(* Authors: Nicolas Ayache and Jean-Christophe Filliātre *)
(* Tactics to call decision procedures *)

(* Works in two steps:

   - first the Coq context and the current goal are translated in
     Polymorphic First-Order Logic (see fol.mli in this directory)

   - then the resulting query is passed to the Why tool that translates
     it to the syntax of the selected prover (Simplify, CVC Lite, haRVey,
     Zenon)
*)

open Util
open Pp
open Libobject
open Summary
open Term
open Tacmach
open Tactics
open Tacticals
open Fol
open Names
open Nameops
open Namegen
open Coqlib
open Hipattern
open Libnames
open Declarations
open Dp_why

let debug = ref false
let set_debug b = debug := b
let trace = ref false
let set_trace b = trace := b
let timeout = ref 10
let set_timeout n = timeout := n

let dp_timeout_obj : int -> obj =
  declare_object
    {(default_object "Dp_timeout") with
       cache_function = (fun (_,x) -> set_timeout x);
       load_function = (fun _ (_,x) -> set_timeout x)}

let dp_timeout x = Lib.add_anonymous_leaf (dp_timeout_obj x)

let dp_debug_obj : bool -> obj =
  declare_object
    {(default_object "Dp_debug") with
       cache_function = (fun (_,x) -> set_debug x);
       load_function = (fun _ (_,x) -> set_debug x)}

let dp_debug x = Lib.add_anonymous_leaf (dp_debug_obj x)

let dp_trace_obj : bool -> obj =
  declare_object
    {(default_object "Dp_trace") with
       cache_function = (fun (_,x) -> set_trace x);
       load_function = (fun _ (_,x) -> set_trace x)}

let dp_trace x = Lib.add_anonymous_leaf (dp_trace_obj x)

let logic_dir = ["Coq";"Logic";"Decidable"]
let coq_modules =
  init_modules @ [logic_dir] @ arith_modules @ zarith_base_modules
    @ [["Coq"; "ZArith"; "BinInt"];
       ["Coq"; "Reals"; "Rdefinitions"];
       ["Coq"; "Reals"; "Raxioms";];
       ["Coq"; "Reals"; "Rbasic_fun";];
       ["Coq"; "Reals"; "R_sqrt";];
       ["Coq"; "Reals"; "Rfunctions";]]
    @ [["Coq"; "omega"; "OmegaLemmas"]]

let constant = gen_constant_in_modules "dp" coq_modules

(* integers constants and operations *)
let coq_Z = lazy (constant "Z")
let coq_Zplus = lazy (constant "Zplus")
let coq_Zmult = lazy (constant "Zmult")
let coq_Zopp = lazy (constant "Zopp")
let coq_Zminus = lazy (constant "Zminus")
let coq_Zdiv = lazy (constant "Zdiv")
let coq_Zs = lazy (constant "Zs")
let coq_Zgt = lazy (constant "Zgt")
let coq_Zle = lazy (constant "Zle")
let coq_Zge = lazy (constant "Zge")
let coq_Zlt = lazy (constant "Zlt")
let coq_Z0 = lazy (constant "Z0")
let coq_Zpos = lazy (constant "Zpos")
let coq_Zneg = lazy (constant "Zneg")
let coq_xH = lazy (constant "xH")
let coq_xI = lazy (constant "xI")
let coq_xO = lazy (constant "xO")
let coq_iff = lazy (constant "iff")

(* real constants and operations *)
let coq_R = lazy (constant "R")
let coq_R0 = lazy (constant "R0")
let coq_R1 = lazy (constant "R1")
let coq_Rgt = lazy (constant "Rgt")
let coq_Rle = lazy (constant "Rle")
let coq_Rge = lazy (constant "Rge")
let coq_Rlt = lazy (constant "Rlt")
let coq_Rplus = lazy (constant "Rplus")
let coq_Rmult = lazy (constant "Rmult")
let coq_Ropp = lazy (constant "Ropp")
let coq_Rminus = lazy (constant "Rminus")
let coq_Rdiv = lazy (constant "Rdiv")
let coq_powerRZ = lazy (constant "powerRZ")

(* not Prop typed expressions *)
exception NotProp

(* not first-order expressions *)
exception NotFO

(* Renaming of Coq globals *)

let global_names = Hashtbl.create 97
let used_names = Hashtbl.create 97

let rename_global r =
  try
    Hashtbl.find global_names r
  with Not_found ->
    let rec loop id =
      if Hashtbl.mem used_names id then
	loop (lift_subscript id)
      else begin
	Hashtbl.add used_names id ();
	let s = string_of_id id in
	Hashtbl.add global_names r s;
	s
      end
    in
    loop (Nametab.basename_of_global r)

let foralls =
  List.fold_right
    (fun (x,t) p -> Forall (x, t, p))

let fresh_var = function
  | Anonymous -> rename_global (VarRef (id_of_string "x"))
  | Name x -> rename_global (VarRef x)

(* coq_rename_vars env [(x1,t1);...;(xn,tn)] renames the xi outside of
   env names, and returns the new variables together with the new
   environment *)
let coq_rename_vars env vars =
  let avoid = ref (Termops.ids_of_named_context (Environ.named_context env)) in
  List.fold_right
    (fun (na,t) (newvars, newenv) ->
       let id = next_name_away na !avoid in
       avoid := id :: !avoid;
       id :: newvars, Environ.push_named (id, None, t) newenv)
    vars ([],env)

(* extract the prenex type quantifications i.e.
   type_quantifiers env (A1:Set)...(Ak:Set)t = A1...An, (env+Ai), t *)
let decomp_type_quantifiers env t =
  let rec loop vars t = match kind_of_term t with
    | Prod (n, a, t) when is_Set a || is_Type a ->
	loop ((n,a) :: vars) t
    | _ ->
	let vars, env = coq_rename_vars env vars in
	let t = substl (List.map mkVar vars) t in
	List.rev vars, env, t
  in
  loop [] t

(* same thing with lambda binders (for axiomatize body) *)
let decomp_type_lambdas env t =
  let rec loop vars t = match kind_of_term t with
    | Lambda (n, a, t) when is_Set a || is_Type a ->
	loop ((n,a) :: vars) t
    | _ ->
	let vars, env = coq_rename_vars env vars in
	let t = substl (List.map mkVar vars) t in
	List.rev vars, env, t
  in
  loop [] t

let decompose_arrows =
  let rec arrows_rec l c = match kind_of_term c with
    | Prod (_,t,c) when not (Termops.dependent (mkRel 1) c) -> arrows_rec (t :: l) c
    | Cast (c,_,_) -> arrows_rec l c
    | _ -> List.rev l, c
  in
  arrows_rec []

let rec eta_expanse t vars env i =
  assert (i >= 0);
  if i = 0 then
    t, vars, env
  else
    match kind_of_term (Typing.type_of env Evd.empty t) with
      | Prod (n, a, b) when not (Termops.dependent (mkRel 1) b) ->
	  let avoid = Termops.ids_of_named_context (Environ.named_context env) in
	  let id = next_name_away n avoid in
	  let env' = Environ.push_named (id, None, a) env in
	  let t' = mkApp (t, [| mkVar id |]) in
	  eta_expanse t' (id :: vars) env' (pred i)
      | _ ->
	  assert false

let rec skip_k_args k cl = match k, cl with
  | 0, _ -> cl
  | _, _ :: cl -> skip_k_args (k-1) cl
  | _, [] -> raise NotFO

(* Coq global references *)

type global = Gnot_fo | Gfo of Fol.decl

let globals = ref Refmap.empty
let globals_stack = ref []

(* synchronization *)
let () =
  Summary.declare_summary "Dp globals"
    { Summary.freeze_function = (fun () -> !globals, !globals_stack);
      Summary.unfreeze_function =
	(fun (g,s) -> globals := g; globals_stack := s);
      Summary.init_function = (fun () -> ()) }

let add_global r d = globals := Refmap.add r d !globals
let mem_global r = Refmap.mem r !globals
let lookup_global r = match Refmap.find r !globals with
  | Gnot_fo -> raise NotFO
  | Gfo d -> d

let locals = Hashtbl.create 97

let lookup_local r =  match Hashtbl.find locals r with
  | Gnot_fo -> raise NotFO
  | Gfo d -> d

let iter_all_constructors i f =
  let _, oib = Global.lookup_inductive i in
  Array.iteri
    (fun j tj -> f j (mkConstruct (i, j+1)))
    oib.mind_nf_lc


(* injection c [t1,...,tn] adds the injection axiom
     forall x1:t1,...,xn:tn,y1:t1,...,yn:tn.
       c(x1,...,xn)=c(y1,...,yn) -> x1=y1 /\ ... /\ xn=yn *)

let injection c l =
  let i = ref 0 in
  let var s = incr i; id_of_string (s ^ string_of_int !i) in
  let xl = List.map (fun t -> rename_global (VarRef (var "x")), t) l in
  i := 0;
  let yl = List.map (fun t -> rename_global (VarRef (var "y")), t) l in
  let f =
    List.fold_right2
      (fun (x,_) (y,_) p -> And (Fatom (Eq (App (x,[]),App (y,[]))), p))
      xl yl True
  in
  let vars = List.map (fun (x,_) -> App(x,[])) in
  let f = Imp (Fatom (Eq (App (c, vars xl), App (c, vars yl))), f) in
  let foralls = List.fold_right (fun (x,t) p -> Forall (x, t, p)) in
  let f = foralls xl (foralls yl f) in
  let ax = Axiom ("injection_" ^ c, f) in
  globals_stack := ax :: !globals_stack

(* rec_names_for c [|n1;...;nk|] builds the list of constant names for
   identifiers n1...nk with the same path as c, if they exist; otherwise
   raises Not_found *)
let rec_names_for c =
  let mp,dp,_ = Names.repr_con c in
  array_map_to_list
    (function
       | Name id ->
	   let c' = Names.make_con mp dp (label_of_id id) in
	   ignore (Global.lookup_constant c');
	   msgnl (Printer.pr_constr (mkConst c'));
	   c'
       | Anonymous ->
	   raise Not_found)

(* abstraction tables *)

let term_abstractions = Hashtbl.create 97

let new_abstraction =
  let r = ref 0 in fun () -> incr r; "abstraction_" ^ string_of_int !r

(* Arithmetic constants *)

exception NotArithConstant

(* translates a closed Coq term p:positive into a FOL term of type int *)

let big_two = Big_int.succ_big_int Big_int.unit_big_int

let rec tr_positive p = match kind_of_term p with
  | Term.Construct _ when p = Lazy.force coq_xH ->
      Big_int.unit_big_int
  | Term.App (f, [|a|]) when f = Lazy.force coq_xI ->
(*
      Plus (Mult (Cst 2, tr_positive a), Cst 1)
*)
      Big_int.succ_big_int (Big_int.mult_big_int big_two (tr_positive a))
  | Term.App (f, [|a|]) when f = Lazy.force coq_xO ->
(*
      Mult (Cst 2, tr_positive a)
*)
      Big_int.mult_big_int big_two (tr_positive a)
  | Term.Cast (p, _, _) ->
      tr_positive p
  | _ ->
      raise NotArithConstant

(* translates a closed Coq term t:Z or R into a FOL term of type int or real *)
let rec tr_arith_constant t = match kind_of_term t with
  | Term.Construct _ when t = Lazy.force coq_Z0 ->
      Cst Big_int.zero_big_int
  | Term.App (f, [|a|]) when f = Lazy.force coq_Zpos ->
      Cst (tr_positive a)
  | Term.App (f, [|a|]) when f = Lazy.force coq_Zneg ->
      Cst (Big_int.minus_big_int (tr_positive a))
  | Term.Const _ when t = Lazy.force coq_R0 ->
      RCst Big_int.zero_big_int
  | Term.Const _ when t = Lazy.force coq_R1 ->
      RCst Big_int.unit_big_int
  | Term.App (f, [|a;b|]) when f = Lazy.force coq_Rplus ->
      let ta = tr_arith_constant a in
      let tb = tr_arith_constant b in
      begin match ta,tb with
        | RCst na, RCst nb -> RCst (Big_int.add_big_int na nb)
        | _ -> raise NotArithConstant
      end
  | Term.App (f, [|a;b|]) when f = Lazy.force coq_Rmult ->
      let ta = tr_arith_constant a in
      let tb = tr_arith_constant b in
      begin match ta,tb with
        | RCst na, RCst nb -> RCst (Big_int.mult_big_int na nb)
        | _ -> raise NotArithConstant
      end
  | Term.App (f, [|a;b|]) when f = Lazy.force coq_powerRZ ->
      tr_powerRZ a b
  | Term.Cast (t, _, _) ->
      tr_arith_constant t
  | _ ->
      raise NotArithConstant

(* translates a constant of the form (powerRZ 2 int_constant) *)
and tr_powerRZ a b =
  (* checking first that a is (R1 + R1) *)
  match kind_of_term a with
  | Term.App (f, [|c;d|]) when f = Lazy.force coq_Rplus ->
      begin
      match kind_of_term c,kind_of_term d with
        | Term.Const _, Term.Const _
            when c = Lazy.force coq_R1 && d = Lazy.force coq_R1 ->
            begin
              match tr_arith_constant b with
                | Cst n -> Power2 n
                | _ -> raise NotArithConstant
            end
        | _ -> raise NotArithConstant
      end
  | _ -> raise NotArithConstant


(* translate a Coq term t:Set into a FOL type expression;
   tv = list of type variables *)
and tr_type tv env t =
  let t = Reductionops.nf_betadeltaiota env Evd.empty t in
  if t = Lazy.force coq_Z then
    Tid ("int", [])
  else if t = Lazy.force coq_R then
    Tid ("real", [])
  else match kind_of_term t with
    | Var x when List.mem x tv ->
	Tvar (string_of_id x)
    | _ ->
	let f, cl = decompose_app t in
	begin try
	  let r = global_of_constr f in
	  match tr_global env r with
	    | DeclType (id, k) ->
		assert (k = List.length cl); (* since t:Set *)
		Tid (id, List.map (tr_type tv env) cl)
	    | _ ->
		raise NotFO
	with
	  | Not_found ->
	      raise NotFO
	  | NotFO ->
	      (* we need to abstract some part of (f cl) *)
	      (*TODO*)
	      raise NotFO
	end

and make_term_abstraction tv env c =
  let ty = Typing.type_of env Evd.empty c in
  let id = new_abstraction () in
  match tr_decl env id ty with
    | DeclFun (id,_,_,_) as _d ->
        raise NotFO
        (* [CM 07/09/2009] deactivated because it generates
           unbound identifiers 'abstraction_<number>'
	begin try
	  Hashtbl.find term_abstractions c
	with Not_found ->
	  Hashtbl.add term_abstractions c id;
	  globals_stack := d :: !globals_stack;
	  id
	end
        *)
    | _ ->
	raise NotFO

(* translate a Coq declaration id:ty in a FOL declaration, that is either
   - a type declaration : DeclType (id, n) where n:int is the type arity
   - a function declaration : DeclFun (id, tl, t) ; that includes constants
   - a predicate declaration : DeclPred (id, tl)
   - an axiom : Axiom (id, p)
 *)
and tr_decl env id ty =
  let tv, env, t = decomp_type_quantifiers env ty in
  if is_Set t || is_Type t then
    DeclType (id, List.length tv)
  else if is_Prop t then
    DeclPred (id, List.length tv, [])
  else
    let s = Typing.type_of env Evd.empty t in
    if is_Prop s then
      Axiom (id, tr_formula tv [] env t)
    else
      let l, t = decompose_arrows t in
      let l = List.map (tr_type tv env) l in
      if is_Prop t then
	DeclPred(id, List.length tv, l)
      else
	let s = Typing.type_of env Evd.empty t in
	if is_Set s || is_Type s then
	  DeclFun (id, List.length tv, l, tr_type tv env t)
	else
	  raise NotFO

(* tr_global(r) = tr_decl(id(r),typeof(r)) + a cache mechanism *)
and tr_global env r = match r with
  | VarRef id ->
      lookup_local id
  | r ->
      try
	lookup_global r
      with Not_found ->
	try
	  let ty = Global.type_of_global r in
	  let id = rename_global r in
	  let d = tr_decl env id ty in
	  (* r can be already declared if it is a constructor *)
	  if not (mem_global r) then begin
	    add_global r (Gfo d);
	    globals_stack := d :: !globals_stack
	  end;
	  begin try axiomatize_body env r id d with NotFO -> () end;
	  d
	with NotFO ->
	  add_global r Gnot_fo;
	  raise NotFO

and axiomatize_body env r id d = match r with
  | VarRef _ ->
      assert false
  | ConstRef c ->
      begin match body_of_constant (Global.lookup_constant c) with
	| Some b ->
	    let b = force b in
	    let axioms =
	      (match d with
		 | DeclPred (id, _, []) ->
		     let tv, env, b = decomp_type_lambdas env b in
		     let value = tr_formula tv [] env b in
		     [id, Iff (Fatom (Pred (id, [])), value)]
		 | DeclFun (id, _, [], _) ->
		     let tv, env, b = decomp_type_lambdas env b in
		     let value = tr_term tv [] env b in
		     [id, Fatom (Eq (Fol.App (id, []), value))]
		 | DeclFun (id, _, l, _) | DeclPred (id, _, l) ->
		     (*Format.eprintf "axiomatize_body %S@." id;*)
		     let b = match kind_of_term b with
		       (* a single recursive function *)
		       | Fix (_, (_,_,[|b|])) ->
			   subst1 (mkConst c) b
                       (* mutually recursive functions *)
		       | Fix ((_,i), (names,_,bodies)) ->
                           (* we only deal with named functions *)
			   begin try
			     let l = rec_names_for c names in
			     substl (List.rev_map mkConst l) bodies.(i)
			   with Not_found ->
			     b
			   end
		       | _ ->
			   b
		     in
		     let tv, env, b = decomp_type_lambdas env b in
		     let vars, t = decompose_lam b in
		     let n = List.length l in
		     let k = List.length vars in
		     assert (k <= n);
		     let vars, env = coq_rename_vars env vars in
		     let t = substl (List.map mkVar vars) t in
		     let t, vars, env = eta_expanse t vars env (n-k) in
		     let vars = List.rev vars in
		     let bv = vars in
		     let vars = List.map (fun x -> string_of_id x) vars in
		     let fol_var x = Fol.App (x, []) in
		     let fol_vars = List.map fol_var vars in
		     let vars = List.combine vars l in
		     begin match d with
		       | DeclFun (_, _, _, ty) ->
			   begin match kind_of_term t with
			     | Case (ci, _, e, br) ->
				 equations_for_case env id vars tv bv ci e br
			     | _ ->
				 let t = tr_term tv bv env t in
				 let ax =
				   add_proof (Fun_def (id, vars, ty, t))
				 in
				 let p = Fatom (Eq (App (id, fol_vars), t)) in
				 [ax, foralls vars p]
			   end
		       | DeclPred _ ->
			   let value = tr_formula tv bv env t in
			   let p = Iff (Fatom (Pred (id, fol_vars)), value) in
			   [id, foralls vars p]
		       | _ ->
			   assert false
		     end
		 | DeclType _ ->
		     raise NotFO
		 | Axiom _ -> assert false)
	    in
	    let axioms = List.map (fun (id,ax) -> Axiom (id, ax)) axioms in
	    globals_stack := axioms @ !globals_stack
	| None ->
	    () (* Coq axiom *)
      end
  | IndRef i ->
      iter_all_constructors i
	(fun _ c ->
	   let rc = global_of_constr c in
	   try
	     begin match tr_global env rc with
	       | DeclFun (_, _, [], _) -> ()
	       | DeclFun (idc, _, al, _) -> injection idc al
	       | _ -> ()
	     end
	   with NotFO ->
	     ())
  | _ -> ()

and equations_for_case env id vars tv bv ci e br = match kind_of_term e with
  | Var x when List.exists (fun (y, _) -> string_of_id x = y) vars ->
      let eqs = ref [] in
      iter_all_constructors ci.ci_ind
	(fun j cj ->
	   try
	     let cjr = global_of_constr cj in
	     begin match tr_global env cjr with
	       | DeclFun (idc, _, l, _) ->
		   let b = br.(j) in
		   let rec_vars, b = decompose_lam b in
		   let rec_vars, env = coq_rename_vars env rec_vars in
		   let coq_rec_vars = List.map mkVar rec_vars in
		   let b = substl coq_rec_vars b in
		   let rec_vars = List.rev rec_vars in
		   let coq_rec_term = applist (cj, List.rev coq_rec_vars) in
		   let b = replace_vars [x, coq_rec_term] b in
		   let bv = bv @ rec_vars in
		   let rec_vars = List.map string_of_id rec_vars in
		   let fol_var x = Fol.App (x, []) in
		   let fol_rec_vars = List.map fol_var rec_vars in
		   let fol_rec_term = App (idc, fol_rec_vars) in
		   let rec_vars = List.combine rec_vars l in
		   let fol_vars = List.map fst vars in
		   let fol_vars = List.map fol_var fol_vars in
		   let fol_vars = List.map (fun y -> match y with
					      | App (id, _) ->
						  if id = string_of_id x
						  then fol_rec_term
						  else y
					      | _ -> y)
				    fol_vars in
		   let vars = vars @ rec_vars in
		   let rec remove l e = match l with
		     | [] -> []
		     | (y, t)::l' -> if y = string_of_id e then l'
		       else (y, t)::(remove l' e) in
		   let vars = remove vars x in
		   let p =
		     Fatom (Eq (App (id, fol_vars),
				tr_term tv bv env b))
		   in
		   eqs := (id ^ "_" ^ idc, foralls vars p) :: !eqs
	       | _ ->
		   assert false end
	   with NotFO ->
	     ());
      !eqs
  | _ ->
      raise NotFO

(* assumption: t:T:Set *)
and tr_term tv bv env t =
      try
	tr_arith_constant t
      with NotArithConstant ->
  match kind_of_term t with
    (* binary operations on integers *)
  | Term.App (f, [|a;b|]) when f = Lazy.force coq_Zplus ->
      Plus (tr_term tv bv env a, tr_term tv bv env b)
  | Term.App (f, [|a;b|]) when f = Lazy.force coq_Zminus ->
      Moins (tr_term tv bv env a, tr_term tv bv env b)
  | Term.App (f, [|a;b|]) when f = Lazy.force coq_Zmult ->
      Mult (tr_term tv bv env a, tr_term tv bv env b)
  | Term.App (f, [|a;b|]) when f = Lazy.force coq_Zdiv ->
      Div (tr_term tv bv env a, tr_term tv bv env b)
  | Term.App (f, [|a|]) when f = Lazy.force coq_Zopp ->
      Opp (tr_term tv bv env a)
    (* binary operations on reals *)
  | Term.App (f, [|a;b|]) when f = Lazy.force coq_Rplus ->
      Plus (tr_term tv bv env a, tr_term tv bv env b)
  | Term.App (f, [|a;b|]) when f = Lazy.force coq_Rminus ->
      Moins (tr_term tv bv env a, tr_term tv bv env b)
  | Term.App (f, [|a;b|]) when f = Lazy.force coq_Rmult ->
      Mult (tr_term tv bv env a, tr_term tv bv env b)
  | Term.App (f, [|a;b|]) when f = Lazy.force coq_Rdiv ->
      Div (tr_term tv bv env a, tr_term tv bv env b)
  | Term.Var id when List.mem id bv ->
      App (string_of_id id, [])
  | _ ->
	let f, cl = decompose_app t in
	begin try
	  let r = global_of_constr f in
	  match tr_global env r with
	    | DeclFun (s, k, _, _) ->
		let cl = skip_k_args k cl in
		Fol.App (s, List.map (tr_term tv bv env) cl)
	    | _ ->
		raise NotFO
	with
	  | Not_found ->
	      raise NotFO
	  | NotFO -> (* we need to abstract some part of (f cl) *)
	      let rec abstract app = function
		| [] ->
		    Fol.App (make_term_abstraction tv env app, [])
		| x :: l as args ->
		    begin try
		      let s = make_term_abstraction tv env app in
		      Fol.App (s, List.map (tr_term tv bv env) args)
		    with NotFO ->
		      abstract (applist (app, [x])) l
		    end
	      in
	      let app,l = match cl with
		| x :: l -> applist (f, [x]), l | [] -> raise NotFO
	      in
	      abstract app l
	end

and quantifiers n a b tv bv env =
  let vars, env = coq_rename_vars env [n,a] in
  let id = match vars with [x] -> x | _ -> assert false in
  let b = subst1 (mkVar id) b in
  let t = tr_type tv env a in
  let bv = id :: bv in
  id, t, bv, env, b

(* assumption: f is of type Prop *)
and tr_formula tv bv env f =
  let c, args = decompose_app f in
  match kind_of_term c, args with
    | Var id, [] ->
	Fatom (Pred (rename_global (VarRef id), []))
    | _, [t;a;b] when c = build_coq_eq () ->
	let ty = Typing.type_of env Evd.empty t in
	if is_Set ty || is_Type ty then
	  let _ = tr_type tv env t in
	  Fatom (Eq (tr_term tv bv env a, tr_term tv bv env b))
	else
	  raise NotFO
          (* comparisons on integers *)
    | _, [a;b] when c = Lazy.force coq_Zle ->
	Fatom (Le (tr_term tv bv env a, tr_term tv bv env b))
    | _, [a;b] when c = Lazy.force coq_Zlt ->
	Fatom (Lt (tr_term tv bv env a, tr_term tv bv env b))
    | _, [a;b] when c = Lazy.force coq_Zge ->
	Fatom (Ge (tr_term tv bv env a, tr_term tv bv env b))
    | _, [a;b] when c = Lazy.force coq_Zgt ->
	Fatom (Gt (tr_term tv bv env a, tr_term tv bv env b))
          (* comparisons on reals *)
    | _, [a;b] when c = Lazy.force coq_Rle ->
	Fatom (Le (tr_term tv bv env a, tr_term tv bv env b))
    | _, [a;b] when c = Lazy.force coq_Rlt ->
	Fatom (Lt (tr_term tv bv env a, tr_term tv bv env b))
    | _, [a;b] when c = Lazy.force coq_Rge ->
	Fatom (Ge (tr_term tv bv env a, tr_term tv bv env b))
    | _, [a;b] when c = Lazy.force coq_Rgt ->
	Fatom (Gt (tr_term tv bv env a, tr_term tv bv env b))
    | _, [] when c = build_coq_False () ->
	False
    | _, [] when c = build_coq_True () ->
	True
    | _, [a] when c = build_coq_not () ->
	Not (tr_formula tv bv env a)
    | _, [a;b] when c = build_coq_and () ->
	And (tr_formula tv bv env a, tr_formula tv bv env b)
    | _, [a;b] when c = build_coq_or () ->
	Or (tr_formula tv bv env a, tr_formula tv bv env b)
    | _, [a;b] when c = Lazy.force coq_iff ->
	Iff (tr_formula tv bv env a, tr_formula tv bv env b)
    | Prod (n, a, b), _ ->
        if is_Prop (Typing.type_of env Evd.empty a) then
	  Imp (tr_formula tv bv env a, tr_formula tv bv env b)
	else
	  let id, t, bv, env, b = quantifiers n a b tv bv env in
	  Forall (string_of_id id, t, tr_formula tv bv env b)
    | _, [_; a] when c = build_coq_ex () ->
	begin match kind_of_term a with
	  | Lambda(n, a, b) ->
	      let id, t, bv, env, b = quantifiers n a b tv bv env in
	      Exists (string_of_id id, t, tr_formula tv bv env b)
	  | _ ->
	      (* unusual case of the shape (ex p) *)
	      raise NotFO (* TODO: we could eta-expanse *)
	end
    | _ ->
	begin try
	  let r = global_of_constr c in
	  match tr_global env r with
	    | DeclPred (s, k, _) ->
		let args = skip_k_args k args in
		Fatom (Pred (s, List.map (tr_term tv bv env) args))
	    | _ ->
		raise NotFO
	with Not_found ->
	  raise NotFO
	end


let tr_goal gl =
  Hashtbl.clear locals;
  let tr_one_hyp (id, ty) =
    try
      let s = rename_global (VarRef id) in
      let d = tr_decl (pf_env gl) s ty in
      Hashtbl.add locals id (Gfo d);
      d
    with NotFO ->
      Hashtbl.add locals id Gnot_fo;
      raise NotFO
  in
  let hyps =
    List.fold_right
      (fun h acc -> try tr_one_hyp h :: acc with NotFO -> acc)
      (pf_hyps_types gl) []
  in
  let c = tr_formula [] [] (pf_env gl) (pf_concl gl) in
  let hyps = List.rev_append !globals_stack (List.rev hyps) in
  hyps, c


type prover = Simplify | Ergo | Yices | CVCLite | Harvey | Zenon | Gwhy | CVC3 | Z3

let remove_files = List.iter (fun f -> try Sys.remove f with _ -> ())

let sprintf = Format.sprintf

let file_contents f =
  let buf = Buffer.create 1024 in
  try
    let c = open_in f in
    begin try
      while true do
	let s = input_line c in Buffer.add_string buf s;
	Buffer.add_char buf '\n'
      done;
      assert false
    with End_of_file ->
      close_in c;
      Buffer.contents buf
    end
  with _ ->
    sprintf "(cannot open %s)" f

let timeout_sys_command cmd =
  if !debug then Format.eprintf "command line: %s@." cmd;
  let out = Filename.temp_file "out" "" in
  let cmd = sprintf "why-cpulimit %d %s > %s 2>&1" !timeout cmd out in
  let ret = Sys.command cmd in
  if !debug then
    Format.eprintf "Output file %s:@.%s@." out (file_contents out);
  ret, out

let timeout_or_failure c cmd out =
  if c = 152 then
    Timeout
  else
    Failure
      (sprintf "command %s failed with output:\n%s " cmd (file_contents out))

let call_prover ?(opt="") file =
  if !debug then Format.eprintf "calling prover on %s@." file;
  let out = Filename.temp_file "out" "" in
  let cmd =
    sprintf "why-dp -timeout %d -batch %s > %s 2>&1" !timeout file out in
  match Sys.command cmd with
      0 -> Valid None
    | 1 -> Failure (sprintf "could not run why-dp\n%s" (file_contents out))
    | 2 -> Invalid
    | 3 -> DontKnow
    | 4 -> Timeout
    | 5 -> Failure (sprintf "prover failed:\n%s" (file_contents out))
    | n -> Failure (sprintf "Unknown exit status of why-dp: %d" n)

let prelude_files = ref ([] : string list)

let set_prelude l = prelude_files := l

let dp_prelude_obj : string list -> obj =
  declare_object
    {(default_object "Dp_prelude") with
       cache_function = (fun (_,x) -> set_prelude x);
       load_function = (fun _ (_,x) -> set_prelude x)}

let dp_prelude x = Lib.add_anonymous_leaf (dp_prelude_obj x)

let why_files f = String.concat " " (!prelude_files @ [f])

let call_simplify fwhy =
  let cmd =
    sprintf "why --simplify %s" (why_files fwhy)
  in
  if Sys.command cmd <> 0 then error ("call to " ^ cmd ^ " failed");
  let fsx = Filename.chop_suffix fwhy ".why" ^ "_why.sx" in
(*
  let cmd =
    sprintf "why-cpulimit %d Simplify %s > out 2>&1 && grep -q -w Valid out"
      !timeout fsx
  in
  let out = Sys.command cmd in
  let r =
    if out = 0 then Valid None else if out = 1 then Invalid else Timeout
  in
*)
  let r = call_prover fsx in
  if not !debug then remove_files [fwhy; fsx];
  r

let call_ergo fwhy =
  let cmd = sprintf "why --alt-ergo %s" (why_files fwhy) in
  if Sys.command cmd <> 0 then error ("call to " ^ cmd ^ " failed");
  let fwhy = Filename.chop_suffix fwhy ".why" ^ "_why.why" in
  (*let ftrace = Filename.temp_file "ergo_trace" "" in*)
  (*NB: why-dp can't handle -cctrace
  let cmd =
    if !trace then
      sprintf "alt-ergo -cctrace %s %s" ftrace fwhy

    else
      sprintf "alt-ergo %s" fwhy
  in*)
  let r = call_prover fwhy in
  if not !debug then remove_files [fwhy; (*out*)];
  r


let call_zenon fwhy =
  let cmd =
    sprintf "why --no-zenon-prelude --zenon %s" (why_files fwhy)
  in
  if Sys.command cmd <> 0 then error ("call to " ^ cmd ^ " failed");
  let fznn = Filename.chop_suffix fwhy ".why" ^ "_why.znn" in
(*  why-dp won't let us having coqterm...
  let out = Filename.temp_file "dp_out" "" in
  let cmd =
    sprintf "timeout %d zenon -ocoqterm %s > %s 2>&1" !timeout fznn out
  in
  let c = Sys.command cmd in
  if not !debug then remove_files [fwhy; fznn];
  if c = 137 then
    Timeout
  else begin
    if c <> 0 then anomaly ("command failed: " ^ cmd);
    if Sys.command (sprintf "grep -q -w Error %s" out) = 0 then
      error "Zenon failed";
    let c = Sys.command (sprintf "grep -q PROOF-FOUND %s" out) in
    if c = 0 then Valid (Some out) else Invalid
  end
 *)
   let r = call_prover fznn in
   if not !debug then remove_files [fwhy; fznn];
   r

let call_smt ~smt fwhy =
  let cmd = 
    sprintf "why -smtlib --encoding sstrat %s" (why_files fwhy)
  in
  if Sys.command cmd <> 0 then error ("call to " ^ cmd ^ " failed");
  let fsmt = Filename.chop_suffix fwhy ".why" ^ "_why.smt" in
  let opt = "-smt-solver " ^ smt in
  let r = call_prover ~opt fsmt in
  if not !debug then remove_files [fwhy; fsmt];
  r

(*
let call_yices fwhy =
  let cmd =
    sprintf "why -smtlib --encoding sstrat %s" (why_files fwhy)
  in
  if Sys.command cmd <> 0 then error ("call to " ^ cmd ^ " failed");
  let fsmt = Filename.chop_suffix fwhy ".why" ^ "_why.smt" in
  let cmd =
    sprintf "why-cpulimit %d yices -pc 0 -smt %s > out 2>&1 && grep -q -w unsat out"
      !timeout fsmt
  in
  let out = Sys.command cmd in
  let r =
    if out = 0 then Valid None else if out = 1 then Invalid else Timeout
  in
  if not !debug then remove_files [fwhy; fsmt];
  r

let call_cvc3 fwhy =
  let cmd =
    sprintf "why -smtlib --encoding sstrat %s" (why_files fwhy)
  in
  if Sys.command cmd <> 0 then error ("call to " ^ cmd ^ " failed");
  let fsmt = Filename.chop_suffix fwhy ".why" ^ "_why.smt" in
  let cmd =
    sprintf "why-cpulimit %d cvc3 -lang smt %s > out 2>&1 && grep -q -w unsat out"
      !timeout fsmt
  in
  let out = Sys.command cmd in
  let r =
    if out = 0 then Valid None else if out = 1 then Invalid else Timeout
  in
  if not !debug then remove_files [fwhy; fsmt];
  r
*)

let call_cvcl fwhy =
  let cmd =
    sprintf "why --cvcl --encoding sstrat %s" (why_files fwhy)
  in
  if Sys.command cmd <> 0 then error ("call to " ^ cmd ^ " failed");
  let fcvc = Filename.chop_suffix fwhy ".why" ^ "_why.cvc" in
(*
  let cmd =
    sprintf "timeout %d cvcl < %s > out 2>&1 && grep -q -w Valid out"
      !timeout fcvc
  in
  let out = Sys.command cmd in
  let r =
    if out = 0 then Valid None else if out = 1 then Invalid else Timeout
  in
*)
  let r = call_prover fcvc in
  if not !debug then remove_files [fwhy; fcvc];
  r

let call_harvey fwhy =
  let cmd =
    sprintf "why --harvey --encoding strat %s" (why_files fwhy)
  in
  if Sys.command cmd <> 0 then error ("call to " ^ cmd ^ " failed");
  let frv = Filename.chop_suffix fwhy ".why" ^ "_why.rv" in
(*
  let out = Sys.command (sprintf "rvc -e -t %s > /dev/null 2>&1" frv) in
  if out <> 0 then anomaly ("call to rvc -e -t " ^ frv ^ " failed");
  let f = Filename.chop_suffix frv ".rv" ^ "-0.baf" in
  let outf = Filename.temp_file "rv" ".out" in
  let out =
    Sys.command (sprintf "timeout %d rv -e\"-T 2000\" %s > %s 2>&1"
		   !timeout f outf)
  in
  let r =
    if out <> 0 then
      Timeout
    else
      let cmd =
	sprintf "grep \"Proof obligation in\" %s | grep -q \"is valid\"" outf
      in
      if Sys.command cmd = 0 then Valid None else Invalid
  in
  if not !debug then remove_files [fwhy; frv; outf];
*)
  let r = call_prover frv in
  if not !debug then remove_files [fwhy; frv];
  r

let call_gwhy fwhy =
  let cmd = sprintf "gwhy %s" (why_files fwhy) in
  if Sys.command cmd <> 0 then ignore (Sys.command (sprintf "emacs %s" fwhy));
  NoAnswer

let ergo_proof_from_file f gl =
  let s =
    let buf = Buffer.create 1024 in
    let c = open_in f in
    try
      while true do Buffer.add_string buf (input_line c) done; assert false
    with End_of_file ->
      close_in c;
      Buffer.contents buf
  in
  let parsed_constr = Pcoq.parse_string Pcoq.Constr.constr s in
  let t = Constrintern.interp_constr (project gl) (pf_env gl) parsed_constr in
  exact_check t gl

let call_prover prover q =
  let fwhy = Filename.temp_file "coq_dp" ".why" in
  Dp_why.output_file fwhy q;
  match prover with
    | Simplify -> call_simplify fwhy
    | Ergo -> call_ergo fwhy
    | CVC3 -> call_smt ~smt:"cvc3" fwhy
    | Yices -> call_smt ~smt:"yices" fwhy
    | Z3 -> call_smt ~smt:"z3" fwhy
    | Zenon -> call_zenon fwhy
    | CVCLite -> call_cvcl fwhy
    | Harvey -> call_harvey fwhy
    | Gwhy -> call_gwhy fwhy

let dp prover gl =
  Coqlib.check_required_library ["Coq";"ZArith";"ZArith"];
  let concl_type = pf_type_of gl (pf_concl gl) in
  if not (is_Prop concl_type) then error "Conclusion is not a Prop";
  try
    let q = tr_goal gl in
    begin match call_prover prover q with
      | Valid (Some f) when prover = Zenon -> Dp_zenon.proof_from_file f gl
      | Valid (Some f) when prover = Ergo -> ergo_proof_from_file f gl
      | Valid _ -> Tactics.admit_as_an_axiom gl
      | Invalid -> error "Invalid"
      | DontKnow -> error "Don't know"
      | Timeout -> error "Timeout"
      | Failure s -> error s
      | NoAnswer -> Tacticals.tclIDTAC gl
    end
  with NotFO ->
    error "Not a first order goal"


let simplify = tclTHEN intros (dp Simplify)
let ergo = tclTHEN intros (dp Ergo)
let cvc3 = tclTHEN intros (dp CVC3)
let yices = tclTHEN intros (dp Yices)
let z3 = tclTHEN intros (dp Z3)
let cvc_lite = tclTHEN intros (dp CVCLite)
let harvey = dp Harvey
let zenon = tclTHEN intros (dp Zenon)
let gwhy = tclTHEN intros (dp Gwhy)

let dp_hint l =
  let env = Global.env () in
  let one_hint (qid,r) =
    if not (mem_global r) then begin
      let ty = Global.type_of_global r in
      let s = Typing.type_of env Evd.empty ty in
      if is_Prop s then
	try
	  let id = rename_global r in
	  let tv, env, ty = decomp_type_quantifiers env ty in
	  let d = Axiom (id, tr_formula tv [] env ty) in
	  add_global r (Gfo d);
	  globals_stack := d :: !globals_stack
	with NotFO ->
	  add_global r Gnot_fo;
	  msg_warning
	    (pr_reference qid ++
	     str " ignored (not a first order proposition)")
	else begin
	  add_global r Gnot_fo;
	  msg_warning
	    (pr_reference qid ++ str " ignored (not a proposition)")
	end
    end
  in
  List.iter one_hint (List.map (fun qid -> qid, Nametab.global qid) l)

let dp_hint_obj : reference list -> obj =
  declare_object
    {(default_object "Dp_hint") with
       cache_function = (fun (_,l) -> dp_hint l);
       load_function = (fun _ (_,l) -> dp_hint l)}

let dp_hint l = Lib.add_anonymous_leaf (dp_hint_obj l)

let dp_predefined qid s =
  let r = Nametab.global qid in
  let ty = Global.type_of_global r in
  let env = Global.env () in
  let id = rename_global r in
  try
    let d = match tr_decl env id ty with
      | DeclType (_, n) -> DeclType (s, n)
      | DeclFun (_, n, tyl, ty) -> DeclFun (s, n, tyl, ty)
      | DeclPred (_, n, tyl) -> DeclPred (s, n, tyl)
      | Axiom _ as d -> d
    in
    match d with
      | Axiom _ ->  msg_warning (str " ignored (axiom)")
      | d -> add_global r (Gfo d)
  with NotFO ->
    msg_warning (str " ignored (not a first order declaration)")

let dp_predefined_obj : reference * string -> obj =
  declare_object
    {(default_object "Dp_predefined") with
       cache_function = (fun (_,(id,s)) -> dp_predefined id s);
       load_function = (fun _ (_,(id,s)) -> dp_predefined id s)}

let dp_predefined id s = Lib.add_anonymous_leaf (dp_predefined_obj (id,s))

let _ = declare_summary "Dp options"
	  { freeze_function =
	      (fun () -> !debug, !trace, !timeout, !prelude_files);
	    unfreeze_function =
	      (fun (d,tr,tm,pr) ->
		debug := d; trace := tr; timeout := tm; prelude_files := pr);
	    init_function =
	      (fun () ->
		debug := false; trace := false; timeout := 10;
		prelude_files := []) }