# Theory Wellfounded

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theory Wellfounded
imports Transitive_Closure
`(*  Title:      HOL/Wellfounded.thy    Author:     Tobias Nipkow    Author:     Lawrence C Paulson    Author:     Konrad Slind    Author:     Alexander Krauss*)header {*Well-founded Recursion*}theory Wellfoundedimports Transitive_Closurebeginsubsection {* Basic Definitions *}definition wf :: "('a * 'a) set => bool" where  "wf r <-> (!P. (!x. (!y. (y,x):r --> P(y)) --> P(x)) --> (!x. P(x)))"definition wfP :: "('a => 'a => bool) => bool" where  "wfP r <-> wf {(x, y). r x y}"lemma wfP_wf_eq [pred_set_conv]: "wfP (λx y. (x, y) ∈ 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 blastlemmas 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 ⊆ A <*> B;      !!x P. [|∀x. (∀y. (y,x) : r --> P y) --> P x;  x : A; x : B |] ==> P x |]  ==>  wf r"  unfolding wf_def by blastlemma wf_induct:     "[| wf(r);                   !!x.[| ALL y. (y,x): r --> P(y) |] ==> P(x)       |]  ==>  P(a)"  unfolding wf_def by blastlemmas 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) blastlemma wf_asym:  assumes "wf r" "(a, x) ∈ r"  obtains "(x, a) ∉ r"  by (drule wf_not_sym[OF assms])lemma wf_not_refl [simp]: "wf r ==> (a, a) ~: r"  by (blast elim: wf_asym)lemma wf_irrefl: assumes "wf r" obtains "(a, a) ∉ r"by (drule wf_not_refl[OF assms])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: class.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 {* Point-free characterization of well-foundedness *}lemma wfE_pf:  assumes wf: "wf R"  assumes a: "A ⊆ R `` A"  shows "A = {}"proof -  { fix x    from wf have "x ∉ A"    proof induct      fix x assume "!!y. (y, x) ∈ R ==> y ∉ A"      then have "x ∉ R `` A" by blast      with a show "x ∉ A" by blast    qed  } thus ?thesis by autoqedlemma wfI_pf:  assumes a: "!!A. A ⊆ R `` A ==> A = {}"  shows "wf R"proof (rule wfUNIVI)  fix P :: "'a => bool" and x  let ?A = "{x. ¬ P x}"  assume "∀x. (∀y. (y, x) ∈ R --> P y) --> P x"  then have "?A ⊆ R `` ?A" by blast  with a show "P x" by blastqedtext{*Minimal-element characterization of well-foundedness*}lemma wfE_min:  assumes wf: "wf R" and Q: "x ∈ Q"  obtains z where "z ∈ Q" "!!y. (y, z) ∈ R ==> y ∉ Q"  using Q wfE_pf[OF wf, of Q] by blastlemma wfI_min:  assumes a: "!!x Q. x ∈ Q ==> ∃z∈Q. ∀y. (y, z) ∈ R --> y ∉ Q"  shows "wf R"proof (rule wfI_pf)  fix A assume b: "A ⊆ R `` A"  { fix x assume "x ∈ A"    from a[OF this] b have "False" by blast  }  thus "A = {}" by blastqedlemma wf_eq_minimal: "wf r = (∀Q x. x∈Q --> (∃z∈Q. ∀y. (y,z)∈r --> y∉Q))"apply autoapply (erule wfE_min, assumption, blast)apply (rule wfI_min, auto)donelemmas wfP_eq_minimal = wf_eq_minimal [to_pred]text{* Well-foundedness of transitive closure *}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 = `!!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 blastqedlemmas 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)  donetext {* Well-foundedness of subsets *}lemma wf_subset: "[| wf(r);  p<=r |] ==> wf(p)"  apply (simp (no_asm_use) add: wf_eq_minimal)  apply fast  donelemmas wfP_subset = wf_subset [to_pred]text {* Well-foundedness of the empty relation *}lemma wf_empty [iff]: "wf {}"  by (simp add: wf_def)lemma wfP_empty [iff]:  "wfP (λx y. False)"proof -  have "wfP bot" by (fact wf_empty [to_pred bot_empty_eq2])  then show ?thesis by (simp add: bot_fun_def)qedlemma wf_Int1: "wf r ==> wf (r Int r')"  apply (erule wf_subset)  apply (rule Int_lower1)  donelemma wf_Int2: "wf r ==> wf (r' Int r)"  apply (erule wf_subset)  apply (rule Int_lower2)  done  text {* Exponentiation *}lemma wf_exp:  assumes "wf (R ^^ n)"  shows "wf R"proof (rule wfI_pf)  fix A assume "A ⊆ R `` A"  then have "A ⊆ (R ^^ n) `` A" by (induct n) force+  with `wf (R ^^ n)`  show "A = {}" by (rule wfE_pf)qedtext {* 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 2apply blast apply (case_tac "y:Q") prefer 2 apply blastapply (rule_tac x = "{z. z:Q & (z,y) : r^*}" in allE) apply assumptionapply (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)donetext{*Well-foundedness of image*}lemma wf_map_pair_image: "[| wf r; inj f |] ==> wf(map_pair 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)donesubsection {* Well-Foundedness Results for Unions *}lemma wf_union_compatible:  assumes "wf R" "wf S"  assumes "R O S ⊆ R"  shows "wf (R ∪ S)"proof (rule wfI_min)  fix x :: 'a and Q   let ?Q' = "{x ∈ Q. ∀y. (y, x) ∈ R --> y ∉ Q}"  assume "x ∈ Q"  obtain a where "a ∈ ?Q'"    by (rule wfE_min [OF `wf R` `x ∈ Q`]) blast  with `wf S`  obtain z where "z ∈ ?Q'" and zmin: "!!y. (y, z) ∈ S ==> y ∉ ?Q'" by (erule wfE_min)  {     fix y assume "(y, z) ∈ S"    then have "y ∉ ?Q'" by (rule zmin)    have "y ∉ Q"    proof       assume "y ∈ Q"      with `y ∉ ?Q'`       obtain w where "(w, y) ∈ R" and "w ∈ Q" by auto      from `(w, y) ∈ R` `(y, z) ∈ S` have "(w, z) ∈ R O S" by (rule relcompI)      with `R O S ⊆ R` have "(w, z) ∈ R" ..      with `z ∈ ?Q'` have "w ∉ Q" by blast       with `w ∈ Q` show False by contradiction    qed  }  with `z ∈ ?Q'` show "∃z∈Q. ∀y. (y, z) ∈ R ∪ S --> y ∉ Q" by blastqedtext {* 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 clarifyapply (drule bspec, assumption)  apply (erule_tac x="{a. a:A & (EX b:A. (b,a) : r i) }" in allE)apply (blast elim!: allE)  donelemma wfP_SUP:  "∀i. wfP (r i) ==> ∀i j. r i ≠ r j --> inf (DomainP (r i)) (RangeP (r j)) = bot ==> wfP (SUPR UNIV r)"  apply (rule wf_UN[to_pred])  apply simp_all  donelemma wf_Union:  "[| ALL r:R. wf r;       ALL r:R. ALL s:R. r ~= s --> Domain r Int Range s = {}    |] ==> wf(Union R)"  using wf_UN[of R "λi. i"] by (simp add: SUP_def)(*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 ∪ S) = wf (R O R ∪ S O R ∪ S)" (is "wf ?A = wf ?B")proof  assume "wf ?A"  with wf_trancl have wfT: "wf (?A^+)" .  moreover have "?B ⊆ ?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 ∈ Q"    with `wf ?B`    obtain z where "z ∈ Q" and "!!y. (y, z) ∈ ?B ==> y ∉ Q"       by (erule wfE_min)    then have A1: "!!y. (y, z) ∈ R O R ==> y ∉ Q"      and A2: "!!y. (y, z) ∈ S O R ==> y ∉ Q"      and A3: "!!y. (y, z) ∈ S ==> y ∉ Q"      by auto        show "∃z∈Q. ∀y. (y, z) ∈ ?A --> y ∉ Q"    proof (cases "∀y. (y, z) ∈ R --> y ∉ Q")      case True      with `z ∈ Q` A3 show ?thesis by blast    next      case False       then obtain z' where "z'∈Q" "(z', z) ∈ R" by blast      have "∀y. (y, z') ∈ ?A --> y ∉ Q"      proof (intro allI impI)        fix y assume "(y, z') ∈ ?A"        then show "y ∉ Q"        proof          assume "(y, z') ∈ R"           then have "(y, z) ∈ R O R" using `(z', z) ∈ R` ..          with A1 show "y ∉ Q" .        next          assume "(y, z') ∈ S"           then have "(y, z) ∈ S O R" using  `(z', z) ∈ R` ..          with A2 show "y ∉ Q" .        qed      qed      with `z' ∈ Q` show ?thesis ..    qed  qedqedlemma wf_comp_self: "wf R = wf (R O R)"  -- {* special case *}  by (rule wf_union_merge [where S = "{}", simplified])subsection {* Acyclic relations *}lemma wf_acyclic: "wf r ==> acyclic r"apply (simp add: acyclic_def)apply (blast elim: wf_trancl [THEN wf_irrefl])donelemmas wfP_acyclicP = wf_acyclic [to_pred]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 simpdonelemma 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])donelemma wf_iff_acyclic_if_finite: "finite r ==> wf r = acyclic r"by (blast intro: finite_acyclic_wf wf_acyclic)subsection {* @{typ nat} is well-founded *}lemma less_nat_rel: "op < = (λm n. n = Suc m)^++"proof (rule ext, rule ext, rule iffI)  fix n m :: nat  assume "m < n"  then show "(λm 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)  qednext  fix n m :: nat  assume "(λm n. n = Suc m)^++ m n"  then show "m < n"    by (induct n)      (simp_all add: less_Suc_eq_le reflexive le_less)qeddefinition  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) ∈ pred_nat^+ <-> m < n"  unfolding less_nat_rel pred_nat_def trancl_def by simplemma pred_nat_trancl_eq_le:  "(m, n) ∈ pred_nat^* <-> m ≤ n"  unfolding less_eq rtrancl_eq_or_trancl by autolemma wf_pred_nat: "wf pred_nat"  apply (unfold wf_def pred_nat_def, clarify)  apply (induct_tac x, blast+)  donelemma 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)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¯¯)"abbreviation  termi :: "('a * 'a) set => 'a set" where  "termi r ≡ acc (r¯)"lemmas accpI = accp.accItext {* Induction rules *}theorem accp_induct:  assumes major: "accp r a"  assumes hyp: "!!x. accp r x ==> ∀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  donetheorems 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  donelemma not_accp_down:  assumes na: "¬ accp R x"  obtains z where "R z x" and "¬ accp R z"proof -  assume a: "!!z. [|R z x; ¬ accp R z|] ==> thesis"  show thesis  proof (cases "∀z. R z x --> accp R z")    case True    hence "!!z. R z x ==> 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 "¬ accp R z"      by auto    with a show thesis .  qedqedlemma accp_downwards_aux: "r⇧*⇧* b a ==> accp r a --> accp r b"  apply (erule rtranclp_induct)   apply blast  apply (blast dest: accp_downward)  donetheorem accp_downwards: "accp r a ==> r⇧*⇧* b a ==> accp r b"  apply (blast dest: accp_downwards_aux)  donetheorem accp_wfPI: "∀x. accp r x ==> wfP r"  apply (rule wfPUNIVI)  apply (rule_tac P=P in accp_induct)   apply blast  apply blast  donetheorem accp_wfPD: "wfP r ==> accp r x"  apply (erule wfP_induct_rule)  apply (rule accp.accI)  apply blast  donetheorem wfP_accp_iff: "wfP r = (∀x. accp r x)"  apply (blast intro: accp_wfPI dest: accp_wfPD)  donetext {* Smaller relations have bigger accessible parts: *}lemma accp_subset:  assumes sub: "R1 ≤ R2"  shows "accp R2 ≤ accp R1"proof (rule predicate1I)  fix x assume "accp R2 x"  then show "accp R1 x"  proof (induct x)    fix x    assume ih: "!!y. R2 y x ==> accp R1 y"    with sub show "accp R1 x"      by (blast intro: accp.accI)  qedqedtext {* This is a generalized induction theorem that works on  subsets of the accessible part. *}lemma accp_subset_induct:  assumes subset: "D ≤ accp R"    and dcl: "!!x z. [|D x; R z x|] ==> D z"    and "D x"    and istep: "!!x. [|D x; (!!z. R z x ==> P z)|] ==> 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 "!!y. R y x ==> D y ==> P y"    with dcl and istep show "P x" by blast  qedqedtext {* 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]lemmas acc_subset_induct = accp_subset_induct [to_set]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 clarifyapply (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 blastdonetext {* Measure functions into @{typ nat} *}definition measure :: "('a => nat) => ('a * 'a)set"where "measure = inv_image less_than"lemma in_measure[simp, code_unfold]: "((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])donelemma wf_if_measure: fixes f :: "'a => nat"shows "(!!x. P x ==> f(g x) < f x) ==> wf {(y,x). P x ∧ y = g x}"apply(insert wf_measure[of f])apply(simp only: measure_def inv_image_def less_than_def less_eq)apply(erule wf_subset)apply autodonetext{* 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) donelemma in_lex_prod[simp]:   "(((a,b),(a',b')): r <*lex*> s) = ((a,a'): r ∨ (a = a' ∧ (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 => nat) => ('a × 'a) set => ('a × 'a) set" (infixr "<*mlex*>" 80)where  "f <*mlex*> R = inv_image (less_than <*lex*> R) (%x. (f x, x))"lemma wf_mlex: "wf R ==> wf (f <*mlex*> R)"unfolding mlex_prod_defby autolemma mlex_less: "f x < f y ==> (x, y) ∈ f <*mlex*> R"unfolding mlex_prod_def by simplemma mlex_leq: "f x ≤ f y ==> (x, y) ∈ R ==> (x, y) ∈ f <*mlex*> R"unfolding mlex_prod_def by autotext {* 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[simp]: "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)donelemma trans_finite_psubset: "trans finite_psubset"by (simp add: finite_psubset_def less_le trans_def, blast)lemma in_finite_psubset[simp]: "(A, B) ∈ finite_psubset = (A < B & finite B)"unfolding finite_psubset_def by autotext {* max- and min-extension of order to finite sets *}inductive_set max_ext :: "('a × 'a) set => ('a set × 'a set) set" for R :: "('a × 'a) set"where  max_extI[intro]: "finite X ==> finite Y ==> Y ≠ {} ==> (!!x. x ∈ X ==> ∃y∈Y. (x, y) ∈ R) ==> (X, Y) ∈ max_ext R"lemma max_ext_wf:  assumes wf: "wf r"  shows "wf (max_ext r)"proof (rule acc_wfI, intro allI)  fix M show "M ∈ acc (max_ext r)" (is "_ ∈ ?W")  proof cases    assume "finite M"    thus ?thesis    proof (induct M)      show "{} ∈ ?W"        by (rule accI) (auto elim: max_ext.cases)    next      fix M a assume "M ∈ ?W" "finite M"      with wf show "insert a M ∈ ?W"      proof (induct arbitrary: M)        fix M a        assume "M ∈ ?W"  and  [intro]: "finite M"        assume hyp: "!!b M. (b, a) ∈ r ==> M ∈ ?W ==> finite M ==> insert b M ∈ ?W"        {          fix N M :: "'a set"          assume "finite N" "finite M"          then          have "[|M ∈ ?W ; (!!y. y ∈ N ==> (y, a) ∈ r)|] ==>  N ∪ M ∈ ?W"            by (induct N arbitrary: M) (auto simp: hyp)        }        note add_less = this                show "insert a M ∈ ?W"        proof (rule accI)          fix N assume Nless: "(N, insert a M) ∈ max_ext r"          hence asm1: "!!x. x ∈ N ==> (x, a) ∈ r ∨ (∃y ∈ M. (x, y) ∈ r)"            by (auto elim!: max_ext.cases)          let ?N1 = "{ n ∈ N. (n, a) ∈ r }"          let ?N2 = "{ n ∈ N. (n, a) ∉ r }"          have N: "?N1 ∪ ?N2 = N" by (rule set_eqI) auto          from Nless have "finite N" by (auto elim: max_ext.cases)          then have finites: "finite ?N1" "finite ?N2" by auto                    have "?N2 ∈ ?W"          proof cases            assume [simp]: "M = {}"            have Mw: "{} ∈ ?W" by (rule accI) (auto elim: max_ext.cases)            from asm1 have "?N2 = {}" by auto            with Mw show "?N2 ∈ ?W" by (simp only:)          next            assume "M ≠ {}"            from asm1 finites have N2: "(?N2, M) ∈ max_ext r"               by (rule_tac max_extI[OF _ _ `M ≠ {}`]) auto            with `M ∈ ?W` show "?N2 ∈ ?W" by (rule acc_downward)          qed          with finites have "?N1 ∪ ?N2 ∈ ?W"             by (rule add_less) simp          then show "N ∈ ?W" by (simp only: N)        qed      qed    qed  next    assume [simp]: "¬ finite M"    show ?thesis      by (rule accI) (auto elim: max_ext.cases)  qedqedlemma max_ext_additive:  "(A, B) ∈ max_ext R ==> (C, D) ∈ max_ext R ==>  (A ∪ C, B ∪ D) ∈ max_ext R"by (force elim!: max_ext.cases)definition min_ext :: "('a × 'a) set => ('a set × 'a set) set"  where  "min_ext r = {(X, Y) | X Y. X ≠ {} ∧ (∀y ∈ Y. (∃x ∈ X. (x, y) ∈ r))}"lemma min_ext_wf:  assumes "wf r"  shows "wf (min_ext r)"proof (rule wfI_min)  fix Q :: "'a set set"  fix x  assume nonempty: "x ∈ Q"  show "∃m ∈ Q. (∀ n. (n, m) ∈ min_ext r --> n ∉ Q)"  proof cases    assume "Q = {{}}" thus ?thesis by (simp add: min_ext_def)  next    assume "Q ≠ {{}}"    with nonempty    obtain e x where "x ∈ Q" "e ∈ x" by force    then have eU: "e ∈ \<Union>Q" by auto    with `wf r`     obtain z where z: "z ∈ \<Union>Q" "!!y. (y, z) ∈ r ==> y ∉ \<Union>Q"       by (erule wfE_min)    from z obtain m where "m ∈ Q" "z ∈ m" by auto    from `m ∈ Q`    show ?thesis    proof (rule, intro bexI allI impI)      fix n      assume smaller: "(n, m) ∈ min_ext r"      with `z ∈ m` obtain y where y: "y ∈ n" "(y, z) ∈ r" by (auto simp: min_ext_def)      then show "n ∉ Q" using z(2) by auto    qed        qedqedtext{* Bounded increase must terminate: *}lemma wf_bounded_measure:fixes ub :: "'a => nat" and f :: "'a => nat"assumes "!!a b. (b,a) : r ==> ub b ≤ ub a & ub a ≥ f b & f b > f a"shows "wf r"apply(rule wf_subset[OF wf_measure[of "%a. ub a - f a"]])apply (auto dest: assms)donelemma wf_bounded_set:fixes ub :: "'a => 'b set" and f :: "'a => 'b set"assumes "!!a b. (b,a) : r ==>  finite(ub a) & ub b ⊆ ub a & ub a ⊇ f b & f b ⊃ f a"shows "wf r"apply(rule wf_bounded_measure[of r "%a. card(ub a)" "%a. card(f a)"])apply(drule assms)apply (blast intro: card_mono finite_subset psubset_card_mono dest: psubset_eq[THEN iffD2])donesubsection {* size of a datatype value *}ML_file "Tools/Function/size.ML"setup Size.setuplemma size_bool [code]:  "size (b::bool) = 0" by (cases b) autolemma nat_size [simp, code]: "size (n::nat) = n"  by (induct n) simp_alldeclare "prod.size" [no_atp]end`