Source file Higher_order.ml
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(** {1 boolean subterms} *)
open Logtk
open Libzipperposition
module BV = CCBV
module T = Term
module Lits = Literals
module IntSet = Set.Make(CCInt)
module IntMap = Util.Int_map
let section = Util.Section.make ~parent:Const.section "ho"
let stat_eq_res = Util.mk_stat "ho.eq_res.steps"
let stat_eq_res_syntactic = Util.mk_stat "ho.eq_res_syntactic.steps"
let stat_ext_neg_lit = Util.mk_stat "ho.extensionality-.steps"
let stat_ext_pos = Util.mk_stat "ho.extensionality+.steps"
let stat_beta = Util.mk_stat "ho.beta_reduce.steps"
let stat_eta_normalize = Util.mk_stat "ho.eta_normalize.steps"
let stat_prim_enum = Util.mk_stat "ho.prim_enum.steps"
let stat_elim_pred = Util.mk_stat "ho.elim_pred.steps"
let stat_ho_unif = Util.mk_stat "ho.unif.calls"
let stat_ho_unif_steps = Util.mk_stat "ho.unif.steps"
let stat_neg_ext = Util.mk_stat "ho.neg_ext_success"
let prof_eq_res = ZProf.make "ho.eq_res"
let prof_eq_res_syn = ZProf.make "ho.eq_res_syntactic"
let prof_ho_unif = ZProf.make "ho.unif"
let k_ext_pos = Flex_state.create_key ()
let k_ext_pos_all_lits = Flex_state.create_key ()
let k_ext_axiom = Flex_state.create_key ()
let k_choice_axiom = Flex_state.create_key ()
let k_elim_pred_var = Flex_state.create_key ()
let k_ext_neg_lit = Flex_state.create_key ()
let k_neg_ext = Flex_state.create_key ()
let k_neg_ext_as_simpl = Flex_state.create_key ()
let k_ext_axiom_penalty = Flex_state.create_key ()
let k_instantiate_choice_ax = Flex_state.create_key ()
let k_elim_leibniz_eq = Flex_state.create_key ()
let k_prune_arg_fun = Flex_state.create_key ()
let k_prim_enum_terms = Flex_state.create_key ()
let k_simple_projection = Flex_state.create_key ()
let k_simple_projection_md = Flex_state.create_key ()
let k_check_lambda_free = Flex_state.create_key ()
let k_purify_applied_vars = Flex_state.create_key()
let k_eta = Flex_state.create_key()
let k_diff_const = Flex_state.create_key()
let k_use_diff_for_neg_ext = Flex_state.create_key()
let k_generalize_choice_trigger = Flex_state.create_key ()
let k_prim_enum_simpl = Flex_state.create_key ()
let k_prim_enum_early_bird = Flex_state.create_key ()
let k_resolve_flex_flex = Flex_state.create_key ()
type prune_kind = [`NoPrune | `OldPrune | `PruneAllCovers | `PruneMaxCover]
module type S = sig
module Env : Env.S
module C : module type of Env.C
(** {5 Registration} *)
val setup : unit -> unit
(** Register rules in the environment *)
val prim_enum_tf : Env.C.t -> Env.C.t list
end
let k_some_ho : bool Flex_state.key = Flex_state.create_key()
let k_enabled : bool Flex_state.key = Flex_state.create_key()
let k_enable_def_unfold : bool Flex_state.key = Flex_state.create_key()
let k_enable_ho_unif : bool Flex_state.key = Flex_state.create_key()
let k_ho_prim_mode :
[ `Combinators | `And | `Or | `Neg | `Quants | `Eq | `TF
| `Full | `Pragmatic | `Simple | `None ] Flex_state.key
= Flex_state.create_key()
let k_ho_prim_max_penalty : int Flex_state.key = Flex_state.create_key()
let k_ground_app_vars : [ `Off | `Fresh | `All ] Flex_state.key = Flex_state.create_key()
module Make(E : Env.S) : S with module Env = E = struct
module Env = E
module C = Env.C
module Ctx = Env.Ctx
module Combs = Combinators.Make(E)
module FV_ext_neg_lit = FV_tree.Make(struct
type t = Literal.t * T.t list
let compare = CCOrd.(pair Literal.compare (list T.compare))
let to_lits (l,_) = Iter.return (Literal.Conv.to_form l)
let labels _ = Util.Int_set.empty
end)
let idx_ext_neg_lit_ : FV_ext_neg_lit.t ref = ref (FV_ext_neg_lit.empty())
let find_skolems_ (lit:Literal.t) : T.t list option =
FV_ext_neg_lit.retrieve_alpha_equiv_c !idx_ext_neg_lit_ (lit, [])
|> Iter.find_map
(fun (lit',skolems) ->
let subst = Literal.variant (lit',0) (lit,1) |> Iter.head in
begin match subst with
| Some (subst,_) ->
let skolems =
List.map
(fun t -> Subst.FO.apply Subst.Renaming.none subst (t,0))
skolems
in
Some skolems
| None -> None
end)
let remove_ff_constraints c =
let module VS = Term.VarSet in
let extract_hd_vars = function
| Literal.Equation(lhs,rhs,false) ->
VS.of_list [T.as_var_exn (T.head_term lhs);
T.as_var_exn (T.head_term rhs)]
| _ -> assert false
in
let is_neg_ff = function
| Literal.Equation(lhs,rhs,false) ->
T.is_var (T.head_term lhs) &&
T.is_var (T.head_term rhs)
| _ -> false
in
let blocked_vars =
CCArray.filter (fun l -> not @@ is_neg_ff l) (C.lits c)
|> Literals.vars
|> VS.of_list
in
let vars_to_remove,_ =
CCArray.fold (fun ((allowed, bl) as acc) lit ->
match lit with
| Literal.Equation(lhs, rhs, false) when is_neg_ff lit ->
let l_var,r_var = CCPair.map_same (fun t -> T.as_var_exn (T.head_term t)) (lhs,rhs) in
if not (VS.mem l_var bl) && not (VS.mem r_var bl) then (
(VS.add_list allowed [l_var;r_var]), bl
) else (
if VS.mem l_var allowed || VS.mem r_var allowed then (
VS.empty, VS.add_list (VS.union allowed bl) [l_var;r_var]
) else (allowed, VS.add_list bl [l_var; r_var])
)
| _ -> acc
) (VS.empty, blocked_vars) (C.lits c) in
if VS.is_empty vars_to_remove then (SimplM.return_same c)
else (
let new_lits =
CCArray.to_list (C.lits c)
|> CCList.filter_map (fun lit ->
if is_neg_ff lit then (
let c_vars = extract_hd_vars lit in
if VS.subset c_vars vars_to_remove then None
else Some lit
) else Some lit) in
let proof =
Proof.Step.simp
~tags:[Proof.Tag.T_ho]
~rule:(Proof.Rule.mk "remove_ff_constraints")
[C.proof_parent c] in
let cl = C.create ~penalty:(C.penalty c) ~trail:(C.trail c) new_lits proof in
SimplM.return_new cl
)
let rec declare_skolems = function
| [] -> ()
| (sym,id) :: rest -> Ctx.declare sym id; declare_skolems rest
let ext_neg_lit (lit:Literal.t) : _ option = match lit with
| Literal.Equation (f, g, false)
when Type.is_fun (T.ty f) &&
not (T.is_var f) &&
not (T.is_var g) &&
not (T.equal f g) ->
let n_ty_params, ty_args, _ = Type.open_poly_fun (T.ty f) in
assert (n_ty_params=0);
let params = match find_skolems_ lit with
| Some l -> l
| None ->
let vars = Literal.vars lit in
let skolems = ref [] in
let l = List.map (fun ty ->
let sk, res = T.mk_fresh_skolem vars ty in
skolems := sk :: !skolems;
res) ty_args in
declare_skolems !skolems;
idx_ext_neg_lit_ := FV_ext_neg_lit.add !idx_ext_neg_lit_ (lit,l);
l
in
let new_lit = Literal.mk_neq (T.app f params) (T.app g params) in
Util.incr_stat stat_ext_neg_lit;
Util.debugf ~section 4
"(@[ho_ext_neg_lit@ :old `%a`@ :new `%a`@])"
(fun k->k Literal.pp lit Literal.pp new_lit);
Some (new_lit,[],[Proof.Tag.T_ho; Proof.Tag.T_ext])
| _ -> None
let ext_pos ?(only_unit=true) (c:C.t): C.t list =
let is_eligible = C.Eligible.always in
if not only_unit || C.lits c |> CCArray.length = 1 then
C.lits c
|> CCArray.mapi (fun i l ->
let l = Literal.map (fun t -> Lambda.eta_reduce ~full:true t) l in
match l with
| Literal.Equation (t1,t2,true)
when is_eligible i l ->
let f1, l1 = T.as_app t1 in
let f2, l2 = T.as_app t2 in
begin match List.rev l1, List.rev l2 with
| last1 :: l1, last2 :: l2 ->
begin match T.view last1, T.view last2 with
| T.Var x, T.Var y
when HVar.equal Type.equal x y &&
not (Type.is_tType (HVar.ty x)) &&
Iter.of_list
[Iter.doubleton f1 f2;
Iter.of_list l1;
Iter.of_list l2]
|> Iter.flatten
|> Iter.flat_map T.Seq.vars
|> Iter.for_all
(fun v' -> not (HVar.equal Type.equal v' x)) ->
let new_lit =
Literal.mk_eq
(T.app f1 (List.rev l1))
(T.app f2 (List.rev l2))
in
let new_lits = C.lits c |> CCArray.to_list |>
List.mapi (fun j l -> if i = j then new_lit else l) in
let proof =
Proof.Step.inference [C.proof_parent c]
~rule:(Proof.Rule.mk "ho_ext_pos")
~tags:[Proof.Tag.T_ho; Proof.Tag.T_ext]
in
let new_c =
C.create new_lits proof ~penalty:(C.penalty c) ~trail:(C.trail c)
in
Util.incr_stat stat_ext_pos;
Util.debugf ~section 4
"(@[ext_pos@ :clause %a@ :yields %a@])"
(fun k->k C.pp c C.pp new_c);
Some new_c
| _,_ -> None
end
| _ -> None
end
| _ -> None)
|> CCArray.filter_map (fun x -> x)
|> CCArray.to_list
else []
let ext_pos_general ?(all_lits = false) (c:C.t) : C.t list =
let eligible = if all_lits then C.Eligible.always else C.Eligible.param c in
let rec ext_pos_lit t s other_lits =
let f, tt = T.as_app t in
let g, ss = T.as_app s in
begin match List.rev tt, List.rev ss with
| last_t :: tl_rev_t, last_s :: tl_rev_s ->
if T.equal last_t last_s && not (T.is_type last_t) then
match T.as_var last_t with
| Some v ->
if not (T.var_occurs ~var:v f)
&& not (T.var_occurs ~var:v g)
&& not (List.exists (T.var_occurs ~var:v) tl_rev_t)
&& not (List.exists (T.var_occurs ~var:v) tl_rev_s)
&& not (List.exists (Literal.var_occurs v) other_lits)
then (
let butlast = (fun l -> CCList.take (List.length l - 1) l) in
let t' = T.app f (butlast tt) in
let s' = T.app g (butlast ss) in
Literal.mk_eq t' s'
:: ext_pos_lit t' s' other_lits
)
else
[]
| None -> []
else []
| _ -> []
end
in
let new_clauses =
C.lits c
|> Iter.of_array |> Util.seq_zipi
|> Iter.filter (fun (idx,lit) -> eligible idx lit)
|> Iter.flat_map_l
(fun (lit_idx,lit) ->
let lit = Literal.map (fun t -> Lambda.eta_reduce t) lit in
match lit with
| Literal.Equation (t, s, true) ->
ext_pos_lit t s (CCArray.except_idx (C.lits c) lit_idx)
|> Iter.of_list
|> Iter.flat_map_l
(fun new_lit ->
let new_lits = new_lit :: CCArray.except_idx (C.lits c) lit_idx in
let proof =
Proof.Step.inference [C.proof_parent c]
~rule:(Proof.Rule.mk "ho_ext_pos_general")
~tags:[Proof.Tag.T_ho; Proof.Tag.T_ext]
in
let new_c =
C.create new_lits proof ~penalty:(C.penalty c) ~trail:(C.trail c)
in
[new_c])
|> Iter.to_list
| _ -> [])
|> Iter.to_rev_list
in
if new_clauses<>[] then (
Util.debugf ~section 4
"(@[ext-pos-general-eq@ :clause %a@ :yields (@[<hv>%a@])@])"
(fun k->k C.pp c (Util.pp_list ~sep:" " C.pp) new_clauses);
);
new_clauses
let neg_ext (c:C.t) : C.t list =
let diff_const = Env.flex_get k_diff_const in
let rec app_diff_exhaustively s t = function
| alpha :: beta :: [] ->
let diff_s_t = T.app diff_const [T.of_ty alpha;T.of_ty beta;s;t] in
T.app s [diff_s_t], T.app t [diff_s_t]
| alpha :: ((beta :: rest) as xs) ->
assert(not @@ CCList.is_empty rest);
let rest', ret = CCList.take_drop (List.length rest -1) rest in
assert(List.length ret = 1);
let new_beta = Type.arrow (beta::rest') (List.hd ret) in
let diff_s_t = T.app diff_const [T.of_ty alpha;T.of_ty new_beta;s;t] in
app_diff_exhaustively (T.app s [diff_s_t]) (T.app t [diff_s_t]) xs
| _ -> invalid_arg "argument must be a function type" in
let is_eligible = C.Eligible.res c in
C.lits c
|> CCArray.mapi (fun i l ->
match l with
| Literal.Equation (lhs,rhs,false)
when is_eligible i l && Type.is_fun @@ T.ty lhs ->
let arg_types, ret_ty = Type.open_fun (T.ty lhs) in
let free_vars = Literal.vars l in
let skolem_decls = ref [] in
let new_lits = CCList.map (fun (j,x) ->
if i!=j then x
else (
let lhs,rhs =
if Env.flex_get k_use_diff_for_neg_ext then (
app_diff_exhaustively lhs rhs (arg_types @ [ret_ty])
)
else (
let skolems = List.map (fun ty ->
let sk, res = T.mk_fresh_skolem free_vars ty in
skolem_decls := sk :: !skolem_decls;
res) arg_types in
T.app lhs skolems, T.app rhs skolems) in
Literal.mk_neq lhs rhs)
) (C.lits c |> Array.mapi (fun j x -> (j,x)) |> Array.to_list) in
declare_skolems !skolem_decls;
let proof =
Proof.Step.inference [C.proof_parent c]
~rule:(Proof.Rule.mk "neg_ext")
~tags:[Proof.Tag.T_ho; Proof.Tag.T_ext; Proof.Tag.T_dont_increase_depth]
in
let new_c =
C.create new_lits proof ~penalty:(C.penalty c) ~trail:(C.trail c) in
Util.debugf 1 ~section "NegExt: @[%a@] => @[%a@].\n" (fun k -> k C.pp c C.pp new_c);
Util.incr_stat stat_neg_ext;
Some new_c
| _ -> None)
|> CCArray.filter_map (fun x -> x)
|> CCArray.to_list
let neg_ext_simpl (c:C.t) : C.t SimplM.t =
let is_eligible = C.Eligible.res c in
let applied_neg_ext = ref false in
let new_lits =
C.lits c
|> CCArray.mapi (fun i l ->
match l with
| Literal.Equation (lhs,rhs,false)
when is_eligible i l && Type.is_fun @@ T.ty lhs ->
let arg_types = Type.expected_args @@ T.ty lhs in
let free_vars = Literal.vars l |> T.VarSet.of_list |> T.VarSet.to_list in
let skolem_decls = ref [] in
let skolems = List.map (fun ty ->
let sk, res = T.mk_fresh_skolem free_vars ty in
skolem_decls := sk :: !skolem_decls;
res) arg_types in
applied_neg_ext := true;
declare_skolems !skolem_decls;
Literal.mk_neq (T.app lhs skolems) (T.app rhs skolems)
| _ -> l) in
if not !applied_neg_ext then SimplM.return_same c
else (
let proof =
Proof.Step.simp [C.proof_parent c]
~rule:(Proof.Rule.mk "neg_ext_simpl")
~tags:[Proof.Tag.T_ho; Proof.Tag.T_ext] in
let c' = C.create_a ~trail:(C.trail c) ~penalty:(C.penalty c) new_lits proof in
SimplM.return_new c'
)
let elim_pred_variable ?(proof_constructor=Proof.Step.inference) (c:C.t) : C.t list =
let find_vars(): _ HVar.t Iter.t =
Literals.vars (C.lits c)
|> CCList.to_iter
|> Iter.filter
(fun v ->
(Type.is_prop @@ Type.returns @@ HVar.ty v) &&
not (Literals.is_shielded v (C.lits c)))
and gather_lits v =
try
Array.fold_left
(fun (others,set,pos_lits) lit ->
begin match lit with
| Literal.Equation (lhs, rhs, sign) when T.equal rhs T.true_->
let f, args = T.as_app lhs in
begin match T.view f with
| T.Var q when HVar.equal Type.equal v q ->
if List.exists (T.var_occurs ~var:v) args then (
raise Exit;
);
others, (args, sign) :: set,
(if sign then [lit] else []) @ pos_lits
| _ -> lit :: others, set, pos_lits
end
| _ -> lit :: others, set, pos_lits
end)
([], [], [])
(C.lits c)
|> CCOpt.return
with Exit -> None
in
let try_elim_var v: _ option =
begin match gather_lits v with
| None
| Some (_, [], _) -> None
| Some (other_lits, constr_l, pos_lits) ->
let pos_args, neg_args =
CCList.partition_map
(fun (args,sign) -> if sign then `Left args else `Right args)
constr_l
in
let subst =
let some_tup = match pos_args, neg_args with
| tup :: _, _ | _, tup :: _ -> tup
| [], [] -> assert false
in
let offset = C.Seq.vars c |> T.Seq.max_var |> succ in
let vars =
List.mapi (fun i t -> HVar.make ~ty:(T.ty t) (i+offset)) some_tup
in
let vars_t = List.map T.var vars in
let body =
neg_args
|> List.map
(fun tup ->
assert (List.length tup = List.length vars);
List.map2 T.Form.eq vars_t tup |> T.Form.and_l)
|> T.Form.or_l
in
Util.debugf ~section 1
"(@[elim-pred-with@ (@[@<1>λ @[%a@].@ %a@])@])"
(fun k->k (Util.pp_list ~sep:" " Type.pp_typed_var) vars T.pp body);
Util.incr_stat stat_elim_pred;
let t = T.fun_of_fvars vars body in
Subst.FO.of_list [((v:>InnerTerm.t HVar.t),0), (t,0)]
in
let renaming = Subst.Renaming.create () in
let new_lits =
let l1 = Literal.apply_subst_list renaming subst (other_lits,0) in
let l2 = Literal.apply_subst_list renaming subst (pos_lits,0) in
l1 @ l2
in
let proof =
proof_constructor ~rule:(Proof.Rule.mk "ho_elim_pred") ~tags:[Proof.Tag.T_ho]
[ C.proof_parent_subst renaming (c,0) subst ]
in
let new_c =
C.create new_lits proof
~penalty:(C.penalty c) ~trail:(C.trail c)
in
Util.debugf ~section 3
"(@[<2>elim_pred_var %a@ :clause %a@ :yields %a@])"
(fun k->k T.pp_var v C.pp c C.pp new_c);
Some new_c
end
in
begin
find_vars()
|> Iter.filter_map try_elim_var
|> Iter.to_rev_list
end
let max_penalty_prim_ = Env.flex_get k_ho_prim_max_penalty
let prim_enum_ ?(proof_constructor = Proof.Step.inference) ~(mode) (c:C.t) : C.t list =
let vars =
Literals.fold_eqn ~both:false ~ord:(Ctx.ord()) ~eligible:(C.Eligible.always)
(C.lits c)
|> Iter.flat_map_l (fun (l,r,_,_) ->
let extract_var t =
let _, body = T.open_fun t in
match T.view body with
| T.Var x -> Some x
| T.App(hd, _) when T.is_var hd -> Some (T.as_var_exn hd)
| _ -> None in
CCOpt.to_list (extract_var l) @ CCOpt.to_list (extract_var r))
|> Iter.filter (fun v -> Type.returns_prop @@ HVar.ty v)
|> T.VarSet.of_iter
in
if not (T.VarSet.is_empty vars) then (
Util.debugf ~section 1 "(@[<hv2>ho.refine@ :clause %a@ :terms {@[%a@]}@])"
(fun k->k C.pp c (Util.pp_iter T.pp_var) (T.VarSet.to_iter vars));
);
let sc_c = 0 in
let offset = C.Seq.vars c |> T.Seq.max_var |> succ in
begin
vars
|> T.VarSet.to_iter
|> Iter.flat_map_l
(fun v -> HO_unif.enum_prop
~enum_cache:(Env.flex_get k_prim_enum_terms)
~signature:(Ctx.signature ())
~mode ~offset (v,sc_c))
|> Iter.map
(fun (subst,penalty) ->
let penalty = if Env.flex_get k_prim_enum_simpl then 0 else penalty in
let renaming = Subst.Renaming.create() in
let lits = Literals.apply_subst renaming subst (C.lits c,sc_c) in
let proof =
Proof.Step.inference ~rule:(Proof.Rule.mk "ho.refine") ~tags:[Proof.Tag.T_ho]
[C.proof_parent_subst renaming (c,sc_c) subst]
in
let new_c =
C.create_a lits proof
~penalty:(C.penalty c + penalty) ~trail:(C.trail c)
in
Util.debugf ~section 1
"(@[<hv2>ho.refine@ :from %a@ :subst %a@ :yields %a@])"
(fun k->k C.pp c Subst.pp subst C.pp new_c);
Util.incr_stat stat_prim_enum;
new_c)
|> Iter.to_rev_list
end
let prim_enum ~(mode) c =
if C.proof_depth c < max_penalty_prim_
then prim_enum_ ~mode c
else []
let prim_enum_tf c =
prim_enum_ ~mode:`TF c
let choice_ops = ref Term.Map.empty
let new_choice_counter = ref 0
let insantiate_choice ?(proof_constructor=Proof.Step.inference) ?(inst_vars=true) ?(choice_ops=choice_ops) c =
let max_var =
ref ((C.Seq.vars c |> Iter.map HVar.id
|> Iter.max |> CCOpt.get_or ~default: 0) + 1) in
let is_choice_subterm t =
match T.view t with
| T.App(hd, [arg]) when T.is_var hd || Term.Map.mem hd !choice_ops ->
let ty = T.ty arg in
Type.is_fun ty && List.length (Type.expected_args ty) = 1 &&
Type.equal (Term.ty t) (List.hd (Type.expected_args ty)) &&
Type.returns_prop ty && T.DB.is_closed t
| _ -> false in
let neg_trigger t =
assert(T.DB.is_closed t);
let arg_ty = List.hd (Type.expected_args (T.ty t)) in
let negated = T.Form.not_ (Lambda.whnf (T.app t [T.bvar ~ty:arg_ty 0])) in
let res = T.fun_ arg_ty negated in
assert(T.DB.is_closed res);
res in
let generalize_trigger t =
assert(T.DB.is_closed t);
let arg_ty = List.hd (Type.expected_args (T.ty t)) in
let applied_to_0 = Lambda.whnf (T.app t [T.bvar ~ty:arg_ty 0]) in
let ty = Type.arrow [Type.prop] Type.prop in
let fresh_var = T.var @@ HVar.fresh_cnt ~counter:max_var ~ty () in
let res = T.fun_ arg_ty (T.app fresh_var [applied_to_0]) in
assert(T.DB.is_closed res);
res in
let choice_inst_of_hd ~def_clause hd arg =
let arg_ty = Term.ty arg in
let ty = List.hd (Type.expected_args arg_ty) in
let x = T.var @@ HVar.fresh_cnt ~counter:max_var ~ty () in
let choice_x = Lambda.whnf (T.app arg [x]) in
let choice_arg = Lambda.snf (T.app arg [T.app hd [arg]]) in
let new_lits = [Literal.mk_prop choice_x false;
Literal.mk_prop choice_arg true] in
let arg_str = CCFormat.sprintf "(%a)" T.TPTP.pp arg in
let parents = match def_clause with
| Some def -> [C.proof_parent def; C.proof_parent c]
| None -> [C.proof_parent c] in
let proof =
proof_constructor ~tags:[Proof.Tag.T_cannot_orphan]
~rule:(Proof.Rule.mk ("inst_choice" ^ arg_str))
parents in
C.create ~penalty:1 ~trail:Trail.empty new_lits proof in
let new_choice_op ty =
let choice_ty_name = "#_choice_" ^
CCInt.to_string (CCRef.get_then_incr new_choice_counter) in
let new_ch_id = ID.make choice_ty_name in
let new_ch_const = T.const new_ch_id ~ty in
Ctx.add_signature (Signature.declare (C.Ctx.signature ()) new_ch_id ty);
Util.debugf 1 "new choice for type %a: %a(%a).\n"
(fun k -> k Type.pp ty T.pp new_ch_const Type.pp (T.ty new_ch_const));
choice_ops := Term.Map.add new_ch_const None !choice_ops;
new_ch_const in
let generate_instances ~def_clause hd arg =
choice_inst_of_hd ~def_clause hd arg
:: choice_inst_of_hd ~def_clause hd (neg_trigger arg)
:: (if not @@ Env.flex_get k_generalize_choice_trigger then []
else [choice_inst_of_hd ~def_clause hd (generalize_trigger arg)]) in
let build_choice_inst t =
match T.view t with
| T.App(hd, [arg]) ->
if Term.is_var hd && inst_vars then (
let hd_ty = Term.ty hd in
let choice_ops =
Term.Map.filter (fun t _ -> Type.equal (Term.ty t) hd_ty) !choice_ops
|> Term.Map.to_list
|> (fun l -> if CCList.is_empty l then [new_choice_op hd_ty, None] else l) in
CCList.flat_map (fun (hd,def_clause) -> generate_instances ~def_clause hd arg)
choice_ops
) else (
match Term.Map.find_opt hd !choice_ops with
| Some def_clause -> generate_instances ~def_clause hd arg
| None -> [])
| _ -> assert (false) in
C.Seq.terms c
|> Iter.flat_map (Term.Seq.subterms ~include_builtin:true)
|> Iter.filter is_choice_subterm
|> Iter.flat_map_l build_choice_inst
|> Iter.to_list
let simple_projection ~penalty_inc ~max_depth c =
if C.proof_depth c > max_depth then []
else (
C.Seq.terms c
|> Iter.flat_map (Term.Seq.subterms ~include_builtin:true ~ignore_head:true)
|> Iter.fold (fun var_map subterm ->
match T.view subterm with
| T.App(hd, args) when T.is_var hd &&
List.exists (fun t ->
not (Type.is_fun (T.ty t)) &&
not (T.is_bvar t))
args ->
let var_arg_tys, var_ret_ty = Type.open_fun (T.ty hd) in
assert (not (Type.is_fun var_ret_ty));
let new_bindings =
CCList.foldi (fun acc idx arg ->
let arg_ty = T.ty arg in
if Type.is_ground arg_ty && Type.equal arg_ty var_ret_ty then (
let db = T.bvar ~ty:arg_ty (List.length var_arg_tys - 1 - idx) in
let binding = T.fun_l var_arg_tys db in
Term.Set.add binding acc
) else acc) Term.Set.empty args in
let old_bindings = Term.Map.get_or hd ~default:Term.Set.empty var_map in
Term.Map.add hd (Term.Set.union old_bindings new_bindings) var_map
| _ -> var_map) Term.Map.empty
|> (fun var_map ->
Term.Map.fold (fun var bindings acc ->
let var = Term.as_var_exn var in
Term.Set.fold (fun binding acc ->
let subst = (Subst.FO.bind' Subst.empty (var, 0) (binding, 0)) in
let proof =
Some
(Proof.Step.inference ~rule:(Proof.Rule.mk "simp.projection")
~tags:[Proof.Tag.T_ho]
[C.proof_parent_subst Subst.Renaming.none (c,0) subst]) in
let res = C.apply_subst ~penalty_inc:(Some penalty_inc) ~proof (c,0) subst in
res :: acc
) bindings acc
) var_map []))
let recognize_choice_ops c =
let extract_not_p_x l = match l with
| Literal.Equation(lhs,rhs,false) when T.equal T.true_ rhs && T.is_app_var lhs ->
begin match T.view lhs with
| T.App(hd, [var]) when T.is_var var -> Some hd
| _ -> None end
| _ -> None in
let extract_p_choice_p p l = match l with
| Literal.Equation(lhs,rhs,true) when T.equal T.true_ rhs ->
begin match T.view lhs with
| T.App(hd, [ch_p]) when T.equal hd p ->
begin match T.view ch_p with
| T.App(sym, [var]) when T.is_const sym && T.equal var p -> Some sym
| _ -> None end
| _ -> None end
| _ -> None in
if C.length c == 2 then (
let px = CCArray.find_map extract_not_p_x (C.lits c) in
match px with
| Some p ->
let p_ch_p = CCArray.find_map (extract_p_choice_p p) (C.lits c) in
begin match p_ch_p with
| Some sym ->
if not (Term.Map.mem sym !choice_ops) then (
choice_ops := Term.Map.add sym (Some c) !choice_ops;
let new_cls =
Env.get_active ()
|> Iter.flat_map_l (fun pas_cl ->
if C.id pas_cl = C.id c then []
else (
insantiate_choice ~inst_vars:false
~choice_ops:(ref (Term.Map.singleton sym (Some c)))
pas_cl
))
|> Iter.map Combs.maybe_conv_lams
in
Env.add_passive new_cls);
C.mark_redundant c;
true
| None -> false end
| None -> false
) else false
let elim_leibniz_eq_ ?(proof_constructor=Proof.Step.inference) c =
let ord = Env.ord () in
let eligible = C.Eligible.always in
let pos_pred_vars, neg_pred_vars, occurrences =
Lits.fold_eqn ~both:false ~ord ~eligible (C.lits c)
|> Iter.fold (fun (pos_vs,neg_vs,occ) (lhs,rhs,sign,_) ->
if Type.is_prop (Term.ty lhs) && Term.is_app_var lhs && T.equal T.true_ rhs then (
let var_hd = Term.as_var_exn (Term.head_term lhs) in
if sign then (Term.VarSet.add var_hd pos_vs, neg_vs, Term.Map.add lhs true occ)
else (pos_vs, Term.VarSet.add var_hd neg_vs, Term.Map.add lhs false occ)
) else (pos_vs, neg_vs, occ)
) (Term.VarSet.empty,Term.VarSet.empty,Term.Map.empty) in
let pos_neg_vars = Term.VarSet.inter pos_pred_vars neg_pred_vars in
let res =
if Term.VarSet.is_empty pos_neg_vars then []
else (
CCList.flat_map (fun (t,sign) ->
let hd, args = T.as_app t in
let var_hd = T.as_var_exn hd in
if Term.VarSet.mem (Term.as_var_exn hd) pos_neg_vars then (
let tyargs, _ = Type.open_fun (Term.ty hd) in
let n = List.length tyargs in
CCList.filter_map (fun (i,arg) ->
if T.var_occurs ~var:var_hd arg then None
else (
let body = (if sign then T.Form.neq else T.Form.eq)
arg (T.bvar ~ty:(T.ty arg) (n-i-1)) in
let subs_term = T.fun_l tyargs body in
(let cached_t = Subst.FO.canonize_all_vars subs_term in
E.flex_add k_prim_enum_terms
(ref (Term.Set.add cached_t !(Env.flex_get k_prim_enum_terms))));
let subst = Subst.FO.bind' (Subst.empty) (var_hd, 0) (subs_term, 0) in
let rule = Proof.Rule.mk ("elim_leibniz_eq_" ^ (if sign then "+" else "-")) in
let tags = [Proof.Tag.T_ho] in
let proof = Some (proof_constructor ~rule ~tags [C.proof_parent c]) in
Some (C.apply_subst ~proof (c,0) subst))
) (CCList.mapi (fun i arg -> (i, arg)) args)
) else []
) (Term.Map.to_list occurrences)) in
res
let elim_leibniz_equality c =
if C.proof_depth c < Env.flex_get k_elim_leibniz_eq then (
elim_leibniz_eq_ c
) else []
let pp_pairs_ out =
let open CCFormat in
Format.fprintf out "(@[<hv>%a@])"
(Util.pp_list ~sep:" " @@ hvbox @@ HO_unif.pp_pair)
let ho_unif_real_ c pairs other_lits : C.t list =
Util.debugf ~section 5
"(@[ho_unif.try@ :pairs (@[<hv>%a@])@ :other_lits %a@])"
(fun k->k pp_pairs_ pairs (Util.pp_list~sep:" " Literal.pp) other_lits);
Util.incr_stat stat_ho_unif;
let offset = C.Seq.vars c |> T.Seq.max_var |> succ in
begin
HO_unif.unif_pairs ?fuel:None (pairs,0) ~offset
|> List.map
(fun (new_pairs, us, penalty, renaming) ->
let subst = Unif_subst.subst us in
let c_guard = Literal.of_unif_subst renaming us in
let new_pairs =
List.map
(fun (env,t,u) ->
assert (env == []);
Literal.mk_constraint t u)
new_pairs
and other_lits =
Literal.apply_subst_list renaming subst (other_lits,0)
in
let all_lits = c_guard @ new_pairs @ other_lits in
let proof =
Proof.Step.inference ~rule:(Proof.Rule.mk "ho_unif") ~tags:[Proof.Tag.T_ho]
[C.proof_parent_subst renaming (c,0) subst]
in
let new_c =
C.create all_lits proof
~trail:(C.trail c) ~penalty:(C.penalty c + penalty)
in
Util.debugf ~section 5
"(@[ho_unif.step@ :pairs (@[%a@])@ :subst %a@ :yields %a@])"
(fun k->k pp_pairs_ pairs Subst.pp subst C.pp new_c);
Util.incr_stat stat_ho_unif_steps;
new_c
)
end
let ho_unif (c:C.t) : C.t list =
if C.lits c |> CCArray.exists Literal.is_ho_constraint then (
let pairs, others =
C.lits c
|> Array.to_list
|> CCList.partition_map
(function
| Literal.Equation (t,u, false) as lit
when Literal.is_ho_constraint lit -> `Left ([],t,u)
| lit -> `Right lit)
in
assert (pairs <> []);
ZProf.enter_prof prof_ho_unif;
let r = ho_unif_real_ c pairs others in
ZProf.exit_prof prof_ho_unif;
r
) else []
let beta_reduce t =
let t' = Lambda.snf t in
if (T.equal t t') then (
Util.debugf ~section 50 "(@[beta_reduce `%a`@ failed `@])" (fun k->k T.pp t );
None)
else (
Util.debugf ~section 50 "(@[beta_reduce `%a`@ :into `%a`@])"
(fun k->k T.pp t T.pp t');
Util.incr_stat stat_beta;
Some t'
)
let eta_normalize () = match Env.flex_get k_eta with
| `Reduce -> Lambda.eta_reduce ~full:true
| `Expand -> Lambda.eta_expand
| `None -> (fun t -> t)
let eta_normalize t =
let t' = eta_normalize () t in
if (T.equal t t') then (
Util.debugf ~section 50 "(@[eta_normalize `%a`@ failed `@])" (fun k->k T.pp t );
None)
else (
Util.debugf ~section 50 "(@[eta_normalize `%a`@ :into `%a`@])"
(fun k->k T.pp t T.pp t');
Util.incr_stat stat_eta_normalize;
Some t'
)
module TVar = struct
type t = Type.t HVar.t
let equal = HVar.equal Type.equal
let hash = HVar.hash
let compare = HVar.compare Type.compare
end
module VarTermMultiMap = CCMultiMap.Make (TVar) (Term)
module VTbl = CCHashtbl.Make(TVar)
let mk_diff_const () =
let diff_id = ID.make("zf_ext_diff") in
ID.set_payload diff_id (ID.Attr_skolem ID.K_normal);
let alpha_var = HVar.make ~ty:Type.tType 0 in
let alpha = Type.var alpha_var in
let beta_var = HVar.make ~ty:Type.tType 1 in
let beta = Type.var beta_var in
let alpha_to_beta = Type.arrow [alpha] beta in
let diff_type = Type.forall_fvars [alpha_var;beta_var] (Type.arrow [alpha_to_beta; alpha_to_beta] alpha) in
let diff = Term.const ~ty:diff_type diff_id in
Env.Ctx.declare diff_id diff_type;
Env.flex_add k_diff_const diff
let mk_extensionality_clause () =
let diff = Env.flex_get k_diff_const in
let alpha_var = HVar.make ~ty:Type.tType 0 in
let alpha = Type.var alpha_var in
let beta_var = HVar.make ~ty:Type.tType 1 in
let beta = Type.var beta_var in
let alpha_to_beta = Type.arrow [alpha] beta in
let x = Term.var (HVar.make ~ty:alpha_to_beta 2) in
let y = Term.var (HVar.make ~ty:alpha_to_beta 3) in
let x_diff = Term.app x [Term.app diff [T.of_ty alpha; T.of_ty beta; x; y]] in
let y_diff = Term.app y [Term.app diff [T.of_ty alpha; T.of_ty beta; x; y]] in
let lits = [Literal.mk_eq x y; Literal.mk_neq x_diff y_diff] in
Env.C.create ~penalty:(Env.flex_get k_ext_axiom_penalty) ~trail:Trail.empty
lits Proof.Step.trivial
let choice_clause =
let choice_id = ID.make("zf_choice") in
let alpha_var = HVar.make ~ty:Type.tType 0 in
let alpha = Type.var alpha_var in
let alpha_to_prop = Type.arrow [alpha] Type.prop in
let choice_type = Type.arrow [alpha_to_prop] alpha in
let choice = Term.const ~ty:choice_type choice_id in
let p = Term.var (HVar.make ~ty:alpha_to_prop 1) in
let x = Term.var (HVar.make ~ty:alpha 2) in
let px = Term.app p [x] in
let p_choice = Term.app p [Term.app choice [p]] in
let lits = [Literal.mk_prop px false; Literal.mk_prop p_choice true] in
Env.C.create ~penalty:1 ~trail:Trail.empty lits Proof.Step.trivial
let early_bird_prim_enum cl var =
assert(T.is_var var);
let offset = C.Seq.vars cl |> T.Seq.max_var |> succ in
let mode = Env.flex_get k_ho_prim_mode in
let sc = 0 in
HO_unif.enum_prop
~enum_cache:(Env.flex_get k_prim_enum_terms)
~signature:(Ctx.signature ()) ~mode ~offset (T.as_var_exn var,sc)
|> CCList.map (fun (subst,p) ->
let renaming = Subst.Renaming.create () in
let lits = Literals.apply_subst renaming subst (C.lits cl, sc) in
let lits = Literals.map (fun t -> Lambda.eta_reduce @@ Lambda.snf t) lits in
let proof =
Proof.Step.inference ~rule:(Proof.Rule.mk "ho.refine.early.bird")
~tags:[Proof.Tag.T_ho; Proof.Tag.T_cannot_orphan]
[C.proof_parent_subst renaming (cl, sc) subst] in
let res = C.create_a lits proof ~penalty:(C.penalty cl + (max (p-1) 0)) ~trail:(C.trail cl) in
let res = Combs.maybe_conv_lams res in
res)
|> CCList.to_iter
|> Env.add_passive
type fixed_arg_status =
| Always of T.t
| Varies
type dupl_arg_status =
| AlwaysSameAs of int
| Unique
(** Removal of fixed/duplicate arguments of variables.
- If within a clause, there exists a variable F that's always applied
to at least i arguments and the ith argument is always the same DB-free term,
we can systematically remove the argument (and repair F's type).
- If within a clause, there exist a variable F, and indices i < j
such that all occurrences of F are applied to at least j arguments and the
ith argument is syntactically same as the jth argument, we can
systematically remove the ith argument (and repair F's type accordingly).
*)
let prune_arg_old c =
let status : (fixed_arg_status * dupl_arg_status) list VTbl.t = VTbl.create 8 in
C.lits c
|> Literals.fold_terms ~vars:true ~ty_args:false ~which:`All ~ord:Ordering.none
~subterms:true ~eligible:(fun _ _ -> true)
|> Iter.iter
(fun (t,_) ->
let head, args = T.as_app t in
match T.as_var head with
| Some var ->
begin match VTbl.get status var with
| Some var_status ->
let update_fas fas arg =
match fas with
| Always u -> if T.equal u arg then Always u else Varies
| Varies -> Varies
in
let rec update_das das arg =
match das with
| AlwaysSameAs j ->
begin
try
if T.equal (List.nth args j) arg
then AlwaysSameAs j
else update_das (snd (List.nth var_status j)) (List.nth args j)
with Failure _ -> Unique
end
| Unique -> Unique
in
let minlen = min (List.length var_status) (List.length args) in
let args = CCList.take minlen args in
let var_status = CCList.take minlen var_status in
VTbl.replace status var (CCList.map (fun ((fas, das), arg) -> update_fas fas arg, update_das das arg) (List.combine var_status args))
| None ->
let rec create_var_status ?(i=0) args : (fixed_arg_status * dupl_arg_status) list =
match args with
| [] -> []
| arg :: args' ->
let fas =
if T.DB.is_closed arg then Always arg else Varies
in
let das = match CCList.find_idx ((Term.equal) arg) args' with
| Some (j, _) -> AlwaysSameAs (i + j + 1)
| None -> Unique
in
(fas, das) :: create_var_status ~i:(i+1) args'
in
VTbl.add status var (create_var_status args)
end
| None -> ()
;
()
);
let subst =
VTbl.to_list status
|> CCList.filter_map (
fun (var, var_status) ->
assert (not (Type.is_tType (HVar.ty var)));
let ty_args, ty_return = Type.open_fun (HVar.ty var) in
let keep = var_status |> CCList.map
(fun (fas, das) ->
fas == Varies && das == Unique
)
in
if CCList.for_all ((=) true) keep
then None
else (
let keep = CCList.(append keep (replicate (length ty_args - length keep) true)) in
let ty_args' = ty_args
|> CCList.combine keep
|> CCList.filter fst
|> CCList.map snd
in
let var' = HVar.cast var ~ty:(Type.arrow ty_args' ty_return) in
let bvars =
CCList.combine keep ty_args
|> List.mapi (fun i (k, ty) -> k, T.bvar ~ty (List.length ty_args - i - 1))
|> CCList.filter fst
|> CCList.map snd
in
let replacement = T.fun_l ty_args (T.app (T.var var') bvars) in
Some ((var,0), (replacement,1))
)
)
|> Subst.FO.of_list'
in
if Subst.is_empty subst
then SimplM.return_same c
else (
let renaming = Subst.Renaming.none in
let new_lits = Lits.apply_subst renaming subst (C.lits c, 0) in
let proof =
Proof.Step.simp
~rule:(Proof.Rule.mk "prune_arg")
~tags:[Proof.Tag.T_ho]
[C.proof_parent_subst renaming (c,0) subst] in
let c' = C.create_a ~trail:(C.trail c) ~penalty:(C.penalty c) new_lits proof in
Util.debugf ~section 3
"@[<>@[%a@]@ @[<2>prune_arg into@ @[%a@]@]@ with @[%a@]@]"
(fun k->k C.pp c C.pp c' Subst.pp subst);
SimplM.return_new c'
)
let prune_arg ~all_covers c =
let get_covers ?(current_sets=[]) head args =
let ty_args, _ = Type.open_fun (T.ty head) in
let missing = CCList.replicate (List.length ty_args - List.length args) None in
let args_opt = List.mapi (fun i a_i ->
assert(Term.DB.is_closed a_i);
assert(CCList.is_empty current_sets ||
List.length current_sets = (List.length args + List.length missing));
if CCList.is_empty current_sets ||
not (Term.Set.is_empty (List.nth current_sets i)) then
(Some (List.mapi (fun j a_j ->
if i = j then None else Some a_j) args))
else None )
args @ missing in
let res = List.mapi (fun i arg_opt ->
if i < List.length args then (
let t = List.nth args i in
begin match arg_opt with
| Some arg_l ->
let res_l = if all_covers then T.cover_with_terms t arg_l
else [t; T.max_cover t arg_l] in
T.Set.of_list res_l
| None -> Term.Set.empty
end)
else Term.Set.empty) args_opt in
res
in
let status = VTbl.create 8 in
let free_vars = Literals.vars (C.lits c) |> T.VarSet.of_list in
C.lits c
|> Literals.map (fun t -> Combs.expand t)
|> Literals.fold_terms ~vars:true ~ty_args:false ~which:`All ~ord:Ordering.none
~subterms:true ~eligible:(fun _ _ -> true)
|> Iter.iter
(fun (t,_) ->
let head, _ = T.as_app t in
match T.as_var head with
| Some var when T.VarSet.mem var free_vars ->
begin match VTbl.get status var with
| Some (current_sets, created_sk) ->
let t, new_sk = T.DB.skolemize_loosely_bound t in
let new_skolems = T.IntMap.bindings new_sk
|> List.map snd |> Term.Set.of_list in
let covers = get_covers ~current_sets head (T.args t) in
assert(List.length current_sets = List.length covers);
let paired = CCList.combine current_sets covers in
let res = List.map (fun (o,n) -> Term.Set.inter o n) paired in
VTbl.replace status var (res, Term.Set.union created_sk new_skolems);
| None ->
let t', created_sk = T.DB.skolemize_loosely_bound t in
let created_sk = T.IntMap.bindings created_sk
|> List.map snd |> Term.Set.of_list in
VTbl.add status var (get_covers head (T.args t'), created_sk);
end
| _ -> ();
()
);
let subst =
VTbl.to_list status
|> CCList.filter_map (fun (var, (args, skolems)) ->
let removed = ref IntSet.empty in
let n = List.length args in
let keep = List.mapi (fun i arg_set ->
let arg_l = Term.Set.to_list arg_set in
let arg_l = List.filter (fun t ->
List.for_all (fun idx ->
not @@ IntSet.mem idx !removed) (T.DB.unbound t) &&
T.Seq.subterms t
|> Iter.for_all (fun subt -> not @@ Term.Set.mem subt skolems))
arg_l in
let res = CCList.is_empty arg_l in
if not res then removed := IntSet.add (n-i-1) !removed;
res) args in
if CCList.for_all ((=) true) keep then None
else (
let ty_args, ty_return = Type.open_fun (HVar.ty var) in
let ty_args' =
CCList.combine keep ty_args
|> CCList.filter fst |> CCList.map snd
in
let var' = HVar.cast var ~ty:(Type.arrow ty_args' ty_return) in
let bvars =
CCList.combine keep ty_args
|> List.mapi (fun i (k, ty) -> k, T.bvar ~ty (List.length ty_args - i - 1))
|> CCList.filter fst|> CCList.map snd
in
let replacement = T.fun_l ty_args (T.app (T.var var') bvars) in
Some ((var,0), (replacement,1))))
|> Subst.FO.of_list'
in
if Subst.is_empty subst
then SimplM.return_same c
else (
let renaming = Subst.Renaming.none in
let new_lits = Lits.apply_subst renaming subst (C.lits c, 0) in
let proof =
Proof.Step.simp
~rule:(Proof.Rule.mk "prune_arg_fun")
~tags:[Proof.Tag.T_ho]
[C.proof_parent_subst renaming (c,0) subst] in
let c' = C.create_a ~trail:(C.trail c) ~penalty:(C.penalty c) new_lits proof in
Util.debugf ~section 3
"@[<>@[%a@]@ @[<2>prune_arg_fun into@ @[%a@]@]@ with @[%a@]@]"
(fun k->k C.pp c C.pp c' Subst.pp subst);
SimplM.return_new c'
)
let groundings = Type.Tbl.create 32
let ground_app_vars ~mode c =
assert(mode != `Off);
let app_var =
C.Seq.vars c
|> Iter.filter (fun v -> Type.is_fun (HVar.ty v) && Type.is_ground (HVar.ty v))
|> Iter.head in
let get_groundings var =
let introduce_new_const ty =
let (f_id, f_ty), f = Term.mk_fresh_skolem [] ty in
C.Ctx.add_signature (Signature.declare (C.Ctx.signature ()) f_id f_ty);
f in
let ty = T.ty var in
match mode with
| `Off -> []
| `Fresh ->
[Type.Tbl.get_or_add groundings ~f:introduce_new_const ~k:ty]
| `All ->
let ids = Signature.find_by_type (C.Ctx.signature ()) ty in
if not (ID.Set.is_empty ids) then List.map (Term.const ~ty) (ID.Set.to_list ids)
else [Type.Tbl.get_or_add groundings ~f:introduce_new_const ~k:ty]
in
CCOpt.map (fun v ->
let inst repl =
assert (T.is_ground repl);
let subst = Subst.FO.bind' Subst.empty (v,0) (repl,0) in
let renaming = Subst.Renaming.none in
let p =
Proof.Step.simp ~rule:(Proof.Rule.mk "ground_app_vars") ~tags:[Proof.Tag.T_ho]
[C.proof_parent_subst renaming (c,0) subst] in
let res = C.apply_subst ~renaming ~proof:(Some p) (c,0) subst in
res in
List.map inst (get_groundings (T.var v))
) app_var
let prim_enum_simpl ~mode c =
if C.proof_depth c < max_penalty_prim_ then (
let proof_constructor = Proof.Step.simp in
if Env.flex_get k_instantiate_choice_ax && recognize_choice_ops c then None
else (
let other_insts =
(if Env.flex_get k_instantiate_choice_ax
then (insantiate_choice ~proof_constructor c) else [])
@ (if C.proof_depth c < Env.flex_get k_elim_leibniz_eq
then elim_leibniz_eq_ ~proof_constructor c else [])
@ (if Env.flex_get k_elim_pred_var
then elim_pred_variable ~proof_constructor c else []) in
let res = other_insts @ prim_enum_ ~proof_constructor ~mode c in
if CCList.is_empty res then None else Some res
)
) else None
let purify_applied_variable c =
let new_lits = ref [] in
let add_lit_ lit = new_lits := lit :: !new_lits in
let cache_replacement_ = T.Tbl.create 8 in
let cache_untouched_ = VTbl.create 8 in
let varidx =
Literals.Seq.terms (C.lits c)
|> Iter.flat_map T.Seq.vars
|> T.Seq.max_var |> succ
|> CCRef.create
in
let replacement_var t =
try T.Tbl.find cache_replacement_ t
with Not_found ->
let head, _ = T.as_app t in
let ty = T.ty head in
let v = T.var_of_int ~ty (CCRef.get_then_incr varidx) in
let lit = Literal.mk_neq v head in
add_lit_ lit;
T.Tbl.add cache_replacement_ t v;
v
in
let same_class t1 t2 =
assert (T.is_var (fst (T.as_app t1)));
assert (T.is_var (fst (T.as_app t2)));
if Env.flex_get k_purify_applied_vars == `Ext
then
T.equal t1 t2
else (
assert (Env.flex_get k_purify_applied_vars == `Int);
match T.view t1, T.view t2 with
| T.Var x, T.Var y when HVar.equal Type.equal x y -> true
| T.App (f, _), T.App (g, _) when T.equal f g -> true
| _ -> false
)
in
let should_purify t v =
try
if same_class t (VTbl.find cache_untouched_ v) then (
Util.debugf ~section 5
"Leaving untouched: %a"
(fun k->k T.pp t);false
) else (
Util.debugf ~section 5
"To purify: %a"
(fun k->k T.pp t);true
)
with Not_found ->
VTbl.add cache_untouched_ v t;
Util.debugf ~section 5
"Add untouched term: %a"
(fun k->k T.pp t);
false
in
let rec purify_term t =
let head, args = T.as_app t in
let res = match T.as_var head with
| Some v ->
if should_purify t v then (
Util.debugf ~section 5
"@[Purifying: %a.@ Untouched is: %a@]"
(fun k->k T.pp t T.pp (VTbl.find cache_untouched_ v));
let v' = replacement_var t in
assert (Type.equal (HVar.ty v) (T.ty v'));
T.app v' (List.map purify_term args)
) else (
T.app head (List.map purify_term args)
)
| None ->
T.app head (List.map purify_term args)
in
assert (Type.equal (T.ty res) (T.ty t));
res
in
let purify_lit lit =
if Literal.for_all T.is_var lit
then lit
else Literal.map purify_term lit
in
let lits' = Array.map purify_lit (C.lits c) in
begin match !new_lits with
| [] -> SimplM.return_same c
| _::_ ->
let all_lits = !new_lits @ (Array.to_list lits') in
let parent = C.proof_parent c in
let proof =
Proof.Step.simp
~rule:(Proof.Rule.mk "ho.purify_applied_variable") ~tags:[Proof.Tag.T_ho]
[parent] in
let new_clause = (C.create ~trail:(C.trail c) ~penalty:(C.penalty c) all_lits proof) in
Util.debugf ~section 5
"@[<hv2>Purified:@ Old: %a@ New: %a@]"
(fun k->k C.pp c C.pp new_clause);
SimplM.return_new new_clause
end
let setup () =
mk_diff_const ();
if not (Env.flex_get k_enabled) then (
Util.debug ~section 1 "HO rules disabled";
) else (
Util.debug ~section 1 "setup HO rules";
Env.Ctx.lost_completeness();
if Env.flex_get k_resolve_flex_flex then (
Env.add_basic_simplify remove_ff_constraints
);
if Env.flex_get k_ground_app_vars != `Off then (
let mode = Env.flex_get k_ground_app_vars in
E.add_cheap_multi_simpl_rule (ground_app_vars ~mode);
E.add_multi_simpl_rule ~priority:1000 (ground_app_vars ~mode)
);
if Env.flex_get k_elim_pred_var then
Env.add_unary_inf "ho_elim_pred_var" elim_pred_variable;
if Env.flex_get k_ext_neg_lit then
Env.add_lit_rule "ho_ext_neg_lit" ext_neg_lit;
if Env.flex_get k_elim_leibniz_eq > 0 then (
Env.add_unary_inf "ho_elim_leibniz_eq" elim_leibniz_equality
);
if Env.flex_get k_instantiate_choice_ax then (
Env.add_redundant recognize_choice_ops;
Env.add_unary_inf "inst_choice" insantiate_choice;
);
if Env.flex_get k_ext_pos then (
Env.add_unary_inf "ho_ext_pos"
(ext_pos_general ~all_lits:(Env.flex_get k_ext_pos_all_lits));
);
if Env.flex_get k_enable_def_unfold then (
Env.add_clause_conversion (
fun c -> match Statement.get_rw_rule c with
| Some _ -> E.CR_drop
| None -> E.CR_skip ));
begin match Env.flex_get k_prune_arg_fun with
| `PruneMaxCover -> Env.add_unary_simplify (prune_arg ~all_covers:false);
| `PruneAllCovers -> Env.add_unary_simplify (prune_arg ~all_covers:true);
| `OldPrune -> Env.add_unary_simplify prune_arg_old;
| `NoPrune -> ();
end;
if Env.flex_get Combinators.k_enable_combinators then (
Env.set_ho_normalization_rule "comb-normalize" Combinators_base.comb_normalize;
) else (
let ho_norm = (fun t -> t |> beta_reduce |> (
fun opt -> match opt with
None -> eta_normalize t
| Some t' ->
match eta_normalize t' with
None -> Some t'
| Some tt -> Some tt))
in
Env.set_ho_normalization_rule "ho_norm" ho_norm);
if(Env.flex_get k_neg_ext) then (
Env.add_unary_inf "neg_ext" neg_ext
)
else if(Env.flex_get k_neg_ext_as_simpl) then (
Env.add_unary_simplify neg_ext_simpl;
);
if(Env.flex_get k_purify_applied_vars != `None) then (
Env.add_unary_simplify purify_applied_variable
);
if Env.flex_get k_enable_ho_unif then (
Env.add_unary_inf "ho_unif" ho_unif;
);
if Env.flex_get k_simple_projection >= 0 then (
let penalty_inc = Env.flex_get k_simple_projection in
let max_depth = Env.flex_get k_simple_projection_md in
Env.add_unary_inf "simple_projection" (simple_projection ~penalty_inc ~max_depth);
);
if not (Env.flex_get k_prim_enum_early_bird) then (
begin match Env.flex_get k_ho_prim_mode with
| `None -> ()
| mode ->
if Env.flex_get k_prim_enum_simpl then (
Env.add_single_step_multi_simpl_rule (prim_enum_simpl ~mode)
) else Env.add_unary_inf "ho_prim_enum" (prim_enum ~mode);
end
) else (
Signal.on Env.on_pred_var_elimination (fun (cl,var) ->
early_bird_prim_enum cl var;
Signal.ContinueListening
)
);
if Env.flex_get k_ext_axiom then
Env.ProofState.PassiveSet.add (Iter.singleton (mk_extensionality_clause ())) ;
if Env.flex_get k_choice_axiom then
Env.ProofState.PassiveSet.add (Iter.singleton choice_clause);
);
()
end
let enabled_ = ref true
let def_unfold_enabled_ = ref false
let force_enabled_ = ref false
let enable_unif_ = ref false
let prim_mode_ = ref `Neg
let prim_max_penalty = ref 2
let set_prim_mode_ =
let l = [
"and", `And;
"or", `Or;
"neg", `Neg;
"eq", `Eq;
"quants", `Quants;
"tf", `TF;
"combs", `Combinators;
"full", `Full;
"pragmatic", `Pragmatic;
"simple", `Simple;
"none", `None;
] in
let set_ s = prim_mode_ := List.assoc s l in
Arg.Symbol (List.map fst l, set_)
let st_contains_ho (st:(_,_,_) Statement.t): bool =
let is_non_atomic_ty ty =
let n_ty_vars, args, _ = Type.open_poly_fun ty in
n_ty_vars > 0 || args<>[]
and has_prop_in_args ty =
let n_ty_vars, args, _ = Type.open_poly_fun ty in
List.exists Type.contains_prop args
in
let has_ho_var () =
Statement.Seq.terms st
|> Iter.flat_map T.Seq.vars
|> Iter.map HVar.ty
|> Iter.exists (fun ty -> is_non_atomic_ty ty || Type.contains_prop ty)
and has_ho_sym () =
Statement.Seq.ty_decls st
|> Iter.exists
(fun (_,ty) -> Type.order ty > 1 || has_prop_in_args ty)
and has_ho_eq() =
Statement.Seq.forms st
|> Iter.exists
(fun c ->
c |> List.exists
(function
| SLiteral.Eq (t,u) | SLiteral.Neq (t,u) ->
T.is_ho_at_root t || T.is_ho_at_root u || is_non_atomic_ty (T.ty t)
| _ -> false))
in
has_ho_sym () || has_ho_var () || has_ho_eq()
let _ext_pos = ref true
let _ext_pos_all_lits = ref false
let _ext_axiom = ref false
let _choice_axiom = ref false
let _elim_pred_var = ref true
let _ext_neg_lit = ref false
let _neg_ext = ref true
let _neg_ext_as_simpl = ref false
let _ext_axiom_penalty = ref 5
let _huet_style = ref false
let _cons_elim = ref true
let _imit_first = ref false
let _compose_subs = ref false
let _var_solve = ref false
let _instantiate_choice_ax = ref false
let _elim_leibniz_eq = ref (-1)
let _prune_arg_fun = ref `NoPrune
let _check_lambda_free = ref `False
let prim_enum_terms = ref Term.Set.empty
let _oracle_composer = ref (OSeq.merge :> (Logtk.Subst.t option OSeq.t OSeq.t -> Logtk.Subst.t option OSeq.t))
let _simple_projection = ref (-1)
let _simple_projection_md = ref 2
let _purify_applied_vars = ref `None
let _eta = ref `Reduce
let _use_diff_for_neg_ext = ref false
let _generalize_choice_trigger = ref false
let _prim_enum_simpl = ref false
let _prim_enum_early_bird = ref false
let _resolve_flex_flex = ref false
let _ground_app_vars = ref `Off
let extension =
let register env =
let module E = (val env : Env.S) in
E.flex_add k_ext_pos !_ext_pos;
E.flex_add k_ext_pos_all_lits !_ext_pos_all_lits;
E.flex_add k_ext_axiom !_ext_axiom;
E.flex_add k_choice_axiom !_choice_axiom;
E.flex_add k_elim_pred_var !_elim_pred_var;
E.flex_add k_ext_neg_lit !_ext_neg_lit;
E.flex_add k_neg_ext !_neg_ext;
E.flex_add k_neg_ext_as_simpl !_neg_ext_as_simpl;
E.flex_add k_ext_axiom_penalty !_ext_axiom_penalty;
E.flex_add k_instantiate_choice_ax !_instantiate_choice_ax;
E.flex_add k_elim_leibniz_eq !_elim_leibniz_eq;
E.flex_add k_prune_arg_fun !_prune_arg_fun;
E.flex_add k_prim_enum_terms prim_enum_terms;
E.flex_add k_simple_projection !_simple_projection;
E.flex_add k_simple_projection_md !_simple_projection_md;
E.flex_add k_check_lambda_free !_check_lambda_free;
E.flex_add k_purify_applied_vars !_purify_applied_vars;
E.flex_add k_eta !_eta;
E.flex_add k_use_diff_for_neg_ext !_use_diff_for_neg_ext;
E.flex_add k_generalize_choice_trigger !_generalize_choice_trigger;
E.flex_add k_prim_enum_simpl !_prim_enum_simpl;
E.flex_add k_prim_enum_early_bird !_prim_enum_early_bird;
E.flex_add k_resolve_flex_flex !_resolve_flex_flex;
E.flex_add k_ground_app_vars !_ground_app_vars;
if E.flex_get k_check_lambda_free = `Only
then E.flex_add Saturate.k_abort_after_fragment_check true;
if E.flex_get k_check_lambda_free != `False then
E.add_fragment_check (fun c ->
E.C.Seq.terms c |> Iter.for_all Term.in_lfho_fragment
);
Signal.on_every E.on_start (fun () ->
if not !Unif._unif_bool
then print_endline
("To remain in the chosen logic fragment, " ^
"unification with booleans has been disabled."));
if E.flex_get k_some_ho || !force_enabled_ then (
let module ET = Make(E) in
ET.setup ()
)
and check_ho vec state =
let is_ho =
CCVector.to_iter vec
|> Iter.exists st_contains_ho
in
if is_ho then (
Util.debug ~section 2 "problem is HO"
);
if !def_unfold_enabled_ then (
CCVector.iter (fun c -> match Statement.get_rw_rule c with
Some (sym, r) -> Util.debugf ~section 1
"@[<2> Adding constant def rule: `@[%a@]`@]"
(fun k->k Rewrite.Rule.pp r);
Rewrite.Defined_cst.declare_or_add sym r;
| _ -> ()) vec
);
state
|> Flex_state.add k_some_ho is_ho
|> Flex_state.add k_enabled !enabled_
|> Flex_state.add PragUnifParams.k_oracle_composer !_oracle_composer
|> Flex_state.add k_enable_def_unfold !def_unfold_enabled_
|> Flex_state.add k_enable_ho_unif (!enabled_ && !enable_unif_)
|> Flex_state.add k_ho_prim_mode (if !enabled_ then !prim_mode_ else `None)
|> Flex_state.add k_ho_prim_max_penalty !prim_max_penalty
in
{ Extensions.default with
Extensions.name = "ho";
post_cnf_actions=[check_ho];
env_actions=[register];
}
let purify_opt =
let set_ n = _purify_applied_vars := n in
let l = [ "ext", `Ext; "int", `Int; "none", `None] in
Arg.Symbol (List.map fst l, fun s -> set_ (List.assoc s l))
let eta_opt =
let set_ n = _eta := n in
let l = [ "reduce", `Reduce; "expand", `Expand; "none", `None] in
Arg.Symbol (List.map fst l, fun s -> set_ (List.assoc s l))
let () =
Options.add_opts
[ "--ho", Arg.Bool (fun b -> enabled_ := b), " enable/disable HO reasoning";
"--force-ho", Arg.Bool (fun b -> force_enabled_ := b), " enable/disable HO reasoning even if the problem is first-order";
"--ho-unif", Arg.Bool (fun v -> enable_unif_ := v), " enable full HO unification";
"--ho-elim-pred-var", Arg.Bool (fun b -> _elim_pred_var := b), " disable predicate variable elimination";
"--ho-prim-enum", set_prim_mode_, " set HO primitive enum mode";
"--ho-prim-max", Arg.Set_int prim_max_penalty, " max penalty for HO primitive enum";
"--ho-prim-enum-simpl", Arg.Bool ((:=) _prim_enum_simpl), " use primitive enumeration as simplification rule";
"--ho-prim-enum-early-bird", Arg.Bool ((:=) _prim_enum_early_bird), " use early-bird primitive enumeration (requires lazy CNF)";
"--ho-resolve-flex-flex", Arg.Bool ((:=) _resolve_flex_flex), " eagerly remove non-essential flex-flex constraints";
"--ho-oracle-composer", Arg.Symbol (["merge";"fair"], (fun s ->
if s = "merge" then _oracle_composer := (OSeq.merge :> (Logtk.Subst.t option OSeq.t OSeq.t -> Logtk.Subst.t option OSeq.t))
else _oracle_composer := UnifFramework.take_fair)), " choose either OSeq.merge or Unif.take_fair as the composer";
"--ho-ext-axiom", Arg.Bool (fun v -> _ext_axiom := v), " enable/disable extensionality axiom";
"--ho-choice-axiom", Arg.Bool (fun v -> _choice_axiom := v), " enable choice axiom";
"--ho-ext-pos", Arg.Bool (fun v -> _ext_pos := v), " enable/disable positive extensionality rule";
"--ho-neg-ext", Arg.Bool (fun v -> _neg_ext := v), " turn NegExt on or off";
"--ho-neg-ext-simpl", Arg.Bool (fun v -> _neg_ext_as_simpl := v), " turn NegExt as simplification rule on or off";
"--ho-ext-pos-all-lits", Arg.Bool (fun v -> _ext_pos_all_lits := v), " turn ExtPos on for all or only eligible literals";
"--ho-prune-arg", Arg.Symbol (["all-covers"; "max-covers"; "old-prune"; "off"], (fun s ->
if s = "all-covers" then _prune_arg_fun := `PruneAllCovers
else if s = "max-covers" then _prune_arg_fun := `PruneMaxCover
else if s = "old-prune" then _prune_arg_fun := `OldPrune
else _prune_arg_fun := `NoPrune)), " choose arg prune mode";
"--ho-ext-neg-lit", Arg.Bool (fun v -> _ext_neg_lit := v), " enable/disable negative extensionality rule on literal level [?]";
"--ho-elim-leibniz", Arg.String (fun v ->
match v with
| "inf" -> _elim_leibniz_eq := max_int
| "off" -> _elim_leibniz_eq := -1
| _ ->
match CCInt.of_string v with
| None -> invalid_arg "number expected for --ho-elim-leibniz"
| Some x -> _elim_leibniz_eq := x
), " enable/disable treatment of Leibniz equality. inf enables it for infinte depth of clauses"
^ "; off disables it; number enables it for a given depth of clause";
"--ho-def-unfold", Arg.Bool (fun v -> def_unfold_enabled_ := v), " enable ho definition unfolding";
"--ho-choice-inst", Arg.Bool (fun v -> _instantiate_choice_ax := v), " enable heuristic Hilbert choice instantiation";
"--ho-simple-projection", Arg.Int (fun v -> _simple_projection := v),
" enable simple projection instantiation." ^
" positive argument is increase in clause penalty for the conclusion; " ^
" negative argument turns the inference off";
"--ho-simple-projection-max-depth", Arg.Set_int _simple_projection_md, " sets the max depth for simple projection";
"--ho-ext-axiom-penalty", Arg.Int (fun p -> _ext_axiom_penalty := p), " penalty for extensionality axiom";
"--ho-purify", purify_opt, " enable purification of applied variables: 'ext' purifies" ^
" whenever a variable is applied to different arguments." ^
" 'int' purifies whenever a variable appears applied and unapplied.";
"--ho-eta", eta_opt, " eta-expansion/reduction";
"--ground-app-vars",
Arg.Symbol (["off"; "fresh"; "all"], (fun kind ->
match kind with
| "off" -> _ground_app_vars := `Off
| "fresh" -> _ground_app_vars := `Fresh
| "all" -> _ground_app_vars := `All
| _ -> assert false)), " ground all applied variables to either all constants of the right type in signature or a fresh constant";
"--ho-use-diff-for-neg-ext", Arg.Bool ((:=) _use_diff_for_neg_ext), " use diff constant for NegExt rule instead of fresh skolem";
"--ho-generalize-choice-trigger", Arg.Bool ((:=) _generalize_choice_trigger), " apply choice trigger to a fresh variable";
"--check-lambda-free", Arg.Symbol (["true";"false";"only"], fun s -> match s with
| "true" -> _check_lambda_free := `True
| "only" -> _check_lambda_free := `Only
| _ -> _check_lambda_free := `False), "check whether problem belongs to lambda-free ('only' will abort after the check)";
];
Params.add_to_mode "ho-complete-basic" (fun () ->
enabled_ := true;
def_unfold_enabled_ := false;
force_enabled_ := true;
_ext_axiom := true;
_ext_neg_lit := false;
_neg_ext := true;
_neg_ext_as_simpl := false;
_ext_pos := true;
_ext_pos_all_lits := false;
prim_mode_ := `None;
_elim_pred_var := false;
enable_unif_ := false;
_prune_arg_fun := `PruneMaxCover;
);
Params.add_to_mode "ho-pragmatic" (fun () ->
enabled_ := true;
def_unfold_enabled_ := false;
force_enabled_ := true;
_ext_axiom := false;
_ext_neg_lit := false;
_neg_ext := true;
_neg_ext_as_simpl := false;
_ext_pos := true;
_ext_pos_all_lits := true;
prim_mode_ := `None;
_elim_pred_var := true;
enable_unif_ := false;
_prune_arg_fun := `PruneMaxCover;
);
Params.add_to_mode "ho-competitive" (fun () ->
enabled_ := true;
def_unfold_enabled_ := false;
force_enabled_ := true;
_ext_axiom := false;
_ext_neg_lit := false;
_neg_ext := true;
_neg_ext_as_simpl := false;
_ext_pos := true;
_ext_pos_all_lits := true;
prim_mode_ := `None;
_elim_pred_var := true;
enable_unif_ := false;
_prune_arg_fun := `PruneMaxCover;
);
Params.add_to_mode "ho-comb-complete" (fun () ->
enabled_ := true;
def_unfold_enabled_ := false;
_resolve_flex_flex := false;
force_enabled_ := true;
_ext_axiom := false;
_ext_neg_lit := false;
_neg_ext := true;
_neg_ext_as_simpl := false;
_ext_pos := true;
_ext_pos_all_lits := true;
prim_mode_ := `None;
_elim_pred_var := true;
enable_unif_ := false;
_prune_arg_fun := `NoPrune;
Unif._allow_pattern_unif := false;
_eta := `None
);
Params.add_to_mode "fo-complete-basic" (fun () ->
enabled_ := false;
_resolve_flex_flex := false;
Unif._allow_pattern_unif := false;
Unif._unif_bool := false;
);
Params.add_to_modes
[ "lambda-free-intensional"
; "lambda-free-extensional"
; "lambda-free-purify-intensional"
; "lambda-free-purify-extensional"] (fun () ->
enabled_ := true;
enable_unif_ := false;
_resolve_flex_flex := false;
force_enabled_ := true;
_elim_pred_var := false;
_neg_ext_as_simpl := false;
_prune_arg_fun := `NoPrune;
prim_mode_ := `None;
_check_lambda_free := `True;
Unif._allow_pattern_unif := false;
Unif._unif_bool := false;
_eta := `None;
);
Params.add_to_modes
[ "lambda-free-extensional"
; "lambda-free-purify-extensional"] (fun () ->
_ext_axiom := true;
_neg_ext := true;
_ext_pos := true;
_ext_pos_all_lits := true;
);
Params.add_to_modes
[ "lambda-free-intensional"
; "lambda-free-purify-intensional"] (fun () ->
_ext_axiom := false;
_neg_ext := false;
_ext_pos := false;
_ext_pos_all_lits := false;
);
Params.add_to_mode "lambda-free-purify-intensional" (fun () ->
_purify_applied_vars := `Int
);
Params.add_to_mode "lambda-free-purify-extensional" (fun () ->
_purify_applied_vars := `Ext
);
Extensions.register extension;