theory Letimports "../Nominal2" beginatom_decl namenominal_datatype trm = Var "name"| App "trm" "trm"| Lam x::"name" t::"trm" bind x in t| Let as::"assn" t::"trm" bind "bn as" in tand assn = ANil| ACons "name" "trm" "assn"binder bnwhere "bn ANil = []"| "bn (ACons x t as) = (atom x) # (bn as)"thm trm_assn.fv_defsthm trm_assn.eq_iff thm trm_assn.bn_defsthm 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_exhaustlemma lets_bla: "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))" by (simp add: trm_assn.eq_iff)lemma lets_ok: "(Let (ACons x (Var y) ANil) (Var x)) = (Let (ACons y (Var y) ANil) (Var y))" apply (simp add: trm_assn.eq_iff Abs_eq_iff ) apply (rule_tac x="(x \<leftrightarrow> y)" in exI) apply (simp_all add: alphas atom_eqvt supp_at_base fresh_star_def trm_assn.bn_defs trm_assn.supp) donelemma lets_ok3: "x \<noteq> y \<Longrightarrow> (Let (ACons x (App (Var y) (Var x)) (ACons y (Var y) ANil)) (App (Var x) (Var y))) \<noteq> (Let (ACons y (App (Var x) (Var y)) (ACons x (Var x) ANil)) (App (Var x) (Var y)))" apply (simp add: trm_assn.eq_iff) donelemma lets_not_ok1: "x \<noteq> y \<Longrightarrow> (Let (ACons x (Var x) (ACons y (Var y) ANil)) (App (Var x) (Var y))) \<noteq> (Let (ACons y (Var x) (ACons x (Var y) ANil)) (App (Var x) (Var y)))" apply (simp add: alphas trm_assn.eq_iff trm_assn.supp fresh_star_def atom_eqvt Abs_eq_iff trm_assn.bn_defs) donelemma lets_nok: "x \<noteq> y \<Longrightarrow> x \<noteq> z \<Longrightarrow> z \<noteq> y \<Longrightarrow> (Let (ACons x (App (Var z) (Var z)) (ACons y (Var z) ANil)) (App (Var x) (Var y))) \<noteq> (Let (ACons y (Var z) (ACons x (App (Var z) (Var z)) ANil)) (App (Var x) (Var y)))" apply (simp add: alphas trm_assn.eq_iff fresh_star_def trm_assn.bn_defs Abs_eq_iff trm_assn.supp trm_assn.distinct) donelemma fixes a b c :: name assumes x: "a \<noteq> c" and y: "b \<noteq> c" shows "\<exists>p.([atom a], Var c) \<approx>lst (op =) supp p ([atom b], Var c)" apply (rule_tac x="(a \<leftrightarrow> b)" in exI) apply (simp add: alphas trm_assn.supp supp_at_base x y fresh_star_def atom_eqvt) by (metis Rep_name_inverse atom_name_def flip_fresh_fresh fresh_atom fresh_perm x y)lemma alpha_bn_refl: "alpha_bn x x"apply (induct x rule: trm_assn.inducts(2))apply (rule TrueI)apply (auto simp add: trm_assn.eq_iff)donelemma alpha_bn_inducts_raw: "\<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 = alpha_bn_inducts_raw[quot_lifted]lemma max_eqvt[eqvt]: "p \<bullet> (max (a :: _ :: pure) b) = max (p \<bullet> a) (p \<bullet> b)" by (simp add: permute_pure)(* TODO: should be provided by nominal *)lemmas [eqvt] = trm_assn.fv_bn_eqvtlemma Abs_lst_fcb2: fixes as bs :: "'a :: fs" and x y :: "'b :: fs" and c::"'c::fs" assumes eq: "[ba as]lst. x = [ba bs]lst. y" and fcb1: "(set (ba as)) \<sharp>* f as x c" and fresh1: "set (ba as) \<sharp>* c" and fresh2: "set (ba 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" and props: "eqvt ba" "inj ba" 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 (ba as))) \<sharp>* (x, c, f as x c, f bs y c)" and fr2: "supp q \<sharp>* ([ba as]lst. x)" and inc: "supp q \<subseteq> (set (ba as)) \<union> q \<bullet> (set (ba as))" using at_set_avoiding3[where xs="set (ba as)" and c="(x, c, f as x c, f bs y c)" and x="[ba as]lst. x"] fin1 fin2 by (auto simp add: supp_Pair finite_supp Abs_fresh_star dest: fresh_star_supp_conv) have "[q \<bullet> (ba as)]lst. (q \<bullet> x) = q \<bullet> ([ba as]lst. x)" by simp also have "\<dots> = [ba as]lst. x" by (simp only: fr2 perm_supp_eq) finally have "[q \<bullet> (ba as)]lst. (q \<bullet> x) = [ba bs]lst. y" using eq by simp then obtain r::perm where qq1: "q \<bullet> x = r \<bullet> y" and qq2: "q \<bullet> (ba as) = r \<bullet> (ba bs)" and qq3: "supp r \<subseteq> (q \<bullet> (set (ba as))) \<union> set (ba 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 qq4: "q \<bullet> as = r \<bullet> bs" using qq2 props unfolding eqvt_def inj_on_def apply(perm_simp) apply(simp) done have "(set (ba as)) \<sharp>* f as x c" by (rule fcb1) then have "q \<bullet> ((set (ba as)) \<sharp>* f as x c)" by (simp add: permute_bool_def) then have "set (q \<bullet> (ba 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> (ba bs)) \<sharp>* f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 qq4 by simp then have "r \<bullet> ((set (ba 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 (ba 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 qq4 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 simpqed(* PROBLEM: the proof needs induction on alpha_bn inside which is not possible... *)nominal_primrec height_trm :: "trm \<Rightarrow> nat"and height_assn :: "assn \<Rightarrow> nat"where "height_trm (Var x) = 1"| "height_trm (App l r) = max (height_trm l) (height_trm r)"| "height_trm (Lam v b) = 1 + (height_trm b)"| "height_trm (Let as b) = max (height_assn as) (height_trm b)"| "height_assn ANil = 0"| "height_assn (ACons v t as) = max (height_trm t) (height_assn as)" apply (simp only: eqvt_def height_trm_height_assn_graph_def) apply (rule, perm_simp, rule, rule TrueI) apply (case_tac x) apply (case_tac a rule: trm_assn.exhaust(1)) apply (auto)[4] apply (drule_tac x="assn" in meta_spec) apply (drule_tac x="trm" in meta_spec) apply (simp add: alpha_bn_refl) apply (case_tac b rule: trm_assn.exhaust(2)) apply (auto) apply (erule Abs_lst1_fcb) apply (simp_all add: pure_fresh) apply (simp add: eqvt_at_def) apply (erule Abs_lst_fcb) apply (simp_all add: pure_fresh) apply (simp_all add: eqvt_at_def eqvts) apply (rule arg_cong) back oopsnominal_primrec subst :: "name \<Rightarrow> trm \<Rightarrow> trm \<Rightarrow> trm"and substa :: "name \<Rightarrow> trm \<Rightarrow> assn \<Rightarrow> assn"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 (substa s t as) (subst s t b)"| "substa s t ANil = ANil"| "substa s t (ACons v t' as) = ACons v (subst v t t') as"(*unfolding eqvt_def subst_substa_graph_def apply (rule, perm_simp)*) defer apply (rule TrueI) apply (case_tac x) apply (case_tac a) apply (rule_tac y="c" and c="(aa,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 (case_tac b) apply (case_tac c rule: trm_assn.exhaust(2)) apply (auto)[2] apply blast apply blast apply auto apply (simp_all add: meta_eq_to_obj_eq[OF subst_def, symmetric, unfolded fun_eq_iff]) apply (simp_all add: meta_eq_to_obj_eq[OF substa_def, symmetric, unfolded fun_eq_iff]) prefer 2 apply (erule_tac Abs_lst_fcb2) oopsend