# Theory OG_Com

theory OG_Com
imports Main
```chapter ‹The Owicki-Gries Method›

section ‹Abstract Syntax›

theory OG_Com imports Main begin

text ‹Type abbreviations for boolean expressions and assertions:›

type_synonym 'a bexp = "'a set"
type_synonym 'a assn = "'a set"

text ‹The syntax of commands is defined by two mutually recursive
datatypes: ‹'a ann_com› for annotated commands and ‹'a
com› for non-annotated commands.›

datatype 'a ann_com =
AnnBasic "('a assn)"  "('a ⇒ 'a)"
| AnnSeq "('a ann_com)"  "('a ann_com)"
| AnnCond1 "('a assn)"  "('a bexp)"  "('a ann_com)"  "('a ann_com)"
| AnnCond2 "('a assn)"  "('a bexp)"  "('a ann_com)"
| AnnWhile "('a assn)"  "('a bexp)"  "('a assn)"  "('a ann_com)"
| AnnAwait "('a assn)"  "('a bexp)"  "('a com)"
and 'a com =
Parallel "('a ann_com option × 'a assn) list"
| Basic "('a ⇒ 'a)"
| Seq "('a com)"  "('a com)"
| Cond "('a bexp)"  "('a com)"  "('a com)"
| While "('a bexp)"  "('a assn)"  "('a com)"

text ‹The function ‹pre› extracts the precondition of an
annotated command:›

primrec pre ::"'a ann_com ⇒ 'a assn"  where
"pre (AnnBasic r f) = r"
| "pre (AnnSeq c1 c2) = pre c1"
| "pre (AnnCond1 r b c1 c2) = r"
| "pre (AnnCond2 r b c) = r"
| "pre (AnnWhile r b i c) = r"
| "pre (AnnAwait r b c) = r"

text ‹Well-formedness predicate for atomic programs:›

primrec atom_com :: "'a com ⇒ bool" where
"atom_com (Parallel Ts) = False"
| "atom_com (Basic f) = True"
| "atom_com (Seq c1 c2) = (atom_com c1 ∧ atom_com c2)"
| "atom_com (Cond b c1 c2) = (atom_com c1 ∧ atom_com c2)"
| "atom_com (While b i c) = atom_com c"

end
```