src/HOL/IMP/ACom.thy

 
 datatype 'a acom =
   SKIP 'a                           ("SKIP {_}" 61) |
   Assign vname aexp 'a              ("(_ ::= _/ {_})" [1000, 61, 0] 61) |
   Seq "('a acom)" "('a acom)"       ("_;//_"  [60, 61] 60) |
-  If bexp "('a acom)" "('a acom)" 'a
-    ("(IF _/ THEN _/ ELSE _//{_})"  [0, 0, 61, 0] 61) |
-  While 'a bexp "('a acom)" 'a
-    ("({_}//WHILE _/ DO (_)//{_})"  [0, 0, 61, 0] 61)
+  If bexp 'a "('a acom)" 'a "('a acom)" 'a
+    ("(IF _/ THEN ({_}/ _)/ ELSE ({_}/ _)//{_})"  [0, 0, 0, 61, 0, 0] 61) |
+  While 'a bexp 'a "('a acom)" 'a
+    ("({_}//WHILE _/ DO ({_}/ _)//{_})"  [0, 0, 0, 61, 0] 61)
 
 fun post :: "'a acom \<Rightarrow>'a" where
 "post (SKIP {P}) = P" |
 "post (x ::= e {P}) = P" |
 "post (c1; c2) = post c2" |
-"post (IF b THEN c1 ELSE c2 {P}) = P" |
-"post ({Inv} WHILE b DO c {P}) = P"
+"post (IF b THEN {P1} c1 ELSE {P2} c2 {P}) = P" |
+"post ({Inv} WHILE b DO {Pc} c {P}) = P"
 
 fun strip :: "'a acom \<Rightarrow> com" where
 "strip (SKIP {P}) = com.SKIP" |
 "strip (x ::= e {P}) = (x ::= e)" |
 "strip (c1;c2) = (strip c1; strip c2)" |
-"strip (IF b THEN c1 ELSE c2 {P}) = (IF b THEN strip c1 ELSE strip c2)" |
-"strip ({Inv} WHILE b DO c {P}) = (WHILE b DO strip c)"
+"strip (IF b THEN {P1} c1 ELSE {P2} c2 {P}) = (IF b THEN strip c1 ELSE strip c2)" |
+"strip ({Inv} WHILE b DO {Pc} c {P}) = (WHILE b DO strip c)"
 
 fun anno :: "'a \<Rightarrow> com \<Rightarrow> 'a acom" where
 "anno a com.SKIP = SKIP {a}" |
 "anno a (x ::= e) = (x ::= e {a})" |
 "anno a (c1;c2) = (anno a c1; anno a c2)" |
 "anno a (IF b THEN c1 ELSE c2) =
-  (IF b THEN anno a c1 ELSE anno a c2 {a})" |
+  (IF b THEN {a} anno a c1 ELSE {a} anno a c2 {a})" |
 "anno a (WHILE b DO c) =
-  ({a} WHILE b DO anno a c {a})"
+  ({a} WHILE b DO {a} anno a c {a})"
 
 fun annos :: "'a acom \<Rightarrow> 'a list" where
 "annos (SKIP {a}) = [a]" |
 "annos (x::=e {a}) = [a]" |
 "annos (C1;C2) = annos C1 @ annos C2" |
-"annos (IF b THEN C1 ELSE C2 {a}) = a #  annos C1 @ annos C2" |
-"annos ({i} WHILE b DO C {a}) = i # a # annos C"
+"annos (IF b THEN {a1} C1 ELSE {a2} C2 {a}) = a1 # a2 #a #  annos C1 @ annos C2" |
+"annos ({i} WHILE b DO {ac} C {a}) = i # ac # a # annos C"
 
 fun map_acom :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a acom \<Rightarrow> 'b acom" where
 "map_acom f (SKIP {P}) = SKIP {f P}" |
 "map_acom f (x ::= e {P}) = (x ::= e {f P})" |
 "map_acom f (c1;c2) = (map_acom f c1; map_acom f c2)" |
-"map_acom f (IF b THEN c1 ELSE c2 {P}) =
-  (IF b THEN map_acom f c1 ELSE map_acom f c2 {f P})" |
-"map_acom f ({Inv} WHILE b DO c {P}) =
-  ({f Inv} WHILE b DO map_acom f c {f P})"
+"map_acom f (IF b THEN {p1} c1 ELSE {p2} c2 {P}) =
+  (IF b THEN {f p1} map_acom f c1 ELSE {f p2} map_acom f c2 {f P})" |
+"map_acom f ({Inv} WHILE b DO {p} c {P}) =
+  ({f Inv} WHILE b DO {f p} map_acom f c {f P})"
 
 
 lemma post_map_acom[simp]: "post(map_acom f c) = f(post c)"
 by (induction c) simp_all
 

  "map_acom f c = c1';c2' \<longleftrightarrow>
  (\<exists>c1 c2. c = c1;c2 \<and> map_acom f c1 = c1' \<and> map_acom f c2 = c2')"
 by (cases c) auto
 
 lemma map_acom_If:
- "map_acom f c = IF b THEN c1' ELSE c2' {S'} \<longleftrightarrow>
- (\<exists>S c1 c2. c = IF b THEN c1 ELSE c2 {S} \<and> map_acom f c1 = c1' \<and> map_acom f c2 = c2' \<and> S' = f S)"
+ "map_acom f c = IF b THEN {p1'} c1' ELSE {p2'} c2' {S'} \<longleftrightarrow>
+ (\<exists>S p1 p2 c1 c2. c = IF b THEN {p1} c1 ELSE {p2} c2 {S} \<and>
+     map_acom f c1 = c1' \<and> map_acom f c2 = c2' \<and> p1' = f p1 \<and> p2' = f p2 \<and> S' = f S)"
 by (cases c) auto
 
 lemma map_acom_While:
- "map_acom f w = {I'} WHILE b DO c' {P'} \<longleftrightarrow>
- (\<exists>I P c. w = {I} WHILE b DO c {P} \<and> map_acom f c = c' \<and> I' = f I \<and> P' = f P)"
+ "map_acom f w = {I'} WHILE b DO {p'} c' {P'} \<longleftrightarrow>
+ (\<exists>I p P c. w = {I} WHILE b DO {p} c {P} \<and> map_acom f c = c' \<and> I' = f I \<and> p' = f p \<and> P' = f P)"
 by (cases w) auto
 
 
 lemma strip_anno[simp]: "strip (anno a c) = c"
 by(induct c) simp_all
 
 lemma strip_eq_SKIP:
-  "strip c = com.SKIP \<longleftrightarrow> (EX P. c = SKIP {P})"
-by (cases c) simp_all
+  "strip C = com.SKIP \<longleftrightarrow> (EX P. C = SKIP {P})"
+by (cases C) simp_all
 
 lemma strip_eq_Assign:
-  "strip c = x::=e \<longleftrightarrow> (EX P. c = x::=e {P})"
-by (cases c) simp_all
+  "strip C = x::=e \<longleftrightarrow> (EX P. C = x::=e {P})"
+by (cases C) simp_all
 
 lemma strip_eq_Seq:
-  "strip c = c1;c2 \<longleftrightarrow> (EX d1 d2. c = d1;d2 & strip d1 = c1 & strip d2 = c2)"
-by (cases c) simp_all
+  "strip C = c1;c2 \<longleftrightarrow> (EX C1 C2. C = C1;C2 & strip C1 = c1 & strip C2 = c2)"
+by (cases C) simp_all
 
 lemma strip_eq_If:
-  "strip c = IF b THEN c1 ELSE c2 \<longleftrightarrow>
-  (EX d1 d2 P. c = IF b THEN d1 ELSE d2 {P} & strip d1 = c1 & strip d2 = c2)"
-by (cases c) simp_all
+  "strip C = IF b THEN c1 ELSE c2 \<longleftrightarrow>
+  (EX p1 p2 C1 C2 P. C = IF b THEN {p1} C1 ELSE {p2} C2 {P} & strip C1 = c1 & strip C2 = c2)"
+by (cases C) simp_all
 
 lemma strip_eq_While:
-  "strip c = WHILE b DO c1 \<longleftrightarrow>
-  (EX I d1 P. c = {I} WHILE b DO d1 {P} & strip d1 = c1)"
-by (cases c) simp_all
-
+  "strip C = WHILE b DO c1 \<longleftrightarrow>
+  (EX I p C1 P. C = {I} WHILE b DO {p} C1 {P} & strip C1 = c1)"
+by (cases C) simp_all
 
 lemma set_annos_anno[simp]: "set (annos (anno a C)) = {a}"
 by(induction C)(auto)
 
 lemma size_annos_same: "strip C1 = strip C2 \<Longrightarrow> size(annos C1) = size(annos C2)"

 apply (auto simp: strip_eq_SKIP strip_eq_Assign strip_eq_Seq strip_eq_If strip_eq_While)
 done
 
 lemmas size_annos_same2 = eqTrueI[OF size_annos_same]
 
-
 end