CristalCaveGem » LustreC

(* Example of a use of Z3 Fixedpoint that doesn't work 

   The file is self-contained and shall be compiled as follow:

   in File _tags, add the simple line

   <**/*>: package(z3)

   Then compile as

   ocamlbuild -use-ocamlfind zustre_test.native

   We obtain the following output:

   $ ./zustre_test.native 

   Registered rules:

     Rule: (forall ((x Int) (y Int)) (=> (= x y) (f x y)))  

     Rule: INIT_STATE

     Rule: (forall ((x Int) (y Int)) (=> (and (f x y) INIT_STATE) (MAIN x y)))

     Rule: (forall ((x Int) (y Int)) (=> (and (not (= x y)) (MAIN x y)) ERR))

   Fatal error: exception Z3.Error("Uninterpreted 'y' in <null>:

   f(#0,#1) :- 

     (= (:var 1) y),

     (= (:var 0) x),

     (= x y).

   ")

*)

let rec fprintf_list ~sep:sep f fmt = function

  | []   -> ()

  | [e]  -> f fmt e

  | x::r -> Format.fprintf fmt "%a%(%)%a" f x sep (fprintf_list ~sep f) r

(* Lustrec.Global references to declare Z3 context and Fixedpoint engine *)

let ctx = ref (Z3.mk_context [])

let fp = ref (Z3.Fixedpoint.mk_fixedpoint !ctx)

(* Shortcuts to basic sorts *)

let bool_sort = Z3.Boolean.mk_sort !ctx

let int_sort = Z3.Arithmetic.Integer.mk_sort !ctx

let real_sort = Z3.Arithmetic.Real.mk_sort !ctx

let _ =

  (* Setting up the fixed point engine *)

  fp := Z3.Fixedpoint.mk_fixedpoint !ctx;

  let module P = Z3.Params in

  let module S = Z3.Symbol in

  let mks = S.mk_string !ctx in

  let params = P.mk_params !ctx in

  P.add_symbol params (mks "engine") (mks "spacer");

  Z3.Fixedpoint.set_parameters !fp params;

  (* Three rules 

     R1: (x = y) => f(x,y)

     R2: INIT and f(x,y) => MAIN(x,y)

     R3: x!=y and MAIN(x,y) => ERR(x,y)

     INIT is assumed as the beginning

     Querying ERR shall be unsat since the only valid MAIN(x,y) are x=y and

     therefore ERR is unsat.

  *)

  (* Simple rule : forall x, y, (x=y => f(x,y)) *)

  let x = Z3.FuncDecl.mk_func_decl_s !ctx "x" [] int_sort in

  let y = Z3.FuncDecl.mk_func_decl_s !ctx "y" [] int_sort in

  let x_expr = Z3.Expr.mk_const_f !ctx x in

  let y_expr = Z3.Expr.mk_const_f !ctx y in

  let decl_f = Z3.FuncDecl.mk_func_decl_s !ctx "f" [int_sort; int_sort] bool_sort in

  Z3.Fixedpoint.register_relation !fp decl_f; (* since f appears in the rhs of a rule *)

  let f_x_y_expr = Z3.Expr.mk_app !ctx decl_f [x_expr; y_expr] in

  let x_eq_y_expr = Z3.Boolean.mk_eq !ctx x_expr y_expr in

  let expr_f_lhs = (* x = y *)

    x_eq_y_expr 

  in

  let expr_f_rhs = f_x_y_expr in

  let expr_f = Z3.Boolean.mk_implies !ctx expr_f_lhs expr_f_rhs in

  (* Adding forall as prefix *)

  let expr_forall_f = Z3.Quantifier.mk_forall_const

    !ctx  (* context *)

    (* [int_sort; int_sort]           (\* sort list*\) *)

    (* [Z3.FuncDecl.get_name x; Z3.FuncDecl.get_name y] (\* symbol list *\) *)

(*    [x_expr; y_expr] Second try with expr list "const" *)

    [Z3.Expr.mk_const_f !ctx x; Z3.Expr.mk_const_f !ctx y]

    expr_f (* expression *)

    None (* quantifier weight, None means 1 *)

    [] (* pattern list ? *)

    [] (* ? *)

    None (* ? *)

    None (* ? *)

  in

  let expr_forall_f = Z3.Quantifier.expr_of_quantifier expr_forall_f in

  Z3.Fixedpoint.add_rule !fp expr_forall_f None;

  (* INIT RULE *)

  let decl_init = Z3.FuncDecl.mk_func_decl_s !ctx "INIT_STATE" [] bool_sort in

  Z3.Fixedpoint.register_relation !fp decl_init; 

  let init_expr = Z3.Expr.mk_app !ctx decl_init [] in

  Z3.Fixedpoint.add_rule !fp init_expr None;

  (* MAIN is defined by two rules : INIT and induction *)

  let decl_main = Z3.FuncDecl.mk_func_decl_s !ctx "MAIN" [int_sort; int_sort] bool_sort in

  Z3.Fixedpoint.register_relation !fp decl_main;

  let main_x_y_expr = Z3.Expr.mk_app !ctx decl_main [x_expr; y_expr] in

  (* Rule 1: forall x, y, INIT_STATE and f(x,y) => MAIN(x,y) : at the beginning x=y *)

  let expr_main1_lhs = (* INIT_STATE and f(x, y) *)

    Z3.Boolean.mk_and !ctx [f_x_y_expr; init_expr] in

  let expr_main1_rhs = main_x_y_expr in

  let expr_main1 = Z3.Boolean.mk_implies !ctx expr_main1_lhs expr_main1_rhs in

  (* Adding forall as prefix *)

  let expr_forall_main1 = Z3.Quantifier.mk_forall_const

    !ctx  (* context *)

    (*

    [int_sort; int_sort]           (* sort list*)

    [Z3.FuncDecl.get_name x; Z3.FuncDecl.get_name y] (* symbol list *)

    *)

    (*    [x_expr; y_expr] Second try with expr list "const" *)

    [Z3.Expr.mk_const_f !ctx x; Z3.Expr.mk_const_f !ctx y]

    expr_main1 (* expression *)

    None (* quantifier weight, None means 1 *)

    [] (* pattern list ? *)

    [] (* ? *)

    None (* ? *)

    None (* ? *)

  in

  let expr_forall_main1 = Z3.Quantifier.expr_of_quantifier expr_forall_main1 in

  Z3.Fixedpoint.add_rule !fp expr_forall_main1 None;

  (* Rule 2: forall x,y, MAIN(x,y) => MAIN(x+1, y+1) *)

  let expr_main2_lhs = main_x_y_expr in

  let plus_one x = Z3.Arithmetic.mk_add !ctx

    [

      x;

      (* Z3.Arithmetic.Integer.mk_const_s !ctx "1" *)

	Z3.Expr.mk_numeral_int !ctx 1 int_sort

    ] in

  let main_x_y_plus_one_expr = Z3.Expr.mk_app !ctx decl_main [plus_one x_expr; plus_one y_expr] in

  let expr_main2_rhs = main_x_y_plus_one_expr in

  let expr_main2 = Z3.Boolean.mk_implies !ctx expr_main2_lhs expr_main2_rhs in

  (* Adding forall as prefix *)

  let expr_forall_main2 = Z3.Quantifier.mk_forall_const

    !ctx  (* context *)

    (*

    [int_sort; int_sort]           (* sort list*)

    [Z3.FuncDecl.get_name x; Z3.FuncDecl.get_name y] (* symbol list *)

    *)

    (*    [x_expr; y_expr] Second try with expr list "const" *)

    [Z3.Expr.mk_const_f !ctx x; Z3.Expr.mk_const_f !ctx y]

    expr_main2 (* expression *)

    None (* quantifier weight, None means 1 *)

    [] (* pattern list ? *)

    [] (* ? *)

    None (* ? *)

    None (* ? *)

  in

  let expr_forall_main2 = Z3.Quantifier.expr_of_quantifier expr_forall_main2 in

  Z3.Fixedpoint.add_rule !fp expr_forall_main2 None;

  (* TODO: not implemented yet since the error is visible without it *)

  (* Query: is it possible to have MAIN(x,y) and x!=y ? *)

  (* This is performed as follow:

     rule (forall x, y, MAIN(x,y) and x!=y => ERR)

  *)

  let decl_err = Z3.FuncDecl.mk_func_decl_s !ctx "ERR" [] bool_sort in

  Z3.Fixedpoint.register_relation !fp decl_err; 

  let err_expr = Z3.Expr.mk_app !ctx decl_err [] in

  let x_diff_y_expr = Z3.Boolean.mk_not !ctx x_eq_y_expr in

  let expr_err_lhs =

    Z3.Boolean.mk_and !ctx [x_diff_y_expr; main_x_y_expr] in

  let expr_err_rhs = err_expr in

  let expr_err = Z3.Boolean.mk_implies !ctx expr_err_lhs expr_err_rhs in

  (* Adding forall as prefix *)

  let expr_forall_err = Z3.Quantifier.mk_forall_const

    !ctx  (* context *)

(*    [int_sort; int_sort]           (* sort list*)

    [Z3.FuncDecl.get_name x; Z3.FuncDecl.get_name y] (* symbol list *)

*)

    (* [x_expr; y_expr] Second try with expr list "const" *)

    [Z3.Expr.mk_const_f !ctx x; Z3.Expr.mk_const_f !ctx y]

    expr_err (* expression *)

    None (* quantifier weight, None means 1 *)   

    [] (* pattern list ? *)

    [] (* ? *)

    None (* ? *)

    None (* ? *)

  in

  let expr_forall_err = Z3.Quantifier.expr_of_quantifier expr_forall_err in

  Z3.Fixedpoint.add_rule !fp expr_forall_err None;

  (* Printing the rules for sanity check *)

  let rules_expr = Z3.Fixedpoint.get_rules !fp in

  Format.eprintf "@[<v 2>Registered rules:@ %a@ @]@."

    (fprintf_list ~sep:"@ "

       (fun fmt e ->

	 (* let e2 = Z3.Quantifier.get_body eq in *)

	 (* let fd = Z3.Expr.get_func_decl e in *)

	 Format.fprintf fmt "Rule: %s@ "

	   (Z3.Expr.to_string e);

       )) rules_expr;

  let res_status = Z3.Fixedpoint.query_r !fp [decl_err] in

  Format.eprintf "Status: %s@." (Z3.Solver.string_of_status res_status);

  let ts = Z3.Tactic.get_tactic_names !ctx in

  List.iter (fun s -> Format.printf "%s@." s) ts;

  ()