1.1 --- a/src/HOL/IMP/Abs_State.thy Wed Mar 06 14:10:07 2013 +0100
1.2 +++ b/src/HOL/IMP/Abs_State.thy Wed Mar 06 16:10:56 2013 +0100
1.3 @@ -60,21 +60,21 @@
1.4 end
1.5
1.6 class semilatticeL = join + L +
1.7 -fixes top :: "vname set \<Rightarrow> 'a"
1.8 -assumes join_ge1 [simp]: "x \<in> L X \<Longrightarrow> y \<in> L X \<Longrightarrow> x \<sqsubseteq> x \<squnion> y"
1.9 -and join_ge2 [simp]: "x \<in> L X \<Longrightarrow> y \<in> L X \<Longrightarrow> y \<sqsubseteq> x \<squnion> y"
1.10 -and join_least[simp]: "x \<sqsubseteq> z \<Longrightarrow> y \<sqsubseteq> z \<Longrightarrow> x \<squnion> y \<sqsubseteq> z"
1.11 -and top[simp]: "x \<in> L X \<Longrightarrow> x \<sqsubseteq> top X"
1.12 -and top_in_L[simp]: "top X \<in> L X"
1.13 +fixes Top :: "vname set \<Rightarrow> 'a"
1.14 +assumes join_ge1 [simp]: "x \<in> L X \<Longrightarrow> y \<in> L X \<Longrightarrow> x \<le> x \<squnion> y"
1.15 +and join_ge2 [simp]: "x \<in> L X \<Longrightarrow> y \<in> L X \<Longrightarrow> y \<le> x \<squnion> y"
1.16 +and join_least[simp]: "x \<le> z \<Longrightarrow> y \<le> z \<Longrightarrow> x \<squnion> y \<le> z"
1.17 +and Top[simp]: "x \<in> L X \<Longrightarrow> x \<le> Top X"
1.18 +and Top_in_L[simp]: "Top X \<in> L X"
1.19 and join_in_L[simp]: "x \<in> L X \<Longrightarrow> y \<in> L X \<Longrightarrow> x \<squnion> y \<in> L X"
1.20
1.21 -notation (input) top ("\<top>\<^bsub>_\<^esub>")
1.22 -notation (latex output) top ("\<top>\<^bsub>\<^raw:\isa{>_\<^raw:}>\<^esub>")
1.23 +notation (input) Top ("\<top>\<^bsub>_\<^esub>")
1.24 +notation (latex output) Top ("\<top>\<^bsub>\<^raw:\isa{>_\<^raw:}>\<^esub>")
1.25
1.26 instantiation option :: (semilatticeL)semilatticeL
1.27 begin
1.28
1.29 -definition top_option where "top c = Some(top c)"
1.30 +definition Top_option where "Top c = Some(Top c)"
1.31
1.32 instance proof
1.33 case goal1 thus ?case by(cases x, simp, cases y, simp_all)
1.34 @@ -83,9 +83,9 @@
1.35 next
1.36 case goal3 thus ?case by(cases z, simp, cases y, simp, cases x, simp_all)
1.37 next
1.38 - case goal4 thus ?case by(cases x, simp_all add: top_option_def)
1.39 + case goal4 thus ?case by(cases x, simp_all add: Top_option_def)
1.40 next
1.41 - case goal5 thus ?case by(simp add: top_option_def)
1.42 + case goal5 thus ?case by(simp add: Top_option_def)
1.43 next
1.44 case goal6 thus ?case by(simp add: L_option_def split: option.splits)
1.45 qed
1.46 @@ -97,41 +97,77 @@
1.47
1.48 hide_type st --"to avoid long names"
1.49
1.50 -text{* A concrete type of state with computable @{text"\<sqsubseteq>"}: *}
1.51 +text{* A concrete type of state with computable @{text"\<le>"}: *}
1.52 +
1.53 +fun st :: "(vname \<Rightarrow> 'a) * vname set \<Rightarrow> bool" where
1.54 +"st (f,X) = (\<forall>x. x \<notin> X \<longrightarrow> f x = undefined)"
1.55
1.56 -datatype 'a st = FunDom "vname \<Rightarrow> 'a" "vname set"
1.57 +typedef 'a st = "{p :: (vname \<Rightarrow> 'a) * vname set. st p}"
1.58 +proof
1.59 + show "(\<lambda>x. undefined,{}) \<in> {p. st p}" by simp
1.60 +qed
1.61 +
1.62 +setup_lifting type_definition_st
1.63 +
1.64 +lift_definition St :: "(vname \<Rightarrow> 'a) \<Rightarrow> vname set \<Rightarrow> 'a st" is
1.65 + "\<lambda>f X. (\<lambda>x. if x \<in> X then f x else undefined, X)"
1.66 +by(simp)
1.67
1.68 -fun "fun" where "fun (FunDom f X) = f"
1.69 -fun dom where "dom (FunDom f X) = X"
1.70 +lift_definition update :: "'a st \<Rightarrow> vname \<Rightarrow> 'a \<Rightarrow> 'a st" is
1.71 + "\<lambda>(f,X) x a. (f(x := a), insert x X)"
1.72 +by(auto)
1.73 +
1.74 +lift_definition "fun" :: "'a st \<Rightarrow> vname \<Rightarrow> 'a" is fst ..
1.75 +
1.76 +lift_definition dom :: "'a st \<Rightarrow> vname set" is snd ..
1.77
1.78 -definition "show_st S = (\<lambda>x. (x, fun S x)) ` dom S"
1.79 +lemma dom_St[simp]: "dom(St f X) = X"
1.80 +by(simp add: St.rep_eq dom.rep_eq)
1.81
1.82 -value [code] "show_st (FunDom (\<lambda>x. 1::int) {''a'',''b''})"
1.83 +lemma fun_St[simp]: "fun(St f X) = (\<lambda>x. if x \<in> X then f x else undefined)"
1.84 +by(simp add: St.rep_eq fun.rep_eq)
1.85 +
1.86 +definition show_st :: "'a st \<Rightarrow> (vname * 'a)set" where
1.87 +"show_st S = (\<lambda>x. (x, fun S x)) ` dom S"
1.88
1.89 definition "show_acom = map_acom (Option.map show_st)"
1.90 definition "show_acom_opt = Option.map show_acom"
1.91
1.92 -definition "update F x y = FunDom ((fun F)(x:=y)) (dom F)"
1.93 +lemma fun_update[simp]: "fun (update S x y) = (fun S)(x:=y)"
1.94 +by transfer auto
1.95
1.96 -lemma fun_update[simp]: "fun (update S x y) = (fun S)(x:=y)"
1.97 -by(rule ext)(auto simp: update_def)
1.98 +lemma dom_update[simp]: "dom (update S x y) = insert x (dom S)"
1.99 +by transfer auto
1.100 +
1.101 +definition \<gamma>_st :: "('a \<Rightarrow> 'b set) \<Rightarrow> 'a st \<Rightarrow> (vname \<Rightarrow> 'b) set" where
1.102 +"\<gamma>_st \<gamma> F = {f. \<forall>x\<in>dom F. f x \<in> \<gamma>(fun F x)}"
1.103
1.104 -lemma dom_update[simp]: "dom (update S x y) = dom S"
1.105 -by(simp add: update_def)
1.106 +lemma ext_st: "dom F = dom G \<Longrightarrow> \<forall>x \<in> dom G. fun F x = fun G x \<Longrightarrow> F=G"
1.107 +apply(induct F rule:Abs_st_induct)
1.108 +apply(induct G rule:Abs_st_induct)
1.109 +apply (auto simp:Abs_st_inject fun_def dom_def Abs_st_inverse simp del: st.simps)
1.110 +apply(rule ext)
1.111 +apply auto
1.112 +done
1.113
1.114 -definition "\<gamma>_st \<gamma> F = {f. \<forall>x\<in>dom F. f x \<in> \<gamma>(fun F x)}"
1.115 -
1.116 -
1.117 -instantiation st :: (preord) preord
1.118 +instantiation st :: (order) order
1.119 begin
1.120
1.121 -definition le_st :: "'a st \<Rightarrow> 'a st \<Rightarrow> bool" where
1.122 -"F \<sqsubseteq> G = (dom F = dom G \<and> (\<forall>x \<in> dom F. fun F x \<sqsubseteq> fun G x))"
1.123 +definition less_eq_st :: "'a st \<Rightarrow> 'a st \<Rightarrow> bool" where
1.124 +"F \<le> (G::'a st) = (dom F = dom G \<and> (\<forall>x \<in> dom F. fun F x \<le> fun G x))"
1.125 +
1.126 +definition less_st where "F < (G::'a st) = (F \<le> G \<and> \<not> G \<le> F)"
1.127
1.128 instance
1.129 proof
1.130 - case goal2 thus ?case by(auto simp: le_st_def)(metis preord_class.le_trans)
1.131 -qed (auto simp: le_st_def)
1.132 + case goal1 show ?case by(rule less_st_def)
1.133 +next
1.134 + case goal2 show ?case by(auto simp: less_eq_st_def)
1.135 +next
1.136 + case goal3 thus ?case by(fastforce simp: less_eq_st_def)
1.137 +next
1.138 + case goal4 thus ?case by (metis less_eq_st_def antisym ext_st)
1.139 +qed
1.140
1.141 end
1.142
1.143 @@ -139,7 +175,7 @@
1.144 begin
1.145
1.146 definition join_st :: "'a st \<Rightarrow> 'a st \<Rightarrow> 'a st" where
1.147 -"F \<squnion> G = FunDom (\<lambda>x. fun F x \<squnion> fun G x) (dom F)"
1.148 +"F \<squnion> (G::'a st) = St (\<lambda>x. fun F x \<squnion> fun G x) (dom F)"
1.149
1.150 instance ..
1.151
1.152 @@ -149,7 +185,7 @@
1.153 begin
1.154
1.155 definition L_st :: "vname set \<Rightarrow> 'a st set" where
1.156 -"L X = {F. dom F = X}"
1.157 +"L(X::vname set) = {F. dom F = X}"
1.158
1.159 instance ..
1.160
1.161 @@ -158,11 +194,10 @@
1.162 instantiation st :: (semilattice) semilatticeL
1.163 begin
1.164
1.165 -definition top_st where "top X = FunDom (\<lambda>x. \<top>) X"
1.166 +definition Top_st :: "vname set \<Rightarrow> 'a st" where "Top(X) = St (\<lambda>x. \<top>) X"
1.167
1.168 instance
1.169 -proof
1.170 -qed (auto simp: le_st_def join_st_def top_st_def L_st_def)
1.171 +proof qed(auto simp add: less_eq_st_def join_st_def Top_st_def L_st_def)
1.172
1.173 end
1.174
1.175 @@ -175,12 +210,12 @@
1.176 But L is not executable. This looping defn makes it look as if it were. *)
1.177
1.178
1.179 -lemma mono_fun: "F \<sqsubseteq> G \<Longrightarrow> x : dom F \<Longrightarrow> fun F x \<sqsubseteq> fun G x"
1.180 -by(auto simp: le_st_def)
1.181 +lemma mono_fun: "F \<le> G \<Longrightarrow> x : dom F \<Longrightarrow> fun F x \<le> fun G x"
1.182 +by(auto simp: less_eq_st_def)
1.183
1.184 lemma mono_update[simp]:
1.185 - "a1 \<sqsubseteq> a2 \<Longrightarrow> S1 \<sqsubseteq> S2 \<Longrightarrow> update S1 x a1 \<sqsubseteq> update S2 x a2"
1.186 -by(auto simp add: le_st_def update_def)
1.187 + "a1 \<le> a2 \<Longrightarrow> S1 \<le> S2 \<Longrightarrow> update S1 x a1 \<le> update S2 x a2"
1.188 +by(auto simp add: less_eq_st_def)
1.189
1.190
1.191 locale Gamma = Val_abs where \<gamma>=\<gamma> for \<gamma> :: "'av::semilattice \<Rightarrow> val set"
1.192 @@ -195,22 +230,22 @@
1.193 abbreviation \<gamma>\<^isub>c :: "'av st option acom \<Rightarrow> state set acom"
1.194 where "\<gamma>\<^isub>c == map_acom \<gamma>\<^isub>o"
1.195
1.196 -lemma gamma_s_Top[simp]: "\<gamma>\<^isub>s (top c) = UNIV"
1.197 -by(auto simp: top_st_def \<gamma>_st_def)
1.198 +lemma gamma_s_Top[simp]: "\<gamma>\<^isub>s (Top X) = UNIV"
1.199 +by(auto simp: Top_st_def \<gamma>_st_def)
1.200
1.201 -lemma gamma_o_Top[simp]: "\<gamma>\<^isub>o (top c) = UNIV"
1.202 -by (simp add: top_option_def)
1.203 +lemma gamma_o_Top[simp]: "\<gamma>\<^isub>o (Top X) = UNIV"
1.204 +by (simp add: Top_option_def)
1.205
1.206 -lemma mono_gamma_s: "f \<sqsubseteq> g \<Longrightarrow> \<gamma>\<^isub>s f \<subseteq> \<gamma>\<^isub>s g"
1.207 -apply(simp add:\<gamma>_st_def subset_iff le_st_def split: if_splits)
1.208 +lemma mono_gamma_s: "f \<le> g \<Longrightarrow> \<gamma>\<^isub>s f \<subseteq> \<gamma>\<^isub>s g"
1.209 +apply(simp add:\<gamma>_st_def subset_iff less_eq_st_def split: if_splits)
1.210 by (metis mono_gamma subsetD)
1.211
1.212 lemma mono_gamma_o:
1.213 - "S1 \<sqsubseteq> S2 \<Longrightarrow> \<gamma>\<^isub>o S1 \<subseteq> \<gamma>\<^isub>o S2"
1.214 -by(induction S1 S2 rule: le_option.induct)(simp_all add: mono_gamma_s)
1.215 + "S1 \<le> S2 \<Longrightarrow> \<gamma>\<^isub>o S1 \<subseteq> \<gamma>\<^isub>o S2"
1.216 +by(induction S1 S2 rule: less_eq_option.induct)(simp_all add: mono_gamma_s)
1.217
1.218 -lemma mono_gamma_c: "C1 \<sqsubseteq> C2 \<Longrightarrow> \<gamma>\<^isub>c C1 \<le> \<gamma>\<^isub>c C2"
1.219 -by (induction C1 C2 rule: le_acom.induct) (simp_all add:mono_gamma_o)
1.220 +lemma mono_gamma_c: "C1 \<le> C2 \<Longrightarrow> \<gamma>\<^isub>c C1 \<le> \<gamma>\<^isub>c C2"
1.221 +by (induction C1 C2 rule: less_eq_acom.induct) (simp_all add:mono_gamma_o)
1.222
1.223 lemma in_gamma_option_iff:
1.224 "x : \<gamma>_option r u \<longleftrightarrow> (\<exists>u'. u = Some u' \<and> x : r u')"