Source file booleans.ml
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(** {1 boolean subterms} *)
open Logtk
open Libzipperposition
module T = Term
module Pos = Position
module US = Unif_subst
type selection_setting = Any | Minimal | Large
type reasoning_kind =
BoolReasoningDisabled | BoolCasesInference | BoolCasesDisabled
| BoolCasesSimplification | BoolCasesKeepParent
| BoolCasesPreprocess
let section = Util.Section.make ~parent:Const.section "booleans"
let k_bool_reasoning = Flex_state.create_key ()
let k_cased_term_selection = Flex_state.create_key ()
let k_quant_rename = Flex_state.create_key ()
let k_interpret_bool_funs = Flex_state.create_key ()
let k_cnf_non_simpl = Flex_state.create_key ()
let k_norm_bools = Flex_state.create_key ()
let k_filter_literals = Flex_state.create_key ()
let k_nnf = Flex_state.create_key ()
let k_elim_bvars = Flex_state.create_key ()
let k_simplify_bools = Flex_state.create_key ()
let selection_to_str = function
| Any -> "any"
| Minimal -> "minimal"
| Large -> "large"
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 *)
end
module Make(E : Env.S) : S with module Env = E = struct
module Env = E
module C = Env.C
module Ctx = Env.Ctx
module Fool = Fool.Make(Env)
module Combs = Combinators.Make(Env)
module HO = Higher_order.Make(Env)
module LazyCNF = Lazy_cnf.Make(Env)
let (=~),(/~) = Literal.mk_eq, Literal.mk_neq
let (@:) = T.app_builtin ~ty:Type.prop
let no a = a =~ T.false_
let yes a = a =~ T.true_
let find_bools c =
let found = ref false in
let subterm_selection = Env.flex_get k_cased_term_selection in
let rec find_in_term ~top t k =
match T.view t with
| T.Const _ when Type.is_prop (T.ty t) && not top -> k t
| T.App(_, args)
| T.AppBuiltin(_, args) ->
let take_subterm =
not top &&
Type.is_prop (T.ty t) &&
not (T.is_true_or_false t) &&
T.DB.is_closed t &&
not (T.is_var (T.head_term t)) &&
not (T.is_bvar (T.head_term t)) &&
(subterm_selection != Minimal ||
Iter.is_empty
(Iter.flat_map (find_in_term ~top:false)
(CCList.to_iter args))) in
let continue =
(subterm_selection = Any || not take_subterm) &&
not (T.is_var (T.head_term t)) &&
not (T.is_bvar (T.head_term t))
in
if take_subterm then k t;
if continue then (
List.iter (fun arg ->
find_in_term ~top:(top && T.is_appbuiltin t) arg k
) args)
| T.Fun (_,body) ->
find_in_term ~top body k
| _ -> () in
let eligible =
match Env.flex_get k_filter_literals with
| `All -> C.Eligible.always
| `Max -> (
match subterm_selection with
| Any -> C.Eligible.res c
| _ -> (fun i lit ->
not (!found) && C.Eligible.res c i lit)) in
Literals.fold_terms ~which:`All
~subterms:false ~eligible ~ord:(C.Ctx.ord ()) (C.lits c)
|> Iter.flat_map (fun (t,_) ->
let res = find_in_term ~top:true t in
if not (Iter.is_empty res) then found := true;
res
)
|> T.Set.of_iter
|> T.Set.to_list
let mk_res ~proof ~old ~repl new_lit c =
C.create ~trail:(C.trail c) ~penalty:(C.penalty c)
(new_lit :: Array.to_list( C.lits c |> Literals.map (T.replace ~old ~by:repl)))
proof
let bool_case_inf (c: C.t) : C.t list =
let proof = Proof.Step.inference [C.proof_parent c]
~rule:(Proof.Rule.mk "bool_inf") ~tags:[Proof.Tag.T_ho] in
Util.debugf 5 ~section "bci(@[%a@])=@." (fun k -> k C.pp c);
find_bools c
|> CCList.fold_left (fun acc old ->
let neg_lit, repl =
if Builtin.compare Builtin.True Builtin.False > 0 then (no old, T.true_)
else (yes old, T.false_) in
(mk_res ~proof ~old ~repl neg_lit c) :: acc
) []
let bool_case_simp (c: C.t) : C.t list option =
let proof = Proof.Step.simp [C.proof_parent c]
~rule:(Proof.Rule.mk"bool_simp") ~tags:[Proof.Tag.T_ho] in
let bool_subterms = find_bools c in
if CCList.is_empty bool_subterms then None
else (
CCOpt.return @@ CCList.fold_left (fun acc old ->
let neg_lit, repl_neg = no old, T.true_ in
let pos_lit, repl_pos = yes old, T.false_ in
(mk_res ~proof ~old ~repl:repl_neg neg_lit c) ::
(mk_res ~proof ~old ~repl:repl_pos pos_lit c) :: acc
) [] bool_subterms)
let simplify_bools t =
let negate t =
match T.view t with
| T.AppBuiltin(((Builtin.Eq|Builtin.Neq) as b), l) ->
let hd = if b = Builtin.Eq then Builtin.Neq else Builtin.Eq in
T.app_builtin ~ty:(T.ty t) hd l
| T.AppBuiltin(Builtin.Not, [s]) -> s
| _ -> T.Form.not_ t in
let simplify_and_or t b l =
let open Term in
let compl_in_l l =
let pos, neg =
CCList.partition_map (fun t ->
match view t with
| AppBuiltin(Builtin.Not, [s]) -> `Right s
| _ -> `Left t) l
|> CCPair.map_same Set.of_list in
not (Set.is_empty (Set.inter pos neg)) in
let res =
assert(b = Builtin.And || b = Builtin.Or);
let netural_el, absorbing_el =
if b = Builtin.And then true_,false_ else (false_,true_) in
let l' = CCList.sort_uniq ~cmp:compare l in
if compl_in_l l || List.exists (equal absorbing_el) l then absorbing_el
else (
let l' = List.filter (fun s -> not (equal s netural_el)) l' in
if List.length l = List.length l' then t
else (
if CCList.is_empty l' then netural_el
else (if List.length l' = 1 then List.hd l'
else app_builtin ~ty:(Type.prop) b l')
))
in
res
in
let rec aux t =
let ty_is_prop t = Type.is_prop (T.ty t) in
match T.view t with
| DB _ | Const _ | Var _ -> t
| Fun(ty, body) ->
let body' = aux body in
assert(Type.equal (T.ty body) (T.ty body'));
if T.equal body body' then t
else T.fun_ ty body'
| App(hd, args) ->
let hd' = aux hd and args' = List.map aux args in
if T.equal hd hd' && T.same_l args args' then t
else T.app hd' args'
| AppBuiltin (Builtin.And, [x])
when T.is_true_or_false x
&& ty_is_prop t
&& List.length (Type.expected_args (T.ty t)) = 1 ->
if T.equal x T.true_ then (
T.fun_ Type.prop (T.bvar ~ty:Type.prop 0)
) else (
assert (T.equal x T.false_);
T.fun_ Type.prop T.false_
)
| AppBuiltin(Builtin.Or, [x])
when T.is_true_or_false x
&& ty_is_prop t
&& List.length (Type.expected_args (T.ty t)) = 1 ->
let prop = Type.prop in
if T.equal x T.true_ then (
T.fun_ prop (T.true_)
) else (
assert (T.equal x T.false_);
T.fun_ prop (T.bvar ~ty:prop 0)
)
| AppBuiltin(Builtin.And, l)
when ty_is_prop t &&
List.length l > 1 ->
let l' = List.map aux l in
let t = if T.same_l l l' then t
else T.app_builtin ~ty:(Type.prop) Builtin.And l' in
simplify_and_or t Builtin.And l'
| AppBuiltin(Builtin.Or, l)
when ty_is_prop t &&
List.length l > 1 ->
let l' = List.map aux l in
let t = if T.same_l l l' then t
else T.app_builtin ~ty:(Type.prop) Builtin.Or l' in
simplify_and_or t Builtin.Or l'
| AppBuiltin(Builtin.Not, [s]) ->
let s' = aux s in
begin match T.view s' with
| AppBuiltin(Builtin.Not, [s'']) -> s''
| _ ->
if T.equal s' T.true_ then T.false_
else if T.equal s' T.false_ then T.true_
else if T.equal s s' then t
else T.app_builtin ~ty:(Type.prop) Builtin.Not [s'] end
| AppBuiltin(Builtin.Imply, [p;c]) ->
let unroll_and p = match T.view p with
| AppBuiltin(And, l) -> T.Set.of_list l
| _ -> T.Set.singleton p in
let unroll_or p = match T.view p with
| AppBuiltin(Or, l) -> T.Set.of_list l
| _ -> T.Set.singleton p in
let is_impl p = match T.view p with
| AppBuiltin(Imply, [l;r]) -> true
| _ -> false in
let unroll_impl p =
assert(is_impl p);
let rec aux acc p =
match T.view p with
| AppBuiltin(Imply, [l;r]) ->
let unrolled_l = unroll_and l in
let acc' = Term.Set.union unrolled_l acc in
if is_impl r then aux acc' r
else (acc', unroll_or r)
| _ -> assert false in
aux Term.Set.empty p
in
let p' = aux p and c' = aux c in
let (premises,conclusions) = unroll_impl (T.Form.imply p' c') in
if not (T.Set.is_empty (T.Set.inter premises conclusions)) then (
T.true_
) else if T.equal p' c' then T.true_
else if T.equal c' (negate p') then c'
else if T.equal p' (negate c') then c'
else if T.equal p' T.true_ then c'
else if T.equal p' T.false_ then T.true_
else if T.equal c' T.false_ then aux (T.Form.not_ p')
else if T.equal c' T.true_ then T.true_
else (
if T.equal p p' && T.equal c c' then t
else T.app_builtin ~ty:(T.ty t) Builtin.Imply [p';c']
)
| AppBuiltin((Builtin.Eq | Builtin.Equiv) as hd, ([a;b]|[_;a;b])) ->
let a',b' = aux a, aux b in
if T.equal a' b' then T.true_
else if T.equal a' T.true_ then b'
else if T.equal b' T.true_ then a'
else if T.equal a' T.false_ then aux (T.Form.not_ b')
else if T.equal b' T.false_ then aux (T.Form.not_ a')
else (
if T.equal a a' && T.equal b b' then t
else T.app_builtin ~ty:(T.ty t) hd [a';b']
)
| AppBuiltin((Builtin.Neq | Builtin.Xor) as hd, ([a;b]|[_;a;b])) ->
let a',b' = aux a, aux b in
if T.equal a' b' then T.false_
else if T.equal a' T.true_ then aux (T.Form.not_ b')
else if T.equal b' T.true_ then aux (T.Form.not_ a')
else if T.equal a' T.false_ then b'
else if T.equal b' T.false_ then a'
else (
if T.equal a a' && T.equal b b' then t
else T.app_builtin ~ty:(T.ty t) hd [a';b']
)
| AppBuiltin((ExistsConst|ForallConst) as b, [g]) ->
let g' = aux g in
let exp_g = Combs.expand g' in
let _, body = T.open_fun exp_g in
assert(Type.is_prop (T.ty body));
if (T.Seq.subterms ~include_builtin:true body
|> Iter.exists T.is_bvar) then (
if T.equal g g' then t
else T.app_builtin ~ty:(T.ty t) b [g']
) else body
| AppBuiltin(hd, args) ->
let args' = List.map aux args in
if T.same_l args args' then t
else T.app_builtin ~ty:(T.ty t) hd args' in
let res = aux t in
assert (T.DB.is_closed res);
res
let simpl_bool_subterms c =
try
let new_lits = Literals.map simplify_bools (C.lits c) in
if Literals.equal (C.lits c) new_lits then (
SimplM.return_same c
) else (
let proof = Proof.Step.simp [C.proof_parent c]
~rule:(Proof.Rule.mk "simplify boolean subterms") in
let new_ = C.create ~trail:(C.trail c) ~penalty:(C.penalty c)
(Array.to_list new_lits) proof in
SimplM.return_new new_
)
with Type.ApplyError err ->
CCFormat.printf "error(%s):@[%a@]@." err C.pp c;
CCFormat.printf "@[%a@]@." Proof.S.pp_tstp (C.proof c);
assert false
let nnf_bools t =
let module F = T.Form in
let rec aux t =
match T.view t with
| Const _ | DB _ | Var _ -> t
| Fun _ ->
let tyargs, body = T.open_fun t in
let body' = aux body in
if T.equal body body' then t
else T.fun_l tyargs body'
| App(hd, l) ->
let hd' = aux hd and l' = List.map aux l in
if T.equal hd hd' && T.same_l l l' then t
else T.app hd' l'
| AppBuiltin (Builtin.Not, [f]) ->
begin match T.view f with
| AppBuiltin(Not, [g]) -> aux g
| AppBuiltin( ((And|Or) as b), l) when List.length l >= 2 ->
let flipped = if b = Builtin.And then F.or_l else F.and_l in
flipped (List.map (fun t -> aux (F.not_ t)) l)
| AppBuiltin( ((ForallConst|ExistsConst) as b), ([g]|[_;g]) ) ->
let flipped =
if b = Builtin.ForallConst then Builtin.ExistsConst
else Builtin.ForallConst in
let g_ty_args, g_body = T.open_fun (Combs.expand g) in
let g_body' = aux @@ F.not_ g_body in
let g' = Lambda.eta_reduce (T.fun_l g_ty_args g_body') in
T.app_builtin ~ty:(T.ty t) flipped [g']
| AppBuiltin( Imply, [g;h] ) ->
F.and_ (aux g) (aux @@ F.not_ h)
| AppBuiltin( ((Equiv|Xor) as b), [g;h] ) ->
let flipped = if b = Equiv then Builtin.Xor else Builtin.Equiv in
aux (T.app_builtin ~ty:(T.ty t) flipped [g;h])
| AppBuiltin(((Eq|Neq) as b), ([_;s;t]|[s;t])) ->
let flipped = if b = Eq then F.neq else F.eq in
flipped (aux s) (aux t)
| _ -> F.not_ (aux f)
end
| AppBuiltin(Imply, [f;g]) -> aux (F.or_ (F.not_ f) g)
| AppBuiltin(Equiv, [f;g]) ->
aux (F.and_ (F.imply f g) (F.imply g f))
| AppBuiltin(Xor, [f;g]) ->
aux (F.and_ (F.or_ f g) (F.or_ (F.not_ f) (F.not_ g)))
| AppBuiltin(b, l) ->
let l' = List.map aux l in
if T.same_l l l' then t
else T.app_builtin ~ty:(T.ty t) b l' in
aux t
let nnf_bool_subters c =
let new_lits = Literals.map nnf_bools (C.lits c) in
if Literals.equal (C.lits c) new_lits then (
SimplM.return_same c
) else (
let proof = Proof.Step.simp [C.proof_parent c]
~rule:(Proof.Rule.mk "nnf boolean subterms") in
let new_ = C.create ~trail:(C.trail c) ~penalty:(C.penalty c)
(Array.to_list new_lits) proof in
SimplM.return_new new_
)
let normalize_bool_terms c =
let new_lits = Literals.map T.normalize_bools (C.lits c) in
if Literals.equal (C.lits c) new_lits then (
SimplM.return_same c
) else (
let proof = Proof.Step.simp [C.proof_parent c]
~rule:(Proof.Rule.mk "normalize subterms") in
let new_ = C.create ~trail:(C.trail c) ~penalty:(C.penalty c)
(Array.to_list new_lits) proof in
SimplM.return_new new_
)
let cnf_otf c : C.t list option =
let idx = CCArray.find_idx (fun l ->
let eq = Literal.View.as_eqn l in
match eq with
| Some (l,r,sign) ->
Type.is_prop (T.ty l) &&
not (T.equal l r) &&
((not (T.equal r T.true_) && not (T.equal r T.false_))
|| T.is_formula l || T.is_formula r)
| None -> false
) (C.lits c) in
let renaming_weight = 40 in
let max_formula_weight =
C.Seq.terms c
|> Iter.filter T.is_formula
|> Iter.map T.size
|> Iter.max in
let opts =
match max_formula_weight with
| None -> [Cnf.DisableRenaming]
| Some m -> if m < renaming_weight then [Cnf.DisableRenaming] else [] in
match idx with
| Some _ ->
let f = Literals.Conv.to_tst (C.lits c) in
let proof = Proof.Step.simp ~rule:(Proof.Rule.mk "cnf_otf") ~tags:[Proof.Tag.T_ho] [C.proof_parent c] in
let trail = C.trail c and penalty = C.penalty c in
let stmt = Statement.assert_ ~proof f in
let cnf_vec = Cnf.convert @@ CCVector.to_iter @@ Cnf.cnf_of ~opts ~ctx:(Ctx.sk_ctx ()) stmt in
CCVector.iter (fun cl ->
Statement.Seq.ty_decls cl
|> Iter.iter (fun (id,ty) -> Ctx.declare id ty)) cnf_vec;
let solved =
if Env.flex_get Lazy_cnf.k_solve_formulas then (
CCOpt.get_or ~default:[] (LazyCNF.solve_bool_formulas c))
else [] in
let clauses = CCVector.map (C.of_statement ~convert_defs:true) cnf_vec
|> CCVector.to_list
|> CCList.flatten
|> List.map (fun c ->
C.create ~penalty ~trail (CCArray.to_list (C.lits c)) proof) in
Util.debugf ~section 1 "cl:@[%a@]@." (fun k-> k C.pp c);
Util.debugf ~section 1 " @[%a@]@." (fun k-> k (CCList.pp C.pp) clauses);
List.iteri (fun i new_c ->
assert((C.proof_depth c) <= C.proof_depth new_c);) clauses;
Some (solved @clauses)
| None -> None
let cnf_infer cl =
CCOpt.get_or ~default:[] (cnf_otf cl)
let elim_bvars c =
C.Seq.terms c
|> Iter.filter_map (fun v ->
if T.is_var v && Type.is_prop (T.ty v) then (
Some (T.as_var_exn v)
) else None)
|> Iter.sort_uniq ~cmp:(HVar.compare Type.compare)
|> Iter.flat_map_l (fun v ->
let subst_true =
Subst.FO.bind' Subst.empty (v, 0) (T.true_, 0) in
let subst_false =
Subst.FO.bind' Subst.empty (v, 0) (T.false_, 0) in
[subst_true; subst_false])
|> Iter.map (fun subst ->
let proof =
Some (Proof.Step.simp
~tags:[Proof.Tag.T_ho] ~rule:(Proof.Rule.mk "elim_bool_vars")
[C.proof_parent c]) in
C.apply_subst ~proof ~penalty_inc:(Some (-1)) (c,0) subst)
|> Iter.to_list
let interpret_boolean_functions c =
let collect_tl_bool_funs t k =
let rec aux t =
let ty_args, ret_ty = Type.open_fun (T.ty t) in
if not (CCList.is_empty ty_args)
&& Type.is_prop ret_ty
&& not (T.is_var t)
&& not (T.is_app_var t) then k t else(
match T.view t with
| App (f, l) ->
List.iter aux l
| AppBuiltin (b,l) when not (Builtin.is_quantifier b) ->
List.iter aux l
| _ -> ())
in
aux t in
let interpret t i =
let ty_args, body = T.open_fun t in
assert(Type.is_prop (Term.ty body));
T.fun_l ty_args i
in
let negate_bool_fun bool_fun =
let ty_args, body = T.open_fun bool_fun in
assert(Type.is_prop (Term.ty body));
T.fun_l ty_args (T.Form.not_ body)
in
let forall_close t =
let ty_args, body = T.open_fun t in
assert(Type.is_prop (T.ty body));
List.fold_right (fun ty acc ->
T.Form.forall (T.fun_ ty acc)
) ty_args body in
Iter.flat_map collect_tl_bool_funs
(C.Seq.terms c
|> Iter.filter (fun t -> not @@ Type.is_fun (T.ty t)))
|> Iter.filter (fun t ->
let cached_t = Subst.FO.canonize_all_vars t in
not (Term.Set.mem cached_t !Higher_order.prim_enum_terms))
|> Iter.sort_uniq ~cmp:Term.compare
|> Iter.fold (fun res t ->
assert(T.DB.is_closed t);
let proof = Proof.Step.inference [C.proof_parent c]
~rule:(Proof.Rule.mk"interpret boolean function") ~tags:[Proof.Tag.T_ho]
in
let t' = Combs.expand t in
let _,t'_body = T.open_fun t' in
if not (T.is_true_or_false t'_body) then (
let as_forall = Literal.mk_prop (forall_close t') false in
let as_neg_forall = Literal.mk_prop (forall_close (negate_bool_fun t')) false in
let forall_cl =
C.create ~trail:(C.trail c) ~penalty:(C.penalty c)
(as_forall :: Array.to_list(C.lits c |> Literals.map(T.replace ~old:t ~by:(interpret t' T.true_))))
proof in
let forall_neg_cl =
C.create ~trail:(C.trail c) ~penalty:(C.penalty c)
(as_neg_forall :: Array.to_list(C.lits c |> Literals.map(T.replace ~old:t ~by:(interpret t' T.false_))))
proof in
Util.debugf ~section 1 "interpret bool(@[%a@]):@.@[%a@] !!> @. @[%a@]@."
(fun k -> k T.pp t Literals.pp (C.lits c) Literals.pp (C.lits forall_cl));
Util.debugf ~section 1 "interpret bool(@[%a@]):@.@[%a@] !!> @. @[%a@]@."
(fun k -> k T.pp t Literals.pp (C.lits c) Literals.pp (C.lits forall_neg_cl));
forall_cl :: forall_neg_cl :: res
) else res)
[]
let normalize_equalities c =
let lits = Array.to_list (C.lits c) in
let normalized = List.map Literal.normalize_eq lits in
if List.exists CCOpt.is_some normalized then (
let new_lits = List.mapi (fun i l_opt ->
CCOpt.get_or ~default:(Array.get (C.lits c) i) l_opt) normalized in
let proof = Proof.Step.simp [C.proof_parent c]
~rule:(Proof.Rule.mk "simplify nested equalities") in
let new_c = C.create ~trail:(C.trail c) ~penalty:(C.penalty c) new_lits proof in
SimplM.return_new new_c
)
else (
SimplM.return_same c
)
let setup () =
Env.add_basic_simplify normalize_equalities;
match Env.flex_get k_bool_reasoning with
| BoolReasoningDisabled -> ()
| _ ->
if Env.flex_get k_simplify_bools then (
Env.add_basic_simplify simpl_bool_subterms
);
if Env.flex_get k_nnf then (
E.add_basic_simplify nnf_bool_subters;
);
if Env.flex_get k_norm_bools then (
Env.add_basic_simplify normalize_bool_terms
);
if Env.flex_get k_elim_bvars then (
Env.add_unary_inf "elim_bvars" elim_bvars;
);
if not !Lazy_cnf.enabled then (
Env.add_multi_simpl_rule ~priority:2 Fool.rw_bool_lits;
if Env.flex_get k_cnf_non_simpl then (
Env.add_unary_inf "cnf otf inf" cnf_infer;
) else Env.add_multi_simpl_rule ~priority:2 cnf_otf);
if (Env.flex_get k_interpret_bool_funs) then (
Env.add_unary_inf "interpret boolean functions" interpret_boolean_functions;
);
if Env.flex_get k_bool_reasoning = BoolCasesInference then (
Env.add_unary_inf "bool_cases" bool_case_inf;
)
else if Env.flex_get k_bool_reasoning = BoolCasesSimplification then (
Env.add_single_step_multi_simpl_rule bool_case_simp;
) else if Env.flex_get k_bool_reasoning = BoolCasesKeepParent then (
let keep_parent c = CCOpt.get_or ~default:[] (bool_case_simp c) in
Env.add_unary_inf "bool_cases_keep_parent" keep_parent;
)
end
open CCFun
open Builtin
open Statement
open TypedSTerm
open CCList
let if_changed proof (mk: ?attrs:Logtk.Statement.attrs -> 'r) s f p =
let fp = f s p in
if fp == [p] then [s] else map(fun x -> mk ~proof:(proof s) x) fp
let map_propositions ~proof f =
CCVector.flat_map_list(fun s -> match Statement.view s with
| Assert p -> if_changed proof assert_ s f p
| Lemma ps -> if_changed proof lemma s (map%f) ps
| Goal p -> if_changed proof goal s f p
| NegatedGoal(ts, ps) -> if_changed proof (neg_goal ~skolems:ts) s (map%f) ps
| _ -> [s]
)
let is_bool t = CCOpt.equal Ty.equal (Some prop) (ty t)
let is_T_F t = match view t with AppBuiltin((True|False),[]) -> true | _ -> false
let rec replaceTST f top t =
let re = replaceTST f in
let ty = ty_exn t in
let transformer = if top then id else f in
transformer
(match view t with
| App(t,ts) ->
app_whnf ~ty (re false t) (map (re false) ts)
| Ite(c,x,y) ->
ite (re false c) (re false x) (re false y)
| Match(t, cases) ->
match_ (re false t) (map (fun (c,vs,e) -> (c,vs, re false e)) cases)
| Let(binds, expr) ->
let_ (map(CCPair.map_snd (re false)) binds) (re false expr)
| Bind(b,x,t) ->
let top = Binder.equal b Binder.Forall || Binder.equal b Binder.Exists in
bind ~ty b x (re top t)
| AppBuiltin(b,ts) ->
let logical = for_all is_bool ts in
app_builtin ~ty b (map (re(top && logical)) ts)
| Multiset ts ->
multiset ~ty (map (re false) ts)
| _ -> t)
let name_quantifiers stmts =
let proof s = Proof.Step.esa [Proof.Parent.from(Statement.as_proof_i s)]
~rule:(Proof.Rule.mk "Quantifier naming")
in
let new_stmts = CCVector.create() in
let changed = ref false in
let if_changed (mk: ?attrs:Logtk.Statement.attrs -> 'r) s r =
if !changed then (changed := false; mk ~proof:(proof s) r) else s in
let if_changed_list (mk: ?attrs:Logtk.Statement.attrs -> 'l) s r =
if !changed then (changed := false; mk ~proof:(proof s) r) else s in
let name_prop_Qs s = replaceTST(fun t -> match TypedSTerm.view t with
| Bind(Binder.Forall,_,_) | Bind(Binder.Exists, _, _) ->
changed := true;
let vars = Var.Set.of_iter (TypedSTerm.Seq.free_vars t) |> Var.Set.to_list in
let qid = ID.gensym() in
let ty = app_builtin ~ty:tType Arrow (prop :: map Var.ty vars) in
let q = const ~ty qid in
let q_vars = app ~ty:prop q (map var vars) in
let proof = Proof.Step.define_internal qid [Proof.Parent.from(Statement.as_proof_i s)] in
let q_typedecl = ty_decl ~proof qid ty in
let definition =
bind_list ~ty:prop Binder.Forall vars
(app_builtin ~ty:prop Builtin.Equiv [q_vars; t])
in
CCVector.push new_stmts q_typedecl;
CCVector.push new_stmts (assert_ ~proof definition);
q_vars
| _ -> t) true
in
stmts |> CCVector.map(fun s ->
match Statement.view s with
| TyDecl(id,t) -> s
| Data ts -> s
| Def defs -> s
| Rewrite _ -> s
| Assert p -> if_changed assert_ s (name_prop_Qs s p)
| Lemma ps -> if_changed_list lemma s (map (name_prop_Qs s) ps)
| Goal p -> if_changed goal s (name_prop_Qs s p)
| NegatedGoal(ts, ps) -> if_changed_list (neg_goal ~skolems:ts) s (map (name_prop_Qs s) ps)
) |> CCVector.append new_stmts;
CCVector.freeze new_stmts
let rec replace old by t =
let r = replace old by in
let ty = ty_exn t in
if TypedSTerm.equal t old then by
else match view t with
| App(f,ps) -> app_whnf ~ty (r f) (map r ps)
| AppBuiltin(f,ps) -> app_builtin ~ty f (map r ps)
| Ite(c,x,y) -> ite (r c) (r x) (r y)
| Let(bs,e) -> let_ (map (CCPair.map_snd r) bs) (r e)
| Bind(b,v,e) -> bind ~ty b v (r e)
| _ -> t
exception Return of TypedSTerm.t
let with_subterm_or_id t f = try
(Seq.subterms_with_bound t (fun(s, var_ctx) ->
match f var_ctx s with
| None -> ()
| Some r -> raise(Return r)));
t
with Return r -> r
let case_bool vs c p =
if is_bool p && not(is_T_F p) && not (TypedSTerm.equal p c) && Var.Set.is_empty(Var.Set.diff (free_vars_set p) vs) then
let ty = prop in
app_builtin ~ty And [
app_builtin ~ty Imply [p; replace p Form.true_ c];
app_builtin ~ty Or [p; replace p Form.false_ c];
]
else c
let rec case_bools_wrt vs t =
with_subterm_or_id t (fun _ s ->
match view s with
| App(f,ps) ->
let t' = fold_left (case_bool vs) t ps in
if TypedSTerm.equal t t' then None else Some(case_bools_wrt vs t')
| _ -> None
)
let eager_cases_far stms =
let proof s = Proof.Step.esa [Proof.Parent.from(Statement.as_proof_i s)]
~rule:(Proof.Rule.mk "eager_cases_far")
in
map_propositions ~proof (fun _ t ->
[with_subterm_or_id t (fun vs s ->
match view s with
| Bind((Forall|Exists) as q, v, b) ->
let b' = case_bools_wrt (Var.Set.add vs v) b in
if TypedSTerm.equal b b' then None else Some(replace s (bind ~ty:prop q v b') t)
| _ -> None)
|> case_bools_wrt Var.Set.empty]) stms
let eager_cases_near stms =
let proof s = Proof.Step.esa [Proof.Parent.from(Statement.as_proof_i s)]
~rule:(Proof.Rule.mk "eager_cases_near")
in
let module T = TypedSTerm in
let find_fool_subterm p =
let rec aux ~top p =
let p_ty = T.ty_exn p in
let return p =
assert(T.Ty.is_prop (T.ty_exn p));
assert(T.closed p);
Some (T.Form.true_, T.Form.false_, p) in
match T.view p with
| AppBuiltin(hd, args)
when not top && T.closed p && T.Ty.is_prop (T.ty_exn p) &&
(Builtin.is_logical_op hd ||
Builtin.equal hd Builtin.Eq ||
Builtin.equal hd Builtin.Neq) ->
return p
| Bind((Binder.Exists | Binder.Forall), var, body)
when not top && T.closed p ->
return p
| Bind(Binder.Lambda, var, body) ->
CCOpt.map (fun (body_t, body_f, s) ->
assert(T.closed s);
(T.fun_l [var] body_t, T.fun_l [var] body_f, s)
) (aux ~top:false body)
| App(hd, args) when not top && T.Ty.is_prop p_ty && T.closed p ->
return p
| Const _ when not top && T.Ty.is_prop p_ty ->
return p
| AppBuiltin(b, args) ->
CCOpt.map (fun (args_t,args_f, s) ->
(T.app_builtin ~ty:p_ty b args_t, T.app_builtin ~ty:p_ty b args_f, s)
) (aux_l args)
| App(hd,args) ->
CCOpt.map (fun (args_t,args_f, s) ->
(T.app ~ty:p_ty hd args_t, T.app ~ty:p_ty hd args_f, s)
) (aux_l args)
| _ -> None
and aux_l = function
| [] -> None
| x :: xs ->
begin match aux ~top:false x with
| Some (x_t, x_f, s) -> Some(x_t::xs, x_f::xs, s)
| None ->
begin match aux_l xs with
| Some (xs_t, xs_f, s) -> Some (x::xs_t, x::xs_f, s)
| None -> None end
end in
let res = aux ~top:true p in
res in
let unroll_fool p =
let rec aux p =
let p_ty = T.ty_exn p in
match T.view p with
| AppBuiltin(((Builtin.Neq|Builtin.Eq) as hd), [a;b]) when not (T.Ty.is_prop (T.ty_exn a)) ->
let cons = if hd = Neq then T.Form.neq else T.Form.eq in
begin match find_fool_subterm a with
| None ->
begin match find_fool_subterm b with
| None -> p
| Some(b_t, b_f, subterm) ->
let subterm' = aux subterm in
let if_true = T.Form.or_ [T.Form.not_ (subterm'); aux @@ cons a b_t] in
let if_false = T.Form.or_ [subterm'; aux @@ cons a b_f] in
T.Form.and_ [if_true; if_false]
end
| Some(a_t, a_f, subterm) ->
let subterm' = aux subterm in
let if_true = T.Form.or_ [T.Form.not_ (subterm'); aux @@ cons a_t b] in
let if_false = T.Form.or_ [subterm'; aux @@ cons a_f b] in
T.Form.and_ [if_true; if_false]
end
| AppBuiltin(hd, args) ->
T.app_builtin ~ty:p_ty hd (List.map aux args)
| App(hd, args) ->
begin match find_fool_subterm p with
| Some(p_t, p_f, subterm) ->
let subterm' = aux subterm in
let if_true = T.Form.or_ [T.Form.not_ (subterm'); aux p_t] in
let if_false = T.Form.or_ [subterm'; aux p_f] in
T.Form.and_ [if_true; if_false]
| None -> p end
| Bind((Binder.Exists | Binder.Forall) as b, var , body) ->
let body' = aux body in
T.bind ~ty:p_ty b var body'
| _ -> p in
let res = aux p in
res in
map_propositions ~proof (fun _ p -> [unroll_fool p]) stms
open Term
let post_eager_cases =
let proof s = Proof.Step.esa [Proof.Parent.from(Statement.as_proof_c s)]
~rule:(Proof.Rule.mk "post_eager_cases")
in
map_propositions ~proof (fun _ c ->
let cased = ref Set.empty in
fold_left(SLiteral.fold(fun res ->
Seq.subterms_depth %> Iter.fold(fun res (s,d) ->
if d = 0 || not(Type.is_prop(ty s)) || is_true_or_false s || is_var s || Set.mem s !cased
|| not (T.DB.is_closed s)
then
res
else(
cased := Set.add s !cased;
let replace_s_by by = map(SLiteral.map ~f:(replace ~old:s ~by)) in
flatten(map(fun c -> [
SLiteral.atom_true s :: replace_s_by false_ c;
SLiteral.atom_false s :: replace_s_by true_ c
]) res))
) res
)) [c] c)
let _bool_reasoning = ref BoolReasoningDisabled
let _quant_rename = ref false
let preprocess_booleans stmts = (match !_bool_reasoning with
| BoolCasesPreprocess -> eager_cases_near
| _ -> id
) (if !_quant_rename then name_quantifiers stmts else stmts)
let preprocess_cnf_booleans stmts = match !_bool_reasoning with
| BoolCasesPreprocess ->
let res = post_eager_cases stmts in
res
| _ -> stmts
let _cased_term_selection = ref Large
let _interpret_bool_funs = ref false
let _cnf_non_simpl = ref false
let _norm_bools = ref false
let _filter_literals = ref `Max
let _nnf = ref false
let _simplify_bools = ref true
let _elim_bvars = ref true
let extension =
let register env =
let module E = (val env : Env.S) in
let module ET = Make(E) in
E.flex_add k_bool_reasoning !_bool_reasoning;
E.flex_add k_cased_term_selection !_cased_term_selection;
E.flex_add k_quant_rename !_quant_rename;
E.flex_add k_interpret_bool_funs !_interpret_bool_funs;
E.flex_add k_cnf_non_simpl !_cnf_non_simpl;
E.flex_add k_norm_bools !_norm_bools;
E.flex_add k_filter_literals !_filter_literals;
E.flex_add k_nnf !_nnf;
E.flex_add k_elim_bvars !_elim_bvars;
E.flex_add k_simplify_bools !_simplify_bools;
ET.setup ()
in
{ Extensions.default with
Extensions.name = "bool";
env_actions=[register];
}
let () =
Options.add_opts
[ "--boolean-reasoning", Arg.Symbol (["off"; "no-cases"; "cases-inf"; "cases-simpl"; "cases-simpl-kp"; "cases-eager"; "cases-preprocess"],
fun s -> _bool_reasoning :=
match s with
| "off" -> BoolReasoningDisabled
| "no-cases" -> BoolCasesDisabled
| "cases-inf" -> BoolCasesInference
| "cases-simpl" -> BoolCasesSimplification
| "cases-simpl-kp" -> BoolCasesKeepParent
| "cases-preprocess" -> BoolCasesPreprocess
| _ -> assert false),
" enable/disable boolean axioms";
"--bool-subterm-selection",
Arg.Symbol(["A"; "M"; "L"], (fun opt -> _cased_term_selection :=
match opt with "A"->Any | "M"->Minimal | "L"->Large
| _ -> assert false)),
" select boolean subterm selection criterion: A for any, M for minimal and L for large";
"--quantifier-renaming"
, Arg.Bool (fun v -> _quant_rename := v)
, " turn the quantifier renaming on or off";
"--disable-simplifying-cnf",
Arg.Set _cnf_non_simpl,
" implement cnf on-the-fly as an inference rule";
"--interpret-bool-funs"
, Arg.Bool (fun v -> _interpret_bool_funs := v)
, " turn interpretation of boolean functions as forall or negation of forall on or off";
"--normalize-bool-terms", Arg.Bool((fun v -> _norm_bools := v)),
" normalize boolean subterms using their weight.";
"--nnf-nested-formulas"
, Arg.Bool (fun v -> _nnf := v)
, " convert nested formulas into negation normal form";
"--simplify-bools"
, Arg.Bool (fun v -> _simplify_bools := v)
, " simplify boolean subterms";
"--elim-bvars"
, Arg.Bool ((:=) _elim_bvars)
, " replace boolean variables by T and F";
"--boolean-reasoning-filter-literals"
, Arg.Symbol(["all"; "max"], (fun v ->
match v with
| "all" -> _filter_literals:=`All
| "max" -> _filter_literals:= `Max
| _ -> assert false;))
, " select on which literals to apply bool reasoning rules"
];
Params.add_to_modes ["ho-complete-basic";
"ho-pragmatic";
"lambda-free-intensional";
"lambda-free-purify-intensional";
"lambda-free-extensional";
"ho-comb-complete";
"lambda-free-purify-extensional";
"fo-complete-basic"] (fun () ->
_bool_reasoning := BoolReasoningDisabled
);
Extensions.register extension