diff -r db0786a521fd -r b4bf3ff4bc91 Nominal/Ex/LetRecB.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Nominal/Ex/LetRecB.thy	Tue Jun 28 14:01:52 2011 +0100
@@ -0,0 +1,160 @@
+theory LetRecB
+imports "../Nominal2"
+begin
+
+atom_decl name
+
+nominal_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 t
+and bp =
+  Bp "name" "trm"
+binder
+  bn::"bp \<Rightarrow> atom list"
+where
+  "bn (Bp x t) = [atom x]"
+
+thm let_rec.distinct
+thm let_rec.induct
+thm let_rec.exhaust
+thm let_rec.fv_defs
+thm let_rec.bn_defs
+thm let_rec.perm_simps
+thm let_rec.eq_iff
+thm let_rec.fv_bn_eqvt
+thm let_rec.size_eqvt
+
+
+lemma 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 simp
+qed
+
+
+lemma 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)
+  --"HERE"
+  thm  Abs_lst_fcb2
+  apply(rule Abs_lst_fcb2)
+     --" does not fit the assumption "
+
+  apply (drule_tac c="()" in Abs_lst_fcb2)
+  prefer 6
+  apply(assumption)
+  apply (drule_tac c="()" in Abs_lst_fcb2)
+  apply (simp add: Abs_eq_iff2)
+  apply (simp add: alphas)
+  apply clarify
+  apply (rule trans)
+  apply(rule_tac p="p" in supp_perm_eq[symmetric])
+  apply (simp add: pure_supp fresh_star_def)
+  apply (simp only: eqvts)
+  apply (simp add: eqvt_at_def)
+  done
+
+termination by lexicographic_order
+
+end
+
+
+