src/HOL/Lambda/WeakNorm.thy

Locally added definition of "int :: nat => int" again to make
code generation work.

(*  Title:      HOL/Lambda/WeakNorm.thy
    ID:         $Id$
    Author:     Stefan Berghofer
    Copyright   2003 TU Muenchen
*)

header {* Weak normalization for simply-typed lambda calculus *}

theory WeakNorm
imports Type Pretty_Int
begin

text {*
Formalization by Stefan Berghofer. Partly based on a paper proof by
Felix Joachimski and Ralph Matthes \cite{Matthes-Joachimski-AML}.
*}


subsection {* Terms in normal form *}

definition
  listall :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool" where
  "listall P xs \<equiv> (\<forall>i. i < length xs \<longrightarrow> P (xs ! i))"

declare listall_def [extraction_expand]

theorem listall_nil: "listall P []"
  by (simp add: listall_def)

theorem listall_nil_eq [simp]: "listall P [] = True"
  by (iprover intro: listall_nil)

theorem listall_cons: "P x \<Longrightarrow> listall P xs \<Longrightarrow> listall P (x # xs)"
  apply (simp add: listall_def)
  apply (rule allI impI)+
  apply (case_tac i)
  apply simp+
  done

theorem listall_cons_eq [simp]: "listall P (x # xs) = (P x \<and> listall P xs)"
  apply (rule iffI)
  prefer 2
  apply (erule conjE)
  apply (erule listall_cons)
  apply assumption
  apply (unfold listall_def)
  apply (rule conjI)
  apply (erule_tac x=0 in allE)
  apply simp
  apply simp
  apply (rule allI)
  apply (erule_tac x="Suc i" in allE)
  apply simp
  done

lemma listall_conj1: "listall (\<lambda>x. P x \<and> Q x) xs \<Longrightarrow> listall P xs"
  by (induct xs) simp_all

lemma listall_conj2: "listall (\<lambda>x. P x \<and> Q x) xs \<Longrightarrow> listall Q xs"
  by (induct xs) simp_all

lemma listall_app: "listall P (xs @ ys) = (listall P xs \<and> listall P ys)"
  apply (induct xs)
   apply (rule iffI, simp, simp)
  apply (rule iffI, simp, simp)
  done

lemma listall_snoc [simp]: "listall P (xs @ [x]) = (listall P xs \<and> P x)"
  apply (rule iffI)
  apply (simp add: listall_app)+
  done

lemma listall_cong [cong, extraction_expand]:
  "xs = ys \<Longrightarrow> listall P xs = listall P ys"
  -- {* Currently needed for strange technical reasons *}
  by (unfold listall_def) simp

inductive2 NF :: "dB \<Rightarrow> bool"
where
  App: "listall NF ts \<Longrightarrow> NF (Var x \<degree>\<degree> ts)"
| Abs: "NF t \<Longrightarrow> NF (Abs t)"
monos listall_def

lemma nat_eq_dec: "\<And>n::nat. m = n \<or> m \<noteq> n"
  apply (induct m)
  apply (case_tac n)
  apply (case_tac [3] n)
  apply (simp only: nat.simps, iprover?)+
  done

lemma nat_le_dec: "\<And>n::nat. m < n \<or> \<not> (m < n)"
  apply (induct m)
  apply (case_tac n)
  apply (case_tac [3] n)
  apply (simp del: simp_thms, iprover?)+
  done

lemma App_NF_D: assumes NF: "NF (Var n \<degree>\<degree> ts)"
  shows "listall NF ts" using NF
  by cases simp_all


subsection {* Properties of @{text NF} *}

lemma Var_NF: "NF (Var n)"
  apply (subgoal_tac "NF (Var n \<degree>\<degree> [])")
   apply simp
  apply (rule NF.App)
  apply simp
  done

lemma subst_terms_NF: "listall NF ts \<Longrightarrow>
    listall (\<lambda>t. \<forall>i j. NF (t[Var i/j])) ts \<Longrightarrow>
    listall NF (map (\<lambda>t. t[Var i/j]) ts)"
  by (induct ts) simp_all

lemma subst_Var_NF: "NF t \<Longrightarrow> NF (t[Var i/j])"
  apply (induct arbitrary: i j set: NF)
  apply simp
  apply (frule listall_conj1)
  apply (drule listall_conj2)
  apply (drule_tac i=i and j=j in subst_terms_NF)
  apply assumption
  apply (rule_tac m=x and n=j in nat_eq_dec [THEN disjE, standard])
  apply simp
  apply (erule NF.App)
  apply (rule_tac m=j and n=x in nat_le_dec [THEN disjE, standard])
  apply simp
  apply (iprover intro: NF.App)
  apply simp
  apply (iprover intro: NF.App)
  apply simp
  apply (iprover intro: NF.Abs)
  done

lemma app_Var_NF: "NF t \<Longrightarrow> \<exists>t'. t \<degree> Var i \<rightarrow>\<^sub>\<beta>\<^sup>* t' \<and> NF t'"
  apply (induct set: NF)
  apply (simplesubst app_last)  --{*Using @{text subst} makes extraction fail*}
  apply (rule exI)
  apply (rule conjI)
  apply (rule rtrancl.rtrancl_refl)
  apply (rule NF.App)
  apply (drule listall_conj1)
  apply (simp add: listall_app)
  apply (rule Var_NF)
  apply (rule exI)
  apply (rule conjI)
  apply (rule rtrancl.rtrancl_into_rtrancl)
  apply (rule rtrancl.rtrancl_refl)
  apply (rule beta)
  apply (erule subst_Var_NF)
  done

lemma lift_terms_NF: "listall NF ts \<Longrightarrow>
    listall (\<lambda>t. \<forall>i. NF (lift t i)) ts \<Longrightarrow>
    listall NF (map (\<lambda>t. lift t i) ts)"
  by (induct ts) simp_all

lemma lift_NF: "NF t \<Longrightarrow> NF (lift t i)"
  apply (induct arbitrary: i set: NF)
  apply (frule listall_conj1)
  apply (drule listall_conj2)
  apply (drule_tac i=i in lift_terms_NF)
  apply assumption
  apply (rule_tac m=x and n=i in nat_le_dec [THEN disjE, standard])
  apply simp
  apply (rule NF.App)
  apply assumption
  apply simp
  apply (rule NF.App)
  apply assumption
  apply simp
  apply (rule NF.Abs)
  apply simp
  done


subsection {* Main theorems *}

lemma norm_list:
  assumes f_compat: "\<And>t t'. t \<rightarrow>\<^sub>\<beta>\<^sup>* t' \<Longrightarrow> f t \<rightarrow>\<^sub>\<beta>\<^sup>* f t'"
  and f_NF: "\<And>t. NF t \<Longrightarrow> NF (f t)"
  and uNF: "NF u" and uT: "e \<turnstile> u : T"
  shows "\<And>Us. e\<langle>i:T\<rangle> \<tturnstile> as : Us \<Longrightarrow>
    listall (\<lambda>t. \<forall>e T' u i. e\<langle>i:T\<rangle> \<turnstile> t : T' \<longrightarrow>
      NF u \<longrightarrow> e \<turnstile> u : T \<longrightarrow> (\<exists>t'. t[u/i] \<rightarrow>\<^sub>\<beta>\<^sup>* t' \<and> NF t')) as \<Longrightarrow>
    \<exists>as'. \<forall>j. Var j \<degree>\<degree> map (\<lambda>t. f (t[u/i])) as \<rightarrow>\<^sub>\<beta>\<^sup>*
      Var j \<degree>\<degree> map f as' \<and> NF (Var j \<degree>\<degree> map f as')"
  (is "\<And>Us. _ \<Longrightarrow> listall ?R as \<Longrightarrow> \<exists>as'. ?ex Us as as'")
proof (induct as rule: rev_induct)
  case (Nil Us)
  with Var_NF have "?ex Us [] []" by simp
  thus ?case ..
next
  case (snoc b bs Us)
  have "e\<langle>i:T\<rangle> \<tturnstile> bs  @ [b] : Us" .
  then obtain Vs W where Us: "Us = Vs @ [W]"
    and bs: "e\<langle>i:T\<rangle> \<tturnstile> bs : Vs" and bT: "e\<langle>i:T\<rangle> \<turnstile> b : W"
    by (rule types_snocE)
  from snoc have "listall ?R bs" by simp
  with bs have "\<exists>bs'. ?ex Vs bs bs'" by (rule snoc)
  then obtain bs' where
    bsred: "\<And>j. Var j \<degree>\<degree> map (\<lambda>t. f (t[u/i])) bs \<rightarrow>\<^sub>\<beta>\<^sup>* Var j \<degree>\<degree> map f bs'"
    and bsNF: "\<And>j. NF (Var j \<degree>\<degree> map f bs')" by iprover
  from snoc have "?R b" by simp
  with bT and uNF and uT have "\<exists>b'. b[u/i] \<rightarrow>\<^sub>\<beta>\<^sup>* b' \<and> NF b'"
    by iprover
  then obtain b' where bred: "b[u/i] \<rightarrow>\<^sub>\<beta>\<^sup>* b'" and bNF: "NF b'"
    by iprover
  from bsNF [of 0] have "listall NF (map f bs')"
    by (rule App_NF_D)
  moreover have "NF (f b')" by (rule f_NF)
  ultimately have "listall NF (map f (bs' @ [b']))"
    by simp
  hence "\<And>j. NF (Var j \<degree>\<degree> map f (bs' @ [b']))" by (rule NF.App)
  moreover from bred have "f (b[u/i]) \<rightarrow>\<^sub>\<beta>\<^sup>* f b'"
    by (rule f_compat)
  with bsred have
    "\<And>j. (Var j \<degree>\<degree> map (\<lambda>t. f (t[u/i])) bs) \<degree> f (b[u/i]) \<rightarrow>\<^sub>\<beta>\<^sup>*
     (Var j \<degree>\<degree> map f bs') \<degree> f b'" by (rule rtrancl_beta_App)
  ultimately have "?ex Us (bs @ [b]) (bs' @ [b'])" by simp
  thus ?case ..
qed

lemma subst_type_NF:
  "\<And>t e T u i. NF t \<Longrightarrow> e\<langle>i:U\<rangle> \<turnstile> t : T \<Longrightarrow> NF u \<Longrightarrow> e \<turnstile> u : U \<Longrightarrow> \<exists>t'. t[u/i] \<rightarrow>\<^sub>\<beta>\<^sup>* t' \<and> NF t'"
  (is "PROP ?P U" is "\<And>t e T u i. _ \<Longrightarrow> PROP ?Q t e T u i U")
proof (induct U)
  fix T t
  let ?R = "\<lambda>t. \<forall>e T' u i.
    e\<langle>i:T\<rangle> \<turnstile> t : T' \<longrightarrow> NF u \<longrightarrow> e \<turnstile> u : T \<longrightarrow> (\<exists>t'. t[u/i] \<rightarrow>\<^sub>\<beta>\<^sup>* t' \<and> NF t')"
  assume MI1: "\<And>T1 T2. T = T1 \<Rightarrow> T2 \<Longrightarrow> PROP ?P T1"
  assume MI2: "\<And>T1 T2. T = T1 \<Rightarrow> T2 \<Longrightarrow> PROP ?P T2"
  assume "NF t"
  thus "\<And>e T' u i. PROP ?Q t e T' u i T"
  proof induct
    fix e T' u i assume uNF: "NF u" and uT: "e \<turnstile> u : T"
    {
      case (App ts x e_ T'_ u_ i_)
      assume "e\<langle>i:T\<rangle> \<turnstile> Var x \<degree>\<degree> ts : T'"
      then obtain Us
	where varT: "e\<langle>i:T\<rangle> \<turnstile> Var x : Us \<Rrightarrow> T'"
	and argsT: "e\<langle>i:T\<rangle> \<tturnstile> ts : Us"
	by (rule var_app_typesE)
      from nat_eq_dec show "\<exists>t'. (Var x \<degree>\<degree> ts)[u/i] \<rightarrow>\<^sub>\<beta>\<^sup>* t' \<and> NF t'"
      proof
	assume eq: "x = i"
	show ?thesis
	proof (cases ts)
	  case Nil
	  with eq have "(Var x \<degree>\<degree> [])[u/i] \<rightarrow>\<^sub>\<beta>\<^sup>* u" by simp
	  with Nil and uNF show ?thesis by simp iprover
	next
	  case (Cons a as)
          with argsT obtain T'' Ts where Us: "Us = T'' # Ts"
	    by (cases Us) (rule FalseE, simp+)
	  from varT and Us have varT: "e\<langle>i:T\<rangle> \<turnstile> Var x : T'' \<Rightarrow> Ts \<Rrightarrow> T'"
	    by simp
          from varT eq have T: "T = T'' \<Rightarrow> Ts \<Rrightarrow> T'" by cases auto
          with uT have uT': "e \<turnstile> u : T'' \<Rightarrow> Ts \<Rrightarrow> T'" by simp
	  from argsT Us Cons have argsT': "e\<langle>i:T\<rangle> \<tturnstile> as : Ts" by simp
	  from argsT Us Cons have argT: "e\<langle>i:T\<rangle> \<turnstile> a : T''" by simp
	  from argT uT refl have aT: "e \<turnstile> a[u/i] : T''" by (rule subst_lemma)
	  from App and Cons have "listall ?R as" by simp (iprover dest: listall_conj2)
	  with lift_preserves_beta' lift_NF uNF uT argsT'
	  have "\<exists>as'. \<forall>j. Var j \<degree>\<degree> map (\<lambda>t. lift (t[u/i]) 0) as \<rightarrow>\<^sub>\<beta>\<^sup>*
            Var j \<degree>\<degree> map (\<lambda>t. lift t 0) as' \<and>
	    NF (Var j \<degree>\<degree> map (\<lambda>t. lift t 0) as')" by (rule norm_list)
	  then obtain as' where
	    asred: "Var 0 \<degree>\<degree> map (\<lambda>t. lift (t[u/i]) 0) as \<rightarrow>\<^sub>\<beta>\<^sup>*
	      Var 0 \<degree>\<degree> map (\<lambda>t. lift t 0) as'"
	    and asNF: "NF (Var 0 \<degree>\<degree> map (\<lambda>t. lift t 0) as')" by iprover
	  from App and Cons have "?R a" by simp
	  with argT and uNF and uT have "\<exists>a'. a[u/i] \<rightarrow>\<^sub>\<beta>\<^sup>* a' \<and> NF a'"
	    by iprover
	  then obtain a' where ared: "a[u/i] \<rightarrow>\<^sub>\<beta>\<^sup>* a'" and aNF: "NF a'" by iprover
	  from uNF have "NF (lift u 0)" by (rule lift_NF)
	  hence "\<exists>u'. lift u 0 \<degree> Var 0 \<rightarrow>\<^sub>\<beta>\<^sup>* u' \<and> NF u'" by (rule app_Var_NF)
	  then obtain u' where ured: "lift u 0 \<degree> Var 0 \<rightarrow>\<^sub>\<beta>\<^sup>* u'" and u'NF: "NF u'"
	    by iprover
	  from T and u'NF have "\<exists>ua. u'[a'/0] \<rightarrow>\<^sub>\<beta>\<^sup>* ua \<and> NF ua"
	  proof (rule MI1)
	    have "e\<langle>0:T''\<rangle> \<turnstile> lift u 0 \<degree> Var 0 : Ts \<Rrightarrow> T'"
	    proof (rule typing.App)
	      from uT' show "e\<langle>0:T''\<rangle> \<turnstile> lift u 0 : T'' \<Rightarrow> Ts \<Rrightarrow> T'" by (rule lift_type)
	      show "e\<langle>0:T''\<rangle> \<turnstile> Var 0 : T''" by (rule typing.Var) simp
	    qed
	    with ured show "e\<langle>0:T''\<rangle> \<turnstile> u' : Ts \<Rrightarrow> T'" by (rule subject_reduction')
	    from ared aT show "e \<turnstile> a' : T''" by (rule subject_reduction')
	  qed
	  then obtain ua where uared: "u'[a'/0] \<rightarrow>\<^sub>\<beta>\<^sup>* ua" and uaNF: "NF ua"
	    by iprover
	  from ared have "(lift u 0 \<degree> Var 0)[a[u/i]/0] \<rightarrow>\<^sub>\<beta>\<^sup>* (lift u 0 \<degree> Var 0)[a'/0]"
	    by (rule subst_preserves_beta2')
	  also from ured have "(lift u 0 \<degree> Var 0)[a'/0] \<rightarrow>\<^sub>\<beta>\<^sup>* u'[a'/0]"
	    by (rule subst_preserves_beta')
	  also note uared
	  finally have "(lift u 0 \<degree> Var 0)[a[u/i]/0] \<rightarrow>\<^sub>\<beta>\<^sup>* ua" .
	  hence uared': "u \<degree> a[u/i] \<rightarrow>\<^sub>\<beta>\<^sup>* ua" by simp
	  from T have "\<exists>r. (Var 0 \<degree>\<degree> map (\<lambda>t. lift t 0) as')[ua/0] \<rightarrow>\<^sub>\<beta>\<^sup>* r \<and> NF r"
	  proof (rule MI2)
	    have "e\<langle>0:Ts \<Rrightarrow> T'\<rangle> \<turnstile> Var 0 \<degree>\<degree> map (\<lambda>t. lift (t[u/i]) 0) as : T'"
	    proof (rule list_app_typeI)
	      show "e\<langle>0:Ts \<Rrightarrow> T'\<rangle> \<turnstile> Var 0 : Ts \<Rrightarrow> T'" by (rule typing.Var) simp
	      from uT argsT' have "e \<tturnstile> map (\<lambda>t. t[u/i]) as : Ts"
		by (rule substs_lemma)
	      hence "e\<langle>0:Ts \<Rrightarrow> T'\<rangle> \<tturnstile> map (\<lambda>t. lift t 0) (map (\<lambda>t. t[u/i]) as) : Ts"
		by (rule lift_types)
	      thus "e\<langle>0:Ts \<Rrightarrow> T'\<rangle> \<tturnstile> map (\<lambda>t. lift (t[u/i]) 0) as : Ts"
		by (simp_all add: map_compose [symmetric] o_def)
	    qed
	    with asred show "e\<langle>0:Ts \<Rrightarrow> T'\<rangle> \<turnstile> Var 0 \<degree>\<degree> map (\<lambda>t. lift t 0) as' : T'"
	      by (rule subject_reduction')
	    from argT uT refl have "e \<turnstile> a[u/i] : T''" by (rule subst_lemma)
	    with uT' have "e \<turnstile> u \<degree> a[u/i] : Ts \<Rrightarrow> T'" by (rule typing.App)
	    with uared' show "e \<turnstile> ua : Ts \<Rrightarrow> T'" by (rule subject_reduction')
	  qed
	  then obtain r where rred: "(Var 0 \<degree>\<degree> map (\<lambda>t. lift t 0) as')[ua/0] \<rightarrow>\<^sub>\<beta>\<^sup>* r"
	    and rnf: "NF r" by iprover
	  from asred have
	    "(Var 0 \<degree>\<degree> map (\<lambda>t. lift (t[u/i]) 0) as)[u \<degree> a[u/i]/0] \<rightarrow>\<^sub>\<beta>\<^sup>*
	    (Var 0 \<degree>\<degree> map (\<lambda>t. lift t 0) as')[u \<degree> a[u/i]/0]"
	    by (rule subst_preserves_beta')
	  also from uared' have "(Var 0 \<degree>\<degree> map (\<lambda>t. lift t 0) as')[u \<degree> a[u/i]/0] \<rightarrow>\<^sub>\<beta>\<^sup>*
	    (Var 0 \<degree>\<degree> map (\<lambda>t. lift t 0) as')[ua/0]" by (rule subst_preserves_beta2')
	  also note rred
	  finally have "(Var 0 \<degree>\<degree> map (\<lambda>t. lift (t[u/i]) 0) as)[u \<degree> a[u/i]/0] \<rightarrow>\<^sub>\<beta>\<^sup>* r" .
	  with rnf Cons eq show ?thesis
	    by (simp add: map_compose [symmetric] o_def) iprover
	qed
      next
	assume neq: "x \<noteq> i"
	from App have "listall ?R ts" by (iprover dest: listall_conj2)
	with TrueI TrueI uNF uT argsT
	have "\<exists>ts'. \<forall>j. Var j \<degree>\<degree> map (\<lambda>t. t[u/i]) ts \<rightarrow>\<^sub>\<beta>\<^sup>* Var j \<degree>\<degree> ts' \<and>
	  NF (Var j \<degree>\<degree> ts')" (is "\<exists>ts'. ?ex ts'")
	  by (rule norm_list [of "\<lambda>t. t", simplified])
	then obtain ts' where NF: "?ex ts'" ..
	from nat_le_dec show ?thesis
	proof
	  assume "i < x"
	  with NF show ?thesis by simp iprover
	next
	  assume "\<not> (i < x)"
	  with NF neq show ?thesis by (simp add: subst_Var) iprover
	qed
      qed
    next
      case (Abs r e_ T'_ u_ i_)
      assume absT: "e\<langle>i:T\<rangle> \<turnstile> Abs r : T'"
      then obtain R S where "e\<langle>0:R\<rangle>\<langle>Suc i:T\<rangle>  \<turnstile> r : S" by (rule abs_typeE) simp
      moreover have "NF (lift u 0)" by (rule lift_NF)
      moreover have "e\<langle>0:R\<rangle> \<turnstile> lift u 0 : T" by (rule lift_type)
      ultimately have "\<exists>t'. r[lift u 0/Suc i] \<rightarrow>\<^sub>\<beta>\<^sup>* t' \<and> NF t'" by (rule Abs)
      thus "\<exists>t'. Abs r[u/i] \<rightarrow>\<^sub>\<beta>\<^sup>* t' \<and> NF t'"
	by simp (iprover intro: rtrancl_beta_Abs NF.Abs)
    }
  qed
qed


-- {* A computationally relevant copy of @{term "e \<turnstile> t : T"} *}
inductive2 rtyping :: "(nat \<Rightarrow> type) \<Rightarrow> dB \<Rightarrow> type \<Rightarrow> bool"  ("_ \<turnstile>\<^sub>R _ : _" [50, 50, 50] 50)
  where
    Var: "e x = T \<Longrightarrow> e \<turnstile>\<^sub>R Var x : T"
  | Abs: "e\<langle>0:T\<rangle> \<turnstile>\<^sub>R t : U \<Longrightarrow> e \<turnstile>\<^sub>R Abs t : (T \<Rightarrow> U)"
  | App: "e \<turnstile>\<^sub>R s : T \<Rightarrow> U \<Longrightarrow> e \<turnstile>\<^sub>R t : T \<Longrightarrow> e \<turnstile>\<^sub>R (s \<degree> t) : U"

lemma rtyping_imp_typing: "e \<turnstile>\<^sub>R t : T \<Longrightarrow> e \<turnstile> t : T"
  apply (induct set: rtyping)
  apply (erule typing.Var)
  apply (erule typing.Abs)
  apply (erule typing.App)
  apply assumption
  done


theorem type_NF:
  assumes "e \<turnstile>\<^sub>R t : T"
  shows "\<exists>t'. t \<rightarrow>\<^sub>\<beta>\<^sup>* t' \<and> NF t'" using prems
proof induct
  case Var
  show ?case by (iprover intro: Var_NF)
next
  case Abs
  thus ?case by (iprover intro: rtrancl_beta_Abs NF.Abs)
next
  case (App e s T U t)
  from App obtain s' t' where
    sred: "s \<rightarrow>\<^sub>\<beta>\<^sup>* s'" and sNF: "NF s'"
    and tred: "t \<rightarrow>\<^sub>\<beta>\<^sup>* t'" and tNF: "NF t'" by iprover
  have "\<exists>u. (Var 0 \<degree> lift t' 0)[s'/0] \<rightarrow>\<^sub>\<beta>\<^sup>* u \<and> NF u"
  proof (rule subst_type_NF)
    have "NF (lift t' 0)" by (rule lift_NF)
    hence "listall NF [lift t' 0]" by (rule listall_cons) (rule listall_nil)
    hence "NF (Var 0 \<degree>\<degree> [lift t' 0])" by (rule NF.App)
    thus "NF (Var 0 \<degree> lift t' 0)" by simp
    show "e\<langle>0:T \<Rightarrow> U\<rangle> \<turnstile> Var 0 \<degree> lift t' 0 : U"
    proof (rule typing.App)
      show "e\<langle>0:T \<Rightarrow> U\<rangle> \<turnstile> Var 0 : T \<Rightarrow> U"
      	by (rule typing.Var) simp
      from tred have "e \<turnstile> t' : T"
      	by (rule subject_reduction') (rule rtyping_imp_typing)
      thus "e\<langle>0:T \<Rightarrow> U\<rangle> \<turnstile> lift t' 0 : T"
      	by (rule lift_type)
    qed
    from sred show "e \<turnstile> s' : T \<Rightarrow> U"
      by (rule subject_reduction') (rule rtyping_imp_typing)
  qed
  then obtain u where ured: "s' \<degree> t' \<rightarrow>\<^sub>\<beta>\<^sup>* u" and unf: "NF u" by simp iprover
  from sred tred have "s \<degree> t \<rightarrow>\<^sub>\<beta>\<^sup>* s' \<degree> t'" by (rule rtrancl_beta_App)
  hence "s \<degree> t \<rightarrow>\<^sub>\<beta>\<^sup>* u" using ured by (rule rtrancl_trans')
  with unf show ?case by iprover
qed


subsection {* Extracting the program *}

declare NF.induct [ind_realizer]
declare rtrancl.induct [ind_realizer irrelevant]
declare rtyping.induct [ind_realizer]
lemmas [extraction_expand] = conj_assoc listall_cons_eq

extract type_NF

lemma rtranclR_rtrancl_eq: "rtranclR r a b = rtrancl r a b"
  apply (rule iffI)
  apply (erule rtranclR.induct)
  apply (rule rtrancl.rtrancl_refl)
  apply (erule rtrancl.rtrancl_into_rtrancl)
  apply assumption
  apply (erule rtrancl.induct)
  apply (rule rtranclR.rtrancl_refl)
  apply (erule rtranclR.rtrancl_into_rtrancl)
  apply assumption
  done

lemma NFR_imp_NF: "NFR nf t \<Longrightarrow> NF t"
  apply (erule NFR.induct)
  apply (rule NF.intros)
  apply (simp add: listall_def)
  apply (erule NF.intros)
  done

text_raw {*
\begin{figure}
\renewcommand{\isastyle}{\scriptsize\it}%
@{thm [display,eta_contract=false,margin=100] subst_type_NF_def}
\renewcommand{\isastyle}{\small\it}%
\caption{Program extracted from @{text subst_type_NF}}
\label{fig:extr-subst-type-nf}
\end{figure}

\begin{figure}
\renewcommand{\isastyle}{\scriptsize\it}%
@{thm [display,margin=100] subst_Var_NF_def}
@{thm [display,margin=100] app_Var_NF_def}
@{thm [display,margin=100] lift_NF_def}
@{thm [display,eta_contract=false,margin=100] type_NF_def}
\renewcommand{\isastyle}{\small\it}%
\caption{Program extracted from lemmas and main theorem}
\label{fig:extr-type-nf}
\end{figure}
*}

text {*
The program corresponding to the proof of the central lemma, which
performs substitution and normalization, is shown in Figure
\ref{fig:extr-subst-type-nf}. The correctness
theorem corresponding to the program @{text "subst_type_NF"} is
@{thm [display,margin=100] subst_type_NF_correctness
  [simplified rtranclR_rtrancl_eq Collect_mem_eq, no_vars]}
where @{text NFR} is the realizability predicate corresponding to
the datatype @{text NFT}, which is inductively defined by the rules
\pagebreak
@{thm [display,margin=90] NFR.App [of ts nfs x] NFR.Abs [of nf t]}

The programs corresponding to the main theorem @{text "type_NF"}, as
well as to some lemmas, are shown in Figure \ref{fig:extr-type-nf}.
The correctness statement for the main function @{text "type_NF"} is
@{thm [display,margin=100] type_NF_correctness
  [simplified rtranclR_rtrancl_eq Collect_mem_eq, no_vars]}
where the realizability predicate @{text "rtypingR"} corresponding to the
computationally relevant version of the typing judgement is inductively
defined by the rules
@{thm [display,margin=100] rtypingR.Var [no_vars]
  rtypingR.Abs [of ty, no_vars] rtypingR.App [of ty e s T U ty' t]}
*}

subsection {* Generating executable code *}

consts_code
  "arbitrary :: 'a"       ("(error \"arbitrary\")")
  "arbitrary :: 'a \<Rightarrow> 'b" ("(fn '_ => error \"arbitrary\")")

code_module Norm
contains
  test = "type_NF"

text {*
The following functions convert between Isabelle's built-in {\tt term}
datatype and the generated {\tt dB} datatype. This allows to
generate example terms using Isabelle's parser and inspect
normalized terms using Isabelle's pretty printer.
*}

ML {*
fun nat_of_int 0 = Norm.zero
  | nat_of_int n = Norm.Suc (nat_of_int (n-1));

fun int_of_nat Norm.zero = 0
  | int_of_nat (Norm.Suc n) = 1 + int_of_nat n;

fun dBtype_of_typ (Type ("fun", [T, U])) =
      Norm.Fun (dBtype_of_typ T, dBtype_of_typ U)
  | dBtype_of_typ (TFree (s, _)) = (case explode s of
        ["'", a] => Norm.Atom (nat_of_int (ord a - 97))
      | _ => error "dBtype_of_typ: variable name")
  | dBtype_of_typ _ = error "dBtype_of_typ: bad type";

fun dB_of_term (Bound i) = Norm.dB_Var (nat_of_int i)
  | dB_of_term (t $ u) = Norm.App (dB_of_term t, dB_of_term u)
  | dB_of_term (Abs (_, _, t)) = Norm.Abs (dB_of_term t)
  | dB_of_term _ = error "dB_of_term: bad term";

fun term_of_dB Ts (Type ("fun", [T, U])) (Norm.Abs dBt) =
      Abs ("x", T, term_of_dB (T :: Ts) U dBt)
  | term_of_dB Ts _ dBt = term_of_dB' Ts dBt
and term_of_dB' Ts (Norm.dB_Var n) = Bound (int_of_nat n)
  | term_of_dB' Ts (Norm.App (dBt, dBu)) =
      let val t = term_of_dB' Ts dBt
      in case fastype_of1 (Ts, t) of
          Type ("fun", [T, U]) => t $ term_of_dB Ts T dBu
        | _ => error "term_of_dB: function type expected"
      end
  | term_of_dB' _ _ = error "term_of_dB: term not in normal form";

fun typing_of_term Ts e (Bound i) =
      Norm.Var (e, nat_of_int i, dBtype_of_typ (List.nth (Ts, i)))
  | typing_of_term Ts e (t $ u) = (case fastype_of1 (Ts, t) of
        Type ("fun", [T, U]) => Norm.rtypingT_App (e, dB_of_term t,
          dBtype_of_typ T, dBtype_of_typ U, dB_of_term u,
          typing_of_term Ts e t, typing_of_term Ts e u)
      | _ => error "typing_of_term: function type expected")
  | typing_of_term Ts e (Abs (s, T, t)) =
      let val dBT = dBtype_of_typ T
      in Norm.rtypingT_Abs (e, dBT, dB_of_term t,
        dBtype_of_typ (fastype_of1 (T :: Ts, t)),
        typing_of_term (T :: Ts) (Norm.shift e Norm.zero dBT) t)
      end
  | typing_of_term _ _ _ = error "typing_of_term: bad term";

fun dummyf _ = error "dummy";
*}

text {*
We now try out the extracted program @{text "type_NF"} on some example terms.
*}

ML {*
val ct1 = @{cterm "%f. ((%f x. f (f (f x))) ((%f x. f (f (f (f x)))) f))"};
val (dB1, _) = Norm.type_NF (typing_of_term [] dummyf (term_of ct1));
val ct1' = cterm_of @{theory} (term_of_dB [] (#T (rep_cterm ct1)) dB1);

val ct2 = @{cterm "%f x. (%x. f x x) ((%x. f x x) ((%x. f x x) ((%x. f x x) ((%x. f x x) ((%x. f x x) x)))))"};
val (dB2, _) = Norm.type_NF (typing_of_term [] dummyf (term_of ct2));
val ct2' = cterm_of @{theory} (term_of_dB [] (#T (rep_cterm ct2)) dB2);
*}

text {*
The same story again for code next generation.
*}

setup {*
  CodegenSerializer.add_undefined "SML" "arbitrary" "(raise Fail \"arbitrary\")"
*}

definition
  int :: "nat \<Rightarrow> int" where
  "int \<equiv> of_nat"

code_gen type_NF nat int in SML

ML {*
structure Norm = ROOT.WeakNorm;
structure Type = ROOT.Type;
structure Lambda = ROOT.Lambda;
structure Nat = ROOT.Nat;

val nat_of_int = ROOT.Integer.nat o IntInf.fromInt;
val int_of_nat = IntInf.toInt o ROOT.WeakNorm.int;

fun dBtype_of_typ (Type ("fun", [T, U])) =
      Type.Fun (dBtype_of_typ T, dBtype_of_typ U)
  | dBtype_of_typ (TFree (s, _)) = (case explode s of
        ["'", a] => Type.Atom (nat_of_int (ord a - 97))
      | _ => error "dBtype_of_typ: variable name")
  | dBtype_of_typ _ = error "dBtype_of_typ: bad type";

fun dB_of_term (Bound i) = Lambda.Var (nat_of_int i)
  | dB_of_term (t $ u) = Lambda.App (dB_of_term t, dB_of_term u)
  | dB_of_term (Abs (_, _, t)) = Lambda.Abs (dB_of_term t)
  | dB_of_term _ = error "dB_of_term: bad term";

fun term_of_dB Ts (Type ("fun", [T, U])) (Lambda.Abs dBt) =
      Abs ("x", T, term_of_dB (T :: Ts) U dBt)
  | term_of_dB Ts _ dBt = term_of_dB' Ts dBt
and term_of_dB' Ts (Lambda.Var n) = Bound (int_of_nat n)
  | term_of_dB' Ts (Lambda.App (dBt, dBu)) =
      let val t = term_of_dB' Ts dBt
      in case fastype_of1 (Ts, t) of
          Type ("fun", [T, U]) => t $ term_of_dB Ts T dBu
        | _ => error "term_of_dB: function type expected"
      end
  | term_of_dB' _ _ = error "term_of_dB: term not in normal form";

fun typing_of_term Ts e (Bound i) =
      Norm.Var (e, nat_of_int i, dBtype_of_typ (nth Ts i))
  | typing_of_term Ts e (t $ u) = (case fastype_of1 (Ts, t) of
        Type ("fun", [T, U]) => Norm.Appa (e, dB_of_term t,
          dBtype_of_typ T, dBtype_of_typ U, dB_of_term u,
          typing_of_term Ts e t, typing_of_term Ts e u)
      | _ => error "typing_of_term: function type expected")
  | typing_of_term Ts e (Abs (s, T, t)) =
      let val dBT = dBtype_of_typ T
      in Norm.Absa (e, dBT, dB_of_term t,
        dBtype_of_typ (fastype_of1 (T :: Ts, t)),
        typing_of_term (T :: Ts) (Type.shift e Nat.Zero_nat dBT) t)
      end
  | typing_of_term _ _ _ = error "typing_of_term: bad term";

fun dummyf _ = error "dummy";
*}

ML {*
val ct1 = @{cterm "%f. ((%f x. f (f (f x))) ((%f x. f (f (f (f x)))) f))"};
val (dB1, _) = ROOT.WeakNorm.type_NF (typing_of_term [] dummyf (term_of ct1));
val ct1' = cterm_of @{theory} (term_of_dB [] (#T (rep_cterm ct1)) dB1);

val ct2 = @{cterm "%f x. (%x. f x x) ((%x. f x x) ((%x. f x x) ((%x. f x x) ((%x. f x x) ((%x. f x x) x)))))"};
val (dB2, _) = ROOT.WeakNorm.type_NF (typing_of_term [] dummyf (term_of ct2));
val ct2' = cterm_of @{theory} (term_of_dB [] (#T (rep_cterm ct2)) dB2);
*}

end