src/HOL/Wellfounded.thy
 author nipkow Fri Aug 28 18:52:41 2009 +0200 (2009-08-28) changeset 32436 10cd49e0c067 parent 32263 8bc0fd4a23a0 child 32461 eee4fa79398f permissions -rw-r--r--
Turned "x <= y ==> sup x y = y" (and relatives) into simp rules
```     1 (*  Author:     Tobias Nipkow
```
```     2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
```
```     3     Author:     Konrad Slind, Alexander Krauss
```
```     4     Copyright   1992-2008  University of Cambridge and TU Muenchen
```
```     5 *)
```
```     6
```
```     7 header {*Well-founded Recursion*}
```
```     8
```
```     9 theory Wellfounded
```
```    10 imports Finite_Set Transitive_Closure
```
```    11 uses ("Tools/Function/size.ML")
```
```    12 begin
```
```    13
```
```    14 subsection {* Basic Definitions *}
```
```    15
```
```    16 inductive
```
```    17   wfrec_rel :: "('a * 'a) set => (('a => 'b) => 'a => 'b) => 'a => 'b => bool"
```
```    18   for R :: "('a * 'a) set"
```
```    19   and F :: "('a => 'b) => 'a => 'b"
```
```    20 where
```
```    21   wfrecI: "ALL z. (z, x) : R --> wfrec_rel R F z (g z) ==>
```
```    22             wfrec_rel R F x (F g x)"
```
```    23
```
```    24 constdefs
```
```    25   wf         :: "('a * 'a)set => bool"
```
```    26   "wf(r) == (!P. (!x. (!y. (y,x):r --> P(y)) --> P(x)) --> (!x. P(x)))"
```
```    27
```
```    28   wfP :: "('a => 'a => bool) => bool"
```
```    29   "wfP r == wf {(x, y). r x y}"
```
```    30
```
```    31   acyclic :: "('a*'a)set => bool"
```
```    32   "acyclic r == !x. (x,x) ~: r^+"
```
```    33
```
```    34   cut        :: "('a => 'b) => ('a * 'a)set => 'a => 'a => 'b"
```
```    35   "cut f r x == (%y. if (y,x):r then f y else undefined)"
```
```    36
```
```    37   adm_wf :: "('a * 'a) set => (('a => 'b) => 'a => 'b) => bool"
```
```    38   "adm_wf R F == ALL f g x.
```
```    39      (ALL z. (z, x) : R --> f z = g z) --> F f x = F g x"
```
```    40
```
```    41   wfrec :: "('a * 'a) set => (('a => 'b) => 'a => 'b) => 'a => 'b"
```
```    42   [code del]: "wfrec R F == %x. THE y. wfrec_rel R (%f x. F (cut f R x) x) x y"
```
```    43
```
```    44 abbreviation acyclicP :: "('a => 'a => bool) => bool" where
```
```    45   "acyclicP r == acyclic {(x, y). r x y}"
```
```    46
```
```    47 lemma wfP_wf_eq [pred_set_conv]: "wfP (\<lambda>x y. (x, y) \<in> r) = wf r"
```
```    48   by (simp add: wfP_def)
```
```    49
```
```    50 lemma wfUNIVI:
```
```    51    "(!!P x. (ALL x. (ALL y. (y,x) : r --> P(y)) --> P(x)) ==> P(x)) ==> wf(r)"
```
```    52   unfolding wf_def by blast
```
```    53
```
```    54 lemmas wfPUNIVI = wfUNIVI [to_pred]
```
```    55
```
```    56 text{*Restriction to domain @{term A} and range @{term B}.  If @{term r} is
```
```    57     well-founded over their intersection, then @{term "wf r"}*}
```
```    58 lemma wfI:
```
```    59  "[| r \<subseteq> A <*> B;
```
```    60      !!x P. [|\<forall>x. (\<forall>y. (y,x) : r --> P y) --> P x;  x : A; x : B |] ==> P x |]
```
```    61   ==>  wf r"
```
```    62   unfolding wf_def by blast
```
```    63
```
```    64 lemma wf_induct:
```
```    65     "[| wf(r);
```
```    66         !!x.[| ALL y. (y,x): r --> P(y) |] ==> P(x)
```
```    67      |]  ==>  P(a)"
```
```    68   unfolding wf_def by blast
```
```    69
```
```    70 lemmas wfP_induct = wf_induct [to_pred]
```
```    71
```
```    72 lemmas wf_induct_rule = wf_induct [rule_format, consumes 1, case_names less, induct set: wf]
```
```    73
```
```    74 lemmas wfP_induct_rule = wf_induct_rule [to_pred, induct set: wfP]
```
```    75
```
```    76 lemma wf_not_sym: "wf r ==> (a, x) : r ==> (x, a) ~: r"
```
```    77   by (induct a arbitrary: x set: wf) blast
```
```    78
```
```    79 (* [| wf r;  ~Z ==> (a,x) : r;  (x,a) ~: r ==> Z |] ==> Z *)
```
```    80 lemmas wf_asym = wf_not_sym [elim_format]
```
```    81
```
```    82 lemma wf_not_refl [simp]: "wf r ==> (a, a) ~: r"
```
```    83   by (blast elim: wf_asym)
```
```    84
```
```    85 (* [| wf r;  (a,a) ~: r ==> PROP W |] ==> PROP W *)
```
```    86 lemmas wf_irrefl = wf_not_refl [elim_format]
```
```    87
```
```    88 lemma wf_wellorderI:
```
```    89   assumes wf: "wf {(x::'a::ord, y). x < y}"
```
```    90   assumes lin: "OFCLASS('a::ord, linorder_class)"
```
```    91   shows "OFCLASS('a::ord, wellorder_class)"
```
```    92 using lin by (rule wellorder_class.intro)
```
```    93   (blast intro: wellorder_axioms.intro wf_induct_rule [OF wf])
```
```    94
```
```    95 lemma (in wellorder) wf:
```
```    96   "wf {(x, y). x < y}"
```
```    97 unfolding wf_def by (blast intro: less_induct)
```
```    98
```
```    99
```
```   100 subsection {* Basic Results *}
```
```   101
```
```   102 text{*transitive closure of a well-founded relation is well-founded! *}
```
```   103 lemma wf_trancl:
```
```   104   assumes "wf r"
```
```   105   shows "wf (r^+)"
```
```   106 proof -
```
```   107   {
```
```   108     fix P and x
```
```   109     assume induct_step: "!!x. (!!y. (y, x) : r^+ ==> P y) ==> P x"
```
```   110     have "P x"
```
```   111     proof (rule induct_step)
```
```   112       fix y assume "(y, x) : r^+"
```
```   113       with `wf r` show "P y"
```
```   114       proof (induct x arbitrary: y)
```
```   115 	case (less x)
```
```   116 	note hyp = `\<And>x' y'. (x', x) : r ==> (y', x') : r^+ ==> P y'`
```
```   117 	from `(y, x) : r^+` show "P y"
```
```   118 	proof cases
```
```   119 	  case base
```
```   120 	  show "P y"
```
```   121 	  proof (rule induct_step)
```
```   122 	    fix y' assume "(y', y) : r^+"
```
```   123 	    with `(y, x) : r` show "P y'" by (rule hyp [of y y'])
```
```   124 	  qed
```
```   125 	next
```
```   126 	  case step
```
```   127 	  then obtain x' where "(x', x) : r" and "(y, x') : r^+" by simp
```
```   128 	  then show "P y" by (rule hyp [of x' y])
```
```   129 	qed
```
```   130       qed
```
```   131     qed
```
```   132   } then show ?thesis unfolding wf_def by blast
```
```   133 qed
```
```   134
```
```   135 lemmas wfP_trancl = wf_trancl [to_pred]
```
```   136
```
```   137 lemma wf_converse_trancl: "wf (r^-1) ==> wf ((r^+)^-1)"
```
```   138   apply (subst trancl_converse [symmetric])
```
```   139   apply (erule wf_trancl)
```
```   140   done
```
```   141
```
```   142
```
```   143 text{*Minimal-element characterization of well-foundedness*}
```
```   144 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))"
```
```   145 proof (intro iffI strip)
```
```   146   fix Q :: "'a set" and x
```
```   147   assume "wf r" and "x \<in> Q"
```
```   148   then show "\<exists>z\<in>Q. \<forall>y. (y, z) \<in> r \<longrightarrow> y \<notin> Q"
```
```   149     unfolding wf_def
```
```   150     by (blast dest: spec [of _ "%x. x\<in>Q \<longrightarrow> (\<exists>z\<in>Q. \<forall>y. (y,z) \<in> r \<longrightarrow> y\<notin>Q)"])
```
```   151 next
```
```   152   assume 1: "\<forall>Q x. x \<in> Q \<longrightarrow> (\<exists>z\<in>Q. \<forall>y. (y, z) \<in> r \<longrightarrow> y \<notin> Q)"
```
```   153   show "wf r"
```
```   154   proof (rule wfUNIVI)
```
```   155     fix P :: "'a \<Rightarrow> bool" and x
```
```   156     assume 2: "\<forall>x. (\<forall>y. (y, x) \<in> r \<longrightarrow> P y) \<longrightarrow> P x"
```
```   157     let ?Q = "{x. \<not> P x}"
```
```   158     have "x \<in> ?Q \<longrightarrow> (\<exists>z \<in> ?Q. \<forall>y. (y, z) \<in> r \<longrightarrow> y \<notin> ?Q)"
```
```   159       by (rule 1 [THEN spec, THEN spec])
```
```   160     then have "\<not> P x \<longrightarrow> (\<exists>z. \<not> P z \<and> (\<forall>y. (y, z) \<in> r \<longrightarrow> P y))" by simp
```
```   161     with 2 have "\<not> P x \<longrightarrow> (\<exists>z. \<not> P z \<and> P z)" by fast
```
```   162     then show "P x" by simp
```
```   163   qed
```
```   164 qed
```
```   165
```
```   166 lemma wfE_min:
```
```   167   assumes "wf R" "x \<in> Q"
```
```   168   obtains z where "z \<in> Q" "\<And>y. (y, z) \<in> R \<Longrightarrow> y \<notin> Q"
```
```   169   using assms unfolding wf_eq_minimal by blast
```
```   170
```
```   171 lemma wfI_min:
```
```   172   "(\<And>x Q. x \<in> Q \<Longrightarrow> \<exists>z\<in>Q. \<forall>y. (y, z) \<in> R \<longrightarrow> y \<notin> Q)
```
```   173   \<Longrightarrow> wf R"
```
```   174   unfolding wf_eq_minimal by blast
```
```   175
```
```   176 lemmas wfP_eq_minimal = wf_eq_minimal [to_pred]
```
```   177
```
```   178 text {* Well-foundedness of subsets *}
```
```   179 lemma wf_subset: "[| wf(r);  p<=r |] ==> wf(p)"
```
```   180   apply (simp (no_asm_use) add: wf_eq_minimal)
```
```   181   apply fast
```
```   182   done
```
```   183
```
```   184 lemmas wfP_subset = wf_subset [to_pred]
```
```   185
```
```   186 text {* Well-foundedness of the empty relation *}
```
```   187 lemma wf_empty [iff]: "wf({})"
```
```   188   by (simp add: wf_def)
```
```   189
```
```   190 lemma wfP_empty [iff]:
```
```   191   "wfP (\<lambda>x y. False)"
```
```   192 proof -
```
```   193   have "wfP bot" by (fact wf_empty [to_pred bot_empty_eq2])
```
```   194   then show ?thesis by (simp add: bot_fun_eq bot_bool_eq)
```
```   195 qed
```
```   196
```
```   197 lemma wf_Int1: "wf r ==> wf (r Int r')"
```
```   198   apply (erule wf_subset)
```
```   199   apply (rule Int_lower1)
```
```   200   done
```
```   201
```
```   202 lemma wf_Int2: "wf r ==> wf (r' Int r)"
```
```   203   apply (erule wf_subset)
```
```   204   apply (rule Int_lower2)
```
```   205   done
```
```   206
```
```   207 text{*Well-foundedness of insert*}
```
```   208 lemma wf_insert [iff]: "wf(insert (y,x) r) = (wf(r) & (x,y) ~: r^*)"
```
```   209 apply (rule iffI)
```
```   210  apply (blast elim: wf_trancl [THEN wf_irrefl]
```
```   211               intro: rtrancl_into_trancl1 wf_subset
```
```   212                      rtrancl_mono [THEN  rev_subsetD])
```
```   213 apply (simp add: wf_eq_minimal, safe)
```
```   214 apply (rule allE, assumption, erule impE, blast)
```
```   215 apply (erule bexE)
```
```   216 apply (rename_tac "a", case_tac "a = x")
```
```   217  prefer 2
```
```   218 apply blast
```
```   219 apply (case_tac "y:Q")
```
```   220  prefer 2 apply blast
```
```   221 apply (rule_tac x = "{z. z:Q & (z,y) : r^*}" in allE)
```
```   222  apply assumption
```
```   223 apply (erule_tac V = "ALL Q. (EX x. x : Q) --> ?P Q" in thin_rl)
```
```   224   --{*essential for speed*}
```
```   225 txt{*Blast with new substOccur fails*}
```
```   226 apply (fast intro: converse_rtrancl_into_rtrancl)
```
```   227 done
```
```   228
```
```   229 text{*Well-foundedness of image*}
```
```   230 lemma wf_prod_fun_image: "[| wf r; inj f |] ==> wf(prod_fun f f ` r)"
```
```   231 apply (simp only: wf_eq_minimal, clarify)
```
```   232 apply (case_tac "EX p. f p : Q")
```
```   233 apply (erule_tac x = "{p. f p : Q}" in allE)
```
```   234 apply (fast dest: inj_onD, blast)
```
```   235 done
```
```   236
```
```   237
```
```   238 subsection {* Well-Foundedness Results for Unions *}
```
```   239
```
```   240 lemma wf_union_compatible:
```
```   241   assumes "wf R" "wf S"
```
```   242   assumes "R O S \<subseteq> R"
```
```   243   shows "wf (R \<union> S)"
```
```   244 proof (rule wfI_min)
```
```   245   fix x :: 'a and Q
```
```   246   let ?Q' = "{x \<in> Q. \<forall>y. (y, x) \<in> R \<longrightarrow> y \<notin> Q}"
```
```   247   assume "x \<in> Q"
```
```   248   obtain a where "a \<in> ?Q'"
```
```   249     by (rule wfE_min [OF `wf R` `x \<in> Q`]) blast
```
```   250   with `wf S`
```
```   251   obtain z where "z \<in> ?Q'" and zmin: "\<And>y. (y, z) \<in> S \<Longrightarrow> y \<notin> ?Q'" by (erule wfE_min)
```
```   252   {
```
```   253     fix y assume "(y, z) \<in> S"
```
```   254     then have "y \<notin> ?Q'" by (rule zmin)
```
```   255
```
```   256     have "y \<notin> Q"
```
```   257     proof
```
```   258       assume "y \<in> Q"
```
```   259       with `y \<notin> ?Q'`
```
```   260       obtain w where "(w, y) \<in> R" and "w \<in> Q" by auto
```
```   261       from `(w, y) \<in> R` `(y, z) \<in> S` have "(w, z) \<in> R O S" by (rule rel_compI)
```
```   262       with `R O S \<subseteq> R` have "(w, z) \<in> R" ..
```
```   263       with `z \<in> ?Q'` have "w \<notin> Q" by blast
```
```   264       with `w \<in> Q` show False by contradiction
```
```   265     qed
```
```   266   }
```
```   267   with `z \<in> ?Q'` show "\<exists>z\<in>Q. \<forall>y. (y, z) \<in> R \<union> S \<longrightarrow> y \<notin> Q" by blast
```
```   268 qed
```
```   269
```
```   270
```
```   271 text {* Well-foundedness of indexed union with disjoint domains and ranges *}
```
```   272
```
```   273 lemma wf_UN: "[| ALL i:I. wf(r i);
```
```   274          ALL i:I. ALL j:I. r i ~= r j --> Domain(r i) Int Range(r j) = {}
```
```   275       |] ==> wf(UN i:I. r i)"
```
```   276 apply (simp only: wf_eq_minimal, clarify)
```
```   277 apply (rename_tac A a, case_tac "EX i:I. EX a:A. EX b:A. (b,a) : r i")
```
```   278  prefer 2
```
```   279  apply force
```
```   280 apply clarify
```
```   281 apply (drule bspec, assumption)
```
```   282 apply (erule_tac x="{a. a:A & (EX b:A. (b,a) : r i) }" in allE)
```
```   283 apply (blast elim!: allE)
```
```   284 done
```
```   285
```
```   286 lemma wfP_SUP:
```
```   287   "\<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)"
```
```   288   by (rule wf_UN [where I=UNIV and r="\<lambda>i. {(x, y). r i x y}", to_pred SUP_UN_eq2 pred_equals_eq])
```
```   289     (simp_all add: bot_fun_eq bot_bool_eq)
```
```   290
```
```   291 lemma wf_Union:
```
```   292  "[| ALL r:R. wf r;
```
```   293      ALL r:R. ALL s:R. r ~= s --> Domain r Int Range s = {}
```
```   294   |] ==> wf(Union R)"
```
```   295 apply (simp add: Union_def)
```
```   296 apply (blast intro: wf_UN)
```
```   297 done
```
```   298
```
```   299 (*Intuition: we find an (R u S)-min element of a nonempty subset A
```
```   300              by case distinction.
```
```   301   1. There is a step a -R-> b with a,b : A.
```
```   302      Pick an R-min element z of the (nonempty) set {a:A | EX b:A. a -R-> b}.
```
```   303      By definition, there is z':A s.t. z -R-> z'. Because z is R-min in the
```
```   304      subset, z' must be R-min in A. Because z' has an R-predecessor, it cannot
```
```   305      have an S-successor and is thus S-min in A as well.
```
```   306   2. There is no such step.
```
```   307      Pick an S-min element of A. In this case it must be an R-min
```
```   308      element of A as well.
```
```   309
```
```   310 *)
```
```   311 lemma wf_Un:
```
```   312      "[| wf r; wf s; Domain r Int Range s = {} |] ==> wf(r Un s)"
```
```   313   using wf_union_compatible[of s r]
```
```   314   by (auto simp: Un_ac)
```
```   315
```
```   316 lemma wf_union_merge:
```
```   317   "wf (R \<union> S) = wf (R O R \<union> S O R \<union> S)" (is "wf ?A = wf ?B")
```
```   318 proof
```
```   319   assume "wf ?A"
```
```   320   with wf_trancl have wfT: "wf (?A^+)" .
```
```   321   moreover have "?B \<subseteq> ?A^+"
```
```   322     by (subst trancl_unfold, subst trancl_unfold) blast
```
```   323   ultimately show "wf ?B" by (rule wf_subset)
```
```   324 next
```
```   325   assume "wf ?B"
```
```   326
```
```   327   show "wf ?A"
```
```   328   proof (rule wfI_min)
```
```   329     fix Q :: "'a set" and x
```
```   330     assume "x \<in> Q"
```
```   331
```
```   332     with `wf ?B`
```
```   333     obtain z where "z \<in> Q" and "\<And>y. (y, z) \<in> ?B \<Longrightarrow> y \<notin> Q"
```
```   334       by (erule wfE_min)
```
```   335     then have A1: "\<And>y. (y, z) \<in> R O R \<Longrightarrow> y \<notin> Q"
```
```   336       and A2: "\<And>y. (y, z) \<in> S O R \<Longrightarrow> y \<notin> Q"
```
```   337       and A3: "\<And>y. (y, z) \<in> S \<Longrightarrow> y \<notin> Q"
```
```   338       by auto
```
```   339
```
```   340     show "\<exists>z\<in>Q. \<forall>y. (y, z) \<in> ?A \<longrightarrow> y \<notin> Q"
```
```   341     proof (cases "\<forall>y. (y, z) \<in> R \<longrightarrow> y \<notin> Q")
```
```   342       case True
```
```   343       with `z \<in> Q` A3 show ?thesis by blast
```
```   344     next
```
```   345       case False
```
```   346       then obtain z' where "z'\<in>Q" "(z', z) \<in> R" by blast
```
```   347
```
```   348       have "\<forall>y. (y, z') \<in> ?A \<longrightarrow> y \<notin> Q"
```
```   349       proof (intro allI impI)
```
```   350         fix y assume "(y, z') \<in> ?A"
```
```   351         then show "y \<notin> Q"
```
```   352         proof
```
```   353           assume "(y, z') \<in> R"
```
```   354           then have "(y, z) \<in> R O R" using `(z', z) \<in> R` ..
```
```   355           with A1 show "y \<notin> Q" .
```
```   356         next
```
```   357           assume "(y, z') \<in> S"
```
```   358           then have "(y, z) \<in> S O R" using  `(z', z) \<in> R` ..
```
```   359           with A2 show "y \<notin> Q" .
```
```   360         qed
```
```   361       qed
```
```   362       with `z' \<in> Q` show ?thesis ..
```
```   363     qed
```
```   364   qed
```
```   365 qed
```
```   366
```
```   367 lemma wf_comp_self: "wf R = wf (R O R)"  -- {* special case *}
```
```   368   by (rule wf_union_merge [where S = "{}", simplified])
```
```   369
```
```   370
```
```   371 subsubsection {* acyclic *}
```
```   372
```
```   373 lemma acyclicI: "ALL x. (x, x) ~: r^+ ==> acyclic r"
```
```   374   by (simp add: acyclic_def)
```
```   375
```
```   376 lemma wf_acyclic: "wf r ==> acyclic r"
```
```   377 apply (simp add: acyclic_def)
```
```   378 apply (blast elim: wf_trancl [THEN wf_irrefl])
```
```   379 done
```
```   380
```
```   381 lemmas wfP_acyclicP = wf_acyclic [to_pred]
```
```   382
```
```   383 lemma acyclic_insert [iff]:
```
```   384      "acyclic(insert (y,x) r) = (acyclic r & (x,y) ~: r^*)"
```
```   385 apply (simp add: acyclic_def trancl_insert)
```
```   386 apply (blast intro: rtrancl_trans)
```
```   387 done
```
```   388
```
```   389 lemma acyclic_converse [iff]: "acyclic(r^-1) = acyclic r"
```
```   390 by (simp add: acyclic_def trancl_converse)
```
```   391
```
```   392 lemmas acyclicP_converse [iff] = acyclic_converse [to_pred]
```
```   393
```
```   394 lemma acyclic_impl_antisym_rtrancl: "acyclic r ==> antisym(r^*)"
```
```   395 apply (simp add: acyclic_def antisym_def)
```
```   396 apply (blast elim: rtranclE intro: rtrancl_into_trancl1 rtrancl_trancl_trancl)
```
```   397 done
```
```   398
```
```   399 (* Other direction:
```
```   400 acyclic = no loops
```
```   401 antisym = only self loops
```
```   402 Goalw [acyclic_def,antisym_def] "antisym( r^* ) ==> acyclic(r - Id)
```
```   403 ==> antisym( r^* ) = acyclic(r - Id)";
```
```   404 *)
```
```   405
```
```   406 lemma acyclic_subset: "[| acyclic s; r <= s |] ==> acyclic r"
```
```   407 apply (simp add: acyclic_def)
```
```   408 apply (blast intro: trancl_mono)
```
```   409 done
```
```   410
```
```   411 text{* Wellfoundedness of finite acyclic relations*}
```
```   412
```
```   413 lemma finite_acyclic_wf [rule_format]: "finite r ==> acyclic r --> wf r"
```
```   414 apply (erule finite_induct, blast)
```
```   415 apply (simp (no_asm_simp) only: split_tupled_all)
```
```   416 apply simp
```
```   417 done
```
```   418
```
```   419 lemma finite_acyclic_wf_converse: "[|finite r; acyclic r|] ==> wf (r^-1)"
```
```   420 apply (erule finite_converse [THEN iffD2, THEN finite_acyclic_wf])
```
```   421 apply (erule acyclic_converse [THEN iffD2])
```
```   422 done
```
```   423
```
```   424 lemma wf_iff_acyclic_if_finite: "finite r ==> wf r = acyclic r"
```
```   425 by (blast intro: finite_acyclic_wf wf_acyclic)
```
```   426
```
```   427
```
```   428 subsection{*Well-Founded Recursion*}
```
```   429
```
```   430 text{*cut*}
```
```   431
```
```   432 lemma cuts_eq: "(cut f r x = cut g r x) = (ALL y. (y,x):r --> f(y)=g(y))"
```
```   433 by (simp add: expand_fun_eq cut_def)
```
```   434
```
```   435 lemma cut_apply: "(x,a):r ==> (cut f r a)(x) = f(x)"
```
```   436 by (simp add: cut_def)
```
```   437
```
```   438 text{*Inductive characterization of wfrec combinator; for details see:
```
```   439 John Harrison, "Inductive definitions: automation and application"*}
```
```   440
```
```   441 lemma wfrec_unique: "[| adm_wf R F; wf R |] ==> EX! y. wfrec_rel R F x y"
```
```   442 apply (simp add: adm_wf_def)
```
```   443 apply (erule_tac a=x in wf_induct)
```
```   444 apply (rule ex1I)
```
```   445 apply (rule_tac g = "%x. THE y. wfrec_rel R F x y" in wfrec_rel.wfrecI)
```
```   446 apply (fast dest!: theI')
```
```   447 apply (erule wfrec_rel.cases, simp)
```
```   448 apply (erule allE, erule allE, erule allE, erule mp)
```
```   449 apply (fast intro: the_equality [symmetric])
```
```   450 done
```
```   451
```
```   452 lemma adm_lemma: "adm_wf R (%f x. F (cut f R x) x)"
```
```   453 apply (simp add: adm_wf_def)
```
```   454 apply (intro strip)
```
```   455 apply (rule cuts_eq [THEN iffD2, THEN subst], assumption)
```
```   456 apply (rule refl)
```
```   457 done
```
```   458
```
```   459 lemma wfrec: "wf(r) ==> wfrec r H a = H (cut (wfrec r H) r a) a"
```
```   460 apply (simp add: wfrec_def)
```
```   461 apply (rule adm_lemma [THEN wfrec_unique, THEN the1_equality], assumption)
```
```   462 apply (rule wfrec_rel.wfrecI)
```
```   463 apply (intro strip)
```
```   464 apply (erule adm_lemma [THEN wfrec_unique, THEN theI'])
```
```   465 done
```
```   466
```
```   467 subsection {* Code generator setup *}
```
```   468
```
```   469 consts_code
```
```   470   "wfrec"   ("\<module>wfrec?")
```
```   471 attach {*
```
```   472 fun wfrec f x = f (wfrec f) x;
```
```   473 *}
```
```   474
```
```   475
```
```   476 subsection {* @{typ nat} is well-founded *}
```
```   477
```
```   478 lemma less_nat_rel: "op < = (\<lambda>m n. n = Suc m)^++"
```
```   479 proof (rule ext, rule ext, rule iffI)
```
```   480   fix n m :: nat
```
```   481   assume "m < n"
```
```   482   then show "(\<lambda>m n. n = Suc m)^++ m n"
```
```   483   proof (induct n)
```
```   484     case 0 then show ?case by auto
```
```   485   next
```
```   486     case (Suc n) then show ?case
```
```   487       by (auto simp add: less_Suc_eq_le le_less intro: tranclp.trancl_into_trancl)
```
```   488   qed
```
```   489 next
```
```   490   fix n m :: nat
```
```   491   assume "(\<lambda>m n. n = Suc m)^++ m n"
```
```   492   then show "m < n"
```
```   493     by (induct n)
```
```   494       (simp_all add: less_Suc_eq_le reflexive le_less)
```
```   495 qed
```
```   496
```
```   497 definition
```
```   498   pred_nat :: "(nat * nat) set" where
```
```   499   "pred_nat = {(m, n). n = Suc m}"
```
```   500
```
```   501 definition
```
```   502   less_than :: "(nat * nat) set" where
```
```   503   "less_than = pred_nat^+"
```
```   504
```
```   505 lemma less_eq: "(m, n) \<in> pred_nat^+ \<longleftrightarrow> m < n"
```
```   506   unfolding less_nat_rel pred_nat_def trancl_def by simp
```
```   507
```
```   508 lemma pred_nat_trancl_eq_le:
```
```   509   "(m, n) \<in> pred_nat^* \<longleftrightarrow> m \<le> n"
```
```   510   unfolding less_eq rtrancl_eq_or_trancl by auto
```
```   511
```
```   512 lemma wf_pred_nat: "wf pred_nat"
```
```   513   apply (unfold wf_def pred_nat_def, clarify)
```
```   514   apply (induct_tac x, blast+)
```
```   515   done
```
```   516
```
```   517 lemma wf_less_than [iff]: "wf less_than"
```
```   518   by (simp add: less_than_def wf_pred_nat [THEN wf_trancl])
```
```   519
```
```   520 lemma trans_less_than [iff]: "trans less_than"
```
```   521   by (simp add: less_than_def trans_trancl)
```
```   522
```
```   523 lemma less_than_iff [iff]: "((x,y): less_than) = (x<y)"
```
```   524   by (simp add: less_than_def less_eq)
```
```   525
```
```   526 lemma wf_less: "wf {(x, y::nat). x < y}"
```
```   527   using wf_less_than by (simp add: less_than_def less_eq [symmetric])
```
```   528
```
```   529
```
```   530 subsection {* Accessible Part *}
```
```   531
```
```   532 text {*
```
```   533  Inductive definition of the accessible part @{term "acc r"} of a
```
```   534  relation; see also \cite{paulin-tlca}.
```
```   535 *}
```
```   536
```
```   537 inductive_set
```
```   538   acc :: "('a * 'a) set => 'a set"
```
```   539   for r :: "('a * 'a) set"
```
```   540   where
```
```   541     accI: "(!!y. (y, x) : r ==> y : acc r) ==> x : acc r"
```
```   542
```
```   543 abbreviation
```
```   544   termip :: "('a => 'a => bool) => 'a => bool" where
```
```   545   "termip r == accp (r\<inverse>\<inverse>)"
```
```   546
```
```   547 abbreviation
```
```   548   termi :: "('a * 'a) set => 'a set" where
```
```   549   "termi r == acc (r\<inverse>)"
```
```   550
```
```   551 lemmas accpI = accp.accI
```
```   552
```
```   553 text {* Induction rules *}
```
```   554
```
```   555 theorem accp_induct:
```
```   556   assumes major: "accp r a"
```
```   557   assumes hyp: "!!x. accp r x ==> \<forall>y. r y x --> P y ==> P x"
```
```   558   shows "P a"
```
```   559   apply (rule major [THEN accp.induct])
```
```   560   apply (rule hyp)
```
```   561    apply (rule accp.accI)
```
```   562    apply fast
```
```   563   apply fast
```
```   564   done
```
```   565
```
```   566 theorems accp_induct_rule = accp_induct [rule_format, induct set: accp]
```
```   567
```
```   568 theorem accp_downward: "accp r b ==> r a b ==> accp r a"
```
```   569   apply (erule accp.cases)
```
```   570   apply fast
```
```   571   done
```
```   572
```
```   573 lemma not_accp_down:
```
```   574   assumes na: "\<not> accp R x"
```
```   575   obtains z where "R z x" and "\<not> accp R z"
```
```   576 proof -
```
```   577   assume a: "\<And>z. \<lbrakk>R z x; \<not> accp R z\<rbrakk> \<Longrightarrow> thesis"
```
```   578
```
```   579   show thesis
```
```   580   proof (cases "\<forall>z. R z x \<longrightarrow> accp R z")
```
```   581     case True
```
```   582     hence "\<And>z. R z x \<Longrightarrow> accp R z" by auto
```
```   583     hence "accp R x"
```
```   584       by (rule accp.accI)
```
```   585     with na show thesis ..
```
```   586   next
```
```   587     case False then obtain z where "R z x" and "\<not> accp R z"
```
```   588       by auto
```
```   589     with a show thesis .
```
```   590   qed
```
```   591 qed
```
```   592
```
```   593 lemma accp_downwards_aux: "r\<^sup>*\<^sup>* b a ==> accp r a --> accp r b"
```
```   594   apply (erule rtranclp_induct)
```
```   595    apply blast
```
```   596   apply (blast dest: accp_downward)
```
```   597   done
```
```   598
```
```   599 theorem accp_downwards: "accp r a ==> r\<^sup>*\<^sup>* b a ==> accp r b"
```
```   600   apply (blast dest: accp_downwards_aux)
```
```   601   done
```
```   602
```
```   603 theorem accp_wfPI: "\<forall>x. accp r x ==> wfP r"
```
```   604   apply (rule wfPUNIVI)
```
```   605   apply (induct_tac P x rule: accp_induct)
```
```   606    apply blast
```
```   607   apply blast
```
```   608   done
```
```   609
```
```   610 theorem accp_wfPD: "wfP r ==> accp r x"
```
```   611   apply (erule wfP_induct_rule)
```
```   612   apply (rule accp.accI)
```
```   613   apply blast
```
```   614   done
```
```   615
```
```   616 theorem wfP_accp_iff: "wfP r = (\<forall>x. accp r x)"
```
```   617   apply (blast intro: accp_wfPI dest: accp_wfPD)
```
```   618   done
```
```   619
```
```   620
```
```   621 text {* Smaller relations have bigger accessible parts: *}
```
```   622
```
```   623 lemma accp_subset:
```
```   624   assumes sub: "R1 \<le> R2"
```
```   625   shows "accp R2 \<le> accp R1"
```
```   626 proof (rule predicate1I)
```
```   627   fix x assume "accp R2 x"
```
```   628   then show "accp R1 x"
```
```   629   proof (induct x)
```
```   630     fix x
```
```   631     assume ih: "\<And>y. R2 y x \<Longrightarrow> accp R1 y"
```
```   632     with sub show "accp R1 x"
```
```   633       by (blast intro: accp.accI)
```
```   634   qed
```
```   635 qed
```
```   636
```
```   637
```
```   638 text {* This is a generalized induction theorem that works on
```
```   639   subsets of the accessible part. *}
```
```   640
```
```   641 lemma accp_subset_induct:
```
```   642   assumes subset: "D \<le> accp R"
```
```   643     and dcl: "\<And>x z. \<lbrakk>D x; R z x\<rbrakk> \<Longrightarrow> D z"
```
```   644     and "D x"
```
```   645     and istep: "\<And>x. \<lbrakk>D x; (\<And>z. R z x \<Longrightarrow> P z)\<rbrakk> \<Longrightarrow> P x"
```
```   646   shows "P x"
```
```   647 proof -
```
```   648   from subset and `D x`
```
```   649   have "accp R x" ..
```
```   650   then show "P x" using `D x`
```
```   651   proof (induct x)
```
```   652     fix x
```
```   653     assume "D x"
```
```   654       and "\<And>y. R y x \<Longrightarrow> D y \<Longrightarrow> P y"
```
```   655     with dcl and istep show "P x" by blast
```
```   656   qed
```
```   657 qed
```
```   658
```
```   659
```
```   660 text {* Set versions of the above theorems *}
```
```   661
```
```   662 lemmas acc_induct = accp_induct [to_set]
```
```   663
```
```   664 lemmas acc_induct_rule = acc_induct [rule_format, induct set: acc]
```
```   665
```
```   666 lemmas acc_downward = accp_downward [to_set]
```
```   667
```
```   668 lemmas not_acc_down = not_accp_down [to_set]
```
```   669
```
```   670 lemmas acc_downwards_aux = accp_downwards_aux [to_set]
```
```   671
```
```   672 lemmas acc_downwards = accp_downwards [to_set]
```
```   673
```
```   674 lemmas acc_wfI = accp_wfPI [to_set]
```
```   675
```
```   676 lemmas acc_wfD = accp_wfPD [to_set]
```
```   677
```
```   678 lemmas wf_acc_iff = wfP_accp_iff [to_set]
```
```   679
```
```   680 lemmas acc_subset = accp_subset [to_set pred_subset_eq]
```
```   681
```
```   682 lemmas acc_subset_induct = accp_subset_induct [to_set pred_subset_eq]
```
```   683
```
```   684
```
```   685 subsection {* Tools for building wellfounded relations *}
```
```   686
```
```   687 text {* Inverse Image *}
```
```   688
```
```   689 lemma wf_inv_image [simp,intro!]: "wf(r) ==> wf(inv_image r (f::'a=>'b))"
```
```   690 apply (simp (no_asm_use) add: inv_image_def wf_eq_minimal)
```
```   691 apply clarify
```
```   692 apply (subgoal_tac "EX (w::'b) . w : {w. EX (x::'a) . x: Q & (f x = w) }")
```
```   693 prefer 2 apply (blast del: allE)
```
```   694 apply (erule allE)
```
```   695 apply (erule (1) notE impE)
```
```   696 apply blast
```
```   697 done
```
```   698
```
```   699 lemma in_inv_image[simp]: "((x,y) : inv_image r f) = ((f x, f y) : r)"
```
```   700   by (auto simp:inv_image_def)
```
```   701
```
```   702 text {* Measure Datatypes into @{typ nat} *}
```
```   703
```
```   704 definition measure :: "('a => nat) => ('a * 'a)set"
```
```   705 where "measure == inv_image less_than"
```
```   706
```
```   707 lemma in_measure[simp]: "((x,y) : measure f) = (f x < f y)"
```
```   708   by (simp add:measure_def)
```
```   709
```
```   710 lemma wf_measure [iff]: "wf (measure f)"
```
```   711 apply (unfold measure_def)
```
```   712 apply (rule wf_less_than [THEN wf_inv_image])
```
```   713 done
```
```   714
```
```   715 text{* Lexicographic combinations *}
```
```   716
```
```   717 definition
```
```   718  lex_prod  :: "[('a*'a)set, ('b*'b)set] => (('a*'b)*('a*'b))set"
```
```   719                (infixr "<*lex*>" 80)
```
```   720 where
```
```   721     "ra <*lex*> rb == {((a,b),(a',b')). (a,a') : ra | a=a' & (b,b') : rb}"
```
```   722
```
```   723 lemma wf_lex_prod [intro!]: "[| wf(ra); wf(rb) |] ==> wf(ra <*lex*> rb)"
```
```   724 apply (unfold wf_def lex_prod_def)
```
```   725 apply (rule allI, rule impI)
```
```   726 apply (simp (no_asm_use) only: split_paired_All)
```
```   727 apply (drule spec, erule mp)
```
```   728 apply (rule allI, rule impI)
```
```   729 apply (drule spec, erule mp, blast)
```
```   730 done
```
```   731
```
```   732 lemma in_lex_prod[simp]:
```
```   733   "(((a,b),(a',b')): r <*lex*> s) = ((a,a'): r \<or> (a = a' \<and> (b, b') : s))"
```
```   734   by (auto simp:lex_prod_def)
```
```   735
```
```   736 text{* @{term "op <*lex*>"} preserves transitivity *}
```
```   737
```
```   738 lemma trans_lex_prod [intro!]:
```
```   739     "[| trans R1; trans R2 |] ==> trans (R1 <*lex*> R2)"
```
```   740 by (unfold trans_def lex_prod_def, blast)
```
```   741
```
```   742 text {* lexicographic combinations with measure Datatypes *}
```
```   743
```
```   744 definition
```
```   745   mlex_prod :: "('a \<Rightarrow> nat) \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" (infixr "<*mlex*>" 80)
```
```   746 where
```
```   747   "f <*mlex*> R = inv_image (less_than <*lex*> R) (%x. (f x, x))"
```
```   748
```
```   749 lemma wf_mlex: "wf R \<Longrightarrow> wf (f <*mlex*> R)"
```
```   750 unfolding mlex_prod_def
```
```   751 by auto
```
```   752
```
```   753 lemma mlex_less: "f x < f y \<Longrightarrow> (x, y) \<in> f <*mlex*> R"
```
```   754 unfolding mlex_prod_def by simp
```
```   755
```
```   756 lemma mlex_leq: "f x \<le> f y \<Longrightarrow> (x, y) \<in> R \<Longrightarrow> (x, y) \<in> f <*mlex*> R"
```
```   757 unfolding mlex_prod_def by auto
```
```   758
```
```   759 text {* proper subset relation on finite sets *}
```
```   760
```
```   761 definition finite_psubset  :: "('a set * 'a set) set"
```
```   762 where "finite_psubset == {(A,B). A < B & finite B}"
```
```   763
```
```   764 lemma wf_finite_psubset[simp]: "wf(finite_psubset)"
```
```   765 apply (unfold finite_psubset_def)
```
```   766 apply (rule wf_measure [THEN wf_subset])
```
```   767 apply (simp add: measure_def inv_image_def less_than_def less_eq)
```
```   768 apply (fast elim!: psubset_card_mono)
```
```   769 done
```
```   770
```
```   771 lemma trans_finite_psubset: "trans finite_psubset"
```
```   772 by (simp add: finite_psubset_def less_le trans_def, blast)
```
```   773
```
```   774 lemma in_finite_psubset[simp]: "(A, B) \<in> finite_psubset = (A < B & finite B)"
```
```   775 unfolding finite_psubset_def by auto
```
```   776
```
```   777 text {* max- and min-extension of order to finite sets *}
```
```   778
```
```   779 inductive_set max_ext :: "('a \<times> 'a) set \<Rightarrow> ('a set \<times> 'a set) set"
```
```   780 for R :: "('a \<times> 'a) set"
```
```   781 where
```
```   782   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"
```
```   783
```
```   784 lemma max_ext_wf:
```
```   785   assumes wf: "wf r"
```
```   786   shows "wf (max_ext r)"
```
```   787 proof (rule acc_wfI, intro allI)
```
```   788   fix M show "M \<in> acc (max_ext r)" (is "_ \<in> ?W")
```
```   789   proof cases
```
```   790     assume "finite M"
```
```   791     thus ?thesis
```
```   792     proof (induct M)
```
```   793       show "{} \<in> ?W"
```
```   794         by (rule accI) (auto elim: max_ext.cases)
```
```   795     next
```
```   796       fix M a assume "M \<in> ?W" "finite M"
```
```   797       with wf show "insert a M \<in> ?W"
```
```   798       proof (induct arbitrary: M)
```
```   799         fix M a
```
```   800         assume "M \<in> ?W"  and  [intro]: "finite M"
```
```   801         assume hyp: "\<And>b M. (b, a) \<in> r \<Longrightarrow> M \<in> ?W \<Longrightarrow> finite M \<Longrightarrow> insert b M \<in> ?W"
```
```   802         {
```
```   803           fix N M :: "'a set"
```
```   804           assume "finite N" "finite M"
```
```   805           then
```
```   806           have "\<lbrakk>M \<in> ?W ; (\<And>y. y \<in> N \<Longrightarrow> (y, a) \<in> r)\<rbrakk> \<Longrightarrow>  N \<union> M \<in> ?W"
```
```   807             by (induct N arbitrary: M) (auto simp: hyp)
```
```   808         }
```
```   809         note add_less = this
```
```   810
```
```   811         show "insert a M \<in> ?W"
```
```   812         proof (rule accI)
```
```   813           fix N assume Nless: "(N, insert a M) \<in> max_ext r"
```
```   814           hence asm1: "\<And>x. x \<in> N \<Longrightarrow> (x, a) \<in> r \<or> (\<exists>y \<in> M. (x, y) \<in> r)"
```
```   815             by (auto elim!: max_ext.cases)
```
```   816
```
```   817           let ?N1 = "{ n \<in> N. (n, a) \<in> r }"
```
```   818           let ?N2 = "{ n \<in> N. (n, a) \<notin> r }"
```
```   819           have N: "?N1 \<union> ?N2 = N" by (rule set_ext) auto
```
```   820           from Nless have "finite N" by (auto elim: max_ext.cases)
```
```   821           then have finites: "finite ?N1" "finite ?N2" by auto
```
```   822
```
```   823           have "?N2 \<in> ?W"
```
```   824           proof cases
```
```   825             assume [simp]: "M = {}"
```
```   826             have Mw: "{} \<in> ?W" by (rule accI) (auto elim: max_ext.cases)
```
```   827
```
```   828             from asm1 have "?N2 = {}" by auto
```
```   829             with Mw show "?N2 \<in> ?W" by (simp only:)
```
```   830           next
```
```   831             assume "M \<noteq> {}"
```
```   832             have N2: "(?N2, M) \<in> max_ext r"
```
```   833               by (rule max_extI[OF _ _ `M \<noteq> {}`]) (insert asm1, auto intro: finites)
```
```   834
```
```   835             with `M \<in> ?W` show "?N2 \<in> ?W" by (rule acc_downward)
```
```   836           qed
```
```   837           with finites have "?N1 \<union> ?N2 \<in> ?W"
```
```   838             by (rule add_less) simp
```
```   839           then show "N \<in> ?W" by (simp only: N)
```
```   840         qed
```
```   841       qed
```
```   842     qed
```
```   843   next
```
```   844     assume [simp]: "\<not> finite M"
```
```   845     show ?thesis
```
```   846       by (rule accI) (auto elim: max_ext.cases)
```
```   847   qed
```
```   848 qed
```
```   849
```
```   850 lemma max_ext_additive:
```
```   851  "(A, B) \<in> max_ext R \<Longrightarrow> (C, D) \<in> max_ext R \<Longrightarrow>
```
```   852   (A \<union> C, B \<union> D) \<in> max_ext R"
```
```   853 by (force elim!: max_ext.cases)
```
```   854
```
```   855
```
```   856 definition
```
```   857   min_ext :: "('a \<times> 'a) set \<Rightarrow> ('a set \<times> 'a set) set"
```
```   858 where
```
```   859   [code del]: "min_ext r = {(X, Y) | X Y. X \<noteq> {} \<and> (\<forall>y \<in> Y. (\<exists>x \<in> X. (x, y) \<in> r))}"
```
```   860
```
```   861 lemma min_ext_wf:
```
```   862   assumes "wf r"
```
```   863   shows "wf (min_ext r)"
```
```   864 proof (rule wfI_min)
```
```   865   fix Q :: "'a set set"
```
```   866   fix x
```
```   867   assume nonempty: "x \<in> Q"
```
```   868   show "\<exists>m \<in> Q. (\<forall> n. (n, m) \<in> min_ext r \<longrightarrow> n \<notin> Q)"
```
```   869   proof cases
```
```   870     assume "Q = {{}}" thus ?thesis by (simp add: min_ext_def)
```
```   871   next
```
```   872     assume "Q \<noteq> {{}}"
```
```   873     with nonempty
```
```   874     obtain e x where "x \<in> Q" "e \<in> x" by force
```
```   875     then have eU: "e \<in> \<Union>Q" by auto
```
```   876     with `wf r`
```
```   877     obtain z where z: "z \<in> \<Union>Q" "\<And>y. (y, z) \<in> r \<Longrightarrow> y \<notin> \<Union>Q"
```
```   878       by (erule wfE_min)
```
```   879     from z obtain m where "m \<in> Q" "z \<in> m" by auto
```
```   880     from `m \<in> Q`
```
```   881     show ?thesis
```
```   882     proof (rule, intro bexI allI impI)
```
```   883       fix n
```
```   884       assume smaller: "(n, m) \<in> min_ext r"
```
```   885       with `z \<in> m` obtain y where y: "y \<in> n" "(y, z) \<in> r" by (auto simp: min_ext_def)
```
```   886       then show "n \<notin> Q" using z(2) by auto
```
```   887     qed
```
```   888   qed
```
```   889 qed
```
```   890
```
```   891
```
```   892 subsection{*Weakly decreasing sequences (w.r.t. some well-founded order)
```
```   893    stabilize.*}
```
```   894
```
```   895 text{*This material does not appear to be used any longer.*}
```
```   896
```
```   897 lemma sequence_trans: "[| ALL i. (f (Suc i), f i) : r^* |] ==> (f (i+k), f i) : r^*"
```
```   898 by (induct k) (auto intro: rtrancl_trans)
```
```   899
```
```   900 lemma wf_weak_decr_stable:
```
```   901   assumes as: "ALL i. (f (Suc i), f i) : r^*" "wf (r^+)"
```
```   902   shows "EX i. ALL k. f (i+k) = f i"
```
```   903 proof -
```
```   904   have lem: "!!x. [| ALL i. (f (Suc i), f i) : r^*; wf (r^+) |]
```
```   905       ==> ALL m. f m = x --> (EX i. ALL k. f (m+i+k) = f (m+i))"
```
```   906   apply (erule wf_induct, clarify)
```
```   907   apply (case_tac "EX j. (f (m+j), f m) : r^+")
```
```   908    apply clarify
```
```   909    apply (subgoal_tac "EX i. ALL k. f ((m+j) +i+k) = f ( (m+j) +i) ")
```
```   910     apply clarify
```
```   911     apply (rule_tac x = "j+i" in exI)
```
```   912     apply (simp add: add_ac, blast)
```
```   913   apply (rule_tac x = 0 in exI, clarsimp)
```
```   914   apply (drule_tac i = m and k = k in sequence_trans)
```
```   915   apply (blast elim: rtranclE dest: rtrancl_into_trancl1)
```
```   916   done
```
```   917
```
```   918   from lem[OF as, THEN spec, of 0, simplified]
```
```   919   show ?thesis by auto
```
```   920 qed
```
```   921
```
```   922 (* special case of the theorem above: <= *)
```
```   923 lemma weak_decr_stable:
```
```   924      "ALL i. f (Suc i) <= ((f i)::nat) ==> EX i. ALL k. f (i+k) = f i"
```
```   925 apply (rule_tac r = pred_nat in wf_weak_decr_stable)
```
```   926 apply (simp add: pred_nat_trancl_eq_le)
```
```   927 apply (intro wf_trancl wf_pred_nat)
```
```   928 done
```
```   929
```
```   930
```
```   931 subsection {* size of a datatype value *}
```
```   932
```
```   933 use "Tools/Function/size.ML"
```
```   934
```
```   935 setup Size.setup
```
```   936
```
```   937 lemma size_bool [code]:
```
```   938   "size (b\<Colon>bool) = 0" by (cases b) auto
```
```   939
```
```   940 lemma nat_size [simp, code]: "size (n\<Colon>nat) = n"
```
```   941   by (induct n) simp_all
```
```   942
```
```   943 declare "prod.size" [noatp]
```
```   944
```
```   945 lemma [code]:
```
```   946   "size (P :: 'a Predicate.pred) = 0" by (cases P) simp
```
```   947
```
```   948 lemma [code]:
```
```   949   "pred_size f P = 0" by (cases P) simp
```
```   950
```
```   951 end
```