theory Classical+
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imports "../Nominal2"+
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begin+
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+
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+
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(* example from Urban's PhD *)+
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+
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atom_decl name+
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atom_decl coname+
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+
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nominal_datatype trm =+
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  Ax "name" "coname"+
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| Cut c::"coname" t1::"trm" n::"name" t2::"trm"             bind n in t1, bind c in t2  +
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     ("Cut <_>._ '(_')._" [100,100,100,100] 100)+
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| NotR n::"name" t::"trm" "coname"                            bind n in t+
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     ("NotR '(_')._ _" [100,100,100] 100)+
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| NotL c::"coname" t::"trm" "name"                            bind c in t   +
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     ("NotL <_>._ _" [100,100,100] 100)+
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| AndR c1::"coname" t1::"trm" c2::"coname" t2::"trm" "coname" bind c1 in t1, bind c2 in t2+
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     ("AndR <_>._ <_>._ _" [100,100,100,100,100] 100)+
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| AndL1 n::"name" t::"trm" "name"                             bind n in t+
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     ("AndL1 '(_')._ _" [100,100,100] 100)+
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| AndL2 n::"name" t::"trm" "name"                             bind n in t+
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     ("AndL2 '(_')._ _" [100,100,100] 100)+
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| OrR1 c::"coname" t::"trm" "coname"                          bind c in t             +
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     ("OrR1 <_>._ _" [100,100,100] 100)+
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| OrR2 c::"coname" t::"trm" "coname"                          bind c in t     +
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     ("OrR2 <_>._ _" [100,100,100] 100)+
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| OrL n1::"name" t1::"trm" n2::"name" t2::"trm" "name"        bind n1 in t1, bind n2 in t2       +
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     ("OrL '(_')._ '(_')._ _" [100,100,100,100,100] 100)+
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| ImpL c::"coname" t1::"trm" n::"name" t2::"trm" "name"       bind c in t1, bind n in t2+
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     ("ImpL <_>._ '(_')._ _" [100,100,100,100,100] 100)+
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| ImpR n::"name" c::"coname" t::"trm" "coname"                bind n c in t+
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     ("ImpR '(_').<_>._ _" [100,100,100,100] 100)+
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+
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thm trm.distinct+
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thm trm.induct+
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thm trm.exhaust+
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thm trm.strong_exhaust+
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thm trm.strong_exhaust[simplified]+
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thm trm.fv_defs+
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thm trm.bn_defs+
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thm trm.perm_simps+
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thm trm.eq_iff+
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thm trm.fv_bn_eqvt+
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thm trm.size_eqvt+
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thm trm.supp+
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thm trm.supp[simplified]+
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+
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lemma Abs_set_fcb2:+
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  fixes as bs :: "atom set"+
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    and x y :: "'b :: fs"+
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    and c::"'c::fs"+
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  assumes eq: "[as]set. x = [bs]set. y"+
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  and fin: "finite as" "finite bs"+
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  and fcb1: "as \<sharp>* f as x c"+
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  and fresh1: "as \<sharp>* c"+
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  and fresh2: "bs \<sharp>* c"+
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  and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f as x c) = f (p \<bullet> as) (p \<bullet> x) c"+
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  and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f bs y c) = f (p \<bullet> bs) (p \<bullet> y) c"+
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  shows "f as x c = f bs y c"+
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proof -+
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  have "supp (as, x, c) supports (f as x c)"+
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    unfolding  supports_def fresh_def[symmetric]+
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    by (simp add: fresh_Pair perm1 fresh_star_def supp_swap swap_fresh_fresh)+
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  then have fin1: "finite (supp (f as x c))"+
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    using fin by (auto intro: supports_finite simp add: finite_supp supp_of_finite_sets supp_Pair)+
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  have "supp (bs, y, c) supports (f bs y c)"+
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    unfolding  supports_def fresh_def[symmetric]+
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    by (simp add: fresh_Pair perm2 fresh_star_def supp_swap swap_fresh_fresh)+
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  then have fin2: "finite (supp (f bs y c))"+
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    using fin by (auto intro: supports_finite simp add: finite_supp supp_of_finite_sets supp_Pair)+
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  obtain q::"perm" where +
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    fr1: "(q \<bullet> as) \<sharp>* (x, c, f as x c, f bs y c)" and +
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    fr2: "supp q \<sharp>* ([as]set. x)" and +
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    inc: "supp q \<subseteq> as \<union> (q \<bullet> as)"+
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    using at_set_avoiding3[where xs="as" and c="(x, c, f as x c, f bs y c)" and x="[as]set. x"]  +
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      fin1 fin2 fin+
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    by (auto simp add: supp_Pair finite_supp Abs_fresh_star dest: fresh_star_supp_conv)+
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  have "[q \<bullet> as]set. (q \<bullet> x) = q \<bullet> ([as]set. x)" by simp+
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  also have "\<dots> = [as]set. x"+
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    by (simp only: fr2 perm_supp_eq)+
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  finally have "[q \<bullet> as]set. (q \<bullet> x) = [bs]set. y" using eq by simp+
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  then obtain r::perm where +
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    qq1: "q \<bullet> x = r \<bullet> y" and +
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    qq2: "q \<bullet> as = r \<bullet> bs" and +
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    qq3: "supp r \<subseteq> (q \<bullet> as) \<union> bs"+
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    apply(drule_tac sym)+
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    apply(simp only: Abs_eq_iff2 alphas)+
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    apply(erule exE)+
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    apply(erule conjE)++
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    apply(drule_tac x="p" in meta_spec)+
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    apply(simp add: set_eqvt)+
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    apply(blast)+
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    done+
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  have "as \<sharp>* f as x c" by (rule fcb1)+
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  then have "q \<bullet> (as \<sharp>* f as x c)"+
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    by (simp add: permute_bool_def)+
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  then have "(q \<bullet> as) \<sharp>* f (q \<bullet> as) (q \<bullet> x) c"+
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    apply(simp add: fresh_star_eqvt set_eqvt)+
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    apply(subst (asm) perm1)+
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    using inc fresh1 fr1+
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    apply(auto simp add: fresh_star_def fresh_Pair)+
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    done+
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  then have "(r \<bullet> bs) \<sharp>* f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 by simp+
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  then have "r \<bullet> (bs \<sharp>* f bs y c)"+
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    apply(simp add: fresh_star_eqvt set_eqvt)+
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    apply(subst (asm) perm2[symmetric])+
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    using qq3 fresh2 fr1+
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    apply(auto simp add: set_eqvt fresh_star_def fresh_Pair)+
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    done+
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  then have fcb2: "bs \<sharp>* f bs y c" by (simp add: permute_bool_def)+
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  have "f as x c = q \<bullet> (f as x c)"+
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    apply(rule perm_supp_eq[symmetric])+
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    using inc fcb1 fr1 by (auto simp add: fresh_star_def)+
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  also have "\<dots> = f (q \<bullet> as) (q \<bullet> x) c" +
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    apply(rule perm1)+
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    using inc fresh1 fr1 by (auto simp add: fresh_star_def)+
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  also have "\<dots> = f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 by simp+
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  also have "\<dots> = r \<bullet> (f bs y c)"+
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    apply(rule perm2[symmetric])+
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    using qq3 fresh2 fr1 by (auto simp add: fresh_star_def)+
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  also have "... = f bs y c"+
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    apply(rule perm_supp_eq)+
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    using qq3 fr1 fcb2 by (auto simp add: fresh_star_def)+
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  finally show ?thesis by simp+
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qed+
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+
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lemma Abs_res_fcb2:+
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  fixes as bs :: "atom set"+
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    and x y :: "'b :: fs"+
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    and c::"'c::fs"+
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  assumes eq: "[as]res. x = [bs]res. y"+
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  and fin: "finite as" "finite bs"+
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  and fcb1: "as \<sharp>* f as x c"+
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  and fresh1: "as \<sharp>* c"+
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  and fresh2: "bs \<sharp>* c"+
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  and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f as x c) = f (p \<bullet> as) (p \<bullet> x) c"+
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  and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f bs y c) = f (p \<bullet> bs) (p \<bullet> y) c"+
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  shows "f as x c = f bs y c"+
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proof -+
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  have "supp (as, x, c) supports (f as x c)"+
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    unfolding  supports_def fresh_def[symmetric]+
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    by (simp add: fresh_Pair perm1 fresh_star_def supp_swap swap_fresh_fresh)+
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  then have fin1: "finite (supp (f as x c))"+
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    using fin by (auto intro: supports_finite simp add: finite_supp supp_of_finite_sets supp_Pair)+
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  have "supp (bs, y, c) supports (f bs y c)"+
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    unfolding  supports_def fresh_def[symmetric]+
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    by (simp add: fresh_Pair perm2 fresh_star_def supp_swap swap_fresh_fresh)+
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  then have fin2: "finite (supp (f bs y c))"+
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    using fin by (auto intro: supports_finite simp add: finite_supp supp_of_finite_sets supp_Pair)+
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  obtain q::"perm" where +
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    fr1: "(q \<bullet> as) \<sharp>* (x, c, f as x c, f bs y c)" and +
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    fr2: "supp q \<sharp>* ([as]res. x)" and +
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    inc: "supp q \<subseteq> as \<union> (q \<bullet> as)"+
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    using at_set_avoiding3[where xs="as" and c="(x, c, f as x c, f bs y c)" and x="[as]res. x"]  +
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      fin1 fin2 fin+
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    by (auto simp add: supp_Pair finite_supp Abs_fresh_star dest: fresh_star_supp_conv)+
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  have "[q \<bullet> as]res. (q \<bullet> x) = q \<bullet> ([as]res. x)" by simp+
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  also have "\<dots> = [as]res. x"+
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    by (simp only: fr2 perm_supp_eq)+
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  finally have "[q \<bullet> as]res. (q \<bullet> x) = [bs]res. y" using eq by simp+
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  then obtain r::perm where +
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    qq1: "q \<bullet> x = r \<bullet> y" and +
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    qq2: "(q \<bullet> as \<inter> supp (q \<bullet> x)) = r \<bullet> (bs \<inter> supp y)" and +
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    qq3: "supp r \<subseteq> bs \<inter> supp y \<union> q \<bullet> as \<inter> supp (q \<bullet> x)"+
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    apply(drule_tac sym)+
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    apply(subst(asm) Abs_eq_res_set)+
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    apply(simp only: Abs_eq_iff2 alphas)+
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    apply(erule exE)+
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    apply(erule conjE)++
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    apply(drule_tac x="p" in meta_spec)+
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    apply(simp add: set_eqvt)+
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    done+
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  have "(as \<inter> supp x) \<sharp>* f (as \<inter> supp x) x c" sorry (* FCB? *)+
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  then have "q \<bullet> ((as \<inter> supp x) \<sharp>* f (as \<inter> supp x) x c)"+
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    by (simp add: permute_bool_def)+
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  then have "(q \<bullet> (as \<inter> supp x)) \<sharp>* f (q \<bullet> (as \<inter> supp x)) (q \<bullet> x) c"+
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    apply(simp add: fresh_star_eqvt set_eqvt)+
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    sorry (* perm? *)+
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  then have "r \<bullet> (bs \<inter> supp y) \<sharp>* f (r \<bullet> (bs \<inter> supp y)) (r \<bullet> y) c" using qq2 +
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    apply (simp add: inter_eqvt)+
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    sorry+
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  (* rest similar reversing it other way around... *)+
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  show ?thesis sorry+
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qed+
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+
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+
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+
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lemma Abs_lst_fcb2:+
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  fixes as bs :: "atom list"+
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    and x y :: "'b :: fs"+
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    and c::"'c::fs"+
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  assumes eq: "[as]lst. x = [bs]lst. y"+
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  and fcb1: "(set as) \<sharp>* f as x c"+
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  and fresh1: "set as \<sharp>* c"+
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  and fresh2: "set bs \<sharp>* c"+
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  and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f as x c) = f (p \<bullet> as) (p \<bullet> x) c"+
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  and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f bs y c) = f (p \<bullet> bs) (p \<bullet> y) c"+
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  shows "f as x c = f bs y c"+
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proof -+
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  have "supp (as, x, c) supports (f as x c)"+
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    unfolding  supports_def fresh_def[symmetric]+
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    by (simp add: fresh_Pair perm1 fresh_star_def supp_swap swap_fresh_fresh)+
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  then have fin1: "finite (supp (f as x c))"+
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    by (auto intro: supports_finite simp add: finite_supp)+
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  have "supp (bs, y, c) supports (f bs y c)"+
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    unfolding  supports_def fresh_def[symmetric]+
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    by (simp add: fresh_Pair perm2 fresh_star_def supp_swap swap_fresh_fresh)+
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  then have fin2: "finite (supp (f bs y c))"+
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    by (auto intro: supports_finite simp add: finite_supp)+
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  obtain q::"perm" where +
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    fr1: "(q \<bullet> (set as)) \<sharp>* (x, c, f as x c, f bs y c)" and +
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    fr2: "supp q \<sharp>* Abs_lst as x" and +
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    inc: "supp q \<subseteq> (set as) \<union> q \<bullet> (set as)"+
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    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"]  +
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      fin1 fin2+
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    by (auto simp add: supp_Pair finite_supp Abs_fresh_star dest: fresh_star_supp_conv)+
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  have "Abs_lst (q \<bullet> as) (q \<bullet> x) = q \<bullet> Abs_lst as x" by simp+
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  also have "\<dots> = Abs_lst as x"+
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    by (simp only: fr2 perm_supp_eq)+
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  finally have "Abs_lst (q \<bullet> as) (q \<bullet> x) = Abs_lst bs y" using eq by simp+
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  then obtain r::perm where +
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    qq1: "q \<bullet> x = r \<bullet> y" and +
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    qq2: "q \<bullet> as = r \<bullet> bs" and +
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    qq3: "supp r \<subseteq> (q \<bullet> (set as)) \<union> set bs"+
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    apply(drule_tac sym)+
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    apply(simp only: Abs_eq_iff2 alphas)+
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    apply(erule exE)+
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    apply(erule conjE)++
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    apply(drule_tac x="p" in meta_spec)+
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    apply(simp add: set_eqvt)+
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    apply(blast)+
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    done+
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  have "(set as) \<sharp>* f as x c" by (rule fcb1)+
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  then have "q \<bullet> ((set as) \<sharp>* f as x c)"+
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    by (simp add: permute_bool_def)+
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  then have "set (q \<bullet> as) \<sharp>* f (q \<bullet> as) (q \<bullet> x) c"+
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    apply(simp add: fresh_star_eqvt set_eqvt)+
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    apply(subst (asm) perm1)+
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    using inc fresh1 fr1+
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    apply(auto simp add: fresh_star_def fresh_Pair)+
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    done+
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  then have "set (r \<bullet> bs) \<sharp>* f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 by simp+
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  then have "r \<bullet> ((set bs) \<sharp>* f bs y c)"+
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    apply(simp add: fresh_star_eqvt set_eqvt)+
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    apply(subst (asm) perm2[symmetric])+
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    using qq3 fresh2 fr1+
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    apply(auto simp add: set_eqvt fresh_star_def fresh_Pair)+
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    done+
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  then have fcb2: "(set bs) \<sharp>* f bs y c" by (simp add: permute_bool_def)+
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  have "f as x c = q \<bullet> (f as x c)"+
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    apply(rule perm_supp_eq[symmetric])+
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    using inc fcb1 fr1 by (auto simp add: fresh_star_def)+
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  also have "\<dots> = f (q \<bullet> as) (q \<bullet> x) c" +
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    apply(rule perm1)+
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    using inc fresh1 fr1 by (auto simp add: fresh_star_def)+
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  also have "\<dots> = f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 by simp+
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  also have "\<dots> = r \<bullet> (f bs y c)"+
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    apply(rule perm2[symmetric])+
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    using qq3 fresh2 fr1 by (auto simp add: fresh_star_def)+
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  also have "... = f bs y c"+
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    apply(rule perm_supp_eq)+
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    using qq3 fr1 fcb2 by (auto simp add: fresh_star_def)+
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  finally show ?thesis by simp+
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qed+
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+
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lemma Abs_lst1_fcb2:+
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  fixes a b :: "atom"+
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    and x y :: "'b :: fs"+
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    and c::"'c :: fs"+
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  assumes e: "(Abs_lst [a] x) = (Abs_lst [b] y)"+
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  and fcb1: "a \<sharp> f a x c"+
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  and fresh: "{a, b} \<sharp>* c"+
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  and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f a x c) = f (p \<bullet> a) (p \<bullet> x) c"+
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  and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f b y c) = f (p \<bullet> b) (p \<bullet> y) c"+
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  shows "f a x c = f b y c"+
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using e+
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apply(drule_tac Abs_lst_fcb2[where c="c" and f="\<lambda>(as::atom list) . f (hd as)"])+
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apply(simp_all)+
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using fcb1 fresh perm1 perm2+
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apply(simp_all add: fresh_star_def)+
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done+
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+
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lemma supp_zero_perm_zero:+
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  shows "supp (p :: perm) = {} \<longleftrightarrow> p = 0"+
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  by (metis supp_perm_singleton supp_zero_perm)+
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+
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lemma permute_atom_list_id:+
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  shows "p \<bullet> l = l \<longleftrightarrow> supp p \<inter> set l = {}"+
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  by (induct l) (auto simp add: supp_Nil supp_perm)+
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+
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lemma permute_length_eq:+
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  shows "p \<bullet> xs = ys \<Longrightarrow> length xs = length ys"+
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  by (auto simp add: length_eqvt[symmetric] permute_pure)+
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+
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lemma Abs_lst_binder_length:+
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  shows "[xs]lst. T = [ys]lst. S \<Longrightarrow> length xs = length ys"+
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  by (auto simp add: Abs_eq_iff alphas length_eqvt[symmetric] permute_pure)+
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+
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lemma Abs_lst_binder_eq:+
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  shows "Abs_lst l T = Abs_lst l S \<longleftrightarrow> T = S"+
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  by (rule, simp_all add: Abs_eq_iff2 alphas)+
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     (metis fresh_star_zero inf_absorb1 permute_atom_list_id supp_perm_eq+
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       supp_zero_perm_zero)+
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+
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lemma in_permute_list:+
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  shows "py \<bullet> p \<bullet> xs = px \<bullet> xs \<Longrightarrow>  x \<in> set xs \<Longrightarrow> py \<bullet> p \<bullet> x = px \<bullet> x"+
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  by (induct xs) auto+
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+
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+
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+
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+
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nominal_primrec +
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  crename :: "trm \<Rightarrow> coname \<Rightarrow> coname \<Rightarrow> trm"  ("_[_\<turnstile>c>_]" [100,100,100] 100) +
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where+
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  "(Ax x a)[d\<turnstile>c>e] = (if a=d then Ax x e else Ax x a)" +
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| "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])" +
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| "(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)" +
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| "atom a \<sharp> (d, e) \<Longrightarrow> (NotL <a>.M x)[d\<turnstile>c>e] = (NotL <a>.(M[d\<turnstile>c>e]) x)" +
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| "\<lbrakk>atom a \<sharp> (d, e); atom b \<sharp> (d, e)\<rbrakk> \<Longrightarrow> (AndR <a>.M <b>.N c)[d\<turnstile>c>e] = +
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          (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)" +
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| "(AndL1 (x).M y)[d\<turnstile>c>e] = AndL1 (x).(M[d\<turnstile>c>e]) y"+
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| "(AndL2 (x).M y)[d\<turnstile>c>e] = AndL2 (x).(M[d\<turnstile>c>e]) y"+
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| "atom a \<sharp> (d, e) \<Longrightarrow> (OrR1 <a>.M b)[d\<turnstile>c>e] = +
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          (if b=d then OrR1 <a>.(M[d\<turnstile>c>e]) e else OrR1 <a>.(M[d\<turnstile>c>e]) b)"+
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| "atom a \<sharp> (d, e) \<Longrightarrow> (OrR2 <a>.M b)[d\<turnstile>c>e] = +
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          (if b=d then OrR2 <a>.(M[d\<turnstile>c>e]) e else OrR2 <a>.(M[d\<turnstile>c>e]) b)"+
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| "(OrL (x).M (y).N z)[d\<turnstile>c>e] = OrL (x).(M[d\<turnstile>c>e]) (y).(N[d\<turnstile>c>e]) z"+
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| "atom a \<sharp> (d, e) \<Longrightarrow> (ImpR (x).<a>.M b)[d\<turnstile>c>e] = +
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          (if b=d then ImpR (x).<a>.(M[d\<turnstile>c>e]) e else ImpR (x).<a>.(M[d\<turnstile>c>e]) b)"+
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| "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"+
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  apply(simp only: eqvt_def)+
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  apply(simp only: crename_graph_def)+
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  apply (rule, perm_simp, rule)+
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  apply(rule TrueI)+
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  -- "covered all cases"+
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  apply(case_tac x)+
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  apply(rule_tac y="a" and c="(b, c)" in trm.strong_exhaust)+
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  apply (simp_all add: fresh_star_def)[12]+
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  apply(metis)++
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  -- "compatibility"+
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  apply(simp_all)+
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  apply(rule conjI)+
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  apply(elim conjE)+
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  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+
−
+
−
+
−
+
−