src/HOL/UNITY/Follows.thy
 author paulson Thu Jan 30 10:35:56 2003 +0100 (2003-01-30) changeset 13796 19f50fa807ae parent 10265 4e004b548049 child 13798 4c1a53627500 permissions -rw-r--r--
converting more UNITY theories to new-style
```     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 The "Follows" relation of Charpentier and Sivilotte
```
```     7 *)
```
```     8
```
```     9 theory Follows = SubstAx + ListOrder + Multiset:
```
```    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 Int Increasing f Int
```
```    16                 Always {s. f s <= g s} Int
```
```    17                 (INT k. {s. k <= g s} LeadsTo {s. k <= f s})"
```
```    18
```
```    19
```
```    20 (*Does this hold for "invariant"?*)
```
```    21 lemma mono_Always_o:
```
```    22      "mono h ==> Always {s. f s <= g s} <= Always {s. h (f s) <= 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       ==> (INT j. {s. j <= g s} LeadsTo {s. j <= f s}) <=
```
```    30           (INT k. {s. k <= h (g s)} LeadsTo {s. k <= 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: "F : (%s. c) Fols (%s. c)"
```
```    39 by (unfold Follows_def, auto)
```
```    40 declare Follows_constant [iff]
```
```    41
```
```    42 lemma mono_Follows_o: "mono h ==> f Fols g <= (h o f) Fols (h o g)"
```
```    43 apply (unfold Follows_def, clarify)
```
```    44 apply (simp add: mono_Increasing_o [THEN [2] rev_subsetD]
```
```    45                  mono_Always_o [THEN [2] rev_subsetD]
```
```    46                  mono_LeadsTo_o [THEN [2] rev_subsetD, THEN INT_D])
```
```    47 done
```
```    48
```
```    49 lemma mono_Follows_apply:
```
```    50      "mono h ==> f Fols g <= (%x. h (f x)) Fols (%x. h (g x))"
```
```    51 apply (drule mono_Follows_o)
```
```    52 apply (force simp add: o_def)
```
```    53 done
```
```    54
```
```    55 lemma Follows_trans:
```
```    56      "[| F : f Fols g;  F: g Fols h |] ==> F : f Fols h"
```
```    57 apply (unfold Follows_def)
```
```    58 apply (simp add: Always_eq_includes_reachable)
```
```    59 apply (blast intro: order_trans LeadsTo_Trans)
```
```    60 done
```
```    61
```
```    62
```
```    63 (** Destructiom rules **)
```
```    64
```
```    65 lemma Follows_Increasing1:
```
```    66      "F : f Fols g ==> F : Increasing f"
```
```    67
```
```    68 apply (unfold Follows_def, blast)
```
```    69 done
```
```    70
```
```    71 lemma Follows_Increasing2:
```
```    72      "F : f Fols g ==> F : Increasing g"
```
```    73 apply (unfold Follows_def, blast)
```
```    74 done
```
```    75
```
```    76 lemma Follows_Bounded:
```
```    77      "F : f Fols g ==> F : Always {s. f s <= g s}"
```
```    78 apply (unfold Follows_def, blast)
```
```    79 done
```
```    80
```
```    81 lemma Follows_LeadsTo:
```
```    82      "F : f Fols g ==> F : {s. k <= g s} LeadsTo {s. k <= f s}"
```
```    83 apply (unfold Follows_def, blast)
```
```    84 done
```
```    85
```
```    86 lemma Follows_LeadsTo_pfixLe:
```
```    87      "F : f Fols g ==> F : {s. k pfixLe g s} LeadsTo {s. k pfixLe f s}"
```
```    88 apply (rule single_LeadsTo_I, clarify)
```
```    89 apply (drule_tac k="g s" in Follows_LeadsTo)
```
```    90 apply (erule LeadsTo_weaken)
```
```    91  apply blast
```
```    92 apply (blast intro: pfixLe_trans prefix_imp_pfixLe)
```
```    93 done
```
```    94
```
```    95 lemma Follows_LeadsTo_pfixGe:
```
```    96      "F : f Fols g ==> F : {s. k pfixGe g s} LeadsTo {s. k pfixGe f s}"
```
```    97 apply (rule single_LeadsTo_I, clarify)
```
```    98 apply (drule_tac k="g s" in Follows_LeadsTo)
```
```    99 apply (erule LeadsTo_weaken)
```
```   100  apply blast
```
```   101 apply (blast intro: pfixGe_trans prefix_imp_pfixGe)
```
```   102 done
```
```   103
```
```   104
```
```   105 lemma Always_Follows1:
```
```   106      "[| F : Always {s. f s = f' s}; F : f Fols g |] ==> F : f' Fols g"
```
```   107
```
```   108 apply (unfold Follows_def Increasing_def Stable_def, auto)
```
```   109 apply (erule_tac [3] Always_LeadsTo_weaken)
```
```   110 apply (erule_tac A = "{s. z <= f s}" and A' = "{s. z <= f s}" in Always_Constrains_weaken, auto)
```
```   111 apply (drule Always_Int_I, assumption)
```
```   112 apply (force intro: Always_weaken)
```
```   113 done
```
```   114
```
```   115 lemma Always_Follows2:
```
```   116      "[| F : Always {s. g s = g' s}; F : f Fols g |] ==> F : f Fols g'"
```
```   117 apply (unfold Follows_def Increasing_def Stable_def, auto)
```
```   118 apply (erule_tac [3] Always_LeadsTo_weaken)
```
```   119 apply (erule_tac A = "{s. z <= g s}" and A' = "{s. z <= g s}" in Always_Constrains_weaken, auto)
```
```   120 apply (drule Always_Int_I, assumption)
```
```   121 apply (force intro: Always_weaken)
```
```   122 done
```
```   123
```
```   124
```
```   125 (** Union properties (with the subset ordering) **)
```
```   126
```
```   127 (*Can replace "Un" by any sup.  But existing max only works for linorders.*)
```
```   128 lemma increasing_Un:
```
```   129     "[| F : increasing f;  F: increasing g |]
```
```   130      ==> F : increasing (%s. (f s) Un (g s))"
```
```   131 apply (unfold increasing_def stable_def constrains_def, auto)
```
```   132 apply (drule_tac x = "f xa" in spec)
```
```   133 apply (drule_tac x = "g xa" in spec)
```
```   134 apply (blast dest!: bspec)
```
```   135 done
```
```   136
```
```   137 lemma Increasing_Un:
```
```   138     "[| F : Increasing f;  F: Increasing g |]
```
```   139      ==> F : Increasing (%s. (f s) Un (g s))"
```
```   140 apply (unfold Increasing_def Stable_def Constrains_def stable_def constrains_def, auto)
```
```   141 apply (drule_tac x = "f xa" in spec)
```
```   142 apply (drule_tac x = "g xa" in spec)
```
```   143 apply (blast dest!: bspec)
```
```   144 done
```
```   145
```
```   146
```
```   147 lemma Always_Un:
```
```   148      "[| F : Always {s. f' s <= f s}; F : Always {s. g' s <= g s} |]
```
```   149       ==> F : Always {s. f' s Un g' s <= f s Un g s}"
```
```   150 apply (simp add: Always_eq_includes_reachable, blast)
```
```   151 done
```
```   152
```
```   153 (*Lemma to re-use the argument that one variable increases (progress)
```
```   154   while the other variable doesn't decrease (safety)*)
```
```   155 lemma Follows_Un_lemma:
```
```   156      "[| F : Increasing f; F : Increasing g;
```
```   157          F : Increasing g'; F : Always {s. f' s <= f s};
```
```   158          ALL k. F : {s. k <= f s} LeadsTo {s. k <= f' s} |]
```
```   159       ==> F : {s. k <= f s Un g s} LeadsTo {s. k <= f' s Un g s}"
```
```   160 apply (rule single_LeadsTo_I)
```
```   161 apply (drule_tac x = "f s" in IncreasingD)
```
```   162 apply (drule_tac x = "g s" in IncreasingD)
```
```   163 apply (rule LeadsTo_weaken)
```
```   164 apply (rule PSP_Stable)
```
```   165 apply (erule_tac x = "f s" in spec)
```
```   166 apply (erule Stable_Int, assumption)
```
```   167 apply blast
```
```   168 apply blast
```
```   169 done
```
```   170
```
```   171 lemma Follows_Un:
```
```   172     "[| F : f' Fols f;  F: g' Fols g |]
```
```   173      ==> F : (%s. (f' s) Un (g' s)) Fols (%s. (f s) Un (g s))"
```
```   174 apply (unfold Follows_def)
```
```   175 apply (simp add: Increasing_Un Always_Un, auto)
```
```   176 apply (rule LeadsTo_Trans)
```
```   177 apply (blast intro: Follows_Un_lemma)
```
```   178 (*Weakening is used to exchange Un's arguments*)
```
```   179 apply (blast intro: Follows_Un_lemma [THEN LeadsTo_weaken])
```
```   180 done
```
```   181
```
```   182
```
```   183 (** Multiset union properties (with the multiset ordering) **)
```
```   184
```
```   185 lemma increasing_union:
```
```   186     "[| F : increasing f;  F: increasing g |]
```
```   187      ==> F : increasing (%s. (f s) + (g s :: ('a::order) multiset))"
```
```   188
```
```   189 apply (unfold increasing_def stable_def constrains_def, auto)
```
```   190 apply (drule_tac x = "f xa" in spec)
```
```   191 apply (drule_tac x = "g xa" in spec)
```
```   192 apply (drule bspec, assumption)
```
```   193 apply (blast intro: union_le_mono order_trans)
```
```   194 done
```
```   195
```
```   196 lemma Increasing_union:
```
```   197     "[| F : Increasing f;  F: Increasing g |]
```
```   198      ==> F : Increasing (%s. (f s) + (g s :: ('a::order) multiset))"
```
```   199 apply (unfold Increasing_def Stable_def Constrains_def stable_def constrains_def, auto)
```
```   200 apply (drule_tac x = "f xa" in spec)
```
```   201 apply (drule_tac x = "g xa" in spec)
```
```   202 apply (drule bspec, assumption)
```
```   203 apply (blast intro: union_le_mono order_trans)
```
```   204 done
```
```   205
```
```   206 lemma Always_union:
```
```   207      "[| F : Always {s. f' s <= f s}; F : Always {s. g' s <= g s} |]
```
```   208       ==> F : Always {s. f' s + g' s <= f s + (g s :: ('a::order) multiset)}"
```
```   209 apply (simp add: Always_eq_includes_reachable)
```
```   210 apply (blast intro: union_le_mono)
```
```   211 done
```
```   212
```
```   213 (*Except the last line, IDENTICAL to the proof script for Follows_Un_lemma*)
```
```   214 lemma Follows_union_lemma:
```
```   215      "[| F : Increasing f; F : Increasing g;
```
```   216          F : Increasing g'; F : Always {s. f' s <= f s};
```
```   217          ALL k::('a::order) multiset.
```
```   218            F : {s. k <= f s} LeadsTo {s. k <= f' s} |]
```
```   219       ==> F : {s. k <= f s + g s} LeadsTo {s. k <= f' s + g s}"
```
```   220 apply (rule single_LeadsTo_I)
```
```   221 apply (drule_tac x = "f s" in IncreasingD)
```
```   222 apply (drule_tac x = "g s" in IncreasingD)
```
```   223 apply (rule LeadsTo_weaken)
```
```   224 apply (rule PSP_Stable)
```
```   225 apply (erule_tac x = "f s" in spec)
```
```   226 apply (erule Stable_Int, assumption)
```
```   227 apply blast
```
```   228 apply (blast intro: union_le_mono order_trans)
```
```   229 done
```
```   230
```
```   231 (*The !! is there to influence to effect of permutative rewriting at the end*)
```
```   232 lemma Follows_union:
```
```   233      "!!g g' ::'b => ('a::order) multiset.
```
```   234         [| F : f' Fols f;  F: g' Fols g |]
```
```   235         ==> F : (%s. (f' s) + (g' s)) Fols (%s. (f s) + (g s))"
```
```   236 apply (unfold Follows_def)
```
```   237 apply (simp add: Increasing_union Always_union, auto)
```
```   238 apply (rule LeadsTo_Trans)
```
```   239 apply (blast intro: Follows_union_lemma)
```
```   240 (*now exchange union's arguments*)
```
```   241 apply (simp add: union_commute)
```
```   242 apply (blast intro: Follows_union_lemma)
```
```   243 done
```
```   244
```
```   245 lemma Follows_setsum:
```
```   246      "!!f ::['c,'b] => ('a::order) multiset.
```
```   247         [| ALL i: I. F : f' i Fols f i;  finite I |]
```
```   248         ==> F : (%s. \<Sum>i:I. f' i s) Fols (%s. \<Sum>i:I. f i s)"
```
```   249 apply (erule rev_mp)
```
```   250 apply (erule finite_induct, simp)
```
```   251 apply (simp add: Follows_union)
```
```   252 done
```
```   253
```
```   254
```
```   255 (*Currently UNUSED, but possibly of interest*)
```
```   256 lemma Increasing_imp_Stable_pfixGe:
```
```   257      "F : Increasing func ==> F : Stable {s. h pfixGe (func s)}"
```
```   258 apply (simp add: Increasing_def Stable_def Constrains_def constrains_def)
```
```   259 apply (blast intro: trans_Ge [THEN trans_genPrefix, THEN transD]
```
```   260                     prefix_imp_pfixGe)
```
```   261 done
```
```   262
```
```   263 (*Currently UNUSED, but possibly of interest*)
```
```   264 lemma LeadsTo_le_imp_pfixGe:
```
```   265      "ALL z. F : {s. z <= f s} LeadsTo {s. z <= g s}
```
```   266       ==> F : {s. z pfixGe f s} LeadsTo {s. z pfixGe g s}"
```
```   267 apply (rule single_LeadsTo_I)
```
```   268 apply (drule_tac x = "f s" in spec)
```
```   269 apply (erule LeadsTo_weaken)
```
```   270  prefer 2
```
```   271  apply (blast intro: trans_Ge [THEN trans_genPrefix, THEN transD]
```
```   272                      prefix_imp_pfixGe, blast)
```
```   273 done
```
```   274
```
```   275 end
```