src/HOL/Tools/numeral_simprocs.ML

tuning

(* 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 prep_simproc: theory -> string * string list * (Proof.context -> term -> thm option)
    -> simproc
  val trans_tac: thm option -> tactic
  val assoc_fold: Proof.context -> cterm -> thm option
  val combine_numerals: Proof.context -> cterm -> thm option
  val eq_cancel_numerals: Proof.context -> cterm -> thm option
  val less_cancel_numerals: Proof.context -> cterm -> thm option
  val le_cancel_numerals: Proof.context -> cterm -> thm option
  val eq_cancel_factor: Proof.context -> cterm -> thm option
  val le_cancel_factor: Proof.context -> cterm -> thm option
  val less_cancel_factor: Proof.context -> cterm -> thm option
  val div_cancel_factor: Proof.context -> cterm -> thm option
  val mod_cancel_factor: Proof.context -> cterm -> thm option
  val dvd_cancel_factor: Proof.context -> cterm -> thm option
  val divide_cancel_factor: Proof.context -> cterm -> thm option
  val eq_cancel_numeral_factor: Proof.context -> cterm -> thm option
  val less_cancel_numeral_factor: Proof.context -> cterm -> thm option
  val le_cancel_numeral_factor: Proof.context -> cterm -> thm option
  val div_cancel_numeral_factor: Proof.context -> cterm -> thm option
  val divide_cancel_numeral_factor: Proof.context -> cterm -> thm option
  val field_combine_numerals: Proof.context -> cterm -> thm option
  val field_cancel_numeral_factors: simproc list
  val num_ss: simpset
  val field_comp_conv: Proof.context -> conv
end;

structure Numeral_Simprocs : NUMERAL_SIMPROCS =
struct

fun prep_simproc thy (name, pats, proc) =
  Simplifier.simproc_global thy name pats proc;

fun trans_tac NONE  = all_tac
  | trans_tac (SOME th) = ALLGOALS (rtac (th RS trans));

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_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.
   UPDATE: this reasoning no longer applies (number_ring is gone)
*)
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} 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 (t, u) =
    case (try HOLogic.dest_number t, try HOLogic.dest_number u) of
      (SOME (_, i), SOME (_, j)) => num_ord (i, j)
    | (SOME _, NONE) => LESS
    | (NONE, SOME _) => GREATER
    | _ => (
      case (t, u) of
        (Abs (_, T, t), Abs(_, U, u)) =>
        (prod_ord numterm_ord Term_Ord.typ_ord ((t, T), (u, 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
            (prod_ord Term_Ord.hd_ord numterms_ord ((f, ts), (g, us)))
          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 =
  simpset_of (put_simpset HOL_basic_ss @{context}
    |> Simplifier.set_termless numtermless);

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

(*Simplify 0+n, n+0, Numeral1*n, n*Numeral1, 1*x, x*1, x/1 *)
val add_0s =  @{thms add_0s};
val mult_1s = @{thms mult_1s divide_numeral_1 mult_1_left mult_1_right mult_minus1 mult_minus1_right divide_1};

(* For post-simplification of the rhs of simproc-generated rules *)
val post_simps =
    [@{thm numeral_1_eq_1},
     @{thm add_0_left}, @{thm add_0_right},
     @{thm mult_zero_left}, @{thm mult_zero_right},
     @{thm mult_1_left}, @{thm mult_1_right},
     @{thm mult_minus1}, @{thm mult_minus1_right}]

val field_post_simps =
    post_simps @ [@{thm divide_zero_left}, @{thm divide_1}]
                      
(*Simplify inverse Numeral1*)
val inverse_1s = [@{thm inverse_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_1_eq_1} RS sym] @
    @{thms add_numeral_left} @
    @{thms add_neg_numeral_left} @
    @{thms mult_numeral_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
    (@{thms add_numeral_left} @
     @{thms add_neg_numeral_left} @
     @{thms numeral_plus_numeral} @
     @{thms add_neg_numeral_simps}) simps;

(*To evaluate binary negations of coefficients*)
val minus_simps = [@{thm minus_zero}, @{thm minus_minus}];

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

(*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_right}, @{thm minus_mult_commute}];

val norm_ss1 =
  simpset_of (put_simpset num_ss @{context}
    addsimps numeral_syms @ add_0s @ mult_1s @
    diff_simps @ minus_simps @ @{thms add_ac})

val norm_ss2 =
  simpset_of (put_simpset num_ss @{context}
    addsimps non_add_simps @ mult_minus_simps)

val norm_ss3 =
  simpset_of (put_simpset num_ss @{context}
    addsimps minus_from_mult_simps @ @{thms add_ac} @ @{thms mult_ac minus_mult_commute})

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

  fun norm_tac ctxt =
    ALLGOALS (simp_tac (put_simpset norm_ss1 ctxt))
    THEN ALLGOALS (simp_tac (put_simpset norm_ss2 ctxt))
    THEN ALLGOALS (simp_tac (put_simpset norm_ss3 ctxt))

  val numeral_simp_ss =
    simpset_of (put_simpset HOL_basic_ss @{context} addsimps add_0s @ simps)
  fun numeral_simp_tac ctxt =
    ALLGOALS (simp_tac (put_simpset numeral_simp_ss ctxt))
  val simplify_meta_eq = Arith_Data.simplify_meta_eq post_simps
  val prove_conv = Arith_Data.prove_conv
end;

structure EqCancelNumerals = CancelNumeralsFun
 (open CancelNumeralsCommon
  val mk_bal   = HOLogic.mk_eq
  val dest_bal = HOLogic.dest_bin @{const_name HOL.eq} 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 mk_bal   = HOLogic.mk_binrel @{const_name Orderings.less}
  val dest_bal = HOLogic.dest_bin @{const_name Orderings.less} 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 mk_bal   = HOLogic.mk_binrel @{const_name Orderings.less_eq}
  val dest_bal = HOLogic.dest_bin @{const_name Orderings.less_eq} dummyT
  val bal_add1 = @{thm le_add_iff1} RS trans
  val bal_add2 = @{thm le_add_iff2} RS trans
);

fun eq_cancel_numerals ctxt ct = EqCancelNumerals.proc ctxt (term_of ct)
fun less_cancel_numerals ctxt ct = LessCancelNumerals.proc ctxt (term_of ct)
fun le_cancel_numerals ctxt ct = LeCancelNumerals.proc ctxt (term_of ct)

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
  val trans_tac = trans_tac

  fun norm_tac ctxt =
    ALLGOALS (simp_tac (put_simpset norm_ss1 ctxt))
    THEN ALLGOALS (simp_tac (put_simpset norm_ss2 ctxt))
    THEN ALLGOALS (simp_tac (put_simpset norm_ss3 ctxt))

  val numeral_simp_ss =
    simpset_of (put_simpset HOL_basic_ss @{context} addsimps add_0s @ simps)
  fun numeral_simp_tac ctxt =
    ALLGOALS (simp_tac (put_simpset numeral_simp_ss ctxt))
  val simplify_meta_eq = Arith_Data.simplify_meta_eq post_simps
end;

structure CombineNumerals = CombineNumeralsFun(CombineNumeralsData);

(*Version for fields, where coefficients can be fractions*)
structure FieldCombineNumeralsData =
struct
  type coeff = int * int
  val iszero = (fn (p, _) => 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
  val trans_tac = trans_tac

  val norm_ss1a =
    simpset_of (put_simpset norm_ss1 @{context} addsimps inverse_1s @ divide_simps)
  fun norm_tac ctxt =
    ALLGOALS (simp_tac (put_simpset norm_ss1a ctxt))
    THEN ALLGOALS (simp_tac (put_simpset norm_ss2 ctxt))
    THEN ALLGOALS (simp_tac (put_simpset norm_ss3 ctxt))

  val numeral_simp_ss =
    simpset_of (put_simpset HOL_basic_ss @{context}
      addsimps add_0s @ simps @ [@{thm add_frac_eq}, @{thm not_False_eq_True}])
  fun numeral_simp_tac ctxt =
    ALLGOALS (simp_tac (put_simpset numeral_simp_ss ctxt))
  val simplify_meta_eq = Arith_Data.simplify_meta_eq field_post_simps
end;

structure FieldCombineNumerals = CombineNumeralsFun(FieldCombineNumeralsData);

fun combine_numerals ctxt ct = CombineNumerals.proc ctxt (term_of ct)

fun field_combine_numerals ctxt ct = FieldCombineNumerals.proc ctxt (term_of ct)


(** 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 = simpset_of (put_simpset HOL_basic_ss @{context} addsimps @{thms mult_ac minus_mult_commute})
  val eq_reflection = eq_reflection
  val is_numeral = can HOLogic.dest_number
end;

structure Semiring_Times_Assoc = Assoc_Fold (Semiring_Times_Assoc_Data);

fun assoc_fold ctxt ct = Semiring_Times_Assoc.proc ctxt (term_of ct)

structure CancelNumeralFactorCommon =
struct
  val mk_coeff = mk_coeff
  val dest_coeff = dest_coeff 1
  val trans_tac = trans_tac

  val norm_ss1 =
    simpset_of (put_simpset HOL_basic_ss @{context} addsimps minus_from_mult_simps @ mult_1s)
  val norm_ss2 =
    simpset_of (put_simpset HOL_basic_ss @{context} addsimps simps @ mult_minus_simps)
  val norm_ss3 =
    simpset_of (put_simpset HOL_basic_ss @{context} addsimps @{thms mult_ac minus_mult_commute})
  fun norm_tac ctxt =
    ALLGOALS (simp_tac (put_simpset norm_ss1 ctxt))
    THEN ALLGOALS (simp_tac (put_simpset norm_ss2 ctxt))
    THEN ALLGOALS (simp_tac (put_simpset norm_ss3 ctxt))

  (* simp_thms are necessary because some of the cancellation rules below
  (e.g. mult_less_cancel_left) introduce various logical connectives *)
  val numeral_simp_ss =
    simpset_of (put_simpset HOL_basic_ss @{context} addsimps simps @ @{thms simp_thms})
  fun numeral_simp_tac ctxt =
    ALLGOALS (simp_tac (put_simpset numeral_simp_ss ctxt))
  val simplify_meta_eq = Arith_Data.simplify_meta_eq
    ([@{thm Nat.add_0}, @{thm Nat.add_0_right}] @ post_simps)
  val prove_conv = Arith_Data.prove_conv
end

(*Version for semiring_div*)
structure DivCancelNumeralFactor = CancelNumeralFactorFun
 (open CancelNumeralFactorCommon
  val mk_bal   = HOLogic.mk_binop @{const_name Divides.div}
  val dest_bal = HOLogic.dest_bin @{const_name Divides.div} dummyT
  val cancel = @{thm div_mult_mult1} RS trans
  val neg_exchanges = false
)

(*Version for fields*)
structure DivideCancelNumeralFactor = CancelNumeralFactorFun
 (open CancelNumeralFactorCommon
  val mk_bal   = HOLogic.mk_binop @{const_name Fields.divide}
  val dest_bal = HOLogic.dest_bin @{const_name Fields.divide} dummyT
  val cancel = @{thm mult_divide_mult_cancel_left} RS trans
  val neg_exchanges = false
)

structure EqCancelNumeralFactor = CancelNumeralFactorFun
 (open CancelNumeralFactorCommon
  val mk_bal   = HOLogic.mk_eq
  val dest_bal = HOLogic.dest_bin @{const_name HOL.eq} dummyT
  val cancel = @{thm mult_cancel_left} RS trans
  val neg_exchanges = false
)

structure LessCancelNumeralFactor = CancelNumeralFactorFun
 (open CancelNumeralFactorCommon
  val mk_bal   = HOLogic.mk_binrel @{const_name Orderings.less}
  val dest_bal = HOLogic.dest_bin @{const_name Orderings.less} dummyT
  val cancel = @{thm mult_less_cancel_left} RS trans
  val neg_exchanges = true
)

structure LeCancelNumeralFactor = CancelNumeralFactorFun
 (open CancelNumeralFactorCommon
  val mk_bal   = HOLogic.mk_binrel @{const_name Orderings.less_eq}
  val dest_bal = HOLogic.dest_bin @{const_name Orderings.less_eq} dummyT
  val cancel = @{thm mult_le_cancel_left} RS trans
  val neg_exchanges = true
)

fun eq_cancel_numeral_factor ctxt ct = EqCancelNumeralFactor.proc ctxt (term_of ct)
fun less_cancel_numeral_factor ctxt ct = LessCancelNumeralFactor.proc ctxt (term_of ct)
fun le_cancel_numeral_factor ctxt ct = LeCancelNumeralFactor.proc ctxt (term_of ct)
fun div_cancel_numeral_factor ctxt ct = DivCancelNumeralFactor.proc ctxt (term_of ct)
fun divide_cancel_numeral_factor ctxt ct = DivideCancelNumeralFactor.proc ctxt (term_of ct)

val field_cancel_numeral_factors =
  map (prep_simproc @{theory})
   [("field_eq_cancel_numeral_factor",
     ["(l::'a::field) * m = n",
      "(l::'a::field) = m * n"],
     EqCancelNumeralFactor.proc),
    ("field_cancel_numeral_factor",
     ["((l::'a::field_inverse_zero) * m) / n",
      "(l::'a::field_inverse_zero) / (m * n)",
      "((numeral v)::'a::field_inverse_zero) / (numeral w)",
      "((numeral v)::'a::field_inverse_zero) / (- numeral w)",
      "((- numeral v)::'a::field_inverse_zero) / (numeral w)",
      "((- numeral v)::'a::field_inverse_zero) / (- numeral w)"],
     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 ctxt cancel_th th =
    simplify_one ctxt (([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 ctxt 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
      val thy = Proof_Context.theory_of ctxt
      fun prove p =
        SOME (Eq_True_elim (Simplifier.asm_rewrite ctxt (Thm.cterm_of thy 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 []
  val trans_tac = trans_tac
  val norm_ss =
    simpset_of (put_simpset HOL_basic_ss @{context} addsimps mult_1s @ @{thms mult_ac minus_mult_commute})
  fun norm_tac ctxt =
    ALLGOALS (simp_tac (put_simpset norm_ss ctxt))
  val simplify_meta_eq  = cancel_simplify_meta_eq 
  fun mk_eq (a, b) = HOLogic.mk_Trueprop (HOLogic.mk_eq (a, b))
end;

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

(*for ordered rings*)
structure LeCancelFactor = ExtractCommonTermFun
 (open CancelFactorCommon
  val mk_bal   = HOLogic.mk_binrel @{const_name Orderings.less_eq}
  val dest_bal = HOLogic.dest_bin @{const_name Orderings.less_eq} 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 mk_bal   = HOLogic.mk_binrel @{const_name Orderings.less}
  val dest_bal = HOLogic.dest_bin @{const_name Orderings.less} 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 mk_bal   = HOLogic.mk_binop @{const_name Divides.div}
  val dest_bal = HOLogic.dest_bin @{const_name Divides.div} dummyT
  fun simp_conv _ _ = SOME @{thm div_mult_mult1_if}
);

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

(*for idoms*)
structure DvdCancelFactor = ExtractCommonTermFun
 (open CancelFactorCommon
  val mk_bal   = HOLogic.mk_binrel @{const_name Rings.dvd}
  val dest_bal = HOLogic.dest_bin @{const_name Rings.dvd} 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 mk_bal   = HOLogic.mk_binop @{const_name Fields.divide}
  val dest_bal = HOLogic.dest_bin @{const_name Fields.divide} dummyT
  fun simp_conv _ _ = SOME @{thm mult_divide_mult_cancel_left_if}
);

fun eq_cancel_factor ctxt ct = EqCancelFactor.proc ctxt (term_of ct)
fun le_cancel_factor ctxt ct = LeCancelFactor.proc ctxt (term_of ct)
fun less_cancel_factor ctxt ct = LessCancelFactor.proc ctxt (term_of ct)
fun div_cancel_factor ctxt ct = DivCancelFactor.proc ctxt (term_of ct)
fun mod_cancel_factor ctxt ct = ModCancelFactor.proc ctxt (term_of ct)
fun dvd_cancel_factor ctxt ct = DvdCancelFactor.proc ctxt (term_of ct)
fun divide_cancel_factor ctxt ct = DivideCancelFactor.proc ctxt (term_of ct)

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

 fun proc phi ctxt 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 ctxt 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 ctxt 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 ctxt 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 ctxt 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 ctxt 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_zero_left},
           @{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_conv_add_uminus}, @{thm "minus_divide_left"},
           @{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)]

val field_comp_ss =
  simpset_of
    (put_simpset HOL_basic_ss @{context}
      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]
      |> Simplifier.add_cong @{thm "if_weak_cong"})

in

fun field_comp_conv ctxt =
  Simplifier.rewrite (put_simpset field_comp_ss ctxt)
  then_conv
  Simplifier.rewrite (put_simpset HOL_basic_ss ctxt addsimps [@{thm numeral_1_eq_1}])

end

end;

(*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";
*)

(*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)";
*)

(*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";
*)