
diff -r 1d9e50934bc5 -r aaef9dec5e1d Nominal/Ex/LetSimple1.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Nominal/Ex/LetSimple1.thy	Sat Jul 02 00:27:47 2011 +0100
@@ -0,0 +1,207 @@
+theory LetSimple1
+imports "../Nominal2" 
+begin
+
+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 Abs_lst1_fcb2:
+  fixes a b :: "atom"
+    and x y :: "'b :: fs"
+    and c::"'c :: fs"
+  assumes e: "(Abs_lst [a] x) = (Abs_lst [b] y)"
+  and fcb1: "a \<sharp> f a x c"
+  and fresh: "{a, b} \<sharp>* c"
+  and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f a x c) = f (p \<bullet> a) (p \<bullet> x) c"
+  and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f b y c) = f (p \<bullet> b) (p \<bullet> y) c"
+  shows "f a x c = f b y c"
+using e
+apply(drule_tac Abs_lst_fcb2[where c="c" and f="\<lambda>(as::atom list) . f (hd as)"])
+apply(simp_all)
+using fcb1 fresh perm1 perm2
+apply(simp_all add: fresh_star_def)
+done
+
+
+atom_decl name
+
+nominal_datatype trm =
+  Var "name"
+| App "trm" "trm"
+| Let x::"name" "trm" t::"trm"   bind x in t
+
+print_theorems
+
+thm trm.fv_defs
+thm trm.eq_iff 
+thm trm.bn_defs
+thm trm.bn_inducts
+thm trm.perm_simps
+thm trm.induct
+thm trm.inducts
+thm trm.distinct
+thm trm.supp
+thm trm.fresh
+thm trm.exhaust
+thm trm.strong_exhaust
+thm trm.perm_bn_simps
+
+nominal_primrec
+    height_trm :: "trm \<Rightarrow> nat"
+where
+  "height_trm (Var x) = 1"
+| "height_trm (App l r) = max (height_trm l) (height_trm r)"
+| "height_trm (Let x t s) = max (height_trm t) (height_trm s)"
+  apply (simp only: eqvt_def height_trm_graph_def)
+  apply (rule, perm_simp, rule, rule TrueI)
+  apply (case_tac x rule: trm.exhaust(1))
+  apply (auto)[3]
+  apply(simp_all)[5]
+  apply (subgoal_tac "height_trm_sumC t = height_trm_sumC ta")
+  apply (subgoal_tac "height_trm_sumC s = height_trm_sumC sa")
+  apply simp
+  apply(simp)
+  apply(erule conjE)
+  apply(erule_tac c="()" in Abs_lst1_fcb2)
+  apply(simp_all add: fresh_star_def pure_fresh)[2]
+  apply(simp_all add: eqvt_at_def)[2]
+  apply(simp)
+  done
+
+termination
+  by lexicographic_order
+
+
+nominal_primrec (invariant "\<lambda>x (y::atom set). finite y")
+  frees_set :: "trm \<Rightarrow> atom set"
+where
+  "frees_set (Var x) = {atom x}"
+| "frees_set (App t1 t2) = frees_set t1 \<union> frees_set t2"
+| "frees_set (Let x t s) = (frees_set s) - {atom x} \<union> (frees_set t)"
+  apply(simp add: eqvt_def frees_set_graph_def)
+  apply(rule, perm_simp, rule)
+  apply(erule frees_set_graph.induct)
+  apply(auto)[3]
+  apply(rule_tac y="x" in trm.exhaust)
+  apply(auto)[3]
+  apply(simp_all)[5]
+  apply(simp)
+  apply(erule conjE)
+  apply(subgoal_tac "frees_set_sumC s - {atom x} = frees_set_sumC sa - {atom xa}")
+  apply(simp)
+  apply(erule_tac c="()" in Abs_lst1_fcb2)
+  apply(simp add: fresh_minus_atom_set)
+  apply(simp add: fresh_star_def fresh_Unit)
+  apply(simp add: Diff_eqvt eqvt_at_def, perm_simp, rule refl)
+  apply(simp add: Diff_eqvt eqvt_at_def, perm_simp, rule refl)
+  done
+
+termination
+  by lexicographic_order
+
+
+nominal_primrec
+  subst :: "trm \<Rightarrow> name \<Rightarrow> trm \<Rightarrow> trm"  ("_ [_ ::= _]" [90, 90, 90] 90)
+where
+  "(Var x)[y ::= s] = (if x = y then s else (Var x))"
+| "(App t1 t2)[y ::= s] = App (t1[y ::= s]) (t2[y ::= s])"
+| "atom x \<sharp> (y, s) \<Longrightarrow> (Let x t t')[y ::= s] = Let x (t[y ::= s]) (t'[y ::= s])"
+  apply(simp add: eqvt_def subst_graph_def)
+  apply (rule, perm_simp, rule)
+  apply(rule TrueI)
+  apply(auto)[1]
+  apply(rule_tac y="a" and c="(aa, b)" in trm.strong_exhaust)
+  apply(blast)+
+  apply(simp_all add: fresh_star_def fresh_Pair_elim)[1]
+  apply(blast)
+  apply(simp_all)[5]
+  apply(simp (no_asm_use))
+  apply(simp)
+  apply(erule conjE)+
+  apply (erule_tac c="(ya,sa)" in Abs_lst1_fcb2)
+  apply(simp add: Abs_fresh_iff)
+  apply(simp add: fresh_star_def fresh_Pair)
+  apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)
+  apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)
+done
+
+termination
+  by lexicographic_order
+
+
+end
\ No newline at end of file




