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theory Follows(* Title: HOL/UNITY/Follows ID: $Id: Follows.thy,v 1.14 2005/06/17 14:13:10 haftmann Exp $ 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 constdefs Follows :: "['a => 'b::{order}, 'a => 'b::{order}] => 'a program set" (infixl "Fols" 65) "f Fols g == Increasing g ∩ Increasing f Int Always {s. f s ≤ g s} Int (\<Inter>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) ==> (\<Inter>j. {s. j ≤ g s} LeadsTo {s. j ≤ f s}) ⊆ (\<Inter>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, 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: union_le_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: union_le_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: union_le_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: union_le_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
lemma mono_Always_o:
mono h ==> Always {s. f s ≤ g s} ⊆ Always {s. h (f s) ≤ h (g s)}
lemma mono_LeadsTo_o:
mono h ==> (INT j. {s. j ≤ g s} LeadsTo {s. j ≤ f s}) ⊆ (INT k. {s. k ≤ h (g s)} LeadsTo {s. k ≤ h (f s)})
lemma Follows_constant:
F ∈ (%s. c) Fols (%s. c)
lemma mono_Follows_o:
mono h ==> f Fols g ⊆ (h o f) Fols (h o g)
lemma mono_Follows_apply:
mono h ==> f Fols g ⊆ (%x. h (f x)) Fols (%x. h (g x))
lemma Follows_trans:
[| F ∈ f Fols g; F ∈ g Fols h |] ==> F ∈ f Fols h
lemma Follows_Increasing1:
F ∈ f Fols g ==> F ∈ Increasing f
lemma Follows_Increasing2:
F ∈ f Fols g ==> F ∈ Increasing g
lemma Follows_Bounded:
F ∈ f Fols g ==> F ∈ Always {s. f s ⊆ g s}
lemma Follows_LeadsTo:
F ∈ f Fols g ==> F ∈ {s. k ≤ g s} LeadsTo {s. k ≤ f s}
lemma Follows_LeadsTo_pfixLe:
F ∈ f Fols g ==> F ∈ {s. k pfixLe g s} LeadsTo {s. k pfixLe f s}
lemma Follows_LeadsTo_pfixGe:
F ∈ f Fols g ==> F ∈ {s. k pfixGe g s} LeadsTo {s. k pfixGe f s}
lemma Always_Follows1:
[| F ∈ Always {s. f s = f' s}; F ∈ f Fols g |] ==> F ∈ f' Fols g
lemma Always_Follows2:
[| F ∈ Always {s. g s = g' s}; F ∈ f Fols g |] ==> F ∈ f Fols g'
lemma increasing_Un:
[| F ∈ increasing f; F ∈ increasing g |] ==> F ∈ increasing (%s. f s ∪ g s)
lemma Increasing_Un:
[| F ∈ Increasing f; F ∈ Increasing g |] ==> F ∈ Increasing (%s. f s ∪ g s)
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}
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}
lemma Follows_Un:
[| F ∈ f' Fols f; F ∈ g' Fols g |] ==> F ∈ (%s. f' s ∪ g' s) Fols (%s. f s ∪ g s)
lemma increasing_union:
[| F ∈ increasing f; F ∈ increasing g |] ==> F ∈ increasing (%s. f s + g s)
lemma Increasing_union:
[| F ∈ Increasing f; F ∈ Increasing g |] ==> F ∈ Increasing (%s. f s + g s)
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}
lemma Follows_union_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}
lemma Follows_union:
[| F ∈ f' Fols f; F ∈ g' Fols g |] ==> F ∈ (%s. f' s + g' s) Fols (%s. f s + g s)
lemma Follows_setsum:
[| ∀i∈I. F ∈ f' i Fols f i; finite I |] ==> F ∈ (%s. ∑i∈I. f' i s) Fols (%s. ∑i∈I. f i s)
lemma Increasing_imp_Stable_pfixGe:
F ∈ Increasing func ==> F ∈ Stable {s. h pfixGe func s}
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}