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src/CCL/Lfp.thy

author | wenzelm |

Mon, 07 Oct 2013 21:24:44 +0200 | |

changeset 54313 | da2e6282a4f5 |

parent 32153 | a0e57fb1b930 |

child 58889 | 5b7a9633cfa8 |

permissions | -rw-r--r-- |

native executable even for Linux, to avoid surprises with file managers opening executable script as text file;

(* Title: CCL/Lfp.thy Author: Lawrence C Paulson, Cambridge University Computer Laboratory Copyright 1992 University of Cambridge *) header {* The Knaster-Tarski Theorem *} theory Lfp imports Set begin definition lfp :: "['a set=>'a set] => 'a set" where -- "least fixed point" "lfp(f) == Inter({u. f(u) <= u})" (* lfp(f) is the greatest lower bound of {u. f(u) <= u} *) lemma lfp_lowerbound: "[| f(A) <= A |] ==> lfp(f) <= A" unfolding lfp_def by blast lemma lfp_greatest: "[| !!u. f(u) <= u ==> A<=u |] ==> A <= lfp(f)" unfolding lfp_def by blast lemma lfp_lemma2: "mono(f) ==> f(lfp(f)) <= lfp(f)" by (rule lfp_greatest, rule subset_trans, drule monoD, rule lfp_lowerbound, assumption+) lemma lfp_lemma3: "mono(f) ==> lfp(f) <= f(lfp(f))" by (rule lfp_lowerbound, frule monoD, drule lfp_lemma2, assumption+) lemma lfp_Tarski: "mono(f) ==> lfp(f) = f(lfp(f))" by (rule equalityI lfp_lemma2 lfp_lemma3 | assumption)+ (*** General induction rule for least fixed points ***) lemma induct: assumes lfp: "a: lfp(f)" and mono: "mono(f)" and indhyp: "!!x. [| x: f(lfp(f) Int {x. P(x)}) |] ==> P(x)" shows "P(a)" apply (rule_tac a = a in Int_lower2 [THEN subsetD, THEN CollectD]) apply (rule lfp [THEN [2] lfp_lowerbound [THEN subsetD]]) apply (rule Int_greatest, rule subset_trans, rule Int_lower1 [THEN mono [THEN monoD]], rule mono [THEN lfp_lemma2], rule CollectI [THEN subsetI], rule indhyp, assumption) done (** Definition forms of lfp_Tarski and induct, to control unfolding **) lemma def_lfp_Tarski: "[| h==lfp(f); mono(f) |] ==> h = f(h)" apply unfold apply (drule lfp_Tarski) apply assumption done lemma def_induct: "[| A == lfp(f); a:A; mono(f); !!x. [| x: f(A Int {x. P(x)}) |] ==> P(x) |] ==> P(a)" apply (rule induct [of concl: P a]) apply simp apply assumption apply blast done (*Monotonicity of lfp!*) lemma lfp_mono: "[| mono(g); !!Z. f(Z)<=g(Z) |] ==> lfp(f) <= lfp(g)" apply (rule lfp_lowerbound) apply (rule subset_trans) apply (erule meta_spec) apply (erule lfp_lemma2) done end