(* Title: CCL/Gfp.ML ID: $Id: Gfp.ML,v 1.7 2005/09/17 15:35:27 wenzelm Exp $ *) (*** Proof of Knaster-Tarski Theorem using gfp ***) (* gfp(f) is the least upper bound of {u. u <= f(u)} *) val prems = goalw (the_context ()) [gfp_def] "[| A <= f(A) |] ==> A <= gfp(f)"; by (rtac (CollectI RS Union_upper) 1); by (resolve_tac prems 1); qed "gfp_upperbound"; val prems = goalw (the_context ()) [gfp_def] "[| !!u. u <= f(u) ==> u<=A |] ==> gfp(f) <= A"; by (REPEAT (ares_tac ([Union_least]@prems) 1)); by (etac CollectD 1); qed "gfp_least"; val [mono] = goal (the_context ()) "mono(f) ==> gfp(f) <= f(gfp(f))"; by (EVERY1 [rtac gfp_least, rtac subset_trans, atac, rtac (mono RS monoD), rtac gfp_upperbound, atac]); qed "gfp_lemma2"; val [mono] = goal (the_context ()) "mono(f) ==> f(gfp(f)) <= gfp(f)"; by (EVERY1 [rtac gfp_upperbound, rtac (mono RS monoD), rtac gfp_lemma2, rtac mono]); qed "gfp_lemma3"; val [mono] = goal (the_context ()) "mono(f) ==> gfp(f) = f(gfp(f))"; by (REPEAT (resolve_tac [equalityI,gfp_lemma2,gfp_lemma3,mono] 1)); qed "gfp_Tarski"; (*** Coinduction rules for greatest fixed points ***) (*weak version*) val prems = goal (the_context ()) "[| a: A; A <= f(A) |] ==> a : gfp(f)"; by (rtac (gfp_upperbound RS subsetD) 1); by (REPEAT (ares_tac prems 1)); qed "coinduct"; val [prem,mono] = goal (the_context ()) "[| A <= f(A) Un gfp(f); mono(f) |] ==> \ \ A Un gfp(f) <= f(A Un gfp(f))"; by (rtac subset_trans 1); by (rtac (mono RS mono_Un) 2); by (rtac (mono RS gfp_Tarski RS subst) 1); by (rtac (prem RS Un_least) 1); by (rtac Un_upper2 1); qed "coinduct2_lemma"; (*strong version, thanks to Martin Coen*) val ainA::prems = goal (the_context ()) "[| a: A; A <= f(A) Un gfp(f); mono(f) |] ==> a : gfp(f)"; by (rtac coinduct 1); by (rtac (prems MRS coinduct2_lemma) 2); by (resolve_tac [ainA RS UnI1] 1); qed "coinduct2"; (*** Even Stronger version of coinduct [by Martin Coen] - instead of the condition A <= f(A) consider A <= (f(A) Un f(f(A)) ...) Un gfp(A) ***) val [prem] = goal (the_context ()) "mono(f) ==> mono(%x. f(x) Un A Un B)"; by (REPEAT (ares_tac [subset_refl, monoI, Un_mono, prem RS monoD] 1)); qed "coinduct3_mono_lemma"; val [prem,mono] = goal (the_context ()) "[| A <= f(lfp(%x. f(x) Un A Un gfp(f))); mono(f) |] ==> \ \ lfp(%x. f(x) Un A Un gfp(f)) <= f(lfp(%x. f(x) Un A 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"; val ainA::prems = goal (the_context ()) "[| a:A; A <= f(lfp(%x. f(x) Un A Un gfp(f))); mono(f) |] ==> a : gfp(f)"; by (rtac coinduct 1); by (rtac (prems MRS coinduct3_lemma) 2); by (resolve_tac (prems RL [coinduct3_mono_lemma RS lfp_Tarski RS ssubst]) 1); by (rtac (ainA RS UnI2 RS UnI1) 1); qed "coinduct3"; (** Definition forms of gfp_Tarski, to control unfolding **) val [rew,mono] = goal (the_context ()) "[| h==gfp(f); mono(f) |] ==> h = f(h)"; by (rewtac rew); by (rtac (mono RS gfp_Tarski) 1); qed "def_gfp_Tarski"; val rew::prems = goal (the_context ()) "[| h==gfp(f); a:A; A <= f(A) |] ==> a: h"; by (rewtac rew); by (REPEAT (ares_tac (prems @ [coinduct]) 1)); qed "def_coinduct"; val rew::prems = goal (the_context ()) "[| h==gfp(f); a:A; A <= f(A) Un h; mono(f) |] ==> a: h"; by (rewtac rew); by (REPEAT (ares_tac (map (rewrite_rule [rew]) prems @ [coinduct2]) 1)); qed "def_coinduct2"; val rew::prems = goal (the_context ()) "[| h==gfp(f); a:A; A <= f(lfp(%x. f(x) Un A Un h)); mono(f) |] ==> a: h"; by (rewtac rew); by (REPEAT (ares_tac (map (rewrite_rule [rew]) prems @ [coinduct3]) 1)); qed "def_coinduct3"; (*Monotonicity of gfp!*) val prems = goal (the_context ()) "[| mono(f); !!Z. f(Z)<=g(Z) |] ==> gfp(f) <= gfp(g)"; by (rtac gfp_upperbound 1); by (rtac subset_trans 1); by (rtac gfp_lemma2 1); by (resolve_tac prems 1); by (resolve_tac prems 1); qed "gfp_mono";