src/HOL/Wellfounded_Recursion.thy
 author krauss Fri Nov 24 13:44:51 2006 +0100 (2006-11-24) changeset 21512 3786eb1b69d6 parent 20592 527563e67194 child 22263 990a638e6f69 permissions -rw-r--r--
Lemma "fundef_default_value" uses predicate instead of set.
```     1 (*  ID:         \$Id\$
```
```     2     Author:     Tobias Nipkow
```
```     3     Copyright   1992  University of Cambridge
```
```     4 *)
```
```     5
```
```     6 header {*Well-founded Recursion*}
```
```     7
```
```     8 theory Wellfounded_Recursion
```
```     9 imports Transitive_Closure
```
```    10 begin
```
```    11
```
```    12 consts
```
```    13   wfrec_rel :: "('a * 'a) set => (('a => 'b) => 'a => 'b) => ('a * 'b) set"
```
```    14
```
```    15 inductive "wfrec_rel R F"
```
```    16 intros
```
```    17   wfrecI: "ALL z. (z, x) : R --> (z, g z) : wfrec_rel R F ==>
```
```    18             (x, F g x) : wfrec_rel R F"
```
```    19
```
```    20 constdefs
```
```    21   wf         :: "('a * 'a)set => bool"
```
```    22   "wf(r) == (!P. (!x. (!y. (y,x):r --> P(y)) --> P(x)) --> (!x. P(x)))"
```
```    23
```
```    24   acyclic :: "('a*'a)set => bool"
```
```    25   "acyclic r == !x. (x,x) ~: r^+"
```
```    26
```
```    27   cut        :: "('a => 'b) => ('a * 'a)set => 'a => 'a => 'b"
```
```    28   "cut f r x == (%y. if (y,x):r then f y else arbitrary)"
```
```    29
```
```    30   adm_wf :: "('a * 'a) set => (('a => 'b) => 'a => 'b) => bool"
```
```    31   "adm_wf R F == ALL f g x.
```
```    32      (ALL z. (z, x) : R --> f z = g z) --> F f x = F g x"
```
```    33
```
```    34   wfrec :: "('a * 'a) set => (('a => 'b) => 'a => 'b) => 'a => 'b"
```
```    35   "wfrec R F == %x. THE y. (x, y) : wfrec_rel R (%f x. F (cut f R x) x)"
```
```    36
```
```    37 axclass wellorder \<subseteq> linorder
```
```    38   wf: "wf {(x,y::'a::ord). x<y}"
```
```    39
```
```    40
```
```    41 lemma wfUNIVI:
```
```    42    "(!!P x. (ALL x. (ALL y. (y,x) : r --> P(y)) --> P(x)) ==> P(x)) ==> wf(r)"
```
```    43 by (unfold wf_def, blast)
```
```    44
```
```    45 text{*Restriction to domain @{term A} and range @{term B}.  If @{term r} is
```
```    46     well-founded over their intersection, then @{term "wf r"}*}
```
```    47 lemma wfI:
```
```    48  "[| r \<subseteq> A <*> B;
```
```    49      !!x P. [|\<forall>x. (\<forall>y. (y,x) : r --> P y) --> P x;  x : A; x : B |] ==> P x |]
```
```    50   ==>  wf r"
```
```    51 by (unfold wf_def, blast)
```
```    52
```
```    53 lemma wf_induct:
```
```    54     "[| wf(r);
```
```    55         !!x.[| ALL y. (y,x): r --> P(y) |] ==> P(x)
```
```    56      |]  ==>  P(a)"
```
```    57 by (unfold wf_def, blast)
```
```    58
```
```    59 lemmas wf_induct_rule = wf_induct [rule_format, consumes 1, case_names less, induct set: wf]
```
```    60
```
```    61 lemma wf_not_sym [rule_format]: "wf(r) ==> ALL x. (a,x):r --> (x,a)~:r"
```
```    62 by (erule_tac a=a in wf_induct, blast)
```
```    63
```
```    64 (* [| wf r;  ~Z ==> (a,x) : r;  (x,a) ~: r ==> Z |] ==> Z *)
```
```    65 lemmas wf_asym = wf_not_sym [elim_format]
```
```    66
```
```    67 lemma wf_not_refl [simp]: "wf(r) ==> (a,a) ~: r"
```
```    68 by (blast elim: wf_asym)
```
```    69
```
```    70 (* [| wf r;  (a,a) ~: r ==> PROP W |] ==> PROP W *)
```
```    71 lemmas wf_irrefl = wf_not_refl [elim_format]
```
```    72
```
```    73 text{*transitive closure of a well-founded relation is well-founded! *}
```
```    74 lemma wf_trancl: "wf(r) ==> wf(r^+)"
```
```    75 apply (subst wf_def, clarify)
```
```    76 apply (rule allE, assumption)
```
```    77   --{*Retains the universal formula for later use!*}
```
```    78 apply (erule mp)
```
```    79 apply (erule_tac a = x in wf_induct)
```
```    80 apply (blast elim: tranclE)
```
```    81 done
```
```    82
```
```    83 lemma wf_converse_trancl: "wf (r^-1) ==> wf ((r^+)^-1)"
```
```    84 apply (subst trancl_converse [symmetric])
```
```    85 apply (erule wf_trancl)
```
```    86 done
```
```    87
```
```    88
```
```    89 subsubsection{*Other simple well-foundedness results*}
```
```    90
```
```    91
```
```    92 text{*Minimal-element characterization of well-foundedness*}
```
```    93 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))"
```
```    94 proof (intro iffI strip)
```
```    95   fix Q::"'a set" and x
```
```    96   assume "wf r" and "x \<in> Q"
```
```    97   thus "\<exists>z\<in>Q. \<forall>y. (y, z) \<in> r \<longrightarrow> y \<notin> Q"
```
```    98     by (unfold wf_def,
```
```    99         blast dest: spec [of _ "%x. x\<in>Q \<longrightarrow> (\<exists>z\<in>Q. \<forall>y. (y,z) \<in> r \<longrightarrow> y\<notin>Q)"])
```
```   100 next
```
```   101   assume "\<forall>Q x. x \<in> Q \<longrightarrow> (\<exists>z\<in>Q. \<forall>y. (y, z) \<in> r \<longrightarrow> y \<notin> Q)"
```
```   102   thus "wf r" by (unfold wf_def, force)
```
```   103 qed
```
```   104
```
```   105 text{*Well-foundedness of subsets*}
```
```   106 lemma wf_subset: "[| wf(r);  p<=r |] ==> wf(p)"
```
```   107 apply (simp (no_asm_use) add: wf_eq_minimal)
```
```   108 apply fast
```
```   109 done
```
```   110
```
```   111 text{*Well-foundedness of the empty relation*}
```
```   112 lemma wf_empty [iff]: "wf({})"
```
```   113 by (simp add: wf_def)
```
```   114
```
```   115 lemma wf_Int1: "wf r ==> wf (r Int r')"
```
```   116 by (erule wf_subset, rule Int_lower1)
```
```   117
```
```   118 lemma wf_Int2: "wf r ==> wf (r' Int r)"
```
```   119 by (erule wf_subset, rule Int_lower2)
```
```   120
```
```   121 text{*Well-foundedness of insert*}
```
```   122 lemma wf_insert [iff]: "wf(insert (y,x) r) = (wf(r) & (x,y) ~: r^*)"
```
```   123 apply (rule iffI)
```
```   124  apply (blast elim: wf_trancl [THEN wf_irrefl]
```
```   125               intro: rtrancl_into_trancl1 wf_subset
```
```   126                      rtrancl_mono [THEN [2] rev_subsetD])
```
```   127 apply (simp add: wf_eq_minimal, safe)
```
```   128 apply (rule allE, assumption, erule impE, blast)
```
```   129 apply (erule bexE)
```
```   130 apply (rename_tac "a", case_tac "a = x")
```
```   131  prefer 2
```
```   132 apply blast
```
```   133 apply (case_tac "y:Q")
```
```   134  prefer 2 apply blast
```
```   135 apply (rule_tac x = "{z. z:Q & (z,y) : r^*}" in allE)
```
```   136  apply assumption
```
```   137 apply (erule_tac V = "ALL Q. (EX x. x : Q) --> ?P Q" in thin_rl)
```
```   138   --{*essential for speed*}
```
```   139 txt{*Blast with new substOccur fails*}
```
```   140 apply (fast intro: converse_rtrancl_into_rtrancl)
```
```   141 done
```
```   142
```
```   143 text{*Well-foundedness of image*}
```
```   144 lemma wf_prod_fun_image: "[| wf r; inj f |] ==> wf(prod_fun f f ` r)"
```
```   145 apply (simp only: wf_eq_minimal, clarify)
```
```   146 apply (case_tac "EX p. f p : Q")
```
```   147 apply (erule_tac x = "{p. f p : Q}" in allE)
```
```   148 apply (fast dest: inj_onD, blast)
```
```   149 done
```
```   150
```
```   151
```
```   152 subsubsection{*Well-Foundedness Results for Unions*}
```
```   153
```
```   154 text{*Well-foundedness of indexed union with disjoint domains and ranges*}
```
```   155
```
```   156 lemma wf_UN: "[| ALL i:I. wf(r i);
```
```   157          ALL i:I. ALL j:I. r i ~= r j --> Domain(r i) Int Range(r j) = {}
```
```   158       |] ==> wf(UN i:I. r i)"
```
```   159 apply (simp only: wf_eq_minimal, clarify)
```
```   160 apply (rename_tac A a, case_tac "EX i:I. EX a:A. EX b:A. (b,a) : r i")
```
```   161  prefer 2
```
```   162  apply force
```
```   163 apply clarify
```
```   164 apply (drule bspec, assumption)
```
```   165 apply (erule_tac x="{a. a:A & (EX b:A. (b,a) : r i) }" in allE)
```
```   166 apply (blast elim!: allE)
```
```   167 done
```
```   168
```
```   169 lemma wf_Union:
```
```   170  "[| ALL r:R. wf r;
```
```   171      ALL r:R. ALL s:R. r ~= s --> Domain r Int Range s = {}
```
```   172   |] ==> wf(Union R)"
```
```   173 apply (simp add: Union_def)
```
```   174 apply (blast intro: wf_UN)
```
```   175 done
```
```   176
```
```   177 (*Intuition: we find an (R u S)-min element of a nonempty subset A
```
```   178              by case distinction.
```
```   179   1. There is a step a -R-> b with a,b : A.
```
```   180      Pick an R-min element z of the (nonempty) set {a:A | EX b:A. a -R-> b}.
```
```   181      By definition, there is z':A s.t. z -R-> z'. Because z is R-min in the
```
```   182      subset, z' must be R-min in A. Because z' has an R-predecessor, it cannot
```
```   183      have an S-successor and is thus S-min in A as well.
```
```   184   2. There is no such step.
```
```   185      Pick an S-min element of A. In this case it must be an R-min
```
```   186      element of A as well.
```
```   187
```
```   188 *)
```
```   189 lemma wf_Un:
```
```   190      "[| wf r; wf s; Domain r Int Range s = {} |] ==> wf(r Un s)"
```
```   191 apply (simp only: wf_eq_minimal, clarify)
```
```   192 apply (rename_tac A a)
```
```   193 apply (case_tac "EX a:A. EX b:A. (b,a) : r")
```
```   194  prefer 2
```
```   195  apply simp
```
```   196  apply (drule_tac x=A in spec)+
```
```   197  apply blast
```
```   198 apply (erule_tac x="{a. a:A & (EX b:A. (b,a) : r) }" in allE)+
```
```   199 apply (blast elim!: allE)
```
```   200 done
```
```   201
```
```   202 subsubsection {*acyclic*}
```
```   203
```
```   204 lemma acyclicI: "ALL x. (x, x) ~: r^+ ==> acyclic r"
```
```   205 by (simp add: acyclic_def)
```
```   206
```
```   207 lemma wf_acyclic: "wf r ==> acyclic r"
```
```   208 apply (simp add: acyclic_def)
```
```   209 apply (blast elim: wf_trancl [THEN wf_irrefl])
```
```   210 done
```
```   211
```
```   212 lemma acyclic_insert [iff]:
```
```   213      "acyclic(insert (y,x) r) = (acyclic r & (x,y) ~: r^*)"
```
```   214 apply (simp add: acyclic_def trancl_insert)
```
```   215 apply (blast intro: rtrancl_trans)
```
```   216 done
```
```   217
```
```   218 lemma acyclic_converse [iff]: "acyclic(r^-1) = acyclic r"
```
```   219 by (simp add: acyclic_def trancl_converse)
```
```   220
```
```   221 lemma acyclic_impl_antisym_rtrancl: "acyclic r ==> antisym(r^*)"
```
```   222 apply (simp add: acyclic_def antisym_def)
```
```   223 apply (blast elim: rtranclE intro: rtrancl_into_trancl1 rtrancl_trancl_trancl)
```
```   224 done
```
```   225
```
```   226 (* Other direction:
```
```   227 acyclic = no loops
```
```   228 antisym = only self loops
```
```   229 Goalw [acyclic_def,antisym_def] "antisym( r^* ) ==> acyclic(r - Id)
```
```   230 ==> antisym( r^* ) = acyclic(r - Id)";
```
```   231 *)
```
```   232
```
```   233 lemma acyclic_subset: "[| acyclic s; r <= s |] ==> acyclic r"
```
```   234 apply (simp add: acyclic_def)
```
```   235 apply (blast intro: trancl_mono)
```
```   236 done
```
```   237
```
```   238
```
```   239 subsection{*Well-Founded Recursion*}
```
```   240
```
```   241 text{*cut*}
```
```   242
```
```   243 lemma cuts_eq: "(cut f r x = cut g r x) = (ALL y. (y,x):r --> f(y)=g(y))"
```
```   244 by (simp add: expand_fun_eq cut_def)
```
```   245
```
```   246 lemma cut_apply: "(x,a):r ==> (cut f r a)(x) = f(x)"
```
```   247 by (simp add: cut_def)
```
```   248
```
```   249 text{*Inductive characterization of wfrec combinator; for details see:
```
```   250 John Harrison, "Inductive definitions: automation and application"*}
```
```   251
```
```   252 lemma wfrec_unique: "[| adm_wf R F; wf R |] ==> EX! y. (x, y) : wfrec_rel R F"
```
```   253 apply (simp add: adm_wf_def)
```
```   254 apply (erule_tac a=x in wf_induct)
```
```   255 apply (rule ex1I)
```
```   256 apply (rule_tac g = "%x. THE y. (x, y) : wfrec_rel R F" in wfrec_rel.wfrecI)
```
```   257 apply (fast dest!: theI')
```
```   258 apply (erule wfrec_rel.cases, simp)
```
```   259 apply (erule allE, erule allE, erule allE, erule mp)
```
```   260 apply (fast intro: the_equality [symmetric])
```
```   261 done
```
```   262
```
```   263 lemma adm_lemma: "adm_wf R (%f x. F (cut f R x) x)"
```
```   264 apply (simp add: adm_wf_def)
```
```   265 apply (intro strip)
```
```   266 apply (rule cuts_eq [THEN iffD2, THEN subst], assumption)
```
```   267 apply (rule refl)
```
```   268 done
```
```   269
```
```   270 lemma wfrec: "wf(r) ==> wfrec r H a = H (cut (wfrec r H) r a) a"
```
```   271 apply (simp add: wfrec_def)
```
```   272 apply (rule adm_lemma [THEN wfrec_unique, THEN the1_equality], assumption)
```
```   273 apply (rule wfrec_rel.wfrecI)
```
```   274 apply (intro strip)
```
```   275 apply (erule adm_lemma [THEN wfrec_unique, THEN theI'])
```
```   276 done
```
```   277
```
```   278
```
```   279 text{** This form avoids giant explosions in proofs.  NOTE USE OF ==*}
```
```   280 lemma def_wfrec: "[| f==wfrec r H;  wf(r) |] ==> f(a) = H (cut f r a) a"
```
```   281 apply auto
```
```   282 apply (blast intro: wfrec)
```
```   283 done
```
```   284
```
```   285
```
```   286 subsection {* Code generator setup *}
```
```   287
```
```   288 consts_code
```
```   289   "wfrec"   ("\<module>wfrec?")
```
```   290 attach {*
```
```   291 fun wfrec f x = f (wfrec f) x;
```
```   292 *}
```
```   293
```
```   294
```
```   295 subsection{*Variants for TFL: the Recdef Package*}
```
```   296
```
```   297 lemma tfl_wf_induct: "ALL R. wf R -->
```
```   298        (ALL P. (ALL x. (ALL y. (y,x):R --> P y) --> P x) --> (ALL x. P x))"
```
```   299 apply clarify
```
```   300 apply (rule_tac r = R and P = P and a = x in wf_induct, assumption, blast)
```
```   301 done
```
```   302
```
```   303 lemma tfl_cut_apply: "ALL f R. (x,a):R --> (cut f R a)(x) = f(x)"
```
```   304 apply clarify
```
```   305 apply (rule cut_apply, assumption)
```
```   306 done
```
```   307
```
```   308 lemma tfl_wfrec:
```
```   309      "ALL M R f. (f=wfrec R M) --> wf R --> (ALL x. f x = M (cut f R x) x)"
```
```   310 apply clarify
```
```   311 apply (erule wfrec)
```
```   312 done
```
```   313
```
```   314 subsection {*LEAST and wellorderings*}
```
```   315
```
```   316 text{* See also @{text wf_linord_ex_has_least} and its consequences in
```
```   317  @{text Wellfounded_Relations.ML}*}
```
```   318
```
```   319 lemma wellorder_Least_lemma [rule_format]:
```
```   320      "P (k::'a::wellorder) --> P (LEAST x. P(x)) & (LEAST x. P(x)) <= k"
```
```   321 apply (rule_tac a = k in wf [THEN wf_induct])
```
```   322 apply (rule impI)
```
```   323 apply (rule classical)
```
```   324 apply (rule_tac s = x in Least_equality [THEN ssubst], auto)
```
```   325 apply (auto simp add: linorder_not_less [symmetric])
```
```   326 done
```
```   327
```
```   328 lemmas LeastI   = wellorder_Least_lemma [THEN conjunct1, standard]
```
```   329 lemmas Least_le = wellorder_Least_lemma [THEN conjunct2, standard]
```
```   330
```
```   331 -- "The following 3 lemmas are due to Brian Huffman"
```
```   332 lemma LeastI_ex: "EX x::'a::wellorder. P x ==> P (Least P)"
```
```   333 apply (erule exE)
```
```   334 apply (erule LeastI)
```
```   335 done
```
```   336
```
```   337 lemma LeastI2:
```
```   338   "[| P (a::'a::wellorder); !!x. P x ==> Q x |] ==> Q (Least P)"
```
```   339 by (blast intro: LeastI)
```
```   340
```
```   341 lemma LeastI2_ex:
```
```   342   "[| EX a::'a::wellorder. P a; !!x. P x ==> Q x |] ==> Q (Least P)"
```
```   343 by (blast intro: LeastI_ex)
```
```   344
```
```   345 lemma not_less_Least: "[| k < (LEAST x. P x) |] ==> ~P (k::'a::wellorder)"
```
```   346 apply (simp (no_asm_use) add: linorder_not_le [symmetric])
```
```   347 apply (erule contrapos_nn)
```
```   348 apply (erule Least_le)
```
```   349 done
```
```   350
```
```   351 ML
```
```   352 {*
```
```   353 val wf_def = thm "wf_def";
```
```   354 val wfUNIVI = thm "wfUNIVI";
```
```   355 val wfI = thm "wfI";
```
```   356 val wf_induct = thm "wf_induct";
```
```   357 val wf_not_sym = thm "wf_not_sym";
```
```   358 val wf_asym = thm "wf_asym";
```
```   359 val wf_not_refl = thm "wf_not_refl";
```
```   360 val wf_irrefl = thm "wf_irrefl";
```
```   361 val wf_trancl = thm "wf_trancl";
```
```   362 val wf_converse_trancl = thm "wf_converse_trancl";
```
```   363 val wf_eq_minimal = thm "wf_eq_minimal";
```
```   364 val wf_subset = thm "wf_subset";
```
```   365 val wf_empty = thm "wf_empty";
```
```   366 val wf_insert = thm "wf_insert";
```
```   367 val wf_UN = thm "wf_UN";
```
```   368 val wf_Union = thm "wf_Union";
```
```   369 val wf_Un = thm "wf_Un";
```
```   370 val wf_prod_fun_image = thm "wf_prod_fun_image";
```
```   371 val acyclicI = thm "acyclicI";
```
```   372 val wf_acyclic = thm "wf_acyclic";
```
```   373 val acyclic_insert = thm "acyclic_insert";
```
```   374 val acyclic_converse = thm "acyclic_converse";
```
```   375 val acyclic_impl_antisym_rtrancl = thm "acyclic_impl_antisym_rtrancl";
```
```   376 val acyclic_subset = thm "acyclic_subset";
```
```   377 val cuts_eq = thm "cuts_eq";
```
```   378 val cut_apply = thm "cut_apply";
```
```   379 val wfrec_unique = thm "wfrec_unique";
```
```   380 val wfrec = thm "wfrec";
```
```   381 val def_wfrec = thm "def_wfrec";
```
```   382 val tfl_wf_induct = thm "tfl_wf_induct";
```
```   383 val tfl_cut_apply = thm "tfl_cut_apply";
```
```   384 val tfl_wfrec = thm "tfl_wfrec";
```
```   385 val LeastI = thm "LeastI";
```
```   386 val Least_le = thm "Least_le";
```
```   387 val not_less_Least = thm "not_less_Least";
```
```   388 *}
```
```   389
```
```   390 end
```