src/HOL/Tools/nat_numeral_simprocs.ML

modules numeral_simprocs, nat_numeral_simprocs; proper structures for numeral simprocs

(* Author: Lawrence C Paulson, Cambridge University Computer Laboratory

Simprocs for nat numerals.
*)

signature NAT_NUMERAL_SIMPROCS =
sig
  val combine_numerals: simproc
  val cancel_numerals: simproc list
  val cancel_factors: simproc list
  val cancel_numeral_factors: simproc list
end;

structure Nat_Numeral_Simprocs =
struct

(*Maps n to #n for n = 0, 1, 2*)
val numeral_syms = [@{thm nat_numeral_0_eq_0} RS sym, @{thm nat_numeral_1_eq_1} RS sym, @{thm numeral_2_eq_2} 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_number n = HOLogic.number_of_const HOLogic.natT $ HOLogic.mk_numeral n;
fun dest_number t = Int.max (0, snd (HOLogic.dest_number t));

fun find_first_numeral past (t::terms) =
        ((dest_number t, t, rev past @ terms)
         handle TERM _ => find_first_numeral (t::past) terms)
  | find_first_numeral past [] = raise TERM("find_first_numeral", []);

val zero = mk_number 0;
val mk_plus = HOLogic.mk_binop @{const_name HOL.plus};

(*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 []        = HOLogic.zero
  | long_mk_sum (t :: ts) = mk_plus (t, mk_sum ts);

val dest_plus = HOLogic.dest_bin @{const_name HOL.plus} HOLogic.natT;


(** Other simproc items **)

val bin_simps =
     [@{thm nat_numeral_0_eq_0} RS sym, @{thm nat_numeral_1_eq_1} RS sym,
      @{thm add_nat_number_of}, @{thm nat_number_of_add_left}, 
      @{thm diff_nat_number_of}, @{thm le_number_of_eq_not_less},
      @{thm mult_nat_number_of}, @{thm nat_number_of_mult_left}, 
      @{thm less_nat_number_of}, 
      @{thm Let_number_of}, @{thm nat_number_of}] @
     @{thms arith_simps} @ @{thms rel_simps} @ @{thms neg_simps};


(*** CancelNumerals simprocs ***)

val one = mk_number 1;
val mk_times = HOLogic.mk_binop @{const_name HOL.times};

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 @{const_name HOL.times} HOLogic.natT;

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_number k, t);

(*Express t as a product of (possibly) a numeral with other factors, sorted*)
fun dest_coeff t =
    let val ts = sort TermOrd.term_ord (dest_prod t)
        val (n, _, ts') = find_first_numeral [] ts
                          handle TERM _ => (1, one, ts)
    in (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 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;


(*Split up a sum into the list of its constituent terms, on the way removing any
  Sucs and counting them.*)
fun dest_Suc_sum (Const ("Suc", _) $ t, (k,ts)) = dest_Suc_sum (t, (k+1,ts))
  | dest_Suc_sum (t, (k,ts)) = 
      let val (t1,t2) = dest_plus t
      in  dest_Suc_sum (t1, dest_Suc_sum (t2, (k,ts)))  end
      handle TERM _ => (k, t::ts);

(*Code for testing whether numerals are already used in the goal*)
fun is_numeral (Const(@{const_name Int.number_of}, _) $ w) = true
  | is_numeral _ = false;

fun prod_has_numeral t = exists is_numeral (dest_prod t);

(*The Sucs found in the term are converted to a binary numeral. If relaxed is false,
  an exception is raised unless the original expression contains at least one
  numeral in a coefficient position.  This prevents nat_combine_numerals from 
  introducing numerals to goals.*)
fun dest_Sucs_sum relaxed t = 
  let val (k,ts) = dest_Suc_sum (t,(0,[]))
  in
     if relaxed orelse exists prod_has_numeral ts then 
       if k=0 then ts
       else mk_number k :: ts
     else raise TERM("Nat_Numeral_Simprocs.dest_Sucs_sum", [t])
  end;


(*Simplify 1*n and n*1 to n*)
val add_0s  = map rename_numerals [@{thm add_0}, @{thm add_0_right}];
val mult_1s = map rename_numerals [@{thm nat_mult_1}, @{thm nat_mult_1_right}];

(*Final simplification: cancel + and *; replace Numeral0 by 0 and Numeral1 by 1*)

(*And these help the simproc return False when appropriate, which helps
  the arith prover.*)
val contra_rules = [@{thm add_Suc}, @{thm add_Suc_right}, @{thm Zero_not_Suc},
  @{thm Suc_not_Zero}, @{thm le_0_eq}];

val simplify_meta_eq =
    Arith_Data.simplify_meta_eq
        ([@{thm nat_numeral_0_eq_0}, @{thm numeral_1_eq_Suc_0}, @{thm add_0}, @{thm add_0_right},
          @{thm mult_0}, @{thm mult_0_right}, @{thm mult_1}, @{thm mult_1_right}] @ contra_rules);


(*** Applying CancelNumeralsFun ***)

structure CancelNumeralsCommon =
  struct
  val mk_sum            = (fn T:typ => mk_sum)
  val dest_sum          = dest_Sucs_sum true
  val mk_coeff          = mk_coeff
  val dest_coeff        = dest_coeff
  val find_first_coeff  = find_first_coeff []
  val trans_tac         = K Arith_Data.trans_tac

  val norm_ss1 = Numeral_Simprocs.num_ss addsimps numeral_syms @ add_0s @ mult_1s @
    [@{thm Suc_eq_add_numeral_1_left}] @ @{thms add_ac}
  val norm_ss2 = Numeral_Simprocs.num_ss addsimps bin_simps @ @{thms add_ac} @ @{thms 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))

  val numeral_simp_ss = HOL_ss addsimps add_0s @ bin_simps;
  fun numeral_simp_tac ss = ALLGOALS (simp_tac (Simplifier.inherit_context ss numeral_simp_ss));
  val simplify_meta_eq  = simplify_meta_eq
  end;


structure EqCancelNumerals = CancelNumeralsFun
 (open CancelNumeralsCommon
  val prove_conv = Arith_Data.prove_conv
  val mk_bal   = HOLogic.mk_eq
  val dest_bal = HOLogic.dest_bin "op =" HOLogic.natT
  val bal_add1 = @{thm nat_eq_add_iff1} RS trans
  val bal_add2 = @{thm nat_eq_add_iff2} RS trans
);

structure LessCancelNumerals = CancelNumeralsFun
 (open CancelNumeralsCommon
  val prove_conv = Arith_Data.prove_conv
  val mk_bal   = HOLogic.mk_binrel @{const_name HOL.less}
  val dest_bal = HOLogic.dest_bin @{const_name HOL.less} HOLogic.natT
  val bal_add1 = @{thm nat_less_add_iff1} RS trans
  val bal_add2 = @{thm nat_less_add_iff2} RS trans
);

structure LeCancelNumerals = CancelNumeralsFun
 (open CancelNumeralsCommon
  val prove_conv = Arith_Data.prove_conv
  val mk_bal   = HOLogic.mk_binrel @{const_name HOL.less_eq}
  val dest_bal = HOLogic.dest_bin @{const_name HOL.less_eq} HOLogic.natT
  val bal_add1 = @{thm nat_le_add_iff1} RS trans
  val bal_add2 = @{thm nat_le_add_iff2} RS trans
);

structure DiffCancelNumerals = CancelNumeralsFun
 (open CancelNumeralsCommon
  val prove_conv = Arith_Data.prove_conv
  val mk_bal   = HOLogic.mk_binop @{const_name HOL.minus}
  val dest_bal = HOLogic.dest_bin @{const_name HOL.minus} HOLogic.natT
  val bal_add1 = @{thm nat_diff_add_eq1} RS trans
  val bal_add2 = @{thm nat_diff_add_eq2} RS trans
);


val cancel_numerals =
  map Arith_Data.prep_simproc
   [("nateq_cancel_numerals",
     ["(l::nat) + m = n", "(l::nat) = m + n",
      "(l::nat) * m = n", "(l::nat) = m * n",
      "Suc m = n", "m = Suc n"],
     K EqCancelNumerals.proc),
    ("natless_cancel_numerals",
     ["(l::nat) + m < n", "(l::nat) < m + n",
      "(l::nat) * m < n", "(l::nat) < m * n",
      "Suc m < n", "m < Suc n"],
     K LessCancelNumerals.proc),
    ("natle_cancel_numerals",
     ["(l::nat) + m <= n", "(l::nat) <= m + n",
      "(l::nat) * m <= n", "(l::nat) <= m * n",
      "Suc m <= n", "m <= Suc n"],
     K LeCancelNumerals.proc),
    ("natdiff_cancel_numerals",
     ["((l::nat) + m) - n", "(l::nat) - (m + n)",
      "(l::nat) * m - n", "(l::nat) - m * n",
      "Suc m - n", "m - Suc n"],
     K DiffCancelNumerals.proc)];


(*** Applying CombineNumeralsFun ***)

structure CombineNumeralsData =
  struct
  type coeff            = int
  val iszero            = (fn x => x = 0)
  val add               = op +
  val mk_sum            = (fn T:typ => long_mk_sum)  (*to work for 2*x + 3*x *)
  val dest_sum          = dest_Sucs_sum false
  val mk_coeff          = mk_coeff
  val dest_coeff        = dest_coeff
  val left_distrib      = @{thm left_add_mult_distrib} RS trans
  val prove_conv        = Arith_Data.prove_conv_nohyps
  val trans_tac         = K Arith_Data.trans_tac

  val norm_ss1 = Numeral_Simprocs.num_ss addsimps numeral_syms @ add_0s @ mult_1s @ [@{thm Suc_eq_add_numeral_1}] @ @{thms add_ac}
  val norm_ss2 = Numeral_Simprocs.num_ss addsimps bin_simps @ @{thms add_ac} @ @{thms 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))

  val numeral_simp_ss = HOL_ss addsimps add_0s @ bin_simps;
  fun numeral_simp_tac ss = ALLGOALS (simp_tac (Simplifier.inherit_context ss numeral_simp_ss))
  val simplify_meta_eq  = simplify_meta_eq
  end;

structure CombineNumerals = CombineNumeralsFun(CombineNumeralsData);

val combine_numerals =
  Arith_Data.prep_simproc ("nat_combine_numerals", ["(i::nat) + j", "Suc (i + j)"], K CombineNumerals.proc);


(*** Applying CancelNumeralFactorFun ***)

structure CancelNumeralFactorCommon =
  struct
  val mk_coeff          = mk_coeff
  val dest_coeff        = dest_coeff
  val trans_tac         = K Arith_Data.trans_tac

  val norm_ss1 = Numeral_Simprocs.num_ss addsimps
    numeral_syms @ add_0s @ mult_1s @ [@{thm Suc_eq_add_numeral_1_left}] @ @{thms add_ac}
  val norm_ss2 = Numeral_Simprocs.num_ss addsimps bin_simps @ @{thms add_ac} @ @{thms 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))

  val numeral_simp_ss = HOL_ss addsimps bin_simps
  fun numeral_simp_tac ss = ALLGOALS (simp_tac (Simplifier.inherit_context ss numeral_simp_ss))
  val simplify_meta_eq  = simplify_meta_eq
  end

structure DivCancelNumeralFactor = CancelNumeralFactorFun
 (open CancelNumeralFactorCommon
  val prove_conv = Arith_Data.prove_conv
  val mk_bal   = HOLogic.mk_binop @{const_name Divides.div}
  val dest_bal = HOLogic.dest_bin @{const_name Divides.div} HOLogic.natT
  val cancel = @{thm nat_mult_div_cancel1} RS trans
  val neg_exchanges = false
)

structure DvdCancelNumeralFactor = CancelNumeralFactorFun
 (open CancelNumeralFactorCommon
  val prove_conv = Arith_Data.prove_conv
  val mk_bal   = HOLogic.mk_binrel @{const_name Ring_and_Field.dvd}
  val dest_bal = HOLogic.dest_bin @{const_name Ring_and_Field.dvd} HOLogic.natT
  val cancel = @{thm nat_mult_dvd_cancel1} RS trans
  val neg_exchanges = false
)

structure EqCancelNumeralFactor = CancelNumeralFactorFun
 (open CancelNumeralFactorCommon
  val prove_conv = Arith_Data.prove_conv
  val mk_bal   = HOLogic.mk_eq
  val dest_bal = HOLogic.dest_bin "op =" HOLogic.natT
  val cancel = @{thm nat_mult_eq_cancel1} RS trans
  val neg_exchanges = false
)

structure LessCancelNumeralFactor = CancelNumeralFactorFun
 (open CancelNumeralFactorCommon
  val prove_conv = Arith_Data.prove_conv
  val mk_bal   = HOLogic.mk_binrel @{const_name HOL.less}
  val dest_bal = HOLogic.dest_bin @{const_name HOL.less} HOLogic.natT
  val cancel = @{thm nat_mult_less_cancel1} RS trans
  val neg_exchanges = true
)

structure LeCancelNumeralFactor = CancelNumeralFactorFun
 (open CancelNumeralFactorCommon
  val prove_conv = Arith_Data.prove_conv
  val mk_bal   = HOLogic.mk_binrel @{const_name HOL.less_eq}
  val dest_bal = HOLogic.dest_bin @{const_name HOL.less_eq} HOLogic.natT
  val cancel = @{thm nat_mult_le_cancel1} RS trans
  val neg_exchanges = true
)

val cancel_numeral_factors =
  map Arith_Data.prep_simproc
   [("nateq_cancel_numeral_factors",
     ["(l::nat) * m = n", "(l::nat) = m * n"],
     K EqCancelNumeralFactor.proc),
    ("natless_cancel_numeral_factors",
     ["(l::nat) * m < n", "(l::nat) < m * n"],
     K LessCancelNumeralFactor.proc),
    ("natle_cancel_numeral_factors",
     ["(l::nat) * m <= n", "(l::nat) <= m * n"],
     K LeCancelNumeralFactor.proc),
    ("natdiv_cancel_numeral_factors",
     ["((l::nat) * m) div n", "(l::nat) div (m * n)"],
     K DivCancelNumeralFactor.proc),
    ("natdvd_cancel_numeral_factors",
     ["((l::nat) * m) dvd n", "(l::nat) dvd (m * n)"],
     K DvdCancelNumeralFactor.proc)];



(*** Applying ExtractCommonTermFun ***)

(*this version ALWAYS includes a trailing one*)
fun long_mk_prod []        = one
  | long_mk_prod (t :: ts) = mk_times (t, mk_prod ts);

(*Find first term that matches u*)
fun find_first_t past u []         = raise TERM("find_first_t", [])
  | find_first_t past u (t::terms) =
        if u aconv t then (rev past @ terms)
        else find_first_t (t::past) u terms
        handle TERM _ => find_first_t (t::past) u terms;

(** Final simplification for the CancelFactor simprocs **)
val simplify_one = Arith_Data.simplify_meta_eq  
  [@{thm mult_1_left}, @{thm mult_1_right}, @{thm div_1}, @{thm numeral_1_eq_Suc_0}];

fun cancel_simplify_meta_eq ss cancel_th th =
    simplify_one ss (([th, cancel_th]) MRS trans);

structure CancelFactorCommon =
  struct
  val mk_sum            = (fn T:typ => long_mk_prod)
  val dest_sum          = dest_prod
  val mk_coeff          = mk_coeff
  val dest_coeff        = dest_coeff
  val find_first        = find_first_t []
  val trans_tac         = K Arith_Data.trans_tac
  val norm_ss = HOL_ss addsimps mult_1s @ @{thms mult_ac}
  fun norm_tac ss = ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss))
  val simplify_meta_eq  = cancel_simplify_meta_eq
  end;

structure EqCancelFactor = ExtractCommonTermFun
 (open CancelFactorCommon
  val prove_conv = Arith_Data.prove_conv
  val mk_bal   = HOLogic.mk_eq
  val dest_bal = HOLogic.dest_bin "op =" HOLogic.natT
  val simp_conv = K(K (SOME @{thm nat_mult_eq_cancel_disj}))
);

structure LessCancelFactor = ExtractCommonTermFun
 (open CancelFactorCommon
  val prove_conv = Arith_Data.prove_conv
  val mk_bal   = HOLogic.mk_binrel @{const_name HOL.less}
  val dest_bal = HOLogic.dest_bin @{const_name HOL.less} HOLogic.natT
  val simp_conv = K(K (SOME @{thm nat_mult_less_cancel_disj}))
);

structure LeCancelFactor = ExtractCommonTermFun
 (open CancelFactorCommon
  val prove_conv = Arith_Data.prove_conv
  val mk_bal   = HOLogic.mk_binrel @{const_name HOL.less_eq}
  val dest_bal = HOLogic.dest_bin @{const_name HOL.less_eq} HOLogic.natT
  val simp_conv = K(K (SOME @{thm nat_mult_le_cancel_disj}))
);

structure DivideCancelFactor = ExtractCommonTermFun
 (open CancelFactorCommon
  val prove_conv = Arith_Data.prove_conv
  val mk_bal   = HOLogic.mk_binop @{const_name Divides.div}
  val dest_bal = HOLogic.dest_bin @{const_name Divides.div} HOLogic.natT
  val simp_conv = K(K (SOME @{thm nat_mult_div_cancel_disj}))
);

structure DvdCancelFactor = ExtractCommonTermFun
 (open CancelFactorCommon
  val prove_conv = Arith_Data.prove_conv
  val mk_bal   = HOLogic.mk_binrel @{const_name Ring_and_Field.dvd}
  val dest_bal = HOLogic.dest_bin @{const_name Ring_and_Field.dvd} HOLogic.natT
  val simp_conv = K(K (SOME @{thm nat_mult_dvd_cancel_disj}))
);

val cancel_factor =
  map Arith_Data.prep_simproc
   [("nat_eq_cancel_factor",
     ["(l::nat) * m = n", "(l::nat) = m * n"],
     K EqCancelFactor.proc),
    ("nat_less_cancel_factor",
     ["(l::nat) * m < n", "(l::nat) < m * n"],
     K LessCancelFactor.proc),
    ("nat_le_cancel_factor",
     ["(l::nat) * m <= n", "(l::nat) <= m * n"],
     K LeCancelFactor.proc),
    ("nat_divide_cancel_factor",
     ["((l::nat) * m) div n", "(l::nat) div (m * n)"],
     K DivideCancelFactor.proc),
    ("nat_dvd_cancel_factor",
     ["((l::nat) * m) dvd n", "(l::nat) dvd (m * n)"],
     K DvdCancelFactor.proc)];

end;


Addsimprocs Nat_Numeral_Simprocs.cancel_numerals;
Addsimprocs [Nat_Numeral_Simprocs.combine_numerals];
Addsimprocs Nat_Numeral_Simprocs.cancel_numeral_factors;
Addsimprocs Nat_Numeral_Simprocs.cancel_factor;


(*examples:
print_depth 22;
set timing;
set trace_simp;
fun test s = (Goal s; by (Simp_tac 1));

(*cancel_numerals*)
test "l +( 2) + (2) + 2 + (l + 2) + (oo  + 2) = (uu::nat)";
test "(2*length xs < 2*length xs + j)";
test "(2*length xs < length xs * 2 + j)";
test "2*u = (u::nat)";
test "2*u = Suc (u)";
test "(i + j + 12 + (k::nat)) - 15 = y";
test "(i + j + 12 + (k::nat)) - 5 = y";
test "Suc u - 2 = y";
test "Suc (Suc (Suc u)) - 2 = y";
test "(i + j + 2 + (k::nat)) - 1 = y";
test "(i + j + 1 + (k::nat)) - 2 = y";

test "(2*x + (u*v) + y) - v*3*u = (w::nat)";
test "(2*x*u*v + 5 + (u*v)*4 + y) - v*u*4 = (w::nat)";
test "(2*x*u*v + (u*v)*4 + y) - v*u = (w::nat)";
test "Suc (Suc (2*x*u*v + u*4 + y)) - u = w";
test "Suc ((u*v)*4) - v*3*u = w";
test "Suc (Suc ((u*v)*3)) - v*3*u = w";

test "(i + j + 12 + (k::nat)) = u + 15 + y";
test "(i + j + 32 + (k::nat)) - (u + 15 + y) = zz";
test "(i + j + 12 + (k::nat)) = u + 5 + y";
(*Suc*)
test "(i + j + 12 + k) = Suc (u + y)";
test "Suc (Suc (Suc (Suc (Suc (u + y))))) <= ((i + j) + 41 + k)";
test "(i + j + 5 + k) < Suc (Suc (Suc (Suc (Suc (u + y)))))";
test "Suc (Suc (Suc (Suc (Suc (u + y))))) - 5 = v";
test "(i + j + 5 + k) = Suc (Suc (Suc (Suc (Suc (Suc (Suc (u + y)))))))";
test "2*y + 3*z + 2*u = Suc (u)";
test "2*y + 3*z + 6*w + 2*y + 3*z + 2*u = Suc (u)";
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::nat)";
test "6 + 2*y + 3*z + 4*u = Suc (vv + 2*u + z)";
test "(2*n*m) < (3*(m*n)) + (u::nat)";

test "(Suc (Suc (Suc (Suc (Suc (Suc (case length (f c) of 0 => 0 | Suc k => k)))))) <= Suc 0)";
 
test "Suc (Suc (Suc (Suc (Suc (Suc (length l1 + length l2)))))) <= length l1";

test "( (Suc (Suc (Suc (Suc (Suc (length (compT P E A ST mxr e) + length l3)))))) <= length (compT P E A ST mxr e))";

test "( (Suc (Suc (Suc (Suc (Suc (length (compT P E A ST mxr e) + length (compT P E (A Un \<A> e) ST mxr c))))))) <= length (compT P E A ST mxr e))";


(*negative numerals: FAIL*)
test "(i + j + -23 + (k::nat)) < u + 15 + y";
test "(i + j + 3 + (k::nat)) < u + -15 + y";
test "(i + j + -12 + (k::nat)) - 15 = y";
test "(i + j + 12 + (k::nat)) - -15 = y";
test "(i + j + -12 + (k::nat)) - -15 = y";

(*combine_numerals*)
test "k + 3*k = (u::nat)";
test "Suc (i + 3) = u";
test "Suc (i + j + 3 + k) = u";
test "k + j + 3*k + j = (u::nat)";
test "Suc (j*i + i + k + 5 + 3*k + i*j*4) = (u::nat)";
test "(2*n*m) + (3*(m*n)) = (u::nat)";
(*negative numerals: FAIL*)
test "Suc (i + j + -3 + k) = u";

(*cancel_numeral_factors*)
test "9*x = 12 * (y::nat)";
test "(9*x) div (12 * (y::nat)) = z";
test "9*x < 12 * (y::nat)";
test "9*x <= 12 * (y::nat)";

(*cancel_factor*)
test "x*k = k*(y::nat)";
test "k = k*(y::nat)";
test "a*(b*c) = (b::nat)";
test "a*(b*c) = d*(b::nat)*(x*a)";

test "x*k < k*(y::nat)";
test "k < k*(y::nat)";
test "a*(b*c) < (b::nat)";
test "a*(b*c) < d*(b::nat)*(x*a)";

test "x*k <= k*(y::nat)";
test "k <= k*(y::nat)";
test "a*(b*c) <= (b::nat)";
test "a*(b*c) <= d*(b::nat)*(x*a)";

test "(x*k) div (k*(y::nat)) = (uu::nat)";
test "(k) div (k*(y::nat)) = (uu::nat)";
test "(a*(b*c)) div ((b::nat)) = (uu::nat)";
test "(a*(b*c)) div (d*(b::nat)*(x*a)) = (uu::nat)";
*)