src/HOL/Wellfounded.thy
 author wenzelm Tue Sep 03 01:12:40 2013 +0200 (2013-09-03) changeset 53374 a14d2a854c02 parent 49945 fb696ff1f086 child 54295 45a5523d4a63 permissions -rw-r--r--
tuned proofs -- clarified flow of facts wrt. calculation;
```     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 begin
```
```    13
```
```    14 subsection {* Basic Definitions *}
```
```    15
```
```    16 definition wf :: "('a * 'a) set => bool" where
```
```    17   "wf r \<longleftrightarrow> (!P. (!x. (!y. (y,x):r --> P(y)) --> P(x)) --> (!x. P(x)))"
```
```    18
```
```    19 definition wfP :: "('a => 'a => bool) => bool" where
```
```    20   "wfP r \<longleftrightarrow> wf {(x, y). r x y}"
```
```    21
```
```    22 lemma wfP_wf_eq [pred_set_conv]: "wfP (\<lambda>x y. (x, y) \<in> r) = wf r"
```
```    23   by (simp add: wfP_def)
```
```    24
```
```    25 lemma wfUNIVI:
```
```    26    "(!!P x. (ALL x. (ALL y. (y,x) : r --> P(y)) --> P(x)) ==> P(x)) ==> wf(r)"
```
```    27   unfolding wf_def by blast
```
```    28
```
```    29 lemmas wfPUNIVI = wfUNIVI [to_pred]
```
```    30
```
```    31 text{*Restriction to domain @{term A} and range @{term B}.  If @{term r} is
```
```    32     well-founded over their intersection, then @{term "wf r"}*}
```
```    33 lemma wfI:
```
```    34  "[| r \<subseteq> A <*> B;
```
```    35      !!x P. [|\<forall>x. (\<forall>y. (y,x) : r --> P y) --> P x;  x : A; x : B |] ==> P x |]
```
```    36   ==>  wf r"
```
```    37   unfolding wf_def by blast
```
```    38
```
```    39 lemma wf_induct:
```
```    40     "[| wf(r);
```
```    41         !!x.[| ALL y. (y,x): r --> P(y) |] ==> P(x)
```
```    42      |]  ==>  P(a)"
```
```    43   unfolding wf_def by blast
```
```    44
```
```    45 lemmas wfP_induct = wf_induct [to_pred]
```
```    46
```
```    47 lemmas wf_induct_rule = wf_induct [rule_format, consumes 1, case_names less, induct set: wf]
```
```    48
```
```    49 lemmas wfP_induct_rule = wf_induct_rule [to_pred, induct set: wfP]
```
```    50
```
```    51 lemma wf_not_sym: "wf r ==> (a, x) : r ==> (x, a) ~: r"
```
```    52   by (induct a arbitrary: x set: wf) blast
```
```    53
```
```    54 lemma wf_asym:
```
```    55   assumes "wf r" "(a, x) \<in> r"
```
```    56   obtains "(x, a) \<notin> r"
```
```    57   by (drule wf_not_sym[OF assms])
```
```    58
```
```    59 lemma wf_not_refl [simp]: "wf r ==> (a, a) ~: r"
```
```    60   by (blast elim: wf_asym)
```
```    61
```
```    62 lemma wf_irrefl: assumes "wf r" obtains "(a, a) \<notin> r"
```
```    63 by (drule wf_not_refl[OF assms])
```
```    64
```
```    65 lemma wf_wellorderI:
```
```    66   assumes wf: "wf {(x::'a::ord, y). x < y}"
```
```    67   assumes lin: "OFCLASS('a::ord, linorder_class)"
```
```    68   shows "OFCLASS('a::ord, wellorder_class)"
```
```    69 using lin by (rule wellorder_class.intro)
```
```    70   (blast intro: class.wellorder_axioms.intro wf_induct_rule [OF wf])
```
```    71
```
```    72 lemma (in wellorder) wf:
```
```    73   "wf {(x, y). x < y}"
```
```    74 unfolding wf_def by (blast intro: less_induct)
```
```    75
```
```    76
```
```    77 subsection {* Basic Results *}
```
```    78
```
```    79 text {* Point-free characterization of well-foundedness *}
```
```    80
```
```    81 lemma wfE_pf:
```
```    82   assumes wf: "wf R"
```
```    83   assumes a: "A \<subseteq> R `` A"
```
```    84   shows "A = {}"
```
```    85 proof -
```
```    86   { fix x
```
```    87     from wf have "x \<notin> A"
```
```    88     proof induct
```
```    89       fix x assume "\<And>y. (y, x) \<in> R \<Longrightarrow> y \<notin> A"
```
```    90       then have "x \<notin> R `` A" by blast
```
```    91       with a show "x \<notin> A" by blast
```
```    92     qed
```
```    93   } thus ?thesis by auto
```
```    94 qed
```
```    95
```
```    96 lemma wfI_pf:
```
```    97   assumes a: "\<And>A. A \<subseteq> R `` A \<Longrightarrow> A = {}"
```
```    98   shows "wf R"
```
```    99 proof (rule wfUNIVI)
```
```   100   fix P :: "'a \<Rightarrow> bool" and x
```
```   101   let ?A = "{x. \<not> P x}"
```
```   102   assume "\<forall>x. (\<forall>y. (y, x) \<in> R \<longrightarrow> P y) \<longrightarrow> P x"
```
```   103   then have "?A \<subseteq> R `` ?A" by blast
```
```   104   with a show "P x" by blast
```
```   105 qed
```
```   106
```
```   107 text{*Minimal-element characterization of well-foundedness*}
```
```   108
```
```   109 lemma wfE_min:
```
```   110   assumes wf: "wf R" and Q: "x \<in> Q"
```
```   111   obtains z where "z \<in> Q" "\<And>y. (y, z) \<in> R \<Longrightarrow> y \<notin> Q"
```
```   112   using Q wfE_pf[OF wf, of Q] by blast
```
```   113
```
```   114 lemma wfI_min:
```
```   115   assumes a: "\<And>x Q. x \<in> Q \<Longrightarrow> \<exists>z\<in>Q. \<forall>y. (y, z) \<in> R \<longrightarrow> y \<notin> Q"
```
```   116   shows "wf R"
```
```   117 proof (rule wfI_pf)
```
```   118   fix A assume b: "A \<subseteq> R `` A"
```
```   119   { fix x assume "x \<in> A"
```
```   120     from a[OF this] b have "False" by blast
```
```   121   }
```
```   122   thus "A = {}" by blast
```
```   123 qed
```
```   124
```
```   125 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))"
```
```   126 apply auto
```
```   127 apply (erule wfE_min, assumption, blast)
```
```   128 apply (rule wfI_min, auto)
```
```   129 done
```
```   130
```
```   131 lemmas wfP_eq_minimal = wf_eq_minimal [to_pred]
```
```   132
```
```   133 text{* Well-foundedness of transitive closure *}
```
```   134
```
```   135 lemma wf_trancl:
```
```   136   assumes "wf r"
```
```   137   shows "wf (r^+)"
```
```   138 proof -
```
```   139   {
```
```   140     fix P and x
```
```   141     assume induct_step: "!!x. (!!y. (y, x) : r^+ ==> P y) ==> P x"
```
```   142     have "P x"
```
```   143     proof (rule induct_step)
```
```   144       fix y assume "(y, x) : r^+"
```
```   145       with `wf r` show "P y"
```
```   146       proof (induct x arbitrary: y)
```
```   147         case (less x)
```
```   148         note hyp = `\<And>x' y'. (x', x) : r ==> (y', x') : r^+ ==> P y'`
```
```   149         from `(y, x) : r^+` show "P y"
```
```   150         proof cases
```
```   151           case base
```
```   152           show "P y"
```
```   153           proof (rule induct_step)
```
```   154             fix y' assume "(y', y) : r^+"
```
```   155             with `(y, x) : r` show "P y'" by (rule hyp [of y y'])
```
```   156           qed
```
```   157         next
```
```   158           case step
```
```   159           then obtain x' where "(x', x) : r" and "(y, x') : r^+" by simp
```
```   160           then show "P y" by (rule hyp [of x' y])
```
```   161         qed
```
```   162       qed
```
```   163     qed
```
```   164   } then show ?thesis unfolding wf_def by blast
```
```   165 qed
```
```   166
```
```   167 lemmas wfP_trancl = wf_trancl [to_pred]
```
```   168
```
```   169 lemma wf_converse_trancl: "wf (r^-1) ==> wf ((r^+)^-1)"
```
```   170   apply (subst trancl_converse [symmetric])
```
```   171   apply (erule wf_trancl)
```
```   172   done
```
```   173
```
```   174 text {* Well-foundedness of subsets *}
```
```   175
```
```   176 lemma wf_subset: "[| wf(r);  p<=r |] ==> wf(p)"
```
```   177   apply (simp (no_asm_use) add: wf_eq_minimal)
```
```   178   apply fast
```
```   179   done
```
```   180
```
```   181 lemmas wfP_subset = wf_subset [to_pred]
```
```   182
```
```   183 text {* Well-foundedness of the empty relation *}
```
```   184
```
```   185 lemma wf_empty [iff]: "wf {}"
```
```   186   by (simp add: wf_def)
```
```   187
```
```   188 lemma wfP_empty [iff]:
```
```   189   "wfP (\<lambda>x y. False)"
```
```   190 proof -
```
```   191   have "wfP bot" by (fact wf_empty [to_pred bot_empty_eq2])
```
```   192   then show ?thesis by (simp add: bot_fun_def)
```
```   193 qed
```
```   194
```
```   195 lemma wf_Int1: "wf r ==> wf (r Int r')"
```
```   196   apply (erule wf_subset)
```
```   197   apply (rule Int_lower1)
```
```   198   done
```
```   199
```
```   200 lemma wf_Int2: "wf r ==> wf (r' Int r)"
```
```   201   apply (erule wf_subset)
```
```   202   apply (rule Int_lower2)
```
```   203   done
```
```   204
```
```   205 text {* Exponentiation *}
```
```   206
```
```   207 lemma wf_exp:
```
```   208   assumes "wf (R ^^ n)"
```
```   209   shows "wf R"
```
```   210 proof (rule wfI_pf)
```
```   211   fix A assume "A \<subseteq> R `` A"
```
```   212   then have "A \<subseteq> (R ^^ n) `` A" by (induct n) force+
```
```   213   with `wf (R ^^ n)`
```
```   214   show "A = {}" by (rule wfE_pf)
```
```   215 qed
```
```   216
```
```   217 text {* Well-foundedness of insert *}
```
```   218
```
```   219 lemma wf_insert [iff]: "wf(insert (y,x) r) = (wf(r) & (x,y) ~: r^*)"
```
```   220 apply (rule iffI)
```
```   221  apply (blast elim: wf_trancl [THEN wf_irrefl]
```
```   222               intro: rtrancl_into_trancl1 wf_subset
```
```   223                      rtrancl_mono [THEN  rev_subsetD])
```
```   224 apply (simp add: wf_eq_minimal, safe)
```
```   225 apply (rule allE, assumption, erule impE, blast)
```
```   226 apply (erule bexE)
```
```   227 apply (rename_tac "a", case_tac "a = x")
```
```   228  prefer 2
```
```   229 apply blast
```
```   230 apply (case_tac "y:Q")
```
```   231  prefer 2 apply blast
```
```   232 apply (rule_tac x = "{z. z:Q & (z,y) : r^*}" in allE)
```
```   233  apply assumption
```
```   234 apply (erule_tac V = "ALL Q. (EX x. x : Q) --> ?P Q" in thin_rl)
```
```   235   --{*essential for speed*}
```
```   236 txt{*Blast with new substOccur fails*}
```
```   237 apply (fast intro: converse_rtrancl_into_rtrancl)
```
```   238 done
```
```   239
```
```   240 text{*Well-foundedness of image*}
```
```   241
```
```   242 lemma wf_map_pair_image: "[| wf r; inj f |] ==> wf(map_pair f f ` r)"
```
```   243 apply (simp only: wf_eq_minimal, clarify)
```
```   244 apply (case_tac "EX p. f p : Q")
```
```   245 apply (erule_tac x = "{p. f p : Q}" in allE)
```
```   246 apply (fast dest: inj_onD, blast)
```
```   247 done
```
```   248
```
```   249
```
```   250 subsection {* Well-Foundedness Results for Unions *}
```
```   251
```
```   252 lemma wf_union_compatible:
```
```   253   assumes "wf R" "wf S"
```
```   254   assumes "R O S \<subseteq> R"
```
```   255   shows "wf (R \<union> S)"
```
```   256 proof (rule wfI_min)
```
```   257   fix x :: 'a and Q
```
```   258   let ?Q' = "{x \<in> Q. \<forall>y. (y, x) \<in> R \<longrightarrow> y \<notin> Q}"
```
```   259   assume "x \<in> Q"
```
```   260   obtain a where "a \<in> ?Q'"
```
```   261     by (rule wfE_min [OF `wf R` `x \<in> Q`]) blast
```
```   262   with `wf S`
```
```   263   obtain z where "z \<in> ?Q'" and zmin: "\<And>y. (y, z) \<in> S \<Longrightarrow> y \<notin> ?Q'" by (erule wfE_min)
```
```   264   {
```
```   265     fix y assume "(y, z) \<in> S"
```
```   266     then have "y \<notin> ?Q'" by (rule zmin)
```
```   267
```
```   268     have "y \<notin> Q"
```
```   269     proof
```
```   270       assume "y \<in> Q"
```
```   271       with `y \<notin> ?Q'`
```
```   272       obtain w where "(w, y) \<in> R" and "w \<in> Q" by auto
```
```   273       from `(w, y) \<in> R` `(y, z) \<in> S` have "(w, z) \<in> R O S" by (rule relcompI)
```
```   274       with `R O S \<subseteq> R` have "(w, z) \<in> R" ..
```
```   275       with `z \<in> ?Q'` have "w \<notin> Q" by blast
```
```   276       with `w \<in> Q` show False by contradiction
```
```   277     qed
```
```   278   }
```
```   279   with `z \<in> ?Q'` show "\<exists>z\<in>Q. \<forall>y. (y, z) \<in> R \<union> S \<longrightarrow> y \<notin> Q" by blast
```
```   280 qed
```
```   281
```
```   282
```
```   283 text {* Well-foundedness of indexed union with disjoint domains and ranges *}
```
```   284
```
```   285 lemma wf_UN: "[| ALL i:I. wf(r i);
```
```   286          ALL i:I. ALL j:I. r i ~= r j --> Domain(r i) Int Range(r j) = {}
```
```   287       |] ==> wf(UN i:I. r i)"
```
```   288 apply (simp only: wf_eq_minimal, clarify)
```
```   289 apply (rename_tac A a, case_tac "EX i:I. EX a:A. EX b:A. (b,a) : r i")
```
```   290  prefer 2
```
```   291  apply force
```
```   292 apply clarify
```
```   293 apply (drule bspec, assumption)
```
```   294 apply (erule_tac x="{a. a:A & (EX b:A. (b,a) : r i) }" in allE)
```
```   295 apply (blast elim!: allE)
```
```   296 done
```
```   297
```
```   298 lemma wfP_SUP:
```
```   299   "\<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)"
```
```   300   apply (rule wf_UN[to_pred])
```
```   301   apply simp_all
```
```   302   done
```
```   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 lemma wf_acyclic: "wf r ==> acyclic r"
```
```   385 apply (simp add: acyclic_def)
```
```   386 apply (blast elim: wf_trancl [THEN wf_irrefl])
```
```   387 done
```
```   388
```
```   389 lemmas wfP_acyclicP = wf_acyclic [to_pred]
```
```   390
```
```   391 text{* Wellfoundedness of finite acyclic relations*}
```
```   392
```
```   393 lemma finite_acyclic_wf [rule_format]: "finite r ==> acyclic r --> wf r"
```
```   394 apply (erule finite_induct, blast)
```
```   395 apply (simp (no_asm_simp) only: split_tupled_all)
```
```   396 apply simp
```
```   397 done
```
```   398
```
```   399 lemma finite_acyclic_wf_converse: "[|finite r; acyclic r|] ==> wf (r^-1)"
```
```   400 apply (erule finite_converse [THEN iffD2, THEN finite_acyclic_wf])
```
```   401 apply (erule acyclic_converse [THEN iffD2])
```
```   402 done
```
```   403
```
```   404 lemma wf_iff_acyclic_if_finite: "finite r ==> wf r = acyclic r"
```
```   405 by (blast intro: finite_acyclic_wf wf_acyclic)
```
```   406
```
```   407
```
```   408 subsection {* @{typ nat} is well-founded *}
```
```   409
```
```   410 lemma less_nat_rel: "op < = (\<lambda>m n. n = Suc m)^++"
```
```   411 proof (rule ext, rule ext, rule iffI)
```
```   412   fix n m :: nat
```
```   413   assume "m < n"
```
```   414   then show "(\<lambda>m n. n = Suc m)^++ m n"
```
```   415   proof (induct n)
```
```   416     case 0 then show ?case by auto
```
```   417   next
```
```   418     case (Suc n) then show ?case
```
```   419       by (auto simp add: less_Suc_eq_le le_less intro: tranclp.trancl_into_trancl)
```
```   420   qed
```
```   421 next
```
```   422   fix n m :: nat
```
```   423   assume "(\<lambda>m n. n = Suc m)^++ m n"
```
```   424   then show "m < n"
```
```   425     by (induct n)
```
```   426       (simp_all add: less_Suc_eq_le reflexive le_less)
```
```   427 qed
```
```   428
```
```   429 definition
```
```   430   pred_nat :: "(nat * nat) set" where
```
```   431   "pred_nat = {(m, n). n = Suc m}"
```
```   432
```
```   433 definition
```
```   434   less_than :: "(nat * nat) set" where
```
```   435   "less_than = pred_nat^+"
```
```   436
```
```   437 lemma less_eq: "(m, n) \<in> pred_nat^+ \<longleftrightarrow> m < n"
```
```   438   unfolding less_nat_rel pred_nat_def trancl_def by simp
```
```   439
```
```   440 lemma pred_nat_trancl_eq_le:
```
```   441   "(m, n) \<in> pred_nat^* \<longleftrightarrow> m \<le> n"
```
```   442   unfolding less_eq rtrancl_eq_or_trancl by auto
```
```   443
```
```   444 lemma wf_pred_nat: "wf pred_nat"
```
```   445   apply (unfold wf_def pred_nat_def, clarify)
```
```   446   apply (induct_tac x, blast+)
```
```   447   done
```
```   448
```
```   449 lemma wf_less_than [iff]: "wf less_than"
```
```   450   by (simp add: less_than_def wf_pred_nat [THEN wf_trancl])
```
```   451
```
```   452 lemma trans_less_than [iff]: "trans less_than"
```
```   453   by (simp add: less_than_def)
```
```   454
```
```   455 lemma less_than_iff [iff]: "((x,y): less_than) = (x<y)"
```
```   456   by (simp add: less_than_def less_eq)
```
```   457
```
```   458 lemma wf_less: "wf {(x, y::nat). x < y}"
```
```   459   using wf_less_than by (simp add: less_than_def less_eq [symmetric])
```
```   460
```
```   461
```
```   462 subsection {* Accessible Part *}
```
```   463
```
```   464 text {*
```
```   465  Inductive definition of the accessible part @{term "acc r"} of a
```
```   466  relation; see also \cite{paulin-tlca}.
```
```   467 *}
```
```   468
```
```   469 inductive_set
```
```   470   acc :: "('a * 'a) set => 'a set"
```
```   471   for r :: "('a * 'a) set"
```
```   472   where
```
```   473     accI: "(!!y. (y, x) : r ==> y : acc r) ==> x : acc r"
```
```   474
```
```   475 abbreviation
```
```   476   termip :: "('a => 'a => bool) => 'a => bool" where
```
```   477   "termip r \<equiv> accp (r\<inverse>\<inverse>)"
```
```   478
```
```   479 abbreviation
```
```   480   termi :: "('a * 'a) set => 'a set" where
```
```   481   "termi r \<equiv> acc (r\<inverse>)"
```
```   482
```
```   483 lemmas accpI = accp.accI
```
```   484
```
```   485 text {* Induction rules *}
```
```   486
```
```   487 theorem accp_induct:
```
```   488   assumes major: "accp r a"
```
```   489   assumes hyp: "!!x. accp r x ==> \<forall>y. r y x --> P y ==> P x"
```
```   490   shows "P a"
```
```   491   apply (rule major [THEN accp.induct])
```
```   492   apply (rule hyp)
```
```   493    apply (rule accp.accI)
```
```   494    apply fast
```
```   495   apply fast
```
```   496   done
```
```   497
```
```   498 theorems accp_induct_rule = accp_induct [rule_format, induct set: accp]
```
```   499
```
```   500 theorem accp_downward: "accp r b ==> r a b ==> accp r a"
```
```   501   apply (erule accp.cases)
```
```   502   apply fast
```
```   503   done
```
```   504
```
```   505 lemma not_accp_down:
```
```   506   assumes na: "\<not> accp R x"
```
```   507   obtains z where "R z x" and "\<not> accp R z"
```
```   508 proof -
```
```   509   assume a: "\<And>z. \<lbrakk>R z x; \<not> accp R z\<rbrakk> \<Longrightarrow> thesis"
```
```   510
```
```   511   show thesis
```
```   512   proof (cases "\<forall>z. R z x \<longrightarrow> accp R z")
```
```   513     case True
```
```   514     hence "\<And>z. R z x \<Longrightarrow> accp R z" by auto
```
```   515     hence "accp R x"
```
```   516       by (rule accp.accI)
```
```   517     with na show thesis ..
```
```   518   next
```
```   519     case False then obtain z where "R z x" and "\<not> accp R z"
```
```   520       by auto
```
```   521     with a show thesis .
```
```   522   qed
```
```   523 qed
```
```   524
```
```   525 lemma accp_downwards_aux: "r\<^sup>*\<^sup>* b a ==> accp r a --> accp r b"
```
```   526   apply (erule rtranclp_induct)
```
```   527    apply blast
```
```   528   apply (blast dest: accp_downward)
```
```   529   done
```
```   530
```
```   531 theorem accp_downwards: "accp r a ==> r\<^sup>*\<^sup>* b a ==> accp r b"
```
```   532   apply (blast dest: accp_downwards_aux)
```
```   533   done
```
```   534
```
```   535 theorem accp_wfPI: "\<forall>x. accp r x ==> wfP r"
```
```   536   apply (rule wfPUNIVI)
```
```   537   apply (rule_tac P=P in accp_induct)
```
```   538    apply blast
```
```   539   apply blast
```
```   540   done
```
```   541
```
```   542 theorem accp_wfPD: "wfP r ==> accp r x"
```
```   543   apply (erule wfP_induct_rule)
```
```   544   apply (rule accp.accI)
```
```   545   apply blast
```
```   546   done
```
```   547
```
```   548 theorem wfP_accp_iff: "wfP r = (\<forall>x. accp r x)"
```
```   549   apply (blast intro: accp_wfPI dest: accp_wfPD)
```
```   550   done
```
```   551
```
```   552
```
```   553 text {* Smaller relations have bigger accessible parts: *}
```
```   554
```
```   555 lemma accp_subset:
```
```   556   assumes sub: "R1 \<le> R2"
```
```   557   shows "accp R2 \<le> accp R1"
```
```   558 proof (rule predicate1I)
```
```   559   fix x assume "accp R2 x"
```
```   560   then show "accp R1 x"
```
```   561   proof (induct x)
```
```   562     fix x
```
```   563     assume ih: "\<And>y. R2 y x \<Longrightarrow> accp R1 y"
```
```   564     with sub show "accp R1 x"
```
```   565       by (blast intro: accp.accI)
```
```   566   qed
```
```   567 qed
```
```   568
```
```   569
```
```   570 text {* This is a generalized induction theorem that works on
```
```   571   subsets of the accessible part. *}
```
```   572
```
```   573 lemma accp_subset_induct:
```
```   574   assumes subset: "D \<le> accp R"
```
```   575     and dcl: "\<And>x z. \<lbrakk>D x; R z x\<rbrakk> \<Longrightarrow> D z"
```
```   576     and "D x"
```
```   577     and istep: "\<And>x. \<lbrakk>D x; (\<And>z. R z x \<Longrightarrow> P z)\<rbrakk> \<Longrightarrow> P x"
```
```   578   shows "P x"
```
```   579 proof -
```
```   580   from subset and `D x`
```
```   581   have "accp R x" ..
```
```   582   then show "P x" using `D x`
```
```   583   proof (induct x)
```
```   584     fix x
```
```   585     assume "D x"
```
```   586       and "\<And>y. R y x \<Longrightarrow> D y \<Longrightarrow> P y"
```
```   587     with dcl and istep show "P x" by blast
```
```   588   qed
```
```   589 qed
```
```   590
```
```   591
```
```   592 text {* Set versions of the above theorems *}
```
```   593
```
```   594 lemmas acc_induct = accp_induct [to_set]
```
```   595
```
```   596 lemmas acc_induct_rule = acc_induct [rule_format, induct set: acc]
```
```   597
```
```   598 lemmas acc_downward = accp_downward [to_set]
```
```   599
```
```   600 lemmas not_acc_down = not_accp_down [to_set]
```
```   601
```
```   602 lemmas acc_downwards_aux = accp_downwards_aux [to_set]
```
```   603
```
```   604 lemmas acc_downwards = accp_downwards [to_set]
```
```   605
```
```   606 lemmas acc_wfI = accp_wfPI [to_set]
```
```   607
```
```   608 lemmas acc_wfD = accp_wfPD [to_set]
```
```   609
```
```   610 lemmas wf_acc_iff = wfP_accp_iff [to_set]
```
```   611
```
```   612 lemmas acc_subset = accp_subset [to_set]
```
```   613
```
```   614 lemmas acc_subset_induct = accp_subset_induct [to_set]
```
```   615
```
```   616
```
```   617 subsection {* Tools for building wellfounded relations *}
```
```   618
```
```   619 text {* Inverse Image *}
```
```   620
```
```   621 lemma wf_inv_image [simp,intro!]: "wf(r) ==> wf(inv_image r (f::'a=>'b))"
```
```   622 apply (simp (no_asm_use) add: inv_image_def wf_eq_minimal)
```
```   623 apply clarify
```
```   624 apply (subgoal_tac "EX (w::'b) . w : {w. EX (x::'a) . x: Q & (f x = w) }")
```
```   625 prefer 2 apply (blast del: allE)
```
```   626 apply (erule allE)
```
```   627 apply (erule (1) notE impE)
```
```   628 apply blast
```
```   629 done
```
```   630
```
```   631 text {* Measure functions into @{typ nat} *}
```
```   632
```
```   633 definition measure :: "('a => nat) => ('a * 'a)set"
```
```   634 where "measure = inv_image less_than"
```
```   635
```
```   636 lemma in_measure[simp, code_unfold]: "((x,y) : measure f) = (f x < f y)"
```
```   637   by (simp add:measure_def)
```
```   638
```
```   639 lemma wf_measure [iff]: "wf (measure f)"
```
```   640 apply (unfold measure_def)
```
```   641 apply (rule wf_less_than [THEN wf_inv_image])
```
```   642 done
```
```   643
```
```   644 lemma wf_if_measure: fixes f :: "'a \<Rightarrow> nat"
```
```   645 shows "(!!x. P x \<Longrightarrow> f(g x) < f x) \<Longrightarrow> wf {(y,x). P x \<and> y = g x}"
```
```   646 apply(insert wf_measure[of f])
```
```   647 apply(simp only: measure_def inv_image_def less_than_def less_eq)
```
```   648 apply(erule wf_subset)
```
```   649 apply auto
```
```   650 done
```
```   651
```
```   652
```
```   653 text{* Lexicographic combinations *}
```
```   654
```
```   655 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
```
```   656   "ra <*lex*> rb = {((a, b), (a', b')). (a, a') \<in> ra \<or> a = a' \<and> (b, b') \<in> rb}"
```
```   657
```
```   658 lemma wf_lex_prod [intro!]: "[| wf(ra); wf(rb) |] ==> wf(ra <*lex*> rb)"
```
```   659 apply (unfold wf_def lex_prod_def)
```
```   660 apply (rule allI, rule impI)
```
```   661 apply (simp (no_asm_use) only: split_paired_All)
```
```   662 apply (drule spec, erule mp)
```
```   663 apply (rule allI, rule impI)
```
```   664 apply (drule spec, erule mp, blast)
```
```   665 done
```
```   666
```
```   667 lemma in_lex_prod[simp]:
```
```   668   "(((a,b),(a',b')): r <*lex*> s) = ((a,a'): r \<or> (a = a' \<and> (b, b') : s))"
```
```   669   by (auto simp:lex_prod_def)
```
```   670
```
```   671 text{* @{term "op <*lex*>"} preserves transitivity *}
```
```   672
```
```   673 lemma trans_lex_prod [intro!]:
```
```   674     "[| trans R1; trans R2 |] ==> trans (R1 <*lex*> R2)"
```
```   675 by (unfold trans_def lex_prod_def, blast)
```
```   676
```
```   677 text {* lexicographic combinations with measure functions *}
```
```   678
```
```   679 definition
```
```   680   mlex_prod :: "('a \<Rightarrow> nat) \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" (infixr "<*mlex*>" 80)
```
```   681 where
```
```   682   "f <*mlex*> R = inv_image (less_than <*lex*> R) (%x. (f x, x))"
```
```   683
```
```   684 lemma wf_mlex: "wf R \<Longrightarrow> wf (f <*mlex*> R)"
```
```   685 unfolding mlex_prod_def
```
```   686 by auto
```
```   687
```
```   688 lemma mlex_less: "f x < f y \<Longrightarrow> (x, y) \<in> f <*mlex*> R"
```
```   689 unfolding mlex_prod_def by simp
```
```   690
```
```   691 lemma mlex_leq: "f x \<le> f y \<Longrightarrow> (x, y) \<in> R \<Longrightarrow> (x, y) \<in> f <*mlex*> R"
```
```   692 unfolding mlex_prod_def by auto
```
```   693
```
```   694 text {* proper subset relation on finite sets *}
```
```   695
```
```   696 definition finite_psubset  :: "('a set * 'a set) set"
```
```   697 where "finite_psubset = {(A,B). A < B & finite B}"
```
```   698
```
```   699 lemma wf_finite_psubset[simp]: "wf(finite_psubset)"
```
```   700 apply (unfold finite_psubset_def)
```
```   701 apply (rule wf_measure [THEN wf_subset])
```
```   702 apply (simp add: measure_def inv_image_def less_than_def less_eq)
```
```   703 apply (fast elim!: psubset_card_mono)
```
```   704 done
```
```   705
```
```   706 lemma trans_finite_psubset: "trans finite_psubset"
```
```   707 by (simp add: finite_psubset_def less_le trans_def, blast)
```
```   708
```
```   709 lemma in_finite_psubset[simp]: "(A, B) \<in> finite_psubset = (A < B & finite B)"
```
```   710 unfolding finite_psubset_def by auto
```
```   711
```
```   712 text {* max- and min-extension of order to finite sets *}
```
```   713
```
```   714 inductive_set max_ext :: "('a \<times> 'a) set \<Rightarrow> ('a set \<times> 'a set) set"
```
```   715 for R :: "('a \<times> 'a) set"
```
```   716 where
```
```   717   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"
```
```   718
```
```   719 lemma max_ext_wf:
```
```   720   assumes wf: "wf r"
```
```   721   shows "wf (max_ext r)"
```
```   722 proof (rule acc_wfI, intro allI)
```
```   723   fix M show "M \<in> acc (max_ext r)" (is "_ \<in> ?W")
```
```   724   proof cases
```
```   725     assume "finite M"
```
```   726     thus ?thesis
```
```   727     proof (induct M)
```
```   728       show "{} \<in> ?W"
```
```   729         by (rule accI) (auto elim: max_ext.cases)
```
```   730     next
```
```   731       fix M a assume "M \<in> ?W" "finite M"
```
```   732       with wf show "insert a M \<in> ?W"
```
```   733       proof (induct arbitrary: M)
```
```   734         fix M a
```
```   735         assume "M \<in> ?W"  and  [intro]: "finite M"
```
```   736         assume hyp: "\<And>b M. (b, a) \<in> r \<Longrightarrow> M \<in> ?W \<Longrightarrow> finite M \<Longrightarrow> insert b M \<in> ?W"
```
```   737         {
```
```   738           fix N M :: "'a set"
```
```   739           assume "finite N" "finite M"
```
```   740           then
```
```   741           have "\<lbrakk>M \<in> ?W ; (\<And>y. y \<in> N \<Longrightarrow> (y, a) \<in> r)\<rbrakk> \<Longrightarrow>  N \<union> M \<in> ?W"
```
```   742             by (induct N arbitrary: M) (auto simp: hyp)
```
```   743         }
```
```   744         note add_less = this
```
```   745
```
```   746         show "insert a M \<in> ?W"
```
```   747         proof (rule accI)
```
```   748           fix N assume Nless: "(N, insert a M) \<in> max_ext r"
```
```   749           hence asm1: "\<And>x. x \<in> N \<Longrightarrow> (x, a) \<in> r \<or> (\<exists>y \<in> M. (x, y) \<in> r)"
```
```   750             by (auto elim!: max_ext.cases)
```
```   751
```
```   752           let ?N1 = "{ n \<in> N. (n, a) \<in> r }"
```
```   753           let ?N2 = "{ n \<in> N. (n, a) \<notin> r }"
```
```   754           have N: "?N1 \<union> ?N2 = N" by (rule set_eqI) auto
```
```   755           from Nless have "finite N" by (auto elim: max_ext.cases)
```
```   756           then have finites: "finite ?N1" "finite ?N2" by auto
```
```   757
```
```   758           have "?N2 \<in> ?W"
```
```   759           proof cases
```
```   760             assume [simp]: "M = {}"
```
```   761             have Mw: "{} \<in> ?W" by (rule accI) (auto elim: max_ext.cases)
```
```   762
```
```   763             from asm1 have "?N2 = {}" by auto
```
```   764             with Mw show "?N2 \<in> ?W" by (simp only:)
```
```   765           next
```
```   766             assume "M \<noteq> {}"
```
```   767             from asm1 finites have N2: "(?N2, M) \<in> max_ext r"
```
```   768               by (rule_tac max_extI[OF _ _ `M \<noteq> {}`]) auto
```
```   769
```
```   770             with `M \<in> ?W` show "?N2 \<in> ?W" by (rule acc_downward)
```
```   771           qed
```
```   772           with finites have "?N1 \<union> ?N2 \<in> ?W"
```
```   773             by (rule add_less) simp
```
```   774           then show "N \<in> ?W" by (simp only: N)
```
```   775         qed
```
```   776       qed
```
```   777     qed
```
```   778   next
```
```   779     assume [simp]: "\<not> finite M"
```
```   780     show ?thesis
```
```   781       by (rule accI) (auto elim: max_ext.cases)
```
```   782   qed
```
```   783 qed
```
```   784
```
```   785 lemma max_ext_additive:
```
```   786  "(A, B) \<in> max_ext R \<Longrightarrow> (C, D) \<in> max_ext R \<Longrightarrow>
```
```   787   (A \<union> C, B \<union> D) \<in> max_ext R"
```
```   788 by (force elim!: max_ext.cases)
```
```   789
```
```   790
```
```   791 definition min_ext :: "('a \<times> 'a) set \<Rightarrow> ('a set \<times> 'a set) set"  where
```
```   792   "min_ext r = {(X, Y) | X Y. X \<noteq> {} \<and> (\<forall>y \<in> Y. (\<exists>x \<in> X. (x, y) \<in> r))}"
```
```   793
```
```   794 lemma min_ext_wf:
```
```   795   assumes "wf r"
```
```   796   shows "wf (min_ext r)"
```
```   797 proof (rule wfI_min)
```
```   798   fix Q :: "'a set set"
```
```   799   fix x
```
```   800   assume nonempty: "x \<in> Q"
```
```   801   show "\<exists>m \<in> Q. (\<forall> n. (n, m) \<in> min_ext r \<longrightarrow> n \<notin> Q)"
```
```   802   proof cases
```
```   803     assume "Q = {{}}" thus ?thesis by (simp add: min_ext_def)
```
```   804   next
```
```   805     assume "Q \<noteq> {{}}"
```
```   806     with nonempty
```
```   807     obtain e x where "x \<in> Q" "e \<in> x" by force
```
```   808     then have eU: "e \<in> \<Union>Q" by auto
```
```   809     with `wf r`
```
```   810     obtain z where z: "z \<in> \<Union>Q" "\<And>y. (y, z) \<in> r \<Longrightarrow> y \<notin> \<Union>Q"
```
```   811       by (erule wfE_min)
```
```   812     from z obtain m where "m \<in> Q" "z \<in> m" by auto
```
```   813     from `m \<in> Q`
```
```   814     show ?thesis
```
```   815     proof (rule, intro bexI allI impI)
```
```   816       fix n
```
```   817       assume smaller: "(n, m) \<in> min_ext r"
```
```   818       with `z \<in> m` obtain y where y: "y \<in> n" "(y, z) \<in> r" by (auto simp: min_ext_def)
```
```   819       then show "n \<notin> Q" using z(2) by auto
```
```   820     qed
```
```   821   qed
```
```   822 qed
```
```   823
```
```   824 text{* Bounded increase must terminate: *}
```
```   825
```
```   826 lemma wf_bounded_measure:
```
```   827 fixes ub :: "'a \<Rightarrow> nat" and f :: "'a \<Rightarrow> nat"
```
```   828 assumes "!!a b. (b,a) : r \<Longrightarrow> ub b \<le> ub a & ub a \<ge> f b & f b > f a"
```
```   829 shows "wf r"
```
```   830 apply(rule wf_subset[OF wf_measure[of "%a. ub a - f a"]])
```
```   831 apply (auto dest: assms)
```
```   832 done
```
```   833
```
```   834 lemma wf_bounded_set:
```
```   835 fixes ub :: "'a \<Rightarrow> 'b set" and f :: "'a \<Rightarrow> 'b set"
```
```   836 assumes "!!a b. (b,a) : r \<Longrightarrow>
```
```   837   finite(ub a) & ub b \<subseteq> ub a & ub a \<supseteq> f b & f b \<supset> f a"
```
```   838 shows "wf r"
```
```   839 apply(rule wf_bounded_measure[of r "%a. card(ub a)" "%a. card(f a)"])
```
```   840 apply(drule assms)
```
```   841 apply (blast intro: card_mono finite_subset psubset_card_mono dest: psubset_eq[THEN iffD2])
```
```   842 done
```
```   843
```
```   844
```
```   845 subsection {* size of a datatype value *}
```
```   846
```
```   847 ML_file "Tools/Function/size.ML"
```
```   848 setup Size.setup
```
```   849
```
```   850 lemma size_bool [code]:
```
```   851   "size (b\<Colon>bool) = 0" by (cases b) auto
```
```   852
```
```   853 lemma nat_size [simp, code]: "size (n\<Colon>nat) = n"
```
```   854   by (induct n) simp_all
```
```   855
```
```   856 declare "prod.size" [no_atp]
```
```   857
```
```   858 end
```