src/HOL/IMP/ACom.thy

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/IMP/ACom.thy	Thu Nov 24 19:58:37 2011 +0100
@@ -0,0 +1,84 @@
+(* Author: Tobias Nipkow *)
+
+theory ACom
+imports Com
+begin
+
+(* is there a better place? *)
+definition "show_state xs s = [(x,s x). x \<leftarrow> xs]"
+
+section "Annotated Commands"
+
+datatype 'a acom =
+  SKIP   'a                           ("SKIP {_}") |
+  Assign vname aexp 'a                ("(_ ::= _/ {_})" [1000, 61, 0] 61) |
+  Semi   "('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)
+
+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"
+
+fun strip :: "'a acom \<Rightarrow> com" where
+"strip (SKIP {a}) = com.SKIP" |
+"strip (x ::= e {a}) = (x ::= e)" |
+"strip (c1;c2) = (strip c1; strip c2)" |
+"strip (IF b THEN c1 ELSE c2 {a}) = (IF b THEN strip c1 ELSE strip c2)" |
+"strip ({a1} WHILE b DO c {a2}) = (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})" |
+"anno a (WHILE b DO c) =
+  ({a} WHILE b DO anno a c {a})"
+
+fun map_acom :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a acom \<Rightarrow> 'b acom" where
+"map_acom f (SKIP {a}) = SKIP {f a}" |
+"map_acom f (x ::= e {a}) = (x ::= e {f a})" |
+"map_acom f (c1;c2) = (map_acom f c1; map_acom f c2)" |
+"map_acom f (IF b THEN c1 ELSE c2 {a}) =
+  (IF b THEN map_acom f c1 ELSE map_acom f c2 {f a})" |
+"map_acom f ({a1} WHILE b DO c {a2}) =
+  ({f a1} WHILE b DO map_acom f c {f a2})"
+
+
+lemma post_map_acom[simp]: "post(map_acom f c) = f(post c)"
+by (induction c) simp_all
+
+lemma strip_acom[simp]: "strip (map_acom f c) = strip c"
+by (induction c) 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
+
+lemma strip_eq_Assign: "strip c = x::=e \<longleftrightarrow> (EX P. c = x::=e {P})"
+by (cases c) simp_all
+
+lemma strip_eq_Semi:
+  "strip c = c1;c2 \<longleftrightarrow> (EX d1 d2. c = d1;d2 & strip d1 = c1 & strip d2 = 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
+
+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
+
+end