(* 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
lemma lfp_lowerbound:
f A ⊆ A ==> lfp f ⊆ A
lemma lfp_greatest:
(!!u. f u ⊆ u ==> A ⊆ u) ==> A ⊆ lfp f
lemma lfp_lemma2:
mono f ==> f (lfp f) ⊆ lfp f
lemma lfp_lemma3:
mono f ==> lfp f ⊆ f (lfp f)
lemma lfp_unfold:
mono f ==> lfp f = f (lfp f)
lemma lfp_induct:
[| a ∈ lfp f; mono f; !!x. x ∈ f (lfp f ∩ {x. P x}) ==> P x |] ==> P a
lemmas lfp_induct2:
[| (a, b) ∈ lfp f; mono f; !!a b. (a, b) ∈ f (lfp f ∩ {(x, y). P x y}) ==> P a b |] ==> P a b
lemmas lfp_induct2:
[| (a, b) ∈ lfp f; mono f; !!a b. (a, b) ∈ f (lfp f ∩ {(x, y). P x y}) ==> P a b |] ==> P a b
lemma lfp_ordinal_induct:
[| mono f; !!S. P S ==> P (f S); !!M. ∀S∈M. P S ==> P (Union M) |] ==> P (lfp f)
lemma def_lfp_unfold:
[| h == lfp f; mono f |] ==> h = f h
lemma def_lfp_induct:
[| A == lfp f; mono f; a ∈ A; !!x. x ∈ f (A ∩ {x. P x}) ==> P x |] ==> P a
lemma lfp_mono:
(!!Z. f Z ⊆ g Z) ==> lfp f ⊆ lfp g
lemma gfp_upperbound:
X ⊆ f X ==> X ⊆ gfp f
lemma gfp_least:
(!!u. u ⊆ f u ==> u ⊆ X) ==> gfp f ⊆ X
lemma gfp_lemma2:
mono f ==> gfp f ⊆ f (gfp f)
lemma gfp_lemma3:
mono f ==> f (gfp f) ⊆ gfp f
lemma gfp_unfold:
mono f ==> gfp f = f (gfp f)
lemma weak_coinduct:
[| a ∈ X; X ⊆ f X |] ==> a ∈ gfp f
lemma weak_coinduct_image:
[| a ∈ X; g ` X ⊆ f (g ` X) |] ==> g a ∈ gfp f
lemma coinduct_lemma:
[| X ⊆ f (X ∪ gfp f); mono f |] ==> X ∪ gfp f ⊆ f (X ∪ gfp f)
lemma coinduct:
[| mono f; a ∈ X; X ⊆ f (X ∪ gfp f) |] ==> a ∈ gfp f
lemma gfp_fun_UnI2:
[| mono f; a ∈ gfp f |] ==> a ∈ f (X ∪ gfp f)
lemma coinduct3_mono_lemma:
mono f ==> mono (%x. f x ∪ X ∪ B)
lemma coinduct3_lemma:
[| X ⊆ f (lfp (%x. f x ∪ X ∪ gfp f)); mono f |] ==> lfp (%x. f x ∪ X ∪ gfp f) ⊆ f (lfp (%x. f x ∪ X ∪ gfp f))
lemma coinduct3:
[| mono f; a ∈ X; X ⊆ f (lfp (%x. f x ∪ X ∪ gfp f)) |] ==> a ∈ gfp f
lemma def_gfp_unfold:
[| A == gfp f; mono f |] ==> A = f A
lemma def_coinduct:
[| A == gfp f; mono f; a ∈ X; X ⊆ f (X ∪ A) |] ==> a ∈ A
lemma def_Collect_coinduct:
[| A == gfp (%w. Collect (P w)); mono (%w. Collect (P w)); a ∈ X; !!z. z ∈ X ==> P (X ∪ A) z |] ==> a ∈ A
lemma def_coinduct3:
[| A == gfp f; mono f; a ∈ X; X ⊆ f (lfp (%x. f x ∪ X ∪ A)) |] ==> a ∈ A
lemma gfp_mono:
(!!Z. f Z ⊆ g Z) ==> gfp f ⊆ gfp g