header {* \chapter{The Owicki-Gries Method} \section{Abstract Syntax} *} theory OG_Com imports Main begin text {* Type abbreviations for boolean expressions and assertions: *} types 'a bexp = "'a set" 'a assn = "'a set" text {* 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. *} 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 @{text pre} extracts the precondition of an annotated command: *} consts pre ::"'a ann_com => 'a assn" primrec "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: *} consts atom_com :: "'a com => bool" primrec "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