(* Title: HOL/Integ/int_arith1.ML ID: $Id: int_arith1.ML,v 1.42 2005/08/01 17:20:26 wenzelm Exp $ Authors: Larry Paulson and Tobias Nipkow Simprocs and decision procedure for linear arithmetic. *) (** Misc ML bindings **) val bin_succ_Pls = thm"bin_succ_Pls"; val bin_succ_Min = thm"bin_succ_Min"; val bin_succ_1 = thm"bin_succ_1"; val bin_succ_0 = thm"bin_succ_0"; val bin_pred_Pls = thm"bin_pred_Pls"; val bin_pred_Min = thm"bin_pred_Min"; val bin_pred_1 = thm"bin_pred_1"; val bin_pred_0 = thm"bin_pred_0"; val bin_minus_Pls = thm"bin_minus_Pls"; val bin_minus_Min = thm"bin_minus_Min"; val bin_minus_1 = thm"bin_minus_1"; val bin_minus_0 = thm"bin_minus_0"; val bin_add_Pls = thm"bin_add_Pls"; val bin_add_Min = thm"bin_add_Min"; 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_mult_Pls = thm"bin_mult_Pls"; val bin_mult_Min = thm"bin_mult_Min"; val bin_mult_1 = thm"bin_mult_1"; val bin_mult_0 = thm"bin_mult_0"; 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 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"; val arith_special = thms"arith_special"; 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) (*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 [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 = [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 simprocs of type nat.*) val num_ss = HOL_ss settermless numtermless; (** Utilities **) fun mk_numeral T n = HOLogic.number_of_const T $ 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 mk_plus = HOLogic.mk_binop "op +"; fun mk_minus t = let val T = Term.fastype_of t in Const ("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 "op +" Term.dummyT; (*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 mk_minus t :: ts; fun dest_sum t = dest_summing (true, t, []); val mk_diff = HOLogic.mk_binop "op -"; val dest_diff = HOLogic.dest_bin "op -" Term.dummyT; val mk_times = HOLogic.mk_binop "op *"; 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 "op *" 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 ("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 simprocs, such as 3 * (5 * x). *) val bin_simps = [numeral_0_eq_0 RS sym, 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 minus_simps = [numeral_m1_eq_minus_1 RS sym, 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 = [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 ss = simplify (Simplifier.inherit_bounds ss 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 = fn _ => trans_tac fun norm_tac ss = let val num_ss' = Simplifier.inherit_bounds ss num_ss in ALLGOALS (simp_tac (num_ss' addsimps numeral_syms @ add_0s @ mult_1s @ diff_simps @ minus_simps @ add_ac)) THEN ALLGOALS (simp_tac (num_ss' addsimps non_add_bin_simps @ mult_minus_simps)) THEN ALLGOALS (simp_tac (num_ss' addsimps minus_from_mult_simps @ add_ac @ mult_ac)) end fun numeral_simp_tac ss = ALLGOALS (simp_tac (Simplifier.inherit_bounds ss 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 =" 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 = Bin_Simprocs.prove_conv val mk_bal = HOLogic.mk_binrel "op <" val dest_bal = HOLogic.dest_bin "op <" 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 = Bin_Simprocs.prove_conv val mk_bal = HOLogic.mk_binrel "op <=" val dest_bal = HOLogic.dest_bin "op <=" Term.dummyT 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::'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"], 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"], 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"], 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 = Bin_Simprocs.prove_conv_nohyps val trans_tac = fn _ => trans_tac fun norm_tac ss = let val num_ss' = Simplifier.inherit_bounds ss num_ss in ALLGOALS (simp_tac (num_ss' addsimps numeral_syms @ add_0s @ mult_1s @ diff_simps @ minus_simps @ add_ac)) THEN ALLGOALS (simp_tac (num_ss' addsimps non_add_bin_simps @ mult_minus_simps)) THEN ALLGOALS (simp_tac (num_ss' addsimps minus_from_mult_simps @ add_ac @ mult_ac)) end fun numeral_simp_tac ss = ALLGOALS (simp_tac (Simplifier.inherit_bounds ss 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::'a::number_ring) + j", "(i::'a::number_ring) - 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 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 ss = HOL_ss val eq_reflection = eq_reflection val thy_ref = Theory.self_ref (the_context ()) val add_ac = mult_ac end; structure Semiring_Times_Assoc = Assoc_Fold (Semiring_Times_Assoc_Data); val assoc_fold_simproc = Bin_Simprocs.prep_simproc ("semiring_assoc_fold", ["(a::'a::comm_semiring_1_cancel) * b"], 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 @ bin_arith_simps @ bin_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_1, 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 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 = [zle_int RS iffD2,int_int_eq RS iffD2] @ inj_thms, lessD = lessD @ [zless_imp_add1_zle], neqE = thm "linorder_neqE_int" :: neqE, simpset = simpset addsimps add_rules addsimprocs simprocs addcongs [if_weak_cong]}), arith_inj_const ("IntDef.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); *)