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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+
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lemma alpha_bn_inducts_raw:+
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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 = 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 alpha_bn_permute:+
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  assumes a: "alpha_bn x y"+
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      and b: "q \<bullet> bn x = r \<bullet> bn y"+
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    shows "alpha_bn (q \<bullet> x) (r \<bullet> y)"+
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proof -+
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  have "alpha_bn x (permute_bn r y)"+
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    by (rule alpha_bn_trans[OF a]) (rule trm_assn.perm_bn_alpha)+
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  then have "alpha_bn (permute_bn r y) x"+
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    by (rule alpha_bn_sym)+
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  then have "alpha_bn (permute_bn r y) (permute_bn q x)"+
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    by (rule alpha_bn_trans) (rule trm_assn.perm_bn_alpha)+
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  then have "alpha_bn (permute_bn q x) (permute_bn r y)"+
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    by (rule alpha_bn_sym)+
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  moreover have "bn (permute_bn q x) = bn (permute_bn r y)"+
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    using b trm_assn.permute_bn by simp+
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  ultimately have "permute_bn q x = permute_bn r y"+
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    using bn_inj by simp+
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*)+
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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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+
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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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+
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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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(* TODO: should be provided by nominal *)+
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lemmas [eqvt] = trm_assn.fv_bn_eqvt+
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thm trm_assn.fv_bn_eqvt+
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+
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+
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thm Abs_lst_fcb+
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+
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lemma Abs_lst_fcb2:+
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  fixes as bs :: "'a :: fs"+
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    and x y :: "'b :: fs"+
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    and c::"'c::fs"+
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  assumes eq: "[ba as]lst. x = [ba bs]lst. y"+
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  and fcb1: "set (ba as) \<sharp>* f as x c"+
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  and fresh1: "set (ba as) \<sharp>* c"+
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  and fresh2: "set (ba 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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(* What we would like in this proof, and lets this proof finish *)+
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  and ba_inj: "\<And>q r. q \<bullet> ba as = r \<bullet> ba bs \<Longrightarrow> q \<bullet> as = r \<bullet> bs"+
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(* What the user can supply with the help of alpha_bn *)+
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(*  and bainj: "ba as = ba bs \<Longrightarrow> as = bs"*)+
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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 (ba as))) \<sharp>* (x, c, f as x c, f bs y c)" and +
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    fr2: "supp q \<sharp>* ([ba as]lst. x)" and +
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    inc: "supp q \<subseteq> (set (ba as)) \<union> q \<bullet> (set (ba as))"+
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    using at_set_avoiding3[where xs="set (ba as)" and c="(x, c, f as x c, f bs y c)" +
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      and x="[ba as]lst. x"]  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 "[q \<bullet> (ba as)]lst. (q \<bullet> x) = q \<bullet> ([ba as]lst. x)" by simp+
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  also have "\<dots> = [ba as]lst. x"+
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    by (simp only: fr2 perm_supp_eq)+
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  finally have "[q \<bullet> (ba as)]lst. (q \<bullet> x) = [ba bs]lst. 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> (ba as) = r \<bullet> (ba bs)" and +
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    qq3: "supp r \<subseteq> (q \<bullet> (set (ba as))) \<union> set (ba 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 (ba as)) \<sharp>* f as x c" by (rule fcb1)+
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  then have "q \<bullet> ((set (ba 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> (ba 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> (ba bs)) \<sharp>* f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 ba_inj +
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    by simp+
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  then have "r \<bullet> ((set (ba 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 (ba 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 ba_inj 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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nominal_primrec+
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    height_trm :: "trm \<Rightarrow> nat"+
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and height_assn :: "assn \<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 (height_assn as) (height_trm b)"+
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| "height_assn ANil = 0"+
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| "height_assn (ACons v t as) = max (height_trm t) (height_assn as)"+
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  apply (simp only: eqvt_def height_trm_height_assn_graph_def)+
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  apply (rule, perm_simp, rule, rule TrueI)+
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  apply (case_tac x)+
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  apply (case_tac a 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 (case_tac b rule: trm_assn.exhaust(2))+
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  apply (auto)[2]+
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  apply(simp_all)+
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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 add: eqvt_at_def)+
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  apply (simp add: eqvt_at_def)+
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  apply assumption+
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  apply(erule conjE)+
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  apply (simp add: meta_eq_to_obj_eq[OF height_trm_def, symmetric, unfolded fun_eq_iff])+
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  apply (simp add: meta_eq_to_obj_eq[OF height_assn_def, symmetric, unfolded fun_eq_iff])+
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  apply (subgoal_tac "eqvt_at height_assn as")+
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  apply (subgoal_tac "eqvt_at height_assn asa")+
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  apply (subgoal_tac "eqvt_at height_trm b")+
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  apply (subgoal_tac "eqvt_at height_trm ba")+
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  apply (thin_tac "eqvt_at height_trm_height_assn_sumC (Inr as)")+
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  apply (thin_tac "eqvt_at height_trm_height_assn_sumC (Inr asa)")+
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  apply (thin_tac "eqvt_at height_trm_height_assn_sumC (Inl b)")+
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  apply (thin_tac "eqvt_at height_trm_height_assn_sumC (Inl ba)")+
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  defer+
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  apply (simp add: eqvt_at_def height_trm_def)+
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  apply (simp add: eqvt_at_def height_trm_def)+
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  apply (simp add: eqvt_at_def height_assn_def)+
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  apply (simp add: eqvt_at_def height_assn_def)+
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  apply (subgoal_tac "height_assn as = height_assn asa")+
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  apply (subgoal_tac "height_trm b = height_trm ba")+
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  apply simp+
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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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  apply assumption+
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  apply (erule_tac c="()" and ba="bn" 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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  apply (rule bn_inj)+
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  prefer 2+
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  apply (simp add: eqvts)+
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  oops+
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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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and substa :: "name \<Rightarrow> trm \<Rightarrow> assn \<Rightarrow> assn"+
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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 (substa s t as) (subst s t b)"+
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| "substa s t ANil = ANil"+
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| "substa s t (ACons v t' as) = ACons v (subst v t t') as"+
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(*unfolding eqvt_def subst_substa_graph_def+
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  apply (rule, perm_simp)*)+
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  defer+
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  apply (rule TrueI)+
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  apply (case_tac x)+
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  apply (case_tac a)+
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  apply (rule_tac y="c" and c="(aa,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 (case_tac b)+
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  apply (case_tac c rule: trm_assn.exhaust(2))+
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  apply (auto)[2]+
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  apply blast+
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  apply blast+
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  apply auto+
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  apply (simp_all add: meta_eq_to_obj_eq[OF subst_def, symmetric, unfolded fun_eq_iff])+
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  apply (simp_all add: meta_eq_to_obj_eq[OF substa_def, symmetric, unfolded fun_eq_iff])+
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  prefer 2+
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  apply (erule_tac c="(sa, ta)" in Abs_lst_fcb2)+
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  apply (simp_all add: fresh_star_Pair)+
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  prefer 6+
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  apply (erule alpha_bn_inducts)+
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 oops+
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+
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end+
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+
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+
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+
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