(* Title: HOL/UNITY/Follows.thy Author: Lawrence C Paulson, Cambridge University Computer Laboratory Copyright 1998 University of Cambridge *) header{*The Follows Relation of Charpentier and Sivilotte*} theory Follows imports SubstAx ListOrder Multiset begin definition Follows :: "['a => 'b::{order}, 'a => 'b::{order}] => 'a program set" (infixl "Fols" 65) where "f Fols g == Increasing g \ Increasing f Int Always {s. f s \ g s} Int (\k. {s. k \ g s} LeadsTo {s. k \ f s})" (*Does this hold for "invariant"?*) lemma mono_Always_o: "mono h ==> Always {s. f s \ g s} \ Always {s. h (f s) \ h (g s)}" apply (simp add: Always_eq_includes_reachable) apply (blast intro: monoD) done lemma mono_LeadsTo_o: "mono (h::'a::order => 'b::order) ==> (\j. {s. j \ g s} LeadsTo {s. j \ f s}) \ (\k. {s. k \ h (g s)} LeadsTo {s. k \ h (f s)})" apply auto apply (rule single_LeadsTo_I) apply (drule_tac x = "g s" in spec) apply (erule LeadsTo_weaken) apply (blast intro: monoD order_trans)+ done lemma Follows_constant [iff]: "F \ (%s. c) Fols (%s. c)" by (simp add: Follows_def) lemma mono_Follows_o: "mono h ==> f Fols g \ (h o f) Fols (h o g)" by (auto simp add: Follows_def mono_Increasing_o [THEN [2] rev_subsetD] mono_Always_o [THEN [2] rev_subsetD] mono_LeadsTo_o [THEN [2] rev_subsetD, THEN INT_D]) lemma mono_Follows_apply: "mono h ==> f Fols g \ (%x. h (f x)) Fols (%x. h (g x))" apply (drule mono_Follows_o) apply (force simp add: o_def) done lemma Follows_trans: "[| F \ f Fols g; F \ g Fols h |] ==> F \ f Fols h" apply (simp add: Follows_def) apply (simp add: Always_eq_includes_reachable) apply (blast intro: order_trans LeadsTo_Trans) done subsection{*Destruction rules*} lemma Follows_Increasing1: "F \ f Fols g ==> F \ Increasing f" by (simp add: Follows_def) lemma Follows_Increasing2: "F \ f Fols g ==> F \ Increasing g" by (simp add: Follows_def) lemma Follows_Bounded: "F \ f Fols g ==> F \ Always {s. f s \ g s}" by (simp add: Follows_def) lemma Follows_LeadsTo: "F \ f Fols g ==> F \ {s. k \ g s} LeadsTo {s. k \ f s}" by (simp add: Follows_def) lemma Follows_LeadsTo_pfixLe: "F \ f Fols g ==> F \ {s. k pfixLe g s} LeadsTo {s. k pfixLe f s}" apply (rule single_LeadsTo_I, clarify) apply (drule_tac k="g s" in Follows_LeadsTo) apply (erule LeadsTo_weaken) apply blast apply (blast intro: pfixLe_trans prefix_imp_pfixLe) done lemma Follows_LeadsTo_pfixGe: "F \ f Fols g ==> F \ {s. k pfixGe g s} LeadsTo {s. k pfixGe f s}" apply (rule single_LeadsTo_I, clarify) apply (drule_tac k="g s" in Follows_LeadsTo) apply (erule LeadsTo_weaken) apply blast apply (blast intro: pfixGe_trans prefix_imp_pfixGe) done lemma Always_Follows1: "[| F \ Always {s. f s = f' s}; F \ f Fols g |] ==> F \ f' Fols g" apply (simp add: Follows_def Increasing_def Stable_def, auto) apply (erule_tac [3] Always_LeadsTo_weaken) apply (erule_tac A = "{s. z \ f s}" and A' = "{s. z \ f s}" in Always_Constrains_weaken, auto) apply (drule Always_Int_I, assumption) apply (force intro: Always_weaken) done lemma Always_Follows2: "[| F \ Always {s. g s = g' s}; F \ f Fols g |] ==> F \ f Fols g'" apply (simp add: Follows_def Increasing_def Stable_def, auto) apply (erule_tac [3] Always_LeadsTo_weaken) apply (erule_tac A = "{s. z \ g s}" and A' = "{s. z \ g s}" in Always_Constrains_weaken, auto) apply (drule Always_Int_I, assumption) apply (force intro: Always_weaken) done subsection{*Union properties (with the subset ordering)*} (*Can replace "Un" by any sup. But existing max only works for linorders.*) lemma increasing_Un: "[| F \ increasing f; F \ increasing g |] ==> F \ increasing (%s. (f s) \ (g s))" apply (simp add: increasing_def stable_def constrains_def, auto) apply (drule_tac x = "f xa" in spec) apply (drule_tac x = "g xa" in spec) apply (blast dest!: bspec) done lemma Increasing_Un: "[| F \ Increasing f; F \ Increasing g |] ==> F \ Increasing (%s. (f s) \ (g s))" apply (auto simp add: Increasing_def Stable_def Constrains_def stable_def constrains_def) apply (drule_tac x = "f xa" in spec) apply (drule_tac x = "g xa" in spec) apply (blast dest!: bspec) done lemma Always_Un: "[| F \ Always {s. f' s \ f s}; F \ Always {s. g' s \ g s} |] ==> F \ Always {s. f' s \ g' s \ f s \ g s}" by (simp add: Always_eq_includes_reachable, blast) (*Lemma to re-use the argument that one variable increases (progress) while the other variable doesn't decrease (safety)*) lemma Follows_Un_lemma: "[| F \ Increasing f; F \ Increasing g; F \ Increasing g'; F \ Always {s. f' s \ f s}; \k. F \ {s. k \ f s} LeadsTo {s. k \ f' s} |] ==> F \ {s. k \ f s \ g s} LeadsTo {s. k \ f' s \ g s}" apply (rule single_LeadsTo_I) apply (drule_tac x = "f s" in IncreasingD) apply (drule_tac x = "g s" in IncreasingD) apply (rule LeadsTo_weaken) apply (rule PSP_Stable) apply (erule_tac x = "f s" in spec) apply (erule Stable_Int, assumption, blast+) done lemma Follows_Un: "[| F \ f' Fols f; F \ g' Fols g |] ==> F \ (%s. (f' s) \ (g' s)) Fols (%s. (f s) \ (g s))" apply (simp add: Follows_def Increasing_Un Always_Un del: Un_subset_iff le_sup_iff, auto) apply (rule LeadsTo_Trans) apply (blast intro: Follows_Un_lemma) (*Weakening is used to exchange Un's arguments*) apply (blast intro: Follows_Un_lemma [THEN LeadsTo_weaken]) done subsection{*Multiset union properties (with the multiset ordering)*} lemma increasing_union: "[| F \ increasing f; F \ increasing g |] ==> F \ increasing (%s. (f s) + (g s :: ('a::order) multiset))" apply (simp add: increasing_def stable_def constrains_def, auto) apply (drule_tac x = "f xa" in spec) apply (drule_tac x = "g xa" in spec) apply (drule bspec, assumption) apply (blast intro: add_mono order_trans) done lemma Increasing_union: "[| F \ Increasing f; F \ Increasing g |] ==> F \ Increasing (%s. (f s) + (g s :: ('a::order) multiset))" apply (auto simp add: Increasing_def Stable_def Constrains_def stable_def constrains_def) apply (drule_tac x = "f xa" in spec) apply (drule_tac x = "g xa" in spec) apply (drule bspec, assumption) apply (blast intro: add_mono order_trans) done lemma Always_union: "[| F \ Always {s. f' s \ f s}; F \ Always {s. g' s \ g s} |] ==> F \ Always {s. f' s + g' s \ f s + (g s :: ('a::order) multiset)}" apply (simp add: Always_eq_includes_reachable) apply (blast intro: add_mono) done (*Except the last line, IDENTICAL to the proof script for Follows_Un_lemma*) lemma Follows_union_lemma: "[| F \ Increasing f; F \ Increasing g; F \ Increasing g'; F \ Always {s. f' s \ f s}; \k::('a::order) multiset. F \ {s. k \ f s} LeadsTo {s. k \ f' s} |] ==> F \ {s. k \ f s + g s} LeadsTo {s. k \ f' s + g s}" apply (rule single_LeadsTo_I) apply (drule_tac x = "f s" in IncreasingD) apply (drule_tac x = "g s" in IncreasingD) apply (rule LeadsTo_weaken) apply (rule PSP_Stable) apply (erule_tac x = "f s" in spec) apply (erule Stable_Int, assumption, blast) apply (blast intro: add_mono order_trans) done (*The !! is there to influence to effect of permutative rewriting at the end*) lemma Follows_union: "!!g g' ::'b => ('a::order) multiset. [| F \ f' Fols f; F \ g' Fols g |] ==> F \ (%s. (f' s) + (g' s)) Fols (%s. (f s) + (g s))" apply (simp add: Follows_def) apply (simp add: Increasing_union Always_union, auto) apply (rule LeadsTo_Trans) apply (blast intro: Follows_union_lemma) (*now exchange union's arguments*) apply (simp add: union_commute) apply (blast intro: Follows_union_lemma) done lemma Follows_setsum: "!!f ::['c,'b] => ('a::order) multiset. [| \i \ I. F \ f' i Fols f i; finite I |] ==> F \ (%s. \i \ I. f' i s) Fols (%s. \i \ I. f i s)" apply (erule rev_mp) apply (erule finite_induct, simp) apply (simp add: Follows_union) done (*Currently UNUSED, but possibly of interest*) lemma Increasing_imp_Stable_pfixGe: "F \ Increasing func ==> F \ Stable {s. h pfixGe (func s)}" apply (simp add: Increasing_def Stable_def Constrains_def constrains_def) apply (blast intro: trans_Ge [THEN trans_genPrefix, THEN transD] prefix_imp_pfixGe) done (*Currently UNUSED, but possibly of interest*) lemma LeadsTo_le_imp_pfixGe: "\z. F \ {s. z \ f s} LeadsTo {s. z \ g s} ==> F \ {s. z pfixGe f s} LeadsTo {s. z pfixGe g s}" apply (rule single_LeadsTo_I) apply (drule_tac x = "f s" in spec) apply (erule LeadsTo_weaken) prefer 2 apply (blast intro: trans_Ge [THEN trans_genPrefix, THEN transD] prefix_imp_pfixGe, blast) done end