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lustrec / src / tools / zustre / zustre_test.ml @ dbab1fe5

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(* Example of a use of Z3 Fixedpoint that doesn't work 
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   The file is self-contained and shall be compiled as follow:
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   in File _tags, add the simple line
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   <**/*>: package(z3)
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   Then compile as
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   ocamlbuild -use-ocamlfind zustre_test.native
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   We obtain the following output:
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   $ ./zustre_test.native 
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   Registered rules:
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     Rule: (forall ((x Int) (y Int)) (=> (= x y) (f x y)))  
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     Rule: INIT_STATE
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     Rule: (forall ((x Int) (y Int)) (=> (and (f x y) INIT_STATE) (MAIN x y)))
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     Rule: (forall ((x Int) (y Int)) (=> (and (not (= x y)) (MAIN x y)) ERR))
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   Fatal error: exception Z3.Error("Uninterpreted 'y' in <null>:
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   f(#0,#1) :- 
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     (= (:var 1) y),
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     (= (:var 0) x),
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     (= x y).
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   ")
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*)
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let rec fprintf_list ~sep:sep f fmt = function
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  | []   -> ()
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  | [e]  -> f fmt e
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  | x::r -> Format.fprintf fmt "%a%(%)%a" f x sep (fprintf_list ~sep f) r
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(* Global references to declare Z3 context and Fixedpoint engine *)
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let ctx = ref (Z3.mk_context [])
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let fp = ref (Z3.Fixedpoint.mk_fixedpoint !ctx)
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(* Shortcuts to basic sorts *)
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let bool_sort = Z3.Boolean.mk_sort !ctx
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let int_sort = Z3.Arithmetic.Integer.mk_sort !ctx
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let real_sort = Z3.Arithmetic.Real.mk_sort !ctx
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let _ =
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  (* Setting up the fixed point engine *)
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  fp := Z3.Fixedpoint.mk_fixedpoint !ctx;
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  let module P = Z3.Params in
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  let module S = Z3.Symbol in
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  let mks = S.mk_string !ctx in
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  let params = P.mk_params !ctx in
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  P.add_symbol params (mks "engine") (mks "spacer");
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  Z3.Fixedpoint.set_parameters !fp params;
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  (* Three rules 
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     R1: (x = y) => f(x,y)
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     R2: INIT and f(x,y) => MAIN(x,y)
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     R3: x!=y and MAIN(x,y) => ERR(x,y)
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     INIT is assumed as the beginning
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     Querying ERR shall be unsat since the only valid MAIN(x,y) are x=y and
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     therefore ERR is unsat.
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  *)
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  (* Simple rule : forall x, y, (x=y => f(x,y)) *)
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  let x = Z3.FuncDecl.mk_func_decl_s !ctx "x" [] int_sort in
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  let y = Z3.FuncDecl.mk_func_decl_s !ctx "y" [] int_sort in
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  let x_expr = Z3.Expr.mk_const_f !ctx x in
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  let y_expr = Z3.Expr.mk_const_f !ctx y in
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  let decl_f = Z3.FuncDecl.mk_func_decl_s !ctx "f" [int_sort; int_sort] bool_sort in
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  Z3.Fixedpoint.register_relation !fp decl_f; (* since f appears in the rhs of a rule *)
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  let f_x_y_expr = Z3.Expr.mk_app !ctx decl_f [x_expr; y_expr] in
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  let x_eq_y_expr = Z3.Boolean.mk_eq !ctx x_expr y_expr in
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  let expr_f_lhs = (* x = y *)
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    x_eq_y_expr 
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  in
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  let expr_f_rhs = f_x_y_expr in
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  let expr_f = Z3.Boolean.mk_implies !ctx expr_f_lhs expr_f_rhs in
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  (* Adding forall as prefix *)
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  let expr_forall_f = Z3.Quantifier.mk_forall_const
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    !ctx  (* context *)
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    (* [int_sort; int_sort]           (\* sort list*\) *)
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    (* [Z3.FuncDecl.get_name x; Z3.FuncDecl.get_name y] (\* symbol list *\) *)
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(*    [x_expr; y_expr] Second try with expr list "const" *)
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    [Z3.Expr.mk_const_f !ctx x; Z3.Expr.mk_const_f !ctx y]
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    expr_f (* expression *)
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    None (* quantifier weight, None means 1 *)
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    [] (* pattern list ? *)
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    [] (* ? *)
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    None (* ? *)
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    None (* ? *)
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  in
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  let expr_forall_f = Z3.Quantifier.expr_of_quantifier expr_forall_f in
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  Z3.Fixedpoint.add_rule !fp expr_forall_f None;
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  (* INIT RULE *)
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  let decl_init = Z3.FuncDecl.mk_func_decl_s !ctx "INIT_STATE" [] bool_sort in
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  Z3.Fixedpoint.register_relation !fp decl_init; 
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  let init_expr = Z3.Expr.mk_app !ctx decl_init [] in
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  Z3.Fixedpoint.add_rule !fp init_expr None;
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  (* MAIN is defined by two rules : INIT and induction *)
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  let decl_main = Z3.FuncDecl.mk_func_decl_s !ctx "MAIN" [int_sort; int_sort] bool_sort in
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  Z3.Fixedpoint.register_relation !fp decl_main;
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  let main_x_y_expr = Z3.Expr.mk_app !ctx decl_main [x_expr; y_expr] in
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  (* Rule 1: forall x, y, INIT_STATE and f(x,y) => MAIN(x,y) : at the beginning x=y *)
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  let expr_main1_lhs = (* INIT_STATE and f(x, y) *)
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    Z3.Boolean.mk_and !ctx [f_x_y_expr; init_expr] in
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  let expr_main1_rhs = main_x_y_expr in
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  let expr_main1 = Z3.Boolean.mk_implies !ctx expr_main1_lhs expr_main1_rhs in
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  (* Adding forall as prefix *)
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  let expr_forall_main1 = Z3.Quantifier.mk_forall_const
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    !ctx  (* context *)
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    (*
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    [int_sort; int_sort]           (* sort list*)
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    [Z3.FuncDecl.get_name x; Z3.FuncDecl.get_name y] (* symbol list *)
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    *)
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    (*    [x_expr; y_expr] Second try with expr list "const" *)
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    [Z3.Expr.mk_const_f !ctx x; Z3.Expr.mk_const_f !ctx y]
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    expr_main1 (* expression *)
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    None (* quantifier weight, None means 1 *)
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    [] (* pattern list ? *)
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    [] (* ? *)
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    None (* ? *)
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    None (* ? *)
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  in
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  let expr_forall_main1 = Z3.Quantifier.expr_of_quantifier expr_forall_main1 in
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  Z3.Fixedpoint.add_rule !fp expr_forall_main1 None;
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  (* Rule 2: forall x,y, MAIN(x,y) => MAIN(x+1, y+1) *)
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  let expr_main2_lhs = main_x_y_expr in
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  let plus_one x = Z3.Arithmetic.mk_add !ctx
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    [
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      x;
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      (* Z3.Arithmetic.Integer.mk_const_s !ctx "1" *)
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	Z3.Expr.mk_numeral_int !ctx 1 int_sort
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    ] in
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  let main_x_y_plus_one_expr = Z3.Expr.mk_app !ctx decl_main [plus_one x_expr; plus_one y_expr] in
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  let expr_main2_rhs = main_x_y_plus_one_expr in
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  let expr_main2 = Z3.Boolean.mk_implies !ctx expr_main2_lhs expr_main2_rhs in
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  (* Adding forall as prefix *)
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  let expr_forall_main2 = Z3.Quantifier.mk_forall_const
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    !ctx  (* context *)
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    (*
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    [int_sort; int_sort]           (* sort list*)
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    [Z3.FuncDecl.get_name x; Z3.FuncDecl.get_name y] (* symbol list *)
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    *)
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    (*    [x_expr; y_expr] Second try with expr list "const" *)
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    [Z3.Expr.mk_const_f !ctx x; Z3.Expr.mk_const_f !ctx y]
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    expr_main2 (* expression *)
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    None (* quantifier weight, None means 1 *)
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    [] (* pattern list ? *)
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    [] (* ? *)
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    None (* ? *)
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    None (* ? *)
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  in
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  let expr_forall_main2 = Z3.Quantifier.expr_of_quantifier expr_forall_main2 in
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  Z3.Fixedpoint.add_rule !fp expr_forall_main2 None;
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  (* TODO: not implemented yet since the error is visible without it *)
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  (* Query: is it possible to have MAIN(x,y) and x!=y ? *)
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  (* This is performed as follow:
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     rule (forall x, y, MAIN(x,y) and x!=y => ERR)
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  *)
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  let decl_err = Z3.FuncDecl.mk_func_decl_s !ctx "ERR" [] bool_sort in
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  Z3.Fixedpoint.register_relation !fp decl_err; 
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  let err_expr = Z3.Expr.mk_app !ctx decl_err [] in
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  let x_diff_y_expr = Z3.Boolean.mk_not !ctx x_eq_y_expr in
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  let expr_err_lhs =
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    Z3.Boolean.mk_and !ctx [x_diff_y_expr; main_x_y_expr] in
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  let expr_err_rhs = err_expr in
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  let expr_err = Z3.Boolean.mk_implies !ctx expr_err_lhs expr_err_rhs in
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  (* Adding forall as prefix *)
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  let expr_forall_err = Z3.Quantifier.mk_forall_const
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    !ctx  (* context *)
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(*    [int_sort; int_sort]           (* sort list*)
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    [Z3.FuncDecl.get_name x; Z3.FuncDecl.get_name y] (* symbol list *)
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*)
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    (* [x_expr; y_expr] Second try with expr list "const" *)
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    [Z3.Expr.mk_const_f !ctx x; Z3.Expr.mk_const_f !ctx y]
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    expr_err (* expression *)
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    None (* quantifier weight, None means 1 *)   
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    [] (* pattern list ? *)
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    [] (* ? *)
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    None (* ? *)
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    None (* ? *)
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  in
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  let expr_forall_err = Z3.Quantifier.expr_of_quantifier expr_forall_err in
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  Z3.Fixedpoint.add_rule !fp expr_forall_err None;
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  (* Printing the rules for sanity check *)
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  let rules_expr = Z3.Fixedpoint.get_rules !fp in
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  Format.eprintf "@[<v 2>Registered rules:@ %a@ @]@."
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    (fprintf_list ~sep:"@ "
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       (fun fmt e ->
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	 (* let e2 = Z3.Quantifier.get_body eq in *)
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	 (* let fd = Z3.Expr.get_func_decl e in *)
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	 Format.fprintf fmt "Rule: %s@ "
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	   (Z3.Expr.to_string e);
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       )) rules_expr;
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  let res_status = Z3.Fixedpoint.query_r !fp [decl_err] in
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  Format.eprintf "Status: %s@." (Z3.Solver.string_of_status res_status)
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