src/HOL/FixedPoint.thy
author krauss
Fri Nov 24 13:44:51 2006 +0100 (2006-11-24)
changeset 21512 3786eb1b69d6
parent 21404 eb85850d3eb7
child 21547 9c9fdf4c2949
permissions -rw-r--r--
Lemma "fundef_default_value" uses predicate instead of set.
     1 (*  Title:      HOL/FixedPoint.thy
     2     ID:         $Id$
     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     4     Author:     Stefan Berghofer, TU Muenchen
     5     Copyright   1992  University of Cambridge
     6 *)
     7 
     8 header{* Fixed Points and the Knaster-Tarski Theorem*}
     9 
    10 theory FixedPoint
    11 imports Product_Type LOrder
    12 begin
    13 
    14 subsection {* Complete lattices *}
    15 (*FIXME Meet \<rightarrow> Inf *)
    16 consts
    17   Meet :: "'a::order set \<Rightarrow> 'a"
    18   Sup :: "'a::order set \<Rightarrow> 'a"
    19 
    20 defs Sup_def: "Sup A == Meet {b. \<forall>a \<in> A. a <= b}"
    21 
    22 definition
    23   SUP :: "('a \<Rightarrow> 'b::order) \<Rightarrow> 'b"  (binder "SUP " 10) where
    24   "SUP x. f x == Sup (f ` UNIV)"
    25 
    26 (*
    27 abbreviation
    28   bot :: "'a::order" where
    29   "bot == Sup {}"
    30 *)
    31 axclass comp_lat < order
    32   Meet_lower: "x \<in> A \<Longrightarrow> Meet A <= x"
    33   Meet_greatest: "(\<And>x. x \<in> A \<Longrightarrow> z <= x) \<Longrightarrow> z <= Meet A"
    34 
    35 theorem Sup_upper: "(x::'a::comp_lat) \<in> A \<Longrightarrow> x <= Sup A"
    36   by (auto simp: Sup_def intro: Meet_greatest)
    37 
    38 theorem Sup_least: "(\<And>x::'a::comp_lat. x \<in> A \<Longrightarrow> x <= z) \<Longrightarrow> Sup A <= z"
    39   by (auto simp: Sup_def intro: Meet_lower)
    40 
    41 text {* A complete lattice is a lattice *}
    42 
    43 lemma is_meet_Meet: "is_meet (\<lambda>(x::'a::comp_lat) y. Meet {x, y})"
    44   by (auto simp: is_meet_def intro: Meet_lower Meet_greatest)
    45 
    46 lemma is_join_Sup: "is_join (\<lambda>(x::'a::comp_lat) y. Sup {x, y})"
    47   by (auto simp: is_join_def intro: Sup_upper Sup_least)
    48 
    49 instance comp_lat < lorder
    50 proof
    51   from is_meet_Meet show "\<exists>m::'a\<Rightarrow>'a\<Rightarrow>'a. is_meet m" by iprover
    52   from is_join_Sup show "\<exists>j::'a\<Rightarrow>'a\<Rightarrow>'a. is_join j" by iprover
    53 qed
    54 
    55 subsubsection {* Properties *}
    56 
    57 lemma mono_join: "mono f \<Longrightarrow> join (f A) (f B) <= f (join A B)"
    58   by (auto simp add: mono_def)
    59 
    60 lemma mono_meet: "mono f \<Longrightarrow> f (meet A B) <= meet (f A) (f B)"
    61   by (auto simp add: mono_def)
    62 
    63 lemma Sup_insert[simp]: "Sup (insert (a::'a::comp_lat) A) = join a (Sup A)"
    64 apply(simp add:Sup_def)
    65 apply(rule order_antisym)
    66  apply(rule Meet_lower)
    67  apply(clarsimp)
    68  apply(rule le_joinI2)
    69  apply(rule Meet_greatest)
    70  apply blast
    71 apply simp
    72 apply rule
    73  apply(rule Meet_greatest)apply blast
    74 apply(rule Meet_greatest)
    75 apply(rule Meet_lower)
    76 apply blast
    77 done
    78 
    79 lemma bot_least[simp]: "Sup{} \<le> (x::'a::comp_lat)"
    80 apply(simp add: Sup_def)
    81 apply(rule Meet_lower)
    82 apply blast
    83 done
    84 (*
    85 lemma Meet_singleton[simp]: "Meet{a} = (a::'a::comp_lat)"
    86 apply(rule order_antisym)
    87  apply(simp add: Meet_lower)
    88 apply(rule Meet_greatest)
    89 apply(simp)
    90 done
    91 *)
    92 lemma le_SupI: "(l::'a::comp_lat) : M \<Longrightarrow> l \<le> Sup M"
    93 apply(simp add:Sup_def)
    94 apply(rule Meet_greatest)
    95 apply(simp)
    96 done
    97 
    98 lemma le_SUPI: "(l::'a::comp_lat) = M i \<Longrightarrow> l \<le> (SUP i. M i)"
    99 apply(simp add:SUP_def)
   100 apply(blast intro:le_SupI)
   101 done
   102 
   103 lemma Sup_leI: "(!!x. x:M \<Longrightarrow> x \<le> u) \<Longrightarrow> Sup M \<le> (u::'a::comp_lat)"
   104 apply(simp add:Sup_def)
   105 apply(rule Meet_lower)
   106 apply(blast)
   107 done
   108 
   109 
   110 lemma SUP_leI: "(!!i. M i \<le> u) \<Longrightarrow> (SUP i. M i) \<le> (u::'a::comp_lat)"
   111 apply(simp add:SUP_def)
   112 apply(blast intro!:Sup_leI)
   113 done
   114 
   115 lemma SUP_const[simp]: "(SUP i. M) = (M::'a::comp_lat)"
   116 by(simp add:SUP_def image_constant_conv join_absorp1)
   117 
   118 
   119 subsection {* Some instances of the type class of complete lattices *}
   120 
   121 subsubsection {* Booleans *}
   122 
   123 instance bool :: ord ..
   124 
   125 defs
   126   le_bool_def: "P <= Q == P \<longrightarrow> Q"
   127   less_bool_def: "P < Q == (P::bool) <= Q \<and> P \<noteq> Q"
   128 
   129 theorem le_boolI: "(P \<Longrightarrow> Q) \<Longrightarrow> P <= Q"
   130   by (simp add: le_bool_def)
   131 
   132 theorem le_boolI': "P \<longrightarrow> Q \<Longrightarrow> P <= Q"
   133   by (simp add: le_bool_def)
   134 
   135 theorem le_boolE: "P <= Q \<Longrightarrow> P \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R"
   136   by (simp add: le_bool_def)
   137 
   138 theorem le_boolD: "P <= Q \<Longrightarrow> P \<longrightarrow> Q"
   139   by (simp add: le_bool_def)
   140 
   141 instance bool :: order
   142   apply intro_classes
   143   apply (unfold le_bool_def less_bool_def)
   144   apply iprover+
   145   done
   146 
   147 defs Meet_bool_def: "Meet A == ALL x:A. x"
   148 
   149 instance bool :: comp_lat
   150   apply intro_classes
   151   apply (unfold Meet_bool_def)
   152   apply (iprover intro!: le_boolI elim: ballE)
   153   apply (iprover intro!: ballI le_boolI elim: ballE le_boolE)
   154   done
   155 
   156 theorem meet_bool_eq: "meet P Q = (P \<and> Q)"
   157   apply (rule order_antisym)
   158   apply (rule le_boolI)
   159   apply (rule conjI)
   160   apply (rule le_boolE)
   161   apply (rule meet_left_le)
   162   apply assumption+
   163   apply (rule le_boolE)
   164   apply (rule meet_right_le)
   165   apply assumption+
   166   apply (rule le_meetI)
   167   apply (rule le_boolI)
   168   apply (erule conjunct1)
   169   apply (rule le_boolI)
   170   apply (erule conjunct2)
   171   done
   172 
   173 theorem join_bool_eq: "join P Q = (P \<or> Q)"
   174   apply (rule order_antisym)
   175   apply (rule join_leI)
   176   apply (rule le_boolI)
   177   apply (erule disjI1)
   178   apply (rule le_boolI)
   179   apply (erule disjI2)
   180   apply (rule le_boolI)
   181   apply (erule disjE)
   182   apply (rule le_boolE)
   183   apply (rule join_left_le)
   184   apply assumption+
   185   apply (rule le_boolE)
   186   apply (rule join_right_le)
   187   apply assumption+
   188   done
   189 
   190 theorem Sup_bool_eq: "Sup A = (EX x:A. x)"
   191   apply (rule order_antisym)
   192   apply (rule Sup_least)
   193   apply (rule le_boolI)
   194   apply (erule bexI, assumption)
   195   apply (rule le_boolI)
   196   apply (erule bexE)
   197   apply (rule le_boolE)
   198   apply (rule Sup_upper)
   199   apply assumption+
   200   done
   201 
   202 subsubsection {* Functions *}
   203 
   204 instance "fun" :: (type, ord) ord ..
   205 
   206 defs
   207   le_fun_def: "f <= g == \<forall>x. f x <= g x"
   208   less_fun_def: "f < g == (f::'a\<Rightarrow>'b::ord) <= g \<and> f \<noteq> g"
   209 
   210 theorem le_funI: "(\<And>x. f x <= g x) \<Longrightarrow> f <= g"
   211   by (simp add: le_fun_def)
   212 
   213 theorem le_funE: "f <= g \<Longrightarrow> (f x <= g x \<Longrightarrow> P) \<Longrightarrow> P"
   214   by (simp add: le_fun_def)
   215 
   216 theorem le_funD: "f <= g \<Longrightarrow> f x <= g x"
   217   by (simp add: le_fun_def)
   218 
   219 text {*
   220 Handy introduction and elimination rules for @{text "\<le>"}
   221 on unary and binary predicates
   222 *}
   223 
   224 lemma predicate1I [intro]:
   225   assumes PQ: "\<And>x. P x \<Longrightarrow> Q x"
   226   shows "P \<le> Q"
   227   apply (rule le_funI)
   228   apply (rule le_boolI)
   229   apply (rule PQ)
   230   apply assumption
   231   done
   232 
   233 lemma predicate1D [elim]: "P \<le> Q \<Longrightarrow> P x \<Longrightarrow> Q x"
   234   apply (erule le_funE)
   235   apply (erule le_boolE)
   236   apply assumption+
   237   done
   238 
   239 lemma predicate2I [intro]:
   240   assumes PQ: "\<And>x y. P x y \<Longrightarrow> Q x y"
   241   shows "P \<le> Q"
   242   apply (rule le_funI)+
   243   apply (rule le_boolI)
   244   apply (rule PQ)
   245   apply assumption
   246   done
   247 
   248 lemma predicate2D [elim]: "P \<le> Q \<Longrightarrow> P x y \<Longrightarrow> Q x y"
   249   apply (erule le_funE)+
   250   apply (erule le_boolE)
   251   apply assumption+
   252   done
   253 
   254 instance "fun" :: (type, order) order
   255   apply intro_classes
   256   apply (rule le_funI)
   257   apply (rule order_refl)
   258   apply (rule le_funI)
   259   apply (erule le_funE)+
   260   apply (erule order_trans)
   261   apply assumption
   262   apply (rule ext)
   263   apply (erule le_funE)+
   264   apply (erule order_antisym)
   265   apply assumption
   266   apply (simp add: less_fun_def)
   267   done
   268 
   269 defs Meet_fun_def: "Meet A == (\<lambda>x. Meet {y. EX f:A. y = f x})"
   270 
   271 instance "fun" :: (type, comp_lat) comp_lat
   272   apply intro_classes
   273   apply (unfold Meet_fun_def)
   274   apply (rule le_funI)
   275   apply (rule Meet_lower)
   276   apply (rule CollectI)
   277   apply (rule bexI)
   278   apply (rule refl)
   279   apply assumption
   280   apply (rule le_funI)
   281   apply (rule Meet_greatest)
   282   apply (erule CollectE)
   283   apply (erule bexE)
   284   apply (iprover elim: le_funE)
   285   done
   286 
   287 theorem meet_fun_eq: "meet f g = (\<lambda>x. meet (f x) (g x))"
   288   apply (rule order_antisym)
   289   apply (rule le_funI)
   290   apply (rule le_meetI)
   291   apply (rule le_funD [OF meet_left_le])
   292   apply (rule le_funD [OF meet_right_le])
   293   apply (rule le_meetI)
   294   apply (rule le_funI)
   295   apply (rule meet_left_le)
   296   apply (rule le_funI)
   297   apply (rule meet_right_le)
   298   done
   299 
   300 theorem join_fun_eq: "join f g = (\<lambda>x. join (f x) (g x))"
   301   apply (rule order_antisym)
   302   apply (rule join_leI)
   303   apply (rule le_funI)
   304   apply (rule join_left_le)
   305   apply (rule le_funI)
   306   apply (rule join_right_le)
   307   apply (rule le_funI)
   308   apply (rule join_leI)
   309   apply (rule le_funD [OF join_left_le])
   310   apply (rule le_funD [OF join_right_le])
   311   done
   312 
   313 theorem Sup_fun_eq: "Sup A = (\<lambda>x. Sup {y::'a::comp_lat. EX f:A. y = f x})"
   314   apply (rule order_antisym)
   315   apply (rule Sup_least)
   316   apply (rule le_funI)
   317   apply (rule Sup_upper)
   318   apply fast
   319   apply (rule le_funI)
   320   apply (rule Sup_least)
   321   apply (erule CollectE)
   322   apply (erule bexE)
   323   apply (drule le_funD [OF Sup_upper])
   324   apply simp
   325   done
   326 
   327 subsubsection {* Sets *}
   328 
   329 defs Meet_set_def: "Meet S == \<Inter>S"
   330 
   331 instance set :: (type) comp_lat
   332   by intro_classes (auto simp add: Meet_set_def)
   333 
   334 theorem meet_set_eq: "meet A B = A \<inter> B"
   335   apply (rule subset_antisym)
   336   apply (rule Int_greatest)
   337   apply (rule meet_left_le)
   338   apply (rule meet_right_le)
   339   apply (rule le_meetI)
   340   apply (rule Int_lower1)
   341   apply (rule Int_lower2)
   342   done
   343 
   344 theorem join_set_eq: "join A B = A \<union> B"
   345   apply (rule subset_antisym)
   346   apply (rule join_leI)
   347   apply (rule Un_upper1)
   348   apply (rule Un_upper2)
   349   apply (rule Un_least)
   350   apply (rule join_left_le)
   351   apply (rule join_right_le)
   352   done
   353 
   354 theorem Sup_set_eq: "Sup S = \<Union>S"
   355   apply (rule subset_antisym)
   356   apply (rule Sup_least)
   357   apply (erule Union_upper)
   358   apply (rule Union_least)
   359   apply (erule Sup_upper)
   360   done
   361 
   362 
   363 subsection {* Least and greatest fixed points *}
   364 
   365 constdefs
   366   lfp :: "(('a::comp_lat) => 'a) => 'a"
   367   "lfp f == Meet {u. f u <= u}"    --{*least fixed point*}
   368 
   369   gfp :: "(('a::comp_lat) => 'a) => 'a"
   370   "gfp f == Sup {u. u <= f u}"    --{*greatest fixed point*}
   371 
   372 
   373 subsection{*Proof of Knaster-Tarski Theorem using @{term lfp}*}
   374 
   375 
   376 text{*@{term "lfp f"} is the least upper bound of 
   377       the set @{term "{u. f(u) \<le> u}"} *}
   378 
   379 lemma lfp_lowerbound: "f A \<le> A ==> lfp f \<le> A"
   380   by (auto simp add: lfp_def intro: Meet_lower)
   381 
   382 lemma lfp_greatest: "(!!u. f u \<le> u ==> A \<le> u) ==> A \<le> lfp f"
   383   by (auto simp add: lfp_def intro: Meet_greatest)
   384 
   385 lemma lfp_lemma2: "mono f ==> f (lfp f) \<le> lfp f"
   386   by (iprover intro: lfp_greatest order_trans monoD lfp_lowerbound)
   387 
   388 lemma lfp_lemma3: "mono f ==> lfp f \<le> f (lfp f)"
   389   by (iprover intro: lfp_lemma2 monoD lfp_lowerbound)
   390 
   391 lemma lfp_unfold: "mono f ==> lfp f = f (lfp f)"
   392   by (iprover intro: order_antisym lfp_lemma2 lfp_lemma3)
   393 
   394 subsection{*General induction rules for least fixed points*}
   395 
   396 theorem lfp_induct:
   397   assumes mono: "mono f" and ind: "f (meet (lfp f) P) <= P"
   398   shows "lfp f <= P"
   399 proof -
   400   have "meet (lfp f) P <= lfp f" by (rule meet_left_le)
   401   with mono have "f (meet (lfp f) P) <= f (lfp f)" ..
   402   also from mono have "f (lfp f) = lfp f" by (rule lfp_unfold [symmetric])
   403   finally have "f (meet (lfp f) P) <= lfp f" .
   404   from this and ind have "f (meet (lfp f) P) <= meet (lfp f) P" by (rule le_meetI)
   405   hence "lfp f <= meet (lfp f) P" by (rule lfp_lowerbound)
   406   also have "meet (lfp f) P <= P" by (rule meet_right_le)
   407   finally show ?thesis .
   408 qed
   409 
   410 lemma lfp_induct_set:
   411   assumes lfp: "a: lfp(f)"
   412       and mono: "mono(f)"
   413       and indhyp: "!!x. [| x: f(lfp(f) Int {x. P(x)}) |] ==> P(x)"
   414   shows "P(a)"
   415   by (rule lfp_induct [THEN subsetD, THEN CollectD, OF mono _ lfp])
   416     (auto simp: meet_set_eq intro: indhyp)
   417 
   418 
   419 text{*Version of induction for binary relations*}
   420 lemmas lfp_induct2 =  lfp_induct_set [of "(a,b)", split_format (complete)]
   421 
   422 
   423 lemma lfp_ordinal_induct: 
   424   assumes mono: "mono f"
   425   shows "[| !!S. P S ==> P(f S); !!M. !S:M. P S ==> P(Union M) |] 
   426          ==> P(lfp f)"
   427 apply(subgoal_tac "lfp f = Union{S. S \<subseteq> lfp f & P S}")
   428  apply (erule ssubst, simp) 
   429 apply(subgoal_tac "Union{S. S \<subseteq> lfp f & P S} \<subseteq> lfp f")
   430  prefer 2 apply blast
   431 apply(rule equalityI)
   432  prefer 2 apply assumption
   433 apply(drule mono [THEN monoD])
   434 apply (cut_tac mono [THEN lfp_unfold], simp)
   435 apply (rule lfp_lowerbound, auto) 
   436 done
   437 
   438 
   439 text{*Definition forms of @{text lfp_unfold} and @{text lfp_induct}, 
   440     to control unfolding*}
   441 
   442 lemma def_lfp_unfold: "[| h==lfp(f);  mono(f) |] ==> h = f(h)"
   443 by (auto intro!: lfp_unfold)
   444 
   445 lemma def_lfp_induct: 
   446     "[| A == lfp(f); mono(f);
   447         f (meet A P) \<le> P
   448      |] ==> A \<le> P"
   449   by (blast intro: lfp_induct)
   450 
   451 lemma def_lfp_induct_set: 
   452     "[| A == lfp(f);  mono(f);   a:A;                    
   453         !!x. [| x: f(A Int {x. P(x)}) |] ==> P(x)         
   454      |] ==> P(a)"
   455   by (blast intro: lfp_induct_set)
   456 
   457 (*Monotonicity of lfp!*)
   458 lemma lfp_mono: "(!!Z. f Z \<le> g Z) ==> lfp f \<le> lfp g"
   459   by (rule lfp_lowerbound [THEN lfp_greatest], blast intro: order_trans)
   460 
   461 
   462 subsection{*Proof of Knaster-Tarski Theorem using @{term gfp}*}
   463 
   464 
   465 text{*@{term "gfp f"} is the greatest lower bound of 
   466       the set @{term "{u. u \<le> f(u)}"} *}
   467 
   468 lemma gfp_upperbound: "X \<le> f X ==> X \<le> gfp f"
   469   by (auto simp add: gfp_def intro: Sup_upper)
   470 
   471 lemma gfp_least: "(!!u. u \<le> f u ==> u \<le> X) ==> gfp f \<le> X"
   472   by (auto simp add: gfp_def intro: Sup_least)
   473 
   474 lemma gfp_lemma2: "mono f ==> gfp f \<le> f (gfp f)"
   475   by (iprover intro: gfp_least order_trans monoD gfp_upperbound)
   476 
   477 lemma gfp_lemma3: "mono f ==> f (gfp f) \<le> gfp f"
   478   by (iprover intro: gfp_lemma2 monoD gfp_upperbound)
   479 
   480 lemma gfp_unfold: "mono f ==> gfp f = f (gfp f)"
   481   by (iprover intro: order_antisym gfp_lemma2 gfp_lemma3)
   482 
   483 subsection{*Coinduction rules for greatest fixed points*}
   484 
   485 text{*weak version*}
   486 lemma weak_coinduct: "[| a: X;  X \<subseteq> f(X) |] ==> a : gfp(f)"
   487 by (rule gfp_upperbound [THEN subsetD], auto)
   488 
   489 lemma weak_coinduct_image: "!!X. [| a : X; g`X \<subseteq> f (g`X) |] ==> g a : gfp f"
   490 apply (erule gfp_upperbound [THEN subsetD])
   491 apply (erule imageI)
   492 done
   493 
   494 lemma coinduct_lemma:
   495      "[| X \<le> f (join X (gfp f));  mono f |] ==> join X (gfp f) \<le> f (join X (gfp f))"
   496   apply (frule gfp_lemma2)
   497   apply (drule mono_join)
   498   apply (rule join_leI)
   499   apply assumption
   500   apply (rule order_trans)
   501   apply (rule order_trans)
   502   apply assumption
   503   apply (rule join_right_le)
   504   apply assumption
   505   done
   506 
   507 text{*strong version, thanks to Coen and Frost*}
   508 lemma coinduct_set: "[| mono(f);  a: X;  X \<subseteq> f(X Un gfp(f)) |] ==> a : gfp(f)"
   509 by (blast intro: weak_coinduct [OF _ coinduct_lemma, simplified join_set_eq])
   510 
   511 lemma coinduct: "[| mono(f); X \<le> f (join X (gfp f)) |] ==> X \<le> gfp(f)"
   512   apply (rule order_trans)
   513   apply (rule join_left_le)
   514   apply (erule gfp_upperbound [OF coinduct_lemma])
   515   apply assumption
   516   done
   517 
   518 lemma gfp_fun_UnI2: "[| mono(f);  a: gfp(f) |] ==> a: f(X Un gfp(f))"
   519 by (blast dest: gfp_lemma2 mono_Un)
   520 
   521 subsection{*Even Stronger Coinduction Rule, by Martin Coen*}
   522 
   523 text{* Weakens the condition @{term "X \<subseteq> f(X)"} to one expressed using both
   524   @{term lfp} and @{term gfp}*}
   525 
   526 lemma coinduct3_mono_lemma: "mono(f) ==> mono(%x. f(x) Un X Un B)"
   527 by (iprover intro: subset_refl monoI Un_mono monoD)
   528 
   529 lemma coinduct3_lemma:
   530      "[| X \<subseteq> f(lfp(%x. f(x) Un X Un gfp(f)));  mono(f) |]
   531       ==> lfp(%x. f(x) Un X Un gfp(f)) \<subseteq> f(lfp(%x. f(x) Un X Un gfp(f)))"
   532 apply (rule subset_trans)
   533 apply (erule coinduct3_mono_lemma [THEN lfp_lemma3])
   534 apply (rule Un_least [THEN Un_least])
   535 apply (rule subset_refl, assumption)
   536 apply (rule gfp_unfold [THEN equalityD1, THEN subset_trans], assumption)
   537 apply (rule monoD, assumption)
   538 apply (subst coinduct3_mono_lemma [THEN lfp_unfold], auto)
   539 done
   540 
   541 lemma coinduct3: 
   542   "[| mono(f);  a:X;  X \<subseteq> f(lfp(%x. f(x) Un X Un gfp(f))) |] ==> a : gfp(f)"
   543 apply (rule coinduct3_lemma [THEN [2] weak_coinduct])
   544 apply (rule coinduct3_mono_lemma [THEN lfp_unfold, THEN ssubst], auto)
   545 done
   546 
   547 
   548 text{*Definition forms of @{text gfp_unfold} and @{text coinduct}, 
   549     to control unfolding*}
   550 
   551 lemma def_gfp_unfold: "[| A==gfp(f);  mono(f) |] ==> A = f(A)"
   552 by (auto intro!: gfp_unfold)
   553 
   554 lemma def_coinduct:
   555      "[| A==gfp(f);  mono(f);  X \<le> f(join X A) |] ==> X \<le> A"
   556 by (iprover intro!: coinduct)
   557 
   558 lemma def_coinduct_set:
   559      "[| A==gfp(f);  mono(f);  a:X;  X \<subseteq> f(X Un A) |] ==> a: A"
   560 by (auto intro!: coinduct_set)
   561 
   562 (*The version used in the induction/coinduction package*)
   563 lemma def_Collect_coinduct:
   564     "[| A == gfp(%w. Collect(P(w)));  mono(%w. Collect(P(w)));   
   565         a: X;  !!z. z: X ==> P (X Un A) z |] ==>  
   566      a : A"
   567 apply (erule def_coinduct_set, auto) 
   568 done
   569 
   570 lemma def_coinduct3:
   571     "[| A==gfp(f); mono(f);  a:X;  X \<subseteq> f(lfp(%x. f(x) Un X Un A)) |] ==> a: A"
   572 by (auto intro!: coinduct3)
   573 
   574 text{*Monotonicity of @{term gfp}!*}
   575 lemma gfp_mono: "(!!Z. f Z \<le> g Z) ==> gfp f \<le> gfp g"
   576   by (rule gfp_upperbound [THEN gfp_least], blast intro: order_trans)
   577 
   578 
   579 
   580 ML
   581 {*
   582 val lfp_def = thm "lfp_def";
   583 val lfp_lowerbound = thm "lfp_lowerbound";
   584 val lfp_greatest = thm "lfp_greatest";
   585 val lfp_unfold = thm "lfp_unfold";
   586 val lfp_induct = thm "lfp_induct";
   587 val lfp_induct2 = thm "lfp_induct2";
   588 val lfp_ordinal_induct = thm "lfp_ordinal_induct";
   589 val def_lfp_unfold = thm "def_lfp_unfold";
   590 val def_lfp_induct = thm "def_lfp_induct";
   591 val def_lfp_induct_set = thm "def_lfp_induct_set";
   592 val lfp_mono = thm "lfp_mono";
   593 val gfp_def = thm "gfp_def";
   594 val gfp_upperbound = thm "gfp_upperbound";
   595 val gfp_least = thm "gfp_least";
   596 val gfp_unfold = thm "gfp_unfold";
   597 val weak_coinduct = thm "weak_coinduct";
   598 val weak_coinduct_image = thm "weak_coinduct_image";
   599 val coinduct = thm "coinduct";
   600 val gfp_fun_UnI2 = thm "gfp_fun_UnI2";
   601 val coinduct3 = thm "coinduct3";
   602 val def_gfp_unfold = thm "def_gfp_unfold";
   603 val def_coinduct = thm "def_coinduct";
   604 val def_Collect_coinduct = thm "def_Collect_coinduct";
   605 val def_coinduct3 = thm "def_coinduct3";
   606 val gfp_mono = thm "gfp_mono";
   607 val le_funI = thm "le_funI";
   608 val le_boolI = thm "le_boolI";
   609 val le_boolI' = thm "le_boolI'";
   610 val meet_fun_eq = thm "meet_fun_eq";
   611 val meet_bool_eq = thm "meet_bool_eq";
   612 val le_funE = thm "le_funE";
   613 val le_boolE = thm "le_boolE";
   614 val le_boolD = thm "le_boolD";
   615 val le_bool_def = thm "le_bool_def";
   616 val le_fun_def = thm "le_fun_def";
   617 *}
   618 
   619 end