theory Let+
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imports "../Nominal2" +
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begin+
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+
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atom_decl name+
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+
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nominal_datatype trm =+
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  Var "name"+
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| App "trm" "trm"+
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| Lam x::"name" t::"trm"  bind  x in t+
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| Let as::"assn" t::"trm"   bind "bn as" in t+
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and assn =+
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  ANil+
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| ACons "name" "trm" "assn"+
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binder+
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  bn+
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where+
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  "bn ANil = []"+
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| "bn (ACons x t as) = (atom x) # (bn as)"+
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+
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print_theorems+
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+
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thm alpha_trm_raw_alpha_assn_raw_alpha_bn_raw.intros+
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thm bn_raw.simps+
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thm permute_bn_raw.simps+
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thm trm_assn.perm_bn_alpha+
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thm trm_assn.permute_bn+
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+
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thm trm_assn.fv_defs+
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thm trm_assn.eq_iff +
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thm trm_assn.bn_defs+
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thm trm_assn.bn_inducts+
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thm trm_assn.perm_simps+
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thm trm_assn.induct+
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thm trm_assn.inducts+
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thm trm_assn.distinct+
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thm trm_assn.supp+
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thm trm_assn.fresh+
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thm trm_assn.exhaust+
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thm trm_assn.strong_exhaust+
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thm trm_assn.perm_bn_simps+
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+
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lemma alpha_bn_inducts_raw[consumes 1]:+
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  "\<lbrakk>alpha_bn_raw a b; P3 ANil_raw ANil_raw;+
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 \<And>trm_raw trm_rawa assn_raw assn_rawa name namea.+
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    \<lbrakk>alpha_trm_raw trm_raw trm_rawa; alpha_bn_raw assn_raw assn_rawa;+
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     P3 assn_raw assn_rawa\<rbrakk>+
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    \<Longrightarrow> P3 (ACons_raw name trm_raw assn_raw)+
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        (ACons_raw namea trm_rawa assn_rawa)\<rbrakk> \<Longrightarrow> P3 a b"+
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  by (erule alpha_trm_raw_alpha_assn_raw_alpha_bn_raw.inducts(3)[of _ _ "\<lambda>x y. True" _ "\<lambda>x y. True", simplified]) auto+
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+
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lemmas alpha_bn_inducts[consumes 1] = alpha_bn_inducts_raw[quot_lifted]+
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+
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+
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+
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lemma alpha_bn_refl: "alpha_bn x x"+
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  by (induct x rule: trm_assn.inducts(2))+
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     (rule TrueI, auto simp add: trm_assn.eq_iff)+
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lemma alpha_bn_sym: "alpha_bn x y \<Longrightarrow> alpha_bn y x"+
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  sorry+
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lemma alpha_bn_trans: "alpha_bn x y \<Longrightarrow> alpha_bn y z \<Longrightarrow> alpha_bn x z"+
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  sorry+
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+
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lemma bn_inj[rule_format]:+
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  assumes a: "alpha_bn x y"+
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  shows "bn x = bn y \<longrightarrow> x = y"+
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  by (rule alpha_bn_inducts[OF a]) (simp_all add: trm_assn.bn_defs)+
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+
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lemma bn_inj2:+
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  assumes a: "alpha_bn x y"+
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  shows "\<And>q r. (q \<bullet> bn x) = (r \<bullet> bn y) \<Longrightarrow> permute_bn q x = permute_bn r y"+
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using a+
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apply(induct rule: alpha_bn_inducts)+
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apply(simp add: trm_assn.perm_bn_simps)+
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apply(simp add: trm_assn.perm_bn_simps)+
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apply(simp add: trm_assn.bn_defs)+
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apply(simp add: atom_eqvt)+
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done+
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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>* c \<Longrightarrow> (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" +
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    apply(rule fcb1)+
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    apply(rule fresh1)+
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    done+
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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[OF fresh1] 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> c \<Longrightarrow> 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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+
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lemma max_eqvt[eqvt]: "p \<bullet> (max (a :: _ :: pure) b) = max (p \<bullet> a) (p \<bullet> b)"+
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  by (simp add: permute_pure)+
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+
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function+
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  apply_assn :: "(trm \<Rightarrow> nat) \<Rightarrow> assn \<Rightarrow> nat"+
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where+
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  "apply_assn f ANil = (0 :: nat)"+
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| "apply_assn f (ACons x t as) = max (f t) (apply_assn f as)"+
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apply(case_tac x)+
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apply(case_tac b rule: trm_assn.exhaust(2))+
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apply(simp_all)+
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apply(blast)+
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done+
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+
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termination by lexicographic_order+
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+
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lemma [eqvt]:+
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  "p \<bullet> (apply_assn f a) = apply_assn (p \<bullet> f) (p \<bullet> a)"+
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  apply(induct f a rule: apply_assn.induct)+
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  apply simp_all+
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  apply(perm_simp)+
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  apply rule+
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  apply(perm_simp)+
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  apply simp+
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  done+
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+
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lemma alpha_bn_apply_assn:+
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  assumes "alpha_bn as bs"+
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  shows "apply_assn f as = apply_assn f bs"+
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  using assms+
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  apply (induct rule: alpha_bn_inducts)+
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  apply simp_all+
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  done+
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+
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nominal_primrec+
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    height_trm :: "trm \<Rightarrow> nat"+
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where+
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  "height_trm (Var x) = 1"+
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| "height_trm (App l r) = max (height_trm l) (height_trm r)"+
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| "height_trm (Lam v b) = 1 + (height_trm b)"+
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| "height_trm (Let as b) = max (apply_assn height_trm as) (height_trm b)"+
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  apply (simp only: eqvt_def height_trm_graph_def)+
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  apply (rule, perm_simp, rule, rule TrueI)+
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  apply (case_tac x rule: trm_assn.exhaust(1))+
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  apply (auto)[4]+
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  apply (drule_tac x="assn" in meta_spec)+
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  apply (drule_tac x="trm" in meta_spec)+
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  apply (simp add: alpha_bn_refl)+
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  apply(simp_all)+
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  apply (erule_tac c="()" in Abs_lst1_fcb2)+
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  apply (simp_all add: pure_fresh fresh_star_def eqvt_at_def)[4]+
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  apply (erule conjE)+
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  apply (subst alpha_bn_apply_assn)+
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  apply assumption+
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  apply (rule arg_cong) back+
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  apply (erule_tac c="()" in Abs_lst_fcb2)+
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  apply (simp_all add: pure_fresh fresh_star_def)[3]+
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  apply (simp_all add: eqvt_at_def)[2]+
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  done+
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+
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definition "height_assn = apply_assn height_trm"+
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+
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function+
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  apply_assn2 :: "(trm \<Rightarrow> trm) \<Rightarrow> assn \<Rightarrow> assn"+
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where+
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  "apply_assn2 f ANil = ANil"+
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| "apply_assn2 f (ACons x t as) = ACons x (f t) (apply_assn2 f as)"+
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  apply(case_tac x)+
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  apply(case_tac b rule: trm_assn.exhaust(2))+
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  apply(simp_all)+
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  apply(blast)+
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  done+
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+
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termination by lexicographic_order+
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+
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lemma [eqvt]:+
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  "p \<bullet> (apply_assn2 f a) = apply_assn2 (p \<bullet> f) (p \<bullet> a)"+
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  apply(induct f a rule: apply_assn2.induct)+
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  apply simp_all+
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  apply(perm_simp)+
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  apply rule+
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  done+
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+
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lemma bn_apply_assn2: "bn (apply_assn2 f as) = bn as"+
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  apply (induct as rule: trm_assn.inducts(2))+
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  apply (rule TrueI)+
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  apply (simp_all add: trm_assn.bn_defs)+
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  done+
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+
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nominal_primrec+
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    subst  :: "name \<Rightarrow> trm \<Rightarrow> trm \<Rightarrow> trm"+
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where+
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  "subst s t (Var x) = (if (s = x) then t else (Var x))"+
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| "subst s t (App l r) = App (subst s t l) (subst s t r)"+
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| "atom v \<sharp> (s, t) \<Longrightarrow> subst s t (Lam v b) = Lam v (subst s t b)"+
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| "set (bn as) \<sharp>* (s, t) \<Longrightarrow> subst s t (Let as b) = Let (apply_assn2 (subst s t) as) (subst s t b)"+
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  apply (simp only: eqvt_def subst_graph_def)+
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  apply (rule, perm_simp, rule)+
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  apply (rule TrueI)+
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  apply (case_tac x)+
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  apply (rule_tac y="c" and c="(a,b)" in trm_assn.strong_exhaust(1))+
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  apply (auto simp add: fresh_star_def)[3]+
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  apply (drule_tac x="assn" in meta_spec)+
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  apply (simp add: Abs1_eq_iff alpha_bn_refl)+
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  apply auto+
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  apply (erule_tac c="(sa, ta)" in Abs_lst1_fcb2)+
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  apply (simp add: Abs_fresh_iff)+
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  apply (simp add: fresh_star_def)+
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  apply (simp_all add: fresh_star_Pair_elim perm_supp_eq eqvt_at_def)[2]+
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  apply (simp add: bn_apply_assn2)+
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  apply (erule_tac c="(sa, ta)" in Abs_lst_fcb2)+
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  apply (simp add: fresh_star_def Abs_fresh_iff)+
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  apply assumption++
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  apply (simp_all add: fresh_star_Pair_elim perm_supp_eq eqvt_at_def trm_assn.fv_bn_eqvt)[2]+
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  apply (erule alpha_bn_inducts)+
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  apply simp_all+
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  done+
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+
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lemma lets_bla:+
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  "x \<noteq> z \<Longrightarrow> y \<noteq> z \<Longrightarrow> x \<noteq> y \<Longrightarrow>(Let (ACons x (Var y) ANil) (Var x)) \<noteq> (Let (ACons x (Var z) ANil) (Var x))"+
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  by (simp add: trm_assn.eq_iff)+
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+
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lemma lets_ok:+
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  "(Let (ACons x (Var y) ANil) (Var x)) = (Let (ACons y (Var y) ANil) (Var y))"+
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  apply (simp add: trm_assn.eq_iff Abs_eq_iff )+
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  apply (rule_tac x="(x \<leftrightarrow> y)" in exI)+
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  apply (simp_all add: alphas atom_eqvt supp_at_base fresh_star_def trm_assn.bn_defs trm_assn.supp)+
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  done+
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+
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lemma lets_ok3:+
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  "x \<noteq> y \<Longrightarrow>+
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   (Let (ACons x (App (Var y) (Var x)) (ACons y (Var y) ANil)) (App (Var x) (Var y))) \<noteq>+
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   (Let (ACons y (App (Var x) (Var y)) (ACons x (Var x) ANil)) (App (Var x) (Var y)))"+
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  apply (simp add: trm_assn.eq_iff)+
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  done+
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+
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lemma lets_not_ok1:+
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  "x \<noteq> y \<Longrightarrow>+
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   (Let (ACons x (Var x) (ACons y (Var y) ANil)) (App (Var x) (Var y))) \<noteq>+
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   (Let (ACons y (Var x) (ACons x (Var y) ANil)) (App (Var x) (Var y)))"+
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  apply (simp add: alphas trm_assn.eq_iff trm_assn.supp fresh_star_def atom_eqvt Abs_eq_iff trm_assn.bn_defs)+
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  done+
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+
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lemma lets_nok:+
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  "x \<noteq> y \<Longrightarrow> x \<noteq> z \<Longrightarrow> z \<noteq> y \<Longrightarrow>+
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   (Let (ACons x (App (Var z) (Var z)) (ACons y (Var z) ANil)) (App (Var x) (Var y))) \<noteq>+
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   (Let (ACons y (Var z) (ACons x (App (Var z) (Var z)) ANil)) (App (Var x) (Var y)))"+
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  apply (simp add: alphas trm_assn.eq_iff fresh_star_def trm_assn.bn_defs Abs_eq_iff trm_assn.supp trm_assn.distinct)+
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  done+
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+
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lemma+
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  fixes a b c :: name+
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  assumes x: "a \<noteq> c" and y: "b \<noteq> c"+
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  shows "\<exists>p.([atom a], Var c) \<approx>lst (op =) supp p ([atom b], Var c)"+
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  apply (rule_tac x="(a \<leftrightarrow> b)" in exI)+
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  apply (simp add: alphas trm_assn.supp supp_at_base x y fresh_star_def atom_eqvt)+
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  by (metis Rep_name_inverse atom_name_def flip_fresh_fresh fresh_atom fresh_perm x y)+
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+
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end+
−