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_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 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)

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