--- a/src/HOL/TLA/Action.thy Fri Jun 26 11:44:22 2015 +0200
+++ b/src/HOL/TLA/Action.thy Fri Jun 26 14:53:15 2015 +0200
@@ -12,40 +12,40 @@
(** abstract syntax **)
-type_synonym 'a trfun = "(state * state) => 'a"
+type_synonym 'a trfun = "(state * state) \<Rightarrow> 'a"
type_synonym action = "bool trfun"
instance prod :: (world, world) world ..
consts
(** abstract syntax **)
- before :: "'a stfun => 'a trfun"
- after :: "'a stfun => 'a trfun"
- unch :: "'a stfun => action"
+ before :: "'a stfun \<Rightarrow> 'a trfun"
+ after :: "'a stfun \<Rightarrow> 'a trfun"
+ unch :: "'a stfun \<Rightarrow> action"
- SqAct :: "[action, 'a stfun] => action"
- AnAct :: "[action, 'a stfun] => action"
- enabled :: "action => stpred"
+ SqAct :: "[action, 'a stfun] \<Rightarrow> action"
+ AnAct :: "[action, 'a stfun] \<Rightarrow> action"
+ enabled :: "action \<Rightarrow> stpred"
(** concrete syntax **)
syntax
(* Syntax for writing action expressions in arbitrary contexts *)
- "_ACT" :: "lift => 'a" ("(ACT _)")
+ "_ACT" :: "lift \<Rightarrow> 'a" ("(ACT _)")
- "_before" :: "lift => lift" ("($_)" [100] 99)
- "_after" :: "lift => lift" ("(_$)" [100] 99)
- "_unchanged" :: "lift => lift" ("(unchanged _)" [100] 99)
+ "_before" :: "lift \<Rightarrow> lift" ("($_)" [100] 99)
+ "_after" :: "lift \<Rightarrow> lift" ("(_$)" [100] 99)
+ "_unchanged" :: "lift \<Rightarrow> lift" ("(unchanged _)" [100] 99)
(*** Priming: same as "after" ***)
- "_prime" :: "lift => lift" ("(_`)" [100] 99)
+ "_prime" :: "lift \<Rightarrow> lift" ("(_`)" [100] 99)
- "_SqAct" :: "[lift, lift] => lift" ("([_]'_(_))" [0,1000] 99)
- "_AnAct" :: "[lift, lift] => lift" ("(<_>'_(_))" [0,1000] 99)
- "_Enabled" :: "lift => lift" ("(Enabled _)" [100] 100)
+ "_SqAct" :: "[lift, lift] \<Rightarrow> lift" ("([_]'_(_))" [0,1000] 99)
+ "_AnAct" :: "[lift, lift] \<Rightarrow> lift" ("(<_>'_(_))" [0,1000] 99)
+ "_Enabled" :: "lift \<Rightarrow> lift" ("(Enabled _)" [100] 100)
translations
- "ACT A" => "(A::state*state => _)"
+ "ACT A" => "(A::state*state \<Rightarrow> _)"
"_before" == "CONST before"
"_after" == "CONST after"
"_prime" => "_after"
@@ -59,16 +59,16 @@
"w |= unchanged f" <= "_unchanged f w"
axiomatization where
- unl_before: "(ACT $v) (s,t) == v s" and
- unl_after: "(ACT v$) (s,t) == v t" and
+ unl_before: "(ACT $v) (s,t) \<equiv> v s" and
+ unl_after: "(ACT v$) (s,t) \<equiv> v t" and
- unchanged_def: "(s,t) |= unchanged v == (v t = v s)"
+ unchanged_def: "(s,t) \<Turnstile> unchanged v \<equiv> (v t = v s)"
defs
- square_def: "ACT [A]_v == ACT (A | unchanged v)"
- angle_def: "ACT <A>_v == ACT (A & \<not> unchanged v)"
+ square_def: "ACT [A]_v \<equiv> ACT (A \<or> unchanged v)"
+ angle_def: "ACT <A>_v \<equiv> ACT (A \<and> \<not> unchanged v)"
- enabled_def: "s |= Enabled A == \<exists>u. (s,u) |= A"
+ enabled_def: "s \<Turnstile> Enabled A \<equiv> \<exists>u. (s,u) \<Turnstile> A"
(* The following assertion specializes "intI" for any world type
@@ -76,22 +76,22 @@
*)
lemma actionI [intro!]:
- assumes "\<And>s t. (s,t) |= A"
- shows "|- A"
+ assumes "\<And>s t. (s,t) \<Turnstile> A"
+ shows "\<turnstile> A"
apply (rule assms intI prod.induct)+
done
-lemma actionD [dest]: "|- A ==> (s,t) |= A"
+lemma actionD [dest]: "\<turnstile> A \<Longrightarrow> (s,t) \<Turnstile> A"
apply (erule intD)
done
lemma pr_rews [int_rewrite]:
- "|- (#c)` = #c"
- "\<And>f. |- f<x>` = f<x` >"
- "\<And>f. |- f<x,y>` = f<x`,y` >"
- "\<And>f. |- f<x,y,z>` = f<x`,y`,z` >"
- "|- (\<forall>x. P x)` = (\<forall>x. (P x)`)"
- "|- (\<exists>x. P x)` = (\<exists>x. (P x)`)"
+ "\<turnstile> (#c)` = #c"
+ "\<And>f. \<turnstile> f<x>` = f<x` >"
+ "\<And>f. \<turnstile> f<x,y>` = f<x`,y` >"
+ "\<And>f. \<turnstile> f<x,y,z>` = f<x`,y`,z` >"
+ "\<turnstile> (\<forall>x. P x)` = (\<forall>x. (P x)`)"
+ "\<turnstile> (\<exists>x. P x)` = (\<exists>x. (P x)`)"
by (rule actionI, unfold unl_after intensional_rews, rule refl)+
@@ -112,7 +112,7 @@
(rewrite_rule ctxt @{thms action_rews} (th RS @{thm actionD}))
handle THM _ => int_unlift ctxt th;
-(* Turn |- A = B into meta-level rewrite rule A == B *)
+(* Turn \<turnstile> A = B into meta-level rewrite rule A == B *)
val action_rewrite = int_rewrite
fun action_use ctxt th =
@@ -132,69 +132,69 @@
(* =========================== square / angle brackets =========================== *)
-lemma idle_squareI: "(s,t) |= unchanged v ==> (s,t) |= [A]_v"
+lemma idle_squareI: "(s,t) \<Turnstile> unchanged v \<Longrightarrow> (s,t) \<Turnstile> [A]_v"
by (simp add: square_def)
-lemma busy_squareI: "(s,t) |= A ==> (s,t) |= [A]_v"
+lemma busy_squareI: "(s,t) \<Turnstile> A \<Longrightarrow> (s,t) \<Turnstile> [A]_v"
by (simp add: square_def)
lemma squareE [elim]:
- "[| (s,t) |= [A]_v; A (s,t) ==> B (s,t); v t = v s ==> B (s,t) |] ==> B (s,t)"
+ "\<lbrakk> (s,t) \<Turnstile> [A]_v; A (s,t) \<Longrightarrow> B (s,t); v t = v s \<Longrightarrow> B (s,t) \<rbrakk> \<Longrightarrow> B (s,t)"
apply (unfold square_def action_rews)
apply (erule disjE)
apply simp_all
done
-lemma squareCI [intro]: "[| v t \<noteq> v s ==> A (s,t) |] ==> (s,t) |= [A]_v"
+lemma squareCI [intro]: "\<lbrakk> v t \<noteq> v s \<Longrightarrow> A (s,t) \<rbrakk> \<Longrightarrow> (s,t) \<Turnstile> [A]_v"
apply (unfold square_def action_rews)
apply (rule disjCI)
apply (erule (1) meta_mp)
done
-lemma angleI [intro]: "\<And>s t. [| A (s,t); v t \<noteq> v s |] ==> (s,t) |= <A>_v"
+lemma angleI [intro]: "\<And>s t. \<lbrakk> A (s,t); v t \<noteq> v s \<rbrakk> \<Longrightarrow> (s,t) \<Turnstile> <A>_v"
by (simp add: angle_def)
-lemma angleE [elim]: "[| (s,t) |= <A>_v; [| A (s,t); v t \<noteq> v s |] ==> R |] ==> R"
+lemma angleE [elim]: "\<lbrakk> (s,t) \<Turnstile> <A>_v; \<lbrakk> A (s,t); v t \<noteq> v s \<rbrakk> \<Longrightarrow> R \<rbrakk> \<Longrightarrow> R"
apply (unfold angle_def action_rews)
apply (erule conjE)
apply simp
done
lemma square_simulation:
- "\<And>f. [| |- unchanged f & \<not>B --> unchanged g;
- |- A & \<not>unchanged g --> B
- |] ==> |- [A]_f --> [B]_g"
+ "\<And>f. \<lbrakk> \<turnstile> unchanged f & \<not>B \<longrightarrow> unchanged g;
+ \<turnstile> A & \<not>unchanged g \<longrightarrow> B
+ \<rbrakk> \<Longrightarrow> \<turnstile> [A]_f \<longrightarrow> [B]_g"
apply clarsimp
apply (erule squareE)
apply (auto simp add: square_def)
done
-lemma not_square: "|- (\<not> [A]_v) = <\<not>A>_v"
+lemma not_square: "\<turnstile> (\<not> [A]_v) = <\<not>A>_v"
by (auto simp: square_def angle_def)
-lemma not_angle: "|- (\<not> <A>_v) = [\<not>A]_v"
+lemma not_angle: "\<turnstile> (\<not> <A>_v) = [\<not>A]_v"
by (auto simp: square_def angle_def)
(* ============================== Facts about ENABLED ============================== *)
-lemma enabledI: "|- A --> $Enabled A"
+lemma enabledI: "\<turnstile> A \<longrightarrow> $Enabled A"
by (auto simp add: enabled_def)
-lemma enabledE: "[| s |= Enabled A; \<And>u. A (s,u) ==> Q |] ==> Q"
+lemma enabledE: "\<lbrakk> s \<Turnstile> Enabled A; \<And>u. A (s,u) \<Longrightarrow> Q \<rbrakk> \<Longrightarrow> Q"
apply (unfold enabled_def)
apply (erule exE)
apply simp
done
-lemma notEnabledD: "|- \<not>$Enabled G --> \<not> G"
+lemma notEnabledD: "\<turnstile> \<not>$Enabled G \<longrightarrow> \<not> G"
by (auto simp add: enabled_def)
(* Monotonicity *)
lemma enabled_mono:
- assumes min: "s |= Enabled F"
- and maj: "|- F --> G"
- shows "s |= Enabled G"
+ assumes min: "s \<Turnstile> Enabled F"
+ and maj: "\<turnstile> F \<longrightarrow> G"
+ shows "s \<Turnstile> Enabled G"
apply (rule min [THEN enabledE])
apply (rule enabledI [action_use])
apply (erule maj [action_use])
@@ -202,50 +202,50 @@
(* stronger variant *)
lemma enabled_mono2:
- assumes min: "s |= Enabled F"
- and maj: "\<And>t. F (s,t) ==> G (s,t)"
- shows "s |= Enabled G"
+ assumes min: "s \<Turnstile> Enabled F"
+ and maj: "\<And>t. F (s,t) \<Longrightarrow> G (s,t)"
+ shows "s \<Turnstile> Enabled G"
apply (rule min [THEN enabledE])
apply (rule enabledI [action_use])
apply (erule maj)
done
-lemma enabled_disj1: "|- Enabled F --> Enabled (F | G)"
+lemma enabled_disj1: "\<turnstile> Enabled F \<longrightarrow> Enabled (F | G)"
by (auto elim!: enabled_mono)
-lemma enabled_disj2: "|- Enabled G --> Enabled (F | G)"
+lemma enabled_disj2: "\<turnstile> Enabled G \<longrightarrow> Enabled (F | G)"
by (auto elim!: enabled_mono)
-lemma enabled_conj1: "|- Enabled (F & G) --> Enabled F"
+lemma enabled_conj1: "\<turnstile> Enabled (F & G) \<longrightarrow> Enabled F"
by (auto elim!: enabled_mono)
-lemma enabled_conj2: "|- Enabled (F & G) --> Enabled G"
+lemma enabled_conj2: "\<turnstile> Enabled (F & G) \<longrightarrow> Enabled G"
by (auto elim!: enabled_mono)
lemma enabled_conjE:
- "[| s |= Enabled (F & G); [| s |= Enabled F; s |= Enabled G |] ==> Q |] ==> Q"
+ "\<lbrakk> s \<Turnstile> Enabled (F & G); \<lbrakk> s \<Turnstile> Enabled F; s \<Turnstile> Enabled G \<rbrakk> \<Longrightarrow> Q \<rbrakk> \<Longrightarrow> Q"
apply (frule enabled_conj1 [action_use])
apply (drule enabled_conj2 [action_use])
apply simp
done
-lemma enabled_disjD: "|- Enabled (F | G) --> Enabled F | Enabled G"
+lemma enabled_disjD: "\<turnstile> Enabled (F | G) \<longrightarrow> Enabled F | Enabled G"
by (auto simp add: enabled_def)
-lemma enabled_disj: "|- Enabled (F | G) = (Enabled F | Enabled G)"
+lemma enabled_disj: "\<turnstile> Enabled (F | G) = (Enabled F | Enabled G)"
apply clarsimp
apply (rule iffI)
apply (erule enabled_disjD [action_use])
apply (erule disjE enabled_disj1 [action_use] enabled_disj2 [action_use])+
done
-lemma enabled_ex: "|- Enabled (\<exists>x. F x) = (\<exists>x. Enabled (F x))"
+lemma enabled_ex: "\<turnstile> Enabled (\<exists>x. F x) = (\<exists>x. Enabled (F x))"
by (force simp add: enabled_def)
(* A rule that combines enabledI and baseE, but generates fewer instantiations *)
lemma base_enabled:
- "[| basevars vs; \<exists>c. \<forall>u. vs u = c --> A(s,u) |] ==> s |= Enabled A"
+ "\<lbrakk> basevars vs; \<exists>c. \<forall>u. vs u = c \<longrightarrow> A(s,u) \<rbrakk> \<Longrightarrow> s \<Turnstile> Enabled A"
apply (erule exE)
apply (erule baseE)
apply (rule enabledI [action_use])
@@ -294,7 +294,7 @@
lemma
assumes "basevars (x,y,z)"
- shows "|- x --> Enabled ($x & (y$ = #False))"
+ shows "\<turnstile> x \<longrightarrow> Enabled ($x & (y$ = #False))"
apply (enabled assms)
apply auto
done