src/HOL/UNITY/Follows.thy
author wenzelm
Fri Mar 28 19:43:54 2008 +0100 (2008-03-28)
changeset 26462 dac4e2bce00d
parent 21710 4e4b7c801142
child 32689 860e1a2317bd
permissions -rw-r--r--
avoid rebinding of existing facts;
     1 (*  Title:      HOL/UNITY/Follows
     2     ID:         $Id$
     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     4     Copyright   1998  University of Cambridge
     5 *)
     6 
     7 header{*The Follows Relation of Charpentier and Sivilotte*}
     8 
     9 theory Follows imports SubstAx ListOrder Multiset begin
    10 
    11 constdefs
    12 
    13   Follows :: "['a => 'b::{order}, 'a => 'b::{order}] => 'a program set"
    14                  (infixl "Fols" 65)
    15    "f Fols g == Increasing g \<inter> Increasing f Int
    16                 Always {s. f s \<le> g s} Int
    17                 (\<Inter>k. {s. k \<le> g s} LeadsTo {s. k \<le> f s})"
    18 
    19 
    20 (*Does this hold for "invariant"?*)
    21 lemma mono_Always_o:
    22      "mono h ==> Always {s. f s \<le> g s} \<subseteq> Always {s. h (f s) \<le> h (g s)}"
    23 apply (simp add: Always_eq_includes_reachable)
    24 apply (blast intro: monoD)
    25 done
    26 
    27 lemma mono_LeadsTo_o:
    28      "mono (h::'a::order => 'b::order)  
    29       ==> (\<Inter>j. {s. j \<le> g s} LeadsTo {s. j \<le> f s}) \<subseteq>  
    30           (\<Inter>k. {s. k \<le> h (g s)} LeadsTo {s. k \<le> h (f s)})"
    31 apply auto
    32 apply (rule single_LeadsTo_I)
    33 apply (drule_tac x = "g s" in spec)
    34 apply (erule LeadsTo_weaken)
    35 apply (blast intro: monoD order_trans)+
    36 done
    37 
    38 lemma Follows_constant [iff]: "F \<in> (%s. c) Fols (%s. c)"
    39 by (simp add: Follows_def)
    40 
    41 lemma mono_Follows_o: "mono h ==> f Fols g \<subseteq> (h o f) Fols (h o g)"
    42 by (auto simp add: Follows_def mono_Increasing_o [THEN [2] rev_subsetD]
    43 		   mono_Always_o [THEN [2] rev_subsetD]
    44 		   mono_LeadsTo_o [THEN [2] rev_subsetD, THEN INT_D])
    45 
    46 lemma mono_Follows_apply:
    47      "mono h ==> f Fols g \<subseteq> (%x. h (f x)) Fols (%x. h (g x))"
    48 apply (drule mono_Follows_o)
    49 apply (force simp add: o_def)
    50 done
    51 
    52 lemma Follows_trans: 
    53      "[| F \<in> f Fols g;  F \<in> g Fols h |] ==> F \<in> f Fols h"
    54 apply (simp add: Follows_def)
    55 apply (simp add: Always_eq_includes_reachable)
    56 apply (blast intro: order_trans LeadsTo_Trans)
    57 done
    58 
    59 
    60 subsection{*Destruction rules*}
    61 
    62 lemma Follows_Increasing1: "F \<in> f Fols g ==> F \<in> Increasing f"
    63 by (simp add: Follows_def)
    64 
    65 lemma Follows_Increasing2: "F \<in> f Fols g ==> F \<in> Increasing g"
    66 by (simp add: Follows_def)
    67 
    68 lemma Follows_Bounded: "F \<in> f Fols g ==> F \<in> Always {s. f s \<le> g s}"
    69 by (simp add: Follows_def)
    70 
    71 lemma Follows_LeadsTo: 
    72      "F \<in> f Fols g ==> F \<in> {s. k \<le> g s} LeadsTo {s. k \<le> f s}"
    73 by (simp add: Follows_def)
    74 
    75 lemma Follows_LeadsTo_pfixLe:
    76      "F \<in> f Fols g ==> F \<in> {s. k pfixLe g s} LeadsTo {s. k pfixLe f s}"
    77 apply (rule single_LeadsTo_I, clarify)
    78 apply (drule_tac k="g s" in Follows_LeadsTo)
    79 apply (erule LeadsTo_weaken)
    80  apply blast 
    81 apply (blast intro: pfixLe_trans prefix_imp_pfixLe)
    82 done
    83 
    84 lemma Follows_LeadsTo_pfixGe:
    85      "F \<in> f Fols g ==> F \<in> {s. k pfixGe g s} LeadsTo {s. k pfixGe f s}"
    86 apply (rule single_LeadsTo_I, clarify)
    87 apply (drule_tac k="g s" in Follows_LeadsTo)
    88 apply (erule LeadsTo_weaken)
    89  apply blast 
    90 apply (blast intro: pfixGe_trans prefix_imp_pfixGe)
    91 done
    92 
    93 
    94 lemma Always_Follows1: 
    95      "[| F \<in> Always {s. f s = f' s}; F \<in> f Fols g |] ==> F \<in> f' Fols g"
    96 
    97 apply (simp add: Follows_def Increasing_def Stable_def, auto)
    98 apply (erule_tac [3] Always_LeadsTo_weaken)
    99 apply (erule_tac A = "{s. z \<le> f s}" and A' = "{s. z \<le> f s}" 
   100        in Always_Constrains_weaken, auto)
   101 apply (drule Always_Int_I, assumption)
   102 apply (force intro: Always_weaken)
   103 done
   104 
   105 lemma Always_Follows2: 
   106      "[| F \<in> Always {s. g s = g' s}; F \<in> f Fols g |] ==> F \<in> f Fols g'"
   107 apply (simp add: Follows_def Increasing_def Stable_def, auto)
   108 apply (erule_tac [3] Always_LeadsTo_weaken)
   109 apply (erule_tac A = "{s. z \<le> g s}" and A' = "{s. z \<le> g s}"
   110        in Always_Constrains_weaken, auto)
   111 apply (drule Always_Int_I, assumption)
   112 apply (force intro: Always_weaken)
   113 done
   114 
   115 
   116 subsection{*Union properties (with the subset ordering)*}
   117 
   118 (*Can replace "Un" by any sup.  But existing max only works for linorders.*)
   119 lemma increasing_Un: 
   120     "[| F \<in> increasing f;  F \<in> increasing g |]  
   121      ==> F \<in> increasing (%s. (f s) \<union> (g s))"
   122 apply (simp add: increasing_def stable_def constrains_def, auto)
   123 apply (drule_tac x = "f xa" in spec)
   124 apply (drule_tac x = "g xa" in spec)
   125 apply (blast dest!: bspec)
   126 done
   127 
   128 lemma Increasing_Un: 
   129     "[| F \<in> Increasing f;  F \<in> Increasing g |]  
   130      ==> F \<in> Increasing (%s. (f s) \<union> (g s))"
   131 apply (auto simp add: Increasing_def Stable_def Constrains_def
   132                       stable_def constrains_def)
   133 apply (drule_tac x = "f xa" in spec)
   134 apply (drule_tac x = "g xa" in spec)
   135 apply (blast dest!: bspec)
   136 done
   137 
   138 
   139 lemma Always_Un:
   140      "[| F \<in> Always {s. f' s \<le> f s}; F \<in> Always {s. g' s \<le> g s} |]  
   141       ==> F \<in> Always {s. f' s \<union> g' s \<le> f s \<union> g s}"
   142 by (simp add: Always_eq_includes_reachable, blast)
   143 
   144 (*Lemma to re-use the argument that one variable increases (progress)
   145   while the other variable doesn't decrease (safety)*)
   146 lemma Follows_Un_lemma:
   147      "[| F \<in> Increasing f; F \<in> Increasing g;  
   148          F \<in> Increasing g'; F \<in> Always {s. f' s \<le> f s}; 
   149          \<forall>k. F \<in> {s. k \<le> f s} LeadsTo {s. k \<le> f' s} |] 
   150       ==> F \<in> {s. k \<le> f s \<union> g s} LeadsTo {s. k \<le> f' s \<union> g s}"
   151 apply (rule single_LeadsTo_I)
   152 apply (drule_tac x = "f s" in IncreasingD)
   153 apply (drule_tac x = "g s" in IncreasingD)
   154 apply (rule LeadsTo_weaken)
   155 apply (rule PSP_Stable)
   156 apply (erule_tac x = "f s" in spec)
   157 apply (erule Stable_Int, assumption, blast+)
   158 done
   159 
   160 lemma Follows_Un: 
   161     "[| F \<in> f' Fols f;  F \<in> g' Fols g |]  
   162      ==> F \<in> (%s. (f' s) \<union> (g' s)) Fols (%s. (f s) \<union> (g s))"
   163 apply (simp add: Follows_def Increasing_Un Always_Un del: Un_subset_iff, auto)
   164 apply (rule LeadsTo_Trans)
   165 apply (blast intro: Follows_Un_lemma)
   166 (*Weakening is used to exchange Un's arguments*)
   167 apply (blast intro: Follows_Un_lemma [THEN LeadsTo_weaken])
   168 done
   169 
   170 
   171 subsection{*Multiset union properties (with the multiset ordering)*}
   172 
   173 lemma increasing_union: 
   174     "[| F \<in> increasing f;  F \<in> increasing g |]  
   175      ==> F \<in> increasing (%s. (f s) + (g s :: ('a::order) multiset))"
   176 apply (simp add: increasing_def stable_def constrains_def, auto)
   177 apply (drule_tac x = "f xa" in spec)
   178 apply (drule_tac x = "g xa" in spec)
   179 apply (drule bspec, assumption) 
   180 apply (blast intro: union_le_mono order_trans)
   181 done
   182 
   183 lemma Increasing_union: 
   184     "[| F \<in> Increasing f;  F \<in> Increasing g |]  
   185      ==> F \<in> Increasing (%s. (f s) + (g s :: ('a::order) multiset))"
   186 apply (auto simp add: Increasing_def Stable_def Constrains_def
   187                       stable_def constrains_def)
   188 apply (drule_tac x = "f xa" in spec)
   189 apply (drule_tac x = "g xa" in spec)
   190 apply (drule bspec, assumption) 
   191 apply (blast intro: union_le_mono order_trans)
   192 done
   193 
   194 lemma Always_union:
   195      "[| F \<in> Always {s. f' s \<le> f s}; F \<in> Always {s. g' s \<le> g s} |]  
   196       ==> F \<in> Always {s. f' s + g' s \<le> f s + (g s :: ('a::order) multiset)}"
   197 apply (simp add: Always_eq_includes_reachable)
   198 apply (blast intro: union_le_mono)
   199 done
   200 
   201 (*Except the last line, IDENTICAL to the proof script for Follows_Un_lemma*)
   202 lemma Follows_union_lemma:
   203      "[| F \<in> Increasing f; F \<in> Increasing g;  
   204          F \<in> Increasing g'; F \<in> Always {s. f' s \<le> f s}; 
   205          \<forall>k::('a::order) multiset.  
   206            F \<in> {s. k \<le> f s} LeadsTo {s. k \<le> f' s} |] 
   207       ==> F \<in> {s. k \<le> f s + g s} LeadsTo {s. k \<le> f' s + g s}"
   208 apply (rule single_LeadsTo_I)
   209 apply (drule_tac x = "f s" in IncreasingD)
   210 apply (drule_tac x = "g s" in IncreasingD)
   211 apply (rule LeadsTo_weaken)
   212 apply (rule PSP_Stable)
   213 apply (erule_tac x = "f s" in spec)
   214 apply (erule Stable_Int, assumption, blast)
   215 apply (blast intro: union_le_mono order_trans)
   216 done
   217 
   218 (*The !! is there to influence to effect of permutative rewriting at the end*)
   219 lemma Follows_union: 
   220      "!!g g' ::'b => ('a::order) multiset.  
   221         [| F \<in> f' Fols f;  F \<in> g' Fols g |]  
   222         ==> F \<in> (%s. (f' s) + (g' s)) Fols (%s. (f s) + (g s))"
   223 apply (simp add: Follows_def)
   224 apply (simp add: Increasing_union Always_union, auto)
   225 apply (rule LeadsTo_Trans)
   226 apply (blast intro: Follows_union_lemma)
   227 (*now exchange union's arguments*)
   228 apply (simp add: union_commute)
   229 apply (blast intro: Follows_union_lemma)
   230 done
   231 
   232 lemma Follows_setsum:
   233      "!!f ::['c,'b] => ('a::order) multiset.  
   234         [| \<forall>i \<in> I. F \<in> f' i Fols f i;  finite I |]  
   235         ==> F \<in> (%s. \<Sum>i \<in> I. f' i s) Fols (%s. \<Sum>i \<in> I. f i s)"
   236 apply (erule rev_mp)
   237 apply (erule finite_induct, simp) 
   238 apply (simp add: Follows_union)
   239 done
   240 
   241 
   242 (*Currently UNUSED, but possibly of interest*)
   243 lemma Increasing_imp_Stable_pfixGe:
   244      "F \<in> Increasing func ==> F \<in> Stable {s. h pfixGe (func s)}"
   245 apply (simp add: Increasing_def Stable_def Constrains_def constrains_def)
   246 apply (blast intro: trans_Ge [THEN trans_genPrefix, THEN transD] 
   247                     prefix_imp_pfixGe)
   248 done
   249 
   250 (*Currently UNUSED, but possibly of interest*)
   251 lemma LeadsTo_le_imp_pfixGe:
   252      "\<forall>z. F \<in> {s. z \<le> f s} LeadsTo {s. z \<le> g s}  
   253       ==> F \<in> {s. z pfixGe f s} LeadsTo {s. z pfixGe g s}"
   254 apply (rule single_LeadsTo_I)
   255 apply (drule_tac x = "f s" in spec)
   256 apply (erule LeadsTo_weaken)
   257  prefer 2
   258  apply (blast intro: trans_Ge [THEN trans_genPrefix, THEN transD] 
   259                      prefix_imp_pfixGe, blast)
   260 done
   261 
   262 end