# HG changeset patch # User paulson # Date 1102432544 -3600 # Node ID 780ea4c697f2a7918051820e663076c85929b076 # Parent 455cfa766dad456661f6959c7c1fd192d424451b converted Gfp to new-style theory diff -r 455cfa766dad -r 780ea4c697f2 src/HOL/Gfp.ML --- a/src/HOL/Gfp.ML Tue Dec 07 16:15:05 2004 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,125 +0,0 @@ -(* Title: HOL/Gfp.ML - ID: $Id$ - Author: Lawrence C Paulson, Cambridge University Computer Laboratory - Copyright 1993 University of Cambridge - -The Knaster-Tarski Theorem for greatest fixed points. -*) - -(*** Proof of Knaster-Tarski Theorem using gfp ***) - -val gfp_def = thm "gfp_def"; - -(* 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_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_unfold"; - -(*** 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"; - -Goal "!!X. [| a : X; g`X <= f (g`X) |] ==> g a : gfp f"; -by (etac (gfp_upperbound RS subsetD) 1); -by (etac imageI 1); -qed "weak_coinduct_image"; - -Goal "[| X <= f(X Un gfp(f)); mono(f) |] ==> \ -\ X Un gfp(f) <= f(X Un gfp(f))"; -by (blast_tac (claset() addDs [gfp_lemma2, 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"; - -Goal "[| mono(f); a: gfp(f) |] ==> a: f(X Un gfp(f))"; -by (blast_tac (claset() addDs [gfp_lemma2, mono_Un]) 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"; - -Goal "[| 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 (etac (coinduct3_mono_lemma RS lfp_lemma3) 1); -by (rtac (Un_least RS Un_least) 1); -by (rtac subset_refl 1); -by (assume_tac 1); -by (rtac (gfp_unfold RS equalityD1 RS subset_trans) 1); -by (assume_tac 1); -by (rtac monoD 1 THEN assume_tac 1); -by (stac (coinduct3_mono_lemma RS lfp_unfold) 1); -by Auto_tac; -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_unfold RS ssubst] 1); -by Auto_tac; -qed "coinduct3"; - - -(** Definition forms of gfp_unfold and coinduct, to control unfolding **) - -Goal "[| A==gfp(f); mono(f) |] ==> A = f(A)"; -by (auto_tac (claset() addSIs [gfp_unfold], simpset())); -qed "def_gfp_unfold"; - -Goal "[| A==gfp(f); mono(f); a:X; X <= f(X Un A) |] ==> a: A"; -by (auto_tac (claset() addSIs [coinduct], simpset())); -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"; - -Goal "[| A==gfp(f); mono(f); a:X; X <= f(lfp(%x. f(x) Un X Un A)) |] \ -\ ==> a: A"; -by (auto_tac (claset() addSIs [coinduct3], simpset())); -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"; diff -r 455cfa766dad -r 780ea4c697f2 src/HOL/Gfp.thy --- a/src/HOL/Gfp.thy Tue Dec 07 16:15:05 2004 +0100 +++ b/src/HOL/Gfp.thy Tue Dec 07 16:15:44 2004 +0100 @@ -1,17 +1,140 @@ -(* Title: HOL/gfp.thy - ID: $Id$ +(* ID: $Id$ Author: Lawrence C Paulson, Cambridge University Computer Laboratory Copyright 1994 University of Cambridge -Greatest fixed points (requires Lfp too!) *) -theory Gfp +header {*Greatest Fixed Points*} + +theory Gfp imports Lfp begin constdefs gfp :: "['a set=>'a set] => 'a set" - "gfp(f) == Union({u. u <= f(u)})" (*greatest fixed point*) + "gfp(f) == Union({u. u \ f(u)})" + + + +subsection{*Proof of Knaster-Tarski Theorem using gfp*} + + +text{*@{term "gfp f"} is the least upper 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 (rules intro: gfp_least subset_trans monoD gfp_upperbound) + +lemma gfp_lemma3: "mono(f) ==> f(gfp(f)) \ gfp(f)" +by (rules intro: gfp_lemma2 monoD gfp_upperbound) + +lemma gfp_unfold: "mono(f) ==> gfp(f) = f(gfp(f))" +by (rules 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 (rules 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 gfp_def = thm "gfp_def"; +val gfp_upperbound = thm "gfp_upperbound"; +val gfp_least = thm "gfp_least"; +val gfp_lemma2 = thm "gfp_lemma2"; +val gfp_lemma3 = thm "gfp_lemma3"; +val gfp_unfold = thm "gfp_unfold"; +val weak_coinduct = thm "weak_coinduct"; +val weak_coinduct_image = thm "weak_coinduct_image"; +val coinduct_lemma = thm "coinduct_lemma"; +val coinduct = thm "coinduct"; +val gfp_fun_UnI2 = thm "gfp_fun_UnI2"; +val coinduct3_mono_lemma = thm "coinduct3_mono_lemma"; +val coinduct3_lemma = thm "coinduct3_lemma"; +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 diff -r 455cfa766dad -r 780ea4c697f2 src/HOL/IsaMakefile --- a/src/HOL/IsaMakefile Tue Dec 07 16:15:05 2004 +0100 +++ b/src/HOL/IsaMakefile Tue Dec 07 16:15:44 2004 +0100 @@ -83,7 +83,7 @@ $(SRC)/TFL/usyntax.ML $(SRC)/TFL/utils.ML \ Datatype.thy Datatype_Universe.ML Datatype_Universe.thy \ Divides.thy Equiv_Relations.thy Extraction.thy Finite_Set.ML Finite_Set.thy \ - Fun.thy Gfp.ML Gfp.thy Hilbert_Choice.thy HOL.ML \ + Fun.thy Gfp.thy Hilbert_Choice.thy HOL.ML \ HOL.thy HOL_lemmas.ML Inductive.thy Infinite_Set.thy Integ/Numeral.thy \ Integ/cooper_dec.ML Integ/cooper_proof.ML \ Integ/IntArith.thy Integ/IntDef.thy \