(* Title: HOL/Wellfounded.thy Author: Tobias Nipkow Author: Lawrence C Paulson Author: Konrad Slind Author: Alexander Krauss *) header {*Well-founded Recursion*} theory Wellfounded imports Transitive_Closure uses ("Tools/Function/size.ML") begin subsection {* 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 (\x 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 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 \ 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 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 lemma 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 auto qed lemma 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 blast qed text{*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 blast lemma 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 blast qed lemma wf_eq_minimal: "wf r = (\Q x. x\Q --> (\z\Q. \y. (y,z)\r --> y\Q))" apply auto apply (erule wfE_min, assumption, blast) apply (rule wfI_min, auto) done lemmas 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 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 {* 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) lemma wfP_empty [iff]: "wfP (\x y. False)" proof - have "wfP bot" by (fact wf_empty [to_pred bot_empty_eq2]) then show ?thesis by (simp add: bot_fun_def) qed 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 {* 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) qed 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_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) done subsection {* 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 rel_compI) 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 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 lemma 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 [where I=UNIV and r="\i. {(x, y). r i x y}", to_pred]) apply (simp_all add: inf_set_def) apply auto done lemma 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 "\i. 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 qed qed lemma 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]) done lemmas 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 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 {* @{typ nat} is well-founded *} lemma less_nat_rel: "op < = (\m n. n = Suc m)^++" proof (rule ext, rule ext, rule iffI) fix n m :: nat assume "m < n" then show "(\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 "(\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) \ pred_nat^+ \ m < n" unfolding less_nat_rel pred_nat_def trancl_def by simp lemma pred_nat_trancl_eq_le: "(m, n) \ pred_nat^* \ m \ 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) lemma less_than_iff [iff]: "((x,y): less_than) = (x '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.accI text {* 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 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: "\ 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 . 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: "\x. accp r x ==> wfP r" apply (rule wfPUNIVI) apply (rule_tac P=P in 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 = (\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 \ 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) qed qed text {* 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 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] 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 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 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 lemma 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 auto 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 \ (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_def by auto lemma mlex_less: "f x < f y \ (x, y) \ f <*mlex*> R" unfolding mlex_prod_def by simp lemma mlex_leq: "f x \ f y \ (x, y) \ R \ (x, y) \ 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[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) done lemma 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 auto text {* 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 \ {}" have N2: "(?N2, M) \ max_ext r" by (rule max_extI[OF _ _ `M \ {}`]) (insert asm1, auto intro: finites) 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) qed qed lemma 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 \ \Q" by auto with `wf r` obtain z where z: "z \ \Q" "\y. (y, z) \ r \ y \ \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 qed qed text{* 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) done lemma 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]) done subsection {* size of a datatype value *} use "Tools/Function/size.ML" setup Size.setup lemma size_bool [code]: "size (b\bool) = 0" by (cases b) auto lemma nat_size [simp, code]: "size (n\nat) = n" by (induct n) simp_all declare "prod.size" [no_atp] lemma [code]: "size (P :: 'a Predicate.pred) = 0" by (cases P) simp lemma [code]: "pred_size f P = 0" by (cases P) simp end