(* Title: HOL/gfp
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1993 University of Cambridge
The Knaster-Tarski Theorem for greatest fixed points.
*)
open Gfp;
(*** Proof of Knaster-Tarski Theorem using gfp ***)
(* gfp(f) is the least upper bound of {u. u <= f(u)} *)
Goalw [gfp_def] "[| X <= f(X) |] ==> X <= gfp(f)";
by (etac (CollectI RS Union_upper) 1);
qed "gfp_upperbound";
val prems = goalw Gfp.thy [gfp_def]
"[| !!u. u <= f(u) ==> u<=X |] ==> gfp(f) <= X";
by (REPEAT (ares_tac ([Union_least]@prems) 1));
by (etac CollectD 1);
qed "gfp_least";
Goal "mono(f) ==> gfp(f) <= f(gfp(f))";
by (EVERY1 [rtac gfp_least, rtac subset_trans, atac,
etac monoD, rtac gfp_upperbound, atac]);
qed "gfp_lemma2";
Goal "mono(f) ==> f(gfp(f)) <= gfp(f)";
by (EVERY1 [rtac gfp_upperbound, rtac monoD, assume_tac,
etac gfp_lemma2]);
qed "gfp_lemma3";
Goal "mono(f) ==> gfp(f) = f(gfp(f))";
by (REPEAT (ares_tac [equalityI,gfp_lemma2,gfp_lemma3] 1));
qed "gfp_Tarski";
(*** Coinduction rules for greatest fixed points ***)
(*weak version*)
Goal "[| a: X; X <= f(X) |] ==> a : gfp(f)";
by (rtac (gfp_upperbound RS subsetD) 1);
by Auto_tac;
qed "weak_coinduct";
val [prem,mono] = goal Gfp.thy
"[| X <= f(X Un gfp(f)); mono(f) |] ==> \
\ X Un gfp(f) <= f(X Un gfp(f))";
by (rtac (prem 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 RS mono_Un) 1);
qed "coinduct_lemma";
(*strong version, thanks to Coen & Frost*)
Goal "[| mono(f); a: X; X <= f(X Un gfp(f)) |] ==> a : gfp(f)";
by (rtac (coinduct_lemma RSN (2, weak_coinduct)) 1);
by (REPEAT (ares_tac [UnI1, Un_least] 1));
qed "coinduct";
val [mono,prem] = goal Gfp.thy
"[| mono(f); a: gfp(f) |] ==> a: f(X Un gfp(f))";
by (rtac (mono RS mono_Un RS subsetD) 1);
by (rtac (mono RS gfp_lemma2 RS subsetD RS UnI2) 1);
by (rtac prem 1);
qed "gfp_fun_UnI2";
(*** Even Stronger version of coinduct [by Martin Coen]
- instead of the condition X <= f(X)
consider X <= (f(X) Un f(f(X)) ...) Un gfp(X) ***)
Goal "mono(f) ==> mono(%x. f(x) Un X Un B)";
by (REPEAT (ares_tac [subset_refl, monoI, Un_mono] 1 ORELSE etac monoD 1));
qed "coinduct3_mono_lemma";
val [prem,mono] = goal Gfp.thy
"[| 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)))";
by (rtac subset_trans 1);
by (rtac (mono RS coinduct3_mono_lemma RS lfp_lemma3) 1);
by (rtac (Un_least RS Un_least) 1);
by (rtac subset_refl 1);
by (rtac prem 1);
by (rtac (mono RS gfp_Tarski RS equalityD1 RS subset_trans) 1);
by (rtac (mono RS monoD) 1);
by (stac (mono RS coinduct3_mono_lemma RS lfp_Tarski) 1);
by (rtac Un_upper2 1);
qed "coinduct3_lemma";
Goal
"[| mono(f); a:X; X <= f(lfp(%x. f(x) Un X Un gfp(f))) |] ==> a : gfp(f)";
by (rtac (coinduct3_lemma RSN (2,weak_coinduct)) 1);
by (resolve_tac [coinduct3_mono_lemma RS lfp_Tarski RS ssubst] 1);
by Auto_tac;
qed "coinduct3";
(** Definition forms of gfp_Tarski and coinduct, to control unfolding **)
val [rew,mono] = goal Gfp.thy "[| A==gfp(f); mono(f) |] ==> A = f(A)";
by (rewtac rew);
by (rtac (mono RS gfp_Tarski) 1);
qed "def_gfp_Tarski";
val rew::prems = goal Gfp.thy
"[| A==gfp(f); mono(f); a:X; X <= f(X Un A) |] ==> a: A";
by (rewtac rew);
by (REPEAT (ares_tac (map (rewrite_rule [rew]) prems @ [coinduct]) 1));
qed "def_coinduct";
(*The version used in the induction/coinduction package*)
val prems = Goal
"[| A == gfp(%w. Collect(P(w))); mono(%w. Collect(P(w))); \
\ a: X; !!z. z: X ==> P (X Un A) z |] ==> \
\ a : A";
by (rtac def_coinduct 1);
by (REPEAT (ares_tac (prems @ [subsetI,CollectI]) 1));
qed "def_Collect_coinduct";
val rew::prems = goal Gfp.thy
"[| A==gfp(f); mono(f); a:X; X <= f(lfp(%x. f(x) Un X Un A)) |] ==> a: A";
by (rewtac rew);
by (REPEAT (ares_tac (map (rewrite_rule [rew]) prems @ [coinduct3]) 1));
qed "def_coinduct3";
(*Monotonicity of gfp!*)
val [prem] = Goal "[| !!Z. f(Z)<=g(Z) |] ==> gfp(f) <= gfp(g)";
by (rtac (gfp_upperbound RS gfp_least) 1);
by (etac (prem RSN (2,subset_trans)) 1);
qed "gfp_mono";