src/HOL/UNITY/Transformers.thy
 changeset 13832 e7649436869c parent 13821 0fd39aa77095 child 13851 f6923453953a
```     1.1 --- a/src/HOL/UNITY/Transformers.thy	Wed Feb 26 10:44:54 2003 +0100
1.2 +++ b/src/HOL/UNITY/Transformers.thy	Wed Feb 26 10:48:00 2003 +0100
1.3 @@ -19,13 +19,15 @@
1.4
1.5  constdefs
1.6    wp :: "[('a*'a) set, 'a set] => 'a set"
1.7 -    --{*Dijkstra's weakest-precondition operator*}
1.8 +    --{*Dijkstra's weakest-precondition operator (for an individual command)*}
1.9      "wp act B == - (act^-1 `` (-B))"
1.10
1.11 -  awp :: "[ 'a program, 'a set] => 'a set"
1.12 +  awp :: "['a program, 'a set] => 'a set"
1.13 +    --{*Dijkstra's weakest-precondition operator (for a program)*}
1.14      "awp F B == (\<Inter>act \<in> Acts F. wp act B)"
1.15
1.16 -  wens :: "[ 'a program, ('a*'a) set, 'a set] => 'a set"
1.17 +  wens :: "['a program, ('a*'a) set, 'a set] => 'a set"
1.18 +    --{*The weakest-ensures transformer*}
1.19      "wens F act B == gfp(\<lambda>X. (wp act B \<inter> awp F (B \<union> X)) \<union> B)"
1.20
1.21  text{*The fundamental theorem for wp*}
1.22 @@ -35,6 +37,13 @@
1.23  lemma Compl_Domain_subset_wp: "- (Domain act) \<subseteq> wp act B"
1.24  by (force simp add: wp_def)
1.25
1.26 +lemma wp_empty [simp]: "wp act {} = - (Domain act)"
1.27 +by (force simp add: wp_def)
1.28 +
1.29 +text{*The identity relation is the skip action*}
1.30 +lemma wp_Id [simp]: "wp Id B = B"
1.32 +
1.33  lemma awp_Int_eq: "awp F (A\<inter>B) = awp F A \<inter> awp F B"
1.34  by (simp add: awp_def wp_def, blast)
1.35
1.36 @@ -55,6 +64,9 @@
1.37  apply (simp add: mono_def wp_def awp_def, blast)
1.38  done
1.39
1.40 +lemma wens_Id [simp]: "wens F Id B = B"
1.41 +by (simp add: wens_def gfp_def wp_def awp_def, blast)
1.42 +
1.43  text{*These two theorems justify the claim that @{term wens} returns the
1.44  weakest assertion satisfying the ensures property*}
1.45  lemma ensures_imp_wens: "F \<in> A ensures B ==> \<exists>act \<in> Acts F. A \<subseteq> wens F act B"
1.46 @@ -93,8 +105,7 @@
1.48  apply (drule constrains_Un [OF Diff_wens_constrains [of F act A]])
1.49  apply (simp add: Un_Int_distrib2 Compl_partition2)
1.50 -apply (erule constrains_weaken)
1.51 - apply blast
1.52 +apply (erule constrains_weaken, blast)
1.53  apply (simp add: Un_subset_iff wens_weakening)
1.54  done
1.55
1.56 @@ -118,6 +129,7 @@
1.57  apply (rule subset_trans [OF _ wens_mono [of "T\<inter>B" B]], auto)
1.58  done
1.59
1.60 +
1.61  subsection{*Defining the Weakest Ensures Set*}
1.62
1.63  consts
1.64 @@ -146,15 +158,10 @@
1.65  lemma wens_set_imp_leadsTo: "A \<in> wens_set F B ==> F \<in> A leadsTo B"
1.66  apply (erule wens_set.induct)
1.69 + apply (blast intro: wens_ensures leadsTo_Trans)
1.71  done
1.72
1.73 -(*????????????????Set.thy Set.all_not_in_conv*)
1.74 -lemma ex_in_conv: "(\<exists>x. x \<in> A) = (A \<noteq> {})"
1.75 -by blast
1.76 -
1.77 -
1.78  lemma leadsTo_imp_wens_set: "F \<in> A leadsTo B ==> \<exists>C \<in> wens_set F B. A \<subseteq> C"
1.80    apply (blast dest!: ensures_imp_wens intro: wens_set.Basis wens_set.Wens)
1.81 @@ -169,13 +176,12 @@
1.82  done
1.83
1.84  text{*Assertion (9): 4.27 in the thesis.*}
1.85 -
1.86  lemma leadsTo_iff_wens_set: "(F \<in> A leadsTo B) = (\<exists>C \<in> wens_set F B. A \<subseteq> C)"
1.88
1.89  text{*This is the result that requires the definition of @{term wens_set} to
1.90 -require @{term W} to be non-empty in the Unio case, for otherwise we should
1.91 -always have @{term "{} \<in> wens_set F B"}.*}
1.92 +  require @{term W} to be non-empty in the Unio case, for otherwise we should
1.93 +  always have @{term "{} \<in> wens_set F B"}.*}
1.94  lemma wens_set_imp_subset: "A \<in> wens_set F B ==> B \<subseteq> A"
1.95  apply (erule wens_set.induct)
1.96    apply (blast intro: wens_weakening [THEN subsetD])+
1.97 @@ -240,7 +246,7 @@
1.98  apply (blast intro: awpG [THEN subsetD] wens_set.Wens)
1.99  done
1.100
1.101 -lemma wens_Union:
1.102 +theorem wens_Union:
1.103    assumes awpF: "T-B \<subseteq> awp F T"
1.104        and awpG: "!!X. X \<in> wens_set F B ==> (T\<inter>X) - B \<subseteq> awp G (T\<inter>X)"
1.105        and major: "X \<in> wens_set F B"
1.106 @@ -260,4 +266,193 @@
1.107  apply (blast intro: wens_set.Union)
1.108  done
1.109
1.111 +  assumes awpF: "T-B \<subseteq> awp F T"
1.112 +      and awpG: "!!X. X \<in> wens_set F B ==> (T\<inter>X) - B \<subseteq> awp G (T\<inter>X)"
1.114 +  shows "F\<squnion>G \<in> T\<inter>A leadsTo B"
1.116 +apply (rule wens_Union [THEN bexE])
1.117 +   apply (rule awpF)
1.118 +  apply (erule awpG)
1.119 + apply assumption
1.121 +done
1.122 +
1.123 +
1.124 +subsection {*The Set @{term "wens_set F B"} for a Single-Assignment Program*}
1.125 +text{*Thesis Section 4.3.3*}
1.126 +
1.127 +text{*We start by proving laws about single-assignment programs*}
1.128 +lemma awp_single_eq [simp]:
1.129 +     "awp (mk_program (init, {act}, allowed)) B = B \<inter> wp act B"
1.130 +by (force simp add: awp_def wp_def)
1.131 +
1.132 +lemma wp_Un_subset: "wp act A \<union> wp act B \<subseteq> wp act (A \<union> B)"
1.133 +by (force simp add: wp_def)
1.134 +
1.135 +lemma wp_Un_eq: "single_valued act ==> wp act (A \<union> B) = wp act A \<union> wp act B"
1.136 +apply (rule equalityI)
1.137 + apply (force simp add: wp_def single_valued_def)
1.138 +apply (rule wp_Un_subset)
1.139 +done
1.140 +
1.141 +lemma wp_UN_subset: "(\<Union>i\<in>I. wp act (A i)) \<subseteq> wp act (\<Union>i\<in>I. A i)"
1.142 +by (force simp add: wp_def)
1.143 +
1.144 +lemma wp_UN_eq:
1.145 +     "[|single_valued act; I\<noteq>{}|]
1.146 +      ==> wp act (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. wp act (A i))"
1.147 +apply (rule equalityI)
1.148 + prefer 2 apply (rule wp_UN_subset)
1.149 + apply (simp add: wp_def Image_INT_eq)
1.150 +done
1.151 +
1.152 +lemma wens_single_eq:
1.153 +     "wens (mk_program (init, {act}, allowed)) act B = B \<union> wp act B"
1.154 +by (simp add: wens_def gfp_def wp_def, blast)
1.155 +
1.156 +
1.157 +text{*Next, we express the @{term "wens_set"} for single-assignment programs*}
1.158 +
1.159 +constdefs
1.160 +  wens_single_finite :: "[('a*'a) set, 'a set, nat] => 'a set"
1.161 +    "wens_single_finite act B k == \<Union>i \<in> atMost k. ((wp act)^i) B"
1.162 +
1.163 +  wens_single :: "[('a*'a) set, 'a set] => 'a set"
1.164 +    "wens_single act B == \<Union>i. ((wp act)^i) B"
1.165 +
1.166 +lemma wens_single_Un_eq:
1.167 +      "single_valued act
1.168 +       ==> wens_single act B \<union> wp act (wens_single act B) = wens_single act B"
1.169 +apply (rule equalityI)
1.170 + apply (simp_all add: Un_upper1 Un_subset_iff)
1.171 +apply (simp add: wens_single_def wp_UN_eq, clarify)
1.172 +apply (rule_tac a="Suc(i)" in UN_I, auto)
1.173 +done
1.174 +
1.175 +lemma atMost_nat_nonempty: "atMost (k::nat) \<noteq> {}"
1.176 +by force
1.177 +
1.178 +lemma wens_single_finite_Suc:
1.179 +      "single_valued act
1.180 +       ==> wens_single_finite act B (Suc k) =
1.181 +           wens_single_finite act B k \<union> wp act (wens_single_finite act B k) "
1.182 +apply (simp add: wens_single_finite_def image_def
1.183 +                 wp_UN_eq [OF _ atMost_nat_nonempty])
1.184 +apply (force elim!: le_SucE)
1.185 +done
1.186 +
1.187 +lemma wens_single_finite_Suc_eq_wens:
1.188 +     "single_valued act
1.189 +       ==> wens_single_finite act B (Suc k) =
1.190 +           wens (mk_program (init, {act}, allowed)) act
1.191 +                (wens_single_finite act B k)"
1.192 +by (simp add: wens_single_finite_Suc wens_single_eq)
1.193 +
1.194 +lemma wens_single_finite_Un_eq:
1.195 +      "single_valued act
1.196 +       ==> wens_single_finite act B k \<union> wp act (wens_single_finite act B k)
1.197 +           \<in> range (wens_single_finite act B)"
1.198 +by (simp add: wens_single_finite_Suc [symmetric])
1.199 +
1.200 +lemma wens_single_eq_Union:
1.201 +      "wens_single act B = \<Union>range (wens_single_finite act B)"
1.202 +by (simp add: wens_single_finite_def wens_single_def, blast)
1.203 +
1.204 +lemma wens_single_finite_eq_Union:
1.205 +     "wens_single_finite act B n = (\<Union>k\<in>atMost n. wens_single_finite act B k)"
1.206 +apply (auto simp add: wens_single_finite_def)
1.207 +apply (blast intro: le_trans)
1.208 +done
1.209 +
1.210 +lemma wens_single_finite_mono:
1.211 +     "m \<le> n ==> wens_single_finite act B m \<subseteq> wens_single_finite act B n"
1.212 +by (force simp add:  wens_single_finite_eq_Union [of act B n])
1.213 +
1.214 +lemma wens_single_finite_subset_wens_single:
1.215 +      "wens_single_finite act B k \<subseteq> wens_single act B"
1.216 +by (simp add: wens_single_eq_Union, blast)
1.217 +
1.218 +lemma subset_wens_single_finite:
1.219 +      "[|W \<subseteq> wens_single_finite act B ` (atMost k); single_valued act; W\<noteq>{}|]
1.220 +       ==> \<exists>m. \<Union>W = wens_single_finite act B m"
1.221 +apply (induct k)
1.222 + apply (simp, blast)
1.223 +apply (auto simp add: atMost_Suc)
1.224 +apply (case_tac "wens_single_finite act B (Suc n) \<in> W")
1.225 + prefer 2 apply blast
1.226 +apply (drule_tac x="Suc n" in spec)
1.227 +apply (erule notE, rule equalityI)
1.228 + prefer 2 apply blast
1.229 +apply (subst wens_single_finite_eq_Union)
1.230 +apply (simp add: atMost_Suc, blast)
1.231 +done
1.232 +
1.233 +text{*lemma for Union case*}
1.234 +lemma Union_eq_wens_single:
1.235 +      "\<lbrakk>\<forall>k. \<not> W \<subseteq> wens_single_finite act B ` {..k};
1.236 +        W \<subseteq> insert (wens_single act B)
1.237 +            (range (wens_single_finite act B))\<rbrakk>
1.238 +       \<Longrightarrow> \<Union>W = wens_single act B"
1.239 +apply (case_tac "wens_single act B \<in> W")
1.240 + apply (blast dest: wens_single_finite_subset_wens_single [THEN subsetD])
1.242 +apply (rule equalityI)
1.243 + apply blast
1.244 +apply (simp add: UN_subset_iff, clarify)
1.245 +apply (subgoal_tac "\<exists>y\<in>W. \<exists>n. y = wens_single_finite act B n & i\<le>n")
1.246 + apply (blast intro: wens_single_finite_mono [THEN subsetD] )
1.247 +apply (drule_tac x=i in spec)
1.248 +apply (force simp add: atMost_def)
1.249 +done
1.250 +
1.251 +lemma wens_set_subset_single:
1.252 +      "single_valued act
1.253 +       ==> wens_set (mk_program (init, {act}, allowed)) B \<subseteq>
1.254 +           insert (wens_single act B) (range (wens_single_finite act B))"
1.255 +apply (rule subsetI)
1.256 +apply (erule wens_set.induct)
1.257 +  txt{*Basis*}
1.258 +  apply (force simp add: wens_single_finite_def)
1.259 + txt{*Wens inductive step*}
1.260 + apply (case_tac "acta = Id", simp)
1.261 + apply (simp add: wens_single_eq)
1.262 + apply (elim disjE)
1.263 + apply (simp add: wens_single_Un_eq)
1.264 + apply (force simp add: wens_single_finite_Un_eq)
1.265 +txt{*Union inductive step*}
1.266 +apply (case_tac "\<exists>k. W \<subseteq> wens_single_finite act B ` (atMost k)")
1.267 + apply (blast dest!: subset_wens_single_finite, simp)
1.268 +apply (rule disjI1 [OF Union_eq_wens_single], blast+)
1.269 +done
1.270 +
1.271 +lemma wens_single_finite_in_wens_set:
1.272 +      "single_valued act \<Longrightarrow>
1.273 +         wens_single_finite act B k
1.274 +         \<in> wens_set (mk_program (init, {act}, allowed)) B"
1.275 +apply (induct_tac k)
1.276 + apply (simp add: wens_single_finite_def wens_set.Basis)
1.278 +                 wens_single_finite_Suc_eq_wens [of act B _ init allowed])
1.279 +done
1.280 +
1.281 +lemma single_subset_wens_set:
1.282 +      "single_valued act
1.283 +       ==> insert (wens_single act B) (range (wens_single_finite act B)) \<subseteq>
1.284 +           wens_set (mk_program (init, {act}, allowed)) B"
1.285 +apply (simp add: wens_single_eq_Union UN_eq)
1.286 +apply (blast intro: wens_set.Union wens_single_finite_in_wens_set)
1.287 +done
1.288 +
1.289 +text{*Theorem (4.29)*}
1.290 +theorem wens_set_single_eq:
1.291 +      "single_valued act
1.292 +       ==> wens_set (mk_program (init, {act}, allowed)) B =
1.293 +           insert (wens_single act B) (range (wens_single_finite act B))"
1.294 +apply (rule equalityI)
1.295 +apply (erule wens_set_subset_single)
1.296 +apply (erule single_subset_wens_set)
1.297 +done
1.298 +
1.299  end
```