--- a/Nominal/Ex/Let.thy	Tue Feb 19 05:38:46 2013 +0000
+++ b/Nominal/Ex/Let.thy	Tue Feb 19 06:58:14 2013 +0000
@@ -41,201 +41,5 @@
 thm trm_assn.strong_exhaust
 thm trm_assn.perm_bn_simps
 
-lemma alpha_bn_inducts_raw[consumes 1]:
-  "\<lbrakk>alpha_bn_raw a b; P3 ANil_raw ANil_raw;
- \<And>trm_raw trm_rawa assn_raw assn_rawa name namea.
-    \<lbrakk>alpha_trm_raw trm_raw trm_rawa; alpha_bn_raw assn_raw assn_rawa;
-     P3 assn_raw assn_rawa\<rbrakk>
-    \<Longrightarrow> P3 (ACons_raw name trm_raw assn_raw)
-        (ACons_raw namea trm_rawa assn_rawa)\<rbrakk> \<Longrightarrow> P3 a b"
-  by (erule alpha_trm_raw_alpha_assn_raw_alpha_bn_raw.inducts(3)[of _ _ "\<lambda>x y. True" _ "\<lambda>x y. True", simplified]) auto
-
-lemmas alpha_bn_inducts[consumes 1] = alpha_bn_inducts_raw[quot_lifted]
-
-
-
-lemma alpha_bn_refl: "alpha_bn x x"
-  by (induct x rule: trm_assn.inducts(2))
-     (rule TrueI, auto simp add: trm_assn.eq_iff)
-lemma alpha_bn_sym: "alpha_bn x y \<Longrightarrow> alpha_bn y x"
-  sorry
-lemma alpha_bn_trans: "alpha_bn x y \<Longrightarrow> alpha_bn y z \<Longrightarrow> alpha_bn x z"
-  sorry
-
-lemma bn_inj[rule_format]:
-  assumes a: "alpha_bn x y"
-  shows "bn x = bn y \<longrightarrow> x = y"
-  by (rule alpha_bn_inducts[OF a]) (simp_all add: trm_assn.bn_defs)
-
-lemma bn_inj2:
-  assumes a: "alpha_bn x y"
-  shows "\<And>q r. (q \<bullet> bn x) = (r \<bullet> bn y) \<Longrightarrow> permute_bn q x = permute_bn r y"
-using a
-apply(induct rule: alpha_bn_inducts)
-apply(simp add: trm_assn.perm_bn_simps)
-apply(simp add: trm_assn.perm_bn_simps)
-apply(simp add: trm_assn.bn_defs)
-done
-
-
-function
-  apply_assn :: "(trm \<Rightarrow> nat) \<Rightarrow> assn \<Rightarrow> nat"
-where
-  "apply_assn f ANil = (0 :: nat)"
-| "apply_assn f (ACons x t as) = max (f t) (apply_assn f as)"
-apply(case_tac x)
-apply(case_tac b rule: trm_assn.exhaust(2))
-apply(simp_all)
-apply(blast)
-done
-
-termination by lexicographic_order
-
-lemma [eqvt]:
-  "p \<bullet> (apply_assn f a) = apply_assn (p \<bullet> f) (p \<bullet> a)"
-  apply(induct f a rule: apply_assn.induct)
-  apply simp
-  apply(simp only: apply_assn.simps trm_assn.perm_simps)
-  apply(perm_simp)
-  apply(simp)
-  done
-
-lemma alpha_bn_apply_assn:
-  assumes "alpha_bn as bs"
-  shows "apply_assn f as = apply_assn f bs"
-  using assms
-  apply (induct rule: alpha_bn_inducts)
-  apply simp_all
-  done
-
-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 (Lam v b) = 1 + (height_trm b)"
-| "height_trm (Let as b) = max (apply_assn height_trm as) (height_trm b)"
-  apply (simp only: eqvt_def height_trm_graph_aux_def)
-  apply (rule, perm_simp, rule, rule TrueI)
-  apply (case_tac x rule: trm_assn.exhaust(1))
-  apply (auto)[4]
-  apply (drule_tac x="assn" in meta_spec)
-  apply (drule_tac x="trm" in meta_spec)
-  apply (simp add: alpha_bn_refl)
-  using [[simproc del: alpha_lst]]
-  apply(simp_all)
-  apply (erule_tac c="()" in Abs_lst1_fcb2)
-  apply (simp_all add: pure_fresh fresh_star_def eqvt_at_def)[4]
-  apply (erule conjE)
-  apply (subst alpha_bn_apply_assn)
-  apply assumption
-  apply (rule arg_cong) back
-  apply (erule_tac c="()" in Abs_lst_fcb2)
-  apply (simp_all add: pure_fresh fresh_star_def)[3]
-  apply (simp_all add: eqvt_at_def)[2]
-  done
-
-definition "height_assn = apply_assn height_trm"
-
-function
-  apply_assn2 :: "(trm \<Rightarrow> trm) \<Rightarrow> assn \<Rightarrow> assn"
-where
-  "apply_assn2 f ANil = ANil"
-| "apply_assn2 f (ACons x t as) = ACons x (f t) (apply_assn2 f as)"
-  apply(case_tac x)
-  apply(case_tac b rule: trm_assn.exhaust(2))
-  apply(simp_all)
-  apply(blast)
-  done
-
-termination by lexicographic_order
-
-lemma [eqvt]:
-  "p \<bullet> (apply_assn2 f a) = apply_assn2 (p \<bullet> f) (p \<bullet> a)"
-  apply(induct f a rule: apply_assn2.induct)
-  apply simp_all
-  done
-
-lemma bn_apply_assn2: "bn (apply_assn2 f as) = bn as"
-  apply (induct as rule: trm_assn.inducts(2))
-  apply (rule TrueI)
-  apply (simp_all add: trm_assn.bn_defs)
-  done
-
-nominal_primrec
-    subst  :: "name \<Rightarrow> trm \<Rightarrow> trm \<Rightarrow> trm"
-where
-  "subst s t (Var x) = (if (s = x) then t else (Var x))"
-| "subst s t (App l r) = App (subst s t l) (subst s t r)"
-| "atom v \<sharp> (s, t) \<Longrightarrow> subst s t (Lam v b) = Lam v (subst s t b)"
-| "set (bn as) \<sharp>* (s, t) \<Longrightarrow> subst s t (Let as b) = Let (apply_assn2 (subst s t) as) (subst s t b)"
-  apply (simp only: eqvt_def subst_graph_aux_def)
-  apply (rule, perm_simp, rule)
-  apply (rule TrueI)
-  apply (case_tac x)
-  apply (rule_tac y="c" and c="(a,b)" in trm_assn.strong_exhaust(1))
-  apply (auto simp add: fresh_star_def)[3]
-  apply (drule_tac x="assn" in meta_spec)
-  apply (simp add: Abs1_eq_iff alpha_bn_refl)
-  apply simp_all[7]
-  prefer 2
-  apply(simp)
-  using [[simproc del: alpha_lst]]
-  apply(simp)
-  apply(erule conjE)+
-  apply (erule_tac c="(sa, ta)" in Abs_lst1_fcb2)
-  apply (simp add: Abs_fresh_iff)
-  apply (simp add: fresh_star_def)
-  apply (simp_all add: fresh_star_Pair_elim perm_supp_eq eqvt_at_def)[2]
-  apply (simp add: bn_apply_assn2)
-  apply(erule conjE)+
-  apply(rule conjI)
-  apply (erule_tac c="(sa, ta)" in Abs_lst_fcb2)
-  apply (simp add: fresh_star_def Abs_fresh_iff)
-  apply assumption+
-  apply (simp_all add: fresh_star_Pair_elim perm_supp_eq eqvt_at_def trm_assn.fv_bn_eqvt)[2]
-  apply (erule alpha_bn_inducts)
-  apply simp_all
-  done
-
-lemma lets_bla:
-  "x \<noteq> z \<Longrightarrow> y \<noteq> z \<Longrightarrow> x \<noteq> y \<Longrightarrow>(Let (ACons x (Var y) ANil) (Var x)) \<noteq> (Let (ACons x (Var z) ANil) (Var x))"
-  by (simp add: trm_assn.eq_iff)
-
-lemma lets_ok:
-  "(Let (ACons x (Var y) ANil) (Var x)) = (Let (ACons y (Var y) ANil) (Var y))"
-  apply (simp add: trm_assn.eq_iff Abs_eq_iff )
-  apply (rule_tac x="(x \<leftrightarrow> y)" in exI)
-  apply (simp_all add: alphas atom_eqvt supp_at_base fresh_star_def trm_assn.bn_defs trm_assn.supp)
-  done
-
-lemma lets_ok3:
-  "x \<noteq> y \<Longrightarrow>
-   (Let (ACons x (App (Var y) (Var x)) (ACons y (Var y) ANil)) (App (Var x) (Var y))) \<noteq>
-   (Let (ACons y (App (Var x) (Var y)) (ACons x (Var x) ANil)) (App (Var x) (Var y)))"
-  apply (simp add: trm_assn.eq_iff)
-  done
-
-lemma lets_not_ok1:
-  "x \<noteq> y \<Longrightarrow>
-   (Let (ACons x (Var x) (ACons y (Var y) ANil)) (App (Var x) (Var y))) \<noteq>
-   (Let (ACons y (Var x) (ACons x (Var y) ANil)) (App (Var x) (Var y)))"
-  apply (simp add: alphas trm_assn.eq_iff trm_assn.supp fresh_star_def atom_eqvt Abs_eq_iff trm_assn.bn_defs)
-  done
-
-lemma lets_nok:
-  "x \<noteq> y \<Longrightarrow> x \<noteq> z \<Longrightarrow> z \<noteq> y \<Longrightarrow>
-   (Let (ACons x (App (Var z) (Var z)) (ACons y (Var z) ANil)) (App (Var x) (Var y))) \<noteq>
-   (Let (ACons y (Var z) (ACons x (App (Var z) (Var z)) ANil)) (App (Var x) (Var y)))"
-  apply (simp add: alphas trm_assn.eq_iff fresh_star_def trm_assn.bn_defs Abs_eq_iff trm_assn.supp trm_assn.distinct)
-  done
-
-lemma
-  fixes a b c :: name
-  assumes x: "a \<noteq> c" and y: "b \<noteq> c"
-  shows "\<exists>p.([atom a], Var c) \<approx>lst (op =) supp p ([atom b], Var c)"
-  apply (rule_tac x="(a \<leftrightarrow> b)" in exI)
-  apply (simp add: alphas trm_assn.supp supp_at_base x y fresh_star_def atom_eqvt)
-  by (metis Rep_name_inverse atom_name_def flip_fresh_fresh fresh_atom fresh_perm x y)
 
 end