diff -r fb201e383f1b -r da575186d492 Nominal/Ex/Lambda_F_T_FCB2.thy
--- a/Nominal/Ex/Lambda_F_T_FCB2.thy	Tue Feb 19 05:38:46 2013 +0000
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,270 +0,0 @@
-theory Lambda
-imports "../Nominal2" 
-begin
-
-atom_decl name
-
-nominal_datatype lam =
-  Var "name"
-| App "lam" "lam"
-| Lam x::"name" l::"lam"  bind x in l ("Lam [_]. _" [100, 100] 100)
-
-lemma fresh_fun_eqvt_app3:
-  assumes a: "eqvt f"
-  and b: "a \<sharp> x" "a \<sharp> y" "a \<sharp> z"
-  shows "a \<sharp> f x y z"
-  using fresh_fun_eqvt_app[OF a b(1)] a b
-  by (metis fresh_fun_app)
-
-lemma fresh_fun_eqvt_app4:
-  assumes a: "eqvt f"
-  and b: "a \<sharp> x" "a \<sharp> y" "a \<sharp> z" "a \<sharp> w"
-  shows "a \<sharp> f x y z w"
-  using fresh_fun_eqvt_app[OF a b(1)] a b
-  by (metis fresh_fun_app)
-
-lemma the_default_pty:
-  assumes f_def: "f == (\<lambda>x::'a. THE_default d (G x))"
-  and unique: "\<exists>!y. G x y"
-  and pty: "\<And>x y. G x y \<Longrightarrow> P x y"
-  shows "P x (f x)"
-  apply(subst f_def)
-  apply (rule ex1E[OF unique])
-  apply (subst THE_default1_equality[OF unique])
-  apply assumption
-  apply (rule pty)
-  apply assumption
-  done
-
-lemma Abs_lst1_fcb2:
-  fixes a b :: "'a :: at"
-    and S T :: "'b :: fs"
-    and c::"'c::fs"
-  assumes e: "(Abs_lst [atom a] T) = (Abs_lst [atom b] S)"
-  and fcb: "\<And>a T. atom a \<sharp> f a T c"
-  and fresh: "{atom a, atom b} \<sharp>* c"
-  and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f a T c) = f (p \<bullet> a) (p \<bullet> T) c"
-  and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f b S c) = f (p \<bullet> b) (p \<bullet> S) c"
-  shows "f a T c = f b S c"
-proof -
-  have fin1: "finite (supp (f a T c))"
-    apply(rule_tac S="supp (a, 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 b S c))"
-    apply(rule_tac S="supp (b, 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 d::"'a::at" where fr: "atom d \<sharp> (a, b, S, T, c, f a T c, f b S c)" 
-    using obtain_fresh'[where x="(a, b, S, T, c, f a T c, f b S c)"]
-    apply(auto simp add: finite_supp supp_Pair fin1 fin2)
-    done
-  have "(a \<leftrightarrow> d) \<bullet> (Abs_lst [atom a] T) = (b \<leftrightarrow> d) \<bullet> (Abs_lst [atom b] S)" 
-    apply(simp (no_asm_use) only: flip_def)
-    apply(subst swap_fresh_fresh)
-    apply(simp add: Abs_fresh_iff)
-    using fr
-    apply(simp add: Abs_fresh_iff)
-    apply(subst swap_fresh_fresh)
-    apply(simp add: Abs_fresh_iff)
-    using fr
-    apply(simp add: Abs_fresh_iff)
-    apply(rule e)
-    done
-  then have "Abs_lst [atom d] ((a \<leftrightarrow> d) \<bullet> T) = Abs_lst [atom d] ((b \<leftrightarrow> d) \<bullet> S)"
-    apply (simp add: swap_atom flip_def)
-    done
-  then have eq: "(a \<leftrightarrow> d) \<bullet> T = (b \<leftrightarrow> d) \<bullet> S"
-    by (simp add: Abs1_eq_iff)
-  have "f a T c = (a \<leftrightarrow> d) \<bullet> f a T c"
-    unfolding flip_def
-    apply(rule sym)
-    apply(rule swap_fresh_fresh)
-    using fcb[where a="a"] 
-    apply(simp)
-    using fr
-    apply(simp add: fresh_Pair)
-    done
-  also have "... = f d ((a \<leftrightarrow> d) \<bullet> T) c"
-    unfolding flip_def
-    apply(subst perm1)
-    using fresh fr
-    apply(simp add: supp_swap fresh_star_def fresh_Pair)
-    apply(simp)
-    done
-  also have "... = f d ((b \<leftrightarrow> d) \<bullet> S) c" using eq by simp
-  also have "... = (b \<leftrightarrow> d) \<bullet> f b S c"
-    unfolding flip_def
-    apply(subst perm2)
-    using fresh fr
-    apply(simp add: supp_swap fresh_star_def fresh_Pair)
-    apply(simp)
-    done
-  also have "... = f b S c"   
-    apply(rule flip_fresh_fresh)
-    using fcb[where a="b"] 
-    apply(simp)
-    using fr
-    apply(simp add: fresh_Pair)
-    done
-  finally show ?thesis by simp
-qed
-
-locale test =
-  fixes f1::"name \<Rightarrow> name list \<Rightarrow> ('a::pt)"
-    and f2::"lam \<Rightarrow> lam \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> name list \<Rightarrow> ('a::pt)"
-    and f3::"name \<Rightarrow> lam \<Rightarrow> 'a \<Rightarrow> name list \<Rightarrow> ('a::pt)"
-  assumes fs: "finite (supp (f1, f2, f3))"
-    and eq: "eqvt f1" "eqvt f2" "eqvt f3"
-    and fcb1: "\<And>l n. atom ` set l \<sharp>* f1 n l"
-    and fcb2: "\<And>l t1 t2 r1 r2. atom ` set l \<sharp>* (r1, r2) \<Longrightarrow> atom ` set l \<sharp>* f2 t1 t2 r1 r2 l"
-    and fcb3: "\<And>t l r. atom ` set (x # l) \<sharp>* r \<Longrightarrow> atom ` set (x # l) \<sharp>* f3 x t r l"
-begin
-
-nominal_primrec (invariant "\<lambda>(x, l) y. atom ` set l \<sharp>* y")
-  f
-where
-  "f (Var x) l = f1 x l"
-| "f (App t1 t2) l = f2 t1 t2 (f t1 l) (f t2 l) l"
-| "atom x \<sharp> l \<Longrightarrow> (f (Lam [x].t) l) = f3 x t (f t (x # l)) l"
-  apply (simp add: eqvt_def f_graph_def)
-  apply (rule, perm_simp)
-  apply (simp only: eq[unfolded eqvt_def])
-  apply (erule f_graph.induct)
-  apply (simp add: fcb1)
-  apply (simp add: fcb2 fresh_star_Pair)
-  apply simp
-  apply (subgoal_tac "atom ` set (xa # l) \<sharp>* f3 xa t (f_sum (t, xa # l)) l")
-  apply (simp add: fresh_star_def)
-  apply (rule fcb3)
-  apply (simp add: fresh_star_def fresh_def)
-  apply simp_all
-  apply(case_tac x)
-  apply(rule_tac y="a" and c="b" in lam.strong_exhaust)
-  apply(auto simp add: fresh_star_def)
-  apply(erule_tac Abs_lst1_fcb2)
---"?"
-  apply (subgoal_tac "atom ` set (a # la) \<sharp>* f3 a T (f_sumC (T, a # la)) la")
-  apply (simp add: fresh_star_def)
-  apply (rule fcb3)
-  apply (simp add: fresh_star_def)
-  apply (rule fresh_fun_eqvt_app4[OF eq(3)])
-  apply (simp add: fresh_at_base)
-  apply assumption
-  apply (erule fresh_eqvt_at)
-  apply (simp add: finite_supp)
-  apply (simp add: fresh_Pair fresh_Cons fresh_at_base)
-  apply assumption
-  apply (subgoal_tac "\<And>p y r l. p \<bullet> (f3 x y r l) = f3 (p \<bullet> x) (p \<bullet> y) (p \<bullet> r) (p \<bullet> l)")
-  apply (subgoal_tac "(atom x \<rightleftharpoons> atom xa) \<bullet> la = la")
-  apply (simp add: eqvt_at_def)
-  apply (simp add: swap_fresh_fresh)
-  apply (simp add: permute_fun_app_eq eq[unfolded eqvt_def])
-  apply simp
-  done
-
-termination
-  by (relation "measure (\<lambda>(x,_). size x)") (auto simp add: lam.size)
-
-end
-
-section {* Locally Nameless Terms *}
-
-nominal_datatype ln = 
-  LNBnd nat
-| LNVar name
-| LNApp ln ln
-| LNLam ln
-
-fun
-  lookup :: "name list \<Rightarrow> nat \<Rightarrow> name \<Rightarrow> ln" 
-where
-  "lookup [] n x = LNVar x"
-| "lookup (y # ys) n x = (if x = y then LNBnd n else (lookup ys (n + 1) x))"
-
-lemma lookup_eqvt[eqvt]:
-  shows "(p \<bullet> lookup xs n x) = lookup (p \<bullet> xs) (p \<bullet> n) (p \<bullet> x)"
-  by (induct xs arbitrary: n) (simp_all add: permute_pure)
-
-lemma fresh_at_list: "atom x \<sharp> xs \<longleftrightarrow> x \<notin> set xs"
-  unfolding fresh_def supp_set[symmetric]
-  apply (induct xs)
-  apply (simp add: supp_set_empty)
-  apply simp
-  apply auto
-  apply (simp_all add: insert_absorb UnI2 finite_set supp_of_finite_insert supp_at_base)
-  done
-
-interpretation trans: test
-  "%x l. lookup l 0 x"
-  "%t1 t2 r1 r2 l. LNApp r1 r2"
-  "%n t r l. LNLam r"
-  apply default
-  apply (auto simp add: pure_fresh supp_Pair)
-  apply (simp_all add: fresh_def supp_def permute_fun_def permute_pure lookup_eqvt)[3]
-  apply (simp_all add: eqvt_def permute_fun_def permute_pure lookup_eqvt)
-  apply (simp add: fresh_star_def)
-  apply (rule_tac x="0 :: nat" in spec)
-  apply (induct_tac l)
-  apply (simp add: ln.fresh pure_fresh)
-  apply (auto simp add: ln.fresh pure_fresh)[1]
-  apply (case_tac "a \<in> set list")
-  apply simp
-  apply (rule_tac f="lookup" in fresh_fun_eqvt_app3)
-  unfolding eqvt_def
-  apply rule
-  using eqvts_raw(35)
-  apply auto[1]
-  apply (simp add: fresh_at_list)
-  apply (simp add: pure_fresh)
-  apply (simp add: fresh_at_base)
-  apply (simp add: fresh_star_Pair fresh_star_def ln.fresh)
-  apply (simp add: fresh_star_def ln.fresh)
-  done
-
-thm trans.f.simps
-
-lemma lam_strong_exhaust2:
-  "\<lbrakk>\<And>name. y = Var name \<Longrightarrow> P; 
-    \<And>lam1 lam2. y = App lam1 lam2 \<Longrightarrow> P;
-    \<And>name lam. \<lbrakk>{atom name} \<sharp>* c; y = Lam [name]. lam\<rbrakk> \<Longrightarrow> P;
-    finite (supp c)\<rbrakk>
-    \<Longrightarrow> P"
-sorry
-
-nominal_primrec
-  g
-where
-  "(~finite (supp (g1, g2, g3))) \<Longrightarrow> g g1 g2 g3 t = t"
-| "finite (supp (g1, g2, g3)) \<Longrightarrow> g g1 g2 g3 (Var x) = g1 x"
-| "finite (supp (g1, g2, g3)) \<Longrightarrow> g g1 g2 g3 (App t1 t2) = g2 t1 t2 (g g1 g2 g3 t1) (g g1 g2 g3 t2)"
-| "finite (supp (g1, g2, g3)) \<Longrightarrow> atom x \<sharp> (g1,g2,g3) \<Longrightarrow> (g g1 g2 g3 (Lam [x].t)) = g3 x t (g g1 g2 g3 t)"
-  apply (simp add: eqvt_def g_graph_def)
-  apply (rule, perm_simp, rule)
-  apply simp_all
-  apply (case_tac x)
-  apply (case_tac "finite (supp (a, b, c))")
-  prefer 2
-  apply simp
-  apply(rule_tac y="d" and c="(a,b,c)" in lam_strong_exhaust2)
-  apply auto
-  apply blast
-  apply (simp add: Abs1_eq_iff fresh_star_def)
-  sorry
-
-termination
-  by (relation "measure (\<lambda>(_,_,_,t). size t)") (simp_all add: lam.size)
-
-end