catala/compiler/verification/z3backend.real.ml

(* This file is part of the Catala compiler, a specification language for tax
   and social benefits computation rules. Copyright (C) 2022 Inria, contributor:
   Aymeric Fromherz <aymeric.fromherz@inria.fr>

   Licensed under the Apache License, Version 2.0 (the "License"); you may not
   use this file except in compliance with the License. You may obtain a copy of
   the License at

   http://www.apache.org/licenses/LICENSE-2.0

   Unless required by applicable law or agreed to in writing, software
   distributed under the License is distributed on an "AS IS" BASIS, WITHOUT
   WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the
   License for the specific language governing permissions and limitations under
   the License. *)

open Catala_utils
open Shared_ast
open Dcalc
open Ast
open Z3
module StringMap = String.Map
module Runtime = Runtime_ocaml.Runtime

type context = {
  ctx_z3 : Z3.context;
  (* The Z3 context, used to create symbols and expressions *)
  ctx_decl : decl_ctx;
  (* The declaration context from the Catala program, containing information to
     precisely pretty print Catala expressions *)
  ctx_funcdecl : (typed expr, FuncDecl.func_decl) Var.Map.t;
  (* A map from Catala function names (represented as variables) to Z3 function
     declarations, used to only define once functions in Z3 queries *)
  ctx_z3vars : (typed expr Var.t * typ) StringMap.t;
  (* A map from strings, corresponding to Z3 symbol names, to the Catala
     variable they represent. Used when to pretty-print Z3 models when a
     counterexample is generated *)
  ctx_z3datatypes : Sort.sort EnumName.Map.t;
  (* A map from Catala enumeration names to the corresponding Z3 sort, from
     which we can retrieve constructors and accessors *)
  ctx_z3matchsubsts : (typed expr, Expr.expr) Var.Map.t;
  (* A map from Catala temporary variables, generated when translating a match,
     to the corresponding enum accessor call as a Z3 expression *)
  ctx_z3structs : Sort.sort StructName.Map.t;
  (* A map from Catala struct names to the corresponding Z3 sort, from which we
     can retrieve the constructor and the accessors *)
  ctx_z3unit : Sort.sort * Expr.expr;
  (* A pair containing the Z3 encodings of the unit type, encoded as a tuple of
     0 elements, and the unit value *)
  ctx_z3constraints : Expr.expr list;
      (* A list of constraints about the created Z3 expressions accumulated
         during their initialization, for instance, that the length of an array
         is an integer which always is greater than 0 *)
}
(** The context contains all the required information to encode a VC represented
    as a Catala term to Z3. The field [ctx_decl] is computed before starting the
    translation to Z3, and are thus unmodified throughout the translation. The
    [ctx_z3] context is an OCaml abstraction on top of an underlying C++
    imperative implementation, it is therefore only created once. Unfortunately,
    the maps [ctx_funcdecl], [ctx_z3vars], [ctx_z3datatypes],
    [ctx_z3matchsubsts], [ctx_z3structs], and [ctx_z3constraints] are computed
    dynamically during the translation requiring us to pass the context around
    in a functional way **)

(** [add_funcdecl] adds the mapping between the Catala variable [v] and the Z3
    function declaration [fd] to the context **)
let add_funcdecl
    (v : typed expr Var.t)
    (fd : FuncDecl.func_decl)
    (ctx : context) : context =
  { ctx with ctx_funcdecl = Var.Map.add v fd ctx.ctx_funcdecl }

(** [add_z3var] adds the mapping between [name] and the Catala variable [v] and
    its typ [ty] to the context **)
let add_z3var (name : string) (v : typed expr Var.t) (ty : typ) (ctx : context)
    : context =
  { ctx with ctx_z3vars = StringMap.add name (v, ty) ctx.ctx_z3vars }

(** [add_z3enum] adds the mapping between the Catala enumeration [enum] and the
    corresponding Z3 datatype [sort] to the context **)
let add_z3enum (enum : EnumName.t) (sort : Sort.sort) (ctx : context) : context
    =
  { ctx with ctx_z3datatypes = EnumName.Map.add enum sort ctx.ctx_z3datatypes }

(** [add_z3matchsubst] adds the mapping between temporary variable [v] and the
    Z3 expression [e] representing an accessor application to the context **)
let add_z3matchsubst (v : typed expr Var.t) (e : Expr.expr) (ctx : context) :
    context =
  { ctx with ctx_z3matchsubsts = Var.Map.add v e ctx.ctx_z3matchsubsts }

(** [add_z3struct] adds the mapping between the Catala struct [s] and the
    corresponding Z3 datatype [sort] to the context **)
let add_z3struct (s : StructName.t) (sort : Sort.sort) (ctx : context) : context
    =
  { ctx with ctx_z3structs = StructName.Map.add s sort ctx.ctx_z3structs }

let add_z3constraint (e : Expr.expr) (ctx : context) : context =
  { ctx with ctx_z3constraints = e :: ctx.ctx_z3constraints }

(** For the Z3 encoding of Catala programs, we define the "day 0" as Jan 1, 1900
    **)
let base_day = Runtime.date_of_numbers 1900 1 1

(** [unique_name] returns the full, unique name corresponding to variable [v],
    as given by Bindlib **)
let unique_name (v : 'e Var.t) : string =
  Format.asprintf "%s_%d" (Bindlib.name_of v) (Bindlib.uid_of v)

(** [date_to_int] translates [date] to an integer corresponding to the number of
    days since Jan 1, 1900 **)
let date_to_int (d : Runtime.date) : int =
  (* Alternatively, could expose this from Runtime as a (noop) coercion, but
     would allow to break abstraction more easily elsewhere *)
  let period = Runtime.Oper.o_sub_dat_dat d base_day in
  let y, m, d = Runtime.duration_to_years_months_days period in
  assert (y = 0 && m = 0);
  d

(** [date_of_year] translates a [year], represented as an integer into an OCaml
    date corresponding to Jan 1st of the same year *)
let date_of_year (year : int) = Runtime.date_of_numbers year 1 1

(** Returns the date (as a string) corresponding to nb days after the base day,
    defined here as Jan 1, 1900 **)
let nb_days_to_date (nb : int) : string =
  Runtime.date_to_string
    (Runtime.Oper.o_add_dat_dur AbortOnRound base_day
       (Runtime.duration_of_numbers 0 0 nb))

(** [print_z3model_expr] pretty-prints the value [e] given by a Z3 model
    according to the Catala type [ty], corresponding to [e] **)
let rec print_z3model_expr (ctx : context) (ty : typ) (e : Expr.expr) : string =
  let print_lit (ty : typ_lit) =
    match ty with
    (* TODO: Print boolean according to current language *)
    | TBool -> Expr.to_string e
    (* TUnit is only used for the absence of an enum constructor argument.
       Hence, when pretty-printing, we print nothing to remain closer from
       Catala sources *)
    | TUnit -> ""
    | TInt -> Expr.to_string e
    | TRat -> Arithmetic.Real.to_decimal_string e Global.options.max_prec_digits
    (* TODO: Print the right money symbol according to language *)
    | TMoney ->
      let z3_str = Expr.to_string e in
      (* The Z3 model returns an integer corresponding to the amount of cents.
         We reformat it as dollars *)
      let to_dollars s =
        Runtime.money_to_string (Runtime.money_of_cents_string s)
      in
      if String.contains z3_str '-' then
        Format.asprintf "-%s $"
          (to_dollars (String.sub z3_str 3 (String.length z3_str - 4)))
      else Format.asprintf "%s $" (to_dollars z3_str)
    (* The Z3 date representation corresponds to the number of days since Jan 1,
       1900. We pretty-print it as the actual date *)
    (* TODO: Use differnt dates conventions depending on the language ? *)
    | TDate -> nb_days_to_date (int_of_string (Expr.to_string e))
    | TDuration -> Format.asprintf "%s days" (Expr.to_string e)
  in

  match Mark.remove ty with
  | TLit ty -> print_lit ty
  | TStruct name ->
    let s = StructName.Map.find name ctx.ctx_decl.ctx_structs in
    let get_fieldname (fn : StructField.t) : string =
      Mark.remove (StructField.get_info fn)
    in
    let fields =
      List.map2
        (fun (fn, ty) e ->
          Format.asprintf "-- %s : %s" (get_fieldname fn)
            (print_z3model_expr ctx ty e))
        (StructField.Map.bindings s)
        (Expr.get_args e)
    in

    let fields_str = String.concat " " fields in

    Format.asprintf "%s { %s }"
      (Mark.remove (StructName.get_info name))
      fields_str
  | TTuple _ ->
    failwith "[Z3 model]: Pretty-printing of unnamed structs not supported"
  | TEnum name ->
    (* The value associated to the enum is a single argument *)
    let e' = List.hd (Expr.get_args e) in
    let fd = Expr.get_func_decl e in
    let fd_name = Symbol.to_string (FuncDecl.get_name fd) in

    let enum_ctrs = EnumName.Map.find name ctx.ctx_decl.ctx_enums in
    let case =
      List.find
        (fun (ctr, _) ->
          (* FIXME: don't match on strings *)
          String.equal fd_name (Mark.remove (EnumConstructor.get_info ctr)))
        (EnumConstructor.Map.bindings enum_ctrs)
    in

    Format.asprintf "%s (%s)" fd_name (print_z3model_expr ctx (snd case) e')
  | TOption _ -> failwith "[Z3 model]: Pretty-printing of options not supported"
  | TArrow _ -> failwith "[Z3 model]: Pretty-printing of arrows not supported"
  | TArray _ ->
    (* For now, only the length of arrays is modeled *)
    Format.asprintf "(length = %s)" (Expr.to_string e)
  | TAny -> failwith "[Z3 model]: Pretty-printing of Any not supported"
  | TClosureEnv ->
    failwith "[Z3 model]: Pretty-printing of closure_env not supported"
  | TDefault _ ->
    failwith "[Z3 model]: Pretty-printing of default terms not supported"

(** [print_model] pretty prints a Z3 model, used to exhibit counter examples
    where verification conditions are not satisfied. The context [ctx] is useful
    to retrieve the mapping between Z3 variables and Catala variables, and to
    retrieve type information about the variables that was lost during the
    translation (e.g., by translating a date to an integer) **)
let print_model (ctx : context) (model : Model.model) : string =
  let decls = Model.get_decls model in
  Format.asprintf "%a"
    (Format.pp_print_list
       ~pp_sep:(fun fmt () -> Format.fprintf fmt "")
       (fun fmt d ->
         if FuncDecl.get_arity d = 0 then
           (* Constant case *)
           match Model.get_const_interp model d with
           (* TODO: Better handling of this case *)
           | None ->
             failwith
               "[Z3 model]: A variable does not have an associated Z3 solution"
           (* Print "name : value\n" *)
           | Some e -> (
             let symbol_name = Symbol.to_string (FuncDecl.get_name d) in
             match StringMap.find_opt symbol_name ctx.ctx_z3vars with
             | None -> ()
             | Some (v, ty) ->
               Format.fprintf fmt "@{<blue>-->@} @{<yellow>%s@} : %s\n"
                 (Bindlib.name_of v)
                 (print_z3model_expr ctx ty e))
         else
           (* Declaration d is a function *)
           match Model.get_func_interp model d with
           (* TODO: Better handling of this case *)
           | None ->
             failwith
               "[Z3 model]: A variable does not have an associated Z3 solution"
           (* Print "name : value\n" *)
           | Some f ->
             let symbol_name = Symbol.to_string (FuncDecl.get_name d) in
             let v, _ = StringMap.find symbol_name ctx.ctx_z3vars in
             Format.fprintf fmt "@{<blue>-->@} @{<yellow>%s@} : %s\n"
               (Bindlib.name_of v)
               (* TODO: Model of a Z3 function should be pretty-printed *)
               (Model.FuncInterp.to_string f)))
    decls

(** [translate_typ_lit] returns the Z3 sort corresponding to the Catala literal
    type [t] **)
let translate_typ_lit (ctx : context) (t : typ_lit) : Sort.sort =
  match t with
  | TBool -> Boolean.mk_sort ctx.ctx_z3
  | TUnit -> fst ctx.ctx_z3unit
  | TInt -> Arithmetic.Integer.mk_sort ctx.ctx_z3
  | TRat -> Arithmetic.Real.mk_sort ctx.ctx_z3
  | TMoney -> Arithmetic.Integer.mk_sort ctx.ctx_z3
  (* Dates are encoded as integers, corresponding to the number of days since
     Jan 1, 1900 *)
  | TDate -> Arithmetic.Integer.mk_sort ctx.ctx_z3
  | TDuration -> Arithmetic.Integer.mk_sort ctx.ctx_z3

(** [translate_typ] returns the Z3 sort correponding to the Catala type [t] **)
let rec translate_typ (ctx : context) (t : naked_typ) : context * Sort.sort =
  match t with
  | TLit t -> ctx, translate_typ_lit ctx t
  | TStruct name -> find_or_create_struct ctx name
  | TTuple _ -> failwith "[Z3 encoding] TTuple type not supported"
  | TEnum e -> find_or_create_enum ctx e
  | TOption _ -> failwith "[Z3 encoding] TOption type not supported"
  | TDefault _ -> failwith "[Z3 encoding] TDefault type not supported"
  | TArrow _ -> failwith "[Z3 encoding] TArrow type not supported"
  | TArray _ ->
    (* For now, we are only encoding the (symbolic) length of an array.
       Ultimately, the type of an array should also contain its elements *)
    ctx, Arithmetic.Integer.mk_sort ctx.ctx_z3
  | TAny -> failwith "[Z3 encoding] TAny type not supported"
  | TClosureEnv -> failwith "[Z3 encoding] TClosureEnv type not supported"

(** [find_or_create_enum] attempts to retrieve the Z3 sort corresponding to the
    Catala enumeration [enum]. If no such sort exists yet, it constructs it by
    creating a Z3 constructor for each Catala constructor of [enum], and adds it
    to the context *)
and find_or_create_enum (ctx : context) (enum : EnumName.t) :
    context * Sort.sort =
  (* Creates a Z3 constructor corresponding to the Catala constructor [c] *)
  let create_constructor (name : EnumConstructor.t) (ty : typ) (ctx : context) :
      context * Datatype.Constructor.constructor =
    let name = Mark.remove (EnumConstructor.get_info name) in
    let ctx, arg_z3_ty = translate_typ ctx (Mark.remove ty) in

    (* The mk_constructor_s Z3 function is not so well documented. From my
       understanding, its argument are: - a string corresponding to the name of
       the constructor - a recognizer as a symbol corresponding to the name
       (unsure why) - a list of symbols corresponding to the arguments of the
       constructor - a list of types, that must be of the same length as the
       list of arguments - a list of sort_refs, of the same length as the list
       of arguments. I'm unsure what this corresponds to *)
    ( ctx,
      Datatype.mk_constructor_s ctx.ctx_z3 name
        (Symbol.mk_string ctx.ctx_z3 name)
        (* We need a name for the argument of the constructor, we arbitrary pick
           the name of the constructor to which we append the special character
           "!" and the integer 0 *)
        [Symbol.mk_string ctx.ctx_z3 (name ^ "!0")]
        (* The type of the argument, translated to a Z3 sort *)
        [Some arg_z3_ty]
        [Sort.get_id arg_z3_ty] )
  in

  match EnumName.Map.find_opt enum ctx.ctx_z3datatypes with
  | Some e -> ctx, e
  | None ->
    let ctrs = EnumName.Map.find enum ctx.ctx_decl.ctx_enums in
    let ctx, z3_ctrs =
      EnumConstructor.Map.fold
        (fun ctr ty (ctx, ctrs) ->
          let ctx, ctr = create_constructor ctr ty ctx in
          ctx, ctr :: ctrs)
        ctrs (ctx, [])
    in
    let z3_enum =
      Datatype.mk_sort_s ctx.ctx_z3
        (Mark.remove (EnumName.get_info enum))
        (List.rev z3_ctrs)
    in
    add_z3enum enum z3_enum ctx, z3_enum

(** [find_or_create_struct] attemps to retrieve the Z3 sort corresponding to the
    struct [s]. If no such sort exists yet, we construct it as a datatype with
    one constructor taking all the fields as arguments, and add it to the
    context *)
and find_or_create_struct (ctx : context) (s : StructName.t) :
    context * Sort.sort =
  match StructName.Map.find_opt s ctx.ctx_z3structs with
  | Some s -> ctx, s
  | None ->
    let s_name = Mark.remove (StructName.get_info s) in
    let fields = StructName.Map.find s ctx.ctx_decl.ctx_structs in
    let z3_fieldnames =
      List.map
        (fun f ->
          Mark.remove (StructField.get_info f) |> Symbol.mk_string ctx.ctx_z3)
        (StructField.Map.keys fields)
    in
    let ctx, z3_fieldtypes_rev =
      StructField.Map.fold
        (fun _ ty (ctx, ftypes) ->
          let ctx, ftype = translate_typ ctx (Mark.remove ty) in
          ctx, ftype :: ftypes)
        fields (ctx, [])
    in
    let z3_fieldtypes = List.rev z3_fieldtypes_rev in
    let z3_sortrefs = List.map Sort.get_id z3_fieldtypes in
    let mk_struct_s = "mk!" ^ s_name in
    let z3_mk_struct =
      Datatype.mk_constructor_s ctx.ctx_z3 mk_struct_s
        (Symbol.mk_string ctx.ctx_z3 mk_struct_s)
        z3_fieldnames
        (List.map (fun x -> Some x) z3_fieldtypes)
        z3_sortrefs
    in

    let z3_struct = Datatype.mk_sort_s ctx.ctx_z3 s_name [z3_mk_struct] in
    add_z3struct s z3_struct ctx, z3_struct

(** [translate_lit] returns the Z3 expression as a literal corresponding to
    [lit] **)
let translate_lit (ctx : context) (l : lit) : Expr.expr =
  match l with
  | LBool b ->
    if b then Boolean.mk_true ctx.ctx_z3 else Boolean.mk_false ctx.ctx_z3
  | LInt n ->
    Arithmetic.Integer.mk_numeral_i ctx.ctx_z3 (Runtime.integer_to_int n)
  | LRat r ->
    Arithmetic.Real.mk_numeral_s ctx.ctx_z3
      (string_of_float (Runtime.decimal_to_float r))
  | LMoney m ->
    let z3_m = Runtime.integer_to_int (Runtime.money_to_cents m) in
    Arithmetic.Integer.mk_numeral_i ctx.ctx_z3 z3_m
  | LUnit -> snd ctx.ctx_z3unit
  (* Encoding a date as an integer corresponding to the number of days since Jan
     1, 1900 *)
  | LDate d -> Arithmetic.Integer.mk_numeral_i ctx.ctx_z3 (date_to_int d)
  | LDuration d ->
    let y, m, d = Runtime.duration_to_years_months_days d in
    if y <> 0 || m <> 0 then
      failwith
        "[Z3 encoding]: Duration literals containing years or months not \
         supported";
    Arithmetic.Integer.mk_numeral_i ctx.ctx_z3 d

(** [find_or_create_funcdecl] attempts to retrieve the Z3 function declaration
    corresponding to the variable [v] and its type [ty]. If no such function
    declaration exists yet, we construct it and add it to the context, thus
    requiring to return a new context *)
let find_or_create_funcdecl (ctx : context) (v : typed expr Var.t) (ty : typ) :
    context * FuncDecl.func_decl =
  match Var.Map.find_opt v ctx.ctx_funcdecl with
  | Some fd -> ctx, fd
  | None -> (
    match Mark.remove ty with
    | TArrow (t1, t2) ->
      let ctx, z3_t1 =
        List.fold_left_map translate_typ ctx (List.map Mark.remove t1)
      in
      let ctx, z3_t2 = translate_typ ctx (Mark.remove t2) in
      let name = unique_name v in
      let fd = FuncDecl.mk_func_decl_s ctx.ctx_z3 name z3_t1 z3_t2 in
      let ctx = add_funcdecl v fd ctx in
      let ctx = add_z3var name v ty ctx in
      ctx, fd
    | TAny ->
      failwith
        "[Z3 Encoding] A function being applied has type TAny, the type was \
         not fully inferred"
    | _ ->
      failwith
        "[Z3 Encoding] Ill-formed VC, a function application does not have a \
         function type")

let is_leap_year = Runtime.is_leap_year
(* Replace with [Dates_calc.Dates.is_leap_year] when existing *)

(** [translate_op] returns the Z3 expression corresponding to the application of
    [op] to the arguments [args] **)
let rec translate_op :
    context -> dcalc operator Mark.pos -> 'm expr list -> context * Expr.expr =
 fun ctx (op, pos) args ->
  let ill_formed () =
    Format.kasprintf failwith
      "[Z3 encoding] Ill-formed operator application: %a" Shared_ast.Expr.format
      (Shared_ast.Expr.eappop ~op:(op, pos)
         ~args:(List.map Shared_ast.Expr.untype args)
         ~tys:[]
         (Untyped { pos })
      |> Shared_ast.Expr.unbox)
  in
  let app f =
    let ctx, args = List.fold_left_map translate_expr ctx args in
    ctx, f ctx.ctx_z3 args
  in
  let app1 f =
    app (fun ctx -> function [a] -> f ctx a | _ -> ill_formed ())
  in
  let app2 f =
    app (fun ctx -> function [a; b] -> f ctx a b | _ -> ill_formed ())
  in
  match op, args with
  | Fold, _ ->
    failwith "[Z3 encoding] ternary operator application not supported"
    (* Special case for GetYear comparisons *)
  | ( Lt_int_int,
      [(EAppOp { op = GetYear, _; args = [e1]; _ }, _); (ELit (LInt n), _)] ) ->
    let n = Runtime.integer_to_int n in
    let ctx, e1 = translate_expr ctx e1 in
    let e2 =
      Arithmetic.Integer.mk_numeral_i ctx.ctx_z3 (date_to_int (date_of_year n))
    in
    (* e2 corresponds to the first day of the year n. GetYear e1 < e2 can thus
       be directly translated as < in the Z3 encoding using the number of
       days *)
    ctx, Arithmetic.mk_lt ctx.ctx_z3 e1 e2
  | ( Lte_int_int,
      [(EAppOp { op = GetYear, _; args = [e1]; _ }, _); (ELit (LInt n), _)] ) ->
    let ctx, e1 = translate_expr ctx e1 in
    let nb_days = if is_leap_year n then 365 else 364 in
    let n = Runtime.integer_to_int n in
    (* We want that the year corresponding to e1 is smaller or equal to n. We
       encode this as the day corresponding to e1 is smaller or equal than the
       last day of the year [n], which is Jan 1st + 365 days if [n] is a leap
       year, Jan 1st + 364 else *)
    let e2 =
      Arithmetic.Integer.mk_numeral_i ctx.ctx_z3
        (date_to_int (date_of_year n) + nb_days)
    in
    ctx, Arithmetic.mk_le ctx.ctx_z3 e1 e2
  | ( Gt_int_int,
      [(EAppOp { op = GetYear, _; args = [e1]; _ }, _); (ELit (LInt n), _)] ) ->
    let ctx, e1 = translate_expr ctx e1 in
    let nb_days = if is_leap_year n then 365 else 364 in
    let n = Runtime.integer_to_int n in
    (* We want that the year corresponding to e1 is greater to n. We encode this
       as the day corresponding to e1 is greater than the last day of the year
       [n], which is Jan 1st + 365 days if [n] is a leap year, Jan 1st + 364
       else *)
    let e2 =
      Arithmetic.Integer.mk_numeral_i ctx.ctx_z3
        (date_to_int (date_of_year n) + nb_days)
    in
    ctx, Arithmetic.mk_gt ctx.ctx_z3 e1 e2
  | ( Gte_int_int,
      [(EAppOp { op = GetYear, _; args = [e1]; _ }, _); (ELit (LInt n), _)] ) ->
    let n = Runtime.integer_to_int n in
    let ctx, e1 = translate_expr ctx e1 in
    let e2 =
      Arithmetic.Integer.mk_numeral_i ctx.ctx_z3 (date_to_int (date_of_year n))
    in
    (* e2 corresponds to the first day of the year n. GetYear e1 >= e2 can thus
       be directly translated as >= in the Z3 encoding using the number of
       days *)
    ctx, Arithmetic.mk_ge ctx.ctx_z3 e1 e2
  | And, _ -> app Boolean.mk_and
  | Or, _ -> app Boolean.mk_or
  | Xor, _ -> app2 Boolean.mk_xor
  | (Add_int_int | Add_rat_rat | Add_mon_mon | Add_dat_dur _ | Add_dur_dur), _
    ->
    app Arithmetic.mk_add
  | ( ( Sub_int_int | Sub_rat_rat | Sub_mon_mon | Sub_dat_dat | Sub_dat_dur
      | Sub_dur_dur ),
      _ ) ->
    app Arithmetic.mk_sub
  | (Mult_int_int | Mult_rat_rat | Mult_mon_rat | Mult_dur_int), _ ->
    app Arithmetic.mk_mul
  | (Div_int_int | Div_rat_rat | Div_mon_rat | Div_mon_mon), _ ->
    app2 Arithmetic.mk_div
  | (Lt_int_int | Lt_rat_rat | Lt_mon_mon | Lt_dat_dat | Lt_dur_dur), _ ->
    app2 Arithmetic.mk_lt
  | (Lte_int_int | Lte_rat_rat | Lte_mon_mon | Lte_dat_dat | Lte_dur_dur), _ ->
    app2 Arithmetic.mk_le
  | (Gt_int_int | Gt_rat_rat | Gt_mon_mon | Gt_dat_dat | Gt_dur_dur), _ ->
    app2 Arithmetic.mk_gt
  | (Gte_int_int | Gte_rat_rat | Gte_mon_mon | Gte_dat_dat | Gte_dur_dur), _ ->
    app2 Arithmetic.mk_ge
  | (Eq_int_int | Eq_rat_rat | Eq_mon_mon | Eq_dat_dat | Eq_dur_dur), _ ->
    failwith "[Z3 encoding] not handling Eq"
  | Map, _ ->
    failwith "[Z3 encoding] application of binary operator Map not supported"
  | Concat, _ ->
    failwith "[Z3 encoding] application of binary operator Concat not supported"
  | Filter, _ ->
    failwith "[Z3 encoding] application of binary operator Filter not supported"
  | Not, _ -> app1 Boolean.mk_not
  (* Omitting the log from the VC *)
  | Log _, [e1] -> translate_expr ctx e1
  | Length, [e1] ->
    (* For now, an array is only its symbolic length. We simply return it *)
    translate_expr ctx e1
  | ToRat_int, _ ->
    failwith
      "[Z3 encoding] application of unary operator ToRat_int not supported"
  | ToRat_mon, _ ->
    failwith
      "[Z3 encoding] application of unary operator ToRat_mon not supported"
  | ToMoney_rat, _ ->
    failwith
      "[Z3 encoding] application of unary operator ToMoney_rat not supported"
  | GetDay, _ ->
    failwith "[Z3 encoding] application of unary operator GetDay not supported"
  | GetMonth, _ ->
    failwith
      "[Z3 encoding] application of unary operator GetMonth not supported"
  | GetYear, _ ->
    failwith
      "[Z3 encoding] GetYear operator only supported in comparisons with \
       literal"
  | FirstDayOfMonth, _ ->
    failwith
      "[Z3 encoding] FirstDayOfMonth operator only supported in comparisons \
       with literal"
  | LastDayOfMonth, _ ->
    failwith
      "[Z3 encoding] LastDayOfMonth operator only supported in comparisons \
       with literal"
  | Round_rat, _ ->
    failwith "[Z3 encoding] Round_rat operator  not implemented yet"
  | Round_mon, _ ->
    failwith "[Z3 encoding] Round_mon operator  not implemented yet"
  | _ -> ill_formed ()

(** [translate_expr] translate the expression [vc] to its corresponding Z3
    expression **)
and translate_expr (ctx : context) (vc : typed expr) : context * Expr.expr =
  let translate_match_arm
      (head : Expr.expr)
      (ctx : context)
      (e : 'm expr * FuncDecl.func_decl list) : context * Expr.expr =
    let e, accessors = e in
    match Mark.remove e with
    | EAbs { binder; _ } ->
      (* Create a fresh Catala variable to substitue and obtain the body *)
      let fresh_v = Var.make "arm!tmp" in
      let fresh_e = EVar fresh_v in

      (* Invariant: Catala enums always have exactly one argument *)
      let accessor = List.hd accessors in
      let proj = Expr.mk_app ctx.ctx_z3 accessor [head] in
      (* The fresh variable should be substituted by a projection into the enum
         in the body, we add this to the context *)
      let ctx = add_z3matchsubst fresh_v proj ctx in

      let body = Bindlib.msubst binder [| fresh_e |] in
      translate_expr ctx body
    (* Invariant: Catala match arms are always lambda*)
    | _ -> failwith "[Z3 encoding] : Arms branches inside VCs should be lambdas"
  in

  match Mark.remove vc with
  | EVar v -> (
    match Var.Map.find_opt v ctx.ctx_z3matchsubsts with
    | None ->
      (* We are in the standard case, where this is a true Catala variable *)
      let (Typed { ty = t; _ }) = Mark.get vc in
      let name = unique_name v in
      let ctx = add_z3var name v t ctx in
      let ctx, ty = translate_typ ctx (Mark.remove t) in
      let z3_var = Expr.mk_const_s ctx.ctx_z3 name ty in
      let ctx =
        match Mark.remove t with
        (* If we are creating a new array, we need to log that its length is
           greater than 0 *)
        | TArray _ ->
          add_z3constraint
            (Arithmetic.mk_ge ctx.ctx_z3 z3_var
               (Arithmetic.Integer.mk_numeral_i ctx.ctx_z3 0))
            ctx
        | _ -> ctx
      in

      ctx, z3_var
    | Some e ->
      (* This variable is a temporary variable generated during VC translation
         of a match. It actually corresponds to applying an accessor to an enum,
         the corresponding Z3 expression was previously stored in the context *)
      ctx, e)
  | EExternal _ -> failwith "[Z3 encoding] EExternal unsupported"
  | EStruct _ -> failwith "[Z3 encoding] EStruct unsupported"
  | EStructAccess { e; field; name } ->
    let ctx, z3_struct = find_or_create_struct ctx name in
    (* This datatype should have only one constructor, corresponding to
       mk_struct. The accessors of this constructor correspond to the field
       accesses *)
    let accessors = List.hd (Datatype.get_accessors z3_struct) in
    let fields = StructName.Map.find name ctx.ctx_decl.ctx_structs in
    let idx_mappings = List.combine (StructField.Map.keys fields) accessors in
    let _, accessor =
      List.find (fun (field1, _) -> StructField.equal field field1) idx_mappings
    in
    let ctx, s = translate_expr ctx e in
    ctx, Expr.mk_app ctx.ctx_z3 accessor [s]
  | ETuple _ -> failwith "[Z3 encoding] ETuple unsupported"
  | ETupleAccess _ -> failwith "[Z3 encoding] ETupleAccess unsupported"
  | EInj { e; cons; name } ->
    (* This node corresponds to creating a value for the enumeration [en], by
       calling the [idx]-th constructor of enum [en], with argument [e] *)
    let ctx, z3_enum = find_or_create_enum ctx name in
    let ctx, z3_arg = translate_expr ctx e in
    let ctrs = Datatype.get_constructors z3_enum in
    let cons_map = EnumName.Map.find name ctx.ctx_decl.ctx_enums in
    (* This should always succeed if the expression is well-typed in dcalc *)
    let idx_mappings = List.combine (EnumConstructor.Map.keys cons_map) ctrs in
    let _, ctr =
      List.find
        (fun (cons1, _) -> EnumConstructor.equal cons cons1)
        idx_mappings
    in
    ctx, Expr.mk_app ctx.ctx_z3 ctr [z3_arg]
  | EMatch { e; cases; name = enum } ->
    (* We will encode a match as a new variable, tmp_v, and add to the
       hypotheses that this variable is equal to the conjunction of all `A? arg
       ==> tmp_v == body`, where `A? arg ==> body` is an arm of the match *)

    (* We use the Var module to ensure that all names for temporary variables
       will be fresh, and thus will not clash in Z3 *)
    let fresh_v = Var.make "z3!match_tmp" in
    let name = unique_name fresh_v in
    let (Typed { ty = match_ty; _ }) = Mark.get vc in
    let ctx, z3_ty = translate_typ ctx (Mark.remove match_ty) in
    let z3_var = Expr.mk_const_s ctx.ctx_z3 name z3_ty in

    let ctx, z3_enum = find_or_create_enum ctx enum in
    let ctx, z3_arg = translate_expr ctx e in
    let _ctx, z3_arms =
      List.fold_left_map
        (translate_match_arm z3_arg)
        ctx
        (List.combine
           (EnumConstructor.Map.values cases)
           (Datatype.get_accessors z3_enum))
    in
    let z3_arms =
      List.map2
        (fun r arm ->
          (* Encodes A? arg ==> z3_var = body *)
          let is_r = Expr.mk_app ctx.ctx_z3 r [z3_arg] in
          let eq = Boolean.mk_eq ctx.ctx_z3 z3_var arm in
          Boolean.mk_implies ctx.ctx_z3 is_r eq)
        (Datatype.get_recognizers z3_enum)
        z3_arms
    in

    (* Add the definition of z3_var to the hypotheses *)
    let ctx = add_z3constraint (Boolean.mk_and ctx.ctx_z3 z3_arms) ctx in
    ctx, z3_var
  | EArray _ -> failwith "[Z3 encoding] EArray unsupported"
  | ELit l -> ctx, translate_lit ctx l
  | EAbs _ -> failwith "[Z3 encoding] EAbs unsupported"
  | EAppOp { op; args; _ } -> translate_op ctx op args
  | EApp { f = head; args; _ } -> (
    match Mark.remove head with
    | EVar v ->
      let (Typed { ty = f_ty; _ }) = Mark.get head in
      let ctx, fd = find_or_create_funcdecl ctx v f_ty in
      (* Fold_right to preserve the order of the arguments: The head argument is
         appended at the head *)
      let ctx, z3_args =
        List.fold_right
          (fun arg (ctx, acc) ->
            let ctx, z3_arg = translate_expr ctx arg in
            ctx, z3_arg :: acc)
          args (ctx, [])
      in
      ctx, Expr.mk_app ctx.ctx_z3 fd z3_args
    | EAbs { binder; _ } ->
      let vars, _ = Bindlib.unmbind binder in
      if Array.length vars != 1 || List.length args != 1 then
        failwith "[Z3 encoding] EAbs not supported beyond let_in"
      else
        let arg = List.hd args in
        let expr = Bindlib.msubst binder [| Mark.remove arg |] in
        translate_expr ctx expr
    | _ ->
      failwith
        "[Z3 encoding] EApp node: Catala function calls should only include \
         operators or function names")
  | EAssert e -> translate_expr ctx e
  | EFatalError _ -> failwith "[Z3 encoding] EFatalError unsupported"
  | EDefault _ -> failwith "[Z3 encoding] EDefault unsupported"
  | EPureDefault _ -> failwith "[Z3 encoding] EPureDefault unsupported"
  | EIfThenElse { cond = e_if; etrue = e_then; efalse = e_else } ->
    (* We rely on Z3's native encoding for ite to encode this node. There might
       be some interesting optimization in the future about when to split this
       node/bubble up the if_then_else, but this is left as future work *)
    let ctx, z3_if = translate_expr ctx e_if in
    let ctx, z3_then = translate_expr ctx e_then in
    let ctx, z3_else = translate_expr ctx e_else in
    ctx, Boolean.mk_ite ctx.ctx_z3 z3_if z3_then z3_else
  | EEmpty -> failwith "[Z3 encoding] 'Empty' literals not supported"
  | EErrorOnEmpty _ -> failwith "[Z3 encoding] ErrorOnEmpty unsupported"
  | _ -> .

(** [create_z3unit] creates a Z3 sort and expression corresponding to the unit
    type and value respectively. Concretely, we represent unit as a tuple with 0
    elements **)
let create_z3unit (ctx : Z3.context) : Z3.context * (Sort.sort * Expr.expr) =
  let unit_sort = Tuple.mk_sort ctx (Symbol.mk_string ctx "unit") [] [] in
  let mk_unit = Tuple.get_mk_decl unit_sort in
  let unit_val = Expr.mk_app ctx mk_unit [] in
  ctx, (unit_sort, unit_val)

module Backend = struct
  type backend_context = context
  type vc_encoding = Z3.Expr.expr

  let print_encoding (vc : vc_encoding) : string = Expr.to_string vc

  type model = Z3.Model.model
  type solver_result = ProvenTrue | ProvenFalse of model option | Unknown

  let solve_vc_encoding (ctx : backend_context) (encoding : vc_encoding) :
      solver_result =
    let solver = Z3.Solver.mk_solver ctx.ctx_z3 None in
    (* We take the negation of the query to check for possible
       counterexamples *)
    let query = Boolean.mk_not ctx.ctx_z3 encoding in
    (* Add all the hypotheses stored in the context *)
    let query_and_hyps = query :: ctx.ctx_z3constraints in
    Z3.Solver.add solver query_and_hyps;
    match Z3.Solver.check solver [] with
    | UNSATISFIABLE -> ProvenTrue
    | SATISFIABLE -> ProvenFalse (Z3.Solver.get_model solver)
    | UNKNOWN -> Unknown

  let print_model (ctx : backend_context) (m : model) : string =
    print_model ctx m

  let is_model_empty (m : model) : bool = Z3.Model.get_decls m = []

  let translate_expr (ctx : backend_context) (e : typed expr) =
    translate_expr ctx e

  let encode_asserts (ctx : backend_context) (e : typed expr) =
    let ctx, vc = translate_expr ctx e in
    add_z3constraint vc ctx

  let init_backend () = Message.debug "Running Z3 version %s" Version.to_string

  let make_context (decl_ctx : decl_ctx) : backend_context =
    let cfg =
      (if Globals.disable_counterexamples () then [] else ["model", "true"])
      @ ["proof", "false"]
    in
    let z3_ctx = mk_context cfg in
    let z3_ctx, z3unit = create_z3unit z3_ctx in
    {
      ctx_z3 = z3_ctx;
      ctx_decl = decl_ctx;
      ctx_funcdecl = Var.Map.empty;
      ctx_z3vars = StringMap.empty;
      ctx_z3datatypes = EnumName.Map.empty;
      ctx_z3matchsubsts = Var.Map.empty;
      ctx_z3structs = StructName.Map.empty;
      ctx_z3unit = z3unit;
      ctx_z3constraints = [];
    }
end

module Io = Io.MakeBackendIO (Backend)