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(* ========================================================================= *) | |
(* Basic definitions and properties of complex numbers. *) | |
(* ========================================================================= *) | |
needs "Library/transc.ml";; | |
prioritize_real();; | |
(* ------------------------------------------------------------------------- *) | |
(* Definition of complex number type. *) | |
(* ------------------------------------------------------------------------- *) | |
let complex_tybij_raw = | |
new_type_definition "complex" ("complex","coords") | |
(prove(`?x:real#real. T`,REWRITE_TAC[]));; | |
let complex_tybij = REWRITE_RULE [] complex_tybij_raw;; | |
(* ------------------------------------------------------------------------- *) | |
(* Real and imaginary parts of a number. *) | |
(* ------------------------------------------------------------------------- *) | |
let RE_DEF = new_definition | |
`Re(z) = FST(coords(z))`;; | |
let IM_DEF = new_definition | |
`Im(z) = SND(coords(z))`;; | |
(* ------------------------------------------------------------------------- *) | |
(* Set up overloading. *) | |
(* ------------------------------------------------------------------------- *) | |
do_list overload_interface | |
["+",`complex_add:complex->complex->complex`; | |
"-",`complex_sub:complex->complex->complex`; | |
"*",`complex_mul:complex->complex->complex`; | |
"/",`complex_div:complex->complex->complex`; | |
"--",`complex_neg:complex->complex`; | |
"pow",`complex_pow:complex->num->complex`; | |
"inv",`complex_inv:complex->complex`];; | |
let prioritize_complex() = prioritize_overload(mk_type("complex",[]));; | |
(* ------------------------------------------------------------------------- *) | |
(* Complex absolute value (modulus). *) | |
(* ------------------------------------------------------------------------- *) | |
make_overloadable "norm" `:A->real`;; | |
overload_interface("norm",`complex_norm:complex->real`);; | |
let complex_norm = new_definition | |
`norm(z) = sqrt(Re(z) pow 2 + Im(z) pow 2)`;; | |
(* ------------------------------------------------------------------------- *) | |
(* Imaginary unit (too inconvenient to use "i"!) *) | |
(* ------------------------------------------------------------------------- *) | |
let ii = new_definition | |
`ii = complex(&0,&1)`;; | |
(* ------------------------------------------------------------------------- *) | |
(* Injection from reals. *) | |
(* ------------------------------------------------------------------------- *) | |
let CX_DEF = new_definition | |
`Cx(a) = complex(a,&0)`;; | |
(* ------------------------------------------------------------------------- *) | |
(* Arithmetic operations. *) | |
(* ------------------------------------------------------------------------- *) | |
let complex_neg = new_definition | |
`--z = complex(--(Re(z)),--(Im(z)))`;; | |
let complex_add = new_definition | |
`w + z = complex(Re(w) + Re(z),Im(w) + Im(z))`;; | |
let complex_sub = new_definition | |
`w - z = w + --z`;; | |
let complex_mul = new_definition | |
`w * z = complex(Re(w) * Re(z) - Im(w) * Im(z), | |
Re(w) * Im(z) + Im(w) * Re(z))`;; | |
let complex_inv = new_definition | |
`inv(z) = complex(Re(z) / (Re(z) pow 2 + Im(z) pow 2), | |
--(Im(z)) / (Re(z) pow 2 + Im(z) pow 2))`;; | |
let complex_div = new_definition | |
`w / z = w * inv(z)`;; | |
let complex_pow = new_recursive_definition num_RECURSION | |
`(x pow 0 = Cx(&1)) /\ | |
(!n. x pow (SUC n) = x * x pow n)`;; | |
(* ------------------------------------------------------------------------- *) | |
(* Various handy rewrites. *) | |
(* ------------------------------------------------------------------------- *) | |
let RE = prove | |
(`(Re(complex(x,y)) = x)`, | |
REWRITE_TAC[RE_DEF; complex_tybij]);; | |
let IM = prove | |
(`Im(complex(x,y)) = y`, | |
REWRITE_TAC[IM_DEF; complex_tybij]);; | |
let COMPLEX = prove | |
(`complex(Re(z),Im(z)) = z`, | |
REWRITE_TAC[IM_DEF; RE_DEF; complex_tybij]);; | |
let COMPLEX_EQ = prove | |
(`!w z. (w = z) <=> (Re(w) = Re(z)) /\ (Im(w) = Im(z))`, | |
REWRITE_TAC[RE_DEF; IM_DEF; GSYM PAIR_EQ] THEN MESON_TAC[complex_tybij]);; | |
(* ------------------------------------------------------------------------- *) | |
(* Crude tactic to automate very simple algebraic equivalences. *) | |
(* ------------------------------------------------------------------------- *) | |
let SIMPLE_COMPLEX_ARITH_TAC = | |
REWRITE_TAC[COMPLEX_EQ; RE; IM; CX_DEF; | |
complex_add; complex_neg; complex_sub; complex_mul] THEN | |
REAL_ARITH_TAC;; | |
let SIMPLE_COMPLEX_ARITH tm = prove(tm,SIMPLE_COMPLEX_ARITH_TAC);; | |
(* ------------------------------------------------------------------------- *) | |
(* Basic algebraic properties that can be proved automatically by this. *) | |
(* ------------------------------------------------------------------------- *) | |
let COMPLEX_ADD_SYM = prove | |
(`!x y. x + y = y + x`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_ADD_ASSOC = prove | |
(`!x y z. x + y + z = (x + y) + z`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_ADD_LID = prove | |
(`!x. Cx(&0) + x = x`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_ADD_LINV = prove | |
(`!x. --x + x = Cx(&0)`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_MUL_SYM = prove | |
(`!x y. x * y = y * x`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_MUL_ASSOC = prove | |
(`!x y z. x * y * z = (x * y) * z`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_MUL_LID = prove | |
(`!x. Cx(&1) * x = x`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_ADD_LDISTRIB = prove | |
(`!x y z. x * (y + z) = x * y + x * z`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_ADD_AC = prove | |
(`(m + n = n + m) /\ ((m + n) + p = m + n + p) /\ (m + n + p = n + m + p)`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_MUL_AC = prove | |
(`(m * n = n * m) /\ ((m * n) * p = m * n * p) /\ (m * n * p = n * m * p)`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_ADD_RID = prove | |
(`!x. x + Cx(&0) = x`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_MUL_RID = prove | |
(`!x. x * Cx(&1) = x`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_ADD_RINV = prove | |
(`!x. x + --x = Cx(&0)`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_ADD_RDISTRIB = prove | |
(`!x y z. (x + y) * z = x * z + y * z`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_EQ_ADD_LCANCEL = prove | |
(`!x y z. (x + y = x + z) <=> (y = z)`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_EQ_ADD_RCANCEL = prove | |
(`!x y z. (x + z = y + z) <=> (x = y)`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_MUL_RZERO = prove | |
(`!x. x * Cx(&0) = Cx(&0)`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_MUL_LZERO = prove | |
(`!x. Cx(&0) * x = Cx(&0)`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_NEG_NEG = prove | |
(`!x. --(--x) = x`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_MUL_RNEG = prove | |
(`!x y. x * --y = --(x * y)`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_MUL_LNEG = prove | |
(`!x y. --x * y = --(x * y)`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_NEG_ADD = prove | |
(`!x y. --(x + y) = --x + --y`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_NEG_0 = prove | |
(`--Cx(&0) = Cx(&0)`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_EQ_ADD_LCANCEL_0 = prove | |
(`!x y. (x + y = x) <=> (y = Cx(&0))`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_EQ_ADD_RCANCEL_0 = prove | |
(`!x y. (x + y = y) <=> (x = Cx(&0))`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_LNEG_UNIQ = prove | |
(`!x y. (x + y = Cx(&0)) <=> (x = --y)`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_RNEG_UNIQ = prove | |
(`!x y. (x + y = Cx(&0)) <=> (y = --x)`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_NEG_LMUL = prove | |
(`!x y. --(x * y) = --x * y`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_NEG_RMUL = prove | |
(`!x y. --(x * y) = x * --y`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_NEG_MUL2 = prove | |
(`!x y. --x * --y = x * y`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_SUB_ADD = prove | |
(`!x y. x - y + y = x`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_SUB_ADD2 = prove | |
(`!x y. y + x - y = x`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_SUB_REFL = prove | |
(`!x. x - x = Cx(&0)`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_SUB_0 = prove | |
(`!x y. (x - y = Cx(&0)) <=> (x = y)`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_NEG_EQ_0 = prove | |
(`!x. (--x = Cx(&0)) <=> (x = Cx(&0))`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_NEG_SUB = prove | |
(`!x y. --(x - y) = y - x`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_ADD_SUB = prove | |
(`!x y. (x + y) - x = y`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_NEG_EQ = prove | |
(`!x y. (--x = y) <=> (x = --y)`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_NEG_MINUS1 = prove | |
(`!x. --x = --Cx(&1) * x`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_SUB_SUB = prove | |
(`!x y. x - y - x = --y`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_ADD2_SUB2 = prove | |
(`!a b c d. (a + b) - (c + d) = a - c + b - d`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_SUB_LZERO = prove | |
(`!x. Cx(&0) - x = --x`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_SUB_RZERO = prove | |
(`!x. x - Cx(&0) = x`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_SUB_LNEG = prove | |
(`!x y. --x - y = --(x + y)`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_SUB_RNEG = prove | |
(`!x y. x - --y = x + y`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_SUB_NEG2 = prove | |
(`!x y. --x - --y = y - x`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_SUB_TRIANGLE = prove | |
(`!a b c. a - b + b - c = a - c`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_EQ_SUB_LADD = prove | |
(`!x y z. (x = y - z) <=> (x + z = y)`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_EQ_SUB_RADD = prove | |
(`!x y z. (x - y = z) <=> (x = z + y)`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_SUB_SUB2 = prove | |
(`!x y. x - (x - y) = y`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_ADD_SUB2 = prove | |
(`!x y. x - (x + y) = --y`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_DIFFSQ = prove | |
(`!x y. (x + y) * (x - y) = x * x - y * y`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_EQ_NEG2 = prove | |
(`!x y. (--x = --y) <=> (x = y)`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_SUB_LDISTRIB = prove | |
(`!x y z. x * (y - z) = x * y - x * z`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_SUB_RDISTRIB = prove | |
(`!x y z. (x - y) * z = x * z - y * z`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_MUL_2 = prove | |
(`!x. &2 * x = x + x`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
(* ------------------------------------------------------------------------- *) | |
(* Homomorphic embedding properties for Cx mapping. *) | |
(* ------------------------------------------------------------------------- *) | |
let CX_INJ = prove | |
(`!x y. (Cx(x) = Cx(y)) <=> (x = y)`, | |
REWRITE_TAC[CX_DEF; COMPLEX_EQ; RE; IM]);; | |
let CX_NEG = prove | |
(`!x. Cx(--x) = --(Cx(x))`, | |
REWRITE_TAC[CX_DEF; complex_neg; RE; IM; REAL_NEG_0]);; | |
let CX_INV = prove | |
(`!x. Cx(inv x) = inv(Cx x)`, | |
GEN_TAC THEN | |
REWRITE_TAC[CX_DEF; complex_inv; RE; IM] THEN | |
REWRITE_TAC[real_div; REAL_NEG_0; REAL_MUL_LZERO] THEN | |
REWRITE_TAC[COMPLEX_EQ; REAL_POW_2; REAL_MUL_RZERO; RE; IM] THEN | |
REWRITE_TAC[REAL_ADD_RID; REAL_INV_MUL] THEN | |
ASM_CASES_TAC `x = &0` THEN | |
ASM_REWRITE_TAC[REAL_INV_0; REAL_MUL_LZERO] THEN | |
REWRITE_TAC[REAL_MUL_ASSOC] THEN | |
GEN_REWRITE_TAC LAND_CONV [GSYM REAL_MUL_LID] THEN | |
AP_THM_TAC THEN AP_TERM_TAC THEN ASM_MESON_TAC[REAL_MUL_RINV]);; | |
let CX_ADD = prove | |
(`!x y. Cx(x + y) = Cx(x) + Cx(y)`, | |
REWRITE_TAC[CX_DEF; complex_add; RE; IM; REAL_ADD_LID]);; | |
let CX_SUB = prove | |
(`!x y. Cx(x - y) = Cx(x) - Cx(y)`, | |
REWRITE_TAC[complex_sub; real_sub; CX_ADD; CX_NEG]);; | |
let CX_MUL = prove | |
(`!x y. Cx(x * y) = Cx(x) * Cx(y)`, | |
REWRITE_TAC[CX_DEF; complex_mul; RE; IM; REAL_MUL_LZERO; REAL_MUL_RZERO] THEN | |
REWRITE_TAC[REAL_SUB_RZERO; REAL_ADD_RID]);; | |
let CX_DIV = prove | |
(`!x y. Cx(x / y) = Cx(x) / Cx(y)`, | |
REWRITE_TAC[complex_div; real_div; CX_MUL; CX_INV]);; | |
let CX_POW = prove | |
(`!x n. Cx(x pow n) = Cx(x) pow n`, | |
GEN_TAC THEN INDUCT_TAC THEN | |
ASM_REWRITE_TAC[complex_pow; real_pow; CX_MUL]);; | |
let CX_ABS = prove | |
(`!x. Cx(abs x) = Cx(norm(Cx(x)))`, | |
REWRITE_TAC[CX_DEF; complex_norm; COMPLEX_EQ; RE; IM] THEN | |
REWRITE_TAC[REAL_POW_2; REAL_MUL_LZERO; REAL_ADD_RID] THEN | |
REWRITE_TAC[GSYM REAL_POW_2; POW_2_SQRT_ABS]);; | |
let COMPLEX_NORM_CX = prove | |
(`!x. norm(Cx(x)) = abs(x)`, | |
REWRITE_TAC[GSYM CX_INJ; CX_ABS]);; | |
(* ------------------------------------------------------------------------- *) | |
(* A convenient lemma that we need a few times below. *) | |
(* ------------------------------------------------------------------------- *) | |
let COMPLEX_ENTIRE = prove | |
(`!x y. (x * y = Cx(&0)) <=> (x = Cx(&0)) \/ (y = Cx(&0))`, | |
REWRITE_TAC[COMPLEX_EQ; complex_mul; RE; IM; CX_DEF; GSYM REAL_SOS_EQ_0] THEN | |
CONV_TAC REAL_RING);; | |
(* ------------------------------------------------------------------------- *) | |
(* Powers. *) | |
(* ------------------------------------------------------------------------- *) | |
let COMPLEX_POW_ADD = prove | |
(`!x m n. x pow (m + n) = x pow m * x pow n`, | |
GEN_TAC THEN INDUCT_TAC THEN | |
ASM_REWRITE_TAC[ADD_CLAUSES; complex_pow; | |
COMPLEX_MUL_LID; COMPLEX_MUL_ASSOC]);; | |
let COMPLEX_POW_POW = prove | |
(`!x m n. (x pow m) pow n = x pow (m * n)`, | |
GEN_TAC THEN GEN_TAC THEN INDUCT_TAC THEN | |
ASM_REWRITE_TAC[complex_pow; MULT_CLAUSES; COMPLEX_POW_ADD]);; | |
let COMPLEX_POW_1 = prove | |
(`!x. x pow 1 = x`, | |
REWRITE_TAC[num_CONV `1`] THEN REWRITE_TAC[complex_pow; COMPLEX_MUL_RID]);; | |
let COMPLEX_POW_2 = prove | |
(`!x. x pow 2 = x * x`, | |
REWRITE_TAC[num_CONV `2`] THEN REWRITE_TAC[complex_pow; COMPLEX_POW_1]);; | |
let COMPLEX_POW_NEG = prove | |
(`!x n. (--x) pow n = if EVEN n then x pow n else --(x pow n)`, | |
GEN_TAC THEN INDUCT_TAC THEN | |
ASM_REWRITE_TAC[complex_pow; EVEN] THEN | |
ASM_CASES_TAC `EVEN n` THEN | |
ASM_REWRITE_TAC[COMPLEX_MUL_RNEG; COMPLEX_MUL_LNEG; COMPLEX_NEG_NEG]);; | |
let COMPLEX_POW_ONE = prove | |
(`!n. Cx(&1) pow n = Cx(&1)`, | |
INDUCT_TAC THEN ASM_REWRITE_TAC[complex_pow; COMPLEX_MUL_LID]);; | |
let COMPLEX_POW_MUL = prove | |
(`!x y n. (x * y) pow n = (x pow n) * (y pow n)`, | |
GEN_TAC THEN GEN_TAC THEN INDUCT_TAC THEN | |
ASM_REWRITE_TAC[complex_pow; COMPLEX_MUL_LID; COMPLEX_MUL_AC]);; | |
let COMPLEX_POW_II_2 = prove | |
(`ii pow 2 = --Cx(&1)`, | |
REWRITE_TAC[ii; COMPLEX_POW_2; complex_mul; CX_DEF; RE; IM; complex_neg] THEN | |
CONV_TAC REAL_RAT_REDUCE_CONV);; | |
let COMPLEX_POW_EQ_0 = prove | |
(`!x n. (x pow n = Cx(&0)) <=> (x = Cx(&0)) /\ ~(n = 0)`, | |
GEN_TAC THEN INDUCT_TAC THEN | |
ASM_REWRITE_TAC[NOT_SUC; complex_pow; COMPLEX_ENTIRE] THENL | |
[SIMPLE_COMPLEX_ARITH_TAC; CONV_TAC TAUT]);; | |
(* ------------------------------------------------------------------------- *) | |
(* Norms (aka "moduli"). *) | |
(* ------------------------------------------------------------------------- *) | |
let COMPLEX_NORM_CX = prove | |
(`!x. norm(Cx x) = abs(x)`, | |
GEN_TAC THEN REWRITE_TAC[complex_norm; CX_DEF; RE; IM] THEN | |
REWRITE_TAC[REAL_POW_2; REAL_MUL_LZERO; REAL_ADD_RID] THEN | |
REWRITE_TAC[GSYM REAL_POW_2; POW_2_SQRT_ABS]);; | |
let COMPLEX_NORM_POS = prove | |
(`!z. &0 <= norm(z)`, | |
SIMP_TAC[complex_norm; SQRT_POS_LE; REAL_POW_2; | |
REAL_LE_SQUARE; REAL_LE_ADD]);; | |
let COMPLEX_ABS_NORM = prove | |
(`!z. abs(norm z) = norm z`, | |
REWRITE_TAC[real_abs; COMPLEX_NORM_POS]);; | |
let COMPLEX_NORM_ZERO = prove | |
(`!z. (norm z = &0) <=> (z = Cx(&0))`, | |
GEN_TAC THEN REWRITE_TAC[complex_norm] THEN | |
GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [GSYM SQRT_0] THEN | |
SIMP_TAC[REAL_POW_2; REAL_LE_SQUARE; REAL_LE_ADD; REAL_POS; SQRT_INJ] THEN | |
REWRITE_TAC[COMPLEX_EQ; RE; IM; CX_DEF] THEN | |
SIMP_TAC[REAL_LE_SQUARE; REAL_ARITH | |
`&0 <= x /\ &0 <= y ==> ((x + y = &0) <=> (x = &0) /\ (y = &0))`] THEN | |
REWRITE_TAC[REAL_ENTIRE]);; | |
let COMPLEX_NORM_NUM = prove | |
(`norm(Cx(&n)) = &n`, | |
REWRITE_TAC[COMPLEX_NORM_CX; REAL_ABS_NUM]);; | |
let COMPLEX_NORM_0 = prove | |
(`norm(Cx(&0)) = &0`, | |
MESON_TAC[COMPLEX_NORM_ZERO]);; | |
let COMPLEX_NORM_NZ = prove | |
(`!z. &0 < norm(z) <=> ~(z = Cx(&0))`, | |
MESON_TAC[COMPLEX_NORM_ZERO; COMPLEX_ABS_NORM; REAL_ABS_NZ]);; | |
let COMPLEX_NORM_NEG = prove | |
(`!z. norm(--z) = norm(z)`, | |
REWRITE_TAC[complex_neg; complex_norm; REAL_POW_2; RE; IM] THEN | |
GEN_TAC THEN AP_TERM_TAC THEN REAL_ARITH_TAC);; | |
let COMPLEX_NORM_MUL = prove | |
(`!w z. norm(w * z) = norm(w) * norm(z)`, | |
REPEAT GEN_TAC THEN | |
REWRITE_TAC[complex_norm; complex_mul; RE; IM] THEN | |
SIMP_TAC[GSYM SQRT_MUL; REAL_POW_2; REAL_LE_ADD; REAL_LE_SQUARE] THEN | |
AP_TERM_TAC THEN REAL_ARITH_TAC);; | |
let COMPLEX_NORM_POW = prove | |
(`!z n. norm(z pow n) = norm(z) pow n`, | |
GEN_TAC THEN INDUCT_TAC THEN | |
ASM_REWRITE_TAC[complex_pow; real_pow; COMPLEX_NORM_NUM; COMPLEX_NORM_MUL]);; | |
let COMPLEX_NORM_INV = prove | |
(`!z. norm(inv z) = inv(norm z)`, | |
GEN_TAC THEN REWRITE_TAC[complex_norm; complex_inv; RE; IM] THEN | |
REWRITE_TAC[REAL_POW_2; real_div] THEN | |
REWRITE_TAC[REAL_ARITH `(r * d) * r * d + (--i * d) * --i * d = | |
(r * r + i * i) * d * d:real`] THEN | |
ASM_CASES_TAC `Re z * Re z + Im z * Im z = &0` THENL | |
[ASM_REWRITE_TAC[REAL_INV_0; SQRT_0; REAL_MUL_LZERO]; ALL_TAC] THEN | |
CONV_TAC SYM_CONV THEN MATCH_MP_TAC REAL_MUL_RINV_UNIQ THEN | |
SIMP_TAC[GSYM SQRT_MUL; REAL_LE_MUL; REAL_LE_INV_EQ; REAL_LE_ADD; | |
REAL_LE_SQUARE] THEN | |
ONCE_REWRITE_TAC[AC REAL_MUL_AC | |
`a * a * b * b:real = (a * b) * (a * b)`] THEN | |
ASM_SIMP_TAC[REAL_MUL_RINV; REAL_MUL_LID; SQRT_1]);; | |
let COMPLEX_NORM_DIV = prove | |
(`!w z. norm(w / z) = norm(w) / norm(z)`, | |
REWRITE_TAC[complex_div; real_div; COMPLEX_NORM_INV; COMPLEX_NORM_MUL]);; | |
let COMPLEX_NORM_TRIANGLE = prove | |
(`!w z. norm(w + z) <= norm(w) + norm(z)`, | |
REPEAT GEN_TAC THEN REWRITE_TAC[complex_norm; complex_add; RE; IM] THEN | |
MATCH_MP_TAC(REAL_ARITH `&0 <= y /\ abs(x) <= abs(y) ==> x <= y`) THEN | |
SIMP_TAC[SQRT_POS_LE; REAL_POW_2; REAL_LE_ADD; REAL_LE_SQUARE; | |
REAL_LE_SQUARE_ABS; SQRT_POW_2] THEN | |
GEN_REWRITE_TAC RAND_CONV[REAL_ARITH | |
`(a + b) * (a + b) = a * a + b * b + &2 * a * b`] THEN | |
REWRITE_TAC[GSYM REAL_POW_2] THEN | |
SIMP_TAC[SQRT_POW_2; REAL_POW_2; REAL_LE_ADD; REAL_LE_SQUARE] THEN | |
REWRITE_TAC[REAL_ARITH | |
`(rw + rz) * (rw + rz) + (iw + iz) * (iw + iz) <= | |
(rw * rw + iw * iw) + (rz * rz + iz * iz) + &2 * other <=> | |
rw * rz + iw * iz <= other`] THEN | |
SIMP_TAC[GSYM SQRT_MUL; REAL_POW_2; REAL_LE_ADD; REAL_LE_SQUARE] THEN | |
MATCH_MP_TAC(REAL_ARITH `&0 <= y /\ abs(x) <= abs(y) ==> x <= y`) THEN | |
SIMP_TAC[SQRT_POS_LE; REAL_POW_2; REAL_LE_ADD; REAL_LE_SQUARE; | |
REAL_LE_SQUARE_ABS; SQRT_POW_2; REAL_LE_MUL] THEN | |
REWRITE_TAC[REAL_ARITH | |
`(rw * rz + iw * iz) * (rw * rz + iw * iz) <= | |
(rw * rw + iw * iw) * (rz * rz + iz * iz) <=> | |
&0 <= (rw * iz - rz * iw) * (rw * iz - rz * iw)`] THEN | |
REWRITE_TAC[REAL_LE_SQUARE]);; | |
let COMPLEX_NORM_TRIANGLE_SUB = prove | |
(`!w z. norm(w) <= norm(w + z) + norm(z)`, | |
MESON_TAC[COMPLEX_NORM_TRIANGLE; COMPLEX_NORM_NEG; COMPLEX_ADD_ASSOC; | |
COMPLEX_ADD_RINV; COMPLEX_ADD_RID]);; | |
let COMPLEX_NORM_ABS_NORM = prove | |
(`!w z. abs(norm w - norm z) <= norm(w - z)`, | |
REPEAT GEN_TAC THEN | |
MATCH_MP_TAC(REAL_ARITH | |
`a - b <= x /\ b - a <= x ==> abs(a - b) <= x:real`) THEN | |
MESON_TAC[COMPLEX_NEG_SUB; COMPLEX_NORM_NEG; REAL_LE_SUB_RADD; complex_sub; | |
COMPLEX_NORM_TRIANGLE_SUB]);; | |
(* ------------------------------------------------------------------------- *) | |
(* Complex conjugate. *) | |
(* ------------------------------------------------------------------------- *) | |
let cnj = new_definition | |
`cnj(z) = complex(Re(z),--(Im(z)))`;; | |
(* ------------------------------------------------------------------------- *) | |
(* Conjugation is an automorphism. *) | |
(* ------------------------------------------------------------------------- *) | |
let CNJ_INJ = prove | |
(`!w z. (cnj(w) = cnj(z)) <=> (w = z)`, | |
REWRITE_TAC[cnj; COMPLEX_EQ; RE; IM; REAL_EQ_NEG2]);; | |
let CNJ_CNJ = prove | |
(`!z. cnj(cnj z) = z`, | |
REWRITE_TAC[cnj; COMPLEX_EQ; RE; IM; REAL_NEG_NEG]);; | |
let CNJ_CX = prove | |
(`!x. cnj(Cx x) = Cx x`, | |
REWRITE_TAC[cnj; COMPLEX_EQ; CX_DEF; REAL_NEG_0; RE; IM]);; | |
let COMPLEX_NORM_CNJ = prove | |
(`!z. norm(cnj z) = norm(z)`, | |
REWRITE_TAC[complex_norm; cnj; REAL_POW_2] THEN | |
REWRITE_TAC[REAL_MUL_LNEG; REAL_MUL_RNEG; RE; IM; REAL_NEG_NEG]);; | |
let CNJ_NEG = prove | |
(`!z. cnj(--z) = --(cnj z)`, | |
REWRITE_TAC[cnj; complex_neg; COMPLEX_EQ; RE; IM]);; | |
let CNJ_INV = prove | |
(`!z. cnj(inv z) = inv(cnj z)`, | |
REWRITE_TAC[cnj; complex_inv; COMPLEX_EQ; RE; IM] THEN | |
REWRITE_TAC[real_div; REAL_NEG_NEG; REAL_POW_2; | |
REAL_MUL_LNEG; REAL_MUL_RNEG]);; | |
let CNJ_ADD = prove | |
(`!w z. cnj(w + z) = cnj(w) + cnj(z)`, | |
REWRITE_TAC[cnj; complex_add; COMPLEX_EQ; RE; IM] THEN | |
REWRITE_TAC[REAL_NEG_ADD; REAL_MUL_LNEG; REAL_MUL_RNEG; REAL_NEG_NEG]);; | |
let CNJ_SUB = prove | |
(`!w z. cnj(w - z) = cnj(w) - cnj(z)`, | |
REWRITE_TAC[complex_sub; CNJ_ADD; CNJ_NEG]);; | |
let CNJ_MUL = prove | |
(`!w z. cnj(w * z) = cnj(w) * cnj(z)`, | |
REWRITE_TAC[cnj; complex_mul; COMPLEX_EQ; RE; IM] THEN | |
REWRITE_TAC[REAL_NEG_ADD; REAL_MUL_LNEG; REAL_MUL_RNEG; REAL_NEG_NEG]);; | |
let CNJ_DIV = prove | |
(`!w z. cnj(w / z) = cnj(w) / cnj(z)`, | |
REWRITE_TAC[complex_div; CNJ_MUL; CNJ_INV]);; | |
let CNJ_POW = prove | |
(`!z n. cnj(z pow n) = cnj(z) pow n`, | |
GEN_TAC THEN INDUCT_TAC THEN | |
ASM_REWRITE_TAC[complex_pow; CNJ_MUL; CNJ_CX]);; | |
(* ------------------------------------------------------------------------- *) | |
(* Conversion of (complex-type) rational constant to ML rational number. *) | |
(* ------------------------------------------------------------------------- *) | |
let is_complex_const = | |
let cx_tm = `Cx` in | |
fun tm -> | |
is_comb tm && | |
let l,r = dest_comb tm in l = cx_tm && is_ratconst r;; | |
let dest_complex_const = | |
let cx_tm = `Cx` in | |
fun tm -> | |
let l,r = dest_comb tm in | |
if l = cx_tm then rat_of_term r | |
else failwith "dest_complex_const";; | |
let mk_complex_const = | |
let cx_tm = `Cx` in | |
fun r -> | |
mk_comb(cx_tm,term_of_rat r);; | |
(* ------------------------------------------------------------------------- *) | |
(* Conversions to perform operations if coefficients are rational constants. *) | |
(* ------------------------------------------------------------------------- *) | |
let COMPLEX_RAT_MUL_CONV = | |
GEN_REWRITE_CONV I [GSYM CX_MUL] THENC RAND_CONV REAL_RAT_MUL_CONV;; | |
let COMPLEX_RAT_ADD_CONV = | |
GEN_REWRITE_CONV I [GSYM CX_ADD] THENC RAND_CONV REAL_RAT_ADD_CONV;; | |
let COMPLEX_RAT_EQ_CONV = | |
GEN_REWRITE_CONV I [CX_INJ] THENC REAL_RAT_EQ_CONV;; | |
let COMPLEX_RAT_POW_CONV = | |
let x_tm = `x:real` | |
and n_tm = `n:num` in | |
let pth = SYM(SPECL [x_tm; n_tm] CX_POW) in | |
fun tm -> | |
let lop,r = dest_comb tm in | |
let op,bod = dest_comb lop in | |
let th1 = INST [rand bod,x_tm; r,n_tm] pth in | |
let tm1,tm2 = dest_comb(concl th1) in | |
if rand tm1 <> tm then failwith "COMPLEX_RAT_POW_CONV" else | |
let tm3,tm4 = dest_comb tm2 in | |
TRANS th1 (AP_TERM tm3 (REAL_RAT_REDUCE_CONV tm4));; | |
(* ------------------------------------------------------------------------- *) | |
(* Instantiate polynomial normalizer. *) | |
(* ------------------------------------------------------------------------- *) | |
let COMPLEX_POLY_CLAUSES = prove | |
(`(!x y z. x + (y + z) = (x + y) + z) /\ | |
(!x y. x + y = y + x) /\ | |
(!x. Cx(&0) + x = x) /\ | |
(!x y z. x * (y * z) = (x * y) * z) /\ | |
(!x y. x * y = y * x) /\ | |
(!x. Cx(&1) * x = x) /\ | |
(!x. Cx(&0) * x = Cx(&0)) /\ | |
(!x y z. x * (y + z) = x * y + x * z) /\ | |
(!x. x pow 0 = Cx(&1)) /\ | |
(!x n. x pow (SUC n) = x * x pow n)`, | |
REWRITE_TAC[complex_pow] THEN SIMPLE_COMPLEX_ARITH_TAC) | |
and COMPLEX_POLY_NEG_CLAUSES = prove | |
(`(!x. --x = Cx(-- &1) * x) /\ | |
(!x y. x - y = x + Cx(-- &1) * y)`, | |
SIMPLE_COMPLEX_ARITH_TAC);; | |
let COMPLEX_POLY_NEG_CONV,COMPLEX_POLY_ADD_CONV,COMPLEX_POLY_SUB_CONV, | |
COMPLEX_POLY_MUL_CONV,COMPLEX_POLY_POW_CONV,COMPLEX_POLY_CONV = | |
SEMIRING_NORMALIZERS_CONV COMPLEX_POLY_CLAUSES COMPLEX_POLY_NEG_CLAUSES | |
(is_complex_const, | |
COMPLEX_RAT_ADD_CONV,COMPLEX_RAT_MUL_CONV,COMPLEX_RAT_POW_CONV) | |
(<);; | |
let COMPLEX_RAT_INV_CONV = | |
GEN_REWRITE_CONV I [GSYM CX_INV] THENC RAND_CONV REAL_RAT_INV_CONV;; | |
let COMPLEX_POLY_CONV = | |
let neg_tm = `(--):complex->complex` | |
and inv_tm = `inv:complex->complex` | |
and add_tm = `(+):complex->complex->complex` | |
and sub_tm = `(-):complex->complex->complex` | |
and mul_tm = `(*):complex->complex->complex` | |
and div_tm = `(/):complex->complex->complex` | |
and pow_tm = `(pow):complex->num->complex` | |
and div_conv = REWR_CONV complex_div in | |
let rec COMPLEX_POLY_CONV tm = | |
if not(is_comb tm) || is_complex_const tm then REFL tm else | |
let lop,r = dest_comb tm in | |
if lop = neg_tm then | |
let th1 = AP_TERM lop (COMPLEX_POLY_CONV r) in | |
TRANS th1 (COMPLEX_POLY_NEG_CONV (rand(concl th1))) | |
else if lop = inv_tm then | |
let th1 = AP_TERM lop (COMPLEX_POLY_CONV r) in | |
TRANS th1 (TRY_CONV COMPLEX_RAT_INV_CONV (rand(concl th1))) | |
else if not(is_comb lop) then REFL tm else | |
let op,l = dest_comb lop in | |
if op = pow_tm then | |
let th1 = AP_THM (AP_TERM op (COMPLEX_POLY_CONV l)) r in | |
TRANS th1 (TRY_CONV COMPLEX_POLY_POW_CONV (rand(concl th1))) | |
else if op = add_tm || op = mul_tm || op = sub_tm then | |
let th1 = MK_COMB(AP_TERM op (COMPLEX_POLY_CONV l), | |
COMPLEX_POLY_CONV r) in | |
let fn = if op = add_tm then COMPLEX_POLY_ADD_CONV | |
else if op = mul_tm then COMPLEX_POLY_MUL_CONV | |
else COMPLEX_POLY_SUB_CONV in | |
TRANS th1 (fn (rand(concl th1))) | |
else if op = div_tm then | |
let th1 = div_conv tm in | |
TRANS th1 (COMPLEX_POLY_CONV (rand(concl th1))) | |
else REFL tm in | |
COMPLEX_POLY_CONV;; | |
(* ------------------------------------------------------------------------- *) | |
(* Complex number version of usual ring procedure. *) | |
(* ------------------------------------------------------------------------- *) | |
let COMPLEX_MUL_LINV = prove | |
(`!z. ~(z = Cx(&0)) ==> (inv(z) * z = Cx(&1))`, | |
REWRITE_TAC[complex_mul; complex_inv; RE; IM; COMPLEX_EQ; CX_DEF] THEN | |
REWRITE_TAC[GSYM REAL_SOS_EQ_0] THEN CONV_TAC REAL_FIELD);; | |
let COMPLEX_MUL_RINV = prove | |
(`!z. ~(z = Cx(&0)) ==> (z * inv(z) = Cx(&1))`, | |
ONCE_REWRITE_TAC[COMPLEX_MUL_SYM] THEN REWRITE_TAC[COMPLEX_MUL_LINV]);; | |
let COMPLEX_RING,complex_ideal_cofactors = | |
let ring_pow_tm = `(pow):complex->num->complex` | |
and COMPLEX_INTEGRAL = prove | |
(`(!x. Cx(&0) * x = Cx(&0)) /\ | |
(!x y z. (x + y = x + z) <=> (y = z)) /\ | |
(!w x y z. (w * y + x * z = w * z + x * y) <=> (w = x) \/ (y = z))`, | |
REWRITE_TAC[COMPLEX_ENTIRE; SIMPLE_COMPLEX_ARITH | |
`(w * y + x * z = w * z + x * y) <=> | |
(w - x) * (y - z) = Cx(&0)`] THEN | |
SIMPLE_COMPLEX_ARITH_TAC) | |
and COMPLEX_RABINOWITSCH = prove | |
(`!x y:complex. ~(x = y) <=> ?z. (x - y) * z = Cx(&1)`, | |
REPEAT GEN_TAC THEN | |
GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [GSYM COMPLEX_SUB_0] THEN | |
MESON_TAC[COMPLEX_MUL_RINV; COMPLEX_MUL_LZERO; | |
SIMPLE_COMPLEX_ARITH `~(Cx(&1) = Cx(&0))`]) | |
and init = ALL_CONV in | |
let pure,ideal = | |
RING_AND_IDEAL_CONV | |
(dest_complex_const,mk_complex_const,COMPLEX_RAT_EQ_CONV, | |
`(--):complex->complex`,`(+):complex->complex->complex`, | |
`(-):complex->complex->complex`,`(inv):complex->complex`, | |
`(*):complex->complex->complex`,`(/):complex->complex->complex`, | |
`(pow):complex->num->complex`, | |
COMPLEX_INTEGRAL,COMPLEX_RABINOWITSCH,COMPLEX_POLY_CONV) in | |
(fun tm -> let th = init tm in EQ_MP (SYM th) (pure(rand(concl th)))), | |
ideal;; | |
(* ------------------------------------------------------------------------- *) | |
(* Most basic properties of inverses. *) | |
(* ------------------------------------------------------------------------- *) | |
let COMPLEX_INV_0 = prove | |
(`inv(Cx(&0)) = Cx(&0)`, | |
REWRITE_TAC[complex_inv; CX_DEF; RE; IM; real_div; REAL_MUL_LZERO; | |
REAL_NEG_0]);; | |
let COMPLEX_INV_MUL = prove | |
(`!w z. inv(w * z) = inv(w) * inv(z)`, | |
REPEAT GEN_TAC THEN | |
MAP_EVERY ASM_CASES_TAC [`w = Cx(&0)`; `z = Cx(&0)`] THEN | |
ASM_REWRITE_TAC[COMPLEX_INV_0; COMPLEX_MUL_LZERO; COMPLEX_MUL_RZERO] THEN | |
REPEAT(POP_ASSUM MP_TAC) THEN | |
REWRITE_TAC[complex_mul; complex_inv; RE; IM; COMPLEX_EQ; CX_DEF] THEN | |
REWRITE_TAC[GSYM REAL_SOS_EQ_0] THEN CONV_TAC REAL_FIELD);; | |
let COMPLEX_INV_1 = prove | |
(`inv(Cx(&1)) = Cx(&1)`, | |
REWRITE_TAC[complex_inv; CX_DEF; RE; IM] THEN | |
CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[REAL_DIV_1]);; | |
let COMPLEX_POW_INV = prove | |
(`!x n. (inv x) pow n = inv(x pow n)`, | |
GEN_TAC THEN INDUCT_TAC THEN | |
ASM_REWRITE_TAC[complex_pow; COMPLEX_INV_1; COMPLEX_INV_MUL]);; | |
let COMPLEX_INV_INV = prove | |
(`!x:complex. inv(inv x) = x`, | |
GEN_TAC THEN ASM_CASES_TAC `x = Cx(&0)` THEN | |
ASM_REWRITE_TAC[COMPLEX_INV_0] THEN | |
POP_ASSUM MP_TAC THEN | |
MAP_EVERY (fun t -> MP_TAC(SPEC t COMPLEX_MUL_RINV)) | |
[`x:complex`; `inv(x):complex`] THEN | |
CONV_TAC COMPLEX_RING);; | |
(* ------------------------------------------------------------------------- *) | |
(* And also field procedure. *) | |
(* ------------------------------------------------------------------------- *) | |
let COMPLEX_FIELD = | |
let prenex_conv = | |
TOP_DEPTH_CONV BETA_CONV THENC | |
PURE_REWRITE_CONV[FORALL_SIMP; EXISTS_SIMP; complex_div; | |
COMPLEX_INV_INV; COMPLEX_INV_MUL; GSYM REAL_POW_INV] THENC | |
NNFC_CONV THENC DEPTH_BINOP_CONV `(/\)` CONDS_CELIM_CONV THENC | |
PRENEX_CONV | |
and setup_conv = NNF_CONV THENC WEAK_CNF_CONV THENC CONJ_CANON_CONV | |
and is_inv = | |
let inv_tm = `inv:complex->complex` | |
and is_div = is_binop `(/):complex->complex->complex` in | |
fun tm -> (is_div tm || (is_comb tm && rator tm = inv_tm)) && | |
not(is_complex_const(rand tm)) | |
and lemma_inv = MESON[COMPLEX_MUL_RINV] | |
`!x. x = Cx(&0) \/ x * inv(x) = Cx(&1)` | |
and dcases = MATCH_MP(TAUT `(p \/ q) /\ (r \/ s) ==> (p \/ r) \/ q /\ s`) in | |
let cases_rule th1 th2 = dcases (CONJ th1 th2) in | |
let BASIC_COMPLEX_FIELD tm = | |
let is_freeinv t = is_inv t && free_in t tm in | |
let itms = setify(map rand (find_terms is_freeinv tm)) in | |
let dth = if itms = [] then TRUTH | |
else end_itlist cases_rule (map (C SPEC lemma_inv) itms) in | |
let tm' = mk_imp(concl dth,tm) in | |
let th1 = setup_conv tm' in | |
let ths = map COMPLEX_RING (conjuncts(rand(concl th1))) in | |
let th2 = EQ_MP (SYM th1) (end_itlist CONJ ths) in | |
MP (EQ_MP (SYM th1) (end_itlist CONJ ths)) dth in | |
fun tm -> | |
let th0 = prenex_conv tm in | |
let tm0 = rand(concl th0) in | |
let avs,bod = strip_forall tm0 in | |
let th1 = setup_conv bod in | |
let ths = map BASIC_COMPLEX_FIELD (conjuncts(rand(concl th1))) in | |
EQ_MP (SYM th0) (GENL avs (EQ_MP (SYM th1) (end_itlist CONJ ths)));; | |
(* ------------------------------------------------------------------------- *) | |
(* Properties of inverses, divisions are now mostly automatic. *) | |
(* ------------------------------------------------------------------------- *) | |
let COMPLEX_POW_DIV = prove | |
(`!x y n. (x / y) pow n = (x pow n) / (y pow n)`, | |
REWRITE_TAC[complex_div; COMPLEX_POW_MUL; COMPLEX_POW_INV]);; | |
let COMPLEX_DIV_REFL = prove | |
(`!x. ~(x = Cx(&0)) ==> (x / x = Cx(&1))`, | |
CONV_TAC COMPLEX_FIELD);; | |
let COMPLEX_EQ_MUL_LCANCEL = prove | |
(`!x y z. (x * y = x * z) <=> (x = Cx(&0)) \/ (y = z)`, | |
CONV_TAC COMPLEX_FIELD);; | |
let COMPLEX_EQ_MUL_RCANCEL = prove | |
(`!x y z. (x * z = y * z) <=> (x = y) \/ (z = Cx(&0))`, | |
CONV_TAC COMPLEX_FIELD);; | |
let COMPLEX_MUL_RINV_UNIQ = prove | |
(`!w z. w * z = Cx(&1) ==> inv w = z`, | |
CONV_TAC COMPLEX_FIELD);; | |
let COMPLEX_MUL_LINV_UNIQ = prove | |
(`!w z. w * z = Cx(&1) ==> inv z = w`, | |
CONV_TAC COMPLEX_FIELD);; | |
let COMPLEX_DIV_LMUL = prove | |
(`!w z. ~(z = Cx(&0)) ==> z * w / z = w`, | |
CONV_TAC COMPLEX_FIELD);; | |
let COMPLEX_DIV_RMUL = prove | |
(`!w z. ~(z = Cx(&0)) ==> w / z * z = w`, | |
CONV_TAC COMPLEX_FIELD);; | |