chapter \<open>The Owicki-Gries Method\<close>
section \<open>Abstract Syntax\<close>
theory OG_Com imports Main begin
text \<open>Type abbreviations for boolean expressions and assertions:\<close>
type_synonym 'a bexp = "'a set"
type_synonym 'a assn = "'a set"
text \<open>The syntax of commands is defined by two mutually recursive
datatypes: @{text "'a ann_com"} for annotated commands and @{text "'a
com"} for non-annotated commands.\<close>
datatype 'a ann_com =
AnnBasic "('a assn)" "('a \<Rightarrow> '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 \<times> 'a assn) list"
| Basic "('a \<Rightarrow> 'a)"
| Seq "('a com)" "('a com)"
| Cond "('a bexp)" "('a com)" "('a com)"
| While "('a bexp)" "('a assn)" "('a com)"
text \<open>The function @{text pre} extracts the precondition of an
annotated command:\<close>
primrec pre ::"'a ann_com \<Rightarrow> '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 \<open>Well-formedness predicate for atomic programs:\<close>
primrec atom_com :: "'a com \<Rightarrow> bool" where
"atom_com (Parallel Ts) = False"
| "atom_com (Basic f) = True"
| "atom_com (Seq c1 c2) = (atom_com c1 \<and> atom_com c2)"
| "atom_com (Cond b c1 c2) = (atom_com c1 \<and> atom_com c2)"
| "atom_com (While b i c) = atom_com c"
end