theory Nominal2_FCB+
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imports "Nominal2_Abs" +
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begin+
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+
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+
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text {*+
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  A tactic which solves all trivial cases in function +
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  definitions, and leaves the others unchanged.+
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*}+
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+
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ML {*+
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val all_trivials : (Proof.context -> Method.method) context_parser =+
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Scan.succeed (fn ctxt =>+
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 let+
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   val tac = TRYALL (SOLVED' (full_simp_tac (simpset_of ctxt)))+
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 in +
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   Method.SIMPLE_METHOD' (K tac)+
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 end)+
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*}+
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+
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method_setup all_trivials = {* all_trivials *} {* solves trivial goals *}+
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+
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+
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lemma Abs_lst1_fcb:+
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  fixes x y :: "'a :: at"+
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    and S T :: "'b :: fs"+
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  assumes e: "[[atom x]]lst. T = [[atom y]]lst. S"+
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  and f1: "\<lbrakk>x \<noteq> y; atom y \<sharp> T; atom x \<sharp> (y \<leftrightarrow> x) \<bullet> T\<rbrakk> \<Longrightarrow> atom x \<sharp> f x T"+
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  and f2: "\<lbrakk>x \<noteq> y; atom y \<sharp> T; atom x \<sharp> (y \<leftrightarrow> x) \<bullet> T\<rbrakk> \<Longrightarrow> atom y \<sharp> f x T"+
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  and p: "\<lbrakk>S = (x \<leftrightarrow> y) \<bullet> T; x \<noteq> y; atom y \<sharp> T; atom x \<sharp> S\<rbrakk> +
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    \<Longrightarrow> (x \<leftrightarrow> y) \<bullet> (f x T) = f y S"+
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  shows "f x T = f y S"+
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  using e+
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  apply(case_tac "atom x \<sharp> S")+
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  apply(simp add: Abs1_eq_iff')+
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  apply(elim conjE disjE)+
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  apply(simp)+
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  apply(rule trans)+
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  apply(rule_tac p="(x \<leftrightarrow> y)" in supp_perm_eq[symmetric])+
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  apply(rule fresh_star_supp_conv)+
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  apply(simp add: flip_def supp_swap fresh_star_def f1 f2)+
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  apply(simp add: flip_commute p)+
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  apply(simp add: Abs1_eq_iff)+
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  done+
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+
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lemma Abs_lst_fcb:+
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  fixes xs ys :: "'a :: fs"+
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    and S T :: "'b :: fs"+
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  assumes e: "(Abs_lst (ba xs) T) = (Abs_lst (ba ys) S)"+
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    and f1: "\<And>x. x \<in> set (ba xs) \<Longrightarrow> x \<sharp> f xs T"+
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    and f2: "\<And>x. \<lbrakk>supp T - set (ba xs) = supp S - set (ba ys); x \<in> set (ba ys)\<rbrakk> \<Longrightarrow> x \<sharp> f xs T"+
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    and eqv: "\<And>p. \<lbrakk>p \<bullet> T = S; p \<bullet> ba xs = ba ys; supp p \<subseteq> set (ba xs) \<union> set (ba ys)\<rbrakk> +
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      \<Longrightarrow> p \<bullet> (f xs T) = f ys S"+
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  shows "f xs T = f ys S"+
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  using e apply -+
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  apply(subst (asm) Abs_eq_iff2)+
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  apply(simp add: alphas)+
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  apply(elim exE conjE)+
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  apply(rule trans)+
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  apply(rule_tac p="p" in supp_perm_eq[symmetric])+
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  apply(rule fresh_star_supp_conv)+
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  apply(drule fresh_star_perm_set_conv)+
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  apply(rule finite_Diff)+
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  apply(rule finite_supp)+
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  apply(subgoal_tac "(set (ba xs) \<union> set (ba ys)) \<sharp>* f xs T")+
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  apply(metis Un_absorb2 fresh_star_Un)+
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  apply(subst fresh_star_Un)+
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  apply(rule conjI)+
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  apply(simp add: fresh_star_def f1)+
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  apply(simp add: fresh_star_def f2)+
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  apply(simp add: eqv)+
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  done+
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+
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lemma Abs_set_fcb:+
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  fixes xs ys :: "'a :: fs"+
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    and S T :: "'b :: fs"+
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  assumes e: "(Abs_set (ba xs) T) = (Abs_set (ba ys) S)"+
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    and f1: "\<And>x. x \<in> ba xs \<Longrightarrow> x \<sharp> f xs T"+
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    and f2: "\<And>x. \<lbrakk>supp T - ba xs = supp S - ba ys; x \<in> ba ys\<rbrakk> \<Longrightarrow> x \<sharp> f xs T"+
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    and eqv: "\<And>p. \<lbrakk>p \<bullet> T = S; p \<bullet> ba xs = ba ys; supp p \<subseteq> ba xs \<union> ba ys\<rbrakk> \<Longrightarrow> p \<bullet> (f xs T) = f ys S"+
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  shows "f xs T = f ys S"+
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  using e apply -+
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  apply(subst (asm) Abs_eq_iff2)+
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  apply(simp add: alphas)+
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  apply(elim exE conjE)+
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  apply(rule trans)+
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  apply(rule_tac p="p" in supp_perm_eq[symmetric])+
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  apply(rule fresh_star_supp_conv)+
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  apply(drule fresh_star_perm_set_conv)+
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  apply(rule finite_Diff)+
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  apply(rule finite_supp)+
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  apply(subgoal_tac "(ba xs \<union> ba ys) \<sharp>* f xs T")+
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  apply(metis Un_absorb2 fresh_star_Un)+
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  apply(subst fresh_star_Un)+
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  apply(rule conjI)+
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  apply(simp add: fresh_star_def f1)+
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  apply(simp add: fresh_star_def f2)+
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  apply(simp add: eqv)+
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  done+
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+
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lemma Abs_res_fcb:+
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  fixes xs ys :: "('a :: at_base) set"+
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    and S T :: "'b :: fs"+
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  assumes e: "(Abs_res (atom ` xs) T) = (Abs_res (atom ` ys) S)"+
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    and f1: "\<And>x. x \<in> atom ` xs \<Longrightarrow> x \<in> supp T \<Longrightarrow> x \<sharp> f xs T"+
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    and f2: "\<And>x. \<lbrakk>supp T - atom ` xs = supp S - atom ` ys; x \<in> atom ` ys; x \<in> supp S\<rbrakk> \<Longrightarrow> x \<sharp> f xs T"+
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    and eqv: "\<And>p. \<lbrakk>p \<bullet> T = S; supp p \<subseteq> atom ` xs \<inter> supp T \<union> atom ` ys \<inter> supp S;+
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      p \<bullet> (atom ` xs \<inter> supp T) = atom ` ys \<inter> supp S\<rbrakk> \<Longrightarrow> p \<bullet> (f xs T) = f ys S"+
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  shows "f xs T = f ys S"+
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  using e apply -+
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  apply(subst (asm) Abs_eq_res_set)+
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  apply(subst (asm) Abs_eq_iff2)+
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  apply(simp add: alphas)+
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  apply(elim exE conjE)+
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  apply(rule trans)+
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  apply(rule_tac p="p" in supp_perm_eq[symmetric])+
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  apply(rule fresh_star_supp_conv)+
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  apply(drule fresh_star_perm_set_conv)+
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  apply(rule finite_Diff)+
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  apply(rule finite_supp)+
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  apply(subgoal_tac "(atom ` xs \<inter> supp T \<union> atom ` ys \<inter> supp S) \<sharp>* f xs T")+
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  apply(metis Un_absorb2 fresh_star_Un)+
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  apply(subst fresh_star_Un)+
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  apply(rule conjI)+
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  apply(simp add: fresh_star_def f1)+
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  apply(subgoal_tac "supp T - atom ` xs = supp S - atom ` ys")+
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  apply(simp add: fresh_star_def f2)+
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  apply(blast)+
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  apply(simp add: eqv)+
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  done+
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+
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+
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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 only: 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 only: 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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+
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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 \<inter> supp x) \<sharp>* f (as \<inter> supp x) 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 \<inter> supp x) x c) = f (p \<bullet> (as \<inter> supp x)) (p \<bullet> x) c"+
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  and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f (bs \<inter> supp y) y c) = f (p \<bullet> (bs \<inter> supp y)) (p \<bullet> y) c"+
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  shows "f (as \<inter> supp x) x c = f (bs \<inter> supp y) y c"+
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proof -+
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  have "supp (as, x, c) supports (f (as \<inter> supp x) 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 inter_eqvt supp_eqvt)+
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  then have fin1: "finite (supp (f (as \<inter> supp x) 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 \<inter> supp y) 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 inter_eqvt supp_eqvt)+
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  then have fin2: "finite (supp (f (bs \<inter> supp y) 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 \<inter> supp x)) \<sharp>* (x, c, f (as \<inter> supp x) x c, f (bs \<inter> supp y) y c)" and +
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    fr2: "supp q \<sharp>* ([as \<inter> supp x]set. x)" and +
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    inc: "supp q \<subseteq> (as \<inter> supp x) \<union> (q \<bullet> (as \<inter> supp x))"+
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    using at_set_avoiding3[where xs="as \<inter> supp x" and c="(x, c, f (as \<inter> supp x) x c, f (bs \<inter> supp y) y c)" +
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      and x="[as \<inter> supp x]set. x"]  +
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      fin1 fin2 fin+
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    apply (auto simp add: supp_Pair finite_supp Abs_fresh_star dest: fresh_star_supp_conv)+
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    done+
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  have "[q \<bullet> (as \<inter> supp x)]set. (q \<bullet> x) = q \<bullet> ([as \<inter> supp x]set. x)" by simp+
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  also have "\<dots> = [as \<inter> supp x]set. x"+
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    by (simp only: fr2 perm_supp_eq)+
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  finally have "[q \<bullet> (as \<inter> supp x)]set. (q \<bullet> x) = [bs \<inter> supp y]set. y" using eq +
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    by(simp add: Abs_eq_res_set)+
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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 x)"+
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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 inter_eqvt supp_eqvt)+
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    done+
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  have "(as \<inter> supp x) \<sharp>* f (as \<inter> supp x) x c" by (rule fcb1)+
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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 only: 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 \<inter> supp y)) \<sharp>* f (r \<bullet> (bs \<inter> supp y)) (r \<bullet> y) c" using qq1 qq2 +
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    apply(perm_simp)+
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    apply simp+
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    done+
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  then have "r \<bullet> ((bs \<inter> supp y) \<sharp>* f (bs \<inter> supp y) y c)"+
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    apply(simp only: 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 \<inter> supp y) \<sharp>* f (bs \<inter> supp y) y c" by (simp add: permute_bool_def)+
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  have "f (as \<inter> supp x) x c = q \<bullet> (f (as \<inter> supp x) x c)"+
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    apply(rule perm_supp_eq[symmetric])+
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    using inc fcb1 fr1 +
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    apply (auto simp add: fresh_star_def)+
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    done+
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  also have "\<dots> = f (q \<bullet> (as \<inter> supp x)) (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 \<inter> supp y)) (r \<bullet> y) c" using qq1 qq2 +
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    apply(perm_simp)+
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    apply simp+
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    done+
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  also have "\<dots> = r \<bullet> (f (bs \<inter> supp y) 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 \<inter> supp y) 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_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+
−
  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)+
−
    apply(simp add: set_eqvt)+
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    apply(blast)+
−
    done+
−
  have "(set as) \<sharp>* f as x c" by (rule fcb1)+
−
  then have "q \<bullet> ((set as) \<sharp>* f as x c)"+
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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"+
−
    apply(simp only: fresh_star_eqvt set_eqvt)+
−
    apply(subst (asm) perm1)+
−
    using inc fresh1 fr1+
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    apply(auto simp add: fresh_star_def fresh_Pair)+
−
    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+
−
  then have "r \<bullet> ((set bs) \<sharp>* f bs y c)"+
−
    apply(simp only: 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)+
−
    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: "[[a]]lst. x = [[b]]lst. 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 Abs_lst1_fcb2':+
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  fixes a b :: "'a::at"+
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    and x y :: "'b :: fs"+
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    and c::"'c :: fs"+
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  assumes e: "[[atom a]]lst. x = [[atom b]]lst. y"+
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  and fcb1: "atom a \<sharp> f a x c"+
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  and fresh: "{atom a, atom 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_lst1_fcb2[where c="c" and f="\<lambda>a . f ((inv atom) a)"])+
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using  fcb1 fresh perm1 perm2+
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apply(simp_all add: fresh_star_def inv_f_f inj_on_def atom_eqvt)+
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done+
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+
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end+
−