(*  ID:         $Id$

     Author:     Tobias Nipkow

     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

     Author:     Konrad Slind, Alexander Krauss

     Copyright   1992-2008  University of Cambridge and TU Muenchen

 *)

 header {*Well-founded Recursion*}

 theory Wellfounded

 imports Finite_Set Nat

 uses ("Tools/function_package/size.ML")

 begin

 subsection {* Basic Definitions *}

 inductive

   wfrec_rel :: "('a * 'a) set => (('a => 'b) => 'a => 'b) => 'a => 'b => bool"

   for R :: "('a * 'a) set"

   and F :: "('a => 'b) => 'a => 'b"

 where

   wfrecI: "ALL z. (z, x) : R --> wfrec_rel R F z (g z) ==>

             wfrec_rel R F x (F g x)"

 constdefs

   wf         :: "('a * 'a)set => bool"

   "wf(r) == (!P. (!x. (!y. (y,x):r --> P(y)) --> P(x)) --> (!x. P(x)))"

   wfP :: "('a => 'a => bool) => bool"

   "wfP r == wf {(x, y). r x y}"

   acyclic :: "('a*'a)set => bool"

   "acyclic r == !x. (x,x) ~: r^+"

   cut        :: "('a => 'b) => ('a * 'a)set => 'a => 'a => 'b"

   "cut f r x == (%y. if (y,x):r then f y else arbitrary)"

   adm_wf :: "('a * 'a) set => (('a => 'b) => 'a => 'b) => bool"

   "adm_wf R F == ALL f g x.

      (ALL z. (z, x) : R --> f z = g z) --> F f x = F g x"

   wfrec :: "('a * 'a) set => (('a => 'b) => 'a => 'b) => 'a => 'b"

   [code func del]: "wfrec R F == %x. THE y. wfrec_rel R (%f x. F (cut f R x) x) x y"

 abbreviation acyclicP :: "('a => 'a => bool) => bool" where

   "acyclicP r == acyclic {(x, y). r x y}"

 lemma wfP_wf_eq [pred_set_conv]: "wfP (\<lambda>x y. (x, y) \<in> r) = wf r"

   by (simp add: wfP_def)

 lemma wfUNIVI: 

    "(!!P x. (ALL x. (ALL y. (y,x) : r --> P(y)) --> P(x)) ==> P(x)) ==> wf(r)"

   unfolding wf_def by blast

 lemmas wfPUNIVI = wfUNIVI [to_pred]

 text{*Restriction to domain @{term A} and range @{term B}.  If @{term r} is

     well-founded over their intersection, then @{term "wf r"}*}

 lemma wfI: 

  "[| r \<subseteq> A <*> B; 

      !!x P. [|\<forall>x. (\<forall>y. (y,x) : r --> P y) --> P x;  x : A; x : B |] ==> P x |]

   ==>  wf r"

   unfolding wf_def by blast

 lemma wf_induct: 

     "[| wf(r);           

         !!x.[| ALL y. (y,x): r --> P(y) |] ==> P(x)  

      |]  ==>  P(a)"

   unfolding wf_def by blast

 lemmas wfP_induct = wf_induct [to_pred]

 lemmas wf_induct_rule = wf_induct [rule_format, consumes 1, case_names less, induct set: wf]

 lemmas wfP_induct_rule = wf_induct_rule [to_pred, induct set: wfP]

 lemma wf_not_sym: "wf r ==> (a, x) : r ==> (x, a) ~: r"

   by (induct a arbitrary: x set: wf) blast

 (* [| wf r;  ~Z ==> (a,x) : r;  (x,a) ~: r ==> Z |] ==> Z *)

 lemmas wf_asym = wf_not_sym [elim_format]

 lemma wf_not_refl [simp]: "wf r ==> (a, a) ~: r"

   by (blast elim: wf_asym)

 (* [| wf r;  (a,a) ~: r ==> PROP W |] ==> PROP W *)

 lemmas wf_irrefl = wf_not_refl [elim_format]

 lemma wf_wellorderI:

   assumes wf: "wf {(x::'a::ord, y). x < y}"

   assumes lin: "OFCLASS('a::ord, linorder_class)"

   shows "OFCLASS('a::ord, wellorder_class)"

 using lin by (rule wellorder_class.intro)

   (blast intro: wellorder_axioms.intro wf_induct_rule [OF wf])

 lemma (in wellorder) wf:

   "wf {(x, y). x < y}"

 unfolding wf_def by (blast intro: less_induct)

 subsection {* Basic Results *}

 text{*transitive closure of a well-founded relation is well-founded! *}

 lemma wf_trancl:

   assumes "wf r"

   shows "wf (r^+)"

 proof -

   {

     fix P and x

     assume induct_step: "!!x. (!!y. (y, x) : r^+ ==> P y) ==> P x"

     have "P x"

     proof (rule induct_step)

       fix y assume "(y, x) : r^+"

       with `wf r` show "P y"

       proof (induct x arbitrary: y)

 	case (less x)

 	note hyp = `\<And>x' y'. (x', x) : r ==> (y', x') : r^+ ==> P y'`

 	from `(y, x) : r^+` show "P y"

 	proof cases

 	  case base

 	  show "P y"

 	  proof (rule induct_step)

 	    fix y' assume "(y', y) : r^+"

 	    with `(y, x) : r` show "P y'" by (rule hyp [of y y'])

 	  qed

 	next

 	  case step

 	  then obtain x' where "(x', x) : r" and "(y, x') : r^+" by simp

 	  then show "P y" by (rule hyp [of x' y])

 	qed

       qed

     qed

   } then show ?thesis unfolding wf_def by blast

 qed

 lemmas wfP_trancl = wf_trancl [to_pred]

 lemma wf_converse_trancl: "wf (r^-1) ==> wf ((r^+)^-1)"

   apply (subst trancl_converse [symmetric])

   apply (erule wf_trancl)

   done

 text{*Minimal-element characterization of well-foundedness*}

 lemma wf_eq_minimal: "wf r = (\<forall>Q x. x\<in>Q --> (\<exists>z\<in>Q. \<forall>y. (y,z)\<in>r --> y\<notin>Q))"

 proof (intro iffI strip)

   fix Q :: "'a set" and x

   assume "wf r" and "x \<in> Q"

   then show "\<exists>z\<in>Q. \<forall>y. (y, z) \<in> r \<longrightarrow> y \<notin> Q"

     unfolding wf_def

     by (blast dest: spec [of _ "%x. x\<in>Q \<longrightarrow> (\<exists>z\<in>Q. \<forall>y. (y,z) \<in> r \<longrightarrow> y\<notin>Q)"]) 

 next

   assume 1: "\<forall>Q x. x \<in> Q \<longrightarrow> (\<exists>z\<in>Q. \<forall>y. (y, z) \<in> r \<longrightarrow> y \<notin> Q)"

   show "wf r"

   proof (rule wfUNIVI)

     fix P :: "'a \<Rightarrow> bool" and x

     assume 2: "\<forall>x. (\<forall>y. (y, x) \<in> r \<longrightarrow> P y) \<longrightarrow> P x"

     let ?Q = "{x. \<not> P x}"

     have "x \<in> ?Q \<longrightarrow> (\<exists>z \<in> ?Q. \<forall>y. (y, z) \<in> r \<longrightarrow> y \<notin> ?Q)"

       by (rule 1 [THEN spec, THEN spec])

     then have "\<not> P x \<longrightarrow> (\<exists>z. \<not> P z \<and> (\<forall>y. (y, z) \<in> r \<longrightarrow> P y))" by simp

     with 2 have "\<not> P x \<longrightarrow> (\<exists>z. \<not> P z \<and> P z)" by fast

     then show "P x" by simp

   qed

 qed

 lemma wfE_min: 

   assumes "wf R" "x \<in> Q"

   obtains z where "z \<in> Q" "\<And>y. (y, z) \<in> R \<Longrightarrow> y \<notin> Q"

   using assms unfolding wf_eq_minimal by blast

 lemma wfI_min:

   "(\<And>x Q. x \<in> Q \<Longrightarrow> \<exists>z\<in>Q. \<forall>y. (y, z) \<in> R \<longrightarrow> y \<notin> Q)

   \<Longrightarrow> wf R"

   unfolding wf_eq_minimal by blast

 lemmas wfP_eq_minimal = wf_eq_minimal [to_pred]

 text {* Well-foundedness of subsets *}

 lemma wf_subset: "[| wf(r);  p<=r |] ==> wf(p)"

   apply (simp (no_asm_use) add: wf_eq_minimal)

   apply fast

   done

 lemmas wfP_subset = wf_subset [to_pred]

 text {* Well-foundedness of the empty relation *}

 lemma wf_empty [iff]: "wf({})"

   by (simp add: wf_def)

 lemmas wfP_empty [iff] =

   wf_empty [to_pred bot_empty_eq2, simplified bot_fun_eq bot_bool_eq]

 lemma wf_Int1: "wf r ==> wf (r Int r')"

   apply (erule wf_subset)

   apply (rule Int_lower1)

   done

 lemma wf_Int2: "wf r ==> wf (r' Int r)"

   apply (erule wf_subset)

   apply (rule Int_lower2)

   done  

 text{*Well-foundedness of insert*}

 lemma wf_insert [iff]: "wf(insert (y,x) r) = (wf(r) & (x,y) ~: r^*)"

 apply (rule iffI)

  apply (blast elim: wf_trancl [THEN wf_irrefl]

               intro: rtrancl_into_trancl1 wf_subset 

                      rtrancl_mono [THEN [2] rev_subsetD])

 apply (simp add: wf_eq_minimal, safe)

 apply (rule allE, assumption, erule impE, blast) 

 apply (erule bexE)

 apply (rename_tac "a", case_tac "a = x")

  prefer 2

 apply blast 

 apply (case_tac "y:Q")

  prefer 2 apply blast

 apply (rule_tac x = "{z. z:Q & (z,y) : r^*}" in allE)

  apply assumption

 apply (erule_tac V = "ALL Q. (EX x. x : Q) --> ?P Q" in thin_rl) 

   --{*essential for speed*}

 txt{*Blast with new substOccur fails*}

 apply (fast intro: converse_rtrancl_into_rtrancl)

 done

 text{*Well-foundedness of image*}

 lemma wf_prod_fun_image: "[| wf r; inj f |] ==> wf(prod_fun f f ` r)"

 apply (simp only: wf_eq_minimal, clarify)

 apply (case_tac "EX p. f p : Q")

 apply (erule_tac x = "{p. f p : Q}" in allE)

 apply (fast dest: inj_onD, blast)

 done

 subsection {* Well-Foundedness Results for Unions *}

 lemma wf_union_compatible:

   assumes "wf R" "wf S"

   assumes "S O R \<subseteq> R"

   shows "wf (R \<union> S)"

 proof (rule wfI_min)

   fix x :: 'a and Q 

   let ?Q' = "{x \<in> Q. \<forall>y. (y, x) \<in> R \<longrightarrow> y \<notin> Q}"

   assume "x \<in> Q"

   obtain a where "a \<in> ?Q'"

     by (rule wfE_min [OF `wf R` `x \<in> Q`]) blast

   with `wf S`

   obtain z where "z \<in> ?Q'" and zmin: "\<And>y. (y, z) \<in> S \<Longrightarrow> y \<notin> ?Q'" by (erule wfE_min)

   { 

     fix y assume "(y, z) \<in> S"

     then have "y \<notin> ?Q'" by (rule zmin)

     have "y \<notin> Q"

     proof 

       assume "y \<in> Q"

       with `y \<notin> ?Q'` 

       obtain w where "(w, y) \<in> R" and "w \<in> Q" by auto

       from `(w, y) \<in> R` `(y, z) \<in> S` have "(w, z) \<in> S O R" by (rule rel_compI)

       with `S O R \<subseteq> R` have "(w, z) \<in> R" ..

       with `z \<in> ?Q'` have "w \<notin> Q" by blast 

       with `w \<in> Q` show False by contradiction

     qed

   }

   with `z \<in> ?Q'` show "\<exists>z\<in>Q. \<forall>y. (y, z) \<in> R \<union> S \<longrightarrow> y \<notin> Q" by blast

 qed

 text {* Well-foundedness of indexed union with disjoint domains and ranges *}

 lemma wf_UN: "[| ALL i:I. wf(r i);  

          ALL i:I. ALL j:I. r i ~= r j --> Domain(r i) Int Range(r j) = {}  

       |] ==> wf(UN i:I. r i)"

 apply (simp only: wf_eq_minimal, clarify)

 apply (rename_tac A a, case_tac "EX i:I. EX a:A. EX b:A. (b,a) : r i")

  prefer 2

  apply force 

 apply clarify

 apply (drule bspec, assumption)  

 apply (erule_tac x="{a. a:A & (EX b:A. (b,a) : r i) }" in allE)

 apply (blast elim!: allE)  

 done

 lemmas wfP_SUP = wf_UN [where I=UNIV and r="\<lambda>i. {(x, y). r i x y}",

   to_pred SUP_UN_eq2 bot_empty_eq pred_equals_eq, simplified, standard]

 lemma wf_Union: 

  "[| ALL r:R. wf r;  

      ALL r:R. ALL s:R. r ~= s --> Domain r Int Range s = {}  

   |] ==> wf(Union R)"

 apply (simp add: Union_def)

 apply (blast intro: wf_UN)

 done

 (*Intuition: we find an (R u S)-min element of a nonempty subset A

              by case distinction.

   1. There is a step a -R-> b with a,b : A.

      Pick an R-min element z of the (nonempty) set {a:A | EX b:A. a -R-> b}.

      By definition, there is z':A s.t. z -R-> z'. Because z is R-min in the

      subset, z' must be R-min in A. Because z' has an R-predecessor, it cannot

      have an S-successor and is thus S-min in A as well.

   2. There is no such step.

      Pick an S-min element of A. In this case it must be an R-min

      element of A as well.

 *)

 lemma wf_Un:

      "[| wf r; wf s; Domain r Int Range s = {} |] ==> wf(r Un s)"

   using wf_union_compatible[of s r] 

   by (auto simp: Un_ac)

 lemma wf_union_merge: 

   "wf (R \<union> S) = wf (R O R \<union> R O S \<union> S)" (is "wf ?A = wf ?B")

 proof

   assume "wf ?A"

   with wf_trancl have wfT: "wf (?A^+)" .

   moreover have "?B \<subseteq> ?A^+"

     by (subst trancl_unfold, subst trancl_unfold) blast

   ultimately show "wf ?B" by (rule wf_subset)

 next

   assume "wf ?B"

   show "wf ?A"

   proof (rule wfI_min)

     fix Q :: "'a set" and x 

     assume "x \<in> Q"

     with `wf ?B`

     obtain z where "z \<in> Q" and "\<And>y. (y, z) \<in> ?B \<Longrightarrow> y \<notin> Q" 

       by (erule wfE_min)

     then have A1: "\<And>y. (y, z) \<in> R O R \<Longrightarrow> y \<notin> Q"

       and A2: "\<And>y. (y, z) \<in> R O S \<Longrightarrow> y \<notin> Q"

       and A3: "\<And>y. (y, z) \<in> S \<Longrightarrow> y \<notin> Q"

       by auto

     show "\<exists>z\<in>Q. \<forall>y. (y, z) \<in> ?A \<longrightarrow> y \<notin> Q"

     proof (cases "\<forall>y. (y, z) \<in> R \<longrightarrow> y \<notin> Q")

       case True

       with `z \<in> Q` A3 show ?thesis by blast

     next

       case False 

       then obtain z' where "z'\<in>Q" "(z', z) \<in> R" by blast

       have "\<forall>y. (y, z') \<in> ?A \<longrightarrow> y \<notin> Q"

       proof (intro allI impI)

         fix y assume "(y, z') \<in> ?A"

         then show "y \<notin> Q"

         proof

           assume "(y, z') \<in> R" 

           then have "(y, z) \<in> R O R" using `(z', z) \<in> R` ..

           with A1 show "y \<notin> Q" .

         next

           assume "(y, z') \<in> S" 

           then have "(y, z) \<in> R O S" using  `(z', z) \<in> R` ..

           with A2 show "y \<notin> Q" .

         qed

       qed

       with `z' \<in> Q` show ?thesis ..

     qed

   qed

 qed

 lemma wf_comp_self: "wf R = wf (R O R)"  -- {* special case *}

   by (rule wf_union_merge [where S = "{}", simplified])

 subsubsection {* acyclic *}

 lemma acyclicI: "ALL x. (x, x) ~: r^+ ==> acyclic r"

   by (simp add: acyclic_def)

 lemma wf_acyclic: "wf r ==> acyclic r"

 apply (simp add: acyclic_def)

 apply (blast elim: wf_trancl [THEN wf_irrefl])

 done

 lemmas wfP_acyclicP = wf_acyclic [to_pred]

 lemma acyclic_insert [iff]:

      "acyclic(insert (y,x) r) = (acyclic r & (x,y) ~: r^*)"

 apply (simp add: acyclic_def trancl_insert)

 apply (blast intro: rtrancl_trans)

 done

 lemma acyclic_converse [iff]: "acyclic(r^-1) = acyclic r"

 by (simp add: acyclic_def trancl_converse)

 lemmas acyclicP_converse [iff] = acyclic_converse [to_pred]

 lemma acyclic_impl_antisym_rtrancl: "acyclic r ==> antisym(r^*)"

 apply (simp add: acyclic_def antisym_def)

 apply (blast elim: rtranclE intro: rtrancl_into_trancl1 rtrancl_trancl_trancl)

 done

 (* Other direction:

 acyclic = no loops

 antisym = only self loops

 Goalw [acyclic_def,antisym_def] "antisym( r^* ) ==> acyclic(r - Id)

 ==> antisym( r^* ) = acyclic(r - Id)";

 *)

 lemma acyclic_subset: "[| acyclic s; r <= s |] ==> acyclic r"

 apply (simp add: acyclic_def)

 apply (blast intro: trancl_mono)

 done

 text{* Wellfoundedness of finite acyclic relations*}

 lemma finite_acyclic_wf [rule_format]: "finite r ==> acyclic r --> wf r"

 apply (erule finite_induct, blast)

 apply (simp (no_asm_simp) only: split_tupled_all)

 apply simp

 done

 lemma finite_acyclic_wf_converse: "[|finite r; acyclic r|] ==> wf (r^-1)"

 apply (erule finite_converse [THEN iffD2, THEN finite_acyclic_wf])

 apply (erule acyclic_converse [THEN iffD2])

 done

 lemma wf_iff_acyclic_if_finite: "finite r ==> wf r = acyclic r"

 by (blast intro: finite_acyclic_wf wf_acyclic)

 subsection{*Well-Founded Recursion*}

 text{*cut*}

 lemma cuts_eq: "(cut f r x = cut g r x) = (ALL y. (y,x):r --> f(y)=g(y))"

 by (simp add: expand_fun_eq cut_def)

 lemma cut_apply: "(x,a):r ==> (cut f r a)(x) = f(x)"

 by (simp add: cut_def)

 text{*Inductive characterization of wfrec combinator; for details see:  

 John Harrison, "Inductive definitions: automation and application"*}

 lemma wfrec_unique: "[| adm_wf R F; wf R |] ==> EX! y. wfrec_rel R F x y"

 apply (simp add: adm_wf_def)

 apply (erule_tac a=x in wf_induct) 

 apply (rule ex1I)

 apply (rule_tac g = "%x. THE y. wfrec_rel R F x y" in wfrec_rel.wfrecI)

 apply (fast dest!: theI')

 apply (erule wfrec_rel.cases, simp)

 apply (erule allE, erule allE, erule allE, erule mp)

 apply (fast intro: the_equality [symmetric])

 done

 lemma adm_lemma: "adm_wf R (%f x. F (cut f R x) x)"

 apply (simp add: adm_wf_def)

 apply (intro strip)

 apply (rule cuts_eq [THEN iffD2, THEN subst], assumption)

 apply (rule refl)

 done

 lemma wfrec: "wf(r) ==> wfrec r H a = H (cut (wfrec r H) r a) a"

 apply (simp add: wfrec_def)

 apply (rule adm_lemma [THEN wfrec_unique, THEN the1_equality], assumption)

 apply (rule wfrec_rel.wfrecI)

 apply (intro strip)

 apply (erule adm_lemma [THEN wfrec_unique, THEN theI'])

 done

 subsection {* Code generator setup *}

 consts_code

   "wfrec"   ("\<module>wfrec?")

 attach {*

 fun wfrec f x = f (wfrec f) x;

 *}

 subsection {* @{typ nat} is well-founded *}

 lemma less_nat_rel: "op < = (\<lambda>m n. n = Suc m)^++"

 proof (rule ext, rule ext, rule iffI)

   fix n m :: nat

   assume "m < n"

   then show "(\<lambda>m n. n = Suc m)^++ m n"

   proof (induct n)

     case 0 then show ?case by auto

   next

     case (Suc n) then show ?case

       by (auto simp add: less_Suc_eq_le le_less intro: tranclp.trancl_into_trancl)

   qed

 next

   fix n m :: nat

   assume "(\<lambda>m n. n = Suc m)^++ m n"

   then show "m < n"

     by (induct n)

       (simp_all add: less_Suc_eq_le reflexive le_less)

 qed

 definition

   pred_nat :: "(nat * nat) set" where

   "pred_nat = {(m, n). n = Suc m}"

 definition

   less_than :: "(nat * nat) set" where

   "less_than = pred_nat^+"

 lemma less_eq: "(m, n) \<in> pred_nat^+ \<longleftrightarrow> m < n"

   unfolding less_nat_rel pred_nat_def trancl_def by simp

 lemma pred_nat_trancl_eq_le:

   "(m, n) \<in> pred_nat^* \<longleftrightarrow> m \<le> n"

   unfolding less_eq rtrancl_eq_or_trancl by auto

 lemma wf_pred_nat: "wf pred_nat"

   apply (unfold wf_def pred_nat_def, clarify)

   apply (induct_tac x, blast+)

   done

 lemma wf_less_than [iff]: "wf less_than"

   by (simp add: less_than_def wf_pred_nat [THEN wf_trancl])

 lemma trans_less_than [iff]: "trans less_than"

   by (simp add: less_than_def trans_trancl)

 lemma less_than_iff [iff]: "((x,y): less_than) = (x<y)"

   by (simp add: less_than_def less_eq)

 lemma wf_less: "wf {(x, y::nat). x < y}"

   using wf_less_than by (simp add: less_than_def less_eq [symmetric])

 subsection {* Accessible Part *}

 text {*

  Inductive definition of the accessible part @{term "acc r"} of a

  relation; see also \cite{paulin-tlca}.

 *}

 inductive_set

   acc :: "('a * 'a) set => 'a set"

   for r :: "('a * 'a) set"

   where

     accI: "(!!y. (y, x) : r ==> y : acc r) ==> x : acc r"

 abbreviation

   termip :: "('a => 'a => bool) => 'a => bool" where

   "termip r == accp (r\<inverse>\<inverse>)"

 abbreviation

   termi :: "('a * 'a) set => 'a set" where

   "termi r == acc (r\<inverse>)"

 lemmas accpI = accp.accI

 text {* Induction rules *}

 theorem accp_induct:

   assumes major: "accp r a"

   assumes hyp: "!!x. accp r x ==> \<forall>y. r y x --> P y ==> P x"

   shows "P a"

   apply (rule major [THEN accp.induct])

   apply (rule hyp)

    apply (rule accp.accI)

    apply fast

   apply fast

   done

 theorems accp_induct_rule = accp_induct [rule_format, induct set: accp]

 theorem accp_downward: "accp r b ==> r a b ==> accp r a"

   apply (erule accp.cases)

   apply fast

   done

 lemma not_accp_down:

   assumes na: "\<not> accp R x"

   obtains z where "R z x" and "\<not> accp R z"

 proof -

   assume a: "\<And>z. \<lbrakk>R z x; \<not> accp R z\<rbrakk> \<Longrightarrow> thesis"

   show thesis

   proof (cases "\<forall>z. R z x \<longrightarrow> accp R z")

     case True

     hence "\<And>z. R z x \<Longrightarrow> accp R z" by auto

     hence "accp R x"

       by (rule accp.accI)

     with na show thesis ..

   next

     case False then obtain z where "R z x" and "\<not> accp R z"

       by auto

     with a show thesis .

   qed

 qed

 lemma accp_downwards_aux: "r\<^sup>*\<^sup>* b a ==> accp r a --> accp r b"

   apply (erule rtranclp_induct)

    apply blast

   apply (blast dest: accp_downward)

   done

 theorem accp_downwards: "accp r a ==> r\<^sup>*\<^sup>* b a ==> accp r b"

   apply (blast dest: accp_downwards_aux)

   done

 theorem accp_wfPI: "\<forall>x. accp r x ==> wfP r"

   apply (rule wfPUNIVI)

   apply (induct_tac P x rule: accp_induct)

    apply blast

   apply blast

   done

 theorem accp_wfPD: "wfP r ==> accp r x"

   apply (erule wfP_induct_rule)

   apply (rule accp.accI)

   apply blast

   done

 theorem wfP_accp_iff: "wfP r = (\<forall>x. accp r x)"

   apply (blast intro: accp_wfPI dest: accp_wfPD)

   done

 text {* Smaller relations have bigger accessible parts: *}

 lemma accp_subset:

   assumes sub: "R1 \<le> R2"

   shows "accp R2 \<le> accp R1"

 proof (rule predicate1I)

   fix x assume "accp R2 x"

   then show "accp R1 x"

   proof (induct x)

     fix x

     assume ih: "\<And>y. R2 y x \<Longrightarrow> accp R1 y"

     with sub show "accp R1 x"

       by (blast intro: accp.accI)

   qed

 qed

 text {* This is a generalized induction theorem that works on

   subsets of the accessible part. *}

 lemma accp_subset_induct:

   assumes subset: "D \<le> accp R"

     and dcl: "\<And>x z. \<lbrakk>D x; R z x\<rbrakk> \<Longrightarrow> D z"

     and "D x"

     and istep: "\<And>x. \<lbrakk>D x; (\<And>z. R z x \<Longrightarrow> P z)\<rbrakk> \<Longrightarrow> P x"

   shows "P x"

 proof -

   from subset and `D x`

   have "accp R x" ..

   then show "P x" using `D x`

   proof (induct x)

     fix x

     assume "D x"

       and "\<And>y. R y x \<Longrightarrow> D y \<Longrightarrow> P y"

     with dcl and istep show "P x" by blast

   qed

 qed

 text {* Set versions of the above theorems *}

 lemmas acc_induct = accp_induct [to_set]

 lemmas acc_induct_rule = acc_induct [rule_format, induct set: acc]

 lemmas acc_downward = accp_downward [to_set]

 lemmas not_acc_down = not_accp_down [to_set]

 lemmas acc_downwards_aux = accp_downwards_aux [to_set]

 lemmas acc_downwards = accp_downwards [to_set]

 lemmas acc_wfI = accp_wfPI [to_set]

 lemmas acc_wfD = accp_wfPD [to_set]

 lemmas wf_acc_iff = wfP_accp_iff [to_set]

 lemmas acc_subset = accp_subset [to_set pred_subset_eq]

 lemmas acc_subset_induct = accp_subset_induct [to_set pred_subset_eq]

 subsection {* Tools for building wellfounded relations *}

 text {* Inverse Image *}

 lemma wf_inv_image [simp,intro!]: "wf(r) ==> wf(inv_image r (f::'a=>'b))"

 apply (simp (no_asm_use) add: inv_image_def wf_eq_minimal)

 apply clarify

 apply (subgoal_tac "EX (w::'b) . w : {w. EX (x::'a) . x: Q & (f x = w) }")

 prefer 2 apply (blast del: allE)

 apply (erule allE)

 apply (erule (1) notE impE)

 apply blast

 done

 lemma in_inv_image[simp]: "((x,y) : inv_image r f) = ((f x, f y) : r)"

   by (auto simp:inv_image_def)

 text {* Measure functions into @{typ nat} *}

 definition measure :: "('a => nat) => ('a * 'a)set"

 where "measure == inv_image less_than"

 lemma in_measure[simp]: "((x,y) : measure f) = (f x < f y)"

   by (simp add:measure_def)

 lemma wf_measure [iff]: "wf (measure f)"

 apply (unfold measure_def)

 apply (rule wf_less_than [THEN wf_inv_image])

 done

 text{* Lexicographic combinations *}

 definition

  lex_prod  :: "[('a*'a)set, ('b*'b)set] => (('a*'b)*('a*'b))set"

                (infixr "<*lex*>" 80)

 where

     "ra <*lex*> rb == {((a,b),(a',b')). (a,a') : ra | a=a' & (b,b') : rb}"

 lemma wf_lex_prod [intro!]: "[| wf(ra); wf(rb) |] ==> wf(ra <*lex*> rb)"

 apply (unfold wf_def lex_prod_def) 

 apply (rule allI, rule impI)

 apply (simp (no_asm_use) only: split_paired_All)

 apply (drule spec, erule mp) 

 apply (rule allI, rule impI)

 apply (drule spec, erule mp, blast) 

 done

 lemma in_lex_prod[simp]: 

   "(((a,b),(a',b')): r <*lex*> s) = ((a,a'): r \<or> (a = a' \<and> (b, b') : s))"

   by (auto simp:lex_prod_def)

 text{* @{term "op <*lex*>"} preserves transitivity *}

 lemma trans_lex_prod [intro!]: 

     "[| trans R1; trans R2 |] ==> trans (R1 <*lex*> R2)"

 by (unfold trans_def lex_prod_def, blast) 

 text {* lexicographic combinations with measure functions *}

 definition 

   mlex_prod :: "('a \<Rightarrow> nat) \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" (infixr "<*mlex*>" 80)

 where

   "f <*mlex*> R = inv_image (less_than <*lex*> R) (%x. (f x, x))"

 lemma wf_mlex: "wf R \<Longrightarrow> wf (f <*mlex*> R)"

 unfolding mlex_prod_def

 by auto

 lemma mlex_less: "f x < f y \<Longrightarrow> (x, y) \<in> f <*mlex*> R"

 unfolding mlex_prod_def by simp

 lemma mlex_leq: "f x \<le> f y \<Longrightarrow> (x, y) \<in> R \<Longrightarrow> (x, y) \<in> f <*mlex*> R"

 unfolding mlex_prod_def by auto

 text {* proper subset relation on finite sets *}

 definition finite_psubset  :: "('a set * 'a set) set"

 where "finite_psubset == {(A,B). A < B & finite B}"

 lemma wf_finite_psubset: "wf(finite_psubset)"

 apply (unfold finite_psubset_def)

 apply (rule wf_measure [THEN wf_subset])

 apply (simp add: measure_def inv_image_def less_than_def less_eq)

 apply (fast elim!: psubset_card_mono)

 done

 lemma trans_finite_psubset: "trans finite_psubset"

 by (simp add: finite_psubset_def less_le trans_def, blast)

 text {*Wellfoundedness of @{text same_fst}*}

 definition

  same_fst :: "('a => bool) => ('a => ('b * 'b)set) => (('a*'b)*('a*'b))set"

 where

     "same_fst P R == {((x',y'),(x,y)) . x'=x & P x & (y',y) : R x}"

    --{*For @{text rec_def} declarations where the first n parameters

        stay unchanged in the recursive call. 

        See @{text "Library/While_Combinator.thy"} for an application.*}

 lemma same_fstI [intro!]:

      "[| P x; (y',y) : R x |] ==> ((x,y'),(x,y)) : same_fst P R"

 by (simp add: same_fst_def)

 lemma wf_same_fst:

   assumes prem: "(!!x. P x ==> wf(R x))"

   shows "wf(same_fst P R)"

 apply (simp cong del: imp_cong add: wf_def same_fst_def)

 apply (intro strip)

 apply (rename_tac a b)

 apply (case_tac "wf (R a)")

  apply (erule_tac a = b in wf_induct, blast)

 apply (blast intro: prem)

 done

 subsection{*Weakly decreasing sequences (w.r.t. some well-founded order) 

    stabilize.*}

 text{*This material does not appear to be used any longer.*}

 lemma lemma1: "[| ALL i. (f (Suc i), f i) : r^* |] ==> (f (i+k), f i) : r^*"

 apply (induct_tac "k", simp_all)

 apply (blast intro: rtrancl_trans)

 done

 lemma lemma2: "[| ALL i. (f (Suc i), f i) : r^*; wf (r^+) |]  

       ==> ALL m. f m = x --> (EX i. ALL k. f (m+i+k) = f (m+i))"

 apply (erule wf_induct, clarify)

 apply (case_tac "EX j. (f (m+j), f m) : r^+")

  apply clarify

  apply (subgoal_tac "EX i. ALL k. f ((m+j) +i+k) = f ( (m+j) +i) ")

   apply clarify

   apply (rule_tac x = "j+i" in exI)

   apply (simp add: add_ac, blast)

 apply (rule_tac x = 0 in exI, clarsimp)

 apply (drule_tac i = m and k = k in lemma1)

 apply (blast elim: rtranclE dest: rtrancl_into_trancl1)

 done

 lemma wf_weak_decr_stable: "[| ALL i. (f (Suc i), f i) : r^*; wf (r^+) |]  

       ==> EX i. ALL k. f (i+k) = f i"

 apply (drule_tac x = 0 in lemma2 [THEN spec], auto)

 done

 (* special case of the theorem above: <= *)

 lemma weak_decr_stable:

      "ALL i. f (Suc i) <= ((f i)::nat) ==> EX i. ALL k. f (i+k) = f i"

 apply (rule_tac r = pred_nat in wf_weak_decr_stable)

 apply (simp add: pred_nat_trancl_eq_le)

 apply (intro wf_trancl wf_pred_nat)

 done

 subsection {* size of a datatype value *}

 use "Tools/function_package/size.ML"

 setup Size.setup

 lemma size_bool [code func]:

   "size (b\<Colon>bool) = 0" by (cases b) auto

 lemma nat_size [simp, code func]: "size (n\<Colon>nat) = n"

   by (induct n) simp_all

 declare "prod.size" [noatp]

 end