src/CCL/Lfp.thy
author wenzelm
Sat Jun 14 23:52:51 2008 +0200 (2008-06-14)
changeset 27221 31328dc30196
parent 21404 eb85850d3eb7
child 32153 a0e57fb1b930
permissions -rw-r--r--
proper context for tactics derived from res_inst_tac;
     1 (*  Title:      CCL/Lfp.thy
     2     ID:         $Id$
     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     4     Copyright   1992  University of Cambridge
     5 *)
     6 
     7 header {* The Knaster-Tarski Theorem *}
     8 
     9 theory Lfp
    10 imports Set
    11 begin
    12 
    13 definition
    14   lfp :: "['a set=>'a set] => 'a set" where -- "least fixed point"
    15   "lfp(f) == Inter({u. f(u) <= u})"
    16 
    17 (* lfp(f) is the greatest lower bound of {u. f(u) <= u} *)
    18 
    19 lemma lfp_lowerbound: "[| f(A) <= A |] ==> lfp(f) <= A"
    20   unfolding lfp_def by blast
    21 
    22 lemma lfp_greatest: "[| !!u. f(u) <= u ==> A<=u |] ==> A <= lfp(f)"
    23   unfolding lfp_def by blast
    24 
    25 lemma lfp_lemma2: "mono(f) ==> f(lfp(f)) <= lfp(f)"
    26   by (rule lfp_greatest, rule subset_trans, drule monoD, rule lfp_lowerbound, assumption+)
    27 
    28 lemma lfp_lemma3: "mono(f) ==> lfp(f) <= f(lfp(f))"
    29   by (rule lfp_lowerbound, frule monoD, drule lfp_lemma2, assumption+)
    30 
    31 lemma lfp_Tarski: "mono(f) ==> lfp(f) = f(lfp(f))"
    32   by (rule equalityI lfp_lemma2 lfp_lemma3 | assumption)+
    33 
    34 
    35 (*** General induction rule for least fixed points ***)
    36 
    37 lemma induct:
    38   assumes lfp: "a: lfp(f)"
    39     and mono: "mono(f)"
    40     and indhyp: "!!x. [| x: f(lfp(f) Int {x. P(x)}) |] ==> P(x)"
    41   shows "P(a)"
    42   apply (rule_tac a = a in Int_lower2 [THEN subsetD, THEN CollectD])
    43   apply (rule lfp [THEN [2] lfp_lowerbound [THEN subsetD]])
    44   apply (rule Int_greatest, rule subset_trans, rule Int_lower1 [THEN mono [THEN monoD]],
    45     rule mono [THEN lfp_lemma2], rule CollectI [THEN subsetI], rule indhyp, assumption)
    46   done
    47 
    48 (** Definition forms of lfp_Tarski and induct, to control unfolding **)
    49 
    50 lemma def_lfp_Tarski: "[| h==lfp(f);  mono(f) |] ==> h = f(h)"
    51   apply unfold
    52   apply (drule lfp_Tarski)
    53   apply assumption
    54   done
    55 
    56 lemma def_induct:
    57   "[| A == lfp(f);  a:A;  mono(f);                     
    58     !!x. [| x: f(A Int {x. P(x)}) |] ==> P(x)         
    59   |] ==> P(a)"
    60   apply (rule induct [of concl: P a])
    61     apply simp
    62    apply assumption
    63   apply blast
    64   done
    65 
    66 (*Monotonicity of lfp!*)
    67 lemma lfp_mono: "[| mono(g);  !!Z. f(Z)<=g(Z) |] ==> lfp(f) <= lfp(g)"
    68   apply (rule lfp_lowerbound)
    69   apply (rule subset_trans)
    70    apply (erule meta_spec)
    71   apply (erule lfp_lemma2)
    72   done
    73 
    74 end