# HG changeset patch # User avigad # Date 1123073271 -7200 # Node ID cffca870816ab5079129963ac832c17f583577ae # Parent 69c415d4488335ebb60575fa36ce4e947b6e37b4 combined Lfp and Gfp to FixedPoint diff -r 69c415d44883 -r cffca870816a src/HOL/FixedPoint.thy --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/HOL/FixedPoint.thy Wed Aug 03 14:47:51 2005 +0200 @@ -0,0 +1,220 @@ +(* Title: HOL/FixedPoint.thy + ID: $Id$ + 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 (rules intro: lfp_greatest subset_trans monoD lfp_lowerbound) + +lemma lfp_lemma3: "mono(f) ==> lfp(f) \ f(lfp(f))" +by (rules intro: lfp_lemma2 monoD lfp_lowerbound) + +lemma lfp_unfold: "mono(f) ==> lfp(f) = f(lfp(f))" +by (rules 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 (rules intro: gfp_least subset_trans monoD gfp_upperbound) + +lemma gfp_lemma3: "mono(f) ==> f(gfp(f)) \ gfp(f)" +by (rules intro: gfp_lemma2 monoD gfp_upperbound) + +lemma gfp_unfold: "mono(f) ==> gfp(f) = f(gfp(f))" +by (rules 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 (rules 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