# Theory ACom_ITP

theory ACom_ITP
imports Com
```(* Author: Tobias Nipkow *)

theory ACom_ITP
imports "../Com"
begin

subsection "Annotated Commands"

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)

fun post :: "'a acom ⇒'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 ⇒ 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)"

fun anno :: "'a ⇒ com ⇒ '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 annos :: "'a acom ⇒ '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"

fun map_acom :: "('a ⇒ 'b) ⇒ 'a acom ⇒ '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})"

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 map_acom_SKIP:
"map_acom f c = SKIP {S'} ⟷ (∃S. c = SKIP {S} ∧ S' = f S)"
by (cases c) auto

lemma map_acom_Assign:
"map_acom f c = x ::= e {S'} ⟷ (∃S. c = x::=e {S} ∧ S' = f S)"
by (cases c) auto

lemma map_acom_Seq:
"map_acom f c = c1';;c2' ⟷
(∃c1 c2. c = c1;;c2 ∧ map_acom f c1 = c1' ∧ map_acom f c2 = c2')"
by (cases c) auto

lemma map_acom_If:
"map_acom f c = IF b THEN c1' ELSE c2' {S'} ⟷
(∃S c1 c2. c = IF b THEN c1 ELSE c2 {S} ∧ map_acom f c1 = c1' ∧ map_acom f c2 = c2' ∧ S' = f S)"
by (cases c) auto

lemma map_acom_While:
"map_acom f w = {I'} WHILE b DO c' {P'} ⟷
(∃I P c. w = {I} WHILE b DO c {P} ∧ map_acom f c = c' ∧ I' = f I ∧ 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 ⟷ (EX P. c = SKIP {P})"
by (cases c) simp_all

lemma strip_eq_Assign:
"strip c = x::=e ⟷ (EX P. c = x::=e {P})"
by (cases c) simp_all

lemma strip_eq_Seq:
"strip c = c1;;c2 ⟷ (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 ⟷
(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 ⟷
(EX I d1 P. c = {I} WHILE b DO d1 {P} & strip d1 = 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 ⟹ size(annos C1) = size(annos C2)"
apply(induct C2 arbitrary: C1)
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
```