src/HOL/FixedPoint.thy

proper signature constraint;
minor tuning;

(*  Title:      HOL/FixedPoint.thy
    ID:         $Id$
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Author:     Stefan Berghofer, TU Muenchen
    Copyright   1992  University of Cambridge
*)

header {* Fixed Points and the Knaster-Tarski Theorem*}

theory FixedPoint
imports Lattices
begin

subsection {* Least and greatest fixed points *}

definition
  lfp :: "('a\<Colon>complete_lattice \<Rightarrow> 'a) \<Rightarrow> 'a" where
  "lfp f = Inf {u. f u \<le> u}"    --{*least fixed point*}

definition
  gfp :: "('a\<Colon>complete_lattice \<Rightarrow> 'a) \<Rightarrow> 'a" where
  "gfp f = Sup {u. u \<le> f u}"    --{*greatest fixed point*}


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) \<le> u}"} *}

lemma lfp_lowerbound: "f A \<le> A ==> lfp f \<le> A"
  by (auto simp add: lfp_def intro: Inf_lower)

lemma lfp_greatest: "(!!u. f u \<le> u ==> A \<le> u) ==> A \<le> lfp f"
  by (auto simp add: lfp_def intro: Inf_greatest)

lemma lfp_lemma2: "mono f ==> f (lfp f) \<le> lfp f"
  by (iprover intro: lfp_greatest order_trans monoD lfp_lowerbound)

lemma lfp_lemma3: "mono f ==> lfp f \<le> f (lfp f)"
  by (iprover intro: lfp_lemma2 monoD lfp_lowerbound)

lemma lfp_unfold: "mono f ==> lfp f = f (lfp f)"
  by (iprover intro: order_antisym lfp_lemma2 lfp_lemma3)

lemma lfp_const: "lfp (\<lambda>x. t) = t"
  by (rule lfp_unfold) (simp add:mono_def)


subsection {* General induction rules for least fixed points *}

theorem lfp_induct:
  assumes mono: "mono f" and ind: "f (inf (lfp f) P) <= P"
  shows "lfp f <= P"
proof -
  have "inf (lfp f) P <= lfp f" by (rule inf_le1)
  with mono have "f (inf (lfp f) P) <= f (lfp f)" ..
  also from mono have "f (lfp f) = lfp f" by (rule lfp_unfold [symmetric])
  finally have "f (inf (lfp f) P) <= lfp f" .
  from this and ind have "f (inf (lfp f) P) <= inf (lfp f) P" by (rule le_infI)
  hence "lfp f <= inf (lfp f) P" by (rule lfp_lowerbound)
  also have "inf (lfp f) P <= P" by (rule inf_le2)
  finally show ?thesis .
qed

lemma lfp_induct_set:
  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)"
  by (rule lfp_induct [THEN subsetD, THEN CollectD, OF mono _ lfp])
    (auto simp: inf_set_eq intro: indhyp)

lemma lfp_ordinal_induct: 
  assumes mono: "mono f"
  and P_f: "!!S. P S ==> P(f S)"
  and P_Union: "!!M. !S:M. P S ==> P(Union M)"
  shows "P(lfp f)"
proof -
  let ?M = "{S. S \<subseteq> lfp f & P S}"
  have "P (Union ?M)" using P_Union by simp
  also have "Union ?M = lfp f"
  proof
    show "Union ?M \<subseteq> lfp f" by blast
    hence "f (Union ?M) \<subseteq> f (lfp f)" by (rule mono [THEN monoD])
    hence "f (Union ?M) \<subseteq> lfp f" using mono [THEN lfp_unfold] by simp
    hence "f (Union ?M) \<in> ?M" using P_f P_Union by simp
    hence "f (Union ?M) \<subseteq> Union ?M" by (rule Union_upper)
    thus "lfp f \<subseteq> Union ?M" by (rule lfp_lowerbound)
  qed
  finally show ?thesis .
qed


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);
        f (inf A P) \<le> P
     |] ==> A \<le> P"
  by (blast intro: lfp_induct)

lemma def_lfp_induct_set: 
    "[| A == lfp(f);  mono(f);   a:A;                    
        !!x. [| x: f(A Int {x. P(x)}) |] ==> P(x)         
     |] ==> P(a)"
  by (blast intro: lfp_induct_set)

(*Monotonicity of lfp!*)
lemma lfp_mono: "(!!Z. f Z \<le> g Z) ==> lfp f \<le> lfp g"
  by (rule lfp_lowerbound [THEN lfp_greatest], blast intro: order_trans)


subsection {* Proof of Knaster-Tarski Theorem using @{term gfp} *}

text{*@{term "gfp f"} is the greatest lower bound of 
      the set @{term "{u. u \<le> f(u)}"} *}

lemma gfp_upperbound: "X \<le> f X ==> X \<le> gfp f"
  by (auto simp add: gfp_def intro: Sup_upper)

lemma gfp_least: "(!!u. u \<le> f u ==> u \<le> X) ==> gfp f \<le> X"
  by (auto simp add: gfp_def intro: Sup_least)

lemma gfp_lemma2: "mono f ==> gfp f \<le> f (gfp f)"
  by (iprover intro: gfp_least order_trans monoD gfp_upperbound)

lemma gfp_lemma3: "mono f ==> f (gfp f) \<le> gfp f"
  by (iprover intro: gfp_lemma2 monoD gfp_upperbound)

lemma gfp_unfold: "mono f ==> gfp f = f (gfp f)"
  by (iprover intro: order_antisym gfp_lemma2 gfp_lemma3)


subsection {* Coinduction rules for greatest fixed points *}

text{*weak version*}
lemma weak_coinduct: "[| a: X;  X \<subseteq> f(X) |] ==> a : gfp(f)"
by (rule gfp_upperbound [THEN subsetD], auto)

lemma weak_coinduct_image: "!!X. [| a : X; g`X \<subseteq> f (g`X) |] ==> g a : gfp f"
apply (erule gfp_upperbound [THEN subsetD])
apply (erule imageI)
done

lemma coinduct_lemma:
     "[| X \<le> f (sup X (gfp f));  mono f |] ==> sup X (gfp f) \<le> f (sup X (gfp f))"
  apply (frule gfp_lemma2)
  apply (drule mono_sup)
  apply (rule le_supI)
  apply assumption
  apply (rule order_trans)
  apply (rule order_trans)
  apply assumption
  apply (rule sup_ge2)
  apply assumption
  done

text{*strong version, thanks to Coen and Frost*}
lemma coinduct_set: "[| mono(f);  a: X;  X \<subseteq> f(X Un gfp(f)) |] ==> a : gfp(f)"
by (blast intro: weak_coinduct [OF _ coinduct_lemma, simplified sup_set_eq])

lemma coinduct: "[| mono(f); X \<le> f (sup X (gfp f)) |] ==> X \<le> gfp(f)"
  apply (rule order_trans)
  apply (rule sup_ge1)
  apply (erule gfp_upperbound [OF coinduct_lemma])
  apply assumption
  done

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 \<subseteq> 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 \<subseteq> f(lfp(%x. f(x) Un X Un gfp(f)));  mono(f) |]
      ==> lfp(%x. f(x) Un X Un gfp(f)) \<subseteq> 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 \<subseteq> 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);  X \<le> f(sup X A) |] ==> X \<le> A"
by (iprover intro!: coinduct)

lemma def_coinduct_set:
     "[| A==gfp(f);  mono(f);  a:X;  X \<subseteq> f(X Un A) |] ==> a: A"
by (auto intro!: coinduct_set)

(*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_set, auto) 
done

lemma def_coinduct3:
    "[| A==gfp(f); mono(f);  a:X;  X \<subseteq> 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 \<le> g Z) ==> gfp f \<le> gfp g"
  by (rule gfp_upperbound [THEN gfp_least], blast intro: order_trans)

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_ordinal_induct = thm "lfp_ordinal_induct";
val def_lfp_unfold = thm "def_lfp_unfold";
val def_lfp_induct = thm "def_lfp_induct";
val def_lfp_induct_set = thm "def_lfp_induct_set";
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";
val le_funI = thm "le_funI";
val le_boolI = thm "le_boolI";
val le_boolI' = thm "le_boolI'";
val inf_fun_eq = thm "inf_fun_eq";
val inf_bool_eq = thm "inf_bool_eq";
val le_funE = thm "le_funE";
val le_funD = thm "le_funD";
val le_boolE = thm "le_boolE";
val le_boolD = thm "le_boolD";
val le_bool_def = thm "le_bool_def";
val le_fun_def = thm "le_fun_def";
*}

end