# Theory RG_Syntax

theory RG_Syntax
imports RG_Hoare Quote_Antiquote
```section ‹Concrete Syntax›

theory RG_Syntax
imports RG_Hoare Quote_Antiquote
begin

abbreviation Skip :: "'a com"  ("SKIP")
where "SKIP ≡ Basic id"

notation Seq  ("(_;;/ _)" [60,61] 60)

syntax
"_Assign"    :: "idt ⇒ 'b ⇒ 'a com"                     ("(´_ :=/ _)" [70, 65] 61)
"_Cond"      :: "'a bexp ⇒ 'a com ⇒ 'a com ⇒ 'a com"   ("(0IF _/ THEN _/ ELSE _/FI)" [0, 0, 0] 61)
"_Cond2"     :: "'a bexp ⇒ 'a com ⇒ 'a com"             ("(0IF _ THEN _ FI)" [0,0] 56)
"_While"     :: "'a bexp ⇒ 'a com ⇒ 'a com"             ("(0WHILE _ /DO _ /OD)"  [0, 0] 61)
"_Await"     :: "'a bexp ⇒ 'a com ⇒ 'a com"             ("(0AWAIT _ /THEN /_ /END)"  [0,0] 61)
"_Atom"      :: "'a com ⇒ 'a com"                        ("(⟨_⟩)" 61)
"_Wait"      :: "'a bexp ⇒ 'a com"                       ("(0WAIT _ END)" 61)

translations
"´x := a" ⇀ "CONST Basic «´(_update_name x (λ_. a))»"
"IF b THEN c1 ELSE c2 FI" ⇀ "CONST Cond ⦃b⦄ c1 c2"
"IF b THEN c FI" ⇌ "IF b THEN c ELSE SKIP FI"
"WHILE b DO c OD" ⇀ "CONST While ⦃b⦄ c"
"AWAIT b THEN c END" ⇌ "CONST Await ⦃b⦄ c"
"⟨c⟩" ⇌ "AWAIT CONST True THEN c END"
"WAIT b END" ⇌ "AWAIT b THEN SKIP END"

nonterminal prgs

syntax
"_PAR"        :: "prgs ⇒ 'a"              ("COBEGIN//_//COEND" 60)
"_prg"        :: "'a ⇒ prgs"              ("_" 57)
"_prgs"       :: "['a, prgs] ⇒ prgs"      ("_//∥//_" [60,57] 57)

translations
"_prg a" ⇀ "[a]"
"_prgs a ps" ⇀ "a # ps"
"_PAR ps" ⇀ "ps"

syntax
"_prg_scheme" :: "['a, 'a, 'a, 'a] ⇒ prgs"  ("SCHEME [_ ≤ _ < _] _" [0,0,0,60] 57)

translations
"_prg_scheme j i k c" ⇌ "(CONST map (λi. c) [j..<k])"

text ‹Translations for variables before and after a transition:›

syntax
"_before" :: "id ⇒ 'a" ("º_")
"_after"  :: "id ⇒ 'a" ("ª_")

translations
"ºx" ⇌ "x ´CONST fst"
"ªx" ⇌ "x ´CONST snd"

print_translation ‹
let
fun quote_tr' f (t :: ts) =
Term.list_comb (f \$ Syntax_Trans.quote_tr' @{syntax_const "_antiquote"} t, ts)
| quote_tr' _ _ = raise Match;

val assert_tr' = quote_tr' (Syntax.const @{syntax_const "_Assert"});

fun bexp_tr' name ((Const (@{const_syntax Collect}, _) \$ t) :: ts) =
quote_tr' (Syntax.const name) (t :: ts)
| bexp_tr' _ _ = raise Match;

fun assign_tr' (Abs (x, _, f \$ k \$ Bound 0) :: ts) =
quote_tr' (Syntax.const @{syntax_const "_Assign"} \$ Syntax_Trans.update_name_tr' f)
(Abs (x, dummyT, Syntax_Trans.const_abs_tr' k) :: ts)
| assign_tr' _ = raise Match;
in
[(@{const_syntax Collect}, K assert_tr'),
(@{const_syntax Basic}, K assign_tr'),
(@{const_syntax Cond}, K (bexp_tr' @{syntax_const "_Cond"})),
(@{const_syntax While}, K (bexp_tr' @{syntax_const "_While"}))]
end
›

end
```