src/HOL/UNITY/ProgressSets.thy
 author hoelzl Fri Feb 19 13:40:50 2016 +0100 (2016-02-19) changeset 62378 85ed00c1fe7c parent 62343 24106dc44def child 63146 f1ecba0272f9 permissions -rw-r--r--
generalize more theorems to support enat and ennreal
```     1 (*  Title:      HOL/UNITY/ProgressSets.thy
```
```     2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
```
```     3     Copyright   2003  University of Cambridge
```
```     4
```
```     5 Progress Sets.  From
```
```     6
```
```     7     David Meier and Beverly Sanders,
```
```     8     Composing Leads-to Properties
```
```     9     Theoretical Computer Science 243:1-2 (2000), 339-361.
```
```    10
```
```    11     David Meier,
```
```    12     Progress Properties in Program Refinement and Parallel Composition
```
```    13     Swiss Federal Institute of Technology Zurich (1997)
```
```    14 *)
```
```    15
```
```    16 section{*Progress Sets*}
```
```    17
```
```    18 theory ProgressSets imports Transformers begin
```
```    19
```
```    20 subsection {*Complete Lattices and the Operator @{term cl}*}
```
```    21
```
```    22 definition lattice :: "'a set set => bool" where
```
```    23    --{*Meier calls them closure sets, but they are just complete lattices*}
```
```    24    "lattice L ==
```
```    25          (\<forall>M. M \<subseteq> L --> \<Inter>M \<in> L) & (\<forall>M. M \<subseteq> L --> \<Union>M \<in> L)"
```
```    26
```
```    27 definition cl :: "['a set set, 'a set] => 'a set" where
```
```    28    --{*short for ``closure''*}
```
```    29    "cl L r == \<Inter>{x. x\<in>L & r \<subseteq> x}"
```
```    30
```
```    31 lemma UNIV_in_lattice: "lattice L ==> UNIV \<in> L"
```
```    32 by (force simp add: lattice_def)
```
```    33
```
```    34 lemma empty_in_lattice: "lattice L ==> {} \<in> L"
```
```    35 by (force simp add: lattice_def)
```
```    36
```
```    37 lemma Union_in_lattice: "[|M \<subseteq> L; lattice L|] ==> \<Union>M \<in> L"
```
```    38 by (simp add: lattice_def)
```
```    39
```
```    40 lemma Inter_in_lattice: "[|M \<subseteq> L; lattice L|] ==> \<Inter>M \<in> L"
```
```    41 by (simp add: lattice_def)
```
```    42
```
```    43 lemma UN_in_lattice:
```
```    44      "[|lattice L; !!i. i\<in>I ==> r i \<in> L|] ==> (\<Union>i\<in>I. r i) \<in> L"
```
```    45 apply (blast intro: Union_in_lattice)
```
```    46 done
```
```    47
```
```    48 lemma INT_in_lattice:
```
```    49      "[|lattice L; !!i. i\<in>I ==> r i \<in> L|] ==> (\<Inter>i\<in>I. r i)  \<in> L"
```
```    50 apply (blast intro: Inter_in_lattice)
```
```    51 done
```
```    52
```
```    53 lemma Un_in_lattice: "[|x\<in>L; y\<in>L; lattice L|] ==> x\<union>y \<in> L"
```
```    54   using Union_in_lattice [of "{x, y}" L] by simp
```
```    55
```
```    56 lemma Int_in_lattice: "[|x\<in>L; y\<in>L; lattice L|] ==> x\<inter>y \<in> L"
```
```    57   using Inter_in_lattice [of "{x, y}" L] by simp
```
```    58
```
```    59 lemma lattice_stable: "lattice {X. F \<in> stable X}"
```
```    60 by (simp add: lattice_def stable_def constrains_def, blast)
```
```    61
```
```    62 text{*The next three results state that @{term "cl L r"} is the minimal
```
```    63  element of @{term L} that includes @{term r}.*}
```
```    64 lemma cl_in_lattice: "lattice L ==> cl L r \<in> L"
```
```    65 apply (simp add: lattice_def cl_def)
```
```    66 apply (erule conjE)
```
```    67 apply (drule spec, erule mp, blast)
```
```    68 done
```
```    69
```
```    70 lemma cl_least: "[|c\<in>L; r\<subseteq>c|] ==> cl L r \<subseteq> c"
```
```    71 by (force simp add: cl_def)
```
```    72
```
```    73 text{*The next three lemmas constitute assertion (4.61)*}
```
```    74 lemma cl_mono: "r \<subseteq> r' ==> cl L r \<subseteq> cl L r'"
```
```    75 by (simp add: cl_def, blast)
```
```    76
```
```    77 lemma subset_cl: "r \<subseteq> cl L r"
```
```    78 by (simp add: cl_def le_Inf_iff)
```
```    79
```
```    80 text{*A reformulation of @{thm subset_cl}*}
```
```    81 lemma clI: "x \<in> r ==> x \<in> cl L r"
```
```    82 by (simp add: cl_def, blast)
```
```    83
```
```    84 text{*A reformulation of @{thm cl_least}*}
```
```    85 lemma clD: "[|c \<in> cl L r; B \<in> L; r \<subseteq> B|] ==> c \<in> B"
```
```    86 by (force simp add: cl_def)
```
```    87
```
```    88 lemma cl_UN_subset: "(\<Union>i\<in>I. cl L (r i)) \<subseteq> cl L (\<Union>i\<in>I. r i)"
```
```    89 by (simp add: cl_def, blast)
```
```    90
```
```    91 lemma cl_Un: "lattice L ==> cl L (r\<union>s) = cl L r \<union> cl L s"
```
```    92 apply (rule equalityI)
```
```    93  prefer 2
```
```    94   apply (simp add: cl_def, blast)
```
```    95 apply (rule cl_least)
```
```    96  apply (blast intro: Un_in_lattice cl_in_lattice)
```
```    97 apply (blast intro: subset_cl [THEN subsetD])
```
```    98 done
```
```    99
```
```   100 lemma cl_UN: "lattice L ==> cl L (\<Union>i\<in>I. r i) = (\<Union>i\<in>I. cl L (r i))"
```
```   101 apply (rule equalityI)
```
```   102  prefer 2 apply (simp add: cl_def, blast)
```
```   103 apply (rule cl_least)
```
```   104  apply (blast intro: UN_in_lattice cl_in_lattice)
```
```   105 apply (blast intro: subset_cl [THEN subsetD])
```
```   106 done
```
```   107
```
```   108 lemma cl_Int_subset: "cl L (r\<inter>s) \<subseteq> cl L r \<inter> cl L s"
```
```   109 by (simp add: cl_def, blast)
```
```   110
```
```   111 lemma cl_idem [simp]: "cl L (cl L r) = cl L r"
```
```   112 by (simp add: cl_def, blast)
```
```   113
```
```   114 lemma cl_ident: "r\<in>L ==> cl L r = r"
```
```   115 by (force simp add: cl_def)
```
```   116
```
```   117 lemma cl_empty [simp]: "lattice L ==> cl L {} = {}"
```
```   118 by (simp add: cl_ident empty_in_lattice)
```
```   119
```
```   120 lemma cl_UNIV [simp]: "lattice L ==> cl L UNIV = UNIV"
```
```   121 by (simp add: cl_ident UNIV_in_lattice)
```
```   122
```
```   123 text{*Assertion (4.62)*}
```
```   124 lemma cl_ident_iff: "lattice L ==> (cl L r = r) = (r\<in>L)"
```
```   125 apply (rule iffI)
```
```   126  apply (erule subst)
```
```   127  apply (erule cl_in_lattice)
```
```   128 apply (erule cl_ident)
```
```   129 done
```
```   130
```
```   131 lemma cl_subset_in_lattice: "[|cl L r \<subseteq> r; lattice L|] ==> r\<in>L"
```
```   132 by (simp add: cl_ident_iff [symmetric] equalityI subset_cl)
```
```   133
```
```   134
```
```   135 subsection {*Progress Sets and the Main Lemma*}
```
```   136 text{*A progress set satisfies certain closure conditions and is a
```
```   137 simple way of including the set @{term "wens_set F B"}.*}
```
```   138
```
```   139 definition closed :: "['a program, 'a set, 'a set,  'a set set] => bool" where
```
```   140    "closed F T B L == \<forall>M. \<forall>act \<in> Acts F. B\<subseteq>M & T\<inter>M \<in> L -->
```
```   141                               T \<inter> (B \<union> wp act M) \<in> L"
```
```   142
```
```   143 definition progress_set :: "['a program, 'a set, 'a set] => 'a set set set" where
```
```   144    "progress_set F T B ==
```
```   145       {L. lattice L & B \<in> L & T \<in> L & closed F T B L}"
```
```   146
```
```   147 lemma closedD:
```
```   148    "[|closed F T B L; act \<in> Acts F; B\<subseteq>M; T\<inter>M \<in> L|]
```
```   149     ==> T \<inter> (B \<union> wp act M) \<in> L"
```
```   150 by (simp add: closed_def)
```
```   151
```
```   152 text{*Note: the formalization below replaces Meier's @{term q} by @{term B}
```
```   153 and @{term m} by @{term X}. *}
```
```   154
```
```   155 text{*Part of the proof of the claim at the bottom of page 97.  It's
```
```   156 proved separately because the argument requires a generalization over
```
```   157 all @{term "act \<in> Acts F"}.*}
```
```   158 lemma lattice_awp_lemma:
```
```   159   assumes TXC:  "T\<inter>X \<in> C" --{*induction hypothesis in theorem below*}
```
```   160       and BsubX:  "B \<subseteq> X"   --{*holds in inductive step*}
```
```   161       and latt: "lattice C"
```
```   162       and TC:   "T \<in> C"
```
```   163       and BC:   "B \<in> C"
```
```   164       and clos: "closed F T B C"
```
```   165     shows "T \<inter> (B \<union> awp F (X \<union> cl C (T\<inter>r))) \<in> C"
```
```   166 apply (simp del: INT_simps add: awp_def INT_extend_simps)
```
```   167 apply (rule INT_in_lattice [OF latt])
```
```   168 apply (erule closedD [OF clos])
```
```   169 apply (simp add: subset_trans [OF BsubX Un_upper1])
```
```   170 apply (subgoal_tac "T \<inter> (X \<union> cl C (T\<inter>r)) = (T\<inter>X) \<union> cl C (T\<inter>r)")
```
```   171  prefer 2 apply (blast intro: TC clD)
```
```   172 apply (erule ssubst)
```
```   173 apply (blast intro: Un_in_lattice latt cl_in_lattice TXC)
```
```   174 done
```
```   175
```
```   176 text{*Remainder of the proof of the claim at the bottom of page 97.*}
```
```   177 lemma lattice_lemma:
```
```   178   assumes TXC:  "T\<inter>X \<in> C" --{*induction hypothesis in theorem below*}
```
```   179       and BsubX:  "B \<subseteq> X"   --{*holds in inductive step*}
```
```   180       and act:  "act \<in> Acts F"
```
```   181       and latt: "lattice C"
```
```   182       and TC:   "T \<in> C"
```
```   183       and BC:   "B \<in> C"
```
```   184       and clos: "closed F T B C"
```
```   185     shows "T \<inter> (wp act X \<inter> awp F (X \<union> cl C (T\<inter>r)) \<union> X) \<in> C"
```
```   186 apply (subgoal_tac "T \<inter> (B \<union> wp act X) \<in> C")
```
```   187  prefer 2 apply (simp add: closedD [OF clos] act BsubX TXC)
```
```   188 apply (drule Int_in_lattice
```
```   189               [OF _ lattice_awp_lemma [OF TXC BsubX latt TC BC clos, of r]
```
```   190                     latt])
```
```   191 apply (subgoal_tac
```
```   192          "T \<inter> (B \<union> wp act X) \<inter> (T \<inter> (B \<union> awp F (X \<union> cl C (T\<inter>r)))) =
```
```   193           T \<inter> (B \<union> wp act X \<inter> awp F (X \<union> cl C (T\<inter>r)))")
```
```   194  prefer 2 apply blast
```
```   195 apply simp
```
```   196 apply (drule Un_in_lattice [OF _ TXC latt])
```
```   197 apply (subgoal_tac
```
```   198          "T \<inter> (B \<union> wp act X \<inter> awp F (X \<union> cl C (T\<inter>r))) \<union> T\<inter>X =
```
```   199           T \<inter> (wp act X \<inter> awp F (X \<union> cl C (T\<inter>r)) \<union> X)")
```
```   200  apply simp
```
```   201 apply (blast intro: BsubX [THEN subsetD])
```
```   202 done
```
```   203
```
```   204
```
```   205 text{*Induction step for the main lemma*}
```
```   206 lemma progress_induction_step:
```
```   207   assumes TXC:  "T\<inter>X \<in> C" --{*induction hypothesis in theorem below*}
```
```   208       and act:  "act \<in> Acts F"
```
```   209       and Xwens: "X \<in> wens_set F B"
```
```   210       and latt: "lattice C"
```
```   211       and  TC:  "T \<in> C"
```
```   212       and  BC:  "B \<in> C"
```
```   213       and clos: "closed F T B C"
```
```   214       and Fstable: "F \<in> stable T"
```
```   215   shows "T \<inter> wens F act X \<in> C"
```
```   216 proof -
```
```   217   from Xwens have BsubX: "B \<subseteq> X"
```
```   218     by (rule wens_set_imp_subset)
```
```   219   let ?r = "wens F act X"
```
```   220   have "?r \<subseteq> (wp act X \<inter> awp F (X\<union>?r)) \<union> X"
```
```   221     by (simp add: wens_unfold [symmetric])
```
```   222   then have "T\<inter>?r \<subseteq> T \<inter> ((wp act X \<inter> awp F (X\<union>?r)) \<union> X)"
```
```   223     by blast
```
```   224   then have "T\<inter>?r \<subseteq> T \<inter> ((wp act X \<inter> awp F (T \<inter> (X\<union>?r))) \<union> X)"
```
```   225     by (simp add: awp_Int_eq Fstable stable_imp_awp_ident, blast)
```
```   226   then have "T\<inter>?r \<subseteq> T \<inter> ((wp act X \<inter> awp F (X \<union> cl C (T\<inter>?r))) \<union> X)"
```
```   227     by (blast intro: awp_mono [THEN  rev_subsetD] subset_cl [THEN subsetD])
```
```   228   then have "cl C (T\<inter>?r) \<subseteq>
```
```   229              cl C (T \<inter> ((wp act X \<inter> awp F (X \<union> cl C (T\<inter>?r))) \<union> X))"
```
```   230     by (rule cl_mono)
```
```   231   then have "cl C (T\<inter>?r) \<subseteq>
```
```   232              T \<inter> ((wp act X \<inter> awp F (X \<union> cl C (T\<inter>?r))) \<union> X)"
```
```   233     by (simp add: cl_ident lattice_lemma [OF TXC BsubX act latt TC BC clos])
```
```   234   then have "cl C (T\<inter>?r) \<subseteq> (wp act X \<inter> awp F (X \<union> cl C (T\<inter>?r))) \<union> X"
```
```   235     by blast
```
```   236   then have "cl C (T\<inter>?r) \<subseteq> ?r"
```
```   237     by (blast intro!: subset_wens)
```
```   238   then have cl_subset: "cl C (T\<inter>?r) \<subseteq> T\<inter>?r"
```
```   239     by (simp add: cl_ident TC
```
```   240                   subset_trans [OF cl_mono [OF Int_lower1]])
```
```   241   show ?thesis
```
```   242     by (rule cl_subset_in_lattice [OF cl_subset latt])
```
```   243 qed
```
```   244
```
```   245 text{*Proved on page 96 of Meier's thesis.  The special case when
```
```   246    @{term "T=UNIV"} states that every progress set for the program @{term F}
```
```   247    and set @{term B} includes the set @{term "wens_set F B"}.*}
```
```   248 lemma progress_set_lemma:
```
```   249      "[|C \<in> progress_set F T B; r \<in> wens_set F B; F \<in> stable T|] ==> T\<inter>r \<in> C"
```
```   250 apply (simp add: progress_set_def, clarify)
```
```   251 apply (erule wens_set.induct)
```
```   252   txt{*Base*}
```
```   253   apply (simp add: Int_in_lattice)
```
```   254  txt{*The difficult @{term wens} case*}
```
```   255  apply (simp add: progress_induction_step)
```
```   256 txt{*Disjunctive case*}
```
```   257 apply (subgoal_tac "(\<Union>U\<in>W. T \<inter> U) \<in> C")
```
```   258  apply simp
```
```   259 apply (blast intro: UN_in_lattice)
```
```   260 done
```
```   261
```
```   262
```
```   263 subsection {*The Progress Set Union Theorem*}
```
```   264
```
```   265 lemma closed_mono:
```
```   266   assumes BB':  "B \<subseteq> B'"
```
```   267       and TBwp: "T \<inter> (B \<union> wp act M) \<in> C"
```
```   268       and B'C:  "B' \<in> C"
```
```   269       and TC:   "T \<in> C"
```
```   270       and latt: "lattice C"
```
```   271   shows "T \<inter> (B' \<union> wp act M) \<in> C"
```
```   272 proof -
```
```   273   from TBwp have "(T\<inter>B) \<union> (T \<inter> wp act M) \<in> C"
```
```   274     by (simp add: Int_Un_distrib)
```
```   275   then have TBBC: "(T\<inter>B') \<union> ((T\<inter>B) \<union> (T \<inter> wp act M)) \<in> C"
```
```   276     by (blast intro: Int_in_lattice Un_in_lattice TC B'C latt)
```
```   277   show ?thesis
```
```   278     by (rule eqelem_imp_iff [THEN iffD1, OF _ TBBC],
```
```   279         blast intro: BB' [THEN subsetD])
```
```   280 qed
```
```   281
```
```   282
```
```   283 lemma progress_set_mono:
```
```   284     assumes BB':  "B \<subseteq> B'"
```
```   285     shows
```
```   286      "[| B' \<in> C;  C \<in> progress_set F T B|]
```
```   287       ==> C \<in> progress_set F T B'"
```
```   288 by (simp add: progress_set_def closed_def closed_mono [OF BB']
```
```   289                  subset_trans [OF BB'])
```
```   290
```
```   291 theorem progress_set_Union:
```
```   292   assumes leadsTo: "F \<in> A leadsTo B'"
```
```   293       and prog: "C \<in> progress_set F T B"
```
```   294       and Fstable: "F \<in> stable T"
```
```   295       and BB':  "B \<subseteq> B'"
```
```   296       and B'C:  "B' \<in> C"
```
```   297       and Gco: "!!X. X\<in>C ==> G \<in> X-B co X"
```
```   298   shows "F\<squnion>G \<in> T\<inter>A leadsTo B'"
```
```   299 apply (insert prog Fstable)
```
```   300 apply (rule leadsTo_Join [OF leadsTo])
```
```   301   apply (force simp add: progress_set_def awp_iff_stable [symmetric])
```
```   302 apply (simp add: awp_iff_constrains)
```
```   303 apply (drule progress_set_mono [OF BB' B'C])
```
```   304 apply (blast intro: progress_set_lemma Gco constrains_weaken_L
```
```   305                     BB' [THEN subsetD])
```
```   306 done
```
```   307
```
```   308
```
```   309 subsection {*Some Progress Sets*}
```
```   310
```
```   311 lemma UNIV_in_progress_set: "UNIV \<in> progress_set F T B"
```
```   312 by (simp add: progress_set_def lattice_def closed_def)
```
```   313
```
```   314
```
```   315
```
```   316 subsubsection {*Lattices and Relations*}
```
```   317 text{*From Meier's thesis, section 4.5.3*}
```
```   318
```
```   319 definition relcl :: "'a set set => ('a * 'a) set" where
```
```   320     -- {*Derived relation from a lattice*}
```
```   321     "relcl L == {(x,y). y \<in> cl L {x}}"
```
```   322
```
```   323 definition latticeof :: "('a * 'a) set => 'a set set" where
```
```   324     -- {*Derived lattice from a relation: the set of upwards-closed sets*}
```
```   325     "latticeof r == {X. \<forall>s t. s \<in> X & (s,t) \<in> r --> t \<in> X}"
```
```   326
```
```   327
```
```   328 lemma relcl_refl: "(a,a) \<in> relcl L"
```
```   329 by (simp add: relcl_def subset_cl [THEN subsetD])
```
```   330
```
```   331 lemma relcl_trans:
```
```   332      "[| (a,b) \<in> relcl L; (b,c) \<in> relcl L; lattice L |] ==> (a,c) \<in> relcl L"
```
```   333 apply (simp add: relcl_def)
```
```   334 apply (blast intro: clD cl_in_lattice)
```
```   335 done
```
```   336
```
```   337 lemma refl_relcl: "lattice L ==> refl (relcl L)"
```
```   338 by (simp add: refl_onI relcl_def subset_cl [THEN subsetD])
```
```   339
```
```   340 lemma trans_relcl: "lattice L ==> trans (relcl L)"
```
```   341 by (blast intro: relcl_trans transI)
```
```   342
```
```   343 lemma lattice_latticeof: "lattice (latticeof r)"
```
```   344 by (auto simp add: lattice_def latticeof_def)
```
```   345
```
```   346 lemma lattice_singletonI:
```
```   347      "[|lattice L; !!s. s \<in> X ==> {s} \<in> L|] ==> X \<in> L"
```
```   348 apply (cut_tac UN_singleton [of X])
```
```   349 apply (erule subst)
```
```   350 apply (simp only: UN_in_lattice)
```
```   351 done
```
```   352
```
```   353 text{*Equation (4.71) of Meier's thesis.  He gives no proof.*}
```
```   354 lemma cl_latticeof:
```
```   355      "[|refl r; trans r|]
```
```   356       ==> cl (latticeof r) X = {t. \<exists>s. s\<in>X & (s,t) \<in> r}"
```
```   357 apply (rule equalityI)
```
```   358  apply (rule cl_least)
```
```   359   apply (simp (no_asm_use) add: latticeof_def trans_def, blast)
```
```   360  apply (simp add: latticeof_def refl_on_def, blast)
```
```   361 apply (simp add: latticeof_def, clarify)
```
```   362 apply (unfold cl_def, blast)
```
```   363 done
```
```   364
```
```   365 text{*Related to (4.71).*}
```
```   366 lemma cl_eq_Collect_relcl:
```
```   367      "lattice L ==> cl L X = {t. \<exists>s. s\<in>X & (s,t) \<in> relcl L}"
```
```   368 apply (cut_tac UN_singleton [of X])
```
```   369 apply (erule subst)
```
```   370 apply (force simp only: relcl_def cl_UN)
```
```   371 done
```
```   372
```
```   373 text{*Meier's theorem of section 4.5.3*}
```
```   374 theorem latticeof_relcl_eq: "lattice L ==> latticeof (relcl L) = L"
```
```   375 apply (rule equalityI)
```
```   376  prefer 2 apply (force simp add: latticeof_def relcl_def cl_def, clarify)
```
```   377 apply (rename_tac X)
```
```   378 apply (rule cl_subset_in_lattice)
```
```   379  prefer 2 apply assumption
```
```   380 apply (drule cl_ident_iff [OF lattice_latticeof, THEN iffD2])
```
```   381 apply (drule equalityD1)
```
```   382 apply (rule subset_trans)
```
```   383  prefer 2 apply assumption
```
```   384 apply (thin_tac "_ \<subseteq> X")
```
```   385 apply (cut_tac A=X in UN_singleton)
```
```   386 apply (erule subst)
```
```   387 apply (simp only: cl_UN lattice_latticeof
```
```   388                   cl_latticeof [OF refl_relcl trans_relcl])
```
```   389 apply (simp add: relcl_def)
```
```   390 done
```
```   391
```
```   392 theorem relcl_latticeof_eq:
```
```   393      "[|refl r; trans r|] ==> relcl (latticeof r) = r"
```
```   394 by (simp add: relcl_def cl_latticeof)
```
```   395
```
```   396
```
```   397 subsubsection {*Decoupling Theorems*}
```
```   398
```
```   399 definition decoupled :: "['a program, 'a program] => bool" where
```
```   400    "decoupled F G ==
```
```   401         \<forall>act \<in> Acts F. \<forall>B. G \<in> stable B --> G \<in> stable (wp act B)"
```
```   402
```
```   403
```
```   404 text{*Rao's Decoupling Theorem*}
```
```   405 lemma stableco: "F \<in> stable A ==> F \<in> A-B co A"
```
```   406 by (simp add: stable_def constrains_def, blast)
```
```   407
```
```   408 theorem decoupling:
```
```   409   assumes leadsTo: "F \<in> A leadsTo B"
```
```   410       and Gstable: "G \<in> stable B"
```
```   411       and dec:     "decoupled F G"
```
```   412   shows "F\<squnion>G \<in> A leadsTo B"
```
```   413 proof -
```
```   414   have prog: "{X. G \<in> stable X} \<in> progress_set F UNIV B"
```
```   415     by (simp add: progress_set_def lattice_stable Gstable closed_def
```
```   416                   stable_Un [OF Gstable] dec [unfolded decoupled_def])
```
```   417   have "F\<squnion>G \<in> (UNIV\<inter>A) leadsTo B"
```
```   418     by (rule progress_set_Union [OF leadsTo prog],
```
```   419         simp_all add: Gstable stableco)
```
```   420   thus ?thesis by simp
```
```   421 qed
```
```   422
```
```   423
```
```   424 text{*Rao's Weak Decoupling Theorem*}
```
```   425 theorem weak_decoupling:
```
```   426   assumes leadsTo: "F \<in> A leadsTo B"
```
```   427       and stable: "F\<squnion>G \<in> stable B"
```
```   428       and dec:     "decoupled F (F\<squnion>G)"
```
```   429   shows "F\<squnion>G \<in> A leadsTo B"
```
```   430 proof -
```
```   431   have prog: "{X. F\<squnion>G \<in> stable X} \<in> progress_set F UNIV B"
```
```   432     by (simp del: Join_stable
```
```   433              add: progress_set_def lattice_stable stable closed_def
```
```   434                   stable_Un [OF stable] dec [unfolded decoupled_def])
```
```   435   have "F\<squnion>G \<in> (UNIV\<inter>A) leadsTo B"
```
```   436     by (rule progress_set_Union [OF leadsTo prog],
```
```   437         simp_all del: Join_stable add: stable,
```
```   438         simp add: stableco)
```
```   439   thus ?thesis by simp
```
```   440 qed
```
```   441
```
```   442 text{*The ``Decoupling via @{term G'} Union Theorem''*}
```
```   443 theorem decoupling_via_aux:
```
```   444   assumes leadsTo: "F \<in> A leadsTo B"
```
```   445       and prog: "{X. G' \<in> stable X} \<in> progress_set F UNIV B"
```
```   446       and GG':  "G \<le> G'"
```
```   447                --{*Beware!  This is the converse of the refinement relation!*}
```
```   448   shows "F\<squnion>G \<in> A leadsTo B"
```
```   449 proof -
```
```   450   from prog have stable: "G' \<in> stable B"
```
```   451     by (simp add: progress_set_def)
```
```   452   have "F\<squnion>G \<in> (UNIV\<inter>A) leadsTo B"
```
```   453     by (rule progress_set_Union [OF leadsTo prog],
```
```   454         simp_all add: stable stableco component_stable [OF GG'])
```
```   455   thus ?thesis by simp
```
```   456 qed
```
```   457
```
```   458
```
```   459 subsection{*Composition Theorems Based on Monotonicity and Commutativity*}
```
```   460
```
```   461 subsubsection{*Commutativity of @{term "cl L"} and assignment.*}
```
```   462 definition commutes :: "['a program, 'a set, 'a set,  'a set set] => bool" where
```
```   463    "commutes F T B L ==
```
```   464        \<forall>M. \<forall>act \<in> Acts F. B \<subseteq> M -->
```
```   465            cl L (T \<inter> wp act M) \<subseteq> T \<inter> (B \<union> wp act (cl L (T\<inter>M)))"
```
```   466
```
```   467
```
```   468 text{*From Meier's thesis, section 4.5.6*}
```
```   469 lemma commutativity1_lemma:
```
```   470   assumes commutes: "commutes F T B L"
```
```   471       and lattice:  "lattice L"
```
```   472       and BL: "B \<in> L"
```
```   473       and TL: "T \<in> L"
```
```   474   shows "closed F T B L"
```
```   475 apply (simp add: closed_def, clarify)
```
```   476 apply (rule ProgressSets.cl_subset_in_lattice [OF _ lattice])
```
```   477 apply (simp add: Int_Un_distrib cl_Un [OF lattice]
```
```   478                  cl_ident Int_in_lattice [OF TL BL lattice] Un_upper1)
```
```   479 apply (subgoal_tac "cl L (T \<inter> wp act M) \<subseteq> T \<inter> (B \<union> wp act (cl L (T \<inter> M)))")
```
```   480  prefer 2
```
```   481  apply (cut_tac commutes, simp add: commutes_def)
```
```   482 apply (erule subset_trans)
```
```   483 apply (simp add: cl_ident)
```
```   484 apply (blast intro: rev_subsetD [OF _ wp_mono])
```
```   485 done
```
```   486
```
```   487 text{*Version packaged with @{thm progress_set_Union}*}
```
```   488 lemma commutativity1:
```
```   489   assumes leadsTo: "F \<in> A leadsTo B"
```
```   490       and lattice:  "lattice L"
```
```   491       and BL: "B \<in> L"
```
```   492       and TL: "T \<in> L"
```
```   493       and Fstable: "F \<in> stable T"
```
```   494       and Gco: "!!X. X\<in>L ==> G \<in> X-B co X"
```
```   495       and commutes: "commutes F T B L"
```
```   496   shows "F\<squnion>G \<in> T\<inter>A leadsTo B"
```
```   497 by (rule progress_set_Union [OF leadsTo _ Fstable subset_refl BL Gco],
```
```   498     simp add: progress_set_def commutativity1_lemma commutes lattice BL TL)
```
```   499
```
```   500
```
```   501
```
```   502 text{*Possibly move to Relation.thy, after @{term single_valued}*}
```
```   503 definition funof :: "[('a*'b)set, 'a] => 'b" where
```
```   504    "funof r == (\<lambda>x. THE y. (x,y) \<in> r)"
```
```   505
```
```   506 lemma funof_eq: "[|single_valued r; (x,y) \<in> r|] ==> funof r x = y"
```
```   507 by (simp add: funof_def single_valued_def, blast)
```
```   508
```
```   509 lemma funof_Pair_in:
```
```   510      "[|single_valued r; x \<in> Domain r|] ==> (x, funof r x) \<in> r"
```
```   511 by (force simp add: funof_eq)
```
```   512
```
```   513 lemma funof_in:
```
```   514      "[|r``{x} \<subseteq> A; single_valued r; x \<in> Domain r|] ==> funof r x \<in> A"
```
```   515 by (force simp add: funof_eq)
```
```   516
```
```   517 lemma funof_imp_wp: "[|funof act t \<in> A; single_valued act|] ==> t \<in> wp act A"
```
```   518 by (force simp add: in_wp_iff funof_eq)
```
```   519
```
```   520
```
```   521 subsubsection{*Commutativity of Functions and Relation*}
```
```   522 text{*Thesis, page 109*}
```
```   523
```
```   524 (*FIXME: this proof is still an ungodly mess*)
```
```   525 text{*From Meier's thesis, section 4.5.6*}
```
```   526 lemma commutativity2_lemma:
```
```   527   assumes dcommutes:
```
```   528       "\<And>act s t. act \<in> Acts F \<Longrightarrow> s \<in> T \<Longrightarrow> (s, t) \<in> relcl L \<Longrightarrow>
```
```   529         s \<in> B | t \<in> B | (funof act s, funof act t) \<in> relcl L"
```
```   530     and determ: "!!act. act \<in> Acts F ==> single_valued act"
```
```   531     and total: "!!act. act \<in> Acts F ==> Domain act = UNIV"
```
```   532     and lattice:  "lattice L"
```
```   533     and BL: "B \<in> L"
```
```   534     and TL: "T \<in> L"
```
```   535     and Fstable: "F \<in> stable T"
```
```   536   shows  "commutes F T B L"
```
```   537 proof -
```
```   538   { fix M and act and t
```
```   539     assume 1: "B \<subseteq> M" "act \<in> Acts F" "t \<in> cl L (T \<inter> wp act M)"
```
```   540     then have "\<exists>s. (s,t) \<in> relcl L \<and> s \<in> T \<inter> wp act M"
```
```   541       by (force simp add: cl_eq_Collect_relcl [OF lattice])
```
```   542     then obtain s where 2: "(s, t) \<in> relcl L" "s \<in> T" "s \<in> wp act M"
```
```   543       by blast
```
```   544     then have 3: "\<forall>u\<in>L. s \<in> u --> t \<in> u"
```
```   545       apply (intro ballI impI)
```
```   546       apply (subst cl_ident [symmetric], assumption)
```
```   547       apply (simp add: relcl_def)
```
```   548       apply (blast intro: cl_mono [THEN  rev_subsetD])
```
```   549       done
```
```   550     with 1 2 Fstable have 4: "funof act s \<in> T\<inter>M"
```
```   551       by (force intro!: funof_in
```
```   552         simp add: wp_def stable_def constrains_def determ total)
```
```   553     with 1 2 3 have 5: "s \<in> B | t \<in> B | (funof act s, funof act t) \<in> relcl L"
```
```   554       by (intro dcommutes) assumption+
```
```   555     with 1 2 3 4 have "t \<in> B | funof act t \<in> cl L (T\<inter>M)"
```
```   556       by (simp add: relcl_def) (blast intro: BL cl_mono [THEN  rev_subsetD])
```
```   557     with 1 2 3 4 5 have "t \<in> B | t \<in> wp act (cl L (T\<inter>M))"
```
```   558       by (blast intro: funof_imp_wp determ)
```
```   559     with 2 3 have "t \<in> T \<and> (t \<in> B \<or> t \<in> wp act (cl L (T \<inter> M)))"
```
```   560       by (blast intro: TL cl_mono [THEN  rev_subsetD])
```
```   561     then have"t \<in> T \<inter> (B \<union> wp act (cl L (T \<inter> M)))"
```
```   562       by simp
```
```   563   }
```
```   564   then show "commutes F T B L" unfolding commutes_def by clarify
```
```   565 qed
```
```   566
```
```   567 text{*Version packaged with @{thm progress_set_Union}*}
```
```   568 lemma commutativity2:
```
```   569   assumes leadsTo: "F \<in> A leadsTo B"
```
```   570       and dcommutes:
```
```   571         "\<forall>act \<in> Acts F.
```
```   572          \<forall>s \<in> T. \<forall>t. (s,t) \<in> relcl L -->
```
```   573                       s \<in> B | t \<in> B | (funof act s, funof act t) \<in> relcl L"
```
```   574       and determ: "!!act. act \<in> Acts F ==> single_valued act"
```
```   575       and total: "!!act. act \<in> Acts F ==> Domain act = UNIV"
```
```   576       and lattice:  "lattice L"
```
```   577       and BL: "B \<in> L"
```
```   578       and TL: "T \<in> L"
```
```   579       and Fstable: "F \<in> stable T"
```
```   580       and Gco: "!!X. X\<in>L ==> G \<in> X-B co X"
```
```   581   shows "F\<squnion>G \<in> T\<inter>A leadsTo B"
```
```   582 apply (rule commutativity1 [OF leadsTo lattice])
```
```   583 apply (simp_all add: Gco commutativity2_lemma dcommutes determ total
```
```   584                      lattice BL TL Fstable)
```
```   585 done
```
```   586
```
```   587
```
```   588 subsection {*Monotonicity*}
```
```   589 text{*From Meier's thesis, section 4.5.7, page 110*}
```
```   590 (*to be continued?*)
```
```   591
```
```   592 end
```