src/HOL/Tools/SMT/z3_interface.ML
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(*  Title:      HOL/Tools/SMT/z3_interface.ML
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    Author:     Sascha Boehme, TU Muenchen
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Interface to Z3 based on a relaxed version of SMT-LIB.
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*)
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signature Z3_INTERFACE =
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sig
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  type builtin_fun = string * typ -> term list -> (string * term list) option
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  val add_builtin_funs: builtin_fun -> Context.generic -> Context.generic
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  val interface: SMT_Solver.interface
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  datatype sym = Sym of string * sym list
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  type mk_builtins = {
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    mk_builtin_typ: sym -> typ option,
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    mk_builtin_num: theory -> int -> typ -> cterm option,
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    mk_builtin_fun: theory -> sym -> cterm list -> cterm option }
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  val add_mk_builtins: mk_builtins -> Context.generic -> Context.generic
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  val mk_builtin_typ: Proof.context -> sym -> typ option
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  val mk_builtin_num: Proof.context -> int -> typ -> cterm option
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  val mk_builtin_fun: Proof.context -> sym -> cterm list -> cterm option
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  val is_builtin_theory_term: Proof.context -> term -> bool
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  val mk_inst_pair: (ctyp -> 'a) -> cterm -> 'a * cterm
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  val destT1: ctyp -> ctyp
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  val destT2: ctyp -> ctyp
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  val instT': cterm -> ctyp * cterm -> cterm
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end
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structure Z3_Interface: Z3_INTERFACE =
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struct
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(** Z3-specific builtins **)
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type builtin_fun = string * typ -> term list -> (string * term list) option
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fun fst_int_ord ((s1, _), (s2, _)) = int_ord (s1, s2)
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structure Builtins = Generic_Data
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(
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  type T = (int * builtin_fun) list
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  val empty = []
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  val extend = I
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  fun merge (bs1, bs2) = OrdList.union fst_int_ord bs2 bs1
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)
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fun add_builtin_funs b =
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  Builtins.map (OrdList.insert fst_int_ord (serial (), b))
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fun get_builtin_funs ctxt c ts =
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  let
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    fun chained [] = NONE
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      | chained (b :: bs) = (case b c ts of SOME x => SOME x | _ => chained bs)
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  in chained (map snd (Builtins.get (Context.Proof ctxt))) end
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fun z3_builtin_fun builtin_fun ctxt c ts =
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  (case builtin_fun ctxt c ts of
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    SOME x => SOME x
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  | _ => get_builtin_funs ctxt c ts)
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(** interface **)
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local
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  val {extra_norm, translate} = SMTLIB_Interface.interface
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  val {prefixes, strict, header, builtins, serialize} = translate
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  val {is_builtin_pred, ...}= the strict
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  val {builtin_typ, builtin_num, builtin_fun} = builtins
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  fun is_int_div_mod @{term "op div :: int => _"} = true
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    | is_int_div_mod @{term "op mod :: int => _"} = true
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    | is_int_div_mod _ = false
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  fun add_div_mod thms =
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    if exists (Term.exists_subterm is_int_div_mod o Thm.prop_of) thms
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    then [@{thm div_by_z3div}, @{thm mod_by_z3mod}] @ thms
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    else thms
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  fun extra_norm' thms = extra_norm (add_div_mod thms)
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  fun z3_builtin_fun' _ (@{const_name z3div}, _) ts = SOME ("div", ts)
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    | z3_builtin_fun' _ (@{const_name z3mod}, _) ts = SOME ("mod", ts)
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    | z3_builtin_fun' ctxt c ts = z3_builtin_fun builtin_fun ctxt c ts
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  val as_propT = (fn @{typ bool} => @{typ prop} | T => T)
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in
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fun is_builtin_num ctxt (T, i) = is_some (builtin_num ctxt T i)
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fun is_builtin_fun ctxt (c as (n, T)) ts =
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  is_some (z3_builtin_fun' ctxt c ts) orelse 
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  is_builtin_pred ctxt (n, Term.strip_type T ||> as_propT |> (op --->))
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val interface = {
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  extra_norm = extra_norm',
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  translate = {
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    prefixes = prefixes,
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    strict = strict,
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    header = header,
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    builtins = {
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      builtin_typ = builtin_typ,
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      builtin_num = builtin_num,
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      builtin_fun = z3_builtin_fun'},
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    serialize = serialize}}
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end
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(** constructors **)
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datatype sym = Sym of string * sym list
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(* additional constructors *)
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type mk_builtins = {
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  mk_builtin_typ: sym -> typ option,
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  mk_builtin_num: theory -> int -> typ -> cterm option,
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  mk_builtin_fun: theory -> sym -> cterm list -> cterm option }
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fun chained _ [] = NONE
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  | chained f (b :: bs) = (case f b of SOME y => SOME y | NONE => chained f bs)
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fun chained_mk_builtin_typ bs sym =
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  chained (fn {mk_builtin_typ=mk, ...} : mk_builtins => mk sym) bs
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fun chained_mk_builtin_num ctxt bs i T =
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  let val thy = ProofContext.theory_of ctxt
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  in chained (fn {mk_builtin_num=mk, ...} : mk_builtins => mk thy i T) bs end
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fun chained_mk_builtin_fun ctxt bs s cts =
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  let val thy = ProofContext.theory_of ctxt
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  in chained (fn {mk_builtin_fun=mk, ...} : mk_builtins => mk thy s cts) bs end
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structure Mk_Builtins = Generic_Data
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(
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  type T = (int * mk_builtins) list
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  val empty = []
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  val extend = I
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  fun merge (bs1, bs2) = OrdList.union fst_int_ord bs2 bs1
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)
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fun add_mk_builtins mk =
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  Mk_Builtins.map (OrdList.insert fst_int_ord (serial (), mk))
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fun get_mk_builtins ctxt = map snd (Mk_Builtins.get (Context.Proof ctxt))
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(* basic and additional constructors *)
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fun mk_builtin_typ _ (Sym ("bool", _)) = SOME @{typ bool}
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  | mk_builtin_typ _ (Sym ("int", _)) = SOME @{typ int}
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  | mk_builtin_typ ctxt sym = chained_mk_builtin_typ (get_mk_builtins ctxt) sym
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fun mk_builtin_num _ i @{typ int} = SOME (Numeral.mk_cnumber @{ctyp int} i)
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  | mk_builtin_num ctxt i T =
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      chained_mk_builtin_num ctxt (get_mk_builtins ctxt) i T
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fun instTs cUs (cTs, ct) = Thm.instantiate_cterm (cTs ~~ cUs, []) ct
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fun instT cU (cT, ct) = instTs [cU] ([cT], ct)
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fun instT' ct = instT (Thm.ctyp_of_term ct)
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fun mk_inst_pair destT cpat = (destT (Thm.ctyp_of_term cpat), cpat)
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val destT1 = hd o Thm.dest_ctyp
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val destT2 = hd o tl o Thm.dest_ctyp
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val mk_true = @{cterm "~False"}
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val mk_false = @{cterm False}
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val mk_not = Thm.capply @{cterm Not}
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val mk_implies = Thm.mk_binop @{cterm "op -->"}
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val mk_iff = Thm.mk_binop @{cterm "op = :: bool => _"}
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fun mk_nary _ cu [] = cu
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  | mk_nary ct _ cts = uncurry (fold_rev (Thm.mk_binop ct)) (split_last cts)
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val eq = mk_inst_pair destT1 @{cpat "op ="}
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fun mk_eq ct cu = Thm.mk_binop (instT' ct eq) ct cu
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val if_term = mk_inst_pair (destT1 o destT2) @{cpat If}
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fun mk_if cc ct cu = Thm.mk_binop (Thm.capply (instT' ct if_term) cc) ct cu
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val nil_term = mk_inst_pair destT1 @{cpat Nil}
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val cons_term = mk_inst_pair destT1 @{cpat Cons}
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fun mk_list cT cts =
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  fold_rev (Thm.mk_binop (instT cT cons_term)) cts (instT cT nil_term)
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val distinct = mk_inst_pair (destT1 o destT1) @{cpat distinct}
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fun mk_distinct [] = mk_true
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  | mk_distinct (cts as (ct :: _)) =
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      Thm.capply (instT' ct distinct) (mk_list (Thm.ctyp_of_term ct) cts)
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val access = mk_inst_pair (Thm.dest_ctyp o destT1) @{cpat fun_app}
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fun mk_access array index =
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  let val cTs = Thm.dest_ctyp (Thm.ctyp_of_term array)
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  in Thm.mk_binop (instTs cTs access) array index end
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val update = mk_inst_pair (Thm.dest_ctyp o destT1) @{cpat fun_upd}
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fun mk_update array index value =
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  let val cTs = Thm.dest_ctyp (Thm.ctyp_of_term array)
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  in Thm.capply (Thm.mk_binop (instTs cTs update) array index) value end
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val mk_uminus = Thm.capply @{cterm "uminus :: int => _"}
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val mk_add = Thm.mk_binop @{cterm "op + :: int => _"}
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val mk_sub = Thm.mk_binop @{cterm "op - :: int => _"}
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val mk_mul = Thm.mk_binop @{cterm "op * :: int => _"}
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val mk_div = Thm.mk_binop @{cterm "z3div :: int => _"}
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val mk_mod = Thm.mk_binop @{cterm "z3mod :: int => _"}
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val mk_lt = Thm.mk_binop @{cterm "op < :: int => _"}
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val mk_le = Thm.mk_binop @{cterm "op <= :: int => _"}
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fun mk_builtin_fun ctxt sym cts =
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  (case (sym, cts) of
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    (Sym ("true", _), []) => SOME mk_true
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  | (Sym ("false", _), []) => SOME mk_false
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  | (Sym ("not", _), [ct]) => SOME (mk_not ct)
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  | (Sym ("and", _), _) => SOME (mk_nary @{cterm "op &"} mk_true cts)
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  | (Sym ("or", _), _) => SOME (mk_nary @{cterm "op |"} mk_false cts)
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  | (Sym ("implies", _), [ct, cu]) => SOME (mk_implies ct cu)
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  | (Sym ("iff", _), [ct, cu]) => SOME (mk_iff ct cu)
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  | (Sym ("~", _), [ct, cu]) => SOME (mk_iff ct cu)
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  | (Sym ("xor", _), [ct, cu]) => SOME (mk_not (mk_iff ct cu))
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  | (Sym ("ite", _), [ct1, ct2, ct3]) => SOME (mk_if ct1 ct2 ct3)
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  | (Sym ("=", _), [ct, cu]) => SOME (mk_eq ct cu)
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  | (Sym ("distinct", _), _) => SOME (mk_distinct cts)
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  | (Sym ("select", _), [ca, ck]) => SOME (mk_access ca ck)
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  | (Sym ("store", _), [ca, ck, cv]) => SOME (mk_update ca ck cv)
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  | _ =>
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    (case (sym, try (#T o Thm.rep_cterm o hd) cts, cts) of
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      (Sym ("+", _), SOME @{typ int}, [ct, cu]) => SOME (mk_add ct cu)
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    | (Sym ("-", _), SOME @{typ int}, [ct]) => SOME (mk_uminus ct)
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    | (Sym ("-", _), SOME @{typ int}, [ct, cu]) => SOME (mk_sub ct cu)
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    | (Sym ("*", _), SOME @{typ int}, [ct, cu]) => SOME (mk_mul ct cu)
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    | (Sym ("div", _), SOME @{typ int}, [ct, cu]) => SOME (mk_div ct cu)
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    | (Sym ("mod", _), SOME @{typ int}, [ct, cu]) => SOME (mk_mod ct cu)
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    | (Sym ("<", _), SOME @{typ int}, [ct, cu]) => SOME (mk_lt ct cu)
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    | (Sym ("<=", _), SOME @{typ int}, [ct, cu]) => SOME (mk_le ct cu)
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    | (Sym (">", _), SOME @{typ int}, [ct, cu]) => SOME (mk_lt cu ct)
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    | (Sym (">=", _), SOME @{typ int}, [ct, cu]) => SOME (mk_le cu ct)
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    | _ => chained_mk_builtin_fun ctxt (get_mk_builtins ctxt) sym cts))
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(** abstraction **)
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fun is_builtin_theory_term ctxt t =
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  (case try HOLogic.dest_number t of
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    SOME n => is_builtin_num ctxt n
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  | NONE =>
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      (case Term.strip_comb t of
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        (Const c, ts) => is_builtin_fun ctxt c ts
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      | _ => false))
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bcd6fce5bf06 layered SMT setup, adapted SMT clients, added further tests, made Z3 proof abstraction configurable
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end