(* Title: ZF/fixedpt.ML
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1992 University of Cambridge
Least and greatest fixed points; the Knaster-Tarski Theorem
Proved in the lattice of subsets of D, namely Pow(D), with Inter as glb
*)
open Fixedpt;
(*** Monotone operators ***)
val prems = Goalw [bnd_mono_def]
"[| h(D)<=D; \
\ !!W X. [| W<=D; X<=D; W<=X |] ==> h(W) <= h(X) \
\ |] ==> bnd_mono(D,h)";
by (REPEAT (ares_tac (prems@[conjI,allI,impI]) 1
ORELSE etac subset_trans 1));
qed "bnd_monoI";
Goalw [bnd_mono_def] "bnd_mono(D,h) ==> h(D) <= D";
by (etac conjunct1 1);
qed "bnd_monoD1";
Goalw [bnd_mono_def] "[| bnd_mono(D,h); W<=X; X<=D |] ==> h(W) <= h(X)";
by (Blast_tac 1);
qed "bnd_monoD2";
val [major,minor] = goal Fixedpt.thy
"[| bnd_mono(D,h); X<=D |] ==> h(X) <= D";
by (rtac (major RS bnd_monoD2 RS subset_trans) 1);
by (rtac (major RS bnd_monoD1) 3);
by (rtac minor 1);
by (rtac subset_refl 1);
qed "bnd_mono_subset";
Goal "[| bnd_mono(D,h); A <= D; B <= D |] ==> \
\ h(A) Un h(B) <= h(A Un B)";
by (REPEAT (ares_tac [Un_upper1, Un_upper2, Un_least] 1
ORELSE etac bnd_monoD2 1));
qed "bnd_mono_Un";
(*Useful??*)
Goal "[| bnd_mono(D,h); A <= D; B <= D |] ==> \
\ h(A Int B) <= h(A) Int h(B)";
by (REPEAT (ares_tac [Int_lower1, Int_lower2, Int_greatest] 1
ORELSE etac bnd_monoD2 1));
qed "bnd_mono_Int";
(**** Proof of Knaster-Tarski Theorem for the lfp ****)
(*lfp is contained in each pre-fixedpoint*)
Goalw [lfp_def]
"[| h(A) <= A; A<=D |] ==> lfp(D,h) <= A";
by (Blast_tac 1);
(*or by (rtac (PowI RS CollectI RS Inter_lower) 1); *)
qed "lfp_lowerbound";
(*Unfolding the defn of Inter dispenses with the premise bnd_mono(D,h)!*)
Goalw [lfp_def,Inter_def] "lfp(D,h) <= D";
by (Blast_tac 1);
qed "lfp_subset";
(*Used in datatype package*)
val [rew] = goal Fixedpt.thy "A==lfp(D,h) ==> A <= D";
by (rewtac rew);
by (rtac lfp_subset 1);
qed "def_lfp_subset";
val prems = Goalw [lfp_def]
"[| h(D) <= D; !!X. [| h(X) <= X; X<=D |] ==> A<=X |] ==> \
\ A <= lfp(D,h)";
by (rtac (Pow_top RS CollectI RS Inter_greatest) 1);
by (REPEAT (ares_tac prems 1 ORELSE eresolve_tac [CollectE,PowD] 1));
qed "lfp_greatest";
val hmono::prems = goal Fixedpt.thy
"[| bnd_mono(D,h); h(A)<=A; A<=D |] ==> h(lfp(D,h)) <= A";
by (rtac (hmono RS bnd_monoD2 RS subset_trans) 1);
by (rtac lfp_lowerbound 1);
by (REPEAT (resolve_tac prems 1));
qed "lfp_lemma1";
Goal "bnd_mono(D,h) ==> h(lfp(D,h)) <= lfp(D,h)";
by (rtac (bnd_monoD1 RS lfp_greatest) 1);
by (rtac lfp_lemma1 2);
by (REPEAT (assume_tac 1));
qed "lfp_lemma2";
val [hmono] = goal Fixedpt.thy
"bnd_mono(D,h) ==> lfp(D,h) <= h(lfp(D,h))";
by (rtac lfp_lowerbound 1);
by (rtac (hmono RS bnd_monoD2) 1);
by (rtac (hmono RS lfp_lemma2) 1);
by (rtac (hmono RS bnd_mono_subset) 2);
by (REPEAT (rtac lfp_subset 1));
qed "lfp_lemma3";
Goal "bnd_mono(D,h) ==> lfp(D,h) = h(lfp(D,h))";
by (REPEAT (ares_tac [equalityI,lfp_lemma2,lfp_lemma3] 1));
qed "lfp_Tarski";
(*Definition form, to control unfolding*)
val [rew,mono] = goal Fixedpt.thy
"[| A==lfp(D,h); bnd_mono(D,h) |] ==> A = h(A)";
by (rewtac rew);
by (rtac (mono RS lfp_Tarski) 1);
qed "def_lfp_Tarski";
(*** General induction rule for least fixedpoints ***)
val [hmono,indstep] = Goal
"[| bnd_mono(D,h); !!x. x : h(Collect(lfp(D,h),P)) ==> P(x) \
\ |] ==> h(Collect(lfp(D,h),P)) <= Collect(lfp(D,h),P)";
by (rtac subsetI 1);
by (rtac CollectI 1);
by (etac indstep 2);
by (rtac (hmono RS lfp_lemma2 RS subsetD) 1);
by (rtac (hmono RS bnd_monoD2 RS subsetD) 1);
by (REPEAT (ares_tac [Collect_subset, lfp_subset] 1));
qed "Collect_is_pre_fixedpt";
(*This rule yields an induction hypothesis in which the components of a
data structure may be assumed to be elements of lfp(D,h)*)
val prems = Goal
"[| bnd_mono(D,h); a : lfp(D,h); \
\ !!x. x : h(Collect(lfp(D,h),P)) ==> P(x) \
\ |] ==> P(a)";
by (rtac (Collect_is_pre_fixedpt RS lfp_lowerbound RS subsetD RS CollectD2) 1);
by (rtac (lfp_subset RS (Collect_subset RS subset_trans)) 3);
by (REPEAT (ares_tac prems 1));
qed "induct";
(*Definition form, to control unfolding*)
val rew::prems = Goal
"[| A == lfp(D,h); bnd_mono(D,h); a:A; \
\ !!x. x : h(Collect(A,P)) ==> P(x) \
\ |] ==> P(a)";
by (rtac induct 1);
by (REPEAT (ares_tac (map (rewrite_rule [rew]) prems) 1));
qed "def_induct";
(*This version is useful when "A" is not a subset of D;
second premise could simply be h(D Int A) <= D or !!X. X<=D ==> h(X)<=D *)
val [hsub,hmono] = goal Fixedpt.thy
"[| h(D Int A) <= A; bnd_mono(D,h) |] ==> lfp(D,h) <= A";
by (rtac (lfp_lowerbound RS subset_trans) 1);
by (rtac (hmono RS bnd_mono_subset RS Int_greatest) 1);
by (REPEAT (resolve_tac [hsub,Int_lower1,Int_lower2] 1));
qed "lfp_Int_lowerbound";
(*Monotonicity of lfp, where h precedes i under a domain-like partial order
monotonicity of h is not strictly necessary; h must be bounded by D*)
val [hmono,imono,subhi] = Goal
"[| bnd_mono(D,h); bnd_mono(E,i); \
\ !!X. X<=D ==> h(X) <= i(X) |] ==> lfp(D,h) <= lfp(E,i)";
by (rtac (bnd_monoD1 RS lfp_greatest) 1);
by (rtac imono 1);
by (rtac (hmono RSN (2, lfp_Int_lowerbound)) 1);
by (rtac (Int_lower1 RS subhi RS subset_trans) 1);
by (rtac (imono RS bnd_monoD2 RS subset_trans) 1);
by (REPEAT (ares_tac [Int_lower2] 1));
qed "lfp_mono";
(*This (unused) version illustrates that monotonicity is not really needed,
but both lfp's must be over the SAME set D; Inter is anti-monotonic!*)
val [isubD,subhi] = Goal
"[| i(D) <= D; !!X. X<=D ==> h(X) <= i(X) |] ==> lfp(D,h) <= lfp(D,i)";
by (rtac lfp_greatest 1);
by (rtac isubD 1);
by (rtac lfp_lowerbound 1);
by (etac (subhi RS subset_trans) 1);
by (REPEAT (assume_tac 1));
qed "lfp_mono2";
(**** Proof of Knaster-Tarski Theorem for the gfp ****)
(*gfp contains each post-fixedpoint that is contained in D*)
Goalw [gfp_def] "[| A <= h(A); A<=D |] ==> A <= gfp(D,h)";
by (rtac (PowI RS CollectI RS Union_upper) 1);
by (REPEAT (assume_tac 1));
qed "gfp_upperbound";
Goalw [gfp_def] "gfp(D,h) <= D";
by (Blast_tac 1);
qed "gfp_subset";
(*Used in datatype package*)
val [rew] = goal Fixedpt.thy "A==gfp(D,h) ==> A <= D";
by (rewtac rew);
by (rtac gfp_subset 1);
qed "def_gfp_subset";
val hmono::prems = Goalw [gfp_def]
"[| bnd_mono(D,h); !!X. [| X <= h(X); X<=D |] ==> X<=A |] ==> \
\ gfp(D,h) <= A";
by (fast_tac (subset_cs addIs ((hmono RS bnd_monoD1)::prems)) 1);
qed "gfp_least";
val hmono::prems = goal Fixedpt.thy
"[| bnd_mono(D,h); A<=h(A); A<=D |] ==> A <= h(gfp(D,h))";
by (rtac (hmono RS bnd_monoD2 RSN (2,subset_trans)) 1);
by (rtac gfp_subset 3);
by (rtac gfp_upperbound 2);
by (REPEAT (resolve_tac prems 1));
qed "gfp_lemma1";
Goal "bnd_mono(D,h) ==> gfp(D,h) <= h(gfp(D,h))";
by (rtac gfp_least 1);
by (rtac gfp_lemma1 2);
by (REPEAT (assume_tac 1));
qed "gfp_lemma2";
val [hmono] = goal Fixedpt.thy
"bnd_mono(D,h) ==> h(gfp(D,h)) <= gfp(D,h)";
by (rtac gfp_upperbound 1);
by (rtac (hmono RS bnd_monoD2) 1);
by (rtac (hmono RS gfp_lemma2) 1);
by (REPEAT (rtac ([hmono, gfp_subset] MRS bnd_mono_subset) 1));
qed "gfp_lemma3";
Goal "bnd_mono(D,h) ==> gfp(D,h) = h(gfp(D,h))";
by (REPEAT (ares_tac [equalityI,gfp_lemma2,gfp_lemma3] 1));
qed "gfp_Tarski";
(*Definition form, to control unfolding*)
val [rew,mono] = goal Fixedpt.thy
"[| A==gfp(D,h); bnd_mono(D,h) |] ==> A = h(A)";
by (rewtac rew);
by (rtac (mono RS gfp_Tarski) 1);
qed "def_gfp_Tarski";
(*** Coinduction rules for greatest fixed points ***)
(*weak version*)
Goal "[| a: X; X <= h(X); X <= D |] ==> a : gfp(D,h)";
by (REPEAT (ares_tac [gfp_upperbound RS subsetD] 1));
qed "weak_coinduct";
val [subs_h,subs_D,mono] = goal Fixedpt.thy
"[| X <= h(X Un gfp(D,h)); X <= D; bnd_mono(D,h) |] ==> \
\ X Un gfp(D,h) <= h(X Un gfp(D,h))";
by (rtac (subs_h RS Un_least) 1);
by (rtac (mono RS gfp_lemma2 RS subset_trans) 1);
by (rtac (Un_upper2 RS subset_trans) 1);
by (rtac ([mono, subs_D, gfp_subset] MRS bnd_mono_Un) 1);
qed "coinduct_lemma";
(*strong version*)
Goal "[| bnd_mono(D,h); a: X; X <= h(X Un gfp(D,h)); X <= D |] ==> \
\ a : gfp(D,h)";
by (rtac weak_coinduct 1);
by (etac coinduct_lemma 2);
by (REPEAT (ares_tac [gfp_subset, UnI1, Un_least] 1));
qed "coinduct";
(*Definition form, to control unfolding*)
val rew::prems = goal Fixedpt.thy
"[| A == gfp(D,h); bnd_mono(D,h); a: X; X <= h(X Un A); X <= D |] ==> \
\ a : A";
by (rewtac rew);
by (rtac coinduct 1);
by (REPEAT (ares_tac (map (rewrite_rule [rew]) prems) 1));
qed "def_coinduct";
(*Lemma used immediately below!*)
val [subsA,XimpP] = Goal
"[| X <= A; !!z. z:X ==> P(z) |] ==> X <= Collect(A,P)";
by (rtac (subsA RS subsetD RS CollectI RS subsetI) 1);
by (assume_tac 1);
by (etac XimpP 1);
qed "subset_Collect";
(*The version used in the induction/coinduction package*)
val prems = Goal
"[| A == gfp(D, %w. Collect(D,P(w))); bnd_mono(D, %w. Collect(D,P(w))); \
\ a: X; X <= D; !!z. z: X ==> P(X Un A, z) |] ==> \
\ a : A";
by (rtac def_coinduct 1);
by (REPEAT (ares_tac (subset_Collect::prems) 1));
qed "def_Collect_coinduct";
(*Monotonicity of gfp!*)
val [hmono,subde,subhi] = Goal
"[| bnd_mono(D,h); D <= E; \
\ !!X. X<=D ==> h(X) <= i(X) |] ==> gfp(D,h) <= gfp(E,i)";
by (rtac gfp_upperbound 1);
by (rtac (hmono RS gfp_lemma2 RS subset_trans) 1);
by (rtac (gfp_subset RS subhi) 1);
by (rtac ([gfp_subset, subde] MRS subset_trans) 1);
qed "gfp_mono";