(* Title: HOL/Integ/int_arith1.ML
ID: $Id$
Authors: Larry Paulson and Tobias Nipkow
Simprocs and decision procedure for linear arithmetic.
*)
(** Misc ML bindings **)
val NCons_Pls = thm"NCons_Pls";
val NCons_Min = thm"NCons_Min";
val NCons_BIT = thm"NCons_BIT";
val number_of_Pls = thm"number_of_Pls";
val number_of_Min = thm"number_of_Min";
val number_of_BIT = thm"number_of_BIT";
val bin_succ_Pls = thm"bin_succ_Pls";
val bin_succ_Min = thm"bin_succ_Min";
val bin_succ_BIT = thm"bin_succ_BIT";
val bin_pred_Pls = thm"bin_pred_Pls";
val bin_pred_Min = thm"bin_pred_Min";
val bin_pred_BIT = thm"bin_pred_BIT";
val bin_minus_Pls = thm"bin_minus_Pls";
val bin_minus_Min = thm"bin_minus_Min";
val bin_minus_BIT = thm"bin_minus_BIT";
val bin_add_Pls = thm"bin_add_Pls";
val bin_add_Min = thm"bin_add_Min";
val bin_mult_Pls = thm"bin_mult_Pls";
val bin_mult_Min = thm"bin_mult_Min";
val bin_mult_BIT = thm"bin_mult_BIT";
val neg_def = thm "neg_def";
val iszero_def = thm "iszero_def";
val not_neg_int = thm "not_neg_int";
val neg_zminus_int = thm "neg_zminus_int";
val zadd_ac = thms "Ring_and_Field.add_ac"
val zmult_ac = thms "Ring_and_Field.mult_ac"
val NCons_Pls_0 = thm"NCons_Pls_0";
val NCons_Pls_1 = thm"NCons_Pls_1";
val NCons_Min_0 = thm"NCons_Min_0";
val NCons_Min_1 = thm"NCons_Min_1";
val bin_succ_1 = thm"bin_succ_1";
val bin_succ_0 = thm"bin_succ_0";
val bin_pred_1 = thm"bin_pred_1";
val bin_pred_0 = thm"bin_pred_0";
val bin_minus_1 = thm"bin_minus_1";
val bin_minus_0 = thm"bin_minus_0";
val bin_add_BIT_11 = thm"bin_add_BIT_11";
val bin_add_BIT_10 = thm"bin_add_BIT_10";
val bin_add_BIT_0 = thm"bin_add_BIT_0";
val bin_add_Pls_right = thm"bin_add_Pls_right";
val bin_add_Min_right = thm"bin_add_Min_right";
val bin_add_BIT_BIT = thm"bin_add_BIT_BIT";
val bin_mult_1 = thm"bin_mult_1";
val bin_mult_0 = thm"bin_mult_0";
val number_of_NCons = thm"number_of_NCons";
val number_of_succ = thm"number_of_succ";
val number_of_pred = thm"number_of_pred";
val number_of_minus = thm"number_of_minus";
val number_of_add = thm"number_of_add";
val diff_number_of_eq = thm"diff_number_of_eq";
val number_of_mult = thm"number_of_mult";
val double_number_of_BIT = thm"double_number_of_BIT";
val int_numeral_0_eq_0 = thm"int_numeral_0_eq_0";
val int_numeral_1_eq_1 = thm"int_numeral_1_eq_1";
val zmult_minus1 = thm"zmult_minus1";
val zmult_minus1_right = thm"zmult_minus1_right";
val zminus_number_of_zmult = thm"zminus_number_of_zmult";
val zminus_1_eq_m1 = thm"zminus_1_eq_m1";
val zero_less_nat_eq = thm"zero_less_nat_eq";
val eq_number_of_eq = thm"eq_number_of_eq";
val iszero_number_of_Pls = thm"iszero_number_of_Pls";
val nonzero_number_of_Min = thm"nonzero_number_of_Min";
val iszero_number_of_BIT = thm"iszero_number_of_BIT";
val iszero_number_of_0 = thm"iszero_number_of_0";
val iszero_number_of_1 = thm"iszero_number_of_1";
val less_number_of_eq_neg = thm"less_number_of_eq_neg";
val not_neg_number_of_Pls = thm"not_neg_number_of_Pls";
val neg_number_of_Min = thm"neg_number_of_Min";
val neg_number_of_BIT = thm"neg_number_of_BIT";
val le_number_of_eq_not_less = thm"le_number_of_eq_not_less";
val zabs_number_of = thm"zabs_number_of";
val zabs_0 = thm"zabs_0";
val zabs_1 = thm"zabs_1";
val number_of_reorient = thm"number_of_reorient";
val add_number_of_left = thm"add_number_of_left";
val mult_number_of_left = thm"mult_number_of_left";
val add_number_of_diff1 = thm"add_number_of_diff1";
val add_number_of_diff2 = thm"add_number_of_diff2";
val less_iff_diff_less_0 = thm"less_iff_diff_less_0";
val eq_iff_diff_eq_0 = thm"eq_iff_diff_eq_0";
val le_iff_diff_le_0 = thm"le_iff_diff_le_0";
val bin_mult_simps = thms"bin_mult_simps";
val NCons_simps = thms"NCons_simps";
val bin_arith_extra_simps = thms"bin_arith_extra_simps";
val bin_arith_simps = thms"bin_arith_simps";
val bin_rel_simps = thms"bin_rel_simps";
val zless_imp_add1_zle = thm "zless_imp_add1_zle";
val combine_common_factor = thm"combine_common_factor";
val eq_add_iff1 = thm"eq_add_iff1";
val eq_add_iff2 = thm"eq_add_iff2";
val less_add_iff1 = thm"less_add_iff1";
val less_add_iff2 = thm"less_add_iff2";
val le_add_iff1 = thm"le_add_iff1";
val le_add_iff2 = thm"le_add_iff2";
structure Bin_Simprocs =
struct
fun prove_conv tacs sg (hyps: thm list) xs (t, u) =
if t aconv u then None
else
let val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (t, u))
in Some (Tactic.prove sg xs [] eq (K (EVERY tacs))) end
fun prove_conv_nohyps tacs sg = prove_conv tacs sg [];
fun prove_conv_nohyps_novars tacs sg = prove_conv tacs sg [] [];
fun prep_simproc (name, pats, proc) =
Simplifier.simproc (Theory.sign_of (the_context())) name pats proc;
fun is_numeral (Const("Numeral.number_of", _) $ w) = true
| is_numeral _ = false
fun simplify_meta_eq f_number_of_eq f_eq =
mk_meta_eq ([f_eq, f_number_of_eq] MRS trans)
structure IntAbstractNumeralsData =
struct
val dest_eq = HOLogic.dest_eq o HOLogic.dest_Trueprop o concl_of
val is_numeral = is_numeral
val numeral_0_eq_0 = int_numeral_0_eq_0
val numeral_1_eq_1 = int_numeral_1_eq_1
val prove_conv = prove_conv_nohyps_novars
fun norm_tac simps = ALLGOALS (simp_tac (HOL_ss addsimps simps))
val simplify_meta_eq = simplify_meta_eq
end
structure IntAbstractNumerals = AbstractNumeralsFun (IntAbstractNumeralsData)
(*For addition, we already have rules for the operand 0.
Multiplication is omitted because there are already special rules for
both 0 and 1 as operands. Unary minus is trivial, just have - 1 = -1.
For the others, having three patterns is a compromise between just having
one (many spurious calls) and having nine (just too many!) *)
val eval_numerals =
map prep_simproc
[("int_add_eval_numerals",
["(m::int) + 1", "(m::int) + number_of v"],
IntAbstractNumerals.proc (number_of_add RS sym)),
("int_diff_eval_numerals",
["(m::int) - 1", "(m::int) - number_of v"],
IntAbstractNumerals.proc diff_number_of_eq),
("int_eq_eval_numerals",
["(m::int) = 0", "(m::int) = 1", "(m::int) = number_of v"],
IntAbstractNumerals.proc eq_number_of_eq),
("int_less_eval_numerals",
["(m::int) < 0", "(m::int) < 1", "(m::int) < number_of v"],
IntAbstractNumerals.proc less_number_of_eq_neg),
("int_le_eval_numerals",
["(m::int) <= 0", "(m::int) <= 1", "(m::int) <= number_of v"],
IntAbstractNumerals.proc le_number_of_eq_not_less)]
(*reorientation simprules using ==, for the following simproc*)
val meta_zero_reorient = zero_reorient RS eq_reflection
val meta_one_reorient = one_reorient RS eq_reflection
val meta_number_of_reorient = number_of_reorient RS eq_reflection
(*reorientation simplification procedure: reorients (polymorphic)
0 = x, 1 = x, nnn = x provided x isn't 0, 1 or a numeral.*)
fun reorient_proc sg _ (_ $ t $ u) =
case u of
Const("0", _) => None
| Const("1", _) => None
| Const("Numeral.number_of", _) $ _ => None
| _ => Some (case t of
Const("0", _) => meta_zero_reorient
| Const("1", _) => meta_one_reorient
| Const("Numeral.number_of", _) $ _ => meta_number_of_reorient)
val reorient_simproc =
prep_simproc ("reorient_simproc", ["0=x", "1=x", "number_of w = x"], reorient_proc)
end;
Addsimprocs Bin_Simprocs.eval_numerals;
Addsimprocs [Bin_Simprocs.reorient_simproc];
structure Int_Numeral_Simprocs =
struct
(*Maps 0 to Numeral0 and 1 to Numeral1 so that arithmetic in simprocs
isn't complicated by the abstract 0 and 1.*)
val numeral_syms = [int_numeral_0_eq_0 RS sym, int_numeral_1_eq_1 RS sym];
val numeral_sym_ss = HOL_ss addsimps numeral_syms;
fun rename_numerals th =
simplify numeral_sym_ss (Thm.transfer (the_context ()) th);
(*Utilities*)
fun mk_numeral n = HOLogic.number_of_const HOLogic.intT $ HOLogic.mk_bin n;
(*Decodes a binary INTEGER*)
fun dest_numeral (Const("0", _)) = 0
| dest_numeral (Const("1", _)) = 1
| dest_numeral (Const("Numeral.number_of", _) $ w) =
(HOLogic.dest_binum w
handle TERM _ => raise TERM("Int_Numeral_Simprocs.dest_numeral:1", [w]))
| dest_numeral t = raise TERM("Int_Numeral_Simprocs.dest_numeral:2", [t]);
fun find_first_numeral past (t::terms) =
((dest_numeral t, rev past @ terms)
handle TERM _ => find_first_numeral (t::past) terms)
| find_first_numeral past [] = raise TERM("find_first_numeral", []);
val zero = mk_numeral 0;
val mk_plus = HOLogic.mk_binop "op +";
val uminus_const = Const ("uminus", HOLogic.intT --> HOLogic.intT);
(*Thus mk_sum[t] yields t+0; longer sums don't have a trailing zero*)
fun mk_sum [] = zero
| mk_sum [t,u] = mk_plus (t, u)
| mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
(*this version ALWAYS includes a trailing zero*)
fun long_mk_sum [] = zero
| long_mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
val dest_plus = HOLogic.dest_bin "op +" HOLogic.intT;
(*decompose additions AND subtractions as a sum*)
fun dest_summing (pos, Const ("op +", _) $ t $ u, ts) =
dest_summing (pos, t, dest_summing (pos, u, ts))
| dest_summing (pos, Const ("op -", _) $ t $ u, ts) =
dest_summing (pos, t, dest_summing (not pos, u, ts))
| dest_summing (pos, t, ts) =
if pos then t::ts else uminus_const$t :: ts;
fun dest_sum t = dest_summing (true, t, []);
val mk_diff = HOLogic.mk_binop "op -";
val dest_diff = HOLogic.dest_bin "op -" HOLogic.intT;
val one = mk_numeral 1;
val mk_times = HOLogic.mk_binop "op *";
fun mk_prod [] = one
| mk_prod [t] = t
| mk_prod (t :: ts) = if t = one then mk_prod ts
else mk_times (t, mk_prod ts);
val dest_times = HOLogic.dest_bin "op *" HOLogic.intT;
fun dest_prod t =
let val (t,u) = dest_times t
in dest_prod t @ dest_prod u end
handle TERM _ => [t];
(*DON'T do the obvious simplifications; that would create special cases*)
fun mk_coeff (k, ts) = mk_times (mk_numeral k, ts);
(*Express t as a product of (possibly) a numeral with other sorted terms*)
fun dest_coeff sign (Const ("uminus", _) $ t) = dest_coeff (~sign) t
| dest_coeff sign t =
let val ts = sort Term.term_ord (dest_prod t)
val (n, ts') = find_first_numeral [] ts
handle TERM _ => (1, ts)
in (sign*n, mk_prod ts') end;
(*Find first coefficient-term THAT MATCHES u*)
fun find_first_coeff past u [] = raise TERM("find_first_coeff", [])
| find_first_coeff past u (t::terms) =
let val (n,u') = dest_coeff 1 t
in if u aconv u' then (n, rev past @ terms)
else find_first_coeff (t::past) u terms
end
handle TERM _ => find_first_coeff (t::past) u terms;
(*Simplify Numeral0+n, n+Numeral0, Numeral1*n, n*Numeral1*)
val add_0s = map rename_numerals [zadd_0, zadd_0_right];
val mult_1s = map rename_numerals [zmult_1, zmult_1_right] @
[zmult_minus1, zmult_minus1_right];
(*To perform binary arithmetic. The "left" rewriting handles patterns
created by the simprocs, such as 3 * (5 * x). *)
val bin_simps = [int_numeral_0_eq_0 RS sym, int_numeral_1_eq_1 RS sym,
add_number_of_left, mult_number_of_left] @
bin_arith_simps @ bin_rel_simps;
(*Binary arithmetic BUT NOT ADDITION since it may collapse adjacent terms
during re-arrangement*)
val non_add_bin_simps =
bin_simps \\ [add_number_of_left, number_of_add RS sym];
(*To evaluate binary negations of coefficients*)
val zminus_simps = NCons_simps @
[zminus_1_eq_m1, number_of_minus RS sym,
bin_minus_1, bin_minus_0, bin_minus_Pls, bin_minus_Min,
bin_pred_1, bin_pred_0, bin_pred_Pls, bin_pred_Min];
(*To let us treat subtraction as addition*)
val diff_simps = [zdiff_def, zminus_zadd_distrib, zminus_zminus];
(*push the unary minus down: - x * y = x * - y *)
val int_minus_mult_eq_1_to_2 =
[zmult_zminus, zmult_zminus_right RS sym] MRS trans |> standard;
(*to extract again any uncancelled minuses*)
val int_minus_from_mult_simps =
[zminus_zminus, zmult_zminus, zmult_zminus_right];
(*combine unary minus with numeric literals, however nested within a product*)
val int_mult_minus_simps =
[zmult_assoc, zmult_zminus RS sym, int_minus_mult_eq_1_to_2];
(*Apply the given rewrite (if present) just once*)
fun trans_tac None = all_tac
| trans_tac (Some th) = ALLGOALS (rtac (th RS trans));
fun simplify_meta_eq rules =
simplify (HOL_basic_ss addeqcongs[eq_cong2] addsimps rules)
o mk_meta_eq;
structure CancelNumeralsCommon =
struct
val mk_sum = mk_sum
val dest_sum = dest_sum
val mk_coeff = mk_coeff
val dest_coeff = dest_coeff 1
val find_first_coeff = find_first_coeff []
val trans_tac = trans_tac
val norm_tac =
ALLGOALS (simp_tac (HOL_ss addsimps numeral_syms@add_0s@mult_1s@
diff_simps@zminus_simps@zadd_ac))
THEN ALLGOALS (simp_tac (HOL_ss addsimps non_add_bin_simps@int_mult_minus_simps))
THEN ALLGOALS (simp_tac (HOL_ss addsimps int_minus_from_mult_simps@
zadd_ac@zmult_ac))
val numeral_simp_tac = ALLGOALS (simp_tac (HOL_ss addsimps add_0s@bin_simps))
val simplify_meta_eq = simplify_meta_eq (add_0s@mult_1s)
end;
structure EqCancelNumerals = CancelNumeralsFun
(open CancelNumeralsCommon
val prove_conv = Bin_Simprocs.prove_conv
val mk_bal = HOLogic.mk_eq
val dest_bal = HOLogic.dest_bin "op =" HOLogic.intT
val bal_add1 = eq_add_iff1 RS trans
val bal_add2 = eq_add_iff2 RS trans
);
structure LessCancelNumerals = CancelNumeralsFun
(open CancelNumeralsCommon
val prove_conv = Bin_Simprocs.prove_conv
val mk_bal = HOLogic.mk_binrel "op <"
val dest_bal = HOLogic.dest_bin "op <" HOLogic.intT
val bal_add1 = less_add_iff1 RS trans
val bal_add2 = less_add_iff2 RS trans
);
structure LeCancelNumerals = CancelNumeralsFun
(open CancelNumeralsCommon
val prove_conv = Bin_Simprocs.prove_conv
val mk_bal = HOLogic.mk_binrel "op <="
val dest_bal = HOLogic.dest_bin "op <=" HOLogic.intT
val bal_add1 = le_add_iff1 RS trans
val bal_add2 = le_add_iff2 RS trans
);
val cancel_numerals =
map Bin_Simprocs.prep_simproc
[("inteq_cancel_numerals",
["(l::int) + m = n", "(l::int) = m + n",
"(l::int) - m = n", "(l::int) = m - n",
"(l::int) * m = n", "(l::int) = m * n"],
EqCancelNumerals.proc),
("intless_cancel_numerals",
["(l::int) + m < n", "(l::int) < m + n",
"(l::int) - m < n", "(l::int) < m - n",
"(l::int) * m < n", "(l::int) < m * n"],
LessCancelNumerals.proc),
("intle_cancel_numerals",
["(l::int) + m <= n", "(l::int) <= m + n",
"(l::int) - m <= n", "(l::int) <= m - n",
"(l::int) * m <= n", "(l::int) <= m * n"],
LeCancelNumerals.proc)];
structure CombineNumeralsData =
struct
val add = op + : int*int -> int
val mk_sum = long_mk_sum (*to work for e.g. 2*x + 3*x *)
val dest_sum = dest_sum
val mk_coeff = mk_coeff
val dest_coeff = dest_coeff 1
val left_distrib = combine_common_factor RS trans
val prove_conv = Bin_Simprocs.prove_conv_nohyps
val trans_tac = trans_tac
val norm_tac =
ALLGOALS (simp_tac (HOL_ss addsimps numeral_syms@add_0s@mult_1s@
diff_simps@zminus_simps@zadd_ac))
THEN ALLGOALS (simp_tac (HOL_ss addsimps non_add_bin_simps@int_mult_minus_simps))
THEN ALLGOALS (simp_tac (HOL_ss addsimps int_minus_from_mult_simps@
zadd_ac@zmult_ac))
val numeral_simp_tac = ALLGOALS
(simp_tac (HOL_ss addsimps add_0s@bin_simps))
val simplify_meta_eq = simplify_meta_eq (add_0s@mult_1s)
end;
structure CombineNumerals = CombineNumeralsFun(CombineNumeralsData);
val combine_numerals =
Bin_Simprocs.prep_simproc
("int_combine_numerals", ["(i::int) + j", "(i::int) - j"], CombineNumerals.proc);
end;
Addsimprocs Int_Numeral_Simprocs.cancel_numerals;
Addsimprocs [Int_Numeral_Simprocs.combine_numerals];
(*examples:
print_depth 22;
set timing;
set trace_simp;
fun test s = (Goal s, by (Simp_tac 1));
test "l + 2 + 2 + 2 + (l + 2) + (oo + 2) = (uu::int)";
test "2*u = (u::int)";
test "(i + j + 12 + (k::int)) - 15 = y";
test "(i + j + 12 + (k::int)) - 5 = y";
test "y - b < (b::int)";
test "y - (3*b + c) < (b::int) - 2*c";
test "(2*x - (u*v) + y) - v*3*u = (w::int)";
test "(2*x*u*v + (u*v)*4 + y) - v*u*4 = (w::int)";
test "(2*x*u*v + (u*v)*4 + y) - v*u = (w::int)";
test "u*v - (x*u*v + (u*v)*4 + y) = (w::int)";
test "(i + j + 12 + (k::int)) = u + 15 + y";
test "(i + j*2 + 12 + (k::int)) = j + 5 + y";
test "2*y + 3*z + 6*w + 2*y + 3*z + 2*u = 2*y' + 3*z' + 6*w' + 2*y' + 3*z' + u + (vv::int)";
test "a + -(b+c) + b = (d::int)";
test "a + -(b+c) - b = (d::int)";
(*negative numerals*)
test "(i + j + -2 + (k::int)) - (u + 5 + y) = zz";
test "(i + j + -3 + (k::int)) < u + 5 + y";
test "(i + j + 3 + (k::int)) < u + -6 + y";
test "(i + j + -12 + (k::int)) - 15 = y";
test "(i + j + 12 + (k::int)) - -15 = y";
test "(i + j + -12 + (k::int)) - -15 = y";
*)
(** Constant folding for integer plus and times **)
(*We do not need
structure Nat_Plus_Assoc = Assoc_Fold (Nat_Plus_Assoc_Data);
structure Int_Plus_Assoc = Assoc_Fold (Int_Plus_Assoc_Data);
because combine_numerals does the same thing*)
structure Int_Times_Assoc_Data : ASSOC_FOLD_DATA =
struct
val ss = HOL_ss
val eq_reflection = eq_reflection
val sg_ref = Sign.self_ref (Theory.sign_of (the_context ()))
val T = HOLogic.intT
val plus = Const ("op *", [HOLogic.intT,HOLogic.intT] ---> HOLogic.intT);
val add_ac = zmult_ac
end;
structure Int_Times_Assoc = Assoc_Fold (Int_Times_Assoc_Data);
Addsimprocs [Int_Times_Assoc.conv];
(** The same for the naturals **)
structure Nat_Times_Assoc_Data : ASSOC_FOLD_DATA =
struct
val ss = HOL_ss
val eq_reflection = eq_reflection
val sg_ref = Sign.self_ref (Theory.sign_of (the_context ()))
val T = HOLogic.natT
val plus = Const ("op *", [HOLogic.natT,HOLogic.natT] ---> HOLogic.natT);
val add_ac = mult_ac
end;
structure Nat_Times_Assoc = Assoc_Fold (Nat_Times_Assoc_Data);
Addsimprocs [Nat_Times_Assoc.conv];
(*** decision procedure for linear arithmetic ***)
(*---------------------------------------------------------------------------*)
(* Linear arithmetic *)
(*---------------------------------------------------------------------------*)
(*
Instantiation of the generic linear arithmetic package for int.
*)
(* Update parameters of arithmetic prover *)
local
(* reduce contradictory <= to False *)
val add_rules =
simp_thms @ bin_arith_simps @ bin_rel_simps @
[int_numeral_0_eq_0, int_numeral_1_eq_1,
minus_zero, diff_minus, left_minus, right_minus,
mult_zero_left, mult_zero_right, mult_1, mult_1_right,
minus_mult_left RS sym, minus_mult_right RS sym,
minus_add_distrib, minus_minus, mult_assoc,
int_0, int_1, int_Suc, zadd_int RS sym, zmult_int RS sym,
le_number_of_eq_not_less];
val simprocs = [Int_Times_Assoc.conv, Int_Numeral_Simprocs.combine_numerals]@
Int_Numeral_Simprocs.cancel_numerals @
Bin_Simprocs.eval_numerals;
in
val int_arith_setup =
[Fast_Arith.map_data (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, simpset} =>
{add_mono_thms = add_mono_thms,
mult_mono_thms = mult_mono_thms,
inj_thms = [zle_int RS iffD2,int_int_eq RS iffD2] @ inj_thms,
lessD = lessD @ [zless_imp_add1_zle],
simpset = simpset addsimps add_rules
addsimprocs simprocs
addcongs [if_weak_cong]}),
arith_inj_const ("IntDef.int", HOLogic.natT --> HOLogic.intT),
arith_discrete ("IntDef.int", true)];
end;
val fast_int_arith_simproc =
Simplifier.simproc (Theory.sign_of (the_context()))
"fast_int_arith" ["(m::int) < n","(m::int) <= n", "(m::int) = n"] Fast_Arith.lin_arith_prover;
Addsimprocs [fast_int_arith_simproc]
(* Some test data
Goal "!!a::int. [| a <= b; c <= d; x+y<z |] ==> a+c <= b+d";
by (fast_arith_tac 1);
Goal "!!a::int. [| a < b; c < d |] ==> a-d+ 2 <= b+(-c)";
by (fast_arith_tac 1);
Goal "!!a::int. [| a < b; c < d |] ==> a+c+ 1 < b+d";
by (fast_arith_tac 1);
Goal "!!a::int. [| a <= b; b+b <= c |] ==> a+a <= c";
by (fast_arith_tac 1);
Goal "!!a::int. [| a+b <= i+j; a<=b; i<=j |] \
\ ==> a+a <= j+j";
by (fast_arith_tac 1);
Goal "!!a::int. [| a+b < i+j; a<b; i<j |] \
\ ==> a+a - - -1 < j+j - 3";
by (fast_arith_tac 1);
Goal "!!a::int. a+b+c <= i+j+k & a<=b & b<=c & i<=j & j<=k --> a+a+a <= k+k+k";
by (arith_tac 1);
Goal "!!a::int. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \
\ ==> a <= l";
by (fast_arith_tac 1);
Goal "!!a::int. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \
\ ==> a+a+a+a <= l+l+l+l";
by (fast_arith_tac 1);
Goal "!!a::int. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \
\ ==> a+a+a+a+a <= l+l+l+l+i";
by (fast_arith_tac 1);
Goal "!!a::int. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \
\ ==> a+a+a+a+a+a <= l+l+l+l+i+l";
by (fast_arith_tac 1);
Goal "!!a::int. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \
\ ==> 6*a <= 5*l+i";
by (fast_arith_tac 1);
*)