(* Title: HOL/FixedPoint.thy ID: $Id: FixedPoint.thy,v 1.2 2005/09/22 21:56:15 nipkow Exp $ Author: Lawrence C Paulson, Cambridge University Computer Laboratory Copyright 1992 University of Cambridge *) header{* Fixed Points and the Knaster-Tarski Theorem*} theory FixedPoint imports Product_Type begin constdefs lfp :: "['a set \ 'a set] \ 'a set" "lfp(f) == Inter({u. f(u) \ u})" --{*least fixed point*} gfp :: "['a set=>'a set] => 'a set" "gfp(f) == Union({u. u \ f(u)})" subsection{*Proof of Knaster-Tarski Theorem using @{term lfp}*} text{*@{term "lfp f"} is the least upper bound of the set @{term "{u. f(u) \ u}"} *} lemma lfp_lowerbound: "f(A) \ A ==> lfp(f) \ A" by (auto simp add: lfp_def) lemma lfp_greatest: "[| !!u. f(u) \ u ==> A\u |] ==> A \ lfp(f)" by (auto simp add: lfp_def) lemma lfp_lemma2: "mono(f) ==> f(lfp(f)) \ lfp(f)" by (iprover intro: lfp_greatest subset_trans monoD lfp_lowerbound) lemma lfp_lemma3: "mono(f) ==> lfp(f) \ f(lfp(f))" by (iprover intro: lfp_lemma2 monoD lfp_lowerbound) lemma lfp_unfold: "mono(f) ==> lfp(f) = f(lfp(f))" by (iprover intro: equalityI lfp_lemma2 lfp_lemma3) subsection{*General induction rules for greatest fixed points*} lemma lfp_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) apply (rule subset_trans [OF Int_lower1 [THEN mono [THEN monoD]] mono [THEN lfp_lemma2]]) apply (blast intro: indhyp) done text{*Version of induction for binary relations*} lemmas lfp_induct2 = lfp_induct [of "(a,b)", split_format (complete)] lemma lfp_ordinal_induct: assumes mono: "mono f" shows "[| !!S. P S ==> P(f S); !!M. !S:M. P S ==> P(Union M) |] ==> P(lfp f)" apply(subgoal_tac "lfp f = Union{S. S \ lfp f & P S}") apply (erule ssubst, simp) apply(subgoal_tac "Union{S. S \ lfp f & P S} \ lfp f") prefer 2 apply blast apply(rule equalityI) prefer 2 apply assumption apply(drule mono [THEN monoD]) apply (cut_tac mono [THEN lfp_unfold], simp) apply (rule lfp_lowerbound, auto) done text{*Definition forms of @{text lfp_unfold} and @{text lfp_induct}, to control unfolding*} lemma def_lfp_unfold: "[| h==lfp(f); mono(f) |] ==> h = f(h)" by (auto intro!: lfp_unfold) lemma def_lfp_induct: "[| A == lfp(f); mono(f); a:A; !!x. [| x: f(A Int {x. P(x)}) |] ==> P(x) |] ==> P(a)" by (blast intro: lfp_induct) (*Monotonicity of lfp!*) lemma lfp_mono: "[| !!Z. f(Z)\g(Z) |] ==> lfp(f) \ lfp(g)" by (rule lfp_lowerbound [THEN lfp_greatest], blast) subsection{*Proof of Knaster-Tarski Theorem using @{term gfp}*} text{*@{term "gfp f"} is the greatest lower bound of the set @{term "{u. u \ f(u)}"} *} lemma gfp_upperbound: "[| X \ f(X) |] ==> X \ gfp(f)" by (auto simp add: gfp_def) lemma gfp_least: "[| !!u. u \ f(u) ==> u\X |] ==> gfp(f) \ X" by (auto simp add: gfp_def) lemma gfp_lemma2: "mono(f) ==> gfp(f) \ f(gfp(f))" by (iprover intro: gfp_least subset_trans monoD gfp_upperbound) lemma gfp_lemma3: "mono(f) ==> f(gfp(f)) \ gfp(f)" by (iprover intro: gfp_lemma2 monoD gfp_upperbound) lemma gfp_unfold: "mono(f) ==> gfp(f) = f(gfp(f))" by (iprover intro: equalityI gfp_lemma2 gfp_lemma3) subsection{*Coinduction rules for greatest fixed points*} text{*weak version*} lemma weak_coinduct: "[| a: X; X \ f(X) |] ==> a : gfp(f)" by (rule gfp_upperbound [THEN subsetD], auto) lemma weak_coinduct_image: "!!X. [| a : X; g`X \ f (g`X) |] ==> g a : gfp f" apply (erule gfp_upperbound [THEN subsetD]) apply (erule imageI) done lemma coinduct_lemma: "[| X \ f(X Un gfp(f)); mono(f) |] ==> X Un gfp(f) \ f(X Un gfp(f))" by (blast dest: gfp_lemma2 mono_Un) text{*strong version, thanks to Coen and Frost*} lemma coinduct: "[| mono(f); a: X; X \ f(X Un gfp(f)) |] ==> a : gfp(f)" by (blast intro: weak_coinduct [OF _ coinduct_lemma]) lemma gfp_fun_UnI2: "[| mono(f); a: gfp(f) |] ==> a: f(X Un gfp(f))" by (blast dest: gfp_lemma2 mono_Un) subsection{*Even Stronger Coinduction Rule, by Martin Coen*} text{* Weakens the condition @{term "X \ f(X)"} to one expressed using both @{term lfp} and @{term gfp}*} lemma coinduct3_mono_lemma: "mono(f) ==> mono(%x. f(x) Un X Un B)" by (iprover intro: subset_refl monoI Un_mono monoD) lemma coinduct3_lemma: "[| X \ f(lfp(%x. f(x) Un X Un gfp(f))); mono(f) |] ==> lfp(%x. f(x) Un X Un gfp(f)) \ f(lfp(%x. f(x) Un X Un gfp(f)))" apply (rule subset_trans) apply (erule coinduct3_mono_lemma [THEN lfp_lemma3]) apply (rule Un_least [THEN Un_least]) apply (rule subset_refl, assumption) apply (rule gfp_unfold [THEN equalityD1, THEN subset_trans], assumption) apply (rule monoD, assumption) apply (subst coinduct3_mono_lemma [THEN lfp_unfold], auto) done lemma coinduct3: "[| mono(f); a:X; X \ f(lfp(%x. f(x) Un X Un gfp(f))) |] ==> a : gfp(f)" apply (rule coinduct3_lemma [THEN [2] weak_coinduct]) apply (rule coinduct3_mono_lemma [THEN lfp_unfold, THEN ssubst], auto) done text{*Definition forms of @{text gfp_unfold} and @{text coinduct}, to control unfolding*} lemma def_gfp_unfold: "[| A==gfp(f); mono(f) |] ==> A = f(A)" by (auto intro!: gfp_unfold) lemma def_coinduct: "[| A==gfp(f); mono(f); a:X; X \ f(X Un A) |] ==> a: A" by (auto intro!: coinduct) (*The version used in the induction/coinduction package*) lemma def_Collect_coinduct: "[| A == gfp(%w. Collect(P(w))); mono(%w. Collect(P(w))); a: X; !!z. z: X ==> P (X Un A) z |] ==> a : A" apply (erule def_coinduct, auto) done lemma def_coinduct3: "[| A==gfp(f); mono(f); a:X; X \ f(lfp(%x. f(x) Un X Un A)) |] ==> a: A" by (auto intro!: coinduct3) text{*Monotonicity of @{term gfp}!*} lemma gfp_mono: "[| !!Z. f(Z)\g(Z) |] ==> gfp(f) \ gfp(g)" by (rule gfp_upperbound [THEN gfp_least], blast) ML {* val lfp_def = thm "lfp_def"; val lfp_lowerbound = thm "lfp_lowerbound"; val lfp_greatest = thm "lfp_greatest"; val lfp_unfold = thm "lfp_unfold"; val lfp_induct = thm "lfp_induct"; val lfp_induct2 = thm "lfp_induct2"; val lfp_ordinal_induct = thm "lfp_ordinal_induct"; val def_lfp_unfold = thm "def_lfp_unfold"; val def_lfp_induct = thm "def_lfp_induct"; val lfp_mono = thm "lfp_mono"; val gfp_def = thm "gfp_def"; val gfp_upperbound = thm "gfp_upperbound"; val gfp_least = thm "gfp_least"; val gfp_unfold = thm "gfp_unfold"; val weak_coinduct = thm "weak_coinduct"; val weak_coinduct_image = thm "weak_coinduct_image"; val coinduct = thm "coinduct"; val gfp_fun_UnI2 = thm "gfp_fun_UnI2"; val coinduct3 = thm "coinduct3"; val def_gfp_unfold = thm "def_gfp_unfold"; val def_coinduct = thm "def_coinduct"; val def_Collect_coinduct = thm "def_Collect_coinduct"; val def_coinduct3 = thm "def_coinduct3"; val gfp_mono = thm "gfp_mono"; *} end