src/HOL/Tools/numeral_simprocs.ML

numeral_simprocs.ML: use HOL_basic_ss instead of HOL_ss for internal normalization proofs of cancel_factor simprocs, to avoid splitting if-then-else

(* Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
   Copyright   2000  University of Cambridge

Simprocs for the (integer) numerals.
*)

(*To quote from Provers/Arith/cancel_numeral_factor.ML:

Cancels common coefficients in balanced expressions:

     u*#m ~~ u'*#m'  ==  #n*u ~~ #n'*u'

where ~~ is an appropriate balancing operation (e.g. =, <=, <, div, /)
and d = gcd(m,m') and n=m/d and n'=m'/d.
*)

signature NUMERAL_SIMPROCS =
sig
  val assoc_fold_simproc: simproc
  val combine_numerals: simproc
  val cancel_numerals: simproc list
  val cancel_factors: simproc list
  val cancel_numeral_factors: simproc list
  val field_combine_numerals: simproc
  val field_cancel_numeral_factors: simproc list
  val num_ss: simpset
  val field_comp_conv: conv
end;

structure Numeral_Simprocs : NUMERAL_SIMPROCS =
struct

val mk_number = Arith_Data.mk_number;
val mk_sum = Arith_Data.mk_sum;
val long_mk_sum = Arith_Data.long_mk_sum;
val dest_sum = Arith_Data.dest_sum;

val mk_diff = HOLogic.mk_binop @{const_name Groups.minus};
val dest_diff = HOLogic.dest_bin @{const_name Groups.minus} Term.dummyT;

val mk_times = HOLogic.mk_binop @{const_name Groups.times};

fun one_of T = Const(@{const_name Groups.one}, T);

(* build product with trailing 1 rather than Numeral 1 in order to avoid the
   unnecessary restriction to type class number_ring
   which is not required for cancellation of common factors in divisions.
*)
fun mk_prod T = 
  let val one = one_of T
  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 []        = one_of T
  | long_mk_prod T (t :: ts) = mk_times (t, mk_prod T ts);

val dest_times = HOLogic.dest_bin @{const_name Groups.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];

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

(*DON'T do the obvious simplifications; that would create special cases*)
fun mk_coeff (k, t) = mk_times (mk_number (Term.fastype_of t) k, t);

(*Express t as a product of (possibly) a numeral with other sorted terms*)
fun dest_coeff sign (Const (@{const_name Groups.uminus}, _) $ t) = dest_coeff (~sign) t
  | dest_coeff sign t =
    let val ts = sort Term_Ord.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;

(*Fractions as pairs of ints. Can't use Rat.rat because the representation
  needs to preserve negative values in the denominator.*)
fun mk_frac (p, q) = if q = 0 then raise Div else (p, q);

(*Don't reduce fractions; sums must be proved by rule add_frac_eq.
  Fractions are reduced later by the cancel_numeral_factor simproc.*)
fun add_frac ((p1, q1), (p2, q2)) = (p1 * q2 + p2 * q1, q1 * q2);

val mk_divide = HOLogic.mk_binop @{const_name Fields.divide};

(*Build term (p / q) * t*)
fun mk_fcoeff ((p, q), t) =
  let val T = Term.fastype_of t
  in mk_times (mk_divide (mk_number T p, mk_number T q), t) end;

(*Express t as a product of a fraction with other sorted terms*)
fun dest_fcoeff sign (Const (@{const_name Groups.uminus}, _) $ t) = dest_fcoeff (~sign) t
  | dest_fcoeff sign (Const (@{const_name Fields.divide}, _) $ t $ u) =
    let val (p, t') = dest_coeff sign t
        val (q, u') = dest_coeff 1 u
    in (mk_frac (p, q), mk_divide (t', u')) end
  | dest_fcoeff sign t =
    let val (p, t') = dest_coeff sign t
        val T = Term.fastype_of t
    in (mk_frac (p, 1), mk_divide (t', one_of T)) end;


(** 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_ord (abs i, abs j) of
    EQUAL => int_ord (Int.sign i, Int.sign j) 
  | ord => ord);

(*This resembles Term_Ord.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 => Term_Ord.typ_ord (T, U) | ord => ord)
  | numterm_ord
     (Const(@{const_name Int.number_of}, _) $ v, Const(@{const_name Int.number_of}, _) $ w) =
     num_ord (HOLogic.dest_numeral v, HOLogic.dest_numeral w)
  | numterm_ord (Const(@{const_name Int.number_of}, _) $ _, _) = LESS
  | numterm_ord (_, Const(@{const_name Int.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 Term_Ord.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);

val num_ss = HOL_ss settermless numtermless;

(*Maps 0 to Numeral0 and 1 to Numeral1 so that arithmetic isn't complicated by the abstract 0 and 1.*)
val numeral_syms = [@{thm numeral_0_eq_0} RS sym, @{thm numeral_1_eq_1} RS sym];

(*Simplify Numeral0+n, n+Numeral0, Numeral1*n, n*Numeral1, 1*x, x*1, x/1 *)
val add_0s =  @{thms add_0s};
val mult_1s = @{thms mult_1s mult_1_left mult_1_right divide_1};

(*Simplify inverse Numeral1, a/Numeral1*)
val inverse_1s = [@{thm inverse_numeral_1}];
val divide_1s = [@{thm divide_numeral_1}];

(*To perform binary arithmetic.  The "left" rewriting handles patterns
  created by the Numeral_Simprocs, such as 3 * (5 * x). *)
val simps = [@{thm numeral_0_eq_0} RS sym, @{thm numeral_1_eq_1} RS sym,
                 @{thm add_number_of_left}, @{thm mult_number_of_left}] @
                @{thms arith_simps} @ @{thms rel_simps};

(*Binary arithmetic BUT NOT ADDITION since it may collapse adjacent terms
  during re-arrangement*)
val non_add_simps =
  subtract Thm.eq_thm [@{thm add_number_of_left}, @{thm number_of_add} RS sym] simps;

(*To evaluate binary negations of coefficients*)
val minus_simps = [@{thm numeral_m1_eq_minus_1} RS sym, @{thm number_of_minus} RS sym] @
                   @{thms minus_bin_simps} @ @{thms pred_bin_simps};

(*To let us treat subtraction as addition*)
val diff_simps = [@{thm diff_minus}, @{thm minus_add_distrib}, @{thm minus_minus}];

(*To let us treat division as multiplication*)
val divide_simps = [@{thm divide_inverse}, @{thm inverse_mult_distrib}, @{thm inverse_inverse_eq}];

(*push the unary minus down*)
val minus_mult_eq_1_to_2 = @{lemma "- (a::'a::ring) * b = a * - b" by simp};

(*to extract again any uncancelled minuses*)
val minus_from_mult_simps =
    [@{thm minus_minus}, @{thm mult_minus_left}, @{thm mult_minus_right}];

(*combine unary minus with numeric literals, however nested within a product*)
val mult_minus_simps =
    [@{thm mult_assoc}, @{thm minus_mult_left}, minus_mult_eq_1_to_2];

val norm_ss1 = num_ss addsimps numeral_syms @ add_0s @ mult_1s @
  diff_simps @ minus_simps @ @{thms add_ac}
val norm_ss2 = num_ss addsimps non_add_simps @ mult_minus_simps
val norm_ss3 = num_ss addsimps minus_from_mult_simps @ @{thms add_ac} @ @{thms mult_ac}

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 []
  fun trans_tac _       = Arith_Data.trans_tac

  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 = Arith_Data.simplify_meta_eq (add_0s @ mult_1s)
  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 @{const_name HOL.eq} Term.dummyT
  val bal_add1 = @{thm eq_add_iff1} RS trans
  val bal_add2 = @{thm 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 Orderings.less}
  val dest_bal = HOLogic.dest_bin @{const_name Orderings.less} Term.dummyT
  val bal_add1 = @{thm less_add_iff1} RS trans
  val bal_add2 = @{thm 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 Orderings.less_eq}
  val dest_bal = HOLogic.dest_bin @{const_name Orderings.less_eq} Term.dummyT
  val bal_add1 = @{thm le_add_iff1} RS trans
  val bal_add2 = @{thm le_add_iff2} RS trans
);

val cancel_numerals =
  map (Arith_Data.prep_simproc @{theory})
   [("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::{linordered_idom,number_ring}) + m < n",
      "(l::'a::{linordered_idom,number_ring}) < m + n",
      "(l::'a::{linordered_idom,number_ring}) - m < n",
      "(l::'a::{linordered_idom,number_ring}) < m - n",
      "(l::'a::{linordered_idom,number_ring}) * m < n",
      "(l::'a::{linordered_idom,number_ring}) < m * n"],
     K LessCancelNumerals.proc),
    ("intle_cancel_numerals",
     ["(l::'a::{linordered_idom,number_ring}) + m <= n",
      "(l::'a::{linordered_idom,number_ring}) <= m + n",
      "(l::'a::{linordered_idom,number_ring}) - m <= n",
      "(l::'a::{linordered_idom,number_ring}) <= m - n",
      "(l::'a::{linordered_idom,number_ring}) * m <= n",
      "(l::'a::{linordered_idom,number_ring}) <= m * n"],
     K LeCancelNumerals.proc)];

structure CombineNumeralsData =
  struct
  type coeff            = int
  val iszero            = (fn x => x = 0)
  val add               = op +
  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      = @{thm combine_common_factor} RS trans
  val prove_conv        = Arith_Data.prove_conv_nohyps
  fun trans_tac _       = Arith_Data.trans_tac

  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 = Arith_Data.simplify_meta_eq (add_0s @ mult_1s)
  end;

structure CombineNumerals = CombineNumeralsFun(CombineNumeralsData);

(*Version for fields, where coefficients can be fractions*)
structure FieldCombineNumeralsData =
  struct
  type coeff            = int * int
  val iszero            = (fn (p, q) => p = 0)
  val add               = add_frac
  val mk_sum            = long_mk_sum
  val dest_sum          = dest_sum
  val mk_coeff          = mk_fcoeff
  val dest_coeff        = dest_fcoeff 1
  val left_distrib      = @{thm combine_common_factor} RS trans
  val prove_conv        = Arith_Data.prove_conv_nohyps
  fun trans_tac _       = Arith_Data.trans_tac

  val norm_ss1a = norm_ss1 addsimps inverse_1s @ divide_simps
  fun norm_tac ss =
    ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss1a))
    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 @ [@{thm add_frac_eq}]
  fun numeral_simp_tac ss = ALLGOALS (simp_tac (Simplifier.inherit_context ss numeral_simp_ss))
  val simplify_meta_eq = Arith_Data.simplify_meta_eq (add_0s @ mult_1s @ divide_1s)
  end;

structure FieldCombineNumerals = CombineNumeralsFun(FieldCombineNumeralsData);

val combine_numerals =
  Arith_Data.prep_simproc @{theory}
    ("int_combine_numerals", 
     ["(i::'a::number_ring) + j", "(i::'a::number_ring) - j"], 
     K CombineNumerals.proc);

val field_combine_numerals =
  Arith_Data.prep_simproc @{theory}
    ("field_combine_numerals", 
     ["(i::'a::{field_inverse_zero, number_ring}) + j",
      "(i::'a::{field_inverse_zero, number_ring}) - j"], 
     K FieldCombineNumerals.proc);

(** 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 @{thms mult_ac}
  val eq_reflection = eq_reflection
  val is_numeral = can HOLogic.dest_number
end;

structure Semiring_Times_Assoc = Assoc_Fold (Semiring_Times_Assoc_Data);

val assoc_fold_simproc =
  Arith_Data.prep_simproc @{theory}
   ("semiring_assoc_fold", ["(a::'a::comm_semiring_1_cancel) * b"],
    K Semiring_Times_Assoc.proc);

structure CancelNumeralFactorCommon =
  struct
  val mk_coeff          = mk_coeff
  val dest_coeff        = dest_coeff 1
  fun trans_tac _       = Arith_Data.trans_tac

  val norm_ss1 = HOL_basic_ss addsimps minus_from_mult_simps @ mult_1s
  val norm_ss2 = HOL_basic_ss addsimps simps @ mult_minus_simps
  val norm_ss3 = HOL_basic_ss addsimps @{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))
    THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss3))

  val numeral_simp_ss = HOL_ss addsimps
    [@{thm eq_number_of_eq}, @{thm less_number_of}, @{thm le_number_of}] @ simps
  fun numeral_simp_tac ss = ALLGOALS (simp_tac (Simplifier.inherit_context ss numeral_simp_ss))
  val simplify_meta_eq = Arith_Data.simplify_meta_eq
    [@{thm Nat.add_0}, @{thm Nat.add_0_right}, @{thm mult_zero_left},
      @{thm mult_zero_right}, @{thm mult_Bit1}, @{thm mult_1_right}];
  end

(*Version for semiring_div*)
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} Term.dummyT
  val cancel = @{thm div_mult_mult1} RS trans
  val neg_exchanges = false
)

(*Version for fields*)
structure DivideCancelNumeralFactor = CancelNumeralFactorFun
 (open CancelNumeralFactorCommon
  val prove_conv = Arith_Data.prove_conv
  val mk_bal   = HOLogic.mk_binop @{const_name Fields.divide}
  val dest_bal = HOLogic.dest_bin @{const_name Fields.divide} Term.dummyT
  val cancel = @{thm mult_divide_mult_cancel_left} 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 @{const_name HOL.eq} Term.dummyT
  val cancel = @{thm mult_cancel_left} 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 Orderings.less}
  val dest_bal = HOLogic.dest_bin @{const_name Orderings.less} Term.dummyT
  val cancel = @{thm mult_less_cancel_left} 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 Orderings.less_eq}
  val dest_bal = HOLogic.dest_bin @{const_name Orderings.less_eq} Term.dummyT
  val cancel = @{thm mult_le_cancel_left} RS trans
  val neg_exchanges = true
)

val cancel_numeral_factors =
  map (Arith_Data.prep_simproc @{theory})
   [("ring_eq_cancel_numeral_factor",
     ["(l::'a::{idom,number_ring}) * m = n",
      "(l::'a::{idom,number_ring}) = m * n"],
     K EqCancelNumeralFactor.proc),
    ("ring_less_cancel_numeral_factor",
     ["(l::'a::{linordered_idom,number_ring}) * m < n",
      "(l::'a::{linordered_idom,number_ring}) < m * n"],
     K LessCancelNumeralFactor.proc),
    ("ring_le_cancel_numeral_factor",
     ["(l::'a::{linordered_idom,number_ring}) * m <= n",
      "(l::'a::{linordered_idom,number_ring}) <= m * n"],
     K LeCancelNumeralFactor.proc),
    ("int_div_cancel_numeral_factors",
     ["((l::'a::{semiring_div,number_ring}) * m) div n",
      "(l::'a::{semiring_div,number_ring}) div (m * n)"],
     K DivCancelNumeralFactor.proc),
    ("divide_cancel_numeral_factor",
     ["((l::'a::{field_inverse_zero,number_ring}) * m) / n",
      "(l::'a::{field_inverse_zero,number_ring}) / (m * n)",
      "((number_of v)::'a::{field_inverse_zero,number_ring}) / (number_of w)"],
     K DivideCancelNumeralFactor.proc)];

val field_cancel_numeral_factors =
  map (Arith_Data.prep_simproc @{theory})
   [("field_eq_cancel_numeral_factor",
     ["(l::'a::{field,number_ring}) * m = n",
      "(l::'a::{field,number_ring}) = m * n"],
     K EqCancelNumeralFactor.proc),
    ("field_cancel_numeral_factor",
     ["((l::'a::{field_inverse_zero,number_ring}) * m) / n",
      "(l::'a::{field_inverse_zero,number_ring}) / (m * n)",
      "((number_of v)::'a::{field_inverse_zero,number_ring}) / (number_of w)"],
     K DivideCancelNumeralFactor.proc)]


(** Declarations for ExtractCommonTerm **)

(*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_by_1}, @{thm numeral_1_eq_1}];

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

local
  val Tp_Eq = Thm.reflexive (Thm.cterm_of @{theory HOL} HOLogic.Trueprop)
  fun Eq_True_elim Eq = 
    Thm.equal_elim (Thm.combination Tp_Eq (Thm.symmetric Eq)) @{thm TrueI}
in
fun sign_conv pos_th neg_th ss t =
  let val T = fastype_of t;
      val zero = Const(@{const_name Groups.zero}, T);
      val less = Const(@{const_name Orderings.less}, [T,T] ---> HOLogic.boolT);
      val pos = less $ zero $ t and neg = less $ t $ zero
      fun prove p =
        Option.map Eq_True_elim (Lin_Arith.simproc ss p)
        handle THM _ => NONE
    in case prove pos of
         SOME th => SOME(th RS pos_th)
       | NONE => (case prove neg of
                    SOME th => SOME(th RS neg_th)
                  | NONE => NONE)
    end;
end

structure CancelFactorCommon =
  struct
  val mk_sum            = long_mk_prod
  val dest_sum          = dest_prod
  val mk_coeff          = mk_coeff
  val dest_coeff        = dest_coeff
  val find_first        = find_first_t []
  fun trans_tac _       = 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;

(*mult_cancel_left requires a ring with no zero divisors.*)
structure EqCancelFactor = ExtractCommonTermFun
 (open CancelFactorCommon
  val prove_conv = Arith_Data.prove_conv
  val mk_bal   = HOLogic.mk_eq
  val dest_bal = HOLogic.dest_bin @{const_name HOL.eq} Term.dummyT
  fun simp_conv _ _ = SOME @{thm mult_cancel_left}
);

(*for ordered rings*)
structure LeCancelFactor = ExtractCommonTermFun
 (open CancelFactorCommon
  val prove_conv = Arith_Data.prove_conv
  val mk_bal   = HOLogic.mk_binrel @{const_name Orderings.less_eq}
  val dest_bal = HOLogic.dest_bin @{const_name Orderings.less_eq} Term.dummyT
  val simp_conv = sign_conv
    @{thm mult_le_cancel_left_pos} @{thm mult_le_cancel_left_neg}
);

(*for ordered rings*)
structure LessCancelFactor = ExtractCommonTermFun
 (open CancelFactorCommon
  val prove_conv = Arith_Data.prove_conv
  val mk_bal   = HOLogic.mk_binrel @{const_name Orderings.less}
  val dest_bal = HOLogic.dest_bin @{const_name Orderings.less} Term.dummyT
  val simp_conv = sign_conv
    @{thm mult_less_cancel_left_pos} @{thm mult_less_cancel_left_neg}
);

(*for semirings with division*)
structure DivCancelFactor = 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} Term.dummyT
  fun simp_conv _ _ = SOME @{thm div_mult_mult1_if}
);

structure ModCancelFactor = ExtractCommonTermFun
 (open CancelFactorCommon
  val prove_conv = Arith_Data.prove_conv
  val mk_bal   = HOLogic.mk_binop @{const_name Divides.mod}
  val dest_bal = HOLogic.dest_bin @{const_name Divides.mod} Term.dummyT
  fun simp_conv _ _ = SOME @{thm mod_mult_mult1}
);

(*for idoms*)
structure DvdCancelFactor = ExtractCommonTermFun
 (open CancelFactorCommon
  val prove_conv = Arith_Data.prove_conv
  val mk_bal   = HOLogic.mk_binrel @{const_name Rings.dvd}
  val dest_bal = HOLogic.dest_bin @{const_name Rings.dvd} Term.dummyT
  fun simp_conv _ _ = SOME @{thm dvd_mult_cancel_left}
);

(*Version for all fields, including unordered ones (type complex).*)
structure DivideCancelFactor = ExtractCommonTermFun
 (open CancelFactorCommon
  val prove_conv = Arith_Data.prove_conv
  val mk_bal   = HOLogic.mk_binop @{const_name Fields.divide}
  val dest_bal = HOLogic.dest_bin @{const_name Fields.divide} Term.dummyT
  fun simp_conv _ _ = SOME @{thm mult_divide_mult_cancel_left_if}
);

val cancel_factors =
  map (Arith_Data.prep_simproc @{theory})
   [("ring_eq_cancel_factor",
     ["(l::'a::idom) * m = n",
      "(l::'a::idom) = m * n"],
     K EqCancelFactor.proc),
    ("linordered_ring_le_cancel_factor",
     ["(l::'a::linordered_ring) * m <= n",
      "(l::'a::linordered_ring) <= m * n"],
     K LeCancelFactor.proc),
    ("linordered_ring_less_cancel_factor",
     ["(l::'a::linordered_ring) * m < n",
      "(l::'a::linordered_ring) < m * n"],
     K LessCancelFactor.proc),
    ("int_div_cancel_factor",
     ["((l::'a::semiring_div) * m) div n", "(l::'a::semiring_div) div (m * n)"],
     K DivCancelFactor.proc),
    ("int_mod_cancel_factor",
     ["((l::'a::semiring_div) * m) mod n", "(l::'a::semiring_div) mod (m * n)"],
     K ModCancelFactor.proc),
    ("dvd_cancel_factor",
     ["((l::'a::idom) * m) dvd n", "(l::'a::idom) dvd (m * n)"],
     K DvdCancelFactor.proc),
    ("divide_cancel_factor",
     ["((l::'a::field_inverse_zero) * m) / n",
      "(l::'a::field_inverse_zero) / (m * n)"],
     K DivideCancelFactor.proc)];

local
 val zr = @{cpat "0"}
 val zT = ctyp_of_term zr
 val geq = @{cpat HOL.eq}
 val eqT = Thm.dest_ctyp (ctyp_of_term geq) |> hd
 val add_frac_eq = mk_meta_eq @{thm "add_frac_eq"}
 val add_frac_num = mk_meta_eq @{thm "add_frac_num"}
 val add_num_frac = mk_meta_eq @{thm "add_num_frac"}

 fun prove_nz ss T t =
    let
      val z = Thm.instantiate_cterm ([(zT,T)],[]) zr
      val eq = Thm.instantiate_cterm ([(eqT,T)],[]) geq
      val th = Simplifier.rewrite (ss addsimps @{thms simp_thms})
           (Thm.capply @{cterm "Trueprop"} (Thm.capply @{cterm "Not"}
                  (Thm.capply (Thm.capply eq t) z)))
    in Thm.equal_elim (Thm.symmetric th) TrueI
    end

 fun proc phi ss ct =
  let
    val ((x,y),(w,z)) =
         (Thm.dest_binop #> (fn (a,b) => (Thm.dest_binop a, Thm.dest_binop b))) ct
    val _ = map (HOLogic.dest_number o term_of) [x,y,z,w]
    val T = ctyp_of_term x
    val [y_nz, z_nz] = map (prove_nz ss T) [y, z]
    val th = instantiate' [SOME T] (map SOME [y,z,x,w]) add_frac_eq
  in SOME (Thm.implies_elim (Thm.implies_elim th y_nz) z_nz)
  end
  handle CTERM _ => NONE | TERM _ => NONE | THM _ => NONE

 fun proc2 phi ss ct =
  let
    val (l,r) = Thm.dest_binop ct
    val T = ctyp_of_term l
  in (case (term_of l, term_of r) of
      (Const(@{const_name Fields.divide},_)$_$_, _) =>
        let val (x,y) = Thm.dest_binop l val z = r
            val _ = map (HOLogic.dest_number o term_of) [x,y,z]
            val ynz = prove_nz ss T y
        in SOME (Thm.implies_elim (instantiate' [SOME T] (map SOME [y,x,z]) add_frac_num) ynz)
        end
     | (_, Const (@{const_name Fields.divide},_)$_$_) =>
        let val (x,y) = Thm.dest_binop r val z = l
            val _ = map (HOLogic.dest_number o term_of) [x,y,z]
            val ynz = prove_nz ss T y
        in SOME (Thm.implies_elim (instantiate' [SOME T] (map SOME [y,z,x]) add_num_frac) ynz)
        end
     | _ => NONE)
  end
  handle CTERM _ => NONE | TERM _ => NONE | THM _ => NONE

 fun is_number (Const(@{const_name Fields.divide},_)$a$b) = is_number a andalso is_number b
   | is_number t = can HOLogic.dest_number t

 val is_number = is_number o term_of

 fun proc3 phi ss ct =
  (case term_of ct of
    Const(@{const_name Orderings.less},_)$(Const(@{const_name Fields.divide},_)$_$_)$_ =>
      let
        val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
        val _ = map is_number [a,b,c]
        val T = ctyp_of_term c
        val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_less_eq"}
      in SOME (mk_meta_eq th) end
  | Const(@{const_name Orderings.less_eq},_)$(Const(@{const_name Fields.divide},_)$_$_)$_ =>
      let
        val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
        val _ = map is_number [a,b,c]
        val T = ctyp_of_term c
        val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_le_eq"}
      in SOME (mk_meta_eq th) end
  | Const(@{const_name HOL.eq},_)$(Const(@{const_name Fields.divide},_)$_$_)$_ =>
      let
        val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
        val _ = map is_number [a,b,c]
        val T = ctyp_of_term c
        val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_eq_eq"}
      in SOME (mk_meta_eq th) end
  | Const(@{const_name Orderings.less},_)$_$(Const(@{const_name Fields.divide},_)$_$_) =>
    let
      val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
        val _ = map is_number [a,b,c]
        val T = ctyp_of_term c
        val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "less_divide_eq"}
      in SOME (mk_meta_eq th) end
  | Const(@{const_name Orderings.less_eq},_)$_$(Const(@{const_name Fields.divide},_)$_$_) =>
    let
      val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
        val _ = map is_number [a,b,c]
        val T = ctyp_of_term c
        val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "le_divide_eq"}
      in SOME (mk_meta_eq th) end
  | Const(@{const_name HOL.eq},_)$_$(Const(@{const_name Fields.divide},_)$_$_) =>
    let
      val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
        val _ = map is_number [a,b,c]
        val T = ctyp_of_term c
        val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "eq_divide_eq"}
      in SOME (mk_meta_eq th) end
  | _ => NONE)
  handle TERM _ => NONE | CTERM _ => NONE | THM _ => NONE

val add_frac_frac_simproc =
       make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + (?w::?'a::field)/?z"}],
                     name = "add_frac_frac_simproc",
                     proc = proc, identifier = []}

val add_frac_num_simproc =
       make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + ?z"}, @{cpat "?z + (?x::?'a::field)/?y"}],
                     name = "add_frac_num_simproc",
                     proc = proc2, identifier = []}

val ord_frac_simproc =
  make_simproc
    {lhss = [@{cpat "(?a::(?'a::{field, ord}))/?b < ?c"},
             @{cpat "(?a::(?'a::{field, ord}))/?b <= ?c"},
             @{cpat "?c < (?a::(?'a::{field, ord}))/?b"},
             @{cpat "?c <= (?a::(?'a::{field, ord}))/?b"},
             @{cpat "?c = ((?a::(?'a::{field, ord}))/?b)"},
             @{cpat "((?a::(?'a::{field, ord}))/ ?b) = ?c"}],
             name = "ord_frac_simproc", proc = proc3, identifier = []}

val ths = [@{thm "mult_numeral_1"}, @{thm "mult_numeral_1_right"},
           @{thm "divide_Numeral1"},
           @{thm "divide_zero"}, @{thm "divide_Numeral0"},
           @{thm "divide_divide_eq_left"}, 
           @{thm "times_divide_eq_left"}, @{thm "times_divide_eq_right"},
           @{thm "times_divide_times_eq"},
           @{thm "divide_divide_eq_right"},
           @{thm "diff_minus"}, @{thm "minus_divide_left"},
           @{thm "Numeral1_eq1_nat"}, @{thm "add_divide_distrib"} RS sym,
           @{thm field_divide_inverse} RS sym, @{thm inverse_divide}, 
           Conv.fconv_rule (Conv.arg_conv (Conv.arg1_conv (Conv.rewr_conv (mk_meta_eq @{thm mult_commute}))))   
           (@{thm field_divide_inverse} RS sym)]

in

val field_comp_conv = (Simplifier.rewrite
(HOL_basic_ss addsimps @{thms "semiring_norm"}
              addsimps ths addsimps @{thms simp_thms}
              addsimprocs field_cancel_numeral_factors
               addsimprocs [add_frac_frac_simproc, add_frac_num_simproc,
                            ord_frac_simproc]
                addcongs [@{thm "if_weak_cong"}]))
then_conv (Simplifier.rewrite (HOL_basic_ss addsimps
  [@{thm numeral_1_eq_1},@{thm numeral_0_eq_0}] @ @{thms numerals(1-2)}))

end

end;

Addsimprocs Numeral_Simprocs.cancel_numerals;
Addsimprocs [Numeral_Simprocs.combine_numerals];
Addsimprocs [Numeral_Simprocs.field_combine_numerals];
Addsimprocs [Numeral_Simprocs.assoc_fold_simproc];

(*examples:
print_depth 22;
set timing;
set simp_trace;
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";
*)

Addsimprocs Numeral_Simprocs.cancel_numeral_factors;

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

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

test "-99*x = 132 * (y::int)";
test "(-99*x) div (132 * (y::int)) = z";
test "-99*x < 132 * (y::int)";
test "-99*x <= 132 * (y::int)";

test "999*x = -396 * (y::int)";
test "(999*x) div (-396 * (y::int)) = z";
test "999*x < -396 * (y::int)";
test "999*x <= -396 * (y::int)";

test "-99*x = -81 * (y::int)";
test "(-99*x) div (-81 * (y::int)) = z";
test "-99*x <= -81 * (y::int)";
test "-99*x < -81 * (y::int)";

test "-2 * x = -1 * (y::int)";
test "-2 * x = -(y::int)";
test "(-2 * x) div (-1 * (y::int)) = z";
test "-2 * x < -(y::int)";
test "-2 * x <= -1 * (y::int)";
test "-x < -23 * (y::int)";
test "-x <= -23 * (y::int)";
*)

(*And the same examples for fields such as rat or real:
test "0 <= (y::rat) * -2";
test "9*x = 12 * (y::rat)";
test "(9*x) / (12 * (y::rat)) = z";
test "9*x < 12 * (y::rat)";
test "9*x <= 12 * (y::rat)";

test "-99*x = 132 * (y::rat)";
test "(-99*x) / (132 * (y::rat)) = z";
test "-99*x < 132 * (y::rat)";
test "-99*x <= 132 * (y::rat)";

test "999*x = -396 * (y::rat)";
test "(999*x) / (-396 * (y::rat)) = z";
test "999*x < -396 * (y::rat)";
test "999*x <= -396 * (y::rat)";

test  "(- ((2::rat) * x) <= 2 * y)";
test "-99*x = -81 * (y::rat)";
test "(-99*x) / (-81 * (y::rat)) = z";
test "-99*x <= -81 * (y::rat)";
test "-99*x < -81 * (y::rat)";

test "-2 * x = -1 * (y::rat)";
test "-2 * x = -(y::rat)";
test "(-2 * x) / (-1 * (y::rat)) = z";
test "-2 * x < -(y::rat)";
test "-2 * x <= -1 * (y::rat)";
test "-x < -23 * (y::rat)";
test "-x <= -23 * (y::rat)";
*)

Addsimprocs Numeral_Simprocs.cancel_factors;


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

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

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

(*And the same examples for fields such as rat or real:
print_depth 22;
set timing;
set simp_trace;
fun test s = (Goal s; by (Asm_simp_tac 1));

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


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

(*FIXME: what do we do about this?*)
test "a*(b*c)/(y*z) = d*(b::rat)*(x*a)/z";
*)