(* 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 (simpset() addsimps [hcomplex_of_complex_zero RS sym]) 1);
qed "hcomplex_numeral_0_eq_0";
Goalw [hcomplex_number_of_def] "Numeral1 = (1::hcomplex)";
by (simp_tac (simpset() addsimps [hcomplex_of_complex_one RS sym]) 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')))";
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]@ hcomplex_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 ****)
(** Combining of literal coefficients in sums of products **)
Goal "(x = y) = (x-y = (0::hcomplex))";
by (simp_tac (simpset() addsimps [hcomplex_diff_eq_eq]) 1);
qed "hcomplex_eq_iff_diff_eq_0";
(** For combine_numerals **)
Goal "i*u + (j*u + k) = (i+j)*u + (k::hcomplex)";
by (asm_simp_tac (simpset() addsimps [hcomplex_add_mult_distrib]
@ hcomplex_add_ac) 1);
qed "left_hcomplex_add_mult_distrib";
(** For cancel_numerals **)
Goal "((x::hcomplex) = u + v) = (x - (u + v) = 0)";
by (auto_tac (claset(),simpset() addsimps [hcomplex_diff_eq_eq]));
qed "hcomplex_eq_add_diff_eq_0";
Goal "((x::hcomplex) = n) = (x - n = 0)";
by (auto_tac (claset(),simpset() addsimps [hcomplex_diff_eq_eq]));
qed "hcomplex_eq_diff_eq_0";
val hcomplex_rel_iff_rel_0_rls = [hcomplex_eq_diff_eq_0,hcomplex_eq_add_diff_eq_0];
Goal "!!i::hcomplex. (i*u + m = j*u + n) = ((i-j)*u + m = n)";
by (auto_tac (claset(), simpset() addsimps [hcomplex_add_mult_distrib,
hcomplex_diff_def] @ hcomplex_add_ac));
by (asm_simp_tac (simpset() addsimps [hcomplex_add_assoc RS sym]) 1);
by (simp_tac (simpset() addsimps [hcomplex_add_assoc]) 1);
qed "hcomplex_eq_add_iff1";
Goal "!!i::hcomplex. (i*u + m = j*u + n) = (m = (j-i)*u + n)";
by (res_inst_tac [("z","i")] eq_Abs_hcomplex 1);
by (res_inst_tac [("z","j")] eq_Abs_hcomplex 1);
by (res_inst_tac [("z","u")] eq_Abs_hcomplex 1);
by (res_inst_tac [("z","m")] eq_Abs_hcomplex 1);
by (res_inst_tac [("z","n")] eq_Abs_hcomplex 1);
by (auto_tac (claset(), simpset() addsimps [hcomplex_diff,hcomplex_add,
hcomplex_mult,complex_eq_add_iff2]));
qed "hcomplex_eq_add_iff2";
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, hcomplex_minus_add_distrib,
minus_minus];
(*push the unary minus down: - x * y = x * - y *)
val hcomplex_minus_mult_eq_1_to_2 =
[hcomplex_minus_mult_eq1 RS sym, hcomplex_minus_mult_eq2] MRS trans
|> standard;
(*to extract again any uncancelled minuses*)
val hcomplex_minus_from_mult_simps =
[minus_minus, hcomplex_minus_mult_eq1 RS sym,
hcomplex_minus_mult_eq2 RS sym];
(*combine unary minus with numeric literals, however nested within a product*)
val hcomplex_mult_minus_simps =
[hcomplex_mult_assoc, hcomplex_minus_mult_eq1, hcomplex_minus_mult_eq_1_to_2];
(*Final simplification: cancel + and * *)
val simplify_meta_eq =
Int_Numeral_Simprocs.simplify_meta_eq
[hcomplex_add_zero_left, hcomplex_add_zero_right,
hcomplex_mult_zero_left, hcomplex_mult_zero_right, hcomplex_mult_one_left,
hcomplex_mult_one_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@hcomplex_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@
hcomplex_add_ac@hcomplex_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 = hcomplex_eq_add_iff1 RS trans
val bal_add2 = hcomplex_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 = left_hcomplex_add_mult_distrib 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@hcomplex_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@
hcomplex_add_ac@hcomplex_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 = hcomplex_mult_ac
end;
structure HComplex_Times_Assoc = Assoc_Fold (HComplex_Times_Assoc_Data);
Addsimprocs [HComplex_Times_Assoc.conv];
Addsimps [hcomplex_of_complex_zero_iff];
(*Simplification of x-y = 0 *)
AddIffs [hcomplex_eq_iff_diff_eq_0 RS sym];
(** 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 ***)
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];