theory Lambda imports "../Nominal2" beginatom_decl namenominal_datatype lam = Var "name"| App "lam" "lam"| Lam x::"name" l::"lam" binds x in l ("Lam [_]. _" [100, 100] 100)nominal_primrec lam_rec :: "(name \<Rightarrow> 'a :: pt) \<Rightarrow> (lam \<Rightarrow> lam \<Rightarrow> 'a) \<Rightarrow> (name \<Rightarrow> lam \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b :: fs \<Rightarrow> lam \<Rightarrow> 'a"where "lam_rec fv fa fl fd P (Var n) = fv n"| "lam_rec fv fa fl fd P (App l r) = fa l r"| "(atom x \<sharp> P \<and> (\<forall>y s. atom y \<sharp> P \<longrightarrow> Lam [x]. t = Lam [y]. s \<longrightarrow> fl x t = fl y s)) \<Longrightarrow> lam_rec fv fa fl fd P (Lam [x]. t) = fl x t"| "(atom x \<sharp> P \<and> \<not>(\<forall>y s. atom y \<sharp> P \<longrightarrow> Lam [x]. t = Lam [y]. s \<longrightarrow> fl x t = fl y s)) \<Longrightarrow> lam_rec fv fa fl fd P (Lam [x]. t) = fd" apply (simp add: eqvt_def lam_rec_graph_def) apply (rule, perm_simp, rule, rule) apply (case_tac x) apply (rule_tac y="f" and c="e" in lam.strong_exhaust) apply metis apply metis unfolding fresh_star_def apply simp apply metis apply simp_all[2] apply (metis (no_types) Pair_inject lam.distinct)+ apply simp apply (metis (no_types) Pair_inject lam.distinct)+ donetermination (eqvt) by lexicographic_orderlemma lam_rec_cong[fundef_cong]: " (\<And>v. l = Var v \<Longrightarrow> fv v = fv' v) \<Longrightarrow> (\<And>t1 t2. l = App t1 t2 \<Longrightarrow> fa t1 t2 = fa' t1 t2) \<Longrightarrow> (\<And>n t. l = Lam [n]. t \<Longrightarrow> fl n t = fl' n t) \<Longrightarrow> lam_rec fv fa fl fd P l = lam_rec fv' fa' fl' fd P l" apply (rule_tac y="l" and c="P" in lam.strong_exhaust) apply auto apply (case_tac "(\<forall>y s. atom y \<sharp> P \<longrightarrow> Lam [name]. lam = Lam [y]. s \<longrightarrow> fl name lam = fl y s)") apply (subst lam_rec.simps) apply (simp add: fresh_star_def) apply (subst lam_rec.simps) apply (simp add: fresh_star_def) using Abs1_eq_iff lam.eq_iff apply metis apply (subst lam_rec.simps(4)) apply (simp add: fresh_star_def) apply (subst lam_rec.simps(4)) apply (simp add: fresh_star_def) using Abs1_eq_iff lam.eq_iff apply metis donenominal_primrec substr where[simp del]: "substr l y s = lam_rec (%x. if x = y then s else (Var x)) (%t1 t2. App (substr t1 y s) (substr t2 y s)) (%x t. Lam [x]. (substr t y s)) (Lam [y]. Var y) (y, s) l"unfolding eqvt_def substr_graph_defapply (rule, perm_simp, rule, rule)by pat_completeness autotermination (eqvt) by lexicographic_orderlemma fresh_fun_eqvt_app3: assumes e: "eqvt f" shows "\<lbrakk>a \<sharp> x; a \<sharp> y; a \<sharp> z\<rbrakk> \<Longrightarrow> a \<sharp> f x y z" using fresh_fun_eqvt_app[OF e] fresh_fun_app by (metis (lifting, full_types))lemma substr_simps: "substr (Var x) y s = (if x = y then s else (Var x))" "substr (App t1 t2) y s = App (substr t1 y s) (substr t2 y s)" "atom x \<sharp> (y, s) \<Longrightarrow> substr (Lam [x]. t) y s = Lam [x]. (substr t y s)" apply (subst substr.simps) apply (simp only: lam_rec.simps) apply (subst substr.simps) apply (simp only: lam_rec.simps) apply (subst substr.simps) apply (subst lam_rec.simps) apply (auto simp add: Abs1_eq_iff substr.eqvt swap_fresh_fresh) apply (rule fresh_fun_eqvt_app3[of substr]) apply (simp add: eqvt_def eqvts_raw) apply (simp_all add: fresh_Pair) doneend