theory Classical
imports "../Nominal2"
begin


(* example from Urban's PhD *)

atom_decl name
atom_decl coname

nominal_datatype trm =
  Ax "name" "coname"
| Cut c::"coname" t1::"trm" n::"name" t2::"trm"             bind n in t1, bind c in t2  
     ("Cut <_>._ '(_')._" [100,100,100,100] 100)
| NotR n::"name" t::"trm" "coname"                            bind n in t
     ("NotR '(_')._ _" [100,100,100] 100)
| NotL c::"coname" t::"trm" "name"                            bind c in t   
     ("NotL <_>._ _" [100,100,100] 100)
| AndR c1::"coname" t1::"trm" c2::"coname" t2::"trm" "coname" bind c1 in t1, bind c2 in t2
     ("AndR <_>._ <_>._ _" [100,100,100,100,100] 100)
| AndL1 n::"name" t::"trm" "name"                             bind n in t
     ("AndL1 '(_')._ _" [100,100,100] 100)
| AndL2 n::"name" t::"trm" "name"                             bind n in t
     ("AndL2 '(_')._ _" [100,100,100] 100)
| OrR1 c::"coname" t::"trm" "coname"                          bind c in t             
     ("OrR1 <_>._ _" [100,100,100] 100)
| OrR2 c::"coname" t::"trm" "coname"                          bind c in t     
     ("OrR2 <_>._ _" [100,100,100] 100)
| OrL n1::"name" t1::"trm" n2::"name" t2::"trm" "name"        bind n1 in t1, bind n2 in t2       
     ("OrL '(_')._ '(_')._ _" [100,100,100,100,100] 100)
| ImpL c::"coname" t1::"trm" n::"name" t2::"trm" "name"       bind c in t1, bind n in t2
     ("ImpL <_>._ '(_')._ _" [100,100,100,100,100] 100)
| ImpR n::"name" c::"coname" t::"trm" "coname"                bind n c in t
     ("ImpR '(_').<_>._ _" [100,100,100,100] 100)

thm trm.distinct
thm trm.induct
thm trm.exhaust
thm trm.strong_exhaust
thm trm.strong_exhaust[simplified]
thm trm.fv_defs
thm trm.bn_defs
thm trm.perm_simps
thm trm.eq_iff
thm trm.fv_bn_eqvt
thm trm.size_eqvt
thm trm.supp
thm trm.supp[simplified]

lemma Abs_set_fcb2:
  fixes as bs :: "atom set"
    and x y :: "'b :: fs"
    and c::"'c::fs"
  assumes eq: "[as]set. x = [bs]set. y"
  and fin: "finite as" "finite bs"
  and fcb1: "as \<sharp>* f as x c"
  and fresh1: "as \<sharp>* c"
  and fresh2: "bs \<sharp>* c"
  and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f as x c) = f (p \<bullet> as) (p \<bullet> x) c"
  and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f bs y c) = f (p \<bullet> bs) (p \<bullet> y) c"
  shows "f as x c = f bs y c"
proof -
  have "supp (as, x, c) supports (f as x c)"
    unfolding  supports_def fresh_def[symmetric]
    by (simp add: fresh_Pair perm1 fresh_star_def supp_swap swap_fresh_fresh)
  then have fin1: "finite (supp (f as x c))"
    using fin by (auto intro: supports_finite simp add: finite_supp supp_of_finite_sets supp_Pair)
  have "supp (bs, y, c) supports (f bs y c)"
    unfolding  supports_def fresh_def[symmetric]
    by (simp add: fresh_Pair perm2 fresh_star_def supp_swap swap_fresh_fresh)
  then have fin2: "finite (supp (f bs y c))"
    using fin by (auto intro: supports_finite simp add: finite_supp supp_of_finite_sets supp_Pair)
  obtain q::"perm" where 
    fr1: "(q \<bullet> as) \<sharp>* (x, c, f as x c, f bs y c)" and 
    fr2: "supp q \<sharp>* ([as]set. x)" and 
    inc: "supp q \<subseteq> as \<union> (q \<bullet> as)"
    using at_set_avoiding3[where xs="as" and c="(x, c, f as x c, f bs y c)" and x="[as]set. x"]  
      fin1 fin2 fin
    by (auto simp add: supp_Pair finite_supp Abs_fresh_star dest: fresh_star_supp_conv)
  have "[q \<bullet> as]set. (q \<bullet> x) = q \<bullet> ([as]set. x)" by simp
  also have "\<dots> = [as]set. x"
    by (simp only: fr2 perm_supp_eq)
  finally have "[q \<bullet> as]set. (q \<bullet> x) = [bs]set. y" using eq by simp
  then obtain r::perm where 
    qq1: "q \<bullet> x = r \<bullet> y" and 
    qq2: "q \<bullet> as = r \<bullet> bs" and 
    qq3: "supp r \<subseteq> (q \<bullet> as) \<union> bs"
    apply(drule_tac sym)
    apply(simp only: Abs_eq_iff2 alphas)
    apply(erule exE)
    apply(erule conjE)+
    apply(drule_tac x="p" in meta_spec)
    apply(simp add: set_eqvt)
    apply(blast)
    done
  have "as \<sharp>* f as x c" by (rule fcb1)
  then have "q \<bullet> (as \<sharp>* f as x c)"
    by (simp add: permute_bool_def)
  then have "(q \<bullet> as) \<sharp>* f (q \<bullet> as) (q \<bullet> x) c"
    apply(simp add: fresh_star_eqvt set_eqvt)
    apply(subst (asm) perm1)
    using inc fresh1 fr1
    apply(auto simp add: fresh_star_def fresh_Pair)
    done
  then have "(r \<bullet> bs) \<sharp>* f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 by simp
  then have "r \<bullet> (bs \<sharp>* f bs y c)"
    apply(simp add: fresh_star_eqvt set_eqvt)
    apply(subst (asm) perm2[symmetric])
    using qq3 fresh2 fr1
    apply(auto simp add: set_eqvt fresh_star_def fresh_Pair)
    done
  then have fcb2: "bs \<sharp>* f bs y c" by (simp add: permute_bool_def)
  have "f as x c = q \<bullet> (f as x c)"
    apply(rule perm_supp_eq[symmetric])
    using inc fcb1 fr1 by (auto simp add: fresh_star_def)
  also have "\<dots> = f (q \<bullet> as) (q \<bullet> x) c" 
    apply(rule perm1)
    using inc fresh1 fr1 by (auto simp add: fresh_star_def)
  also have "\<dots> = f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 by simp
  also have "\<dots> = r \<bullet> (f bs y c)"
    apply(rule perm2[symmetric])
    using qq3 fresh2 fr1 by (auto simp add: fresh_star_def)
  also have "... = f bs y c"
    apply(rule perm_supp_eq)
    using qq3 fr1 fcb2 by (auto simp add: fresh_star_def)
  finally show ?thesis by simp
qed

lemma Abs_res_fcb2:
  fixes as bs :: "atom set"
    and x y :: "'b :: fs"
    and c::"'c::fs"
  assumes eq: "[as]res. x = [bs]res. y"
  and fin: "finite as" "finite bs"
  and fcb1: "as \<sharp>* f as x c"
  and fresh1: "as \<sharp>* c"
  and fresh2: "bs \<sharp>* c"
  and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f as x c) = f (p \<bullet> as) (p \<bullet> x) c"
  and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f bs y c) = f (p \<bullet> bs) (p \<bullet> y) c"
  shows "f as x c = f bs y c"
proof -
  have "supp (as, x, c) supports (f as x c)"
    unfolding  supports_def fresh_def[symmetric]
    by (simp add: fresh_Pair perm1 fresh_star_def supp_swap swap_fresh_fresh)
  then have fin1: "finite (supp (f as x c))"
    using fin by (auto intro: supports_finite simp add: finite_supp supp_of_finite_sets supp_Pair)
  have "supp (bs, y, c) supports (f bs y c)"
    unfolding  supports_def fresh_def[symmetric]
    by (simp add: fresh_Pair perm2 fresh_star_def supp_swap swap_fresh_fresh)
  then have fin2: "finite (supp (f bs y c))"
    using fin by (auto intro: supports_finite simp add: finite_supp supp_of_finite_sets supp_Pair)
  obtain q::"perm" where 
    fr1: "(q \<bullet> as) \<sharp>* (x, c, f as x c, f bs y c)" and 
    fr2: "supp q \<sharp>* ([as]res. x)" and 
    inc: "supp q \<subseteq> as \<union> (q \<bullet> as)"
    using at_set_avoiding3[where xs="as" and c="(x, c, f as x c, f bs y c)" and x="[as]res. x"]  
      fin1 fin2 fin
    by (auto simp add: supp_Pair finite_supp Abs_fresh_star dest: fresh_star_supp_conv)
  have "[q \<bullet> as]res. (q \<bullet> x) = q \<bullet> ([as]res. x)" by simp
  also have "\<dots> = [as]res. x"
    by (simp only: fr2 perm_supp_eq)
  finally have "[q \<bullet> as]res. (q \<bullet> x) = [bs]res. y" using eq by simp
  then obtain r::perm where 
    qq1: "q \<bullet> x = r \<bullet> y" and 
    qq2: "(q \<bullet> as \<inter> supp (q \<bullet> x)) = r \<bullet> (bs \<inter> supp y)" and 
    qq3: "supp r \<subseteq> bs \<inter> supp y \<union> q \<bullet> as \<inter> supp (q \<bullet> x)"
    apply(drule_tac sym)
    apply(subst(asm) Abs_eq_res_set)
    apply(simp only: Abs_eq_iff2 alphas)
    apply(erule exE)
    apply(erule conjE)+
    apply(drule_tac x="p" in meta_spec)
    apply(simp add: set_eqvt)
    done
  have "(as \<inter> supp x) \<sharp>* f (as \<inter> supp x) x c" sorry (* FCB? *)
  then have "q \<bullet> ((as \<inter> supp x) \<sharp>* f (as \<inter> supp x) x c)"
    by (simp add: permute_bool_def)
  then have "(q \<bullet> (as \<inter> supp x)) \<sharp>* f (q \<bullet> (as \<inter> supp x)) (q \<bullet> x) c"
    apply(simp add: fresh_star_eqvt set_eqvt)
    sorry (* perm? *)
  then have "r \<bullet> (bs \<inter> supp y) \<sharp>* f (r \<bullet> (bs \<inter> supp y)) (r \<bullet> y) c" using qq2 
    apply (simp add: inter_eqvt)
    sorry
  (* rest similar reversing it other way around... *)
  show ?thesis sorry
qed



lemma Abs_lst_fcb2:
  fixes as bs :: "atom list"
    and x y :: "'b :: fs"
    and c::"'c::fs"
  assumes eq: "[as]lst. x = [bs]lst. y"
  and fcb1: "(set as) \<sharp>* f as x c"
  and fresh1: "set as \<sharp>* c"
  and fresh2: "set bs \<sharp>* c"
  and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f as x c) = f (p \<bullet> as) (p \<bullet> x) c"
  and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f bs y c) = f (p \<bullet> bs) (p \<bullet> y) c"
  shows "f as x c = f bs y c"
proof -
  have "supp (as, x, c) supports (f as x c)"
    unfolding  supports_def fresh_def[symmetric]
    by (simp add: fresh_Pair perm1 fresh_star_def supp_swap swap_fresh_fresh)
  then have fin1: "finite (supp (f as x c))"
    by (auto intro: supports_finite simp add: finite_supp)
  have "supp (bs, y, c) supports (f bs y c)"
    unfolding  supports_def fresh_def[symmetric]
    by (simp add: fresh_Pair perm2 fresh_star_def supp_swap swap_fresh_fresh)
  then have fin2: "finite (supp (f bs y c))"
    by (auto intro: supports_finite simp add: finite_supp)
  obtain q::"perm" where 
    fr1: "(q \<bullet> (set as)) \<sharp>* (x, c, f as x c, f bs y c)" and 
    fr2: "supp q \<sharp>* Abs_lst as x" and 
    inc: "supp q \<subseteq> (set as) \<union> q \<bullet> (set as)"
    using at_set_avoiding3[where xs="set as" and c="(x, c, f as x c, f bs y c)" and x="[as]lst. x"]  
      fin1 fin2
    by (auto simp add: supp_Pair finite_supp Abs_fresh_star dest: fresh_star_supp_conv)
  have "Abs_lst (q \<bullet> as) (q \<bullet> x) = q \<bullet> Abs_lst as x" by simp
  also have "\<dots> = Abs_lst as x"
    by (simp only: fr2 perm_supp_eq)
  finally have "Abs_lst (q \<bullet> as) (q \<bullet> x) = Abs_lst bs y" using eq by simp
  then obtain r::perm where 
    qq1: "q \<bullet> x = r \<bullet> y" and 
    qq2: "q \<bullet> as = r \<bullet> bs" and 
    qq3: "supp r \<subseteq> (q \<bullet> (set as)) \<union> set bs"
    apply(drule_tac sym)
    apply(simp only: Abs_eq_iff2 alphas)
    apply(erule exE)
    apply(erule conjE)+
    apply(drule_tac x="p" in meta_spec)
    apply(simp add: set_eqvt)
    apply(blast)
    done
  have "(set as) \<sharp>* f as x c" by (rule fcb1)
  then have "q \<bullet> ((set as) \<sharp>* f as x c)"
    by (simp add: permute_bool_def)
  then have "set (q \<bullet> as) \<sharp>* f (q \<bullet> as) (q \<bullet> x) c"
    apply(simp add: fresh_star_eqvt set_eqvt)
    apply(subst (asm) perm1)
    using inc fresh1 fr1
    apply(auto simp add: fresh_star_def fresh_Pair)
    done
  then have "set (r \<bullet> bs) \<sharp>* f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 by simp
  then have "r \<bullet> ((set bs) \<sharp>* f bs y c)"
    apply(simp add: fresh_star_eqvt set_eqvt)
    apply(subst (asm) perm2[symmetric])
    using qq3 fresh2 fr1
    apply(auto simp add: set_eqvt fresh_star_def fresh_Pair)
    done
  then have fcb2: "(set bs) \<sharp>* f bs y c" by (simp add: permute_bool_def)
  have "f as x c = q \<bullet> (f as x c)"
    apply(rule perm_supp_eq[symmetric])
    using inc fcb1 fr1 by (auto simp add: fresh_star_def)
  also have "\<dots> = f (q \<bullet> as) (q \<bullet> x) c" 
    apply(rule perm1)
    using inc fresh1 fr1 by (auto simp add: fresh_star_def)
  also have "\<dots> = f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 by simp
  also have "\<dots> = r \<bullet> (f bs y c)"
    apply(rule perm2[symmetric])
    using qq3 fresh2 fr1 by (auto simp add: fresh_star_def)
  also have "... = f bs y c"
    apply(rule perm_supp_eq)
    using qq3 fr1 fcb2 by (auto simp add: fresh_star_def)
  finally show ?thesis by simp
qed

lemma Abs_lst1_fcb2:
  fixes a b :: "atom"
    and x y :: "'b :: fs"
    and c::"'c :: fs"
  assumes e: "(Abs_lst [a] x) = (Abs_lst [b] y)"
  and fcb1: "a \<sharp> f a x c"
  and fresh: "{a, b} \<sharp>* c"
  and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f a x c) = f (p \<bullet> a) (p \<bullet> x) c"
  and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f b y c) = f (p \<bullet> b) (p \<bullet> y) c"
  shows "f a x c = f b y c"
using e
apply(drule_tac Abs_lst_fcb2[where c="c" and f="\<lambda>(as::atom list) . f (hd as)"])
apply(simp_all)
using fcb1 fresh perm1 perm2
apply(simp_all add: fresh_star_def)
done

lemma supp_zero_perm_zero:
  shows "supp (p :: perm) = {} \<longleftrightarrow> p = 0"
  by (metis supp_perm_singleton supp_zero_perm)

lemma permute_atom_list_id:
  shows "p \<bullet> l = l \<longleftrightarrow> supp p \<inter> set l = {}"
  by (induct l) (auto simp add: supp_Nil supp_perm)

lemma permute_length_eq:
  shows "p \<bullet> xs = ys \<Longrightarrow> length xs = length ys"
  by (auto simp add: length_eqvt[symmetric] permute_pure)

lemma Abs_lst_binder_length:
  shows "[xs]lst. T = [ys]lst. S \<Longrightarrow> length xs = length ys"
  by (auto simp add: Abs_eq_iff alphas length_eqvt[symmetric] permute_pure)

lemma Abs_lst_binder_eq:
  shows "Abs_lst l T = Abs_lst l S \<longleftrightarrow> T = S"
  by (rule, simp_all add: Abs_eq_iff2 alphas)
     (metis fresh_star_zero inf_absorb1 permute_atom_list_id supp_perm_eq
       supp_zero_perm_zero)

lemma in_permute_list:
  shows "py \<bullet> p \<bullet> xs = px \<bullet> xs \<Longrightarrow>  x \<in> set xs \<Longrightarrow> py \<bullet> p \<bullet> x = px \<bullet> x"
  by (induct xs) auto




nominal_primrec 
  crename :: "trm \<Rightarrow> coname \<Rightarrow> coname \<Rightarrow> trm"  ("_[_\<turnstile>c>_]" [100,100,100] 100) 
where
  "(Ax x a)[d\<turnstile>c>e] = (if a=d then Ax x e else Ax x a)" 
| "atom a \<sharp> (d, e) \<Longrightarrow> (Cut <a>.M (x).N)[d\<turnstile>c>e] = Cut <a>.(M[d\<turnstile>c>e]) (x).(N[d\<turnstile>c>e])" 
| "(NotR (x).M a)[d\<turnstile>c>e] = (if a=d then NotR (x).(M[d\<turnstile>c>e]) e else NotR (x).(M[d\<turnstile>c>e]) a)" 
| "atom a \<sharp> (d, e) \<Longrightarrow> (NotL <a>.M x)[d\<turnstile>c>e] = (NotL <a>.(M[d\<turnstile>c>e]) x)" 
| "\<lbrakk>atom a \<sharp> (d, e); atom b \<sharp> (d, e)\<rbrakk> \<Longrightarrow> (AndR <a>.M <b>.N c)[d\<turnstile>c>e] = 
          (if c=d then AndR <a>.(M[d\<turnstile>c>e]) <b>.(N[d \<turnstile>c>e]) e else AndR <a>.(M[d\<turnstile>c>e]) <b>.(N[d\<turnstile>c>e]) c)" 
| "(AndL1 (x).M y)[d\<turnstile>c>e] = AndL1 (x).(M[d\<turnstile>c>e]) y"
| "(AndL2 (x).M y)[d\<turnstile>c>e] = AndL2 (x).(M[d\<turnstile>c>e]) y"
| "atom a \<sharp> (d, e) \<Longrightarrow> (OrR1 <a>.M b)[d\<turnstile>c>e] = 
          (if b=d then OrR1 <a>.(M[d\<turnstile>c>e]) e else OrR1 <a>.(M[d\<turnstile>c>e]) b)"
| "atom a \<sharp> (d, e) \<Longrightarrow> (OrR2 <a>.M b)[d\<turnstile>c>e] = 
          (if b=d then OrR2 <a>.(M[d\<turnstile>c>e]) e else OrR2 <a>.(M[d\<turnstile>c>e]) b)"
| "(OrL (x).M (y).N z)[d\<turnstile>c>e] = OrL (x).(M[d\<turnstile>c>e]) (y).(N[d\<turnstile>c>e]) z"
| "atom a \<sharp> (d, e) \<Longrightarrow> (ImpR (x).<a>.M b)[d\<turnstile>c>e] = 
          (if b=d then ImpR (x).<a>.(M[d\<turnstile>c>e]) e else ImpR (x).<a>.(M[d\<turnstile>c>e]) b)"
| "atom a \<sharp> (d, e) \<Longrightarrow> (ImpL <a>.M (x).N y)[d\<turnstile>c>e] = ImpL <a>.(M[d\<turnstile>c>e]) (x).(N[d\<turnstile>c>e]) y"
  apply(simp only: eqvt_def)
  apply(simp only: crename_graph_def)
  apply (rule, perm_simp, rule)
  apply(rule TrueI)
  -- "covered all cases"
  apply(case_tac x)
  apply(rule_tac y="a" and c="(b, c)" in trm.strong_exhaust)
  apply (simp_all add: fresh_star_def)[12]
  apply(metis)+
  -- "compatibility"
  apply(simp_all)
  apply(rule conjI)
  apply(elim conjE)
  apply(erule_tac c="(da,ea)" in Abs_lst1_fcb2)
  apply(simp add: Abs_fresh_iff)
  apply(simp add: fresh_at_base fresh_star_def fresh_Pair)
  apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)
  apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)
  apply(elim conjE)
  apply(erule_tac c="(da,ea)" in Abs_lst1_fcb2)
  apply(simp add: Abs_fresh_iff)
  apply(simp add: fresh_at_base fresh_star_def fresh_Pair)
  apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)
  apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)
  apply(elim conjE)
  apply(subgoal_tac "eqvt_at crename_sumC (M, da, ea)")
  apply(subgoal_tac "eqvt_at crename_sumC (Ma, da, ea)")
  apply(erule Abs_lst1_fcb2)
  apply(simp add: Abs_fresh_iff)
  apply(simp add: fresh_at_base fresh_star_def fresh_Pair)
  apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)
  apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)
  apply(blast)
  apply(blast)
  apply(elim conjE)
  apply(erule_tac c="(da,ea)" in Abs_lst1_fcb2)
  apply(simp add: Abs_fresh_iff)
  apply(simp add: fresh_at_base fresh_star_def fresh_Pair)
  apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)
  apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)
  apply(rule conjI)
  apply(elim conjE)
  apply(subgoal_tac "eqvt_at crename_sumC (M, da, ea)")
  apply(subgoal_tac "eqvt_at crename_sumC (Ma, da, ea)")
  apply(erule Abs_lst1_fcb2)
  apply(simp add: Abs_fresh_iff)
  apply(simp add: fresh_at_base fresh_star_def fresh_Pair)
  apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)
  apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)
  apply(blast)
  apply(blast)
  apply(erule conjE)+
  apply(subgoal_tac "eqvt_at crename_sumC (N, da, ea)")
  apply(subgoal_tac "eqvt_at crename_sumC (Na, da, ea)")
  apply(erule Abs_lst1_fcb2)
  apply(simp add: Abs_fresh_iff)
  apply(simp add: fresh_at_base fresh_star_def fresh_Pair)
  apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)
  apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)
  apply(blast)
  apply(blast)
  apply(elim conjE)
  apply(erule_tac c="(da,ea)" in Abs_lst1_fcb2)
  apply(simp add: Abs_fresh_iff)
  apply(simp add: fresh_at_base fresh_star_def fresh_Pair)
  apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)
  apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)
  apply(elim conjE)
  apply(erule_tac c="(da,ea)" in Abs_lst1_fcb2)
  apply(simp add: Abs_fresh_iff)
  apply(simp add: fresh_at_base fresh_star_def fresh_Pair)
  apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)
  apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)
  apply(elim conjE)
  apply(subgoal_tac "eqvt_at crename_sumC (M, da, ea)")
  apply(subgoal_tac "eqvt_at crename_sumC (Ma, da, ea)")
  apply(erule Abs_lst1_fcb2)
  apply(simp add: Abs_fresh_iff)
  apply(simp add: fresh_at_base fresh_star_def fresh_Pair)
  apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)
  apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)
  apply(blast)
  apply(blast)
  apply(erule conjE)+
  apply(subgoal_tac "eqvt_at crename_sumC (M, da, ea)")
  apply(subgoal_tac "eqvt_at crename_sumC (Ma, da, ea)")
  apply(erule Abs_lst1_fcb2)
  apply(simp add: Abs_fresh_iff)
  apply(simp add: fresh_at_base fresh_star_def fresh_Pair)
  apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)
  apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)
  apply(blast)
  apply(blast)
  apply(rule conjI)
  apply(erule conjE)+
  apply(subgoal_tac "eqvt_at crename_sumC (M, da, ea)")
  apply(subgoal_tac "eqvt_at crename_sumC (Ma, da, ea)")
  apply(erule Abs_lst1_fcb2)
  apply(simp add: Abs_fresh_iff)
  apply(simp add: fresh_at_base fresh_star_def fresh_Pair)
  apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)
  apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)
  apply(blast)
  apply(blast)
  apply(erule conjE)+
  apply(subgoal_tac "eqvt_at crename_sumC (N, da, ea)")
  apply(subgoal_tac "eqvt_at crename_sumC (Na, da, ea)")
  apply(erule Abs_lst1_fcb2)
  apply(simp add: Abs_fresh_iff)
  apply(simp add: fresh_at_base fresh_star_def fresh_Pair)
  apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)
  apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)
  apply(blast)
  apply(blast)
  defer
  apply(erule conjE)+
  apply(rule conjI)
  apply(subgoal_tac "eqvt_at crename_sumC (M, da, ea)")
  apply(subgoal_tac "eqvt_at crename_sumC (Ma, da, ea)")
  apply(erule Abs_lst1_fcb2)
  apply(simp add: Abs_fresh_iff)
  apply(simp add: fresh_at_base fresh_star_def fresh_Pair)
  apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)
  apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)
  apply(blast)
  apply(blast)  
  apply(subgoal_tac "eqvt_at crename_sumC (N, da, ea)")
  apply(subgoal_tac "eqvt_at crename_sumC (Na, da, ea)")
  apply(erule Abs_lst1_fcb2)
  apply(simp add: Abs_fresh_iff)
  apply(simp add: fresh_at_base fresh_star_def fresh_Pair)
  apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)
  apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)
  apply(blast)
  apply(blast)
  apply(subgoal_tac "eqvt_at crename_sumC (M, da, ea)")
  apply(subgoal_tac "eqvt_at crename_sumC (Ma, da, ea)")
  apply(erule conjE)+
  apply(erule Abs_lst_fcb2)
  apply(simp add: Abs_fresh_star)
  apply(simp add: Abs_fresh_star)
  apply(simp add: fresh_at_base fresh_star_def fresh_Pair)
  apply(simp add: fresh_at_base fresh_star_def fresh_Pair)
  apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)
  apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)
  apply(blast)
  apply(blast)
  done



end