(* Title: HOL/arith_data.ML
ID: $Id$
Author: Markus Wenzel, Stefan Berghofer and Tobias Nipkow
Various arithmetic proof procedures.
*)
(*---------------------------------------------------------------------------*)
(* 1. Cancellation of common terms *)
(*---------------------------------------------------------------------------*)
structure NatArithUtils =
struct
(** abstract syntax of structure nat: 0, Suc, + **)
(* mk_sum, mk_norm_sum *)
val mk_plus = HOLogic.mk_binop @{const_name HOL.plus};
fun mk_sum [] = HOLogic.zero
| mk_sum [t] = t
| mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
(*normal form of sums: Suc (... (Suc (a + (b + ...))))*)
fun mk_norm_sum ts =
let val (ones, sums) = List.partition (equal HOLogic.Suc_zero) ts in
funpow (length ones) HOLogic.mk_Suc (mk_sum sums)
end;
(* dest_sum *)
val dest_plus = HOLogic.dest_bin @{const_name HOL.plus} HOLogic.natT;
fun dest_sum tm =
if HOLogic.is_zero tm then []
else
(case try HOLogic.dest_Suc tm of
SOME t => HOLogic.Suc_zero :: dest_sum t
| NONE =>
(case try dest_plus tm of
SOME (t, u) => dest_sum t @ dest_sum u
| NONE => [tm]));
(** generic proof tools **)
(* prove conversions *)
fun prove_conv expand_tac norm_tac ss tu = (* FIXME avoid standard *)
mk_meta_eq (standard (Goal.prove (Simplifier.the_context ss) [] []
(HOLogic.mk_Trueprop (HOLogic.mk_eq tu))
(K (EVERY [expand_tac, norm_tac ss]))));
(* rewriting *)
fun simp_all_tac rules =
let val ss0 = HOL_ss addsimps rules
in fn ss => ALLGOALS (simp_tac (Simplifier.inherit_context ss ss0)) end;
fun prep_simproc (name, pats, proc) =
Simplifier.simproc (the_context ()) name pats proc;
end; (* NatArithUtils *)
signature ARITH_DATA =
sig
val nat_cancel_sums_add: simproc list
val nat_cancel_sums: simproc list
end;
structure ArithData: ARITH_DATA =
struct
open NatArithUtils;
(** cancel common summands **)
structure Sum =
struct
val mk_sum = mk_norm_sum;
val dest_sum = dest_sum;
val prove_conv = prove_conv;
val norm_tac1 = simp_all_tac [@{thm "add_Suc"}, @{thm "add_Suc_right"},
@{thm "add_0"}, @{thm "add_0_right"}];
val norm_tac2 = simp_all_tac @{thms add_ac};
fun norm_tac ss = norm_tac1 ss THEN norm_tac2 ss;
end;
fun gen_uncancel_tac rule ct =
rtac (instantiate' [] [NONE, SOME ct] (rule RS @{thm subst_equals})) 1;
(* nat eq *)
structure EqCancelSums = CancelSumsFun
(struct
open Sum;
val mk_bal = HOLogic.mk_eq;
val dest_bal = HOLogic.dest_bin "op =" HOLogic.natT;
val uncancel_tac = gen_uncancel_tac @{thm "nat_add_left_cancel"};
end);
(* nat less *)
structure LessCancelSums = CancelSumsFun
(struct
open Sum;
val mk_bal = HOLogic.mk_binrel @{const_name Orderings.less};
val dest_bal = HOLogic.dest_bin @{const_name Orderings.less} HOLogic.natT;
val uncancel_tac = gen_uncancel_tac @{thm "nat_add_left_cancel_less"};
end);
(* nat le *)
structure LeCancelSums = CancelSumsFun
(struct
open Sum;
val mk_bal = HOLogic.mk_binrel @{const_name Orderings.less_eq};
val dest_bal = HOLogic.dest_bin @{const_name Orderings.less_eq} HOLogic.natT;
val uncancel_tac = gen_uncancel_tac @{thm "nat_add_left_cancel_le"};
end);
(* nat diff *)
structure DiffCancelSums = CancelSumsFun
(struct
open Sum;
val mk_bal = HOLogic.mk_binop @{const_name HOL.minus};
val dest_bal = HOLogic.dest_bin @{const_name HOL.minus} HOLogic.natT;
val uncancel_tac = gen_uncancel_tac @{thm "diff_cancel"};
end);
(** prepare nat_cancel simprocs **)
val nat_cancel_sums_add = map prep_simproc
[("nateq_cancel_sums",
["(l::nat) + m = n", "(l::nat) = m + n", "Suc m = n", "m = Suc n"],
K EqCancelSums.proc),
("natless_cancel_sums",
["(l::nat) + m < n", "(l::nat) < m + n", "Suc m < n", "m < Suc n"],
K LessCancelSums.proc),
("natle_cancel_sums",
["(l::nat) + m <= n", "(l::nat) <= m + n", "Suc m <= n", "m <= Suc n"],
K LeCancelSums.proc)];
val nat_cancel_sums = nat_cancel_sums_add @
[prep_simproc ("natdiff_cancel_sums",
["((l::nat) + m) - n", "(l::nat) - (m + n)", "Suc m - n", "m - Suc n"],
K DiffCancelSums.proc)];
end; (* ArithData *)
open ArithData;
(*---------------------------------------------------------------------------*)
(* 2. Linear arithmetic *)
(*---------------------------------------------------------------------------*)
(* Parameters data for general linear arithmetic functor *)
structure LA_Logic: LIN_ARITH_LOGIC =
struct
val ccontr = ccontr;
val conjI = conjI;
val notI = notI;
val sym = sym;
val not_lessD = @{thm linorder_not_less} RS iffD1;
val not_leD = @{thm linorder_not_le} RS iffD1;
val le0 = thm "le0";
fun mk_Eq thm = (thm RS Eq_FalseI) handle THM _ => (thm RS Eq_TrueI);
val mk_Trueprop = HOLogic.mk_Trueprop;
fun atomize thm = case #prop(rep_thm thm) of
Const("Trueprop",_) $ (Const("op &",_) $ _ $ _) =>
atomize(thm RS conjunct1) @ atomize(thm RS conjunct2)
| _ => [thm];
fun neg_prop ((TP as Const("Trueprop",_)) $ (Const("Not",_) $ t)) = TP $ t
| neg_prop ((TP as Const("Trueprop",_)) $ t) = TP $ (HOLogic.Not $t)
| neg_prop t = raise TERM ("neg_prop", [t]);
fun is_False thm =
let val _ $ t = #prop(rep_thm thm)
in t = Const("False",HOLogic.boolT) end;
fun is_nat(t) = fastype_of1 t = HOLogic.natT;
fun mk_nat_thm sg t =
let val ct = cterm_of sg t and cn = cterm_of sg (Var(("n",0),HOLogic.natT))
in instantiate ([],[(cn,ct)]) le0 end;
end; (* LA_Logic *)
(* arith theory data *)
datatype arithtactic = ArithTactic of {name: string, tactic: int -> tactic, id: stamp};
fun mk_arith_tactic name tactic = ArithTactic {name = name, tactic = tactic, id = stamp ()};
fun eq_arith_tactic (ArithTactic {id = id1, ...}, ArithTactic {id = id2, ...}) = (id1 = id2);
structure ArithTheoryData = TheoryDataFun
(
type T = {splits: thm list,
inj_consts: (string * typ) list,
discrete: string list,
tactics: arithtactic list};
val empty = {splits = [], inj_consts = [], discrete = [], tactics = []};
val copy = I;
val extend = I;
fun merge _ ({splits= splits1, inj_consts= inj_consts1, discrete= discrete1, tactics= tactics1},
{splits= splits2, inj_consts= inj_consts2, discrete= discrete2, tactics= tactics2}) =
{splits = Library.merge Thm.eq_thm_prop (splits1, splits2),
inj_consts = Library.merge (op =) (inj_consts1, inj_consts2),
discrete = Library.merge (op =) (discrete1, discrete2),
tactics = Library.merge eq_arith_tactic (tactics1, tactics2)};
);
val arith_split_add = Thm.declaration_attribute (fn thm =>
Context.mapping (ArithTheoryData.map (fn {splits,inj_consts,discrete,tactics} =>
{splits= insert Thm.eq_thm_prop thm splits, inj_consts= inj_consts, discrete= discrete, tactics= tactics})) I);
fun arith_discrete d = ArithTheoryData.map (fn {splits,inj_consts,discrete,tactics} =>
{splits = splits, inj_consts = inj_consts, discrete = insert (op =) d discrete, tactics= tactics});
fun arith_inj_const c = ArithTheoryData.map (fn {splits,inj_consts,discrete,tactics} =>
{splits = splits, inj_consts = insert (op =) c inj_consts, discrete = discrete, tactics= tactics});
fun arith_tactic_add tac = ArithTheoryData.map (fn {splits,inj_consts,discrete,tactics} =>
{splits= splits, inj_consts= inj_consts, discrete= discrete, tactics= insert eq_arith_tactic tac tactics});
signature HOL_LIN_ARITH_DATA =
sig
include LIN_ARITH_DATA
val fast_arith_split_limit : int ref
end;
structure LA_Data_Ref: HOL_LIN_ARITH_DATA =
struct
(* internal representation of linear (in-)equations *)
type decompT = ((term * Rat.rat) list * Rat.rat * string * (term * Rat.rat) list * Rat.rat * bool);
(* Decomposition of terms *)
fun nT (Type ("fun", [N, _])) = (N = HOLogic.natT)
| nT _ = false;
fun add_atom (t : term) (m : Rat.rat) (p : (term * Rat.rat) list, i : Rat.rat) :
(term * Rat.rat) list * Rat.rat =
case AList.lookup (op =) p t of NONE => ((t, m) :: p, i)
| SOME n => (AList.update (op =) (t, Rat.add n m) p, i);
exception Zero;
fun rat_of_term (numt, dent) =
let
val num = HOLogic.dest_numeral numt
val den = HOLogic.dest_numeral dent
in
if den = 0 then raise Zero else Rat.rat_of_quotient (num, den)
end;
(* Warning: in rare cases number_of encloses a non-numeral,
in which case dest_numeral raises TERM; hence all the handles below.
Same for Suc-terms that turn out not to be numerals -
although the simplifier should eliminate those anyway ...
*)
fun number_of_Sucs (Const ("Suc", _) $ n) : int =
number_of_Sucs n + 1
| number_of_Sucs t =
if HOLogic.is_zero t then 0 else raise TERM ("number_of_Sucs", []);
(* decompose nested multiplications, bracketing them to the right and combining
all their coefficients
*)
fun demult (inj_consts : (string * typ) list) : term * Rat.rat -> term option * Rat.rat =
let
fun demult ((mC as Const (@{const_name HOL.times}, _)) $ s $ t, m) = (
(case s of
Const ("Numeral.number_class.number_of", _) $ n =>
demult (t, Rat.mult m (Rat.rat_of_int (HOLogic.dest_numeral n)))
| Const (@{const_name HOL.uminus}, _) $ (Const ("Numeral.number_class.number_of", _) $ n) =>
demult (t, Rat.mult m (Rat.rat_of_int (~(HOLogic.dest_numeral n))))
| Const (@{const_name Suc}, _) $ _ =>
demult (t, Rat.mult m (Rat.rat_of_int (HOLogic.dest_nat s)))
| Const (@{const_name HOL.times}, _) $ s1 $ s2 =>
demult (mC $ s1 $ (mC $ s2 $ t), m)
| Const (@{const_name HOL.divide}, _) $ numt $ (Const ("Numeral.number_class.number_of", _) $ dent) =>
let
val den = HOLogic.dest_numeral dent
in
if den = 0 then
raise Zero
else
demult (mC $ numt $ t, Rat.mult m (Rat.inv (Rat.rat_of_int den)))
end
| _ =>
atomult (mC, s, t, m)
) handle TERM _ => atomult (mC, s, t, m)
)
| demult (atom as Const(@{const_name HOL.divide}, _) $ t $ (Const ("Numeral.number_class.number_of", _) $ dent), m) =
(let
val den = HOLogic.dest_numeral dent
in
if den = 0 then
raise Zero
else
demult (t, Rat.mult m (Rat.inv (Rat.rat_of_int den)))
end
handle TERM _ => (SOME atom, m))
| demult (Const (@{const_name HOL.zero}, _), m) = (NONE, Rat.zero)
| demult (Const (@{const_name HOL.one}, _), m) = (NONE, m)
| demult (t as Const ("Numeral.number_class.number_of", _) $ n, m) =
((NONE, Rat.mult m (Rat.rat_of_int (HOLogic.dest_numeral n)))
handle TERM _ => (SOME t, m))
| demult (Const (@{const_name HOL.uminus}, _) $ t, m) = demult (t, Rat.neg m)
| demult (t as Const f $ x, m) =
(if member (op =) inj_consts f then SOME x else SOME t, m)
| demult (atom, m) = (SOME atom, m)
and
atomult (mC, atom, t, m) = (
case demult (t, m) of (NONE, m') => (SOME atom, m')
| (SOME t', m') => (SOME (mC $ atom $ t'), m')
)
in demult end;
fun decomp0 (inj_consts : (string * typ) list) (rel : string, lhs : term, rhs : term) :
((term * Rat.rat) list * Rat.rat * string * (term * Rat.rat) list * Rat.rat) option =
let
(* Turn term into list of summand * multiplicity plus a constant *)
fun poly (Const (@{const_name HOL.plus}, _) $ s $ t, m : Rat.rat, pi : (term * Rat.rat) list * Rat.rat) =
poly (s, m, poly (t, m, pi))
| poly (all as Const (@{const_name HOL.minus}, T) $ s $ t, m, pi) =
if nT T then add_atom all m pi else poly (s, m, poly (t, Rat.neg m, pi))
| poly (all as Const (@{const_name HOL.uminus}, T) $ t, m, pi) =
if nT T then add_atom all m pi else poly (t, Rat.neg m, pi)
| poly (Const (@{const_name HOL.zero}, _), _, pi) =
pi
| poly (Const (@{const_name HOL.one}, _), m, (p, i)) =
(p, Rat.add i m)
| poly (Const (@{const_name Suc}, _) $ t, m, (p, i)) =
poly (t, m, (p, Rat.add i m))
| poly (all as Const (@{const_name HOL.times}, _) $ _ $ _, m, pi as (p, i)) =
(case demult inj_consts (all, m) of
(NONE, m') => (p, Rat.add i m')
| (SOME u, m') => add_atom u m' pi)
| poly (all as Const (@{const_name HOL.divide}, _) $ _ $ _, m, pi as (p, i)) =
(case demult inj_consts (all, m) of
(NONE, m') => (p, Rat.add i m')
| (SOME u, m') => add_atom u m' pi)
| poly (all as Const ("Numeral.number_class.number_of", Type(_,[_,T])) $ t, m, pi as (p, i)) =
(let val k = HOLogic.dest_numeral t
val k2 = if k < 0 andalso T = HOLogic.natT then 0 else k
in (p, Rat.add i (Rat.mult m (Rat.rat_of_int k2))) end
handle TERM _ => add_atom all m pi)
| poly (all as Const f $ x, m, pi) =
if f mem inj_consts then poly (x, m, pi) else add_atom all m pi
| poly (all, m, pi) =
add_atom all m pi
val (p, i) = poly (lhs, Rat.one, ([], Rat.zero))
val (q, j) = poly (rhs, Rat.one, ([], Rat.zero))
in
case rel of
@{const_name Orderings.less} => SOME (p, i, "<", q, j)
| @{const_name Orderings.less_eq} => SOME (p, i, "<=", q, j)
| "op =" => SOME (p, i, "=", q, j)
| _ => NONE
end handle Zero => NONE;
fun of_lin_arith_sort sg (U : typ) : bool =
Type.of_sort (Sign.tsig_of sg) (U, ["Ring_and_Field.ordered_idom"])
fun allows_lin_arith sg (discrete : string list) (U as Type (D, [])) : bool * bool =
if of_lin_arith_sort sg U then
(true, D mem discrete)
else (* special cases *)
if D mem discrete then (true, true) else (false, false)
| allows_lin_arith sg discrete U =
(of_lin_arith_sort sg U, false);
fun decomp_typecheck (sg, discrete, inj_consts) (T : typ, xxx) : decompT option =
case T of
Type ("fun", [U, _]) =>
(case allows_lin_arith sg discrete U of
(true, d) =>
(case decomp0 inj_consts xxx of
NONE => NONE
| SOME (p, i, rel, q, j) => SOME (p, i, rel, q, j, d))
| (false, _) =>
NONE)
| _ => NONE;
fun negate (SOME (x, i, rel, y, j, d)) = SOME (x, i, "~" ^ rel, y, j, d)
| negate NONE = NONE;
fun decomp_negation data
((Const ("Trueprop", _)) $ (Const (rel, T) $ lhs $ rhs)) : decompT option =
decomp_typecheck data (T, (rel, lhs, rhs))
| decomp_negation data ((Const ("Trueprop", _)) $
(Const ("Not", _) $ (Const (rel, T) $ lhs $ rhs))) =
negate (decomp_typecheck data (T, (rel, lhs, rhs)))
| decomp_negation data _ =
NONE;
fun decomp sg : term -> decompT option =
let
val {discrete, inj_consts, ...} = ArithTheoryData.get sg
in
decomp_negation (sg, discrete, inj_consts)
end;
fun domain_is_nat (_ $ (Const (_, T) $ _ $ _)) = nT T
| domain_is_nat (_ $ (Const ("Not", _) $ (Const (_, T) $ _ $ _))) = nT T
| domain_is_nat _ = false;
fun number_of (n, T) = HOLogic.mk_number T n;
(*---------------------------------------------------------------------------*)
(* code that performs certain goal transformations for linear arithmetic *)
(*---------------------------------------------------------------------------*)
(* A "do nothing" variant of pre_decomp and pre_tac:
fun pre_decomp sg Ts termitems = [termitems];
fun pre_tac i = all_tac;
*)
(*---------------------------------------------------------------------------*)
(* the following code performs splitting of certain constants (e.g. min, *)
(* max) in a linear arithmetic problem; similar to what split_tac later does *)
(* to the proof state *)
(*---------------------------------------------------------------------------*)
val fast_arith_split_limit = ref 9;
(* checks if splitting with 'thm' is implemented *)
fun is_split_thm (thm : thm) : bool =
case concl_of thm of _ $ (_ $ (_ $ lhs) $ _) => (
(* Trueprop $ ((op =) $ (?P $ lhs) $ rhs) *)
case head_of lhs of
Const (a, _) => member (op =) [@{const_name Orderings.max},
@{const_name Orderings.min},
@{const_name HOL.abs},
@{const_name HOL.minus},
"IntDef.nat",
"Divides.div_class.mod",
"Divides.div_class.div"] a
| _ => (warning ("Lin. Arith.: wrong format for split rule " ^
Display.string_of_thm thm);
false))
| _ => (warning ("Lin. Arith.: wrong format for split rule " ^
Display.string_of_thm thm);
false);
(* substitute new for occurrences of old in a term, incrementing bound *)
(* variables as needed when substituting inside an abstraction *)
fun subst_term ([] : (term * term) list) (t : term) = t
| subst_term pairs t =
(case AList.lookup (op aconv) pairs t of
SOME new =>
new
| NONE =>
(case t of Abs (a, T, body) =>
let val pairs' = map (pairself (incr_boundvars 1)) pairs
in Abs (a, T, subst_term pairs' body) end
| t1 $ t2 =>
subst_term pairs t1 $ subst_term pairs t2
| _ => t));
(* approximates the effect of one application of split_tac (followed by NNF *)
(* normalization) on the subgoal represented by '(Ts, terms)'; returns a *)
(* list of new subgoals (each again represented by a typ list for bound *)
(* variables and a term list for premises), or NONE if split_tac would fail *)
(* on the subgoal *)
(* FIXME: currently only the effect of certain split theorems is reproduced *)
(* (which is why we need 'is_split_thm'). A more canonical *)
(* implementation should analyze the right-hand side of the split *)
(* theorem that can be applied, and modify the subgoal accordingly. *)
(* Or even better, the splitter should be extended to provide *)
(* splitting on terms as well as splitting on theorems (where the *)
(* former can have a faster implementation as it does not need to be *)
(* proof-producing). *)
fun split_once_items (sg : theory) (Ts : typ list, terms : term list) :
(typ list * term list) list option =
let
(* takes a list [t1, ..., tn] to the term *)
(* tn' --> ... --> t1' --> False , *)
(* where ti' = HOLogic.dest_Trueprop ti *)
(* term list -> term *)
fun REPEAT_DETERM_etac_rev_mp terms' =
fold (curry HOLogic.mk_imp) (map HOLogic.dest_Trueprop terms') HOLogic.false_const
val split_thms = filter is_split_thm (#splits (ArithTheoryData.get sg))
val cmap = Splitter.cmap_of_split_thms split_thms
val splits = Splitter.split_posns cmap sg Ts (REPEAT_DETERM_etac_rev_mp terms)
in
if length splits > !fast_arith_split_limit then (
tracing ("fast_arith_split_limit exceeded (current value is " ^
string_of_int (!fast_arith_split_limit) ^ ")");
NONE
) else (
case splits of [] =>
(* split_tac would fail: no possible split *)
NONE
| ((_, _, _, split_type, split_term) :: _) => (
(* ignore all but the first possible split *)
case strip_comb split_term of
(* ?P (max ?i ?j) = ((?i <= ?j --> ?P ?j) & (~ ?i <= ?j --> ?P ?i)) *)
(Const (@{const_name Orderings.max}, _), [t1, t2]) =>
let
val rev_terms = rev terms
val terms1 = map (subst_term [(split_term, t1)]) rev_terms
val terms2 = map (subst_term [(split_term, t2)]) rev_terms
val t1_leq_t2 = Const (@{const_name Orderings.less_eq},
split_type --> split_type --> HOLogic.boolT) $ t1 $ t2
val not_t1_leq_t2 = HOLogic.Not $ t1_leq_t2
val not_false = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
val subgoal1 = (HOLogic.mk_Trueprop t1_leq_t2) :: terms2 @ [not_false]
val subgoal2 = (HOLogic.mk_Trueprop not_t1_leq_t2) :: terms1 @ [not_false]
in
SOME [(Ts, subgoal1), (Ts, subgoal2)]
end
(* ?P (min ?i ?j) = ((?i <= ?j --> ?P ?i) & (~ ?i <= ?j --> ?P ?j)) *)
| (Const (@{const_name Orderings.min}, _), [t1, t2]) =>
let
val rev_terms = rev terms
val terms1 = map (subst_term [(split_term, t1)]) rev_terms
val terms2 = map (subst_term [(split_term, t2)]) rev_terms
val t1_leq_t2 = Const (@{const_name Orderings.less_eq},
split_type --> split_type --> HOLogic.boolT) $ t1 $ t2
val not_t1_leq_t2 = HOLogic.Not $ t1_leq_t2
val not_false = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
val subgoal1 = (HOLogic.mk_Trueprop t1_leq_t2) :: terms1 @ [not_false]
val subgoal2 = (HOLogic.mk_Trueprop not_t1_leq_t2) :: terms2 @ [not_false]
in
SOME [(Ts, subgoal1), (Ts, subgoal2)]
end
(* ?P (abs ?a) = ((0 <= ?a --> ?P ?a) & (?a < 0 --> ?P (- ?a))) *)
| (Const (@{const_name HOL.abs}, _), [t1]) =>
let
val rev_terms = rev terms
val terms1 = map (subst_term [(split_term, t1)]) rev_terms
val terms2 = map (subst_term [(split_term, Const (@{const_name HOL.uminus},
split_type --> split_type) $ t1)]) rev_terms
val zero = Const (@{const_name HOL.zero}, split_type)
val zero_leq_t1 = Const (@{const_name Orderings.less_eq},
split_type --> split_type --> HOLogic.boolT) $ zero $ t1
val t1_lt_zero = Const (@{const_name Orderings.less},
split_type --> split_type --> HOLogic.boolT) $ t1 $ zero
val not_false = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
val subgoal1 = (HOLogic.mk_Trueprop zero_leq_t1) :: terms1 @ [not_false]
val subgoal2 = (HOLogic.mk_Trueprop t1_lt_zero) :: terms2 @ [not_false]
in
SOME [(Ts, subgoal1), (Ts, subgoal2)]
end
(* ?P (?a - ?b) = ((?a < ?b --> ?P 0) & (ALL d. ?a = ?b + d --> ?P d)) *)
| (Const (@{const_name HOL.minus}, _), [t1, t2]) =>
let
(* "d" in the above theorem becomes a new bound variable after NNF *)
(* transformation, therefore some adjustment of indices is necessary *)
val rev_terms = rev terms
val zero = Const (@{const_name HOL.zero}, split_type)
val d = Bound 0
val terms1 = map (subst_term [(split_term, zero)]) rev_terms
val terms2 = map (subst_term [(incr_boundvars 1 split_term, d)])
(map (incr_boundvars 1) rev_terms)
val t1' = incr_boundvars 1 t1
val t2' = incr_boundvars 1 t2
val t1_lt_t2 = Const (@{const_name Orderings.less},
split_type --> split_type --> HOLogic.boolT) $ t1 $ t2
val t1_eq_t2_plus_d = Const ("op =", split_type --> split_type --> HOLogic.boolT) $ t1' $
(Const (@{const_name HOL.plus},
split_type --> split_type --> split_type) $ t2' $ d)
val not_false = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
val subgoal1 = (HOLogic.mk_Trueprop t1_lt_t2) :: terms1 @ [not_false]
val subgoal2 = (HOLogic.mk_Trueprop t1_eq_t2_plus_d) :: terms2 @ [not_false]
in
SOME [(Ts, subgoal1), (split_type :: Ts, subgoal2)]
end
(* ?P (nat ?i) = ((ALL n. ?i = int n --> ?P n) & (?i < 0 --> ?P 0)) *)
| (Const ("IntDef.nat", _), [t1]) =>
let
val rev_terms = rev terms
val zero_int = Const (@{const_name HOL.zero}, HOLogic.intT)
val zero_nat = Const (@{const_name HOL.zero}, HOLogic.natT)
val n = Bound 0
val terms1 = map (subst_term [(incr_boundvars 1 split_term, n)])
(map (incr_boundvars 1) rev_terms)
val terms2 = map (subst_term [(split_term, zero_nat)]) rev_terms
val t1' = incr_boundvars 1 t1
val t1_eq_int_n = Const ("op =", HOLogic.intT --> HOLogic.intT --> HOLogic.boolT) $ t1' $
(Const ("Nat.of_nat", HOLogic.natT --> HOLogic.intT) $ n)
val t1_lt_zero = Const (@{const_name Orderings.less},
HOLogic.intT --> HOLogic.intT --> HOLogic.boolT) $ t1 $ zero_int
val not_false = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
val subgoal1 = (HOLogic.mk_Trueprop t1_eq_int_n) :: terms1 @ [not_false]
val subgoal2 = (HOLogic.mk_Trueprop t1_lt_zero) :: terms2 @ [not_false]
in
SOME [(HOLogic.natT :: Ts, subgoal1), (Ts, subgoal2)]
end
(* "?P ((?n::nat) mod (number_of ?k)) =
((number_of ?k = 0 --> ?P ?n) & (~ (number_of ?k = 0) -->
(ALL i j. j < number_of ?k --> ?n = number_of ?k * i + j --> ?P j))) *)
| (Const ("Divides.div_class.mod", Type ("fun", [Type ("nat", []), _])), [t1, t2]) =>
let
val rev_terms = rev terms
val zero = Const (@{const_name HOL.zero}, split_type)
val i = Bound 1
val j = Bound 0
val terms1 = map (subst_term [(split_term, t1)]) rev_terms
val terms2 = map (subst_term [(incr_boundvars 2 split_term, j)])
(map (incr_boundvars 2) rev_terms)
val t1' = incr_boundvars 2 t1
val t2' = incr_boundvars 2 t2
val t2_eq_zero = Const ("op =",
split_type --> split_type --> HOLogic.boolT) $ t2 $ zero
val t2_neq_zero = HOLogic.mk_not (Const ("op =",
split_type --> split_type --> HOLogic.boolT) $ t2' $ zero)
val j_lt_t2 = Const (@{const_name Orderings.less},
split_type --> split_type--> HOLogic.boolT) $ j $ t2'
val t1_eq_t2_times_i_plus_j = Const ("op =", split_type --> split_type --> HOLogic.boolT) $ t1' $
(Const (@{const_name HOL.plus}, split_type --> split_type --> split_type) $
(Const (@{const_name HOL.times},
split_type --> split_type --> split_type) $ t2' $ i) $ j)
val not_false = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
val subgoal1 = (HOLogic.mk_Trueprop t2_eq_zero) :: terms1 @ [not_false]
val subgoal2 = (map HOLogic.mk_Trueprop
[t2_neq_zero, j_lt_t2, t1_eq_t2_times_i_plus_j])
@ terms2 @ [not_false]
in
SOME [(Ts, subgoal1), (split_type :: split_type :: Ts, subgoal2)]
end
(* "?P ((?n::nat) div (number_of ?k)) =
((number_of ?k = 0 --> ?P 0) & (~ (number_of ?k = 0) -->
(ALL i j. j < number_of ?k --> ?n = number_of ?k * i + j --> ?P i))) *)
| (Const ("Divides.div_class.div", Type ("fun", [Type ("nat", []), _])), [t1, t2]) =>
let
val rev_terms = rev terms
val zero = Const (@{const_name HOL.zero}, split_type)
val i = Bound 1
val j = Bound 0
val terms1 = map (subst_term [(split_term, zero)]) rev_terms
val terms2 = map (subst_term [(incr_boundvars 2 split_term, i)])
(map (incr_boundvars 2) rev_terms)
val t1' = incr_boundvars 2 t1
val t2' = incr_boundvars 2 t2
val t2_eq_zero = Const ("op =",
split_type --> split_type --> HOLogic.boolT) $ t2 $ zero
val t2_neq_zero = HOLogic.mk_not (Const ("op =",
split_type --> split_type --> HOLogic.boolT) $ t2' $ zero)
val j_lt_t2 = Const (@{const_name Orderings.less},
split_type --> split_type--> HOLogic.boolT) $ j $ t2'
val t1_eq_t2_times_i_plus_j = Const ("op =", split_type --> split_type --> HOLogic.boolT) $ t1' $
(Const (@{const_name HOL.plus}, split_type --> split_type --> split_type) $
(Const (@{const_name HOL.times},
split_type --> split_type --> split_type) $ t2' $ i) $ j)
val not_false = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
val subgoal1 = (HOLogic.mk_Trueprop t2_eq_zero) :: terms1 @ [not_false]
val subgoal2 = (map HOLogic.mk_Trueprop
[t2_neq_zero, j_lt_t2, t1_eq_t2_times_i_plus_j])
@ terms2 @ [not_false]
in
SOME [(Ts, subgoal1), (split_type :: split_type :: Ts, subgoal2)]
end
(* "?P ((?n::int) mod (number_of ?k)) =
((iszero (number_of ?k) --> ?P ?n) &
(neg (number_of (uminus ?k)) -->
(ALL i j. 0 <= j & j < number_of ?k & ?n = number_of ?k * i + j --> ?P j)) &
(neg (number_of ?k) -->
(ALL i j. number_of ?k < j & j <= 0 & ?n = number_of ?k * i + j --> ?P j))) *)
| (Const ("Divides.div_class.mod",
Type ("fun", [Type ("IntDef.int", []), _])), [t1, t2 as (number_of $ k)]) =>
let
val rev_terms = rev terms
val zero = Const (@{const_name HOL.zero}, split_type)
val i = Bound 1
val j = Bound 0
val terms1 = map (subst_term [(split_term, t1)]) rev_terms
val terms2_3 = map (subst_term [(incr_boundvars 2 split_term, j)])
(map (incr_boundvars 2) rev_terms)
val t1' = incr_boundvars 2 t1
val (t2' as (_ $ k')) = incr_boundvars 2 t2
val iszero_t2 = Const ("IntDef.iszero", split_type --> HOLogic.boolT) $ t2
val neg_minus_k = Const ("IntDef.neg", split_type --> HOLogic.boolT) $
(number_of $
(Const (@{const_name HOL.uminus},
HOLogic.intT --> HOLogic.intT) $ k'))
val zero_leq_j = Const (@{const_name Orderings.less_eq},
split_type --> split_type --> HOLogic.boolT) $ zero $ j
val j_lt_t2 = Const (@{const_name Orderings.less},
split_type --> split_type--> HOLogic.boolT) $ j $ t2'
val t1_eq_t2_times_i_plus_j = Const ("op =", split_type --> split_type --> HOLogic.boolT) $ t1' $
(Const (@{const_name HOL.plus}, split_type --> split_type --> split_type) $
(Const (@{const_name HOL.times},
split_type --> split_type --> split_type) $ t2' $ i) $ j)
val neg_t2 = Const ("IntDef.neg", split_type --> HOLogic.boolT) $ t2'
val t2_lt_j = Const (@{const_name Orderings.less},
split_type --> split_type--> HOLogic.boolT) $ t2' $ j
val j_leq_zero = Const (@{const_name Orderings.less_eq},
split_type --> split_type --> HOLogic.boolT) $ j $ zero
val not_false = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
val subgoal1 = (HOLogic.mk_Trueprop iszero_t2) :: terms1 @ [not_false]
val subgoal2 = (map HOLogic.mk_Trueprop [neg_minus_k, zero_leq_j])
@ hd terms2_3
:: (if tl terms2_3 = [] then [not_false] else [])
@ (map HOLogic.mk_Trueprop [j_lt_t2, t1_eq_t2_times_i_plus_j])
@ (if tl terms2_3 = [] then [] else tl terms2_3 @ [not_false])
val subgoal3 = (map HOLogic.mk_Trueprop [neg_t2, t2_lt_j])
@ hd terms2_3
:: (if tl terms2_3 = [] then [not_false] else [])
@ (map HOLogic.mk_Trueprop [j_leq_zero, t1_eq_t2_times_i_plus_j])
@ (if tl terms2_3 = [] then [] else tl terms2_3 @ [not_false])
val Ts' = split_type :: split_type :: Ts
in
SOME [(Ts, subgoal1), (Ts', subgoal2), (Ts', subgoal3)]
end
(* "?P ((?n::int) div (number_of ?k)) =
((iszero (number_of ?k) --> ?P 0) &
(neg (number_of (uminus ?k)) -->
(ALL i. (EX j. 0 <= j & j < number_of ?k & ?n = number_of ?k * i + j) --> ?P i)) &
(neg (number_of ?k) -->
(ALL i. (EX j. number_of ?k < j & j <= 0 & ?n = number_of ?k * i + j) --> ?P i))) *)
| (Const ("Divides.div_class.div",
Type ("fun", [Type ("IntDef.int", []), _])), [t1, t2 as (number_of $ k)]) =>
let
val rev_terms = rev terms
val zero = Const (@{const_name HOL.zero}, split_type)
val i = Bound 1
val j = Bound 0
val terms1 = map (subst_term [(split_term, zero)]) rev_terms
val terms2_3 = map (subst_term [(incr_boundvars 2 split_term, i)])
(map (incr_boundvars 2) rev_terms)
val t1' = incr_boundvars 2 t1
val (t2' as (_ $ k')) = incr_boundvars 2 t2
val iszero_t2 = Const ("IntDef.iszero", split_type --> HOLogic.boolT) $ t2
val neg_minus_k = Const ("IntDef.neg", split_type --> HOLogic.boolT) $
(number_of $
(Const (@{const_name HOL.uminus},
HOLogic.intT --> HOLogic.intT) $ k'))
val zero_leq_j = Const (@{const_name Orderings.less_eq},
split_type --> split_type --> HOLogic.boolT) $ zero $ j
val j_lt_t2 = Const (@{const_name Orderings.less},
split_type --> split_type--> HOLogic.boolT) $ j $ t2'
val t1_eq_t2_times_i_plus_j = Const ("op =",
split_type --> split_type --> HOLogic.boolT) $ t1' $
(Const (@{const_name HOL.plus}, split_type --> split_type --> split_type) $
(Const (@{const_name HOL.times},
split_type --> split_type --> split_type) $ t2' $ i) $ j)
val neg_t2 = Const ("IntDef.neg", split_type --> HOLogic.boolT) $ t2'
val t2_lt_j = Const (@{const_name Orderings.less},
split_type --> split_type--> HOLogic.boolT) $ t2' $ j
val j_leq_zero = Const (@{const_name Orderings.less_eq},
split_type --> split_type --> HOLogic.boolT) $ j $ zero
val not_false = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
val subgoal1 = (HOLogic.mk_Trueprop iszero_t2) :: terms1 @ [not_false]
val subgoal2 = (HOLogic.mk_Trueprop neg_minus_k)
:: terms2_3
@ not_false
:: (map HOLogic.mk_Trueprop
[zero_leq_j, j_lt_t2, t1_eq_t2_times_i_plus_j])
val subgoal3 = (HOLogic.mk_Trueprop neg_t2)
:: terms2_3
@ not_false
:: (map HOLogic.mk_Trueprop
[t2_lt_j, j_leq_zero, t1_eq_t2_times_i_plus_j])
val Ts' = split_type :: split_type :: Ts
in
SOME [(Ts, subgoal1), (Ts', subgoal2), (Ts', subgoal3)]
end
(* this will only happen if a split theorem can be applied for which no *)
(* code exists above -- in which case either the split theorem should be *)
(* implemented above, or 'is_split_thm' should be modified to filter it *)
(* out *)
| (t, ts) => (
warning ("Lin. Arith.: split rule for " ^ Sign.string_of_term sg t ^
" (with " ^ Int.toString (length ts) ^
" argument(s)) not implemented; proof reconstruction is likely to fail");
NONE
))
)
end;
(* remove terms that do not satisfy 'p'; change the order of the remaining *)
(* terms in the same way as filter_prems_tac does *)
fun filter_prems_tac_items (p : term -> bool) (terms : term list) : term list =
let
fun filter_prems (t, (left, right)) =
if p t then (left, right @ [t]) else (left @ right, [])
val (left, right) = foldl filter_prems ([], []) terms
in
right @ left
end;
(* return true iff TRY (etac notE) THEN eq_assume_tac would succeed on a *)
(* subgoal that has 'terms' as premises *)
fun negated_term_occurs_positively (terms : term list) : bool =
List.exists
(fn (Trueprop $ (Const ("Not", _) $ t)) => member (op aconv) terms (Trueprop $ t)
| _ => false)
terms;
fun pre_decomp sg (Ts : typ list, terms : term list) : (typ list * term list) list =
let
(* repeatedly split (including newly emerging subgoals) until no further *)
(* splitting is possible *)
fun split_loop ([] : (typ list * term list) list) = ([] : (typ list * term list) list)
| split_loop (subgoal::subgoals) = (
case split_once_items sg subgoal of
SOME new_subgoals => split_loop (new_subgoals @ subgoals)
| NONE => subgoal :: split_loop subgoals
)
fun is_relevant t = isSome (decomp sg t)
(* filter_prems_tac is_relevant: *)
val relevant_terms = filter_prems_tac_items is_relevant terms
(* split_tac, NNF normalization: *)
val split_goals = split_loop [(Ts, relevant_terms)]
(* necessary because split_once_tac may normalize terms: *)
val beta_eta_norm = map (apsnd (map (Envir.eta_contract o Envir.beta_norm))) split_goals
(* TRY (etac notE) THEN eq_assume_tac: *)
val result = List.filter (not o negated_term_occurs_positively o snd) beta_eta_norm
in
result
end;
(* takes the i-th subgoal [| A1; ...; An |] ==> B to *)
(* An --> ... --> A1 --> B, performs splitting with the given 'split_thms' *)
(* (resulting in a different subgoal P), takes P to ~P ==> False, *)
(* performs NNF-normalization of ~P, and eliminates conjunctions, *)
(* disjunctions and existential quantifiers from the premises, possibly (in *)
(* the case of disjunctions) resulting in several new subgoals, each of the *)
(* general form [| Q1; ...; Qm |] ==> False. Fails if more than *)
(* !fast_arith_split_limit splits are possible. *)
local
val nnf_simpset =
empty_ss setmkeqTrue mk_eq_True
setmksimps (mksimps mksimps_pairs)
addsimps [imp_conv_disj, iff_conv_conj_imp, de_Morgan_disj, de_Morgan_conj,
not_all, not_ex, not_not]
fun prem_nnf_tac i st =
full_simp_tac (Simplifier.theory_context (Thm.theory_of_thm st) nnf_simpset) i st
in
fun split_once_tac (split_thms : thm list) (i : int) : tactic =
let
fun cond_split_tac i st =
let
val subgoal = Logic.nth_prem (i, Thm.prop_of st)
val Ts = rev (map snd (Logic.strip_params subgoal))
val concl = HOLogic.dest_Trueprop (Logic.strip_assums_concl subgoal)
val cmap = Splitter.cmap_of_split_thms split_thms
val splits = Splitter.split_posns cmap (theory_of_thm st) Ts concl
in
if length splits > !fast_arith_split_limit then
no_tac st
else
split_tac split_thms i st
end
in
EVERY' [
REPEAT_DETERM o etac rev_mp,
cond_split_tac,
rtac ccontr,
prem_nnf_tac,
TRY o REPEAT_ALL_NEW (DETERM o (eresolve_tac [conjE, exE] ORELSE' etac disjE))
] i
end
end; (* local *)
(* remove irrelevant premises, then split the i-th subgoal (and all new *)
(* subgoals) by using 'split_once_tac' repeatedly. Beta-eta-normalize new *)
(* subgoals and finally attempt to solve them by finding an immediate *)
(* contradiction (i.e. a term and its negation) in their premises. *)
fun pre_tac i st =
let
val sg = theory_of_thm st
val split_thms = filter is_split_thm (#splits (ArithTheoryData.get sg))
fun is_relevant t = isSome (decomp sg t)
in
DETERM (
TRY (filter_prems_tac is_relevant i)
THEN (
(TRY o REPEAT_ALL_NEW (split_once_tac split_thms))
THEN_ALL_NEW
(CONVERSION Drule.beta_eta_conversion
THEN'
(TRY o (etac notE THEN' eq_assume_tac)))
) i
) st
end;
end; (* LA_Data_Ref *)
structure Fast_Arith =
Fast_Lin_Arith(structure LA_Logic=LA_Logic and LA_Data=LA_Data_Ref);
val fast_arith_tac = Fast_Arith.lin_arith_tac false;
val fast_ex_arith_tac = Fast_Arith.lin_arith_tac;
val trace_arith = Fast_Arith.trace;
val fast_arith_neq_limit = Fast_Arith.fast_arith_neq_limit;
val fast_arith_split_limit = LA_Data_Ref.fast_arith_split_limit;
(* reduce contradictory <= to False.
Most of the work is done by the cancel tactics. *)
val init_lin_arith_data =
Fast_Arith.map_data (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, ...} =>
{add_mono_thms = add_mono_thms @
@{thms add_mono_thms_ordered_semiring} @ @{thms add_mono_thms_ordered_field},
mult_mono_thms = mult_mono_thms,
inj_thms = inj_thms,
lessD = lessD @ [thm "Suc_leI"],
neqE = [@{thm linorder_neqE_nat}, @{thm linorder_neqE_ordered_idom}],
simpset = HOL_basic_ss
addsimps [@{thm "monoid_add_class.zero_plus.add_0_left"}, @{thm "monoid_add_class.zero_plus.add_0_right"},
@{thm "Zero_not_Suc"}, @{thm "Suc_not_Zero"}, @{thm "le_0_eq"}, @{thm "One_nat_def"},
@{thm "order_less_irrefl"}, @{thm "zero_neq_one"}, @{thm "zero_less_one"},
@{thm "zero_le_one"}, @{thm "zero_neq_one"} RS not_sym, @{thm "not_one_le_zero"},
@{thm "not_one_less_zero"}]
addsimprocs [ab_group_add_cancel.sum_conv, ab_group_add_cancel.rel_conv]
(*abel_cancel helps it work in abstract algebraic domains*)
addsimprocs nat_cancel_sums_add}) #>
arith_discrete "nat";
val fast_nat_arith_simproc =
Simplifier.simproc (the_context ()) "fast_nat_arith"
["(m::nat) < n","(m::nat) <= n", "(m::nat) = n"] Fast_Arith.lin_arith_prover;
(* Because of fast_nat_arith_simproc, the arithmetic solver is really only
useful to detect inconsistencies among the premises for subgoals which are
*not* themselves (in)equalities, because the latter activate
fast_nat_arith_simproc anyway. However, it seems cheaper to activate the
solver all the time rather than add the additional check. *)
(* arith proof method *)
local
fun raw_arith_tac ex i st =
(* FIXME: K true should be replaced by a sensible test (perhaps "isSome o
decomp sg"? -- but note that the test is applied to terms already before
they are split/normalized) to speed things up in case there are lots of
irrelevant terms involved; elimination of min/max can be optimized:
(max m n + k <= r) = (m+k <= r & n+k <= r)
(l <= min m n + k) = (l <= m+k & l <= n+k)
*)
refute_tac (K true)
(* Splitting is also done inside fast_arith_tac, but not completely -- *)
(* split_tac may use split theorems that have not been implemented in *)
(* fast_arith_tac (cf. pre_decomp and split_once_items above), and *)
(* fast_arith_split_limit may trigger. *)
(* Therefore splitting outside of fast_arith_tac may allow us to prove *)
(* some goals that fast_arith_tac alone would fail on. *)
(REPEAT_DETERM o split_tac (#splits (ArithTheoryData.get (Thm.theory_of_thm st))))
(fast_ex_arith_tac ex)
i st;
fun arith_theory_tac i st =
let
val tactics = #tactics (ArithTheoryData.get (Thm.theory_of_thm st))
in
FIRST' (map (fn ArithTactic {tactic, ...} => tactic) tactics) i st
end;
in
val simple_arith_tac = FIRST' [fast_arith_tac,
ObjectLogic.full_atomize_tac THEN' (REPEAT_DETERM o rtac impI) THEN' raw_arith_tac true];
val arith_tac = FIRST' [fast_arith_tac,
ObjectLogic.full_atomize_tac THEN' (REPEAT_DETERM o rtac impI) THEN' raw_arith_tac true,
arith_theory_tac];
val silent_arith_tac = FIRST' [fast_arith_tac,
ObjectLogic.full_atomize_tac THEN' (REPEAT_DETERM o rtac impI) THEN' raw_arith_tac false,
arith_theory_tac];
fun arith_method prems =
Method.METHOD (fn facts => HEADGOAL (Method.insert_tac (prems @ facts) THEN' arith_tac));
end;
(* antisymmetry:
combines x <= y (or ~(y < x)) and y <= x (or ~(x < y)) into x = y
local
val antisym = mk_meta_eq order_antisym
val not_lessD = @{thm linorder_not_less} RS iffD1
fun prp t thm = (#prop(rep_thm thm) = t)
in
fun antisym_eq prems thm =
let
val r = #prop(rep_thm thm);
in
case r of
Tr $ ((c as Const(@{const_name Orderings.less_eq},T)) $ s $ t) =>
let val r' = Tr $ (c $ t $ s)
in
case Library.find_first (prp r') prems of
NONE =>
let val r' = Tr $ (HOLogic.Not $ (Const(@{const_name Orderings.less},T) $ s $ t))
in case Library.find_first (prp r') prems of
NONE => []
| SOME thm' => [(thm' RS not_lessD) RS (thm RS antisym)]
end
| SOME thm' => [thm' RS (thm RS antisym)]
end
| Tr $ (Const("Not",_) $ (Const(@{const_name Orderings.less},T) $ s $ t)) =>
let val r' = Tr $ (Const(@{const_name Orderings.less_eq},T) $ s $ t)
in
case Library.find_first (prp r') prems of
NONE =>
let val r' = Tr $ (HOLogic.Not $ (Const(@{const_name Orderings.less},T) $ t $ s))
in case Library.find_first (prp r') prems of
NONE => []
| SOME thm' =>
[(thm' RS not_lessD) RS ((thm RS not_lessD) RS antisym)]
end
| SOME thm' => [thm' RS ((thm RS not_lessD) RS antisym)]
end
| _ => []
end
handle THM _ => []
end;
*)
(* theory setup *)
val arith_setup =
init_lin_arith_data #>
(fn thy => (Simplifier.change_simpset_of thy (fn ss => ss
addsimprocs (nat_cancel_sums @ [fast_nat_arith_simproc])
addSolver (mk_solver' "lin. arith." Fast_Arith.cut_lin_arith_tac)); thy)) #>
Method.add_methods
[("arith", (arith_method o fst) oo Method.syntax Args.bang_facts,
"decide linear arithmethic")] #>
Attrib.add_attributes [("arith_split", Attrib.no_args arith_split_add,
"declaration of split rules for arithmetic procedure")];