(* ========================================================================= *)
(* The type "real^2" regarded as the complex numbers.                        *)
(*                                                                           *)
(*              (c) Copyright, John Harrison 1998-2008                       *)
(*              (c) Copyright, Valentina Bruno 2010                          *)
(* ========================================================================= *)

needs "Multivariate/integration.ml";;

new_type_abbrev("complex",`:real^2`);;

let prioritize_complex() =
  overload_interface("--",`vector_neg:complex->complex`);
  overload_interface("+",`vector_add:complex->complex->complex`);
  overload_interface("-",`vector_sub:complex->complex->complex`);
  overload_interface("*",`complex_mul:complex->complex->complex`);
  overload_interface("/",`complex_div:complex->complex->complex`);
  overload_interface("pow",`complex_pow:complex->num->complex`);
  overload_interface("inv",`complex_inv:complex->complex`);;

prioritize_complex();;

(* ------------------------------------------------------------------------- *)
(* Real and imaginary parts of a number.                                     *)
(* ------------------------------------------------------------------------- *)

let RE_DEF = new_definition
  `Re(z:complex) = z$1`;;
let IM_DEF = new_definition
  `Im(z:complex) = z$2`;;
(* ------------------------------------------------------------------------- *) (* Real injection and imaginary unit. *) (* ------------------------------------------------------------------------- *)
let complex = new_definition
 `complex(x,y) = vector[x;y]:complex`;;
let CX_DEF = new_definition
 `Cx(a) = complex(a,&0)`;;
let ii = new_definition
  `ii = complex(&0,&1)`;;
(* ------------------------------------------------------------------------- *) (* Complex multiplication. *) (* ------------------------------------------------------------------------- *)
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 = define
  `(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; VECTOR_2]);;
let IM = 
prove (`Im(complex(x,y)) = y`,
REWRITE_TAC[IM_DEF; complex; VECTOR_2]);;
let COMPLEX_EQ = 
prove (`!w z. (w = z) <=> (Re(w) = Re(z)) /\ (Im(w) = Im(z))`,
SIMP_TAC[CART_EQ; FORALL_2; DIMINDEX_2; RE_DEF; IM_DEF]);;
let COMPLEX = 
prove (`!z. complex(Re(z),Im(z)) = z`,
REWRITE_TAC[COMPLEX_EQ; RE; IM]);;
let COMPLEX_EQ_0 = 
prove (`z = Cx(&0) <=> Re(z) pow 2 + Im(z) pow 2 = &0`,
REWRITE_TAC[COMPLEX_EQ; CX_DEF; RE; IM] THEN EQ_TAC THEN SIMP_TAC[] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN DISCH_THEN(MP_TAC o MATCH_MP (REAL_ARITH `!x y:real. x + y = &0 ==> &0 <= x /\ &0 <= y ==> x = &0 /\ y = &0`)) THEN REWRITE_TAC[REAL_POW_2; REAL_LE_SQUARE; REAL_ENTIRE]);;
let FORALL_COMPLEX = 
prove (`(!z. P z) <=> (!x y. P(complex(x,y)))`,
MESON_TAC[COMPLEX]);;
let EXISTS_COMPLEX = 
prove (`(?z. P z) <=> (?x y. P(complex(x,y)))`,
MESON_TAC[COMPLEX]);;
(* ------------------------------------------------------------------------- *) (* Pseudo-definitions of other general vector concepts over R^2. *) (* ------------------------------------------------------------------------- *)
let complex_neg = 
prove (`--z = complex(--(Re(z)),--(Im(z)))`,
REWRITE_TAC[COMPLEX_EQ; RE; IM] THEN REWRITE_TAC[RE_DEF; IM_DEF] THEN SIMP_TAC[VECTOR_NEG_COMPONENT; DIMINDEX_2; ARITH]);;
let complex_add = 
prove (`w + z = complex(Re(w) + Re(z),Im(w) + Im(z))`,
REWRITE_TAC[COMPLEX_EQ; RE; IM] THEN REWRITE_TAC[RE_DEF; IM_DEF] THEN SIMP_TAC[VECTOR_ADD_COMPONENT; DIMINDEX_2; ARITH]);;
let complex_sub = VECTOR_ARITH `(w:complex) - z = w + --z`;;
let complex_norm = 
prove (`norm(z) = sqrt(Re(z) pow 2 + Im(z) pow 2)`,
REWRITE_TAC[vector_norm; dot; RE_DEF; IM_DEF; SUM_2; DIMINDEX_2] THEN AP_TERM_TAC THEN REAL_ARITH_TAC);;
let COMPLEX_SQNORM = 
prove (`norm(z) pow 2 = Re(z) pow 2 + Im(z) pow 2`,
REWRITE_TAC[NORM_POW_2; dot; RE_DEF; IM_DEF; SUM_2; DIMINDEX_2] THEN REAL_ARITH_TAC);;
(* ------------------------------------------------------------------------- *) (* 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; complex_inv; complex_div] THEN CONV_TAC REAL_FIELD;; 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. Cx(&2) * x = x + x`,
SIMPLE_COMPLEX_ARITH_TAC);;
(* ------------------------------------------------------------------------- *) (* Sometimes here we need to tweak non-zeroness assertions. *) (* ------------------------------------------------------------------------- *)
let II_NZ = 
prove (`~(ii = Cx(&0))`,
REWRITE_TAC[ii] THEN SIMPLE_COMPLEX_ARITH_TAC);;
let COMPLEX_MUL_LINV = 
prove (`!z. ~(z = Cx(&0)) ==> (inv(z) * z = Cx(&1))`,
REWRITE_TAC[COMPLEX_EQ_0] THEN SIMPLE_COMPLEX_ARITH_TAC);;
let COMPLEX_ENTIRE = 
prove (`!x y. (x * y = Cx(&0)) <=> (x = Cx(&0)) \/ (y = Cx(&0))`,
REWRITE_TAC[COMPLEX_EQ_0] THEN SIMPLE_COMPLEX_ARITH_TAC);;
let COMPLEX_MUL_RINV = 
prove (`!z. ~(z = Cx(&0)) ==> (z * inv(z) = Cx(&1))`,
REWRITE_TAC[COMPLEX_EQ_0] THEN SIMPLE_COMPLEX_ARITH_TAC);;
let COMPLEX_DIV_REFL = 
prove (`!x. ~(x = Cx(&0)) ==> (x / x = Cx(&1))`,
REWRITE_TAC[COMPLEX_EQ_0] THEN 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_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_INV = 
prove (`!x. Cx(inv x) = inv(Cx x)`,
GEN_TAC THEN REWRITE_TAC[CX_DEF; complex_inv; RE; IM; COMPLEX_EQ] THEN ASM_CASES_TAC `x = &0` THEN ASM_REWRITE_TAC[] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN POP_ASSUM MP_TAC THEN CONV_TAC REAL_FIELD);;
let CX_MUL = 
prove (`!x y. Cx(x * y) = Cx(x) * Cx(y)`,
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_DIV = 
prove (`!x y. Cx(x / y) = Cx(x) / Cx(y)`,
REWRITE_TAC[complex_div; real_div; CX_MUL; CX_INV]);;
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]);;
(* ------------------------------------------------------------------------- *) (* Some "linear" things hold for Re and Im too. *) (* ------------------------------------------------------------------------- *)
let RE_CX = 
prove (`!x. Re(Cx x) = x`,
REWRITE_TAC[RE; CX_DEF]);;
let RE_NEG = 
prove (`!x. Re(--x) = --Re(x)`,
REWRITE_TAC[complex_neg; RE]);;
let RE_ADD = 
prove (`!x y. Re(x + y) = Re(x) + Re(y)`,
REWRITE_TAC[complex_add; RE]);;
let RE_SUB = 
prove (`!x y. Re(x - y) = Re(x) - Re(y)`,
REWRITE_TAC[complex_sub; real_sub; RE_ADD; RE_NEG]);;
let IM_CX = 
prove (`!x. Im(Cx x) = &0`,
REWRITE_TAC[IM; CX_DEF]);;
let IM_NEG = 
prove (`!x. Im(--x) = --Im(x)`,
REWRITE_TAC[complex_neg; IM]);;
let IM_ADD = 
prove (`!x y. Im(x + y) = Im(x) + Im(y)`,
REWRITE_TAC[complex_add; IM]);;
let IM_SUB = 
prove (`!x y. Im(x - y) = Im(x) - Im(y)`,
REWRITE_TAC[complex_sub; real_sub; IM_ADD; IM_NEG]);;
(* ------------------------------------------------------------------------- *) (* An "expansion" theorem into the traditional notation. *) (* ------------------------------------------------------------------------- *)
let COMPLEX_EXPAND = 
prove (`!z. z = Cx(Re z) + ii * Cx(Im z)`,
REWRITE_TAC[ii] THEN SIMPLE_COMPLEX_ARITH_TAC);;
let COMPLEX_TRAD = 
prove (`!x y. complex(x,y) = Cx(x) + ii * Cx(y)`,
REWRITE_TAC[ii] THEN SIMPLE_COMPLEX_ARITH_TAC);;
(* ------------------------------------------------------------------------- *) (* Real and complex parts of ii and multiples. *) (* ------------------------------------------------------------------------- *)
let RE_II = 
prove (`Re ii = &0`,
REWRITE_TAC[ii] THEN SIMPLE_COMPLEX_ARITH_TAC);;
let IM_II = 
prove (`Im ii = &1`,
REWRITE_TAC[ii] THEN SIMPLE_COMPLEX_ARITH_TAC);;
let RE_MUL_II = 
prove (`!z. Re(z * ii) = --(Im z) /\ Re(ii * z) = --(Im z)`,
REWRITE_TAC[ii] THEN SIMPLE_COMPLEX_ARITH_TAC);;
let IM_MUL_II = 
prove (`!z. Im(z * ii) = Re z /\ Im(ii * z) = Re z`,
REWRITE_TAC[ii] THEN SIMPLE_COMPLEX_ARITH_TAC);;
let COMPLEX_NORM_II = 
prove (`norm ii = &1`,
REWRITE_TAC[complex_norm; RE_II; IM_II] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[SQRT_1]);;
(* ------------------------------------------------------------------------- *) (* Limited "multiplicative" theorems for Re and Im. *) (* ------------------------------------------------------------------------- *)
let RE_CMUL = 
prove (`!a z. Re(a % z) = a * Re z`,
SIMP_TAC[RE_DEF; VECTOR_MUL_COMPONENT; DIMINDEX_2; ARITH]);;
let IM_CMUL = 
prove (`!a z. Im(a % z) = a * Im z`,
SIMP_TAC[IM_DEF; VECTOR_MUL_COMPONENT; DIMINDEX_2; ARITH]);;
let RE_MUL_CX = 
prove (`!x z. Re(Cx(x) * z) = x * Re z /\ Re(z * Cx(x)) = Re z * x`,
SIMPLE_COMPLEX_ARITH_TAC);;
let IM_MUL_CX = 
prove (`!x z. Im(Cx(x) * z) = x * Im z /\ Im(z * Cx(x)) = Im z * x`,
SIMPLE_COMPLEX_ARITH_TAC);;
let RE_DIV_CX = 
prove (`!z x. Re(z / Cx(x)) = Re(z) / x`,
REWRITE_TAC[complex_div; real_div; GSYM CX_INV; RE_MUL_CX]);;
let IM_DIV_CX = 
prove (`!z x. Im(z / Cx(x)) = Im(z) / x`,
REWRITE_TAC[complex_div; real_div; GSYM CX_INV; IM_MUL_CX]);;
(* ------------------------------------------------------------------------- *) (* Syntax constructors etc. for complex constants. *) (* ------------------------------------------------------------------------- *) 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 for arithmetic on complex constants. *) (* ------------------------------------------------------------------------- *) let COMPLEX_RAT_EQ_CONV = GEN_REWRITE_CONV I [CX_INJ] THENC REAL_RAT_EQ_CONV;; 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_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));; (* ------------------------------------------------------------------------- *) (* Complex 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) (<);; (* ------------------------------------------------------------------------- *) (* Extend it to handle "inv" and division, by constants after normalization. *) (* ------------------------------------------------------------------------- *) let COMPLEX_RAT_INV_CONV = REWR_CONV(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) or is_ratconst 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 or op = mul_tm or 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_RING,complex_ideal_cofactors =
let 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 COMPLEX_IIII = prove (`ii * ii + Cx(&1) = Cx(&0)`, REWRITE_TAC[ii; CX_DEF; complex_mul; complex_add; RE; IM] THEN AP_TERM_TAC THEN BINOP_TAC THEN REAL_ARITH_TAC) in let ring,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) and ii_tm = `ii` and iiii_tm = concl COMPLEX_IIII in (fun tm -> if free_in ii_tm tm then MP (ring (mk_imp(iiii_tm,tm))) COMPLEX_IIII else ring tm), ideal;; (* ------------------------------------------------------------------------- *) (* Most basic properties of inverses. *) (* ------------------------------------------------------------------------- *)
let COMPLEX_INV_0 = prove
 (`inv(Cx(&0)) = Cx(&0)`,
  SIMPLE_COMPLEX_ARITH_TAC);;
let COMPLEX_INV_1 = 
prove (`inv(Cx(&1)) = Cx(&1)`,
SIMPLE_COMPLEX_ARITH_TAC);;
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_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);;
let COMPLEX_INV_DIV = 
prove (`!w z:complex. inv(w / z) = z / w`,
REWRITE_TAC[complex_div; COMPLEX_INV_MUL; COMPLEX_INV_INV] THEN REWRITE_TAC[COMPLEX_MUL_AC]);;
(* ------------------------------------------------------------------------- *) (* And also field procedure. *) (* ------------------------------------------------------------------------- *)
let COMPLEX_EQ_MUL_LCANCEL = 
prove (`!x y z. (x * y = x * z) <=> (x = Cx(&0)) \/ (y = z)`,
CONV_TAC COMPLEX_RING);;
let COMPLEX_EQ_MUL_RCANCEL = 
prove (`!x y z. (x * z = y * z) <=> (x = y) \/ (z = Cx(&0))`,
CONV_TAC COMPLEX_RING);;
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 COMPLEX_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 or (is_comb tm & rator tm = inv_tm)) & not(is_ratconst(rand tm)) 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 hyps = map (fun t -> SPEC t COMPLEX_MUL_RINV) itms in let tm' = itlist (fun th t -> mk_imp(concl th,t)) hyps tm in let th1 = setup_conv tm' in let cjs = conjuncts(rand(concl th1)) in let ths = map COMPLEX_RING cjs in let th2 = EQ_MP (SYM th1) (end_itlist CONJ ths) in rev_itlist (C MP) hyps th2 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)));; (* ------------------------------------------------------------------------- *) (* More trivial lemmas. *) (* ------------------------------------------------------------------------- *)
let COMPLEX_DIV_1 = 
prove (`!z. z / Cx(&1) = z`,
CONV_TAC COMPLEX_FIELD);;
let COMPLEX_DIV_LMUL = 
prove (`!x y. ~(y = Cx(&0)) ==> y * x / y = x`,
CONV_TAC COMPLEX_FIELD);;
let COMPLEX_DIV_RMUL = 
prove (`!x y. ~(y = Cx(&0)) ==> x / y * y = x`,
CONV_TAC COMPLEX_FIELD);;
let COMPLEX_INV_EQ_0 = 
prove (`!x. inv x = Cx(&0) <=> x = Cx(&0)`,
GEN_TAC THEN ASM_CASES_TAC `x = Cx(&0)` THEN ASM_REWRITE_TAC[COMPLEX_INV_0] THEN POP_ASSUM MP_TAC THEN CONV_TAC COMPLEX_FIELD);;
let COMPLEX_INV_NEG = 
prove (`!x:complex. inv(--x) = --(inv x)`,
GEN_TAC THEN ASM_CASES_TAC `x = Cx(&0)` THEN ASM_REWRITE_TAC[COMPLEX_INV_0; COMPLEX_NEG_0] THEN POP_ASSUM MP_TAC THEN CONV_TAC COMPLEX_FIELD);;
let COMPLEX_NEG_INV = 
prove (`!x:complex. --(inv x) = inv(--x)`,
REWRITE_TAC[COMPLEX_INV_NEG]);;
let COMPLEX_INV_EQ_1 = 
prove (`!x. inv x = Cx(&1) <=> x = Cx(&1)`,
GEN_TAC THEN ASM_CASES_TAC `x = Cx(&0)` THEN ASM_REWRITE_TAC[COMPLEX_INV_0] THEN POP_ASSUM MP_TAC THEN CONV_TAC COMPLEX_FIELD);;
let COMPLEX_DIV_EQ_0 = 
prove (`!w z. w / z = Cx(&0) <=> w = Cx(&0) \/ z = Cx(&0)`,
(* ------------------------------------------------------------------------- *) (* 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_DIV = 
prove (`!x y n. (x / y) pow n = (x pow n) / (y pow n)`,
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]);;
let COMPLEX_POW_ZERO = 
prove (`!n. Cx(&0) pow n = if n = 0 then Cx(&1) else Cx(&0)`,
INDUCT_TAC THEN REWRITE_TAC[complex_pow; COMPLEX_MUL_LZERO; NOT_SUC]);;
let COMPLEX_INV_II = 
prove (`inv ii = --ii`,
CONV_TAC COMPLEX_FIELD);;
let COMPLEX_DIV_POW = 
prove (`!x:complex n k:num. ~(x= Cx(&0)) /\ k <= n /\ ~(k = 0) ==> x pow (n-k) = x pow n / x pow k`,
REPEAT STRIP_TAC THEN SUBGOAL_THEN `x:complex pow (n - k) * x pow k = x pow n / x pow k * x pow k` (fun th-> ASM_MESON_TAC [th;COMPLEX_POW_EQ_0;COMPLEX_EQ_MUL_RCANCEL]) THEN ASM_SIMP_TAC[GSYM COMPLEX_POW_ADD;SUB_ADD] THEN MP_TAC (MESON [COMPLEX_POW_EQ_0;ASSUME `~(k = 0)`; ASSUME `~(x = Cx(&0))`] `~(x pow k = Cx(&0))`) THEN ASM_SIMP_TAC[COMPLEX_DIV_RMUL]);;
(* ------------------------------------------------------------------------- *) (* Norms (aka "moduli"). *) (* ------------------------------------------------------------------------- *)
let COMPLEX_VEC_0 = 
prove (`vec 0 = Cx(&0)`,
SIMP_TAC[CART_EQ; VEC_COMPONENT; CX_DEF; complex; DIMINDEX_2; FORALL_2; VECTOR_2]);;
let COMPLEX_NORM_ZERO = 
prove (`!z. (norm z = &0) <=> (z = Cx(&0))`,
REWRITE_TAC[NORM_EQ_0; COMPLEX_VEC_0]);;
let COMPLEX_NORM_NUM = 
prove (`!n. 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))`,
REWRITE_TAC[NORM_POS_LT; COMPLEX_VEC_0]);;
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)`,
let COMPLEX_NORM_TRIANGLE_SUB = 
prove (`!w z. norm(w) <= norm(w + z) + norm(z)`,
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; NORM_NEG; REAL_LE_SUB_RADD; complex_sub; COMPLEX_NORM_TRIANGLE_SUB]);;
let COMPLEX_POW_EQ_1 = 
prove (`!z n. z pow n = Cx(&1) ==> norm(z) = &1 \/ n = 0`,
REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o AP_TERM `norm:complex->real`) THEN SIMP_TAC[COMPLEX_NORM_POW; COMPLEX_NORM_CX; REAL_POW_EQ_1; REAL_ABS_NUM] THEN SIMP_TAC[REAL_ABS_NORM] THEN CONV_TAC TAUT);;
(* ------------------------------------------------------------------------- *) (* 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]);;
let RE_CNJ = 
prove (`!z. Re(cnj z) = Re z`,
REWRITE_TAC[cnj; RE]);;
let IM_CNJ = 
prove (`!z. Im(cnj z) = --Im z`,
REWRITE_TAC[cnj; IM]);;
let CNJ_EQ_CX = 
prove (`!x z. cnj z = Cx x <=> z = Cx x`,
REWRITE_TAC[COMPLEX_EQ; RE_CNJ; IM_CNJ; RE_CX; IM_CX] THEN CONV_TAC REAL_RING);;
let CNJ_EQ_0 = 
prove (`!z. cnj z = Cx(&0) <=> z = Cx(&0)`,
REWRITE_TAC[CNJ_EQ_CX]);;
let COMPLEX_ADD_CNJ = 
prove (`(!z. z + cnj z = Cx(&2 * Re z)) /\ (!z. cnj z + z = Cx(&2 * Re z))`,
REWRITE_TAC[COMPLEX_EQ; RE_CX; IM_CX; RE_ADD; IM_ADD; RE_CNJ; IM_CNJ] THEN REAL_ARITH_TAC);;
let CNJ_II = 
prove (`cnj ii = --ii`,
REWRITE_TAC[cnj; ii; RE; IM; complex_neg; REAL_NEG_0]);;
let CX_RE_CNJ = 
prove (`!z. Cx(Re z) = (z + cnj z) / Cx(&2)`,
REWRITE_TAC[COMPLEX_EQ; RE_DIV_CX; IM_DIV_CX; RE_CX; IM_CX] THEN REWRITE_TAC[RE_ADD; IM_ADD; RE_CNJ; IM_CNJ] THEN REAL_ARITH_TAC);;
let CX_IM_CNJ = 
prove (`!z. Cx(Im z) = --ii * (z - cnj z) / Cx(&2)`,
REWRITE_TAC[COMPLEX_EQ; RE_DIV_CX; IM_DIV_CX; RE_CX; IM_CX; COMPLEX_MUL_LNEG; RE_NEG; IM_NEG; RE_MUL_II; IM_MUL_II] THEN REWRITE_TAC[RE_SUB; IM_SUB; RE_CNJ; IM_CNJ] THEN REAL_ARITH_TAC);;
let FORALL_CNJ = 
prove (`(!z. P(cnj z)) <=> (!z. P z)`,
MESON_TAC[CNJ_CNJ]);;
let EXISTS_CNJ = 
prove (`(?z. P(cnj z)) <=> (?z. P z)`,
MESON_TAC[CNJ_CNJ]);;
(* ------------------------------------------------------------------------- *) (* Slightly ad hoc theorems relating multiplication, inverse and conjugation *) (* ------------------------------------------------------------------------- *)
let COMPLEX_NORM_POW_2 = 
prove (`!z. Cx(norm z) pow 2 = z * cnj z`,
GEN_TAC THEN REWRITE_TAC [GSYM CX_POW; COMPLEX_SQNORM] THEN REWRITE_TAC [cnj; complex_mul; CX_DEF; RE; IM; COMPLEX_EQ] THEN CONV_TAC REAL_RING);;
let COMPLEX_MUL_CNJ = 
prove (`!z. cnj z * z = Cx(norm(z)) pow 2 /\ z * cnj z = Cx(norm(z)) pow 2`,
GEN_TAC THEN REWRITE_TAC[COMPLEX_MUL_SYM] THEN REWRITE_TAC[cnj; complex_mul; RE; IM; GSYM CX_POW; COMPLEX_SQNORM] THEN REWRITE_TAC[CX_DEF] THEN AP_TERM_TAC THEN BINOP_TAC THEN CONV_TAC REAL_RING);;
let COMPLEX_INV_CNJ = 
prove (`!z. inv z = cnj z / Cx(norm z) pow 2`,
GEN_TAC THEN ASM_CASES_TAC `z = Cx(&0)` THENL [ASM_REWRITE_TAC[CNJ_CX; complex_div; COMPLEX_INV_0; COMPLEX_MUL_LZERO]; MATCH_MP_TAC(COMPLEX_FIELD `x * y = z /\ ~(x = Cx(&0)) /\ ~(z = Cx(&0)) ==> inv x = y / z`) THEN ASM_REWRITE_TAC[COMPLEX_MUL_CNJ; GSYM CX_POW; CX_INJ; REAL_POW_EQ_0] THEN ASM_REWRITE_TAC[COMPLEX_NORM_ZERO; ARITH]]);;
let COMPLEX_DIV_CNJ = 
prove (`!a b. a / b = (a * cnj b) / Cx(norm b) pow 2`,
REPEAT GEN_TAC THEN REWRITE_TAC[complex_div; GSYM COMPLEX_MUL_ASSOC] THEN AP_TERM_TAC THEN GEN_REWRITE_TAC LAND_CONV [COMPLEX_INV_CNJ] THEN REWRITE_TAC[complex_div]);;
let RE_COMPLEX_DIV_EQ_0 = 
prove (`!a b. Re(a / b) = &0 <=> Re(a * cnj b) = &0`,
REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[COMPLEX_DIV_CNJ] THEN REWRITE_TAC[complex_div; GSYM CX_POW; GSYM CX_INV] THEN REWRITE_TAC[RE_MUL_CX; REAL_INV_EQ_0; REAL_POW_EQ_0; ARITH; REAL_ENTIRE; COMPLEX_NORM_ZERO] THEN ASM_CASES_TAC `b = Cx(&0)` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[CNJ_CX; COMPLEX_MUL_RZERO; RE_CX]);;
let IM_COMPLEX_DIV_EQ_0 = 
prove (`!a b. Im(a / b) = &0 <=> Im(a * cnj b) = &0`,
REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[COMPLEX_DIV_CNJ] THEN REWRITE_TAC[complex_div; GSYM CX_POW; GSYM CX_INV] THEN REWRITE_TAC[IM_MUL_CX; REAL_INV_EQ_0; REAL_POW_EQ_0; ARITH; REAL_ENTIRE; COMPLEX_NORM_ZERO] THEN ASM_CASES_TAC `b = Cx(&0)` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[CNJ_CX; COMPLEX_MUL_RZERO; IM_CX]);;
let RE_COMPLEX_DIV_GT_0 = 
prove (`!a b. &0 < Re(a / b) <=> &0 < Re(a * cnj b)`,
REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[COMPLEX_DIV_CNJ] THEN REWRITE_TAC[complex_div; GSYM CX_POW; GSYM CX_INV] THEN REWRITE_TAC[RE_MUL_CX; REAL_INV_EQ_0; REAL_POW_EQ_0; ARITH; REAL_ENTIRE; COMPLEX_NORM_ZERO] THEN ASM_CASES_TAC `b = Cx(&0)` THEN ASM_REWRITE_TAC[CNJ_CX; COMPLEX_MUL_RZERO; RE_CX; REAL_MUL_LZERO] THEN REWRITE_TAC[REAL_ARITH `&0 < a * x <=> &0 * x < a * x`] THEN ASM_SIMP_TAC[REAL_LT_RMUL_EQ; REAL_LT_INV_EQ; REAL_POW_LT; ARITH; COMPLEX_NORM_NZ]);;
let IM_COMPLEX_DIV_GT_0 = 
prove (`!a b. &0 < Im(a / b) <=> &0 < Im(a * cnj b)`,
REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[COMPLEX_DIV_CNJ] THEN REWRITE_TAC[complex_div; GSYM CX_POW; GSYM CX_INV] THEN REWRITE_TAC[IM_MUL_CX; REAL_INV_EQ_0; REAL_POW_EQ_0; ARITH; REAL_ENTIRE; COMPLEX_NORM_ZERO] THEN ASM_CASES_TAC `b = Cx(&0)` THEN ASM_REWRITE_TAC[CNJ_CX; COMPLEX_MUL_RZERO; IM_CX; REAL_MUL_LZERO] THEN REWRITE_TAC[REAL_ARITH `&0 < a * x <=> &0 * x < a * x`] THEN ASM_SIMP_TAC[REAL_LT_RMUL_EQ; REAL_LT_INV_EQ; REAL_POW_LT; ARITH; COMPLEX_NORM_NZ]);;
let RE_COMPLEX_DIV_GE_0 = 
prove (`!a b. &0 <= Re(a / b) <=> &0 <= Re(a * cnj b)`,
REWRITE_TAC[REAL_ARITH `&0 <= x <=> &0 < x \/ x = &0`] THEN REWRITE_TAC[RE_COMPLEX_DIV_GT_0; RE_COMPLEX_DIV_EQ_0]);;
let IM_COMPLEX_DIV_GE_0 = 
prove (`!a b. &0 <= Im(a / b) <=> &0 <= Im(a * cnj b)`,
REWRITE_TAC[REAL_ARITH `&0 <= x <=> &0 < x \/ x = &0`] THEN REWRITE_TAC[IM_COMPLEX_DIV_GT_0; IM_COMPLEX_DIV_EQ_0]);;
let RE_COMPLEX_DIV_LE_0 = 
prove (`!a b. Re(a / b) <= &0 <=> Re(a * cnj b) <= &0`,
REWRITE_TAC[GSYM REAL_NOT_LT; RE_COMPLEX_DIV_GT_0]);;
let IM_COMPLEX_DIV_LE_0 = 
prove (`!a b. Im(a / b) <= &0 <=> Im(a * cnj b) <= &0`,
REWRITE_TAC[GSYM REAL_NOT_LT; IM_COMPLEX_DIV_GT_0]);;
let RE_COMPLEX_DIV_LT_0 = 
prove (`!a b. Re(a / b) < &0 <=> Re(a * cnj b) < &0`,
REWRITE_TAC[GSYM REAL_NOT_LE; RE_COMPLEX_DIV_GE_0]);;
let IM_COMPLEX_DIV_LT_0 = 
prove (`!a b. Im(a / b) < &0 <=> Im(a * cnj b) < &0`,
REWRITE_TAC[GSYM REAL_NOT_LE; IM_COMPLEX_DIV_GE_0]);;
let IM_COMPLEX_INV_GE_0 = 
prove (`!z. &0 <= Im(inv z) <=> Im(z) <= &0`,
GEN_TAC THEN MP_TAC(ISPECL [`Cx(&1)`; `z:complex`] IM_COMPLEX_DIV_GE_0) THEN REWRITE_TAC[complex_div; COMPLEX_MUL_LID; IM_CNJ] THEN REAL_ARITH_TAC);;
let IM_COMPLEX_INV_LE_0 = 
prove (`!z. Im(inv z) <= &0 <=> &0 <= Im(z)`,
let IM_COMPLEX_INV_GT_0 = 
prove (`!z. &0 < Im(inv z) <=> Im(z) < &0`,
REWRITE_TAC[REAL_ARITH `&0 < a <=> ~(a <= &0)`; IM_COMPLEX_INV_LE_0] THEN REAL_ARITH_TAC);;
let IM_COMPLEX_INV_LT_0 = 
prove (`!z. Im(inv z) < &0 <=> &0 < Im(z)`,
REWRITE_TAC[REAL_ARITH `a < &0 <=> ~(&0 <= a)`; IM_COMPLEX_INV_GE_0] THEN REAL_ARITH_TAC);;
let IM_COMPLEX_INV_EQ_0 = 
prove (`!z. Im(inv z) = &0 <=> Im(z) = &0`,
SIMP_TAC[GSYM REAL_LE_ANTISYM; IM_COMPLEX_INV_LE_0; IM_COMPLEX_INV_GE_0] THEN REAL_ARITH_TAC);;
let REAL_SGN_RE_COMPLEX_DIV = 
prove (`!w z. real_sgn(Re(w / z)) = real_sgn(Re(w * cnj z))`,
REWRITE_TAC[real_sgn; RE_COMPLEX_DIV_GT_0; RE_COMPLEX_DIV_GE_0; REAL_ARITH `x < &0 <=> ~(&0 <= x)`]);;
let REAL_SGN_IM_COMPLEX_DIV = 
prove (`!w z. real_sgn(Im(w / z)) = real_sgn(Im(w * cnj z))`,
REWRITE_TAC[real_sgn; IM_COMPLEX_DIV_GT_0; IM_COMPLEX_DIV_GE_0; REAL_ARITH `x < &0 <=> ~(&0 <= x)`]);;
(* ------------------------------------------------------------------------- *) (* Norm versus components for complex numbers. *) (* ------------------------------------------------------------------------- *)
let COMPLEX_NORM_GE_RE_IM = 
prove (`!z. abs(Re(z)) <= norm(z) /\ abs(Im(z)) <= norm(z)`,
GEN_TAC THEN ONCE_REWRITE_TAC[GSYM POW_2_SQRT_ABS] THEN REWRITE_TAC[complex_norm] THEN CONJ_TAC THEN MATCH_MP_TAC SQRT_MONO_LE THEN ASM_SIMP_TAC[REAL_LE_ADDR; REAL_LE_ADDL; REAL_POW_2; REAL_LE_SQUARE]);;
let COMPLEX_NORM_LE_RE_IM = 
prove (`!z. norm(z) <= abs(Re z) + abs(Im z)`,
GEN_TAC THEN MP_TAC(ISPEC `z:complex` NORM_LE_L1) THEN REWRITE_TAC[DIMINDEX_2; SUM_2; RE_DEF; IM_DEF]);;
let COMPLEX_L1_LE_NORM = 
prove (`!z. sqrt(&2) / &2 * (abs(Re z) + abs(Im z)) <= norm z`,
GEN_TAC THEN MATCH_MP_TAC REAL_LE_LCANCEL_IMP THEN EXISTS_TAC `sqrt(&2)` THEN SIMP_TAC[REAL_ARITH `x * x / &2 * y = (x pow 2) / &2 * y`; SQRT_POW_2; REAL_POS; SQRT_POS_LT; REAL_OF_NUM_LT; ARITH] THEN MP_TAC(ISPEC `z:complex` L1_LE_NORM) THEN REWRITE_TAC[DIMINDEX_2; SUM_2; RE_DEF; IM_DEF] THEN REAL_ARITH_TAC);;
(* ------------------------------------------------------------------------- *) (* Complex square roots. *) (* ------------------------------------------------------------------------- *)
let csqrt = new_definition
  `csqrt(z) = if Im(z) = &0 then
                if &0 <= Re(z) then complex(sqrt(Re(z)),&0)
                else complex(&0,sqrt(--Re(z)))
              else complex(sqrt((norm(z) + Re(z)) / &2),
                           (Im(z) / abs(Im(z))) *
                           sqrt((norm(z) - Re(z)) / &2))`;;
let CSQRT = 
prove (`!z. csqrt(z) pow 2 = z`,
GEN_TAC THEN REWRITE_TAC[COMPLEX_POW_2; csqrt] THEN COND_CASES_TAC THENL [COND_CASES_TAC THEN ASM_REWRITE_TAC[CX_DEF; complex_mul; RE; IM; REAL_MUL_RZERO; REAL_MUL_LZERO; REAL_SUB_LZERO; REAL_SUB_RZERO; REAL_ADD_LID; COMPLEX_EQ] THEN REWRITE_TAC[REAL_NEG_EQ; GSYM REAL_POW_2] THEN ASM_SIMP_TAC[SQRT_POW_2; REAL_ARITH `~(&0 <= x) ==> &0 <= --x`]; ALL_TAC] THEN REWRITE_TAC[complex_mul; RE; IM] THEN ONCE_REWRITE_TAC[REAL_ARITH `(s * s - (i * s') * (i * s') = s * s - (i * i) * (s' * s')) /\ (s * i * s' + (i * s')* s = &2 * i * s * s')`] THEN REWRITE_TAC[GSYM REAL_POW_2] THEN SUBGOAL_THEN `&0 <= norm(z) + Re(z) /\ &0 <= norm(z) - Re(z)` STRIP_ASSUME_TAC THENL [MP_TAC(SPEC `z:complex` COMPLEX_NORM_GE_RE_IM) THEN REAL_ARITH_TAC; ALL_TAC] THEN ASM_SIMP_TAC[REAL_LE_DIV; REAL_POS; GSYM SQRT_MUL; SQRT_POW_2] THEN REWRITE_TAC[COMPLEX_EQ; RE; IM] THEN CONJ_TAC THENL [ASM_SIMP_TAC[REAL_POW_DIV; REAL_POW2_ABS; REAL_POW_EQ_0; REAL_DIV_REFL] THEN REWRITE_TAC[real_div; REAL_MUL_LID; GSYM REAL_SUB_RDISTRIB] THEN REWRITE_TAC[REAL_ARITH `(m + r) - (m - r) = r * &2`] THEN REWRITE_TAC[GSYM REAL_MUL_ASSOC] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[REAL_MUL_RID]; ALL_TAC] THEN REWRITE_TAC[real_div] THEN ONCE_REWRITE_TAC[AC REAL_MUL_AC `(a * b) * a' * b = (a * a') * (b * b:real)`] THEN REWRITE_TAC[REAL_DIFFSQ] THEN REWRITE_TAC[complex_norm; GSYM REAL_POW_2] THEN SIMP_TAC[SQRT_POW_2; REAL_LE_ADD; REWRITE_RULE[GSYM REAL_POW_2] REAL_LE_SQUARE] THEN REWRITE_TAC[REAL_ADD_SUB; GSYM REAL_POW_MUL] THEN REWRITE_TAC[POW_2_SQRT_ABS] THEN REWRITE_TAC[REAL_ABS_MUL; REAL_ABS_INV; REAL_ABS_NUM] THEN ONCE_REWRITE_TAC[AC REAL_MUL_AC `&2 * (i * a') * a * h = i * (&2 * h) * a * a'`] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[REAL_MUL_LID; GSYM real_div] THEN ASM_SIMP_TAC[REAL_DIV_REFL; REAL_ABS_ZERO; REAL_MUL_RID]);;
let CX_SQRT = 
prove (`!x. &0 <= x ==> Cx(sqrt x) = csqrt(Cx x)`,
SIMP_TAC[csqrt; IM_CX; RE_CX; COMPLEX_EQ; RE; IM]);;
let CSQRT_CX = 
prove (`!x. &0 <= x ==> csqrt(Cx x) = Cx(sqrt x)`,
SIMP_TAC[CX_SQRT]);;
let CSQRT_0 = 
prove (`csqrt(Cx(&0)) = Cx(&0)`,
SIMP_TAC[CSQRT_CX; REAL_POS; SQRT_0]);;
let CSQRT_1 = 
prove (`csqrt(Cx(&1)) = Cx(&1)`,
SIMP_TAC[CSQRT_CX; REAL_POS; SQRT_1]);;
let CSQRT_PRINCIPAL = 
prove (`!z. &0 < Re(csqrt(z)) \/ Re(csqrt(z)) = &0 /\ &0 <= Im(csqrt(z))`,
GEN_TAC THEN REWRITE_TAC[csqrt] THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[RE; IM]) THENL [FIRST_ASSUM(MP_TAC o MATCH_MP SQRT_POS_LE) THEN REAL_ARITH_TAC; DISJ2_TAC THEN REWRITE_TAC[real_ge] THEN MATCH_MP_TAC SQRT_POS_LE THEN ASM_REAL_ARITH_TAC; DISJ1_TAC THEN MATCH_MP_TAC SQRT_POS_LT THEN MATCH_MP_TAC(REAL_ARITH `abs(y) < x ==> &0 < (x + y) / &2`) THEN REWRITE_TAC[complex_norm] THEN REWRITE_TAC[GSYM POW_2_SQRT_ABS] THEN MATCH_MP_TAC SQRT_MONO_LT THEN REWRITE_TAC[REAL_POW_2; REAL_LE_SQUARE; REAL_LT_ADDR] THEN REWRITE_TAC[REAL_ARITH `&0 < x <=> &0 <= x /\ ~(x = &0)`] THEN ASM_REWRITE_TAC[REAL_LE_SQUARE; REAL_ENTIRE]]);;
let RE_CSQRT = 
prove (`!z. &0 <= Re(csqrt z)`,
MP_TAC CSQRT_PRINCIPAL THEN MATCH_MP_TAC MONO_FORALL THEN REAL_ARITH_TAC);;
let CSQRT_UNIQUE = 
prove (`!s z. s pow 2 = z /\ (&0 < Re s \/ Re s = &0 /\ &0 <= Im s) ==> csqrt z = s`,
REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN ASSUME_TAC) THEN FIRST_X_ASSUM(SUBST_ALL_TAC o SYM) THEN MP_TAC(SPEC `(s:complex) pow 2` CSQRT) THEN SIMP_TAC[COMPLEX_RING `a pow 2 = b pow 2 <=> a = b \/ a = --b:complex`] THEN STRIP_TAC THEN ASM_REWRITE_TAC[COMPLEX_RING `--z = z <=> z = Cx(&0)`] THEN FIRST_ASSUM(MP_TAC o AP_TERM `Re`) THEN FIRST_X_ASSUM(MP_TAC o AP_TERM `Im`) THEN REWRITE_TAC[RE_NEG; IM_NEG; COMPLEX_EQ; RE_CX; IM_CX] THEN MP_TAC(SPEC `(s:complex) pow 2` CSQRT_PRINCIPAL) THEN POP_ASSUM MP_TAC THEN REAL_ARITH_TAC);;
let POW_2_CSQRT = 
prove (`!z. &0 < Re z \/ Re(z) = &0 /\ &0 <= Im(z) ==> csqrt(z pow 2) = z`,
MESON_TAC[CSQRT_UNIQUE]);;
let CSQRT_EQ_0 = 
prove (`!z. csqrt z = Cx(&0) <=> z = Cx(&0)`,
GEN_TAC THEN MP_TAC (SPEC `z:complex` CSQRT) THEN CONV_TAC COMPLEX_RING);;
(* ------------------------------------------------------------------------- *) (* A few more complex-specific cases of vector notions. *) (* ------------------------------------------------------------------------- *)
let COMPLEX_CMUL = 
prove (`!c x. c % x = Cx(c) * x`,
SIMP_TAC[CART_EQ; VECTOR_MUL_COMPONENT; CX_DEF; complex; complex_mul; DIMINDEX_2; FORALL_2; IM_DEF; RE_DEF; VECTOR_2] THEN REAL_ARITH_TAC);;
let LINEAR_COMPLEX_MUL = 
prove (`!c. linear (\x. c * x)`,
REWRITE_TAC[linear; COMPLEX_CMUL] THEN CONV_TAC COMPLEX_RING);;
let BILINEAR_COMPLEX_MUL = 
prove (`bilinear( * )`,
REWRITE_TAC[bilinear; linear; COMPLEX_CMUL] THEN CONV_TAC COMPLEX_RING);;
let LINEAR_CNJ = 
prove (`linear cnj`,
REWRITE_TAC[linear; COMPLEX_CMUL; CNJ_ADD; CNJ_MUL; CNJ_CX]);;
(* ------------------------------------------------------------------------- *) (* Complex-specific theorems about sums. *) (* ------------------------------------------------------------------------- *)
let RE_VSUM = 
prove (`!f s. FINITE s ==> Re(vsum s f) = sum s (\x. Re(f x))`,
SIMP_TAC[RE_DEF; VSUM_COMPONENT; DIMINDEX_2; ARITH]);;
let IM_VSUM = 
prove (`!f s. FINITE s ==> Im(vsum s f) = sum s (\x. Im(f x))`,
SIMP_TAC[IM_DEF; VSUM_COMPONENT; DIMINDEX_2; ARITH]);;
let VSUM_COMPLEX_LMUL = 
prove (`!c f s. FINITE(s) ==> vsum s (\x. c * f x) = c * vsum s f`,
GEN_TAC THEN GEN_TAC THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN SIMP_TAC[VSUM_CLAUSES; COMPLEX_VEC_0; COMPLEX_MUL_RZERO] THEN SIMPLE_COMPLEX_ARITH_TAC);;
let VSUM_COMPLEX_RMUL = 
prove (`!c f s. FINITE(s) ==> vsum s (\x. f x * c) = vsum s f * c`,
ONCE_REWRITE_TAC[COMPLEX_MUL_SYM] THEN REWRITE_TAC[VSUM_COMPLEX_LMUL]);;
let VSUM_CX = 
prove (`!f:A->real s. FINITE s ==> vsum s (\a. Cx(f a)) = Cx(sum s f)`,
GEN_TAC THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN SIMP_TAC[SUM_CLAUSES; VSUM_CLAUSES; COMPLEX_VEC_0; CX_ADD]);;
let CNJ_VSUM = 
prove (`!f s. FINITE s ==> cnj(vsum s f) = vsum s (\x. cnj(f x))`,
GEN_TAC THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN SIMP_TAC[VSUM_CLAUSES; CNJ_ADD; CNJ_CX; COMPLEX_VEC_0]);;
let VSUM_CX_NUMSEG = 
prove (`!f m n. vsum (m..n) (\a. Cx(f a)) = Cx(sum (m..n) f)`,
SIMP_TAC[VSUM_CX; FINITE_NUMSEG]);;
let COMPLEX_SUB_POW = 
prove (`!x y n. 1 <= n ==> x pow n - y pow n = (x - y) * vsum(0..n-1) (\i. x pow i * y pow (n - 1 - i))`,
SIMP_TAC[GSYM VSUM_COMPLEX_LMUL; FINITE_NUMSEG] THEN REWRITE_TAC[COMPLEX_RING `(x - y) * (a * b):complex = (x * a) * b - a * (y * b)`] THEN SIMP_TAC[GSYM complex_pow; ADD1; ARITH_RULE `1 <= n /\ x <= n - 1 ==> n - 1 - x = n - (x + 1) /\ SUC(n - 1 - x) = n - x`] THEN REWRITE_TAC[VSUM_DIFFS_ALT; LE_0] THEN SIMP_TAC[SUB_0; SUB_ADD; SUB_REFL; complex_pow; COMPLEX_MUL_LID; COMPLEX_MUL_RID]);;
let COMPLEX_SUB_POW_R1 = 
prove (`!x n. 1 <= n ==> x pow n - Cx(&1) = (x - Cx(&1)) * vsum(0..n-1) (\i. x pow i)`,
REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o SPECL [`x:complex`; `Cx(&1)`] o MATCH_MP COMPLEX_SUB_POW) THEN REWRITE_TAC[COMPLEX_POW_ONE; COMPLEX_MUL_RID]);;
let COMPLEX_SUB_POW_L1 = 
prove (`!x n. 1 <= n ==> Cx(&1) - x pow n = (Cx(&1) - x) * vsum(0..n-1) (\i. x pow i)`,
ONCE_REWRITE_TAC[GSYM COMPLEX_NEG_SUB] THEN SIMP_TAC[COMPLEX_SUB_POW_R1] THEN REWRITE_TAC[COMPLEX_MUL_LNEG]);;
(* ------------------------------------------------------------------------- *) (* The complex numbers that are real (zero imaginary part). *) (* ------------------------------------------------------------------------- *)
let real = new_definition
 `real z <=> Im z = &0`;;
let REAL = 
prove (`!z. real z <=> Cx(Re z) = z`,
REWRITE_TAC[COMPLEX_EQ; real; CX_DEF; RE; IM] THEN REAL_ARITH_TAC);;
let REAL_CNJ = 
prove (`!z. real z <=> cnj z = z`,
REWRITE_TAC[real; cnj; COMPLEX_EQ; RE; IM] THEN REAL_ARITH_TAC);;
let REAL_IMP_CNJ = 
prove (`!z. real z ==> cnj z = z`,
REWRITE_TAC[REAL_CNJ]);;
let REAL_EXISTS = 
prove (`!z. real z <=> ?x. z = Cx x`,
MESON_TAC[REAL; real; IM_CX]);;
let FORALL_REAL = 
prove (`(!z. real z ==> P z) <=> (!x. P(Cx x))`,
MESON_TAC[REAL_EXISTS]);;
let EXISTS_REAL = 
prove (`(?z. real z /\ P z) <=> (?x. P(Cx x))`,
MESON_TAC[REAL_EXISTS]);;
let REAL_CX = 
prove (`!x. real(Cx x)`,
REWRITE_TAC[REAL_CNJ; CNJ_CX]);;
let REAL_MUL_CX = 
prove (`!x z. real(Cx x * z) <=> x = &0 \/ real z`,
REWRITE_TAC[real; IM_MUL_CX; REAL_ENTIRE]);;
let REAL_ADD = 
prove (`!w z. real w /\ real z ==> real(w + z)`,
SIMP_TAC[REAL_CNJ; CNJ_ADD]);;
let REAL_NEG = 
prove (`!z. real z ==> real(--z)`,
SIMP_TAC[REAL_CNJ; CNJ_NEG]);;
let REAL_SUB = 
prove (`!w z. real w /\ real z ==> real(w - z)`,
SIMP_TAC[REAL_CNJ; CNJ_SUB]);;
let REAL_MUL = 
prove (`!w z. real w /\ real z ==> real(w * z)`,
SIMP_TAC[REAL_CNJ; CNJ_MUL]);;
let REAL_POW = 
prove (`!z n. real z ==> real(z pow n)`,
SIMP_TAC[REAL_CNJ; CNJ_POW]);;
let REAL_INV = 
prove (`!z. real z ==> real(inv z)`,
SIMP_TAC[REAL_CNJ; CNJ_INV]);;
let REAL_INV_EQ = 
prove (`!z. real(inv z) = real z`,
MESON_TAC[REAL_INV; COMPLEX_INV_INV]);;
let REAL_DIV = 
prove (`!w z. real w /\ real z ==> real(w / z)`,
SIMP_TAC[REAL_CNJ; CNJ_DIV]);;
let REAL_VSUM = 
prove (`!f s. FINITE s /\ (!a. a IN s ==> real(f a)) ==> real(vsum s f)`,
SIMP_TAC[CNJ_VSUM; REAL_CNJ]);;
let REAL_SEGMENT = 
prove (`!a b x. x IN segment[a,b] /\ real a /\ real b ==> real x`,
SIMP_TAC[segment; IN_ELIM_THM; real; COMPLEX_EQ; LEFT_AND_EXISTS_THM; LEFT_IMP_EXISTS_THM; IM_ADD; IM_CMUL] THEN REAL_ARITH_TAC);;
let IN_SEGMENT_CX = 
prove (`!a b x. Cx(x) IN segment[Cx(a),Cx(b)] <=> a <= x /\ x <= b \/ b <= x /\ x <= a`,
REPEAT STRIP_TAC THEN REWRITE_TAC[segment; IN_ELIM_THM] THEN REWRITE_TAC[COMPLEX_CMUL; GSYM CX_ADD; CX_INJ; GSYM CX_MUL] THEN ASM_CASES_TAC `a:real = b` THENL [ASM_REWRITE_TAC[REAL_ARITH `(&1 - u) * b + u * b = b`] THEN ASM_CASES_TAC `x:real = b` THEN ASM_REWRITE_TAC[REAL_LE_ANTISYM] THEN EXISTS_TAC `&0` THEN REWRITE_TAC[REAL_POS]; ALL_TAC] THEN EQ_TAC THENL [DISCH_THEN(X_CHOOSE_THEN `u:real` (CONJUNCTS_THEN2 STRIP_ASSUME_TAC SUBST1_TAC)) THEN REWRITE_TAC[REAL_ARITH `a <= (&1 - u) * a + u * b <=> &0 <= u * (b - a)`; REAL_ARITH `b <= (&1 - u) * a + u * b <=> &0 <= (&1 - u) * (a - b)`; REAL_ARITH `(&1 - u) * a + u * b <= a <=> &0 <= u * (a - b)`; REAL_ARITH `(&1 - u) * a + u * b <= b <=> &0 <= (&1 - u) * (b - a)`] THEN DISJ_CASES_TAC(REAL_ARITH `a <= b \/ b <= a`) THENL [DISJ1_TAC; DISJ2_TAC] THEN CONJ_TAC THEN MATCH_MP_TAC REAL_LE_MUL THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN STRIP_TAC THENL [SUBGOAL_THEN `&0 < b - a` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; EXISTS_TAC `(x - a:real) / (b - a)`]; SUBGOAL_THEN `&0 < a - b` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; EXISTS_TAC `(a - x:real) / (a - b)`]] THEN (CONJ_TAC THENL [ALL_TAC; UNDISCH_TAC `~(a:real = b)` THEN CONV_TAC REAL_FIELD]) THEN ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_LE_RDIV_EQ] THEN ASM_REAL_ARITH_TAC);;
let IN_SEGMENT_CX_GEN = 
prove (`!a b x. x IN segment[Cx a,Cx b] <=> Im(x) = &0 /\ (a <= Re x /\ Re x <= b \/ b <= Re x /\ Re x <= a)`,
REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM real] THEN ASM_CASES_TAC `real x` THENL [FIRST_X_ASSUM(SUBST1_TAC o SYM o REWRITE_RULE[REAL]) THEN REWRITE_TAC[IN_SEGMENT_CX; REAL_CX; RE_CX] THEN REAL_ARITH_TAC; ASM_MESON_TAC[REAL_SEGMENT; REAL_CX]]);;
let RE_POS_SEGMENT = 
prove (`!a b x. x IN segment[a,b] /\ &0 < Re a /\ &0 < Re b ==> &0 < Re x`,
SIMP_TAC[segment; IN_ELIM_THM; real; COMPLEX_EQ; LEFT_AND_EXISTS_THM; LEFT_IMP_EXISTS_THM; RE_ADD; RE_CMUL] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC(REAL_ARITH `&0 <= x /\ &0 <= y /\ ~(x = &0 /\ y = &0) ==> &0 < x + y`) THEN ASM_SIMP_TAC[REAL_LE_MUL; REAL_SUB_LE; REAL_LT_IMP_LE; REAL_ENTIRE] THEN ASM_REAL_ARITH_TAC);;
let CONVEX_REAL = 
prove (`convex real`,
REWRITE_TAC[convex; IN; COMPLEX_CMUL] THEN SIMP_TAC[REAL_ADD; REAL_MUL; REAL_CX]);;
let IMAGE_CX = 
prove (`!s. IMAGE Cx s = {z | real z /\ Re(z) IN s}`,
REWRITE_TAC[EXTENSION; IN_ELIM_THM; IN_IMAGE] THEN MESON_TAC[RE_CX; REAL]);;
(* ------------------------------------------------------------------------- *) (* Useful bound-type theorems for real quantities. *) (* ------------------------------------------------------------------------- *)
let REAL_NORM = 
prove (`!z. real z ==> norm(z) = abs(Re z)`,
SIMP_TAC[real; complex_norm] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[POW_2_SQRT_ABS; REAL_ADD_RID]);;
let REAL_NORM_POS = 
prove (`!z. real z /\ &0 <= Re z ==> norm(z) = Re(z)`,
SIMP_TAC[REAL_NORM] THEN REAL_ARITH_TAC);;
let COMPLEX_NORM_VSUM_SUM_RE = 
prove (`!f s. FINITE s /\ (!x. x IN s ==> real(f x) /\ &0 <= Re(f x)) ==> norm(vsum s f) = sum s (\x. Re(f x))`,
SIMP_TAC[GSYM RE_VSUM] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_NORM_POS THEN ASM_SIMP_TAC[REAL_VSUM; RE_VSUM; SUM_POS_LE]);;
let COMPLEX_NORM_VSUM_BOUND = 
prove (`!s f:A->complex g:A->complex. FINITE s /\ (!x. x IN s ==> real(g x) /\ norm(f x) <= Re(g x)) ==> norm(vsum s f) <= norm(vsum s g)`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `sum s (\x. norm((f:A->complex) x))` THEN ASM_SIMP_TAC[VSUM_NORM] THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `sum s (\x. Re((g:A->complex) x))` THEN ASM_SIMP_TAC[SUM_LE] THEN MATCH_MP_TAC(REAL_ARITH `x:real = y ==> y <= x`) THEN MATCH_MP_TAC COMPLEX_NORM_VSUM_SUM_RE THEN ASM_MESON_TAC[REAL_LE_TRANS; NORM_POS_LE]);;
let COMPLEX_NORM_VSUM_BOUND_SUBSET = 
prove (`!f:A->complex g:A->complex s t. FINITE s /\ t SUBSET s /\ (!x. x IN s ==> real(g x) /\ norm(f x) <= Re(g x)) ==> norm(vsum t f) <= norm(vsum s g)`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `norm(vsum t (g:A->complex))` THEN CONJ_TAC THENL [ASM_MESON_TAC[COMPLEX_NORM_VSUM_BOUND; SUBSET; FINITE_SUBSET];ALL_TAC] THEN SUBGOAL_THEN `norm(vsum t (g:A->complex)) = sum t (\x. Re(g x)) /\ norm(vsum s g) = sum s (\x. Re(g x))` (CONJUNCTS_THEN SUBST1_TAC) THENL [CONJ_TAC THEN MATCH_MP_TAC COMPLEX_NORM_VSUM_SUM_RE; MATCH_MP_TAC SUM_SUBSET THEN REWRITE_TAC[IN_DIFF]] THEN ASM_MESON_TAC[REAL_LE_TRANS; NORM_POS_LE; FINITE_SUBSET; SUBSET]);;
(* ------------------------------------------------------------------------- *) (* Geometric progression. *) (* ------------------------------------------------------------------------- *)
let VSUM_GP_BASIC = 
prove (`!x n. (Cx(&1) - x) * vsum(0..n) (\i. x pow i) = Cx(&1) - x pow (SUC n)`,
GEN_TAC THEN INDUCT_TAC THEN REWRITE_TAC[VSUM_CLAUSES_NUMSEG] THEN REWRITE_TAC[complex_pow; COMPLEX_MUL_RID; LE_0] THEN ASM_REWRITE_TAC[COMPLEX_ADD_LDISTRIB; complex_pow] THEN SIMPLE_COMPLEX_ARITH_TAC);;
let VSUM_GP_MULTIPLIED = 
prove (`!x m n. m <= n ==> ((Cx(&1) - x) * vsum(m..n) (\i. x pow i) = x pow m - x pow (SUC n))`,
REPEAT STRIP_TAC THEN ASM_SIMP_TAC[VSUM_OFFSET_0; COMPLEX_POW_ADD; FINITE_NUMSEG; COMPLEX_MUL_ASSOC; VSUM_GP_BASIC; VSUM_COMPLEX_RMUL] THEN REWRITE_TAC[COMPLEX_SUB_RDISTRIB; GSYM COMPLEX_POW_ADD; COMPLEX_MUL_LID] THEN ASM_SIMP_TAC[ARITH_RULE `m <= n ==> (SUC(n - m) + m = SUC n)`]);;
let VSUM_GP = 
prove (`!x m n. vsum(m..n) (\i. x pow i) = if n < m then Cx(&0) else if x = Cx(&1) then Cx(&((n + 1) - m)) else (x pow m - x pow (SUC n)) / (Cx(&1) - x)`,
REPEAT GEN_TAC THEN DISJ_CASES_TAC(ARITH_RULE `n < m \/ ~(n < m) /\ m <= n:num`) THEN ASM_SIMP_TAC[VSUM_TRIV_NUMSEG; COMPLEX_VEC_0] THEN COND_CASES_TAC THENL [ASM_REWRITE_TAC[COMPLEX_POW_ONE; VSUM_CONST_NUMSEG; COMPLEX_MUL_RID]; ALL_TAC] THEN REWRITE_TAC[COMPLEX_CMUL; COMPLEX_MUL_RID] THEN MATCH_MP_TAC(COMPLEX_FIELD `~(z = Cx(&1)) /\ (Cx(&1) - z) * x = y ==> x = y / (Cx(&1) - z)`) THEN ASM_SIMP_TAC[COMPLEX_DIV_LMUL; COMPLEX_SUB_0; VSUM_GP_MULTIPLIED]);;
let VSUM_GP_OFFSET = 
prove (`!x m n. vsum(m..m+n) (\i. x pow i) = if x = Cx(&1) then Cx(&n) + Cx(&1) else x pow m * (Cx(&1) - x pow (SUC n)) / (Cx(&1) - x)`,
REPEAT GEN_TAC THEN REWRITE_TAC[VSUM_GP; ARITH_RULE `~(m + n < m:num)`] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THENL [REWRITE_TAC[REAL_OF_NUM_ADD; GSYM CX_ADD] THEN AP_TERM_TAC THEN AP_TERM_TAC THEN ARITH_TAC; REWRITE_TAC[complex_div; complex_pow; COMPLEX_POW_ADD] THEN SIMPLE_COMPLEX_ARITH_TAC]);;
(* ------------------------------------------------------------------------- *) (* Basics about polynomial functions: extremal behaviour and root counts. *) (* ------------------------------------------------------------------------- *)
let COMPLEX_SUB_POLYFUN = 
prove (`!a x y n. 1 <= n ==> vsum(0..n) (\i. a i * x pow i) - vsum(0..n) (\i. a i * y pow i) = (x - y) * vsum(0..n-1) (\j. vsum(j+1..n) (\i. a i * y pow (i - j - 1)) * x pow j)`,
REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM VSUM_SUB_NUMSEG; GSYM COMPLEX_SUB_LDISTRIB] THEN GEN_REWRITE_TAC LAND_CONV [MATCH_MP VSUM_CLAUSES_LEFT (SPEC_ALL LE_0)] THEN REWRITE_TAC[COMPLEX_SUB_REFL; complex_pow; COMPLEX_MUL_RZERO; COMPLEX_ADD_LID] THEN SIMP_TAC[COMPLEX_SUB_POW; ADD_CLAUSES] THEN ONCE_REWRITE_TAC[COMPLEX_RING `a * x * s:complex = x * a * s`] THEN SIMP_TAC[VSUM_COMPLEX_LMUL; FINITE_NUMSEG] THEN AP_TERM_TAC THEN SIMP_TAC[GSYM VSUM_COMPLEX_LMUL; GSYM VSUM_COMPLEX_RMUL; FINITE_NUMSEG; VSUM_VSUM_PRODUCT; FINITE_NUMSEG] THEN MATCH_MP_TAC VSUM_EQ_GENERAL_INVERSES THEN REPEAT(EXISTS_TAC `\(x:num,y:num). (y,x)`) THEN REWRITE_TAC[FORALL_IN_GSPEC; IN_ELIM_PAIR_THM; IN_NUMSEG] THEN REWRITE_TAC[ARITH_RULE `a - b - c:num = a - (b + c)`; ADD_SYM] THEN REWRITE_TAC[COMPLEX_MUL_AC] THEN ARITH_TAC);;
let COMPLEX_SUB_POLYFUN_ALT = 
prove (`!a x y n. 1 <= n ==> vsum(0..n) (\i. a i * x pow i) - vsum(0..n) (\i. a i * y pow i) = (x - y) * vsum(0..n-1) (\j. vsum(0..n-j-1) (\k. a(j+k+1) * y pow k) * x pow j)`,
REPEAT STRIP_TAC THEN ASM_SIMP_TAC[COMPLEX_SUB_POLYFUN] THEN AP_TERM_TAC THEN MATCH_MP_TAC VSUM_EQ_NUMSEG THEN X_GEN_TAC `j:num` THEN REPEAT STRIP_TAC THEN REWRITE_TAC[] THEN AP_THM_TAC THEN AP_TERM_TAC THEN MATCH_MP_TAC VSUM_EQ_GENERAL_INVERSES THEN MAP_EVERY EXISTS_TAC [`\i. i - (j + 1)`; `\k. j + k + 1`] THEN REWRITE_TAC[IN_NUMSEG] THEN REPEAT STRIP_TAC THEN TRY(BINOP_TAC THEN AP_TERM_TAC) THEN ASM_ARITH_TAC);;
let COMPLEX_POLYFUN_LINEAR_FACTOR = 
prove (`!a c n. ?b. !z. vsum(0..n) (\i. c(i) * z pow i) = (z - a) * vsum(0..n-1) (\i. b(i) * z pow i) + vsum(0..n) (\i. c(i) * a pow i)`,
REPEAT GEN_TAC THEN REWRITE_TAC[GSYM COMPLEX_EQ_SUB_RADD] THEN ASM_CASES_TAC `n = 0` THENL [EXISTS_TAC `\i:num. Cx(&0)` THEN ASM_SIMP_TAC[VSUM_SING; NUMSEG_SING; complex_pow; COMPLEX_MUL_LZERO] THEN REWRITE_TAC[COMPLEX_SUB_REFL; GSYM COMPLEX_VEC_0; VSUM_0] THEN REWRITE_TAC[COMPLEX_VEC_0; COMPLEX_MUL_RZERO]; ASM_SIMP_TAC[COMPLEX_SUB_POLYFUN; LE_1] THEN EXISTS_TAC `\j. vsum (j + 1..n) (\i. c i * a pow (i - j - 1))` THEN REWRITE_TAC[]]);;
let COMPLEX_POLYFUN_LINEAR_FACTOR_ROOT = 
prove (`!a c n. vsum(0..n) (\i. c(i) * a pow i) = Cx(&0) ==> ?b. !z. vsum(0..n) (\i. c(i) * z pow i) = (z - a) * vsum(0..n-1) (\i. b(i) * z pow i)`,
let COMPLEX_POLYFUN_EXTREMAL_LEMMA = 
prove (`!c n e. &0 < e ==> ?M. !z. M <= norm(z) ==> norm(vsum(0..n) (\i. c(i) * z pow i)) <= e * norm(z) pow (n + 1)`,
GEN_TAC THEN INDUCT_TAC THEN SIMP_TAC[VSUM_CLAUSES_NUMSEG; LE_0] THEN REPEAT STRIP_TAC THENL [REWRITE_TAC[ADD_CLAUSES; complex_pow; REAL_POW_1; COMPLEX_MUL_RID] THEN EXISTS_TAC `norm(c 0:complex) / e` THEN ASM_SIMP_TAC[REAL_LE_LDIV_EQ] THEN REWRITE_TAC[REAL_MUL_AC]; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o C MATCH_MP (REAL_ARITH `&0 < &1 / &2`)) THEN DISCH_THEN(X_CHOOSE_TAC `M:real`) THEN EXISTS_TAC `max M ((&1 / &2 + norm(c(n+1):complex)) / e)` THEN X_GEN_TAC `z:complex` THEN REWRITE_TAC[REAL_MAX_LE] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `z:complex`) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(NORM_ARITH `a + norm(y) <= b ==> norm(x) <= a ==> norm(x + y) <= b`) THEN SIMP_TAC[ADD1; COMPLEX_NORM_MUL; COMPLEX_NORM_POW; GSYM REAL_ADD_RDISTRIB; ARITH_RULE `(n + 1) + 1 = 1 + n + 1`] THEN GEN_REWRITE_TAC (RAND_CONV o RAND_CONV) [REAL_POW_ADD] THEN REWRITE_TAC[REAL_MUL_ASSOC] THEN MATCH_MP_TAC REAL_LE_RMUL THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN ASM_SIMP_TAC[GSYM REAL_LE_LDIV_EQ; REAL_POW_LE; NORM_POS_LE; REAL_POW_1]);;
let COMPLEX_POLYFUN_EXTREMAL = 
prove (`!c n. (!k. k IN 1..n ==> c(k) = Cx(&0)) \/ !B. eventually (\z. norm(vsum(0..n) (\i. c(i) * z pow i)) >= B) at_infinity`,
GEN_TAC THEN MATCH_MP_TAC num_WF THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN ASM_CASES_TAC `n = 0` THEN ASM_REWRITE_TAC[NUMSEG_CLAUSES; ARITH; NOT_IN_EMPTY] THEN MP_TAC(ARITH_RULE `0 <= n`) THEN SIMP_TAC[GSYM NUMSEG_RREC] THEN DISCH_THEN(K ALL_TAC) THEN ASM_CASES_TAC `c(n:num) = Cx(&0)` THENL [FIRST_X_ASSUM(MP_TAC o SPEC `n - 1`) THEN ANTS_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN ASM_SIMP_TAC[GSYM NUMSEG_RREC; LE_1] THEN SIMP_TAC[IN_INSERT; VSUM_CLAUSES; FINITE_NUMSEG; IN_NUMSEG] THEN ASM_REWRITE_TAC[COMPLEX_MUL_LZERO; COMPLEX_ADD_LID; COND_ID] THEN ASM_MESON_TAC[]; DISJ2_TAC THEN MP_TAC(ISPECL [`c:num->complex`; `n - 1`; `norm(c(n:num):complex) / &2`] COMPLEX_POLYFUN_EXTREMAL_LEMMA) THEN ASM_SIMP_TAC[SUB_ADD; LE_1] THEN ASM_SIMP_TAC[COMPLEX_NORM_NZ; REAL_LT_DIV; REAL_OF_NUM_LT; ARITH] THEN SIMP_TAC[IN_INSERT; VSUM_CLAUSES; FINITE_NUMSEG; IN_NUMSEG] THEN ASM_SIMP_TAC[ARITH_RULE `~(n = 0) ==> ~(n <= n - 1)`] THEN DISCH_THEN(X_CHOOSE_TAC `M:real`) THEN X_GEN_TAC `B:real` THEN REWRITE_TAC[EVENTUALLY_AT_INFINITY] THEN EXISTS_TAC `max M (max (&1) ((abs B + &1) / (norm(c(n:num):complex) / &2)))` THEN X_GEN_TAC `z:complex` THEN REWRITE_TAC[real_ge; REAL_MAX_LE] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `z:complex`) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(NORM_ARITH `abs b + &1 <= norm(y) - a ==> norm(x) <= a ==> b <= norm(y + x)`) THEN REWRITE_TAC[COMPLEX_NORM_MUL; COMPLEX_NORM_POW] THEN REWRITE_TAC[REAL_ARITH `c * x - c / &2 * x = x * c / &2`] THEN ASM_SIMP_TAC[GSYM REAL_LE_LDIV_EQ; COMPLEX_NORM_NZ; REAL_LT_DIV; REAL_OF_NUM_LT; ARITH] THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `norm(z:complex) pow 1` THEN CONJ_TAC THENL [ASM_REWRITE_TAC[REAL_POW_1]; ALL_TAC] THEN MATCH_MP_TAC REAL_POW_MONO THEN ASM_SIMP_TAC[LE_1]]);;
let COMPLEX_POLYFUN_ROOTBOUND = 
prove (`!n c. ~(!i. i IN 0..n ==> c(i) = Cx(&0)) ==> FINITE {z | vsum(0..n) (\i. c(i) * z pow i) = Cx(&0)} /\ CARD {z | vsum(0..n) (\i. c(i) * z pow i) = Cx(&0)} <= n`,
REWRITE_TAC[TAUT `~a ==> b <=> a \/ b`] THEN INDUCT_TAC THEN GEN_TAC THENL [SIMP_TAC[NUMSEG_SING; VSUM_SING; IN_SING; complex_pow] THEN ASM_CASES_TAC `c 0 = Cx(&0)` THEN ASM_REWRITE_TAC[COMPLEX_MUL_RID] THEN REWRITE_TAC[EMPTY_GSPEC; FINITE_RULES; CARD_CLAUSES; LE_REFL]; ALL_TAC] THEN ASM_CASES_TAC `{z | vsum(0..SUC n) (\i. c(i) * z pow i) = Cx(&0)} = {}` THEN ASM_REWRITE_TAC[FINITE_RULES; CARD_CLAUSES; LE_0] THEN FIRST_X_ASSUM(X_CHOOSE_THEN `a:complex` MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN REWRITE_TAC[IN_ELIM_THM] THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP COMPLEX_POLYFUN_LINEAR_FACTOR_ROOT) THEN DISCH_THEN(X_CHOOSE_TAC `b:num->complex`) THEN ASM_REWRITE_TAC[COMPLEX_ENTIRE; COMPLEX_SUB_0; SUC_SUB1; SET_RULE `{z | z = a \/ P z} = a INSERT {z | P z}`] THEN FIRST_X_ASSUM(MP_TAC o SPEC `b:num->complex`) THEN STRIP_TAC THEN ASM_SIMP_TAC[CARD_CLAUSES; FINITE_RULES] THENL [DISJ1_TAC; ASM_ARITH_TAC] THEN MP_TAC(SPECL [`c:num->complex`; `SUC n`] COMPLEX_POLYFUN_EXTREMAL) THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MP_TAC o SPEC `Cx(&0)`) THEN ASM_SIMP_TAC[SUC_SUB1; COMPLEX_MUL_LZERO] THEN SIMP_TAC[COMPLEX_POW_ZERO; COND_RAND; COMPLEX_MUL_RZERO] THEN ASM_SIMP_TAC[VSUM_0; GSYM COMPLEX_VEC_0; VSUM_DELTA; IN_NUMSEG; LE_0] THEN REWRITE_TAC[COMPLEX_VEC_0; COMPLEX_MUL_RZERO; COMPLEX_NORM_NUM] THEN REWRITE_TAC[COMPLEX_MUL_RID; real_ge; EVENTUALLY_AT_INFINITY] THEN REPEAT STRIP_TAC THENL [ASM_MESON_TAC[LE_1]; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o SPEC `&1`) THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN MATCH_MP_TAC(TAUT `~a ==> a ==> b`) THEN REWRITE_TAC[NOT_EXISTS_THM; NOT_FORALL_THM] THEN X_GEN_TAC `b:real` THEN MP_TAC(SPEC `b:real` (INST_TYPE [`:2`,`:N`] VECTOR_CHOOSE_SIZE)) THEN ASM_MESON_TAC[NORM_POS_LE; REAL_LE_TOTAL; REAL_LE_TRANS]);;
let COMPLEX_POLYFUN_FINITE_ROOTS = 
prove (`!n c. FINITE {x | vsum(0..n) (\i. c i * x pow i) = Cx(&0)} <=> ?i. i IN 0..n /\ ~(c i = Cx(&0))`,
REPEAT GEN_TAC THEN REWRITE_TAC[TAUT `a /\ ~b <=> ~(a ==> b)`] THEN REWRITE_TAC[GSYM NOT_FORALL_THM] THEN EQ_TAC THEN SIMP_TAC[COMPLEX_POLYFUN_ROOTBOUND] THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN SIMP_TAC[COMPLEX_MUL_LZERO] THEN SIMP_TAC[GSYM COMPLEX_VEC_0; VSUM_0] THEN REWRITE_TAC[SET_RULE `{x | T} = (:complex)`; GSYM INFINITE; EUCLIDEAN_SPACE_INFINITE]);;
let COMPLEX_POLYFUN_EQ_0 = 
prove (`!n c. (!z. vsum(0..n) (\i. c i * z pow i) = Cx(&0)) <=> (!i. i IN 0..n ==> c i = Cx(&0))`,
REPEAT GEN_TAC THEN EQ_TAC THEN DISCH_TAC THENL [GEN_REWRITE_TAC I [TAUT `p <=> ~ ~p`] THEN DISCH_THEN(MP_TAC o MATCH_MP COMPLEX_POLYFUN_ROOTBOUND) THEN ASM_REWRITE_TAC[EUCLIDEAN_SPACE_INFINITE; GSYM INFINITE; DE_MORGAN_THM; SET_RULE `{x | T} = (:complex)`]; ASM_SIMP_TAC[IN_NUMSEG; LE_0; COMPLEX_MUL_LZERO] THEN REWRITE_TAC[GSYM COMPLEX_VEC_0; VSUM_0]]);;
let COMPLEX_POLYFUN_EQ_CONST = 
prove (`!n c k. (!z. vsum(0..n) (\i. c i * z pow i) = k) <=> c 0 = k /\ (!i. i IN 1..n ==> c i = Cx(&0))`,
REPEAT GEN_TAC THEN MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `!x. vsum(0..n) (\i. (if i = 0 then c 0 - k else c i) * x pow i) = Cx(&0)` THEN CONJ_TAC THENL [SIMP_TAC[VSUM_CLAUSES_LEFT; LE_0; complex_pow; COMPLEX_MUL_RID] THEN REWRITE_TAC[COMPLEX_RING `(c - k) + s = Cx(&0) <=> c + s = k`] THEN AP_TERM_TAC THEN ABS_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN AP_TERM_TAC THEN MATCH_MP_TAC VSUM_EQ THEN GEN_TAC THEN REWRITE_TAC[IN_NUMSEG] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[ARITH]; REWRITE_TAC[COMPLEX_POLYFUN_EQ_0; IN_NUMSEG; LE_0] THEN GEN_REWRITE_TAC LAND_CONV [MESON[] `(!n. P n) <=> P 0 /\ (!n. ~(n = 0) ==> P n)`] THEN SIMP_TAC[LE_0; COMPLEX_SUB_0] THEN MESON_TAC[LE_1]]);;
(* ------------------------------------------------------------------------- *) (* Complex products. *) (* ------------------------------------------------------------------------- *)
let cproduct = new_definition
  `cproduct = iterate (( * ):complex->complex->complex)`;;
let NEUTRAL_COMPLEX_MUL = 
prove (`neutral(( * ):complex->complex->complex) = Cx(&1)`,
REWRITE_TAC[neutral] THEN MATCH_MP_TAC SELECT_UNIQUE THEN MESON_TAC[COMPLEX_MUL_LID; COMPLEX_MUL_RID]);;
let MONOIDAL_COMPLEX_MUL = 
prove (`monoidal(( * ):complex->complex->complex)`,
REWRITE_TAC[monoidal; NEUTRAL_COMPLEX_MUL] THEN SIMPLE_COMPLEX_ARITH_TAC);;
let CPRODUCT_CLAUSES = 
prove (`(!f. cproduct {} f = Cx(&1)) /\ (!x f s. FINITE(s) ==> (cproduct (x INSERT s) f = if x IN s then cproduct s f else f(x) * cproduct s f))`,
REWRITE_TAC[cproduct; GSYM NEUTRAL_COMPLEX_MUL] THEN ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN MATCH_MP_TAC ITERATE_CLAUSES THEN REWRITE_TAC[MONOIDAL_COMPLEX_MUL]);;
let CPRODUCT_EQ_0 = 
prove (`!f s. FINITE s ==> (cproduct s f = Cx(&0) <=> ?x. x IN s /\ f(x) = Cx(&0))`,
GEN_TAC THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN SIMP_TAC[CPRODUCT_CLAUSES; COMPLEX_ENTIRE; IN_INSERT; CX_INJ; REAL_OF_NUM_EQ; ARITH; NOT_IN_EMPTY] THEN MESON_TAC[]);;
let CPRODUCT_INV = 
prove (`!f s. FINITE s ==> cproduct s (\x. inv(f x)) = inv(cproduct s f)`,
GEN_TAC THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN SIMP_TAC[CPRODUCT_CLAUSES; COMPLEX_INV_1; COMPLEX_INV_MUL]);;
let CPRODUCT_MUL = 
prove (`!f g s. FINITE s ==> cproduct s (\x. f x * g x) = cproduct s f * cproduct s g`,
GEN_TAC THEN GEN_TAC THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN SIMP_TAC[CPRODUCT_CLAUSES; COMPLEX_MUL_AC; COMPLEX_MUL_LID]);;
let CPRODUCT_EQ_1 = 
prove (`!f s. (!x:A. x IN s ==> (f(x) = Cx(&1))) ==> (cproduct s f = Cx(&1))`,
REWRITE_TAC[cproduct; GSYM NEUTRAL_COMPLEX_MUL] THEN SIMP_TAC[ITERATE_EQ_NEUTRAL; MONOIDAL_COMPLEX_MUL]);;
let CPRODUCT_1 = 
prove (`!s. cproduct s (\n. Cx(&1)) = Cx(&1)`,
SIMP_TAC[CPRODUCT_EQ_1]);;
let CPRODUCT_POW = 
prove (`!f s n. FINITE s ==> cproduct s (\x. f x pow n) = (cproduct s f) pow n`,
GEN_TAC THEN GEN_TAC THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM] THEN DISCH_TAC THEN INDUCT_TAC THEN ASM_SIMP_TAC[complex_pow; CPRODUCT_MUL; CPRODUCT_1]);;
let NORM_CPRODUCT = 
prove (`!f s. FINITE s ==> norm(cproduct s f) = product s (\x. norm(f x))`,
let CPRODUCT_EQ = 
prove (`!f g s. (!x. x IN s ==> (f x = g x)) ==> cproduct s f = cproduct s g`,
REWRITE_TAC[cproduct] THEN MATCH_MP_TAC ITERATE_EQ THEN REWRITE_TAC[MONOIDAL_COMPLEX_MUL]);;