--- a/Nominal/Ex/Let.thy	Sun Jul 17 11:33:09 2011 +0100
+++ b/Nominal/Ex/Let.thy	Mon Jul 18 00:21:51 2011 +0100
@@ -79,9 +79,6 @@
 done
 
 
-lemma max_eqvt[eqvt]: "p \<bullet> (max (a :: _ :: pure) b) = max (p \<bullet> a) (p \<bullet> b)"
-  by (simp add: permute_pure)
-
 function
   apply_assn :: "(trm \<Rightarrow> nat) \<Rightarrow> assn \<Rightarrow> nat"
 where

--- a/Nominal/Ex/LetRec.thy	Sun Jul 17 11:33:09 2011 +0100
+++ b/Nominal/Ex/LetRec.thy	Mon Jul 18 00:21:51 2011 +0100
@@ -27,8 +27,6 @@
 thm let_rec.fv_bn_eqvt
 thm let_rec.size_eqvt
 
-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"
@@ -56,8 +54,9 @@
   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)
+  apply(simp add: eqvt_at_def)
+  apply(perm_simp)
+  apply (simp add: permute_fun_def)
   done
 
 termination by lexicographic_order

--- a/Nominal/Ex/LetRecB.thy	Sun Jul 17 11:33:09 2011 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,196 +0,0 @@
-theory LetRecB
-imports "../Nominal2"
-begin
-
-atom_decl name
-
-nominal_datatype let_rec:
- trm =
-  Var "name"
-| App "trm" "trm"
-| Lam x::"name" t::"trm"     binds x in t
-| Let_Rec bp::"bp" t::"trm"  binds "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 :: "'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 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)
-  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)
-  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="()" in Abs_lst_fcb2)
-  apply (simp add: fresh_star_def pure_fresh)
-  apply (simp add: fresh_star_def pure_fresh)
-  apply (simp add: fresh_star_def pure_fresh)
-  apply (simp add: eqvt_at_def)
-  apply (simp add: eqvt_at_def)
-  defer defer
-  apply (subgoal_tac "(\<lambda>as x c. height_bp (fst (bp, b))) as x c = (\<lambda>as x c. height_bp (fst (bpa, ba))) as x c")
-  apply simp
-  apply (erule_tac c="()" in Abs_lst_fcb2)
-  apply (simp add: fresh_star_def pure_fresh)
-  apply (simp add: fresh_star_def pure_fresh)
-  apply (simp add: fresh_star_def pure_fresh)
-  apply (simp add: fresh_star_def pure_fresh)
-  apply (simp add: eqvt_at_def)
-  apply (simp add: eqvt_at_def)
---""
-  apply(simp_all add: eqvt_def inj_on_def)
-  apply(perm_simp)
-  apply(simp)
-  apply(perm_simp)
-  apply(simp)
-  done
-
-termination by lexicographic_order
-
-end
-
-
-

--- a/Nominal/Ex/LetSimple2.thy	Sun Jul 17 11:33:09 2011 +0100
+++ b/Nominal/Ex/LetSimple2.thy	Mon Jul 18 00:21:51 2011 +0100
@@ -62,8 +62,6 @@
 
 lemma k: "A \<Longrightarrow> A \<and> A" by blast
 
-lemma max_eqvt[eqvt]: "p \<bullet> (max (a :: _ :: pure) b) = max (p \<bullet> a) (p \<bullet> b)"
-  by (simp add: permute_pure)
 
 
 section {* definition with helper functions *}
@@ -261,6 +259,7 @@
   apply(simp add: eqvt_at_def perm_supp_eq)
   done
 
+termination by lexicographic_order
 
 lemma ww1:
   shows "finite (fv_trm t)"
@@ -324,7 +323,7 @@
 where
   "(Var x)[y ::trm2= s] = (if x = y then s else (Var x))"
 | "(App t1 t2)[y ::trm2= s] = App (t1[y ::trm2= s]) (t2[y ::trm2= s])"
-| "(set (bn as)) \<sharp>* (y, s) \<Longrightarrow> (Let as t)[y ::trm2= s] = Let (ast[y ::assn2= s]) (t[y ::trm2= s])"
+| "(set (bn as)) \<sharp>* (y, s, fv_bn as) \<Longrightarrow> (Let as t)[y ::trm2= s] = Let (ast[y ::assn2= s]) (t[y ::trm2= s])"
 | "(Assn x t)[y ::assn2= s] = Assn x (t[y ::trm2= s])"
 apply(subgoal_tac "\<And>p x r. subst_trm2_subst_assn2_graph x r \<Longrightarrow> subst_trm2_subst_assn2_graph (p \<bullet> x) (p \<bullet> r)")
 apply(simp add: eqvt_def)
@@ -348,9 +347,23 @@
   apply(simp)
   apply(case_tac a)
   apply(simp)
-  apply(rule_tac y="aa" and c="(b, c)" in trm_assn.strong_exhaust(1))
+  apply(rule_tac y="aa" and c="(b, c, aa)" in trm_assn.strong_exhaust(1))
   apply(blast)+
   apply(simp)
+  apply(drule_tac x="assn" in meta_spec)
+  apply(drule_tac x="b" in meta_spec)
+  apply(drule_tac x="c" in meta_spec)
+  apply(drule_tac x="trm" in meta_spec)
+  apply(simp add: trm_assn.alpha_refl)
+  apply(rotate_tac 5)
+  apply(drule meta_mp)
+  apply(simp add: fresh_star_Pair)
+  apply(simp add: fresh_star_def trm_assn.fresh)
+  apply(simp add: fresh_def)
+  apply(subst supp_finite_atom_set)
+  apply(simp)
+  apply(simp)
+  apply(simp)
   apply(case_tac b)
   apply(simp)
   apply(rule_tac y="a" in trm_assn.exhaust(2))
@@ -362,7 +375,6 @@
   apply(simp)
   prefer 2
   apply(simp)
-  thm Inl_inject
   apply(drule Inl_inject)
   apply(rule arg_cong)
   back
@@ -376,39 +388,20 @@
   apply (thin_tac "eqvt_at subst_trm2_subst_assn2_sumC (Inr (asta, ya, sa))")
   apply (thin_tac "eqvt_at subst_trm2_subst_assn2_sumC (Inl (t, y, s))")
   apply (thin_tac "eqvt_at subst_trm2_subst_assn2_sumC (Inl (ta, ya, sa))")
-  defer
-  apply(simp add: Abs_eq_iff alphas)
-  apply(clarify)
-  apply(rule eqvt_at_perm)
   apply(simp)
-  apply(simp add: subst_trm2_def)
-  apply(simp add: eqvt_at_def)
-  defer
-  defer
-  defer
-  defer
-  defer
-  apply(rule conjI)
-  apply (subgoal_tac "subst_assn2 ast ya sa= subst_assn2 asta ya sa")
-  apply (subgoal_tac "subst_trm2 t ya sa = subst_trm2 ta ya sa")
+  (* HERE *)
+  apply (subgoal_tac "subst_assn2 ast y s= subst_assn2 asta ya sa")
+  apply (subgoal_tac "subst_trm2 t y s = subst_trm2 ta ya sa")
+  apply(simp)
   apply(simp)
   apply(erule_tac conjE)+
   apply(erule alpha_bn_cases)
   apply(simp add: trm_assn.bn_defs)
   apply(rotate_tac 7)
-  apply(drule k)
-  apply(erule conjE)
-  apply(subst (asm) Abs1_eq_iff)
-  apply(rule sort_of_atom_eq)
-  apply(rule sort_of_atom_eq)
-  apply(erule disjE)
-  apply(simp)
-  apply(rotate_tac 12)
-  apply(drule sym)
-  apply(rule sym)
   apply (erule_tac c="(ya,sa)" in Abs_lst1_fcb2)
   apply(erule fresh_eqvt_at)
   
+  
   thm fresh_eqvt_at
   apply(simp add: Abs_fresh_iff)
   apply(simp add: fresh_star_def fresh_Pair)