--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/ex/NBE.thy Tue Jun 26 13:02:28 2007 +0200
@@ -0,0 +1,865 @@
+(* ID: $Id$
+ Author: Klaus Aehlig, Tobias Nipkow
+ Work in progress
+*)
+
+theory NBE imports Main ExecutableSet begin
+
+ML"set quick_and_dirty"
+
+declare Let_def[simp]
+
+consts_code undefined ("(raise Match)")
+
+(*typedecl const_name*)
+types lam_var_name = nat
+ ml_var_name = nat
+ const_name = nat
+
+datatype tm = Ct const_name | Vt lam_var_name | Lam tm | At tm tm
+ | term_of ml (* function 'to_term' *)
+and ml = (* rep of universal datatype *)
+ C const_name "ml list" | V lam_var_name "ml list"
+ | Fun ml "ml list" nat
+ | "apply" ml ml (* function 'apply' *)
+ (* ML *)
+ | V_ML ml_var_name | A_ML ml "ml list" | Lam_ML ml
+ | CC const_name (* ref to compiled code *)
+
+lemma [simp]: "x \<in> set vs \<Longrightarrow> size x < Suc (ml_list_size1 vs)"
+by (induct vs) auto
+lemma [simp]: "x \<in> set vs \<Longrightarrow> size x < Suc (ml_list_size2 vs)"
+by (induct vs) auto
+lemma [simp]:"x \<in> set vs \<Longrightarrow> size x < Suc (size v + ml_list_size3 vs)"
+by (induct vs) auto
+lemma [simp]: "x \<in> set vs \<Longrightarrow> size x < Suc (size v + ml_list_size4 vs)"
+by (induct vs) auto
+
+locale Vars =
+ fixes r s t:: tm
+ and rs ss ts :: "tm list"
+ and u v w :: ml
+ and us vs ws :: "ml list"
+ and nm :: const_name
+ and x :: lam_var_name
+ and X :: ml_var_name
+
+consts Pure_tms :: "tm set"
+inductive Pure_tms
+intros
+"Ct s : Pure_tms"
+"Vt x : Pure_tms"
+"t : Pure_tms ==> Lam t : Pure_tms"
+"s : Pure_tms ==> t : Pure_tms ==> At s t : Pure_tms"
+
+consts
+ R :: "(const_name * tm list * tm)set" (* reduction rules *)
+ compR :: "(const_name * ml list * ml)set" (* compiled reduction rules *)
+ tRed :: "(tm * tm)set" (* beta + R reduction on pure terms *)
+ tRed_list :: "(tm list * tm list) set"
+
+fun
+ lift_tm :: "nat \<Rightarrow> tm \<Rightarrow> tm" ("lift") and
+ lift_ml :: "nat \<Rightarrow> ml \<Rightarrow> ml" ("lift")
+where
+"lift i (Ct nm) = Ct nm" |
+"lift i (Vt x) = Vt(if x < i then x else x+1)" |
+"lift i (Lam t) = Lam (lift (i+1) t)" |
+"lift i (At s t) = At (lift i s) (lift i t)" |
+"lift i (term_of v) = term_of (lift i v)" |
+
+"lift i (C nm vs) = C nm (map (lift i) vs)" |
+"lift i (V x vs) = V (if x < i then x else x+1) (map (lift i) vs)" |
+"lift i (Fun v vs n) = Fun (lift i v) (map (lift i) vs) n" |
+"lift i (apply u v) = apply (lift i u) (lift i v)" |
+"lift i (V_ML X) = V_ML X" |
+"lift i (A_ML v vs) = A_ML (lift i v) (map (lift i) vs)" |
+"lift i (Lam_ML v) = Lam_ML (lift i v)" |
+"lift i (CC nm) = CC nm"
+(*
+termination
+apply (relation "measure (sum_case (%(i,t). size t) (%(i,v). size v))")
+apply auto
+*)
+
+fun
+ lift_tm_ML :: "nat \<Rightarrow> tm \<Rightarrow> tm" ("lift\<^bsub>ML\<^esub>") and
+ lift_ml_ML :: "nat \<Rightarrow> ml \<Rightarrow> ml" ("lift\<^bsub>ML\<^esub>")
+where
+"lift\<^bsub>ML\<^esub> i (Ct nm) = Ct nm" |
+"lift\<^bsub>ML\<^esub> i (Vt x) = Vt x" |
+"lift\<^bsub>ML\<^esub> i (Lam t) = Lam (lift\<^bsub>ML\<^esub> i t)" |
+"lift\<^bsub>ML\<^esub> i (At s t) = At (lift\<^bsub>ML\<^esub> i s) (lift\<^bsub>ML\<^esub> i t)" |
+"lift\<^bsub>ML\<^esub> i (term_of v) = term_of (lift\<^bsub>ML\<^esub> i v)" |
+
+"lift\<^bsub>ML\<^esub> i (C nm vs) = C nm (map (lift\<^bsub>ML\<^esub> i) vs)" |
+"lift\<^bsub>ML\<^esub> i (V x vs) = V x (map (lift\<^bsub>ML\<^esub> i) vs)" |
+"lift\<^bsub>ML\<^esub> i (Fun v vs n) = Fun (lift\<^bsub>ML\<^esub> i v) (map (lift\<^bsub>ML\<^esub> i) vs) n" |
+"lift\<^bsub>ML\<^esub> i (apply u v) = apply (lift\<^bsub>ML\<^esub> i u) (lift\<^bsub>ML\<^esub> i v)" |
+"lift\<^bsub>ML\<^esub> i (V_ML X) = V_ML (if X < i then X else X+1)" |
+"lift\<^bsub>ML\<^esub> i (A_ML v vs) = A_ML (lift\<^bsub>ML\<^esub> i v) (map (lift\<^bsub>ML\<^esub> i) vs)" |
+"lift\<^bsub>ML\<^esub> i (Lam_ML v) = Lam_ML (lift\<^bsub>ML\<^esub> (i+1) v)" |
+"lift\<^bsub>ML\<^esub> i (CC nm) = CC nm"
+(*
+termination
+ by (relation "measure (sum_case (%(i,t). size t) (%(i,v). size v))") auto
+*)
+constdefs
+ cons :: "tm \<Rightarrow> (nat \<Rightarrow> tm) \<Rightarrow> (nat \<Rightarrow> tm)" (infix "##" 65)
+"t##f \<equiv> \<lambda>i. case i of 0 \<Rightarrow> t | Suc j \<Rightarrow> lift 0 (f j)"
+ cons_ML :: "ml \<Rightarrow> (nat \<Rightarrow> ml) \<Rightarrow> (nat \<Rightarrow> ml)" (infix "##" 65)
+"v##f \<equiv> \<lambda>i. case i of 0 \<Rightarrow> v::ml | Suc j \<Rightarrow> lift\<^bsub>ML\<^esub> 0 (f j)"
+
+(* Only for pure terms! *)
+consts subst :: "(nat \<Rightarrow> tm) \<Rightarrow> tm \<Rightarrow> tm"
+primrec
+"subst f (Ct nm) = Ct nm"
+"subst f (Vt x) = f x"
+"subst f (Lam t) = Lam (subst (Vt 0 ## f) t)"
+"subst f (At s t) = At (subst f s) (subst f t)"
+
+lemma size_lift[simp]: shows
+ "size(lift i t) = size(t::tm)" and "size(lift i (v::ml)) = size v"
+and "ml_list_size1 (map (lift i) vs) = ml_list_size1 vs"
+and "ml_list_size2 (map (lift i) vs) = ml_list_size2 vs"
+and "ml_list_size3 (map (lift i) vs) = ml_list_size3 vs"
+and "ml_list_size4 (map (lift i) vs) = ml_list_size4 vs"
+by (induct arbitrary: i and i and i and i and i and i rule: tm_ml.inducts)
+ simp_all
+
+lemma size_lift_ML[simp]: shows
+ "size(lift\<^bsub>ML\<^esub> i t) = size(t::tm)" and "size(lift\<^bsub>ML\<^esub> i (v::ml)) = size v"
+and "ml_list_size1 (map (lift\<^bsub>ML\<^esub> i) vs) = ml_list_size1 vs"
+and "ml_list_size2 (map (lift\<^bsub>ML\<^esub> i) vs) = ml_list_size2 vs"
+and "ml_list_size3 (map (lift\<^bsub>ML\<^esub> i) vs) = ml_list_size3 vs"
+and "ml_list_size4 (map (lift\<^bsub>ML\<^esub> i) vs) = ml_list_size4 vs"
+by (induct arbitrary: i and i and i and i and i and i rule: tm_ml.inducts)
+ simp_all
+
+fun
+ subst_ml_ML :: "(nat \<Rightarrow> ml) \<Rightarrow> ml \<Rightarrow> ml" ("subst\<^bsub>ML\<^esub>") and
+ subst_tm_ML :: "(nat \<Rightarrow> ml) \<Rightarrow> tm \<Rightarrow> tm" ("subst\<^bsub>ML\<^esub>")
+where
+"subst\<^bsub>ML\<^esub> f (Ct nm) = Ct nm" |
+"subst\<^bsub>ML\<^esub> f (Vt x) = Vt x" |
+"subst\<^bsub>ML\<^esub> f (Lam t) = Lam (subst\<^bsub>ML\<^esub> (lift 0 o f) t)" |
+"subst\<^bsub>ML\<^esub> f (At s t) = At (subst\<^bsub>ML\<^esub> f s) (subst\<^bsub>ML\<^esub> f t)" |
+"subst\<^bsub>ML\<^esub> f (term_of v) = term_of (subst\<^bsub>ML\<^esub> f v)" |
+
+"subst\<^bsub>ML\<^esub> f (C nm vs) = C nm (map (subst\<^bsub>ML\<^esub> f) vs)" |
+"subst\<^bsub>ML\<^esub> f (V x vs) = V x (map (subst\<^bsub>ML\<^esub> f) vs)" |
+"subst\<^bsub>ML\<^esub> f (Fun v vs n) = Fun (subst\<^bsub>ML\<^esub> f v) (map (subst\<^bsub>ML\<^esub> f) vs) n" |
+"subst\<^bsub>ML\<^esub> f (apply u v) = apply (subst\<^bsub>ML\<^esub> f u) (subst\<^bsub>ML\<^esub> f v)" |
+"subst\<^bsub>ML\<^esub> f (V_ML X) = f X" |
+"subst\<^bsub>ML\<^esub> f (A_ML v vs) = A_ML (subst\<^bsub>ML\<^esub> f v) (map (subst\<^bsub>ML\<^esub> f) vs)" |
+"subst\<^bsub>ML\<^esub> f (Lam_ML v) = Lam_ML (subst\<^bsub>ML\<^esub> (V_ML 0 ## f) v)" |
+"subst\<^bsub>ML\<^esub> f (CC nm) = CC nm"
+
+(* FIXME currrently needed for code generator *)
+lemmas [code] = lift_tm_ML.simps lift_ml_ML.simps
+lemmas [code] = lift_tm.simps lift_ml.simps
+lemmas [code] = subst_tm_ML.simps subst_ml_ML.simps
+
+abbreviation
+ subst_decr :: "nat \<Rightarrow> tm \<Rightarrow> nat \<Rightarrow> tm" where
+ "subst_decr k t == %n. if n<k then Vt n else if n=k then t else Vt(n - 1)"
+abbreviation
+ subst_decr_ML :: "nat \<Rightarrow> ml \<Rightarrow> nat \<Rightarrow> ml" where
+ "subst_decr_ML k v == %n. if n<k then V_ML n else if n=k then v else V_ML(n - 1)"
+abbreviation
+ subst1 :: "tm \<Rightarrow> tm \<Rightarrow> nat \<Rightarrow> tm" ("(_/[_'/_])" [300, 0, 0] 300) where
+ "s[t/k] == subst (subst_decr k t) s"
+abbreviation
+ subst1_ML :: "ml \<Rightarrow> ml \<Rightarrow> nat \<Rightarrow> ml" ("(_/[_'/_])" [300, 0, 0] 300) where
+ "u[v/k] == subst\<^bsub>ML\<^esub> (subst_decr_ML k v) u"
+
+
+lemma size_subst_ML[simp]: shows
+ "(!x. size(f x) = 0) \<longrightarrow> size(subst\<^bsub>ML\<^esub> f t) = size(t::tm)" and
+"(!x. size(f x) = 0) \<longrightarrow> size(subst\<^bsub>ML\<^esub> f (v::ml)) = size v"
+and "(!x. size(f x) = 0) \<longrightarrow> ml_list_size1 (map (subst\<^bsub>ML\<^esub> f) vs) = ml_list_size1 vs"
+and "(!x. size(f x) = 0) \<longrightarrow> ml_list_size2 (map (subst\<^bsub>ML\<^esub> f) vs) = ml_list_size2 vs"
+and "(!x. size(f x) = 0) \<longrightarrow> ml_list_size3 (map (subst\<^bsub>ML\<^esub> f) vs) = ml_list_size3 vs"
+and "(!x. size(f x) = 0) \<longrightarrow> ml_list_size4 (map (subst\<^bsub>ML\<^esub> f) vs) = ml_list_size4 vs"
+apply (induct arbitrary: f and f and f and f and f and f rule: tm_ml.inducts)
+apply (simp_all add:cons_ML_def split:nat.split)
+done
+
+lemma lift_lift: includes Vars shows
+ "i < k+1 \<Longrightarrow> lift (Suc k) (lift i t) = lift i (lift k t)"
+and "i < k+1 \<Longrightarrow> lift (Suc k) (lift i v) = lift i (lift k v)"
+apply(induct t and v arbitrary: i and i rule:lift_tm_lift_ml.induct)
+apply(simp_all add:map_compose[symmetric])
+done
+
+corollary lift_o_lift: shows
+ "i < k+1 \<Longrightarrow> lift_tm (Suc k) o (lift_tm i) = lift_tm i o lift_tm k" and
+ "i < k+1 \<Longrightarrow> lift_ml (Suc k) o (lift_ml i) = lift_ml i o lift_ml k"
+by(rule ext, simp add:lift_lift)+
+
+lemma lift_lift_ML: includes Vars shows
+ "i < k+1 \<Longrightarrow> lift\<^bsub>ML\<^esub> (Suc k) (lift\<^bsub>ML\<^esub> i t) = lift\<^bsub>ML\<^esub> i (lift\<^bsub>ML\<^esub> k t)"
+and "i < k+1 \<Longrightarrow> lift\<^bsub>ML\<^esub> (Suc k) (lift\<^bsub>ML\<^esub> i v) = lift\<^bsub>ML\<^esub> i (lift\<^bsub>ML\<^esub> k v)"
+apply(induct t and v arbitrary: i and i rule:lift_tm_ML_lift_ml_ML.induct)
+apply(simp_all add:map_compose[symmetric])
+done
+
+
+lemma lift_lift_ML_comm: includes Vars shows
+ "lift j (lift\<^bsub>ML\<^esub> i t) = lift\<^bsub>ML\<^esub> i (lift j t)" and
+ "lift j (lift\<^bsub>ML\<^esub> i v) = lift\<^bsub>ML\<^esub> i (lift j v)"
+apply(induct t and v arbitrary: i j and i j rule:lift_tm_ML_lift_ml_ML.induct)
+apply(simp_all add:map_compose[symmetric])
+done
+
+lemma [simp]:
+ "V_ML 0 ## subst_decr_ML k v = subst_decr_ML (Suc k) (lift\<^bsub>ML\<^esub> 0 v)"
+by(rule ext)(simp add:cons_ML_def split:nat.split)
+
+lemma [simp]: "lift 0 o subst_decr_ML k v = subst_decr_ML k (lift 0 v)"
+by(rule ext)(simp add:cons_ML_def split:nat.split)
+
+lemma subst_lift_id[simp]: includes Vars shows
+ "subst\<^bsub>ML\<^esub> (subst_decr_ML k v) (lift\<^bsub>ML\<^esub> k t) = t" and "(lift\<^bsub>ML\<^esub> k u)[v/k] = u"
+apply(induct k t and k u arbitrary: v and v rule: lift_tm_ML_lift_ml_ML.induct)
+apply (simp_all add:map_idI map_compose[symmetric])
+apply (simp cong:if_cong)
+done
+
+abbreviation
+ tred :: "[tm, tm] => bool" (infixl "\<rightarrow>" 50) where
+ "s \<rightarrow> t == (s, t) \<in> tRed"
+abbreviation
+ treds :: "[tm, tm] => bool" (infixl "\<rightarrow>*" 50) where
+ "s \<rightarrow>* t == (s, t) \<in> tRed^*"
+abbreviation
+ treds_list :: "[tm list, tm list] \<Rightarrow> bool" (infixl "\<rightarrow>*" 50) where
+ "ss \<rightarrow>* ts == (ss, ts) \<in> tRed_list"
+
+inductive tRed
+intros
+"At (Lam t) s \<rightarrow> t[s/0]"
+"(nm,ts,t) : R ==> foldl At (Ct nm) (map (subst rs) ts) \<rightarrow> subst rs t"
+"t \<rightarrow> t' ==> Lam t \<rightarrow> Lam t'"
+"s \<rightarrow> s' ==> At s t \<rightarrow> At s' t"
+"t \<rightarrow> t' ==> At s t \<rightarrow> At s t'"
+
+inductive tRed_list
+intros
+"[] \<rightarrow>* []"
+"ts \<rightarrow>* ts' ==> t \<rightarrow>* t' ==> t#ts \<rightarrow>* t'#ts'"
+
+declare tRed_list.intros[simp]
+
+lemma tRed_list_refl[simp]: includes Vars shows "ts \<rightarrow>* ts"
+by(induct ts) auto
+
+consts
+ Red :: "(ml * ml)set"
+ Redl :: "(ml list * ml list)set"
+ Redt :: "(tm * tm)set"
+
+abbreviation
+ red :: "[ml, ml] => bool" (infixl "\<Rightarrow>" 50) where
+ "s \<Rightarrow> t == (s, t) \<in> Red"
+abbreviation
+ redl :: "[ml list, ml list] => bool" (infixl "\<Rightarrow>" 50) where
+ "s \<Rightarrow> t == (s, t) \<in> Redl"
+abbreviation
+ redt :: "[tm, tm] => bool" (infixl "\<Rightarrow>" 50) where
+ "s \<Rightarrow> t == (s, t) \<in> Redt"
+abbreviation
+ reds :: "[ml, ml] => bool" (infixl "\<Rightarrow>*" 50) where
+ "s \<Rightarrow>* t == (s, t) \<in> Red^*"
+abbreviation
+ redts :: "[tm, tm] => bool" (infixl "\<Rightarrow>*" 50) where
+ "s \<Rightarrow>* t == (s, t) \<in> Redt^*"
+
+
+fun ML_closed :: "nat \<Rightarrow> ml \<Rightarrow> bool"
+and ML_closed_t :: "nat \<Rightarrow> tm \<Rightarrow> bool" where
+"ML_closed i (C nm vs) = (ALL v:set vs. ML_closed i v)" |
+"ML_closed i (V nm vs) = (ALL v:set vs. ML_closed i v)" |
+"ML_closed i (Fun f vs n) = (ML_closed i f & (ALL v:set vs. ML_closed i v))" |
+"ML_closed i (A_ML v vs) = (ML_closed i v & (ALL v:set vs. ML_closed i v))" |
+"ML_closed i (apply v w) = (ML_closed i v & ML_closed i w)" |
+"ML_closed i (CC nm) = True" |
+"ML_closed i (V_ML X) = (X<i)" |
+"ML_closed i (Lam_ML v) = ML_closed (i+1) v" |
+"ML_closed_t i (term_of v) = ML_closed i v" |
+"ML_closed_t i (At r s) = (ML_closed_t i r & ML_closed_t i s)" |
+"ML_closed_t i (Lam t) = (ML_closed_t i t)" |
+"ML_closed_t i v = True"
+thm ML_closed.simps ML_closed_t.simps
+
+inductive Red Redt Redl
+intros
+(* ML *)
+"A_ML (Lam_ML u) [v] \<Rightarrow> u[v/0]"
+(* compiled rules *)
+"(nm,vs,v) : compR ==> ALL i. ML_closed 0 (f i) \<Longrightarrow> A_ML (CC nm) (map (subst\<^bsub>ML\<^esub> f) vs) \<Rightarrow> subst\<^bsub>ML\<^esub> f v"
+(* apply *)
+apply_Fun1: "apply (Fun f vs (Suc 0)) v \<Rightarrow> A_ML f (vs @ [v])"
+apply_Fun2: "n > 0 ==>
+ apply (Fun f vs (Suc n)) v \<Rightarrow> Fun f (vs @ [v]) n"
+apply_C: "apply (C nm vs) v \<Rightarrow> C nm (vs @ [v])"
+apply_V: "apply (V x vs) v \<Rightarrow> V x (vs @ [v])"
+(* term_of *)
+term_of_C: "term_of (C nm vs) \<Rightarrow> foldl At (Ct nm) (map term_of vs)"
+term_of_V: "term_of (V x vs) \<Rightarrow> foldl At (Vt x) (map term_of vs)"
+term_of_Fun: "term_of(Fun vf vs n) \<Rightarrow>
+ Lam (term_of ((apply (lift 0 (Fun vf vs n)) (V_ML 0))[V 0 []/0]))"
+(* Context *)
+ctxt_Lam: "t \<Rightarrow> t' ==> Lam t \<Rightarrow> Lam t'"
+ctxt_At1: "s \<Rightarrow> s' ==> At s t \<Rightarrow> At s' t"
+ctxt_At2: "t \<Rightarrow> t' ==> At s t \<Rightarrow> At s t'"
+ctxt_term_of: "v \<Rightarrow> v' ==> term_of v \<Rightarrow> term_of v'"
+ctxt_C: "vs \<Rightarrow> vs' ==> C nm vs \<Rightarrow> C nm vs'"
+ctxt_V: "vs \<Rightarrow> vs' ==> V x vs \<Rightarrow> V x vs'"
+ctxt_Fun1: "f \<Rightarrow> f' ==> Fun f vs n \<Rightarrow> Fun f' vs n"
+ctxt_Fun3: "vs \<Rightarrow> vs' ==> Fun f vs n \<Rightarrow> Fun f vs' n"
+ctxt_apply1: "s \<Rightarrow> s' ==> apply s t \<Rightarrow> apply s' t"
+ctxt_apply2: "t \<Rightarrow> t' ==> apply s t \<Rightarrow> apply s t'"
+ctxt_A_ML1: "f \<Rightarrow> f' ==> A_ML f vs \<Rightarrow> A_ML f' vs"
+ctxt_A_ML2: "vs \<Rightarrow> vs' ==> A_ML f vs \<Rightarrow> A_ML f vs'"
+ctxt_list1: "v \<Rightarrow> v' ==> v#vs \<Rightarrow> v'#vs"
+ctxt_list2: "vs \<Rightarrow> vs' ==> v#vs \<Rightarrow> v#vs'"
+
+
+consts
+ ar :: "const_name \<Rightarrow> nat"
+
+axioms
+ar_pos: "ar nm > 0"
+
+types env = "ml list"
+
+consts eval :: "tm \<Rightarrow> env \<Rightarrow> ml"
+primrec
+"eval (Vt x) e = e!x"
+"eval (Ct nm) e = Fun (CC nm) [] (ar nm)"
+"eval (At s t) e = apply (eval s e) (eval t e)"
+"eval (Lam t) e = Fun (Lam_ML (eval t ((V_ML 0) # map (lift\<^bsub>ML\<^esub> 0) e))) [] 1"
+
+fun size' :: "ml \<Rightarrow> nat" where
+"size' (C nm vs) = (\<Sum>v\<leftarrow>vs. size' v)+1" |
+"size' (V nm vs) = (\<Sum>v\<leftarrow>vs. size' v)+1" |
+"size' (Fun f vs n) = (size' f + (\<Sum>v\<leftarrow>vs. size' v))+1" |
+"size' (A_ML v vs) = (size' v + (\<Sum>v\<leftarrow>vs. size' v))+1" |
+"size' (apply v w) = (size' v + size' w)+1" |
+"size' (CC nm) = 1" |
+"size' (V_ML X) = 1" |
+"size' (Lam_ML v) = size' v + 1"
+
+lemma listsum_size'[simp]:
+ "v \<in> set vs \<Longrightarrow> size' v < Suc(listsum (map size' vs))"
+sorry
+
+corollary cor_listsum_size'[simp]:
+ "v \<in> set vs \<Longrightarrow> size' v < Suc(m + listsum (map size' vs))"
+using listsum_size'[of v vs] by arith
+
+lemma
+size_subst_ML[simp]: includes Vars assumes A: "!i. size(f i) = 0"
+shows "size(subst\<^bsub>ML\<^esub> f t) = size(t)"
+and "size(subst\<^bsub>ML\<^esub> f v) = size(v)"
+and "ml_list_size1 (map (subst\<^bsub>ML\<^esub> f) vs) = ml_list_size1 vs"
+and "ml_list_size2 (map (subst\<^bsub>ML\<^esub> f) vs) = ml_list_size2 vs"
+and "ml_list_size3 (map (subst\<^bsub>ML\<^esub> f) vs) = ml_list_size3 vs"
+and "ml_list_size4 (map (subst\<^bsub>ML\<^esub> f) vs) = ml_list_size4 vs"
+by (induct rule: tm_ml.inducts) (simp_all add: A cons_ML_def split:nat.split)
+
+lemma [simp]:
+ "\<forall>i j. size'(f i) = size'(V_ML j) \<Longrightarrow> size' (subst\<^bsub>ML\<^esub> f v) = size' v"
+sorry
+
+lemma [simp]: "size' (lift i v) = size' v"
+sorry
+
+(* the kernel function as in Section 4.1 of "Operational aspects\<dots>" *)
+
+function kernel :: "ml \<Rightarrow> tm" ("_!" 300) where
+"(C nm vs)! = foldl At (Ct nm) (map kernel vs)" |
+"(Lam_ML v)! = Lam (((lift 0 v)[V 0 []/0])!)" |
+"(Fun f vs n)! = foldl At (f!) (map kernel vs)" |
+"(A_ML v vs)! = foldl At (v!) (map kernel vs)" |
+"(apply v w)! = At (v!) (w!)" |
+"(CC nm)! = Ct nm" |
+"(V x vs)! = foldl At (Vt x) (map kernel vs)" |
+"(V_ML X)! = undefined"
+by pat_completeness auto
+termination by(relation "measure size'") auto
+
+consts kernelt :: "tm \<Rightarrow> tm" ("_!" 300)
+primrec
+"(Ct nm)! = Ct nm"
+"(term_of v)! = v!"
+"(Vt x)! = Vt x"
+"(At s t)! = At (s!) (t!)"
+"(Lam t)! = Lam (t!)"
+
+abbreviation
+ kernels :: "ml list \<Rightarrow> tm list" ("_!" 300) where
+ "vs ! == map kernel vs"
+
+(* soundness of the code generator *)
+axioms
+compiler_correct:
+"(nm, vs, v) : compR ==> ALL i. ML_closed 0 (f i) \<Longrightarrow> (nm, (map (subst\<^bsub>ML\<^esub> f) vs)!, (subst\<^bsub>ML\<^esub> f v)!) : R"
+
+
+consts
+ free_vars :: "tm \<Rightarrow> lam_var_name set"
+primrec
+"free_vars (Ct nm) = {}"
+"free_vars (Vt x) = {x}"
+"free_vars (Lam t) = {i. EX j : free_vars t. j = i+1}"
+"free_vars (At s t) = free_vars s \<union> free_vars t"
+
+lemma [simp]: "t : Pure_tms \<Longrightarrow> lift\<^bsub>ML\<^esub> k t = t"
+by (erule Pure_tms.induct) simp_all
+
+lemma kernel_pure: includes Vars assumes "t : Pure_tms" shows "t! = t"
+using assms by (induct) simp_all
+
+lemma lift_eval:
+ "t : Pure_tms \<Longrightarrow> ALL e k. (ALL i : free_vars t. i < size e) --> lift k (eval t e) = eval t (map (lift k) e)"
+apply(induct set:Pure_tms)
+apply simp_all
+apply clarsimp
+apply(erule_tac x = "V_ML 0 # map (lift\<^bsub>ML\<^esub> 0) e" in allE)
+apply simp
+apply(erule impE)
+ apply clarsimp
+ apply(case_tac i)apply simp
+ apply simp
+apply (simp add: map_compose[symmetric])
+apply (simp add: o_def lift_lift_ML_comm)
+done
+
+lemma lift_ML_eval[rule_format]:
+ "t : Pure_tms \<Longrightarrow> ALL e k. (ALL i : free_vars t. i < size e) --> lift\<^bsub>ML\<^esub> k (eval t e) = eval t (map (lift\<^bsub>ML\<^esub> k) e)"
+apply(induct set:Pure_tms)
+apply simp_all
+apply clarsimp
+apply(erule_tac x = "V_ML 0 # map (lift\<^bsub>ML\<^esub> 0) e" in allE)
+apply simp
+apply(erule impE)
+ apply clarsimp
+ apply(case_tac i)apply simp
+ apply simp
+apply (simp add: map_compose[symmetric])
+apply (simp add:o_def lift_lift_ML)
+done
+
+lemma [simp]: includes Vars shows "(v ## f) 0 = v"
+by(simp add:cons_ML_def)
+
+lemma [simp]: includes Vars shows "(v ## f) (Suc n) = lift\<^bsub>ML\<^esub> 0 (f n)"
+by(simp add:cons_ML_def)
+
+lemma lift_o_shift: "lift k o (V_ML 0 ## f) = (V_ML 0 ## (lift k \<circ> f))"
+apply(rule ext)
+apply (simp add:cons_ML_def lift_lift_ML_comm split:nat.split)
+done
+
+lemma lift_subst_ML: shows
+ "lift_tm k (subst\<^bsub>ML\<^esub> f t) = subst\<^bsub>ML\<^esub> (lift_ml k o f) (lift_tm k t)" and
+ "lift_ml k (subst\<^bsub>ML\<^esub> f v) = subst\<^bsub>ML\<^esub> (lift_ml k o f) (lift_ml k v)"
+apply (induct t and v arbitrary: f k and f k rule: lift_tm_lift_ml.induct)
+apply (simp_all add:map_compose[symmetric] o_assoc lift_o_lift lift_o_shift)
+done
+
+corollary lift_subst_ML1: "\<forall>v k. lift_ml 0 (u[v/k]) = (lift_ml 0 u)[lift 0 v/k]"
+apply(rule measure_induct[where f = "size" and a = u])
+apply(case_tac x)
+apply(simp_all add:lift_lift map_compose[symmetric] lift_subst_ML)
+apply(subst lift_lift_ML_comm)apply simp
+done
+
+lemma lift_ML_lift_ML: includes Vars shows
+ "i < k+1 \<Longrightarrow> lift\<^bsub>ML\<^esub> (Suc k) (lift\<^bsub>ML\<^esub> i t) = lift\<^bsub>ML\<^esub> i (lift\<^bsub>ML\<^esub> k t)"
+and "i < k+1 \<Longrightarrow> lift\<^bsub>ML\<^esub> (Suc k) (lift\<^bsub>ML\<^esub> i v) = lift\<^bsub>ML\<^esub> i (lift\<^bsub>ML\<^esub> k v)"
+apply (induct k t and k v arbitrary: i k and i k
+ rule: lift_tm_ML_lift_ml_ML.induct)
+apply(simp_all add:map_compose[symmetric])
+done
+
+corollary lift_ML_o_lift_ML: shows
+ "i < k+1 \<Longrightarrow> lift_tm_ML (Suc k) o (lift_tm_ML i) = lift_tm_ML i o lift_tm_ML k" and
+ "i < k+1 \<Longrightarrow> lift_ml_ML (Suc k) o (lift_ml_ML i) = lift_ml_ML i o lift_ml_ML k"
+by(rule ext, simp add:lift_ML_lift_ML)+
+
+abbreviation insrt where
+"insrt k f == (%i. if i<k then lift_ml_ML k (f i) else if i=k then V_ML k else lift_ml_ML k (f(i - 1)))"
+
+lemma subst_insrt_lift: includes Vars shows
+ "subst\<^bsub>ML\<^esub> (insrt k f) (lift\<^bsub>ML\<^esub> k t) = lift\<^bsub>ML\<^esub> k (subst\<^bsub>ML\<^esub> f t)" and
+ "subst\<^bsub>ML\<^esub> (insrt k f) (lift\<^bsub>ML\<^esub> k v) = lift\<^bsub>ML\<^esub> k (subst\<^bsub>ML\<^esub> f v)"
+apply (induct k t and k v arbitrary: f k and f k rule: lift_tm_ML_lift_ml_ML.induct)
+apply (simp_all add:map_compose[symmetric] o_assoc lift_o_lift lift_o_shift)
+ apply(subgoal_tac "lift 0 \<circ> insrt k f = insrt k (lift 0 \<circ> f)")
+ apply simp
+ apply(rule ext)
+ apply (simp add:lift_lift_ML_comm)
+apply(subgoal_tac "V_ML 0 ## insrt k f = insrt (Suc k) (V_ML 0 ## f)")
+ apply simp
+ apply(rule ext)
+ apply (simp add:lift_ML_lift_ML cons_ML_def split:nat.split)
+done
+
+corollary subst_cons_lift: includes Vars shows
+ "subst\<^bsub>ML\<^esub> (V_ML 0 ## f) o (lift_ml_ML 0) = lift_ml_ML 0 o (subst_ml_ML f)"
+apply(rule ext)
+apply(simp add: cons_ML_def subst_insrt_lift[symmetric])
+apply(subgoal_tac "nat_case (V_ML 0) (\<lambda>j. lift\<^bsub>ML\<^esub> 0 (f j)) = (\<lambda>i. if i = 0 then V_ML 0 else lift\<^bsub>ML\<^esub> 0 (f (i - 1)))")
+ apply simp
+apply(rule ext, simp split:nat.split)
+done
+
+lemma subst_eval[rule_format]: "t : Pure_tms \<Longrightarrow>
+ ALL f e. (ALL i : free_vars t. i < size e) \<longrightarrow> subst\<^bsub>ML\<^esub> f (eval t e) = eval t (map (subst\<^bsub>ML\<^esub> f) e)"
+apply(induct set:Pure_tms)
+apply simp_all
+apply clarsimp
+apply(erule_tac x="V_ML 0 ## f" in allE)
+apply(erule_tac x= "(V_ML 0 # map (lift\<^bsub>ML\<^esub> 0) e)" in allE)
+apply(erule impE)
+ apply clarsimp
+ apply(case_tac i)apply simp
+ apply simp
+apply (simp add:subst_cons_lift map_compose[symmetric])
+done
+
+
+theorem kernel_eval[rule_format]: includes Vars shows
+ "t : Pure_tms ==>
+ ALL e. (ALL i : free_vars t. i < size e) \<longrightarrow> (ALL i < size e. e!i = V i []) --> (eval t e)! = t!"
+apply(induct set:Pure_tms)
+apply simp_all
+apply clarsimp
+apply(subst lift_eval) apply simp
+ apply clarsimp
+ apply(case_tac i)apply simp
+ apply simp
+apply(subst subst_eval) apply simp
+ apply clarsimp
+ apply(case_tac i)apply simp
+ apply simp
+apply(erule_tac x="map (subst\<^bsub>ML\<^esub> (\<lambda>n. if n = 0 then V 0 [] else V_ML (n - 1)))
+ (map (lift 0) (V_ML 0 # map (lift\<^bsub>ML\<^esub> 0) e))" in allE)
+apply(erule impE)
+apply(clarsimp)
+ apply(case_tac i)apply simp
+ apply simp
+apply(erule impE)
+apply(clarsimp)
+ apply(case_tac i)apply simp
+ apply simp
+apply simp
+done
+
+(*
+lemma subst_ML_compose:
+ "subst_ml_ML f2 (subst_ml_ML f1 v) = subst_ml_ML (%i. subst_ml_ML f2 (f1 i)) v"
+sorry
+*)
+
+lemma map_eq_iff_nth:
+ "(map f xs = map g xs) = (!i<size xs. f(xs!i) = g(xs!i))"
+sorry
+
+lemma [simp]: includes Vars shows "ML_closed k v \<Longrightarrow> lift\<^bsub>ML\<^esub> k v = v"
+sorry
+lemma [simp]: includes Vars shows "ML_closed 0 v \<Longrightarrow> subst\<^bsub>ML\<^esub> f v = v"
+sorry
+lemma [simp]: includes Vars shows "ML_closed k v \<Longrightarrow> ML_closed k (lift m v)"
+sorry
+
+lemma red_Lam[simp]: includes Vars shows "t \<rightarrow>* t' ==> Lam t \<rightarrow>* Lam t'"
+apply(induct rule:rtrancl_induct)
+apply(simp_all)
+apply(blast intro: rtrancl_into_rtrancl tRed.intros)
+done
+
+lemma red_At1[simp]: includes Vars shows "t \<rightarrow>* t' ==> At t s \<rightarrow>* At t' s"
+apply(induct rule:rtrancl_induct)
+apply(simp_all)
+apply(blast intro: rtrancl_into_rtrancl tRed.intros)
+done
+
+lemma red_At2[simp]: includes Vars shows "t \<rightarrow>* t' ==> At s t \<rightarrow>* At s t'"
+apply(induct rule:rtrancl_induct)
+apply(simp_all)
+apply(blast intro:rtrancl_into_rtrancl tRed.intros)
+done
+
+lemma tRed_list_foldl_At:
+ "ts \<rightarrow>* ts' \<Longrightarrow> s \<rightarrow>* s' \<Longrightarrow> foldl At s ts \<rightarrow>* foldl At s' ts'"
+apply(induct arbitrary:s s' rule:tRed_list.induct)
+apply simp
+apply simp
+apply(blast dest: red_At1 red_At2 intro:rtrancl_trans)
+done
+
+lemma [trans]: "s = t \<Longrightarrow> t \<rightarrow> t' \<Longrightarrow> s \<rightarrow> t'"
+by simp
+
+
+lemma subst_foldl[simp]:
+ "subst f (foldl At s ts) = foldl At (subst f s) (map (subst f) ts)"
+by (induct ts arbitrary: s) auto
+
+
+lemma foldl_At_size: "size ts = size ts' \<Longrightarrow>
+ foldl At s ts = foldl At s' ts' \<longleftrightarrow> s = s' & ts = ts'"
+by (induct arbitrary: s s' rule:list_induct2) simp_all
+
+consts depth_At :: "tm \<Rightarrow> nat"
+primrec
+"depth_At(Ct cn) = 0"
+"depth_At(Vt x) = 0"
+"depth_At(Lam t) = 0"
+"depth_At(At s t) = depth_At s + 1"
+"depth_At(term_of v) = 0"
+
+lemma depth_At_foldl:
+ "depth_At(foldl At s ts) = depth_At s + size ts"
+by (induct ts arbitrary: s) simp_all
+
+lemma foldl_At_eq_length:
+ "foldl At s ts = foldl At s ts' \<Longrightarrow> length ts = length ts'"
+apply(subgoal_tac "depth_At(foldl At s ts) = depth_At(foldl At s ts')")
+apply(erule thin_rl)
+ apply (simp add:depth_At_foldl)
+apply simp
+done
+
+lemma foldl_At_eq[simp]: "foldl At s ts = foldl At s ts' \<longleftrightarrow> ts = ts'"
+apply(rule)
+ prefer 2 apply simp
+apply(blast dest:foldl_At_size foldl_At_eq_length)
+done
+
+lemma [simp]: "foldl At s ts ! = foldl At (s!) (map kernelt ts)"
+by (induct ts arbitrary: s) simp_all
+
+lemma [simp]: "(kernelt \<circ> term_of) = kernel"
+by(rule ext) simp
+
+lemma shift_subst_decr:
+ "Vt 0 ## subst_decr k t = subst_decr (Suc k) (lift 0 t)"
+apply(rule ext)
+apply (simp add:cons_def split:nat.split)
+done
+
+lemma [simp]: "lift k (foldl At s ts) = foldl At (lift k s) (map (lift k) ts)"
+by(induct ts arbitrary:s) simp_all
+
+subsection "Horrible detour"
+
+definition "liftn n == lift_ml 0 ^ n"
+
+lemma [simp]: "liftn n (C i vs) = C i (map (liftn n) vs)"
+apply(unfold liftn_def)
+apply(induct n)
+apply (simp_all add: map_compose[symmetric])
+done
+
+lemma [simp]: "liftn n (CC nm) = CC nm"
+apply(unfold liftn_def)
+apply(induct n)
+apply (simp_all add: map_compose[symmetric])
+done
+
+lemma [simp]: "liftn n (apply v w) = apply (liftn n v) (liftn n w)"
+apply(unfold liftn_def)
+apply(induct n)
+apply (simp_all add: map_compose[symmetric])
+done
+
+lemma [simp]: "liftn n (A_ML v vs) = A_ML (liftn n v) (map (liftn n) vs)"
+apply(unfold liftn_def)
+apply(induct n)
+apply (simp_all add: map_compose[symmetric])
+done
+
+lemma [simp]:
+ "liftn n (Fun v vs i) = Fun (liftn n v) (map (liftn n) vs) i"
+apply(unfold liftn_def)
+apply(induct n)
+apply (simp_all add: map_compose[symmetric] id_def)
+done
+
+lemma [simp]: "liftn n (Lam_ML v) = Lam_ML (liftn n v)"
+apply(unfold liftn_def)
+apply(induct n)
+apply (simp_all add: map_compose[symmetric] id_def)
+done
+
+lemma liftn_liftn_add: "liftn m (liftn n v) = liftn (m+n) v"
+by(simp add:liftn_def funpow_add)
+
+lemma [simp]: "liftn n (V_ML k) = V_ML k"
+apply(unfold liftn_def)
+apply(induct n)
+apply (simp_all)
+done
+
+lemma liftn_lift_ML_comm: "liftn n (lift\<^bsub>ML\<^esub> 0 v) = lift\<^bsub>ML\<^esub> 0 (liftn n v)"
+apply(unfold liftn_def)
+apply(induct n)
+apply (simp_all add:lift_lift_ML_comm)
+done
+
+lemma liftn_cons: "liftn n ((V_ML 0 ## f) x) = (V_ML 0 ## (liftn n o f)) x"
+apply(simp add:cons_ML_def liftn_lift_ML_comm split:nat.split)
+done
+
+text{* End of horrible detour *}
+
+lemma kernel_subst1:
+"ML_closed 1 u \<Longrightarrow> ML_closed 0 v \<Longrightarrow> kernel( u[v/0]) = (kernel((lift 0 u)[V 0 []/0]))[kernel v/0]"
+sorry
+
+lemma includes Vars shows foldl_Pure[simp]:
+ "t : Pure_tms \<Longrightarrow> \<forall>t\<in>set ts. t : Pure_tms \<Longrightarrow>
+ (!!s t. s : Pure_tms \<Longrightarrow> t : Pure_tms \<Longrightarrow> f s t : Pure_tms) \<Longrightarrow>
+ foldl f t ts \<in> Pure_tms"
+by(induct ts arbitrary: t) simp_all
+
+declare Pure_tms.intros[simp]
+
+lemma includes Vars shows "ML_closed 0 v \<Longrightarrow> kernel v : Pure_tms"
+apply(induct rule:kernel.induct)
+apply simp_all
+apply(rule Pure_tms.intros);
+(* "ML_closed (Suc k) v \<Longrightarrow> ML_closed k (lift 0 v)" *)
+sorry
+
+lemma subst_Vt: includes Vars shows "subst Vt = id"
+sorry
+(*
+apply(rule ext)
+apply(induct_tac x)
+apply simp_all
+
+done
+*)
+(* klappt noch nicht ganz *)
+theorem Red_sound: includes Vars
+ shows "v \<Rightarrow> v' \<Longrightarrow> ML_closed 0 v \<Longrightarrow> v! \<rightarrow>* v'! & ML_closed 0 v'"
+ and "t \<Rightarrow> t' \<Longrightarrow> ML_closed_t 0 t \<Longrightarrow> kernelt t \<rightarrow>* kernelt t' & ML_closed_t 0 t'"
+ and "(vs :: ml list) \<Rightarrow> vs' \<Longrightarrow> !v : set vs . ML_closed 0 v \<Longrightarrow> map kernel vs \<rightarrow>* map kernel vs' & (! v':set vs'. ML_closed 0 v')"
+proof(induct rule:Red_Redt_Redl.inducts)
+ fix u v
+ let ?v = "A_ML (Lam_ML u) [v]"
+ assume cl: "ML_closed 0 (A_ML (Lam_ML u) [v])"
+ let ?u' = "(lift_ml 0 u)[V 0 []/0]"
+ have "?v! = At (Lam ((?u')!)) (v !)" by simp
+ also have "\<dots> \<rightarrow> (?u' !)[v!/0]" (is "_ \<rightarrow> ?R") by(rule tRed.intros)
+ also have "?R = u[v/0]!" using cl
+apply(cut_tac u = "u" and v = "v" in kernel_subst1)
+apply(simp_all)
+done
+ finally have "kernel(A_ML (Lam_ML u) [v]) \<rightarrow>* kernel(u[v/0])" (is ?A) by(rule r_into_rtrancl)
+ moreover have "ML_closed 0 (u[v/0])" (is "?C") using cl apply simp sorry
+ ultimately show "?A & ?C" ..
+next
+ case term_of_C thus ?case apply (auto simp:map_compose[symmetric])sorry
+next
+ fix f :: "nat \<Rightarrow> ml" and nm vs v
+ assume f: "\<forall>i. ML_closed 0 (f i)" and compR: "(nm, vs, v) \<in> compR"
+ note tRed.intros(2)[OF compiler_correct[OF compR f], of Vt,simplified map_compose[symmetric]]
+ hence red: "foldl At (Ct nm) (map (kernel o subst\<^bsub>ML\<^esub> f) vs) \<rightarrow>
+ (subst\<^bsub>ML\<^esub> f v)!" (is "_ \<rightarrow> ?R") apply(simp add:map_compose) sorry
+ have "A_ML (CC nm) (map (subst\<^bsub>ML\<^esub> f) vs)! =
+ foldl At (Ct nm) (map (kernel o subst\<^bsub>ML\<^esub> f) vs)" by (simp add:map_compose)
+ also(* have "map (kernel o subst\<^bsub>ML\<^esub> f) vs = map (subst (kernel o f)) (vs!)"
+ using closed_subst_kernel(2)[OF compiled_V_free1[OF compR]]
+ by (simp add:map_compose[symmetric])
+ also*) note red
+ (*also have "?R = subst\<^bsub>ML\<^esub> f v!"
+ using closed_subst_kernel(2)[OF compiled_V_free2[OF compR]] by simp*)
+ finally have "A_ML (CC nm) (map (subst\<^bsub>ML\<^esub> f) vs)! \<rightarrow>* subst\<^bsub>ML\<^esub> f v!" (is "?A")
+ by(rule r_into_rtrancl) (*
+ also have "?l = (subst\<^bsub>ML\<^esub> fa (A_ML (CC nm) (map (subst\<^bsub>ML\<^esub> f) vs)))!" (is "_ = ?l'") sorry
+ also have "?r = subst\<^bsub>ML\<^esub> fa (subst\<^bsub>ML\<^esub> f v)!" (is "_ = ?r'") sorry
+ finally have "?l' \<rightarrow>* ?r'" (is ?A) . *)
+ moreover have "ML_closed 0 (subst\<^bsub>ML\<^esub> f v)" (is "?C") using prems sorry
+ ultimately show "?A & ?C" ..
+next
+ case term_of_V thus ?case apply (auto simp:map_compose[symmetric]) sorry
+next
+ case (term_of_Fun n vf vs)
+ hence "term_of (Fun vf vs n)! \<rightarrow>*
+ Lam (term_of (apply (lift 0 (Fun vf vs n)) (V_ML 0)[V 0 []/0]))!" sorry
+ moreover
+ have "ML_closed_t 0
+ (Lam (term_of (apply (lift 0 (Fun vf vs n)) (V_ML 0)[V 0 []/0])))" sorry
+ ultimately show ?case ..
+next
+ case apply_Fun1 thus ?case by simp
+next
+ case apply_Fun2 thus ?case by simp
+next
+ case apply_C thus ?case by simp
+next
+ case apply_V thus ?case by simp
+next
+ case ctxt_Lam thus ?case by(auto)
+next
+ case ctxt_At1 thus ?case by(auto)
+next
+ case ctxt_At2 thus ?case by (auto)
+next
+ case ctxt_term_of thus ?case by (auto)
+next
+ case ctxt_C thus ?case by (fastsimp simp:tRed_list_foldl_At)
+next
+ case ctxt_V thus ?case by (fastsimp simp:tRed_list_foldl_At)
+next
+ case ctxt_Fun1 thus ?case by (fastsimp simp:tRed_list_foldl_At)
+next
+ case ctxt_Fun3 thus ?case by (fastsimp simp:tRed_list_foldl_At)
+next
+ case ctxt_apply1 thus ?case by auto
+next
+ case ctxt_apply2 thus ?case by auto
+next
+ case ctxt_A_ML1 thus ?case by (fastsimp simp:tRed_list_foldl_At)
+next
+ case ctxt_A_ML2 thus ?case by (fastsimp simp:tRed_list_foldl_At)
+next
+ case ctxt_list1 thus ?case by simp
+next
+ case ctxt_list2 thus ?case by simp
+qed
+
+
+inductive_cases tRedE: "Ct n \<rightarrow> u"
+thm tRedE
+
+lemma [simp]: "Ct n = foldl At t ts \<longleftrightarrow> t = Ct n & ts = []"
+by (induct ts arbitrary:t) auto
+
+corollary kernel_inv: includes Vars shows
+ "(t :: tm) \<Rightarrow>* t' ==> ML_closed_t 0 t ==> t! \<rightarrow>* t'!"
+sorry
+
+theorem includes Vars
+assumes t: "t : Pure_tms" and t': "t' : Pure_tms" and
+ closed: "free_vars t = {}" and reds: "term_of (eval t []) \<Rightarrow>* t'"
+ shows "t \<rightarrow>* t' "
+proof -
+ have ML_cl: "ML_closed_t 0 (term_of (eval t []))" sorry
+ have "(eval t [])! = t!"
+ using kernel_eval[OF t, where e="[]"] closed by simp
+ hence "(term_of (eval t []))! = t!" by simp
+ moreover have "term_of (eval t [])! \<rightarrow>* t'!"
+ using kernel_inv[OF reds ML_cl] by auto
+ ultimately have "t! \<rightarrow>* t'!" by simp
+ thus ?thesis using kernel_pure t t' by auto
+qed
+
+end