) = \<phi> \<one>\<^bsub>G\<^esub> e"
+    using H_3 A_3 group.l_inv
+    by fastforce 
+  hence H_8: "\<And>e. e \<in> X \<Longrightarrow> \<phi> (inv\<^bsub>G\<^esub> g) (\<phi> g e) = e"
+    using H_0
+    by (simp add: A_3 group_action.orbit_sym_aux)
+  thus "(a, b) \<in> R"
+    using A_1 A_2 H_4
+    by simp
+qed
+
+lemma equiv_equivar_class_some_eq:
+  assumes
+    A_0: "equiv X R" and
+    A_1: "w \<in> X // R" and
+    A_2: "g \<in> carrier G"
+  shows "([ \<phi> ]\<^sub>R) g w = R `` {(SOME z'. z' \<in> \<phi> g ` w)}"
+proof-
+  obtain z where H_z: "w = R `` {z} \<and> z \<in> w"
+    by (metis A_0 A_1 equiv_class_self quotientE)
+  have H_0: "\<And>x. (\<phi> g z, x) \<in> R \<Longrightarrow> x \<in> \<phi> g ` {y. (z, y) \<in> R}"
+  proof-
+    fix y
+    assume
+      A1_0: "(\<phi> g z, y) \<in> R"
+    obtain y' where H2_y': "y' = \<phi> (inv\<^bsub>G\<^esub> g) y \<and> y' \<in> X" 
+      using eq_var_rel_axioms
+      apply (clarsimp simp add: eq_var_rel_def group_action_def group_hom_def) 
+      by (meson A_0 eq_var_rel_axioms A_2 A1_0 equiv_class_eq_iff eq_var_rel.is_eq_var_rel
+          group.inv_closed element_image) 
+    from A_1 A_2 H2_y' have H2_0: "\<phi> g y' = y"
+      apply (clarsimp simp add: eq_var_rel_def eq_var_rel_axioms_def)
+      using A_2 A1_0 group_action.bij_prop1[where G = "G" and E = "X" and \<phi> = "\<phi>"]
+      by (metis A_0 alt_group_act_is_grp_act alt_grp_act_axioms equiv_class_eq_iff orbit_sym_aux)
+    from A_1 A_2 A1_0 have H2_1: "(z, y') \<in> R"
+      by (metis H2_0 A_0 A_2 H2_y' H_z equiv_class_eq_iff is_eq_var_rel_rev
+          quotient_eq_iff make_op_def)
+    thus "y \<in> \<phi> g ` {v. (z, v) \<in> R}" 
+      using H2_0
+      by (auto simp add: image_def)
+  qed
+  have H_1: "\<phi> g ` (R `` {z}) = R `` {\<phi> g z}"
+    apply (clarsimp simp add: Relation.Image_def)
+    apply (rule subset_antisym; simp add: Set.subset_eq; rule allI; rule impI)
+    using eq_var_rel_axioms A_2 eq_var_rel.is_eq_var_rel
+     apply simp
+    by (simp add: H_0)
+  have H_2: "\<phi> g ` w \<in> X // R"
+    using eq_var_rel_axioms A_1 A_2 H_1 
+    by (metis A_0 H_z equiv_class_eq_iff quotientI quotient_eq_iff element_image)
+  thus "([\<phi>]\<^sub>R) g w = R `` {SOME z'. z' \<in> \<phi> g ` w}"
+    using A_0 A_1 A_2
+    apply (clarsimp simp add: Image_def)
+    apply (intro subset_antisym)
+     apply (clarify)
+    using A_0 H_z imageI insert_absorb insert_not_empty some_in_eq some_equiv_class_id 
+     apply (smt (z3) A_1 Eps_cong Image_singleton_iff equiv_Eps_in)
+    apply (clarify)
+    by (smt (z3) Eps_cong equiv_Eps_in image_iff in_quotient_imp_closed quotient_eq_iff)
+qed
+
+lemma ec_er_closed_under_action:
+  assumes
+    A_0: "equiv X R" and
+    A_1: "g \<in> carrier G" and
+    A_2: "w \<in> X//R"
+  shows "\<phi> g ` w \<in> X // R"
+proof-
+  obtain z where H_z: "R `` {z} = w \<and> z \<in> X"
+    by (metis A_2 quotientE)
+  have H_0: "equiv X R \<Longrightarrow> g \<in> carrier G \<Longrightarrow> w \<in> X // R \<Longrightarrow>
+      {y. (\<phi> g z, y) \<in> R} \<subseteq> {y. \<exists>x. (z, x) \<in> R \<and> y = \<phi> g x}"
+  proof (clarify)
+    fix x
+    assume
+      A1_0: "equiv X R" and
+      A1_1: "g \<in> carrier G" and
+      A1_2: "w \<in> X // R" and
+      A1_3: "(\<phi> g z, x) \<in> R"
+    obtain x' where H2_x': "x = \<phi> g x' \<and> x' \<in> X" 
+      using group_action_axioms
+      by (metis A1_1 is_subrel A1_3 SigmaD2 group_action.bij_prop1 subsetD)
+    thus "\<exists>y. (z, y) \<in> R \<and> x = \<phi> g y"
+      using is_eq_var_rel_rev[where g = "g" and a = "z" and b = "x'"] A1_3
+      by (auto simp add: eq_var_rel_def eq_var_rel_axioms_def A1_1 A1_2 group_action_axioms H_z
+          H2_x')
+  qed
+  have H_1: "\<phi> g ` R `` {z} = R `` {\<phi> g z}"
+    using A_0 A_1 A_2
+    apply (clarsimp simp add: eq_var_rel_axioms_def eq_var_rel_def
+        Image_def image_def)
+    apply (intro subset_antisym)
+     apply (auto)[1]
+    by (rule H_0)
+  thus "\<phi> g ` w \<in> X // R"
+    using H_1 H_z
+    by (metis A_1 quotientI element_image)
+qed
+
+text \<open>
+The following lemma corresponds to the first part of lemma 3.5 from \cite{bojanczyk2014automata}:
+\<close>
+
+lemma quot_act_wd:
+  "\<lbrakk>equiv X R; x \<in> X; g \<in> carrier G\<rbrakk> \<Longrightarrow> g \<odot>\<^bsub>[\<phi>]\<^sub>R\<^esub> (R `` {x}) = (R `` {g \<odot>\<^bsub>\<phi>\<^esub> x})"
+  apply (clarsimp simp add: eq_var_rel_def eq_var_rel_axioms_def)
+  apply (rule conjI; rule impI)
+   apply (smt (verit, best) Eps_cong Image_singleton_iff eq_var_rel.is_eq_var_rel'
+      eq_var_rel_axioms equiv_Eps_in equiv_class_eq)
+  by (simp add: quotientI)+
+
+text \<open>
+The following lemma corresponds to the second part of lemma 3.5 from \cite{bojanczyk2014automata}:
+\<close>
+
+lemma quot_act_is_grp_act:
+  "equiv X R \<Longrightarrow> alt_grp_act G (X // R) ([\<phi>]\<^sub>R)"
+proof-
+  assume A_0: "equiv X R"
+  have H_0: "\<And>x. Group.group G \<Longrightarrow>
+         Group.group (BijGroup X) \<Longrightarrow>
+         R \<subseteq> X \<times> X \<Longrightarrow>
+         \<phi> \<in> carrier G \<rightarrow> carrier (BijGroup X) \<Longrightarrow>
+         \<forall>x\<in>carrier G. \<forall>y\<in>carrier G. \<phi> (x \<otimes>\<^bsub>G\<^esub> y) = \<phi> x \<otimes>\<^bsub>BijGroup X\<^esub> \<phi> y \<Longrightarrow>
+         x \<in> carrier G \<Longrightarrow> (\<lambda>xa\<in>X // R. R `` {\<phi> x (SOME z. z \<in> xa)}) \<in> carrier (BijGroup (X // R))"
+  proof-
+    fix g
+    assume
+      A1_0: "Group.group G" and
+      A1_1: "Group.group (BijGroup X)" and
+      A1_2: "\<phi> \<in> carrier G \<rightarrow> carrier (BijGroup X)" and
+      A1_3: "\<forall>x\<in>carrier G. \<forall>y\<in>carrier G. \<phi> (x \<otimes>\<^bsub>G\<^esub> y) = \<phi> x \<otimes>\<^bsub>BijGroup X\<^esub> \<phi> y" and
+      A1_4: "g \<in> carrier G"
+    have H_0: "group_action G X \<phi>"
+      apply (clarsimp simp add: group_action_def group_hom_def group_hom_axioms_def)
+      apply (simp add: A1_0 A1_1)+
+      apply (simp add: hom_def)
+      apply (rule conjI)
+      using A1_2
+       apply blast
+      by (simp add: A1_3)
+    have H1_0: "\<And>x y. \<lbrakk>x \<in> X // R; y \<in> X // R; R `` {\<phi> g (SOME z. z \<in> x)} =
+                                                     R `` {\<phi> g (SOME z. z \<in> y)}\<rbrakk> \<Longrightarrow> x \<subseteq> y"
+    proof (clarify; rename_tac a)
+      fix x y a
+      assume
+        A2_0: "x \<in> X // R" and
+        A2_1: "y \<in> X // R" and
+        A2_2: "R `` {\<phi> g (SOME z. z \<in> x)} = R `` {\<phi> g (SOME z. z \<in> y)}" and
+        A2_3: "a \<in> x"
+      obtain b where H2_b: "R `` {b} = y \<and> b \<in> X" 
+        by (metis A2_1 quotientE)
+      obtain a' b' where H2_a'_b': "a'\<in> x \<and> b'\<in> y \<and> R `` {\<phi> g a'} = R `` {\<phi> g b'}"
+        by (metis A_0 A2_1 A2_2 A2_3 equiv_Eps_in some_eq_imp)
+      from H2_a'_b' have H2_2: "(\<phi> g a', \<phi> g b') \<in> R"
+        by (metis A_0 A1_4 A2_1 Image_singleton_iff eq_var_rel.is_eq_var_rel' eq_var_rel_axioms
+            quotient_eq_iff)
+      hence H2_0: "(\<phi> (inv\<^bsub>G\<^esub> g) (\<phi> g a'), \<phi> (inv\<^bsub>G\<^esub> g) (\<phi> g b')) \<in> R"
+        by (simp add: A1_0 is_eq_var_rel A1_4)
+      have H2_1: "a' \<in> X \<and> b' \<in> X" 
+        using A_0 A2_0 A2_1 H2_a'_b' in_quotient_imp_subset
+        by blast
+      hence H2_2: "(a', b') \<in> R"
+        using H2_0
+        by (metis A1_4 H_0 group_action.orbit_sym_aux)
+      have H2_3: "(a, a') \<in> R" 
+        by (meson A_0 A2_0 A2_3 H2_a'_b' quotient_eq_iff)
+      hence H2_4: "(b', a) \<in> R" 
+        using H2_2
+        by (metis A_0 A2_0 A2_1 A2_3 H2_a'_b' quotient_eqI quotient_eq_iff)
+      thus "a \<in> y"
+        by (metis A_0 A2_1 H2_a'_b' in_quotient_imp_closed)
+    qed
+    have H1_1: "\<And>x. x \<in> X // R \<Longrightarrow> \<exists>xa\<in>X // R. x = R `` {\<phi> g (SOME z. z \<in> xa)}"
+    proof -
+      fix x
+      assume
+        A2_0: "x \<in> X // R"
+      have H2_0: "\<And>e. R `` {e} \<in> X // R \<Longrightarrow> R `` {e} \<subseteq> R `` {\<phi> g (\<phi> (inv\<^bsub>G\<^esub> g) e)}"
+      proof (rule subsetI)
+        fix e y
+        assume
+          A3_0: "R `` {e} \<in> X // R" and
+          A3_1: "y \<in> R `` {e}"
+        have H3_0: "y \<in> X" 
+          using A3_1 is_subrel
+          by blast
+        from H_0 have H3_1: "\<phi> g (\<phi> (inv\<^bsub>G\<^esub> g) y) = y"
+          by (metis (no_types, lifting) A1_0 A1_4 H3_0 group.inv_closed group.inv_inv
+              group_action.orbit_sym_aux)
+        from A3_1 have H3_2: "(e, y) \<in> R"
+          by simp
+        hence H3_3: "((\<phi> (inv\<^bsub>G\<^esub> g) e), (\<phi> (inv\<^bsub>G\<^esub> g) y)) \<in> R"
+          using is_eq_var_rel A1_4 A1_0
+          by simp
+        hence H3_4: "(\<phi> g (\<phi> (inv\<^bsub>G\<^esub> g) e), \<phi> g (\<phi> (inv\<^bsub>G\<^esub> g) y)) \<in> R"
+          using is_eq_var_rel A1_4 A1_0
+          by simp
+        hence H3_5: "(\<phi> g (\<phi> (inv\<^bsub>G\<^esub> g) e), y) \<in> R"
+          using H3_1
+          by simp
+        thus "y \<in> R `` {\<phi> g (\<phi> (inv\<^bsub>G\<^esub> g) e)}"
+          by simp
+      qed
+      hence H2_1: "\<And>e. R `` {e} \<in> X // R \<Longrightarrow> R `` {e} = R `` {\<phi> g (\<phi> (inv\<^bsub>G\<^esub> g) e)}" 
+        by (metis A_0 proj_def proj_in_iff equiv_class_eq_iff subset_equiv_class)
+      have H2_2: "\<And>e f. R `` {e} \<in> X // R \<Longrightarrow> R `` {f} \<in> X // R \<Longrightarrow>
+        R `` {e} = R `` {f} \<Longrightarrow> \<forall>f'\<in> R `` {f}. R `` {e} = R `` {f'}"
+        by (metis A_0 Image_singleton_iff equiv_class_eq)
+      have H2_3: "x \<in> X // R \<Longrightarrow> \<exists>e\<in>X. x = R `` {e}" 
+        by (meson quotientE)
+      have H2_4: "\<And>e. R `` {e} \<in> X // R \<Longrightarrow> R `` {e} = R `` {\<phi> g (\<phi> (inv\<^bsub>G\<^esub> g) e)} \<and>
+        (\<phi> (inv\<^bsub>G\<^esub> g) e) \<in> R `` {\<phi> (inv\<^bsub>G\<^esub> g) e}"
+        by (smt (z3) A_0 A1_0 A1_4 H_0 H2_1 Image_singleton_iff equiv_class_eq_iff
+            group.inv_closed group_action.element_image in_quotient_imp_non_empty
+            subset_empty subset_emptyI)
+      have H2_5: "\<And>e. R `` {e} \<in> X // R \<Longrightarrow> \<forall>z\<in>R `` {\<phi> (inv\<^bsub>G\<^esub> g) e}. (\<phi> (inv\<^bsub>G\<^esub> g) e, z) \<in> R"
+        by simp
+      hence H2_6: "\<And>e. R `` {e} \<in> X // R \<Longrightarrow>
+        \<forall>z\<in>R `` {\<phi> (inv\<^bsub>G\<^esub> g) e}. (\<phi> g (\<phi> (inv\<^bsub>G\<^esub> g) e), \<phi> g z) \<in> R"
+        using is_eq_var_rel' A1_4 A1_0
+        by blast
+      hence H2_7: "\<And>e. R `` {e} \<in> X // R \<Longrightarrow> \<forall>z\<in>R `` {\<phi> (inv\<^bsub>G\<^esub> g) e}. (e, \<phi> g z) \<in> R"
+        using H2_1
+        by blast
+      hence H2_8: "\<And>e. R `` {e} \<in> X // R \<Longrightarrow> \<forall>z\<in>R `` {\<phi> (inv\<^bsub>G\<^esub> g) e}. R `` {e} = R `` {\<phi> g z}"
+        by (meson A_0 equiv_class_eq_iff)
+      have H2_9: "\<And>e. R `` {e} \<in> X // R \<Longrightarrow>
+      R `` {e} = R `` {\<phi> g (SOME z. z \<in> R `` {\<phi> (inv\<^bsub>G\<^esub> g) e})}"
+      proof-
+        fix e
+        assume
+          A3_0: "R `` {e} \<in> X // R"
+        show "R `` {e} = R `` {\<phi> g (SOME z. z \<in> R `` {\<phi> (inv\<^bsub>G\<^esub> g) e})}"
+          apply (rule someI2[where Q = "\<lambda>z. R `` {e} = R `` {\<phi> g z}" and
+                P = "\<lambda>z. z \<in> R `` {\<phi> (inv\<^bsub>G\<^esub> g) e}" and a = "\<phi> (inv\<^bsub>G\<^esub> g) e"])
+          using A3_0 H2_4
+           apply blast
+          using A3_0 H2_8
+          by auto
+      qed
+      have H2_10: "\<forall>e. (R `` {e} \<in> X // R \<longrightarrow>
+      (R `` {e} = R `` {\<phi> g (SOME z. z \<in> R `` {\<phi> (inv\<^bsub>G\<^esub> g) e})}))"
+        using H2_9
+        by auto
+      hence H2_11: "\<forall>e. (R `` {e} \<in> X // R \<longrightarrow>
+      (\<exists>xa\<in>X // R. R `` {e} = R `` {\<phi> g (SOME z. z \<in> xa)}))"
+        using H2_8
+        apply clarsimp
+        by (smt (verit, best) A_0 H2_3 H2_5 H2_4 equiv_Eps_in equiv_class_eq_iff quotientI)
+      have H2_12: "\<And>x. x \<in> X // R \<Longrightarrow> \<exists>e\<in>X. x = R `` {e} "
+        by (meson quotientE)
+      have H2_13: "\<And>x. x \<in> X // R \<Longrightarrow> \<exists>xa\<in>X // R. x = R `` {\<phi> g (SOME z. z \<in> xa)}"
+        using H2_11 H2_12
+        by blast
+      show "\<exists>xa\<in>X // R. x = R `` {\<phi> g (SOME z. z \<in> xa)}"
+        by (simp add: A2_0 H2_13)
+    qed
+    show "(\<lambda>x\<in>X // R. R `` {\<phi> g (SOME z. z \<in> x)}) \<in> carrier (BijGroup (X // R))"
+      apply (clarsimp simp add: BijGroup_def Bij_def bij_betw_def)
+      subgoal
+        apply (clarsimp simp add: inj_on_def)
+        apply (rule conjI)
+         apply (clarsimp)
+         apply (rule subset_antisym)
+          apply (simp add: H1_0)
+         apply (simp add: \<open>\<And>y x. \<lbrakk>x \<in> X // R;
+          y \<in> X // R; R `` {\<phi> g (SOME z. z \<in> x)} = R `` {\<phi> g (SOME z. z \<in> y)}\<rbrakk> \<Longrightarrow> x \<subseteq> y\<close>)
+        apply (rule subset_antisym; clarify)
+        subgoal for x y
+          by (metis A_0 is_eq_var_rel' A1_4 Eps_cong equiv_Eps_preserves equiv_class_eq_iff
+              quotientI) 
+        apply (clarsimp simp add: Set.image_def)
+        by (simp add: H1_1)
+      done
+  qed
+  have H_1: "\<And>x y. \<lbrakk>Group.group G; Group.group (BijGroup X); R \<subseteq> X \<times> X;
+            \<phi> \<in> carrier G \<rightarrow> carrier (BijGroup X);
+           \<forall>x\<in>carrier G. \<forall>y\<in>carrier G. \<phi> (x \<otimes>\<^bsub>G\<^esub> y) = \<phi> x \<otimes>\<^bsub>BijGroup X\<^esub> \<phi> y;
+           x \<in> carrier G; y \<in> carrier G; x \<otimes>\<^bsub>G\<^esub> y \<in> carrier G\<rbrakk> \<Longrightarrow>
+           (\<lambda>xa\<in>X // R. R `` {(\<phi> x \<otimes>\<^bsub>BijGroup X\<^esub> \<phi> y) (SOME z. z \<in> xa)}) =
+           (\<lambda>xa\<in>X // R. R `` {\<phi> x (SOME z. z \<in> xa)}) \<otimes>\<^bsub>BijGroup (X // R)\<^esub>
+           (\<lambda>x\<in>X // R. R `` {\<phi> y (SOME z. z \<in> x)})"
+  proof -
+    fix x y 
+    assume
+      A1_1: "Group.group G" and
+      A1_2: "Group.group (BijGroup X)" and
+      A1_3: "\<phi> \<in> carrier G \<rightarrow> carrier (BijGroup X)" and
+      A1_4: "\<forall>x\<in>carrier G. \<forall>y\<in>carrier G. \<phi> (x \<otimes>\<^bsub>G\<^esub> y) = \<phi> x \<otimes>\<^bsub>BijGroup X\<^esub> \<phi> y" and
+      A1_5: "x \<in> carrier G" and
+      A1_6: "y \<in> carrier G" and
+      A1_7: "x \<otimes>\<^bsub>G\<^esub> y \<in> carrier G"
+    have H1_0: "\<And>w::'X set. w \<in> X // R \<Longrightarrow>
+    R `` {(\<phi> x \<otimes>\<^bsub>BijGroup X\<^esub> \<phi> y) (SOME z. z \<in> w)} =
+    ((\<lambda>v\<in>X // R. R `` {\<phi> x (SOME z. z \<in> v)}) \<otimes>\<^bsub>BijGroup (X // R)\<^esub>
+    (\<lambda>x\<in>X // R. R `` {\<phi> y (SOME z. z \<in> x)})) w"
+    proof-
+      fix w
+      assume
+        A2_0: "w \<in> X // R" 
+      have H2_4: "\<phi> y ` w \<in> X // R"
+        using ec_er_closed_under_action[where w = "w" and g = "y"]
+        by (clarsimp simp add: group_hom_axioms_def hom_def A_0 A1_1 A1_2 is_eq_var_rel' A1_3 A1_4
+            A1_6 A2_0) 
+      hence H2_1: "R `` {(\<phi> x \<otimes>\<^bsub>BijGroup X\<^esub> \<phi> y) (SOME z. z \<in> w)} =
+      R `` {\<phi> (x \<otimes>\<^bsub>G\<^esub> y) (SOME z. z \<in> w)}"
+        using A1_4 A1_5 A1_6
+        by auto
+      also have H2_2: "... = R `` {SOME z. z \<in> \<phi> (x \<otimes>\<^bsub>G\<^esub> y) ` w}"
+        using A1_7 equiv_equivar_class_some_eq[where w = "w" and g = "x \<otimes>\<^bsub>G\<^esub> y"]
+        by (clarsimp simp add: A1_7 A_0 A2_0 group_action_def group_hom_def group_hom_axioms_def
+            hom_def)
+      also have H2_3: "... = R `` {SOME z. z \<in> \<phi> x ` \<phi> y ` w}"
+        apply (rule meta_mp[of "\<not>(\<exists>x. x \<in> w \<and> x \<notin> X)"])
+        using A1_1 is_eq_var_rel' A1_3 A1_4 A1_5 A1_6 A2_0
+         apply (clarsimp simp add: image_def BijGroup_def restrict_def compose_def Pi_def)
+         apply (smt (z3) Eps_cong)
+        apply (clarify) 
+        using A_0 A2_0 in_quotient_imp_subset
+        by auto
+      also have H2_5: "... = R `` {\<phi> x (SOME z. z \<in> \<phi> y ` w)}"
+        using equiv_equivar_class_some_eq[where w = "\<phi> y ` w" and g = "x"]
+        apply (clarsimp simp add: A_0 group_action_def group_hom_def group_hom_axioms_def hom_def)
+        by (simp add: A1_1 A1_2 is_eq_var_rel' A1_3 A1_4 A1_5 H2_4)
+      also have H2_6: "... = R `` {\<phi> x (SOME z. z \<in> R``{(SOME z'. z' \<in> \<phi> y ` w)})}"
+        using H2_4 nested_somes[where w = "\<phi> y ` w" and X = "X" and R = "R"] A_0
+        by presburger
+      also have H2_7: "... = R `` {\<phi> x (SOME z. z \<in> R `` {\<phi> y (SOME z'. z' \<in> w)})}"
+        using equiv_equivar_class_some_eq[where g = "y" and w = "w"] H2_6
+        by (simp add: A_0 group_action_def
+            group_hom_def group_hom_axioms_def hom_def A1_1 A1_2 is_eq_var_rel' A1_3 A1_4 A2_0 A1_6) 
+      also have H2_9: "... = ((\<lambda>v\<in>X // R. R `` {\<phi> x (SOME z. z \<in> v)}) \<otimes>\<^bsub>BijGroup (X // R)\<^esub>
+      (\<lambda>x\<in>X // R. R `` {\<phi> y (SOME z. z \<in> x)})) w"
+      proof-
+        have H3_0: "\<And>u. R `` {\<phi> y (SOME z. z \<in> w)} \<in> X // R \<Longrightarrow> u \<in> carrier G \<Longrightarrow>
+        (\<lambda>v\<in>X // R. R `` {\<phi> u (SOME z. z \<in> v)}) \<in> Bij (X // R)"
+        proof -
+          fix u
+          assume 
+            A4_0: "R `` {\<phi> y (SOME z. z \<in> w)} \<in> X // R" and
+            A4_1: "u \<in> carrier G"
+          have H4_0: "\<forall>g \<in> carrier G.
+          (\<lambda>x\<in>X // R. R `` {\<phi> g (SOME z. z \<in> x)}) \<in> carrier (BijGroup (X // R))"
+            by (simp add: A_0 A1_1 A1_2 A1_3 A1_4 H_0 is_subrel)
+          thus "(\<lambda>v\<in>X // R. R `` {\<phi> u (SOME z. z \<in> v)}) \<in> Bij (X // R)"
+            by (auto simp add: BijGroup_def A4_1)
+        qed
+        have H3_1: "R `` {\<phi> y (SOME z. z \<in> w)} \<in> X // R"
+        proof-
+          have H4_0: "\<phi> y ` w \<in> X // R"
+            using ec_er_closed_under_action
+            by (simp add: H2_4)
+          hence H4_1: "R `` {(SOME z. z \<in> \<phi> y ` w)} = \<phi> y ` w"
+            apply (clarsimp simp add: image_def) 
+            apply (rule subset_antisym)
+            using A_0 equiv_Eps_in in_quotient_imp_closed
+             apply fastforce
+            using A_0 equiv_Eps_in quotient_eq_iff
+            by fastforce
+          have H4_2: "R `` {\<phi> y (SOME z. z \<in> w)} = R `` {(SOME z. z \<in> \<phi> y ` w)}"
+            using equiv_equivar_class_some_eq[where g = "y" and w = "w"]
+            by (metis A_0 A2_0 H4_0 H4_1 equiv_Eps_in imageI some_equiv_class_id)
+          from H4_0 H4_1 H4_2 show "R `` {\<phi> y (SOME z. z \<in> w)} \<in> X // R"
+            by auto
+        qed
+        show ?thesis
+          apply (rule meta_mp[of "R `` {\<phi> y (SOME z. z \<in> w)} \<in> X // R"])
+           apply (rule meta_mp[of "\<forall>u \<in> carrier G.
+         (\<lambda>v\<in>X // R. R `` {\<phi> u (SOME z. z \<in> v)}) \<in> Bij (X // R)"])
+          using A2_0 A1_5 A1_6
+            apply ( simp add: BijGroup_def compose_def)
+           apply clarify
+          by (simp add: H3_0 H3_1)+
+      qed
+      finally show "R `` {(\<phi> x \<otimes>\<^bsub>BijGroup X\<^esub> \<phi> y) (SOME z. z \<in> w)} =
+      ((\<lambda>v\<in>X // R. R `` {\<phi> x (SOME z. z \<in> v)}) \<otimes>\<^bsub>BijGroup (X // R)\<^esub>
+      (\<lambda>x\<in>X // R. R `` {\<phi> y (SOME z. z \<in> x)})) w"
+        by simp
+    qed
+    have H1_1: "\<And>w::'X set. w \<notin> X // R \<Longrightarrow>
+    ((\<lambda>v\<in>X // R. R `` {\<phi> x (SOME z. z \<in> v)}) \<otimes>\<^bsub>BijGroup (X // R)\<^esub>
+    (\<lambda>x\<in>X // R. R `` {\<phi> y (SOME z. z \<in> x)})) w = undefined"
+    proof -
+      fix w
+      assume
+        A2_0: "w \<notin> X // R" 
+      have H2_0: "\<And>u. u\<in> carrier G \<Longrightarrow> (\<lambda>v\<in>X // R. R `` {\<phi> u (SOME z. z \<in> v)}) \<in> Bij (X // R)"
+        using H_0
+        apply (clarsimp simp add: A_0 A1_1 A1_2 is_eq_var_rel' A1_3 A1_4 is_subrel)
+        by (simp add: BijGroup_def)
+      hence H2_1: "(\<lambda>x'\<in>X // R. R `` {\<phi> y (SOME z. z \<in> x')}) \<in> Bij (X // R)"
+        using A1_6
+        by auto
+      from H2_0 have H2_2: "(\<lambda>x'\<in>X // R. R `` {\<phi> x (SOME z. z \<in> x')}) \<in> Bij (X // R)"
+        by (simp add: A1_5)
+      thus "((\<lambda>v\<in>X // R. R `` {\<phi> x (SOME z. z \<in> v)}) \<otimes>\<^bsub>BijGroup (X // R)\<^esub>
+      (\<lambda>x\<in>X // R. R `` {\<phi> y (SOME z. z \<in> x)})) w = undefined"
+        using H2_1 H2_2
+        by (auto simp add: BijGroup_def compose_def A2_0)
+    qed
+    from H1_0 H1_1 have "\<And>w. (\<lambda>xa\<in>X // R. R `` {(\<phi> x \<otimes>\<^bsub>BijGroup X\<^esub> \<phi> y) (SOME z. z \<in> xa)}) w =
+    ((\<lambda>xa\<in>X // R. R `` {\<phi> x (SOME z. z \<in> xa)}) \<otimes>\<^bsub>BijGroup (X // R)\<^esub>
+    (\<lambda>x'\<in>X // R. R `` {\<phi> y (SOME z. z \<in> x')})) w"
+      by auto
+    thus "(\<lambda>xa\<in>X // R. R `` {(\<phi> x \<otimes>\<^bsub>BijGroup X\<^esub> \<phi> y) (SOME z. z \<in> xa)}) =
+    (\<lambda>xa\<in>X // R. R `` {\<phi> x (SOME z. z \<in> xa)}) \<otimes>\<^bsub>BijGroup (X // R)\<^esub>
+    (\<lambda>x\<in>X // R. R `` {\<phi> y (SOME z. z \<in> x)})"
+      by (simp add: restrict_def)
+  qed
+  show ?thesis
+    apply (clarsimp simp add: group_action_def group_hom_def)
+    using eq_var_rel_axioms
+    apply (clarsimp simp add: eq_var_rel_def eq_var_rel_axioms_def
+        group_action_def group_hom_def)
+    apply (rule conjI)
+     apply (simp add: group_BijGroup)
+    apply (clarsimp simp add: group_hom_axioms_def hom_def)
+    apply (intro conjI)
+     apply (rule funcsetI; simp)
+     apply (simp add: H_0)
+    apply (clarify; rule conjI; intro impI)
+     apply (simp add: H_1)
+    by (auto simp add: group.is_monoid monoid.m_closed)
+qed
+end
+
+locale eq_var_func = GA_0: alt_grp_act G X \<phi> + GA_1: alt_grp_act G Y \<psi> 
+  for
+    G :: "('grp, 'b) monoid_scheme" and
+    X :: "'X set" and
+    \<phi> and
+    Y :: "'Y set" and
+    \<psi> +
+  fixes
+    f :: "'X \<Rightarrow> 'Y" 
+  assumes
+    is_ext_func_bet:
+    "f \<in> (X \<rightarrow>\<^sub>E Y)" and
+    is_eq_var_func:
+    "\<And>a g. a \<in> X \<Longrightarrow> g \<in> carrier G \<Longrightarrow> f (g \<odot>\<^bsub>\<phi>\<^esub> a) = g \<odot>\<^bsub>\<psi>\<^esub> (f a)"
+begin
+
+lemma is_eq_var_func' [simp]:
+  "a \<in> X \<Longrightarrow> g \<in> carrier G \<Longrightarrow> f (\<phi> g a) = \<psi> g (f a)"
+  using is_eq_var_func
+  by auto
+
+end
+
+lemma G_set_equiv:
+  "alt_grp_act G A \<phi> \<Longrightarrow> eq_var_subset G A \<phi> A"
+  by (auto simp add: eq_var_subset_def eq_var_subset_axioms_def group_action_def
+      group_hom_def group_hom_axioms_def hom_def BijGroup_def Bij_def bij_betw_def)
+
+subsection \<open>
+Basic ($G$)-Automata Theory
+\<close>
+
+
+locale language = 
+  fixes A :: "'alpha set" and
+    L
+  assumes
+    is_lang: "L \<subseteq> A\<^sup>\<star>"
+
+locale G_lang = alt_grp_act G A \<phi> + language A L 
+  for
+    G :: "('grp, 'b) monoid_scheme" and
+    A :: "'alpha set" (structure) and
+    \<phi> L +
+  assumes
+    L_is_equivar:
+    "eq_var_subset G (A\<^sup>\<star>) (induced_star_map \<phi>) L"
+begin
+lemma G_lang_is_lang[simp]: "language A L"
+  by (simp add: language_axioms)
+end
+
+sublocale G_lang \<subseteq> language
+  by simp
+
+fun give_input :: "('state \<Rightarrow> 'alpha \<Rightarrow> 'state) \<Rightarrow> 'state \<Rightarrow> 'alpha list \<Rightarrow> 'state"
+  where "give_input trans_func s Nil = s"
+  |   "give_input trans_func s (a#as) = give_input trans_func (trans_func s a) as"
+
+adhoc_overloading
+  star give_input
+
+locale det_aut =
+  fixes
+    labels :: "'alpha set" and
+    states :: "'state set" and
+    init_state :: "'state" and
+    fin_states :: "'state set" and
+    trans_func :: "'state \<Rightarrow> 'alpha \<Rightarrow> 'state" ("\<delta>")
+  assumes
+    init_state_is_a_state:
+    "init_state \<in> states" and
+    fin_states_are_states:
+    "fin_states \<subseteq> states" and
+    trans_func_ext:
+    "(\<lambda>(state, label). trans_func state label) \<in> (states \<times> labels) \<rightarrow>\<^sub>E states" 
+begin
+
+lemma trans_func_well_def:
+  "\<And>state label. state \<in> states \<Longrightarrow> label \<in> labels \<Longrightarrow> (\<delta> state label) \<in> states"
+  using trans_func_ext
+  by auto
+
+lemma give_input_closed:
+  "input \<in> (labels\<^sup>\<star>) \<Longrightarrow> s \<in> states \<Longrightarrow> (\<delta>\<^sup>\<star>) s input \<in> states"
+  apply (induction input arbitrary: s)
+  by (auto simp add: trans_func_well_def)
+
+lemma input_under_concat:
+  "w \<in> labels\<^sup>\<star> \<Longrightarrow> v \<in> labels\<^sup>\<star> \<Longrightarrow> (\<delta>\<^sup>\<star>) s (w @ v) = (\<delta>\<^sup>\<star>) ((\<delta>\<^sup>\<star>) s w) v"
+  apply (induction w arbitrary: s)
+  by auto
+
+lemma eq_pres_under_concat:
+  assumes
+    "w \<in> labels\<^sup>\<star>" and
+    "w' \<in> labels\<^sup>\<star>" and
+    "s \<in> states" and
+    "(\<delta>\<^sup>\<star>) s w = (\<delta>\<^sup>\<star>) s w'"
+  shows "\<forall>v \<in> labels\<^sup>\<star>. (\<delta>\<^sup>\<star>) s (w @ v) = (\<delta>\<^sup>\<star>) s (w' @ v)"
+  using input_under_concat[where w = w and s = s] input_under_concat[where w = w' and s = s] assms
+  by auto
+
+lemma trans_to_charact:
+  "\<And>s a w. \<lbrakk>s \<in> states; a \<in> labels; w \<in> labels\<^sup>\<star>; s = (\<delta>\<^sup>\<star>) i w\<rbrakk> \<Longrightarrow> (\<delta>\<^sup>\<star>) i (w @ [a]) = \<delta> s a"
+proof-
+  fix s a w
+  assume
+    A_0: "s \<in> states" and
+    A_1: "a \<in> labels" and
+    A_2: "w \<in> labels\<^sup>\<star>" and
+    A_3: "s = (\<delta>\<^sup>\<star>) i w"
+  have H_0: "trans_func s a = (\<delta>\<^sup>\<star>) s [a]"
+    by auto
+  from A_2 A_3 H_0 have H_1: "(\<delta>\<^sup>\<star>) s [a] = (\<delta>\<^sup>\<star>) ((\<delta>\<^sup>\<star>) i w) [a]"
+    by simp
+  from A_1 A_2 have H_2: "(\<delta>\<^sup>\<star>) ((\<delta>\<^sup>\<star>) i w) [a] = (\<delta>\<^sup>\<star>) i (w @ [a])"
+    using input_under_concat
+    by force
+  show "(\<delta>\<^sup>\<star>) i (w @ [a]) = \<delta> s a"
+    using A_1 H_0 A_3 H_1 H_2
+    by force
+qed
+
+end
+
+locale aut_hom = Aut0: det_aut A S\<^sub>0 i\<^sub>0 F\<^sub>0 \<delta>\<^sub>0 + Aut1: det_aut A S\<^sub>1 i\<^sub>1 F\<^sub>1 \<delta>\<^sub>1 for
+  A :: "'alpha set" and
+  S\<^sub>0 :: "'states_0 set" and
+  i\<^sub>0 and F\<^sub>0 and \<delta>\<^sub>0 and
+  S\<^sub>1 :: "'states_1 set" and
+  i\<^sub>1 and F\<^sub>1 and \<delta>\<^sub>1 +
+fixes f :: "'states_0 \<Rightarrow> 'states_1"
+assumes
+  hom_is_ext:
+  "f \<in> S\<^sub>0 \<rightarrow>\<^sub>E S\<^sub>1" and
+  pres_init:
+  "f i\<^sub>0 = i\<^sub>1" and
+  pres_final:
+  "s \<in> F\<^sub>0 \<longleftrightarrow> f s \<in> F\<^sub>1 \<and> s \<in> S\<^sub>0" and
+  pres_trans:
+  "s\<^sub>0 \<in> S\<^sub>0 \<Longrightarrow> a \<in> A \<Longrightarrow> f (\<delta>\<^sub>0 s\<^sub>0 a) = \<delta>\<^sub>1 (f s\<^sub>0) a"
+begin
+
+lemma hom_translation:
+  "input \<in> (A\<^sup>\<star>) \<Longrightarrow> s \<in> S\<^sub>0 \<Longrightarrow> (f ((\<delta>\<^sub>0\<^sup>\<star>) s input)) = ((\<delta>\<^sub>1\<^sup>\<star>) (f s) input)"
+  apply (induction input arbitrary: s)
+  by (auto simp add: Aut0.trans_func_well_def pres_trans)
+
+lemma recognise_same_lang:
+  "input \<in> A\<^sup>\<star> \<Longrightarrow> ((\<delta>\<^sub>0\<^sup>\<star>) i\<^sub>0 input) \<in> F\<^sub>0 \<longleftrightarrow> ((\<delta>\<^sub>1\<^sup>\<star>) i\<^sub>1 input) \<in> F\<^sub>1"
+  using hom_translation[where input = input and s = i\<^sub>0]
+  apply (clarsimp  simp add: Aut0.init_state_is_a_state pres_init pres_final)
+  apply (induction input)
+   apply (clarsimp simp add: Aut0.init_state_is_a_state)
+  using Aut0.give_input_closed Aut0.init_state_is_a_state
+  by blast
+
+end
+
+locale aut_epi = aut_hom + 
+  assumes
+    is_epi: "f ` S\<^sub>0 = S\<^sub>1"
+
+locale det_G_aut =
+  is_aut:       det_aut A S i F \<delta> +
+  labels_a_G_set:   alt_grp_act G A \<phi> + 
+  states_a_G_set:   alt_grp_act G S \<psi> +
+  accepting_is_eq_var: eq_var_subset G S \<psi> F +
+  init_is_eq_var:   eq_var_subset G S \<psi> "{i}" +
+  trans_is_eq_var:   eq_var_func G "S \<times> A"
+  "\<lambda>g\<in>carrier G. \<lambda>(s, a) \<in> (S \<times> A). (\<psi> g s, \<phi> g a)"
+  S "\<psi>" "(\<lambda>(s, a) \<in> (S \<times> A). \<delta> s a)"
+  for A :: "'alpha set" (structure) and
+    S :: "'states set" and
+    i F \<delta> and
+    G :: "('grp, 'b) monoid_scheme" and
+    \<phi> \<psi>
+begin
+
+adhoc_overloading
+  star labels_a_G_set.induced_star_map
+
+lemma give_input_eq_var:
+  "eq_var_func G
+ (A\<^sup>\<star> \<times> S) (\<lambda>g\<in>carrier G. \<lambda>(w, s) \<in> (A\<^sup>\<star> \<times> S). ((\<phi>\<^sup>\<star>) g w, \<psi> g s))
+ S \<psi>
+ (\<lambda>(w, s) \<in> (A\<^sup>\<star> \<times> S). (\<delta>\<^sup>\<star>) s w)"
+proof-
+  have H_0: "\<And>a w s g.
+       (\<And>s. s \<in> S \<Longrightarrow> (\<phi>\<^sup>\<star>) g w \<in> A\<^sup>\<star> \<and> \<psi> g s \<in> S \<Longrightarrow>
+       (\<delta>\<^sup>\<star>) (\<psi> g s) ((\<phi>\<^sup>\<star>) g w) = \<psi> g ((\<delta>\<^sup>\<star>) s w)) \<Longrightarrow>
+       s \<in> S \<Longrightarrow>
+       g \<in> carrier G \<Longrightarrow>
+       a \<in> A \<Longrightarrow> \<forall>x\<in>set w. x \<in> A \<Longrightarrow> \<psi> g s \<in> S \<Longrightarrow> \<forall>x\<in>set ((\<phi>\<^sup>\<star>) g (a # w)). x \<in> A \<Longrightarrow>
+       (\<delta>\<^sup>\<star>) (\<psi> g s) ((\<phi>\<^sup>\<star>) g (a # w)) = \<psi> g ((\<delta>\<^sup>\<star>) (\<delta> s a) w)"
+  proof-
+    fix a w s g
+    assume
+      A_IH: "(\<And>s. s \<in> S \<Longrightarrow>
+             (\<phi>\<^sup>\<star>) g w \<in> A\<^sup>\<star> \<and> \<psi> g s \<in> S \<Longrightarrow>
+             (\<delta>\<^sup>\<star>) (\<psi> g s) ((\<phi>\<^sup>\<star>) g w) = \<psi> g ((\<delta>\<^sup>\<star>) s w))" and
+      A_0: "s \<in> S" and
+      A_1: "\<psi> g s \<in> S" and
+      A_2: "\<forall>x\<in>set ((\<phi>\<^sup>\<star>) g (a # w)). x \<in> A" and
+      A_3: "g \<in> carrier G" and
+      A_4: " a \<in> A" and
+      A_5: "\<forall>x\<in>set w. x \<in> A"
+    have H_0: "((\<phi>\<^sup>\<star>) g (a # w)) = (\<phi> g a) # (\<phi>\<^sup>\<star>) g w"
+      using A_4 A_5 A_3
+      by auto
+    hence H_1: "(\<delta>\<^sup>\<star>) (\<psi> g s) ((\<phi>\<^sup>\<star>) g (a # w))
+       = (\<delta>\<^sup>\<star>) (\<psi> g s) ((\<phi> g a) # (\<phi>\<^sup>\<star>) g w)"
+      by simp
+    have H_2: "...  = (\<delta>\<^sup>\<star>) ((\<delta>\<^sup>\<star>) (\<psi> g s) [\<phi> g a]) ((\<phi>\<^sup>\<star>) g w)"
+      using is_aut.input_under_concat
+      by simp
+    have H_3: "(\<delta>\<^sup>\<star>) (\<psi> g s) [\<phi> g a] = \<psi> g (\<delta> s a)"
+      using trans_is_eq_var.eq_var_func_axioms A_4 A_5 A_0 A_1 A_3 apply (clarsimp simp del:
+          GMN_simps simp add: eq_var_func_def eq_var_func_axioms_def make_op_def)
+      apply (rule meta_mp[of "\<psi> g s \<in> S \<and> \<phi> g a \<in> A \<and> s \<in> S \<and> a \<in> A"])
+       apply presburger
+      apply (clarify)
+      using labels_a_G_set.element_image
+      by presburger
+    have H_4: "(\<delta>\<^sup>\<star>) (\<psi> g (\<delta> s a)) ((\<phi>\<^sup>\<star>) g w) = \<psi> g ((\<delta>\<^sup>\<star>) (\<delta> s a) w)"
+      apply (rule A_IH[where s1 = "\<delta> s a"])
+      subgoal
+        using A_4 A_5 A_0
+        by (auto simp add: is_aut.trans_func_well_def)
+      using A_4 A_5 A_0 A_3 \<open>\<delta> s a \<in> S\<close> states_a_G_set.element_image
+      by (metis A_2 Cons_in_lists_iff H_0 in_listsI)
+    show "(\<delta>\<^sup>\<star>) (\<psi> g s) ((\<phi>\<^sup>\<star>) g (a # w)) = \<psi> g ((\<delta>\<^sup>\<star>) (\<delta> s a) w)"
+      using H_0 H_1 H_2 H_3 H_4
+      by presburger
+  qed
+  show ?thesis
+    apply (subst eq_var_func_def)
+    apply (subst eq_var_func_axioms_def)
+    apply (rule conjI)
+     apply (rule prod_group_act[where G = G and A = "A\<^sup>\<star>" and \<phi> = "(\<phi>\<^sup>\<star>)"
+          and B = S     and \<psi> = \<psi>])
+    using labels_a_G_set.lists_a_Gset
+      apply blast
+     apply (simp add: states_a_G_set.group_action_axioms)
+    apply (rule conjI)
+     apply (simp add: states_a_G_set.group_action_axioms)
+    apply (rule conjI)
+     apply (subst extensional_funcset_def)
+     apply (subst restrict_def)
+     apply (subst Pi_def)
+     apply (subst extensional_def)
+     apply (auto simp add: in_listsI is_aut.give_input_closed)[1]
+    apply (subst restrict_def)
+    apply (clarsimp simp del: GMN_simps simp add: make_op_def)
+    apply (rule conjI; intro impI)
+    subgoal for w s g
+      apply (induction w arbitrary: s) 
+       apply simp
+      apply (clarsimp simp del: GMN_simps)
+      by (simp add: H_0 del: GMN_simps)
+    apply clarsimp
+    by (metis (no_types, lifting) image_iff in_lists_conv_set labels_a_G_set.surj_prop list.set_map
+        states_a_G_set.element_image)
+qed
+
+definition
+  accepted_words :: "'alpha list set"
+  where "accepted_words = {w. w \<in> A\<^sup>\<star> \<and> ((\<delta>\<^sup>\<star>) i w) \<in> F}"
+
+lemma induced_g_lang:
+  "G_lang G A \<phi> accepted_words"
+proof-
+  have H_0: "\<And>g w. g \<in> carrier G \<Longrightarrow> w \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i w \<in> F \<Longrightarrow> map (\<phi> g) w \<in> A\<^sup>\<star>"
+    apply (clarsimp)
+    using labels_a_G_set.element_image
+    by blast
+  have H_1: "\<And>g w. g \<in> carrier G \<Longrightarrow> w \<in> A\<^sup>\<star> \<Longrightarrow> (\<delta>\<^sup>\<star>) i w \<in> F \<Longrightarrow> (\<delta>\<^sup>\<star>) i (map (\<phi> g) w) \<in> F"
+  proof -
+    fix g w
+    assume
+      A_0: "g \<in> carrier G" and
+      A_1: "w \<in> A\<^sup>\<star>" and
+      A_2: "(\<delta>\<^sup>\<star>) i w \<in> F"
+    have H1_0: "\<psi> g ((\<delta>\<^sup>\<star>) i w) \<in> F"
+      using accepting_is_eq_var.eq_var_subset_axioms
+        A_0 A_2 accepting_is_eq_var.is_equivar
+      by blast
+    have H1_1: "\<psi> g i = i"
+      using init_is_eq_var.eq_var_subset_axioms A_0
+        init_is_eq_var.is_equivar
+      by auto
+    have H1_2: "\<And>w g. \<lbrakk>g \<in> carrier G; w \<in> A\<^sup>\<star>; (\<delta>\<^sup>\<star>) i w \<in> F\<rbrakk> \<Longrightarrow> (\<phi>\<^sup>\<star>) g w \<in> A\<^sup>\<star>"
+      using H_0
+      by auto
+    from A_1 have H1_3: "w \<in> A\<^sup>\<star>"
+      by auto
+    show "(\<delta>\<^sup>\<star>) i (map (\<phi> g) w) \<in> F"
+      using give_input_eq_var A_0 A_1 H1_1 H1_3
+      apply (clarsimp simp del: GMN_simps simp add: eq_var_func_def eq_var_func_axioms_def
+          make_op_def)
+      using A_2 H1_0 is_aut.init_state_is_a_state H1_2
+      by (smt (verit, best) H1_3 labels_a_G_set.induced_star_map_def restrict_apply)
+  qed
+  show ?thesis
+    apply (clarsimp simp del: GMN_simps simp add: G_lang_def accepted_words_def G_lang_axioms_def)
+    apply (rule conjI)
+    using labels_a_G_set.alt_grp_act_axioms
+     apply (auto)[1]
+    apply (intro conjI)
+     apply (simp add: language.intro)
+    apply (rule alt_grp_act.eq_var_one_direction)
+    using labels_a_G_set.alt_grp_act_axioms labels_a_G_set.lists_a_Gset
+      apply blast
+     apply (clarsimp )
+    apply (clarsimp)
+    by (simp add: H_0 H_1 in_listsI)
+qed
+end
+
+locale reach_det_aut =
+  det_aut A S i F \<delta>  
+  for A :: "'alpha set" (structure) and
+    S :: "'states set" and
+    i F \<delta> +
+  assumes
+    is_reachable:
+    "s \<in> S \<Longrightarrow> \<exists>input \<in> A\<^sup>\<star>. (\<delta>\<^sup>\<star>) i input = s"
+
+locale reach_det_G_aut =
+  det_G_aut A S i F \<delta> G \<phi> \<psi> + reach_det_aut A S i F \<delta> 
+  for A :: "'alpha set" (structure) and
+    S :: "'states set" and
+    i and F and \<delta> and
+    G :: "('grp, 'b) monoid_scheme" and
+    \<phi> \<psi> 
+begin
+
+text \<open>
+To avoid duplicate variant of "star":
+\<close>
+
+no_adhoc_overloading
+  star labels_a_G_set.induced_star_map
+end
+
+sublocale reach_det_G_aut \<subseteq> reach_det_aut
+  using reach_det_aut_axioms
+  by simp
+
+locale G_aut_hom = Aut0: reach_det_G_aut A S\<^sub>0 i\<^sub>0 F\<^sub>0 \<delta>\<^sub>0 G \<phi> \<psi>\<^sub>0 +
+  Aut1: reach_det_G_aut A S\<^sub>1 i\<^sub>1 F\<^sub>1 \<delta>\<^sub>1 G \<phi> \<psi>\<^sub>1 +
+  hom_f: aut_hom A S\<^sub>0 i\<^sub>0 F\<^sub>0 \<delta>\<^sub>0 S\<^sub>1 i\<^sub>1 F\<^sub>1 \<delta>\<^sub>1 f +
+  eq_var_f: eq_var_func G S\<^sub>0 \<psi>\<^sub>0 S\<^sub>1 \<psi>\<^sub>1 f for
+  A :: "'alpha set" and
+  S\<^sub>0 :: "'states_0 set" and
+  i\<^sub>0 and F\<^sub>0 and \<delta>\<^sub>0 and
+  S\<^sub>1 :: "'states_1 set" and
+  i\<^sub>1 and F\<^sub>1 and \<delta>\<^sub>1 and
+  G :: "('grp, 'b) monoid_scheme" and
+  \<phi> \<psi>\<^sub>0 \<psi>\<^sub>1 f 
+
+locale G_aut_epi = G_aut_hom + 
+  assumes
+    is_epi: "f ` S\<^sub>0 = S\<^sub>1"
+
+locale det_aut_rec_lang = det_aut A S i F \<delta>  + language A L
+  for A :: "'alpha set" (structure) and
+    S :: "'states set" and
+    i F \<delta> L + 
+  assumes
+    is_recognised:
+    "w \<in> L \<longleftrightarrow> w \<in> A\<^sup>\<star> \<and> ((\<delta>\<^sup>\<star>) i w) \<in> F"
+
+locale det_G_aut_rec_lang = det_G_aut A S i F \<delta> G \<phi> \<psi> + det_aut_rec_lang A S i F \<delta> L 
+  for A :: "'alpha set" (structure) and
+    S :: "'states set" and
+    i F \<delta> and
+    G :: "('grp, 'b) monoid_scheme" and
+    \<phi> \<psi> L
+begin
+
+lemma lang_is_G_lang: "G_lang G A \<phi> L"
+proof-
+  have H0: "L = accepted_words"
+    apply (simp add: accepted_words_def)
+    apply (subst is_recognised [symmetric])
+    by simp
+  show "G_lang G A \<phi> L"
+    apply (subst H0)
+    apply (rule det_G_aut.induced_g_lang[of A S i F \<delta> G \<phi> \<psi>])
+    by (simp add: det_G_aut_axioms)
+qed  
+
+text \<open>
+To avoid ambiguous parse trees:
+\<close>
+
+no_notation trans_is_eq_var.GA_0.induced_quot_map ("[_]\<^sub>_\<index>" 60)
+no_notation states_a_G_set.induced_quot_map ("[_]\<^sub>_\<index>" 60)
+
+end
+
+locale reach_det_aut_rec_lang = reach_det_aut A S i F \<delta> + det_aut_rec_lang A S i F \<delta> L
+  for A :: "'alpha set" and
+    S :: "'states set" and
+    i F \<delta> and
+    L :: "'alpha list set"
+
+locale reach_det_G_aut_rec_lang = det_G_aut_rec_lang A S i F \<delta> G \<phi> \<psi> L +
+  reach_det_G_aut A S i F \<delta> G \<phi> \<psi>
+  for A :: "'alpha set" and
+    S :: "'states set" and
+    i F \<delta> and
+    G :: "('grp, 'b) monoid_scheme" and
+    \<phi> \<psi> and
+    L :: "'alpha list set"
+
+sublocale reach_det_G_aut_rec_lang \<subseteq> det_G_aut_rec_lang
+  apply (simp add: det_G_aut_rec_lang_def)
+  using reach_det_G_aut_rec_lang_axioms
+  by (simp add: det_G_aut_axioms det_aut_rec_lang_axioms) 
+
+locale det_G_aut_recog_G_lang = det_G_aut_rec_lang A S i F \<delta> G \<phi> \<psi> L + G_lang G A \<phi> L
+  for A :: "'alpha set" (structure) and
+    S :: "'states set" and
+    i F \<delta> and
+    G :: "('grp, 'b) monoid_scheme" and
+    \<phi> \<psi> and
+    L :: "'alpha list set"
+
+sublocale det_G_aut_rec_lang \<subseteq> det_G_aut_recog_G_lang
+  apply (simp add: det_G_aut_recog_G_lang_def)
+  apply (rule conjI)
+   apply (simp add: det_G_aut_rec_lang_axioms)
+  by (simp add: lang_is_G_lang)
+
+locale reach_det_G_aut_rec_G_lang = reach_det_G_aut_rec_lang A S i F \<delta> G \<phi> \<psi> L + G_lang G A \<phi> L
+  for A :: "'alpha set" (structure) and
+    S :: "'states set" and
+    i F \<delta> and
+    G :: "('grp, 'b) monoid_scheme" and
+    \<phi> \<psi> L 
+
+sublocale reach_det_G_aut_rec_lang \<subseteq> reach_det_G_aut_rec_G_lang
+  apply (simp add: reach_det_G_aut_rec_G_lang_def)
+  apply (rule conjI)
+   apply (simp add: reach_det_G_aut_rec_lang_axioms)
+  by (simp add: lang_is_G_lang)
+
+lemma (in reach_det_G_aut)
+  "reach_det_G_aut_rec_lang A S i F \<delta> G \<phi> \<psi> accepted_words"
+  apply (clarsimp simp del: simp add: reach_det_G_aut_rec_lang_def
+      det_G_aut_rec_lang_def det_aut_rec_lang_axioms_def)
+  apply (intro conjI)
+    apply (simp add: det_G_aut_axioms)
+   apply (clarsimp simp add: reach_det_G_aut_axioms accepted_words_def reach_det_aut_rec_lang_def)
+   apply (simp add: det_aut_rec_lang_def det_aut_rec_lang_axioms.intro is_aut.det_aut_axioms
+      language_def)
+  by (simp add: reach_det_G_aut_axioms)
+
+lemma (in det_G_aut) action_on_input:
+  "\<And> g w. g \<in> carrier G \<Longrightarrow> w \<in> A\<^sup>\<star> \<Longrightarrow> \<psi> g ((\<delta>\<^sup>\<star>) i w) = (\<delta>\<^sup>\<star>) i ((\<phi>\<^sup>\<star>) g w)"
+proof-
+  fix g w
+  assume
+    A_0: "g \<in> carrier G" and
+    A_1: "w \<in> A\<^sup>\<star>"
+  have H_0: "(\<delta>\<^sup>\<star>) (\<psi> g i) ((\<phi>\<^sup>\<star>) g w) = (\<delta>\<^sup>\<star>) i ((\<phi>\<^sup>\<star>) g w)"
+    using A_0 init_is_eq_var.is_equivar
+    by fastforce
+  have H_1: "\<psi> g ((\<delta>\<^sup>\<star>) i w) = (\<delta>\<^sup>\<star>) (\<psi> g i) ((\<phi>\<^sup>\<star>) g w)"
+    using A_0 A_1 give_input_eq_var
+    apply (clarsimp simp del: GMN_simps simp add: eq_var_func_axioms_def eq_var_func_def
+        make_op_def)
+    apply (rule meta_mp[of "((\<phi>\<^sup>\<star>) g w) \<in> A\<^sup>\<star> \<and> \<psi> g i \<in> S"]) 
+    using is_aut.init_state_is_a_state A_1
+     apply presburger
+    using det_G_aut_axioms
+    apply (clarsimp simp add: det_G_aut_def) 
+    apply (rule conjI; rule impI; rule conjI)
+    using labels_a_G_set.element_image
+       apply fastforce
+    using is_aut.init_state_is_a_state states_a_G_set.element_image
+    by blast+
+  show "\<psi> g ((\<delta>\<^sup>\<star>) i w) = (\<delta>\<^sup>\<star>) i ((\<phi>\<^sup>\<star>) g w)"
+    using H_0 H_1
+    by simp
+qed
+
+definition (in det_G_aut)
+  reachable_states :: "'states set" ("S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h")
+  where "S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h = {s . \<exists> w \<in> A\<^sup>\<star>. (\<delta>\<^sup>\<star>) i w = s}"
+
+definition (in det_G_aut)
+  reachable_trans :: "'states \<Rightarrow> 'alpha \<Rightarrow> 'states" ("\<delta>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h")
+  where "\<delta>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h s a = (\<lambda>(s', a') \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<times> A. \<delta> s' a') (s, a)"
+
+definition (in det_G_aut)
+  reachable_action :: "'grp \<Rightarrow> 'states \<Rightarrow> 'states" ("\<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h")
+  where "\<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g s = (\<lambda>(g', s') \<in> carrier G \<times> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h. \<psi> g' s') (g, s)"
+
+lemma (in det_G_aut) reachable_action_is_restict:
+  "\<And>g s. g \<in> carrier G \<Longrightarrow> s \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<Longrightarrow> \<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g s = \<psi> g s"
+  by (auto simp add: reachable_action_def reachable_states_def)
+
+lemma (in det_G_aut_rec_lang) reach_det_aut_is_det_aut_rec_L:
+  "reach_det_G_aut_rec_lang A S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h i (F \<inter> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h) \<delta>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h G \<phi> \<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h L"
+proof-
+  have H_0: "(\<lambda>(x, y). \<delta>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h x y) \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<times> A \<rightarrow>\<^sub>E S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h"
+  proof-
+    have H1_0: "(\<lambda>(x, y). \<delta> x y) \<in> extensional (S \<times> A)"
+      using is_aut.trans_func_ext
+      by (simp add: PiE_iff)
+    have H1_1: "(\<lambda>(s', a') \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<times> A. \<delta> s' a') \<in> extensional (S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<times> A)"
+      using H1_0
+      by simp
+    have H1_2: "(\<lambda>(s', a')\<in>S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<times> A. \<delta> s' a') = (\<lambda>(x, y). \<delta>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h x y)"
+      by (auto simp add: reachable_trans_def)
+    show "(\<lambda>(x, y). \<delta>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h x y) \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<times> A \<rightarrow>\<^sub>E S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h"
+      apply (clarsimp simp add: PiE_iff)
+      apply (rule conjI)
+       apply (clarify)
+      using reachable_trans_def
+       apply (simp add: reachable_states_def)[1]
+       apply (metis Cons_in_lists_iff append_Nil2 append_in_lists_conv is_aut.give_input_closed
+          is_aut.init_state_is_a_state is_aut.trans_to_charact)
+      using H1_1 H1_2
+      by simp
+  qed
+  have H_1: "\<And>g. g \<in> carrier G \<Longrightarrow>
+    (\<And>s. \<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g s = (if s \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h then case (g, s) of (x, xa) \<Rightarrow> \<psi> x xa else undefined)) \<Longrightarrow>
+    bij_betw (\<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g) S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h"
+  proof-
+    fix g
+    assume
+      A1_0: "g \<in> carrier G" and
+      A1_1: "(\<And>s. \<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g s =
+        (if s \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h
+        then case (g, s) of (x, xa) \<Rightarrow> \<psi> x xa
+        else undefined))" 
+    have H1_0: "\<And>r. r \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<Longrightarrow> (\<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g) r \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h"
+      using A1_0
+      apply (clarsimp simp add: reachable_states_def reachable_action_def)
+      apply (rule meta_mp[of "\<And>w. w \<in> A\<^sup>\<star> \<Longrightarrow> ((\<phi>\<^sup>\<star>) g w) \<in> A\<^sup>\<star>"])
+      using action_on_input[where g = g]
+       apply (metis in_listsI) 
+      by (metis alt_group_act_is_grp_act group_action.element_image labels_a_G_set.lists_a_Gset)
+    have H1_1: "\<And>f T U. bij_betw f T T \<Longrightarrow> f ` U = U \<Longrightarrow> U \<subseteq> T \<Longrightarrow> bij_betw (restrict f U) U U"
+      apply (clarsimp simp add: bij_betw_def inj_on_def image_def)
+      by (meson in_mono)
+    have H1_2: "\<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g = restrict (\<psi> g) S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h"
+      using reachable_action_def A1_0
+      by (auto simp add: restrict_def)
+    have H1_3: "bij_betw (\<psi> g) S S \<Longrightarrow> (\<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g) ` S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h = S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h
+      \<Longrightarrow> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<subseteq> S \<Longrightarrow> bij_betw (\<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g) S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h"
+      by (metis H1_2 bij_betw_imp_inj_on inj_on_imp_bij_betw inj_on_restrict_eq inj_on_subset)
+    have H1_4: "\<And>w s. s = (\<delta>\<^sup>\<star>) i w \<Longrightarrow>
+         \<forall>x\<in>set w. x \<in> A \<Longrightarrow>
+         \<exists>x. (\<exists>w\<in> A\<^sup>\<star>. (\<delta>\<^sup>\<star>) i w = x) \<and> (\<delta>\<^sup>\<star>) i w = \<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g x"
+    proof-
+      fix w s
+      assume
+        A2_0: "\<forall>x\<in>set w. x \<in> A" and
+        A2_1: "s = (\<delta>\<^sup>\<star>) i w"
+      have H2_0: "(inv \<^bsub>G\<^esub> g) \<in> carrier G"
+        apply (rule meta_mp[of "group G"])
+        using A1_0
+         apply simp
+        using det_G_aut_rec_lang_axioms
+        by (auto simp add: det_G_aut_rec_lang_def
+            det_aut_rec_lang_axioms_def det_G_aut_def group_action_def group_hom_def)
+      have H2_1: "\<psi> (inv \<^bsub>G\<^esub> g) s = (\<delta>\<^sup>\<star>) i ((\<phi>\<^sup>\<star>) (inv \<^bsub>G\<^esub> g) w)"
+        apply (simp del: GMN_simps add: A2_1)
+        apply (rule action_on_input[where g = "(inv \<^bsub>G\<^esub> g)" and w = w]) 
+        using H2_0 A2_0
+        by auto
+      have H2_2: "((\<phi>\<^sup>\<star>) (inv \<^bsub>G\<^esub> g) w) \<in> A\<^sup>\<star>"
+        using A2_0 H2_0 det_G_aut_rec_lang_axioms
+        apply (clarsimp)
+        using labels_a_G_set.surj_prop list.set_map
+        by fastforce
+      have H2_3: "\<exists>w\<in>A\<^sup>\<star>. (\<delta>\<^sup>\<star>) i w = \<psi> (inv \<^bsub>G\<^esub> g) s" 
+        by (metis H2_1 H2_2)
+      from H2_3 have H2_4: "\<psi> (inv \<^bsub>G\<^esub> g) s \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h"
+        by (simp add: reachable_states_def)
+      have H2_5: "\<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g (\<psi> (inv \<^bsub>G\<^esub> g) s) = \<psi> g (\<psi> (inv \<^bsub>G\<^esub> g) s)"
+        apply (rule reachable_action_is_restict)
+        using A1_0 H2_4
+        by simp+
+      have H2_6: "(\<delta>\<^sup>\<star>) i w = \<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g (\<psi> (inv \<^bsub>G\<^esub> g) s)"
+        apply (simp add: H2_5 A2_1) 
+        by (metis A1_0 A2_0 in_listsI A2_1 H2_5 is_aut.give_input_closed
+            is_aut.init_state_is_a_state states_a_G_set.bij_prop1 states_a_G_set.orbit_sym_aux)
+      show " \<exists>x. (\<exists>w\<in>A\<^sup>\<star>. (\<delta>\<^sup>\<star>) i w = x) \<and> (\<delta>\<^sup>\<star>) i w = \<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g x"
+        using H2_3 H2_6
+        by blast
+    qed
+    show "bij_betw (\<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g) S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h"
+      apply (rule H1_3)
+        apply (simp add: A1_0 bij_betw_def states_a_G_set.inj_prop states_a_G_set.surj_prop)
+       apply (clarsimp simp add: image_def H1_0)
+       apply (rule subset_antisym; simp add: Set.subset_eq; clarify)
+      using H1_0 
+        apply auto[1]
+      subgoal for s
+        apply (clarsimp simp add: reachable_states_def)
+        by (simp add: H1_4)
+      apply (simp add: reachable_states_def Set.subset_eq; rule allI; rule impI)
+      using is_aut.give_input_closed is_aut.init_state_is_a_state
+      by auto
+  qed
+  have H_2: "group G"
+    using det_G_aut_rec_lang_axioms
+    by (auto simp add: det_G_aut_rec_lang_def det_G_aut_def group_action_def
+        group_hom_def)
+  have H_3: "\<And>g. g \<in> carrier G \<Longrightarrow> \<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g \<in> carrier (BijGroup S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h)"
+    subgoal for g
+      using reachable_action_def[where g = g]
+      apply (simp add: BijGroup_def Bij_def extensional_def)
+      by (simp add: H_1)
+    done
+  have H_4: "\<And>x y. x \<in> carrier G \<Longrightarrow> y \<in> carrier G \<Longrightarrow> \<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h (x \<otimes>\<^bsub>G\<^esub> y) = \<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h x \<otimes>\<^bsub>BijGroup S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h\<^esub>
+                                                                           \<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h y"
+  proof -
+    fix g h
+    assume
+      A1_0: "g \<in> carrier G" and
+      A1_1: "h \<in> carrier G"
+    have H1_0: "\<And>g . g \<in> carrier G \<Longrightarrow> \<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g = restrict (\<psi> g) S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h"
+      using reachable_action_def
+      by (auto simp add: restrict_def)
+    from H1_0 have H1_1: "\<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h (g \<otimes>\<^bsub>G\<^esub> h) = restrict (\<psi> (g \<otimes>\<^bsub>G\<^esub> h)) S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h"
+      by (simp add: A1_0 A1_1 H_2 group.subgroup_self subgroup.m_closed)
+    have H1_2: "\<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g \<otimes>\<^bsub>BijGroup S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h\<^esub> \<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h h = 
+    (restrict (\<psi> g) S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h) \<otimes>\<^bsub>BijGroup S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h\<^esub>
+    (restrict (\<psi> h) S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h)"
+      using A1_0 A1_1 H1_0
+      by simp
+    have H1_3: "\<And>g. g \<in> carrier G \<Longrightarrow> \<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g \<in> carrier (BijGroup S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h)"
+      by (simp add: H_3)
+    have H1_4: "\<And>x y. x\<in>carrier G \<Longrightarrow> y\<in>carrier G \<Longrightarrow> \<psi> (x \<otimes>\<^bsub>G\<^esub> y) = \<psi> x \<otimes>\<^bsub>BijGroup S\<^esub> \<psi> y"
+      using det_G_aut_axioms
+      by (simp add: det_G_aut_def group_action_def group_hom_def group_hom_axioms_def hom_def)
+    hence H1_5: "\<psi> (g \<otimes>\<^bsub>G\<^esub> h) = \<psi> g \<otimes>\<^bsub>BijGroup S\<^esub> \<psi> h"
+      using A1_0 A1_1
+      by simp
+    have H1_6: "(\<lambda>x. if x \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h
+                   then if (if x \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h
+                            then \<psi> h x
+                            else undefined) \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h
+                        then \<psi> g (if x \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h
+                                  then \<psi> h x
+                                  else undefined)
+                        else undefined
+                   else undefined) =
+              (\<lambda>x. if x \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h
+                   then \<psi> g (\<psi> h x) 
+                   else undefined)" 
+      apply (rule meta_mp[of "\<And>x. x \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<Longrightarrow> (\<psi> h x) \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h"])
+      using H1_3[where g1 = h] A1_1 H1_0
+      by (auto simp add: A1_1 BijGroup_def Bij_def bij_betw_def) 
+    have H1_7: "... = (\<lambda>x. if x \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h
+                          then if x \<in> S
+                               then \<psi> g (\<psi> h x)
+                               else undefined
+                          else undefined)"
+      apply (clarsimp simp add: reachable_states_def) 
+      by (metis is_aut.give_input_closed is_aut.init_state_is_a_state)
+    have H1_8: "(restrict (\<psi> g) S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h) \<otimes>\<^bsub>BijGroup S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h\<^esub> (restrict (\<psi> h) S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h) =
+     restrict (\<psi> (g \<otimes>\<^bsub>G\<^esub> h)) S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h"
+      apply (rule meta_mp[of "\<And>g. g \<in> carrier G \<Longrightarrow> restrict (\<psi> g) S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<in> Bij S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<and>
+      \<psi> g \<in> Bij S"])
+       apply (clarsimp simp add: H1_5 BijGroup_def; intro conjI; intro impI)
+      subgoal
+        using A1_0 A1_1
+        apply (clarsimp simp add: compose_def restrict_def)
+        by (simp add: H1_6 H1_7) 
+       apply (simp add: A1_0 A1_1)+
+      subgoal for g
+        using H1_3[where g1 = g] H1_0[of g]
+        by (simp add: BijGroup_def states_a_G_set.bij_prop0)
+      done
+    show "\<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h (g \<otimes>\<^bsub>G\<^esub> h) =
+      \<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g \<otimes>\<^bsub>BijGroup S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h\<^esub> \<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h h"
+      by (simp add: H1_1 H1_2 H1_8)
+  qed
+  have H_5: "\<And>w' w g. g \<in> carrier G \<Longrightarrow>
+          (\<delta>\<^sup>\<star>) i w \<in> F \<Longrightarrow> \<forall>x\<in>set w. x \<in> A \<Longrightarrow> (\<delta>\<^sup>\<star>) i w' = (\<delta>\<^sup>\<star>) i w \<Longrightarrow> \<forall>x\<in>set w'. x \<in> A \<Longrightarrow>
+          \<exists>w'\<in> A\<^sup>\<star>. (\<delta>\<^sup>\<star>) i w' = \<psi> g ((\<delta>\<^sup>\<star>) i w)"
+  proof -
+    fix w' w g
+    assume
+      A1_0: "g \<in> carrier G" and
+      A1_1: "(\<delta>\<^sup>\<star>) i w \<in> F" and
+      A1_2: "\<forall>x\<in>set w. x \<in> A" and
+      A1_3: "(\<delta>\<^sup>\<star>) i w' = (\<delta>\<^sup>\<star>) i w" and
+      A1_4: "\<forall>x\<in>set w. x \<in> A"
+    from A1_1 A1_2 have H1_0: "((\<delta>\<^sup>\<star>) i w) \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h"
+      using reachable_states_def
+      by auto
+    have H1_1: "\<psi> g ((\<delta>\<^sup>\<star>) i w) = ((\<delta>\<^sup>\<star>) i ((\<phi>\<^sup>\<star>) g w))"
+      using give_input_eq_var
+      apply (clarsimp simp add: eq_var_func_def eq_var_func_axioms_def simp del: GMN_simps)
+      using A1_0 A1_2 action_on_input
+      by blast
+    have H1_2: "(\<phi>\<^sup>\<star>) g w \<in> A\<^sup>\<star>"
+      using A1_0 A1_2
+      by (metis in_listsI alt_group_act_is_grp_act group_action.element_image
+          labels_a_G_set.lists_a_Gset)
+    show "\<exists>wa\<in>A\<^sup>\<star>. (\<delta>\<^sup>\<star>) i wa = \<psi> g ((\<delta>\<^sup>\<star>) i w)"
+      by (metis H1_1 H1_2)
+  qed
+  have H_6: "alt_grp_act G S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h"
+    apply (clarsimp simp add: group_action_def group_hom_def group_hom_axioms_def hom_def)
+    apply (intro conjI)
+       apply (simp add: H_2)
+    subgoal 
+      by (simp add: group_BijGroup)
+     apply clarify
+     apply (simp add: H_3)
+    by (simp add: H_4)
+  have H_7: "\<And>g w. g \<in> carrier G \<Longrightarrow> (\<delta>\<^sup>\<star>) i w \<in> F \<Longrightarrow> \<forall>x\<in>set w. x \<in> A \<Longrightarrow>
+            \<exists>x. x \<in> F \<and> (\<exists>w\<in>A\<^sup>\<star>. (\<delta>\<^sup>\<star>) i w = x) \<and> (\<delta>\<^sup>\<star>) i w = \<psi> g x"
+  proof-
+    fix g w
+    assume
+      A1_0: "g \<in> carrier G" and
+      A1_1: "(\<delta>\<^sup>\<star>) i w \<in> F" and
+      A1_2: "\<forall>x\<in>set w. x \<in> A"
+    have H1_0: "(inv \<^bsub>G\<^esub> g) \<in> carrier G" 
+      by (meson A1_0 group.inv_closed group_hom.axioms(1) labels_a_G_set.group_hom)
+    have H1_1: "((\<delta>\<^sup>\<star>) i w) \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h"
+      using A1_1 A1_2 reachable_states_def
+      by auto
+    have H1_2: "\<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h (inv \<^bsub>G\<^esub> g) ((\<delta>\<^sup>\<star>) i w) = \<psi> (inv \<^bsub>G\<^esub> g) ((\<delta>\<^sup>\<star>) i w)"
+      apply (rule reachable_action_is_restict)
+      using H1_0 H1_1
+      by auto
+    have H1_3: "\<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g (\<psi> (inv \<^bsub>G\<^esub> g) ((\<delta>\<^sup>\<star>) i w)) = ((\<delta>\<^sup>\<star>) i w)" 
+      by (smt (verit) A1_0 H1_1 H_6 H1_2
+          alt_group_act_is_grp_act group_action.bij_prop1 group_action.orbit_sym_aux)
+    have H1_4: "\<psi> (inv \<^bsub>G\<^esub> g) ((\<delta>\<^sup>\<star>) i w) \<in> F"
+      using A1_1 H1_0 accepting_is_eq_var.is_equivar
+      by blast
+    have H1_5: "\<psi> (inv \<^bsub>G\<^esub> g) ((\<delta>\<^sup>\<star>) i w) \<in> F \<and> (\<delta>\<^sup>\<star>) i w = \<psi> g (\<psi> (inv \<^bsub>G\<^esub> g) ((\<delta>\<^sup>\<star>) i w))"
+      using H1_4 H1_3 A1_0 A1_1 H1_0 H1_1 reachable_action_is_restict
+      by (metis H_6 alt_group_act_is_grp_act
+          group_action.element_image)
+    have H1_6: "\<psi> (inv \<^bsub>G\<^esub> g) ((\<delta>\<^sup>\<star>) i w) = ((\<delta>\<^sup>\<star>) i ((\<phi>\<^sup>\<star>) (inv \<^bsub>G\<^esub> g) w))"
+      using give_input_eq_var
+      apply (clarsimp simp add: eq_var_func_def eq_var_func_axioms_def simp del: GMN_simps)
+      using A1_2 H1_0 action_on_input
+      by blast
+    have H1_7: "(\<phi>\<^sup>\<star>) (inv \<^bsub>G\<^esub> g) w \<in> A\<^sup>\<star>"
+      by (metis A1_2 in_listsI H1_0 alt_group_act_is_grp_act group_action.element_image
+          labels_a_G_set.lists_a_Gset)
+    thus "\<exists>x. x \<in> F \<and> (\<exists>w\<in>A\<^sup>\<star>. (\<delta>\<^sup>\<star>) i w = x) \<and> (\<delta>\<^sup>\<star>) i w = \<psi> g x"
+      using H1_5 H1_6 H1_7
+      by metis
+  qed
+  have H_8: "\<And>r a g . r \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<Longrightarrow> a \<in> A \<Longrightarrow> \<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g r \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<and> \<phi> g a \<in> A \<Longrightarrow> g \<in> carrier G \<Longrightarrow>
+            \<delta>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h (\<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g r) (\<phi> g a) = \<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g (\<delta>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h r a)"
+  proof-
+    fix r a g
+    assume
+      A1_0: "r \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h" and
+      A1_1: "a \<in> A" and
+      A1_2: "\<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g r \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<and> \<phi> g a \<in> A" and
+      A1_3: "g \<in> carrier G"
+    have H1_0: "r \<in> S \<and> \<psi> g r \<in> S"
+      apply (rule conjI)
+      subgoal
+        using A1_0
+        apply (clarsimp simp add: reachable_states_def) 
+        by (simp add: in_listsI is_aut.give_input_closed is_aut.init_state_is_a_state)
+      using \<open>r \<in> S\<close> A1_3 states_a_G_set.element_image
+      by blast
+    have H1_1: "\<And>a b g . a \<in> S \<and> b \<in> A \<Longrightarrow> g \<in> carrier G \<Longrightarrow>
+        (if \<psi> g a \<in> S \<and> \<phi> g b \<in> A then \<delta> (\<psi> g a) (\<phi> g b) else undefined) =
+        \<psi> g (\<delta> a b)"
+      using det_G_aut_axioms A1_0 A1_1 A1_3
+      apply (clarsimp simp add: det_G_aut_def eq_var_func_def eq_var_func_axioms_def) 
+      by presburger+
+    hence H1_2: "\<psi> g (\<delta> r a) = (\<delta> (\<psi> g r) (\<phi> g a))"
+      using H1_1[where a1 = r and b1 = a and g1 = g] H1_0 A1_1 A1_2 A1_3
+      by simp
+    have H1_3: "\<And>a w. a \<in> A \<Longrightarrow> w \<in> A\<^sup>\<star> \<Longrightarrow> \<exists>w'\<in> A\<^sup>\<star>. (\<delta>\<^sup>\<star>) i w' = \<delta> ((\<delta>\<^sup>\<star>) i w) a"
+    proof -
+      fix a w
+      assume
+        A2_0: "a \<in> A" and
+        A2_1: "w \<in> A\<^sup>\<star>" 
+      have H2_0: "(w @ [a]) \<in> A\<^sup>\<star> \<and> (w @ [a]) \<in> A\<^sup>\<star> \<Longrightarrow> (\<delta>\<^sup>\<star>) i (w @ [a]) = \<delta> ((\<delta>\<^sup>\<star>) i w) a"
+        by (simp add: is_aut.give_input_closed is_aut.trans_to_charact
+            is_aut.init_state_is_a_state)
+      show "\<exists>w'\<in>A\<^sup>\<star>. (\<delta>\<^sup>\<star>) i w' = \<delta> ((\<delta>\<^sup>\<star>) i w) a"
+        using H2_0
+        apply clarsimp 
+        by (metis A2_0 A2_1 append_in_lists_conv lists.Cons lists.Nil)
+    qed
+    have H1_4: "\<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g (\<delta>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h r a) = \<psi> g (\<delta> r a)"
+      apply (clarsimp simp add: reachable_action_def reachable_trans_def)
+      using A1_0 A1_1 A1_3 H1_0 H1_3 
+      using reachable_states_def by fastforce
+    have H1_5:"\<psi> g r = \<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g r"
+      using A1_0 A1_3
+      by (auto simp add: reachable_action_def)
+    hence H1_6: "\<psi> g r \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h"
+      using A1_2
+      by simp
+    have H1_7: "\<delta>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h (\<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g r) (\<phi> g a) = \<delta> (\<psi> g r) (\<phi> g a)"
+      using A1_0 A1_1 A1_2 A1_3
+      by (auto simp del: simp add:reachable_trans_def reachable_action_def )
+    show "\<delta>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h (\<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g r) (\<phi> g a) = \<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g (\<delta>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h r a)"
+      using H1_2 H1_4 H1_7
+      by auto
+  qed
+  have H_9: "\<And>a w s. \<lbrakk>(\<And>s. s \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<Longrightarrow> (\<delta>\<^sup>\<star>) s w = (\<delta>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h\<^sup>\<star>) s w);
+       a \<in> A \<and> (\<forall>x\<in>set w. x \<in> A); s \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h\<rbrakk> \<Longrightarrow> (\<delta>\<^sup>\<star>) (\<delta> s a) w = (\<delta>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h\<^sup>\<star>) (\<delta>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h s a) w"
+  proof-
+    fix a w s
+    assume 
+      A1_IH: "(\<And>s. s \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<Longrightarrow> (\<delta>\<^sup>\<star>) s w = (\<delta>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h\<^sup>\<star>) s w)" and
+      A1_0: "a \<in> A \<and> (\<forall>x\<in>set w. x \<in> A)" and
+      A1_1: "s \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h" 
+    have H1_0: "\<delta>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h s a = \<delta> s a"
+      using A1_1
+      apply (clarsimp simp add: reachable_trans_def)
+      apply (rule meta_mp[of "det_aut A S i F \<delta>"])
+      using det_aut.trans_func_ext[where labels = A and states = S and
+          init_state = i and fin_states = F and trans_func = \<delta>]
+       apply (simp add: extensional_def)
+      by (auto simp add: A1_0)
+    show "(\<delta>\<^sup>\<star>) (\<delta> s a) w = (\<delta>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h\<^sup>\<star>) (\<delta>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h s a) w"
+      apply (simp add: H1_0)
+      apply (rule A1_IH[where s1 = "\<delta> s a"])
+      using A1_0 A1_1
+      apply (simp add: reachable_states_def)
+      by (metis Cons_in_lists_iff append_Nil2 append_in_lists_conv is_aut.give_input_closed
+          is_aut.init_state_is_a_state is_aut.trans_to_charact)
+  qed
+  show ?thesis
+    apply (clarsimp simp del: GMN_simps simp add: reach_det_G_aut_rec_lang_def
+        det_G_aut_rec_lang_def det_G_aut_def det_aut_def)
+    apply (intro conjI)
+    subgoal
+      apply (simp add: reachable_states_def) 
+      by (meson give_input.simps(1) lists.Nil)
+           apply (simp add: H_0)
+    using labels_a_G_set.alt_grp_act_axioms
+          apply (auto)[1]
+         apply (rule H_6)
+    subgoal
+      apply (clarsimp simp add: eq_var_subset_def eq_var_subset_axioms_def)
+      apply (rule conjI)
+      using H_6
+       apply (auto)[1]
+      apply (simp del: add: reachable_states_def)[1] 
+      apply (clarify; rule subset_antisym; simp add: Set.subset_eq; clarify)
+       apply (rule conjI)
+      subgoal for g _ w
+        apply (clarsimp simp add: reachable_action_def reachable_states_def) 
+        using accepting_is_eq_var.is_equivar
+        by blast
+      subgoal for g _ w
+        apply (clarsimp simp add: reachable_action_def reachable_states_def)
+        apply (rule conjI; clarify)
+         apply (auto)[2]
+        by (simp add: H_5)
+      apply (clarsimp simp add: reachable_states_def Int_def reachable_action_def ) 
+      apply (clarsimp simp add: image_def)
+      by (simp add: H_7)
+    subgoal
+      apply (clarsimp simp add: eq_var_subset_def)
+      apply (rule conjI)
+      using H_6
+       apply (auto)[1]
+      apply (clarsimp simp add: eq_var_subset_axioms_def) 
+      apply (simp add: \<open>i \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h\<close>)
+      apply (simp add: reachable_action_def)
+      using \<open>i \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h\<close> init_is_eq_var.is_equivar
+      by fastforce
+    subgoal
+      apply (clarsimp simp add: eq_var_func_def eq_var_func_axioms_def) 
+      apply (intro conjI)
+      using H_6 alt_grp_act.axioms
+        labels_a_G_set.group_action_axioms prod_group_act labels_a_G_set.alt_grp_act_axioms
+         apply blast
+      using H_6
+        apply (auto)[1]
+       apply (rule funcsetI; clarsimp)
+      subgoal for s a
+        apply (clarsimp simp add: reachable_states_def reachable_trans_def)
+        by (metis Cons_in_lists_iff append_Nil2 append_in_lists_conv in_listsI
+            is_aut.give_input_closed is_aut.init_state_is_a_state is_aut.trans_to_charact)
+      apply (intro allI; clarify; rule conjI; intro impI)
+       apply (simp add: H_8)
+      using G_set_equiv H_6 eq_var_subset.is_equivar
+        labels_a_G_set.element_image
+      by fastforce
+     apply (rule meta_mp[of "\<And>w s. w \<in> A\<^sup>\<star> \<Longrightarrow> s \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<Longrightarrow> (\<delta>\<^sup>\<star>) s w = (\<delta>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h\<^sup>\<star>) s w"])
+    subgoal
+      using det_G_aut_rec_lang_axioms
+      apply (clarsimp simp add: det_aut_rec_lang_axioms_def det_aut_rec_lang_def
+          det_G_aut_rec_lang_def det_aut_def)
+      apply (intro conjI)
+      using \<open>i \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h\<close> 
+        apply blast 
+      using H_0
+       apply blast
+      by (metis (mono_tags, lifting) \<open>i \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h\<close> mem_Collect_eq reachable_states_def)
+    subgoal for w s
+      apply (induction w arbitrary: s)
+       apply (clarsimp)
+      apply (simp add: in_lists_conv_set)
+      by (simp add: H_9)
+    apply (clarsimp simp add: reach_det_G_aut_def det_G_aut_def det_aut_def)
+    apply (intro conjI)
+           apply (simp add: \<open>i \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h\<close>)
+          apply (simp add: H_0)
+         apply (simp add: labels_a_G_set.group_action_axioms)
+    using \<open>alt_grp_act G S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h\<close>
+        apply (auto)[1]
+       apply (simp add: \<open>eq_var_subset G S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h (F \<inter> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h)\<close>)
+      apply (simp add: \<open>eq_var_subset G S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h {i}\<close>)
+    using \<open>eq_var_func G (S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<times> A) (\<lambda>g\<in>carrier G. \<lambda>(s, a)\<in>S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<times> A. (\<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g s, \<phi> g a))
+  S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h (\<lambda>(x, y)\<in>S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<times> A. \<delta>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h x y)\<close>
+     apply blast
+    apply (simp add: reach_det_aut_axioms_def reach_det_aut_def reachable_states_def)
+    apply (rule meta_mp[of "\<And>s input. s \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<Longrightarrow> input \<in> A\<^sup>\<star> \<Longrightarrow>
+   (\<delta>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h\<^sup>\<star>) s input = (\<delta>\<^sup>\<star>) s input"])
+    using \<open>i \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h\<close>
+     apply (metis (no_types, lifting) \<open>(\<And>w s. \<lbrakk>w \<in> A\<^sup>\<star>; s \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h\<rbrakk> \<Longrightarrow>
+    (\<delta>\<^sup>\<star>) s w = (\<delta>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h\<^sup>\<star>) s w) \<Longrightarrow> det_aut_rec_lang A S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h i (F \<inter> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h) \<delta>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h L\<close> det_aut_rec_lang_def
+        reachable_states_def)
+    by (simp add: \<open>\<And>w s. \<lbrakk>w \<in> A\<^sup>\<star>; s \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h\<rbrakk> \<Longrightarrow> (\<delta>\<^sup>\<star>) s w = (\<delta>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h\<^sup>\<star>) s w\<close>)
+qed
+
+subsection \<open>
+Syntactic Automaton
+\<close>
+context language begin
+
+definition
+  rel_MN :: "('alpha list \<times> 'alpha list) set" ("\<equiv>\<^sub>M\<^sub>N")
+  where "rel_MN = {(w, w') \<in> (A\<^sup>\<star>)\<times>(A\<^sup>\<star>). (\<forall>v \<in> A\<^sup>\<star>. (w @ v) \<in> L \<longleftrightarrow> (w' @ v) \<in> L)}"
+
+lemma MN_rel_equival:
+  "equiv (A\<^sup>\<star>) rel_MN"
+  by (auto simp add: rel_MN_def equiv_def refl_on_def sym_def trans_def)
+
+abbreviation
+  MN_equiv
+  where "MN_equiv \<equiv> A\<^sup>\<star> // rel_MN" 
+
+definition
+  alt_natural_map_MN :: "'alpha list \<Rightarrow> 'alpha list set " ("[_]\<^sub>M\<^sub>N")
+  where "[w]\<^sub>M\<^sub>N = rel_MN `` {w}"
+
+definition
+  MN_trans_func :: "('alpha list set) \<Rightarrow> 'alpha \<Rightarrow> 'alpha list set" ("\<delta>\<^sub>M\<^sub>N")
+  where "MN_trans_func W' a' =
+     (\<lambda>(W,a) \<in> MN_equiv \<times> A. rel_MN `` {(SOME w. w \<in> W) @ [a]}) (W', a')"
+
+abbreviation
+  MN_init_state
+  where "MN_init_state \<equiv> [Nil::'alpha list]\<^sub>M\<^sub>N"
+
+abbreviation
+  MN_fin_states
+  where "MN_fin_states \<equiv> {v. \<exists>w\<in>L. v = [w]\<^sub>M\<^sub>N}"
+
+lemmas 
+  alt_natural_map_MN_def [simp, GMN_simps]
+  MN_trans_func_def [simp, GMN_simps]
+end
+
+context G_lang begin
+adhoc_overloading
+  star induced_star_map
+
+lemma MN_quot_act_wd:
+  "w' \<in> [w]\<^sub>M\<^sub>N \<Longrightarrow> \<forall>g \<in> carrier G. (g \<odot> \<^bsub>\<phi>\<^sup>\<star>\<^esub> w') \<in> [g \<odot> \<^bsub>\<phi>\<^sup>\<star>\<^esub> w]\<^sub>M\<^sub>N"
+proof-
+  assume A_0: "w' \<in> [w]\<^sub>M\<^sub>N"
+  have H_0: "\<And>g. \<lbrakk>(w, w') \<in> \<equiv>\<^sub>M\<^sub>N; g \<in> carrier G; group_hom G (BijGroup A) \<phi>;
+  group_hom G (BijGroup (A\<^sup>\<star>)) (\<lambda>g\<in>carrier G. restrict (map (\<phi> g)) (A\<^sup>\<star>)); L \<subseteq> A\<^sup>\<star>;
+  \<forall>g\<in>carrier G. map (\<phi> g) ` (L \<inter> A\<^sup>\<star>) \<union> (\<lambda>x. undefined) ` (L \<inter> {x. x \<notin> A\<^sup>\<star>}) = L;
+    \<forall>x\<in>set w. x \<in> A; w' \<in> A\<^sup>\<star>\<rbrakk> \<Longrightarrow> (map (\<phi> g) w, map (\<phi> g) w') \<in> \<equiv>\<^sub>M\<^sub>N"
+  proof-
+    fix g
+    assume
+      A1_0: "(w, w') \<in> \<equiv>\<^sub>M\<^sub>N" and
+      A1_1: "g \<in> carrier G" and
+      A1_2: "group_hom G (BijGroup A) \<phi>" and
+      A1_3: "group_hom G (BijGroup (lists A)) (\<lambda>g\<in>carrier G. restrict (map (\<phi> g)) (lists A))" and
+      A1_4: "L \<subseteq> lists A" and
+      A1_5: "\<forall>g\<in>carrier G.
+           map (\<phi> g) ` (L \<inter> lists A) \<union> (\<lambda>x. undefined) ` (L \<inter> {x. x \<notin> lists A}) = L" and
+      A1_6: "\<forall>x\<in>set w. x \<in> A" and
+      A1_7: "w' \<in> A\<^sup>\<star>"
+    have H1_0: "\<And>v w w'. \<lbrakk>g \<in> carrier G; group_hom G (BijGroup A) \<phi>;
+    group_hom G (BijGroup (lists A)) (\<lambda>g\<in>carrier G. restrict (map (\<phi> g)) (lists A));
+    L \<subseteq> lists A; \<forall>g\<in>carrier G.
+    {y. \<exists>x\<in>L \<inter> lists A. y = map (\<phi> g) x} \<union> {y. y = undefined \<and> (\<exists>x. x \<in> L \<and> x \<notin> lists A)} = L;
+    \<forall>x\<in>set w. x \<in> A; \<forall>v\<in>lists A. (w @ v \<in> L) = (w' @ v \<in> L); \<forall>x\<in>set w'. x \<in> A; \<forall>x\<in>set v. x \<in> A;
+    map (\<phi> g) w @ v \<in> L\<rbrakk> \<Longrightarrow> map (\<phi> g) w' @ v \<in> L"
+    proof -
+      fix v w w'
+      assume
+        A2_0: "g \<in> carrier G" and
+        A2_1: "L \<subseteq> A\<^sup>\<star>" and
+        A2_2: "group_hom G (BijGroup A) \<phi>" and
+        A2_3: "group_hom G (BijGroup (A\<^sup>\<star>)) (\<lambda>g\<in>carrier G. restrict (map (\<phi> g)) (A\<^sup>\<star>))" and
+        A2_4: "\<forall>g\<in>carrier G. {y. \<exists>x\<in>L \<inter> A\<^sup>\<star>. y = map (\<phi> g) x} \<union>
+        {y. y = undefined \<and> (\<exists>x. x \<in> L \<and> x \<notin> A\<^sup>\<star>)} = L" and
+        A2_5: "\<forall>x\<in>set w. x \<in> A" and
+        A2_6: "\<forall>x\<in>set w'. x \<in> A" and
+        A2_7: "\<forall>v\<in>A\<^sup>\<star>. (w @ v \<in> L) = (w' @ v \<in> L)" and
+        A2_8: "\<forall>x\<in>set v. x \<in> A" and
+        A2_9: "map (\<phi> g) w @ v \<in> L" 
+      have H2_0: "\<forall>g\<in>carrier G. {y. \<exists>x\<in>L \<inter> A\<^sup>\<star>. y = map (\<phi> g) x} = L"
+        using A2_1 A2_4 subset_eq
+        by (smt (verit, ccfv_SIG) Collect_mono_iff sup.orderE)
+      hence H2_1: "\<forall>g\<in>carrier G. {y. \<exists>x\<in>L. y = map (\<phi> g) x} = L"
+        using A2_1 inf.absorb_iff1
+        by (smt (verit, ccfv_SIG) Collect_cong)
+      hence H2_2: "\<forall>g\<in>carrier G.\<forall>x\<in>L. map (\<phi> g) x \<in> L"
+        by auto
+      from A2_2 have H2_3: "\<forall>h\<in>carrier G. \<forall>a\<in>A. (\<phi> h) a \<in> A"
+        by (auto simp add: group_hom_def BijGroup_def group_hom_axioms_def hom_def Bij_def
+            bij_betw_def)
+      from A2_8 have H2_4: "v \<in> lists A"
+        by (simp add: in_listsI)
+      hence H2_5: "\<forall>h\<in>carrier G. map (\<phi> h) v \<in> lists A"
+        using H2_3
+        by fastforce
+      hence H2_6: "\<forall>h\<in>carrier G. (w @ (map (\<phi> h) v) \<in> L) = (w' @ (map (\<phi> h) v) \<in> L)"
+        using A2_7
+        by force
+      hence H2_7: "(w @ (map (\<phi> (inv\<^bsub>G\<^esub> g)) v) \<in> L) = (w' @ (map (\<phi> (inv\<^bsub>G\<^esub> g)) v) \<in> L)"
+        using A2_0
+        by (meson A2_7 A2_1 append_in_lists_conv in_mono)
+      have "(map (\<phi> g) w) \<in> (A\<^sup>\<star>)"
+        using A2_0 A2_2 A2_5 H2_3
+        by (auto simp add: group_hom_def group_hom_axioms_def hom_def BijGroup_def Bij_def
+            bij_betw_def)
+      hence H2_8: "\<forall>w\<in>A\<^sup>\<star>. \<forall>g\<in>carrier G. map (\<phi> (inv\<^bsub>G\<^esub> g)) ((map (\<phi> g) w) @ v) =
+      w @ (map (\<phi> (inv\<^bsub>G\<^esub> g)) v)"
+        using act_maps_n_distrib triv_act_map A2_0 A2_2 A2_3 H2_4
+        apply (clarsimp)
+        by (smt (verit, del_insts) comp_apply group_action.intro group_action.orbit_sym_aux map_idI)
+      have H2_9: "map (\<phi> (inv\<^bsub>G\<^esub> g)) ((map (\<phi> g) w) @ v) \<in> L"
+        using A2_9 H2_1 H2_2 A2_1
+        apply clarsimp
+        by (metis A2_0 A2_2 group.inv_closed group_hom.axioms(1) list.map_comp map_append)
+      hence H2_10: "w @ (map (\<phi> (inv\<^bsub>G\<^esub> g)) v) \<in> L"
+        using H2_8 A2_0
+        by (auto simp add: A2_5 in_listsI)
+      hence H2_11: "w' @ (map (\<phi> (inv\<^bsub>G\<^esub> g)) v) \<in> L"
+        using H2_7
+        by simp
+      hence H2_12: "map (\<phi> (inv\<^bsub>G\<^esub> g)) ((map (\<phi> g) w') @ v) \<in> L"
+        using A2_0 H2_8 A2_1 subsetD
+        by (metis append_in_lists_conv)
+      have H2_13: "\<forall>g\<in>carrier G. restrict (map (\<phi> g)) (A\<^sup>\<star>) \<in> Bij (A\<^sup>\<star>)"
+        using alt_grp_act.lists_a_Gset[where G = "G" and X = "A" and \<phi> = "\<phi>"] A1_3 
+        by (auto simp add: group_action_def
+            group_hom_def group_hom_axioms_def Pi_def hom_def BijGroup_def)
+      have H2_14: "\<forall>g\<in>carrier G. restrict (map (\<phi> g)) L ` L = L"
+        using H2_2
+        apply (clarsimp simp add: Set.image_def)
+        using H2_1
+        by blast
+      have H2_15: "map (\<phi> g) w' \<in> lists A"
+        using A2_0 A2_1 H2_13 H2_2 
+        by (metis H2_11 append_in_lists_conv image_eqI lists_image subset_eq surj_prop)
+      have H2_16: "inv\<^bsub>G\<^esub> g \<in> carrier G" 
+        by (metis A2_0 A2_2 group.inv_closed group_hom.axioms(1))
+      thus "map (\<phi> g) w' @ v \<in> L" 
+        using A2_0 A2_1 A2_2 H2_4 H2_12 H2_13 H2_14 H2_15 H2_16 group.inv_closed group_hom.axioms(1)
+          alt_grp_act.lists_a_Gset[where G = "G" and X = "A" and \<phi> = "\<phi>"] 
+          pre_image_lemma[where S = "L" and T = "A\<^sup>\<star>" and f = "map (\<phi> (inv\<^bsub>G\<^esub> g))" and
+            x = "((map (\<phi> g) w') @ v)"]
+        apply (clarsimp simp add: group_action_def) 
+        by (smt (verit, best) A2_1 FuncSet.restrict_restrict H2_14 H2_15 H2_16 H2_4
+            append_in_lists_conv inf.absorb_iff2 map_append map_map pre_image_lemma restrict_apply'
+            restrict_apply') 
+    qed
+    show "(map (\<phi> g) w, map (\<phi> g) w') \<in> \<equiv>\<^sub>M\<^sub>N"
+      apply (clarsimp simp add: rel_MN_def Set.image_def)
+      apply (intro conjI)
+      using A1_1 A1_6 group_action.surj_prop group_action_axioms
+        apply fastforce
+      using A1_1 A1_7 image_iff surj_prop
+       apply fastforce
+      apply (clarify; rule iffI)
+      subgoal for v
+        apply (rule H1_0[where v1 ="v" and w1 = "w" and w'1 = "w'"])
+        using A1_0 A1_1 A1_2 A1_3 A1_4 A1_5 A1_6 A1_7
+        by (auto simp add: rel_MN_def Set.image_def)
+      apply (rule H1_0[where w1 = "w'" and w'1 = "w"])
+      using A1_0 A1_1 A1_2 A1_3 A1_4 A1_5 A1_6 A1_7
+      by (auto simp add: rel_MN_def Set.image_def)
+  qed
+  show ?thesis
+    using G_lang_axioms A_0
+    apply (clarsimp simp add: G_lang_def G_lang_axioms_def eq_var_subset_def
+        eq_var_subset_axioms_def alt_grp_act_def group_action_def)
+    apply (intro conjI; clarify)
+     apply (rule conjI; rule impI)
+      apply (simp add: H_0)
+    by (auto simp add: rel_MN_def)
+qed
+
+text \<open>
+The following lemma corresponds to lemma 3.4 from \cite{bojanczyk2014automata}:
+\<close>
+
+lemma MN_rel_eq_var:
+  "eq_var_rel G (A\<^sup>\<star>) (\<phi>\<^sup>\<star>) \<equiv>\<^sub>M\<^sub>N"
+  apply (clarsimp simp add: eq_var_rel_def alt_grp_act_def eq_var_rel_axioms_def)
+  apply (intro conjI)
+    apply (metis L_is_equivar alt_grp_act.axioms eq_var_subset.axioms(1) induced_star_map_def)
+  using L_is_equivar
+   apply (simp add: rel_MN_def eq_var_subset_def eq_var_subset_axioms_def)
+   apply fastforce
+  apply (clarify)
+  apply (intro conjI; rule impI; rule conjI; rule impI)
+     apply (simp add: in_lists_conv_set)
+     apply (clarsimp simp add: rel_MN_def)
+     apply (intro conjI)
+       apply (clarsimp simp add: rel_MN_def)
+  subgoal for w v g w'
+    using L_is_equivar
+    apply (clarsimp simp add: restrict_def eq_var_subset_def eq_var_subset_axioms_def)
+    by (meson element_image)
+      apply(metis image_mono in_listsI in_mono list.set_map lists_mono subset_code(1) surj_prop)
+     apply (clarify; rule iffI)
+  subgoal for w v g u
+    using G_lang_axioms MN_quot_act_wd[where w = "w" and w'= "v"]
+    by (auto simp add: rel_MN_def G_lang_def G_lang_axioms_def
+        eq_var_subset_def eq_var_subset_axioms_def Set.subset_eq element_image) 
+  subgoal for w v g u
+    using G_lang_axioms MN_quot_act_wd[where w = "w" and w'= "v"]
+    by (auto simp add: rel_MN_def G_lang_def G_lang_axioms_def
+        eq_var_subset_def eq_var_subset_axioms_def Set.subset_eq element_image) 
+  using G_lang_axioms MN_quot_act_wd
+  by (auto simp add: rel_MN_def G_lang_def G_lang_axioms_def
+      eq_var_subset_def eq_var_subset_axioms_def Set.subset_eq element_image)
+
+lemma quot_act_wd_alt_notation:
+  "w \<in> A\<^sup>\<star> \<Longrightarrow> g \<in> carrier G \<Longrightarrow> g \<odot>\<^bsub>[\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>\<^esub> ([w]\<^sub>M\<^sub>N) = ([g \<odot>\<^bsub>\<phi>\<^sup>\<star>\<^esub> w]\<^sub>M\<^sub>N)"
+  using eq_var_rel.quot_act_wd[where G = G and \<phi> = "\<phi>\<^sup>\<star>" and X = "A\<^sup>\<star>" and R = "\<equiv>\<^sub>M\<^sub>N" and x = w
+      and g = g]
+  by (simp del: GMN_simps add: alt_natural_map_MN_def MN_rel_eq_var MN_rel_equival)
+
+lemma MN_trans_func_characterization:
+  "v \<in> (A\<^sup>\<star>) \<Longrightarrow> a \<in> A \<Longrightarrow> \<delta>\<^sub>M\<^sub>N [v]\<^sub>M\<^sub>N a = [v @ [a]]\<^sub>M\<^sub>N"
+proof-
+  assume
+    A_0: "v \<in> (A\<^sup>\<star>)" and
+    A_1: "a \<in> A"
+  have H_0: "\<And>u. u \<in> [v]\<^sub>M\<^sub>N \<Longrightarrow> (u @ [a]) \<in> [v @ [a]]\<^sub>M\<^sub>N"
+    by (auto simp add: rel_MN_def A_1 A_0)
+  hence H_1: "(SOME w. (v, w) \<in> \<equiv>\<^sub>M\<^sub>N) \<in> [v]\<^sub>M\<^sub>N \<Longrightarrow> ((SOME w. (v, w) \<in> \<equiv>\<^sub>M\<^sub>N) @ [a]) \<in> [v @ [a]]\<^sub>M\<^sub>N"
+    by auto
+  from A_0 have "(v, v) \<in> \<equiv>\<^sub>M\<^sub>N \<and> v \<in> [v]\<^sub>M\<^sub>N"
+    by (auto simp add: rel_MN_def)
+  hence H_2: "(SOME w. (v, w) \<in> \<equiv>\<^sub>M\<^sub>N) \<in> [v]\<^sub>M\<^sub>N"
+    apply (clarsimp simp add: rel_MN_def)
+    apply (rule conjI)
+     apply (smt (verit, ccfv_SIG) A_0 in_listsD verit_sko_ex_indirect)
+    by (smt (verit, del_insts) A_0 in_listsI tfl_some)
+  hence H_3: " ((SOME w. (v, w) \<in> \<equiv>\<^sub>M\<^sub>N) @ [a]) \<in> [v @ [a]]\<^sub>M\<^sub>N"
+    using H_1
+    by simp
+  thus "\<delta>\<^sub>M\<^sub>N [v]\<^sub>M\<^sub>N a = [v @ [a]]\<^sub>M\<^sub>N"
+    using A_0 A_1 MN_rel_equival
+    apply (clarsimp simp add: equiv_def)
+    apply (rule conjI; rule impI)
+     apply (metis MN_rel_equival equiv_class_eq)
+    by (simp add: A_0 quotientI)
+qed
+
+lemma MN_trans_eq_var_func :
+  "eq_var_func G
+ (MN_equiv \<times> A) (\<lambda>g\<in>carrier G. \<lambda>(W, a) \<in> (MN_equiv \<times> A). (([(\<phi>\<^sup>\<star>)]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>) g W, \<phi> g a))
+  MN_equiv ([(\<phi>\<^sup>\<star>)]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>)
+  (\<lambda>(w, a) \<in> MN_equiv \<times> A. \<delta>\<^sub>M\<^sub>N w a)"
+proof-
+  have H_0: "alt_grp_act G MN_equiv ([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>)"
+    using MN_rel_eq_var MN_rel_equival eq_var_rel.quot_act_is_grp_act 
+      alt_group_act_is_grp_act restrict_apply
+    by fastforce
+  have H_1: "\<And>a b g.
+       a \<in> MN_equiv \<Longrightarrow>
+       b \<in> A \<Longrightarrow>
+       (([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>) g a \<in> MN_equiv \<and> \<phi> g b \<in> A \<longrightarrow>
+       g \<in> carrier G \<longrightarrow> \<delta>\<^sub>M\<^sub>N (([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>) g a) (\<phi> g b) = ([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>) g (\<delta>\<^sub>M\<^sub>N a b)) \<and>
+       ((([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>) g a \<in> MN_equiv \<longrightarrow> \<phi> g b \<notin> A) \<longrightarrow>
+        g \<in> carrier G \<longrightarrow> undefined = ([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>) g (\<delta>\<^sub>M\<^sub>N a b))"
+  proof -
+    fix C a g
+    assume
+      A1_0: "C \<in> MN_equiv" and
+      A1_1: "a \<in> A"
+    have H1_0: "g \<in> carrier G \<Longrightarrow> \<phi> g a \<in> A" 
+      by (meson A1_1 element_image)
+    from A1_0 obtain c where H1_c: "[c]\<^sub>M\<^sub>N = C \<and> c \<in> A\<^sup>\<star>"
+      by (auto simp add: quotient_def) 
+    have H1_1: "g \<in> carrier G \<Longrightarrow> \<delta>\<^sub>M\<^sub>N (([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>) g C) (\<phi> g a) = ([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>) g (\<delta>\<^sub>M\<^sub>N [c]\<^sub>M\<^sub>N a)"
+    proof-
+      assume 
+        A2_0: "g \<in> carrier G" 
+      have H2_0: "\<phi> g a \<in> A"
+        using H1_0 A2_0
+        by simp
+      have H2_1: "(\<phi>\<^sup>\<star>) g \<in> Bij (A\<^sup>\<star>)" using G_lang_axioms lists_a_Gset A2_0
+        apply (clarsimp simp add: G_lang_def G_lang_axioms_def group_action_def
+            group_hom_def hom_def group_hom_axioms_def BijGroup_def image_def)  
+        by (meson Pi_iff restrict_Pi_cancel)
+      hence H2_2: "(\<phi>\<^sup>\<star>) g c \<in> (A\<^sup>\<star>)"
+        using H1_c
+        apply (clarsimp simp add: Bij_def bij_betw_def inj_on_def Image_def image_def)
+        apply (rule conjI; rule impI; clarify)
+        using surj_prop
+         apply fastforce
+        using A2_0
+        by blast
+      from H1_c have H2_1: "([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>) g (\<equiv>\<^sub>M\<^sub>N `` {c}) = ([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>) g C"
+        by auto
+      also have H2_2: "([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>) g C = [(\<phi>\<^sup>\<star>) g c]\<^sub>M\<^sub>N"
+        using eq_var_rel.quot_act_wd[where R = "\<equiv>\<^sub>M\<^sub>N" and G = G and X = "A\<^sup>\<star>" and \<phi> = "\<phi>\<^sup>\<star>" and g = g
+            and x = c]
+        by (clarsimp simp del: GMN_simps simp add: alt_natural_map_MN_def make_op_def MN_rel_eq_var
+            MN_rel_equival H1_c A2_0 H2_1) 
+      hence H2_3: "\<delta>\<^sub>M\<^sub>N (([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>) g C) (\<phi> g a) = \<delta>\<^sub>M\<^sub>N ([(\<phi>\<^sup>\<star>) g c]\<^sub>M\<^sub>N) (\<phi> g a)" 
+        using H2_2 
+        by simp
+      also have H2_4: "... = [((\<phi>\<^sup>\<star>) g c) @ [(\<phi> g a)]]\<^sub>M\<^sub>N"
+        using MN_trans_func_characterization[where v = "(\<phi>\<^sup>\<star>) g c" and a = "\<phi> g a"] H1_c  A2_0
+          G_set_equiv H2_0 eq_var_subset.is_equivar insert_iff lists_a_Gset
+        by blast
+      also have H2_5: "... = [(\<phi>\<^sup>\<star>) g (c @ [a])]\<^sub>M\<^sub>N"
+        using A2_0 H1_c A1_1
+        by auto
+      also have H2_6: "... = ([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>) g [(c @ [a])]\<^sub>M\<^sub>N"
+        apply (rule meta_mp[of "c @ [a] \<in> A\<^sup>\<star>"])
+        using eq_var_rel.quot_act_wd[where R = "\<equiv>\<^sub>M\<^sub>N" and G = G and X = "A\<^sup>\<star>" and \<phi> = "\<phi>\<^sup>\<star>" and g = g
+            and x = "c @ [a]"]
+         apply (clarsimp simp del: GMN_simps simp add: make_op_def MN_rel_eq_var MN_rel_equival H1_c
+            A2_0 H2_1) 
+        using H1_c A1_1
+        by auto 
+      also have H2_7: "... = ([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>) g (\<delta>\<^sub>M\<^sub>N [c]\<^sub>M\<^sub>N a)"
+        using MN_trans_func_characterization[where v = "c" and a = "a"] H1_c A1_1
+        by metis
+      finally show "\<delta>\<^sub>M\<^sub>N (([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>) g C) (\<phi> g a) = ([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>) g (\<delta>\<^sub>M\<^sub>N [c]\<^sub>M\<^sub>N a)"
+        using H2_1 
+        by metis
+    qed
+    show "(([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>) g C \<in> MN_equiv \<and> \<phi> g a \<in> A \<longrightarrow>
+    g \<in> carrier G \<longrightarrow>
+    \<delta>\<^sub>M\<^sub>N (([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>) g C) (\<phi> g a) =
+    ([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>) g (\<delta>\<^sub>M\<^sub>N C a)) \<and>
+    ((([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>) g C \<in> MN_equiv \<longrightarrow> \<phi> g a \<notin> A) \<longrightarrow>
+    g \<in> carrier G \<longrightarrow> undefined = ([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>) g (\<delta>\<^sub>M\<^sub>N C a))"
+      apply (rule conjI; clarify)
+      using H1_1 H1_c
+       apply blast
+      by (metis A1_0 H1_0 H_0 alt_group_act_is_grp_act
+          group_action.element_image)
+  qed
+  show ?thesis
+    apply (subst eq_var_func_def)
+    apply (subst eq_var_func_axioms_def)
+    apply (rule conjI)
+    subgoal 
+      apply (rule prod_group_act[where G = G and A = "MN_equiv" and \<phi> = "[(\<phi>\<^sup>\<star>)]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>"
+            and B = A     and \<psi> = \<phi>])
+       apply (rule H_0)
+      using G_lang_axioms
+      by (auto simp add: G_lang_def G_lang_axioms_def)
+    apply (rule conjI)
+    subgoal
+      using MN_rel_eq_var MN_rel_equival eq_var_rel.quot_act_is_grp_act
+      using alt_group_act_is_grp_act restrict_apply
+      by fastforce
+    apply (rule conjI)
+    subgoal
+      apply (subst extensional_funcset_def)
+      apply (subst restrict_def)
+      apply (subst Pi_def)
+      apply (subst extensional_def)
+      apply (clarsimp)
+      by (metis MN_rel_equival append_in_lists_conv equiv_Eps_preserves lists.Cons lists.Nil
+          quotientI)
+    apply (subst restrict_def)
+    apply (clarsimp simp del: GMN_simps simp add: make_op_def)
+    by (simp add: H_1 del: GMN_simps)
+qed
+
+lemma MN_quot_act_on_empty_str:
+  "\<And>g. \<lbrakk>g \<in> carrier G; ([], x) \<in> \<equiv>\<^sub>M\<^sub>N\<rbrakk> \<Longrightarrow> x \<in> map (\<phi> g) ` \<equiv>\<^sub>M\<^sub>N `` {[]}"
+proof-
+  fix g
+  assume
+    A_0: "g \<in> carrier G" and
+    A_1: "([], x) \<in> \<equiv>\<^sub>M\<^sub>N"
+  from A_1 have H_0: "x \<in> (A\<^sup>\<star>)"
+    by (auto simp add: rel_MN_def)
+  from A_0 H_0 have H_1: "x = (\<phi>\<^sup>\<star>) g ((\<phi>\<^sup>\<star>) (inv \<^bsub>G\<^esub> g) x)"
+    by (smt (verit) alt_grp_act_def group_action.bij_prop1 group_action.orbit_sym_aux lists_a_Gset) 
+  have H_2: "inv \<^bsub>G \<^esub> g \<in> carrier G"
+    using A_0 MN_rel_eq_var
+    by (auto simp add: eq_var_rel_def eq_var_rel_axioms_def group_action_def group_hom_def)
+  have H_3: "([], (\<phi>\<^sup>\<star>) (inv \<^bsub>G\<^esub> g) x) \<in> \<equiv>\<^sub>M\<^sub>N"
+    using A_0 A_1 H_0 MN_rel_eq_var
+    apply (clarsimp simp add: eq_var_rel_def eq_var_rel_axioms_def) 
+    apply (rule conjI; clarify)
+     apply (smt (verit, best) H_0 list.simps(8) lists.Nil)
+    using H_2
+    by simp
+  hence H_4: "\<exists>y\<in>\<equiv>\<^sub>M\<^sub>N `` {[]}. x = map (\<phi> g) y"
+    using A_0 H_0 H_1 H_2
+    apply clarsimp
+    by (metis H_0 Image_singleton_iff insert_iff insert_image lists_image surj_prop)
+  thus "x \<in> map (\<phi> g) ` \<equiv>\<^sub>M\<^sub>N `` {[]}"
+    by (auto simp add: image_def)
+qed
+
+lemma MN_init_state_equivar:
+  "eq_var_subset G (A\<^sup>\<star>) (\<phi>\<^sup>\<star>) MN_init_state"
+  apply (rule alt_grp_act.eq_var_one_direction)
+  using lists_a_Gset
+    apply (auto)[1]
+   apply (clarsimp)
+  subgoal for w a
+    by (auto simp add: rel_MN_def)
+  apply (simp add: Set.subset_eq; clarify)
+  apply (clarsimp simp add: image_def Image_def Int_def) 
+  apply (erule disjE)
+  subgoal for g w
+    using MN_rel_eq_var
+    apply (clarsimp simp add: eq_var_rel_def eq_var_rel_axioms_def)
+    by (metis (full_types, opaque_lifting) in_listsI list.simps(8) lists.Nil)
+  by (auto simp add: \<open>\<And>a w. \<lbrakk>([], w) \<in> \<equiv>\<^sub>M\<^sub>N; a \<in> set w\<rbrakk> \<Longrightarrow> a \<in> A\<close>) 
+
+lemma MN_init_state_equivar_v2:
+  "eq_var_subset G (MN_equiv) ([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star> \<^esub>) {MN_init_state}"
+proof-
+  have H_0: "\<forall>g\<in>carrier G. (\<phi>\<^sup>\<star>) g ` MN_init_state = MN_init_state \<Longrightarrow>
+             \<forall>g\<in>carrier G. ([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>) g MN_init_state = MN_init_state"
+  proof (clarify)
+    fix g 
+    assume
+      A_0: "g \<in> carrier G" 
+    have H_0: "\<And>x. [x]\<^sub>M\<^sub>N = \<equiv>\<^sub>M\<^sub>N `` {x}"
+      by simp
+    have H_1: "([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>) g [[]]\<^sub>M\<^sub>N = [(\<phi>\<^sup>\<star>) g []]\<^sub>M\<^sub>N"
+      using eq_var_rel.quot_act_wd[where R = "\<equiv>\<^sub>M\<^sub>N" and G = G and X = "A\<^sup>\<star>" and \<phi> = "\<phi>\<^sup>\<star>" and g = g
+          and x = "[]"] MN_rel_eq_var MN_rel_equival
+      by (clarsimp simp del: GMN_simps simp add: H_0 make_op_def A_0) 
+    from A_0 H_1 show "([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>) g [[]]\<^sub>M\<^sub>N = [[]]\<^sub>M\<^sub>N"
+      by auto
+  qed
+  show ?thesis
+    using MN_init_state_equivar
+    apply (clarsimp simp add: eq_var_subset_def simp del: GMN_simps)
+    apply (rule conjI)
+    subgoal
+      by (metis MN_rel_eq_var MN_rel_equival eq_var_rel.quot_act_is_grp_act)
+    apply (clarsimp del: subset_antisym simp del: GMN_simps simp add: eq_var_subset_axioms_def)
+    apply (rule conjI)
+     apply (auto simp add: quotient_def)[1]
+    by (simp add: H_0 del: GMN_simps)
+qed
+
+lemma MN_final_state_equiv:
+  "eq_var_subset G (MN_equiv) ([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star> \<^esub>) MN_fin_states"
+proof-
+  have H_0: "\<And>g x w. g \<in> carrier G \<Longrightarrow> w \<in> L \<Longrightarrow> \<exists>wa\<in>L. ([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>) g [w]\<^sub>M\<^sub>N = [wa]\<^sub>M\<^sub>N"
+  proof-
+    fix g w
+    assume
+      A1_0: "g \<in> carrier G" and 
+      A1_1: "w \<in> L" 
+    have H1_0: "\<And>v. v \<in> L \<Longrightarrow> (\<phi>\<^sup>\<star>) g v \<in> L"
+      using A1_0 G_lang_axioms
+      apply (clarsimp simp add: G_lang_def G_lang_axioms_def eq_var_subset_def
+          eq_var_subset_axioms_def)
+      by blast
+    hence H1_1: "(\<phi>\<^sup>\<star>) g w \<in> L"
+      using A1_1
+      by simp
+    from A1_1 have H1_2: "\<And>v. v \<in> [w]\<^sub>M\<^sub>N \<Longrightarrow> v \<in> L"
+      apply (clarsimp simp add: rel_MN_def) 
+      by (metis lists.simps self_append_conv)
+    have H1_3: "([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>) g [w]\<^sub>M\<^sub>N = [(\<phi>\<^sup>\<star>) g w]\<^sub>M\<^sub>N"
+      using eq_var_rel.quot_act_wd[where R = "\<equiv>\<^sub>M\<^sub>N" and G = G and X = "A\<^sup>\<star>" and \<phi> = "\<phi>\<^sup>\<star>" and g = g
+          and x = "w"] MN_rel_eq_var MN_rel_equival G_lang_axioms
+      by (clarsimp simp add: A1_0 A1_1 G_lang_axioms_def G_lang_def eq_var_subset_def
+          eq_var_subset_axioms_def subset_eq) 
+    show "\<exists>wa\<in>L. ([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>) g [w]\<^sub>M\<^sub>N = [wa]\<^sub>M\<^sub>N"
+      using H1_1 H1_3
+      by blast
+  qed
+  show ?thesis
+    apply (rule alt_grp_act.eq_var_one_direction) 
+    using MN_init_state_equivar_v2 eq_var_subset.axioms(1)
+      apply blast
+     apply (clarsimp)
+    subgoal for w
+      using G_lang_axioms
+      by (auto simp add: quotient_def G_lang_axioms_def G_lang_def eq_var_subset_def
+          eq_var_subset_axioms_def)
+    apply (simp add: Set.subset_eq del: GMN_simps; clarify)
+    by (simp add: H_0 del: GMN_simps)
+qed
+
+interpretation syntac_aut :
+  det_aut "A" "MN_equiv" "MN_init_state" "MN_fin_states" "MN_trans_func"
+proof-
+  have H_0: "\<And>state label. state \<in> MN_equiv \<Longrightarrow> label \<in> A \<Longrightarrow> \<delta>\<^sub>M\<^sub>N state label \<in> MN_equiv"
+  proof -
+    fix state label
+    assume
+      A_0: "state \<in> MN_equiv" and
+      A_1: "label \<in> A" 
+    obtain w where H_w: "state = [w]\<^sub>M\<^sub>N \<and> w \<in> A\<^sup>\<star>"
+      by (metis A_0 alt_natural_map_MN_def quotientE)
+    have H_0: "\<delta>\<^sub>M\<^sub>N [w]\<^sub>M\<^sub>N label = [w @ [label]]\<^sub>M\<^sub>N"
+      using MN_trans_func_characterization[where v = w and a = label] H_w A_1
+      by simp
+    have H_1: "\<And>v. v \<in> A\<^sup>\<star> \<Longrightarrow> [v]\<^sub>M\<^sub>N \<in> MN_equiv"
+      by (simp add: in_listsI quotientI)
+    show "\<delta>\<^sub>M\<^sub>N state label \<in> MN_equiv"
+      using H_w H_0 H_1 
+      by (simp add: A_1)
+  qed
+  show "det_aut A MN_equiv MN_init_state MN_fin_states \<delta>\<^sub>M\<^sub>N"
+    apply (clarsimp simp del: GMN_simps simp add: det_aut_def alt_natural_map_MN_def)
+    apply (intro conjI)
+      apply (auto simp add: quotient_def)[1]
+    using G_lang_axioms
+     apply (auto simp add: quotient_def G_lang_axioms_def G_lang_def
+        eq_var_subset_def eq_var_subset_axioms_def)[1]
+    apply (auto simp add: extensional_def PiE_iff simp del: MN_trans_func_def)[1]
+        apply (simp add: H_0 del: GMN_simps)
+    by auto
+qed
+
+corollary syth_aut_is_det_aut:
+  "det_aut A MN_equiv MN_init_state MN_fin_states \<delta>\<^sub>M\<^sub>N"
+  using local.syntac_aut.det_aut_axioms
+  by simp
+
+lemma give_input_transition_func:
+  "w \<in> (A\<^sup>\<star>) \<Longrightarrow> \<forall>v \<in> (A\<^sup>\<star>). [v @ w]\<^sub>M\<^sub>N = (\<delta>\<^sub>M\<^sub>N\<^sup>\<star>) [v]\<^sub>M\<^sub>N w"
+proof-
+  assume 
+    A_0: "w \<in> A\<^sup>\<star>"
+  have H_0: "\<And> a w v. \<lbrakk>a \<in> A; w \<in> A\<^sup>\<star>; \<forall>v\<in>A\<^sup>\<star>. [v @ w]\<^sub>M\<^sub>N = (\<delta>\<^sub>M\<^sub>N\<^sup>\<star>) [v]\<^sub>M\<^sub>N w; v \<in> A\<^sup>\<star>\<rbrakk> \<Longrightarrow>
+            [v @ a # w]\<^sub>M\<^sub>N = (\<delta>\<^sub>M\<^sub>N\<^sup>\<star>) [v]\<^sub>M\<^sub>N (a # w)"
+  proof-
+    fix a w v
+    assume
+      A1_IH: "\<forall>v\<in> A\<^sup>\<star>. [v @ w]\<^sub>M\<^sub>N = (\<delta>\<^sub>M\<^sub>N\<^sup>\<star>) [v]\<^sub>M\<^sub>N w" and
+      A1_0: "a \<in> A" and
+      A1_1: "v \<in> A\<^sup>\<star>" and
+      A1_2: "w \<in> A\<^sup>\<star>"
+    from A1_IH A1_1 A1_2 have H1_1: "[v @ w]\<^sub>M\<^sub>N = (\<delta>\<^sub>M\<^sub>N\<^sup>\<star>) [v]\<^sub>M\<^sub>N w"
+      by auto
+    have H1_2: "[(v @ [a]) @ w]\<^sub>M\<^sub>N = (\<delta>\<^sub>M\<^sub>N\<^sup>\<star>) [v @ [a]]\<^sub>M\<^sub>N w"
+      apply (rule meta_mp[of "(v @ [a]) \<in> (A\<^sup>\<star>)"])
+      using A1_IH A1_2 H1_1
+       apply blast 
+      using A1_0 A1_1
+      by auto
+    have H1_3: "\<delta>\<^sub>M\<^sub>N [v]\<^sub>M\<^sub>N a = [v @ [a]]\<^sub>M\<^sub>N"
+      using MN_trans_func_characterization[where a = a] A1_0 A1_1
+      by auto
+    hence H1_4: "[v @ a # w]\<^sub>M\<^sub>N = (\<delta>\<^sub>M\<^sub>N\<^sup>\<star>) [v @ [a]]\<^sub>M\<^sub>N w"
+      using H1_2
+      by auto
+    also have H1_5: "... = (\<delta>\<^sub>M\<^sub>N\<^sup>\<star>) (\<delta>\<^sub>M\<^sub>N [v]\<^sub>M\<^sub>N a) w"
+      using H1_4 H1_3 A1_1
+      by auto 
+    thus "[v @ a # w]\<^sub>M\<^sub>N = (\<delta>\<^sub>M\<^sub>N\<^sup>\<star>) [v]\<^sub>M\<^sub>N (a # w)"
+      using calculation
+      by auto
+  qed
+  from A_0 show ?thesis
+    apply (induction w)
+     apply (auto)[1] 
+    by (simp add: H_0 del: GMN_simps)
+qed
+  
+
+lemma MN_unique_init_state:
+  "w \<in> (A\<^sup>\<star>) \<Longrightarrow> [w]\<^sub>M\<^sub>N = (\<delta>\<^sub>M\<^sub>N\<^sup>\<star>) [Nil]\<^sub>M\<^sub>N w"
+  using give_input_transition_func[where w = w] 
+  by (metis append_self_conv2 lists.Nil)
+
+lemma fin_states_rep_by_lang:
+  "w \<in> A\<^sup>\<star> \<Longrightarrow> [w]\<^sub>M\<^sub>N \<in> MN_fin_states \<Longrightarrow> w \<in> L"
+proof-
+  assume
+    A_0: "w \<in> A\<^sup>\<star>" and
+    A_1: "[w]\<^sub>M\<^sub>N \<in> MN_fin_states"
+  from A_1 have H_0: "\<exists>w'\<in>[w]\<^sub>M\<^sub>N. w' \<in> L"
+    apply (clarsimp) 
+    by (metis A_0 MN_rel_equival equiv_class_self proj_def proj_in_iff)
+  from H_0 obtain w' where H_w': "w'\<in>[w]\<^sub>M\<^sub>N \<and> w' \<in> L" 
+    by auto
+  have H_1: "\<And>v. v \<in> A\<^sup>\<star> \<Longrightarrow> w'@v \<in> L \<Longrightarrow> w@v \<in> L"
+    using H_w' A_1 A_0
+    by (auto simp add: rel_MN_def)
+  show "w \<in> L"
+    using H_1 H_w'
+    apply clarify 
+    by (metis append_Nil2 lists.Nil)
+qed
+
+text \<open>
+The following lemma corresponds to lemma 3.6 from \cite{bojanczyk2014automata}:
+\<close>
+
+lemma syntactic_aut_det_G_aut:
+  "det_G_aut A MN_equiv MN_init_state MN_fin_states MN_trans_func G \<phi> ([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>)"
+  apply (clarsimp simp add: det_G_aut_def simp del: GMN_simps)
+  apply (intro conjI)
+  using syth_aut_is_det_aut
+       apply (auto)[1]
+  using alt_grp_act_axioms
+      apply (auto)[1]
+  using MN_init_state_equivar_v2 eq_var_subset.axioms(1)
+     apply blast
+  using MN_final_state_equiv
+    apply presburger
+  using MN_init_state_equivar_v2
+   apply presburger
+  using MN_trans_eq_var_func
+  by linarith
+
+lemma syntactic_aut_det_G_aut_rec_L:
+  "det_G_aut_rec_lang A MN_equiv MN_init_state MN_fin_states MN_trans_func G \<phi>
+ ([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>) L"
+  apply (clarsimp simp add: det_G_aut_rec_lang_def det_aut_rec_lang_axioms_def
+      det_aut_rec_lang_def simp del: GMN_simps)
+  apply (intro conjI)
+  using syntactic_aut_det_G_aut syth_aut_is_det_aut
+    apply (auto)[1]
+  using syntactic_aut_det_G_aut syth_aut_is_det_aut
+   apply (auto)[1]
+  apply (rule allI; rule iffI)
+   apply (rule conjI)
+  using L_is_equivar eq_var_subset.is_subset image_iff image_mono insert_image insert_subset
+    apply blast
+  using MN_unique_init_state L_is_equivar eq_var_subset.is_subset
+   apply blast
+  using MN_unique_init_state fin_states_rep_by_lang in_lists_conv_set 
+  by (smt (verit) mem_Collect_eq)
+
+lemma syntact_aut_is_reach_aut_rec_lang:
+  "reach_det_G_aut_rec_lang A MN_equiv MN_init_state MN_fin_states MN_trans_func G \<phi>
+ ([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>) L"
+  apply (clarsimp simp del: GMN_simps simp add: reach_det_G_aut_rec_lang_def
+      det_G_aut_rec_lang_def det_aut_rec_lang_axioms_def reach_det_G_aut_def
+      reach_det_aut_def reach_det_aut_axioms_def det_G_aut_def det_aut_rec_lang_def)
+  apply (intro conjI)
+  using syth_aut_is_det_aut
+                 apply blast
+  using alt_grp_act_axioms
+                apply (auto)[1]
+  subgoal
+    using MN_init_state_equivar_v2 eq_var_subset.axioms(1)
+    by blast
+  using MN_final_state_equiv
+              apply presburger
+  using MN_init_state_equivar_v2
+  subgoal
+    by presburger
+  using MN_trans_eq_var_func
+            apply linarith
+  using syth_aut_is_det_aut
+           apply (auto)[1]
+          apply (metis (mono_tags, lifting) G_lang.MN_unique_init_state G_lang_axioms
+      det_G_aut_rec_lang_def det_aut_rec_lang.is_recognised syntactic_aut_det_G_aut_rec_L)
+  using syth_aut_is_det_aut
+         apply (auto)[1]
+  using alt_grp_act_axioms
+        apply (auto)[1]
+  using \<open>alt_grp_act G MN_equiv ([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>)\<close>
+       apply blast
+  using \<open>eq_var_subset G MN_equiv ([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>) MN_fin_states\<close>
+      apply blast
+  using \<open>eq_var_subset G MN_equiv ([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>) {MN_init_state}\<close>
+     apply blast
+  using MN_trans_eq_var_func
+    apply blast
+  using syth_aut_is_det_aut
+   apply auto[1]
+  by (metis MN_unique_init_state alt_natural_map_MN_def quotientE)
+end
+
+subsection \<open>
+Proving the Myhill-Nerode Theorem for $G$-Automata
+\<close>
+context det_G_aut begin
+no_adhoc_overloading
+  star labels_a_G_set.induced_star_map
+end
+
+context reach_det_G_aut_rec_lang begin
+adhoc_overloading
+  star labels_a_G_set.induced_star_map
+
+definition
+  states_to_words :: "'states \<Rightarrow> 'alpha list"
+  where "states_to_words = (\<lambda>s \<in> S. SOME w. w \<in> A\<^sup>\<star> \<and> ((\<delta>\<^sup>\<star>) i w = s))"
+
+definition
+  words_to_syth_states :: "'alpha list \<Rightarrow> 'alpha list set"
+  where "words_to_syth_states w = [w]\<^sub>M\<^sub>N"
+
+definition
+  induced_epi:: "'states \<Rightarrow> 'alpha list set"
+  where "induced_epi = compose S words_to_syth_states states_to_words"
+
+lemma induced_epi_wd1:
+  "s \<in> S \<Longrightarrow> \<exists>w. w \<in> A\<^sup>\<star> \<and> ((\<delta>\<^sup>\<star>) i w = s)"
+  using reach_det_G_aut_rec_lang_axioms is_reachable
+  by auto
+
+lemma induced_epi_wd2:
+  "w \<in> A\<^sup>\<star> \<Longrightarrow> w' \<in> A\<^sup>\<star> \<Longrightarrow> (\<delta>\<^sup>\<star>) i w = (\<delta>\<^sup>\<star>) i w' \<Longrightarrow> [w]\<^sub>M\<^sub>N = [w']\<^sub>M\<^sub>N"
+proof- 
+  assume
+    A_0: "w \<in> A\<^sup>\<star>" and
+    A_1: "w' \<in> A\<^sup>\<star>" and
+    A_2: "(\<delta>\<^sup>\<star>) i w = (\<delta>\<^sup>\<star>) i w'"
+  have H_0: "\<And>v. v \<in> A\<^sup>\<star> \<Longrightarrow> w @ v \<in> L \<longleftrightarrow> w' @ v \<in> L"
+    apply clarify 
+    by (smt (z3) A_0 A_1 A_2 append_in_lists_conv is_aut.eq_pres_under_concat
+        is_aut.init_state_is_a_state is_lang is_recognised subsetD)+
+  show "[w]\<^sub>M\<^sub>N = [w']\<^sub>M\<^sub>N "
+    apply (simp add: rel_MN_def)
+    using H_0 A_0 A_1
+    by auto
+qed
+
+lemma states_to_words_on_final:
+  "states_to_words \<in> (F \<rightarrow> L)"
+proof-
+  have H_0: "\<And>x. x \<in> F \<Longrightarrow> x \<in> S \<Longrightarrow> (SOME w. w \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i w = x) \<in> L"
+  proof-
+    fix s
+    assume
+      A1_0: "s \<in> F"
+    have H1_0: "\<exists>w. w \<in> lists A \<and> (\<delta>\<^sup>\<star>) i w = s"
+      using A1_0 is_reachable 
+      by (metis is_aut.fin_states_are_states subsetD)
+    have H1_1: "\<And>w. w \<in> lists A \<and> (\<delta>\<^sup>\<star>) i w = s \<Longrightarrow> w \<in> L"
+      using A1_0 is_recognised
+      by auto
+    show "(SOME w. w \<in> lists A \<and> (\<delta>\<^sup>\<star>) i w = s) \<in> L "
+      by (metis (mono_tags, lifting) H1_0 H1_1 someI_ex)
+  qed
+  show ?thesis
+    apply (clarsimp simp add: states_to_words_def)
+    apply (rule conjI; rule impI)
+     apply ( simp add: H_0)
+    using is_aut.fin_states_are_states
+    by blast
+qed
+
+
+lemma induced_epi_eq_var:
+  "eq_var_func G S \<psi> MN_equiv ([(\<phi>\<^sup>\<star>)]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>) induced_epi"
+proof-
+  have H_0: "\<And> s g. \<lbrakk>s \<in> S; g \<in> carrier G; \<psi> g s \<in> S\<rbrakk> \<Longrightarrow>
+    words_to_syth_states (states_to_words (\<psi> g s)) =
+    ([(\<phi>\<^sup>\<star>)]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>) g (words_to_syth_states (states_to_words s))"
+  proof-
+    fix s g 
+    assume
+      A1_0: "s \<in> S" and
+      A1_1: "g \<in> carrier G" and
+      A1_2: "\<psi> g s \<in> S"
+    have H1_0: "([(\<phi>\<^sup>\<star>)]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>) g (words_to_syth_states (states_to_words s)) =
+     [(\<phi>\<^sup>\<star>) g (SOME w. w \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i w = s)]\<^sub>M\<^sub>N"
+      apply (clarsimp simp del: GMN_simps simp add: words_to_syth_states_def
+          states_to_words_def A1_0)
+      apply (rule meta_mp[of "(SOME w. w \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i w = s) \<in> A\<^sup>\<star>"])
+      using quot_act_wd_alt_notation[where w = "(SOME w. w \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i w = s)" and g = g] A1_1
+       apply simp
+      using A1_0
+      by (metis (mono_tags, lifting) induced_epi_wd1 some_eq_imp)
+    have H1_1: "\<And>g s' w'. \<lbrakk>s'\<in> S; w'\<in> A\<^sup>\<star>; g \<in> carrier G; (\<phi>\<^sup>\<star>) g w' \<in> A\<^sup>\<star> \<and> \<psi> g s' \<in> S\<rbrakk>
+               \<Longrightarrow> (\<delta>\<^sup>\<star>) (\<psi> g s') ((\<phi>\<^sup>\<star>) g w') = \<psi> g ((\<delta>\<^sup>\<star>) s' w')"
+      using give_input_eq_var
+      apply (clarsimp simp del: GMN_simps simp add: eq_var_func_axioms_def eq_var_func_def
+          make_op_def)
+      by (meson in_listsI)
+    have H1_2: "{w. w \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i w = \<psi> g s} =
+     {w'. \<exists>w \<in> A\<^sup>\<star>. (\<phi>\<^sup>\<star>) g w = w' \<and> (\<delta>\<^sup>\<star>) i w = s}"
+    proof (rule subset_antisym; clarify)
+      fix w'
+      assume
+        A2_0: "(\<delta>\<^sup>\<star>) i w' = \<psi> g s" and
+        A2_1: "\<forall>x\<in>set w'. x \<in> A"
+      have H2_0: "(inv \<^bsub>G\<^esub> g) \<in> carrier G"
+        by (meson A1_1 group.inv_closed group_hom.axioms(1) states_a_G_set.group_hom)
+      have H2_1: "(\<phi>\<^sup>\<star>) g ((\<phi>\<^sup>\<star>) (inv \<^bsub>G\<^esub> g) w') = w'"
+        by (smt (verit) A1_1 A2_1 alt_group_act_is_grp_act group_action.bij_prop1
+            group_action.orbit_sym_aux in_listsI labels_a_G_set.lists_a_Gset)
+      have H2_2: "\<And>g w. g \<in> carrier G \<Longrightarrow> w \<in> A\<^sup>\<star> \<Longrightarrow> (\<delta>\<^sup>\<star>) i ((\<phi>\<^sup>\<star>) g w) = (\<delta>\<^sup>\<star>) (\<psi> g i) ((\<phi>\<^sup>\<star>) g w)"
+        using init_is_eq_var.eq_var_subset_axioms init_is_eq_var.is_equivar
+        by auto
+      have H2_3: "\<And>g w. g \<in> carrier G \<Longrightarrow> w \<in> A\<^sup>\<star> \<Longrightarrow> (\<delta>\<^sup>\<star>) (\<psi> g i) ((\<phi>\<^sup>\<star>) g w) = \<psi> g ((\<delta>\<^sup>\<star>) i w)"
+        apply (rule H1_1[where s'1 = i]) 
+           apply (simp add: A2_1 in_lists_conv_set H2_0 is_aut.init_state_is_a_state)+
+        using is_aut.init_state_is_a_state labels_a_G_set.element_image
+          states_a_G_set.element_image
+        by blast
+      have H2_4: "\<psi> (inv \<^bsub>G\<^esub> g) ((\<delta>\<^sup>\<star>) i w') = s"
+        using A2_0 H2_0
+        by (simp add: A1_0 A1_1 states_a_G_set.orbit_sym_aux)
+      have H2_5: "(\<delta>\<^sup>\<star>) i ((\<phi>\<^sup>\<star>) (inv \<^bsub>G\<^esub> g) w') = s"
+        apply (rule meta_mp[of "w'\<in> A\<^sup>\<star>"])
+        using H2_0 H2_1 H2_4 A2_1 H2_2 H2_3
+         apply presburger 
+        using A2_1
+        by auto
+      have H2_6: "(\<phi>\<^sup>\<star>) (inv\<^bsub>G\<^esub> g) w' \<in> lists A "
+        using H2_0 A2_1 
+        by (metis alt_group_act_is_grp_act group_action.element_image in_listsI
+            labels_a_G_set.lists_a_Gset)
+      thus "\<exists>w\<in>lists A. (\<phi>\<^sup>\<star>) g w = w' \<and> (\<delta>\<^sup>\<star>) i w = s"
+        using H2_1 H2_5 H2_6
+        by blast
+    next
+      fix x w
+      assume
+        A2_0: "\<forall>x\<in>set w. x \<in> A" and
+        A2_1: "s = (\<delta>\<^sup>\<star>) i w"
+      show "(\<phi>\<^sup>\<star>) g w \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i ((\<phi>\<^sup>\<star>) g w) = \<psi> g ((\<delta>\<^sup>\<star>) i w) "
+        apply (rule conjI)
+         apply (rule meta_mp[of "(inv \<^bsub>G\<^esub> g) \<in> carrier G"])
+        using alt_group_act_is_grp_act group_action.element_image in_listsI
+          labels_a_G_set.lists_a_Gset
+          apply (metis A1_1 A2_0)
+         apply (meson A1_1 group.inv_closed group_hom.axioms(1) states_a_G_set.group_hom)
+        apply (rule meta_mp[of "\<psi> g i = i"])
+        using H1_1[where s'1 = i and g1 = "g"]
+         apply (metis A1_1 A2_0 action_on_input in_listsI)
+        using init_is_eq_var.eq_var_subset_axioms init_is_eq_var.is_equivar
+        by (simp add: A1_1)
+    qed
+    have H1_3: "\<exists>w. w \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i w = s" 
+      using A1_0 is_reachable
+      by auto
+    have H1_4: "\<exists>w. w \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i w = \<psi> g s"
+      using A1_2 induced_epi_wd1
+      by auto
+    have H1_5: "[(\<phi>\<^sup>\<star>) g (SOME w. w \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i w = s)]\<^sub>M\<^sub>N = [SOME w. w \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i w = \<psi> g s]\<^sub>M\<^sub>N" 
+    proof (rule subset_antisym; clarify)
+      fix w'
+      assume
+        A2_0: "w' \<in> [(\<phi>\<^sup>\<star>) g (SOME w. w \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i w = s)]\<^sub>M\<^sub>N"
+      have H2_0: "\<And>w. w \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i w = s \<Longrightarrow> w' \<in> [(\<phi>\<^sup>\<star>) g w]\<^sub>M\<^sub>N"
+        using A2_0 H1_3 H1_2 H1_4 induced_epi_wd2 mem_Collect_eq tfl_some 
+        by (smt (verit, best))  
+      obtain w'' where H2_w'': "w' \<in> [(\<phi>\<^sup>\<star>) g w'']\<^sub>M\<^sub>N \<and> w'' \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i w'' = s"
+        using A2_0 H1_3 tfl_some
+        by (metis (mono_tags, lifting))
+      from H1_2 H2_w'' have H2_1: "(\<delta>\<^sup>\<star>) i ((\<phi>\<^sup>\<star>) g w'') = \<psi> g s"
+        by blast
+      have H2_2: "\<And>w. w \<in> A\<^sup>\<star> \<Longrightarrow> (\<delta>\<^sup>\<star>) i w = \<psi> g s \<Longrightarrow> w' \<in> [w]\<^sub>M\<^sub>N"
+      proof -
+        fix w''
+        assume
+          A3_0: "w'' \<in> A\<^sup>\<star>" and
+          A3_1: "(\<delta>\<^sup>\<star>) i w'' = \<psi> g s"
+        have H3_0: "(inv \<^bsub>G\<^esub>g) \<in> carrier G" 
+          by (metis A1_1 group.inv_closed group_hom.axioms(1) states_a_G_set.group_hom)
+        from A3_0 H3_0 have H3_1: "(\<phi>\<^sup>\<star>) (inv \<^bsub>G\<^esub> g) w'' \<in> A\<^sup>\<star>"
+          by (metis alt_grp_act.axioms group_action.element_image 
+              labels_a_G_set.lists_a_Gset)
+        have H3_2: "\<And>g w. g \<in> carrier G \<Longrightarrow> w \<in> A\<^sup>\<star> \<Longrightarrow> (\<delta>\<^sup>\<star>) i ((\<phi>\<^sup>\<star>) g w) = (\<delta>\<^sup>\<star>) (\<psi> g i) ((\<phi>\<^sup>\<star>) g w)"
+          using init_is_eq_var.eq_var_subset_axioms init_is_eq_var.is_equivar
+          by auto
+        have H3_3: "\<And>g w. g \<in> carrier G \<Longrightarrow> w \<in> A\<^sup>\<star> \<Longrightarrow> (\<delta>\<^sup>\<star>) (\<psi> g i) ((\<phi>\<^sup>\<star>) g w) = \<psi> g ((\<delta>\<^sup>\<star>) i w)"
+          apply (rule H1_1[where s'1 = i]) 
+             apply (simp add: A3_1 in_lists_conv_set H2_1 is_aut.init_state_is_a_state)+
+          using is_aut.init_state_is_a_state labels_a_G_set.element_image
+            states_a_G_set.element_image
+          by blast
+        have H3_4: "s = (\<delta>\<^sup>\<star>) i ((\<phi>\<^sup>\<star>) (inv \<^bsub>G\<^esub> g) w'')"
+          using A3_0 A3_1 H3_0 H3_2 H3_3 A1_0 A1_1 states_a_G_set.orbit_sym_aux
+          by auto
+        from H3_4 show " w' \<in> [w'']\<^sub>M\<^sub>N"
+          by (metis (mono_tags, lifting) A1_1 G_set_equiv H2_1 H2_w'' \<open>(\<delta>\<^sup>\<star>) i w'' = \<psi> g s\<close> A3_0
+              eq_var_subset.is_equivar image_eqI induced_epi_wd2
+              labels_a_G_set.lists_a_Gset)
+      qed
+      from H2_2 show "w' \<in> [SOME w. w \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i w = \<psi> g s]\<^sub>M\<^sub>N"
+        by (smt (verit) H1_4 some_eq_ex)
+    next
+      fix w'
+      assume
+        A2_0: "w' \<in> [SOME w. w \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i w = \<psi> g s]\<^sub>M\<^sub>N"
+      obtain w'' where H2_w'': "w' \<in> [(\<phi>\<^sup>\<star>) g w'']\<^sub>M\<^sub>N \<and> w'' \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i w'' = s"
+        using A2_0 H1_3 tfl_some 
+        by (smt (verit) H1_2 mem_Collect_eq)
+      from H1_2 H2_w'' have H2_0: "(\<delta>\<^sup>\<star>) i ((\<phi>\<^sup>\<star>) g w'') = \<psi> g s"
+        by blast
+      have H2_1: "\<And>w. w \<in> A\<^sup>\<star> \<Longrightarrow> (\<delta>\<^sup>\<star>) i w = s \<Longrightarrow> w' \<in> [(\<phi>\<^sup>\<star>) g w]\<^sub>M\<^sub>N"
+      proof -
+        fix w''
+        assume
+          A3_0: "w'' \<in> A\<^sup>\<star>" and
+          A3_1: "(\<delta>\<^sup>\<star>) i w'' = s"
+        have H3_0: "(inv \<^bsub>G\<^esub>g) \<in> carrier G" 
+          by (metis A1_1 group.inv_closed group_hom.axioms(1) states_a_G_set.group_hom)
+        have H3_1: "(\<phi>\<^sup>\<star>) (inv \<^bsub>G\<^esub> g) w'' \<in> A\<^sup>\<star>"
+          using A3_0 H3_0 
+          by (metis alt_group_act_is_grp_act group_action.element_image labels_a_G_set.lists_a_Gset)
+        have H3_2: "\<And>g w. g \<in> carrier G \<Longrightarrow> w \<in> A\<^sup>\<star> \<Longrightarrow> (\<delta>\<^sup>\<star>) i ((\<phi>\<^sup>\<star>) g w) =
+       (\<delta>\<^sup>\<star>) (\<psi> g i) ((\<phi>\<^sup>\<star>) g w)"
+          using init_is_eq_var.eq_var_subset_axioms init_is_eq_var.is_equivar
+          by auto
+        have H3_3: "\<And>g w. g \<in> carrier G \<Longrightarrow> w \<in> A\<^sup>\<star> \<Longrightarrow> (\<delta>\<^sup>\<star>) (\<psi> g i) ((\<phi>\<^sup>\<star>) g w) = 
+       \<psi> g ((\<delta>\<^sup>\<star>) i w)"
+          apply (rule H1_1[where s'1 = i]) 
+             apply (simp add: A3_1 in_lists_conv_set H2_0 is_aut.init_state_is_a_state)+
+          using is_aut.init_state_is_a_state labels_a_G_set.element_image
+            states_a_G_set.element_image
+          by blast
+        have H3_4: "\<psi> (inv \<^bsub>G\<^esub> g) s = (\<delta>\<^sup>\<star>) i ((\<phi>\<^sup>\<star>) (inv \<^bsub>G\<^esub> g) w'')"
+          using A3_0 A3_1 H3_0 H3_2 H3_3
+          by auto
+        show "w' \<in> [(\<phi>\<^sup>\<star>) g w'']\<^sub>M\<^sub>N "
+          using H3_4 H3_1 
+          by (smt (verit, del_insts) A1_1 A3_0 A3_1 in_listsI H3_2 H3_3
+              \<open>\<And>thesis. (\<And>w''. w' \<in> [(\<phi>\<^sup>\<star>) g w'']\<^sub>M\<^sub>N \<and> w'' \<in> A\<^sup>\<star> \<and>
+              (\<delta>\<^sup>\<star>) i w'' = s \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
+              alt_group_act_is_grp_act group_action.surj_prop image_eqI induced_epi_wd2
+              labels_a_G_set.lists_a_Gset)
+      qed
+      show "w' \<in> [(\<phi>\<^sup>\<star>) g (SOME w. w \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i w = s)]\<^sub>M\<^sub>N"
+        using H2_1 H1_3
+        by (metis (mono_tags, lifting) someI)
+    qed
+    show "words_to_syth_states (states_to_words (\<psi> g s)) =
+    ([(\<phi>\<^sup>\<star>)]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>) g (words_to_syth_states (states_to_words s))"
+      using H1_5
+      apply (clarsimp simp del: GMN_simps simp add: words_to_syth_states_def states_to_words_def)
+      apply (intro conjI; clarify; rule conjI)
+      using H1_0
+         apply (auto del: subset_antisym simp del: GMN_simps simp add: words_to_syth_states_def
+          states_to_words_def)[1]
+      using A1_2
+        apply blast 
+      using A1_0
+       apply blast
+      using A1_0
+      by blast
+  qed
+  show ?thesis
+    apply (clarsimp del: subset_antisym simp del: GMN_simps simp add: eq_var_func_def
+        eq_var_func_axioms_def)
+    apply (intro conjI)
+    subgoal
+      using states_a_G_set.alt_grp_act_axioms
+      by auto
+      apply (metis MN_rel_eq_var MN_rel_equival eq_var_rel.quot_act_is_grp_act)
+     apply (clarsimp simp add: FuncSet.extensional_funcset_def Pi_def)
+     apply (rule conjI)
+      apply (clarify)
+    subgoal for s
+      using is_reachable[where s = s]
+      apply (clarsimp simp add: induced_epi_def compose_def states_to_words_def
+          words_to_syth_states_def)
+      by (smt (verit) \<open>s \<in> S \<Longrightarrow> \<exists>input\<in>A\<^sup>\<star>. (\<delta>\<^sup>\<star>) i input = s\<close> alt_natural_map_MN_def
+          lists_eq_set quotientI rel_MN_def singleton_conv someI)
+     apply (clarsimp simp del: GMN_simps simp add: induced_epi_def make_op_def
+        compose_def)
+    apply (clarify)
+    apply (clarsimp simp del: GMN_simps simp add: induced_epi_def compose_def make_op_def)
+    apply (rule conjI; rule impI)
+     apply (simp add: H_0)
+    using states_a_G_set.element_image
+    by blast
+qed
+
+text \<open>
+The following lemma corresponds to lemma 3.7 from \cite{bojanczyk2014automata}:
+\<close>
+
+lemma reach_det_G_aut_rec_lang:
+  "G_aut_epi A S i F \<delta> MN_equiv MN_init_state MN_fin_states \<delta>\<^sub>M\<^sub>N G \<phi> \<psi> ([(\<phi>\<^sup>\<star>)]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>) induced_epi"
+proof-
+  have H_0: "\<And>s. s \<in> MN_equiv \<Longrightarrow> \<exists>input\<in> A\<^sup>\<star>. (\<delta>\<^sub>M\<^sub>N\<^sup>\<star>) MN_init_state input = s"
+  proof-
+    fix s
+    assume
+      A_0: "s \<in> MN_equiv"
+    from A_0 have H_0: "\<exists>w. w \<in> A\<^sup>\<star> \<and> s = [w]\<^sub>M\<^sub>N"
+      by (auto simp add: quotient_def)
+    show "\<exists>input\<in>A\<^sup>\<star>. (\<delta>\<^sub>M\<^sub>N\<^sup>\<star>) MN_init_state input = s"
+      using H_0 
+      by (metis MN_unique_init_state)
+  qed
+  have H_1: "\<And>s\<^sub>0 a. s\<^sub>0 \<in> S \<Longrightarrow> a \<in> A \<Longrightarrow> induced_epi (\<delta> s\<^sub>0 a) = \<delta>\<^sub>M\<^sub>N (induced_epi s\<^sub>0) a"
+  proof-
+    fix s\<^sub>0 a
+    assume
+      A1_0: "s\<^sub>0 \<in> S" and
+      A1_1: "a \<in> A" 
+    obtain w where H1_w: "w \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i w = s\<^sub>0" 
+      using A1_0 induced_epi_wd1
+      by auto
+    have H1_0: "[SOME w. w \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i w = s\<^sub>0]\<^sub>M\<^sub>N = [w]\<^sub>M\<^sub>N"
+      by (metis (mono_tags, lifting) H1_w induced_epi_wd2 some_eq_imp)
+    have H1_1: "(\<delta>\<^sup>\<star>) i (SOME w. w \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i w = \<delta> s\<^sub>0 a) = (\<delta>\<^sup>\<star>) i (w @ [a])"
+      using A1_0 A1_1 H1_w is_aut.trans_to_charact[where s = s\<^sub>0 and a = a and w = w] 
+      by (smt (verit, del_insts) induced_epi_wd1 is_aut.trans_func_well_def tfl_some)
+    have H1_2: "w @ [a] \<in> A\<^sup>\<star>" using H1_w A1_1 by auto
+    have H1_3: "[(SOME w. w \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i w = s\<^sub>0) @ [a]]\<^sub>M\<^sub>N = [w @ [a]]\<^sub>M\<^sub>N" 
+      by (metis (mono_tags, lifting) A1_1 H1_0 H1_w MN_trans_func_characterization someI)
+    have H1_4: "... = [SOME w. w \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i w = \<delta> s\<^sub>0 a]\<^sub>M\<^sub>N"
+      apply (rule sym)
+      apply (rule induced_epi_wd2[where w = "SOME w. w \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i w = \<delta> s\<^sub>0 a"
+            and w'= "w @ [a]"])
+        apply (metis (mono_tags, lifting) A1_0 A1_1 H1_w some_eq_imp H1_2 is_aut.trans_to_charact)
+       apply (rule H1_2)
+      using H1_1
+      by simp
+    show "induced_epi (\<delta> s\<^sub>0 a) = \<delta>\<^sub>M\<^sub>N (induced_epi s\<^sub>0) a"
+      apply (clarsimp del: subset_antisym simp del: GMN_simps simp add: induced_epi_def
+          words_to_syth_states_def states_to_words_def compose_def is_aut.trans_func_well_def)
+      using A1_1 H1_w H1_0 H1_3 H1_4 MN_trans_func_characterization A1_0
+        is_aut.trans_func_well_def
+      by presburger
+  qed
+  have H_2: "induced_epi ` S = MN_equiv"
+  proof-
+    have H1_0: "\<forall>s \<in> S. \<exists>v\<in> A\<^sup>\<star>. (\<delta>\<^sup>\<star>) i v = s \<and> [SOME w. w \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i w = s]\<^sub>M\<^sub>N = [v]\<^sub>M\<^sub>N"
+      by (smt (verit) is_reachable tfl_some)
+    have H1_1: "\<And>v. v\<in> A\<^sup>\<star> \<Longrightarrow> (\<delta>\<^sup>\<star>) i v \<in> S" 
+      using is_aut.give_input_closed
+      by (auto simp add: is_aut.init_state_is_a_state)
+    show ?thesis
+      apply (clarsimp simp del: GMN_simps simp add: induced_epi_def words_to_syth_states_def
+          states_to_words_def compose_def image_def)
+      using H1_0 H1_1
+      apply (clarsimp)
+      apply (rule subset_antisym; simp del: GMN_simps add: Set.subset_eq)
+       apply (metis (no_types, lifting) quotientI)
+      by (metis (no_types, lifting) alt_natural_map_MN_def induced_epi_wd2 quotientE)
+  qed
+  show ?thesis
+    apply (simp del: GMN_simps add: G_aut_epi_def G_aut_epi_axioms_def)
+    apply (rule conjI)
+    subgoal
+      apply (clarsimp simp del: GMN_simps simp add: G_aut_hom_def aut_hom_def reach_det_G_aut_def
+          is_reachable det_G_aut_def reach_det_aut_def reach_det_aut_axioms_def)
+      apply (intro conjI)
+                        apply (simp add: is_aut.det_aut_axioms)
+      using labels_a_G_set.alt_grp_act_axioms
+                       apply (auto)[1]
+      using states_a_G_set.alt_grp_act_axioms
+                      apply blast
+                     apply (simp add: accepting_is_eq_var.eq_var_subset_axioms)
+      using init_is_eq_var.eq_var_subset_axioms
+                    apply (auto)[1]
+                   apply (simp add: trans_is_eq_var.eq_var_func_axioms)
+                  apply (simp add: is_aut.det_aut_axioms)
+      using syth_aut_is_det_aut
+                 apply simp
+      using labels_a_G_set.alt_grp_act_axioms
+                apply (auto)[1]
+               apply (metis MN_rel_eq_var MN_rel_equival eq_var_rel.quot_act_is_grp_act)
+      using MN_final_state_equiv
+              apply presburger 
+      using MN_init_state_equivar_v2
+             apply presburger 
+      using MN_trans_eq_var_func
+            apply blast
+      using syth_aut_is_det_aut
+           apply auto[1]
+          apply (clarify)
+          apply (simp add: H_0 del: GMN_simps)
+         apply (simp add: is_aut.det_aut_axioms)
+      using syth_aut_is_det_aut
+        apply blast
+       apply (clarsimp del: subset_antisym simp del: GMN_simps simp add: aut_hom_axioms_def
+          FuncSet.extensional_funcset_def Pi_def extensional_def)[1]
+       apply (intro conjI)
+           apply (clarify)
+           apply (simp add: induced_epi_def)
+           apply (simp add: induced_epi_def words_to_syth_states_def states_to_words_def
+          compose_def)
+           apply (rule meta_mp[of "(\<delta>\<^sup>\<star>) i Nil = i"])
+      using induced_epi_wd2[where w = "Nil"]
+            apply (auto simp add: is_aut.init_state_is_a_state del: subset_antisym)[2] 
+      subgoal for x
+        apply (rule quotientI)
+        using is_reachable[where s = x] someI[where P = "\<lambda>w. w \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i w = x"]      
+        by blast
+          apply (auto simp add: induced_epi_def words_to_syth_states_def states_to_words_def
+          compose_def)[1]
+         apply (simp add: induced_epi_def states_to_words_def compose_def
+          is_aut.init_state_is_a_state)
+         apply (metis (mono_tags, lifting) \<open>\<And>w'. \<lbrakk>[] \<in> A\<^sup>\<star>; w' \<in> A\<^sup>\<star>;
+      (\<delta>\<^sup>\<star>) i [] = (\<delta>\<^sup>\<star>) i w'\<rbrakk> \<Longrightarrow> MN_init_state = [w']\<^sub>M\<^sub>N\<close>
+          alt_natural_map_MN_def give_input.simps(1) lists.Nil some_eq_imp
+          words_to_syth_states_def)
+        apply (clarify)
+      subgoal for s 
+        apply (rule iffI)
+         apply (smt (verit) Pi_iff compose_eq in_mono induced_epi_def is_aut.fin_states_are_states
+            states_to_words_on_final words_to_syth_states_def)
+        apply (clarsimp simp del: GMN_simps simp add: induced_epi_def words_to_syth_states_def
+            states_to_words_def compose_def)
+        apply (rule meta_mp[of "(SOME w. w \<in> A\<^sup>\<star> \<and> (\<delta>\<^sup>\<star>) i w = s) \<in> L"])
+         apply (smt (verit) induced_epi_wd1 is_recognised someI)
+        using fin_states_rep_by_lang is_reachable mem_Collect_eq
+        by (metis (mono_tags, lifting))
+       apply (clarsimp simp del: GMN_simps)
+       apply (simp add: H_1)
+      using induced_epi_eq_var
+      by blast
+    by (simp add: H_2)
+qed
+
+end
+
+lemma (in det_G_aut) finite_reachable:
+  "finite (orbits G S \<psi>) \<Longrightarrow> finite (orbits G S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h)"
+proof-
+  assume
+    A_0: "finite (orbits G S \<psi>)"
+  have H_0: "S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<subseteq> S" 
+    apply (clarsimp simp add: reachable_states_def) 
+    by (simp add: in_listsI is_aut.give_input_closed is_aut.init_state_is_a_state)
+  have H_1: "{{\<psi> g x |g. g \<in> carrier G} |x. x \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h} \<subseteq>
+    {{\<psi> g x |g. g \<in> carrier G} |x. x \<in> S}"
+    by (smt (verit, best) Collect_mono_iff H_0 subsetD)
+  have H_2: "\<And>x. x \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<Longrightarrow>
+     {\<psi> g x |g. g \<in> carrier G} = {\<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g x |g. g \<in> carrier G}"
+    using reachable_action_is_restict
+    by (metis)
+  hence H_3: "{{\<psi> g x |g. g \<in> carrier G} |x. x \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h} =
+    {{\<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h g x |g. g \<in> carrier G} |x. x \<in> S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h}" 
+    by blast
+  show "finite (orbits G S\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h \<psi>\<^sub>r\<^sub>e\<^sub>a\<^sub>c\<^sub>h)"
+    using A_0 apply (clarsimp simp add: orbits_def orbit_def)
+    using Finite_Set.finite_subset H_1 H_3
+    by auto
+qed
+
+lemma (in det_G_aut)
+  orbs_pos_card: "finite (orbits G S \<psi>) \<Longrightarrow> card (orbits G S \<psi>) > 0"
+  apply (clarsimp simp add: card_gt_0_iff orbits_def)
+  using is_aut.init_state_is_a_state
+  by auto
+
+lemma (in reach_det_G_aut_rec_lang) MN_B2T:
+  assumes
+    Fin: "finite (orbits G S \<psi>)"
+  shows
+    "finite (orbits G (language.MN_equiv A L) (([(\<phi>\<^sup>\<star>)]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>)))"
+proof- 
+  have H_0: "finite {{\<psi> g x |g. g \<in> carrier G} |x. x \<in> S}"
+    using Fin
+    by (auto simp add: orbits_def orbit_def)
+  have H_1: "induced_epi ` S = MN_equiv"
+    using reach_det_G_aut_rec_lang
+    by (auto simp del: GMN_simps simp add: G_aut_epi_def G_aut_epi_axioms_def)
+  have H_2: "\<And>B f. finite B \<Longrightarrow> finite {f b| b. b \<in> B} "
+    by auto
+  have H_3: "finite {{\<psi> g x |g. g \<in> carrier G} |x. x \<in> S} \<Longrightarrow>
+      finite {induced_epi ` b |b. b \<in> {{\<psi> g x |g. g \<in> carrier G} |x. x \<in> S}}"
+    using H_2[where f1 = "(\<lambda>x. induced_epi ` x)" and B1 = "{{\<psi> g x |g. g \<in> carrier G} |x. x \<in> S}"]
+    by auto
+  have H_4: "\<And>s. s \<in> S \<Longrightarrow> \<exists>b. {induced_epi (\<psi> g s) |g. g \<in> carrier G}
+              = {y. \<exists>x\<in>b. y = induced_epi x} \<and> (\<exists>x. b = {\<psi> g x |g. g \<in> carrier G} \<and> x \<in> S)"
+  proof-
+    fix s
+    assume
+      A2_0: "s \<in> S"
+    have H2_0: "{induced_epi (\<psi> g s) |g. g \<in> carrier G} = {y. \<exists>x \<in> {\<psi> g s |g. g \<in> carrier G}. y =
+    induced_epi x}"
+      by blast
+    have H2_1: "(\<exists>x. {\<psi> g s |g. g \<in> carrier G} = {\<psi> g x |g. g \<in> carrier G} \<and> x \<in> S)"
+      using A2_0 
+      by auto
+    show "\<exists>b. {induced_epi (\<psi> g s) |g. g \<in> carrier G} =
+    {y. \<exists>x\<in>b. y = induced_epi x} \<and> (\<exists>x. b = {\<psi> g x |g. g \<in> carrier G} \<and> x \<in> S)"
+      using A2_0 H2_0 H2_1 
+      by meson
+  qed
+  have H_5: "{induced_epi ` b |b. b \<in> {{\<psi> g x |g. g \<in> carrier G} |x. x \<in> S}} =
+  {{induced_epi (\<psi> g s) | g . g \<in> carrier G} |s. s \<in> S}"
+    apply (clarsimp simp add: image_def)
+    apply (rule subset_antisym; simp add: Set.subset_eq; clarify)
+     apply auto[1]
+    apply (simp)
+    by (simp add: H_4)
+  from H_3 H_5 have H_6: "finite {{\<psi> g x |g. g \<in> carrier G} |x. x \<in> S} \<Longrightarrow>
+      finite {{induced_epi (\<psi> g s) | g . g \<in> carrier G} |s. s \<in> S}"
+    by metis
+  have H_7: "finite {{induced_epi (\<psi> g x) |g. g \<in> carrier G} |x. x \<in> S}"
+    apply (rule H_6)
+    by (simp add: H_0)
+  have H_8: "\<And>x. x \<in> S \<Longrightarrow> {induced_epi (\<psi> g x) |g. g \<in> carrier G} =
+  {([(\<phi>\<^sup>\<star>)]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>) g (induced_epi x) |g. g \<in> carrier G}"
+    using induced_epi_eq_var
+    apply (simp del: GMN_simps add: eq_var_func_def eq_var_func_axioms_def make_op_def)
+    by blast
+  hence H_9: "{{induced_epi (\<psi> g x) |g. g \<in> carrier G} |x. x \<in> S} =
+  {{([(\<phi>\<^sup>\<star>)]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>) g (induced_epi x) |g. g \<in> carrier G} |x. x \<in> S}"
+    by blast
+  have H_10: "\<And>f g X B C. g ` B = C \<Longrightarrow>
+             {{f x (g b)|x. x\<in>X}|b. b \<in> B} = {{f x c|x. x \<in> X}|c. c \<in> C}"
+    by auto
+  have H_11: "{{([(\<phi>\<^sup>\<star>)]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>) g (induced_epi x) |g. g \<in> carrier G} |x. x \<in> S} =
+  {{([(\<phi>\<^sup>\<star>)]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>) g W |g. g \<in> carrier G} |W. W \<in> MN_equiv}"
+    apply (rule H_10[where f2 = "([(\<phi>\<^sup>\<star>)]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>)" and X2 = "carrier G" and g2 = induced_epi
+          and B2 = S and C2 = MN_equiv])
+    using H_1
+    by simp
+  have H_12: "{{([(\<phi>\<^sup>\<star>)]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>) g W |g. g \<in> carrier G} |W. W \<in> MN_equiv} =
+  orbits G (language.MN_equiv A L) (([(\<phi>\<^sup>\<star>)]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>))"
+    by (auto simp add: orbits_def orbit_def)
+  show "finite (orbits G (language.MN_equiv A L) (([(\<phi>\<^sup>\<star>)]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>)))"
+    using H_9 H_11 H_12 H_7
+    by presburger
+qed
+
+context det_G_aut_rec_lang begin
+
+text \<open>
+To avoid duplicate variant of "star":
+\<close>
+
+no_adhoc_overloading
+  star labels_a_G_set.induced_star_map
+
+end
+
+context det_G_aut_rec_lang begin
+adhoc_overloading
+  star labels_a_G_set.induced_star_map
+end
+
+
+lemma (in det_G_aut_rec_lang) MN_prep:
+  "\<exists>S'. \<exists>\<delta>'. \<exists>F'. \<exists>\<psi>'.
+  (reach_det_G_aut_rec_lang A S' i F' \<delta>' G \<phi> \<psi>' L \<and>
+  (finite (orbits G S \<psi>) \<longrightarrow> finite (orbits G S' \<psi>')))"
+  by (meson G_lang_axioms finite_reachable reach_det_G_aut_rec_lang.intro
+      reach_det_aut_is_det_aut_rec_L)
+
+lemma (in det_G_aut_rec_lang) MN_fin_orbs_imp_fin_states:
+  assumes
+    Fin: "finite (orbits G S \<psi>)"
+  shows
+    "finite (orbits G (language.MN_equiv A L) (([(\<phi>\<^sup>\<star>)]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>)))"
+  using MN_prep 
+  by (metis assms reach_det_G_aut_rec_lang.MN_B2T)
+
+text \<open>
+The following theorem corresponds to theorem 3.8 from \cite{bojanczyk2014automata}, i.e. the
+Myhill-Nerode theorem for G-automata.
+The left to right direction (see statement below) of the typical Myhill-Nerode theorem would
+qantify over types (if some condition holds, then there exists some automaton accepting the
+language). As it is not possible to qantify over types in this way, the equivalence is spit into
+two directions. In the left to right direction, the explicit type of the syntactic automaton is
+used. In the right to left direction some type, 's, is fixed.
+As the two directions are split, the opertunity was taken to strengthen the right to left direction:
+We do not assume the given automaton is reachable.
+
+This splitting of the directions will be present in all other Myhill-Nerode theorems that will be
+proved in this document.
+\<close>
+
+theorem (in G_lang) G_Myhill_Nerode :
+  assumes
+    "finite (orbits G A \<phi>)"
+  shows
+    G_Myhill_Nerode_LR: "finite (orbits G MN_equiv ([(\<phi>\<^sup>\<star>)]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>)) \<Longrightarrow>
+     (\<exists>S F :: 'alpha list set set. \<exists>i :: 'alpha list set. \<exists>\<delta>. \<exists>\<psi>.
+     reach_det_G_aut_rec_lang A S i F \<delta> G \<phi> \<psi> L \<and> finite (orbits G S \<psi>))" and
+    G_Myhill_Nerode_RL: "(\<exists>S F :: 's set. \<exists>i :: 's. \<exists>\<delta>. \<exists>\<psi>.
+     det_G_aut_rec_lang A S i F \<delta> G \<phi> \<psi> L \<and> finite (orbits G S \<psi>))
+     \<Longrightarrow> finite (orbits G MN_equiv ([(\<phi>\<^sup>\<star>)]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>))"
+  subgoal
+    using syntact_aut_is_reach_aut_rec_lang 
+    by blast
+  by (metis det_G_aut_rec_lang.MN_fin_orbs_imp_fin_states)
+
+subsection \<open>
+Proving the standard Myhill-Nerode Theorem
+\<close>
+
+text \<open>
+Any automaton is a $G$-automaton with respect to the trivial group and action,
+hence the standard Myhill-Nerode theorem is a special case of the $G$-Myhill-Nerode theorem.
+\<close>
+
+interpretation triv_act:
+  alt_grp_act "singleton_group (undefined)" X "(\<lambda>x \<in> {undefined}. one (BijGroup X))"
+  apply (simp add: group_action_def group_hom_def group_hom_axioms_def)
+  apply (intro conjI)
+   apply (simp add: group_BijGroup)
+  using trivial_hom 
+  by (smt (verit) carrier_singleton_group group.hom_restrict group_BijGroup restrict_apply
+      singleton_group)
+
+lemma (in det_aut) triv_G_aut:
+  fixes triv_G
+  assumes H_triv_G: "triv_G = (singleton_group (undefined))"
+  shows "det_G_aut labels states init_state fin_states \<delta> 
+  triv_G (\<lambda>x \<in> {undefined}. one (BijGroup labels)) (\<lambda>x \<in> {undefined}. one (BijGroup states))"
+  apply (simp add: det_G_aut_def group_hom_def group_hom_axioms_def
+      eq_var_subset_def eq_var_subset_axioms_def eq_var_func_def eq_var_func_axioms_def)
+  apply (intro conjI)
+             apply (rule det_aut_axioms)
+            apply (simp add: assms triv_act.group_action_axioms)+
+  using fin_states_are_states
+          apply (auto)[1]
+         apply (clarify; rule conjI; rule impI)
+          apply (simp add: H_triv_G BijGroup_def image_def)
+  using fin_states_are_states
+          apply auto[1]
+         apply (simp add: H_triv_G BijGroup_def image_def)
+        apply (simp add: assms triv_act.group_action_axioms)
+       apply (simp add: init_state_is_a_state)
+      apply (clarify; rule conjI; rule impI)
+       apply (simp add: H_triv_G BijGroup_def image_def init_state_is_a_state)+ 
+     apply (clarsimp simp add: group_action_def BijGroup_def hom_def group_hom_def
+      group_hom_axioms_def)
+     apply (rule conjI)
+      apply (smt (verit) BijGroup_def Bij_imp_funcset Id_compose SigmaE case_prod_conv
+      group_BijGroup id_Bij restrict_ext restrict_extensional)
+     apply (rule meta_mp[of "undefined \<otimes>\<^bsub>singleton_group undefined\<^esub> undefined = undefined"])
+      apply (auto)[1]
+     apply (metis carrier_singleton_group comm_groupE(1) singletonD singletonI
+      singleton_abelian_group)
+    apply (simp add: assms triv_act.group_action_axioms)
+   apply (auto simp add: trans_func_well_def)[1] 
+  by (clarsimp simp add: BijGroup_def trans_func_well_def H_triv_G) 
+
+lemma triv_orbits:
+  "orbits (singleton_group (undefined)) S (\<lambda>x \<in> {undefined}. one (BijGroup S)) =
+   {{s} |s. s \<in> S}"
+  apply (simp add: BijGroup_def singleton_group_def orbits_def orbit_def)
+  by auto
+
+lemma fin_triv_orbs:
+  "finite (orbits (singleton_group (undefined)) S (\<lambda>x \<in> {undefined}. one (BijGroup S))) = finite S"
+  apply (subst triv_orbits)
+  apply (rule meta_mp[of "bij_betw (\<lambda>s \<in> S. {s}) S {{s} |s. s \<in> S}"])
+  using bij_betw_finite
+   apply (auto)[1]
+  by (auto simp add: bij_betw_def image_def)
+
+context language begin
+
+interpretation triv_G_lang:
+  G_lang "singleton_group (undefined)" A "(\<lambda>x \<in> {undefined}. one (BijGroup A))" L
+  apply (simp add: G_lang_def G_lang_axioms_def eq_var_subset_def eq_var_subset_axioms_def)
+  apply (intro conjI)
+      apply (simp add: triv_act.group_action_axioms)
+     apply (simp add: language_axioms)
+  using triv_act.lists_a_Gset
+    apply fastforce
+   apply (rule is_lang)
+  apply (clarsimp simp add: BijGroup_def image_def)
+  apply (rule subset_antisym; simp add: Set.subset_eq; clarify)
+  using is_lang
+   apply (auto simp add: map_idI)[1] 
+  using is_lang map_idI 
+  by (metis in_listsD in_mono inf.absorb_iff1 restrict_apply)
+
+definition triv_G :: "'grp monoid"
+  where "triv_G = (singleton_group (undefined))"
+
+definition triv_act :: "'grp \<Rightarrow> 'alpha \<Rightarrow> 'alpha"
+  where "triv_act = (\<lambda>x \<in> {undefined}. \<one>\<^bsub>BijGroup A\<^esub>)"
+
+corollary standard_Myhill_Nerode:
+  assumes
+    H_fin_alph: "finite A"
+  shows
+    standard_Myhill_Nerode_LR: "finite MN_equiv \<Longrightarrow>
+     (\<exists>S F :: 'alpha list set set. \<exists>i :: 'alpha list set. \<exists>\<delta>.
+     reach_det_aut_rec_lang A S i F \<delta> L \<and> finite S)" and
+    standard_Myhill_Nerode_RL: "(\<exists>S F :: 's set. \<exists>i :: 's. \<exists>\<delta>.
+     det_aut_rec_lang A S i F \<delta> L \<and> finite S) \<Longrightarrow> finite MN_equiv"
+proof-
+  assume
+    A_0: "finite MN_equiv"
+  have H_0: "reach_det_aut_rec_lang A MN_equiv MN_init_state MN_fin_states \<delta>\<^sub>M\<^sub>N L"
+    using triv_G_lang.syntact_aut_is_reach_aut_rec_lang
+    apply (clarsimp simp add: reach_det_G_aut_rec_lang_def det_G_aut_rec_lang_def
+        reach_det_aut_rec_lang_def reach_det_aut_def reach_det_aut_axioms_def det_G_aut_def)
+    by (smt (z3) alt_natural_map_MN_def quotientE triv_G_lang.MN_unique_init_state)
+  show "\<exists>S F:: 'alpha list set set. \<exists>i :: 'alpha list set. \<exists>\<delta>.
+  reach_det_aut_rec_lang A S i F \<delta> L \<and> finite S"
+    using A_0 H_0 
+    by auto
+next
+  assume
+    A_0: "\<exists>S F:: 's set. \<exists>i :: 's. \<exists>\<delta>. det_aut_rec_lang A S i F \<delta> L \<and> finite S"
+  obtain S F :: "'s set" and i :: "'s"  and \<delta>
+    where H_MN: "det_aut_rec_lang A S i F \<delta> L \<and> finite S"
+    using A_0
+    by auto
+  have H_0: "det_G_aut A S i F \<delta> triv_G (\<lambda>x\<in>{undefined}. \<one>\<^bsub>BijGroup A\<^esub>)
+   (\<lambda>x\<in>{undefined}. \<one>\<^bsub>BijGroup S\<^esub>)"
+    apply (rule det_aut.triv_G_aut[of A S i F \<delta> triv_G])
+    using H_MN
+     apply (simp add: det_aut_rec_lang_def)
+    by (rule triv_G_def)
+  have H_1: "det_G_aut_rec_lang A S i F \<delta> triv_G (\<lambda>x\<in>{undefined}. \<one>\<^bsub>BijGroup A\<^esub>)
+   (\<lambda>x\<in>{undefined}. \<one>\<^bsub>BijGroup S\<^esub>) L"
+    by (auto simp add: det_G_aut_rec_lang_def H_0 H_MN)
+  have H_2: "(\<exists>S F:: 's set. \<exists>i :: 's. \<exists>\<delta> \<psi>.
+         det_G_aut_rec_lang A S i F \<delta> (singleton_group undefined) (\<lambda>x\<in>{undefined}. \<one>\<^bsub>BijGroup A\<^esub>)
+         \<psi> L \<and> finite (orbits (singleton_group undefined) S \<psi>))"
+    using H_1
+    by (metis H_MN fin_triv_orbs triv_G_def)
+  have H_3: "finite (orbits triv_G A triv_act)"
+    apply (subst triv_G_def; subst triv_act_def; subst fin_triv_orbs[of A])
+    by (rule H_fin_alph)
+  have H_4:"finite (orbits triv_G MN_equiv (triv_act.induced_quot_map (A\<^sup>\<star>)
+     (triv_act.induced_star_map A triv_act) \<equiv>\<^sub>M\<^sub>N))"
+    using H_3
+    apply (simp add: triv_G_def triv_act_def del: GMN_simps)
+    using triv_G_lang.G_Myhill_Nerode H_2
+    by blast    
+  have H_5:"triv_act.induced_star_map A triv_act = (\<lambda>x \<in> {undefined}. \<one>\<^bsub>BijGroup (A\<^sup>\<star>)\<^esub>)"
+    apply (simp add: BijGroup_def restrict_def fun_eq_iff triv_act_def)
+    by (clarsimp simp add: list.map_ident_strong)
+  have H_6: "(triv_act.induced_quot_map (A\<^sup>\<star>) (triv_act.induced_star_map A
+  triv_act) \<equiv>\<^sub>M\<^sub>N) = (\<lambda>x \<in> {undefined}. \<one>\<^bsub>BijGroup MN_equiv\<^esub>)"
+    apply (subst H_5)
+    apply (simp add: BijGroup_def fun_eq_iff Image_def)
+    apply (rule allI; rule conjI; intro impI)
+     apply (smt (verit) Collect_cong Collect_mem_eq Eps_cong MN_rel_equival equiv_Eps_in
+        in_quotient_imp_closed quotient_eq_iff)
+    using MN_rel_equival equiv_Eps_preserves
+    by auto
+  show "finite MN_equiv"
+    apply (subst fin_triv_orbs [symmetric]; subst H_6 [symmetric]; subst triv_G_def [symmetric])
+    by (rule H_4)
+qed
+end
+
+section \<open>
+Myhill-Nerode Theorem for Nominal $G$-Automata
+\<close>
+
+subsection \<open>
+Data Symmetries, Supports and Nominal Actions
+\<close>
+
+text \<open>
+The following locale corresponds to the definition 2.2 from \cite{bojanczyk2014automata}.
+Note that we let $G$ be an arbitrary group instead of a subgroup of \texttt{BijGroup} $D$,
+but assume there is a homomoprhism $\pi: G \to \texttt{BijGroup} D$.
+By \texttt{group\_hom.img\_is\_subgroup} this is an equivalent definition:
+\<close>
+
+locale data_symm = group_action G D \<pi>
+  for 
+    G :: "('grp, 'b) monoid_scheme" and
+    D :: "'D set" ("\<bbbD>") and
+    \<pi> 
+
+text \<open>
+The following locales corresponds to definition 4.3 from \cite{bojanczyk2014automata}:
+\<close>
+
+locale supports = data_symm G D \<pi> + alt_grp_act G X \<phi> 
+  for
+    G :: "('grp, 'b) monoid_scheme" and
+    D :: "'D set" ("\<bbbD>") and
+    \<pi> and
+    X :: "'X set" (structure) and
+    \<phi> +
+  fixes
+    C :: "'D set" and
+    x :: "'X"
+  assumes
+    is_in_set:
+    "x \<in> X" and
+    is_subset:
+    "C \<subseteq> \<bbbD>" and
+    supports:
+    "g \<in> carrier G \<Longrightarrow> (\<forall>c. c \<in> C \<longrightarrow> \<pi> g c = c) \<Longrightarrow> g \<odot>\<^bsub>\<phi>\<^esub> x = x"
+begin
+
+text \<open>
+The following lemma corresponds to lemma 4.9 from \cite{bojanczyk2014automata}:
+\<close>
+
+lemma image_supports:
+  "\<And>g. g \<in> carrier G \<Longrightarrow> supports G D \<pi> X \<phi> (\<pi> g ` C) (g \<odot>\<^bsub>\<phi>\<^esub> x)"
+proof-
+  fix g
+  assume 
+    A_0: "g \<in> carrier G"
+  have H_0: "\<And>h. data_symm G \<bbbD> \<pi> \<Longrightarrow>
+         group_action G X \<phi> \<Longrightarrow>
+         x \<in> X \<Longrightarrow>
+         C \<subseteq> \<bbbD> \<Longrightarrow>
+         \<forall>g. g \<in> carrier G \<longrightarrow> (\<forall>c. c \<in> C \<longrightarrow> \<pi> g c = c) \<longrightarrow> \<phi> g x = x \<Longrightarrow>
+         h \<in> carrier G \<Longrightarrow> \<forall>c. c \<in> \<pi> g ` C \<longrightarrow> \<pi> h c = c \<Longrightarrow>
+         \<phi> h (\<phi> g x) = \<phi> g x"
+  proof-
+    fix h
+    assume
+      A1_0: "data_symm G \<bbbD> \<pi>" and
+      A1_1: "group_action G X \<phi>" and
+      A1_2: "\<forall>g. g \<in> carrier G \<longrightarrow> (\<forall>c. c \<in> C \<longrightarrow> \<pi> g c = c) \<longrightarrow> \<phi> g x = x" and
+      A1_3: "h \<in> carrier G" and
+      A1_4: "\<forall>c. c \<in> \<pi> g ` C \<longrightarrow> \<pi> h c = c"
+    have H1_0: "\<And>g. g \<in> carrier G \<Longrightarrow> (\<forall>c. c \<in> C \<longrightarrow> \<pi> g c = c) \<Longrightarrow> \<phi> g x = x"
+      using A1_2
+      by auto 
+    have H1_1: "\<forall>c. c \<in> C \<longrightarrow> \<pi> ((inv\<^bsub>G\<^esub> g) \<otimes>\<^bsub>G\<^esub> h \<otimes>\<^bsub>G\<^esub> g) c = c \<Longrightarrow>
+    \<phi> ((inv\<^bsub>G\<^esub> g) \<otimes>\<^bsub>G\<^esub> h \<otimes>\<^bsub>G\<^esub> g) x = x"
+      apply (rule H1_0[of "((inv\<^bsub>G\<^esub> g) \<otimes>\<^bsub>G\<^esub> h \<otimes>\<^bsub>G\<^esub> g)"])
+       apply (meson A_0 A1_3 group.subgroupE(3) group.subgroup_self group_hom group_hom.axioms(1)
+          subgroup.m_closed)
+      by simp
+    have H2: "\<pi> (((inv\<^bsub>G\<^esub> g) \<otimes>\<^bsub>G\<^esub> h) \<otimes>\<^bsub>G\<^esub> g) = compose \<bbbD> (\<pi> ((inv\<^bsub>G\<^esub> g) \<otimes>\<^bsub>G\<^esub> h)) (\<pi> g)"
+      using A1_0
+      apply (clarsimp simp add: data_symm_def group_action_def BijGroup_def group_hom_def
+          group_hom_axioms_def hom_def restrict_def)
+      apply (rule meta_mp[of "\<pi> g \<in> Bij \<bbbD> \<and> \<pi> ((inv\<^bsub>G\<^esub> g) \<otimes>\<^bsub>G\<^esub> h) \<in> Bij \<bbbD>"])
+       apply (smt (verit) A_0 A1_3 data_symm.axioms data_symm_axioms group.inv_closed
+          group.surj_const_mult group_action.bij_prop0 image_eqI)
+      apply (rule conjI)
+      using A_0
+       apply blast
+      by (meson A_0 A1_3 data_symm.axioms data_symm_axioms group.subgroupE(3) group.subgroupE(4)
+          group.subgroup_self group_action.bij_prop0)
+    also have H1_3: "... = compose \<bbbD> (compose \<bbbD> (\<pi> (inv\<^bsub>G\<^esub> g) ) (\<pi> h)) (\<pi> g)"
+      using A1_0
+      apply (clarsimp simp add: data_symm_def group_action_def BijGroup_def comp_def
+          group_hom_def group_hom_axioms_def hom_def restrict_def)
+      apply (rule meta_mp[of "\<pi> (inv\<^bsub>G\<^esub> g) \<in> Bij \<bbbD> \<and> \<pi> h \<in> Bij \<bbbD>"])
+       apply (simp add: A_0 A1_3)
+      apply (rule conjI)
+       apply (simp add: A_0 Pi_iff)
+      using A1_3
+      by blast
+    also have H1_4: "... = compose \<bbbD> ((\<pi> (inv\<^bsub>G\<^esub> g)) \<circ> (\<pi> h)) (\<pi> g)"
+      using A1_0
+      apply (clarsimp simp add: data_symm_def group_action_def BijGroup_def comp_def group_hom_def
+          group_hom_axioms_def hom_def restrict_def compose_def)
+      using A_0 A1_3  
+      by (meson data_symm.axioms data_symm_axioms group.inv_closed group_action.element_image)
+    also have H1_5: "... = (\<lambda>d \<in> \<bbbD>. ((\<pi> (inv\<^bsub>G\<^esub> g)) \<circ> (\<pi> h) \<circ> (\<pi> g)) d)"
+      by (simp add: compose_def)
+    have H1_6: "\<And>c. c \<in> C \<Longrightarrow> ((\<pi> h) \<circ> (\<pi> g)) c = (\<pi> g) c"
+      using A1_4
+      by auto
+    have H1_7: "\<And>c. c \<in> C \<Longrightarrow> ((\<pi> (inv\<^bsub>G\<^esub> g)) \<circ> (\<pi> h) \<circ> (\<pi> g)) c = c"
+      using H1_6 A1_0
+      apply (simp add: data_symm_def group_action_def BijGroup_def compose_def group_hom_def
+          group_hom_axioms_def hom_def)
+      by (meson A_0 data_symm.axioms data_symm_axioms group_action.orbit_sym_aux is_subset subsetD)
+    have H1_8: "\<forall>c. c \<in> C \<longrightarrow> \<pi> ((inv\<^bsub>G\<^esub> g) \<otimes>\<^bsub>G\<^esub> h \<otimes>\<^bsub>G\<^esub> g) c = c"
+      using H1_7 H1_5
+      by (metis calculation is_subset restrict_apply' subsetD)
+    have H1_9: "\<phi> ((inv\<^bsub>G\<^esub> g) \<otimes>\<^bsub>G\<^esub> h \<otimes>\<^bsub>G\<^esub> g) x = x"
+      using H1_8
+      by (simp add: H1_1)
+    hence H1_10: "\<phi> ((inv\<^bsub>G\<^esub> g) \<otimes>\<^bsub>G\<^esub> h \<otimes>\<^bsub>G\<^esub> g) x = \<phi> ((inv\<^bsub>G\<^esub> g) \<otimes>\<^bsub>G\<^esub> (h \<otimes>\<^bsub>G\<^esub> g)) x"
+      by (smt (verit, ccfv_SIG) A_0 A1_3 group.inv_closed group.subgroupE(4) group.subgroup_self
+          group_action.composition_rule group_action.element_image group_action_axioms group_hom
+          group_hom.axioms(1) is_in_set)
+    have H1_11: "... = ((\<phi> (inv\<^bsub>G\<^esub> g)) \<circ> (\<phi> (h \<otimes>\<^bsub>G\<^esub> g))) x"
+      using A_0 A1_3 group.subgroupE(4) group.subgroup_self group_action.composition_rule
+        group_action_axioms group_hom group_hom.axioms(1) is_in_set
+      by fastforce
+    have H1_12: "... = ((the_inv_into X (\<phi> g)) \<circ> (\<phi> (h \<otimes>\<^bsub>G\<^esub> g))) x"
+      using A1_1
+      apply (simp add: group_action_def)
+      by (smt (verit) A_0 A1_3 group.inv_closed group.subgroupE(4) group.subgroup_self
+          group_action.element_image group_action.inj_prop group_action.orbit_sym_aux
+          group_action_axioms group_hom.axioms(1) is_in_set the_inv_into_f_f)
+    have H1_13: "((the_inv_into X (\<phi> g)) \<circ> (\<phi> (h \<otimes>\<^bsub>G\<^esub> g))) x = x" 
+      using H1_9 H1_10 H1_11 H1_12
+      by auto
+    hence H1_14: "(\<phi> (h \<otimes>\<^bsub>G\<^esub> g)) x = \<phi> g x"
+      using H1_13
+      by (metis A_0 A1_3 comp_apply composition_rule element_image f_the_inv_into_f inj_prop is_in_set
+          surj_prop)
+    show "\<phi> h (\<phi> g x) = \<phi> g x"
+      using A1_3 A1_2 A_0 H1_14 composition_rule
+      by (simp add: is_in_set)
+  qed
+  show "supports G D \<pi> X \<phi> (\<pi> g ` C) (g \<odot>\<^bsub>\<phi>\<^esub> x)"
+    using supports_axioms
+    apply (clarsimp simp add: supports_def supports_axioms_def)
+    apply (intro conjI)
+    using element_image is_in_set A_0
+      apply blast
+     apply (metis A_0 data_symm_def group_action.surj_prop image_mono is_subset)
+    apply (rule allI; intro impI)
+    apply (rename_tac h)
+    by (simp add: H_0)
+qed
+end
+
+locale nominal = data_symm G D \<pi> + alt_grp_act G X \<phi>
+  for
+    G :: "('grp, 'b) monoid_scheme" and
+    D :: "'D set" ("\<bbbD>") and
+    \<pi> and
+    X :: "'X set" (structure) and
+    \<phi> +
+  assumes
+    is_nominal:
+    "\<And>g x. g \<in> carrier G \<Longrightarrow> x \<in> X \<Longrightarrow> \<exists>C. C \<subseteq> \<bbbD> \<and> finite C \<and> supports G \<bbbD> \<pi> X \<phi> C x"
+
+locale nominal_det_G_aut = det_G_aut + 
+  nominal G D \<pi> A \<phi> + nominal G D \<pi> S \<psi>
+  for
+    D :: "'D set" ("\<bbbD>") and
+    \<pi> 
+
+text \<open>
+The following lemma corresponds to lemma 4.8 from \cite{bojanczyk2014automata}:
+\<close>
+
+lemma (in eq_var_func) supp_el_pres:
+  "supports G D \<pi> X \<phi> C x \<Longrightarrow> supports G D \<pi> Y \<psi> C (f x)"
+  apply (clarsimp simp add: supports_def supports_axioms_def)
+  apply (rule conjI)
+  using eq_var_func_axioms
+   apply (simp add: eq_var_func_def eq_var_func_axioms_def)
+  apply (intro conjI)
+  using is_ext_func_bet 
+   apply blast
+  apply clarify
+  by (metis is_eq_var_func')
+
+lemma (in nominal) support_union_lem:
+  fixes f sup_C col
+  assumes H_f: "f = (\<lambda>x. (SOME C. C \<subseteq> \<bbbD> \<and> finite C \<and> supports G \<bbbD> \<pi> X \<phi> C x))"
+    and H_col: "col \<subseteq> X \<and> finite col"
+    and H_sup_C: "sup_C = \<Union>{Cx. Cx \<in> f ` col}"
+  shows "\<And>x. x \<in> col \<Longrightarrow> sup_C \<subseteq> \<bbbD> \<and> finite sup_C \<and> supports G \<bbbD> \<pi> X \<phi> sup_C x"
+proof - 
+  fix x
+  assume A_0: "x \<in> col"
+  have H_0: "\<And>x. x \<in> X \<Longrightarrow> \<exists>C. C \<subseteq> \<bbbD> \<and> finite C \<and> supports G \<bbbD> \<pi> X \<phi> C x"
+    using nominal_axioms 
+    apply (clarsimp simp add: nominal_def nominal_axioms_def)
+    using stabilizer_one_closed stabilizer_subset 
+    by blast
+  have H_1: "\<And>x. x \<in> col \<Longrightarrow> f x \<subseteq> \<bbbD> \<and> finite (f x) \<and> supports G \<bbbD> \<pi> X \<phi> (f x) x"
+    apply (subst H_f)
+    using someI_ex H_col H_f H_0
+    by (metis (no_types, lifting) in_mono)
+  have H_2: "sup_C \<subseteq> \<bbbD>"
+    using H_1 
+    by (simp add: H_sup_C UN_least)
+  have H_3: "finite sup_C"
+    using H_1 H_col H_sup_C
+    by simp
+  have H_4: "f x \<subseteq> sup_C"
+    using H_1 H_sup_C A_0
+    by blast
+  have H_5: "\<And>g c. \<lbrakk>g \<in> carrier G; (c \<in> sup_C \<longrightarrow> \<pi> g c = c); c \<in> (f x)\<rbrakk> \<Longrightarrow> \<pi> g c = c"
+    using H_4 H_1 A_0
+    by (auto simp add: image_def supports_def supports_axioms_def) 
+  have H_6: "supports G \<bbbD> \<pi> X \<phi> sup_C x"
+    apply (simp add: supports_def supports_axioms_def)
+    apply (intro conjI)
+        apply (simp add: data_symm_axioms)
+    using A_0 H_1 supports.axioms(2)
+       apply fastforce
+    using H_col A_0
+      apply blast
+     apply (rule H_2)
+    apply (clarify)
+    using supports_axioms_def[of G D \<pi> X \<phi> sup_C]
+    apply (clarsimp)
+    using H_1 A_0
+    apply (clarsimp simp add: supports_def supports_axioms_def)
+    using A_0 H_5
+    by presburger
+  show "sup_C \<subseteq> \<bbbD> \<and> finite sup_C \<and> supports G \<bbbD> \<pi> X \<phi> sup_C x"
+    using H_2 H_3 H_6 by auto
+qed
+
+lemma (in nominal) set_of_list_nom:
+  "nominal G D \<pi> (X\<^sup>\<star>) (\<phi>\<^sup>\<star>)"
+proof-
+  have H_0: "\<And>g x. g \<in> carrier G \<Longrightarrow> \<forall>x\<in>set x. x \<in> X \<Longrightarrow>
+            \<exists>C\<subseteq>\<bbbD>. finite C \<and> supports G \<bbbD> \<pi> (X\<^sup>\<star>) (\<phi>\<^sup>\<star>) C x"
+  proof-
+    fix g w
+    assume
+      A1_0: "g \<in> carrier G" and
+      A1_1: "\<forall>x\<in>set w. x \<in> X"
+    have H1_0: "\<And>x. x \<in> X \<Longrightarrow> \<exists>C\<subseteq>\<bbbD>. finite C \<and> supports G \<bbbD> \<pi> X \<phi> C x"
+      using A1_0 is_nominal by force
+    define f where H1_f: "f = (\<lambda>x. (SOME C. C \<subseteq> \<bbbD> \<and>  finite C \<and> supports G \<bbbD> \<pi> X \<phi> C x))"
+    define sup_C :: "'D set " where H1_sup_C: "sup_C = \<Union>{Cx. Cx \<in> f ` set w}"
+    have H1_1: "\<And>x. x \<in> set w \<Longrightarrow> sup_C \<subseteq> \<bbbD> \<and> finite sup_C \<and> supports G \<bbbD> \<pi> X \<phi> sup_C x"
+      apply (rule support_union_lem[where f = f and col = "set w"])
+         apply (rule H1_f)
+      using A1_0
+        apply (simp add: A1_1 subset_code(1))
+       apply (rule H1_sup_C)
+      by simp
+    have H1_2: "supports G \<bbbD> \<pi> (X\<^sup>\<star>) (\<phi>\<^sup>\<star>) sup_C w"
+      apply (clarsimp simp add: supports_def supports_axioms_def simp del: GMN_simps)
+      apply (intro conjI) 
+          apply (simp add: data_symm_axioms)
+      using lists_a_Gset
+         apply (auto)[1]
+        apply (simp add: A1_1 in_listsI)
+      using H1_1 H1_sup_C
+       apply blast
+      apply (rule allI; intro impI)
+      apply clarsimp
+      apply (rule conjI)
+      using H1_1
+      by (auto simp add: supports_def supports_axioms_def map_idI)
+    show "\<exists>C\<subseteq>\<bbbD>. finite C \<and> supports G \<bbbD> \<pi> (X\<^sup>\<star>) (\<phi>\<^sup>\<star>) C w"
+      using nominal_axioms_def
+      apply (clarsimp simp add: nominal_def simp del: GMN_simps)
+      using H1_1 H1_2
+      by (metis Collect_empty_eq H1_sup_C Union_empty empty_iff image_empty infinite_imp_nonempty
+          subset_empty subset_emptyI supports.is_subset)
+  qed
+  show ?thesis
+    apply (clarsimp simp add: nominal_def nominal_axioms_def simp del: GMN_simps)
+    apply (intro conjI)
+    using group.subgroupE(2) group.subgroup_self group_hom group_hom.axioms(1)
+      apply (simp add: data_symm_axioms)
+     apply (rule lists_a_Gset)
+    apply (clarify)
+    by (simp add: H_0 del: GMN_simps)
+qed
+
+subsection \<open>
+Proving the Myhill-Nerode Theorem for Nominal $G$-Automata
+\<close>
+
+context det_G_aut begin
+adhoc_overloading
+  star labels_a_G_set.induced_star_map
+end
+
+lemma (in det_G_aut) input_to_init_eqvar:
+  "eq_var_func G (A\<^sup>\<star>) (\<phi>\<^sup>\<star>) S \<psi> (\<lambda>w\<in>A\<^sup>\<star>. (\<delta>\<^sup>\<star>) i w)"
+proof-
+  have H_0: "\<And>a g. \<lbrakk>\<forall>x\<in>set a. x \<in> A; map (\<phi> g) a \<in> A\<^sup>\<star>; g \<in> carrier G\<rbrakk> \<Longrightarrow>
+            (\<delta>\<^sup>\<star>) i (map (\<phi> g) a) = \<psi> g ((\<delta>\<^sup>\<star>) i a)"
+  proof-
+    fix w g
+    assume
+      A1_0: "\<forall>x\<in>set w. x \<in> A" and
+      A1_1: "map (\<phi> g) w \<in> A\<^sup>\<star>" and
+      A1_2: "g \<in> carrier G"
+    have H1_0: "(\<delta>\<^sup>\<star>) (\<psi> g i) (map (\<phi> g) w) = \<psi> g ((\<delta>\<^sup>\<star>) i w)"
+      using give_input_eq_var
+      apply (clarsimp simp add: eq_var_func_axioms_def eq_var_func_def)
+      using A1_0 A1_1 A1_2 in_listsI is_aut.init_state_is_a_state states_a_G_set.element_image 
+      by (smt (verit, del_insts))
+    have H1_1: "(\<psi> g i) = i"
+      using A1_2 is_aut.init_state_is_a_state init_is_eq_var.is_equivar
+      by force
+    show "(\<delta>\<^sup>\<star>) i (map (\<phi> g) w) = \<psi> g ((\<delta>\<^sup>\<star>) i w)"
+      using H1_0 H1_1
+      by simp
+  qed
+  show ?thesis
+    apply (clarsimp simp add: eq_var_func_def eq_var_func_axioms_def)
+    apply (intro conjI)
+    using labels_a_G_set.lists_a_Gset
+       apply fastforce
+      apply (simp add: states_a_G_set.group_action_axioms del: GMN_simps)
+     apply (simp add: in_listsI is_aut.give_input_closed is_aut.init_state_is_a_state)
+    apply clarify
+    apply (rule conjI; intro impI)
+     apply (simp add: H_0)
+    using labels_a_G_set.surj_prop
+    by fastforce
+qed
+
+lemma (in reach_det_G_aut) input_to_init_surj:
+  "(\<lambda>w\<in>A\<^sup>\<star>. (\<delta>\<^sup>\<star>) i w) ` (A\<^sup>\<star>) = S" 
+  using reach_det_G_aut_axioms
+  apply (clarsimp simp add: image_def reach_det_G_aut_def reach_det_aut_def
+      reach_det_aut_axioms_def)
+  using is_aut.give_input_closed is_aut.init_state_is_a_state
+  by blast
+
+context reach_det_G_aut begin
+adhoc_overloading
+  star labels_a_G_set.induced_star_map
+end
+
+text \<open>
+The following lemma corresponds to proposition 5.1 from \cite{bojanczyk2014automata}:
+\<close>
+
+proposition (in reach_det_G_aut) alpha_nom_imp_states_nom:
+  "nominal G D \<pi> A \<phi> \<Longrightarrow> nominal G D \<pi> S \<psi>"
+proof-
+  assume
+    A_0: "nominal G D \<pi> A \<phi>"
+  have H_0: "\<And>g x. \<lbrakk>g \<in> carrier G; data_symm G D \<pi>; group_action G A \<phi>;
+             \<forall>x. x \<in> A \<longrightarrow> (\<exists>C\<subseteq>D. finite C \<and> supports G D \<pi> A \<phi> C x); x \<in> S\<rbrakk>
+           \<Longrightarrow> \<exists>C\<subseteq>D. finite C \<and> supports G D \<pi> S \<psi> C x"
+  proof -
+    fix g s
+    assume
+      A1_0: "g \<in> carrier G" and
+      A1_1: "data_symm G D \<pi>" and
+      A1_2: "group_action G A \<phi>" and
+      A1_3: "\<forall>x. x \<in> A \<longrightarrow> (\<exists>C\<subseteq>D. finite C \<and> supports G D \<pi> A \<phi> C x)" and
+      A1_4: "s \<in> S"
+    have H1_0: "\<And>x. x \<in> (A\<^sup>\<star>) \<Longrightarrow> \<exists>C\<subseteq>D. finite C \<and> supports G D \<pi> (A\<^sup>\<star>) (\<phi>\<^sup>\<star>) C x"
+      using nominal.set_of_list_nom[of G D \<pi> A \<phi>] A1_2
+      apply (clarsimp simp add: nominal_def)
+      by (metis A1_0 A1_1 A1_3 in_listsI labels_a_G_set.induced_star_map_def nominal_axioms_def)   
+    define f where H1_f: "f = (\<lambda>w\<in>A\<^sup>\<star>. (\<delta>\<^sup>\<star>) i w)"      
+    obtain w where H1_w0: "s = f w" and H1_w1: "w \<in> (A\<^sup>\<star>)"
+      using input_to_init_surj A1_4
+      apply (simp add: H1_f image_def)
+      by (metis is_reachable)
+    obtain C where H1_C: "finite C \<and> supports G D \<pi> (A\<^sup>\<star>) (labels_a_G_set.induced_star_map \<phi>) C w"
+      by (meson H1_0 H1_w0 H1_w1)
+    have H1_2: "supports G D \<pi> S \<psi> C s"
+      apply (subst H1_w0)
+      apply (rule eq_var_func.supp_el_pres[where X = "A\<^sup>\<star>" and \<phi> = "\<phi>\<^sup>\<star>"])
+       apply (subst H1_f)
+       apply (rule det_G_aut.input_to_init_eqvar[of A S i F \<delta> G \<phi> \<psi>])
+      using reach_det_G_aut_axioms
+       apply (simp add: reach_det_G_aut_def) 
+      using H1_C
+      by simp
+    show "\<exists>C\<subseteq>D. finite C \<and> supports G D \<pi> S \<psi> C s"
+      using H1_2 H1_C 
+      by (meson supports.is_subset)
+  qed
+  show ?thesis
+    apply (rule meta_mp[of "(\<exists>g. g \<in> carrier G)"])
+    subgoal 
+      using A_0 apply (clarsimp simp add: nominal_def nominal_axioms_def)
+      apply (rule conjI)
+      subgoal for g
+        by (clarsimp simp add: states_a_G_set.group_action_axioms)
+      apply clarify
+      by (simp add: H_0)
+    by (metis bot.extremum_unique ex_in_conv is_aut.init_state_is_a_state
+        states_a_G_set.stabilizer_one_closed states_a_G_set.stabilizer_subset)
+qed
+
+text \<open>
+The following theorem corresponds to theorem 5.2 from \cite{bojanczyk2014automata}:
+\<close>
+
+theorem (in G_lang) Nom_G_Myhill_Nerode:
+  assumes
+    orb_fin: "finite (orbits G A \<phi>)" and
+    nom: "nominal G D \<pi> A \<phi>"
+  shows
+    Nom_G_Myhill_Nerode_LR: "finite (orbits G MN_equiv ([(\<phi>\<^sup>\<star>)]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>)) \<Longrightarrow>
+     (\<exists>S F :: 'alpha list set set. \<exists>i :: 'alpha list set. \<exists>\<delta>. \<exists>\<psi>.
+     reach_det_G_aut_rec_lang A S i F \<delta> G \<phi> \<psi> L \<and> finite (orbits G S \<psi>)
+     \<and> nominal_det_G_aut A S i F \<delta> G \<phi> \<psi> D \<pi>)" and
+    Nom_G_Myhill_Nerode_RL: "(\<exists>S F :: 's set. \<exists>i :: 's. \<exists>\<delta>. \<exists>\<psi>.
+     det_G_aut_rec_lang A S i F \<delta> G \<phi> \<psi> L \<and> finite (orbits G S \<psi>)
+     \<and> nominal_det_G_aut A S i F \<delta> G \<phi> \<psi> D \<pi>)
+    \<Longrightarrow> finite (orbits G MN_equiv ([(\<phi>\<^sup>\<star>)]\<^sub>\<equiv>\<^sub>M\<^sub>N \<^bsub>A\<^sup>\<star>\<^esub>))"
+proof-
+  assume
+    A_0: "finite (orbits G MN_equiv ([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>))"
+  obtain S F :: "'alpha list set set" and i :: "'alpha list set" and \<delta> \<psi>
+    where H_MN: "reach_det_G_aut_rec_lang A S i F \<delta> G \<phi> \<psi> L \<and> finite (orbits G S \<psi>)"
+    using A_0 orb_fin G_Myhill_Nerode_LR
+    by blast
+  have H_0: "nominal G D \<pi> S \<psi>"
+    using H_MN
+    apply (clarsimp simp del: GMN_simps simp add: reach_det_G_aut_rec_lang_def)
+    using nom reach_det_G_aut.alpha_nom_imp_states_nom 
+    by metis
+  show "\<exists>S F :: 'alpha list set set. \<exists>i :: 'alpha list set. \<exists>\<delta>. \<exists>\<psi>.
+       reach_det_G_aut_rec_lang A S i F \<delta> G \<phi> \<psi> L \<and>
+       finite (orbits G S \<psi>) \<and> nominal_det_G_aut A S i F \<delta> G \<phi> \<psi> D \<pi>"
+    apply (simp add: nominal_det_G_aut_def reach_det_G_aut_rec_lang_def)
+    using nom H_MN H_0
+    apply (clarsimp simp add: reach_det_G_aut_rec_lang_def reach_det_G_aut_def
+        reach_det_aut_axioms_def)
+    by blast
+next
+  assume A0: "\<exists>S F i \<delta> \<psi>. det_G_aut_rec_lang A S i F \<delta> G \<phi> \<psi> L \<and> finite (orbits G S \<psi>)
+                \<and> nominal_det_G_aut A S i F \<delta> G \<phi> \<psi> D \<pi>"
+  show "finite (orbits G MN_equiv ([\<phi>\<^sup>\<star>]\<^sub>\<equiv>\<^sub>M\<^sub>N\<^bsub>A\<^sup>\<star>\<^esub>))"
+    using A0 orb_fin
+    by (meson G_Myhill_Nerode_RL)
+qed
 end
\ No newline at end of file