haftmann@29580: (*  Author:     Tobias Nipkow
krauss@26748:     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
krauss@26748:     Author:     Konrad Slind, Alexander Krauss
krauss@26748:     Copyright   1992-2008  University of Cambridge and TU Muenchen
krauss@26748: *)
krauss@26748: 
krauss@26748: header {*Well-founded Recursion*}
krauss@26748: 
krauss@26748: theory Wellfounded
haftmann@29609: imports Finite_Set Transitive_Closure Nat
krauss@26748: uses ("Tools/function_package/size.ML")
krauss@26748: begin
krauss@26748: 
krauss@26976: subsection {* Basic Definitions *}
krauss@26976: 
krauss@26748: inductive
krauss@26748:   wfrec_rel :: "('a * 'a) set => (('a => 'b) => 'a => 'b) => 'a => 'b => bool"
krauss@26748:   for R :: "('a * 'a) set"
krauss@26748:   and F :: "('a => 'b) => 'a => 'b"
krauss@26748: where
krauss@26748:   wfrecI: "ALL z. (z, x) : R --> wfrec_rel R F z (g z) ==>
krauss@26748:             wfrec_rel R F x (F g x)"
krauss@26748: 
krauss@26748: constdefs
krauss@26748:   wf         :: "('a * 'a)set => bool"
krauss@26748:   "wf(r) == (!P. (!x. (!y. (y,x):r --> P(y)) --> P(x)) --> (!x. P(x)))"
krauss@26748: 
krauss@26748:   wfP :: "('a => 'a => bool) => bool"
krauss@26748:   "wfP r == wf {(x, y). r x y}"
krauss@26748: 
krauss@26748:   acyclic :: "('a*'a)set => bool"
krauss@26748:   "acyclic r == !x. (x,x) ~: r^+"
krauss@26748: 
krauss@26748:   cut        :: "('a => 'b) => ('a * 'a)set => 'a => 'a => 'b"
haftmann@28524:   "cut f r x == (%y. if (y,x):r then f y else undefined)"
krauss@26748: 
krauss@26748:   adm_wf :: "('a * 'a) set => (('a => 'b) => 'a => 'b) => bool"
krauss@26748:   "adm_wf R F == ALL f g x.
krauss@26748:      (ALL z. (z, x) : R --> f z = g z) --> F f x = F g x"
krauss@26748: 
krauss@26748:   wfrec :: "('a * 'a) set => (('a => 'b) => 'a => 'b) => 'a => 'b"
haftmann@28562:   [code del]: "wfrec R F == %x. THE y. wfrec_rel R (%f x. F (cut f R x) x) x y"
krauss@26748: 
krauss@26748: abbreviation acyclicP :: "('a => 'a => bool) => bool" where
krauss@26748:   "acyclicP r == acyclic {(x, y). r x y}"
krauss@26748: 
krauss@26748: lemma wfP_wf_eq [pred_set_conv]: "wfP (\<lambda>x y. (x, y) \<in> r) = wf r"
krauss@26748:   by (simp add: wfP_def)
krauss@26748: 
krauss@26748: lemma wfUNIVI: 
krauss@26748:    "(!!P x. (ALL x. (ALL y. (y,x) : r --> P(y)) --> P(x)) ==> P(x)) ==> wf(r)"
krauss@26748:   unfolding wf_def by blast
krauss@26748: 
krauss@26748: lemmas wfPUNIVI = wfUNIVI [to_pred]
krauss@26748: 
krauss@26748: text{*Restriction to domain @{term A} and range @{term B}.  If @{term r} is
krauss@26748:     well-founded over their intersection, then @{term "wf r"}*}
krauss@26748: lemma wfI: 
krauss@26748:  "[| r \<subseteq> A <*> B; 
krauss@26748:      !!x P. [|\<forall>x. (\<forall>y. (y,x) : r --> P y) --> P x;  x : A; x : B |] ==> P x |]
krauss@26748:   ==>  wf r"
krauss@26748:   unfolding wf_def by blast
krauss@26748: 
krauss@26748: lemma wf_induct: 
krauss@26748:     "[| wf(r);           
krauss@26748:         !!x.[| ALL y. (y,x): r --> P(y) |] ==> P(x)  
krauss@26748:      |]  ==>  P(a)"
krauss@26748:   unfolding wf_def by blast
krauss@26748: 
krauss@26748: lemmas wfP_induct = wf_induct [to_pred]
krauss@26748: 
krauss@26748: lemmas wf_induct_rule = wf_induct [rule_format, consumes 1, case_names less, induct set: wf]
krauss@26748: 
krauss@26748: lemmas wfP_induct_rule = wf_induct_rule [to_pred, induct set: wfP]
krauss@26748: 
krauss@26748: lemma wf_not_sym: "wf r ==> (a, x) : r ==> (x, a) ~: r"
krauss@26748:   by (induct a arbitrary: x set: wf) blast
krauss@26748: 
krauss@26748: (* [| wf r;  ~Z ==> (a,x) : r;  (x,a) ~: r ==> Z |] ==> Z *)
krauss@26748: lemmas wf_asym = wf_not_sym [elim_format]
krauss@26748: 
krauss@26748: lemma wf_not_refl [simp]: "wf r ==> (a, a) ~: r"
krauss@26748:   by (blast elim: wf_asym)
krauss@26748: 
krauss@26748: (* [| wf r;  (a,a) ~: r ==> PROP W |] ==> PROP W *)
krauss@26748: lemmas wf_irrefl = wf_not_refl [elim_format]
krauss@26748: 
haftmann@27823: lemma wf_wellorderI:
haftmann@27823:   assumes wf: "wf {(x::'a::ord, y). x < y}"
haftmann@27823:   assumes lin: "OFCLASS('a::ord, linorder_class)"
haftmann@27823:   shows "OFCLASS('a::ord, wellorder_class)"
haftmann@27823: using lin by (rule wellorder_class.intro)
haftmann@27823:   (blast intro: wellorder_axioms.intro wf_induct_rule [OF wf])
haftmann@27823: 
haftmann@27823: lemma (in wellorder) wf:
haftmann@27823:   "wf {(x, y). x < y}"
haftmann@27823: unfolding wf_def by (blast intro: less_induct)
haftmann@27823: 
haftmann@27823: 
krauss@26976: subsection {* Basic Results *}
krauss@26976: 
krauss@26748: text{*transitive closure of a well-founded relation is well-founded! *}
krauss@26748: lemma wf_trancl:
krauss@26748:   assumes "wf r"
krauss@26748:   shows "wf (r^+)"
krauss@26748: proof -
krauss@26748:   {
krauss@26748:     fix P and x
krauss@26748:     assume induct_step: "!!x. (!!y. (y, x) : r^+ ==> P y) ==> P x"
krauss@26748:     have "P x"
krauss@26748:     proof (rule induct_step)
krauss@26748:       fix y assume "(y, x) : r^+"
krauss@26748:       with `wf r` show "P y"
krauss@26748:       proof (induct x arbitrary: y)
krauss@26748: 	case (less x)
krauss@26748: 	note hyp = `\<And>x' y'. (x', x) : r ==> (y', x') : r^+ ==> P y'`
krauss@26748: 	from `(y, x) : r^+` show "P y"
krauss@26748: 	proof cases
krauss@26748: 	  case base
krauss@26748: 	  show "P y"
krauss@26748: 	  proof (rule induct_step)
krauss@26748: 	    fix y' assume "(y', y) : r^+"
krauss@26748: 	    with `(y, x) : r` show "P y'" by (rule hyp [of y y'])
krauss@26748: 	  qed
krauss@26748: 	next
krauss@26748: 	  case step
krauss@26748: 	  then obtain x' where "(x', x) : r" and "(y, x') : r^+" by simp
krauss@26748: 	  then show "P y" by (rule hyp [of x' y])
krauss@26748: 	qed
krauss@26748:       qed
krauss@26748:     qed
krauss@26748:   } then show ?thesis unfolding wf_def by blast
krauss@26748: qed
krauss@26748: 
krauss@26748: lemmas wfP_trancl = wf_trancl [to_pred]
krauss@26748: 
krauss@26748: lemma wf_converse_trancl: "wf (r^-1) ==> wf ((r^+)^-1)"
krauss@26748:   apply (subst trancl_converse [symmetric])
krauss@26748:   apply (erule wf_trancl)
krauss@26748:   done
krauss@26748: 
krauss@26748: 
krauss@26748: text{*Minimal-element characterization of well-foundedness*}
krauss@26748: 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))"
krauss@26748: proof (intro iffI strip)
krauss@26748:   fix Q :: "'a set" and x
krauss@26748:   assume "wf r" and "x \<in> Q"
krauss@26748:   then show "\<exists>z\<in>Q. \<forall>y. (y, z) \<in> r \<longrightarrow> y \<notin> Q"
krauss@26748:     unfolding wf_def
krauss@26748:     by (blast dest: spec [of _ "%x. x\<in>Q \<longrightarrow> (\<exists>z\<in>Q. \<forall>y. (y,z) \<in> r \<longrightarrow> y\<notin>Q)"]) 
krauss@26748: next
krauss@26748:   assume 1: "\<forall>Q x. x \<in> Q \<longrightarrow> (\<exists>z\<in>Q. \<forall>y. (y, z) \<in> r \<longrightarrow> y \<notin> Q)"
krauss@26748:   show "wf r"
krauss@26748:   proof (rule wfUNIVI)
krauss@26748:     fix P :: "'a \<Rightarrow> bool" and x
krauss@26748:     assume 2: "\<forall>x. (\<forall>y. (y, x) \<in> r \<longrightarrow> P y) \<longrightarrow> P x"
krauss@26748:     let ?Q = "{x. \<not> P x}"
krauss@26748:     have "x \<in> ?Q \<longrightarrow> (\<exists>z \<in> ?Q. \<forall>y. (y, z) \<in> r \<longrightarrow> y \<notin> ?Q)"
krauss@26748:       by (rule 1 [THEN spec, THEN spec])
krauss@26748:     then have "\<not> P x \<longrightarrow> (\<exists>z. \<not> P z \<and> (\<forall>y. (y, z) \<in> r \<longrightarrow> P y))" by simp
krauss@26748:     with 2 have "\<not> P x \<longrightarrow> (\<exists>z. \<not> P z \<and> P z)" by fast
krauss@26748:     then show "P x" by simp
krauss@26748:   qed
krauss@26748: qed
krauss@26748: 
krauss@26748: lemma wfE_min: 
krauss@26748:   assumes "wf R" "x \<in> Q"
krauss@26748:   obtains z where "z \<in> Q" "\<And>y. (y, z) \<in> R \<Longrightarrow> y \<notin> Q"
krauss@26748:   using assms unfolding wf_eq_minimal by blast
krauss@26748: 
krauss@26748: lemma wfI_min:
krauss@26748:   "(\<And>x Q. x \<in> Q \<Longrightarrow> \<exists>z\<in>Q. \<forall>y. (y, z) \<in> R \<longrightarrow> y \<notin> Q)
krauss@26748:   \<Longrightarrow> wf R"
krauss@26748:   unfolding wf_eq_minimal by blast
krauss@26748: 
krauss@26748: lemmas wfP_eq_minimal = wf_eq_minimal [to_pred]
krauss@26748: 
krauss@26748: text {* Well-foundedness of subsets *}
krauss@26748: lemma wf_subset: "[| wf(r);  p<=r |] ==> wf(p)"
krauss@26748:   apply (simp (no_asm_use) add: wf_eq_minimal)
krauss@26748:   apply fast
krauss@26748:   done
krauss@26748: 
krauss@26748: lemmas wfP_subset = wf_subset [to_pred]
krauss@26748: 
krauss@26748: text {* Well-foundedness of the empty relation *}
krauss@26748: lemma wf_empty [iff]: "wf({})"
krauss@26748:   by (simp add: wf_def)
krauss@26748: 
krauss@26748: lemmas wfP_empty [iff] =
krauss@26748:   wf_empty [to_pred bot_empty_eq2, simplified bot_fun_eq bot_bool_eq]
krauss@26748: 
krauss@26748: lemma wf_Int1: "wf r ==> wf (r Int r')"
krauss@26748:   apply (erule wf_subset)
krauss@26748:   apply (rule Int_lower1)
krauss@26748:   done
krauss@26748: 
krauss@26748: lemma wf_Int2: "wf r ==> wf (r' Int r)"
krauss@26748:   apply (erule wf_subset)
krauss@26748:   apply (rule Int_lower2)
krauss@26748:   done  
krauss@26748: 
krauss@26748: text{*Well-foundedness of insert*}
krauss@26748: lemma wf_insert [iff]: "wf(insert (y,x) r) = (wf(r) & (x,y) ~: r^*)"
krauss@26748: apply (rule iffI)
krauss@26748:  apply (blast elim: wf_trancl [THEN wf_irrefl]
krauss@26748:               intro: rtrancl_into_trancl1 wf_subset 
krauss@26748:                      rtrancl_mono [THEN [2] rev_subsetD])
krauss@26748: apply (simp add: wf_eq_minimal, safe)
krauss@26748: apply (rule allE, assumption, erule impE, blast) 
krauss@26748: apply (erule bexE)
krauss@26748: apply (rename_tac "a", case_tac "a = x")
krauss@26748:  prefer 2
krauss@26748: apply blast 
krauss@26748: apply (case_tac "y:Q")
krauss@26748:  prefer 2 apply blast
krauss@26748: apply (rule_tac x = "{z. z:Q & (z,y) : r^*}" in allE)
krauss@26748:  apply assumption
krauss@26748: apply (erule_tac V = "ALL Q. (EX x. x : Q) --> ?P Q" in thin_rl) 
krauss@26748:   --{*essential for speed*}
krauss@26748: txt{*Blast with new substOccur fails*}
krauss@26748: apply (fast intro: converse_rtrancl_into_rtrancl)
krauss@26748: done
krauss@26748: 
krauss@26748: text{*Well-foundedness of image*}
krauss@26748: lemma wf_prod_fun_image: "[| wf r; inj f |] ==> wf(prod_fun f f ` r)"
krauss@26748: apply (simp only: wf_eq_minimal, clarify)
krauss@26748: apply (case_tac "EX p. f p : Q")
krauss@26748: apply (erule_tac x = "{p. f p : Q}" in allE)
krauss@26748: apply (fast dest: inj_onD, blast)
krauss@26748: done
krauss@26748: 
krauss@26748: 
krauss@26976: subsection {* Well-Foundedness Results for Unions *}
krauss@26748: 
krauss@26748: lemma wf_union_compatible:
krauss@26748:   assumes "wf R" "wf S"
krauss@26748:   assumes "S O R \<subseteq> R"
krauss@26748:   shows "wf (R \<union> S)"
krauss@26748: proof (rule wfI_min)
krauss@26748:   fix x :: 'a and Q 
krauss@26748:   let ?Q' = "{x \<in> Q. \<forall>y. (y, x) \<in> R \<longrightarrow> y \<notin> Q}"
krauss@26748:   assume "x \<in> Q"
krauss@26748:   obtain a where "a \<in> ?Q'"
krauss@26748:     by (rule wfE_min [OF `wf R` `x \<in> Q`]) blast
krauss@26748:   with `wf S`
krauss@26748:   obtain z where "z \<in> ?Q'" and zmin: "\<And>y. (y, z) \<in> S \<Longrightarrow> y \<notin> ?Q'" by (erule wfE_min)
krauss@26748:   { 
krauss@26748:     fix y assume "(y, z) \<in> S"
krauss@26748:     then have "y \<notin> ?Q'" by (rule zmin)
krauss@26748: 
krauss@26748:     have "y \<notin> Q"
krauss@26748:     proof 
krauss@26748:       assume "y \<in> Q"
krauss@26748:       with `y \<notin> ?Q'` 
krauss@26748:       obtain w where "(w, y) \<in> R" and "w \<in> Q" by auto
krauss@26748:       from `(w, y) \<in> R` `(y, z) \<in> S` have "(w, z) \<in> S O R" by (rule rel_compI)
krauss@26748:       with `S O R \<subseteq> R` have "(w, z) \<in> R" ..
krauss@26748:       with `z \<in> ?Q'` have "w \<notin> Q" by blast 
krauss@26748:       with `w \<in> Q` show False by contradiction
krauss@26748:     qed
krauss@26748:   }
krauss@26748:   with `z \<in> ?Q'` show "\<exists>z\<in>Q. \<forall>y. (y, z) \<in> R \<union> S \<longrightarrow> y \<notin> Q" by blast
krauss@26748: qed
krauss@26748: 
krauss@26748: 
krauss@26748: text {* Well-foundedness of indexed union with disjoint domains and ranges *}
krauss@26748: 
krauss@26748: lemma wf_UN: "[| ALL i:I. wf(r i);  
krauss@26748:          ALL i:I. ALL j:I. r i ~= r j --> Domain(r i) Int Range(r j) = {}  
krauss@26748:       |] ==> wf(UN i:I. r i)"
krauss@26748: apply (simp only: wf_eq_minimal, clarify)
krauss@26748: apply (rename_tac A a, case_tac "EX i:I. EX a:A. EX b:A. (b,a) : r i")
krauss@26748:  prefer 2
krauss@26748:  apply force 
krauss@26748: apply clarify
krauss@26748: apply (drule bspec, assumption)  
krauss@26748: apply (erule_tac x="{a. a:A & (EX b:A. (b,a) : r i) }" in allE)
krauss@26748: apply (blast elim!: allE)  
krauss@26748: done
krauss@26748: 
krauss@26748: lemmas wfP_SUP = wf_UN [where I=UNIV and r="\<lambda>i. {(x, y). r i x y}",
berghofe@26803:   to_pred SUP_UN_eq2 bot_empty_eq pred_equals_eq, simplified, standard]
krauss@26748: 
krauss@26748: lemma wf_Union: 
krauss@26748:  "[| ALL r:R. wf r;  
krauss@26748:      ALL r:R. ALL s:R. r ~= s --> Domain r Int Range s = {}  
krauss@26748:   |] ==> wf(Union R)"
krauss@26748: apply (simp add: Union_def)
krauss@26748: apply (blast intro: wf_UN)
krauss@26748: done
krauss@26748: 
krauss@26748: (*Intuition: we find an (R u S)-min element of a nonempty subset A
krauss@26748:              by case distinction.
krauss@26748:   1. There is a step a -R-> b with a,b : A.
krauss@26748:      Pick an R-min element z of the (nonempty) set {a:A | EX b:A. a -R-> b}.
krauss@26748:      By definition, there is z':A s.t. z -R-> z'. Because z is R-min in the
krauss@26748:      subset, z' must be R-min in A. Because z' has an R-predecessor, it cannot
krauss@26748:      have an S-successor and is thus S-min in A as well.
krauss@26748:   2. There is no such step.
krauss@26748:      Pick an S-min element of A. In this case it must be an R-min
krauss@26748:      element of A as well.
krauss@26748: 
krauss@26748: *)
krauss@26748: lemma wf_Un:
krauss@26748:      "[| wf r; wf s; Domain r Int Range s = {} |] ==> wf(r Un s)"
krauss@26748:   using wf_union_compatible[of s r] 
krauss@26748:   by (auto simp: Un_ac)
krauss@26748: 
krauss@26748: lemma wf_union_merge: 
krauss@26748:   "wf (R \<union> S) = wf (R O R \<union> R O S \<union> S)" (is "wf ?A = wf ?B")
krauss@26748: proof
krauss@26748:   assume "wf ?A"
krauss@26748:   with wf_trancl have wfT: "wf (?A^+)" .
krauss@26748:   moreover have "?B \<subseteq> ?A^+"
krauss@26748:     by (subst trancl_unfold, subst trancl_unfold) blast
krauss@26748:   ultimately show "wf ?B" by (rule wf_subset)
krauss@26748: next
krauss@26748:   assume "wf ?B"
krauss@26748: 
krauss@26748:   show "wf ?A"
krauss@26748:   proof (rule wfI_min)
krauss@26748:     fix Q :: "'a set" and x 
krauss@26748:     assume "x \<in> Q"
krauss@26748: 
krauss@26748:     with `wf ?B`
krauss@26748:     obtain z where "z \<in> Q" and "\<And>y. (y, z) \<in> ?B \<Longrightarrow> y \<notin> Q" 
krauss@26748:       by (erule wfE_min)
krauss@26748:     then have A1: "\<And>y. (y, z) \<in> R O R \<Longrightarrow> y \<notin> Q"
krauss@26748:       and A2: "\<And>y. (y, z) \<in> R O S \<Longrightarrow> y \<notin> Q"
krauss@26748:       and A3: "\<And>y. (y, z) \<in> S \<Longrightarrow> y \<notin> Q"
krauss@26748:       by auto
krauss@26748:     
krauss@26748:     show "\<exists>z\<in>Q. \<forall>y. (y, z) \<in> ?A \<longrightarrow> y \<notin> Q"
krauss@26748:     proof (cases "\<forall>y. (y, z) \<in> R \<longrightarrow> y \<notin> Q")
krauss@26748:       case True
krauss@26748:       with `z \<in> Q` A3 show ?thesis by blast
krauss@26748:     next
krauss@26748:       case False 
krauss@26748:       then obtain z' where "z'\<in>Q" "(z', z) \<in> R" by blast
krauss@26748: 
krauss@26748:       have "\<forall>y. (y, z') \<in> ?A \<longrightarrow> y \<notin> Q"
krauss@26748:       proof (intro allI impI)
krauss@26748:         fix y assume "(y, z') \<in> ?A"
krauss@26748:         then show "y \<notin> Q"
krauss@26748:         proof
krauss@26748:           assume "(y, z') \<in> R" 
krauss@26748:           then have "(y, z) \<in> R O R" using `(z', z) \<in> R` ..
krauss@26748:           with A1 show "y \<notin> Q" .
krauss@26748:         next
krauss@26748:           assume "(y, z') \<in> S" 
krauss@26748:           then have "(y, z) \<in> R O S" using  `(z', z) \<in> R` ..
krauss@26748:           with A2 show "y \<notin> Q" .
krauss@26748:         qed
krauss@26748:       qed
krauss@26748:       with `z' \<in> Q` show ?thesis ..
krauss@26748:     qed
krauss@26748:   qed
krauss@26748: qed
krauss@26748: 
krauss@26748: lemma wf_comp_self: "wf R = wf (R O R)"  -- {* special case *}
krauss@26748:   by (rule wf_union_merge [where S = "{}", simplified])
krauss@26748: 
krauss@26748: 
krauss@26748: subsubsection {* acyclic *}
krauss@26748: 
krauss@26748: lemma acyclicI: "ALL x. (x, x) ~: r^+ ==> acyclic r"
krauss@26748:   by (simp add: acyclic_def)
krauss@26748: 
krauss@26748: lemma wf_acyclic: "wf r ==> acyclic r"
krauss@26748: apply (simp add: acyclic_def)
krauss@26748: apply (blast elim: wf_trancl [THEN wf_irrefl])
krauss@26748: done
krauss@26748: 
krauss@26748: lemmas wfP_acyclicP = wf_acyclic [to_pred]
krauss@26748: 
krauss@26748: lemma acyclic_insert [iff]:
krauss@26748:      "acyclic(insert (y,x) r) = (acyclic r & (x,y) ~: r^*)"
krauss@26748: apply (simp add: acyclic_def trancl_insert)
krauss@26748: apply (blast intro: rtrancl_trans)
krauss@26748: done
krauss@26748: 
krauss@26748: lemma acyclic_converse [iff]: "acyclic(r^-1) = acyclic r"
krauss@26748: by (simp add: acyclic_def trancl_converse)
krauss@26748: 
krauss@26748: lemmas acyclicP_converse [iff] = acyclic_converse [to_pred]
krauss@26748: 
krauss@26748: lemma acyclic_impl_antisym_rtrancl: "acyclic r ==> antisym(r^*)"
krauss@26748: apply (simp add: acyclic_def antisym_def)
krauss@26748: apply (blast elim: rtranclE intro: rtrancl_into_trancl1 rtrancl_trancl_trancl)
krauss@26748: done
krauss@26748: 
krauss@26748: (* Other direction:
krauss@26748: acyclic = no loops
krauss@26748: antisym = only self loops
krauss@26748: Goalw [acyclic_def,antisym_def] "antisym( r^* ) ==> acyclic(r - Id)
krauss@26748: ==> antisym( r^* ) = acyclic(r - Id)";
krauss@26748: *)
krauss@26748: 
krauss@26748: lemma acyclic_subset: "[| acyclic s; r <= s |] ==> acyclic r"
krauss@26748: apply (simp add: acyclic_def)
krauss@26748: apply (blast intro: trancl_mono)
krauss@26748: done
krauss@26748: 
krauss@26748: text{* Wellfoundedness of finite acyclic relations*}
krauss@26748: 
krauss@26748: lemma finite_acyclic_wf [rule_format]: "finite r ==> acyclic r --> wf r"
krauss@26748: apply (erule finite_induct, blast)
krauss@26748: apply (simp (no_asm_simp) only: split_tupled_all)
krauss@26748: apply simp
krauss@26748: done
krauss@26748: 
krauss@26748: lemma finite_acyclic_wf_converse: "[|finite r; acyclic r|] ==> wf (r^-1)"
krauss@26748: apply (erule finite_converse [THEN iffD2, THEN finite_acyclic_wf])
krauss@26748: apply (erule acyclic_converse [THEN iffD2])
krauss@26748: done
krauss@26748: 
krauss@26748: lemma wf_iff_acyclic_if_finite: "finite r ==> wf r = acyclic r"
krauss@26748: by (blast intro: finite_acyclic_wf wf_acyclic)
krauss@26748: 
krauss@26748: 
krauss@26748: subsection{*Well-Founded Recursion*}
krauss@26748: 
krauss@26748: text{*cut*}
krauss@26748: 
krauss@26748: lemma cuts_eq: "(cut f r x = cut g r x) = (ALL y. (y,x):r --> f(y)=g(y))"
krauss@26748: by (simp add: expand_fun_eq cut_def)
krauss@26748: 
krauss@26748: lemma cut_apply: "(x,a):r ==> (cut f r a)(x) = f(x)"
krauss@26748: by (simp add: cut_def)
krauss@26748: 
krauss@26748: text{*Inductive characterization of wfrec combinator; for details see:  
krauss@26748: John Harrison, "Inductive definitions: automation and application"*}
krauss@26748: 
krauss@26748: lemma wfrec_unique: "[| adm_wf R F; wf R |] ==> EX! y. wfrec_rel R F x y"
krauss@26748: apply (simp add: adm_wf_def)
krauss@26748: apply (erule_tac a=x in wf_induct) 
krauss@26748: apply (rule ex1I)
krauss@26748: apply (rule_tac g = "%x. THE y. wfrec_rel R F x y" in wfrec_rel.wfrecI)
krauss@26748: apply (fast dest!: theI')
krauss@26748: apply (erule wfrec_rel.cases, simp)
krauss@26748: apply (erule allE, erule allE, erule allE, erule mp)
krauss@26748: apply (fast intro: the_equality [symmetric])
krauss@26748: done
krauss@26748: 
krauss@26748: lemma adm_lemma: "adm_wf R (%f x. F (cut f R x) x)"
krauss@26748: apply (simp add: adm_wf_def)
krauss@26748: apply (intro strip)
krauss@26748: apply (rule cuts_eq [THEN iffD2, THEN subst], assumption)
krauss@26748: apply (rule refl)
krauss@26748: done
krauss@26748: 
krauss@26748: lemma wfrec: "wf(r) ==> wfrec r H a = H (cut (wfrec r H) r a) a"
krauss@26748: apply (simp add: wfrec_def)
krauss@26748: apply (rule adm_lemma [THEN wfrec_unique, THEN the1_equality], assumption)
krauss@26748: apply (rule wfrec_rel.wfrecI)
krauss@26748: apply (intro strip)
krauss@26748: apply (erule adm_lemma [THEN wfrec_unique, THEN theI'])
krauss@26748: done
krauss@26748: 
krauss@26748: subsection {* Code generator setup *}
krauss@26748: 
krauss@26748: consts_code
krauss@26748:   "wfrec"   ("\<module>wfrec?")
krauss@26748: attach {*
krauss@26748: fun wfrec f x = f (wfrec f) x;
krauss@26748: *}
krauss@26748: 
krauss@26748: 
krauss@26748: subsection {* @{typ nat} is well-founded *}
krauss@26748: 
krauss@26748: lemma less_nat_rel: "op < = (\<lambda>m n. n = Suc m)^++"
krauss@26748: proof (rule ext, rule ext, rule iffI)
krauss@26748:   fix n m :: nat
krauss@26748:   assume "m < n"
krauss@26748:   then show "(\<lambda>m n. n = Suc m)^++ m n"
krauss@26748:   proof (induct n)
krauss@26748:     case 0 then show ?case by auto
krauss@26748:   next
krauss@26748:     case (Suc n) then show ?case
krauss@26748:       by (auto simp add: less_Suc_eq_le le_less intro: tranclp.trancl_into_trancl)
krauss@26748:   qed
krauss@26748: next
krauss@26748:   fix n m :: nat
krauss@26748:   assume "(\<lambda>m n. n = Suc m)^++ m n"
krauss@26748:   then show "m < n"
krauss@26748:     by (induct n)
krauss@26748:       (simp_all add: less_Suc_eq_le reflexive le_less)
krauss@26748: qed
krauss@26748: 
krauss@26748: definition
krauss@26748:   pred_nat :: "(nat * nat) set" where
krauss@26748:   "pred_nat = {(m, n). n = Suc m}"
krauss@26748: 
krauss@26748: definition
krauss@26748:   less_than :: "(nat * nat) set" where
krauss@26748:   "less_than = pred_nat^+"
krauss@26748: 
krauss@26748: lemma less_eq: "(m, n) \<in> pred_nat^+ \<longleftrightarrow> m < n"
krauss@26748:   unfolding less_nat_rel pred_nat_def trancl_def by simp
krauss@26748: 
krauss@26748: lemma pred_nat_trancl_eq_le:
krauss@26748:   "(m, n) \<in> pred_nat^* \<longleftrightarrow> m \<le> n"
krauss@26748:   unfolding less_eq rtrancl_eq_or_trancl by auto
krauss@26748: 
krauss@26748: lemma wf_pred_nat: "wf pred_nat"
krauss@26748:   apply (unfold wf_def pred_nat_def, clarify)
krauss@26748:   apply (induct_tac x, blast+)
krauss@26748:   done
krauss@26748: 
krauss@26748: lemma wf_less_than [iff]: "wf less_than"
krauss@26748:   by (simp add: less_than_def wf_pred_nat [THEN wf_trancl])
krauss@26748: 
krauss@26748: lemma trans_less_than [iff]: "trans less_than"
krauss@26748:   by (simp add: less_than_def trans_trancl)
krauss@26748: 
krauss@26748: lemma less_than_iff [iff]: "((x,y): less_than) = (x<y)"
krauss@26748:   by (simp add: less_than_def less_eq)
krauss@26748: 
krauss@26748: lemma wf_less: "wf {(x, y::nat). x < y}"
krauss@26748:   using wf_less_than by (simp add: less_than_def less_eq [symmetric])
krauss@26748: 
krauss@26748: 
krauss@26748: subsection {* Accessible Part *}
krauss@26748: 
krauss@26748: text {*
krauss@26748:  Inductive definition of the accessible part @{term "acc r"} of a
krauss@26748:  relation; see also \cite{paulin-tlca}.
krauss@26748: *}
krauss@26748: 
krauss@26748: inductive_set
krauss@26748:   acc :: "('a * 'a) set => 'a set"
krauss@26748:   for r :: "('a * 'a) set"
krauss@26748:   where
krauss@26748:     accI: "(!!y. (y, x) : r ==> y : acc r) ==> x : acc r"
krauss@26748: 
krauss@26748: abbreviation
krauss@26748:   termip :: "('a => 'a => bool) => 'a => bool" where
krauss@26748:   "termip r == accp (r\<inverse>\<inverse>)"
krauss@26748: 
krauss@26748: abbreviation
krauss@26748:   termi :: "('a * 'a) set => 'a set" where
krauss@26748:   "termi r == acc (r\<inverse>)"
krauss@26748: 
krauss@26748: lemmas accpI = accp.accI
krauss@26748: 
krauss@26748: text {* Induction rules *}
krauss@26748: 
krauss@26748: theorem accp_induct:
krauss@26748:   assumes major: "accp r a"
krauss@26748:   assumes hyp: "!!x. accp r x ==> \<forall>y. r y x --> P y ==> P x"
krauss@26748:   shows "P a"
krauss@26748:   apply (rule major [THEN accp.induct])
krauss@26748:   apply (rule hyp)
krauss@26748:    apply (rule accp.accI)
krauss@26748:    apply fast
krauss@26748:   apply fast
krauss@26748:   done
krauss@26748: 
krauss@26748: theorems accp_induct_rule = accp_induct [rule_format, induct set: accp]
krauss@26748: 
krauss@26748: theorem accp_downward: "accp r b ==> r a b ==> accp r a"
krauss@26748:   apply (erule accp.cases)
krauss@26748:   apply fast
krauss@26748:   done
krauss@26748: 
krauss@26748: lemma not_accp_down:
krauss@26748:   assumes na: "\<not> accp R x"
krauss@26748:   obtains z where "R z x" and "\<not> accp R z"
krauss@26748: proof -
krauss@26748:   assume a: "\<And>z. \<lbrakk>R z x; \<not> accp R z\<rbrakk> \<Longrightarrow> thesis"
krauss@26748: 
krauss@26748:   show thesis
krauss@26748:   proof (cases "\<forall>z. R z x \<longrightarrow> accp R z")
krauss@26748:     case True
krauss@26748:     hence "\<And>z. R z x \<Longrightarrow> accp R z" by auto
krauss@26748:     hence "accp R x"
krauss@26748:       by (rule accp.accI)
krauss@26748:     with na show thesis ..
krauss@26748:   next
krauss@26748:     case False then obtain z where "R z x" and "\<not> accp R z"
krauss@26748:       by auto
krauss@26748:     with a show thesis .
krauss@26748:   qed
krauss@26748: qed
krauss@26748: 
krauss@26748: lemma accp_downwards_aux: "r\<^sup>*\<^sup>* b a ==> accp r a --> accp r b"
krauss@26748:   apply (erule rtranclp_induct)
krauss@26748:    apply blast
krauss@26748:   apply (blast dest: accp_downward)
krauss@26748:   done
krauss@26748: 
krauss@26748: theorem accp_downwards: "accp r a ==> r\<^sup>*\<^sup>* b a ==> accp r b"
krauss@26748:   apply (blast dest: accp_downwards_aux)
krauss@26748:   done
krauss@26748: 
krauss@26748: theorem accp_wfPI: "\<forall>x. accp r x ==> wfP r"
krauss@26748:   apply (rule wfPUNIVI)
krauss@26748:   apply (induct_tac P x rule: accp_induct)
krauss@26748:    apply blast
krauss@26748:   apply blast
krauss@26748:   done
krauss@26748: 
krauss@26748: theorem accp_wfPD: "wfP r ==> accp r x"
krauss@26748:   apply (erule wfP_induct_rule)
krauss@26748:   apply (rule accp.accI)
krauss@26748:   apply blast
krauss@26748:   done
krauss@26748: 
krauss@26748: theorem wfP_accp_iff: "wfP r = (\<forall>x. accp r x)"
krauss@26748:   apply (blast intro: accp_wfPI dest: accp_wfPD)
krauss@26748:   done
krauss@26748: 
krauss@26748: 
krauss@26748: text {* Smaller relations have bigger accessible parts: *}
krauss@26748: 
krauss@26748: lemma accp_subset:
krauss@26748:   assumes sub: "R1 \<le> R2"
krauss@26748:   shows "accp R2 \<le> accp R1"
berghofe@26803: proof (rule predicate1I)
krauss@26748:   fix x assume "accp R2 x"
krauss@26748:   then show "accp R1 x"
krauss@26748:   proof (induct x)
krauss@26748:     fix x
krauss@26748:     assume ih: "\<And>y. R2 y x \<Longrightarrow> accp R1 y"
krauss@26748:     with sub show "accp R1 x"
krauss@26748:       by (blast intro: accp.accI)
krauss@26748:   qed
krauss@26748: qed
krauss@26748: 
krauss@26748: 
krauss@26748: text {* This is a generalized induction theorem that works on
krauss@26748:   subsets of the accessible part. *}
krauss@26748: 
krauss@26748: lemma accp_subset_induct:
krauss@26748:   assumes subset: "D \<le> accp R"
krauss@26748:     and dcl: "\<And>x z. \<lbrakk>D x; R z x\<rbrakk> \<Longrightarrow> D z"
krauss@26748:     and "D x"
krauss@26748:     and istep: "\<And>x. \<lbrakk>D x; (\<And>z. R z x \<Longrightarrow> P z)\<rbrakk> \<Longrightarrow> P x"
krauss@26748:   shows "P x"
krauss@26748: proof -
krauss@26748:   from subset and `D x`
krauss@26748:   have "accp R x" ..
krauss@26748:   then show "P x" using `D x`
krauss@26748:   proof (induct x)
krauss@26748:     fix x
krauss@26748:     assume "D x"
krauss@26748:       and "\<And>y. R y x \<Longrightarrow> D y \<Longrightarrow> P y"
krauss@26748:     with dcl and istep show "P x" by blast
krauss@26748:   qed
krauss@26748: qed
krauss@26748: 
krauss@26748: 
krauss@26748: text {* Set versions of the above theorems *}
krauss@26748: 
krauss@26748: lemmas acc_induct = accp_induct [to_set]
krauss@26748: 
krauss@26748: lemmas acc_induct_rule = acc_induct [rule_format, induct set: acc]
krauss@26748: 
krauss@26748: lemmas acc_downward = accp_downward [to_set]
krauss@26748: 
krauss@26748: lemmas not_acc_down = not_accp_down [to_set]
krauss@26748: 
krauss@26748: lemmas acc_downwards_aux = accp_downwards_aux [to_set]
krauss@26748: 
krauss@26748: lemmas acc_downwards = accp_downwards [to_set]
krauss@26748: 
krauss@26748: lemmas acc_wfI = accp_wfPI [to_set]
krauss@26748: 
krauss@26748: lemmas acc_wfD = accp_wfPD [to_set]
krauss@26748: 
krauss@26748: lemmas wf_acc_iff = wfP_accp_iff [to_set]
krauss@26748: 
berghofe@26803: lemmas acc_subset = accp_subset [to_set pred_subset_eq]
krauss@26748: 
berghofe@26803: lemmas acc_subset_induct = accp_subset_induct [to_set pred_subset_eq]
krauss@26748: 
krauss@26748: 
krauss@26748: subsection {* Tools for building wellfounded relations *}
krauss@26748: 
krauss@26748: text {* Inverse Image *}
krauss@26748: 
krauss@26748: lemma wf_inv_image [simp,intro!]: "wf(r) ==> wf(inv_image r (f::'a=>'b))"
krauss@26748: apply (simp (no_asm_use) add: inv_image_def wf_eq_minimal)
krauss@26748: apply clarify
krauss@26748: apply (subgoal_tac "EX (w::'b) . w : {w. EX (x::'a) . x: Q & (f x = w) }")
krauss@26748: prefer 2 apply (blast del: allE)
krauss@26748: apply (erule allE)
krauss@26748: apply (erule (1) notE impE)
krauss@26748: apply blast
krauss@26748: done
krauss@26748: 
krauss@26748: lemma in_inv_image[simp]: "((x,y) : inv_image r f) = ((f x, f y) : r)"
krauss@26748:   by (auto simp:inv_image_def)
krauss@26748: 
krauss@26748: text {* Measure functions into @{typ nat} *}
krauss@26748: 
krauss@26748: definition measure :: "('a => nat) => ('a * 'a)set"
krauss@26748: where "measure == inv_image less_than"
krauss@26748: 
krauss@26748: lemma in_measure[simp]: "((x,y) : measure f) = (f x < f y)"
krauss@26748:   by (simp add:measure_def)
krauss@26748: 
krauss@26748: lemma wf_measure [iff]: "wf (measure f)"
krauss@26748: apply (unfold measure_def)
krauss@26748: apply (rule wf_less_than [THEN wf_inv_image])
krauss@26748: done
krauss@26748: 
krauss@26748: text{* Lexicographic combinations *}
krauss@26748: 
krauss@26748: definition
krauss@26748:  lex_prod  :: "[('a*'a)set, ('b*'b)set] => (('a*'b)*('a*'b))set"
krauss@26748:                (infixr "<*lex*>" 80)
krauss@26748: where
krauss@26748:     "ra <*lex*> rb == {((a,b),(a',b')). (a,a') : ra | a=a' & (b,b') : rb}"
krauss@26748: 
krauss@26748: lemma wf_lex_prod [intro!]: "[| wf(ra); wf(rb) |] ==> wf(ra <*lex*> rb)"
krauss@26748: apply (unfold wf_def lex_prod_def) 
krauss@26748: apply (rule allI, rule impI)
krauss@26748: apply (simp (no_asm_use) only: split_paired_All)
krauss@26748: apply (drule spec, erule mp) 
krauss@26748: apply (rule allI, rule impI)
krauss@26748: apply (drule spec, erule mp, blast) 
krauss@26748: done
krauss@26748: 
krauss@26748: lemma in_lex_prod[simp]: 
krauss@26748:   "(((a,b),(a',b')): r <*lex*> s) = ((a,a'): r \<or> (a = a' \<and> (b, b') : s))"
krauss@26748:   by (auto simp:lex_prod_def)
krauss@26748: 
krauss@26748: text{* @{term "op <*lex*>"} preserves transitivity *}
krauss@26748: 
krauss@26748: lemma trans_lex_prod [intro!]: 
krauss@26748:     "[| trans R1; trans R2 |] ==> trans (R1 <*lex*> R2)"
krauss@26748: by (unfold trans_def lex_prod_def, blast) 
krauss@26748: 
krauss@26748: text {* lexicographic combinations with measure functions *}
krauss@26748: 
krauss@26748: definition 
krauss@26748:   mlex_prod :: "('a \<Rightarrow> nat) \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" (infixr "<*mlex*>" 80)
krauss@26748: where
krauss@26748:   "f <*mlex*> R = inv_image (less_than <*lex*> R) (%x. (f x, x))"
krauss@26748: 
krauss@26748: lemma wf_mlex: "wf R \<Longrightarrow> wf (f <*mlex*> R)"
krauss@26748: unfolding mlex_prod_def
krauss@26748: by auto
krauss@26748: 
krauss@26748: lemma mlex_less: "f x < f y \<Longrightarrow> (x, y) \<in> f <*mlex*> R"
krauss@26748: unfolding mlex_prod_def by simp
krauss@26748: 
krauss@26748: lemma mlex_leq: "f x \<le> f y \<Longrightarrow> (x, y) \<in> R \<Longrightarrow> (x, y) \<in> f <*mlex*> R"
krauss@26748: unfolding mlex_prod_def by auto
krauss@26748: 
krauss@26748: text {* proper subset relation on finite sets *}
krauss@26748: 
krauss@26748: definition finite_psubset  :: "('a set * 'a set) set"
krauss@26748: where "finite_psubset == {(A,B). A < B & finite B}"
krauss@26748: 
krauss@28260: lemma wf_finite_psubset[simp]: "wf(finite_psubset)"
krauss@26748: apply (unfold finite_psubset_def)
krauss@26748: apply (rule wf_measure [THEN wf_subset])
krauss@26748: apply (simp add: measure_def inv_image_def less_than_def less_eq)
krauss@26748: apply (fast elim!: psubset_card_mono)
krauss@26748: done
krauss@26748: 
krauss@26748: lemma trans_finite_psubset: "trans finite_psubset"
berghofe@26803: by (simp add: finite_psubset_def less_le trans_def, blast)
krauss@26748: 
krauss@28260: lemma in_finite_psubset[simp]: "(A, B) \<in> finite_psubset = (A < B & finite B)"
krauss@28260: unfolding finite_psubset_def by auto
krauss@26748: 
krauss@28735: text {* max- and min-extension of order to finite sets *}
krauss@28735: 
krauss@28735: inductive_set max_ext :: "('a \<times> 'a) set \<Rightarrow> ('a set \<times> 'a set) set" 
krauss@28735: for R :: "('a \<times> 'a) set"
krauss@28735: where
krauss@28735:   max_extI[intro]: "finite X \<Longrightarrow> finite Y \<Longrightarrow> Y \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> \<exists>y\<in>Y. (x, y) \<in> R) \<Longrightarrow> (X, Y) \<in> max_ext R"
krauss@28735: 
krauss@28735: lemma max_ext_wf:
krauss@28735:   assumes wf: "wf r"
krauss@28735:   shows "wf (max_ext r)"
krauss@28735: proof (rule acc_wfI, intro allI)
krauss@28735:   fix M show "M \<in> acc (max_ext r)" (is "_ \<in> ?W")
krauss@28735:   proof cases
krauss@28735:     assume "finite M"
krauss@28735:     thus ?thesis
krauss@28735:     proof (induct M)
krauss@28735:       show "{} \<in> ?W"
krauss@28735:         by (rule accI) (auto elim: max_ext.cases)
krauss@28735:     next
krauss@28735:       fix M a assume "M \<in> ?W" "finite M"
krauss@28735:       with wf show "insert a M \<in> ?W"
krauss@28735:       proof (induct arbitrary: M)
krauss@28735:         fix M a
krauss@28735:         assume "M \<in> ?W"  and  [intro]: "finite M"
krauss@28735:         assume hyp: "\<And>b M. (b, a) \<in> r \<Longrightarrow> M \<in> ?W \<Longrightarrow> finite M \<Longrightarrow> insert b M \<in> ?W"
krauss@28735:         {
krauss@28735:           fix N M :: "'a set"
krauss@28735:           assume "finite N" "finite M"
krauss@28735:           then
krauss@28735:           have "\<lbrakk>M \<in> ?W ; (\<And>y. y \<in> N \<Longrightarrow> (y, a) \<in> r)\<rbrakk> \<Longrightarrow>  N \<union> M \<in> ?W"
krauss@28735:             by (induct N arbitrary: M) (auto simp: hyp)
krauss@28735:         }
krauss@28735:         note add_less = this
krauss@28735:         
krauss@28735:         show "insert a M \<in> ?W"
krauss@28735:         proof (rule accI)
krauss@28735:           fix N assume Nless: "(N, insert a M) \<in> max_ext r"
krauss@28735:           hence asm1: "\<And>x. x \<in> N \<Longrightarrow> (x, a) \<in> r \<or> (\<exists>y \<in> M. (x, y) \<in> r)"
krauss@28735:             by (auto elim!: max_ext.cases)
krauss@28735: 
krauss@28735:           let ?N1 = "{ n \<in> N. (n, a) \<in> r }"
krauss@28735:           let ?N2 = "{ n \<in> N. (n, a) \<notin> r }"
krauss@28735:           have N: "?N1 \<union> ?N2 = N" by (rule set_ext) auto
krauss@28735:           from Nless have "finite N" by (auto elim: max_ext.cases)
krauss@28735:           then have finites: "finite ?N1" "finite ?N2" by auto
krauss@28735:           
krauss@28735:           have "?N2 \<in> ?W"
krauss@28735:           proof cases
krauss@28735:             assume [simp]: "M = {}"
krauss@28735:             have Mw: "{} \<in> ?W" by (rule accI) (auto elim: max_ext.cases)
krauss@28735: 
krauss@28735:             from asm1 have "?N2 = {}" by auto
krauss@28735:             with Mw show "?N2 \<in> ?W" by (simp only:)
krauss@28735:           next
krauss@28735:             assume "M \<noteq> {}"
krauss@28735:             have N2: "(?N2, M) \<in> max_ext r" 
krauss@28735:               by (rule max_extI[OF _ _ `M \<noteq> {}`]) (insert asm1, auto intro: finites)
krauss@28735:             
krauss@28735:             with `M \<in> ?W` show "?N2 \<in> ?W" by (rule acc_downward)
krauss@28735:           qed
krauss@28735:           with finites have "?N1 \<union> ?N2 \<in> ?W" 
krauss@28735:             by (rule add_less) simp
krauss@28735:           then show "N \<in> ?W" by (simp only: N)
krauss@28735:         qed
krauss@28735:       qed
krauss@28735:     qed
krauss@28735:   next
krauss@28735:     assume [simp]: "\<not> finite M"
krauss@28735:     show ?thesis
krauss@28735:       by (rule accI) (auto elim: max_ext.cases)
krauss@28735:   qed
krauss@28735: qed
krauss@28735: 
krauss@29125: lemma max_ext_additive: 
krauss@29125:  "(A, B) \<in> max_ext R \<Longrightarrow> (C, D) \<in> max_ext R \<Longrightarrow>
krauss@29125:   (A \<union> C, B \<union> D) \<in> max_ext R"
krauss@29125: by (force elim!: max_ext.cases)
krauss@29125: 
krauss@28735: 
krauss@28735: definition
krauss@28735:   min_ext :: "('a \<times> 'a) set \<Rightarrow> ('a set \<times> 'a set) set" 
krauss@28735: where
krauss@28735:   [code del]: "min_ext r = {(X, Y) | X Y. X \<noteq> {} \<and> (\<forall>y \<in> Y. (\<exists>x \<in> X. (x, y) \<in> r))}"
krauss@28735: 
krauss@28735: lemma min_ext_wf:
krauss@28735:   assumes "wf r"
krauss@28735:   shows "wf (min_ext r)"
krauss@28735: proof (rule wfI_min)
krauss@28735:   fix Q :: "'a set set"
krauss@28735:   fix x
krauss@28735:   assume nonempty: "x \<in> Q"
krauss@28735:   show "\<exists>m \<in> Q. (\<forall> n. (n, m) \<in> min_ext r \<longrightarrow> n \<notin> Q)"
krauss@28735:   proof cases
krauss@28735:     assume "Q = {{}}" thus ?thesis by (simp add: min_ext_def)
krauss@28735:   next
krauss@28735:     assume "Q \<noteq> {{}}"
krauss@28735:     with nonempty
krauss@28735:     obtain e x where "x \<in> Q" "e \<in> x" by force
krauss@28735:     then have eU: "e \<in> \<Union>Q" by auto
krauss@28735:     with `wf r` 
krauss@28735:     obtain z where z: "z \<in> \<Union>Q" "\<And>y. (y, z) \<in> r \<Longrightarrow> y \<notin> \<Union>Q" 
krauss@28735:       by (erule wfE_min)
krauss@28735:     from z obtain m where "m \<in> Q" "z \<in> m" by auto
krauss@28735:     from `m \<in> Q`
krauss@28735:     show ?thesis
krauss@28735:     proof (rule, intro bexI allI impI)
krauss@28735:       fix n
krauss@28735:       assume smaller: "(n, m) \<in> min_ext r"
krauss@28735:       with `z \<in> m` obtain y where y: "y \<in> n" "(y, z) \<in> r" by (auto simp: min_ext_def)
krauss@28735:       then show "n \<notin> Q" using z(2) by auto
krauss@28735:     qed      
krauss@28735:   qed
krauss@28735: qed
krauss@26748: 
krauss@26748: text {*Wellfoundedness of @{text same_fst}*}
krauss@26748: 
krauss@26748: definition
krauss@26748:  same_fst :: "('a => bool) => ('a => ('b * 'b)set) => (('a*'b)*('a*'b))set"
krauss@26748: where
krauss@26748:     "same_fst P R == {((x',y'),(x,y)) . x'=x & P x & (y',y) : R x}"
krauss@26748:    --{*For @{text rec_def} declarations where the first n parameters
krauss@28735:        stay unchanged in the recursive call. *}
krauss@26748: 
krauss@26748: lemma same_fstI [intro!]:
krauss@26748:      "[| P x; (y',y) : R x |] ==> ((x,y'),(x,y)) : same_fst P R"
krauss@26748: by (simp add: same_fst_def)
krauss@26748: 
krauss@26748: lemma wf_same_fst:
krauss@26748:   assumes prem: "(!!x. P x ==> wf(R x))"
krauss@26748:   shows "wf(same_fst P R)"
krauss@26748: apply (simp cong del: imp_cong add: wf_def same_fst_def)
krauss@26748: apply (intro strip)
krauss@26748: apply (rename_tac a b)
krauss@26748: apply (case_tac "wf (R a)")
krauss@26748:  apply (erule_tac a = b in wf_induct, blast)
krauss@26748: apply (blast intro: prem)
krauss@26748: done
krauss@26748: 
krauss@26748: 
krauss@26748: subsection{*Weakly decreasing sequences (w.r.t. some well-founded order) 
krauss@26748:    stabilize.*}
krauss@26748: 
krauss@26748: text{*This material does not appear to be used any longer.*}
krauss@26748: 
krauss@28845: lemma sequence_trans: "[| ALL i. (f (Suc i), f i) : r^* |] ==> (f (i+k), f i) : r^*"
krauss@28845: by (induct k) (auto intro: rtrancl_trans)
krauss@26748: 
krauss@28845: lemma wf_weak_decr_stable: 
krauss@28845:   assumes as: "ALL i. (f (Suc i), f i) : r^*" "wf (r^+)"
krauss@28845:   shows "EX i. ALL k. f (i+k) = f i"
krauss@28845: proof -
krauss@28845:   have lem: "!!x. [| ALL i. (f (Suc i), f i) : r^*; wf (r^+) |]  
krauss@26748:       ==> ALL m. f m = x --> (EX i. ALL k. f (m+i+k) = f (m+i))"
krauss@28845:   apply (erule wf_induct, clarify)
krauss@28845:   apply (case_tac "EX j. (f (m+j), f m) : r^+")
krauss@28845:    apply clarify
krauss@28845:    apply (subgoal_tac "EX i. ALL k. f ((m+j) +i+k) = f ( (m+j) +i) ")
krauss@28845:     apply clarify
krauss@28845:     apply (rule_tac x = "j+i" in exI)
krauss@28845:     apply (simp add: add_ac, blast)
krauss@28845:   apply (rule_tac x = 0 in exI, clarsimp)
krauss@28845:   apply (drule_tac i = m and k = k in sequence_trans)
krauss@28845:   apply (blast elim: rtranclE dest: rtrancl_into_trancl1)
krauss@28845:   done
krauss@26748: 
krauss@28845:   from lem[OF as, THEN spec, of 0, simplified] 
krauss@28845:   show ?thesis by auto
krauss@28845: qed
krauss@26748: 
krauss@26748: (* special case of the theorem above: <= *)
krauss@26748: lemma weak_decr_stable:
krauss@26748:      "ALL i. f (Suc i) <= ((f i)::nat) ==> EX i. ALL k. f (i+k) = f i"
krauss@26748: apply (rule_tac r = pred_nat in wf_weak_decr_stable)
krauss@26748: apply (simp add: pred_nat_trancl_eq_le)
krauss@26748: apply (intro wf_trancl wf_pred_nat)
krauss@26748: done
krauss@26748: 
krauss@26748: 
krauss@26748: subsection {* size of a datatype value *}
krauss@26748: 
krauss@26748: use "Tools/function_package/size.ML"
krauss@26748: 
krauss@26748: setup Size.setup
krauss@26748: 
haftmann@28562: lemma size_bool [code]:
haftmann@27823:   "size (b\<Colon>bool) = 0" by (cases b) auto
haftmann@27823: 
haftmann@28562: lemma nat_size [simp, code]: "size (n\<Colon>nat) = n"
krauss@26748:   by (induct n) simp_all
krauss@26748: 
haftmann@27823: declare "prod.size" [noatp]
krauss@26748: 
haftmann@30430: lemma [code]:
haftmann@30430:   "size (P :: 'a Predicate.pred) = 0" by (cases P) simp
haftmann@30430: 
haftmann@30430: lemma [code]:
haftmann@30430:   "pred_size f P = 0" by (cases P) simp
haftmann@30430: 
krauss@26748: end