src/HOL/HOLCF/IOA/meta_theory/TL.thy

--- a/src/HOL/HOLCF/IOA/meta_theory/TL.thy	Wed Dec 30 21:23:38 2015 +0100
+++ b/src/HOL/HOLCF/IOA/meta_theory/TL.thy	Wed Dec 30 21:35:21 2015 +0100
@@ -22,17 +22,12 @@
 
 unlift     ::  "'a lift => 'a"
 
-Init         ::"'a predicate => 'a temporal"          ("<_>" [0] 1000)
+Init       :: "'a predicate => 'a temporal"          ("<_>" [0] 1000)
 
-Box          ::"'a temporal => 'a temporal"   ("[] (_)" [80] 80)
-Diamond      ::"'a temporal => 'a temporal"   ("<> (_)" [80] 80)
-Next         ::"'a temporal => 'a temporal"
-Leadsto      ::"'a temporal => 'a temporal => 'a temporal"  (infixr "~>" 22)
-
-notation (xsymbols)
-  Box  ("\<box> (_)" [80] 80) and
-  Diamond  ("\<diamond> (_)" [80] 80) and
-  Leadsto  (infixr "\<leadsto>" 22)
+Box        :: "'a temporal => 'a temporal"   ("\<box> (_)" [80] 80)
+Diamond    :: "'a temporal => 'a temporal"   ("\<diamond> (_)" [80] 80)
+Next       :: "'a temporal => 'a temporal"
+Leadsto    :: "'a temporal => 'a temporal => 'a temporal"  (infixr "\<leadsto>" 22)
 
 defs
 
@@ -50,37 +45,37 @@
   "tsuffix s2 s == s2 ~= nil & s2 ~= UU & suffix s2 s"
 
 Box_def:
-  "([] P) s == ! s2. tsuffix s2 s --> P s2"
+  "(\<box>P) s == ! s2. tsuffix s2 s --> P s2"
 
 Next_def:
   "(Next P) s == if (TL$s=UU | TL$s=nil) then (P s) else P (TL$s)"
 
 Diamond_def:
-  "<> P == .~ ([] (.~ P))"
+  "\<diamond>P == .~ (\<box>(.~ P))"
 
 Leadsto_def:
-   "P ~> Q == ([] (P .--> (<> Q)))"
+   "P \<leadsto> Q == (\<box>(P .--> (\<diamond>Q)))"
 
 validT_def:
   "validT P == ! s. s~=UU & s~=nil --> (s |= P)"
 
 
-lemma simple: "[] <> (.~ P) = (.~ <> [] P)"
+lemma simple: "\<box>\<diamond>(.~ P) = (.~ \<diamond>\<box>P)"
 apply (rule ext)
 apply (simp add: Diamond_def NOT_def Box_def)
 done
 
-lemma Boxnil: "nil |= [] P"
+lemma Boxnil: "nil |= \<box>P"
 apply (simp add: satisfies_def Box_def tsuffix_def suffix_def nil_is_Conc)
 done
 
-lemma Diamondnil: "~(nil |= <> P)"
+lemma Diamondnil: "~(nil |= \<diamond>P)"
 apply (simp add: Diamond_def satisfies_def NOT_def)
 apply (cut_tac Boxnil)
 apply (simp add: satisfies_def)
 done
 
-lemma Diamond_def2: "(<> F) s = (? s2. tsuffix s2 s & F s2)"
+lemma Diamond_def2: "(\<diamond>F) s = (? s2. tsuffix s2 s & F s2)"
 apply (simp add: Diamond_def NOT_def Box_def)
 done
 
@@ -94,7 +89,7 @@
 apply auto
 done
 
-lemma reflT: "s~=UU & s~=nil --> (s |= [] F .--> F)"
+lemma reflT: "s~=UU & s~=nil --> (s |= \<box>F .--> F)"
 apply (simp add: satisfies_def IMPLIES_def Box_def)
 apply (rule impI)+
 apply (erule_tac x = "s" in allE)
@@ -110,7 +105,7 @@
 apply (simp (no_asm) add: Conc_assoc)
 done
 
-lemma transT: "s |= [] F .--> [] [] F"
+lemma transT: "s |= \<box>F .--> \<box>\<box>F"
 apply (simp (no_asm) add: satisfies_def IMPLIES_def Box_def tsuffix_def)
 apply auto
 apply (drule suffix_trans)
@@ -120,14 +115,14 @@
 done
 
 
-lemma normalT: "s |= [] (F .--> G) .--> [] F .--> [] G"
+lemma normalT: "s |= \<box>(F .--> G) .--> \<box>F .--> \<box>G"
 apply (simp (no_asm) add: satisfies_def IMPLIES_def Box_def)
 done
 
 
 subsection "TLA Rules by Lamport"
 
-lemma STL1a: "validT P ==> validT ([] P)"
+lemma STL1a: "validT P ==> validT (\<box>P)"
 apply (simp add: validT_def satisfies_def Box_def tsuffix_def)
 done
 
@@ -135,13 +130,13 @@
 apply (simp add: valid_def validT_def satisfies_def Init_def)
 done
 
-lemma STL1: "valid P ==> validT ([] (Init P))"
+lemma STL1: "valid P ==> validT (\<box>(Init P))"
 apply (rule STL1a)
 apply (erule STL1b)
 done
 
 (* Note that unlift and HD is not at all used !!! *)
-lemma STL4: "valid (P .--> Q)  ==> validT ([] (Init P) .--> [] (Init Q))"
+lemma STL4: "valid (P .--> Q)  ==> validT (\<box>(Init P) .--> \<box>(Init Q))"
 apply (simp add: valid_def validT_def satisfies_def IMPLIES_def Box_def Init_def)
 done
 
@@ -161,13 +156,13 @@
 
 declare split_if [split del]
 lemma LTL1: 
-   "s~=UU & s~=nil --> (s |= [] F .--> (F .& (Next ([] F))))"
+   "s~=UU & s~=nil --> (s |= \<box>F .--> (F .& (Next (\<box>F))))"
 apply (unfold Next_def satisfies_def NOT_def IMPLIES_def AND_def Box_def)
 apply auto
-(* []F .--> F *)
+(* \<box>F .--> F *)
 apply (erule_tac x = "s" in allE)
 apply (simp add: tsuffix_def suffix_refl)
-(* []F .--> Next [] F *)
+(* \<box>F .--> Next \<box>F *)
 apply (simp split add: split_if)
 apply auto
 apply (drule tsuffix_TL2)