Added an exception handler and error msg.
(* Title: ComplexBin.ML
Author: Jacques D. Fleuriot
Copyright: 2001 University of Edinburgh
Descrition: Binary arithmetic for the complex numbers
*)
(** complex_of_real (coercion from real to complex) **)
Goal "complex_of_real (number_of w) = number_of w";
by (simp_tac (simpset() addsimps [complex_number_of_def]) 1);
qed "complex_number_of";
Addsimps [complex_number_of];
Goalw [complex_number_of_def] "Numeral0 = (0::complex)";
by (Simp_tac 1);
qed "complex_numeral_0_eq_0";
Goalw [complex_number_of_def] "Numeral1 = (1::complex)";
by (Simp_tac 1);
qed "complex_numeral_1_eq_1";
(** Addition **)
Goal "(number_of v :: complex) + number_of v' = number_of (bin_add v v')";
by (simp_tac
(HOL_ss addsimps [complex_number_of_def,
complex_of_real_add, add_real_number_of]) 1);
qed "add_complex_number_of";
Addsimps [add_complex_number_of];
(** Subtraction **)
Goalw [complex_number_of_def]
"- (number_of w :: complex) = number_of (bin_minus w)";
by (simp_tac
(HOL_ss addsimps [minus_real_number_of, complex_of_real_minus RS sym]) 1);
qed "minus_complex_number_of";
Addsimps [minus_complex_number_of];
Goalw [complex_number_of_def, complex_diff_def]
"(number_of v :: complex) - number_of w = number_of (bin_add v (bin_minus w))";
by (Simp_tac 1);
qed "diff_complex_number_of";
Addsimps [diff_complex_number_of];
(** Multiplication **)
Goal "(number_of v :: complex) * number_of v' = number_of (bin_mult v v')";
by (simp_tac
(HOL_ss addsimps [complex_number_of_def,
complex_of_real_mult, mult_real_number_of]) 1);
qed "mult_complex_number_of";
Addsimps [mult_complex_number_of];
Goal "(2::complex) = 1 + 1";
by (simp_tac (simpset() addsimps [complex_numeral_1_eq_1 RS sym]) 1);
val lemma = result();
(*For specialist use: NOT as default simprules*)
Goal "2 * z = (z+z::complex)";
by (simp_tac (simpset () addsimps [lemma, complex_add_mult_distrib]) 1);
qed "complex_mult_2";
Goal "z * 2 = (z+z::complex)";
by (stac complex_mult_commute 1 THEN rtac complex_mult_2 1);
qed "complex_mult_2_right";
(** Equals (=) **)
Goal "((number_of v :: complex) = number_of v') = \
\ iszero (number_of (bin_add v (bin_minus v')))";
by (simp_tac
(HOL_ss addsimps [complex_number_of_def,
complex_of_real_eq_iff, eq_real_number_of]) 1);
qed "eq_complex_number_of";
Addsimps [eq_complex_number_of];
(*** New versions of existing theorems involving 0, 1 ***)
Goal "- 1 = (-1::complex)";
by (simp_tac (simpset() addsimps [complex_numeral_1_eq_1 RS sym]) 1);
qed "complex_minus_1_eq_m1";
Goal "-1 * z = -(z::complex)";
by (simp_tac (simpset() addsimps [complex_minus_1_eq_m1 RS sym]) 1);
qed "complex_mult_minus1";
Goal "z * -1 = -(z::complex)";
by (stac complex_mult_commute 1 THEN rtac complex_mult_minus1 1);
qed "complex_mult_minus1_right";
Addsimps [complex_mult_minus1,complex_mult_minus1_right];
(*Maps 0 to Numeral0 and 1 to Numeral1 and -Numeral1 to -1*)
val complex_numeral_ss =
hypreal_numeral_ss addsimps [complex_numeral_0_eq_0 RS sym, complex_numeral_1_eq_1 RS sym,
complex_minus_1_eq_m1];
fun rename_numerals th =
asm_full_simplify complex_numeral_ss (Thm.transfer (the_context ()) th);
(*Now insert some identities previously stated for 0 and 1c*)
Addsimps [complex_numeral_0_eq_0,complex_numeral_1_eq_1];
Goal "number_of v + (number_of w + z) = (number_of(bin_add v w) + z::complex)";
by (auto_tac (claset(),simpset() addsimps [complex_add_assoc RS sym]));
qed "complex_add_number_of_left";
Goal "number_of v *(number_of w * z) = (number_of(bin_mult v w) * z::complex)";
by (simp_tac (simpset() addsimps [complex_mult_assoc RS sym]) 1);
qed "complex_mult_number_of_left";
Goalw [complex_diff_def]
"number_of v + (number_of w - c) = number_of(bin_add v w) - (c::complex)";
by (rtac complex_add_number_of_left 1);
qed "complex_add_number_of_diff1";
Goal "number_of v + (c - number_of w) = \
\ number_of (bin_add v (bin_minus w)) + (c::complex)";
by (auto_tac (claset(),simpset() addsimps [complex_diff_def]@ complex_add_ac));
qed "complex_add_number_of_diff2";
Addsimps [complex_add_number_of_left, complex_mult_number_of_left,
complex_add_number_of_diff1, complex_add_number_of_diff2];
(**** Simprocs for numeric literals ****)
(** Combining of literal coefficients in sums of products **)
Goal "(x = y) = (x-y = (0::complex))";
by (simp_tac (simpset() addsimps [complex_diff_eq_eq]) 1);
qed "complex_eq_iff_diff_eq_0";
(** For combine_numerals **)
Goal "i*u + (j*u + k) = (i+j)*u + (k::complex)";
by (asm_simp_tac (simpset() addsimps [complex_add_mult_distrib]
@ complex_add_ac) 1);
qed "left_complex_add_mult_distrib";
(** For cancel_numerals **)
Goal "((x::complex) = u + v) = (x - (u + v) = 0)";
by (auto_tac (claset(),simpset() addsimps [complex_diff_eq_eq]));
qed "complex_eq_add_diff_eq_0";
Goal "((x::complex) = n) = (x - n = 0)";
by (auto_tac (claset(),simpset() addsimps [complex_diff_eq_eq]));
qed "complex_eq_diff_eq_0";
val complex_rel_iff_rel_0_rls = [complex_eq_diff_eq_0,complex_eq_add_diff_eq_0];
Goal "!!i::complex. (i*u + m = j*u + n) = ((i-j)*u + m = n)";
by (auto_tac (claset(), simpset() addsimps [complex_add_mult_distrib,
complex_diff_def] @ complex_add_ac));
by (asm_simp_tac (simpset() addsimps [complex_add_assoc RS sym]) 1);
by (simp_tac (simpset() addsimps [complex_add_assoc]) 1);
qed "complex_eq_add_iff1";
Goal "!!i::complex. (i*u + m = j*u + n) = (m = (j-i)*u + n)";
by (simp_tac (simpset() addsimps [ complex_eq_add_iff1]) 1);
by (auto_tac (claset(), simpset() addsimps [complex_diff_def,
complex_add_mult_distrib]@ complex_add_ac));
qed "complex_eq_add_iff2";
structure Complex_Numeral_Simprocs =
struct
(*Maps 0 to Numeral0 and 1 to Numeral1 so that arithmetic in simprocs
isn't complicated by the abstract 0 and 1.*)
val numeral_syms = [complex_numeral_0_eq_0 RS sym, complex_numeral_1_eq_1 RS sym];
(*Utilities*)
val complexT = Type("Complex.complex",[]);
fun mk_numeral n = HOLogic.number_of_const complexT $ HOLogic.mk_bin n;
val dest_numeral = Real_Numeral_Simprocs.dest_numeral;
val find_first_numeral = Real_Numeral_Simprocs.find_first_numeral;
val zero = mk_numeral 0;
val mk_plus = HOLogic.mk_binop "op +";
val uminus_const = Const ("uminus", complexT --> complexT);
(*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 +" complexT;
(*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 -" complexT;
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 *" complexT;
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 [complex_add_zero_left, complex_add_zero_right];
val mult_plus_1s = map rename_numerals [complex_mult_one_left, complex_mult_one_right];
val mult_minus_1s = map rename_numerals
[complex_mult_minus1, complex_mult_minus1_right];
val mult_1s = mult_plus_1s @ mult_minus_1s;
(*To perform binary arithmetic*)
val bin_simps =
[complex_numeral_0_eq_0 RS sym, complex_numeral_1_eq_1 RS sym,
add_complex_number_of, complex_add_number_of_left,
minus_complex_number_of, diff_complex_number_of, mult_complex_number_of,
complex_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 \\ [complex_add_number_of_left, add_complex_number_of];
(*To evaluate binary negations of coefficients*)
val complex_minus_simps = NCons_simps @
[complex_minus_1_eq_m1,minus_complex_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 = [complex_diff_def, complex_minus_add_distrib,
complex_minus_minus];
(* push the unary minus down: - x * y = x * - y *)
val complex_minus_mult_eq_1_to_2 =
[complex_minus_mult_eq1 RS sym, complex_minus_mult_eq2] MRS trans
|> standard;
(*to extract again any uncancelled minuses*)
val complex_minus_from_mult_simps =
[complex_minus_minus, complex_minus_mult_eq1 RS sym,
complex_minus_mult_eq2 RS sym];
(*combine unary minus with numeric literals, however nested within a product*)
val complex_mult_minus_simps =
[complex_mult_assoc, complex_minus_mult_eq1, complex_minus_mult_eq_1_to_2];
(*Final simplification: cancel + and * *)
val simplify_meta_eq =
Int_Numeral_Simprocs.simplify_meta_eq
[complex_add_zero_left, complex_add_zero_right,
complex_mult_zero_left, complex_mult_zero_right, complex_mult_one_left,
complex_mult_one_right];
val prep_simproc = Real_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@
complex_minus_simps@complex_add_ac))
THEN ALLGOALS (simp_tac (HOL_ss addsimps non_add_bin_simps@complex_mult_minus_simps))
THEN ALLGOALS
(simp_tac (HOL_ss addsimps complex_minus_from_mult_simps@
complex_add_ac@complex_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 =" complexT
val bal_add1 = complex_eq_add_iff1 RS trans
val bal_add2 = complex_eq_add_iff2 RS trans
);
val cancel_numerals =
map prep_simproc
[("complexeq_cancel_numerals",
["(l::complex) + m = n", "(l::complex) = m + n",
"(l::complex) - m = n", "(l::complex) = m - n",
"(l::complex) * m = n", "(l::complex) = 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_complex_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@
complex_minus_simps@complex_add_ac))
THEN ALLGOALS (simp_tac (HOL_ss addsimps non_add_bin_simps@complex_mult_minus_simps))
THEN ALLGOALS (simp_tac (HOL_ss addsimps complex_minus_from_mult_simps@
complex_add_ac@complex_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 ("complex_combine_numerals",
["(i::complex) + j", "(i::complex) - 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
[complex_mult_one_left, complex_mult_one_right]
(([th, cancel_th]) MRS trans);
(*** Making constant folding work for 0 and 1 too ***)
structure ComplexAbstractNumeralsData =
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 = complex_numeral_0_eq_0
val numeral_1_eq_1 = complex_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 ComplexAbstractNumerals = AbstractNumeralsFun (ComplexAbstractNumeralsData)
(*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
[("complex_add_eval_numerals",
["(m::complex) + 1", "(m::complex) + number_of v"],
ComplexAbstractNumerals.proc add_complex_number_of),
("complex_diff_eval_numerals",
["(m::complex) - 1", "(m::complex) - number_of v"],
ComplexAbstractNumerals.proc diff_complex_number_of),
("complex_eq_eval_numerals",
["(m::complex) = 0", "(m::complex) = 1", "(m::complex) = number_of v"],
ComplexAbstractNumerals.proc eq_complex_number_of)]
end;
Addsimprocs Complex_Numeral_Simprocs.eval_numerals;
Addsimprocs Complex_Numeral_Simprocs.cancel_numerals;
Addsimprocs [Complex_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::complex)";
test " 2*u = (u::complex)";
test "(i + j + 12 + (k::complex)) - 15 = y";
test "(i + j + 12 + (k::complex)) - 5 = y";
test "( 2*x - (u*v) + y) - v* 3*u = (w::complex)";
test "( 2*x*u*v + (u*v)* 4 + y) - v*u* 4 = (w::complex)";
test "( 2*x*u*v + (u*v)* 4 + y) - v*u = (w::complex)";
test "u*v - (x*u*v + (u*v)* 4 + y) = (w::complex)";
test "(i + j + 12 + (k::complex)) = u + 15 + y";
test "(i + j* 2 + 12 + (k::complex)) = 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::complex)";
test "a + -(b+c) + b = (d::complex)";
test "a + -(b+c) - b = (d::complex)";
(*negative numerals*)
test "(i + j + -2 + (k::complex)) - (u + 5 + y) = zz";
test "(i + j + -12 + (k::complex)) - 15 = y";
test "(i + j + 12 + (k::complex)) - -15 = y";
test "(i + j + -12 + (k::complex)) - -15 = y";
*)
(** Constant folding for complex plus and times **)
structure Complex_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 = Complex_Numeral_Simprocs.complexT
val plus = Const ("op *", [T,T] ---> T)
val add_ac = complex_mult_ac
end;
structure Complex_Times_Assoc = Assoc_Fold (Complex_Times_Assoc_Data);
Addsimprocs [Complex_Times_Assoc.conv];
Addsimps [complex_of_real_zero_iff];
(*Simplification of x-y = 0 *)
AddIffs [complex_eq_iff_diff_eq_0 RS sym];
(*** Real and imaginary stuff ***)
Goalw [complex_number_of_def]
"((number_of xa :: complex) + ii * number_of ya = \
\ number_of xb + ii * number_of yb) = \
\ (((number_of xa :: complex) = number_of xb) & \
\ ((number_of ya :: complex) = number_of yb))";
by (auto_tac (claset(), HOL_ss addsimps [complex_eq_cancel_iff]));
qed "complex_number_of_eq_cancel_iff";
Addsimps [complex_number_of_eq_cancel_iff];
Goalw [complex_number_of_def]
"((number_of xa :: complex) + number_of ya * ii = \
\ number_of xb + number_of yb * ii) = \
\ (((number_of xa :: complex) = number_of xb) & \
\ ((number_of ya :: complex) = number_of yb))";
by (auto_tac (claset(), HOL_ss addsimps [complex_eq_cancel_iffA]));
qed "complex_number_of_eq_cancel_iffA";
Addsimps [complex_number_of_eq_cancel_iffA];
Goalw [complex_number_of_def]
"((number_of xa :: complex) + number_of ya * ii = \
\ number_of xb + ii * number_of yb) = \
\ (((number_of xa :: complex) = number_of xb) & \
\ ((number_of ya :: complex) = number_of yb))";
by (auto_tac (claset(), HOL_ss addsimps [complex_eq_cancel_iffB]));
qed "complex_number_of_eq_cancel_iffB";
Addsimps [complex_number_of_eq_cancel_iffB];
Goalw [complex_number_of_def]
"((number_of xa :: complex) + ii * number_of ya = \
\ number_of xb + number_of yb * ii) = \
\ (((number_of xa :: complex) = number_of xb) & \
\ ((number_of ya :: complex) = number_of yb))";
by (auto_tac (claset(), HOL_ss addsimps [complex_eq_cancel_iffC]));
qed "complex_number_of_eq_cancel_iffC";
Addsimps [complex_number_of_eq_cancel_iffC];
Goalw [complex_number_of_def]
"((number_of xa :: complex) + ii * number_of ya = \
\ number_of xb) = \
\ (((number_of xa :: complex) = number_of xb) & \
\ ((number_of ya :: complex) = 0))";
by (auto_tac (claset(), HOL_ss addsimps [complex_eq_cancel_iff2,
complex_of_real_zero_iff]));
qed "complex_number_of_eq_cancel_iff2";
Addsimps [complex_number_of_eq_cancel_iff2];
Goalw [complex_number_of_def]
"((number_of xa :: complex) + number_of ya * ii = \
\ number_of xb) = \
\ (((number_of xa :: complex) = number_of xb) & \
\ ((number_of ya :: complex) = 0))";
by (auto_tac (claset(), HOL_ss addsimps [complex_eq_cancel_iff2a,
complex_of_real_zero_iff]));
qed "complex_number_of_eq_cancel_iff2a";
Addsimps [complex_number_of_eq_cancel_iff2a];
Goalw [complex_number_of_def]
"((number_of xa :: complex) + ii * number_of ya = \
\ ii * number_of yb) = \
\ (((number_of xa :: complex) = 0) & \
\ ((number_of ya :: complex) = number_of yb))";
by (auto_tac (claset(), HOL_ss addsimps [complex_eq_cancel_iff3,
complex_of_real_zero_iff]));
qed "complex_number_of_eq_cancel_iff3";
Addsimps [complex_number_of_eq_cancel_iff3];
Goalw [complex_number_of_def]
"((number_of xa :: complex) + number_of ya * ii= \
\ ii * number_of yb) = \
\ (((number_of xa :: complex) = 0) & \
\ ((number_of ya :: complex) = number_of yb))";
by (auto_tac (claset(), HOL_ss addsimps [complex_eq_cancel_iff3a,
complex_of_real_zero_iff]));
qed "complex_number_of_eq_cancel_iff3a";
Addsimps [complex_number_of_eq_cancel_iff3a];
Goalw [complex_number_of_def] "cnj (number_of v :: complex) = number_of v";
by (rtac complex_cnj_complex_of_real 1);
qed "complex_number_of_cnj";
Addsimps [complex_number_of_cnj];
Goalw [complex_number_of_def]
"cmod(number_of v :: complex) = abs (number_of v :: real)";
by (auto_tac (claset(), HOL_ss addsimps [complex_mod_complex_of_real]));
qed "complex_number_of_cmod";
Addsimps [complex_number_of_cmod];
Goalw [complex_number_of_def]
"Re(number_of v :: complex) = number_of v";
by (auto_tac (claset(), HOL_ss addsimps [Re_complex_of_real]));
qed "complex_number_of_Re";
Addsimps [complex_number_of_Re];
Goalw [complex_number_of_def]
"Im(number_of v :: complex) = 0";
by (auto_tac (claset(), HOL_ss addsimps [Im_complex_of_real]));
qed "complex_number_of_Im";
Addsimps [complex_number_of_Im];
Goalw [expi_def]
"expi((2::complex) * complex_of_real pi * ii) = 1";
by (auto_tac (claset(),simpset() addsimps [complex_Re_mult_eq,
complex_Im_mult_eq,cis_def]));
qed "expi_two_pi_i";
Addsimps [expi_two_pi_i];
(*examples:
print_depth 22;
set timing;
set trace_simp;
fun test s = (Goal s, by (Simp_tac 1));
test "23 * ii + 45 * ii= (x::complex)";
test "5 * ii + 12 - 45 * ii= (x::complex)";
test "5 * ii + 40 - 12 * ii + 9 = (x::complex) + 89 * ii";
test "5 * ii + 40 - 12 * ii + 9 - 78 = (x::complex) + 89 * ii";
test "l + 10 * ii + 90 + 3*l + 9 + 45 * ii= (x::complex)";
test "87 + 10 * ii + 90 + 3*7 + 9 + 45 * ii= (x::complex)";
*)