(* 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 succ_Pls = thm "succ_Pls";
val succ_Min = thm "succ_Min";
val succ_1 = thm "succ_1";
val succ_0 = thm "succ_0";
val pred_Pls = thm "pred_Pls";
val pred_Min = thm "pred_Min";
val pred_1 = thm "pred_1";
val pred_0 = thm "pred_0";
val minus_Pls = thm "minus_Pls";
val minus_Min = thm "minus_Min";
val minus_1 = thm "minus_1";
val minus_0 = thm "minus_0";
val add_Pls = thm "add_Pls";
val add_Min = thm "add_Min";
val add_BIT_11 = thm "add_BIT_11";
val add_BIT_10 = thm "add_BIT_10";
val add_BIT_0 = thm "add_BIT_0";
val add_Pls_right = thm "add_Pls_right";
val add_Min_right = thm "add_Min_right";
val mult_Pls = thm "mult_Pls";
val mult_Min = thm "mult_Min";
val mult_num1 = thm "mult_num1";
val mult_num0 = thm "mult_num0";
val neg_def = thm "neg_def";
val iszero_def = thm "iszero_def";
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 numeral_0_eq_0 = thm "numeral_0_eq_0";
val numeral_1_eq_1 = thm "numeral_1_eq_1";
val numeral_m1_eq_minus_1 = thm "numeral_m1_eq_minus_1";
val mult_minus1 = thm "mult_minus1";
val mult_minus1_right = thm "mult_minus1_right";
val minus_number_of_mult = thm "minus_number_of_mult";
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 le_number_of_eq = thm "le_number_of_eq";
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 abs_number_of = thm "abs_number_of";
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 arith_extra_simps = thms "arith_extra_simps";
val arith_simps = thms "arith_simps";
val rel_simps = thms "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";
val arith_special = thms "arith_special";
structure Int_Numeral_Base_Simprocs =
struct
fun prove_conv tacs ctxt (_: thm list) (t, u) =
if t aconv u then NONE
else
let val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (t, u))
in SOME (Goal.prove ctxt [] [] eq (K (EVERY tacs))) end
fun prove_conv_nohyps 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)
(*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;
Addsimps arith_special;
Addsimprocs [Int_Numeral_Base_Simprocs.reorient_simproc];
structure Int_Numeral_Simprocs =
struct
(*Maps 0 to Numeral0 and 1 to Numeral1 so that arithmetic in Int_Numeral_Base_Simprocs
isn't complicated by the abstract 0 and 1.*)
val numeral_syms = [numeral_0_eq_0 RS sym, numeral_1_eq_1 RS sym];
(** New term ordering so that AC-rewriting brings numerals to the front **)
(*Order integers by absolute value and then by sign. The standard integer
ordering is not well-founded.*)
fun num_ord (i,j) =
(case IntInf.compare (IntInf.abs i, IntInf.abs j) of
EQUAL => int_ord (IntInf.sign i, IntInf.sign j)
| ord => ord);
(*This resembles Term.term_ord, but it puts binary numerals before other
non-atomic terms.*)
local open Term
in
fun numterm_ord (Abs (_, T, t), Abs(_, U, u)) =
(case numterm_ord (t, u) of EQUAL => typ_ord (T, U) | ord => ord)
| numterm_ord
(Const("Numeral.number_of", _) $ v, Const("Numeral.number_of", _) $ w) =
num_ord (HOLogic.dest_binum v, HOLogic.dest_binum w)
| numterm_ord (Const("Numeral.number_of", _) $ _, _) = LESS
| numterm_ord (_, Const("Numeral.number_of", _) $ _) = GREATER
| numterm_ord (t, u) =
(case int_ord (size_of_term t, size_of_term u) of
EQUAL =>
let val (f, ts) = strip_comb t and (g, us) = strip_comb u in
(case hd_ord (f, g) of EQUAL => numterms_ord (ts, us) | ord => ord)
end
| ord => ord)
and numterms_ord (ts, us) = list_ord numterm_ord (ts, us)
end;
fun numtermless tu = (numterm_ord tu = LESS);
(*Defined in this file, but perhaps needed only for Int_Numeral_Base_Simprocs of type nat.*)
val num_ss = HOL_ss settermless numtermless;
(** Utilities **)
fun mk_numeral T n = HOLogic.number_of_const T $ HOLogic.mk_binum 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 mk_plus = HOLogic.mk_binop "HOL.plus";
fun mk_minus t =
let val T = Term.fastype_of t
in Const ("HOL.uminus", T --> T) $ t
end;
(*Thus mk_sum[t] yields t+0; longer sums don't have a trailing zero*)
fun mk_sum T [] = mk_numeral T 0
| mk_sum T [t,u] = mk_plus (t, u)
| mk_sum T (t :: ts) = mk_plus (t, mk_sum T ts);
(*this version ALWAYS includes a trailing zero*)
fun long_mk_sum T [] = mk_numeral T 0
| long_mk_sum T (t :: ts) = mk_plus (t, mk_sum T ts);
val dest_plus = HOLogic.dest_bin "HOL.plus" Term.dummyT;
(*decompose additions AND subtractions as a sum*)
fun dest_summing (pos, Const ("HOL.plus", _) $ t $ u, ts) =
dest_summing (pos, t, dest_summing (pos, u, ts))
| dest_summing (pos, Const ("HOL.minus", _) $ t $ u, ts) =
dest_summing (pos, t, dest_summing (not pos, u, ts))
| dest_summing (pos, t, ts) =
if pos then t::ts else mk_minus t :: ts;
fun dest_sum t = dest_summing (true, t, []);
val mk_diff = HOLogic.mk_binop "HOL.minus";
val dest_diff = HOLogic.dest_bin "HOL.minus" Term.dummyT;
val mk_times = HOLogic.mk_binop "HOL.times";
fun mk_prod T =
let val one = mk_numeral T 1
fun mk [] = one
| mk [t] = t
| mk (t :: ts) = if t = one then mk ts else mk_times (t, mk ts)
in mk end;
(*This version ALWAYS includes a trailing one*)
fun long_mk_prod T [] = mk_numeral T 1
| long_mk_prod T (t :: ts) = mk_times (t, mk_prod T ts);
val dest_times = HOLogic.dest_bin "HOL.times" Term.dummyT;
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, t) = mk_times (mk_numeral (Term.fastype_of t) k, t);
(*Express t as a product of (possibly) a numeral with other sorted terms*)
fun dest_coeff sign (Const ("HOL.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 (Term.fastype_of t) 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 = thms "add_0s";
val mult_1s = thms "mult_1s";
(*To perform binary arithmetic. The "left" rewriting handles patterns
created by the Int_Numeral_Base_Simprocs, such as 3 * (5 * x). *)
val simps = [numeral_0_eq_0 RS sym, numeral_1_eq_1 RS sym,
add_number_of_left, mult_number_of_left] @
arith_simps @ rel_simps;
(*Binary arithmetic BUT NOT ADDITION since it may collapse adjacent terms
during re-arrangement*)
val non_add_simps =
subtract Thm.eq_thm [add_number_of_left, number_of_add RS sym] simps;
(*To evaluate binary negations of coefficients*)
val minus_simps = [numeral_m1_eq_minus_1 RS sym, number_of_minus RS sym,
minus_1, minus_0, minus_Pls, minus_Min,
pred_1, pred_0, pred_Pls, pred_Min];
(*To let us treat subtraction as addition*)
val diff_simps = [diff_minus, minus_add_distrib, minus_minus];
(*push the unary minus down: - x * y = x * - y *)
val minus_mult_eq_1_to_2 =
[minus_mult_left RS sym, minus_mult_right] MRS trans |> standard;
(*to extract again any uncancelled minuses*)
val minus_from_mult_simps =
[minus_minus, minus_mult_left RS sym, minus_mult_right RS sym];
(*combine unary minus with numeric literals, however nested within a product*)
val mult_minus_simps =
[mult_assoc, minus_mult_left, 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 =
let val ss0 = HOL_basic_ss addeqcongs [eq_cong2] addsimps rules
in fn ss => simplify (Simplifier.inherit_context ss ss0) o mk_meta_eq end
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 = fn _ => trans_tac
val norm_ss1 = num_ss addsimps numeral_syms @ add_0s @ mult_1s @
diff_simps @ minus_simps @ add_ac
val norm_ss2 = num_ss addsimps non_add_simps @ mult_minus_simps
val norm_ss3 = num_ss addsimps minus_from_mult_simps @ add_ac @ mult_ac
fun norm_tac ss =
ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss1))
THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss2))
THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss3))
val numeral_simp_ss = HOL_ss addsimps add_0s @ simps
fun numeral_simp_tac ss = ALLGOALS (simp_tac (Simplifier.inherit_context ss numeral_simp_ss))
val simplify_meta_eq = simplify_meta_eq (add_0s @ mult_1s)
end;
structure EqCancelNumerals = CancelNumeralsFun
(open CancelNumeralsCommon
val prove_conv = Int_Numeral_Base_Simprocs.prove_conv
val mk_bal = HOLogic.mk_eq
val dest_bal = HOLogic.dest_bin "op =" Term.dummyT
val bal_add1 = eq_add_iff1 RS trans
val bal_add2 = eq_add_iff2 RS trans
);
structure LessCancelNumerals = CancelNumeralsFun
(open CancelNumeralsCommon
val prove_conv = Int_Numeral_Base_Simprocs.prove_conv
val mk_bal = HOLogic.mk_binrel "Orderings.less"
val dest_bal = HOLogic.dest_bin "Orderings.less" Term.dummyT
val bal_add1 = less_add_iff1 RS trans
val bal_add2 = less_add_iff2 RS trans
);
structure LeCancelNumerals = CancelNumeralsFun
(open CancelNumeralsCommon
val prove_conv = Int_Numeral_Base_Simprocs.prove_conv
val mk_bal = HOLogic.mk_binrel "Orderings.less_eq"
val dest_bal = HOLogic.dest_bin "Orderings.less_eq" Term.dummyT
val bal_add1 = le_add_iff1 RS trans
val bal_add2 = le_add_iff2 RS trans
);
val cancel_numerals =
map Int_Numeral_Base_Simprocs.prep_simproc
[("inteq_cancel_numerals",
["(l::'a::number_ring) + m = n",
"(l::'a::number_ring) = m + n",
"(l::'a::number_ring) - m = n",
"(l::'a::number_ring) = m - n",
"(l::'a::number_ring) * m = n",
"(l::'a::number_ring) = m * n"],
K EqCancelNumerals.proc),
("intless_cancel_numerals",
["(l::'a::{ordered_idom,number_ring}) + m < n",
"(l::'a::{ordered_idom,number_ring}) < m + n",
"(l::'a::{ordered_idom,number_ring}) - m < n",
"(l::'a::{ordered_idom,number_ring}) < m - n",
"(l::'a::{ordered_idom,number_ring}) * m < n",
"(l::'a::{ordered_idom,number_ring}) < m * n"],
K LessCancelNumerals.proc),
("intle_cancel_numerals",
["(l::'a::{ordered_idom,number_ring}) + m <= n",
"(l::'a::{ordered_idom,number_ring}) <= m + n",
"(l::'a::{ordered_idom,number_ring}) - m <= n",
"(l::'a::{ordered_idom,number_ring}) <= m - n",
"(l::'a::{ordered_idom,number_ring}) * m <= n",
"(l::'a::{ordered_idom,number_ring}) <= m * n"],
K LeCancelNumerals.proc)];
structure CombineNumeralsData =
struct
val add = IntInf.+
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 = Int_Numeral_Base_Simprocs.prove_conv_nohyps
val trans_tac = fn _ => trans_tac
val norm_ss1 = num_ss addsimps numeral_syms @ add_0s @ mult_1s @
diff_simps @ minus_simps @ add_ac
val norm_ss2 = num_ss addsimps non_add_simps @ mult_minus_simps
val norm_ss3 = num_ss addsimps minus_from_mult_simps @ add_ac @ mult_ac
fun norm_tac ss =
ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss1))
THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss2))
THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss3))
val numeral_simp_ss = HOL_ss addsimps add_0s @ simps
fun numeral_simp_tac ss = ALLGOALS (simp_tac (Simplifier.inherit_context ss numeral_simp_ss))
val simplify_meta_eq = simplify_meta_eq (add_0s @ mult_1s)
end;
structure CombineNumerals = CombineNumeralsFun(CombineNumeralsData);
val combine_numerals =
Int_Numeral_Base_Simprocs.prep_simproc
("int_combine_numerals",
["(i::'a::number_ring) + j", "(i::'a::number_ring) - j"],
K 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 multiplication in semirings **)
(*We do not need folding for addition: combine_numerals does the same thing*)
structure Semiring_Times_Assoc_Data : ASSOC_FOLD_DATA =
struct
val assoc_ss = HOL_ss addsimps mult_ac
val eq_reflection = eq_reflection
end;
structure Semiring_Times_Assoc = Assoc_Fold (Semiring_Times_Assoc_Data);
val assoc_fold_simproc =
Int_Numeral_Base_Simprocs.prep_simproc
("semiring_assoc_fold", ["(a::'a::comm_semiring_1_cancel) * b"],
K Semiring_Times_Assoc.proc);
Addsimprocs [assoc_fold_simproc];
(*** 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 @ arith_simps @ rel_simps @ arith_special @
[neg_le_iff_le, numeral_0_eq_0, numeral_1_eq_1,
minus_zero, diff_minus, left_minus, right_minus,
mult_zero_left, mult_zero_right, mult_num1, mult_1_right,
minus_mult_left RS sym, minus_mult_right RS sym,
minus_add_distrib, minus_minus, mult_assoc,
of_nat_0, of_nat_1, of_nat_Suc, of_nat_add, of_nat_mult,
of_int_0, of_int_1, of_int_add, of_int_mult, int_eq_of_nat]
val nat_inj_thms = [zle_int RS iffD2,
int_int_eq RS iffD2]
val Int_Numeral_Base_Simprocs = [assoc_fold_simproc, Int_Numeral_Simprocs.combine_numerals]@
Int_Numeral_Simprocs.cancel_numerals
in
val int_arith_setup =
Fast_Arith.map_data (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, neqE, simpset} =>
{add_mono_thms = add_mono_thms,
mult_mono_thms = [mult_strict_left_mono,mult_left_mono] @ mult_mono_thms,
inj_thms = nat_inj_thms @ inj_thms,
lessD = lessD @ [zless_imp_add1_zle],
neqE = neqE,
simpset = simpset addsimps add_rules
addsimprocs Int_Numeral_Base_Simprocs
addcongs [if_weak_cong]}) #>
arith_inj_const ("NatArith.of_nat", HOLogic.natT --> HOLogic.intT) #>
arith_inj_const ("IntDef.int", HOLogic.natT --> HOLogic.intT) #>
arith_discrete "IntDef.int"
end;
val fast_int_arith_simproc =
Simplifier.simproc (Theory.sign_of (the_context()))
"fast_int_arith"
["(m::'a::{ordered_idom,number_ring}) < n",
"(m::'a::{ordered_idom,number_ring}) <= n",
"(m::'a::{ordered_idom,number_ring}) = 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);
*)