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|
(**************************************************************************)
(* *)
(* SMTCoq *)
(* Copyright (C) 2011 - 2019 *)
(* *)
(* See file "AUTHORS" for the list of authors *)
(* *)
(* This file is distributed under the terms of the CeCILL-C licence *)
(* *)
(**************************************************************************)
open Util
open SmtMisc
open CoqTerms
open SmtBtype
module type ATOM =
sig
type t
val index : t -> int
val equal : t -> t -> bool
val is_bool_type : t -> bool
val is_bv_type : t -> bool
val to_smt : Format.formatter -> t -> unit
val logic : t -> logic
end
type fop =
| Ftrue
| Ffalse
| Fand
| For
| Fxor
| Fimp
| Fiff
| Fite
| Fnot2 of int
| Fforall of (string * btype) list
type ('a,'f) gen_pform =
| Fatom of 'a
| Fapp of fop * 'f array
| FbbT of 'a * 'f list
module type FORM =
sig
type hatom
type t
type pform = (hatom, t) gen_pform
val pform_true : pform
val pform_false : pform
val equal : t -> t -> bool
val to_lit : t -> int
val index : t -> int
val pform : t -> pform
val neg : t -> t
val is_pos : t -> bool
val is_neg : t -> bool
val to_smt : ?debug:bool ->
Format.formatter -> t -> unit
val logic : t -> logic
(* Building formula from positive formula *)
exception NotWellTyped of pform
type reify
val create : unit -> reify
val clear : reify -> unit
val get : ?declare:bool -> reify -> pform -> t
(** Given a coq term, build the corresponding formula *)
val of_coq : (Structures.constr -> hatom) -> reify -> Structures.constr -> t
val hash_hform : (hatom -> hatom) -> reify -> t -> t
(* Flattening of [Fand] and [For], removing of [Fnot2] *)
val flatten : reify -> t -> t
(** Turn n-ary [Fand] and [For] into their right-associative
counter-parts *)
val right_assoc : reify -> t -> t
(** Producing Coq terms *)
val to_coq : t -> Structures.constr
val pform_tbl : reify -> pform array
val to_array : reify -> 'a -> (pform -> 'a) -> int * 'a array
val interp_tbl : reify -> Structures.constr * Structures.constr
val nvars : reify -> int
(* Producing a Coq term corresponding to the interpretation
of a formula *)
(* [interp_atom] map [hatom] to coq term, it is better if it produce
shared terms. *)
val interp_to_coq :
(hatom -> Structures.constr) -> (int, Structures.constr) Hashtbl.t ->
t -> Structures.constr
(* Unstratified terms *)
type atom_form_lit =
| Atom of hatom
| Form of pform
| Lit of t
val lit_of_atom_form_lit : reify -> bool * atom_form_lit -> t
end
module Make (Atom:ATOM) =
struct
type hatom = Atom.t
type pform = (Atom.t, t) gen_pform
and hpform = pform gen_hashed
and t =
| Pos of hpform
| Neg of hpform
let pform_true = Fapp (Ftrue,[||])
let pform_false = Fapp (Ffalse,[||])
let equal h1 h2 =
match h1, h2 with
| Pos hp1, Pos hp2 -> hp1.index == hp2.index
| Neg hp1, Neg hp2 -> hp1.index == hp2.index
| _, _ -> false
let index = function
| Pos hp -> hp.index
| Neg hp -> hp.index
let to_lit = function
| Pos hp -> hp.index * 2
| Neg hp -> hp.index * 2 + 1
let neg = function
| Pos hp -> Neg hp
| Neg hp -> Pos hp
let is_pos = function
| Pos _ -> true
| _ -> false
let is_neg = function
| Neg _ -> true
| _ -> false
let pform = function
| Pos hp -> hp.hval
| Neg hp -> hp.hval
let rec to_smt ?debug:(debug=false) fmt = function
| Pos hp ->
if debug then Format.fprintf fmt "%s" (string_of_int hp.index ^ ":");
to_smt_pform fmt hp.hval
| Neg hp ->
if debug then Format.fprintf fmt "%s" (string_of_int hp.index ^ ":");
Format.fprintf fmt "(not ";
to_smt_pform fmt hp.hval;
Format.fprintf fmt ")"
and to_smt_pform fmt = function
| Fatom a -> Atom.to_smt fmt a
| Fapp (op,args) -> to_smt_op fmt op args
(* This is an intermediate object of proofs, it correspond to nothing in SMT *)
| FbbT (a, l) ->
Format.fprintf fmt "(bbT %a [" Atom.to_smt a;
let fi = ref true in
List.iter (fun f -> Format.fprintf fmt "%s%a"
(if !fi then "" else "; ")
(to_smt ~debug:false) f; fi := false) l;
Format.fprintf fmt "])"
and to_smt_op fmt op args =
let (s1,s2) = if ((Array.length args = 0) || (match op with Fnot2 _ -> true | _ -> false)) then ("","") else ("(",")") in
Format.fprintf fmt "%s" s1;
(match op with
| Ftrue -> Format.fprintf fmt "true"
| Ffalse -> Format.fprintf fmt "false"
| Fand -> Format.fprintf fmt "and"
| For -> Format.fprintf fmt "or"
| Fxor -> Format.fprintf fmt "xor"
| Fimp -> Format.fprintf fmt "=>"
| Fiff -> Format.fprintf fmt "="
| Fite -> Format.fprintf fmt "ite"
| Fnot2 _ -> ()
| Fforall l ->
(Format.fprintf fmt "forall (";
to_smt_args fmt l;
Format.fprintf fmt ")")
);
Array.iter (fun h -> Format.fprintf fmt " "; to_smt fmt h) args;
Format.fprintf fmt "%s" s2
and to_smt_args fmt = function
| [] -> Format.fprintf fmt " "
| (s, t)::rem ->
(Format.fprintf fmt " (%s " s;
SmtBtype.to_smt fmt t;
Format.fprintf fmt ")";
to_smt_args fmt rem)
let rec logic_pform = function
| Fatom a -> Atom.logic a
| Fapp (_, args) ->
Array.fold_left (fun l f ->
SL.union (logic f) l
) SL.empty args
| FbbT _ -> SL.singleton LBitvectors
and logic = function
| Pos hp | Neg hp -> logic_pform hp.hval
let dumbed_down op =
let dumbed_down_bt = function
| Tindex it -> Tindex (dummy_indexed_type (indexed_type_index it))
| x -> x in
match op with
| Fforall l -> Fforall (List.map (fun (x, bt) -> x, dumbed_down_bt bt) l)
| x -> x
module HashedForm =
struct
type t = pform
let equal pf1 pf2 =
match pf1, pf2 with
| Fatom ha1, Fatom ha2 -> Atom.equal ha1 ha2
| Fapp(op1,args1), Fapp(op2,args2) ->
dumbed_down op1 = dumbed_down op2 &&
Array.length args1 == Array.length args2 &&
(try
for i = 0 to Array.length args1 - 1 do
if not (equal args1.(i) args2.(i)) then raise Not_found
done;
true
with Not_found -> false)
| FbbT(ha1, l1), FbbT(ha2, l2) ->
(try
Atom.equal ha1 ha2 &&
List.for_all2 (fun i j -> equal i j) l1 l2
with | Invalid_argument _ -> false)
| _, _ -> false
let hash pf =
match pf with
| Fatom ha1 -> Atom.index ha1 * 2
| Fapp(op, args) ->
let hash_args =
match Array.length args with
| 0 -> 0
| 1 -> to_lit args.(0)
| 2 -> (to_lit args.(1)) lsl 2 + to_lit args.(0)
| _ ->
(to_lit args.(2)) lsl 4 + (to_lit args.(1)) lsl 2 +
to_lit args.(0) in
(hash_args * 10 + Hashtbl.hash (dumbed_down op)) * 2 + 1
| FbbT(ha, l) ->
let hash_args =
match l with
| [] -> 0
| [a0] -> to_lit a0
| [a0;a1] -> (to_lit a1) lsl 2 + to_lit a0
| a0::a1::a2::_ ->
(to_lit a2) lsl 4 + (to_lit a1) lsl 2 + to_lit a0 in
(hash_args * 10 + Atom.index ha) * 2 + 1
end
module HashForm = Hashtbl.Make (HashedForm)
type reify = {
mutable count : int;
tbl : t HashForm.t
}
exception NotWellTyped of pform
let check pf =
match pf with
| Fatom ha -> if not (Atom.is_bool_type ha) then
raise (Format.eprintf "nwt: %a" to_smt_pform pf;
NotWellTyped pf)
| Fapp (op, args) ->
(match op with
| Ftrue | Ffalse ->
if Array.length args <> 0 then
raise (Format.eprintf "nwt: %a" to_smt_pform pf;
NotWellTyped pf)
| Fnot2 _ ->
if Array.length args <> 1 then
raise (Format.eprintf "nwt: %a" to_smt_pform pf;
NotWellTyped pf)
| Fand | For -> ()
| Fxor | Fimp | Fiff ->
if Array.length args <> 2 then
raise (Format.eprintf "nwt: %a" to_smt_pform pf;
NotWellTyped pf)
| Fite ->
if Array.length args <> 3 then
raise (Format.eprintf "nwt: %a" to_smt_pform pf;
NotWellTyped pf)
| Fforall l -> ()
)
| FbbT (ha, l) -> if not (Atom.is_bv_type ha) then
raise (Format.eprintf "nwt: %a" to_smt_pform pf;
NotWellTyped pf)
let declare reify pf =
check pf;
let res = Pos {index = reify.count; hval = pf} in
HashForm.add reify.tbl pf res;
reify.count <- reify.count + 1;
res
let create () =
let reify =
{ count = 0;
tbl = HashForm.create 17 } in
let _ = declare reify pform_true in
let _ = declare reify pform_false in
reify
let clear r =
r.count <- 0;
HashForm.clear r.tbl;
let _ = declare r pform_true in
let _ = declare r pform_false in
()
let get ?declare:(decl=true) reify pf =
if decl then
try HashForm.find reify.tbl pf
with Not_found -> declare reify pf
else Pos {index = -1; hval = pf}
(** Given a coq term, build the corresponding formula *)
type coq_cst =
| CCtrue
| CCfalse
| CCnot
| CCand
| CCor
| CCxor
| CCimp
| CCiff
| CCifb
| CCunknown
module ConstrHash = struct
type t = Structures.constr
let equal = Structures.eq_constr
let hash = Structures.hash_constr
end
module ConstrHashtbl = Hashtbl.Make(ConstrHash)
let op_tbl () =
let tbl = ConstrHashtbl.create 29 in
let add (c1,c2) = ConstrHashtbl.add tbl (Lazy.force c1) c2 in
List.iter add
[
ctrue,CCtrue; cfalse,CCfalse;
candb,CCand; corb,CCor; cxorb,CCxor; cimplb,CCimp; cnegb,CCnot;
ceqb,CCiff; cifb,CCifb ];
tbl
let op_tbl = lazy (op_tbl ())
let empty_args = [||]
let of_coq atom_of_coq reify c =
let op_tbl = Lazy.force op_tbl in
let get_cst c =
try ConstrHashtbl.find op_tbl c with Not_found -> CCunknown in
let rec mk_hform h =
let c, args = Structures.decompose_app h in
match get_cst c with
| CCtrue -> get reify (Fapp(Ftrue,empty_args))
| CCfalse -> get reify (Fapp(Ffalse,empty_args))
| CCnot -> mk_fnot 1 args
| CCand -> mk_fand [] args
| CCor -> mk_for [] args
| CCxor -> op2 (fun l -> Fapp(Fxor,l)) args
| CCiff -> op2 (fun l -> Fapp(Fiff,l)) args
| CCimp ->
(match args with
| [b1;b2] ->
let l1 = mk_hform b1 in
let l2 = mk_hform b2 in
get reify (Fapp (Fimp, [|l1;l2|]))
| _ -> Structures.error "SmtForm.Form.of_coq: wrong number of arguments for implb")
| CCifb ->
(* We should also be able to reify if then else *)
begin match args with
| [b1;b2;b3] ->
let l1 = mk_hform b1 in
let l2 = mk_hform b2 in
let l3 = mk_hform b3 in
get reify (Fapp (Fite, [|l1;l2;l3|]))
| _ -> Structures.error "SmtForm.Form.of_coq: wrong number of arguments for ifb"
end
| _ ->
let a = atom_of_coq h in
get reify (Fatom a)
and op2 f args =
match args with
| [b1;b2] ->
let l1 = mk_hform b1 in
let l2 = mk_hform b2 in
get reify (f [|l1; l2|])
| _ -> Structures.error "SmtForm.Form.of_coq: wrong number of arguments"
and mk_fnot i args =
match args with
| [t] ->
let c,args = Structures.decompose_app t in
if Structures.eq_constr c (Lazy.force cnegb) then
mk_fnot (i+1) args
else
let q,r = i lsr 1 , i land 1 in
let l = mk_hform t in
let l = if r = 0 then l else neg l in
if q = 0 then l
else get reify (Fapp(Fnot2 q, [|l|]))
| _ -> Structures.error "SmtForm.Form.mk_hform: wrong number of arguments for negb"
and mk_fand acc args =
match args with
| [t1;t2] ->
let l2 = mk_hform t2 in
let c, args = Structures.decompose_app t1 in
if Structures.eq_constr c (Lazy.force candb) then
mk_fand (l2::acc) args
else
let l1 = mk_hform t1 in
get reify (Fapp(Fand, Array.of_list (l1::l2::acc)))
| _ -> Structures.error "SmtForm.Form.mk_hform: wrong number of arguments for andb"
and mk_for acc args =
match args with
| [t1;t2] ->
let l2 = mk_hform t2 in
let c, args = Structures.decompose_app t1 in
if Structures.eq_constr c (Lazy.force corb) then
mk_for (l2::acc) args
else
let l1 = mk_hform t1 in
get reify (Fapp(For, Array.of_list (l1::l2::acc)))
| _ -> Structures.error "SmtForm.Form.mk_hform: wrong number of arguments for orb" in
mk_hform c
let hash_hform hash_hatom rf_quant hf =
let rec mk_hform = function
| Pos hp -> Pos (mk_hpform hp)
| Neg hp -> Neg (mk_hpform hp)
and mk_hpform {index = _; hval = hv} =
let new_hv = match hv with
| Fatom a -> Fatom (hash_hatom a)
| Fapp (fop, arr) -> Fapp (fop, Array.map mk_hform arr)
| FbbT (a, l) -> FbbT (hash_hatom a, List.map mk_hform l)
in
match get rf_quant new_hv with Pos x | Neg x -> x in
mk_hform hf
(** Flattening of Fand and For, removing of Fnot2 *)
let set_sign f f' =
if is_pos f then f' else neg f'
let rec flatten reify f =
match pform f with
| Fatom _ | FbbT _ -> f
| Fapp(Fnot2 _,args) -> set_sign f (flatten reify args.(0))
| Fapp(Fand, args) -> set_sign f (flatten_and reify [] (Array.to_list args))
| Fapp(For,args) -> set_sign f (flatten_or reify [] (Array.to_list args))
| Fapp(op,args) ->
(* TODO change Fimp into For ? *)
set_sign f (get reify (Fapp(op, Array.map (flatten reify) args)))
and flatten_and reify acc args =
match args with
| [] -> get reify (Fapp(Fand, Array.of_list (List.rev acc)))
| a::args ->
(* TODO change (not For) and (not Fimp) into Fand *)
match pform a with
| Fapp(Fand, args') when is_pos a ->
let args = Array.fold_right (fun a args -> a::args) args' args in
flatten_and reify acc args
| _ -> flatten_and reify (flatten reify a :: acc) args
and flatten_or reify acc args =
(* TODO change Fimp and (not Fand) into For *)
match args with
| [] -> get reify (Fapp(For, Array.of_list (List.rev acc)))
| a::args ->
match pform a with
| Fapp(For, args') when is_pos a ->
let args = Array.fold_right (fun a args -> a::args) args' args in
flatten_or reify acc args
| _ -> flatten_or reify (flatten reify a :: acc) args
let rec right_assoc reify f =
match pform f with
| Fapp(Fand, args) when Array.length args > 2 ->
let a = args.(0) in
let rargs = Array.sub args 1 (Array.length args - 1) in
let f' = right_assoc reify (get reify (Fapp (Fand, rargs))) in
set_sign f (get reify (Fapp (Fand, [|a; f'|])))
| Fapp(For, args) when Array.length args > 2 ->
let a = args.(0) in
let rargs = Array.sub args 1 (Array.length args - 1) in
let f' = right_assoc reify (get reify (Fapp (For, rargs))) in
set_sign f (get reify (Fapp (For, [|a; f'|])))
| _ -> f
(** Producing Coq terms *)
let to_coq hf = let i = to_lit hf in
if i < 0 then failwith "This formula should'nt be in Coq"
else mkInt i
let args_to_coq args =
let cargs = Array.make (Array.length args + 1) (mkInt 0) in
Array.iteri (fun i hf -> cargs.(i) <- to_coq hf) args;
Structures.mkArray (Lazy.force cint, cargs)
let pf_to_coq = function
| Fatom a -> mklApp cFatom [|mkInt (Atom.index a)|]
| Fapp(op,args) ->
(match op with
| Ftrue -> Lazy.force cFtrue
| Ffalse -> Lazy.force cFfalse
| Fand -> mklApp cFand [| args_to_coq args|]
| For -> mklApp cFor [| args_to_coq args|]
| Fimp -> mklApp cFimp [| args_to_coq args|]
| Fxor -> mklApp cFxor (Array.map to_coq args)
| Fiff -> mklApp cFiff (Array.map to_coq args)
| Fite -> mklApp cFite (Array.map to_coq args)
| Fnot2 i -> mklApp cFnot2 [|mkInt i; to_coq args.(0)|]
| Fforall _ -> failwith "pf_to_coq on forall")
| FbbT(a, l) -> mklApp cFbbT
[|mkInt (Atom.index a);
List.fold_right (fun f l -> mklApp ccons [|Lazy.force cint; to_coq f; l|]) l (mklApp cnil [|Lazy.force cint|])|]
let pform_tbl reify =
let t = Array.make reify.count pform_true in
let set _ h =
match h with
| Pos hp -> t.(hp.index) <- hp.hval
| _ -> assert false in
HashForm.iter set reify.tbl;
t
let to_array reify dft f =
let t = Array.make (reify.count + 1) dft in
let set _ h =
match h with
| Pos hp -> t.(hp.index) <- f hp.hval
| _ -> assert false in
HashForm.iter set reify.tbl;
(reify.count, t)
let interp_tbl reify =
let (i,t) = to_array reify (Lazy.force cFtrue) pf_to_coq in
(mkInt i, Structures.mkArray (Lazy.force cform, t))
let nvars reify = reify.count
(* Producing a Coq term corresponding to the interpretation of a formula *)
(* [interp_atom] map [Atom.t] to coq term, it is better if it produce
shared terms. *)
let interp_to_coq interp_atom form_tbl f =
let rec interp_form f =
let l = to_lit f in
try Hashtbl.find form_tbl l
with Not_found ->
if is_neg f then
let pc = interp_form (neg f) in
let nc = mklApp cnegb [|pc|] in
Hashtbl.add form_tbl l nc;
nc
else
let pc =
match pform f with
| Fatom a -> interp_atom a
| Fapp(op, args) ->
(match op with
| Ftrue -> Lazy.force ctrue
| Ffalse -> Lazy.force cfalse
| Fand -> interp_args candb args
| For -> interp_args corb args
| Fxor -> interp_args cxorb args
| Fimp ->
let r = ref (interp_form args.(Array.length args - 1)) in
for i = Array.length args - 2 downto 0 do
r := mklApp cimplb [|interp_form args.(i); !r|]
done;
!r
| Fiff -> interp_args ceqb args
| Fite ->
(* TODO with if here *)
mklApp cifb (Array.map interp_form args)
| Fnot2 n ->
(let r = ref (interp_form args.(0)) in
for _ = 1 to n do
r := mklApp cnegb [|!r|]
done;
!r)
| Fforall _ -> failwith "interp_to_coq on forall")
| FbbT(a, l) ->
mklApp cbv_eq
[|mkN (List.length l);
interp_atom a;
mklApp cof_bits [|List.fold_right (fun f l -> mklApp ccons [|Lazy.force cbool; interp_form f; l|]) l (mklApp cnil [|Lazy.force cbool|])|]|]
in
Hashtbl.add form_tbl l pc;
pc
and interp_args op args =
let r = ref (interp_form args.(0)) in
for i = 1 to Array.length args - 1 do
r := mklApp op [|!r;interp_form args.(i)|]
done;
!r in
interp_form f
(* Unstratified terms *)
type atom_form_lit =
| Atom of hatom
| Form of pform
| Lit of t
let lit_of_atom_form_lit rf = function
| decl, Atom a -> get ~declare:decl rf (Fatom a)
| decl, Form f -> begin match f with
| Fapp (Fforall _, _) when decl -> failwith "decl is true on a forall"
| _ -> get ~declare:decl rf f end
| decl, Lit l -> l
end
|