File ‹Tools/SMT/smt_real.ML›
structure SMT_Real: sig end =
struct
fun smtlib_logic _ ts =
if exists (Term.exists_type (Term.exists_subtype (equal \<^typ>‹real›))) ts
then SOME "AUFLIRA"
else NONE
local
fun real_num _ i = SOME (string_of_int i ^ ".0")
fun is_linear [t] = SMT_Util.is_number t
| is_linear [t, u] = SMT_Util.is_number t orelse SMT_Util.is_number u
| is_linear _ = false
fun mk_times ts = Term.list_comb (\<^Const>‹times \<^Type>‹real››, ts)
fun times _ _ ts = if is_linear ts then SOME ("*", 2, ts, mk_times) else NONE
in
fun real_type_parser (SMTLIB.Sym "Real", []) = SOME \<^typ>‹Real.real›
| real_type_parser _ = NONE
fun real_term_parser (SMTLIB.Dec (i, 0), []) = SOME (HOLogic.mk_number \<^typ>‹Real.real› i)
| real_term_parser (SMTLIB.Sym "/", [t1, t2]) =
SOME (\<^term>‹Rings.divide :: real => _› $ t1 $ t2)
| real_term_parser (SMTLIB.Sym "to_real", [t]) = SOME (\<^term>‹Int.of_int :: int => _› $ t)
| real_term_parser _ = NONE
fun abstract abs t =
(case t of
(c as \<^term>‹Rings.divide :: real => _›) $ t1 $ t2 =>
abs t1 ##>> abs t2 #>> (fn (u1, u2) => SOME (c $ u1 $ u2))
| (c as \<^term>‹Int.of_int :: int => _›) $ t =>
abs t #>> (fn u => SOME (c $ u))
| _ => pair NONE)
val _ = Theory.setup (Context.theory_map (
SMTLIB_Proof.add_type_parser real_type_parser #>
SMTLIB_Proof.add_term_parser real_term_parser #>
SMT_Replay_Methods.add_arith_abstracter abstract))
val setup_builtins =
SMT_Builtin.add_builtin_typ SMTLIB_Interface.smtlibC
(\<^typ>‹real›, K (SOME ("Real", [])), real_num) #>
fold (SMT_Builtin.add_builtin_fun' SMTLIB_Interface.smtlibC) [
(\<^Const>‹less \<^Type>‹real››, "<"),
(\<^Const>‹less_eq \<^Type>‹real››, "<="),
(\<^Const>‹uminus \<^Type>‹real››, "-"),
(\<^Const>‹plus \<^Type>‹real››, "+"),
(\<^Const>‹minus \<^Type>‹real››, "-") ] #>
SMT_Builtin.add_builtin_fun SMTLIB_Interface.smtlibC
(Term.dest_Const \<^Const>‹times \<^Type>‹real››, times)
end
local
fun smt_mk_builtin_typ (Z3_Interface.Sym ("Real", _)) = SOME \<^typ>‹real›
| smt_mk_builtin_typ (Z3_Interface.Sym ("real", _)) = SOME \<^typ>‹real›
| smt_mk_builtin_typ _ = NONE
fun smt_mk_builtin_num _ i T =
if T = \<^typ>‹real› then SOME (Numeral.mk_cnumber \<^ctyp>‹real› i)
else NONE
fun mk_nary _ cu [] = cu
| mk_nary ct _ cts = uncurry (fold_rev (Thm.mk_binop ct)) (split_last cts)
val mk_uminus = Thm.apply \<^cterm>‹uminus :: real ⇒ _›
val add = \<^cterm>‹(+) :: real ⇒ _›
val real0 = Numeral.mk_cnumber \<^ctyp>‹real› 0
val mk_sub = Thm.mk_binop \<^cterm>‹(-) :: real ⇒ _›
val mk_mul = Thm.mk_binop \<^cterm>‹(*) :: real ⇒ _›
val mk_div = Thm.mk_binop \<^cterm>‹(/) :: real ⇒ _›
val mk_lt = Thm.mk_binop \<^cterm>‹(<) :: real ⇒ _›
val mk_le = Thm.mk_binop \<^cterm>‹(≤) :: real ⇒ _›
fun smt_mk_builtin_fun (Z3_Interface.Sym ("-", _)) [ct] = SOME (mk_uminus ct)
| smt_mk_builtin_fun (Z3_Interface.Sym ("+", _)) cts = SOME (mk_nary add real0 cts)
| smt_mk_builtin_fun (Z3_Interface.Sym ("-", _)) [ct, cu] = SOME (mk_sub ct cu)
| smt_mk_builtin_fun (Z3_Interface.Sym ("*", _)) [ct, cu] = SOME (mk_mul ct cu)
| smt_mk_builtin_fun (Z3_Interface.Sym ("/", _)) [ct, cu] = SOME (mk_div ct cu)
| smt_mk_builtin_fun (Z3_Interface.Sym ("<", _)) [ct, cu] = SOME (mk_lt ct cu)
| smt_mk_builtin_fun (Z3_Interface.Sym ("<=", _)) [ct, cu] = SOME (mk_le ct cu)
| smt_mk_builtin_fun (Z3_Interface.Sym (">", _)) [ct, cu] = SOME (mk_lt cu ct)
| smt_mk_builtin_fun (Z3_Interface.Sym (">=", _)) [ct, cu] = SOME (mk_le cu ct)
| smt_mk_builtin_fun _ _ = NONE
in
val smt_mk_builtins = {
mk_builtin_typ = smt_mk_builtin_typ,
mk_builtin_num = smt_mk_builtin_num,
mk_builtin_fun = (fn _ => fn sym => fn cts =>
(case try (Thm.typ_of_cterm o hd) cts of
SOME \<^typ>‹real› => smt_mk_builtin_fun sym cts
| _ => NONE)) }
end
val real_linarith_proc =
Simplifier.make_simproc \<^context> "fast_real_arith"
{lhss = [\<^term>‹(m::real) < n›, \<^term>‹(m::real) ≤ n›, \<^term>‹(m::real) = n›],
proc = K Lin_Arith.simproc}
val _ = Theory.setup (Context.theory_map (
SMTLIB_Interface.add_logic (10, smtlib_logic) #>
setup_builtins #>
Z3_Interface.add_mk_builtins smt_mk_builtins #>
SMT_Replay.add_simproc real_linarith_proc))
end;