theory Letimports "../Nominal2" beginatom_decl namenominal_datatype trm = Var "name"| App "trm" "trm"| Lam x::"name" t::"trm" binds x in t| Let as::"assn" t::"trm" binds "bn as" in tand assn = ANil| ACons "name" "trm" "assn"binder bnwhere "bn ANil = []"| "bn (ACons x t as) = (atom x) # (bn as)"print_theoremsthm alpha_trm_raw_alpha_assn_raw_alpha_bn_raw.introsthm bn_raw.simpsthm permute_bn_raw.simpsthm trm_assn.perm_bn_alphathm trm_assn.permute_bnthm trm_assn.fv_defsthm trm_assn.eq_iff thm trm_assn.bn_defsthm trm_assn.bn_inductsthm trm_assn.perm_simpsthm trm_assn.inductthm trm_assn.inductsthm trm_assn.distinctthm trm_assn.suppthm trm_assn.freshthm trm_assn.exhaustthm trm_assn.strong_exhaustthm trm_assn.perm_bn_simpslemma alpha_bn_inducts_raw[consumes 1]: "\<lbrakk>alpha_bn_raw a b; P3 ANil_raw ANil_raw; \<And>trm_raw trm_rawa assn_raw assn_rawa name namea. \<lbrakk>alpha_trm_raw trm_raw trm_rawa; alpha_bn_raw assn_raw assn_rawa; P3 assn_raw assn_rawa\<rbrakk> \<Longrightarrow> P3 (ACons_raw name trm_raw assn_raw) (ACons_raw namea trm_rawa assn_rawa)\<rbrakk> \<Longrightarrow> P3 a b" by (erule alpha_trm_raw_alpha_assn_raw_alpha_bn_raw.inducts(3)[of _ _ "\<lambda>x y. True" _ "\<lambda>x y. True", simplified]) autolemmas alpha_bn_inducts[consumes 1] = alpha_bn_inducts_raw[quot_lifted]lemma alpha_bn_refl: "alpha_bn x x" by (induct x rule: trm_assn.inducts(2)) (rule TrueI, auto simp add: trm_assn.eq_iff)lemma alpha_bn_sym: "alpha_bn x y \<Longrightarrow> alpha_bn y x" sorrylemma alpha_bn_trans: "alpha_bn x y \<Longrightarrow> alpha_bn y z \<Longrightarrow> alpha_bn x z" sorrylemma bn_inj[rule_format]: assumes a: "alpha_bn x y" shows "bn x = bn y \<longrightarrow> x = y" by (rule alpha_bn_inducts[OF a]) (simp_all add: trm_assn.bn_defs)lemma bn_inj2: assumes a: "alpha_bn x y" shows "\<And>q r. (q \<bullet> bn x) = (r \<bullet> bn y) \<Longrightarrow> permute_bn q x = permute_bn r y"using aapply(induct rule: alpha_bn_inducts)apply(simp add: trm_assn.perm_bn_simps)apply(simp add: trm_assn.perm_bn_simps)apply(simp add: trm_assn.bn_defs)apply(simp add: atom_eqvt)donelemma 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>* c \<Longrightarrow> (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" apply(rule fcb1) apply(rule fresh1) done 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[OF fresh1] 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> c \<Longrightarrow> 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)donefunction apply_assn2 :: "(trm \<Rightarrow> trm) \<Rightarrow> assn \<Rightarrow> assn"where "apply_assn2 f ANil = ANil"| "apply_assn2 f (ACons x t as) = ACons x (f t) (apply_assn2 f as)" apply(case_tac x) apply(case_tac b rule: trm_assn.exhaust(2)) apply(simp_all) apply(blast) donetermination by lexicographic_orderlemma apply_assn_eqvt[eqvt]: "p \<bullet> (apply_assn2 f a) = apply_assn2 (p \<bullet> f) (p \<bullet> a)" apply(induct f a rule: apply_assn2.induct) apply simp_all apply(perm_simp) apply rule donelemma fixes x y :: "'a :: fs" shows "[a # as]lst. x = [b # bs]lst. y \<Longrightarrow> [[a]]lst. [as]lst. x = [[b]]lst. [bs]lst. y" apply (simp add: Abs_eq_iff) apply (elim exE) apply (rule_tac x="p" in exI) apply (simp add: alphas) apply clarify apply rule apply (simp add: supp_Abs) apply blast apply (simp add: supp_Abs fresh_star_def) apply blast donelemma assumes neq: "a \<noteq> b" "sort_of a = sort_of b" shows "[[a]]lst. [[a]]lst. a = [[a]]lst. [[b]]lst. b \<and> [[a, a]]lst. a \<noteq> [[a, b]]lst. b" apply (simp add: Abs1_eq_iff) apply rule apply (simp add: Abs_eq_iff alphas supp_atom fresh_star_def) apply (rule_tac x="(a \<rightleftharpoons> b)" in exI) apply (simp add: neq) apply (simp add: Abs_eq_iff alphas supp_atom fresh_star_def neq) donenominal_primrec subst :: "name \<Rightarrow> trm \<Rightarrow> trm \<Rightarrow> trm"where "subst s t (Var x) = (if (s = x) then t else (Var x))"| "subst s t (App l r) = App (subst s t l) (subst s t r)"| "atom v \<sharp> (s, t) \<Longrightarrow> subst s t (Lam v b) = Lam v (subst s t b)"| "set (bn as) \<sharp>* (s, t) \<Longrightarrow> subst s t (Let as b) = Let (apply_assn2 (subst s t) as) (subst s t b)" apply (simp only: eqvt_def subst_graph_def) apply (rule, perm_simp, rule) apply (rule TrueI) apply (case_tac x) apply (rule_tac y="c" and c="(a,b)" in trm_assn.strong_exhaust(1)) apply (auto simp add: fresh_star_def)[3] apply (drule_tac x="assn" in meta_spec) apply (simp add: Abs1_eq_iff alpha_bn_refl) apply auto[7] prefer 2 apply(simp) thm subst_sumC_def thm THE_default_def thm theI apply (erule_tac c="(sa, ta)" in Abs_lst1_fcb2) apply (simp add: Abs_fresh_iff) apply (simp add: fresh_star_def) apply (simp_all add: fresh_star_Pair_elim perm_supp_eq eqvt_at_def)[2] apply (subgoal_tac "apply_assn2 (\<lambda>x2\<Colon>trm. subst_sumC (sa, ta, x2)) asa = apply_assn2 (\<lambda>x2\<Colon>trm. subst_sumC (sa, ta, x2)) as") prefer 2 apply (erule alpha_bn_inducts) apply simp apply (simp only: apply_assn2.simps) apply simp--"We know nothing about names; not true; but we can apply fcb2" defer apply (simp only: ) apply (erule_tac c="(sa, ta)" in Abs_lst_fcb2)--"We again need induction for fcb assumption; this time true" apply (induct_tac as rule: trm_assn.inducts(2)) apply (rule TrueI)+ apply (simp_all add: trm_assn.bn_defs fresh_star_def)[2] apply (auto simp add: Abs_fresh_iff)[1] apply assumption+--"But eqvt is not going to be true as well" apply (simp add: fresh_star_Pair_elim perm_supp_eq eqvt_at_def trm_assn.fv_bn_eqvt) apply (simp add: apply_assn_eqvt) apply (drule sym) apply (subgoal_tac "p \<bullet> (\<lambda>x2\<Colon>trm. subst_sumC (sa, ta, x2)) = (\<lambda>x2\<Colon>trm. subst_sumC (sa, ta, x2))") apply (simp) apply (erule alpha_bn_inducts) apply simp apply simp apply (simp add: trm_assn.bn_defs)--"Again we cannot relate 'namea' with 'p \<bullet> name'" prefer 4 apply (erule alpha_bn_inducts) apply simp_all[2] oopsend