theory Nominal2_FCBimports "Nominal2_Abs" begintext {* A tactic which solves all trivial cases in function definitions, and leaves the others unchanged.*}ML {*val all_trivials : (Proof.context -> Method.method) context_parser =Scan.succeed (fn ctxt => let val tac = TRYALL (SOLVED' (full_simp_tac (simpset_of ctxt))) in Method.SIMPLE_METHOD' (K tac) end)*}method_setup all_trivials = {* all_trivials *} {* solves trivial goals *}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]) donelemma 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) donelemma 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) donelemma 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) donelemma 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 simpqedlemma 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 \<inter> supp x) \<sharp>* f (as \<inter> supp x) 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 \<inter> supp x) x c) = f (p \<bullet> (as \<inter> supp x)) (p \<bullet> x) c" 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" shows "f (as \<inter> supp x) x c = f (bs \<inter> supp y) y c"proof - have "supp (as, x, c) supports (f (as \<inter> supp x) x c)" unfolding supports_def fresh_def[symmetric] by (simp add: fresh_Pair perm1 fresh_star_def supp_swap swap_fresh_fresh inter_eqvt supp_eqvt) then have fin1: "finite (supp (f (as \<inter> supp x) 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 \<inter> supp y) y c)" unfolding supports_def fresh_def[symmetric] by (simp add: fresh_Pair perm2 fresh_star_def supp_swap swap_fresh_fresh inter_eqvt supp_eqvt) then have fin2: "finite (supp (f (bs \<inter> supp y) 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 \<inter> supp x)) \<sharp>* (x, c, f (as \<inter> supp x) x c, f (bs \<inter> supp y) y c)" and fr2: "supp q \<sharp>* ([as \<inter> supp x]set. x)" and inc: "supp q \<subseteq> (as \<inter> supp x) \<union> (q \<bullet> (as \<inter> supp x))" 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)" and x="[as \<inter> supp x]set. x"] fin1 fin2 fin apply (auto simp add: supp_Pair finite_supp Abs_fresh_star dest: fresh_star_supp_conv) done have "[q \<bullet> (as \<inter> supp x)]set. (q \<bullet> x) = q \<bullet> ([as \<inter> supp x]set. x)" by simp also have "\<dots> = [as \<inter> supp x]set. x" by (simp only: fr2 perm_supp_eq) finally have "[q \<bullet> (as \<inter> supp x)]set. (q \<bullet> x) = [bs \<inter> supp y]set. y" using eq by(simp add: Abs_eq_res_set) 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 x)" 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 inter_eqvt supp_eqvt) done have "(as \<inter> supp x) \<sharp>* f (as \<inter> supp x) x c" by (rule fcb1) 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) apply(subst (asm) perm1) using inc fresh1 fr1 apply(auto simp add: fresh_star_def fresh_Pair) done then have "(r \<bullet> (bs \<inter> supp y)) \<sharp>* f (r \<bullet> (bs \<inter> supp y)) (r \<bullet> y) c" using qq1 qq2 apply(perm_simp) apply simp done then have "r \<bullet> ((bs \<inter> supp y) \<sharp>* f (bs \<inter> supp y) 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 \<inter> supp y) \<sharp>* f (bs \<inter> supp y) y c" by (simp add: permute_bool_def) have "f (as \<inter> supp x) x c = q \<bullet> (f (as \<inter> supp x) x c)" apply(rule perm_supp_eq[symmetric]) using inc fcb1 fr1 apply (auto simp add: fresh_star_def) done also have "\<dots> = f (q \<bullet> (as \<inter> supp x)) (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 \<inter> supp y)) (r \<bullet> y) c" using qq1 qq2 apply(perm_simp) apply simp done also have "\<dots> = r \<bullet> (f (bs \<inter> supp y) y c)" apply(rule perm2[symmetric]) using qq3 fresh2 fr1 by (auto simp add: fresh_star_def) also have "... = f (bs \<inter> supp y) y c" apply(rule perm_supp_eq) using qq3 fr1 fcb2 by (auto simp add: fresh_star_def) finally show ?thesis by simpqedlemma 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 simpqedlemma 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 eapply(drule_tac Abs_lst_fcb2[where c="c" and f="\<lambda>(as::atom list) . f (hd as)"])apply(simp_all)using fcb1 fresh perm1 perm2apply(simp_all add: fresh_star_def)donelemma 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 eapply(drule_tac Abs_lst1_fcb2[where c="c" and f="\<lambda>a . f ((inv atom) a)"])using fcb1 fresh perm1 perm2apply(simp_all add: fresh_star_def inv_f_f inj_on_def atom_eqvt)doneend