--- a/Nominal/Ex/Classical_Test.thy	Tue Jun 28 12:36:34 2011 +0900
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,115 +0,0 @@
-theory Classical
-imports "../Nominal2"
-begin
-
-lemma supp_zero_perm_zero:
-  shows "supp (p :: perm) = {} \<longleftrightarrow> p = 0"
-  by (metis supp_perm_singleton supp_zero_perm)
-
-lemma permute_atom_list_id:
-  shows "p \<bullet> l = l \<longleftrightarrow> supp p \<inter> set l = {}"
-  by (induct l) (auto simp add: supp_Nil supp_perm)
-
-lemma permute_length_eq:
-  shows "p \<bullet> xs = ys \<Longrightarrow> length xs = length ys"
-  by (auto simp add: length_eqvt[symmetric] permute_pure)
-
-lemma Abs_lst_binder_length:
-  shows "[xs]lst. T = [ys]lst. S \<Longrightarrow> length xs = length ys"
-  by (auto simp add: Abs_eq_iff alphas length_eqvt[symmetric] permute_pure)
-
-lemma Abs_lst_binder_eq:
-  shows "Abs_lst l T = Abs_lst l S \<longleftrightarrow> T = S"
-  by (rule, simp_all add: Abs_eq_iff2 alphas)
-     (metis fresh_star_zero inf_absorb1 permute_atom_list_id supp_perm_eq
-       supp_zero_perm_zero)
-
-lemma in_permute_list:
-  shows "py \<bullet> p \<bullet> xs = px \<bullet> xs \<Longrightarrow>  x \<in> set xs \<Longrightarrow> py \<bullet> p \<bullet> x = px \<bullet> x"
-  by (induct xs) auto
-
-lemma obtain_atom_list:
-  assumes eq: "p \<bullet> xs = ys"
-      and fin: "finite (supp c)"
-      and sorts: "map sort_of xs = map sort_of ys"
-  shows "\<exists>ds px py. (set ds \<sharp>* c) \<and> (px \<bullet> xs = ds) \<and> (py \<bullet> ys = ds)
-    \<and> (supp px - set xs) \<sharp>* c \<and> (supp py - set ys) \<sharp>* c"
-  sorry
-
-lemma Abs_lst_fcb2:
-  fixes S T :: "'b :: fs"
-    and c::"'c::fs"
-  assumes e: "[xs]lst. T = [ys]lst. S"
-  and sorts: "map sort_of xs = map sort_of ys"
-  and fcb1: "\<And>x. x \<in> set xs \<Longrightarrow> x \<sharp> f xs T c"
-  and fcb2: "\<And>x. x \<in> set ys \<Longrightarrow> x \<sharp> f ys S c"
-  and fresh: "(set xs \<union> set ys) \<sharp>* c"
-  and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f xs T c) = f (p \<bullet> xs) (p \<bullet> T) c"
-  and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f ys S c) = f (p \<bullet> ys) (p \<bullet> S) c"
-  shows "f xs T c = f ys S c"
-proof -
-  have fin1: "finite (supp (f xs T c))"
-    apply(rule_tac S="supp (xs, T, c)" in supports_finite)
-    apply(simp add: supports_def)
-    apply(simp add: fresh_def[symmetric])
-    apply(clarify)
-    apply(subst perm1)
-    apply(simp add: supp_swap fresh_star_def)
-    apply(simp add: swap_fresh_fresh fresh_Pair)
-    apply(simp add: finite_supp)
-    done
-  have fin2: "finite (supp (f ys S c))"
-    apply(rule_tac S="supp (ys, S, c)" in supports_finite)
-    apply(simp add: supports_def)
-    apply(simp add: fresh_def[symmetric])
-    apply(clarify)
-    apply(subst perm2)
-    apply(simp add: supp_swap fresh_star_def)
-    apply(simp add: swap_fresh_fresh fresh_Pair)
-    apply(simp add: finite_supp)
-    done
-  obtain p :: perm where xs_ys: "p \<bullet> xs = ys" using e
-    by (auto simp add: Abs_eq_iff alphas)
-  obtain ds::"atom list" and px and py
-    where fr: "set ds \<sharp>* (xs, ys, S, T, c, f xs T c, f ys S c)"
-    and pxd: "px \<bullet> xs = ds"     and pyd: "py \<bullet> ys = ds"
-    and spx: "(supp px - set xs) \<sharp>* (xs, ys, S, T, c, f xs T c, f ys S c)"
-    and spy: "(supp py - set ys) \<sharp>* (xs, ys, S, T, c, f xs T c, f ys S c)"
-    using obtain_atom_list[OF xs_ys, of "(xs, ys, S, T, c, f xs T c, f ys S c)"]
-    sorts by (auto simp add: finite_supp supp_Pair fin1 fin2)
-  have "px \<bullet> (Abs_lst xs T) = py \<bullet> (Abs_lst ys S)"
-    apply (subst perm_supp_eq)
-    using spx apply (auto simp add: fresh_star_def Abs_fresh_iff)[1]
-    apply (subst perm_supp_eq)
-    using spy apply (auto simp add: fresh_star_def Abs_fresh_iff)[1]
-    by(rule e)
-  then have "Abs_lst ds (px \<bullet> T) = Abs_lst ds (py \<bullet> S)" by (simp add: pxd pyd)
-  then have eq: "px \<bullet> T = py \<bullet> S" by (simp add: Abs_lst_binder_eq)
-  have "f xs T c = px \<bullet> f xs T c"
-    apply(rule perm_supp_eq[symmetric])
-    using spx unfolding fresh_star_def
-    apply (intro ballI)
-    by (case_tac "a \<in> set xs") (simp_all add: fcb1)
-  also have "... = f (px \<bullet> xs) (px \<bullet> T) c"
-    apply(rule perm1)
-    using spx fresh unfolding fresh_star_def
-    apply (intro ballI)
-    by (case_tac "a \<in> set xs") (simp_all add: fcb1)
-  also have "... = f (py \<bullet> ys) (py \<bullet> S) c" using eq pxd pyd by simp
-  also have "... = py \<bullet> f ys S c"
-    apply(rule perm2[symmetric])
-    using spy fresh unfolding fresh_star_def
-    apply (intro ballI)
-    by (case_tac "a \<in> set ys") (simp_all add: fcb1)
-  also have "... = f ys S c"
-    apply(rule perm_supp_eq)
-    using spy unfolding fresh_star_def
-    apply (intro ballI)
-    by (case_tac "a \<in> set ys") (simp_all add: fcb2)
-  finally show ?thesis by simp
-qed
-
-end
-
-
-

--- /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
+
+
+