src/HOL/Wellfounded.thy
author hoelzl
Thu Sep 15 12:40:08 2011 -0400 (2011-09-15)
changeset 44937 22c0857b8aab
parent 44921 58eef4843641
child 45012 060f76635bfe
permissions -rw-r--r--
removed further legacy rules from Complete_Lattices
     1 (*  Title:      HOL/Wellfounded.thy
     2     Author:     Tobias Nipkow
     3     Author:     Lawrence C Paulson
     4     Author:     Konrad Slind
     5     Author:     Alexander Krauss
     6 *)
     7 
     8 header {*Well-founded Recursion*}
     9 
    10 theory Wellfounded
    11 imports Transitive_Closure
    12 uses ("Tools/Function/size.ML")
    13 begin
    14 
    15 subsection {* Basic Definitions *}
    16 
    17 definition wf :: "('a * 'a) set => bool" where
    18   "wf(r) == (!P. (!x. (!y. (y,x):r --> P(y)) --> P(x)) --> (!x. P(x)))"
    19 
    20 definition wfP :: "('a => 'a => bool) => bool" where
    21   "wfP r == wf {(x, y). r x y}"
    22 
    23 lemma wfP_wf_eq [pred_set_conv]: "wfP (\<lambda>x y. (x, y) \<in> r) = wf r"
    24   by (simp add: wfP_def)
    25 
    26 lemma wfUNIVI: 
    27    "(!!P x. (ALL x. (ALL y. (y,x) : r --> P(y)) --> P(x)) ==> P(x)) ==> wf(r)"
    28   unfolding wf_def by blast
    29 
    30 lemmas wfPUNIVI = wfUNIVI [to_pred]
    31 
    32 text{*Restriction to domain @{term A} and range @{term B}.  If @{term r} is
    33     well-founded over their intersection, then @{term "wf r"}*}
    34 lemma wfI: 
    35  "[| r \<subseteq> A <*> B; 
    36      !!x P. [|\<forall>x. (\<forall>y. (y,x) : r --> P y) --> P x;  x : A; x : B |] ==> P x |]
    37   ==>  wf r"
    38   unfolding wf_def by blast
    39 
    40 lemma wf_induct: 
    41     "[| wf(r);           
    42         !!x.[| ALL y. (y,x): r --> P(y) |] ==> P(x)  
    43      |]  ==>  P(a)"
    44   unfolding wf_def by blast
    45 
    46 lemmas wfP_induct = wf_induct [to_pred]
    47 
    48 lemmas wf_induct_rule = wf_induct [rule_format, consumes 1, case_names less, induct set: wf]
    49 
    50 lemmas wfP_induct_rule = wf_induct_rule [to_pred, induct set: wfP]
    51 
    52 lemma wf_not_sym: "wf r ==> (a, x) : r ==> (x, a) ~: r"
    53   by (induct a arbitrary: x set: wf) blast
    54 
    55 lemma wf_asym:
    56   assumes "wf r" "(a, x) \<in> r"
    57   obtains "(x, a) \<notin> r"
    58   by (drule wf_not_sym[OF assms])
    59 
    60 lemma wf_not_refl [simp]: "wf r ==> (a, a) ~: r"
    61   by (blast elim: wf_asym)
    62 
    63 lemma wf_irrefl: assumes "wf r" obtains "(a, a) \<notin> r"
    64 by (drule wf_not_refl[OF assms])
    65 
    66 lemma wf_wellorderI:
    67   assumes wf: "wf {(x::'a::ord, y). x < y}"
    68   assumes lin: "OFCLASS('a::ord, linorder_class)"
    69   shows "OFCLASS('a::ord, wellorder_class)"
    70 using lin by (rule wellorder_class.intro)
    71   (blast intro: class.wellorder_axioms.intro wf_induct_rule [OF wf])
    72 
    73 lemma (in wellorder) wf:
    74   "wf {(x, y). x < y}"
    75 unfolding wf_def by (blast intro: less_induct)
    76 
    77 
    78 subsection {* Basic Results *}
    79 
    80 text {* Point-free characterization of well-foundedness *}
    81 
    82 lemma wfE_pf:
    83   assumes wf: "wf R"
    84   assumes a: "A \<subseteq> R `` A"
    85   shows "A = {}"
    86 proof -
    87   { fix x
    88     from wf have "x \<notin> A"
    89     proof induct
    90       fix x assume "\<And>y. (y, x) \<in> R \<Longrightarrow> y \<notin> A"
    91       then have "x \<notin> R `` A" by blast
    92       with a show "x \<notin> A" by blast
    93     qed
    94   } thus ?thesis by auto
    95 qed
    96 
    97 lemma wfI_pf:
    98   assumes a: "\<And>A. A \<subseteq> R `` A \<Longrightarrow> A = {}"
    99   shows "wf R"
   100 proof (rule wfUNIVI)
   101   fix P :: "'a \<Rightarrow> bool" and x
   102   let ?A = "{x. \<not> P x}"
   103   assume "\<forall>x. (\<forall>y. (y, x) \<in> R \<longrightarrow> P y) \<longrightarrow> P x"
   104   then have "?A \<subseteq> R `` ?A" by blast
   105   with a show "P x" by blast
   106 qed
   107 
   108 text{*Minimal-element characterization of well-foundedness*}
   109 
   110 lemma wfE_min:
   111   assumes wf: "wf R" and Q: "x \<in> Q"
   112   obtains z where "z \<in> Q" "\<And>y. (y, z) \<in> R \<Longrightarrow> y \<notin> Q"
   113   using Q wfE_pf[OF wf, of Q] by blast
   114 
   115 lemma wfI_min:
   116   assumes a: "\<And>x Q. x \<in> Q \<Longrightarrow> \<exists>z\<in>Q. \<forall>y. (y, z) \<in> R \<longrightarrow> y \<notin> Q"
   117   shows "wf R"
   118 proof (rule wfI_pf)
   119   fix A assume b: "A \<subseteq> R `` A"
   120   { fix x assume "x \<in> A"
   121     from a[OF this] b have "False" by blast
   122   }
   123   thus "A = {}" by blast
   124 qed
   125 
   126 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))"
   127 apply auto
   128 apply (erule wfE_min, assumption, blast)
   129 apply (rule wfI_min, auto)
   130 done
   131 
   132 lemmas wfP_eq_minimal = wf_eq_minimal [to_pred]
   133 
   134 text{* Well-foundedness of transitive closure *}
   135 
   136 lemma wf_trancl:
   137   assumes "wf r"
   138   shows "wf (r^+)"
   139 proof -
   140   {
   141     fix P and x
   142     assume induct_step: "!!x. (!!y. (y, x) : r^+ ==> P y) ==> P x"
   143     have "P x"
   144     proof (rule induct_step)
   145       fix y assume "(y, x) : r^+"
   146       with `wf r` show "P y"
   147       proof (induct x arbitrary: y)
   148         case (less x)
   149         note hyp = `\<And>x' y'. (x', x) : r ==> (y', x') : r^+ ==> P y'`
   150         from `(y, x) : r^+` show "P y"
   151         proof cases
   152           case base
   153           show "P y"
   154           proof (rule induct_step)
   155             fix y' assume "(y', y) : r^+"
   156             with `(y, x) : r` show "P y'" by (rule hyp [of y y'])
   157           qed
   158         next
   159           case step
   160           then obtain x' where "(x', x) : r" and "(y, x') : r^+" by simp
   161           then show "P y" by (rule hyp [of x' y])
   162         qed
   163       qed
   164     qed
   165   } then show ?thesis unfolding wf_def by blast
   166 qed
   167 
   168 lemmas wfP_trancl = wf_trancl [to_pred]
   169 
   170 lemma wf_converse_trancl: "wf (r^-1) ==> wf ((r^+)^-1)"
   171   apply (subst trancl_converse [symmetric])
   172   apply (erule wf_trancl)
   173   done
   174 
   175 text {* Well-foundedness of subsets *}
   176 
   177 lemma wf_subset: "[| wf(r);  p<=r |] ==> wf(p)"
   178   apply (simp (no_asm_use) add: wf_eq_minimal)
   179   apply fast
   180   done
   181 
   182 lemmas wfP_subset = wf_subset [to_pred]
   183 
   184 text {* Well-foundedness of the empty relation *}
   185 
   186 lemma wf_empty [iff]: "wf {}"
   187   by (simp add: wf_def)
   188 
   189 lemma wfP_empty [iff]:
   190   "wfP (\<lambda>x y. False)"
   191 proof -
   192   have "wfP bot" by (fact wf_empty [to_pred bot_empty_eq2])
   193   then show ?thesis by (simp add: bot_fun_def)
   194 qed
   195 
   196 lemma wf_Int1: "wf r ==> wf (r Int r')"
   197   apply (erule wf_subset)
   198   apply (rule Int_lower1)
   199   done
   200 
   201 lemma wf_Int2: "wf r ==> wf (r' Int r)"
   202   apply (erule wf_subset)
   203   apply (rule Int_lower2)
   204   done  
   205 
   206 text {* Exponentiation *}
   207 
   208 lemma wf_exp:
   209   assumes "wf (R ^^ n)"
   210   shows "wf R"
   211 proof (rule wfI_pf)
   212   fix A assume "A \<subseteq> R `` A"
   213   then have "A \<subseteq> (R ^^ n) `` A" by (induct n) force+
   214   with `wf (R ^^ n)`
   215   show "A = {}" by (rule wfE_pf)
   216 qed
   217 
   218 text {* Well-foundedness of insert *}
   219 
   220 lemma wf_insert [iff]: "wf(insert (y,x) r) = (wf(r) & (x,y) ~: r^*)"
   221 apply (rule iffI)
   222  apply (blast elim: wf_trancl [THEN wf_irrefl]
   223               intro: rtrancl_into_trancl1 wf_subset 
   224                      rtrancl_mono [THEN [2] rev_subsetD])
   225 apply (simp add: wf_eq_minimal, safe)
   226 apply (rule allE, assumption, erule impE, blast) 
   227 apply (erule bexE)
   228 apply (rename_tac "a", case_tac "a = x")
   229  prefer 2
   230 apply blast 
   231 apply (case_tac "y:Q")
   232  prefer 2 apply blast
   233 apply (rule_tac x = "{z. z:Q & (z,y) : r^*}" in allE)
   234  apply assumption
   235 apply (erule_tac V = "ALL Q. (EX x. x : Q) --> ?P Q" in thin_rl) 
   236   --{*essential for speed*}
   237 txt{*Blast with new substOccur fails*}
   238 apply (fast intro: converse_rtrancl_into_rtrancl)
   239 done
   240 
   241 text{*Well-foundedness of image*}
   242 
   243 lemma wf_map_pair_image: "[| wf r; inj f |] ==> wf(map_pair f f ` r)"
   244 apply (simp only: wf_eq_minimal, clarify)
   245 apply (case_tac "EX p. f p : Q")
   246 apply (erule_tac x = "{p. f p : Q}" in allE)
   247 apply (fast dest: inj_onD, blast)
   248 done
   249 
   250 
   251 subsection {* Well-Foundedness Results for Unions *}
   252 
   253 lemma wf_union_compatible:
   254   assumes "wf R" "wf S"
   255   assumes "R O S \<subseteq> R"
   256   shows "wf (R \<union> S)"
   257 proof (rule wfI_min)
   258   fix x :: 'a and Q 
   259   let ?Q' = "{x \<in> Q. \<forall>y. (y, x) \<in> R \<longrightarrow> y \<notin> Q}"
   260   assume "x \<in> Q"
   261   obtain a where "a \<in> ?Q'"
   262     by (rule wfE_min [OF `wf R` `x \<in> Q`]) blast
   263   with `wf S`
   264   obtain z where "z \<in> ?Q'" and zmin: "\<And>y. (y, z) \<in> S \<Longrightarrow> y \<notin> ?Q'" by (erule wfE_min)
   265   { 
   266     fix y assume "(y, z) \<in> S"
   267     then have "y \<notin> ?Q'" by (rule zmin)
   268 
   269     have "y \<notin> Q"
   270     proof 
   271       assume "y \<in> Q"
   272       with `y \<notin> ?Q'` 
   273       obtain w where "(w, y) \<in> R" and "w \<in> Q" by auto
   274       from `(w, y) \<in> R` `(y, z) \<in> S` have "(w, z) \<in> R O S" by (rule rel_compI)
   275       with `R O S \<subseteq> R` have "(w, z) \<in> R" ..
   276       with `z \<in> ?Q'` have "w \<notin> Q" by blast 
   277       with `w \<in> Q` show False by contradiction
   278     qed
   279   }
   280   with `z \<in> ?Q'` show "\<exists>z\<in>Q. \<forall>y. (y, z) \<in> R \<union> S \<longrightarrow> y \<notin> Q" by blast
   281 qed
   282 
   283 
   284 text {* Well-foundedness of indexed union with disjoint domains and ranges *}
   285 
   286 lemma wf_UN: "[| ALL i:I. wf(r i);  
   287          ALL i:I. ALL j:I. r i ~= r j --> Domain(r i) Int Range(r j) = {}  
   288       |] ==> wf(UN i:I. r i)"
   289 apply (simp only: wf_eq_minimal, clarify)
   290 apply (rename_tac A a, case_tac "EX i:I. EX a:A. EX b:A. (b,a) : r i")
   291  prefer 2
   292  apply force 
   293 apply clarify
   294 apply (drule bspec, assumption)  
   295 apply (erule_tac x="{a. a:A & (EX b:A. (b,a) : r i) }" in allE)
   296 apply (blast elim!: allE)  
   297 done
   298 
   299 lemma wfP_SUP:
   300   "\<forall>i. wfP (r i) \<Longrightarrow> \<forall>i j. r i \<noteq> r j \<longrightarrow> inf (DomainP (r i)) (RangeP (r j)) = bot \<Longrightarrow> wfP (SUPR UNIV r)"
   301   by (rule wf_UN [where I=UNIV and r="\<lambda>i. {(x, y). r i x y}", to_pred SUP_UN_eq2])
   302     (simp_all add: Collect_def)
   303 
   304 lemma wf_Union: 
   305  "[| ALL r:R. wf r;  
   306      ALL r:R. ALL s:R. r ~= s --> Domain r Int Range s = {}  
   307   |] ==> wf(Union R)"
   308   using wf_UN[of R "\<lambda>i. i"] by (simp add: SUP_def)
   309 
   310 (*Intuition: we find an (R u S)-min element of a nonempty subset A
   311              by case distinction.
   312   1. There is a step a -R-> b with a,b : A.
   313      Pick an R-min element z of the (nonempty) set {a:A | EX b:A. a -R-> b}.
   314      By definition, there is z':A s.t. z -R-> z'. Because z is R-min in the
   315      subset, z' must be R-min in A. Because z' has an R-predecessor, it cannot
   316      have an S-successor and is thus S-min in A as well.
   317   2. There is no such step.
   318      Pick an S-min element of A. In this case it must be an R-min
   319      element of A as well.
   320 
   321 *)
   322 lemma wf_Un:
   323      "[| wf r; wf s; Domain r Int Range s = {} |] ==> wf(r Un s)"
   324   using wf_union_compatible[of s r] 
   325   by (auto simp: Un_ac)
   326 
   327 lemma wf_union_merge: 
   328   "wf (R \<union> S) = wf (R O R \<union> S O R \<union> S)" (is "wf ?A = wf ?B")
   329 proof
   330   assume "wf ?A"
   331   with wf_trancl have wfT: "wf (?A^+)" .
   332   moreover have "?B \<subseteq> ?A^+"
   333     by (subst trancl_unfold, subst trancl_unfold) blast
   334   ultimately show "wf ?B" by (rule wf_subset)
   335 next
   336   assume "wf ?B"
   337 
   338   show "wf ?A"
   339   proof (rule wfI_min)
   340     fix Q :: "'a set" and x 
   341     assume "x \<in> Q"
   342 
   343     with `wf ?B`
   344     obtain z where "z \<in> Q" and "\<And>y. (y, z) \<in> ?B \<Longrightarrow> y \<notin> Q" 
   345       by (erule wfE_min)
   346     then have A1: "\<And>y. (y, z) \<in> R O R \<Longrightarrow> y \<notin> Q"
   347       and A2: "\<And>y. (y, z) \<in> S O R \<Longrightarrow> y \<notin> Q"
   348       and A3: "\<And>y. (y, z) \<in> S \<Longrightarrow> y \<notin> Q"
   349       by auto
   350     
   351     show "\<exists>z\<in>Q. \<forall>y. (y, z) \<in> ?A \<longrightarrow> y \<notin> Q"
   352     proof (cases "\<forall>y. (y, z) \<in> R \<longrightarrow> y \<notin> Q")
   353       case True
   354       with `z \<in> Q` A3 show ?thesis by blast
   355     next
   356       case False 
   357       then obtain z' where "z'\<in>Q" "(z', z) \<in> R" by blast
   358 
   359       have "\<forall>y. (y, z') \<in> ?A \<longrightarrow> y \<notin> Q"
   360       proof (intro allI impI)
   361         fix y assume "(y, z') \<in> ?A"
   362         then show "y \<notin> Q"
   363         proof
   364           assume "(y, z') \<in> R" 
   365           then have "(y, z) \<in> R O R" using `(z', z) \<in> R` ..
   366           with A1 show "y \<notin> Q" .
   367         next
   368           assume "(y, z') \<in> S" 
   369           then have "(y, z) \<in> S O R" using  `(z', z) \<in> R` ..
   370           with A2 show "y \<notin> Q" .
   371         qed
   372       qed
   373       with `z' \<in> Q` show ?thesis ..
   374     qed
   375   qed
   376 qed
   377 
   378 lemma wf_comp_self: "wf R = wf (R O R)"  -- {* special case *}
   379   by (rule wf_union_merge [where S = "{}", simplified])
   380 
   381 
   382 subsection {* Acyclic relations *}
   383 
   384 definition acyclic :: "('a * 'a) set => bool" where
   385   "acyclic r == !x. (x,x) ~: r^+"
   386 
   387 abbreviation acyclicP :: "('a => 'a => bool) => bool" where
   388   "acyclicP r == acyclic {(x, y). r x y}"
   389 
   390 lemma acyclicI: "ALL x. (x, x) ~: r^+ ==> acyclic r"
   391   by (simp add: acyclic_def)
   392 
   393 lemma wf_acyclic: "wf r ==> acyclic r"
   394 apply (simp add: acyclic_def)
   395 apply (blast elim: wf_trancl [THEN wf_irrefl])
   396 done
   397 
   398 lemmas wfP_acyclicP = wf_acyclic [to_pred]
   399 
   400 lemma acyclic_insert [iff]:
   401      "acyclic(insert (y,x) r) = (acyclic r & (x,y) ~: r^*)"
   402 apply (simp add: acyclic_def trancl_insert)
   403 apply (blast intro: rtrancl_trans)
   404 done
   405 
   406 lemma acyclic_converse [iff]: "acyclic(r^-1) = acyclic r"
   407 by (simp add: acyclic_def trancl_converse)
   408 
   409 lemmas acyclicP_converse [iff] = acyclic_converse [to_pred]
   410 
   411 lemma acyclic_impl_antisym_rtrancl: "acyclic r ==> antisym(r^*)"
   412 apply (simp add: acyclic_def antisym_def)
   413 apply (blast elim: rtranclE intro: rtrancl_into_trancl1 rtrancl_trancl_trancl)
   414 done
   415 
   416 (* Other direction:
   417 acyclic = no loops
   418 antisym = only self loops
   419 Goalw [acyclic_def,antisym_def] "antisym( r^* ) ==> acyclic(r - Id)
   420 ==> antisym( r^* ) = acyclic(r - Id)";
   421 *)
   422 
   423 lemma acyclic_subset: "[| acyclic s; r <= s |] ==> acyclic r"
   424 apply (simp add: acyclic_def)
   425 apply (blast intro: trancl_mono)
   426 done
   427 
   428 text{* Wellfoundedness of finite acyclic relations*}
   429 
   430 lemma finite_acyclic_wf [rule_format]: "finite r ==> acyclic r --> wf r"
   431 apply (erule finite_induct, blast)
   432 apply (simp (no_asm_simp) only: split_tupled_all)
   433 apply simp
   434 done
   435 
   436 lemma finite_acyclic_wf_converse: "[|finite r; acyclic r|] ==> wf (r^-1)"
   437 apply (erule finite_converse [THEN iffD2, THEN finite_acyclic_wf])
   438 apply (erule acyclic_converse [THEN iffD2])
   439 done
   440 
   441 lemma wf_iff_acyclic_if_finite: "finite r ==> wf r = acyclic r"
   442 by (blast intro: finite_acyclic_wf wf_acyclic)
   443 
   444 
   445 subsection {* @{typ nat} is well-founded *}
   446 
   447 lemma less_nat_rel: "op < = (\<lambda>m n. n = Suc m)^++"
   448 proof (rule ext, rule ext, rule iffI)
   449   fix n m :: nat
   450   assume "m < n"
   451   then show "(\<lambda>m n. n = Suc m)^++ m n"
   452   proof (induct n)
   453     case 0 then show ?case by auto
   454   next
   455     case (Suc n) then show ?case
   456       by (auto simp add: less_Suc_eq_le le_less intro: tranclp.trancl_into_trancl)
   457   qed
   458 next
   459   fix n m :: nat
   460   assume "(\<lambda>m n. n = Suc m)^++ m n"
   461   then show "m < n"
   462     by (induct n)
   463       (simp_all add: less_Suc_eq_le reflexive le_less)
   464 qed
   465 
   466 definition
   467   pred_nat :: "(nat * nat) set" where
   468   "pred_nat = {(m, n). n = Suc m}"
   469 
   470 definition
   471   less_than :: "(nat * nat) set" where
   472   "less_than = pred_nat^+"
   473 
   474 lemma less_eq: "(m, n) \<in> pred_nat^+ \<longleftrightarrow> m < n"
   475   unfolding less_nat_rel pred_nat_def trancl_def by simp
   476 
   477 lemma pred_nat_trancl_eq_le:
   478   "(m, n) \<in> pred_nat^* \<longleftrightarrow> m \<le> n"
   479   unfolding less_eq rtrancl_eq_or_trancl by auto
   480 
   481 lemma wf_pred_nat: "wf pred_nat"
   482   apply (unfold wf_def pred_nat_def, clarify)
   483   apply (induct_tac x, blast+)
   484   done
   485 
   486 lemma wf_less_than [iff]: "wf less_than"
   487   by (simp add: less_than_def wf_pred_nat [THEN wf_trancl])
   488 
   489 lemma trans_less_than [iff]: "trans less_than"
   490   by (simp add: less_than_def)
   491 
   492 lemma less_than_iff [iff]: "((x,y): less_than) = (x<y)"
   493   by (simp add: less_than_def less_eq)
   494 
   495 lemma wf_less: "wf {(x, y::nat). x < y}"
   496   using wf_less_than by (simp add: less_than_def less_eq [symmetric])
   497 
   498 
   499 subsection {* Accessible Part *}
   500 
   501 text {*
   502  Inductive definition of the accessible part @{term "acc r"} of a
   503  relation; see also \cite{paulin-tlca}.
   504 *}
   505 
   506 inductive_set
   507   acc :: "('a * 'a) set => 'a set"
   508   for r :: "('a * 'a) set"
   509   where
   510     accI: "(!!y. (y, x) : r ==> y : acc r) ==> x : acc r"
   511 
   512 abbreviation
   513   termip :: "('a => 'a => bool) => 'a => bool" where
   514   "termip r == accp (r\<inverse>\<inverse>)"
   515 
   516 abbreviation
   517   termi :: "('a * 'a) set => 'a set" where
   518   "termi r == acc (r\<inverse>)"
   519 
   520 lemmas accpI = accp.accI
   521 
   522 text {* Induction rules *}
   523 
   524 theorem accp_induct:
   525   assumes major: "accp r a"
   526   assumes hyp: "!!x. accp r x ==> \<forall>y. r y x --> P y ==> P x"
   527   shows "P a"
   528   apply (rule major [THEN accp.induct])
   529   apply (rule hyp)
   530    apply (rule accp.accI)
   531    apply fast
   532   apply fast
   533   done
   534 
   535 theorems accp_induct_rule = accp_induct [rule_format, induct set: accp]
   536 
   537 theorem accp_downward: "accp r b ==> r a b ==> accp r a"
   538   apply (erule accp.cases)
   539   apply fast
   540   done
   541 
   542 lemma not_accp_down:
   543   assumes na: "\<not> accp R x"
   544   obtains z where "R z x" and "\<not> accp R z"
   545 proof -
   546   assume a: "\<And>z. \<lbrakk>R z x; \<not> accp R z\<rbrakk> \<Longrightarrow> thesis"
   547 
   548   show thesis
   549   proof (cases "\<forall>z. R z x \<longrightarrow> accp R z")
   550     case True
   551     hence "\<And>z. R z x \<Longrightarrow> accp R z" by auto
   552     hence "accp R x"
   553       by (rule accp.accI)
   554     with na show thesis ..
   555   next
   556     case False then obtain z where "R z x" and "\<not> accp R z"
   557       by auto
   558     with a show thesis .
   559   qed
   560 qed
   561 
   562 lemma accp_downwards_aux: "r\<^sup>*\<^sup>* b a ==> accp r a --> accp r b"
   563   apply (erule rtranclp_induct)
   564    apply blast
   565   apply (blast dest: accp_downward)
   566   done
   567 
   568 theorem accp_downwards: "accp r a ==> r\<^sup>*\<^sup>* b a ==> accp r b"
   569   apply (blast dest: accp_downwards_aux)
   570   done
   571 
   572 theorem accp_wfPI: "\<forall>x. accp r x ==> wfP r"
   573   apply (rule wfPUNIVI)
   574   apply (rule_tac P=P in accp_induct)
   575    apply blast
   576   apply blast
   577   done
   578 
   579 theorem accp_wfPD: "wfP r ==> accp r x"
   580   apply (erule wfP_induct_rule)
   581   apply (rule accp.accI)
   582   apply blast
   583   done
   584 
   585 theorem wfP_accp_iff: "wfP r = (\<forall>x. accp r x)"
   586   apply (blast intro: accp_wfPI dest: accp_wfPD)
   587   done
   588 
   589 
   590 text {* Smaller relations have bigger accessible parts: *}
   591 
   592 lemma accp_subset:
   593   assumes sub: "R1 \<le> R2"
   594   shows "accp R2 \<le> accp R1"
   595 proof (rule predicate1I)
   596   fix x assume "accp R2 x"
   597   then show "accp R1 x"
   598   proof (induct x)
   599     fix x
   600     assume ih: "\<And>y. R2 y x \<Longrightarrow> accp R1 y"
   601     with sub show "accp R1 x"
   602       by (blast intro: accp.accI)
   603   qed
   604 qed
   605 
   606 
   607 text {* This is a generalized induction theorem that works on
   608   subsets of the accessible part. *}
   609 
   610 lemma accp_subset_induct:
   611   assumes subset: "D \<le> accp R"
   612     and dcl: "\<And>x z. \<lbrakk>D x; R z x\<rbrakk> \<Longrightarrow> D z"
   613     and "D x"
   614     and istep: "\<And>x. \<lbrakk>D x; (\<And>z. R z x \<Longrightarrow> P z)\<rbrakk> \<Longrightarrow> P x"
   615   shows "P x"
   616 proof -
   617   from subset and `D x`
   618   have "accp R x" ..
   619   then show "P x" using `D x`
   620   proof (induct x)
   621     fix x
   622     assume "D x"
   623       and "\<And>y. R y x \<Longrightarrow> D y \<Longrightarrow> P y"
   624     with dcl and istep show "P x" by blast
   625   qed
   626 qed
   627 
   628 
   629 text {* Set versions of the above theorems *}
   630 
   631 lemmas acc_induct = accp_induct [to_set]
   632 
   633 lemmas acc_induct_rule = acc_induct [rule_format, induct set: acc]
   634 
   635 lemmas acc_downward = accp_downward [to_set]
   636 
   637 lemmas not_acc_down = not_accp_down [to_set]
   638 
   639 lemmas acc_downwards_aux = accp_downwards_aux [to_set]
   640 
   641 lemmas acc_downwards = accp_downwards [to_set]
   642 
   643 lemmas acc_wfI = accp_wfPI [to_set]
   644 
   645 lemmas acc_wfD = accp_wfPD [to_set]
   646 
   647 lemmas wf_acc_iff = wfP_accp_iff [to_set]
   648 
   649 lemmas acc_subset = accp_subset [to_set pred_subset_eq]
   650 
   651 lemmas acc_subset_induct = accp_subset_induct [to_set pred_subset_eq]
   652 
   653 
   654 subsection {* Tools for building wellfounded relations *}
   655 
   656 text {* Inverse Image *}
   657 
   658 lemma wf_inv_image [simp,intro!]: "wf(r) ==> wf(inv_image r (f::'a=>'b))"
   659 apply (simp (no_asm_use) add: inv_image_def wf_eq_minimal)
   660 apply clarify
   661 apply (subgoal_tac "EX (w::'b) . w : {w. EX (x::'a) . x: Q & (f x = w) }")
   662 prefer 2 apply (blast del: allE)
   663 apply (erule allE)
   664 apply (erule (1) notE impE)
   665 apply blast
   666 done
   667 
   668 text {* Measure functions into @{typ nat} *}
   669 
   670 definition measure :: "('a => nat) => ('a * 'a)set"
   671 where "measure == inv_image less_than"
   672 
   673 lemma in_measure[simp]: "((x,y) : measure f) = (f x < f y)"
   674   by (simp add:measure_def)
   675 
   676 lemma wf_measure [iff]: "wf (measure f)"
   677 apply (unfold measure_def)
   678 apply (rule wf_less_than [THEN wf_inv_image])
   679 done
   680 
   681 lemma wf_if_measure: fixes f :: "'a \<Rightarrow> nat"
   682 shows "(!!x. P x \<Longrightarrow> f(g x) < f x) \<Longrightarrow> wf {(y,x). P x \<and> y = g x}"
   683 apply(insert wf_measure[of f])
   684 apply(simp only: measure_def inv_image_def less_than_def less_eq)
   685 apply(erule wf_subset)
   686 apply auto
   687 done
   688 
   689 
   690 text{* Lexicographic combinations *}
   691 
   692 definition lex_prod :: "('a \<times>'a) set \<Rightarrow> ('b \<times> 'b) set \<Rightarrow> (('a \<times> 'b) \<times> ('a \<times> 'b)) set" (infixr "<*lex*>" 80) where
   693   "ra <*lex*> rb = {((a, b), (a', b')). (a, a') \<in> ra \<or> a = a' \<and> (b, b') \<in> rb}"
   694 
   695 lemma wf_lex_prod [intro!]: "[| wf(ra); wf(rb) |] ==> wf(ra <*lex*> rb)"
   696 apply (unfold wf_def lex_prod_def) 
   697 apply (rule allI, rule impI)
   698 apply (simp (no_asm_use) only: split_paired_All)
   699 apply (drule spec, erule mp) 
   700 apply (rule allI, rule impI)
   701 apply (drule spec, erule mp, blast) 
   702 done
   703 
   704 lemma in_lex_prod[simp]: 
   705   "(((a,b),(a',b')): r <*lex*> s) = ((a,a'): r \<or> (a = a' \<and> (b, b') : s))"
   706   by (auto simp:lex_prod_def)
   707 
   708 text{* @{term "op <*lex*>"} preserves transitivity *}
   709 
   710 lemma trans_lex_prod [intro!]: 
   711     "[| trans R1; trans R2 |] ==> trans (R1 <*lex*> R2)"
   712 by (unfold trans_def lex_prod_def, blast) 
   713 
   714 text {* lexicographic combinations with measure functions *}
   715 
   716 definition 
   717   mlex_prod :: "('a \<Rightarrow> nat) \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" (infixr "<*mlex*>" 80)
   718 where
   719   "f <*mlex*> R = inv_image (less_than <*lex*> R) (%x. (f x, x))"
   720 
   721 lemma wf_mlex: "wf R \<Longrightarrow> wf (f <*mlex*> R)"
   722 unfolding mlex_prod_def
   723 by auto
   724 
   725 lemma mlex_less: "f x < f y \<Longrightarrow> (x, y) \<in> f <*mlex*> R"
   726 unfolding mlex_prod_def by simp
   727 
   728 lemma mlex_leq: "f x \<le> f y \<Longrightarrow> (x, y) \<in> R \<Longrightarrow> (x, y) \<in> f <*mlex*> R"
   729 unfolding mlex_prod_def by auto
   730 
   731 text {* proper subset relation on finite sets *}
   732 
   733 definition finite_psubset  :: "('a set * 'a set) set"
   734 where "finite_psubset == {(A,B). A < B & finite B}"
   735 
   736 lemma wf_finite_psubset[simp]: "wf(finite_psubset)"
   737 apply (unfold finite_psubset_def)
   738 apply (rule wf_measure [THEN wf_subset])
   739 apply (simp add: measure_def inv_image_def less_than_def less_eq)
   740 apply (fast elim!: psubset_card_mono)
   741 done
   742 
   743 lemma trans_finite_psubset: "trans finite_psubset"
   744 by (simp add: finite_psubset_def less_le trans_def, blast)
   745 
   746 lemma in_finite_psubset[simp]: "(A, B) \<in> finite_psubset = (A < B & finite B)"
   747 unfolding finite_psubset_def by auto
   748 
   749 text {* max- and min-extension of order to finite sets *}
   750 
   751 inductive_set max_ext :: "('a \<times> 'a) set \<Rightarrow> ('a set \<times> 'a set) set" 
   752 for R :: "('a \<times> 'a) set"
   753 where
   754   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"
   755 
   756 lemma max_ext_wf:
   757   assumes wf: "wf r"
   758   shows "wf (max_ext r)"
   759 proof (rule acc_wfI, intro allI)
   760   fix M show "M \<in> acc (max_ext r)" (is "_ \<in> ?W")
   761   proof cases
   762     assume "finite M"
   763     thus ?thesis
   764     proof (induct M)
   765       show "{} \<in> ?W"
   766         by (rule accI) (auto elim: max_ext.cases)
   767     next
   768       fix M a assume "M \<in> ?W" "finite M"
   769       with wf show "insert a M \<in> ?W"
   770       proof (induct arbitrary: M)
   771         fix M a
   772         assume "M \<in> ?W"  and  [intro]: "finite M"
   773         assume hyp: "\<And>b M. (b, a) \<in> r \<Longrightarrow> M \<in> ?W \<Longrightarrow> finite M \<Longrightarrow> insert b M \<in> ?W"
   774         {
   775           fix N M :: "'a set"
   776           assume "finite N" "finite M"
   777           then
   778           have "\<lbrakk>M \<in> ?W ; (\<And>y. y \<in> N \<Longrightarrow> (y, a) \<in> r)\<rbrakk> \<Longrightarrow>  N \<union> M \<in> ?W"
   779             by (induct N arbitrary: M) (auto simp: hyp)
   780         }
   781         note add_less = this
   782         
   783         show "insert a M \<in> ?W"
   784         proof (rule accI)
   785           fix N assume Nless: "(N, insert a M) \<in> max_ext r"
   786           hence asm1: "\<And>x. x \<in> N \<Longrightarrow> (x, a) \<in> r \<or> (\<exists>y \<in> M. (x, y) \<in> r)"
   787             by (auto elim!: max_ext.cases)
   788 
   789           let ?N1 = "{ n \<in> N. (n, a) \<in> r }"
   790           let ?N2 = "{ n \<in> N. (n, a) \<notin> r }"
   791           have N: "?N1 \<union> ?N2 = N" by (rule set_eqI) auto
   792           from Nless have "finite N" by (auto elim: max_ext.cases)
   793           then have finites: "finite ?N1" "finite ?N2" by auto
   794           
   795           have "?N2 \<in> ?W"
   796           proof cases
   797             assume [simp]: "M = {}"
   798             have Mw: "{} \<in> ?W" by (rule accI) (auto elim: max_ext.cases)
   799 
   800             from asm1 have "?N2 = {}" by auto
   801             with Mw show "?N2 \<in> ?W" by (simp only:)
   802           next
   803             assume "M \<noteq> {}"
   804             have N2: "(?N2, M) \<in> max_ext r" 
   805               by (rule max_extI[OF _ _ `M \<noteq> {}`]) (insert asm1, auto intro: finites)
   806             
   807             with `M \<in> ?W` show "?N2 \<in> ?W" by (rule acc_downward)
   808           qed
   809           with finites have "?N1 \<union> ?N2 \<in> ?W" 
   810             by (rule add_less) simp
   811           then show "N \<in> ?W" by (simp only: N)
   812         qed
   813       qed
   814     qed
   815   next
   816     assume [simp]: "\<not> finite M"
   817     show ?thesis
   818       by (rule accI) (auto elim: max_ext.cases)
   819   qed
   820 qed
   821 
   822 lemma max_ext_additive: 
   823  "(A, B) \<in> max_ext R \<Longrightarrow> (C, D) \<in> max_ext R \<Longrightarrow>
   824   (A \<union> C, B \<union> D) \<in> max_ext R"
   825 by (force elim!: max_ext.cases)
   826 
   827 
   828 definition min_ext :: "('a \<times> 'a) set \<Rightarrow> ('a set \<times> 'a set) set"  where
   829   "min_ext r = {(X, Y) | X Y. X \<noteq> {} \<and> (\<forall>y \<in> Y. (\<exists>x \<in> X. (x, y) \<in> r))}"
   830 
   831 lemma min_ext_wf:
   832   assumes "wf r"
   833   shows "wf (min_ext r)"
   834 proof (rule wfI_min)
   835   fix Q :: "'a set set"
   836   fix x
   837   assume nonempty: "x \<in> Q"
   838   show "\<exists>m \<in> Q. (\<forall> n. (n, m) \<in> min_ext r \<longrightarrow> n \<notin> Q)"
   839   proof cases
   840     assume "Q = {{}}" thus ?thesis by (simp add: min_ext_def)
   841   next
   842     assume "Q \<noteq> {{}}"
   843     with nonempty
   844     obtain e x where "x \<in> Q" "e \<in> x" by force
   845     then have eU: "e \<in> \<Union>Q" by auto
   846     with `wf r` 
   847     obtain z where z: "z \<in> \<Union>Q" "\<And>y. (y, z) \<in> r \<Longrightarrow> y \<notin> \<Union>Q" 
   848       by (erule wfE_min)
   849     from z obtain m where "m \<in> Q" "z \<in> m" by auto
   850     from `m \<in> Q`
   851     show ?thesis
   852     proof (rule, intro bexI allI impI)
   853       fix n
   854       assume smaller: "(n, m) \<in> min_ext r"
   855       with `z \<in> m` obtain y where y: "y \<in> n" "(y, z) \<in> r" by (auto simp: min_ext_def)
   856       then show "n \<notin> Q" using z(2) by auto
   857     qed      
   858   qed
   859 qed
   860 
   861 text{* Bounded increase must terminate: *}
   862 
   863 lemma wf_bounded_measure:
   864 fixes ub :: "'a \<Rightarrow> nat" and f :: "'a \<Rightarrow> nat"
   865 assumes "!!a b. (b,a) : r \<Longrightarrow> ub b \<le> ub a & ub a \<ge> f b & f b > f a"
   866 shows "wf r"
   867 apply(rule wf_subset[OF wf_measure[of "%a. ub a - f a"]])
   868 apply (auto dest: assms)
   869 done
   870 
   871 lemma wf_bounded_set:
   872 fixes ub :: "'a \<Rightarrow> 'b set" and f :: "'a \<Rightarrow> 'b set"
   873 assumes "!!a b. (b,a) : r \<Longrightarrow>
   874   finite(ub a) & ub b \<subseteq> ub a & ub a \<supseteq> f b & f b \<supset> f a"
   875 shows "wf r"
   876 apply(rule wf_bounded_measure[of r "%a. card(ub a)" "%a. card(f a)"])
   877 apply(drule assms)
   878 apply (blast intro: card_mono finite_subset psubset_card_mono dest: psubset_eq[THEN iffD2])
   879 done
   880 
   881 
   882 subsection {* size of a datatype value *}
   883 
   884 use "Tools/Function/size.ML"
   885 
   886 setup Size.setup
   887 
   888 lemma size_bool [code]:
   889   "size (b\<Colon>bool) = 0" by (cases b) auto
   890 
   891 lemma nat_size [simp, code]: "size (n\<Colon>nat) = n"
   892   by (induct n) simp_all
   893 
   894 declare "prod.size" [no_atp]
   895 
   896 lemma [code]:
   897   "size (P :: 'a Predicate.pred) = 0" by (cases P) simp
   898 
   899 lemma [code]:
   900   "pred_size f P = 0" by (cases P) simp
   901 
   902 end