src/HOL/Complex/NSComplexBin.ML
author paulson
Tue Dec 23 14:45:47 2003 +0100 (2003-12-23)
changeset 14320 fb7a114826be
parent 14318 7dbd3988b15b
child 14335 9c0b5e081037
permissions -rw-r--r--
tidying up hcomplex arithmetic
     1 (*  Title:      NSComplexBin.ML
     2     Author:     Jacques D. Fleuriot
     3     Copyright:  2001 University of Edinburgh
     4     Descrition: Binary arithmetic for the nonstandard complex numbers
     5 *)
     6 
     7 (** hcomplex_of_complex (coercion from complex to nonstandard complex) **)
     8 
     9 Goal "hcomplex_of_complex (number_of w) = number_of w";
    10 by (simp_tac (simpset() addsimps [hcomplex_number_of_def]) 1);
    11 qed "hcomplex_number_of";
    12 Addsimps [hcomplex_number_of];
    13 
    14 Goalw [hypreal_of_real_def]
    15      "hcomplex_of_hypreal (hypreal_of_real x) = \
    16 \     hcomplex_of_complex(complex_of_real x)";
    17 by (simp_tac (simpset() addsimps [hcomplex_of_hypreal,
    18     hcomplex_of_complex_def,complex_of_real_def]) 1);
    19 qed "hcomplex_of_hypreal_eq_hcomplex_of_complex";
    20 
    21 Goalw [complex_number_of_def,hypreal_number_of_def] 
    22   "hcomplex_of_complex (number_of w) = hcomplex_of_hypreal(number_of w)";
    23 by (rtac (hcomplex_of_hypreal_eq_hcomplex_of_complex RS sym) 1);
    24 qed "hcomplex_hypreal_number_of";
    25 
    26 Goalw [hcomplex_number_of_def] "Numeral0 = (0::hcomplex)";
    27 by(Simp_tac 1);
    28 qed "hcomplex_numeral_0_eq_0";
    29 
    30 Goalw [hcomplex_number_of_def] "Numeral1 = (1::hcomplex)";
    31 by(Simp_tac 1);
    32 qed "hcomplex_numeral_1_eq_1";
    33 
    34 (*
    35 Goal "z + hcnj z = \
    36 \     hcomplex_of_hypreal (2 * hRe(z))";
    37 by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
    38 by (auto_tac (claset(),HOL_ss addsimps [hRe,hcnj,hcomplex_add,
    39     hypreal_mult,hcomplex_of_hypreal,complex_add_cnj]));
    40 qed "hcomplex_add_hcnj";
    41 
    42 Goal "z - hcnj z = \
    43 \     hcomplex_of_hypreal (hypreal_of_real #2 * hIm(z)) * iii";
    44 by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
    45 by (auto_tac (claset(),simpset() addsimps [hIm,hcnj,hcomplex_diff,
    46     hypreal_of_real_def,hypreal_mult,hcomplex_of_hypreal,
    47     complex_diff_cnj,iii_def,hcomplex_mult]));
    48 qed "hcomplex_diff_hcnj";
    49 *)
    50 
    51 (** Addition **)
    52 
    53 Goal "(number_of v :: hcomplex) + number_of v' = number_of (bin_add v v')";
    54 by (simp_tac
    55     (HOL_ss addsimps [hcomplex_number_of_def, 
    56                       hcomplex_of_complex_add RS sym, add_complex_number_of]) 1);
    57 qed "add_hcomplex_number_of";
    58 Addsimps [add_hcomplex_number_of];
    59 
    60 
    61 (** Subtraction **)
    62 
    63 Goalw [hcomplex_number_of_def]
    64      "- (number_of w :: hcomplex) = number_of (bin_minus w)";
    65 by (simp_tac
    66     (HOL_ss addsimps [minus_complex_number_of, hcomplex_of_complex_minus RS sym]) 1);
    67 qed "minus_hcomplex_number_of";
    68 Addsimps [minus_hcomplex_number_of];
    69 
    70 Goalw [hcomplex_number_of_def, hcomplex_diff_def]
    71      "(number_of v :: hcomplex) - number_of w = \
    72 \     number_of (bin_add v (bin_minus w))";
    73 by (Simp_tac 1); 
    74 qed "diff_hcomplex_number_of";
    75 Addsimps [diff_hcomplex_number_of];
    76 
    77 
    78 (** Multiplication **)
    79 
    80 Goal "(number_of v :: hcomplex) * number_of v' = number_of (bin_mult v v')";
    81 by (simp_tac
    82     (HOL_ss addsimps [hcomplex_number_of_def, 
    83 	              hcomplex_of_complex_mult RS sym, mult_complex_number_of]) 1);
    84 qed "mult_hcomplex_number_of";
    85 Addsimps [mult_hcomplex_number_of];
    86 
    87 Goal "(2::hcomplex) = 1 + 1";
    88 by (simp_tac (simpset() addsimps [hcomplex_numeral_1_eq_1 RS sym]) 1);
    89 val lemma = result();
    90 
    91 (*For specialist use: NOT as default simprules*)
    92 Goal "2 * z = (z+z::hcomplex)";
    93 by (simp_tac (simpset() addsimps [lemma, hcomplex_add_mult_distrib]) 1);
    94 qed "hcomplex_mult_2";
    95 
    96 Goal "z * 2 = (z+z::hcomplex)";
    97 by (stac hcomplex_mult_commute 1 THEN rtac hcomplex_mult_2 1);
    98 qed "hcomplex_mult_2_right";
    99 
   100 (** Equals (=) **)
   101 
   102 Goal "((number_of v :: hcomplex) = number_of v') = \
   103 \     iszero (number_of (bin_add v (bin_minus v')))";
   104 by (simp_tac
   105     (HOL_ss addsimps [hcomplex_number_of_def, 
   106 	              hcomplex_of_complex_eq_iff, eq_complex_number_of]) 1);
   107 qed "eq_hcomplex_number_of";
   108 Addsimps [eq_hcomplex_number_of];
   109 
   110 (*** New versions of existing theorems involving 0, 1hc ***)
   111 
   112 Goal "- 1 = (-1::hcomplex)";
   113 by (simp_tac (simpset() addsimps [hcomplex_numeral_1_eq_1 RS sym]) 1);
   114 qed "hcomplex_minus_1_eq_m1";
   115 
   116 Goal "-1 * z = -(z::hcomplex)";
   117 by (simp_tac (simpset() addsimps [hcomplex_minus_1_eq_m1 RS sym]) 1);
   118 qed "hcomplex_mult_minus1";
   119 
   120 Goal "z * -1 = -(z::hcomplex)";
   121 by (stac hcomplex_mult_commute 1 THEN rtac hcomplex_mult_minus1 1);
   122 qed "hcomplex_mult_minus1_right";
   123 
   124 Addsimps [hcomplex_mult_minus1,hcomplex_mult_minus1_right];
   125 
   126 (*Maps 0 to Numeral0 and 1 to Numeral1 and -Numeral1 to -1*)
   127 val hcomplex_numeral_ss = 
   128     complex_numeral_ss addsimps [hcomplex_numeral_0_eq_0 RS sym, hcomplex_numeral_1_eq_1 RS sym, 
   129 		                 hcomplex_minus_1_eq_m1];
   130 
   131 fun rename_numerals th = 
   132     asm_full_simplify hcomplex_numeral_ss (Thm.transfer (the_context ()) th);
   133 
   134 
   135 (*Now insert some identities previously stated for 0 and 1hc*)
   136 
   137 
   138 Addsimps [hcomplex_numeral_0_eq_0,hcomplex_numeral_1_eq_1];
   139 
   140 Goal "number_of v + (number_of w + z) = (number_of(bin_add v w) + z::hcomplex)";
   141 by (auto_tac (claset(),simpset() addsimps [hcomplex_add_assoc RS sym]));
   142 qed "hcomplex_add_number_of_left";
   143 
   144 Goal "number_of v *(number_of w * z) = (number_of(bin_mult v w) * z::hcomplex)";
   145 by (simp_tac (simpset() addsimps [hcomplex_mult_assoc RS sym]) 1);
   146 qed "hcomplex_mult_number_of_left";
   147 
   148 Goalw [hcomplex_diff_def]
   149     "number_of v + (number_of w - c) = number_of(bin_add v w) - (c::hcomplex)";
   150 by (rtac hcomplex_add_number_of_left 1);
   151 qed "hcomplex_add_number_of_diff1";
   152 
   153 Goal "number_of v + (c - number_of w) = \
   154 \     number_of (bin_add v (bin_minus w)) + (c::hcomplex)";
   155 by (auto_tac (claset(),simpset() addsimps [hcomplex_diff_def]@ hcomplex_add_ac));
   156 qed "hcomplex_add_number_of_diff2";
   157 
   158 Addsimps [hcomplex_add_number_of_left, hcomplex_mult_number_of_left,
   159 	  hcomplex_add_number_of_diff1, hcomplex_add_number_of_diff2]; 
   160 
   161 
   162 (**** Simprocs for numeric literals ****)
   163 
   164 (** Combining of literal coefficients in sums of products **)
   165 
   166 Goal "(x = y) = (x-y = (0::hcomplex))";
   167 by (simp_tac (simpset() addsimps [hcomplex_diff_eq_eq]) 1);   
   168 qed "hcomplex_eq_iff_diff_eq_0";
   169 
   170 (** For combine_numerals **)
   171 
   172 Goal "i*u + (j*u + k) = (i+j)*u + (k::hcomplex)";
   173 by (asm_simp_tac (simpset() addsimps [hcomplex_add_mult_distrib]
   174     @ hcomplex_add_ac) 1);
   175 qed "left_hcomplex_add_mult_distrib";
   176 
   177 (** For cancel_numerals **)
   178 
   179 Goal "((x::hcomplex) = u + v) = (x - (u + v) = 0)";
   180 by (auto_tac (claset(),simpset() addsimps [hcomplex_diff_eq_eq]));
   181 qed "hcomplex_eq_add_diff_eq_0";
   182 
   183 Goal "((x::hcomplex) = n) = (x - n = 0)";
   184 by (auto_tac (claset(),simpset() addsimps [hcomplex_diff_eq_eq]));
   185 qed "hcomplex_eq_diff_eq_0";
   186 
   187 val hcomplex_rel_iff_rel_0_rls = [hcomplex_eq_diff_eq_0,hcomplex_eq_add_diff_eq_0];
   188 
   189 Goal "!!i::hcomplex. (i*u + m = j*u + n) = ((i-j)*u + m = n)";
   190 by (auto_tac (claset(), simpset() addsimps [hcomplex_add_mult_distrib,
   191     hcomplex_diff_def] @ hcomplex_add_ac));
   192 by (asm_simp_tac (simpset() addsimps [hcomplex_add_assoc RS sym]) 1);
   193 by (simp_tac (simpset() addsimps [hcomplex_add_assoc]) 1);
   194 qed "hcomplex_eq_add_iff1";
   195 
   196 Goal "!!i::hcomplex. (i*u + m = j*u + n) = (m = (j-i)*u + n)";
   197 by (res_inst_tac [("z","i")] eq_Abs_hcomplex 1);
   198 by (res_inst_tac [("z","j")] eq_Abs_hcomplex 1);
   199 by (res_inst_tac [("z","u")] eq_Abs_hcomplex 1);
   200 by (res_inst_tac [("z","m")] eq_Abs_hcomplex 1);
   201 by (res_inst_tac [("z","n")] eq_Abs_hcomplex 1);
   202 by (auto_tac (claset(), simpset() addsimps [hcomplex_diff,hcomplex_add,
   203     hcomplex_mult,complex_eq_add_iff2]));
   204 qed "hcomplex_eq_add_iff2";
   205 
   206 
   207 structure HComplex_Numeral_Simprocs =
   208 struct
   209 
   210 (*Utilities*)
   211 
   212 val hcomplexT = Type("NSComplex.hcomplex",[]);
   213 
   214 fun mk_numeral n = HOLogic.number_of_const hcomplexT $ HOLogic.mk_bin n;
   215 
   216 val dest_numeral = Complex_Numeral_Simprocs.dest_numeral;
   217 
   218 val find_first_numeral = Complex_Numeral_Simprocs.find_first_numeral;
   219 
   220 val zero = mk_numeral 0;
   221 val mk_plus = HOLogic.mk_binop "op +";
   222 
   223 val uminus_const = Const ("uminus", hcomplexT --> hcomplexT);
   224 
   225 (*Thus mk_sum[t] yields t+0; longer sums don't have a trailing zero*)
   226 fun mk_sum []        = zero
   227   | mk_sum [t,u]     = mk_plus (t, u)
   228   | mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
   229 
   230 (*this version ALWAYS includes a trailing zero*)
   231 fun long_mk_sum []        = zero
   232   | long_mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
   233 
   234 val dest_plus = HOLogic.dest_bin "op +" hcomplexT;
   235 
   236 (*decompose additions AND subtractions as a sum*)
   237 fun dest_summing (pos, Const ("op +", _) $ t $ u, ts) =
   238         dest_summing (pos, t, dest_summing (pos, u, ts))
   239   | dest_summing (pos, Const ("op -", _) $ t $ u, ts) =
   240         dest_summing (pos, t, dest_summing (not pos, u, ts))
   241   | dest_summing (pos, t, ts) =
   242 	if pos then t::ts else uminus_const$t :: ts;
   243 
   244 fun dest_sum t = dest_summing (true, t, []);
   245 
   246 val mk_diff = HOLogic.mk_binop "op -";
   247 val dest_diff = HOLogic.dest_bin "op -" hcomplexT;
   248 
   249 val one = mk_numeral 1;
   250 val mk_times = HOLogic.mk_binop "op *";
   251 
   252 fun mk_prod [] = one
   253   | mk_prod [t] = t
   254   | mk_prod (t :: ts) = if t = one then mk_prod ts
   255                         else mk_times (t, mk_prod ts);
   256 
   257 val dest_times = HOLogic.dest_bin "op *" hcomplexT;
   258 
   259 fun dest_prod t =
   260       let val (t,u) = dest_times t 
   261       in  dest_prod t @ dest_prod u  end
   262       handle TERM _ => [t];
   263 
   264 (*DON'T do the obvious simplifications; that would create special cases*) 
   265 fun mk_coeff (k, ts) = mk_times (mk_numeral k, ts);
   266 
   267 (*Express t as a product of (possibly) a numeral with other sorted terms*)
   268 fun dest_coeff sign (Const ("uminus", _) $ t) = dest_coeff (~sign) t
   269   | dest_coeff sign t =
   270     let val ts = sort Term.term_ord (dest_prod t)
   271 	val (n, ts') = find_first_numeral [] ts
   272                           handle TERM _ => (1, ts)
   273     in (sign*n, mk_prod ts') end;
   274 
   275 (*Find first coefficient-term THAT MATCHES u*)
   276 fun find_first_coeff past u [] = raise TERM("find_first_coeff", []) 
   277   | find_first_coeff past u (t::terms) =
   278 	let val (n,u') = dest_coeff 1 t
   279 	in  if u aconv u' then (n, rev past @ terms)
   280 			  else find_first_coeff (t::past) u terms
   281 	end
   282 	handle TERM _ => find_first_coeff (t::past) u terms;
   283 
   284 
   285 
   286 (*Simplify Numeral0+n, n+Numeral0, Numeral1*n, n*Numeral1*)
   287 val add_0s = map rename_numerals [hcomplex_add_zero_left, hcomplex_add_zero_right];
   288 val mult_plus_1s = map rename_numerals [hcomplex_mult_one_left, hcomplex_mult_one_right];
   289 val mult_minus_1s = map rename_numerals [hcomplex_mult_minus1, hcomplex_mult_minus1_right];
   290 val mult_1s = mult_plus_1s @ mult_minus_1s;
   291 
   292 (*To perform binary arithmetic*)
   293 val bin_simps =
   294     [hcomplex_numeral_0_eq_0 RS sym, hcomplex_numeral_1_eq_1 RS sym,
   295      add_hcomplex_number_of, hcomplex_add_number_of_left, 
   296      minus_hcomplex_number_of, diff_hcomplex_number_of, mult_hcomplex_number_of, 
   297      hcomplex_mult_number_of_left] @ bin_arith_simps @ bin_rel_simps;
   298 
   299 (*Binary arithmetic BUT NOT ADDITION since it may collapse adjacent terms
   300   during re-arrangement*)
   301 val non_add_bin_simps = 
   302     bin_simps \\ [hcomplex_add_number_of_left, add_hcomplex_number_of];
   303 
   304 (*To evaluate binary negations of coefficients*)
   305 val hcomplex_minus_simps = NCons_simps @
   306                    [hcomplex_minus_1_eq_m1,minus_hcomplex_number_of, 
   307 		    bin_minus_1, bin_minus_0, bin_minus_Pls, bin_minus_Min,
   308 		    bin_pred_1, bin_pred_0, bin_pred_Pls, bin_pred_Min];
   309 
   310 
   311 (*To let us treat subtraction as addition*)
   312 val diff_simps = [hcomplex_diff_def, hcomplex_minus_add_distrib, 
   313                   minus_minus];
   314 
   315 (*push the unary minus down: - x * y = x * - y *)
   316 val hcomplex_minus_mult_eq_1_to_2 = 
   317     [hcomplex_minus_mult_eq1 RS sym, hcomplex_minus_mult_eq2] MRS trans 
   318     |> standard;
   319 
   320 (*to extract again any uncancelled minuses*)
   321 val hcomplex_minus_from_mult_simps = 
   322     [minus_minus, hcomplex_minus_mult_eq1 RS sym, 
   323      hcomplex_minus_mult_eq2 RS sym];
   324 
   325 (*combine unary minus with numeric literals, however nested within a product*)
   326 val hcomplex_mult_minus_simps =
   327     [hcomplex_mult_assoc, hcomplex_minus_mult_eq1, hcomplex_minus_mult_eq_1_to_2];
   328 
   329 (*Final simplification: cancel + and *  *)
   330 val simplify_meta_eq = 
   331     Int_Numeral_Simprocs.simplify_meta_eq
   332          [hcomplex_add_zero_left, hcomplex_add_zero_right,
   333  	  hcomplex_mult_zero_left, hcomplex_mult_zero_right, hcomplex_mult_one_left, 
   334           hcomplex_mult_one_right];
   335 
   336 val prep_simproc = Complex_Numeral_Simprocs.prep_simproc;
   337 
   338 
   339 structure CancelNumeralsCommon =
   340   struct
   341   val mk_sum    	= mk_sum
   342   val dest_sum		= dest_sum
   343   val mk_coeff		= mk_coeff
   344   val dest_coeff	= dest_coeff 1
   345   val find_first_coeff	= find_first_coeff []
   346   val trans_tac         = Real_Numeral_Simprocs.trans_tac
   347   val norm_tac = 
   348      ALLGOALS (simp_tac (HOL_ss addsimps add_0s@mult_1s@diff_simps@
   349                                          hcomplex_minus_simps@hcomplex_add_ac))
   350      THEN ALLGOALS (simp_tac (HOL_ss addsimps non_add_bin_simps@hcomplex_mult_minus_simps))
   351      THEN ALLGOALS
   352               (simp_tac (HOL_ss addsimps hcomplex_minus_from_mult_simps@
   353                                          hcomplex_add_ac@hcomplex_mult_ac))
   354   val numeral_simp_tac	= ALLGOALS (simp_tac (HOL_ss addsimps add_0s@bin_simps))
   355   val simplify_meta_eq  = simplify_meta_eq
   356   end;
   357 
   358 
   359 structure EqCancelNumerals = CancelNumeralsFun
   360  (open CancelNumeralsCommon
   361   val prove_conv = Bin_Simprocs.prove_conv
   362   val mk_bal   = HOLogic.mk_eq
   363   val dest_bal = HOLogic.dest_bin "op =" hcomplexT
   364   val bal_add1 = hcomplex_eq_add_iff1 RS trans
   365   val bal_add2 = hcomplex_eq_add_iff2 RS trans
   366 );
   367 
   368 
   369 val cancel_numerals = 
   370   map prep_simproc
   371    [("hcomplexeq_cancel_numerals",
   372       ["(l::hcomplex) + m = n", "(l::hcomplex) = m + n", 
   373 		"(l::hcomplex) - m = n", "(l::hcomplex) = m - n", 
   374 		"(l::hcomplex) * m = n", "(l::hcomplex) = m * n"], 
   375      EqCancelNumerals.proc)];
   376 
   377 structure CombineNumeralsData =
   378   struct
   379   val add		= op + : int*int -> int 
   380   val mk_sum    	= long_mk_sum    (*to work for e.g. #2*x + #3*x *)
   381   val dest_sum		= dest_sum
   382   val mk_coeff		= mk_coeff
   383   val dest_coeff	= dest_coeff 1
   384   val left_distrib	= left_hcomplex_add_mult_distrib RS trans
   385   val prove_conv	= Bin_Simprocs.prove_conv_nohyps
   386   val trans_tac         = Real_Numeral_Simprocs.trans_tac
   387   val norm_tac = 
   388      ALLGOALS (simp_tac (HOL_ss addsimps add_0s@mult_1s@diff_simps@
   389                                          hcomplex_minus_simps@hcomplex_add_ac))
   390      THEN ALLGOALS (simp_tac (HOL_ss addsimps non_add_bin_simps@hcomplex_mult_minus_simps))
   391      THEN ALLGOALS (simp_tac (HOL_ss addsimps hcomplex_minus_from_mult_simps@
   392                                               hcomplex_add_ac@hcomplex_mult_ac))
   393   val numeral_simp_tac	= ALLGOALS 
   394                     (simp_tac (HOL_ss addsimps add_0s@bin_simps))
   395   val simplify_meta_eq  = simplify_meta_eq
   396   end;
   397 
   398 structure CombineNumerals = CombineNumeralsFun(CombineNumeralsData);
   399 
   400 val combine_numerals = 
   401     prep_simproc ("hcomplex_combine_numerals",
   402 		  ["(i::hcomplex) + j", "(i::hcomplex) - j"],
   403 		  CombineNumerals.proc);
   404 
   405 (** Declarations for ExtractCommonTerm **)
   406 
   407 (*this version ALWAYS includes a trailing one*)
   408 fun long_mk_prod []        = one
   409   | long_mk_prod (t :: ts) = mk_times (t, mk_prod ts);
   410 
   411 (*Find first term that matches u*)
   412 fun find_first past u []         = raise TERM("find_first", []) 
   413   | find_first past u (t::terms) =
   414 	if u aconv t then (rev past @ terms)
   415         else find_first (t::past) u terms
   416 	handle TERM _ => find_first (t::past) u terms;
   417 
   418 (*Final simplification: cancel + and *  *)
   419 fun cancel_simplify_meta_eq cancel_th th = 
   420     Int_Numeral_Simprocs.simplify_meta_eq 
   421         [hcomplex_mult_one_left, hcomplex_mult_one_right] 
   422         (([th, cancel_th]) MRS trans);
   423 
   424 (*** Making constant folding work for 0 and 1 too ***)
   425 
   426 structure HComplexAbstractNumeralsData =
   427   struct
   428   val dest_eq         = HOLogic.dest_eq o HOLogic.dest_Trueprop o concl_of
   429   val is_numeral      = Bin_Simprocs.is_numeral
   430   val numeral_0_eq_0  = hcomplex_numeral_0_eq_0
   431   val numeral_1_eq_1  = hcomplex_numeral_1_eq_1
   432   val prove_conv      = Bin_Simprocs.prove_conv_nohyps_novars
   433   fun norm_tac simps  = ALLGOALS (simp_tac (HOL_ss addsimps simps))
   434   val simplify_meta_eq = Bin_Simprocs.simplify_meta_eq
   435   end
   436 
   437 structure HComplexAbstractNumerals = AbstractNumeralsFun (HComplexAbstractNumeralsData)
   438 
   439 (*For addition, we already have rules for the operand 0.
   440   Multiplication is omitted because there are already special rules for
   441   both 0 and 1 as operands.  Unary minus is trivial, just have - 1 = -1.
   442   For the others, having three patterns is a compromise between just having
   443   one (many spurious calls) and having nine (just too many!) *)
   444 val eval_numerals =
   445   map prep_simproc
   446    [("hcomplex_add_eval_numerals",
   447      ["(m::hcomplex) + 1", "(m::hcomplex) + number_of v"],
   448      HComplexAbstractNumerals.proc add_hcomplex_number_of),
   449     ("hcomplex_diff_eval_numerals",
   450      ["(m::hcomplex) - 1", "(m::hcomplex) - number_of v"],
   451      HComplexAbstractNumerals.proc diff_hcomplex_number_of),
   452     ("hcomplex_eq_eval_numerals",
   453      ["(m::hcomplex) = 0", "(m::hcomplex) = 1", "(m::hcomplex) = number_of v"],
   454      HComplexAbstractNumerals.proc eq_hcomplex_number_of)]
   455 
   456 end;
   457 
   458 Addsimprocs HComplex_Numeral_Simprocs.eval_numerals;
   459 Addsimprocs HComplex_Numeral_Simprocs.cancel_numerals;
   460 Addsimprocs [HComplex_Numeral_Simprocs.combine_numerals];
   461 
   462 
   463 (*examples:
   464 print_depth 22;
   465 set timing;
   466 set trace_simp;
   467 fun test s = (Goal s, by (Simp_tac 1)); 
   468 
   469 test "l +  2 +  2 +  2 + (l +  2) + (oo +  2) = (uu::hcomplex)";
   470 test " 2*u = (u::hcomplex)";
   471 test "(i + j + 12 + (k::hcomplex)) - 15 = y";
   472 test "(i + j + 12 + (k::hcomplex)) -  5 = y";
   473 
   474 test "( 2*x - (u*v) + y) - v* 3*u = (w::hcomplex)";
   475 test "( 2*x*u*v + (u*v)* 4 + y) - v*u* 4 = (w::hcomplex)";
   476 test "( 2*x*u*v + (u*v)* 4 + y) - v*u = (w::hcomplex)";
   477 test "u*v - (x*u*v + (u*v)* 4 + y) = (w::hcomplex)";
   478 
   479 test "(i + j + 12 + (k::hcomplex)) = u + 15 + y";
   480 test "(i + j* 2 + 12 + (k::hcomplex)) = j +  5 + y";
   481 
   482 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)";
   483 
   484 test "a + -(b+c) + b = (d::hcomplex)";
   485 test "a + -(b+c) - b = (d::hcomplex)";
   486 
   487 (*negative numerals*)
   488 test "(i + j +  -2 + (k::hcomplex)) - (u +  5 + y) = zz";
   489 
   490 test "(i + j +  -12 + (k::hcomplex)) - 15 = y";
   491 test "(i + j + 12 + (k::hcomplex)) -  -15 = y";
   492 test "(i + j +  -12 + (k::hcomplex)) - -15 = y";
   493 *)
   494 
   495 (** Constant folding for hcomplex plus and times **)
   496 
   497 structure HComplex_Times_Assoc_Data : ASSOC_FOLD_DATA =
   498 struct
   499   val ss		= HOL_ss
   500   val eq_reflection	= eq_reflection
   501   val sg_ref    = Sign.self_ref (Theory.sign_of (the_context ()))
   502   val T	     = HComplex_Numeral_Simprocs.hcomplexT
   503   val plus   = Const ("op *", [T,T] ---> T)
   504   val add_ac = hcomplex_mult_ac
   505 end;
   506 
   507 structure HComplex_Times_Assoc = Assoc_Fold (HComplex_Times_Assoc_Data);
   508 
   509 Addsimprocs [HComplex_Times_Assoc.conv];
   510 
   511 Addsimps [hcomplex_of_complex_zero_iff];
   512 
   513 (*Simplification of  x-y = 0 *)
   514 
   515 AddIffs [hcomplex_eq_iff_diff_eq_0 RS sym];
   516 
   517 (** extra thms **)
   518 
   519 Goal "(hcnj z = 0) = (z = 0)";
   520 by (auto_tac (claset(),simpset() addsimps [hcomplex_hcnj_zero_iff]));
   521 qed "hcomplex_hcnj_num_zero_iff";
   522 Addsimps [hcomplex_hcnj_num_zero_iff];
   523 
   524 Goal "0 = Abs_hcomplex (hcomplexrel `` {%n. 0})";
   525 by (simp_tac (simpset() addsimps [hcomplex_zero_def RS meta_eq_to_obj_eq RS sym]) 1);
   526 qed "hcomplex_zero_num";
   527 
   528 Goal "1 =  Abs_hcomplex (hcomplexrel `` {%n. 1})";
   529 by (simp_tac (simpset() addsimps [hcomplex_one_def RS meta_eq_to_obj_eq RS sym]) 1);
   530 qed "hcomplex_one_num";
   531 
   532 (*** Real and imaginary stuff ***)
   533 
   534 Goalw [hcomplex_number_of_def] 
   535   "((number_of xa :: hcomplex) + iii * number_of ya = \
   536 \       number_of xb + iii * number_of yb) = \
   537 \  (((number_of xa :: hcomplex) = number_of xb) & \
   538 \   ((number_of ya :: hcomplex) = number_of yb))";
   539 by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iff,
   540      hcomplex_hypreal_number_of]));
   541 qed "hcomplex_number_of_eq_cancel_iff";
   542 Addsimps [hcomplex_number_of_eq_cancel_iff];
   543 
   544 Goalw [hcomplex_number_of_def] 
   545   "((number_of xa :: hcomplex) + number_of ya * iii = \
   546 \       number_of xb + number_of yb * iii) = \
   547 \  (((number_of xa :: hcomplex) = number_of xb) & \
   548 \   ((number_of ya :: hcomplex) = number_of yb))";
   549 by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iffA,
   550     hcomplex_hypreal_number_of]));
   551 qed "hcomplex_number_of_eq_cancel_iffA";
   552 Addsimps [hcomplex_number_of_eq_cancel_iffA];
   553 
   554 Goalw [hcomplex_number_of_def] 
   555   "((number_of xa :: hcomplex) + number_of ya * iii = \
   556 \       number_of xb + iii * number_of yb) = \
   557 \  (((number_of xa :: hcomplex) = number_of xb) & \
   558 \   ((number_of ya :: hcomplex) = number_of yb))";
   559 by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iffB,
   560     hcomplex_hypreal_number_of]));
   561 qed "hcomplex_number_of_eq_cancel_iffB";
   562 Addsimps [hcomplex_number_of_eq_cancel_iffB];
   563 
   564 Goalw [hcomplex_number_of_def] 
   565   "((number_of xa :: hcomplex) + iii * number_of ya = \
   566 \       number_of xb + number_of yb * iii) = \
   567 \  (((number_of xa :: hcomplex) = number_of xb) & \
   568 \   ((number_of ya :: hcomplex) = number_of yb))";
   569 by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iffC,
   570      hcomplex_hypreal_number_of]));
   571 qed "hcomplex_number_of_eq_cancel_iffC";
   572 Addsimps [hcomplex_number_of_eq_cancel_iffC];
   573 
   574 Goalw [hcomplex_number_of_def] 
   575   "((number_of xa :: hcomplex) + iii * number_of ya = \
   576 \       number_of xb) = \
   577 \  (((number_of xa :: hcomplex) = number_of xb) & \
   578 \   ((number_of ya :: hcomplex) = 0))";
   579 by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iff2,
   580     hcomplex_hypreal_number_of,hcomplex_of_hypreal_zero_iff]));
   581 qed "hcomplex_number_of_eq_cancel_iff2";
   582 Addsimps [hcomplex_number_of_eq_cancel_iff2];
   583 
   584 Goalw [hcomplex_number_of_def] 
   585   "((number_of xa :: hcomplex) + number_of ya * iii = \
   586 \       number_of xb) = \
   587 \  (((number_of xa :: hcomplex) = number_of xb) & \
   588 \   ((number_of ya :: hcomplex) = 0))";
   589 by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iff2a,
   590     hcomplex_hypreal_number_of,hcomplex_of_hypreal_zero_iff]));
   591 qed "hcomplex_number_of_eq_cancel_iff2a";
   592 Addsimps [hcomplex_number_of_eq_cancel_iff2a];
   593 
   594 Goalw [hcomplex_number_of_def] 
   595   "((number_of xa :: hcomplex) + iii * number_of ya = \
   596 \    iii * number_of yb) = \
   597 \  (((number_of xa :: hcomplex) = 0) & \
   598 \   ((number_of ya :: hcomplex) = number_of yb))";
   599 by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iff3,
   600     hcomplex_hypreal_number_of,hcomplex_of_hypreal_zero_iff]));
   601 qed "hcomplex_number_of_eq_cancel_iff3";
   602 Addsimps [hcomplex_number_of_eq_cancel_iff3];
   603 
   604 Goalw [hcomplex_number_of_def] 
   605   "((number_of xa :: hcomplex) + number_of ya * iii= \
   606 \    iii * number_of yb) = \
   607 \  (((number_of xa :: hcomplex) = 0) & \
   608 \   ((number_of ya :: hcomplex) = number_of yb))";
   609 by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iff3a,
   610     hcomplex_hypreal_number_of,hcomplex_of_hypreal_zero_iff]));
   611 qed "hcomplex_number_of_eq_cancel_iff3a";
   612 Addsimps [hcomplex_number_of_eq_cancel_iff3a];
   613 
   614 Goalw [hcomplex_number_of_def] "hcnj (number_of v :: hcomplex) = number_of v";
   615 by (rtac (hcomplex_hypreal_number_of RS ssubst) 1);
   616 by (rtac hcomplex_hcnj_hcomplex_of_hypreal 1);
   617 qed "hcomplex_number_of_hcnj";
   618 Addsimps [hcomplex_number_of_hcnj];
   619 
   620 Goalw [hcomplex_number_of_def] 
   621       "hcmod(number_of v :: hcomplex) = abs (number_of v :: hypreal)";
   622 by (rtac (hcomplex_hypreal_number_of RS ssubst) 1);
   623 by (auto_tac (claset(), HOL_ss addsimps [hcmod_hcomplex_of_hypreal]));
   624 qed "hcomplex_number_of_hcmod";
   625 Addsimps [hcomplex_number_of_hcmod];
   626 
   627 Goalw [hcomplex_number_of_def] 
   628       "hRe(number_of v :: hcomplex) = number_of v";
   629 by (rtac (hcomplex_hypreal_number_of RS ssubst) 1);
   630 by (auto_tac (claset(), HOL_ss addsimps [hRe_hcomplex_of_hypreal]));
   631 qed "hcomplex_number_of_hRe";
   632 Addsimps [hcomplex_number_of_hRe];
   633 
   634 Goalw [hcomplex_number_of_def] 
   635       "hIm(number_of v :: hcomplex) = 0";
   636 by (rtac (hcomplex_hypreal_number_of RS ssubst) 1);
   637 by (auto_tac (claset(), HOL_ss addsimps [hIm_hcomplex_of_hypreal]));
   638 qed "hcomplex_number_of_hIm";
   639 Addsimps [hcomplex_number_of_hIm];
   640 
   641 
   642