theory LetRecBimports "../Nominal2"beginatom_decl namenominal_datatype let_rec: trm = Var "name"| App "trm" "trm"| Lam x::"name" t::"trm" bind x in t| Let_Rec bp::"bp" t::"trm" bind "bn bp" in bp tand bp = Bp "name" "trm"binder bn::"bp \<Rightarrow> atom list"where "bn (Bp x t) = [atom x]"thm let_rec.distinctthm let_rec.inductthm let_rec.exhaustthm let_rec.fv_defsthm let_rec.bn_defsthm let_rec.perm_simpsthm let_rec.eq_iffthm let_rec.fv_bn_eqvtthm let_rec.size_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 simpqedlemma max_eqvt[eqvt]: "p \<bullet> (max (a :: _ :: pure) b) = max (p \<bullet> a) (p \<bullet> b)" by (simp add: permute_pure)nominal_primrec height_trm :: "trm \<Rightarrow> nat"and height_bp :: "bp \<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_Rec bp b) = max (height_bp bp) (height_trm b)"| "height_bp (Bp v t) = height_trm t" --"eqvt" apply (simp only: eqvt_def height_trm_height_bp_graph_def) apply (rule, perm_simp, rule, rule TrueI) --"completeness" apply (case_tac x) apply (case_tac a rule: let_rec.exhaust(1)) apply (auto)[4] apply (case_tac b rule: let_rec.exhaust(2)) apply blast apply(simp_all) apply (erule_tac c="()" in Abs_lst_fcb2) apply (simp_all add: fresh_star_def pure_fresh)[3] apply (simp add: eqvt_at_def) apply (simp add: eqvt_at_def) apply(simp add: eqvt_def) apply(perm_simp) apply(simp) apply(simp add: inj_on_def) --"The following could be done by nominal" apply (simp add: meta_eq_to_obj_eq[OF height_trm_def, symmetric, unfolded fun_eq_iff]) apply (simp add: meta_eq_to_obj_eq[OF height_bp_def, symmetric, unfolded fun_eq_iff]) apply (subgoal_tac "eqvt_at height_bp bp") apply (subgoal_tac "eqvt_at height_bp bpa") apply (subgoal_tac "eqvt_at height_trm b") apply (subgoal_tac "eqvt_at height_trm ba") apply (thin_tac "eqvt_at height_trm_height_bp_sumC (Inr bp)") apply (thin_tac "eqvt_at height_trm_height_bp_sumC (Inr bpa)") apply (thin_tac "eqvt_at height_trm_height_bp_sumC (Inl b)") apply (thin_tac "eqvt_at height_trm_height_bp_sumC (Inl ba)") defer apply (simp add: eqvt_at_def height_trm_def) apply (simp add: eqvt_at_def height_trm_def) apply (simp add: eqvt_at_def height_bp_def) apply (simp add: eqvt_at_def height_bp_def) --"I'd like to apply FCB here, but the following fails" apply (subgoal_tac "height_bp bp = height_bp bpa") apply (subgoal_tac "height_trm b = height_trm ba") apply simp apply (subgoal_tac "(\<lambda>as x c. height_trm (snd (bp, b))) as x c = (\<lambda>as x c. height_trm (snd (bpa, ba))) as x c") apply simp apply (erule_tac c="()" and ba="bn" and f="\<lambda>as x c. height_trm (snd x)" in Abs_lst_fcb2) ... donetermination by lexicographic_orderend