diff --git a/src/main/main.ml b/src/main/main.ml
index 7a0ad1c8..ada5dd9f 100644
--- a/src/main/main.ml
+++ b/src/main/main.ml
@@ -123,10 +123,11 @@ let main () =
   (* process statements *)
   let res =
     try
+      let hyps = Vec.make_empty [] in
       E.fold_l
         (fun () ->
            Process.process_stmt
-             ~gc:!gc ~restarts:!restarts ~pp_cnf:!p_cnf
+             ~hyps ~gc:!gc ~restarts:!restarts ~pp_cnf:!p_cnf
              ~time:!time_limit ~memory:!size_limit
              ?dot_proof ~pp_model:!p_model ~check:!check ~progress:!p_progress
              solver)
diff --git a/src/sat/Solver_intf.ml b/src/sat/Solver_intf.ml
index 26fea6fd..e41e17cc 100644
--- a/src/sat/Solver_intf.ml
+++ b/src/sat/Solver_intf.ml
@@ -152,7 +152,9 @@ module type S = sig
     val compare : t -> t -> int
     val equal : t -> t -> bool
     val get_formula : t -> formula
+    val is_true : t -> bool
 
+    val dummy : t
     val make : solver -> formula -> t
 
     val pp : t printer
diff --git a/src/smt/Congruence_closure.ml b/src/smt/Congruence_closure.ml
index fdd8b6ea..ed960bf3 100644
--- a/src/smt/Congruence_closure.ml
+++ b/src/smt/Congruence_closure.ml
@@ -587,26 +587,65 @@ let final_check cc : unit =
 (* model: map each uninterpreted equiv class to some ID *)
 let mk_model (cc:t) (m:Model.t) : Model.t =
   (* populate [repr -> value] table *)
-  let tbl = Equiv_class.Tbl.create 32 in
+  let t_tbl = Equiv_class.Tbl.create 32 in
+  (* type -> default value *)
+  let ty_tbl = Ty.Tbl.create 8 in
   Term.Tbl.values cc.tbl
     (fun r ->
        if is_root_ r then (
-         let v = match Model.eval m r.n_term with
+         let t = r.n_term in
+         let v = match Model.eval m t with
            | Some v -> v
            | None ->
              Value.mk_elt
-               (ID.makef "v_%d" @@ Term.id r.n_term)
+               (ID.makef "v_%d" @@ Term.id t)
                (Term.ty r.n_term)
          in
-         Equiv_class.Tbl.add tbl r v
+         if not @@ Ty.Tbl.mem ty_tbl (Term.ty t) then (
+           Ty.Tbl.add ty_tbl (Term.ty t) v; (* also give a value to this type *)
+         );
+         Equiv_class.Tbl.add t_tbl r v
        ));
-  (* now map every uninterpreted term to its representative's value *)
-  Term.Tbl.to_seq cc.tbl
-  |> Sequence.fold
-    (fun m (t,r) ->
-       if Model.mem t m then m
-       else (
-         let v = Equiv_class.Tbl.find tbl r in
-         Model.add t v m
-       ))
-    m
+  (* now map every uninterpreted term to its representative's value, and
+     create function tables *)
+  let m, funs =
+    Term.Tbl.to_seq cc.tbl
+    |> Sequence.fold
+      (fun (m,funs) (t,r) ->
+         let r = find cc r in (* get representative *)
+         match Term.view t with
+         | _ when Model.mem t m -> m, funs
+         | App_cst (c, args) ->
+           if Model.mem t m then m, funs
+           else if Cst.is_undefined c && IArray.length args > 0 then (
+             (* update signature of [c] *)
+             let ty = Term.ty t in
+             let v = Equiv_class.Tbl.find t_tbl r in
+             let args =
+               args
+               |> IArray.map (fun t -> Equiv_class.Tbl.find t_tbl @@ find_tn cc t)
+               |> IArray.to_list
+             in
+             let ty, l = Cst.Map.get_or c funs ~default:(ty,[]) in
+             m, Cst.Map.add c (ty, (args,v)::l) funs
+           ) else (
+             let v = Equiv_class.Tbl.find t_tbl r in
+             Model.add t v m, funs
+           )
+         | _ ->
+           let v = Equiv_class.Tbl.find t_tbl r in
+           Model.add t v m, funs)
+      (m,Cst.Map.empty)
+  in
+  (* get or make a default value for this type *)
+  let get_ty_default (ty:Ty.t) : Value.t =
+    Ty.Tbl.get_or_add ty_tbl ~k:ty
+      ~f:(fun ty -> Value.mk_elt (ID.makef "ty_%d" @@ Ty.id ty) ty)
+  in
+  let funs =
+    Cst.Map.map
+      (fun (ty,l) ->
+         Model.Fun_interpretation.make ~default:(get_ty_default ty) l)
+      funs
+  in
+  Model.add_funs funs m
diff --git a/src/smt/Cst.ml b/src/smt/Cst.ml
index 44c410ce..ddd7fe71 100644
--- a/src/smt/Cst.ml
+++ b/src/smt/Cst.ml
@@ -12,6 +12,8 @@ let as_undefined (c:t) = match view c with
   | Cst_undef ty -> Some (c,ty)
   | Cst_def _ -> None
 
+let[@inline] is_undefined c = match view c with Cst_undef _ -> true | _ -> false
+
 let as_undefined_exn (c:t) = match as_undefined c with
   | Some tup -> tup
   | None -> assert false
diff --git a/src/smt/Cst.mli b/src/smt/Cst.mli
index 08d118eb..b6d98dc8 100644
--- a/src/smt/Cst.mli
+++ b/src/smt/Cst.mli
@@ -11,6 +11,7 @@ val compare : t -> t -> int
 val hash : t -> int
 val as_undefined : t -> (t * Ty.Fun.t) option
 val as_undefined_exn : t -> t * Ty.Fun.t
+val is_undefined : t -> bool
 
 val mk_undef : ID.t -> Ty.Fun.t -> t
 val mk_undef_const : ID.t -> Ty.t -> t
diff --git a/src/smt/Model.ml b/src/smt/Model.ml
index 2407eaea..b3057484 100644
--- a/src/smt/Model.ml
+++ b/src/smt/Model.ml
@@ -5,11 +5,58 @@
 
 open Solver_types
 
+module Val_map = struct
+  module M = CCIntMap
+  module Key = struct
+    type t = Value.t list
+    let equal = CCList.equal Value.equal
+    let hash = Hash.list Value.hash
+  end
+  type key = Key.t
+
+  type 'a t = (key * 'a) list M.t
+
+  let empty = M.empty
+  let is_empty m = M.cardinal m = 0
+
+  let cardinal = M.cardinal
+
+  let find k m =
+    try Some (CCList.assoc ~eq:Key.equal k @@ M.find_exn (Key.hash k) m)
+    with Not_found -> None
+
+  let add k v m =
+    let h = Key.hash k in
+    let l = M.find h m |> CCOpt.get_or ~default:[] in
+    let l = CCList.Assoc.set ~eq:Key.equal k v l in
+    M.add h l m
+
+  let to_seq m yield = M.iter (fun _ l -> List.iter yield l) m
+end
+
+module Fun_interpretation = struct
+  type t = {
+    cases: Value.t Val_map.t;
+    default: Value.t;
+  }
+
+  let default fi = fi.default
+  let cases_list fi = Val_map.to_seq fi.cases |> Sequence.to_rev_list
+
+  let make ~default l : t =
+    let m = List.fold_left (fun m (k,v) -> Val_map.add k v m) Val_map.empty l in
+    { cases=m; default }
+end
+
 type t = {
   values: Value.t Term.Map.t;
+  funs: Fun_interpretation.t Cst.Map.t;
 }
 
-let empty : t = {values=Term.Map.empty}
+let empty : t = {
+  values=Term.Map.empty;
+  funs=Cst.Map.empty;
+}
 
 let[@inline] mem t m = Term.Map.mem t m.values
 let[@inline] find t m = Term.Map.get t m.values
@@ -23,7 +70,13 @@ let add t v m : t =
     );
     m
   | exception Not_found ->
-    {values=Term.Map.add t v m.values}
+    {m with values=Term.Map.add t v m.values}
+
+let add_fun c v m : t =
+  match Cst.Map.find c m.funs with
+  | _ -> Error.errorf "@[Model: function %a already has an interpretation@]" Cst.pp c
+  | exception Not_found ->
+    {m with funs=Cst.Map.add c v m.funs}
 
 (* merge two models *)
 let merge m1 m2 : t =
@@ -36,17 +89,36 @@ let merge m1 m2 : t =
             Error.errorf "@[Model: incompatible values for term %a@ :previous %a@ :new %a@]"
               Term.pp t Value.pp v1 Value.pp v2
           ))
+  and funs =
+    Cst.Map.merge_safe m1.funs m2.funs
+      ~f:(fun c o -> match o with
+        | `Left v | `Right v -> Some v
+        | `Both _ ->
+          Error.errorf "cannot merge the two interpretations of function %a" Cst.pp c)
   in
-  {values}
+  {values; funs}
 
-let pp out (m:t) =
-  let pp_tv out (t,v) = Fmt.fprintf out "(@[%a@ -> %a@])" Term.pp t Value.pp v in
-  Fmt.fprintf out "(@[model@ %a@])"
-    (Fmt.seq ~sep:Fmt.(return "@ ") pp_tv) (Term.Map.to_seq m.values)
+let add_funs fs m : t = merge {values=Term.Map.empty; funs=fs} m
+
+let pp out {values; funs} =
+  let module FI = Fun_interpretation in
+  let pp_tv out (t,v) = Fmt.fprintf out "(@[%a@ %a@])" Term.pp t Value.pp v in
+  let pp_fun_entry out (vals,ret) =
+    Format.fprintf out "(@[%a@ %a@])" (Fmt.Dump.list Value.pp) vals Value.pp ret
+  in
+  let pp_fun out (c, fi: Cst.t * FI.t) =
+    Format.fprintf out "(@[<hov>%a :default %a@ %a@])"
+      Cst.pp c Value.pp fi.FI.default
+      (Fmt.list ~sep:(Fmt.return "@ ") pp_fun_entry) (FI.cases_list fi) 
+  in
+  Fmt.fprintf out "(@[model@ @[:terms (@[<hv>%a@])@]@ @[:funs (@[<hv>%a@])@]@])"
+    (Fmt.seq ~sep:Fmt.(return "@ ") pp_tv) (Term.Map.to_seq values)
+    (Fmt.seq ~sep:Fmt.(return "@ ") pp_fun) (Cst.Map.to_seq funs)
 
 exception No_value
 
 let eval (m:t) (t:Term.t) : Value.t option =
+  let module FI = Fun_interpretation in
   let rec aux t = match Term.view t with
     | Bool b -> Value.bool b
     | If (a,b,c) ->
@@ -64,8 +136,17 @@ let eval (m:t) (t:Term.t) : Value.t option =
           let args = IArray.map aux args in
           udef.eval args
         | Cst_undef _ ->
-          Log.debugf 5 (fun k->k "(@[model.eval.undef@ %a@])" Term.pp t);
-          raise No_value (* no particular interpretation *)
+          begin match Cst.Map.find c m.funs with
+            | fi ->
+              let args = IArray.map aux args |> IArray.to_list in
+              begin match Val_map.find args fi.FI.cases with
+                | None -> fi.FI.default
+                | Some v -> v
+              end
+            | exception Not_found ->
+              Log.debugf 5 (fun k->k "(@[model.eval.undef@ %a@])" Term.pp t);
+              raise No_value (* no particular interpretation *)
+          end
       end
   in
   try Some (aux t)
diff --git a/src/smt/Model.mli b/src/smt/Model.mli
index e0ac5ebc..c6ac4c04 100644
--- a/src/smt/Model.mli
+++ b/src/smt/Model.mli
@@ -3,14 +3,44 @@
 
 (** {1 Model} *)
 
+module Val_map : sig
+  type key = Value.t list
+  type 'a t
+  val empty : 'a t
+  val is_empty : _ t -> bool
+  val cardinal : _ t -> int
+  val find : key -> 'a t -> 'a option
+  val add : key -> 'a -> 'a t -> 'a t
+end
+
+module Fun_interpretation : sig
+  type t = {
+    cases: Value.t Val_map.t;
+    default: Value.t;
+  }
+
+  val default : t -> Value.t
+  val cases_list : t -> (Value.t list * Value.t) list
+
+  val make :
+    default:Value.t ->
+    (Value.t list * Value.t) list ->
+    t
+end
+
 type t = {
   values: Value.t Term.Map.t;
+  funs: Fun_interpretation.t Cst.Map.t;
 }
 
 val empty : t
 
 val add : Term.t -> Value.t -> t -> t
 
+val add_fun : Cst.t -> Fun_interpretation.t -> t -> t
+
+val add_funs : Fun_interpretation.t Cst.Map.t -> t -> t
+
 val mem : Term.t -> t -> bool
 
 val find : Term.t -> t -> Value.t option
diff --git a/src/smt/Solver.ml b/src/smt/Solver.ml
index 3a14b70f..3f8ab8ab 100644
--- a/src/smt/Solver.ml
+++ b/src/smt/Solver.ml
@@ -202,7 +202,9 @@ let[@inline] assume_eq self t u expl : unit =
 let[@inline] assume_distinct self l ~neq lit : unit =
   Congruence_closure.assert_distinct (cc self) l lit ~neq
 
-let check_model (s:t) = Sat_solver.check_model s.solver
+let check_model (s:t) : unit =
+  Log.debug 1 "(smt.solver.check-model)";
+  Sat_solver.check_model s.solver
 
 (* TODO: main loop with iterative deepening of the unrolling  limit
    (not the value depth limit) *)
diff --git a/src/smt/Solver_types.ml b/src/smt/Solver_types.ml
index b9f4d4ed..6ea45fe3 100644
--- a/src/smt/Solver_types.ml
+++ b/src/smt/Solver_types.ml
@@ -130,6 +130,7 @@ and value =
       view: value_custom_view;
       pp: value_custom_view Fmt.printer;
       eq: value_custom_view -> value_custom_view -> bool;
+      hash: value_custom_view -> int;
     } (** Custom value *)
 
 and value_custom_view = ..
@@ -192,6 +193,11 @@ let eq_value a b = match a, b with
   | V_bool _, _ | V_element _, _ | V_custom _, _
     -> false
 
+let hash_value a = match a with
+  | V_bool a -> Hash.bool a
+  | V_element e -> ID.hash e.id
+  | V_custom x -> x.hash x.view
+
 let pp_value out = function
   | V_bool b -> Fmt.bool out b
   | V_element e -> ID.pp out e.id
diff --git a/src/smt/Ty.ml b/src/smt/Ty.ml
index 7b944667..f175044e 100644
--- a/src/smt/Ty.ml
+++ b/src/smt/Ty.ml
@@ -5,7 +5,8 @@ type t = ty
 type view = Solver_types.ty_view
 type def = Solver_types.ty_def
 
-let view t = t.ty_view
+let[@inline] id t = t.ty_id
+let[@inline] view t = t.ty_view
 
 let equal = eq_ty
 let[@inline] compare a b = CCInt.compare a.ty_id b.ty_id
diff --git a/src/smt/Ty.mli b/src/smt/Ty.mli
index 6ee5413c..7976e1ed 100644
--- a/src/smt/Ty.mli
+++ b/src/smt/Ty.mli
@@ -7,6 +7,7 @@ type t = Solver_types.ty
 type view = Solver_types.ty_view
 type def = Solver_types.ty_def
 
+val id : t -> int
 val view : t -> view
 
 val prop : t
diff --git a/src/smtlib/Process.ml b/src/smtlib/Process.ml
index c9749348..81616e74 100644
--- a/src/smtlib/Process.ml
+++ b/src/smtlib/Process.ml
@@ -217,6 +217,47 @@ end
 let conv_ty = Conv.conv_ty
 let conv_term = Conv.conv_term
 
+(* check SMT model *)
+let check_smt_model (solver:Solver.Sat_solver.t) (hyps:_ Vec.t) (m:Model.t) : unit =
+  Log.debug 1 "(smt.check-smt-model)";
+  let open Solver_types in
+  let module S = Solver.Sat_solver in
+  let check_atom (lit:Lit.t) : bool option =
+    Log.debugf 5 (fun k->k "(@[smt.check-smt-model.atom@ %a@])" Lit.pp lit);
+    let a = S.Atom.make solver lit in
+    let is_true = S.Atom.is_true a in
+    let is_false = S.Atom.is_true (S.Atom.neg a) in
+    let sat_value = if is_true then Some true else if is_false then Some false else None in
+    begin match Lit.as_atom lit with
+    | None -> assert false
+    | Some (t, sign) ->
+      match Model.eval m t with
+      | Some (V_bool b) ->
+        let b = if sign then b else not b in
+        if (is_true || is_false) && ((b && is_false) || (not b && is_true)) then (
+          Error.errorf "(@[check-model.error@ :atom %a@ :model-val %B@ :sat-val %B@])"
+            S.Atom.pp a b (if is_true then true else not is_false)
+        )
+      | Some v ->
+        Error.errorf "(@[check-model.error@ :atom %a@ :non-bool-value %a@])"
+          S.Atom.pp a Value.pp v
+      | None ->
+        if is_true || is_false then (
+          Error.errorf "(@[check-model.error@ :atom %a@ :no-smt-value@ :sat-val %B@])"
+            S.Atom.pp a is_true
+        );
+    end;
+    sat_value
+  in
+  let check_c c =
+    let bs = List.map check_atom c in
+    if List.for_all (function Some true -> false | _ -> true) bs then (
+      Error.errorf "(@[check-model.error.none-true@ :clause %a@ :vals %a@])"
+        (Fmt.Dump.list Lit.pp) c Fmt.(Dump.list @@ Dump.option bool) bs
+    );
+  in
+  Vec.iter check_c hyps
+
 (* call the solver to check-sat *)
 let solve
     ?gc:_
@@ -225,11 +266,13 @@ let solve
     ?(pp_model=false)
     ?(check=false)
     ?time:_ ?memory:_ ?progress:_
-    ~assumptions s : unit =
+    ?hyps
+    ~assumptions
+    s : unit =
   let t1 = Sys.time() in
   let res =
     Solver.solve ~assumptions s
-    (* ?gc ?restarts ?time ?memory ?progress s *)
+    (* ?gc ?restarts ?time ?memory ?progress *)
   in
   let t2 = Sys.time () in
   begin match res with
@@ -237,7 +280,10 @@ let solve
       if pp_model then (
         Format.printf "(@[<hv1>model@ %a@])@." Model.pp  m
       );
-      if check then Solver.check_model s;
+      if check then (
+        Solver.check_model s;
+        CCOpt.iter (fun h -> check_smt_model (Solver.solver s) h m) hyps;
+      );
       let t3 = Sys.time () -. t2 in
       Format.printf "Sat (%.3f/%.3f/%.3f)@." t1 (t2-.t1) t3;
     | Solver.Unsat p ->
@@ -271,6 +317,7 @@ let mk_iatom =
 
 (* process a single statement *)
 let process_stmt
+    ?hyps
     ?gc ?restarts ?(pp_cnf=false) ?dot_proof ?pp_model ?check
     ?time ?memory ?progress
     (solver:Solver.t)
@@ -301,8 +348,10 @@ let process_stmt
       Log.debug 1 "exit";
       raise Exit
     | A.CheckSat ->
-      solve ?gc ?restarts ?dot_proof ?check ?pp_model ?time ?memory ?progress
-        solver ~assumptions:[];
+      solve
+        ?gc ?restarts ?dot_proof ?check ?pp_model ?time ?memory ?progress
+        ~assumptions:[] ?hyps
+        solver;
       E.return()
     | A.TyDecl (id,n) ->
       decl_sort id n;
@@ -318,13 +367,13 @@ let process_stmt
       if pp_cnf then (
         Format.printf "(@[<hv1>assert@ %a@])@." Term.pp t
       );
-      (* TODO
-      hyps := clauses @ !hyps;
-         *)
-      Solver.assume solver (IArray.singleton (Lit.atom t));
+      let atom = Lit.atom t in
+      CCOpt.iter (fun h -> Vec.push h [atom]) hyps;
+      Solver.assume solver (IArray.singleton atom);
       E.return()
     | A.Assert_bool l ->
       let c = List.rev_map (mk_iatom tst) l in
+      CCOpt.iter (fun h -> Vec.push h c) hyps;
       Solver.assume solver (IArray.of_list c);
       E.return ()
     | A.Goal (_, _) ->
diff --git a/src/smtlib/Process.mli b/src/smtlib/Process.mli
index 978cc1b2..46c20333 100644
--- a/src/smtlib/Process.mli
+++ b/src/smtlib/Process.mli
@@ -12,6 +12,7 @@ val conv_ty : Ast.Ty.t -> Ty.t
 val conv_term : Term.state -> Ast.term -> Term.t
 
 val process_stmt :
+  ?hyps:Lit.t list Vec.t ->
   ?gc:bool ->
   ?restarts:bool ->
   ?pp_cnf:bool ->
diff --git a/src/util/Vec.ml b/src/util/Vec.ml
index 67106d0d..c7df6b56 100644
--- a/src/util/Vec.ml
+++ b/src/util/Vec.ml
@@ -146,6 +146,8 @@ let append a b =
   grow_to_at_least a (size a + size b);
   iter (push a) b
 
+let append_l v l = List.iter (push v) l
+
 let fold f acc t =
   let rec _fold f acc t i =
     if i=t.sz
diff --git a/src/util/Vec.mli b/src/util/Vec.mli
index 0d9b6cef..c7497020 100644
--- a/src/util/Vec.mli
+++ b/src/util/Vec.mli
@@ -66,6 +66,8 @@ val push : 'a t -> 'a -> unit
 val append : 'a t -> 'a t -> unit
 (** [append v1 v2] pushes all elements of [v2] into [v1] *)
 
+val append_l : 'a t -> 'a list -> unit
+
 val last : 'a t -> 'a
 (** Last element, or
     @raise Invalid_argument if the vector is empty *)