# HG changeset patch # User Christian Urban <urbanc@in.tum.de> # Date 1309832342 -7200 # Node ID 8648ae682442e13142747b194287d3d006ded10c # Parent 09834ba7ce590ab7432bac4fd9796a046a8c3d22 all FCB lemmas diff -r 09834ba7ce59 -r 8648ae682442 Nominal/Nominal2_FCB.thy --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/Nominal/Nominal2_FCB.thy Tue Jul 05 04:19:02 2011 +0200 @@ -0,0 +1,370 @@ +theory Nominal2_FCB +imports "Nominal2_Abs" +begin + + +lemma Abs_lst1_fcb: + fixes x y :: "'a :: at_base" + and S T :: "'b :: fs" + assumes e: "(Abs_lst [atom x] T) = (Abs_lst [atom y] S)" + and f1: "\<lbrakk>x \<noteq> y; atom y \<sharp> T; atom x \<sharp> (atom y \<rightleftharpoons> atom x) \<bullet> T\<rbrakk> \<Longrightarrow> atom x \<sharp> f x T" + and f2: "\<lbrakk>x \<noteq> y; atom y \<sharp> T; atom x \<sharp> (atom y \<rightleftharpoons> atom x) \<bullet> T\<rbrakk> \<Longrightarrow> atom y \<sharp> f x T" + and p: "\<lbrakk>S = (atom x \<rightleftharpoons> atom y) \<bullet> T; x \<noteq> y; atom y \<sharp> T; atom x \<sharp> S\<rbrakk> + \<Longrightarrow> (atom x \<rightleftharpoons> atom y) \<bullet> (f x T) = f y S" + and s: "sort_of (atom x) = sort_of (atom y)" + shows "f x T = f y S" + using e + apply(case_tac "atom x \<sharp> S") + apply(simp add: Abs1_eq_iff'[OF s s]) + apply(elim conjE disjE) + apply(simp) + apply(rule trans) + apply(rule_tac p="(atom x \<rightleftharpoons> atom y)" in supp_perm_eq[symmetric]) + apply(rule fresh_star_supp_conv) + apply(simp add: supp_swap fresh_star_def s f1 f2) + apply(simp add: swap_commute p) + apply(simp add: Abs1_eq_iff[OF s s]) + done + +lemma Abs_lst_fcb: + fixes xs ys :: "'a :: fs" + and S T :: "'b :: fs" + assumes e: "(Abs_lst (ba xs) T) = (Abs_lst (ba ys) S)" + and f1: "\<And>x. x \<in> set (ba xs) \<Longrightarrow> x \<sharp> f xs T" + 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" + 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> + \<Longrightarrow> p \<bullet> (f xs T) = f ys S" + shows "f xs T = f ys S" + using e apply - + apply(subst (asm) Abs_eq_iff2) + apply(simp add: alphas) + apply(elim exE conjE) + apply(rule trans) + apply(rule_tac p="p" in supp_perm_eq[symmetric]) + apply(rule fresh_star_supp_conv) + apply(drule fresh_star_perm_set_conv) + apply(rule finite_Diff) + apply(rule finite_supp) + apply(subgoal_tac "(set (ba xs) \<union> set (ba ys)) \<sharp>* f xs T") + apply(metis Un_absorb2 fresh_star_Un) + apply(subst fresh_star_Un) + apply(rule conjI) + apply(simp add: fresh_star_def f1) + apply(simp add: fresh_star_def f2) + apply(simp add: eqv) + done + +lemma Abs_set_fcb: + fixes xs ys :: "'a :: fs" + and S T :: "'b :: fs" + assumes e: "(Abs_set (ba xs) T) = (Abs_set (ba ys) S)" + and f1: "\<And>x. x \<in> ba xs \<Longrightarrow> x \<sharp> f xs T" + and f2: "\<And>x. \<lbrakk>supp T - ba xs = supp S - ba ys; x \<in> ba ys\<rbrakk> \<Longrightarrow> x \<sharp> f xs T" + 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" + shows "f xs T = f ys S" + using e apply - + apply(subst (asm) Abs_eq_iff2) + apply(simp add: alphas) + apply(elim exE conjE) + apply(rule trans) + apply(rule_tac p="p" in supp_perm_eq[symmetric]) + apply(rule fresh_star_supp_conv) + apply(drule fresh_star_perm_set_conv) + apply(rule finite_Diff) + apply(rule finite_supp) + apply(subgoal_tac "(ba xs \<union> ba ys) \<sharp>* f xs T") + apply(metis Un_absorb2 fresh_star_Un) + apply(subst fresh_star_Un) + apply(rule conjI) + apply(simp add: fresh_star_def f1) + apply(simp add: fresh_star_def f2) + apply(simp add: eqv) + done + +lemma Abs_res_fcb: + fixes xs ys :: "('a :: at_base) set" + and S T :: "'b :: fs" + assumes e: "(Abs_res (atom ` xs) T) = (Abs_res (atom ` ys) S)" + and f1: "\<And>x. x \<in> atom ` xs \<Longrightarrow> x \<in> supp T \<Longrightarrow> x \<sharp> f xs T" + 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" + and eqv: "\<And>p. \<lbrakk>p \<bullet> T = S; supp p \<subseteq> atom ` xs \<inter> supp T \<union> atom ` ys \<inter> supp S; + p \<bullet> (atom ` xs \<inter> supp T) = atom ` ys \<inter> supp S\<rbrakk> \<Longrightarrow> p \<bullet> (f xs T) = f ys S" + shows "f xs T = f ys S" + using e apply - + apply(subst (asm) Abs_eq_res_set) + apply(subst (asm) Abs_eq_iff2) + apply(simp add: alphas) + apply(elim exE conjE) + apply(rule trans) + apply(rule_tac p="p" in supp_perm_eq[symmetric]) + apply(rule fresh_star_supp_conv) + apply(drule fresh_star_perm_set_conv) + apply(rule finite_Diff) + apply(rule finite_supp) + apply(subgoal_tac "(atom ` xs \<inter> supp T \<union> atom ` ys \<inter> supp S) \<sharp>* f xs T") + apply(metis Un_absorb2 fresh_star_Un) + apply(subst fresh_star_Un) + apply(rule conjI) + apply(simp add: fresh_star_def f1) + apply(subgoal_tac "supp T - atom ` xs = supp S - atom ` ys") + apply(simp add: fresh_star_def f2) + apply(blast) + apply(simp add: eqv) + done + + + +lemma Abs_set_fcb2: + fixes as bs :: "atom set" + and x y :: "'b :: fs" + and c::"'c::fs" + assumes eq: "[as]set. x = [bs]set. y" + and fin: "finite as" "finite bs" + and fcb1: "as \<sharp>* f as x c" + and fresh1: "as \<sharp>* c" + and fresh2: "bs \<sharp>* c" + and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f as x c) = f (p \<bullet> as) (p \<bullet> x) c" + and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f bs y c) = f (p \<bullet> bs) (p \<bullet> y) c" + shows "f as x c = f bs y c" +proof - + have "supp (as, x, c) supports (f as x c)" + unfolding supports_def fresh_def[symmetric] + by (simp add: fresh_Pair perm1 fresh_star_def supp_swap swap_fresh_fresh) + then have fin1: "finite (supp (f as x c))" + using fin by (auto intro: supports_finite simp add: finite_supp supp_of_finite_sets supp_Pair) + have "supp (bs, y, c) supports (f bs y c)" + unfolding supports_def fresh_def[symmetric] + by (simp add: fresh_Pair perm2 fresh_star_def supp_swap swap_fresh_fresh) + then have fin2: "finite (supp (f bs y c))" + using fin by (auto intro: supports_finite simp add: finite_supp supp_of_finite_sets supp_Pair) + obtain q::"perm" where + fr1: "(q \<bullet> as) \<sharp>* (x, c, f as x c, f bs y c)" and + fr2: "supp q \<sharp>* ([as]set. x)" and + inc: "supp q \<subseteq> as \<union> (q \<bullet> as)" + using at_set_avoiding3[where xs="as" and c="(x, c, f as x c, f bs y c)" and x="[as]set. x"] + fin1 fin2 fin + by (auto simp add: supp_Pair finite_supp Abs_fresh_star dest: fresh_star_supp_conv) + have "[q \<bullet> as]set. (q \<bullet> x) = q \<bullet> ([as]set. x)" by simp + also have "\<dots> = [as]set. x" + by (simp only: fr2 perm_supp_eq) + finally have "[q \<bullet> as]set. (q \<bullet> x) = [bs]set. y" using eq by simp + then obtain r::perm where + qq1: "q \<bullet> x = r \<bullet> y" and + qq2: "q \<bullet> as = r \<bullet> bs" and + qq3: "supp r \<subseteq> (q \<bullet> as) \<union> bs" + apply(drule_tac sym) + apply(simp only: Abs_eq_iff2 alphas) + apply(erule exE) + apply(erule conjE)+ + apply(drule_tac x="p" in meta_spec) + apply(simp add: set_eqvt) + apply(blast) + done + have "as \<sharp>* f as x c" by (rule fcb1) + then have "q \<bullet> (as \<sharp>* f as x c)" + by (simp add: permute_bool_def) + then have "(q \<bullet> as) \<sharp>* f (q \<bullet> as) (q \<bullet> x) c" + apply(simp add: fresh_star_eqvt set_eqvt) + apply(subst (asm) perm1) + using inc fresh1 fr1 + apply(auto simp add: fresh_star_def fresh_Pair) + done + then have "(r \<bullet> bs) \<sharp>* f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 by simp + then have "r \<bullet> (bs \<sharp>* f bs y c)" + apply(simp add: fresh_star_eqvt set_eqvt) + apply(subst (asm) perm2[symmetric]) + using qq3 fresh2 fr1 + apply(auto simp add: set_eqvt fresh_star_def fresh_Pair) + done + then have fcb2: "bs \<sharp>* f bs y c" by (simp add: permute_bool_def) + have "f as x c = q \<bullet> (f as x c)" + apply(rule perm_supp_eq[symmetric]) + using inc fcb1 fr1 by (auto simp add: fresh_star_def) + also have "\<dots> = f (q \<bullet> as) (q \<bullet> x) c" + apply(rule perm1) + using inc fresh1 fr1 by (auto simp add: fresh_star_def) + also have "\<dots> = f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 by simp + also have "\<dots> = r \<bullet> (f bs y c)" + apply(rule perm2[symmetric]) + using qq3 fresh2 fr1 by (auto simp add: fresh_star_def) + also have "... = f bs y c" + apply(rule perm_supp_eq) + using qq3 fr1 fcb2 by (auto simp add: fresh_star_def) + finally show ?thesis by simp +qed + + +text {* NOT DONE +lemma Abs_res_fcb2: + fixes as bs :: "atom set" + and x y :: "'b :: fs" + and c::"'c::fs" + assumes eq: "[as]res. x = [bs]res. y" + and fin: "finite as" "finite bs" + and fcb1: "as \<sharp>* f as x c" + and fresh1: "as \<sharp>* c" + and fresh2: "bs \<sharp>* c" + and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f as x c) = f (p \<bullet> as) (p \<bullet> x) c" + and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f bs y c) = f (p \<bullet> bs) (p \<bullet> y) c" + shows "f as x c = f bs y c" +proof - + have "supp (as, x, c) supports (f as x c)" + unfolding supports_def fresh_def[symmetric] + by (simp add: fresh_Pair perm1 fresh_star_def supp_swap swap_fresh_fresh) + then have fin1: "finite (supp (f as x c))" + using fin by (auto intro: supports_finite simp add: finite_supp supp_of_finite_sets supp_Pair) + have "supp (bs, y, c) supports (f bs y c)" + unfolding supports_def fresh_def[symmetric] + by (simp add: fresh_Pair perm2 fresh_star_def supp_swap swap_fresh_fresh) + then have fin2: "finite (supp (f bs y c))" + using fin by (auto intro: supports_finite simp add: finite_supp supp_of_finite_sets supp_Pair) + obtain q::"perm" where + fr1: "(q \<bullet> as) \<sharp>* (x, c, f as x c, f bs y c)" and + fr2: "supp q \<sharp>* ([as]res. x)" and + inc: "supp q \<subseteq> as \<union> (q \<bullet> as)" + using at_set_avoiding3[where xs="as" and c="(x, c, f as x c, f bs y c)" and x="[as]res. x"] + fin1 fin2 fin + by (auto simp add: supp_Pair finite_supp Abs_fresh_star dest: fresh_star_supp_conv) + have "[q \<bullet> as]res. (q \<bullet> x) = q \<bullet> ([as]res. x)" by simp + also have "\<dots> = [as]res. x" + by (simp only: fr2 perm_supp_eq) + finally have "[q \<bullet> as]res. (q \<bullet> x) = [bs]res. y" using eq by simp + then obtain r::perm where + qq1: "q \<bullet> x = r \<bullet> y" and + qq2: "(q \<bullet> as \<inter> supp (q \<bullet> x)) = r \<bullet> (bs \<inter> supp y)" and + qq3: "supp r \<subseteq> bs \<inter> supp y \<union> q \<bullet> as \<inter> supp (q \<bullet> x)" + apply(drule_tac sym) + apply(subst(asm) Abs_eq_res_set) + apply(simp only: Abs_eq_iff2 alphas) + apply(erule exE) + apply(erule conjE)+ + apply(drule_tac x="p" in meta_spec) + apply(simp add: set_eqvt) + done + have "(as \<inter> supp x) \<sharp>* f (as \<inter> supp x) x c" sorry (* FCB? *) + then have "q \<bullet> ((as \<inter> supp x) \<sharp>* f (as \<inter> supp x) x c)" + by (simp add: permute_bool_def) + then have "(q \<bullet> (as \<inter> supp x)) \<sharp>* f (q \<bullet> (as \<inter> supp x)) (q \<bullet> x) c" + apply(simp add: fresh_star_eqvt set_eqvt) + sorry (* perm? *) + then have "r \<bullet> (bs \<inter> supp y) \<sharp>* f (r \<bullet> (bs \<inter> supp y)) (r \<bullet> y) c" using qq2 + apply (simp add: inter_eqvt) + sorry + (* rest similar reversing it other way around... *) + show ?thesis sorry +qed +*} + + +lemma Abs_lst_fcb2: + fixes as bs :: "atom list" + and x y :: "'b :: fs" + and c::"'c::fs" + assumes eq: "[as]lst. x = [bs]lst. y" + and fcb1: "(set as) \<sharp>* f as x c" + and fresh1: "set as \<sharp>* c" + and fresh2: "set bs \<sharp>* c" + and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f as x c) = f (p \<bullet> as) (p \<bullet> x) c" + and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f bs y c) = f (p \<bullet> bs) (p \<bullet> y) c" + shows "f as x c = f bs y c" +proof - + have "supp (as, x, c) supports (f as x c)" + unfolding supports_def fresh_def[symmetric] + by (simp add: fresh_Pair perm1 fresh_star_def supp_swap swap_fresh_fresh) + then have fin1: "finite (supp (f as x c))" + by (auto intro: supports_finite simp add: finite_supp) + have "supp (bs, y, c) supports (f bs y c)" + unfolding supports_def fresh_def[symmetric] + by (simp add: fresh_Pair perm2 fresh_star_def supp_swap swap_fresh_fresh) + then have fin2: "finite (supp (f bs y c))" + by (auto intro: supports_finite simp add: finite_supp) + obtain q::"perm" where + fr1: "(q \<bullet> (set as)) \<sharp>* (x, c, f as x c, f bs y c)" and + fr2: "supp q \<sharp>* Abs_lst as x" and + inc: "supp q \<subseteq> (set as) \<union> q \<bullet> (set as)" + using at_set_avoiding3[where xs="set as" and c="(x, c, f as x c, f bs y c)" and x="[as]lst. x"] + fin1 fin2 + by (auto simp add: supp_Pair finite_supp Abs_fresh_star dest: fresh_star_supp_conv) + have "Abs_lst (q \<bullet> as) (q \<bullet> x) = q \<bullet> Abs_lst as x" by simp + also have "\<dots> = Abs_lst as x" + by (simp only: fr2 perm_supp_eq) + finally have "Abs_lst (q \<bullet> as) (q \<bullet> x) = Abs_lst bs y" using eq by simp + then obtain r::perm where + qq1: "q \<bullet> x = r \<bullet> y" and + qq2: "q \<bullet> as = r \<bullet> bs" and + qq3: "supp r \<subseteq> (q \<bullet> (set as)) \<union> set bs" + apply(drule_tac sym) + apply(simp only: Abs_eq_iff2 alphas) + apply(erule exE) + apply(erule conjE)+ + apply(drule_tac x="p" in meta_spec) + apply(simp add: set_eqvt) + apply(blast) + done + have "(set as) \<sharp>* f as x c" by (rule fcb1) + then have "q \<bullet> ((set as) \<sharp>* f as x c)" + by (simp add: permute_bool_def) + then have "set (q \<bullet> as) \<sharp>* f (q \<bullet> as) (q \<bullet> x) c" + apply(simp add: fresh_star_eqvt set_eqvt) + apply(subst (asm) perm1) + using inc fresh1 fr1 + apply(auto simp add: fresh_star_def fresh_Pair) + done + then have "set (r \<bullet> bs) \<sharp>* f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 by simp + then have "r \<bullet> ((set bs) \<sharp>* f bs y c)" + apply(simp add: fresh_star_eqvt set_eqvt) + apply(subst (asm) perm2[symmetric]) + using qq3 fresh2 fr1 + apply(auto simp add: set_eqvt fresh_star_def fresh_Pair) + done + then have fcb2: "(set bs) \<sharp>* f bs y c" by (simp add: permute_bool_def) + have "f as x c = q \<bullet> (f as x c)" + apply(rule perm_supp_eq[symmetric]) + using inc fcb1 fr1 by (auto simp add: fresh_star_def) + also have "\<dots> = f (q \<bullet> as) (q \<bullet> x) c" + apply(rule perm1) + using inc fresh1 fr1 by (auto simp add: fresh_star_def) + also have "\<dots> = f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 by simp + also have "\<dots> = r \<bullet> (f bs y c)" + apply(rule perm2[symmetric]) + using qq3 fresh2 fr1 by (auto simp add: fresh_star_def) + also have "... = f bs y c" + apply(rule perm_supp_eq) + using qq3 fr1 fcb2 by (auto simp add: fresh_star_def) + finally show ?thesis by simp +qed + +lemma Abs_lst1_fcb2: + fixes a b :: "atom" + and x y :: "'b :: fs" + and c::"'c :: fs" + assumes e: "(Abs_lst [a] x) = (Abs_lst [b] y)" + and fcb1: "a \<sharp> f a x c" + and fresh: "{a, b} \<sharp>* c" + and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f a x c) = f (p \<bullet> a) (p \<bullet> x) c" + and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f b y c) = f (p \<bullet> b) (p \<bullet> y) c" + shows "f a x c = f b y c" +using e +apply(drule_tac Abs_lst_fcb2[where c="c" and f="\<lambda>(as::atom list) . f (hd as)"]) +apply(simp_all) +using fcb1 fresh perm1 perm2 +apply(simp_all add: fresh_star_def) +done + +lemma Abs_lst1_fcb2': + fixes a b :: "'a::at" + and x y :: "'b :: fs" + and c::"'c :: fs" + assumes e: "(Abs_lst [atom a] x) = (Abs_lst [atom b] y)" + and fcb1: "atom a \<sharp> f a x c" + and fresh: "{atom a, atom b} \<sharp>* c" + and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f a x c) = f (p \<bullet> a) (p \<bullet> x) c" + and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f b y c) = f (p \<bullet> b) (p \<bullet> y) c" + shows "f a x c = f b y c" +using e +apply(drule_tac Abs_lst1_fcb2[where c="c" and f="\<lambda>a . f ((inv atom) a)"]) +using fcb1 fresh perm1 perm2 +apply(simp_all add: fresh_star_def inv_f_f inj_on_def atom_eqvt) +done + +end \ No newline at end of file