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theory Wellfounded(* 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 Transitive_Closure 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 undefined)" 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 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 (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 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 (* [| 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 = `!!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 = (∀Q x. x∈Q --> (∃z∈Q. ∀y. (y,z)∈r --> y∉Q))" proof (intro iffI strip) fix Q :: "'a set" and x assume "wf r" and "x ∈ Q" then show "∃z∈Q. ∀y. (y, z) ∈ r --> y ∉ Q" unfolding wf_def by (blast dest: spec [of _ "%x. x∈Q --> (∃z∈Q. ∀y. (y,z) ∈ r --> y∉Q)"]) next assume 1: "∀Q x. x ∈ Q --> (∃z∈Q. ∀y. (y, z) ∈ r --> y ∉ Q)" show "wf r" proof (rule wfUNIVI) fix P :: "'a => bool" and x assume 2: "∀x. (∀y. (y, x) ∈ r --> P y) --> P x" let ?Q = "{x. ¬ P x}" have "x ∈ ?Q --> (∃z ∈ ?Q. ∀y. (y, z) ∈ r --> y ∉ ?Q)" by (rule 1 [THEN spec, THEN spec]) then have "¬ P x --> (∃z. ¬ P z ∧ (∀y. (y, z) ∈ r --> P y))" by simp with 2 have "¬ P x --> (∃z. ¬ P z ∧ P z)" by fast then show "P x" by simp qed qed lemma wfE_min: assumes "wf R" "x ∈ Q" obtains z where "z ∈ Q" "!!y. (y, z) ∈ R ==> y ∉ Q" using assms unfolding wf_eq_minimal by blast lemma wfI_min: "(!!x Q. x ∈ Q ==> ∃z∈Q. ∀y. (y, z) ∈ R --> y ∉ Q) ==> 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 ⊆ 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) ∈ S O R" by (rule rel_compI) with `S O R ⊆ 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 lemmas wfP_SUP = wf_UN [where I=UNIV and r="λi. {(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 ∪ S) = wf (R O R ∪ R O S ∪ 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) ∈ R O S ==> 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) ∈ R O S" 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]) 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 < = (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) qed next 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) 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 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¯¯)" 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** 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** 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 (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 = (∀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 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 ∨ (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_ext) 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 [code del]: "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 qed qed 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. *} 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 sequence_trans: "[| ALL i. (f (Suc i), f i) : r^* |] ==> (f (i+k), f i) : r^*" by (induct k) (auto intro: rtrancl_trans) lemma wf_weak_decr_stable: assumes as: "ALL i. (f (Suc i), f i) : r^*" "wf (r^+)" shows "EX i. ALL k. f (i+k) = f i" proof - have lem: "!!x. [| 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 sequence_trans) apply (blast elim: rtranclE dest: rtrancl_into_trancl1) done from lem[OF as, THEN spec, of 0, simplified] show ?thesis by auto qed (* 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]: "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" [noatp] lemma [code]: "size (P :: 'a Predicate.pred) = 0" by (cases P) simp lemma [code]: "pred_size f P = 0" by (cases P) simp end