(*  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 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 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 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";

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 Int.compare (abs i, abs j) of

            EQUAL => Int.compare (Int.sign i, Int.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.compare (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 = NCons_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 =

    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@minus_simps@add_ac))

     THEN ALLGOALS (simp_tac (HOL_ss addsimps non_add_bin_simps@mult_minus_simps))

     THEN ALLGOALS (simp_tac (HOL_ss addsimps minus_from_mult_simps@

                                              add_ac@mult_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 =" 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               = 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@minus_simps@add_ac))

     THEN ALLGOALS (simp_tac (HOL_ss addsimps non_add_bin_simps@mult_minus_simps))

     THEN ALLGOALS (simp_tac (HOL_ss addsimps minus_from_mult_simps@

                                              add_ac@mult_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::'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 sg_ref = Sign.self_ref (Theory.sign_of (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,

     zero_neq_one, zero_less_one, zero_le_one, 

     zero_neq_one RS not_sym, not_one_le_zero, not_one_less_zero];

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, 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.of_nat", HOLogic.natT --> HOLogic.intT),

  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::'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);

*)