| author | haftmann |
| Thu, 08 Jul 2010 16:19:24 +0200 | |
| changeset 37744 | 3daaf23b9ab4 |
| parent 32153 | a0e57fb1b930 |
| child 58889 | 5b7a9633cfa8 |
| permissions | -rw-r--r-- |
| 17456 | 1 |
(* Title: CCL/Lfp.thy |
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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Copyright 1992 University of Cambridge |
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*) |
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header {* The Knaster-Tarski Theorem *}
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theory Lfp |
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imports Set |
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begin |
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definition |
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more robust syntax for definition/abbreviation/notation;
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lfp :: "['a set=>'a set] => 'a set" where -- "least fixed point" |
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"lfp(f) == Inter({u. f(u) <= u})"
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(* lfp(f) is the greatest lower bound of {u. f(u) <= u} *)
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lemma lfp_lowerbound: "[| f(A) <= A |] ==> lfp(f) <= A" |
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unfolding lfp_def by blast |
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lemma lfp_greatest: "[| !!u. f(u) <= u ==> A<=u |] ==> A <= lfp(f)" |
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unfolding lfp_def by blast |
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lemma lfp_lemma2: "mono(f) ==> f(lfp(f)) <= lfp(f)" |
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by (rule lfp_greatest, rule subset_trans, drule monoD, rule lfp_lowerbound, assumption+) |
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lemma lfp_lemma3: "mono(f) ==> lfp(f) <= f(lfp(f))" |
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by (rule lfp_lowerbound, frule monoD, drule lfp_lemma2, assumption+) |
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lemma lfp_Tarski: "mono(f) ==> lfp(f) = f(lfp(f))" |
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by (rule equalityI lfp_lemma2 lfp_lemma3 | assumption)+ |
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(*** General induction rule for least fixed points ***) |
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lemma induct: |
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assumes lfp: "a: lfp(f)" |
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and mono: "mono(f)" |
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and indhyp: "!!x. [| x: f(lfp(f) Int {x. P(x)}) |] ==> P(x)"
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shows "P(a)" |
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apply (rule_tac a = a in Int_lower2 [THEN subsetD, THEN CollectD]) |
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apply (rule lfp [THEN [2] lfp_lowerbound [THEN subsetD]]) |
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apply (rule Int_greatest, rule subset_trans, rule Int_lower1 [THEN mono [THEN monoD]], |
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rule mono [THEN lfp_lemma2], rule CollectI [THEN subsetI], rule indhyp, assumption) |
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done |
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(** Definition forms of lfp_Tarski and induct, to control unfolding **) |
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lemma def_lfp_Tarski: "[| h==lfp(f); mono(f) |] ==> h = f(h)" |
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apply unfold |
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apply (drule lfp_Tarski) |
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apply assumption |
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done |
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lemma def_induct: |
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"[| A == lfp(f); a:A; mono(f); |
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!!x. [| x: f(A Int {x. P(x)}) |] ==> P(x)
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|] ==> P(a)" |
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apply (rule induct [of concl: P a]) |
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apply simp |
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apply assumption |
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apply blast |
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done |
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(*Monotonicity of lfp!*) |
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lemma lfp_mono: "[| mono(g); !!Z. f(Z)<=g(Z) |] ==> lfp(f) <= lfp(g)" |
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apply (rule lfp_lowerbound) |
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apply (rule subset_trans) |
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apply (erule meta_spec) |
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apply (erule lfp_lemma2) |
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done |
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end |