(*  Title:      NSComplexBin.ML

    Author:     Jacques D. Fleuriot

    Copyright:  2001 University of Edinburgh

    Descrition: Binary arithmetic for the nonstandard complex numbers

*)

(** hcomplex_of_complex (coercion from complex to nonstandard complex) **)

Goal "hcomplex_of_complex (number_of w) = number_of w";

by (simp_tac (simpset() addsimps [hcomplex_number_of_def]) 1);

qed "hcomplex_number_of";

Addsimps [hcomplex_number_of];

Goalw [hypreal_of_real_def]

     "hcomplex_of_hypreal (hypreal_of_real x) = \

\     hcomplex_of_complex(complex_of_real x)";

by (simp_tac (simpset() addsimps [hcomplex_of_hypreal,

    hcomplex_of_complex_def,complex_of_real_def]) 1);

qed "hcomplex_of_hypreal_eq_hcomplex_of_complex";

Goalw [complex_number_of_def,hypreal_number_of_def] 

  "hcomplex_of_complex (number_of w) = hcomplex_of_hypreal(number_of w)";

by (rtac (hcomplex_of_hypreal_eq_hcomplex_of_complex RS sym) 1);

qed "hcomplex_hypreal_number_of";

Goalw [hcomplex_number_of_def] "Numeral0 = (0::hcomplex)";

by(Simp_tac 1);

qed "hcomplex_numeral_0_eq_0";

Goalw [hcomplex_number_of_def] "Numeral1 = (1::hcomplex)";

by(Simp_tac 1);

qed "hcomplex_numeral_1_eq_1";

(*

Goal "z + hcnj z = \

\     hcomplex_of_hypreal (2 * hRe(z))";

by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);

by (auto_tac (claset(),HOL_ss addsimps [hRe,hcnj,hcomplex_add,

    hypreal_mult,hcomplex_of_hypreal,complex_add_cnj]));

qed "hcomplex_add_hcnj";

Goal "z - hcnj z = \

\     hcomplex_of_hypreal (hypreal_of_real #2 * hIm(z)) * iii";

by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);

by (auto_tac (claset(),simpset() addsimps [hIm,hcnj,hcomplex_diff,

    hypreal_of_real_def,hypreal_mult,hcomplex_of_hypreal,

    complex_diff_cnj,iii_def,hcomplex_mult]));

qed "hcomplex_diff_hcnj";

*)

(** Addition **)

Goal "(number_of v :: hcomplex) + number_of v' = number_of (bin_add v v')";

by (simp_tac

    (HOL_ss addsimps [hcomplex_number_of_def, 

                      hcomplex_of_complex_add RS sym, add_complex_number_of]) 1);

qed "add_hcomplex_number_of";

Addsimps [add_hcomplex_number_of];

(** Subtraction **)

Goalw [hcomplex_number_of_def]

     "- (number_of w :: hcomplex) = number_of (bin_minus w)";

by (simp_tac

    (HOL_ss addsimps [minus_complex_number_of, hcomplex_of_complex_minus RS sym]) 1);

qed "minus_hcomplex_number_of";

Addsimps [minus_hcomplex_number_of];

Goalw [hcomplex_number_of_def, hcomplex_diff_def]

     "(number_of v :: hcomplex) - number_of w = \

\     number_of (bin_add v (bin_minus w))";

by (Simp_tac 1); 

qed "diff_hcomplex_number_of";

Addsimps [diff_hcomplex_number_of];

(** Multiplication **)

Goal "(number_of v :: hcomplex) * number_of v' = number_of (bin_mult v v')";

by (simp_tac

    (HOL_ss addsimps [hcomplex_number_of_def, 

	              hcomplex_of_complex_mult RS sym, mult_complex_number_of]) 1);

qed "mult_hcomplex_number_of";

Addsimps [mult_hcomplex_number_of];

Goal "(2::hcomplex) = 1 + 1";

by (simp_tac (simpset() addsimps [hcomplex_numeral_1_eq_1 RS sym]) 1);

val lemma = result();

(*For specialist use: NOT as default simprules*)

Goal "2 * z = (z+z::hcomplex)";

by (simp_tac (simpset() addsimps [lemma, hcomplex_add_mult_distrib]) 1);

qed "hcomplex_mult_2";

Goal "z * 2 = (z+z::hcomplex)";

by (stac hcomplex_mult_commute 1 THEN rtac hcomplex_mult_2 1);

qed "hcomplex_mult_2_right";

(** Equals (=) **)

Goal "((number_of v :: hcomplex) = number_of v') = \

\     iszero (number_of (bin_add v (bin_minus v')) :: int)";

by (simp_tac

    (HOL_ss addsimps [hcomplex_number_of_def, 

	              hcomplex_of_complex_eq_iff, eq_complex_number_of]) 1);

qed "eq_hcomplex_number_of";

Addsimps [eq_hcomplex_number_of];

(*** New versions of existing theorems involving 0, 1hc ***)

Goal "- 1 = (-1::hcomplex)";

by (simp_tac (simpset() addsimps [hcomplex_numeral_1_eq_1 RS sym]) 1);

qed "hcomplex_minus_1_eq_m1";

Goal "-1 * z = -(z::hcomplex)";

by (simp_tac (simpset() addsimps [hcomplex_minus_1_eq_m1 RS sym]) 1);

qed "hcomplex_mult_minus1";

Goal "z * -1 = -(z::hcomplex)";

by (stac hcomplex_mult_commute 1 THEN rtac hcomplex_mult_minus1 1);

qed "hcomplex_mult_minus1_right";

Addsimps [hcomplex_mult_minus1,hcomplex_mult_minus1_right];

(*Maps 0 to Numeral0 and 1 to Numeral1 and -Numeral1 to -1*)

val hcomplex_numeral_ss = 

    complex_numeral_ss addsimps [hcomplex_numeral_0_eq_0 RS sym, hcomplex_numeral_1_eq_1 RS sym, 

		                 hcomplex_minus_1_eq_m1];

fun rename_numerals th = 

    asm_full_simplify hcomplex_numeral_ss (Thm.transfer (the_context ()) th);

(*Now insert some identities previously stated for 0 and 1hc*)

Addsimps [hcomplex_numeral_0_eq_0,hcomplex_numeral_1_eq_1];

Goal "number_of v + (number_of w + z) = (number_of(bin_add v w) + z::hcomplex)";

by (auto_tac (claset(),simpset() addsimps [hcomplex_add_assoc RS sym]));

qed "hcomplex_add_number_of_left";

Goal "number_of v *(number_of w * z) = (number_of(bin_mult v w) * z::hcomplex)";

by (simp_tac (simpset() addsimps [hcomplex_mult_assoc RS sym]) 1);

qed "hcomplex_mult_number_of_left";

Goalw [hcomplex_diff_def]

    "number_of v + (number_of w - c) = number_of(bin_add v w) - (c::hcomplex)";

by (rtac hcomplex_add_number_of_left 1);

qed "hcomplex_add_number_of_diff1";

Goal "number_of v + (c - number_of w) = \

\     number_of (bin_add v (bin_minus w)) + (c::hcomplex)";

by (auto_tac (claset(),simpset() addsimps [hcomplex_diff_def]@ add_ac));

qed "hcomplex_add_number_of_diff2";

Addsimps [hcomplex_add_number_of_left, hcomplex_mult_number_of_left,

	  hcomplex_add_number_of_diff1, hcomplex_add_number_of_diff2]; 

(**** Simprocs for numeric literals ****)

structure HComplex_Numeral_Simprocs =

struct

(*Utilities*)

val hcomplexT = Type("NSComplex.hcomplex",[]);

fun mk_numeral n = HOLogic.number_of_const hcomplexT $ HOLogic.mk_bin n;

val dest_numeral = Complex_Numeral_Simprocs.dest_numeral;

val find_first_numeral = Complex_Numeral_Simprocs.find_first_numeral;

val zero = mk_numeral 0;

val mk_plus = HOLogic.mk_binop "op +";

val uminus_const = Const ("uminus", hcomplexT --> hcomplexT);

(*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 []        = zero

  | long_mk_sum (t :: ts) = mk_plus (t, mk_sum ts);

val dest_plus = HOLogic.dest_bin "op +" hcomplexT;

(*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 uminus_const$t :: ts;

fun dest_sum t = dest_summing (true, t, []);

val mk_diff = HOLogic.mk_binop "op -";

val dest_diff = HOLogic.dest_bin "op -" hcomplexT;

val one = mk_numeral 1;

val mk_times = HOLogic.mk_binop "op *";

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 "op *" hcomplexT;

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, ts) = mk_times (mk_numeral k, ts);

(*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 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 = map rename_numerals [hcomplex_add_zero_left, hcomplex_add_zero_right];

val mult_plus_1s = map rename_numerals [hcomplex_mult_one_left, hcomplex_mult_one_right];

val mult_minus_1s = map rename_numerals [hcomplex_mult_minus1, hcomplex_mult_minus1_right];

val mult_1s = mult_plus_1s @ mult_minus_1s;

(*To perform binary arithmetic*)

val bin_simps =

    [hcomplex_numeral_0_eq_0 RS sym, hcomplex_numeral_1_eq_1 RS sym,

     add_hcomplex_number_of, hcomplex_add_number_of_left, 

     minus_hcomplex_number_of, diff_hcomplex_number_of, mult_hcomplex_number_of, 

     hcomplex_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 \\ [hcomplex_add_number_of_left, add_hcomplex_number_of];

(*To evaluate binary negations of coefficients*)

val hcomplex_minus_simps = NCons_simps @

                   [hcomplex_minus_1_eq_m1,minus_hcomplex_number_of, 

		    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 = [hcomplex_diff_def, minus_add_distrib, minus_minus];

(*push the unary minus down: - x * y = x * - y *)

val hcomplex_minus_mult_eq_1_to_2 = 

    [minus_mult_left RS sym, minus_mult_right] MRS trans 

    |> standard;

(*to extract again any uncancelled minuses*)

val hcomplex_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 hcomplex_mult_minus_simps =

    [hcomplex_mult_assoc, minus_mult_left, hcomplex_minus_mult_eq_1_to_2];

(*Final simplification: cancel + and *  *)

val simplify_meta_eq = 

    Int_Numeral_Simprocs.simplify_meta_eq

         [add_zero_left, add_zero_right,

 	  mult_zero_left, mult_zero_right, mult_1, mult_1_right];

val prep_simproc = Complex_Numeral_Simprocs.prep_simproc;

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         = Real_Numeral_Simprocs.trans_tac

  val norm_tac = 

     ALLGOALS (simp_tac (HOL_ss addsimps add_0s@mult_1s@diff_simps@

                                         hcomplex_minus_simps@add_ac))

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

     THEN ALLGOALS

              (simp_tac (HOL_ss addsimps hcomplex_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

  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 =" hcomplexT

  val bal_add1 = eq_add_iff1 RS trans

  val bal_add2 = eq_add_iff2 RS trans

);

val cancel_numerals = 

  map prep_simproc

   [("hcomplexeq_cancel_numerals",

      ["(l::hcomplex) + m = n", "(l::hcomplex) = m + n", 

		"(l::hcomplex) - m = n", "(l::hcomplex) = m - n", 

		"(l::hcomplex) * m = n", "(l::hcomplex) = m * n"], 

     EqCancelNumerals.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         = Real_Numeral_Simprocs.trans_tac

  val norm_tac = 

     ALLGOALS (simp_tac (HOL_ss addsimps add_0s@mult_1s@diff_simps@

                                         hcomplex_minus_simps@add_ac))

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

     THEN ALLGOALS (simp_tac (HOL_ss addsimps hcomplex_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

  end;

structure CombineNumerals = CombineNumeralsFun(CombineNumeralsData);

val combine_numerals = 

    prep_simproc ("hcomplex_combine_numerals",

		  ["(i::hcomplex) + j", "(i::hcomplex) - j"],

		  CombineNumerals.proc);

(** Declarations for ExtractCommonTerm **)

(*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 past u []         = raise TERM("find_first", []) 

  | find_first past u (t::terms) =

	if u aconv t then (rev past @ terms)

        else find_first (t::past) u terms

	handle TERM _ => find_first (t::past) u terms;

(*Final simplification: cancel + and *  *)

fun cancel_simplify_meta_eq cancel_th th = 

    Int_Numeral_Simprocs.simplify_meta_eq 

        [hcomplex_mult_one_left, hcomplex_mult_one_right] 

        (([th, cancel_th]) MRS trans);

(*** Making constant folding work for 0 and 1 too ***)

structure HComplexAbstractNumeralsData =

  struct

  val dest_eq         = HOLogic.dest_eq o HOLogic.dest_Trueprop o concl_of

  val is_numeral      = Bin_Simprocs.is_numeral

  val numeral_0_eq_0  = hcomplex_numeral_0_eq_0

  val numeral_1_eq_1  = hcomplex_numeral_1_eq_1

  val prove_conv      = Bin_Simprocs.prove_conv_nohyps_novars

  fun norm_tac simps  = ALLGOALS (simp_tac (HOL_ss addsimps simps))

  val simplify_meta_eq = Bin_Simprocs.simplify_meta_eq

  end

structure HComplexAbstractNumerals = AbstractNumeralsFun (HComplexAbstractNumeralsData)

(*For addition, we already have rules for the operand 0.

  Multiplication is omitted because there are already special rules for

  both 0 and 1 as operands.  Unary minus is trivial, just have - 1 = -1.

  For the others, having three patterns is a compromise between just having

  one (many spurious calls) and having nine (just too many!) *)

val eval_numerals =

  map prep_simproc

   [("hcomplex_add_eval_numerals",

     ["(m::hcomplex) + 1", "(m::hcomplex) + number_of v"],

     HComplexAbstractNumerals.proc add_hcomplex_number_of),

    ("hcomplex_diff_eval_numerals",

     ["(m::hcomplex) - 1", "(m::hcomplex) - number_of v"],

     HComplexAbstractNumerals.proc diff_hcomplex_number_of),

    ("hcomplex_eq_eval_numerals",

     ["(m::hcomplex) = 0", "(m::hcomplex) = 1", "(m::hcomplex) = number_of v"],

     HComplexAbstractNumerals.proc eq_hcomplex_number_of)]

end;

Addsimprocs HComplex_Numeral_Simprocs.eval_numerals;

Addsimprocs HComplex_Numeral_Simprocs.cancel_numerals;

Addsimprocs [HComplex_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::hcomplex)";

test " 2*u = (u::hcomplex)";

test "(i + j + 12 + (k::hcomplex)) - 15 = y";

test "(i + j + 12 + (k::hcomplex)) -  5 = y";

test "( 2*x - (u*v) + y) - v* 3*u = (w::hcomplex)";

test "( 2*x*u*v + (u*v)* 4 + y) - v*u* 4 = (w::hcomplex)";

test "( 2*x*u*v + (u*v)* 4 + y) - v*u = (w::hcomplex)";

test "u*v - (x*u*v + (u*v)* 4 + y) = (w::hcomplex)";

test "(i + j + 12 + (k::hcomplex)) = u + 15 + y";

test "(i + j* 2 + 12 + (k::hcomplex)) = 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::hcomplex)";

test "a + -(b+c) + b = (d::hcomplex)";

test "a + -(b+c) - b = (d::hcomplex)";

(*negative numerals*)

test "(i + j +  -2 + (k::hcomplex)) - (u +  5 + y) = zz";

test "(i + j +  -12 + (k::hcomplex)) - 15 = y";

test "(i + j + 12 + (k::hcomplex)) -  -15 = y";

test "(i + j +  -12 + (k::hcomplex)) - -15 = y";

*)

(** Constant folding for hcomplex plus and times **)

structure HComplex_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 T	     = HComplex_Numeral_Simprocs.hcomplexT

  val plus   = Const ("op *", [T,T] ---> T)

  val add_ac = mult_ac

end;

structure HComplex_Times_Assoc = Assoc_Fold (HComplex_Times_Assoc_Data);

Addsimprocs [HComplex_Times_Assoc.conv];

Addsimps [hcomplex_of_complex_zero_iff];

(** extra thms **)

Goal "(hcnj z = 0) = (z = 0)";

by (auto_tac (claset(),simpset() addsimps [hcomplex_hcnj_zero_iff]));

qed "hcomplex_hcnj_num_zero_iff";

Addsimps [hcomplex_hcnj_num_zero_iff];

Goal "0 = Abs_hcomplex (hcomplexrel `` {%n. 0})";

by (simp_tac (simpset() addsimps [hcomplex_zero_def RS meta_eq_to_obj_eq RS sym]) 1);

qed "hcomplex_zero_num";

Goal "1 =  Abs_hcomplex (hcomplexrel `` {%n. 1})";

by (simp_tac (simpset() addsimps [hcomplex_one_def RS meta_eq_to_obj_eq RS sym]) 1);

qed "hcomplex_one_num";

(*** Real and imaginary stuff ***)

(*Convert???

Goalw [hcomplex_number_of_def] 

  "((number_of xa :: hcomplex) + iii * number_of ya = \

\       number_of xb + iii * number_of yb) = \

\  (((number_of xa :: hcomplex) = number_of xb) & \

\   ((number_of ya :: hcomplex) = number_of yb))";

by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iff,

     hcomplex_hypreal_number_of]));

qed "hcomplex_number_of_eq_cancel_iff";

Addsimps [hcomplex_number_of_eq_cancel_iff];

Goalw [hcomplex_number_of_def] 

  "((number_of xa :: hcomplex) + number_of ya * iii = \

\       number_of xb + number_of yb * iii) = \

\  (((number_of xa :: hcomplex) = number_of xb) & \

\   ((number_of ya :: hcomplex) = number_of yb))";

by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iffA,

    hcomplex_hypreal_number_of]));

qed "hcomplex_number_of_eq_cancel_iffA";

Addsimps [hcomplex_number_of_eq_cancel_iffA];

Goalw [hcomplex_number_of_def] 

  "((number_of xa :: hcomplex) + number_of ya * iii = \

\       number_of xb + iii * number_of yb) = \

\  (((number_of xa :: hcomplex) = number_of xb) & \

\   ((number_of ya :: hcomplex) = number_of yb))";

by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iffB,

    hcomplex_hypreal_number_of]));

qed "hcomplex_number_of_eq_cancel_iffB";

Addsimps [hcomplex_number_of_eq_cancel_iffB];

Goalw [hcomplex_number_of_def] 

  "((number_of xa :: hcomplex) + iii * number_of ya = \

\       number_of xb + number_of yb * iii) = \

\  (((number_of xa :: hcomplex) = number_of xb) & \

\   ((number_of ya :: hcomplex) = number_of yb))";

by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iffC,

     hcomplex_hypreal_number_of]));

qed "hcomplex_number_of_eq_cancel_iffC";

Addsimps [hcomplex_number_of_eq_cancel_iffC];

Goalw [hcomplex_number_of_def] 

  "((number_of xa :: hcomplex) + iii * number_of ya = \

\       number_of xb) = \

\  (((number_of xa :: hcomplex) = number_of xb) & \

\   ((number_of ya :: hcomplex) = 0))";

by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iff2,

    hcomplex_hypreal_number_of,hcomplex_of_hypreal_zero_iff]));

qed "hcomplex_number_of_eq_cancel_iff2";

Addsimps [hcomplex_number_of_eq_cancel_iff2];

Goalw [hcomplex_number_of_def] 

  "((number_of xa :: hcomplex) + number_of ya * iii = \

\       number_of xb) = \

\  (((number_of xa :: hcomplex) = number_of xb) & \

\   ((number_of ya :: hcomplex) = 0))";

by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iff2a,

    hcomplex_hypreal_number_of,hcomplex_of_hypreal_zero_iff]));

qed "hcomplex_number_of_eq_cancel_iff2a";

Addsimps [hcomplex_number_of_eq_cancel_iff2a];

Goalw [hcomplex_number_of_def] 

  "((number_of xa :: hcomplex) + iii * number_of ya = \

\    iii * number_of yb) = \

\  (((number_of xa :: hcomplex) = 0) & \

\   ((number_of ya :: hcomplex) = number_of yb))";

by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iff3,

    hcomplex_hypreal_number_of,hcomplex_of_hypreal_zero_iff]));

qed "hcomplex_number_of_eq_cancel_iff3";

Addsimps [hcomplex_number_of_eq_cancel_iff3];

Goalw [hcomplex_number_of_def] 

  "((number_of xa :: hcomplex) + number_of ya * iii= \

\    iii * number_of yb) = \

\  (((number_of xa :: hcomplex) = 0) & \

\   ((number_of ya :: hcomplex) = number_of yb))";

by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iff3a,

    hcomplex_hypreal_number_of,hcomplex_of_hypreal_zero_iff]));

qed "hcomplex_number_of_eq_cancel_iff3a";

Addsimps [hcomplex_number_of_eq_cancel_iff3a];

*)

Goalw [hcomplex_number_of_def] "hcnj (number_of v :: hcomplex) = number_of v";

by (rtac (hcomplex_hypreal_number_of RS ssubst) 1);

by (rtac hcomplex_hcnj_hcomplex_of_hypreal 1);

qed "hcomplex_number_of_hcnj";

Addsimps [hcomplex_number_of_hcnj];

Goalw [hcomplex_number_of_def] 

      "hcmod(number_of v :: hcomplex) = abs (number_of v :: hypreal)";

by (rtac (hcomplex_hypreal_number_of RS ssubst) 1);

by (auto_tac (claset(), HOL_ss addsimps [hcmod_hcomplex_of_hypreal]));

qed "hcomplex_number_of_hcmod";

Addsimps [hcomplex_number_of_hcmod];

Goalw [hcomplex_number_of_def] 

      "hRe(number_of v :: hcomplex) = number_of v";

by (rtac (hcomplex_hypreal_number_of RS ssubst) 1);

by (auto_tac (claset(), HOL_ss addsimps [hRe_hcomplex_of_hypreal]));

qed "hcomplex_number_of_hRe";

Addsimps [hcomplex_number_of_hRe];

Goalw [hcomplex_number_of_def] 

      "hIm(number_of v :: hcomplex) = 0";

by (rtac (hcomplex_hypreal_number_of RS ssubst) 1);

by (auto_tac (claset(), HOL_ss addsimps [hIm_hcomplex_of_hypreal]));

qed "hcomplex_number_of_hIm";

Addsimps [hcomplex_number_of_hIm];