(* ========================================================================= *)
(* Complex transcendentals and their real counterparts.                      *)
(*                                                                           *)
(*              (c) Copyright, John Harrison 1998-2008                       *)
(* ========================================================================= *)

needs "Multivariate/determinants.ml";;
needs "Multivariate/canal.ml";;

prioritize_complex();;

(* ------------------------------------------------------------------------- *)
(* The complex exponential function.                                         *)
(* ------------------------------------------------------------------------- *)

let cexp = new_definition
 `cexp z = infsum (from 0) (\n. z pow n / Cx(&(FACT n)))`;;
let CEXP_0 = 
prove (`cexp(Cx(&0)) = Cx(&1)`,
REWRITE_TAC[cexp] THEN MATCH_MP_TAC INFSUM_UNIQUE THEN MP_TAC(ISPECL [`\i. Cx(&0) pow i / Cx(&(FACT i))`; `{0}`; `from 0`] SERIES_FINITE_SUPPORT) THEN SIMP_TAC[FROM_0; INTER_UNIV; FINITE_INSERT; FINITE_RULES] THEN ANTS_TAC THENL [INDUCT_TAC THEN REWRITE_TAC[IN_SING; NOT_SUC] THEN REWRITE_TAC[complex_div; complex_pow; COMPLEX_MUL_LZERO; COMPLEX_VEC_0]; REWRITE_TAC[VSUM_SING; FACT; COMPLEX_DIV_1; complex_pow]]);;
let CEXP_CONVERGES_UNIFORMLY_CAUCHY = 
prove (`!R e. &0 < e /\ &0 < R ==> ?N. !m n z. m >= N /\ norm(z) <= R ==> norm(vsum(m..n) (\i. z pow i / Cx(&(FACT i)))) < e`,
REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`&1 / &2`; `\i. Cx(R) pow i / Cx(&(FACT i))`; `from 0`] SERIES_RATIO) THEN REWRITE_TAC[SERIES_CAUCHY; LEFT_FORALL_IMP_THM] THEN MP_TAC(SPEC `&2 * norm(Cx(R))` REAL_ARCH_SIMPLE) THEN REWRITE_TAC[COMPLEX_NORM_CX; COMPLEX_NORM_DIV; COMPLEX_NORM_POW] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN MATCH_MP_TAC(TAUT `(a ==> b) /\ (c ==> d) ==> a ==> (b ==> c) ==> d`) THEN CONJ_TAC THENL [MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `N:num` THEN DISCH_TAC THEN X_GEN_TAC `n:num` THEN REWRITE_TAC[GE] THEN DISCH_TAC THEN SIMP_TAC[FACT; real_pow; GSYM REAL_OF_NUM_MUL; real_div; REAL_INV_MUL] THEN REWRITE_TAC[REAL_ARITH `(z * zn) * (is * ik) <= (&1 * inv(&2)) * zn * ik <=> &0 <= (&1 - (&2 * z) * is) * zn * ik`] THEN MATCH_MP_TAC REAL_LE_MUL THEN SIMP_TAC[REAL_LE_MUL; REAL_POS; REAL_POW_LE; REAL_SUB_LE; REAL_LE_INV_EQ; REAL_ABS_POS] THEN ASM_SIMP_TAC[GSYM real_div; REAL_LE_LDIV_EQ; REAL_OF_NUM_LT; LT_0] THEN REPEAT(POP_ASSUM MP_TAC) THEN REWRITE_TAC[GSYM REAL_OF_NUM_LE; GSYM REAL_OF_NUM_SUC] THEN REAL_ARITH_TAC; DISCH_THEN(MP_TAC o SPEC `e:real`) THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `N:num` THEN REWRITE_TAC[FROM_0; INTER_UNIV] THEN REPEAT(MATCH_MP_TAC MONO_FORALL THEN GEN_TAC) THEN DISCH_THEN(fun th -> REPEAT STRIP_TAC THEN MP_TAC th) THEN ASM_SIMP_TAC[GSYM CX_DIV; GSYM CX_POW; VSUM_CX_NUMSEG; COMPLEX_NORM_CX] THEN MATCH_MP_TAC(REAL_ARITH `x <= y ==> y < e ==> x < e`) THEN SUBGOAL_THEN `abs (sum (m..n) (\i. R pow i / &(FACT i))) = sum (m..n) (\i. R pow i / &(FACT i))` SUBST1_TAC THENL [REWRITE_TAC[REAL_ABS_REFL] THEN MATCH_MP_TAC SUM_POS_LE_NUMSEG THEN ASM_SIMP_TAC[REAL_LT_IMP_LE;REAL_LT_DIV; REAL_OF_NUM_LT; FACT_LT; REAL_POW_LT]; ALL_TAC] THEN MATCH_MP_TAC VSUM_NORM_LE THEN REWRITE_TAC[IN_NUMSEG; FINITE_NUMSEG] THEN X_GEN_TAC `i:num` THEN REWRITE_TAC[COMPLEX_NORM_DIV; COMPLEX_NORM_POW; COMPLEX_NORM_CX] THEN SIMP_TAC[REAL_ABS_NUM; REAL_LE_DIV2_EQ; REAL_OF_NUM_LT; FACT_LT] THEN ASM_SIMP_TAC[REAL_POW_LE2; NORM_POS_LE]]);;
let CEXP_CONVERGES = 
prove (`!z. ((\n. z pow n / Cx(&(FACT n))) sums cexp(z)) (from 0)`,
GEN_TAC THEN REWRITE_TAC[cexp; SUMS_INFSUM; summable; SERIES_CAUCHY] THEN REWRITE_TAC[FROM_0; INTER_UNIV] THEN MP_TAC(SPEC `norm(z:complex) + &1` CEXP_CONVERGES_UNIFORMLY_CAUCHY) THEN SIMP_TAC[REAL_ARITH `&0 <= x ==> &0 < x + &1`; NORM_POS_LE] THEN MESON_TAC[REAL_ARITH `x <= x + &1`]);;
let CEXP_CONVERGES_UNIQUE = 
prove (`!w z. ((\n. z pow n / Cx(&(FACT n))) sums w) (from 0) <=> w = cexp(z)`,
REPEAT GEN_TAC THEN EQ_TAC THEN SIMP_TAC[CEXP_CONVERGES] THEN DISCH_THEN(MP_TAC o C CONJ (SPEC `z:complex` CEXP_CONVERGES)) THEN REWRITE_TAC[SERIES_UNIQUE]);;
let CEXP_CONVERGES_UNIFORMLY = 
prove (`!R e. &0 < R /\ &0 < e ==> ?N. !n z. n >= N /\ norm(z) < R ==> norm(vsum(0..n) (\i. z pow i / Cx(&(FACT i))) - cexp(z)) <= e`,
REPEAT STRIP_TAC THEN MP_TAC(SPECL [`R:real`; `e / &2`] CEXP_CONVERGES_UNIFORMLY_CAUCHY) THEN ASM_REWRITE_TAC[REAL_HALF] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `N:num` THEN DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`n:num`; `z:complex`] THEN STRIP_TAC THEN MP_TAC(SPEC `z:complex` CEXP_CONVERGES) THEN REWRITE_TAC[sums; LIM_SEQUENTIALLY; FROM_0; INTER_UNIV; dist] THEN DISCH_THEN(MP_TAC o SPEC `e / &2`) THEN ASM_REWRITE_TAC[REAL_HALF] THEN DISCH_THEN(X_CHOOSE_THEN `M:num` (MP_TAC o SPEC `n + M + 1`)) THEN FIRST_X_ASSUM(MP_TAC o SPECL [`n + 1`; `n + M + 1`; `z:complex`]) THEN ASM_SIMP_TAC[ARITH_RULE `(n >= N ==> n + 1 >= N) /\ M <= n + M + 1`] THEN ASM_SIMP_TAC[REAL_LT_IMP_LE; VSUM_ADD_SPLIT; LE_0] THEN CONV_TAC(ONCE_DEPTH_CONV(ALPHA_CONV `i:num`)) THEN NORM_ARITH_TAC);;
let HAS_COMPLEX_DERIVATIVE_CEXP = 
prove (`!z. (cexp has_complex_derivative cexp(z)) (at z)`,
REPEAT GEN_TAC THEN MP_TAC(ISPECL [`ball(Cx(&0),norm(z:complex) + &1)`; `\n z. z pow n / Cx(&(FACT n))`; `\n z. if n = 0 then Cx(&0) else z pow (n-1) / Cx(&(FACT(n-1)))`; `cexp:complex->complex`; `(from 0)`] HAS_COMPLEX_DERIVATIVE_SERIES) THEN REWRITE_TAC[CONVEX_BALL; OPEN_BALL; IN_BALL; dist] THEN SIMP_TAC[HAS_COMPLEX_DERIVATIVE_WITHIN_OPEN; OPEN_BALL; IN_BALL; dist; COMPLEX_SUB_LZERO; COMPLEX_SUB_RZERO; NORM_NEG] THEN ANTS_TAC THEN REPEAT CONJ_TAC THENL [X_GEN_TAC `n:num` THEN REPEAT STRIP_TAC THEN COMPLEX_DIFF_TAC THEN SPEC_TAC(`n:num`,`n:num`) THEN INDUCT_TAC THEN REWRITE_TAC[ARITH; complex_div; COMPLEX_MUL_LZERO] THEN MP_TAC(SPECL [`&n + &1`; `&0`] CX_INJ) THEN REWRITE_TAC[NOT_SUC; SUC_SUB1; GSYM REAL_OF_NUM_SUC; FACT; CX_ADD; CX_MUL; GSYM REAL_OF_NUM_MUL; COMPLEX_INV_MUL] THEN REWRITE_TAC[REAL_ARITH `~(&n + &1 = &0)`] THEN ABBREV_TAC `a = inv(Cx(&(FACT n)))` THEN CONV_TAC COMPLEX_FIELD; REPEAT STRIP_TAC THEN MP_TAC(SPECL [`norm(z:complex) + &1`; `e:real`] CEXP_CONVERGES_UNIFORMLY) THEN ASM_SIMP_TAC[NORM_POS_LE; REAL_ARITH `&0 <= x ==> &0 < x + &1`] THEN DISCH_THEN(X_CHOOSE_TAC `N:num`) THEN EXISTS_TAC `N + 1` THEN MAP_EVERY X_GEN_TAC [`n:num`; `w:complex`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`n - 1`; `w:complex`]) THEN ASM_SIMP_TAC[ARITH_RULE `n >= m + 1 ==> n - 1 >= m`] THEN REWRITE_TAC[FROM_0; INTER_UNIV] THEN MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN AP_TERM_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN SUBGOAL_THEN `0..n = 0 INSERT (IMAGE SUC (0..n-1))` SUBST1_TAC THENL [REWRITE_TAC[EXTENSION; IN_INSERT; IN_IMAGE; IN_NUMSEG] THEN INDUCT_TAC THEN REWRITE_TAC[LE_0; NOT_SUC; SUC_INJ; UNWIND_THM1] THEN UNDISCH_TAC `n >= N + 1` THEN ARITH_TAC; ALL_TAC] THEN SIMP_TAC[VSUM_CLAUSES; FINITE_IMAGE; FINITE_NUMSEG] THEN REWRITE_TAC[IN_IMAGE; NOT_SUC; COMPLEX_ADD_LID] THEN SIMP_TAC[VSUM_IMAGE; FINITE_NUMSEG; SUC_INJ] THEN MATCH_MP_TAC VSUM_EQ THEN SIMP_TAC[IN_NUMSEG; NOT_SUC; o_THM; SUC_SUB1]; MAP_EVERY EXISTS_TAC [`Cx(&0)`; `cexp(Cx(&0))`] THEN REWRITE_TAC[CEXP_CONVERGES; COMPLEX_NORM_0] THEN SIMP_TAC[REAL_ARITH `&0 <= z ==> &0 < z + &1`; NORM_POS_LE]; DISCH_THEN(X_CHOOSE_THEN `g:complex->complex` MP_TAC) THEN REWRITE_TAC[CEXP_CONVERGES_UNIQUE] THEN STRIP_TAC THEN MATCH_MP_TAC HAS_COMPLEX_DERIVATIVE_TRANSFORM_AT THEN MAP_EVERY EXISTS_TAC [`g:complex->complex`; `&1`] THEN REWRITE_TAC[REAL_LT_01] THEN CONJ_TAC THENL [ALL_TAC; FIRST_X_ASSUM(MP_TAC o SPEC `z:complex`) THEN ANTS_TAC THENL [REAL_ARITH_TAC; SIMP_TAC[]]] THEN POP_ASSUM MP_TAC THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `w:complex` THEN MATCH_MP_TAC MONO_IMP THEN SIMP_TAC[] THEN NORM_ARITH_TAC]);;
let COMPLEX_DIFFERENTIABLE_AT_CEXP = 
prove (`!z. cexp complex_differentiable at z`,
REWRITE_TAC[complex_differentiable] THEN MESON_TAC[HAS_COMPLEX_DERIVATIVE_CEXP]);;
let COMPLEX_DIFFERENTIABLE_WITHIN_CEXP = 
prove (`!s z. cexp complex_differentiable (at z within s)`,
let CONTINUOUS_AT_CEXP = 
prove (`!z. cexp continuous at z`,
let CONTINUOUS_WITHIN_CEXP = 
prove (`!s z. cexp continuous (at z within s)`,
let CONTINUOUS_ON_CEXP = 
prove (`!s. cexp continuous_on s`,
let HOLOMORPHIC_ON_CEXP = 
prove (`!s. cexp holomorphic_on s`,
(* ------------------------------------------------------------------------- *) (* Add it to the database. *) (* ------------------------------------------------------------------------- *) add_complex_differentiation_theorems (CONJUNCTS(REWRITE_RULE[FORALL_AND_THM] (MATCH_MP HAS_COMPLEX_DERIVATIVE_CHAIN_UNIV HAS_COMPLEX_DERIVATIVE_CEXP)));; (* ------------------------------------------------------------------------- *) (* Hence the main results. *) (* ------------------------------------------------------------------------- *)
let CEXP_ADD_MUL = 
prove (`!w z. cexp(w + z) * cexp(--z) = cexp(w)`,
GEN_TAC THEN ONCE_REWRITE_TAC[SET_RULE `(!x. P x) <=> (!x. x IN UNIV ==> P x)`] THEN MATCH_MP_TAC HAS_COMPLEX_DERIVATIVE_ZERO_UNIQUE THEN EXISTS_TAC `Cx(&0)` THEN REWRITE_TAC[OPEN_UNIV; CONVEX_UNIV; IN_UNIV] THEN REWRITE_TAC[COMPLEX_ADD_RID; COMPLEX_NEG_0; CEXP_0; COMPLEX_MUL_RID] THEN GEN_TAC THEN COMPLEX_DIFF_TAC THEN CONV_TAC COMPLEX_RING);;
let CEXP_NEG_RMUL = 
prove (`!z. cexp(z) * cexp(--z) = Cx(&1)`,
MP_TAC(SPEC `Cx(&0)` CEXP_ADD_MUL) THEN MATCH_MP_TAC MONO_FORALL THEN SIMP_TAC[COMPLEX_ADD_LID; CEXP_0]);;
let CEXP_NEG_LMUL = 
prove (`!z. cexp(--z) * cexp(z) = Cx(&1)`,
ONCE_REWRITE_TAC[COMPLEX_MUL_SYM] THEN REWRITE_TAC[CEXP_NEG_RMUL]);;
let CEXP_NEG = 
prove (`!z. cexp(--z) = inv(cexp z)`,
MP_TAC CEXP_NEG_LMUL THEN MATCH_MP_TAC MONO_FORALL THEN CONV_TAC COMPLEX_FIELD);;
let CEXP_ADD = 
prove (`!w z. cexp(w + z) = cexp(w) * cexp(z)`,
REPEAT GEN_TAC THEN MP_TAC(SPECL [`w:complex`; `z:complex`] CEXP_ADD_MUL) THEN MP_TAC(SPEC `z:complex` CEXP_NEG_LMUL) THEN CONV_TAC COMPLEX_FIELD);;
let CEXP_SUB = 
prove (`!w z. cexp(w - z) = cexp(w) / cexp(z)`,
REPEAT GEN_TAC THEN REWRITE_TAC[complex_sub; complex_div; CEXP_ADD; CEXP_NEG]);;
let CEXP_NZ = 
prove (`!z. ~(cexp(z) = Cx(&0))`,
MP_TAC CEXP_NEG_LMUL THEN MATCH_MP_TAC MONO_FORALL THEN CONV_TAC COMPLEX_FIELD);;
let CEXP_N = 
prove (`!n x. cexp(Cx(&n) * x) = cexp(x) pow n`,
INDUCT_TAC THEN REWRITE_TAC[GSYM REAL_OF_NUM_SUC; CX_ADD] THEN REWRITE_TAC[COMPLEX_MUL_LZERO; complex_pow; CEXP_0] THEN ASM_REWRITE_TAC[COMPLEX_ADD_RDISTRIB; CEXP_ADD; COMPLEX_MUL_LID] THEN REWRITE_TAC[COMPLEX_MUL_AC]);;
let CEXP_VSUM = 
prove (`!f s. FINITE s ==> cexp(vsum s f) = cproduct s (\x. cexp(f x))`,
GEN_TAC THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN SIMP_TAC[VSUM_CLAUSES; CPRODUCT_CLAUSES; CEXP_ADD; COMPLEX_VEC_0; CEXP_0]);;
let LIM_CEXP_MINUS_1 = 
prove (`((\z. (cexp(z) - Cx(&1)) / z) --> Cx(&1)) (at (Cx(&0)))`,
MP_TAC(COMPLEX_DIFF_CONV `((\z. cexp(z) - Cx(&1)) has_complex_derivative f') (at(Cx(&0)))`) THEN REWRITE_TAC[HAS_COMPLEX_DERIVATIVE_AT; CEXP_0; COMPLEX_SUB_REFL] THEN REWRITE_TAC[COMPLEX_MUL_LID; COMPLEX_SUB_RZERO]);;
(* ------------------------------------------------------------------------- *) (* Crude bounds on complex exponential function, usable to get tighter ones. *) (* ------------------------------------------------------------------------- *)
let CEXP_BOUND_BLEMMA = 
prove (`!B. (!z. norm(z) <= &1 / &2 ==> norm(cexp z) <= B) ==> !z. norm(z) <= &1 / &2 ==> norm(cexp z) <= &1 + B / &2`,
REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`cexp`; `cexp`; `cball(Cx(&0),&1 / &2)`; `B:real`] COMPLEX_DIFFERENTIABLE_BOUND) THEN ASM_SIMP_TAC[CONVEX_CBALL; IN_CBALL; dist; COMPLEX_SUB_LZERO; NORM_NEG; HAS_COMPLEX_DERIVATIVE_AT_WITHIN; HAS_COMPLEX_DERIVATIVE_CEXP] THEN DISCH_THEN(MP_TAC o SPECL [`z:complex`; `Cx(&0)`]) THEN REWRITE_TAC[COMPLEX_NORM_0; CEXP_0; COMPLEX_SUB_RZERO] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(NORM_ARITH `norm(y) = &1 /\ d <= e ==> norm(x - y) <= d ==> norm(x) <= &1 + e`) THEN REWRITE_TAC[COMPLEX_NORM_CX; real_div; REAL_ABS_NUM] THEN MATCH_MP_TAC REAL_LE_LMUL THEN FIRST_X_ASSUM(MP_TAC o SPEC `Cx(&0)`) THEN REWRITE_TAC[COMPLEX_NORM_CX] THEN POP_ASSUM MP_TAC THEN NORM_ARITH_TAC);;
let CEXP_BOUND_HALF = 
prove (`!z. norm(z) <= &1 / &2 ==> norm(cexp z) <= &2`,
REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`IMAGE cexp (cball(Cx(&0),&1 / &2))`; `Cx(&0)`] DISTANCE_ATTAINS_SUP) THEN SIMP_TAC[COMPACT_CONTINUOUS_IMAGE; COMPACT_CBALL; CONTINUOUS_ON_CEXP; IMAGE_EQ_EMPTY; CBALL_EQ_EMPTY; FORALL_IN_IMAGE; EXISTS_IN_IMAGE; IN_CBALL; dist; COMPLEX_SUB_LZERO; NORM_NEG] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN DISCH_THEN(X_CHOOSE_THEN `w:complex` STRIP_ASSUME_TAC) THEN FIRST_ASSUM(MP_TAC o SPEC `w:complex` o MATCH_MP CEXP_BOUND_BLEMMA) THEN FIRST_X_ASSUM(MP_TAC o SPEC `z:complex`) THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC);;
let CEXP_BOUND_LEMMA = 
prove (`!z. norm(z) <= &1 / &2 ==> norm(cexp z) <= &1 + &2 * norm(z)`,
REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`cexp`; `cexp`; `cball(Cx(&0),&1 / &2)`; `&2`] COMPLEX_DIFFERENTIABLE_BOUND) THEN ASM_SIMP_TAC[CONVEX_CBALL; IN_CBALL; dist; COMPLEX_SUB_LZERO; NORM_NEG; HAS_COMPLEX_DERIVATIVE_AT_WITHIN; HAS_COMPLEX_DERIVATIVE_CEXP; CEXP_BOUND_HALF] THEN DISCH_THEN(MP_TAC o SPECL [`z:complex`; `Cx(&0)`]) THEN REWRITE_TAC[COMPLEX_NORM_0; CEXP_0; COMPLEX_SUB_RZERO] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(NORM_ARITH `norm(y) = &1 ==> norm(x - y) <= d ==> norm(x) <= &1 + d`) THEN REWRITE_TAC[COMPLEX_NORM_CX; REAL_ABS_NUM]);;
(* ------------------------------------------------------------------------- *) (* Complex trig functions. *) (* ------------------------------------------------------------------------- *)
let ccos = new_definition
  `ccos z = (cexp(ii * z) + cexp(--ii * z)) / Cx(&2)`;;
let csin = new_definition
  `csin z = (cexp(ii * z) - cexp(--ii * z)) / (Cx(&2) * ii)`;;
let CSIN_0 = 
prove (`csin(Cx(&0)) = Cx(&0)`,
REWRITE_TAC[csin; COMPLEX_MUL_RZERO; COMPLEX_SUB_REFL] THEN CONV_TAC COMPLEX_FIELD);;
let CCOS_0 = 
prove (`ccos(Cx(&0)) = Cx(&1)`,
REWRITE_TAC[ccos; COMPLEX_MUL_RZERO; CEXP_0] THEN CONV_TAC COMPLEX_FIELD);;
let CSIN_CIRCLE = 
prove (`!z. csin(z) pow 2 + ccos(z) pow 2 = Cx(&1)`,
GEN_TAC THEN REWRITE_TAC[csin; ccos] THEN MP_TAC(SPEC `ii * z` CEXP_NEG_LMUL) THEN REWRITE_TAC[COMPLEX_MUL_LNEG] THEN CONV_TAC COMPLEX_FIELD);;
let CSIN_ADD = 
prove (`!w z. csin(w + z) = csin(w) * ccos(z) + ccos(w) * csin(z)`,
REPEAT GEN_TAC THEN REWRITE_TAC[csin; ccos; COMPLEX_ADD_LDISTRIB; CEXP_ADD] THEN CONV_TAC COMPLEX_FIELD);;
let CCOS_ADD = 
prove (`!w z. ccos(w + z) = ccos(w) * ccos(z) - csin(w) * csin(z)`,
REPEAT GEN_TAC THEN REWRITE_TAC[csin; ccos; COMPLEX_ADD_LDISTRIB; CEXP_ADD] THEN CONV_TAC COMPLEX_FIELD);;
let CSIN_NEG = 
prove (`!z. csin(--z) = --(csin(z))`,
REWRITE_TAC[csin; COMPLEX_MUL_LNEG; COMPLEX_MUL_RNEG; COMPLEX_NEG_NEG] THEN CONV_TAC COMPLEX_FIELD);;
let CCOS_NEG = 
prove (`!z. ccos(--z) = ccos(z)`,
REWRITE_TAC[ccos; COMPLEX_MUL_LNEG; COMPLEX_MUL_RNEG; COMPLEX_NEG_NEG] THEN CONV_TAC COMPLEX_FIELD);;
let CSIN_DOUBLE = 
prove (`!z. csin(Cx(&2) * z) = Cx(&2) * csin(z) * ccos(z)`,
REWRITE_TAC[COMPLEX_RING `Cx(&2) * x = x + x`; CSIN_ADD] THEN CONV_TAC COMPLEX_RING);;
let CCOS_DOUBLE = 
prove (`!z. ccos(Cx(&2) * z) = (ccos(z) pow 2) - (csin(z) pow 2)`,
REWRITE_TAC[COMPLEX_RING `Cx(&2) * x = x + x`; CCOS_ADD] THEN CONV_TAC COMPLEX_RING);;
let CSIN_SUB = 
prove (`!w z. csin(w - z) = csin(w) * ccos(z) - ccos(w) * csin(z)`,
let CCOS_SUB = 
prove (`!w z. ccos(w - z) = ccos(w) * ccos(z) + csin(w) * csin(z)`,
let COMPLEX_MUL_CSIN_CSIN = 
prove (`!w z. csin(w) * csin(z) = (ccos(w - z) - ccos(w + z)) / Cx(&2)`,
REWRITE_TAC[CCOS_ADD; CCOS_SUB] THEN CONV_TAC COMPLEX_RING);;
let COMPLEX_MUL_CSIN_CCOS = 
prove (`!w z. csin(w) * ccos(z) = (csin(w + z) + csin(w - z)) / Cx(&2)`,
REWRITE_TAC[CSIN_ADD; CSIN_SUB] THEN CONV_TAC COMPLEX_RING);;
let COMPLEX_MUL_CCOS_CSIN = 
prove (`!w z. ccos(w) * csin(z) = (csin(w + z) - csin(w - z)) / Cx(&2)`,
REWRITE_TAC[CSIN_ADD; CSIN_SUB] THEN CONV_TAC COMPLEX_RING);;
let COMPLEX_MUL_CCOS_CCOS = 
prove (`!w z. ccos(w) * ccos(z) = (ccos(w - z) + ccos(w + z)) / Cx(&2)`,
REWRITE_TAC[CCOS_ADD; CCOS_SUB] THEN CONV_TAC COMPLEX_RING);;
let COMPLEX_ADD_CSIN = 
prove (`!w z. csin(w) + csin(z) = Cx(&2) * csin((w + z) / Cx(&2)) * ccos((w - z) / Cx(&2))`,
SIMP_TAC[COMPLEX_MUL_CSIN_CCOS; COMPLEX_RING `Cx(&2) * x / Cx(&2) = x`] THEN REPEAT GEN_TAC THEN BINOP_TAC THEN AP_TERM_TAC THEN CONV_TAC COMPLEX_RING);;
let COMPLEX_SUB_CSIN = 
prove (`!w z. csin(w) - csin(z) = Cx(&2) * csin((w - z) / Cx(&2)) * ccos((w + z) / Cx(&2))`,
SIMP_TAC[COMPLEX_MUL_CSIN_CCOS; COMPLEX_RING `Cx(&2) * x / Cx(&2) = x`] THEN REPEAT GEN_TAC THEN REWRITE_TAC[complex_sub; GSYM CSIN_NEG] THEN BINOP_TAC THEN AP_TERM_TAC THEN CONV_TAC COMPLEX_RING);;
let COMPLEX_ADD_CCOS = 
prove (`!w z. ccos(w) + ccos(z) = Cx(&2) * ccos((w + z) / Cx(&2)) * ccos((w - z) / Cx(&2))`,
SIMP_TAC[COMPLEX_MUL_CCOS_CCOS; COMPLEX_RING `Cx(&2) * x / Cx(&2) = x`] THEN REPEAT GEN_TAC THEN GEN_REWRITE_TAC LAND_CONV [COMPLEX_ADD_SYM] THEN BINOP_TAC THEN AP_TERM_TAC THEN CONV_TAC COMPLEX_RING);;
let COMPLEX_SUB_CCOS = 
prove (`!w z. ccos(w) - ccos(z) = Cx(&2) * csin((w + z) / Cx(&2)) * csin((z - w) / Cx(&2))`,
SIMP_TAC[COMPLEX_MUL_CSIN_CSIN; COMPLEX_RING `Cx(&2) * x / Cx(&2) = x`] THEN REPEAT GEN_TAC THEN BINOP_TAC THEN AP_TERM_TAC THEN CONV_TAC COMPLEX_RING);;
let CCOS_DOUBLE_CCOS = 
prove (`!z. ccos(Cx(&2) * z) = Cx(&2) * ccos z pow 2 - Cx(&1)`,
GEN_TAC THEN REWRITE_TAC[COMPLEX_RING `Cx(&2) * x = x + x`; CCOS_ADD] THEN MP_TAC(SPEC `z:complex` CSIN_CIRCLE) THEN CONV_TAC COMPLEX_RING);;
let CCOS_DOUBLE_CSIN = 
prove (`!z. ccos(Cx(&2) * z) = Cx(&1) - Cx(&2) * csin z pow 2`,
GEN_TAC THEN REWRITE_TAC[COMPLEX_RING `Cx(&2) * x = x + x`; CCOS_ADD] THEN MP_TAC(SPEC `z:complex` CSIN_CIRCLE) THEN CONV_TAC COMPLEX_RING);;
(* ------------------------------------------------------------------------- *) (* Euler and de Moivre formulas. *) (* ------------------------------------------------------------------------- *)
let CEXP_EULER = 
prove (`!z. cexp(ii * z) = ccos(z) + ii * csin(z)`,
REWRITE_TAC[ccos; csin] THEN CONV_TAC COMPLEX_FIELD);;
let DEMOIVRE = 
prove (`!z n. (ccos z + ii * csin z) pow n = ccos(Cx(&n) * z) + ii * csin(Cx(&n) * z)`,
REWRITE_TAC[GSYM CEXP_EULER; GSYM CEXP_N] THEN REWRITE_TAC[COMPLEX_MUL_AC]);;
(* ------------------------------------------------------------------------- *) (* Real exponential function. Same names as old Library/transc.ml. *) (* ------------------------------------------------------------------------- *)
let exp = new_definition `exp(x) = Re(cexp(Cx x))`;;
let CNJ_CEXP = 
prove (`!z. cnj(cexp z) = cexp(cnj z)`,
GEN_TAC THEN MATCH_MP_TAC SERIES_UNIQUE THEN MAP_EVERY EXISTS_TAC [`\n. cnj(z pow n / Cx(&(FACT n)))`; `from 0`] THEN CONJ_TAC THENL [REWRITE_TAC[SUMS_CNJ; CEXP_CONVERGES]; REWRITE_TAC[CNJ_DIV; CNJ_CX; CNJ_POW; CEXP_CONVERGES]]);;
let REAL_EXP = 
prove (`!z. real z ==> real(cexp z)`,
SIMP_TAC[REAL_CNJ; CNJ_CEXP]);;
let CX_EXP = 
prove (`!x. Cx(exp x) = cexp(Cx x)`,
REWRITE_TAC[exp] THEN MESON_TAC[REAL; REAL_CX; REAL_EXP]);;
let REAL_EXP_ADD = 
prove (`!x y. exp(x + y) = exp(x) * exp(y)`,
REWRITE_TAC[GSYM CX_INJ; CX_MUL; CX_EXP; CX_ADD; CEXP_ADD]);;
let REAL_EXP_0 = 
prove (`exp(&0) = &1`,
REWRITE_TAC[GSYM CX_INJ; CX_EXP; CEXP_0]);;
let REAL_EXP_ADD_MUL = 
prove (`!x y. exp(x + y) * exp(--x) = exp(y)`,
ONCE_REWRITE_TAC[REAL_ADD_SYM] THEN REWRITE_TAC[GSYM CX_INJ; CX_MUL; CX_EXP; CX_ADD; CX_NEG; CEXP_ADD_MUL]);;
let REAL_EXP_NEG_MUL = 
prove (`!x. exp(x) * exp(--x) = &1`,
REWRITE_TAC[GSYM CX_INJ; CX_MUL; CX_EXP; CX_NEG; CEXP_NEG_RMUL]);;
let REAL_EXP_NEG_MUL2 = 
prove (`!x. exp(--x) * exp(x) = &1`,
REWRITE_TAC[GSYM CX_INJ; CX_MUL; CX_EXP; CX_NEG; CEXP_NEG_LMUL]);;
let REAL_EXP_NEG = 
prove (`!x. exp(--x) = inv(exp(x))`,
REWRITE_TAC[GSYM CX_INJ; CX_INV; CX_EXP; CX_NEG; CEXP_NEG]);;
let REAL_EXP_N = 
prove (`!n x. exp(&n * x) = exp(x) pow n`,
REWRITE_TAC[GSYM CX_INJ; CX_EXP; CX_POW; CX_MUL; CEXP_N]);;
let REAL_EXP_SUB = 
prove (`!x y. exp(x - y) = exp(x) / exp(y)`,
REWRITE_TAC[GSYM CX_INJ; CX_SUB; CX_DIV; CX_EXP; CEXP_SUB]);;
let REAL_EXP_NZ = 
prove (`!x. ~(exp(x) = &0)`,
REWRITE_TAC[GSYM CX_INJ; CX_EXP; CEXP_NZ]);;
let REAL_EXP_POS_LE = 
prove (`!x. &0 <= exp(x)`,
GEN_TAC THEN SUBST1_TAC(REAL_ARITH `x = x / &2 + x / &2`) THEN REWRITE_TAC[REAL_EXP_ADD; REAL_LE_SQUARE]);;
let REAL_EXP_POS_LT = 
prove (`!x. &0 < exp(x)`,
let REAL_EXP_LE_X = 
prove (`!x. &0 <= x ==> &1 + x <= exp(x)`,
REPEAT STRIP_TAC THEN REWRITE_TAC[exp; RE_DEF] THEN MATCH_MP_TAC(MATCH_MP (ONCE_REWRITE_RULE[TAUT `a /\ b /\ c ==> d <=> b ==> a /\ c ==> d`] LIM_COMPONENT_LBOUND) (REWRITE_RULE[sums] (SPEC `Cx x` CEXP_CONVERGES))) THEN SIMP_TAC[DIMINDEX_2; ARITH; TRIVIAL_LIMIT_SEQUENTIALLY; VSUM_COMPONENT; EVENTUALLY_SEQUENTIALLY; FROM_0; INTER_UNIV] THEN REWRITE_TAC[GSYM CX_DIV; GSYM RE_DEF; RE_CX; GSYM CX_POW] THEN EXISTS_TAC `1` THEN SIMP_TAC[SUM_CLAUSES_LEFT; LE_0; ADD_CLAUSES] THEN CONV_TAC NUM_REDUCE_CONV THEN SIMP_TAC[real_pow; REAL_POW_1; REAL_DIV_1; REAL_LE_ADDR; REAL_ADD_ASSOC] THEN ASM_SIMP_TAC[SUM_POS_LE_NUMSEG; REAL_LE_DIV; REAL_POW_LE; REAL_POS]);;
let REAL_EXP_LT_1 = 
prove (`!x. &0 < x ==> &1 < exp(x)`,
MP_TAC REAL_EXP_LE_X THEN MATCH_MP_TAC MONO_FORALL THEN REAL_ARITH_TAC);;
let REAL_EXP_MONO_IMP = 
prove (`!x y. x < y ==> exp(x) < exp(y)`,
REPEAT GEN_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM REAL_SUB_LT] THEN DISCH_THEN(MP_TAC o MATCH_MP REAL_EXP_LT_1) THEN SIMP_TAC[REAL_EXP_SUB; REAL_LT_RDIV_EQ; REAL_EXP_POS_LT; REAL_MUL_LID]);;
let REAL_EXP_MONO_LT = 
prove (`!x y. exp(x) < exp(y) <=> x < y`,
REPEAT GEN_TAC THEN MATCH_MP_TAC(REAL_ARITH `(x < y ==> f < g) /\ (x = y ==> f = g) /\ (y < x ==> g < f) ==> (f < g <=> x < y)`) THEN SIMP_TAC[REAL_EXP_MONO_IMP]);;
let REAL_EXP_MONO_LE = 
prove (`!x y. exp(x) <= exp(y) <=> x <= y`,
REWRITE_TAC[GSYM REAL_NOT_LT; REAL_EXP_MONO_LT]);;
let REAL_EXP_INJ = 
prove (`!x y. (exp(x) = exp(y)) <=> (x = y)`,
REWRITE_TAC[GSYM REAL_LE_ANTISYM; REAL_EXP_MONO_LE]);;
let REAL_EXP_EQ_1 = 
prove (`!x. exp(x) = &1 <=> x = &0`,
ONCE_REWRITE_TAC[GSYM REAL_EXP_0] THEN REWRITE_TAC[REAL_EXP_INJ]);;
let REAL_ABS_EXP = 
prove (`!x. abs(exp x) = exp x`,
REWRITE_TAC[real_abs; REAL_EXP_POS_LE]);;
let REAL_EXP_SUM = 
prove (`!f s. FINITE s ==> exp(sum s f) = product s (\x. exp(f x))`,
GEN_TAC THEN MATCH_MP_TAC FINITE_INDUCT_STRONG THEN SIMP_TAC[SUM_CLAUSES; PRODUCT_CLAUSES; REAL_EXP_ADD; REAL_EXP_0]);;
let REAL_EXP_BOUND_LEMMA = 
prove (`!x. &0 <= x /\ x <= inv(&2) ==> exp(x) <= &1 + &2 * x`,
REPEAT STRIP_TAC THEN MP_TAC(SPEC `Cx x` CEXP_BOUND_LEMMA) THEN REWRITE_TAC[GSYM CX_EXP; COMPLEX_NORM_CX; RE_CX] THEN ASM_REAL_ARITH_TAC);;
(* ------------------------------------------------------------------------- *) (* Real trig functions, their reality, derivatives of complex versions. *) (* ------------------------------------------------------------------------- *)
let sin = new_definition `sin(x) = Re(csin(Cx x))`;;
let cos = new_definition `cos(x) = Re(ccos(Cx x))`;;
let CNJ_CSIN = 
prove (`!z. cnj(csin z) = csin(cnj z)`,
REWRITE_TAC[csin; CNJ_DIV; CNJ_SUB; CNJ_MUL; CNJ_CX; CNJ_CEXP; CNJ_NEG; CNJ_II; COMPLEX_NEG_NEG] THEN CONV_TAC COMPLEX_FIELD);;
let CNJ_CCOS = 
prove (`!z. cnj(ccos z) = ccos(cnj z)`,
let REAL_SIN = 
prove (`!z. real z ==> real(csin z)`,
SIMP_TAC[REAL_CNJ; CNJ_CSIN]);;
let REAL_COS = 
prove (`!z. real z ==> real(ccos z)`,
SIMP_TAC[REAL_CNJ; CNJ_CCOS]);;
let CX_SIN = 
prove (`!x. Cx(sin x) = csin(Cx x)`,
REWRITE_TAC[sin] THEN MESON_TAC[REAL; REAL_CX; REAL_SIN]);;
let CX_COS = 
prove (`!x. Cx(cos x) = ccos(Cx x)`,
REWRITE_TAC[cos] THEN MESON_TAC[REAL; REAL_CX; REAL_COS]);;
let HAS_COMPLEX_DERIVATIVE_CSIN = 
prove (`!z. (csin has_complex_derivative ccos z) (at z)`,
GEN_TAC THEN GEN_REWRITE_TAC (RATOR_CONV o LAND_CONV) [GSYM ETA_AX] THEN REWRITE_TAC[csin; ccos] THEN COMPLEX_DIFF_TAC THEN CONV_TAC COMPLEX_FIELD);;
let COMPLEX_DIFFERENTIABLE_AT_CSIN = 
prove (`!z. csin complex_differentiable at z`,
REWRITE_TAC[complex_differentiable] THEN MESON_TAC[HAS_COMPLEX_DERIVATIVE_CSIN]);;
let COMPLEX_DIFFERENTIABLE_WITHIN_CSIN = 
prove (`!s z. csin complex_differentiable (at z within s)`,
add_complex_differentiation_theorems (CONJUNCTS(REWRITE_RULE[FORALL_AND_THM] (MATCH_MP HAS_COMPLEX_DERIVATIVE_CHAIN_UNIV HAS_COMPLEX_DERIVATIVE_CSIN)));;
let HAS_COMPLEX_DERIVATIVE_CCOS = 
prove (`!z. (ccos has_complex_derivative --csin z) (at z)`,
GEN_TAC THEN GEN_REWRITE_TAC (RATOR_CONV o LAND_CONV) [GSYM ETA_AX] THEN REWRITE_TAC[csin; ccos] THEN COMPLEX_DIFF_TAC THEN CONV_TAC COMPLEX_FIELD);;
let COMPLEX_DIFFERENTIABLE_AT_CCOS = 
prove (`!z. ccos complex_differentiable at z`,
REWRITE_TAC[complex_differentiable] THEN MESON_TAC[HAS_COMPLEX_DERIVATIVE_CCOS]);;
let COMPLEX_DIFFERENTIABLE_WITHIN_CCOS = 
prove (`!s z. ccos complex_differentiable (at z within s)`,
add_complex_differentiation_theorems (CONJUNCTS(REWRITE_RULE[FORALL_AND_THM] (MATCH_MP HAS_COMPLEX_DERIVATIVE_CHAIN_UNIV HAS_COMPLEX_DERIVATIVE_CCOS)));;
let CONTINUOUS_AT_CSIN = 
prove (`!z. csin continuous at z`,
let CONTINUOUS_WITHIN_CSIN = 
prove (`!s z. csin continuous (at z within s)`,
let CONTINUOUS_ON_CSIN = 
prove (`!s. csin continuous_on s`,
let HOLOMORPHIC_ON_CSIN = 
prove (`!s. csin holomorphic_on s`,
let CONTINUOUS_AT_CCOS = 
prove (`!z. ccos continuous at z`,
let CONTINUOUS_WITHIN_CCOS = 
prove (`!s z. ccos continuous (at z within s)`,
let CONTINUOUS_ON_CCOS = 
prove (`!s. ccos continuous_on s`,
let HOLOMORPHIC_ON_CCOS = 
prove (`!s. ccos holomorphic_on s`,
(* ------------------------------------------------------------------------- *) (* Slew of theorems for compatibility with old transc.ml file. *) (* ------------------------------------------------------------------------- *)
let SIN_0 = 
prove (`sin(&0) = &0`,
REWRITE_TAC[GSYM CX_INJ; CX_SIN; CSIN_0]);;
let COS_0 = 
prove (`cos(&0) = &1`,
REWRITE_TAC[GSYM CX_INJ; CX_COS; CCOS_0]);;
let SIN_CIRCLE = 
prove (`!x. (sin(x) pow 2) + (cos(x) pow 2) = &1`,
REWRITE_TAC[GSYM CX_INJ; CX_COS; CX_SIN; CX_ADD; CX_POW; CSIN_CIRCLE]);;
let SIN_ADD = 
prove (`!x y. sin(x + y) = sin(x) * cos(y) + cos(x) * sin(y)`,
REWRITE_TAC[GSYM CX_INJ; CX_COS; CX_SIN; CX_ADD; CX_MUL; CSIN_ADD]);;
let COS_ADD = 
prove (`!x y. cos(x + y) = cos(x) * cos(y) - sin(x) * sin(y)`,
REWRITE_TAC[GSYM CX_INJ; CX_COS; CX_SIN; CX_ADD; CX_SUB; CX_MUL; CCOS_ADD]);;
let SIN_NEG = 
prove (`!x. sin(--x) = --(sin(x))`,
REWRITE_TAC[GSYM CX_INJ; CX_SIN; CX_NEG; CSIN_NEG]);;
let COS_NEG = 
prove (`!x. cos(--x) = cos(x)`,
REWRITE_TAC[GSYM CX_INJ; CX_COS; CX_NEG; CCOS_NEG]);;
let SIN_DOUBLE = 
prove (`!x. sin(&2 * x) = &2 * sin(x) * cos(x)`,
REWRITE_TAC[GSYM CX_INJ; CX_SIN; CX_COS; CX_MUL; CSIN_DOUBLE]);;
let COS_DOUBLE = 
prove (`!x. cos(&2 * x) = (cos(x) pow 2) - (sin(x) pow 2)`,
let COS_DOUBLE_COS = 
prove (`!x. cos(&2 * x) = &2 * cos(x) pow 2 - &1`,
MP_TAC SIN_CIRCLE THEN MATCH_MP_TAC MONO_FORALL THEN REWRITE_TAC[COS_DOUBLE] THEN REAL_ARITH_TAC);;
let (SIN_BOUND,COS_BOUND) = (CONJ_PAIR o prove) (`(!x. abs(sin x) <= &1) /\ (!x. abs(cos x) <= &1)`, CONJ_TAC THEN GEN_TAC THEN ONCE_REWRITE_TAC[GSYM REAL_ABS_NUM] THEN ONCE_REWRITE_TAC[REAL_LE_SQUARE_ABS] THEN MP_TAC(SPEC `x:real` SIN_CIRCLE) THEN MAP_EVERY (MP_TAC o C SPEC REAL_LE_SQUARE) [`sin x`; `cos x`] THEN REAL_ARITH_TAC);;
let SIN_BOUNDS = 
prove (`!x. --(&1) <= sin(x) /\ sin(x) <= &1`,
MP_TAC SIN_BOUND THEN MATCH_MP_TAC MONO_FORALL THEN REAL_ARITH_TAC);;
let COS_BOUNDS = 
prove (`!x. --(&1) <= cos(x) /\ cos(x) <= &1`,
MP_TAC COS_BOUND THEN MATCH_MP_TAC MONO_FORALL THEN REAL_ARITH_TAC);;
let COS_ABS = 
prove (`!x. cos(abs x) = cos(x)`,
REWRITE_TAC[real_abs] THEN MESON_TAC[COS_NEG]);;
let SIN_SUB = 
prove (`!w z. sin(w - z) = sin(w) * cos(z) - cos(w) * sin(z)`,
REWRITE_TAC[GSYM CX_INJ; CX_SIN; CX_COS; CX_SUB; CX_MUL; CSIN_SUB]);;
let COS_SUB = 
prove (`!w z. cos(w - z) = cos(w) * cos(z) + sin(w) * sin(z)`,
REWRITE_TAC[GSYM CX_INJ; CX_SIN; CX_COS; CX_SUB; CX_ADD; CX_MUL; CCOS_SUB]);;
let REAL_MUL_SIN_SIN = 
prove (`!x y. sin(x) * sin(y) = (cos(x - y) - cos(x + y)) / &2`,
REWRITE_TAC[GSYM CX_INJ; CX_COS; CX_SIN; CX_ADD; CX_SUB; CX_MUL; CX_DIV] THEN REWRITE_TAC[COMPLEX_MUL_CSIN_CSIN]);;
let REAL_MUL_SIN_COS = 
prove (`!x y. sin(x) * cos(y) = (sin(x + y) + sin(x - y)) / &2`,
REWRITE_TAC[GSYM CX_INJ; CX_COS; CX_SIN; CX_ADD; CX_SUB; CX_MUL; CX_DIV] THEN REWRITE_TAC[COMPLEX_MUL_CSIN_CCOS]);;
let REAL_MUL_COS_SIN = 
prove (`!x y. cos(x) * sin(y) = (sin(x + y) - sin(x - y)) / &2`,
REWRITE_TAC[GSYM CX_INJ; CX_COS; CX_SIN; CX_ADD; CX_SUB; CX_MUL; CX_DIV] THEN REWRITE_TAC[COMPLEX_MUL_CCOS_CSIN]);;
let REAL_MUL_COS_COS = 
prove (`!x y. cos(x) * cos(y) = (cos(x - y) + cos(x + y)) / &2`,
REWRITE_TAC[GSYM CX_INJ; CX_COS; CX_SIN; CX_ADD; CX_SUB; CX_MUL; CX_DIV] THEN REWRITE_TAC[COMPLEX_MUL_CCOS_CCOS]);;
let REAL_ADD_SIN = 
prove (`!x y. sin(x) + sin(y) = &2 * sin((x + y) / &2) * cos((x - y) / &2)`,
REWRITE_TAC[GSYM CX_INJ; CX_COS; CX_SIN; CX_ADD; CX_SUB; CX_MUL; CX_DIV] THEN REWRITE_TAC[COMPLEX_ADD_CSIN]);;
let REAL_SUB_SIN = 
prove (`!x y. sin(x) - sin(y) = &2 * sin((x - y) / &2) * cos((x + y) / &2)`,
REWRITE_TAC[GSYM CX_INJ; CX_COS; CX_SIN; CX_ADD; CX_SUB; CX_MUL; CX_DIV] THEN REWRITE_TAC[COMPLEX_SUB_CSIN]);;
let REAL_ADD_COS = 
prove (`!x y. cos(x) + cos(y) = &2 * cos((x + y) / &2) * cos((x - y) / &2)`,
REWRITE_TAC[GSYM CX_INJ; CX_COS; CX_SIN; CX_ADD; CX_SUB; CX_MUL; CX_DIV] THEN REWRITE_TAC[COMPLEX_ADD_CCOS]);;
let REAL_SUB_COS = 
prove (`!x y. cos(x) - cos(y) = &2 * sin((x + y) / &2) * sin((y - x) / &2)`,
REWRITE_TAC[GSYM CX_INJ; CX_COS; CX_SIN; CX_ADD; CX_SUB; CX_MUL; CX_DIV] THEN REWRITE_TAC[COMPLEX_SUB_CCOS]);;
let COS_DOUBLE_SIN = 
prove (`!x. cos(&2 * x) = &1 - &2 * sin x pow 2`,
GEN_TAC THEN REWRITE_TAC[REAL_RING `&2 * x = x + x`; COS_ADD] THEN MP_TAC(SPEC `x:real` SIN_CIRCLE) THEN CONV_TAC REAL_RING);;
(* ------------------------------------------------------------------------- *) (* Get a nice real/imaginary separation in Euler's formula. *) (* ------------------------------------------------------------------------- *)
let EULER = 
prove (`!z. cexp(z) = Cx(exp(Re z)) * (Cx(cos(Im z)) + ii * Cx(sin(Im z)))`,
GEN_TAC THEN GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [COMPLEX_EXPAND] THEN REWRITE_TAC[CEXP_ADD; CEXP_EULER; GSYM CX_SIN; GSYM CX_COS; GSYM CX_EXP]);;
let RE_CEXP = 
prove (`!z. Re(cexp z) = exp(Re z) * cos(Im z)`,
REWRITE_TAC[EULER; RE_ADD; RE_MUL_CX; RE_MUL_II; IM_CX; RE_CX] THEN REAL_ARITH_TAC);;
let IM_CEXP = 
prove (`!z. Im(cexp z) = exp(Re z) * sin(Im z)`,
REWRITE_TAC[EULER; IM_ADD; IM_MUL_CX; IM_MUL_II; IM_CX; RE_CX] THEN REAL_ARITH_TAC);;
let RE_CSIN = 
prove (`!z. Re(csin z) = (exp(Im z) + exp(--(Im z))) / &2 * sin(Re z)`,
GEN_TAC THEN REWRITE_TAC[csin] THEN SIMP_TAC[COMPLEX_FIELD `x / (Cx(&2) * ii) = ii * --(x / Cx(&2))`] THEN REWRITE_TAC[IM_MUL_II; IM_DIV_CX; RE_NEG; IM_SUB; IM_CEXP; RE_MUL_II; COMPLEX_MUL_LNEG; IM_NEG] THEN REWRITE_TAC[REAL_NEG_NEG; SIN_NEG] THEN CONV_TAC REAL_RING);;
let IM_CSIN = 
prove (`!z. Im(csin z) = (exp(Im z) - exp(--(Im z))) / &2 * cos(Re z)`,
GEN_TAC THEN REWRITE_TAC[csin] THEN SIMP_TAC[COMPLEX_FIELD `x / (Cx(&2) * ii) = ii * --(x / Cx(&2))`] THEN REWRITE_TAC[IM_MUL_II; RE_DIV_CX; RE_NEG; RE_SUB; RE_CEXP; RE_MUL_II; COMPLEX_MUL_LNEG; IM_NEG] THEN REWRITE_TAC[REAL_NEG_NEG; COS_NEG] THEN CONV_TAC REAL_RING);;
let RE_CCOS = 
prove (`!z. Re(ccos z) = (exp(Im z) + exp(--(Im z))) / &2 * cos(Re z)`,
GEN_TAC THEN REWRITE_TAC[ccos] THEN REWRITE_TAC[RE_DIV_CX; RE_ADD; RE_CEXP; COMPLEX_MUL_LNEG; RE_MUL_II; IM_MUL_II; RE_NEG; IM_NEG; COS_NEG] THEN REWRITE_TAC[REAL_NEG_NEG] THEN CONV_TAC REAL_RING);;
let IM_CCOS = 
prove (`!z. Im(ccos z) = (exp(--(Im z)) - exp(Im z)) / &2 * sin(Re z)`,
GEN_TAC THEN REWRITE_TAC[ccos] THEN REWRITE_TAC[IM_DIV_CX; IM_ADD; IM_CEXP; COMPLEX_MUL_LNEG; RE_MUL_II; IM_MUL_II; RE_NEG; IM_NEG; SIN_NEG] THEN REWRITE_TAC[REAL_NEG_NEG] THEN CONV_TAC REAL_RING);;
(* ------------------------------------------------------------------------- *) (* Some special intermediate value theorems over the reals. *) (* ------------------------------------------------------------------------- *)
let IVT_INCREASING_RE = 
prove (`!f a b y. a <= b /\ (!x. a <= x /\ x <= b ==> f continuous at (Cx x)) /\ Re(f(Cx a)) <= y /\ y <= Re(f(Cx b)) ==> ?x. a <= x /\ x <= b /\ Re(f(Cx x)) = y`,
REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`(f:complex->complex) o Cx o drop`; `lift a`; `lift b`; `y:real`; `1`] IVT_INCREASING_COMPONENT_1) THEN REWRITE_TAC[EXISTS_DROP; GSYM drop; LIFT_DROP; o_THM; GSYM RE_DEF] THEN ASM_REWRITE_TAC[IN_INTERVAL_1; GSYM CONJ_ASSOC; LIFT_DROP] THEN DISCH_THEN MATCH_MP_TAC THEN REWRITE_TAC[DIMINDEX_2; ARITH] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC CONTINUOUS_AT_COMPOSE THEN ASM_SIMP_TAC[o_THM] THEN REWRITE_TAC[continuous_at; o_THM] THEN REWRITE_TAC[dist; GSYM CX_SUB; GSYM DROP_SUB; COMPLEX_NORM_CX] THEN REWRITE_TAC[GSYM ABS_DROP] THEN MESON_TAC[]);;
let IVT_DECREASING_RE = 
prove (`!f a b y. a <= b /\ (!x. a <= x /\ x <= b ==> f continuous at (Cx x)) /\ Re(f(Cx b)) <= y /\ y <= Re(f(Cx a)) ==> ?x. a <= x /\ x <= b /\ Re(f(Cx x)) = y`,
REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[GSYM REAL_EQ_NEG2] THEN REWRITE_TAC[GSYM RE_NEG] THEN MATCH_MP_TAC IVT_INCREASING_RE THEN ASM_SIMP_TAC[CONTINUOUS_NEG; RE_NEG; REAL_LE_NEG2]);;
let IVT_INCREASING_IM = 
prove (`!f a b y. a <= b /\ (!x. a <= x /\ x <= b ==> f continuous at (Cx x)) /\ Im(f(Cx a)) <= y /\ y <= Im(f(Cx b)) ==> ?x. a <= x /\ x <= b /\ Im(f(Cx x)) = y`,
REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[GSYM REAL_EQ_NEG2] THEN REWRITE_TAC[SYM(CONJUNCT2(SPEC_ALL RE_MUL_II))] THEN MATCH_MP_TAC IVT_DECREASING_RE THEN ASM_SIMP_TAC[CONTINUOUS_COMPLEX_MUL; ETA_AX; CONTINUOUS_CONST] THEN ASM_REWRITE_TAC[RE_MUL_II; REAL_LE_NEG2]);;
let IVT_DECREASING_IM = 
prove (`!f a b y. a <= b /\ (!x. a <= x /\ x <= b ==> f continuous at (Cx x)) /\ Im(f(Cx b)) <= y /\ y <= Im(f(Cx a)) ==> ?x. a <= x /\ x <= b /\ Im(f(Cx x)) = y`,
REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[GSYM REAL_EQ_NEG2] THEN REWRITE_TAC[GSYM IM_NEG] THEN MATCH_MP_TAC IVT_INCREASING_IM THEN ASM_SIMP_TAC[CONTINUOUS_NEG; IM_NEG; REAL_LE_NEG2]);;
(* ------------------------------------------------------------------------- *) (* Some minimal properties of real logs help to define complex logs. *) (* ------------------------------------------------------------------------- *)
let log_def = new_definition
 `log y = @x. exp(x) = y`;;
let EXP_LOG = 
prove (`!x. &0 < x ==> exp(log x) = x`,
REPEAT STRIP_TAC THEN REWRITE_TAC[log_def] THEN CONV_TAC SELECT_CONV THEN SUBGOAL_THEN `?y. --inv(x) <= y /\ y <= x /\ Re(cexp(Cx y)) = x` MP_TAC THENL [ALL_TAC; MESON_TAC[CX_EXP; RE_CX]] THEN MATCH_MP_TAC IVT_INCREASING_RE THEN SIMP_TAC[GSYM CX_EXP; RE_CX; CONTINUOUS_AT_CEXP] THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC(REAL_ARITH `&0 < x /\ &0 < y ==> --y <= x`) THEN ASM_SIMP_TAC[REAL_LT_INV_EQ]; ONCE_REWRITE_TAC[GSYM REAL_INV_INV] THEN MATCH_MP_TAC REAL_LE_INV2 THEN ASM_REWRITE_TAC[REAL_EXP_NEG; REAL_INV_INV; REAL_LT_INV_EQ]; ALL_TAC] THEN MATCH_MP_TAC(REAL_ARITH `&1 + x <= y ==> x <= y`) THEN ASM_SIMP_TAC[REAL_EXP_LE_X; REAL_LE_INV_EQ; REAL_LT_IMP_LE]);;
let LOG_EXP = 
prove (`!x. log(exp x) = x`,
GEN_TAC THEN ONCE_REWRITE_TAC[GSYM REAL_EXP_INJ] THEN SIMP_TAC[EXP_LOG; REAL_EXP_POS_LT]);;
let REAL_EXP_LOG = 
prove (`!x. (exp(log x) = x) <=> &0 < x`,
MESON_TAC[EXP_LOG; REAL_EXP_POS_LT]);;
let LOG_MUL = 
prove (`!x y. &0 < x /\ &0 < y ==> (log(x * y) = log(x) + log(y))`,
REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[GSYM REAL_EXP_INJ] THEN ASM_SIMP_TAC[REAL_EXP_ADD; REAL_LT_MUL; EXP_LOG]);;
let LOG_INJ = 
prove (`!x y. &0 < x /\ &0 < y ==> (log(x) = log(y) <=> x = y)`,
REPEAT STRIP_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM REAL_EXP_INJ] THEN ASM_SIMP_TAC[EXP_LOG]);;
let LOG_1 = 
prove (`log(&1) = &0`,
ONCE_REWRITE_TAC[GSYM REAL_EXP_INJ] THEN REWRITE_TAC[REAL_EXP_0; REAL_EXP_LOG; REAL_LT_01]);;
let LOG_INV = 
prove (`!x. &0 < x ==> (log(inv x) = --(log x))`,
REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[GSYM REAL_EXP_INJ] THEN ASM_SIMP_TAC[REAL_EXP_NEG; EXP_LOG; REAL_LT_INV_EQ]);;
let LOG_DIV = 
prove (`!x y. &0 < x /\ &0 < y ==> log(x / y) = log(x) - log(y)`,
let LOG_MONO_LT = 
prove (`!x y. &0 < x /\ &0 < y ==> (log(x) < log(y) <=> x < y)`,
REPEAT STRIP_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM REAL_EXP_MONO_LT] THEN ASM_SIMP_TAC[EXP_LOG]);;
let LOG_MONO_LT_IMP = 
prove (`!x y. &0 < x /\ x < y ==> log(x) < log(y)`,
MESON_TAC[LOG_MONO_LT; REAL_LT_TRANS]);;
let LOG_MONO_LE = 
prove (`!x y. &0 < x /\ &0 < y ==> (log(x) <= log(y) <=> x <= y)`,
REPEAT STRIP_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM REAL_EXP_MONO_LE] THEN ASM_SIMP_TAC[EXP_LOG]);;
let LOG_MONO_LE_IMP = 
prove (`!x y. &0 < x /\ x <= y ==> log(x) <= log(y)`,
let LOG_POW = 
prove (`!n x. &0 < x ==> (log(x pow n) = &n * log(x))`,
REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[GSYM REAL_EXP_INJ] THEN ASM_SIMP_TAC[REAL_EXP_N; EXP_LOG; REAL_POW_LT]);;
let LOG_LE = 
prove (`!x. &0 <= x ==> log(&1 + x) <= x`,
REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[GSYM REAL_EXP_MONO_LE] THEN ASM_SIMP_TAC[EXP_LOG; REAL_ARITH `&0 <= x ==> &0 < &1 + x`; REAL_EXP_LE_X]);;
let LOG_LT_X = 
prove (`!x. &0 < x ==> log(x) < x`,
REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[GSYM REAL_EXP_MONO_LT] THEN ASM_SIMP_TAC[EXP_LOG] THEN MP_TAC(SPEC `x:real` REAL_EXP_LE_X) THEN POP_ASSUM MP_TAC THEN REAL_ARITH_TAC);;
let LOG_POS = 
prove (`!x. &1 <= x ==> &0 <= log(x)`,
REWRITE_TAC[GSYM LOG_1] THEN SIMP_TAC[LOG_MONO_LE; ARITH_RULE `&1 <= x ==> &0 < x`; REAL_LT_01]);;
let LOG_POS_LT = 
prove (`!x. &1 < x ==> &0 < log(x)`,
REWRITE_TAC[GSYM LOG_1] THEN SIMP_TAC[LOG_MONO_LT; ARITH_RULE `&1 < x ==> &0 < x`; REAL_LT_01]);;
(* ------------------------------------------------------------------------- *) (* Deduce periodicity just from derivative and zero values. *) (* ------------------------------------------------------------------------- *)
let SIN_NEARZERO = 
prove (`?x. &0 < x /\ !y. &0 < y /\ y <= x ==> &0 < sin(y)`,
MP_TAC(SPEC `&1 / &2` (CONJUNCT2 (REWRITE_RULE[has_complex_derivative; HAS_DERIVATIVE_AT_ALT] (ISPEC `Cx(&0)` HAS_COMPLEX_DERIVATIVE_CSIN)))) THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[CSIN_0; COMPLEX_SUB_RZERO; CCOS_0; COMPLEX_MUL_LZERO; COMPLEX_MUL_LID] THEN DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `d / &2` THEN ASM_REWRITE_TAC[REAL_HALF] THEN X_GEN_TAC `y:real` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `Cx y`) THEN ASM_REWRITE_TAC[GSYM CX_SIN; COMPLEX_NORM_CX; GSYM CX_SUB] THEN ASM_REAL_ARITH_TAC);;
let SIN_NONTRIVIAL = 
prove (`?x. &0 < x /\ ~(sin x = &0)`,
let COS_NONTRIVIAL = 
prove (`?x. &0 < x /\ ~(cos x = &1)`,
MP_TAC SIN_NONTRIVIAL THEN MATCH_MP_TAC MONO_EXISTS THEN MP_TAC SIN_CIRCLE THEN MATCH_MP_TAC MONO_FORALL THEN CONV_TAC REAL_FIELD);;
let COS_DOUBLE_BOUND = 
prove (`!x. &0 <= cos x ==> &2 * (&1 - cos x) <= &1 - cos(&2 * x)`,
REWRITE_TAC[COS_DOUBLE_COS] THEN REWRITE_TAC[REAL_ARITH `&2 * (&1 - a) <= &1 - (&2 * b - &1) <=> b <= &1 * a`] THEN SIMP_TAC[REAL_POW_2; REAL_LE_RMUL; COS_BOUNDS]);;
let COS_GOESNEGATIVE_LEMMA = 
prove (`!x. cos(x) < &1 ==> ?n. cos(&2 pow n * x) < &0`,
GEN_TAC THEN DISCH_TAC THEN MATCH_MP_TAC(TAUT `(~p ==> p) ==> p`) THEN REWRITE_TAC[NOT_EXISTS_THM; REAL_NOT_LT] THEN DISCH_TAC THEN SUBGOAL_THEN `!n. &2 pow n * (&1 - cos x) <= &1 - cos(&2 pow n * x)` ASSUME_TAC THENL [INDUCT_TAC THEN REWRITE_TAC[real_pow; REAL_MUL_LID; REAL_LE_REFL] THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `&2 * (&1 - cos(&2 pow n * x))` THEN ASM_SIMP_TAC[GSYM REAL_MUL_ASSOC; REAL_LE_LMUL; REAL_POS; COS_DOUBLE_BOUND]; MP_TAC(ISPEC `&1 / (&1 - cos(x))` REAL_ARCH_POW2) THEN ASM_SIMP_TAC[REAL_LT_LDIV_EQ; REAL_SUB_LT] THEN DISCH_THEN(X_CHOOSE_THEN `n:num` MP_TAC) THEN REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `n:num`)) THEN REAL_ARITH_TAC]);;
let COS_GOESNEGATIVE = 
prove (`?x. &0 < x /\ cos(x) < &0`,
X_CHOOSE_TAC `x:real` COS_NONTRIVIAL THEN MP_TAC(SPEC `x:real` COS_GOESNEGATIVE_LEMMA) THEN ANTS_TAC THENL [MP_TAC(SPEC `x:real` COS_BOUNDS) THEN ASM_REAL_ARITH_TAC; ASM_MESON_TAC[REAL_LT_MUL; REAL_POW_LT; REAL_ARITH `&0 < &2`]]);;
let COS_HASZERO = 
prove (`?x. &0 < x /\ cos(x) = &0`,
X_CHOOSE_THEN `z:real` STRIP_ASSUME_TAC COS_GOESNEGATIVE THEN SUBGOAL_THEN `?x. &0 <= x /\ x <= z /\ Re(ccos(Cx x)) = &0` MP_TAC THENL [MATCH_MP_TAC IVT_DECREASING_RE THEN ASM_SIMP_TAC[GSYM CX_COS; RE_CX; REAL_LT_IMP_LE; COS_0; REAL_POS] THEN MESON_TAC[HAS_COMPLEX_DERIVATIVE_IMP_CONTINUOUS_AT; HAS_COMPLEX_DERIVATIVE_CCOS]; MATCH_MP_TAC MONO_EXISTS THEN REWRITE_TAC[GSYM CX_COS; RE_CX] THEN MESON_TAC[COS_0; REAL_LE_LT; REAL_ARITH `~(&1 = &0)`]]);;
let SIN_HASZERO = 
prove (`?x. &0 < x /\ sin(x) = &0`,
X_CHOOSE_THEN `x:real` STRIP_ASSUME_TAC COS_HASZERO THEN EXISTS_TAC `&2 * x` THEN ASM_SIMP_TAC[SIN_DOUBLE] THEN ASM_REAL_ARITH_TAC);;
let SIN_HASZERO_MINIMAL = 
prove (`?p. &0 < p /\ sin p = &0 /\ !x. &0 < x /\ x < p ==> ~(sin x = &0)`,
X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC SIN_NEARZERO THEN MP_TAC(ISPECL [`{z | z IN IMAGE Cx {x | x >= e} /\ csin z IN {Cx(&0)}}`; `Cx(&0)`] DISTANCE_ATTAINS_INF) THEN ANTS_TAC THENL [ALL_TAC; REWRITE_TAC[IN_ELIM_THM; GSYM CONJ_ASSOC; IMP_CONJ] THEN REWRITE_TAC[FORALL_IN_IMAGE; EXISTS_IN_IMAGE] THEN REWRITE_TAC[IN_ELIM_THM; IN_SING; real_ge; GSYM CX_COS; CX_INJ] THEN REWRITE_TAC[dist; GSYM CX_SUB; GSYM CX_SIN; CX_INJ; COMPLEX_NORM_CX] THEN MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN REWRITE_TAC[REAL_ARITH `abs(&0 - x) = abs x`] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ALL_TAC; X_GEN_TAC `x:real` THEN REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `x:real`))] THEN ASM_REAL_ARITH_TAC] THEN X_CHOOSE_TAC `a:real` SIN_HASZERO THEN CONJ_TAC THENL [ALL_TAC; REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN EXISTS_TAC `Cx a` THEN ASM_REWRITE_TAC[IN_SING; IN_IMAGE; IN_ELIM_THM; GSYM CX_SIN] THEN ASM_MESON_TAC[REAL_ARITH `x >= w \/ x <= w`; REAL_LT_REFL]] THEN MATCH_MP_TAC CONTINUOUS_CLOSED_PREIMAGE THEN REWRITE_TAC[CONTINUOUS_ON_CSIN; CLOSED_SING] THEN SUBGOAL_THEN `IMAGE Cx {x | x >= e} = {z | Im(z) = &0} INTER {z | Re(z) >= e}` (fun th -> SIMP_TAC[th; CLOSED_INTER; CLOSED_HALFSPACE_IM_EQ; CLOSED_HALFSPACE_RE_GE]) THEN REWRITE_TAC[EXTENSION; IN_IMAGE; IN_INTER; IN_ELIM_THM] THEN REWRITE_TAC[FORALL_COMPLEX; COMPLEX_EQ; RE; IM; RE_CX; IM_CX] THEN MESON_TAC[]);;
let pi = new_definition
 `pi = @p. &0 < p /\ sin(p) = &0 /\ !x. &0 < x /\ x < p ==> ~(sin(x) = &0)`;;
let PI_WORKS = 
prove (`&0 < pi /\ sin(pi) = &0 /\ !x. &0 < x /\ x < pi ==> ~(sin x = &0)`,
REWRITE_TAC[pi] THEN CONV_TAC SELECT_CONV THEN REWRITE_TAC[SIN_HASZERO_MINIMAL]);;
(* ------------------------------------------------------------------------- *) (* Now more relatively easy consequences. *) (* ------------------------------------------------------------------------- *)
let PI_POS = 
prove (`&0 < pi`,
REWRITE_TAC[PI_WORKS]);;
let PI_POS_LE = 
prove (`&0 <= pi`,
REWRITE_TAC[REAL_LE_LT; PI_POS]);;
let PI_NZ = 
prove (`~(pi = &0)`,
SIMP_TAC[PI_POS; REAL_LT_IMP_NZ]);;
let REAL_ABS_PI = 
prove (`abs pi = pi`,
REWRITE_TAC[real_abs; PI_POS_LE]);;
let SIN_PI = 
prove (`sin(pi) = &0`,
REWRITE_TAC[PI_WORKS]);;
let SIN_POS_PI = 
prove (`!x. &0 < x /\ x < pi ==> &0 < sin(x)`,
REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[GSYM REAL_NOT_LE] THEN DISCH_TAC THEN X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC SIN_NEARZERO THEN MP_TAC(ISPECL [`csin`; `e:real`; `x:real`; `&0`] IVT_DECREASING_RE) THEN ASM_SIMP_TAC[NOT_IMP; CONTINUOUS_AT_CSIN; GSYM CX_SIN; RE_CX; SIN_0] THEN ASM_MESON_TAC[REAL_LE_TOTAL; REAL_LET_ANTISYM; PI_WORKS; REAL_LET_TRANS; REAL_LTE_TRANS]);;
let COS_PI2 = 
prove (`cos(pi / &2) = &0`,
MP_TAC(SYM(SPEC `pi / &2` SIN_DOUBLE)) THEN REWRITE_TAC[REAL_HALF; SIN_PI; REAL_ENTIRE; REAL_OF_NUM_EQ; ARITH] THEN MATCH_MP_TAC(REAL_ARITH `&0 < y ==> y = &0 \/ z = &0 ==> z = &0`) THEN MATCH_MP_TAC SIN_POS_PI THEN MP_TAC PI_POS THEN REAL_ARITH_TAC);;
let COS_PI = 
prove (`cos(pi) = -- &1`,
ONCE_REWRITE_TAC[REAL_ARITH `pi = &2 * pi / &2`] THEN REWRITE_TAC[COS_DOUBLE_COS; COS_PI2] THEN REAL_ARITH_TAC);;
let SIN_PI2 = 
prove (`sin(pi / &2) = &1`,
MP_TAC(SPEC `pi / &2` SIN_CIRCLE) THEN REWRITE_TAC[COS_PI2; REAL_POW_2; REAL_ADD_RID; REAL_MUL_LZERO] THEN REWRITE_TAC[REAL_RING `x * x = &1 <=> x = &1 \/ x = -- &1`] THEN MP_TAC(SPEC `pi / &2` SIN_POS_PI) THEN MP_TAC PI_POS THEN REAL_ARITH_TAC);;
let SIN_COS = 
prove (`!x. sin(x) = cos(pi / &2 - x)`,
REWRITE_TAC[COS_SUB; COS_PI2; SIN_PI2] THEN REAL_ARITH_TAC);;
let COS_SIN = 
prove (`!x. cos(x) = sin(pi / &2 - x)`,
REWRITE_TAC[SIN_SUB; COS_PI2; SIN_PI2] THEN REAL_ARITH_TAC);;
let SIN_PERIODIC_PI = 
prove (`!x. sin(x + pi) = --(sin(x))`,
REWRITE_TAC[SIN_ADD; SIN_PI; COS_PI] THEN REAL_ARITH_TAC);;
let COS_PERIODIC_PI = 
prove (`!x. cos(x + pi) = --(cos(x))`,
REWRITE_TAC[COS_ADD; SIN_PI; COS_PI] THEN REAL_ARITH_TAC);;
let SIN_PERIODIC = 
prove (`!x. sin(x + &2 * pi) = sin(x)`,
REWRITE_TAC[REAL_MUL_2; REAL_ADD_ASSOC; SIN_PERIODIC_PI; REAL_NEG_NEG]);;
let COS_PERIODIC = 
prove (`!x. cos(x + &2 * pi) = cos(x)`,
REWRITE_TAC[REAL_MUL_2; REAL_ADD_ASSOC; COS_PERIODIC_PI; REAL_NEG_NEG]);;
let SIN_NPI = 
prove (`!n. sin(&n * pi) = &0`,
INDUCT_TAC THEN ASM_REWRITE_TAC[GSYM REAL_OF_NUM_SUC; REAL_MUL_LID; REAL_ADD_RDISTRIB; REAL_NEG_0; SIN_PERIODIC_PI; REAL_MUL_LZERO; SIN_0]);;
let COS_NPI = 
prove (`!n. cos(&n * pi) = --(&1) pow n`,
INDUCT_TAC THEN ASM_REWRITE_TAC[real_pow; REAL_MUL_LZERO; COS_0; COS_PERIODIC_PI; REAL_MUL_LID; REAL_MUL_LNEG; GSYM REAL_OF_NUM_SUC; REAL_ADD_RDISTRIB]);;
let COS_POS_PI2 = 
prove (`!x. &0 < x /\ x < pi / &2 ==> &0 < cos(x)`,
REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[GSYM REAL_NOT_LE] THEN DISCH_TAC THEN MP_TAC(ISPECL [`ccos`; `&0`; `x:real`; `&0`] IVT_DECREASING_RE) THEN ASM_SIMP_TAC[CONTINUOUS_AT_CCOS; REAL_LT_IMP_LE; GSYM CX_COS; RE_CX] THEN REWRITE_TAC[COS_0; REAL_POS] THEN DISCH_THEN(X_CHOOSE_TAC `y:real`) THEN MP_TAC(SPEC `y:real` SIN_DOUBLE) THEN ASM_REWRITE_TAC[REAL_MUL_RZERO] THEN MATCH_MP_TAC(last(CONJUNCTS PI_WORKS)) THEN REPEAT(POP_ASSUM MP_TAC) THEN ASM_CASES_TAC `y = &0` THEN ASM_REWRITE_TAC[COS_0] THEN POP_ASSUM MP_TAC THEN REAL_ARITH_TAC);;
let SIN_POS_PI2 = 
prove (`!x. &0 < x /\ x < pi / &2 ==> &0 < sin(x)`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC SIN_POS_PI THEN ASM_REAL_ARITH_TAC);;
let COS_POS_PI = 
prove (`!x. --(pi / &2) < x /\ x < pi / &2 ==> &0 < cos(x)`,
GEN_TAC THEN MP_TAC(SPEC `abs x` COS_POS_PI2) THEN REWRITE_TAC[COS_ABS] THEN ASM_CASES_TAC `x = &0` THEN ASM_REWRITE_TAC[COS_0] THEN POP_ASSUM MP_TAC THEN REAL_ARITH_TAC);;
let COS_POS_PI_LE = 
prove (`!x. --(pi / &2) <= x /\ x <= pi / &2 ==> &0 <= cos(x)`,
REWRITE_TAC[REAL_LE_LT] THEN MESON_TAC[COS_PI2; COS_NEG; COS_POS_PI]);;
let SIN_POS_PI_LE = 
prove (`!x. &0 <= x /\ x <= pi ==> &0 <= sin(x)`,
REWRITE_TAC[REAL_LE_LT] THEN MESON_TAC[SIN_0; SIN_PI; SIN_POS_PI]);;
let SIN_PIMUL_EQ_0 = 
prove (`!n. sin(n * pi) = &0 <=> integer(n)`,
SUBGOAL_THEN `!n. integer n ==> sin(n * pi) = &0 /\ ~(cos(n * pi) = &0)` ASSUME_TAC THENL [REWRITE_TAC[INTEGER_CASES] THEN GEN_TAC THEN STRIP_TAC THEN CONJ_TAC THEN ASM_SIMP_TAC[REAL_MUL_LNEG; COS_NPI; SIN_NPI; SIN_NEG; COS_NEG; REAL_POW_EQ_0] THEN REAL_ARITH_TAC; ALL_TAC] THEN GEN_TAC THEN EQ_TAC THEN ASM_SIMP_TAC[] THEN SUBST1_TAC(last(CONJUNCTS(SPEC `n:real` FLOOR_FRAC))) THEN ASM_SIMP_TAC[REAL_ADD_RDISTRIB; FLOOR; SIN_ADD; REAL_MUL_LZERO] THEN ASM_SIMP_TAC[REAL_ADD_LID; REAL_ENTIRE; FLOOR] THEN DISCH_TAC THEN MP_TAC(SPEC `frac n * pi` SIN_POS_PI) THEN ASM_SIMP_TAC[REAL_LT_REFL; GSYM REAL_LT_RDIV_EQ; GSYM REAL_LT_LDIV_EQ; PI_POS; REAL_DIV_REFL; REAL_LT_IMP_NZ] THEN MP_TAC(SPEC `n:real` FLOOR_FRAC) THEN ASM_CASES_TAC `frac n = &0` THEN ASM_REWRITE_TAC[FLOOR; REAL_ADD_RID] THEN ASM_REAL_ARITH_TAC);;
let SIN_EQ_0 = 
prove (`!x. sin(x) = &0 <=> ?n. integer n /\ x = n * pi`,
GEN_TAC THEN MP_TAC(SPEC `x / pi` SIN_PIMUL_EQ_0) THEN SIMP_TAC[REAL_DIV_RMUL; REAL_LT_IMP_NZ; GSYM REAL_EQ_LDIV_EQ; PI_POS] THEN ONCE_REWRITE_TAC[CONJ_SYM] THEN REWRITE_TAC[UNWIND_THM1]);;
let COS_EQ_0 = 
prove (`!x. cos(x) = &0 <=> ?n. integer n /\ x = (n + &1 / &2) * pi`,
GEN_TAC THEN REWRITE_TAC[COS_SIN; SIN_EQ_0] THEN EQ_TAC THEN DISCH_THEN(X_CHOOSE_THEN `n:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `--n:real` THEN ASM_REWRITE_TAC[INTEGER_NEG] THEN ASM_REAL_ARITH_TAC);;
let SIN_ZERO_PI = 
prove (`!x. sin(x) = &0 <=> (?n. x = &n * pi) \/ (?n. x = --(&n * pi))`,
REWRITE_TAC[SIN_EQ_0; INTEGER_CASES] THEN MESON_TAC[REAL_MUL_LNEG]);;
let COS_ZERO_PI = 
prove (`!x. cos(x) = &0 <=> (?n. x = (&n + &1 / &2) * pi) \/ (?n. x = --((&n + &1 / &2) * pi))`,
GEN_TAC THEN REWRITE_TAC[COS_EQ_0; INTEGER_CASES; RIGHT_OR_DISTRIB] THEN REWRITE_TAC[EXISTS_OR_THM; LEFT_AND_EXISTS_THM] THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN SIMP_TAC[UNWIND_THM2] THEN EQ_TAC THEN DISCH_THEN(DISJ_CASES_THEN (X_CHOOSE_THEN `n:num` SUBST1_TAC)) THENL [DISJ1_TAC THEN EXISTS_TAC `n:num`; ASM_CASES_TAC `n = 0` THENL [DISJ1_TAC THEN EXISTS_TAC `0`; DISJ2_TAC THEN EXISTS_TAC `n - 1`]; DISJ1_TAC THEN EXISTS_TAC `n:num`; DISJ2_TAC THEN EXISTS_TAC `n + 1`] THEN ASM_SIMP_TAC[GSYM REAL_OF_NUM_SUB; GSYM REAL_OF_NUM_ADD; ARITH_RULE `1 <= n <=> ~(n = 0)`] THEN REAL_ARITH_TAC);;
let SIN_ZERO = 
prove (`!x. (sin(x) = &0) <=> (?n. EVEN n /\ x = &n * (pi / &2)) \/ (?n. EVEN n /\ x = --(&n * (pi / &2)))`,
REWRITE_TAC[SIN_ZERO_PI; EVEN_EXISTS; LEFT_AND_EXISTS_THM] THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN REWRITE_TAC[UNWIND_THM2] THEN SIMP_TAC[GSYM REAL_OF_NUM_MUL; REAL_ARITH `(&2 * x) * y / &2 = x * y`]);;
let COS_ZERO = 
prove (`!x. cos(x) = &0 <=> (?n. ~EVEN n /\ (x = &n * (pi / &2))) \/ (?n. ~EVEN n /\ (x = --(&n * (pi / &2))))`,
REWRITE_TAC[COS_ZERO_PI; NOT_EVEN; ODD_EXISTS; LEFT_AND_EXISTS_THM] THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN REWRITE_TAC[UNWIND_THM2] THEN SIMP_TAC[GSYM REAL_OF_NUM_MUL; GSYM REAL_OF_NUM_SUC; REAL_ARITH `(&2 * x + &1) * y / &2 = (x + &1 / &2) * y`]);;
let COS_ONE_2PI = 
prove (`!x. (cos(x) = &1) <=> (?n. x = &n * &2 * pi) \/ (?n. x = --(&n * &2 * pi))`,
GEN_TAC THEN EQ_TAC THEN DISCH_TAC THENL [FIRST_ASSUM(MP_TAC o SPEC `sin(x)` o MATCH_MP (REAL_RING `c = &1 ==> !s. s pow 2 + c pow 2 = &1 ==> s = &0`)) THEN REWRITE_TAC[SIN_ZERO_PI; SIN_CIRCLE] THEN DISCH_THEN(DISJ_CASES_THEN(X_CHOOSE_THEN `n:num` SUBST_ALL_TAC)) THEN POP_ASSUM MP_TAC THEN REWRITE_TAC[COS_NEG; COS_NPI; REAL_POW_NEG] THEN COND_CASES_TAC THEN REWRITE_TAC[REAL_POW_ONE] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN FIRST_X_ASSUM(MP_TAC o REWRITE_RULE[EVEN_EXISTS]) THEN REWRITE_TAC[OR_EXISTS_THM] THEN MATCH_MP_TAC MONO_EXISTS THEN SIMP_TAC[GSYM REAL_OF_NUM_MUL] THEN REAL_ARITH_TAC; FIRST_X_ASSUM (DISJ_CASES_THEN CHOOSE_TAC) THEN ASM_REWRITE_TAC[COS_NEG; REAL_MUL_ASSOC; REAL_OF_NUM_MUL; COS_NPI; REAL_POW_NEG; EVEN_MULT; ARITH; REAL_POW_ONE]]);;
let SIN_COS_SQRT = 
prove (`!x. &0 <= sin(x) ==> (sin(x) = sqrt(&1 - (cos(x) pow 2)))`,
REPEAT STRIP_TAC THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC SQRT_UNIQUE THEN ASM_REWRITE_TAC[SIN_CIRCLE; REAL_EQ_SUB_LADD]);;
let SIN_EQ_0_PI = 
prove (`!x. --pi < x /\ x < pi /\ sin(x) = &0 ==> x = &0`,
GEN_TAC THEN REWRITE_TAC[SIN_EQ_0; CONJ_ASSOC] THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC (X_CHOOSE_THEN `n:real` STRIP_ASSUME_TAC)) THEN ASM_REWRITE_TAC[REAL_ARITH `--p < n * p /\ n * p < p <=> -- &1 * p < n * p /\ n * p < &1 * p`] THEN SIMP_TAC[REAL_ENTIRE; REAL_LT_IMP_NZ; REAL_LT_RMUL_EQ; PI_POS] THEN MP_TAC(SPEC `n:real` REAL_ABS_INTEGER_LEMMA) THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC);;
let COS_TREBLE_COS = 
prove (`!x. cos(&3 * x) = &4 * cos(x) pow 3 - &3 * cos x`,
GEN_TAC THEN REWRITE_TAC[COS_ADD; REAL_ARITH `&3 * x = &2 * x + x`] THEN REWRITE_TAC[SIN_DOUBLE; COS_DOUBLE_COS] THEN MP_TAC(SPEC `x:real` SIN_CIRCLE) THEN CONV_TAC REAL_RING);;
let COS_PI6 = 
prove (`cos(pi / &6) = sqrt(&3) / &2`,
MP_TAC(ISPEC `pi / &6` COS_TREBLE_COS) THEN REWRITE_TAC[REAL_ARITH `&3 * x / &6 = x / &2`; COS_PI2] THEN REWRITE_TAC[REAL_RING `&0 = &4 * c pow 3 - &3 * c <=> c = &0 \/ (&2 * c) pow 2 = &3`] THEN SUBGOAL_THEN `&0 < cos(pi / &6)` ASSUME_TAC THENL [MATCH_MP_TAC COS_POS_PI THEN MP_TAC PI_POS THEN REAL_ARITH_TAC; DISCH_THEN(DISJ_CASES_THEN MP_TAC) THENL [ASM_MESON_TAC[REAL_LT_REFL]; ALL_TAC] THEN DISCH_THEN(MP_TAC o AP_TERM `sqrt`) THEN ASM_SIMP_TAC[POW_2_SQRT; REAL_LE_MUL; REAL_LT_IMP_LE; REAL_POS] THEN REAL_ARITH_TAC]);;
let SIN_PI6 = 
prove (`sin(pi / &6) = &1 / &2`,
MP_TAC(SPEC `pi / &6` SIN_CIRCLE) THEN REWRITE_TAC[COS_PI6] THEN SIMP_TAC[REAL_POW_DIV; SQRT_POW_2; REAL_POS] THEN MATCH_MP_TAC(REAL_FIELD `~(s + &1 / &2 = &0) ==> s pow 2 + &3 / &2 pow 2 = &1 ==> s = &1 / &2`) THEN MATCH_MP_TAC(REAL_ARITH `&0 < x ==> ~(x + &1 / &2 = &0)`) THEN MATCH_MP_TAC SIN_POS_PI THEN MP_TAC PI_POS THEN REAL_ARITH_TAC);;
let SIN_POS_PI_REV = 
prove (`!x. &0 <= x /\ x <= &2 * pi /\ &0 < sin x ==> &0 < x /\ x < pi`,
GEN_TAC THEN ASM_CASES_TAC `x = &0` THEN ASM_REWRITE_TAC[SIN_0; REAL_LT_REFL] THEN ASM_CASES_TAC `x = pi` THEN ASM_REWRITE_TAC[SIN_PI; REAL_LT_REFL] THEN ASM_CASES_TAC `x = &2 * pi` THEN ASM_REWRITE_TAC[SIN_NPI; REAL_LT_REFL] THEN REPEAT STRIP_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[GSYM REAL_NOT_LE] THEN DISCH_TAC THEN SUBGOAL_THEN `&0 < sin(&2 * pi - x)` MP_TAC THENL [MATCH_MP_TAC SIN_POS_PI THEN ASM_REAL_ARITH_TAC; REWRITE_TAC[SIN_SUB; SIN_NPI; COS_NPI] THEN ASM_REAL_ARITH_TAC]);;
(* ------------------------------------------------------------------------- *) (* Prove totality of trigs. *) (* ------------------------------------------------------------------------- *)
let SIN_TOTAL_POS = 
prove (`!y. &0 <= y /\ y <= &1 ==> ?x. &0 <= x /\ x <= pi / &2 /\ sin(x) = y`,
REPEAT STRIP_TAC THEN MP_TAC(SPECL [`csin`; `&0`; `pi / &2`; `y:real`] IVT_INCREASING_RE) THEN ASM_REWRITE_TAC[GSYM CX_SIN; RE_CX; SIN_0; SIN_PI2] THEN SIMP_TAC[CONTINUOUS_AT_CSIN; PI_POS; REAL_ARITH `&0 < x ==> &0 <= x / &2`]);;
let SINCOS_TOTAL_PI2 = 
prove (`!x y. &0 <= x /\ &0 <= y /\ x pow 2 + y pow 2 = &1 ==> ?t. &0 <= t /\ t <= pi / &2 /\ x = cos t /\ y = sin t`,
REPEAT STRIP_TAC THEN MP_TAC(SPEC `y:real` SIN_TOTAL_POS) THEN ANTS_TAC THENL [ASM_REWRITE_TAC[] THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `x pow 2 + y pow 2 = &1 ==> (&1 < y ==> &1 pow 2 < y pow 2) /\ &0 <= x * x ==> y <= &1`)) THEN SIMP_TAC[REAL_LE_SQUARE; REAL_POW_LT2; REAL_POS; ARITH]; MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `t:real` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN SUBGOAL_THEN `x = cos t \/ x = --(cos t)` MP_TAC THENL [MP_TAC(SPEC `t:real` SIN_CIRCLE); MP_TAC(SPEC `t:real` COS_POS_PI_LE)] THEN REPEAT(POP_ASSUM MP_TAC) THEN CONV_TAC REAL_FIELD]);;
let SINCOS_TOTAL_PI = 
prove (`!x y. &0 <= y /\ x pow 2 + y pow 2 = &1 ==> ?t. &0 <= t /\ t <= pi /\ x = cos t /\ y = sin t`,
REPEAT STRIP_TAC THEN DISJ_CASES_TAC(REAL_ARITH `&0 <= x \/ &0 <= --x`) THENL [MP_TAC(SPECL [`x:real`; `y:real`] SINCOS_TOTAL_PI2); MP_TAC(SPECL [`--x:real`; `y:real`] SINCOS_TOTAL_PI2)] THEN ASM_REWRITE_TAC[REAL_POW_NEG; ARITH] THEN DISCH_THEN(X_CHOOSE_THEN `t:real` STRIP_ASSUME_TAC) THENL [EXISTS_TAC `t:real`; EXISTS_TAC `pi - t`] THEN ASM_REWRITE_TAC[SIN_SUB; COS_SUB; SIN_PI; COS_PI] THEN ASM_REAL_ARITH_TAC);;
let SINCOS_TOTAL_2PI = 
prove (`!x y. x pow 2 + y pow 2 = &1 ==> ?t. &0 <= t /\ t < &2 * pi /\ x = cos t /\ y = sin t`,
REPEAT STRIP_TAC THEN ASM_CASES_TAC `x = &1 /\ y = &0` THENL [EXISTS_TAC `&0` THEN ASM_REWRITE_TAC[SIN_0; COS_0] THEN MP_TAC PI_POS THEN REAL_ARITH_TAC; ALL_TAC] THEN DISJ_CASES_TAC(REAL_ARITH `&0 <= y \/ &0 <= --y`) THENL [MP_TAC(SPECL [`x:real`; `y:real`] SINCOS_TOTAL_PI); MP_TAC(SPECL [`x:real`; `--y:real`] SINCOS_TOTAL_PI)] THEN ASM_REWRITE_TAC[REAL_POW_NEG; ARITH] THEN DISCH_THEN(X_CHOOSE_THEN `t:real` STRIP_ASSUME_TAC) THENL [EXISTS_TAC `t:real`; EXISTS_TAC `&2 * pi - t`] THEN ASM_REWRITE_TAC[SIN_SUB; COS_SUB; SIN_NPI; COS_NPI] THENL [MP_TAC PI_POS THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN REPEAT(POP_ASSUM MP_TAC) THEN ASM_CASES_TAC `t = &0` THEN ASM_REWRITE_TAC[SIN_0; COS_0] THEN POP_ASSUM MP_TAC THEN REAL_ARITH_TAC);;
let CIRCLE_SINCOS = 
prove (`!x y. x pow 2 + y pow 2 = &1 ==> ?t. x = cos(t) /\ y = sin(t)`,
MESON_TAC[SINCOS_TOTAL_2PI]);;
(* ------------------------------------------------------------------------- *) (* Polar representation. *) (* ------------------------------------------------------------------------- *)
let CX_PI_NZ = 
prove (`~(Cx pi = Cx(&0))`,
SIMP_TAC[CX_INJ; REAL_LT_IMP_NZ; PI_POS]);;
let COMPLEX_UNIMODULAR_POLAR = 
prove (`!z. (norm z = &1) ==> ?x. z = complex(cos(x),sin(x))`,
GEN_TAC THEN DISCH_THEN(MP_TAC o C AP_THM `2` o AP_TERM `(pow):real->num->real`) THEN REWRITE_TAC[complex_norm] THEN SIMP_TAC[REAL_POW_2; REWRITE_RULE[REAL_POW_2] SQRT_POW_2; REAL_LE_SQUARE; REAL_LE_ADD] THEN REWRITE_TAC[GSYM REAL_POW_2; REAL_MUL_LID] THEN DISCH_THEN(X_CHOOSE_TAC `t:real` o MATCH_MP CIRCLE_SINCOS) THEN EXISTS_TAC `t:real` THEN ASM_REWRITE_TAC[COMPLEX_EQ; RE; IM]);;
let SIN_INTEGER_2PI = 
prove (`!n. integer n ==> sin((&2 * pi) * n) = &0`,
REWRITE_TAC[SIN_EQ_0; REAL_ARITH `(&2 * pi) * n = (&2 * n) * pi`] THEN MESON_TAC[INTEGER_CLOSED]);;
let SIN_INTEGER_PI = 
prove (`!n. integer n ==> sin (n * pi) = &0`,
REWRITE_TAC[INTEGER_CASES] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[REAL_MUL_LNEG; SIN_NPI; SIN_NEG; REAL_NEG_0]);;
let COS_INTEGER_2PI = 
prove (`!n. integer n ==> cos((&2 * pi) * n) = &1`,
REWRITE_TAC[INTEGER_CASES; REAL_ARITH `(&2 * pi) * n = (&2 * n) * pi`] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[REAL_MUL_RNEG; REAL_OF_NUM_MUL] THEN SIMP_TAC[COS_NEG; COS_NPI; REAL_POW_NEG; REAL_MUL_LNEG; ARITH; EVEN_MULT; REAL_POW_ONE]);;
let SINCOS_PRINCIPAL_VALUE = 
prove (`!x. ?y. (--pi < y /\ y <= pi) /\ (sin(y) = sin(x) /\ cos(y) = cos(x))`,
GEN_TAC THEN EXISTS_TAC `pi - (&2 * pi) * frac((pi - x) / (&2 * pi))` THEN CONJ_TAC THENL [SIMP_TAC[REAL_ARITH `--p < p - x <=> x < (&2 * p) * &1`; REAL_ARITH `p - x <= p <=> (&2 * p) * &0 <= x`; REAL_LT_LMUL_EQ; REAL_LE_LMUL_EQ; REAL_LT_MUL; PI_POS; REAL_OF_NUM_LT; ARITH; FLOOR_FRAC]; REWRITE_TAC[FRAC_FLOOR; REAL_SUB_LDISTRIB] THEN SIMP_TAC[REAL_DIV_LMUL; REAL_ENTIRE; REAL_OF_NUM_EQ; ARITH; REAL_LT_IMP_NZ; PI_POS; REAL_ARITH `a - (a - b - c):real = b + c`; SIN_ADD; COS_ADD] THEN SIMP_TAC[FLOOR_FRAC; SIN_INTEGER_2PI; COS_INTEGER_2PI] THEN CONV_TAC REAL_RING]);;
let CEXP_COMPLEX = 
prove (`!r t. cexp(complex(r,t)) = Cx(exp r) * complex(cos t,sin t)`,
REPEAT GEN_TAC THEN GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [COMPLEX_EXPAND] THEN REWRITE_TAC[RE; IM; CEXP_ADD; CEXP_EULER; CX_EXP] THEN REWRITE_TAC[COMPLEX_TRAD; CX_SIN; CX_COS]);;
let NORM_COSSIN = 
prove (`!t. norm(complex(cos t,sin t)) = &1`,
REWRITE_TAC[complex_norm; RE; IM] THEN ONCE_REWRITE_TAC[REAL_ADD_SYM] THEN REWRITE_TAC[SIN_CIRCLE; SQRT_1]);;
let NORM_CEXP = 
prove (`!z. norm(cexp z) = exp(Re z)`,
REWRITE_TAC[FORALL_COMPLEX; CEXP_COMPLEX; COMPLEX_NORM_MUL] THEN REWRITE_TAC[NORM_COSSIN; RE; COMPLEX_NORM_CX] THEN MP_TAC REAL_EXP_POS_LT THEN MATCH_MP_TAC MONO_FORALL THEN REAL_ARITH_TAC);;
let NORM_CEXP_II = 
prove (`!t. norm (cexp (ii * Cx t)) = &1`,
let NORM_CEXP_IMAGINARY = 
prove (`!z. norm(cexp z) = &1 ==> Re(z) = &0`,
REWRITE_TAC[NORM_CEXP; REAL_EXP_EQ_1]);;
let CEXP_EQ_1 = 
prove (`!z. cexp z = Cx(&1) <=> Re(z) = &0 /\ ?n. integer n /\ Im(z) = &2 * n * pi`,
REWRITE_TAC[FORALL_COMPLEX; CEXP_COMPLEX; RE; IM] THEN MAP_EVERY X_GEN_TAC [`x:real`; `y:real`] THEN EQ_TAC THENL [DISCH_TAC THEN FIRST_ASSUM(MP_TAC o AP_TERM `norm:complex->real`) THEN SIMP_TAC[COMPLEX_NORM_MUL; CX_EXP; NORM_CEXP; RE_CX; COMPLEX_NORM_CX] THEN REWRITE_TAC[NORM_COSSIN; REAL_ABS_NUM; REAL_ABS_EXP; REAL_MUL_RID] THEN REWRITE_TAC[REAL_EXP_EQ_1] THEN DISCH_THEN SUBST_ALL_TAC THEN POP_ASSUM MP_TAC THEN REWRITE_TAC[REAL_EXP_0; COMPLEX_MUL_LID] THEN REWRITE_TAC[COMPLEX_EQ; RE; IM; RE_CX; IM_CX] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [SIN_EQ_0]) THEN DISCH_THEN(X_CHOOSE_THEN `m:real` (CONJUNCTS_THEN2 ASSUME_TAC SUBST_ALL_TAC)) THEN EXISTS_TAC `m / &2` THEN CONJ_TAC THENL [ALL_TAC; REAL_ARITH_TAC] THEN ONCE_REWRITE_TAC[GSYM INTEGER_ABS] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE LAND_CONV [GSYM COS_ABS]) THEN REWRITE_TAC[REAL_ABS_MUL; REAL_ABS_DIV; REAL_ABS_NUM] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [integer]) THEN DISCH_THEN(X_CHOOSE_THEN `n:num` SUBST_ALL_TAC) THEN SIMP_TAC[real_abs; PI_POS; REAL_LT_IMP_LE; COS_NPI] THEN REWRITE_TAC[REAL_POW_NEG; REAL_POW_ONE] THEN COND_CASES_TAC THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN FIRST_X_ASSUM(CHOOSE_THEN SUBST1_TAC o REWRITE_RULE[EVEN_EXISTS]) THEN REWRITE_TAC[GSYM REAL_OF_NUM_MUL; REAL_ARITH `(&2 * x) / &2 = x`] THEN REWRITE_TAC[INTEGER_CLOSED]; DISCH_THEN(CONJUNCTS_THEN2 SUBST1_TAC (X_CHOOSE_TAC `n:real`)) THEN ASM_SIMP_TAC[REAL_EXP_0; COMPLEX_MUL_LID] THEN ONCE_REWRITE_TAC[REAL_ARITH `&2 * x * y = (&2 * y) * x`] THEN ASM_SIMP_TAC[SIN_INTEGER_2PI; COS_INTEGER_2PI] THEN SIMPLE_COMPLEX_ARITH_TAC]);;
let CEXP_EQ = 
prove (`!w z. cexp w = cexp z <=> ?n. integer n /\ w = z + Cx(&2 * n * pi) * ii`,
SIMP_TAC[CEXP_NZ; COMPLEX_FIELD `~(z = Cx(&0)) ==> (w = z <=> w / z = Cx(&1))`] THEN REWRITE_TAC[GSYM CEXP_SUB; CEXP_EQ_1; RE_SUB; IM_SUB; REAL_SUB_0] THEN SIMP_TAC[COMPLEX_EQ; RE_ADD; IM_ADD; RE_MUL_II; IM_MUL_II; RE_CX; IM_CX] THEN REWRITE_TAC[REAL_NEG_0; REAL_ADD_RID; REAL_EQ_SUB_RADD] THEN MESON_TAC[REAL_ADD_SYM]);;
let COMPLEX_EQ_CEXP = 
prove (`!w z. abs(Im w - Im z) < &2 * pi /\ cexp w = cexp z ==> w = z`,
SIMP_TAC[CEXP_NZ; GSYM CEXP_SUB; CEXP_EQ_1; COMPLEX_FIELD `~(a = Cx(&0)) /\ ~(b = Cx(&0)) ==> (a = b <=> a / b = Cx(&1))`] THEN REPEAT GEN_TAC THEN DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC) THEN FIRST_X_ASSUM(X_CHOOSE_THEN `n:real` STRIP_ASSUME_TAC) THEN UNDISCH_TAC `abs(Im w - Im z) < &2 * pi` THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN DISCH_TAC THEN ASM_REWRITE_TAC[GSYM IM_SUB; REAL_ABS_MUL; REAL_ABS_PI; REAL_ABS_NUM] THEN SIMP_TAC[REAL_MUL_ASSOC; REAL_LT_RMUL_EQ; PI_POS] THEN MATCH_MP_TAC(REAL_ARITH `&1 <= x ==> ~(&2 * x < &2)`) THEN MATCH_MP_TAC REAL_ABS_INTEGER_LEMMA THEN ASM_REWRITE_TAC[] THEN DISCH_THEN SUBST_ALL_TAC THEN UNDISCH_TAC `~(w:complex = z)` THEN REWRITE_TAC[] THEN ONCE_REWRITE_TAC[GSYM COMPLEX_SUB_0] THEN ASM_REWRITE_TAC[COMPLEX_EQ; RE_CX; IM_CX; REAL_MUL_LZERO; REAL_MUL_RZERO]);;
let CEXP_INTEGER_2PI = 
prove (`!n. integer n ==> cexp(Cx(&2 * n * pi) * ii) = Cx(&1)`,
REWRITE_TAC[CEXP_EQ_1; IM_MUL_II; RE_MUL_II; RE_CX; IM_CX] THEN REWRITE_TAC[REAL_NEG_0] THEN MESON_TAC[]);;
let SIN_COS_EQ = 
prove (`!x y. sin y = sin x /\ cos y = cos x <=> ?n. integer n /\ y = x + &2 * n * pi`,
REPEAT GEN_TAC THEN MP_TAC(ISPECL [`ii * Cx y`; `ii * Cx x`] CEXP_EQ) THEN REWRITE_TAC[CEXP_EULER; GSYM CX_SIN; GSYM CX_COS] THEN REWRITE_TAC[COMPLEX_RING `ii * y = ii * x + z * ii <=> y = x + z`] THEN REWRITE_TAC[GSYM CX_ADD; CX_INJ] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN REWRITE_TAC[COMPLEX_EQ; RE_ADD; IM_ADD; RE_MUL_II; IM_MUL_II; RE_CX; IM_CX; REAL_NEG_0; REAL_ADD_LID; REAL_ADD_RID] THEN MESON_TAC[]);;
let SIN_COS_INJ = 
prove (`!x y. sin x = sin y /\ cos x = cos y /\ abs(x - y) < &2 * pi ==> x = y`,
REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[GSYM CX_INJ] THEN MATCH_MP_TAC(COMPLEX_RING `ii * x = ii * y ==> x = y`) THEN MATCH_MP_TAC COMPLEX_EQ_CEXP THEN ASM_REWRITE_TAC[CEXP_EULER; GSYM CX_SIN; GSYM CX_COS] THEN ASM_REWRITE_TAC[IM_MUL_II; RE_CX]);;
let CEXP_II_NE_1 = 
prove (`!x. &0 < x /\ x < &2 * pi ==> ~(cexp(ii * Cx x) = Cx(&1))`,
GEN_TAC THEN STRIP_TAC THEN REWRITE_TAC[CEXP_EQ_1] THEN REWRITE_TAC[RE_MUL_II; IM_CX; IM_MUL_II; IM_CX; REAL_NEG_0; RE_CX] THEN DISCH_THEN(X_CHOOSE_THEN `n:real` (CONJUNCTS_THEN2 ASSUME_TAC SUBST_ALL_TAC)) THEN UNDISCH_TAC `&0 < &2 * n * pi` THEN ASM_CASES_TAC `n = &0` THEN ASM_REWRITE_TAC[REAL_MUL_LZERO; REAL_MUL_RZERO; REAL_LT_REFL] THEN MP_TAC(ISPEC `n:real` REAL_ABS_INTEGER_LEMMA) THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP (REAL_ARITH `&2 * n * pi < &2 * pi ==> &0 < (&1 - n) * &2 * pi`)) THEN ASM_SIMP_TAC[REAL_LT_MUL_EQ; PI_POS; REAL_LT_MUL; REAL_OF_NUM_LT; ARITH] THEN ASM_REAL_ARITH_TAC);;
let CSIN_EQ_0 = 
prove (`!z. csin z = Cx(&0) <=> ?n. integer n /\ z = Cx(n * pi)`,
GEN_TAC THEN REWRITE_TAC[csin; COMPLEX_MUL_LNEG; CEXP_NEG] THEN SIMP_TAC[CEXP_NZ; COMPLEX_FIELD `~(z = Cx(&0)) ==> ((z - inv z) / (Cx(&2) * ii) = Cx(&0) <=> z pow 2 = Cx(&1))`] THEN REWRITE_TAC[GSYM CEXP_N; CEXP_EQ_1] THEN REWRITE_TAC[RE_MUL_CX; IM_MUL_CX; RE_MUL_II; IM_MUL_II] THEN REWRITE_TAC[COMPLEX_EQ; IM_CX; RE_CX; RIGHT_AND_EXISTS_THM] THEN EQ_TAC THEN MATCH_MP_TAC MONO_EXISTS THEN SIMP_TAC[] THEN REAL_ARITH_TAC);;
let CCOS_EQ_0 = 
prove (`!z. ccos z = Cx(&0) <=> ?n. integer n /\ z = Cx((n + &1 / &2) * pi)`,
GEN_TAC THEN MP_TAC(SPEC `z - Cx(pi / &2)` CSIN_EQ_0) THEN REWRITE_TAC[CSIN_SUB; GSYM CX_SIN; GSYM CX_COS; SIN_PI2; COS_PI2] THEN SIMP_TAC[COMPLEX_RING `s * Cx(&0) - c * Cx(&1) = Cx(&0) <=> c = Cx(&0)`] THEN REWRITE_TAC[REAL_ADD_RDISTRIB; COMPLEX_EQ_SUB_RADD; CX_ADD] THEN REWRITE_TAC[REAL_ARITH `&1 / &2 * x = x / &2`]);;
let CCOS_EQ_1 = 
prove (`!z. ccos z = Cx(&1) <=> ?n. integer n /\ z = Cx(&2 * n * pi)`,
GEN_TAC THEN GEN_REWRITE_TAC (LAND_CONV o LAND_CONV o RAND_CONV) [COMPLEX_RING `z = Cx(&2) * z / Cx(&2)`] THEN REWRITE_TAC[CCOS_DOUBLE_CSIN; COMPLEX_RING `a - Cx(&2) * s pow 2 = a <=> s = Cx(&0)`] THEN REWRITE_TAC[CSIN_EQ_0; CX_MUL] THEN EQ_TAC THEN MATCH_MP_TAC MONO_EXISTS THEN SIMP_TAC[] THEN CONV_TAC COMPLEX_RING);;
let CSIN_EQ_1 = 
prove (`!z. csin z = Cx(&1) <=> ?n. integer n /\ z = Cx((&2 * n + &1 / &2) * pi)`,
GEN_TAC THEN MP_TAC(SPEC `z - Cx(pi / &2)` CCOS_EQ_1) THEN REWRITE_TAC[CCOS_SUB; GSYM CX_SIN; GSYM CX_COS; SIN_PI2; COS_PI2] THEN SIMP_TAC[COMPLEX_RING `s * Cx(&0) + c * Cx(&1) = Cx(&1) <=> c = Cx(&1)`] THEN REWRITE_TAC[REAL_ADD_RDISTRIB; COMPLEX_EQ_SUB_RADD; CX_ADD] THEN REWRITE_TAC[REAL_MUL_ASSOC; REAL_ARITH `&1 / &2 * x = x / &2`]);;
let CSIN_EQ_MINUS1 = 
prove (`!z. csin z = --Cx(&1) <=> ?n. integer n /\ z = Cx((&2 * n + &3 / &2) * pi)`,
GEN_TAC THEN REWRITE_TAC[COMPLEX_RING `z:complex = --w <=> --z = w`] THEN REWRITE_TAC[GSYM CSIN_NEG; CSIN_EQ_1] THEN REWRITE_TAC[COMPLEX_RING `--z:complex = w <=> z = --w`] THEN EQ_TAC THEN DISCH_THEN(X_CHOOSE_THEN `n:real` STRIP_ASSUME_TAC) THEN ASM_REWRITE_TAC[GSYM CX_NEG; CX_INJ] THEN EXISTS_TAC `--(n + &1)` THEN ASM_SIMP_TAC[INTEGER_CLOSED] THEN REAL_ARITH_TAC);;
let CCOS_EQ_MINUS1 = 
prove (`!z. ccos z = --Cx(&1) <=> ?n. integer n /\ z = Cx((&2 * n + &1) * pi)`,
GEN_TAC THEN MP_TAC(SPEC `z - Cx(pi / &2)` CSIN_EQ_1) THEN REWRITE_TAC[CSIN_SUB; GSYM CX_SIN; GSYM CX_COS; SIN_PI2; COS_PI2] THEN SIMP_TAC[COMPLEX_RING `s * Cx(&0) - c * Cx(&1) = Cx(&1) <=> c = --Cx(&1)`] THEN REWRITE_TAC[REAL_ADD_RDISTRIB; COMPLEX_EQ_SUB_RADD; GSYM CX_ADD] THEN DISCH_TAC THEN EQ_TAC THEN MATCH_MP_TAC MONO_EXISTS THEN SIMP_TAC[CX_INJ] THEN REAL_ARITH_TAC);;
let COS_EQ_1 = 
prove (`!x. cos x = &1 <=> ?n. integer n /\ x = &2 * n * pi`,
REWRITE_TAC[GSYM CX_INJ; CX_COS; CCOS_EQ_1]);;
let SIN_EQ_1 = 
prove (`!x. sin x = &1 <=> ?n. integer n /\ x = (&2 * n + &1 / &2) * pi`,
REWRITE_TAC[GSYM CX_INJ; CX_SIN; CSIN_EQ_1]);;
let SIN_EQ_MINUS1 = 
prove (`!x. sin x = --(&1) <=> ?n. integer n /\ x = (&2 * n + &3 / &2) * pi`,
REWRITE_TAC[GSYM CX_INJ; CX_NEG; CX_SIN; CSIN_EQ_MINUS1]);;
let COS_EQ_MINUS1 = 
prove (`!x. cos x = --(&1) <=> ?n. integer n /\ x = (&2 * n + &1) * pi`,
REWRITE_TAC[GSYM CX_INJ; CX_NEG; CX_COS; CCOS_EQ_MINUS1]);;
let DIST_CEXP_II_1 = 
prove (`!z. norm(cexp(ii * Cx t) - Cx(&1)) = &2 * abs(sin(t / &2))`,
GEN_TAC THEN REWRITE_TAC[NORM_EQ_SQUARE] THEN CONJ_TAC THENL [REAL_ARITH_TAC; REWRITE_TAC[GSYM NORM_POW_2]] THEN REWRITE_TAC[CEXP_EULER; COMPLEX_SQNORM; GSYM CX_COS; GSYM CX_SIN] THEN REWRITE_TAC[IM_ADD; RE_ADD; IM_SUB; RE_SUB; IM_MUL_II; RE_MUL_II] THEN REWRITE_TAC[RE_CX; IM_CX; REAL_POW2_ABS; REAL_POW_MUL] THEN MP_TAC(ISPEC `t / &2` COS_DOUBLE_SIN) THEN REWRITE_TAC[REAL_ARITH `&2 * t / &2 = t`] THEN MP_TAC(SPEC `t:real` SIN_CIRCLE) THEN CONV_TAC REAL_RING);;
let CX_SINH = 
prove (`Cx((exp x - inv(exp x)) / &2) = --ii * csin(ii * Cx x)`,
REWRITE_TAC[csin; COMPLEX_RING `--ii * ii * z = z /\ ii * ii * z = --z`] THEN REWRITE_TAC[CEXP_NEG; GSYM CX_EXP; GSYM CX_INV; CX_SUB; CX_DIV] THEN CONV_TAC COMPLEX_FIELD);;
let CX_COSH = 
prove (`Cx((exp x + inv(exp x)) / &2) = ccos(ii * Cx x)`,
REWRITE_TAC[ccos; COMPLEX_RING `--ii * ii * z = z /\ ii * ii * z = --z`] THEN REWRITE_TAC[CEXP_NEG; GSYM CX_EXP; GSYM CX_INV; CX_ADD; CX_DIV] THEN CONV_TAC COMPLEX_FIELD);;
let NORM_CCOS_POW_2 = 
prove (`!z. norm(ccos z) pow 2 = cos(Re z) pow 2 + (exp(Im z) - inv(exp(Im z))) pow 2 / &4`,
REWRITE_TAC[FORALL_COMPLEX; RE; IM] THEN REWRITE_TAC[COMPLEX_TRAD; CCOS_ADD; COMPLEX_SQNORM] THEN SIMP_TAC[RE_SUB; IM_SUB; GSYM CX_COS; GSYM CX_SIN; IM_MUL_CX; RE_MUL_CX] THEN REWRITE_TAC[ccos; csin; CEXP_NEG; COMPLEX_FIELD `--ii * ii * z = z /\ ii * ii * z = --z /\ z / (Cx(&2) * ii) = --(ii * z / Cx(&2))`] THEN REWRITE_TAC[RE_ADD; RE_SUB; IM_ADD; IM_SUB; RE_MUL_II; IM_MUL_II; RE_DIV_CX; IM_DIV_CX; RE_NEG; IM_NEG] THEN REWRITE_TAC[GSYM CX_EXP; GSYM CX_INV; IM_CX; RE_CX] THEN MAP_EVERY X_GEN_TAC [`x:real`; `y:real`] THEN MP_TAC(SPEC `x:real` SIN_CIRCLE) THEN MP_TAC(SPEC `y:real` REAL_EXP_NZ) THEN CONV_TAC REAL_FIELD);;
let NORM_CSIN_POW_2 = 
prove (`!z. norm(csin z) pow 2 = (exp(&2 * Im z) + inv(exp(&2 * Im z)) - &2 * cos(&2 * Re z)) / &4`,
REWRITE_TAC[FORALL_COMPLEX; RE; IM] THEN REWRITE_TAC[COMPLEX_TRAD; CSIN_ADD; COMPLEX_SQNORM] THEN SIMP_TAC[RE_ADD; IM_ADD; GSYM CX_SIN; GSYM CX_SIN; IM_MUL_CX; RE_MUL_CX; GSYM CX_COS] THEN REWRITE_TAC[ccos; csin; CEXP_NEG; COMPLEX_FIELD `--ii * ii * z = z /\ ii * ii * z = --z /\ z / (Cx(&2) * ii) = --(ii * z / Cx(&2))`] THEN REWRITE_TAC[RE_ADD; RE_SUB; IM_ADD; IM_SUB; RE_MUL_II; IM_MUL_II; RE_DIV_CX; IM_DIV_CX; RE_NEG; IM_NEG] THEN REWRITE_TAC[GSYM CX_EXP; GSYM CX_INV; IM_CX; RE_CX] THEN REWRITE_TAC[REAL_EXP_N; COS_DOUBLE] THEN MAP_EVERY X_GEN_TAC [`x:real`; `y:real`] THEN MP_TAC(SPEC `x:real` SIN_CIRCLE) THEN MP_TAC(SPEC `y:real` REAL_EXP_NZ) THEN CONV_TAC REAL_FIELD);;
let CSIN_EQ = 
prove (`!w z. csin w = csin z <=> ?n. integer n /\ (w = z + Cx(&2 * n * pi) \/ w = --z + Cx((&2 * n + &1) * pi))`,
REPEAT GEN_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM COMPLEX_SUB_0] THEN REWRITE_TAC[COMPLEX_SUB_CSIN; COMPLEX_ENTIRE; CSIN_EQ_0; CCOS_EQ_0] THEN REWRITE_TAC[CX_INJ; REAL_OF_NUM_EQ; ARITH_EQ; OR_EXISTS_THM] THEN AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN X_GEN_TAC `n:real` THEN ASM_CASES_TAC `integer(n)` THEN ASM_REWRITE_TAC[COMPLEX_FIELD `a / Cx(&2) = b <=> a = Cx(&2) * b`] THEN REWRITE_TAC[GSYM CX_MUL; REAL_ARITH `&2 * (n + &1 / &2) * pi = (&2 * n + &1) * pi`] THEN CONV_TAC COMPLEX_RING);;
let CCOS_EQ = 
prove (`!w z. ccos(w) = ccos(z) <=> ?n. integer n /\ (w = z + Cx(&2 * n * pi) \/ w = --z + Cx(&2 * n * pi))`,
REPEAT GEN_TAC THEN CONV_TAC(LAND_CONV SYM_CONV) THEN GEN_REWRITE_TAC LAND_CONV [GSYM COMPLEX_SUB_0] THEN REWRITE_TAC[COMPLEX_SUB_CCOS; COMPLEX_ENTIRE; CSIN_EQ_0] THEN REWRITE_TAC[CX_INJ; REAL_OF_NUM_EQ; ARITH_EQ; OR_EXISTS_THM] THEN AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN X_GEN_TAC `n:real` THEN ASM_CASES_TAC `integer(n)` THEN ASM_REWRITE_TAC[CX_MUL] THEN CONV_TAC COMPLEX_RING);;
let SIN_EQ = 
prove (`!x y. sin x = sin y <=> ?n. integer n /\ (x = y + &2 * n * pi \/ x = --y + (&2 * n + &1) * pi)`,
REWRITE_TAC[GSYM CX_INJ; CX_SIN; CSIN_EQ] THEN REWRITE_TAC[GSYM CX_ADD; GSYM CX_NEG; CX_INJ]);;
let COS_EQ = 
prove (`!x y. cos x = cos y <=> ?n. integer n /\ (x = y + &2 * n * pi \/ x = --y + &2 * n * pi)`,
REWRITE_TAC[GSYM CX_INJ; CX_COS; CCOS_EQ] THEN REWRITE_TAC[GSYM CX_ADD; GSYM CX_NEG; CX_INJ]);;
let NORM_CCOS_LE = 
prove (`!z. norm(ccos z) <= exp(norm z)`,
GEN_TAC THEN REWRITE_TAC[ccos] THEN REWRITE_TAC[COMPLEX_NORM_DIV; COMPLEX_NORM_CX; REAL_ABS_NUM] THEN REWRITE_TAC[REAL_ARITH `x / &2 <= y <=> x <= &2 * y`] THEN MATCH_MP_TAC(NORM_ARITH `norm(a) + norm(b) <= d ==> norm(a + b) <= d`) THEN REWRITE_TAC[NORM_CEXP; COMPLEX_MUL_LNEG; RE_NEG; REAL_EXP_NEG] THEN REWRITE_TAC[COMPLEX_NORM_CX; RE_MUL_II; REAL_ABS_NUM] THEN MATCH_MP_TAC(REAL_ARITH `exp(&0) = &1 /\ (exp(&0) <= w \/ exp(&0) <= z) /\ (w <= u /\ z <= u) ==> w + z <= &2 * u`) THEN REWRITE_TAC[GSYM REAL_EXP_NEG; REAL_EXP_MONO_LE] THEN REWRITE_TAC[REAL_EXP_0] THEN MP_TAC(SPEC `z:complex` COMPLEX_NORM_GE_RE_IM) THEN REAL_ARITH_TAC);;
let NORM_CCOS_PLUS1_LE = 
prove (`!z. norm(Cx(&1) + ccos z) <= &2 * exp(norm z)`,
GEN_TAC THEN REWRITE_TAC[ccos] THEN REWRITE_TAC[COMPLEX_NORM_DIV; COMPLEX_NORM_CX; REAL_ABS_NUM; COMPLEX_RING `Cx(&1) + (z + z') / Cx(&2) = (Cx(&2) + z + z') / Cx(&2)`] THEN REWRITE_TAC[REAL_ARITH `x / &2 <= &2 * y <=> x <= &4 * y`] THEN MATCH_MP_TAC(NORM_ARITH `norm(a) + norm(b) + norm(c) <= d ==> norm(a + b + c) <= d`) THEN REWRITE_TAC[NORM_CEXP; COMPLEX_MUL_LNEG; RE_NEG; REAL_EXP_NEG] THEN REWRITE_TAC[COMPLEX_NORM_CX; RE_MUL_II; REAL_ABS_NUM] THEN MATCH_MP_TAC(REAL_ARITH `exp(&0) = &1 /\ (exp(&0) <= w \/ exp(&0) <= z) /\ (w <= u /\ z <= u) ==> &2 + w + z <= &4 * u`) THEN REWRITE_TAC[GSYM REAL_EXP_NEG; REAL_EXP_MONO_LE] THEN REWRITE_TAC[REAL_EXP_0] THEN MP_TAC(SPEC `z:complex` COMPLEX_NORM_GE_RE_IM) THEN REAL_ARITH_TAC);;
(* ------------------------------------------------------------------------- *) (* Taylor series for complex exponential. *) (* ------------------------------------------------------------------------- *)
let TAYLOR_CEXP = 
prove (`!n z. norm(cexp z - vsum(0..n) (\k. z pow k / Cx(&(FACT k)))) <= exp(abs(Re z)) * (norm z) pow (n + 1) / &(FACT n)`,
REPEAT GEN_TAC THEN MP_TAC(ISPECL [`\k:num. cexp`; `n:num`; `segment[Cx(&0),z]`; `exp(abs(Re z))`] COMPLEX_TAYLOR) THEN REWRITE_TAC[CONVEX_SEGMENT; NORM_CEXP; REAL_EXP_MONO_LE] THEN ANTS_TAC THENL [REWRITE_TAC[IN_SEGMENT] THEN REPEAT STRIP_TAC THENL [GEN_REWRITE_TAC(RATOR_CONV o LAND_CONV) [GSYM ETA_AX] THEN COMPLEX_DIFF_TAC THEN REWRITE_TAC[COMPLEX_MUL_LID]; ASM_REWRITE_TAC[GSYM COMPLEX_VEC_0; VECTOR_MUL_RZERO] THEN REWRITE_TAC[VECTOR_ADD_LID; COMPLEX_CMUL; COMPLEX_NORM_MUL] THEN GEN_REWRITE_TAC RAND_CONV [GSYM REAL_MUL_LID] THEN MATCH_MP_TAC(REAL_ARITH `abs x <= a ==> x <= a`) THEN REWRITE_TAC[RE_MUL_CX; REAL_ABS_MUL] THEN MATCH_MP_TAC REAL_LE_RMUL THEN ASM_REAL_ARITH_TAC]; DISCH_THEN(MP_TAC o SPECL [`Cx(&0)`; `z:complex`]) THEN SIMP_TAC[ENDS_IN_SEGMENT; COMPLEX_SUB_RZERO; CEXP_0; COMPLEX_MUL_LID]]);;
(* ------------------------------------------------------------------------- *) (* Approximation to e. *) (* ------------------------------------------------------------------------- *)
let E_APPROX_32 = 
prove (`abs(exp(&1) - &11674931555 / &4294967296) <= inv(&2 pow 32)`,
let lemma = prove
   (`abs(e - x) <= e * d
     ==> &0 <= d /\ d < &1
         ==> abs(e - x) <= x * d / (&1 - d)`,
    DISCH_THEN(fun th -> STRIP_TAC THEN ASSUME_TAC th THEN MP_TAC th) THEN
    MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN
    ASM_SIMP_TAC[REAL_ARITH `e * d <= x * d / i <=> d * e <= d * x / i`] THEN
    MATCH_MP_TAC REAL_LE_LMUL THEN ASM_REWRITE_TAC[] THEN
   ASM_SIMP_TAC[REAL_LE_RDIV_EQ; REAL_SUB_LT] THEN ASM_REAL_ARITH_TAC) in
  MP_TAC(ISPECL [`14`; `Cx(&1)`] TAYLOR_CEXP) THEN
  SIMP_TAC[GSYM CX_EXP; RE_CX; COMPLEX_NORM_CX; REAL_ABS_NUM;
           REAL_POW_ONE; GSYM CX_DIV; GSYM CX_POW] THEN
  CONV_TAC(ONCE_DEPTH_CONV EXPAND_VSUM_CONV) THEN
  REWRITE_TAC[GSYM CX_ADD; GSYM CX_SUB; COMPLEX_NORM_CX] THEN
  CONV_TAC NUM_REDUCE_CONV THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN
  DISCH_THEN(MP_TAC o MATCH_MP lemma) THEN REAL_ARITH_TAC);;
(* ------------------------------------------------------------------------- *) (* Taylor series for complex sine and cosine. *) (* ------------------------------------------------------------------------- *)
let TAYLOR_CSIN_RAW = 
prove (`!n z. norm(csin z - vsum(0..n) (\k. if ODD k then --ii * (ii * z) pow k / Cx(&(FACT k)) else Cx(&0))) <= exp(abs(Im z)) * (norm z) pow (n + 1) / &(FACT n)`,
MP_TAC TAYLOR_CEXP THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN X_GEN_TAC `z:complex` THEN REWRITE_TAC[csin] THEN REWRITE_TAC[COMPLEX_FIELD `a / (Cx(&2) * ii) - b = (a - Cx(&2) * ii * b) / (Cx(&2) * ii)`] THEN FIRST_ASSUM(fun th -> MP_TAC(SPEC `ii * z` th) THEN MP_TAC(SPEC `--ii * z` th)) THEN REWRITE_TAC[COMPLEX_MUL_LNEG; RE_NEG; REAL_ABS_NEG; RE_MUL_II] THEN REWRITE_TAC[COMPLEX_NORM_DIV; COMPLEX_NORM_MUL; COMPLEX_NORM_CX; NORM_NEG; COMPLEX_NORM_II; REAL_ABS_NUM; REAL_MUL_RID; REAL_MUL_LID; REAL_ARITH `x / &2 <= y <=> x <= &2 * y`] THEN MATCH_MP_TAC(NORM_ARITH `sp - sn = s2 ==> norm(en - sn) <= d ==> norm(ep - sp) <= d ==> norm(ep - en - s2) <= &2 * d`) THEN SIMP_TAC[GSYM VSUM_SUB; GSYM VSUM_COMPLEX_LMUL; FINITE_NUMSEG] THEN MATCH_MP_TAC VSUM_EQ THEN X_GEN_TAC `k:num` THEN DISCH_TAC THEN REWRITE_TAC[COMPLEX_POW_NEG; GSYM NOT_EVEN] THEN ASM_CASES_TAC `EVEN k` THEN ASM_REWRITE_TAC[COMPLEX_SUB_REFL; COMPLEX_MUL_RZERO] THEN REWRITE_TAC[COMPLEX_RING `Cx(&2) * ii * --(ii * z) = Cx(&2) * z`] THEN SIMPLE_COMPLEX_ARITH_TAC);;
let TAYLOR_CSIN = 
prove (`!n z. norm(csin z - vsum(0..n) (\k. --Cx(&1) pow k * z pow (2 * k + 1) / Cx(&(FACT(2 * k + 1))))) <= exp(abs(Im z)) * norm(z) pow (2 * n + 3) / &(FACT(2 * n + 2))`,
REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`SUC(2 * n + 1)`; `z:complex`] TAYLOR_CSIN_RAW) THEN SIMP_TAC[VSUM_CLAUSES_NUMSEG; VSUM_PAIR_0; ODD_ADD; ODD_MULT; ARITH_ODD; LE_0; ODD; COMPLEX_ADD_LID; COMPLEX_ADD_RID] THEN SIMP_TAC[ARITH_RULE `SUC(2 * n + 1) = 2 * n + 2`; GSYM ADD_ASSOC; ARITH] THEN MATCH_MP_TAC(NORM_ARITH `s = t ==> norm(x - s) <= e ==> norm(x - t) <= e`) THEN MATCH_MP_TAC VSUM_EQ THEN X_GEN_TAC `k:num` THEN STRIP_TAC THEN REWRITE_TAC[COMPLEX_POW_MUL; complex_div; COMPLEX_MUL_ASSOC] THEN AP_THM_TAC THEN AP_TERM_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[COMPLEX_POW_ADD; GSYM COMPLEX_POW_POW] THEN REWRITE_TAC[COMPLEX_POW_II_2] THEN CONV_TAC COMPLEX_RING);;
let TAYLOR_CCOS_RAW = 
prove (`!n z. norm(ccos z - vsum(0..n) (\k. if EVEN k then (ii * z) pow k / Cx(&(FACT k)) else Cx(&0))) <= exp(abs(Im z)) * (norm z) pow (n + 1) / &(FACT n)`,
MP_TAC TAYLOR_CEXP THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN X_GEN_TAC `z:complex` THEN REWRITE_TAC[ccos] THEN REWRITE_TAC[COMPLEX_FIELD `a / Cx(&2) - b = (a - Cx(&2) * b) / Cx(&2)`] THEN FIRST_ASSUM(fun th -> MP_TAC(SPEC `ii * z` th) THEN MP_TAC(SPEC `--ii * z` th)) THEN REWRITE_TAC[COMPLEX_MUL_LNEG; RE_NEG; REAL_ABS_NEG; RE_MUL_II] THEN REWRITE_TAC[COMPLEX_NORM_DIV; COMPLEX_NORM_MUL; COMPLEX_NORM_CX; NORM_NEG; COMPLEX_NORM_II; REAL_ABS_NUM; REAL_MUL_RID; REAL_MUL_LID; REAL_ARITH `x / &2 <= y <=> x <= &2 * y`] THEN MATCH_MP_TAC(NORM_ARITH `sp + sn = s2 ==> norm(en - sn) <= d ==> norm(ep - sp) <= d ==> norm((ep + en) - s2) <= &2 * d`) THEN SIMP_TAC[GSYM VSUM_ADD; GSYM VSUM_COMPLEX_LMUL; FINITE_NUMSEG] THEN MATCH_MP_TAC VSUM_EQ THEN X_GEN_TAC `k:num` THEN DISCH_TAC THEN REWRITE_TAC[COMPLEX_POW_NEG; GSYM NOT_EVEN] THEN ASM_CASES_TAC `EVEN k` THEN ASM_REWRITE_TAC[COMPLEX_ADD_RINV; COMPLEX_MUL_RZERO] THEN SIMPLE_COMPLEX_ARITH_TAC);;
let TAYLOR_CCOS = 
prove (`!n z. norm(ccos z - vsum(0..n) (\k. --Cx(&1) pow k * z pow (2 * k) / Cx(&(FACT(2 * k))))) <= exp(abs(Im z)) * norm(z) pow (2 * n + 2) / &(FACT(2 * n + 1))`,
REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`2 * n + 1`; `z:complex`] TAYLOR_CCOS_RAW) THEN SIMP_TAC[VSUM_PAIR_0; EVEN_ADD; EVEN_MULT; ARITH_EVEN; LE_0; EVEN; COMPLEX_ADD_LID; COMPLEX_ADD_RID] THEN SIMP_TAC[ARITH_RULE `(2 * n + 1) + 1 = 2 * n + 2`] THEN MATCH_MP_TAC(NORM_ARITH `s = t ==> norm(x - s) <= e ==> norm(x - t) <= e`) THEN MATCH_MP_TAC VSUM_EQ THEN X_GEN_TAC `k:num` THEN STRIP_TAC THEN REWRITE_TAC[COMPLEX_POW_MUL; complex_div; COMPLEX_MUL_ASSOC] THEN AP_THM_TAC THEN AP_TERM_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[GSYM COMPLEX_POW_POW; COMPLEX_POW_II_2]);;
(* ------------------------------------------------------------------------- *) (* The argument of a complex number, where 0 <= arg(z) < 2 pi *) (* ------------------------------------------------------------------------- *)
let Arg_DEF = new_definition
 `Arg z = if z = Cx(&0) then &0
          else @t. &0 <= t /\ t < &2 * pi /\
                   z = Cx(norm(z)) * cexp(ii * Cx t)`;;
let ARG_0 = 
prove (`Arg(Cx(&0)) = &0`,
REWRITE_TAC[Arg_DEF]);;
let ARG = 
prove (`!z. &0 <= Arg(z) /\ Arg(z) < &2 * pi /\ z = Cx(norm z) * cexp(ii * Cx(Arg z))`,
GEN_TAC THEN REWRITE_TAC[Arg_DEF] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[COMPLEX_NORM_0; COMPLEX_MUL_LZERO] THEN SIMP_TAC[REAL_LE_REFL; REAL_LT_MUL; PI_POS; REAL_ARITH `&0 < &2`] THEN CONV_TAC SELECT_CONV THEN MP_TAC(SPECL [`Re(z) / norm z`; `Im(z) / norm z`] SINCOS_TOTAL_2PI) THEN ASM_SIMP_TAC[COMPLEX_SQNORM; COMPLEX_NORM_ZERO; REAL_FIELD `~(z = &0) /\ x pow 2 + y pow 2 = z pow 2 ==> (x / z) pow 2 + (y / z) pow 2 = &1`] THEN MATCH_MP_TAC MONO_EXISTS THEN ASM_SIMP_TAC[COMPLEX_NORM_ZERO; REAL_FIELD `~(z = &0) ==> (x / z = y <=> x = z * y)`] THEN REWRITE_TAC[COMPLEX_EQ; RE_MUL_CX; IM_MUL_CX; CEXP_EULER; RE_ADD; IM_ADD; RE_MUL_II; IM_MUL_II; GSYM CX_SIN; GSYM CX_COS; RE_CX; IM_CX] THEN REAL_ARITH_TAC);;
let COMPLEX_NORM_EQ_1_CEXP = 
prove (`!z. norm z = &1 <=> (?t. z = cexp(ii * Cx t))`,
GEN_TAC THEN EQ_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC [NORM_CEXP; RE_MUL_II; IM_CX; REAL_NEG_0; REAL_EXP_0] THEN MP_TAC (SPEC `z:complex` ARG) THEN ASM_REWRITE_TAC [COMPLEX_MUL_LID] THEN MESON_TAC[]);;
let ARG_UNIQUE = 
prove (`!a r z. &0 < r /\ Cx r * cexp(ii * Cx a) = z /\ &0 <= a /\ a < &2 * pi ==> Arg z = a`,
REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[GSYM CX_INJ] THEN MATCH_MP_TAC(COMPLEX_RING `ii * x = ii * y ==> x = y`) THEN MATCH_MP_TAC COMPLEX_EQ_CEXP THEN CONJ_TAC THENL [REWRITE_TAC[IM_MUL_II; RE_CX] THEN MATCH_MP_TAC(REAL_ARITH `&0 <= x /\ x < p /\ &0 <= y /\ y < p ==> abs(x - y) < p`) THEN ASM_SIMP_TAC[ARG]; MATCH_MP_TAC(COMPLEX_RING `!a b. Cx a = Cx b /\ ~(Cx b = Cx(&0)) /\ Cx a * w = Cx b * z ==> w = z`) THEN MAP_EVERY EXISTS_TAC [`norm(z:complex)`; `r:real`] THEN ASM_REWRITE_TAC[GSYM ARG] THEN ASM_SIMP_TAC[CX_INJ; REAL_LT_IMP_NZ] THEN EXPAND_TAC "z" THEN REWRITE_TAC[NORM_CEXP_II; COMPLEX_NORM_MUL; COMPLEX_NORM_CX] THEN ASM_REAL_ARITH_TAC]);;
let ARG_MUL_CX = 
prove (`!r z. &0 < r ==> Arg(Cx r * z) = Arg(z)`,
REPEAT STRIP_TAC THEN ASM_CASES_TAC `z = Cx(&0)` THEN ASM_REWRITE_TAC[COMPLEX_MUL_RZERO] THEN MATCH_MP_TAC ARG_UNIQUE THEN EXISTS_TAC `r * norm(z:complex)` THEN ASM_REWRITE_TAC[CX_MUL; GSYM COMPLEX_MUL_ASSOC; GSYM ARG] THEN ASM_SIMP_TAC[REAL_LT_MUL; COMPLEX_NORM_NZ]);;
let ARG_DIV_CX = 
prove (`!r z. &0 < r ==> Arg(z / Cx r) = Arg(z)`,
REWRITE_TAC[ONCE_REWRITE_RULE[COMPLEX_MUL_SYM] complex_div] THEN SIMP_TAC[GSYM CX_INV; ARG_MUL_CX; REAL_LT_INV_EQ]);;
let ARG_LT_NZ = 
prove (`!z. &0 < Arg z <=> ~(Arg z = &0)`,
MP_TAC ARG THEN MATCH_MP_TAC MONO_FORALL THEN REAL_ARITH_TAC);;
let ARG_LE_PI = 
prove (`!z. Arg z <= pi <=> &0 <= Im z`,
GEN_TAC THEN ASM_CASES_TAC `z = Cx(&0)` THENL [ASM_REWRITE_TAC[Arg_DEF; IM_CX; REAL_LE_REFL; PI_POS_LE]; ALL_TAC] THEN GEN_REWRITE_TAC (funpow 3 RAND_CONV) [ARG] THEN ASM_SIMP_TAC[IM_MUL_CX; CEXP_EULER; REAL_LE_MUL_EQ; COMPLEX_NORM_NZ] THEN REWRITE_TAC[IM_ADD; GSYM CX_SIN; GSYM CX_COS; IM_CX; IM_MUL_II; RE_CX] THEN REWRITE_TAC[REAL_ADD_LID] THEN EQ_TAC THEN SIMP_TAC[ARG; SIN_POS_PI_LE] THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN DISCH_TAC THEN SUBGOAL_THEN `&0 < sin(&2 * pi - Arg z)` MP_TAC THENL [MATCH_MP_TAC SIN_POS_PI THEN MP_TAC(SPEC `z:complex` ARG) THEN ASM_REAL_ARITH_TAC; REWRITE_TAC[SIN_SUB; SIN_NPI; COS_NPI] THEN REAL_ARITH_TAC]);;
let ARG_LT_PI = 
prove (`!z. &0 < Arg z /\ Arg z < pi <=> &0 < Im z`,
GEN_TAC THEN ASM_CASES_TAC `z = Cx(&0)` THENL [ASM_REWRITE_TAC[Arg_DEF; IM_CX; REAL_LT_REFL; PI_POS_LE]; ALL_TAC] THEN GEN_REWRITE_TAC (funpow 3 RAND_CONV) [ARG] THEN ASM_SIMP_TAC[IM_MUL_CX; CEXP_EULER; REAL_LT_MUL_EQ; COMPLEX_NORM_NZ] THEN REWRITE_TAC[IM_ADD; GSYM CX_SIN; GSYM CX_COS; IM_CX; IM_MUL_II; RE_CX] THEN REWRITE_TAC[REAL_ADD_LID] THEN EQ_TAC THEN SIMP_TAC[SIN_POS_PI] THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN ASM_CASES_TAC `Arg z = &0` THEN ASM_REWRITE_TAC[SIN_0; REAL_LT_REFL] THEN ASM_SIMP_TAC[ARG; REAL_ARITH `~(x = &0) ==> (&0 < x <=> &0 <= x)`] THEN DISCH_TAC THEN SUBGOAL_THEN `&0 <= sin(&2 * pi - Arg z)` MP_TAC THENL [MATCH_MP_TAC SIN_POS_PI_LE THEN MP_TAC(SPEC `z:complex` ARG) THEN ASM_REAL_ARITH_TAC; REWRITE_TAC[SIN_SUB; SIN_NPI; COS_NPI] THEN REAL_ARITH_TAC]);;
let ARG_EQ_0 = 
prove (`!z. Arg z = &0 <=> real z /\ &0 <= Re z`,
GEN_TAC THEN ASM_CASES_TAC `z = Cx(&0)` THENL [ASM_REWRITE_TAC[REAL_CX; RE_CX; Arg_DEF; REAL_LE_REFL]; ALL_TAC] THEN CONV_TAC(RAND_CONV(SUBS_CONV[last(CONJUNCTS(SPEC `z:complex` ARG))])) THEN ASM_SIMP_TAC[RE_MUL_CX; REAL_MUL_CX; REAL_LE_MUL_EQ; COMPLEX_NORM_NZ] THEN ASM_REWRITE_TAC[COMPLEX_NORM_ZERO; CEXP_EULER] THEN REWRITE_TAC[real; RE_ADD; IM_ADD; RE_MUL_II; IM_MUL_II; GSYM CX_SIN; GSYM CX_COS; RE_CX; IM_CX] THEN REWRITE_TAC[REAL_ADD_RID; REAL_ADD_LID; REAL_NEG_0] THEN EQ_TAC THEN SIMP_TAC[SIN_0; COS_0; REAL_POS] THEN ASM_CASES_TAC `Arg z = pi` THENL [ASM_REWRITE_TAC[COS_PI] THEN REAL_ARITH_TAC; ALL_TAC] THEN MP_TAC(SPEC `z:complex` ARG) THEN REWRITE_TAC[CONJ_ASSOC] THEN DISCH_THEN(MP_TAC o CONJUNCT1) THEN DISCH_THEN(STRIP_ASSUME_TAC o MATCH_MP (REAL_ARITH `&0 <= x /\ x < &2 * pi ==> --pi < x /\ x < pi \/ --pi < x - pi /\ x - pi < pi`)) THEN REPEAT STRIP_TAC THEN ASM_SIMP_TAC[SIN_EQ_0_PI] THEN UNDISCH_TAC `~(Arg z = pi)` THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN DISCH_TAC THEN REWRITE_TAC[REAL_ARITH `x = pi <=> x - pi = &0`] THEN MATCH_MP_TAC SIN_EQ_0_PI THEN ASM_REWRITE_TAC[SIN_SUB; SIN_PI] THEN REAL_ARITH_TAC);;
let ARG_NUM = 
prove (`!n. Arg(Cx(&n)) = &0`,
REWRITE_TAC[ARG_EQ_0; REAL_CX; RE_CX; REAL_POS]);;
let ARG_EQ_PI = 
prove (`!z. Arg z = pi <=> real z /\ Re z < &0`,
SIMP_TAC[ARG; PI_POS; REAL_ARITH `&0 < pi /\ &0 <= z ==> (z = pi <=> z <= pi /\ ~(z = &0) /\ ~(&0 < z /\ z < pi))`] THEN REWRITE_TAC[ARG_EQ_0; ARG; ARG_LT_PI; ARG_LE_PI; real] THEN REAL_ARITH_TAC);;
let ARG_EQ_0_PI = 
prove (`!z. Arg z = &0 \/ Arg z = pi <=> real z`,
REWRITE_TAC[ARG_EQ_0; ARG_EQ_PI; real] THEN REAL_ARITH_TAC);;
let ARG_INV = 
prove (`!z. ~(real z /\ &0 <= Re z) ==> Arg(inv z) = &2 * pi - Arg z`,
GEN_TAC THEN ASM_CASES_TAC `z = Cx(&0)` THEN ASM_REWRITE_TAC[REAL_CX; RE_CX; REAL_LE_REFL] THEN REWRITE_TAC[real] THEN STRIP_TAC THEN MATCH_MP_TAC ARG_UNIQUE THEN EXISTS_TAC `inv(norm(z:complex))` THEN ASM_SIMP_TAC[COMPLEX_NORM_NZ; REAL_LT_INV_EQ] THEN REWRITE_TAC[CX_SUB; CX_MUL; COMPLEX_SUB_LDISTRIB; CEXP_SUB] THEN SUBST1_TAC(SPEC `Cx(&2) * Cx pi` CEXP_EULER) THEN REWRITE_TAC[GSYM CX_MUL; GSYM CX_SIN; GSYM CX_COS] THEN REWRITE_TAC[SIN_NPI; COS_NPI; COMPLEX_MUL_RZERO; COMPLEX_ADD_RID] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[complex_div; COMPLEX_MUL_LID; CX_INV; GSYM COMPLEX_INV_MUL] THEN REWRITE_TAC[GSYM ARG] THEN MP_TAC(SPEC `z:complex` ARG_EQ_0) THEN ASM_REWRITE_TAC[real] THEN MP_TAC(SPEC `z:complex` ARG) THEN REAL_ARITH_TAC);;
let ARG_EQ = 
prove (`!w z. ~(w = Cx(&0)) /\ ~(z = Cx(&0)) ==> (Arg w = Arg z <=> ?x. &0 < x /\ w = Cx(x) * z)`,
REPEAT STRIP_TAC THEN EQ_TAC THENL [ALL_TAC; STRIP_TAC THEN ASM_SIMP_TAC[ARG_MUL_CX]] THEN DISCH_TAC THEN MAP_EVERY (MP_TAC o CONJUNCT2 o CONJUNCT2 o C SPEC ARG) [`z:complex`; `w:complex`] THEN ASM_REWRITE_TAC[IMP_IMP] THEN DISCH_THEN(fun th -> CONV_TAC(SUBS_CONV(CONJUNCTS th))) THEN EXISTS_TAC `norm(w:complex) / norm(z:complex)` THEN ASM_SIMP_TAC[REAL_LT_DIV; COMPLEX_NORM_NZ; CX_DIV] THEN REWRITE_TAC[COMPLEX_MUL_ASSOC] THEN AP_THM_TAC THEN AP_TERM_TAC THEN ASM_SIMP_TAC[COMPLEX_DIV_RMUL; COMPLEX_NORM_ZERO; CX_INJ]);;
let ARG_INV_EQ_0 = 
prove (`!z. Arg(inv z) = &0 <=> Arg z = &0`,
GEN_TAC THEN REWRITE_TAC[ARG_EQ_0; REAL_INV_EQ] THEN MATCH_MP_TAC(TAUT `(a ==> (b <=> c)) ==> (a /\ b <=> a /\ c)`) THEN REWRITE_TAC[real] THEN DISCH_TAC THEN ASM_REWRITE_TAC[complex_inv; RE] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[REAL_ADD_RID] THEN ASM_CASES_TAC `Re z = &0` THEN ASM_REWRITE_TAC[real_div; REAL_MUL_LZERO] THEN ASM_SIMP_TAC[REAL_FIELD `~(x = &0) ==> x * inv(x pow 2) = inv x`] THEN REWRITE_TAC[REAL_LE_INV_EQ]);;
let ARG_LE_DIV_SUM = 
prove (`!w z. ~(w = Cx(&0)) /\ ~(z = Cx(&0)) /\ Arg(w) <= Arg(z) ==> Arg(z) = Arg(w) + Arg(z / w)`,
REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[REAL_ARITH `a:real = b + c <=> c = a - b`] THEN MATCH_MP_TAC ARG_UNIQUE THEN EXISTS_TAC `norm(z / w)`THEN ASM_SIMP_TAC[ARG; REAL_ARITH `&0 <= a /\ a < &2 * pi /\ &0 <= b /\ b <= a ==> a - b < &2 * pi`] THEN ASM_REWRITE_TAC[REAL_SUB_LE] THEN ASM_SIMP_TAC[COMPLEX_NORM_DIV; CX_DIV] THEN ASM_SIMP_TAC[REAL_LT_DIV; COMPLEX_NORM_NZ] THEN REWRITE_TAC[COMPLEX_SUB_LDISTRIB; CEXP_SUB; CX_SUB] THEN REWRITE_TAC[complex_div] THEN ONCE_REWRITE_TAC[COMPLEX_RING `(a * b) * (c * d):complex = (a * c) * (b * d)`] THEN REWRITE_TAC[GSYM COMPLEX_INV_MUL] THEN ASM_SIMP_TAC[GSYM ARG]);;
let ARG_LE_DIV_SUM_EQ = 
prove (`!w z. ~(w = Cx(&0)) /\ ~(z = Cx(&0)) ==> (Arg(w) <= Arg(z) <=> Arg(z) = Arg(w) + Arg(z / w))`,
let REAL_SUB_ARG = 
prove (`!w z. ~(w = Cx(&0)) /\ ~(z = Cx(&0)) ==> Arg w - Arg z = if Arg(z) <= Arg(w) then Arg(w / z) else Arg(w / z) - &2 * pi`,
REPEAT STRIP_TAC THEN COND_CASES_TAC THENL [MP_TAC(ISPECL [`z:complex`; `w:complex`] ARG_LE_DIV_SUM) THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC; MP_TAC(ISPECL [`w:complex`; `z:complex`] ARG_LE_DIV_SUM) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; DISCH_THEN SUBST1_TAC] THEN REWRITE_TAC[REAL_ARITH `a - (a + b):real = --b`] THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [GSYM COMPLEX_INV_DIV] THEN MATCH_MP_TAC(REAL_ARITH `x = &2 * pi - y ==> --x = y - &2 * pi`) THEN MATCH_MP_TAC ARG_INV THEN REWRITE_TAC[GSYM ARG_EQ_0] THEN ONCE_REWRITE_TAC[GSYM COMPLEX_INV_DIV] THEN REWRITE_TAC[ARG_INV_EQ_0] THEN MP_TAC(ISPECL [`w:complex`; `z:complex`] ARG_LE_DIV_SUM) THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC]);;
let REAL_ADD_ARG = 
prove (`!w z. ~(w = Cx(&0)) /\ ~(z = Cx(&0)) ==> Arg(w) + Arg(z) = if Arg w + Arg z < &2 * pi then Arg(w * z) else Arg(w * z) + &2 * pi`,
REPEAT STRIP_TAC THEN MP_TAC(SPECL [`w * z:complex`; `z:complex`] REAL_SUB_ARG) THEN MP_TAC(SPECL [`z:complex`; `w * z:complex`] ARG_LE_DIV_SUM_EQ) THEN ASM_SIMP_TAC[COMPLEX_ENTIRE; COMPLEX_FIELD `~(z = Cx(&0)) ==> (w * z) / z = w`] THEN ASM_CASES_TAC `Arg (w * z) = Arg z + Arg w` THEN ASM_REWRITE_TAC[] THENL [ASM_MESON_TAC[ARG; REAL_ADD_SYM]; SIMP_TAC[REAL_ARITH `wz - z = w - &2 * pi <=> w + z = wz + &2 * pi`] THEN REWRITE_TAC[REAL_ARITH `w + p < p <=> ~(&0 <= w)`; ARG]]);;
let ARG_MUL = 
prove (`!w z. ~(w = Cx(&0)) /\ ~(z = Cx(&0)) ==> Arg(w * z) = if Arg w + Arg z < &2 * pi then Arg w + Arg z else (Arg w + Arg z) - &2 * pi`,
REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP REAL_ADD_ARG) THEN REAL_ARITH_TAC);;
let ARG_CNJ = 
prove (`!z. Arg(cnj z) = if real z /\ &0 <= Re z then Arg z else &2 * pi - Arg z`,
GEN_TAC THEN ASM_CASES_TAC `z = Cx(&0)` THEN ASM_REWRITE_TAC[CNJ_CX; ARG_0; REAL_CX; RE_CX; REAL_LE_REFL] THEN COND_CASES_TAC THEN ASM_SIMP_TAC[REAL_IMP_CNJ] THEN MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `Arg(inv z)` THEN CONJ_TAC THENL [REWRITE_TAC[COMPLEX_INV_CNJ] THEN ASM_SIMP_TAC[GSYM CX_POW; ARG_DIV_CX; REAL_POW_LT; COMPLEX_NORM_NZ]; ASM_SIMP_TAC[ARG_INV]]);;
let ARG_REAL = 
prove (`!z. real z ==> Arg z = if &0 <= Re z then &0 else pi`,
REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[ARG_EQ_PI; ARG_EQ_0] THEN ASM_REAL_ARITH_TAC);;
let ARG_CEXP = 
prove (`!z. &0 <= Im z /\ Im z < &2 * pi ==> Arg(cexp(z)) = Im z`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC ARG_UNIQUE THEN EXISTS_TAC `exp(Re z)` THEN ASM_REWRITE_TAC[CX_EXP; GSYM CEXP_ADD; REAL_EXP_POS_LT] THEN REWRITE_TAC[GSYM COMPLEX_EXPAND]);;
(* ------------------------------------------------------------------------- *) (* Properties of 2-D rotations, and their interpretation using cexp. *) (* ------------------------------------------------------------------------- *)
let rotate2d = new_definition
 `(rotate2d:real->real^2->real^2) t x =
        vector[x$1 * cos(t) - x$2 * sin(t);
               x$1 * sin(t) + x$2 * cos(t)]`;;
let LINEAR_ROTATE2D = 
prove (`!t. linear(rotate2d t)`,
SIMP_TAC[linear; CART_EQ; DIMINDEX_2; FORALL_2; VECTOR_2; VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT; rotate2d] THEN REAL_ARITH_TAC);;
let ROTATE2D_ADD_VECTORS = 
prove (`!t w z. rotate2d t (w + z) = rotate2d t w + rotate2d t z`,
let ROTATE2D_SUB = 
prove (`!t w z. rotate2d t (w - z) = rotate2d t w - rotate2d t z`,
let NORM_ROTATE2D = 
prove (`!t z. norm(rotate2d t z) = norm z`,
REWRITE_TAC[NORM_EQ; rotate2d; DIMINDEX_2; DOT_2; VECTOR_2] THEN REPEAT GEN_TAC THEN MP_TAC(ISPEC `t:real` SIN_CIRCLE) THEN CONV_TAC REAL_RING);;
let ROTATE2D_0 = 
prove (`!t. rotate2d t (Cx(&0)) = Cx(&0)`,
let ROTATE2D_EQ_0 = 
prove (`!t z. rotate2d t z = Cx(&0) <=> z = Cx(&0)`,
REWRITE_TAC[GSYM COMPLEX_NORM_ZERO; NORM_ROTATE2D]);;
let ROTATE2D_ZERO = 
prove (`!z. rotate2d (&0) z = z`,
REWRITE_TAC[rotate2d; SIN_0; COS_0] THEN REWRITE_TAC[CART_EQ; DIMINDEX_2; FORALL_2; VECTOR_2] THEN REAL_ARITH_TAC);;
let ORTHOGONAL_TRANSFORMATION_ROTATE2D = 
prove (`!t. orthogonal_transformation(rotate2d t)`,
let ROTATE2D_POLAR = 
prove (`!r t s. rotate2d t (vector[r * cos(s); r * sin(s)]) = vector[r * cos(t + s); r * sin(t + s)]`,
SIMP_TAC[rotate2d; DIMINDEX_2; VECTOR_2; CART_EQ; FORALL_2] THEN REWRITE_TAC[SIN_ADD; COS_ADD] THEN REAL_ARITH_TAC);;
let MATRIX_ROTATE2D = 
prove (`!t. matrix(rotate2d t) = vector[vector[cos t;--(sin t)]; vector[sin t; cos t]]`,
SIMP_TAC[MATRIX_EQ; MATRIX_WORKS; LINEAR_ROTATE2D] THEN SIMP_TAC[matrix_vector_mul; rotate2d; CART_EQ; DIMINDEX_2; FORALL_2; LAMBDA_BETA; VECTOR_2; ARITH; SUM_2] THEN REAL_ARITH_TAC);;
let DET_MATRIX_ROTATE2D = 
prove (`!t. det(matrix(rotate2d t)) = &1`,
GEN_TAC THEN REWRITE_TAC[MATRIX_ROTATE2D; DET_2; VECTOR_2] THEN MP_TAC(SPEC `t:real` SIN_CIRCLE) THEN REAL_ARITH_TAC);;
let ROTATION_ROTATE2D = 
prove (`!f. orthogonal_transformation f /\ det(matrix f) = &1 ==> ?t. &0 <= t /\ t < &2 * pi /\ f = rotate2d t`,
REWRITE_TAC[ORTHOGONAL_TRANSFORMATION_MATRIX] THEN REWRITE_TAC[matrix_mul; orthogonal_matrix; transp] THEN SIMP_TAC[DIMINDEX_2; SUM_2; FORALL_2; LAMBDA_BETA; ARITH; CART_EQ; mat; DET_2] THEN REPEAT STRIP_TAC THEN SUBGOAL_THEN `(matrix f)$1$1 pow 2 + (matrix f)$2$1 pow 2 = &1 /\ (matrix f)$1$2 = --((matrix f)$2$1) /\ (matrix f:real^2^2)$2$2 = (matrix f)$1$1` STRIP_ASSUME_TAC THENL [REPEAT(FIRST_X_ASSUM(MP_TAC o SYM)) THEN CONV_TAC REAL_RING; FIRST_X_ASSUM(MP_TAC o MATCH_MP SINCOS_TOTAL_2PI) THEN MATCH_MP_TAC MONO_EXISTS THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC LINEAR_EQ_MATRIX THEN ASM_REWRITE_TAC[LINEAR_ROTATE2D; MATRIX_ROTATE2D] THEN ASM_SIMP_TAC[CART_EQ; DIMINDEX_2; FORALL_2; VECTOR_2]]);;
let ROTATE2D_ADD = 
prove (`!s t x. rotate2d (s + t) x = rotate2d s (rotate2d t x)`,
SIMP_TAC[CART_EQ; rotate2d; LAMBDA_BETA; DIMINDEX_2; ARITH; FORALL_2; VECTOR_2] THEN REWRITE_TAC[SIN_ADD; COS_ADD] THEN REAL_ARITH_TAC);;
let ROTATE2D_COMPLEX = 
prove (`!t z. rotate2d t z = cexp(ii * Cx t) * z`,
REPEAT GEN_TAC THEN GEN_REWRITE_TAC RAND_CONV [complex_mul] THEN REWRITE_TAC[CEXP_EULER; rotate2d; GSYM CX_SIN; GSYM CX_COS; RE_ADD; IM_ADD; RE_MUL_II; IM_MUL_II; IM_CX; RE_CX] THEN REWRITE_TAC[CART_EQ; FORALL_2; VECTOR_2; DIMINDEX_2] THEN REWRITE_TAC[GSYM RE_DEF; GSYM IM_DEF; RE; IM] THEN REAL_ARITH_TAC);;
let ROTATE2D_PI2 = 
prove (`!z. rotate2d (pi / &2) z = ii * z`,
REWRITE_TAC[ROTATE2D_COMPLEX; CEXP_EULER; SIN_PI2; COS_PI2; GSYM CX_SIN; GSYM CX_COS] THEN CONV_TAC COMPLEX_RING);;
let ROTATE2D_PI = 
prove (`!z. rotate2d pi z = --z`,
REWRITE_TAC[ROTATE2D_COMPLEX; CEXP_EULER; SIN_PI; COS_PI; GSYM CX_SIN; GSYM CX_COS] THEN CONV_TAC COMPLEX_RING);;
let ROTATE2D_NPI = 
prove (`!n z. rotate2d (&n * pi) z = --Cx(&1) pow n * z`,
REWRITE_TAC[ROTATE2D_COMPLEX; CEXP_EULER; SIN_NPI; COS_NPI; GSYM CX_SIN; GSYM CX_COS; CX_NEG; CX_POW] THEN CONV_TAC COMPLEX_RING);;
let ROTATE2D_2PI = 
prove (`!z. rotate2d (&2 * pi) z = z`,
REWRITE_TAC[ROTATE2D_NPI] THEN CONV_TAC COMPLEX_RING);;
let ARG_ROTATE2D = 
prove (`!t z. ~(z = Cx(&0)) /\ &0 <= t + Arg z /\ t + Arg z < &2 * pi ==> Arg(rotate2d t z) = t + Arg z`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC ARG_UNIQUE THEN EXISTS_TAC `norm(z:complex)` THEN ASM_SIMP_TAC[ARG; ROTATE2D_COMPLEX; REAL_LE_ADD; COMPLEX_NORM_NZ] THEN GEN_REWRITE_TAC (RAND_CONV o RAND_CONV) [ARG] THEN REWRITE_TAC[CX_ADD; COMPLEX_ADD_LDISTRIB; CEXP_ADD] THEN REWRITE_TAC[COMPLEX_MUL_AC]);;
let ARG_ROTATE2D_UNIQUE = 
prove (`!t a z. ~(z = Cx(&0)) /\ Arg(rotate2d t z) = a ==> ?n. integer n /\ t = &2 * n * pi + (a - Arg z)`,
REPEAT STRIP_TAC THEN MP_TAC(last(CONJUNCTS(ISPEC `rotate2d t z` ARG))) THEN ASM_REWRITE_TAC[NORM_ROTATE2D] THEN REWRITE_TAC[ROTATE2D_COMPLEX] THEN GEN_REWRITE_TAC (LAND_CONV o LAND_CONV o RAND_CONV) [ARG] THEN ASM_REWRITE_TAC[COMPLEX_RING `a * z * b = z * c <=> z = Cx(&0) \/ a * b = c`; CX_INJ; COMPLEX_NORM_ZERO; GSYM CEXP_ADD; CEXP_EQ] THEN MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN MATCH_MP_TAC MONO_AND THEN REWRITE_TAC[GSYM CX_ADD; GSYM CX_SUB; CX_INJ; COMPLEX_RING `ii * t + ii * z = ii * a + n * ii <=> t = n + (a - z)`]);;
let ARG_ROTATE2D_UNIQUE_2PI = 
prove (`!s t z. ~(z = Cx(&0)) /\ &0 <= s /\ s < &2 * pi /\ &0 <= t /\ t < &2 * pi /\ Arg(rotate2d s z) = Arg(rotate2d t z) ==> s = t`,
REPEAT STRIP_TAC THEN ABBREV_TAC `a = Arg(rotate2d t z)` THEN MP_TAC(ISPECL [`s:real`; `a:real`; `z:complex`] ARG_ROTATE2D_UNIQUE) THEN MP_TAC(ISPECL [`t:real`; `a:real`; `z:complex`] ARG_ROTATE2D_UNIQUE) THEN ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC SIN_COS_INJ THEN REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL [ASM_REWRITE_TAC[SIN_COS_EQ; REAL_RING `x + az:real = (y + az) + z <=> x - y = z`] THEN REWRITE_TAC[GSYM REAL_SUB_LDISTRIB; GSYM REAL_SUB_RDISTRIB] THEN ASM_MESON_TAC[INTEGER_CLOSED]; ASM_REAL_ARITH_TAC]);;
let COMPLEX_DIV_ROTATION = 
prove (`!f w z. orthogonal_transformation f /\ det(matrix f) = &1 ==> f w / f z = w / z`,
REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP ROTATION_ROTATE2D) THEN DISCH_THEN(X_CHOOSE_THEN `t:real` STRIP_ASSUME_TAC) THEN ASM_REWRITE_TAC[ROTATE2D_COMPLEX] THEN SIMP_TAC[complex_div; COMPLEX_INV_MUL; CEXP_NZ; COMPLEX_FIELD `~(a = Cx(&0)) ==> (a * w) * (inv a * z) = w * z`]);;
let th = 
prove (`!f w z. linear f /\ (!x. norm(f x) = norm x) /\ (2 <= dimindex(:2) ==> det(matrix f) = &1) ==> f w / f z = w / z`,
REWRITE_TAC[CONJ_ASSOC; GSYM ORTHOGONAL_TRANSFORMATION; DIMINDEX_2; LE_REFL; COMPLEX_DIV_ROTATION]) in add_linear_invariants [th];;
let th = 
prove (`!f t z. linear f /\ (!x. norm(f x) = norm x) /\ (2 <= dimindex(:2) ==> det(matrix f) = &1) ==> rotate2d t (f z) = f(rotate2d t z)`,
REWRITE_TAC[DIMINDEX_2; LE_REFL] THEN REPEAT STRIP_TAC THEN MP_TAC(SPEC `f:complex->complex` ROTATION_ROTATE2D) THEN ASM_REWRITE_TAC[ORTHOGONAL_TRANSFORMATION] THEN DISCH_THEN(X_CHOOSE_THEN `s:real` STRIP_ASSUME_TAC) THEN ASM_REWRITE_TAC[GSYM ROTATE2D_ADD] THEN REWRITE_TAC[REAL_ADD_SYM]) in add_linear_invariants [th];;
let ROTATION_ROTATE2D_EXISTS_GEN = 
prove (`!x y. ?t. &0 <= t /\ t < &2 * pi /\ norm(y) % rotate2d t x = norm(x) % y`,
REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`norm(y:real^2) % x:real^2`; `norm(x:real^2) % y:real^2`] ROTATION_EXISTS) THEN ASM_REWRITE_TAC[DIMINDEX_2; NORM_MUL; ARITH; REAL_ABS_NORM; EQT_INTRO(SPEC_ALL REAL_MUL_SYM); CONJ_ASSOC] THEN DISCH_THEN(X_CHOOSE_THEN `f:real^2->real^2` (CONJUNCTS_THEN ASSUME_TAC)) THEN FIRST_ASSUM(MP_TAC o MATCH_MP ROTATION_ROTATE2D) THEN MATCH_MP_TAC MONO_EXISTS THEN FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN REPEAT STRIP_TAC THEN ASM_SIMP_TAC[LINEAR_CMUL; LINEAR_ROTATE2D]);;
let ROTATION_ROTATE2D_EXISTS = 
prove (`!x y. norm x = norm y ==> ?t. &0 <= t /\ t < &2 * pi /\ rotate2d t x = y`,
REPEAT GEN_TAC THEN ASM_CASES_TAC `norm(y:complex) = &0` THENL [ASM_REWRITE_TAC[] THEN DISCH_TAC THEN EXISTS_TAC `&0` THEN SIMP_TAC[REAL_LT_MUL; PI_POS; REAL_OF_NUM_LT; ARITH; REAL_LE_REFL] THEN ASM_MESON_TAC[COMPLEX_NORM_ZERO; ROTATE2D_0]; DISCH_TAC THEN MP_TAC(ISPECL [`x:complex`; `y:complex`] ROTATION_ROTATE2D_EXISTS_GEN) THEN ASM_REWRITE_TAC[VECTOR_MUL_LCANCEL]]);;
let ROTATION_ROTATE2D_EXISTS_ORTHOGONAL = 
prove (`!e1 e2. norm(e1) = &1 /\ norm(e2) = &1 /\ orthogonal e1 e2 ==> e1 = rotate2d (pi / &2) e2 \/ e2 = rotate2d (pi / &2) e1`,
REWRITE_TAC[NORM_EQ_1; orthogonal] THEN SIMP_TAC[DOT_2; CART_EQ; FORALL_2; DIMINDEX_2; rotate2d; VECTOR_2] THEN REWRITE_TAC[COS_PI2; SIN_PI2; REAL_MUL_RZERO; REAL_ADD_RID; REAL_SUB_LZERO; REAL_SUB_RZERO; REAL_MUL_RID] THEN CONV_TAC REAL_RING);;
let ROTATION_ROTATE2D_EXISTS_ORTHOGONAL_ORIENTED = 
prove (`!e1 e2. norm(e1) = &1 /\ norm(e2) = &1 /\ orthogonal e1 e2 /\ &0 < e1$1 * e2$2 - e1$2 * e2$1 ==> e2 = rotate2d (pi / &2) e1`,
REPEAT GEN_TAC THEN REWRITE_TAC[CONJ_ASSOC] THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN REWRITE_TAC[GSYM CONJ_ASSOC] THEN DISCH_TAC THEN FIRST_ASSUM(DISJ_CASES_THEN SUBST_ALL_TAC o MATCH_MP ROTATION_ROTATE2D_EXISTS_ORTHOGONAL) THEN REWRITE_TAC[] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM REAL_NOT_LE]) THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN DISCH_THEN(K ALL_TAC) THEN SIMP_TAC[DOT_2; CART_EQ; FORALL_2; DIMINDEX_2; rotate2d; VECTOR_2] THEN REWRITE_TAC[COS_PI2; SIN_PI2; REAL_MUL_RZERO; REAL_ADD_RID; REAL_SUB_LZERO; REAL_SUB_RZERO; REAL_MUL_RID] THEN REWRITE_TAC[REAL_ARITH `--x * x - y * y <= &0 <=> &0 <= x * x + y * y`] THEN MATCH_MP_TAC REAL_LE_ADD THEN REWRITE_TAC[REAL_LE_SQUARE]);;
let ROTATE2D_EQ = 
prove (`!t x y. rotate2d t x = rotate2d t y <=> x = y`,
let ROTATE2D_SUB_ARG = 
prove (`!w z. ~(w = Cx(&0)) /\ ~(z = Cx(&0)) ==> rotate2d(Arg w - Arg z) = rotate2d(Arg(w / z))`,
REPEAT STRIP_TAC THEN ASM_SIMP_TAC[REAL_SUB_ARG] THEN COND_CASES_TAC THEN REWRITE_TAC[real_sub; ROTATE2D_ADD; FUN_EQ_THM] THEN GEN_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[ROTATE2D_COMPLEX] THEN REWRITE_TAC[EULER; RE_MUL_II; IM_MUL_II; RE_CX; IM_CX; COS_NEG; SIN_NEG] THEN REWRITE_TAC[SIN_NPI; COS_NPI; REAL_EXP_NEG; REAL_EXP_0; CX_NEG] THEN REWRITE_TAC[COMPLEX_NEG_0; COMPLEX_MUL_RZERO; COMPLEX_ADD_RID] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[COMPLEX_MUL_LID]);;
let ROTATION_MATRIX_ROTATE2D = 
prove (`!t. rotation_matrix(matrix(rotate2d t))`,
SIMP_TAC[ROTATION_MATRIX_2; MATRIX_ROTATE2D; VECTOR_2] THEN MESON_TAC[SIN_CIRCLE; REAL_ADD_SYM]);;
let ROTATION_MATRIX_ROTATE2D_EQ = 
prove (`!A:real^2^2. rotation_matrix A <=> ?t. A = matrix(rotate2d t)`,
GEN_TAC THEN EQ_TAC THEN SIMP_TAC[LEFT_IMP_EXISTS_THM; ROTATION_MATRIX_ROTATE2D] THEN REWRITE_TAC[ROTATION_MATRIX_2; MATRIX_ROTATE2D] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP SINCOS_TOTAL_2PI) THEN MATCH_MP_TAC MONO_EXISTS THEN REPEAT STRIP_TAC THEN REWRITE_TAC[CART_EQ; DIMINDEX_2; FORALL_2; VECTOR_2] THEN ASM_REAL_ARITH_TAC);;
(* ------------------------------------------------------------------------- *) (* Homotopy staying within the set of orthogonal transformations *) (* ------------------------------------------------------------------------- *)
let NULLHOMOTOPIC_ORTHOGONAL_TRANSFORMATION = 
prove (`!f:real^N->real^N. orthogonal_transformation f /\ det(matrix f) = &1 ==> homotopic_with orthogonal_transformation ((:real^N),(:real^N)) f I`,
let lemma0 = prove
   (`!a x:real^N.
          2 <= dimindex(:N) /\ a IN span {basis 1,basis 2}
          ==> reflect_along (vector[a$1; a$2]:real^2) (lambda i. x$i) =
              (lambda i. reflect_along a x$i)`,
    REPEAT STRIP_TAC THEN
    SIMP_TAC[CART_EQ; LAMBDA_BETA; reflect_along; VECTOR_SUB_COMPONENT;
             VECTOR_MUL_COMPONENT; DIMINDEX_2; FORALL_2; VECTOR_2; ARITH] THEN
    CONJ_TAC THEN AP_TERM_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN
    AP_TERM_TAC THEN BINOP_TAC THEN REWRITE_TAC[dot] THEN
    CONV_TAC SYM_CONV THEN MATCH_MP_TAC SUM_EQ_SUPERSET THEN
    ASM_SIMP_TAC[FINITE_NUMSEG; IN_NUMSEG; FORALL_2; DIMINDEX_2; LAMBDA_BETA;
                 ARITH; VECTOR_2; SUBSET_NUMSEG] THEN
    REWRITE_TAC[ARITH_RULE
     `(1 <= i /\ i <= n) /\ ~(1 <= i /\ i <= 2) <=>
      1 <= i /\ 3 <= i /\ i <= n`] THEN
    FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [SPAN_2]) THEN
    REWRITE_TAC[IN_ELIM_THM; IN_UNIV] THEN
    STRIP_TAC THEN
    ASM_REWRITE_TAC[VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT] THEN
    SIMP_TAC[BASIS_COMPONENT] THEN
    REPEAT STRIP_TAC THEN
    REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[REAL_MUL_RZERO]) THEN
    ASM_ARITH_TAC) in
  let lemma1 = prove
   (`!a b:real^2 r.
          ~(a = vec 0) /\ ~(b = vec 0)
          ==> homotopic_with orthogonal_transformation ((:real^2),(:real^2))
                             (reflect_along a o reflect_along b) I`,
    REPEAT STRIP_TAC THEN
    MP_TAC(SPEC `reflect_along (a:real^2) o reflect_along b`
          ROTATION_ROTATE2D) THEN
    ANTS_TAC THENL
     [REPEAT(FIRST_X_ASSUM(MP_TAC o
        MATCH_MP ROTOINVERSION_MATRIX_REFLECT_ALONG)) THEN
      REWRITE_TAC[rotoinversion_matrix] THEN
      SIMP_TAC[ORTHOGONAL_MATRIX_MATRIX;
               ORTHGOONAL_TRANSFORMATION_REFLECT_ALONG;
               ORTHOGONAL_TRANSFORMATION_COMPOSE; MATRIX_COMPOSE;
               LINEAR_REFLECT_ALONG; DET_MUL] THEN
      CONV_TAC REAL_RAT_REDUCE_CONV;
      DISCH_THEN(X_CHOOSE_THEN `t:real` STRIP_ASSUME_TAC) THEN
      ONCE_REWRITE_TAC[HOMOTOPIC_WITH_SYM] THEN
      ASM_REWRITE_TAC[homotopic_with] THEN
      EXISTS_TAC `\z. rotate2d (drop(fstcart z) * t) (sndcart z)` THEN
      SIMP_TAC[ORTHOGONAL_TRANSFORMATION_ROTATE2D; SNDCART_PASTECART;
               ETA_AX; FSTCART_PASTECART; DROP_VEC; I_THM; NORM_ROTATE2D;
               REAL_MUL_LZERO; REAL_MUL_LID; SUBSET; FORALL_IN_IMAGE; IN_UNIV;
               FORALL_IN_PCROSS; IN_SPHERE_0; ROTATE2D_ZERO] THEN
      REWRITE_TAC[ROTATE2D_COMPLEX] THEN
      MATCH_MP_TAC CONTINUOUS_ON_COMPLEX_MUL THEN
      SIMP_TAC[LINEAR_CONTINUOUS_ON; LINEAR_SNDCART] THEN
      GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN
      MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN
      REWRITE_TAC[CONTINUOUS_ON_CEXP; CX_MUL] THEN
      ONCE_REWRITE_TAC[COMPLEX_RING `ii * x * t = (ii * t) * x`] THEN
      MATCH_MP_TAC CONTINUOUS_ON_COMPLEX_LMUL THEN
      MATCH_MP_TAC CONTINUOUS_ON_CX_DROP THEN
      SIMP_TAC[LINEAR_CONTINUOUS_ON; LINEAR_FSTCART]]) in
  let lemma2 = prove
   (`!a b:real^N r.
          2 <= dimindex(:N) /\
          ~(a = vec 0) /\ ~(b = vec 0) /\
          {a,b} SUBSET span {basis 1,basis 2}
          ==> homotopic_with orthogonal_transformation ((:real^N),(:real^N))
                             (reflect_along a o reflect_along b) I`,
    REPEAT STRIP_TAC THEN
    SUBGOAL_THEN
      `homotopic_with orthogonal_transformation
        ((:real^N),(:real^N))
        ((\z. (lambda i. if i <= 2 then (fstcart z)$i
                         else (sndcart z)$i):real^N) o
         (\z. pastecart
               (((reflect_along (vector [(a:real^N)$1; a$2]) o
                 reflect_along (vector [(b:real^N)$1; b$2]))
                  :real^2->real^2)(fstcart z))
               (sndcart z)) o
         (\z:real^N. pastecart ((lambda i. z$i) :real^2) z))
        ((\z. (lambda i. if i <= 2 then (fstcart z)$i
                         else (sndcart z)$i):real^N) o
         I o
         (\z:real^N. pastecart ((lambda i. z$i) :real^2) z))`
    MP_TAC THENL
     [MATCH_MP_TAC HOMOTOPIC_WITH_COMPOSE_CONTINUOUS_LEFT THEN
      EXISTS_TAC `(:real^2) PCROSS (:real^N)` THEN
      REWRITE_TAC[SUBSET_UNIV] THEN CONJ_TAC THENL
       [ALL_TAC;
        MATCH_MP_TAC LINEAR_CONTINUOUS_ON THEN
        ONCE_REWRITE_TAC[LINEAR_COMPONENTWISE] THEN
        SIMP_TAC[LAMBDA_BETA] THEN X_GEN_TAC `i:num` THEN
        STRIP_TAC THEN ASM_CASES_TAC `i <= 2` THEN ASM_REWRITE_TAC[] THEN
        REWRITE_TAC[linear; FSTCART_ADD; FSTCART_CMUL;
                            SNDCART_ADD; SNDCART_CMUL] THEN
        REWRITE_TAC[VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT] THEN
        REWRITE_TAC[LIFT_ADD; LIFT_CMUL]] THEN
      MATCH_MP_TAC HOMOTOPIC_WITH_COMPOSE_CONTINUOUS_RIGHT THEN
      EXISTS_TAC `(:real^2) PCROSS (:real^N)` THEN
      REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_UNIV; PASTECART_IN_PCROSS] THEN
      CONJ_TAC THENL
       [ALL_TAC;
        MATCH_MP_TAC LINEAR_CONTINUOUS_ON THEN
        MATCH_MP_TAC LINEAR_PASTECART THEN REWRITE_TAC[LINEAR_ID] THEN
        SIMP_TAC[linear; CART_EQ; LAMBDA_BETA; VECTOR_ADD_COMPONENT;
                 VECTOR_MUL_COMPONENT]] THEN
      SUBGOAL_THEN
       `I = \z:real^(2,N)finite_sum. pastecart (fstcart z) (sndcart z)`
      SUBST1_TAC THENL
       [REWRITE_TAC[PASTECART_FST_SND; I_DEF]; ALL_TAC] THEN
      MATCH_MP_TAC HOMOTOPIC_WITH_PCROSS THEN
      EXISTS_TAC `orthogonal_transformation:(real^2->real^2)->bool` THEN
      EXISTS_TAC `\f:real^N->real^N. f = I` THEN REPEAT CONJ_TAC THENL
       [REWRITE_TAC[GSYM I_DEF; ETA_AX] THEN MATCH_MP_TAC lemma1 THEN
        FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [INSERT_SUBSET]) THEN
        REWRITE_TAC[SING_SUBSET; SPAN_2; IN_ELIM_THM; IN_UNIV] THEN
        DISCH_THEN(REPEAT_TCL STRIP_THM_THEN SUBST_ALL_TAC) THEN
        POP_ASSUM_LIST(MP_TAC o end_itlist CONJ o rev) THEN
        REWRITE_TAC[CART_EQ; VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT;
                    DIMINDEX_2; FORALL_2; VECTOR_2] THEN
        SIMP_TAC[BASIS_COMPONENT; ARITH; DIMINDEX_2; VEC_COMPONENT;
                 DIMINDEX_GE_1; LE_REFL] THEN
        MATCH_MP_TAC(TAUT
         `(r ==> q) /\ (s ==> p) ==> a /\ ~p /\ ~q ==> ~s /\ ~r`) THEN
        SIMP_TAC[REAL_MUL_RZERO; REAL_MUL_LZERO; REAL_MUL_RID;
                 REAL_ADD_LID; REAL_ADD_RID];
        REWRITE_TAC[HOMOTOPIC_WITH_REFL; SUBSET_UNIV; I_DEF] THEN
        REWRITE_TAC[CONTINUOUS_ON_ID];
        SIMP_TAC[o_DEF; FSTCART_PASTECART; SNDCART_PASTECART;
                 LAMBDA_BETA; DIMINDEX_2; ARITH; I_THM] THEN
        REWRITE_TAC[ORTHOGONAL_TRANSFORMATION; NORM_EQ] THEN
        X_GEN_TAC `f:real^2->real^2` THEN GEN_TAC THEN STRIP_TAC THEN
        CONJ_TAC THENL
         [FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [linear]) THEN
          SIMP_TAC[linear; CART_EQ; LAMBDA_BETA; VECTOR_ADD_COMPONENT;
                   VECTOR_MUL_COMPONENT; DIMINDEX_2; ARITH] THEN
          MATCH_MP_TAC MONO_AND THEN CONJ_TAC THEN
          DISCH_THEN(ASSUME_TAC o GSYM) THEN GEN_TAC THEN
          GEN_TAC THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN
          COND_CASES_TAC THEN ASM_SIMP_TAC[] THEN
          AP_THM_TAC THEN AP_TERM_TAC THEN AP_TERM_TAC THEN
          SIMP_TAC[CART_EQ; LAMBDA_BETA; VECTOR_ADD_COMPONENT;
                   VECTOR_MUL_COMPONENT];
          X_GEN_TAC `v:real^N` THEN REWRITE_TAC[dot; GSYM REAL_POW_2] THEN
          SUBGOAL_THEN `dimindex(:N) = 2 + (dimindex(:N) - 2)` SUBST1_TAC THENL
           [ASM_ARITH_TAC; ALL_TAC] THEN
          ASM_SIMP_TAC[SUM_ADD_SPLIT; ARITH_RULE `1 <= n + 1`] THEN
          BINOP_TAC THENL
           [RULE_ASSUM_TAC(REWRITE_RULE[dot; DIMINDEX_2; GSYM REAL_POW_2]) THEN
            FIRST_X_ASSUM(MP_TAC o SPEC `(lambda i. (v:real^N)$i):real^2`) THEN
            MATCH_MP_TAC EQ_IMP THEN BINOP_TAC THEN
            MATCH_MP_TAC SUM_EQ_NUMSEG THEN
            FIRST_ASSUM(MP_TAC o MATCH_MP (ARITH_RULE
             `2 <= n ==> !i. i <= 2 ==> i <= n`)) THEN
            SIMP_TAC[LAMBDA_BETA; DIMINDEX_2];
            ASM_SIMP_TAC[ARITH_RULE `2 <= n ==> 2 + n - 2 = n`] THEN
            MATCH_MP_TAC SUM_EQ_NUMSEG THEN
            SIMP_TAC[ARITH_RULE `2 + 1 <= i ==> 1 <= i`;
                     LAMBDA_BETA; DIMINDEX_2] THEN
            REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
            ASM_ARITH_TAC]]];
      MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ_ALT] HOMOTOPIC_WITH_EQ) THEN
      REWRITE_TAC[IN_UNIV; GSYM FUN_EQ_THM] THEN
      SIMP_TAC[o_DEF; FSTCART_PASTECART; SNDCART_PASTECART;
               LAMBDA_BETA; DIMINDEX_2; ARITH; I_THM] THEN
      RULE_ASSUM_TAC(REWRITE_RULE[INSERT_SUBSET; EMPTY_SUBSET]) THEN
      ASM_SIMP_TAC[lemma0] THEN
      SIMP_TAC[CART_EQ; LAMBDA_BETA; DIMINDEX_2; ARITH; COND_ID] THEN
      MAP_EVERY X_GEN_TAC [`x:real^N`; `i:num`] THEN STRIP_TAC THEN
      COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
      SUBGOAL_THEN `(a:real^N)$i = &0 /\ (b:real^N)$i = &0` ASSUME_TAC THENL
       [FIRST_X_ASSUM(CONJUNCTS_THEN MP_TAC) THEN
        REWRITE_TAC[SPAN_2; IN_ELIM_THM; IN_UNIV] THEN REPEAT STRIP_TAC THEN
        ASM_SIMP_TAC[VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT;
                     BASIS_COMPONENT] THEN
        REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[]) THEN
        (REAL_ARITH_TAC ORELSE ASM_ARITH_TAC);
        ASM_REWRITE_TAC[reflect_along; VECTOR_SUB_COMPONENT; REAL_MUL_RZERO;
                        VECTOR_MUL_COMPONENT; REAL_SUB_RZERO]]]) in
  let lemma3 = prove
   (`!a b:real^N r.
          ~(a = vec 0) /\ ~(b = vec 0)
          ==> homotopic_with orthogonal_transformation ((:real^N),(:real^N))
                             (reflect_along a o reflect_along b) I`,
    REPEAT STRIP_TAC THEN ASM_CASES_TAC `dimindex(:N) = 1` THENL
     [ASM_SIMP_TAC[o_DEF; I_DEF; REFLECT_ALONG_1D; VECTOR_NEG_NEG] THEN
      REWRITE_TAC[HOMOTOPIC_WITH_REFL; SUBSET_UNIV; CONTINUOUS_ON_ID] THEN
      REWRITE_TAC[ORTHOGONAL_TRANSFORMATION_ID];
      FIRST_X_ASSUM(MP_TAC o MATCH_MP(ARITH_RULE
       `~(n = 1) ==> 1 <= n ==> 2 <= n`)) THEN
      REWRITE_TAC[DIMINDEX_GE_1] THEN DISCH_TAC] THEN
    MP_TAC(ISPECL [`span{a:real^N,b}`; `span{basis 1:real^N,basis 2}`]
          ORTHOGONAL_TRANSFORMATION_INTO_SUBSPACE) THEN
    REWRITE_TAC[SUBSPACE_SPAN; DIM_SPAN] THEN ANTS_TAC THENL
     [ASM_REWRITE_TAC[DIM_INSERT; SPAN_SING; SPAN_EMPTY;
                      IN_SING; DIM_EMPTY] THEN
      MATCH_MP_TAC(ARITH_RULE `m <= 2 /\ n = 2 ==> m <= n`) THEN
      CONJ_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN
      ASM_SIMP_TAC[BASIS_NONZERO; ARITH] THEN
      REWRITE_TAC[IN_ELIM_THM; IN_UNIV] THEN
      COND_CASES_TAC THEN REWRITE_TAC[] THEN
      FIRST_X_ASSUM(CHOOSE_THEN (MP_TAC o AP_TERM `(\x:real^N. x$1)`)) THEN
      ASM_SIMP_TAC[BASIS_COMPONENT; VECTOR_MUL_COMPONENT;
                   ARITH; DIMINDEX_GE_1] THEN
      REAL_ARITH_TAC;
      DISCH_THEN(X_CHOOSE_THEN `f:real^N->real^N` STRIP_ASSUME_TAC) THEN
      MP_TAC(ISPEC `f:real^N->real^N` ORTHOGONAL_TRANSFORMATION_INVERSE_o) THEN
      ASM_REWRITE_TAC[] THEN
      DISCH_THEN(X_CHOOSE_THEN `g:real^N->real^N` STRIP_ASSUME_TAC)] THEN
    SUBGOAL_THEN
     `homotopic_with orthogonal_transformation ((:real^N),(:real^N))
       (g o (f o (reflect_along a o reflect_along b) o (g:real^N->real^N)) o f)
       (g o (f o I o (g:real^N->real^N)) o f)`
    MP_TAC THENL
     [ALL_TAC;
      ASM_REWRITE_TAC[o_ASSOC] THEN ASM_REWRITE_TAC[GSYM o_ASSOC; I_O_ID]] THEN
    MATCH_MP_TAC HOMOTOPIC_WITH_COMPOSE_CONTINUOUS_LEFT THEN
    EXISTS_TAC `(:real^N)` THEN REWRITE_TAC[SUBSET_UNIV] THEN
    ASM_SIMP_TAC[ORTHOGONAL_TRANSFORMATION_LINEAR; LINEAR_CONTINUOUS_ON] THEN
    MATCH_MP_TAC HOMOTOPIC_WITH_COMPOSE_CONTINUOUS_RIGHT THEN
    EXISTS_TAC `(:real^N)` THEN REWRITE_TAC[SUBSET_UNIV] THEN
    ASM_SIMP_TAC[ORTHOGONAL_TRANSFORMATION_LINEAR; LINEAR_CONTINUOUS_ON] THEN
    ASM_REWRITE_TAC[I_O_ID] THEN
    MP_TAC(ISPEC `f:real^N->real^N` REFLECT_ALONG_LINEAR_IMAGE) THEN
    ASM_REWRITE_TAC[GSYM ORTHOGONAL_TRANSFORMATION] THEN
    DISCH_THEN(ASSUME_TAC o GSYM) THEN
    SUBGOAL_THEN
     `!h:real^N->real^N.
          orthogonal_transformation (g o h o (f:real^N->real^N)) <=>
          orthogonal_transformation h`
     (fun th -> REWRITE_TAC[th; ETA_AX])
    THENL
     [GEN_TAC THEN EQ_TAC THEN
      ASM_SIMP_TAC[ORTHOGONAL_TRANSFORMATION_COMPOSE] THEN
      DISCH_TAC THEN
      SUBGOAL_THEN `h:real^N->real^N = f o (g o h o f) o (g:real^N->real^N)`
      SUBST1_TAC THENL
       [ALL_TAC; ASM_SIMP_TAC[ORTHOGONAL_TRANSFORMATION_COMPOSE]] THEN
      ASM_REWRITE_TAC[o_ASSOC] THEN ASM_REWRITE_TAC[GSYM o_ASSOC; I_O_ID];
      ALL_TAC] THEN
    SUBGOAL_THEN
     `(f:real^N->real^N) o (reflect_along a o reflect_along b) o g =
      reflect_along (f a) o reflect_along (f b)`
    SUBST1_TAC THENL
     [RULE_ASSUM_TAC(REWRITE_RULE[FUN_EQ_THM; o_THM; I_THM]) THEN
      ASM_REWRITE_TAC[o_DEF];
      MATCH_MP_TAC lemma2 THEN RULE_ASSUM_TAC
       (REWRITE_RULE[GSYM NORM_EQ_0; ORTHOGONAL_TRANSFORMATION]) THEN
      ASM_REWRITE_TAC[GSYM NORM_EQ_0] THEN
      FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT]
         SUBSET_TRANS)) THEN
      ASM_SIMP_TAC[GSYM SPAN_LINEAR_IMAGE; IMAGE_CLAUSES] THEN
      REWRITE_TAC[SPAN_INC]]) in
  GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
  MP_TAC(ISPECL [`f:real^N->real^N`; `dimindex(:N)`]
        ORTHOGONAL_TRANSFORMATION_GENERATED_BY_REFLECTIONS) THEN
  ASM_REWRITE_TAC[ARITH_RULE `n:num <= a + n`] THEN
  DISCH_THEN(X_CHOOSE_THEN `l:(real^N)list` STRIP_ASSUME_TAC) THEN
  UNDISCH_TAC `ALL (\v:real^N. ~(v = vec 0)) l` THEN
  UNDISCH_TAC `orthogonal_transformation(f:real^N->real^N)` THEN
  MATCH_MP_TAC(TAUT `r /\ (p /\ q ==> s) ==> r ==> p ==> q ==> s`) THEN
  ASM_REWRITE_TAC[IMP_IMP] THEN
  SPEC_TAC(`l:(real^N)list`,`l:(real^N)list`) THEN
  POP_ASSUM_LIST(K ALL_TAC) THEN GEN_TAC THEN
  WF_INDUCT_TAC `LENGTH(l:(real^N)list)` THEN POP_ASSUM MP_TAC THEN
  SPEC_TAC(`l:(real^N)list`,`l:(real^N)list`) THEN
  MATCH_MP_TAC list_INDUCT THEN
  REWRITE_TAC[ALL; ITLIST; HOMOTOPIC_WITH_REFL] THEN
  REWRITE_TAC[REWRITE_RULE[GSYM I_DEF] CONTINUOUS_ON_ID;
              ORTHOGONAL_TRANSFORMATION_I; SUBSET_UNIV] THEN
  X_GEN_TAC `a:real^N` THEN MATCH_MP_TAC list_INDUCT THEN
  REWRITE_TAC[ALL; ITLIST; I_O_ID; DET_MATRIX_REFLECT_ALONG] THEN
  REWRITE_TAC[ORTHGOONAL_TRANSFORMATION_REFLECT_ALONG] THEN
  CONJ_TAC THENL [MESON_TAC[REAL_ARITH `~(-- &1 = &1)`]; ALL_TAC] THEN
  MAP_EVERY X_GEN_TAC [`b:real^N`; `l:(real^N)list`] THEN
  REPLICATE_TAC 2 (DISCH_THEN(K ALL_TAC)) THEN
  DISCH_THEN(MP_TAC o SPEC `l:(real^N)list`) THEN
  REWRITE_TAC[LENGTH; ARITH_RULE `n < SUC(SUC n)`] THEN
  SIMP_TAC[LINEAR_COMPOSE; LINEAR_REFLECT_ALONG; MATRIX_COMPOSE;
     ORTHGOONAL_TRANSFORMATION_REFLECT_ALONG;
     ORTHOGONAL_TRANSFORMATION_COMPOSE; ORTHOGONAL_TRANSFORMATION_LINEAR] THEN
  DISCH_THEN(fun th ->
    DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC) THEN MP_TAC th) THEN
  ASM_SIMP_TAC[DET_MUL; DET_MATRIX_REFLECT_ALONG; REAL_ARITH
   `-- &1 * -- &1 * x = x`] THEN
  DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
  DISCH_THEN(fun th -> DISCH_TAC THEN MP_TAC th) THEN
  ASM_REWRITE_TAC[] THEN
  MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] HOMOTOPIC_WITH_TRANS) THEN
  GEN_REWRITE_TAC RAND_CONV [MESON[I_O_ID] `f = I o f`] THEN
  REWRITE_TAC[o_ASSOC] THEN
  MATCH_MP_TAC HOMOTOPIC_WITH_COMPOSE_CONTINUOUS_RIGHT THEN
  EXISTS_TAC `(:real^N)` THEN REWRITE_TAC[SUBSET_UNIV] THEN
  ASM_SIMP_TAC[LINEAR_CONTINUOUS_ON; ORTHOGONAL_TRANSFORMATION_LINEAR] THEN
  ABBREV_TAC `g = ITLIST (\v:real^N h. reflect_along v o h) l I` THEN
  SUBGOAL_THEN
   `(\f:real^N->real^N.
        orthogonal_transformation (f o g)) = orthogonal_transformation`
  SUBST1_TAC THENL [ALL_TAC; MATCH_MP_TAC lemma3 THEN ASM_REWRITE_TAC[]] THEN
  REWRITE_TAC[FUN_EQ_THM] THEN X_GEN_TAC `f:real^N->real^N` THEN
  EQ_TAC THEN ASM_SIMP_TAC[ORTHOGONAL_TRANSFORMATION_COMPOSE] THEN
  DISCH_TAC THEN
  MP_TAC(ISPEC `g:real^N->real^N` ORTHOGONAL_TRANSFORMATION_INVERSE_o) THEN
  ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `h:real^N->real^N` THEN
  STRIP_TAC THEN
  SUBGOAL_THEN `f = ((f:real^N->real^N) o (g:real^N->real^N)) o h`
  SUBST1_TAC THENL
   [ASM_REWRITE_TAC[GSYM o_ASSOC; I_O_ID];
    ASM_SIMP_TAC[ORTHOGONAL_TRANSFORMATION_COMPOSE]]);;
let HOMOTOPIC_SPECIAL_ORTHOGONAL_TRANSFORMATIONS, HOMOTOPIC_ORTHOGONAL_TRANSFORMATIONS = (CONJ_PAIR o prove) (`(!f g. homotopic_with (\h. orthogonal_transformation h /\ det(matrix h) = det(matrix f)) ((:real^N),(:real^N)) f g <=> homotopic_with orthogonal_transformation ((:real^N),(:real^N)) f g) /\ !f g. homotopic_with orthogonal_transformation ((:real^N),(:real^N)) f g <=> orthogonal_transformation f /\ orthogonal_transformation g /\ det(matrix f) = det(matrix g)`, REWRITE_TAC[AND_FORALL_THM] THEN REPEAT GEN_TAC THEN MATCH_MP_TAC(TAUT `(u ==> s) /\ (s ==> t) /\ (t ==> u) ==> (u <=> t) /\ (t <=> s)`) THEN REPEAT CONJ_TAC THENL [DISCH_THEN(MP_TAC o MATCH_MP HOMOTOPIC_WITH_IMP_PROPERTY) THEN MESON_TAC[]; STRIP_TAC THEN MP_TAC(ISPEC `g:real^N->real^N` ORTHOGONAL_TRANSFORMATION_INVERSE_o) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `h:real^N->real^N` STRIP_ASSUME_TAC) THEN SUBGOAL_THEN `(f:real^N->real^N) = g o (h:real^N->real^N) o f /\ g = g o I` (fun th -> ONCE_REWRITE_TAC[th]) THENL [ASM_REWRITE_TAC[o_ASSOC; I_O_ID]; ALL_TAC] THEN MATCH_MP_TAC HOMOTOPIC_WITH_COMPOSE_CONTINUOUS_LEFT THEN EXISTS_TAC `(:real^N)` THEN REWRITE_TAC[SUBSET_UNIV] THEN ASM_SIMP_TAC[ORTHOGONAL_TRANSFORMATION_LINEAR; LINEAR_CONTINUOUS_ON] THEN SUBGOAL_THEN `!k:real^N->real^N. orthogonal_transformation (g o k) <=> orthogonal_transformation k` (fun th -> REWRITE_TAC[th; ETA_AX]) THENL [GEN_TAC THEN EQ_TAC THEN ASM_SIMP_TAC[ORTHOGONAL_TRANSFORMATION_COMPOSE] THEN DISCH_THEN (MP_TAC o SPEC `h:real^N->real^N` o MATCH_MP (ONCE_REWRITE_RULE [IMP_CONJ_ALT] ORTHOGONAL_TRANSFORMATION_COMPOSE)) THEN ASM_SIMP_TAC[o_ASSOC; I_O_ID]; MATCH_MP_TAC NULLHOMOTOPIC_ORTHOGONAL_TRANSFORMATION THEN REPEAT(FIRST_X_ASSUM(MP_TAC o AP_TERM `\f:real^N->real^N. det(matrix f)`)) THEN ASM_SIMP_TAC[MATRIX_COMPOSE; ORTHOGONAL_TRANSFORMATION_LINEAR; ORTHOGONAL_TRANSFORMATION_COMPOSE; DET_MUL; MATRIX_I; DET_I]]; REWRITE_TAC[homotopic_with] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `k:real^(1,N)finite_sum->real^N` THEN STRIP_TAC THEN ASM_SIMP_TAC[] THEN MP_TAC(ISPECL [`\t. lift( det(matrix((k:real^(1,N)finite_sum->real^N) o pastecart t)))`; `interval[vec 0:real^1,vec 1]`] CONTINUOUS_DISCRETE_RANGE_CONSTANT) THEN REWRITE_TAC[CONNECTED_INTERVAL] THEN ANTS_TAC THENL [CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_LIFT_DET THEN SIMP_TAC[matrix; LAMBDA_BETA; o_DEF] THEN MAP_EVERY X_GEN_TAC [`i:num`; `j:num`] THEN STRIP_TAC THEN MATCH_MP_TAC CONTINUOUS_ON_LIFT_COMPONENT_COMPOSE THEN ASM_REWRITE_TAC[] THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN SIMP_TAC[CONTINUOUS_ON_PASTECART; CONTINUOUS_ON_CONST; CONTINUOUS_ON_ID] THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] CONTINUOUS_ON_SUBSET)) THEN SIMP_TAC[SUBSET; FORALL_IN_IMAGE; PASTECART_IN_PCROSS; IN_UNIV]; X_GEN_TAC `t:real^1` THEN DISCH_TAC THEN EXISTS_TAC `&1` THEN REWRITE_TAC[REAL_LT_01] THEN X_GEN_TAC `u:real^1` THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN REWRITE_TAC[GSYM LIFT_SUB; NORM_LIFT; LIFT_EQ] THEN SUBGOAL_THEN `orthogonal_transformation ((k:real^(1,N)finite_sum->real^N) o pastecart t) /\ orthogonal_transformation (k o pastecart u)` MP_TAC THENL [ASM_SIMP_TAC[o_DEF]; ALL_TAC] THEN DISCH_THEN(CONJUNCTS_THEN (STRIP_ASSUME_TAC o MATCH_MP DET_ORTHOGONAL_MATRIX o MATCH_MP ORTHOGONAL_MATRIX_MATRIX)) THEN ASM_REWRITE_TAC[] THEN CONV_TAC REAL_RAT_REDUCE_CONV]; REWRITE_TAC[o_DEF; LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `a:real^1` THEN DISCH_TAC THEN REPEAT(FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM FUN_EQ_THM])) THEN REPEAT(DISCH_THEN(SUBST1_TAC o SYM)) THEN ASM_SIMP_TAC[ENDS_IN_UNIT_INTERVAL; GSYM LIFT_EQ]]]);; (* ------------------------------------------------------------------------- *) (* Complex tangent function. *) (* ------------------------------------------------------------------------- *)
let ctan = new_definition
 `ctan z = csin z / ccos z`;;
let CTAN_0 = 
prove (`ctan(Cx(&0)) = Cx(&0)`,
REWRITE_TAC[ctan; CSIN_0; CCOS_0; COMPLEX_DIV_1]);;
let CTAN_NEG = 
prove (`!z. ctan(--z) = --(ctan z)`,
let CTAN_ADD = 
prove (`!w z. ~(ccos(w) = Cx(&0)) /\ ~(ccos(z) = Cx(&0)) /\ ~(ccos(w + z) = Cx(&0)) ==> ctan(w + z) = (ctan w + ctan z) / (Cx(&1) - ctan(w) * ctan(z))`,
REPEAT GEN_TAC THEN REWRITE_TAC[ctan; CSIN_ADD; CCOS_ADD] THEN CONV_TAC COMPLEX_FIELD);;
let CTAN_DOUBLE = 
prove (`!z. ~(ccos(z) = Cx(&0)) /\ ~(ccos(Cx(&2) * z) = Cx(&0)) ==> ctan(Cx(&2) * z) = (Cx(&2) * ctan z) / (Cx(&1) - ctan(z) pow 2)`,
let CTAN_SUB = 
prove (`!w z. ~(ccos(w) = Cx(&0)) /\ ~(ccos(z) = Cx(&0)) /\ ~(ccos(w - z) = Cx(&0)) ==> ctan(w - z) = (ctan w - ctan z) / (Cx(&1) + ctan(w) * ctan(z))`,
let COMPLEX_ADD_CTAN = 
prove (`!w z. ~(ccos(w) = Cx(&0)) /\ ~(ccos(z) = Cx(&0)) ==> ctan(w) + ctan(z) = csin(w + z) / (ccos(w) * ccos(z))`,
REWRITE_TAC[ctan; CSIN_ADD] THEN CONV_TAC COMPLEX_FIELD);;
let COMPLEX_SUB_CTAN = 
prove (`!w z. ~(ccos(w) = Cx(&0)) /\ ~(ccos(z) = Cx(&0)) ==> ctan(w) - ctan(z) = csin(w - z) / (ccos(w) * ccos(z))`,
REWRITE_TAC[ctan; CSIN_SUB] THEN CONV_TAC COMPLEX_FIELD);;
(* ------------------------------------------------------------------------- *) (* Analytic properties of tangent function. *) (* ------------------------------------------------------------------------- *)
let HAS_COMPLEX_DERIVATIVE_CTAN = 
prove (`!z. ~(ccos z = Cx(&0)) ==> (ctan has_complex_derivative (inv(ccos(z) pow 2))) (at z)`,
REPEAT STRIP_TAC THEN GEN_REWRITE_TAC (RATOR_CONV o LAND_CONV) [GSYM ETA_AX] THEN REWRITE_TAC[ctan] THEN COMPLEX_DIFF_TAC THEN MP_TAC(SPEC `z:complex` CSIN_CIRCLE) THEN POP_ASSUM MP_TAC THEN CONV_TAC COMPLEX_FIELD);;
let COMPLEX_DIFFERENTIABLE_AT_CTAN = 
prove (`!z. ~(ccos z = Cx(&0)) ==> ctan complex_differentiable at z`,
REWRITE_TAC[complex_differentiable] THEN MESON_TAC[HAS_COMPLEX_DERIVATIVE_CTAN]);;
let COMPLEX_DIFFERENTIABLE_WITHIN_CTAN = 
prove (`!s z. ~(ccos z = Cx(&0)) ==> ctan complex_differentiable (at z within s)`,
add_complex_differentiation_theorems (CONJUNCTS(REWRITE_RULE[FORALL_AND_THM] (MATCH_MP HAS_COMPLEX_DERIVATIVE_CHAIN HAS_COMPLEX_DERIVATIVE_CTAN)));;
let CONTINUOUS_AT_CTAN = 
prove (`!z. ~(ccos z = Cx(&0)) ==> ctan continuous at z`,
let CONTINUOUS_WITHIN_CTAN = 
prove (`!s z. ~(ccos z = Cx(&0)) ==> ctan continuous (at z within s)`,
let CONTINUOUS_ON_CTAN = 
prove (`!s. (!z. z IN s ==> ~(ccos z = Cx(&0))) ==> ctan continuous_on s`,
let HOLOMORPHIC_ON_CTAN = 
prove (`!s. (!z. z IN s ==> ~(ccos z = Cx(&0))) ==> ctan holomorphic_on s`,
(* ------------------------------------------------------------------------- *) (* Real tangent function. *) (* ------------------------------------------------------------------------- *)
let tan_def = new_definition
 `tan(x) = Re(ctan(Cx x))`;;
let CNJ_CTAN = 
prove (`!z. cnj(ctan z) = ctan(cnj z)`,
REWRITE_TAC[ctan; CNJ_DIV; CNJ_CSIN; CNJ_CCOS]);;
let REAL_TAN = 
prove (`!z. real z ==> real(ctan z)`,
SIMP_TAC[REAL_CNJ; CNJ_CTAN]);;
let CX_TAN = 
prove (`!x. Cx(tan x) = ctan(Cx x)`,
REWRITE_TAC[tan_def] THEN MESON_TAC[REAL; REAL_CX; REAL_TAN]);;
let tan = 
prove (`!x. tan x = sin x / cos x`,
REWRITE_TAC[GSYM CX_INJ; CX_DIV; CX_TAN; CX_SIN; CX_COS; ctan]);;
let TAN_0 = 
prove (`tan(&0) = &0`,
REWRITE_TAC[GSYM CX_INJ; CX_TAN; CTAN_0]);;
let TAN_PI = 
prove (`tan(pi) = &0`,
REWRITE_TAC[tan; SIN_PI; real_div; REAL_MUL_LZERO]);;
let TAN_NPI = 
prove (`!n. tan(&n * pi) = &0`,
REWRITE_TAC[tan; SIN_NPI; real_div; REAL_MUL_LZERO]);;
let TAN_NEG = 
prove (`!x. tan(--x) = --(tan x)`,
REWRITE_TAC[GSYM CX_INJ; CX_TAN; CX_NEG; CTAN_NEG]);;
let TAN_PERIODIC_PI = 
prove (`!x. tan(x + pi) = tan(x)`,
let TAN_PERIODIC_NPI = 
prove (`!x n. tan(x + &n * pi) = tan(x)`,
GEN_TAC THEN INDUCT_TAC THEN REWRITE_TAC[REAL_MUL_LZERO; REAL_ADD_RID] THEN REWRITE_TAC[GSYM REAL_OF_NUM_SUC; REAL_ADD_RDISTRIB; REAL_MUL_LID] THEN ASM_REWRITE_TAC[REAL_ADD_ASSOC; TAN_PERIODIC_PI]);;
let TAN_ADD = 
prove (`!x y. ~(cos(x) = &0) /\ ~(cos(y) = &0) /\ ~(cos(x + y) = &0) ==> tan(x + y) = (tan(x) + tan(y)) / (&1 - tan(x) * tan(y))`,
REWRITE_TAC[GSYM CX_INJ; CX_TAN; CX_SIN; CX_COS; CTAN_ADD; CX_DIV; CX_ADD; CX_SUB; CX_MUL]);;
let TAN_SUB = 
prove (`!x y. ~(cos(x) = &0) /\ ~(cos(y) = &0) /\ ~(cos(x - y) = &0) ==> tan(x - y) = (tan(x) - tan(y)) / (&1 + tan(x) * tan(y))`,
let TAN_DOUBLE = 
prove (`!x. ~(cos(x) = &0) /\ ~(cos(&2 * x) = &0) ==> tan(&2 * x) = (&2 * tan(x)) / (&1 - (tan(x) pow 2))`,
SIMP_TAC[REAL_MUL_2; TAN_ADD; REAL_POW_2]);;
let REAL_ADD_TAN = 
prove (`!x y. ~(cos(x) = &0) /\ ~(cos(y) = &0) ==> tan(x) + tan(y) = sin(x + y) / (cos(x) * cos(y))`,
REWRITE_TAC[GSYM CX_INJ; CX_TAN; CX_SIN; CX_COS; CX_MUL; CX_ADD; CX_DIV] THEN REWRITE_TAC[COMPLEX_ADD_CTAN]);;
let REAL_SUB_TAN = 
prove (`!x y. ~(cos(x) = &0) /\ ~(cos(y) = &0) ==> tan(x) - tan(y) = sin(x - y) / (cos(x) * cos(y))`,
REWRITE_TAC[GSYM CX_INJ; CX_TAN; CX_SIN; CX_COS; CX_MUL; CX_SUB; CX_DIV] THEN REWRITE_TAC[COMPLEX_SUB_CTAN]);;
let TAN_PI4 = 
prove (`tan(pi / &4) = &1`,
REWRITE_TAC[tan; SIN_COS; REAL_ARITH `p / &2 - p / &4 = p / &4`] THEN MATCH_MP_TAC REAL_DIV_REFL THEN REWRITE_TAC[COS_EQ_0; PI_NZ; REAL_FIELD `p / &4 = (n + &1 / &2) * p <=> p = &0 \/ n = -- &1 / &4`] THEN ONCE_REWRITE_TAC[CONJ_SYM] THEN REWRITE_TAC[UNWIND_THM2] THEN DISCH_THEN(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] REAL_ABS_INTEGER_LEMMA)) THEN REAL_ARITH_TAC);;
let TAN_POS_PI2 = 
prove (`!x. &0 < x /\ x < pi / &2 ==> &0 < tan x`,
REPEAT STRIP_TAC THEN REWRITE_TAC[tan] THEN MATCH_MP_TAC REAL_LT_DIV THEN CONJ_TAC THENL [MATCH_MP_TAC SIN_POS_PI; MATCH_MP_TAC COS_POS_PI] THEN ASM_REAL_ARITH_TAC);;
let TAN_POS_PI2_LE = 
prove (`!x. &0 <= x /\ x < pi / &2 ==> &0 <= tan x`,
REWRITE_TAC[REAL_LE_LT] THEN MESON_TAC[TAN_0; TAN_POS_PI2]);;
let COS_TAN = 
prove (`!x. abs(x) < pi / &2 ==> cos(x) = &1 / sqrt(&1 + tan(x) pow 2)`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC(REAL_FIELD `sqrt(s) pow 2 = s /\ c pow 2 * s = &1 /\ ~(&1 + c * sqrt s = &0) ==> c = &1 / sqrt s`) THEN SUBGOAL_THEN `&0 < &1 + tan x pow 2` ASSUME_TAC THENL [MP_TAC(SPEC `tan x` REAL_LE_SQUARE) THEN REAL_ARITH_TAC; ALL_TAC] THEN ASM_SIMP_TAC[SQRT_POW_2; REAL_LT_IMP_LE] THEN CONJ_TAC THENL [REWRITE_TAC[tan] THEN MATCH_MP_TAC(REAL_FIELD `s pow 2 + c pow 2 = &1 /\ &0 < c ==> c pow 2 * (&1 + (s / c) pow 2) = &1`) THEN ASM_SIMP_TAC[SIN_CIRCLE; COS_POS_PI; REAL_BOUNDS_LT]; MATCH_MP_TAC(REAL_ARITH `&0 < x ==> ~(&1 + x = &0)`) THEN ASM_SIMP_TAC[SIN_CIRCLE; COS_POS_PI; REAL_BOUNDS_LT; SQRT_POS_LT; REAL_LT_MUL]]);;
let SIN_TAN = 
prove (`!x. abs(x) < pi / &2 ==> sin(x) = tan(x) / sqrt(&1 + tan(x) pow 2)`,
REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[REAL_ARITH `a / b = a * &1 / b`] THEN ASM_SIMP_TAC[GSYM COS_TAN] THEN ASM_SIMP_TAC[tan; REAL_DIV_RMUL; REAL_LT_IMP_NZ; COS_POS_PI; REAL_BOUNDS_LT]);;
(* ------------------------------------------------------------------------- *) (* Monotonicity theorems for the basic trig functions. *) (* ------------------------------------------------------------------------- *)
let SIN_MONO_LT = 
prove (`!x y. --(pi / &2) <= x /\ x < y /\ y <= pi / &2 ==> sin(x) < sin(y)`,
REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[GSYM REAL_SUB_LT] THEN REWRITE_TAC[REAL_SUB_SIN; REAL_ARITH `&0 < &2 * x <=> &0 < x`] THEN MATCH_MP_TAC REAL_LT_MUL THEN CONJ_TAC THENL [MATCH_MP_TAC SIN_POS_PI; MATCH_MP_TAC COS_POS_PI] THEN ASM_REAL_ARITH_TAC);;
let SIN_MONO_LE = 
prove (`!x y. --(pi / &2) <= x /\ x <= y /\ y <= pi / &2 ==> sin(x) <= sin(y)`,
MESON_TAC[SIN_MONO_LT; REAL_LE_LT]);;
let SIN_MONO_LT_EQ = 
prove (`!x y. --(pi / &2) <= x /\ x <= pi / &2 /\ --(pi / &2) <= y /\ y <= pi / &2 ==> (sin(x) < sin(y) <=> x < y)`,
let SIN_MONO_LE_EQ = 
prove (`!x y. --(pi / &2) <= x /\ x <= pi / &2 /\ --(pi / &2) <= y /\ y <= pi / &2 ==> (sin(x) <= sin(y) <=> x <= y)`,
let SIN_INJ_PI = 
prove (`!x y. --(pi / &2) <= x /\ x <= pi / &2 /\ --(pi / &2) <= y /\ y <= pi / &2 /\ sin(x) = sin(y) ==> x = y`,
REWRITE_TAC[GSYM REAL_LE_ANTISYM] THEN MESON_TAC[SIN_MONO_LE_EQ]);;
let COS_MONO_LT = 
prove (`!x y. &0 <= x /\ x < y /\ y <= pi ==> cos(y) < cos(x)`,
REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[GSYM REAL_SUB_LT] THEN REWRITE_TAC[REAL_SUB_COS; REAL_ARITH `&0 < &2 * x <=> &0 < x`] THEN MATCH_MP_TAC REAL_LT_MUL THEN CONJ_TAC THEN MATCH_MP_TAC SIN_POS_PI THEN ASM_REAL_ARITH_TAC);;
let COS_MONO_LE = 
prove (`!x y. &0 <= x /\ x <= y /\ y <= pi ==> cos(y) <= cos(x)`,
MESON_TAC[COS_MONO_LT; REAL_LE_LT]);;
let COS_MONO_LT_EQ = 
prove (`!x y. &0 <= x /\ x <= pi /\ &0 <= y /\ y <= pi ==> (cos(x) < cos(y) <=> y < x)`,
let COS_MONO_LE_EQ = 
prove (`!x y. &0 <= x /\ x <= pi /\ &0 <= y /\ y <= pi ==> (cos(x) <= cos(y) <=> y <= x)`,
let COS_INJ_PI = 
prove (`!x y. &0 <= x /\ x <= pi /\ &0 <= y /\ y <= pi /\ cos(x) = cos(y) ==> x = y`,
REWRITE_TAC[GSYM REAL_LE_ANTISYM] THEN MESON_TAC[COS_MONO_LE_EQ]);;
let TAN_MONO_LT = 
prove (`!x y. --(pi / &2) < x /\ x < y /\ y < pi / &2 ==> tan(x) < tan(y)`,
REPEAT STRIP_TAC THEN GEN_REWRITE_TAC I [GSYM REAL_SUB_LT] THEN SUBGOAL_THEN `&0 < cos(x) /\ &0 < cos(y)` STRIP_ASSUME_TAC THENL [CONJ_TAC THEN MATCH_MP_TAC COS_POS_PI; ASM_SIMP_TAC[REAL_LT_IMP_NZ; REAL_SUB_TAN] THEN MATCH_MP_TAC REAL_LT_DIV THEN ASM_SIMP_TAC[REAL_LT_MUL] THEN MATCH_MP_TAC SIN_POS_PI] THEN ASM_REAL_ARITH_TAC);;
let TAN_MONO_LE = 
prove (`!x y. --(pi / &2) < x /\ x <= y /\ y < pi / &2 ==> tan(x) <= tan(y)`,
REWRITE_TAC[REAL_LE_LT] THEN MESON_TAC[TAN_MONO_LT]);;
let TAN_MONO_LT_EQ = 
prove (`!x y. --(pi / &2) < x /\ x < pi / &2 /\ --(pi / &2) < y /\ y < pi / &2 ==> (tan(x) < tan(y) <=> x < y)`,
let TAN_MONO_LE_EQ = 
prove (`!x y. --(pi / &2) < x /\ x < pi / &2 /\ --(pi / &2) < y /\ y < pi / &2 ==> (tan(x) <= tan(y) <=> x <= y)`,
let TAN_BOUND_PI2 = 
prove (`!x. abs(x) < pi / &4 ==> abs(tan x) < &1`,
REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM TAN_PI4] THEN REWRITE_TAC[GSYM TAN_NEG; REAL_ARITH `abs(x) < a <=> --a < x /\ x < a`] THEN CONJ_TAC THEN MATCH_MP_TAC TAN_MONO_LT THEN POP_ASSUM MP_TAC THEN REAL_ARITH_TAC);;
let TAN_COT = 
prove (`!x. tan(pi / &2 - x) = inv(tan x)`,
REWRITE_TAC[tan; SIN_SUB; COS_SUB; SIN_PI2; COS_PI2; REAL_INV_DIV] THEN GEN_TAC THEN BINOP_TAC THEN REAL_ARITH_TAC);;
(* ------------------------------------------------------------------------- *) (* Approximation to pi. *) (* ------------------------------------------------------------------------- *)
let SIN_PI6_STRADDLE = 
prove (`!a b. &0 <= a /\ a <= b /\ b <= &4 /\ sin(a / &6) <= &1 / &2 /\ &1 / &2 <= sin(b / &6) ==> a <= pi /\ pi <= b`,
REPEAT GEN_TAC THEN STRIP_TAC THEN MP_TAC(SPECL [`pi / &6`; `b / &6`] SIN_MONO_LE_EQ) THEN MP_TAC(SPECL [`a / &6`; `pi / &6`] SIN_MONO_LE_EQ) THEN ASM_REWRITE_TAC[SIN_PI6] THEN SUBGOAL_THEN `!x. &0 < x /\ x < &7 / &5 ==> &0 < sin x` MP_TAC THENL [REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`0`; `Cx(x)`] TAYLOR_CSIN) THEN REWRITE_TAC[VSUM_SING_NUMSEG] THEN CONV_TAC NUM_REDUCE_CONV THEN REWRITE_TAC[COMPLEX_DIV_1; COMPLEX_POW_1; complex_pow] THEN REWRITE_TAC[COMPLEX_MUL_LID; GSYM CX_SIN; GSYM CX_SUB] THEN REWRITE_TAC[IM_CX; COMPLEX_NORM_CX; REAL_ABS_NUM; REAL_EXP_0] THEN MATCH_MP_TAC(REAL_ARITH `e + d < a ==> abs(s - a) <= d ==> e < s`) THEN ASM_SIMP_TAC[real_abs; real_pow; REAL_MUL_LID; REAL_LT_IMP_LE] THEN SIMP_TAC[REAL_ARITH `&0 + x pow 3 / &2 < x <=> x * x pow 2 < x * &2`] THEN ASM_SIMP_TAC[REAL_LT_LMUL_EQ] THEN MATCH_MP_TAC REAL_LT_TRANS THEN EXISTS_TAC `(&7 / &5) pow 2` THEN ASM_SIMP_TAC[REAL_POW_LT2; ARITH_EQ; REAL_LT_IMP_LE] THEN CONV_TAC REAL_RAT_REDUCE_CONV; DISCH_THEN(MP_TAC o SPEC `pi`) THEN SIMP_TAC[SIN_PI; REAL_LT_REFL; PI_POS; REAL_NOT_LT] THEN ASM_REAL_ARITH_TAC]);;
let PI_APPROX_32 = 
prove (`abs(pi - &13493037705 / &4294967296) <= inv(&2 pow 32)`,
REWRITE_TAC[REAL_ARITH `abs(x - a) <= e <=> a - e <= x /\ x <= a + e`] THEN MATCH_MP_TAC SIN_PI6_STRADDLE THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN CONJ_TAC THENL [MP_TAC(SPECL [`5`; `Cx(&1686629713 / &3221225472)`] TAYLOR_CSIN); MP_TAC(SPECL [`5`; `Cx(&6746518853 / &12884901888)`] TAYLOR_CSIN)] THEN CONV_TAC(ONCE_DEPTH_CONV EXPAND_VSUM_CONV) THEN CONV_TAC NUM_REDUCE_CONV THEN REWRITE_TAC[GSYM CX_POW; GSYM CX_DIV; GSYM CX_ADD; GSYM CX_SUB; GSYM CX_NEG; GSYM CX_MUL; GSYM CX_SIN; COMPLEX_NORM_CX] THEN REWRITE_TAC[IM_CX; REAL_ABS_NUM; REAL_EXP_0] THEN REAL_ARITH_TAC);;
let PI2_BOUNDS = 
prove (`&0 < pi / &2 /\ pi / &2 < &2`,
MP_TAC PI_APPROX_32 THEN REAL_ARITH_TAC);;
(* ------------------------------------------------------------------------- *) (* Complex logarithms (the conventional principal value). *) (* ------------------------------------------------------------------------- *)
let clog = new_definition
 `clog z = @w. cexp(w) = z /\ --pi < Im(w) /\ Im(w) <= pi`;;
let EXISTS_COMPLEX' = 
prove (`!P. (?z. P (Re z) (Im z)) <=> ?x y. P x y`,
MESON_TAC[RE; IM; COMPLEX]);;
let CLOG_WORKS = 
prove (`!z. ~(z = Cx(&0)) ==> cexp(clog z) = z /\ --pi < Im(clog z) /\ Im(clog z) <= pi`,
GEN_TAC THEN DISCH_TAC THEN REWRITE_TAC[clog] THEN CONV_TAC SELECT_CONV THEN MP_TAC(SPEC `z / Cx(norm z)` COMPLEX_UNIMODULAR_POLAR) THEN ANTS_TAC THENL [ASM_SIMP_TAC[COMPLEX_NORM_DIV; COMPLEX_NORM_CX] THEN ASM_SIMP_TAC[REAL_ABS_NORM; REAL_DIV_REFL; COMPLEX_NORM_ZERO]; ALL_TAC] THEN DISCH_THEN(X_CHOOSE_THEN `x:real` STRIP_ASSUME_TAC) THEN MP_TAC(SPEC `x:real` SINCOS_PRINCIPAL_VALUE) THEN DISCH_THEN(X_CHOOSE_THEN `y:real` STRIP_ASSUME_TAC) THEN EXISTS_TAC `complex(log(norm(z:complex)),y)` THEN ASM_REWRITE_TAC[RE; IM; CEXP_COMPLEX] THEN REPEAT(FIRST_X_ASSUM(SUBST_ALL_TAC o SYM)) THEN ASM_SIMP_TAC[EXP_LOG; COMPLEX_NORM_NZ; COMPLEX_DIV_LMUL; COMPLEX_NORM_ZERO; CX_INJ]);;
let CEXP_CLOG = 
prove (`!z. ~(z = Cx(&0)) ==> cexp(clog z) = z`,
SIMP_TAC[CLOG_WORKS]);;
let CLOG_CEXP = 
prove (`!z. --pi < Im(z) /\ Im(z) <= pi ==> clog(cexp z) = z`,
REPEAT STRIP_TAC THEN REWRITE_TAC[clog] THEN MATCH_MP_TAC SELECT_UNIQUE THEN X_GEN_TAC `w:complex` THEN EQ_TAC THEN ASM_SIMP_TAC[] THEN REWRITE_TAC[CEXP_EQ] THEN REWRITE_TAC[IMP_CONJ] THEN DISCH_THEN(X_CHOOSE_THEN `n:real` (CONJUNCTS_THEN2 ASSUME_TAC SUBST1_TAC)) THEN REWRITE_TAC[IM_ADD; IM_MUL_II; RE_CX] THEN REPEAT STRIP_TAC THEN ASM_CASES_TAC `n = &0` THEN ASM_REWRITE_TAC[REAL_MUL_LZERO] THEN REWRITE_TAC[REAL_MUL_RZERO; COMPLEX_ADD_RID; COMPLEX_MUL_LZERO] THEN SUBGOAL_THEN `abs(n * pi) < &1 * pi` MP_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN ASM_SIMP_TAC[REAL_ABS_MUL; REAL_LT_RMUL_EQ; PI_POS; REAL_ABS_PI] THEN ASM_MESON_TAC[REAL_ABS_INTEGER_LEMMA; REAL_NOT_LT]);;
let CLOG_EQ = 
prove (`!w z. ~(w = Cx(&0)) /\ ~(z = Cx(&0)) ==> (clog w = clog z <=> w = z)`,
MESON_TAC[CEXP_CLOG]);;
let CLOG_UNIQUE = 
prove (`!w z. --pi < Im(z) /\ Im(z) <= pi /\ cexp(z) = w ==> clog w = z`,
MESON_TAC[CLOG_CEXP]);;
let RE_CLOG = 
prove (`!z. ~(z = Cx(&0)) ==> Re(clog z) = log(norm z)`,
REPEAT STRIP_TAC THEN FIRST_X_ASSUM (MP_TAC o AP_TERM `norm:complex->real` o MATCH_MP CEXP_CLOG) THEN REWRITE_TAC[NORM_CEXP] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN REWRITE_TAC[LOG_EXP]);;
let EXISTS_COMPLEX_ROOT = 
prove (`!a n. ~(n = 0) ==> ?z. z pow n = a`,
REPEAT STRIP_TAC THEN ASM_CASES_TAC `a = Cx(&0)` THENL [EXISTS_TAC `Cx(&0)` THEN ASM_REWRITE_TAC[COMPLEX_POW_ZERO]; EXISTS_TAC `cexp(clog(a) / Cx(&n))` THEN REWRITE_TAC[GSYM CEXP_N] THEN ASM_SIMP_TAC[COMPLEX_DIV_LMUL; CX_INJ; REAL_OF_NUM_EQ; CEXP_CLOG]]);;
(* ------------------------------------------------------------------------- *) (* Derivative of clog away from the branch cut. *) (* ------------------------------------------------------------------------- *)
let HAS_COMPLEX_DERIVATIVE_CLOG = 
prove (`!z. (Im(z) = &0 ==> &0 < Re(z)) ==> (clog has_complex_derivative inv(z)) (at z)`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC HAS_COMPLEX_DERIVATIVE_INVERSE_STRONG_X THEN EXISTS_TAC `cexp` THEN EXISTS_TAC `{w | --pi < Im(w) /\ Im(w) < pi}` THEN REWRITE_TAC[IN_ELIM_THM] THEN ASM_CASES_TAC `z = Cx(&0)` THENL [FIRST_X_ASSUM SUBST_ALL_TAC THEN POP_ASSUM MP_TAC THEN ASM_REWRITE_TAC[RE_CX; IM_CX; REAL_LT_REFL]; ALL_TAC] THEN ASM_SIMP_TAC[CONTINUOUS_ON_CEXP; CEXP_CLOG; CLOG_CEXP; REAL_LT_IMP_LE] THEN REPEAT CONJ_TAC THENL [REWRITE_TAC[SET_RULE `{x | p x /\ q x} = {x | p x} INTER {x | q x}`] THEN MATCH_MP_TAC OPEN_INTER THEN REWRITE_TAC[REAL_ARITH `--x < w <=> w > --x`] THEN REWRITE_TAC[OPEN_HALFSPACE_IM_LT; OPEN_HALFSPACE_IM_GT]; ASM_SIMP_TAC[CLOG_WORKS]; ASM_SIMP_TAC[CLOG_WORKS; REAL_LT_LE] THEN DISCH_THEN(fun th -> FIRST_X_ASSUM(SUBST_ALL_TAC o SYM o MATCH_MP CEXP_CLOG) THEN POP_ASSUM MP_TAC THEN ASSUME_TAC th) THEN ASM_REWRITE_TAC[EULER; COS_PI; SIN_PI; COMPLEX_MUL_RZERO] THEN REWRITE_TAC[COMPLEX_ADD_RID; CX_NEG; COMPLEX_MUL_RNEG] THEN REWRITE_TAC[COMPLEX_MUL_RID; IM_NEG; IM_CX; RE_NEG; RE_CX] THEN MP_TAC(SPEC `Re(clog z)` REAL_EXP_POS_LT) THEN REAL_ARITH_TAC; ASM_MESON_TAC[HAS_COMPLEX_DERIVATIVE_CEXP; CEXP_CLOG]]);;
let COMPLEX_DIFFERENTIABLE_AT_CLOG = 
prove (`!z. (Im(z) = &0 ==> &0 < Re(z)) ==> clog complex_differentiable at z`,
REWRITE_TAC[complex_differentiable] THEN MESON_TAC[HAS_COMPLEX_DERIVATIVE_CLOG]);;
let COMPLEX_DIFFERENTIABLE_WITHIN_CLOG = 
prove (`!s z. (Im(z) = &0 ==> &0 < Re(z)) ==> clog complex_differentiable (at z within s)`,
add_complex_differentiation_theorems (CONJUNCTS(REWRITE_RULE[FORALL_AND_THM] (MATCH_MP HAS_COMPLEX_DERIVATIVE_CHAIN HAS_COMPLEX_DERIVATIVE_CLOG)));;
let CONTINUOUS_AT_CLOG = 
prove (`!z. (Im(z) = &0 ==> &0 < Re(z)) ==> clog continuous at z`,
let CONTINUOUS_WITHIN_CLOG = 
prove (`!s z. (Im(z) = &0 ==> &0 < Re(z)) ==> clog continuous (at z within s)`,
let CONTINUOUS_ON_CLOG = 
prove (`!s. (!z. z IN s /\ Im(z) = &0 ==> &0 < Re(z)) ==> clog continuous_on s`,
let HOLOMORPHIC_ON_CLOG = 
prove (`!s. (!z. z IN s /\ Im(z) = &0 ==> &0 < Re(z)) ==> clog holomorphic_on s`,
(* ------------------------------------------------------------------------- *) (* Relation to real log. *) (* ------------------------------------------------------------------------- *)
let CX_LOG = 
prove (`!z. &0 < z ==> Cx(log z) = clog(Cx z)`,
REPEAT STRIP_TAC THEN FIRST_ASSUM(fun th -> GEN_REWRITE_TAC (RAND_CONV o ONCE_DEPTH_CONV) [SYM(MATCH_MP EXP_LOG th)]) THEN REWRITE_TAC[CX_EXP] THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC CLOG_CEXP THEN REWRITE_TAC[IM_CX] THEN MP_TAC PI_POS THEN REAL_ARITH_TAC);;
(* ------------------------------------------------------------------------- *) (* Quadrant-type results for clog. *) (* ------------------------------------------------------------------------- *)
let RE_CLOG_POS_LT = 
prove (`!z. ~(z = Cx(&0)) ==> (abs(Im(clog z)) < pi / &2 <=> &0 < Re(z))`,
GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP CLOG_WORKS) THEN DISCH_THEN(CONJUNCTS_THEN2 (fun th -> GEN_REWRITE_TAC (RAND_CONV o ONCE_DEPTH_CONV) [SYM th]) MP_TAC) THEN SIMP_TAC[RE_CEXP; REAL_LT_MUL_EQ; REAL_EXP_POS_LT] THEN SPEC_TAC(`clog z`,`z:complex`) THEN GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP (REAL_ARITH `--p < x /\ x <= p ==> --(p / &2) < x /\ x < p / &2 \/ --(p / &2) <= p + x /\ p + x <= p / &2 \/ --(p / &2) <= x - p /\ x - p <= p / &2`)) THEN DISCH_THEN(REPEAT_TCL DISJ_CASES_THEN ASSUME_TAC) THEN (FIRST_ASSUM(MP_TAC o MATCH_MP COS_POS_PI) ORELSE FIRST_ASSUM(MP_TAC o MATCH_MP COS_POS_PI_LE)) THEN REPEAT(POP_ASSUM MP_TAC) THEN REWRITE_TAC[COS_ADD; COS_SUB; COS_PI; SIN_PI] THEN REAL_ARITH_TAC);;
let RE_CLOG_POS_LE = 
prove (`!z. ~(z = Cx(&0)) ==> (abs(Im(clog z)) <= pi / &2 <=> &0 <= Re(z))`,
GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP CLOG_WORKS) THEN DISCH_THEN(CONJUNCTS_THEN2 (fun th -> GEN_REWRITE_TAC (RAND_CONV o ONCE_DEPTH_CONV) [SYM th]) MP_TAC) THEN SIMP_TAC[RE_CEXP; REAL_LE_MUL_EQ; REAL_EXP_POS_LT] THEN SPEC_TAC(`clog z`,`z:complex`) THEN GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP (REAL_ARITH `--p < x /\ x <= p ==> --(p / &2) <= x /\ x <= p / &2 \/ --(p / &2) < p + x /\ p + x < p / &2 \/ --(p / &2) < x - p /\ x - p < p / &2`)) THEN DISCH_THEN(REPEAT_TCL DISJ_CASES_THEN ASSUME_TAC) THEN (FIRST_ASSUM(MP_TAC o MATCH_MP COS_POS_PI) ORELSE FIRST_ASSUM(MP_TAC o MATCH_MP COS_POS_PI_LE)) THEN REPEAT(POP_ASSUM MP_TAC) THEN REWRITE_TAC[COS_ADD; COS_SUB; COS_PI; SIN_PI] THEN REAL_ARITH_TAC);;
let IM_CLOG_POS_LT = 
prove (`!z. ~(z = Cx(&0)) ==> (&0 < Im(clog z) /\ Im(clog z) < pi <=> &0 < Im(z))`,
GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP CLOG_WORKS) THEN DISCH_THEN(CONJUNCTS_THEN2 (fun th -> GEN_REWRITE_TAC (RAND_CONV o ONCE_DEPTH_CONV) [SYM th]) MP_TAC) THEN SIMP_TAC[IM_CEXP; REAL_LT_MUL_EQ; REAL_EXP_POS_LT] THEN SPEC_TAC(`clog z`,`z:complex`) THEN GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP (REAL_ARITH `--p < x /\ x <= p ==> &0 < x /\ x < p \/ &0 <= x + p /\ x + p <= p \/ &0 <= x - p /\ x - p <= p`)) THEN DISCH_THEN(REPEAT_TCL DISJ_CASES_THEN ASSUME_TAC) THEN (FIRST_ASSUM(MP_TAC o MATCH_MP SIN_POS_PI) ORELSE FIRST_ASSUM(MP_TAC o MATCH_MP SIN_POS_PI_LE)) THEN REPEAT(POP_ASSUM MP_TAC) THEN REWRITE_TAC[SIN_ADD; SIN_SUB; COS_PI; SIN_PI] THEN REAL_ARITH_TAC);;
let IM_CLOG_POS_LE = 
prove (`!z. ~(z = Cx(&0)) ==> (&0 <= Im(clog z) <=> &0 <= Im(z))`,
GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP CLOG_WORKS) THEN DISCH_THEN(CONJUNCTS_THEN2 (fun th -> GEN_REWRITE_TAC (RAND_CONV o ONCE_DEPTH_CONV) [SYM th]) MP_TAC) THEN SIMP_TAC[IM_CEXP; REAL_LE_MUL_EQ; REAL_EXP_POS_LT] THEN SPEC_TAC(`clog z`,`z:complex`) THEN GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP (REAL_ARITH `--p < x /\ x <= p ==> &0 <= x /\ x <= p \/ &0 < x + p /\ x + p < p \/ &0 < p - x /\ p - x < p`)) THEN DISCH_THEN(REPEAT_TCL DISJ_CASES_THEN ASSUME_TAC) THEN (FIRST_ASSUM(MP_TAC o MATCH_MP SIN_POS_PI) ORELSE FIRST_ASSUM(MP_TAC o MATCH_MP SIN_POS_PI_LE)) THEN REPEAT(POP_ASSUM MP_TAC) THEN REWRITE_TAC[SIN_ADD; SIN_SUB; COS_PI; SIN_PI] THEN REAL_ARITH_TAC);;
let RE_CLOG_POS_LT_IMP = 
prove (`!z. &0 < Re(z) ==> abs(Im(clog z)) < pi / &2`,
GEN_TAC THEN ASM_CASES_TAC `z = Cx(&0)` THEN ASM_SIMP_TAC[RE_CLOG_POS_LT; RE_CX; REAL_LT_REFL]);;
let IM_CLOG_POS_LT_IMP = 
prove (`!z. &0 < Im(z) ==> &0 < Im(clog z) /\ Im(clog z) < pi`,
GEN_TAC THEN ASM_CASES_TAC `z = Cx(&0)` THEN ASM_SIMP_TAC[IM_CLOG_POS_LT; IM_CX; REAL_LT_REFL]);;
let IM_CLOG_EQ_0 = 
prove (`!z. ~(z = Cx(&0)) ==> (Im(clog z) = &0 <=> &0 < Re(z) /\ Im(z) = &0)`,
REPEAT STRIP_TAC THEN GEN_REWRITE_TAC (RAND_CONV o ONCE_DEPTH_CONV) [REAL_ARITH `z = &0 <=> &0 <= z /\ ~(&0 < z)`] THEN ASM_SIMP_TAC[GSYM RE_CLOG_POS_LT; GSYM IM_CLOG_POS_LE; GSYM IM_CLOG_POS_LT] THEN MP_TAC PI_POS THEN REAL_ARITH_TAC);;
let IM_CLOG_EQ_PI = 
prove (`!z. ~(z = Cx(&0)) ==> (Im(clog z) = pi <=> Re(z) < &0 /\ Im(z) = &0)`,
SIMP_TAC[PI_POS; RE_CLOG_POS_LE; IM_CLOG_POS_LE; IM_CLOG_POS_LT; CLOG_WORKS; REAL_ARITH `&0 < pi ==> (x = pi <=> (&0 <= x /\ x <= pi) /\ ~(abs x <= pi / &2) /\ ~(&0 < x /\ x < pi))`] THEN REAL_ARITH_TAC);;
(* ------------------------------------------------------------------------- *) (* Various properties. *) (* ------------------------------------------------------------------------- *)
let CNJ_CLOG = 
prove (`!z. (Im z = &0 ==> &0 < Re z) ==> cnj(clog z) = clog(cnj z)`,
GEN_TAC THEN ASM_CASES_TAC `z = Cx(&0)` THEN ASM_REWRITE_TAC[RE_CX; IM_CX; REAL_LT_REFL] THEN DISCH_TAC THEN MATCH_MP_TAC COMPLEX_EQ_CEXP THEN REWRITE_TAC[GSYM CNJ_CEXP] THEN ASM_SIMP_TAC[CEXP_CLOG; CNJ_EQ_CX; IM_CNJ] THEN MATCH_MP_TAC(REAL_ARITH `(--p < x /\ x <= p) /\ (--p < y /\ y <= p) /\ ~(x = p /\ y = p) ==> abs(--x - y) < &2 * p`) THEN ASM_SIMP_TAC[IM_CLOG_EQ_PI; CNJ_EQ_CX; CLOG_WORKS] THEN ASM_REAL_ARITH_TAC);;
let CLOG_INV = 
prove (`!z. (Im(z) = &0 ==> &0 < Re z) ==> clog(inv z) = --(clog z)`,
GEN_TAC THEN ASM_CASES_TAC `z = Cx(&0)` THEN ASM_REWRITE_TAC[RE_CX; IM_CX; REAL_LT_REFL] THEN STRIP_TAC THEN MATCH_MP_TAC COMPLEX_EQ_CEXP THEN ASM_SIMP_TAC[CEXP_CLOG; CEXP_NEG; COMPLEX_INV_EQ_0] THEN REWRITE_TAC[IM_NEG; REAL_SUB_RNEG] THEN MATCH_MP_TAC(REAL_ARITH `--pi < x /\ x <= pi /\ --pi < y /\ y <= pi /\ ~(x = pi /\ y = pi) ==> abs(x + y) < &2 * pi`) THEN ASM_SIMP_TAC[CLOG_WORKS; COMPLEX_INV_EQ_0; IM_CLOG_EQ_PI] THEN UNDISCH_TAC `Im z = &0 ==> &0 < Re z` THEN ASM_CASES_TAC `Im z = &0` THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC);;
let CLOG_1 = 
prove (`clog(Cx(&1)) = Cx(&0)`,
REWRITE_TAC[GSYM CEXP_0] THEN MATCH_MP_TAC CLOG_CEXP THEN REWRITE_TAC[IM_CX] THEN MP_TAC PI_POS THEN REAL_ARITH_TAC);;
let CLOG_NEG_1 = 
prove (`clog(--Cx(&1)) = ii * Cx pi`,
MATCH_MP_TAC COMPLEX_EQ_CEXP THEN REWRITE_TAC[GSYM CX_NEG] THEN SIMP_TAC[CEXP_EULER; GSYM CX_COS; GSYM CX_SIN; IM_MUL_II; IM_CX; RE_CX] THEN REWRITE_TAC[COS_PI; SIN_PI; COMPLEX_MUL_RZERO; COMPLEX_ADD_RID] THEN SIMP_TAC[CLOG_WORKS; COMPLEX_RING `~(Cx(-- &1) = Cx(&0))`; REAL_ARITH `--pi < x /\ x <= pi ==> abs(x - pi) < &2 * pi`]);;
let CLOG_II = 
prove (`clog ii = ii * Cx(pi / &2)`,
MP_TAC(SPEC `ii * Cx(pi / &2)` CLOG_CEXP) THEN SIMP_TAC[CEXP_EULER; GSYM CX_COS; GSYM CX_SIN; IM_MUL_II; IM_CX; RE_CX] THEN REWRITE_TAC[COS_PI2; SIN_PI2] THEN ANTS_TAC THENL [MP_TAC PI_POS THEN REAL_ARITH_TAC; REWRITE_TAC[COMPLEX_ADD_LID; COMPLEX_MUL_RID]]);;
let CLOG_NEG_II = 
prove (`clog(--ii) = --ii * Cx(pi / &2)`,
GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [COMPLEX_FIELD `--ii = inv ii`] THEN SIMP_TAC[CLOG_INV; RE_II; IM_II; REAL_OF_NUM_EQ; ARITH; CLOG_II] THEN REWRITE_TAC[COMPLEX_MUL_LNEG]);;
(* ------------------------------------------------------------------------- *) (* Relation between square root and exp/log, and hence its derivative. *) (* ------------------------------------------------------------------------- *)
let CSQRT_CEXP_CLOG = 
prove (`!z. ~(z = Cx(&0)) ==> csqrt z = cexp(clog(z) / Cx(&2))`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC CSQRT_UNIQUE THEN REWRITE_TAC[GSYM CEXP_N; RE_CEXP; IM_CEXP] THEN ASM_SIMP_TAC[COMPLEX_DIV_LMUL; CX_INJ; REAL_OF_NUM_EQ; ARITH; CEXP_CLOG] THEN SIMP_TAC[REAL_LT_MUL_EQ; REAL_EXP_POS_LT; REAL_LE_MUL_EQ] THEN REWRITE_TAC[REAL_ENTIRE; REAL_EXP_NZ; IM_DIV_CX] THEN FIRST_ASSUM(STRIP_ASSUME_TAC o CONJUNCT2 o MATCH_MP CLOG_WORKS) THEN FIRST_X_ASSUM(DISJ_CASES_TAC o REWRITE_RULE[REAL_LE_LT]) THENL [DISJ1_TAC THEN MATCH_MP_TAC COS_POS_PI THEN ASM_REAL_ARITH_TAC; DISJ2_TAC THEN ASM_REWRITE_TAC[COS_PI2; SIN_PI2; REAL_POS]]);;
let CNJ_CSQRT = 
prove (`!z. (Im z = &0 ==> &0 <= Re(z)) ==> cnj(csqrt z) = csqrt(cnj z)`,
GEN_TAC THEN ASM_CASES_TAC `z = Cx(&0)` THEN ASM_REWRITE_TAC[CSQRT_0; CNJ_CX] THEN DISCH_TAC THEN SUBGOAL_THEN `Im z = &0 ==> &0 < Re(z)` ASSUME_TAC THENL [REPEAT(POP_ASSUM MP_TAC) THEN REWRITE_TAC[COMPLEX_EQ; IM_CX; RE_CX] THEN REAL_ARITH_TAC; ASM_REWRITE_TAC[RE_CX; IM_CX; REAL_LT_REFL] THEN ASM_SIMP_TAC[CSQRT_CEXP_CLOG; CNJ_CEXP; CNJ_CLOG; CNJ_DIV; CNJ_EQ_CX; CNJ_CX]]);;
let HAS_COMPLEX_DERIVATIVE_CSQRT = 
prove (`!z. (Im z = &0 ==> &0 < Re(z)) ==> (csqrt has_complex_derivative inv(Cx(&2) * csqrt z)) (at z)`,
GEN_TAC THEN ASM_CASES_TAC `z = Cx(&0)` THEN ASM_REWRITE_TAC[IM_CX; RE_CX; REAL_LT_REFL] THEN DISCH_TAC THEN MATCH_MP_TAC HAS_COMPLEX_DERIVATIVE_TRANSFORM_AT THEN MAP_EVERY EXISTS_TAC [`\z. cexp(clog(z) / Cx(&2))`; `norm(z:complex)`] THEN ASM_REWRITE_TAC[COMPLEX_NORM_NZ] THEN CONJ_TAC THENL [REPEAT STRIP_TAC THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC CSQRT_CEXP_CLOG THEN REWRITE_TAC[GSYM COMPLEX_VEC_0] THEN REPEAT(POP_ASSUM MP_TAC) THEN NORM_ARITH_TAC; COMPLEX_DIFF_TAC THEN ASM_SIMP_TAC[GSYM CSQRT_CEXP_CLOG] THEN UNDISCH_TAC `~(z = Cx(&0))` THEN MP_TAC(SPEC `z:complex` CSQRT) THEN CONV_TAC COMPLEX_FIELD]);;
let COMPLEX_DIFFERENTIABLE_AT_CSQRT = 
prove (`!z. (Im z = &0 ==> &0 < Re(z)) ==> csqrt complex_differentiable at z`,
REWRITE_TAC[complex_differentiable] THEN MESON_TAC[HAS_COMPLEX_DERIVATIVE_CSQRT]);;
let COMPLEX_DIFFERENTIABLE_WITHIN_CSQRT = 
prove (`!s z. (Im z = &0 ==> &0 < Re(z)) ==> csqrt complex_differentiable (at z within s)`,
add_complex_differentiation_theorems (CONJUNCTS(REWRITE_RULE[FORALL_AND_THM] (MATCH_MP HAS_COMPLEX_DERIVATIVE_CHAIN HAS_COMPLEX_DERIVATIVE_CSQRT)));;
let CONTINUOUS_AT_CSQRT = 
prove (`!z. (Im z = &0 ==> &0 < Re(z)) ==> csqrt continuous at z`,
let CONTINUOUS_WITHIN_CSQRT = 
prove (`!s z. (Im z = &0 ==> &0 < Re(z)) ==> csqrt continuous (at z within s)`,
let CONTINUOUS_ON_CSQRT = 
prove (`!s. (!z. z IN s /\ Im z = &0 ==> &0 < Re(z)) ==> csqrt continuous_on s`,
let HOLOMORPHIC_ON_CSQRT = 
prove (`!s. (!z. z IN s /\ Im(z) = &0 ==> &0 < Re(z)) ==> csqrt holomorphic_on s`,
let CONTINUOUS_WITHIN_CSQRT_POSREAL = 
prove (`!z. csqrt continuous (at z within {w | real w /\ &0 <= Re(w)})`,
REPEAT STRIP_TAC THEN ASM_CASES_TAC `Im z = &0 ==> &0 < Re(z)` THENL [ASM_SIMP_TAC[CONTINUOUS_WITHIN_CSQRT]; ALL_TAC] THEN POP_ASSUM MP_TAC THEN REWRITE_TAC[NOT_IMP; REAL_NOT_LT] THEN REWRITE_TAC[REAL_ARITH `x <= &0 <=> x < &0 \/ x = &0`] THEN STRIP_TAC THENL [MATCH_MP_TAC CONTINUOUS_WITHIN_CLOSED_NONTRIVIAL THEN REWRITE_TAC[SET_RULE `{x | P x /\ Q x} = {x | P x} INTER {x | Q x}`] THEN SIMP_TAC[CLOSED_REAL_SET; CLOSED_INTER; IN_INTER; IN_ELIM_THM; REWRITE_RULE[real_ge] CLOSED_HALFSPACE_RE_GE] THEN ASM_REAL_ARITH_TAC; SUBGOAL_THEN `z = Cx(&0)` SUBST_ALL_TAC THENL [ASM_REWRITE_TAC[COMPLEX_EQ; RE_CX; IM_CX]; ALL_TAC] THEN REWRITE_TAC[continuous_within] THEN REWRITE_TAC[IN_ELIM_THM; IMP_CONJ; FORALL_REAL; RE_CX] THEN SIMP_TAC[GSYM CX_SQRT; REAL_LE_REFL] THEN SIMP_TAC[dist; GSYM CX_SUB; COMPLEX_NORM_CX; SQRT_0; REAL_SUB_RZERO] THEN X_GEN_TAC `e:real` THEN STRIP_TAC THEN EXISTS_TAC `(e:real) pow 2` THEN ASM_SIMP_TAC[REAL_POW_LT] THEN REPEAT STRIP_TAC THEN SUBGOAL_THEN `e = sqrt(e pow 2)` SUBST1_TAC THENL [ASM_SIMP_TAC[POW_2_SQRT; REAL_LT_IMP_LE]; ASM_SIMP_TAC[real_abs; SQRT_POS_LE]] THEN MATCH_MP_TAC SQRT_MONO_LT THEN ASM_REAL_ARITH_TAC]);;
(* ------------------------------------------------------------------------- *) (* Complex powers. *) (* ------------------------------------------------------------------------- *) parse_as_infix("cpow",(24,"left"));;
let cpow = new_definition
 `w cpow z = if w = Cx(&0) then Cx(&0)
             else cexp(z * clog w)`;;
let CPOW_0 = 
prove (`!z. Cx(&0) cpow z = Cx(&0)`,
REWRITE_TAC[cpow]);;
let CPOW_N = 
prove (`!z. z cpow (Cx(&n)) = if z = Cx(&0) then Cx(&0) else z pow n`,
GEN_TAC THEN REWRITE_TAC[cpow] THEN COND_CASES_TAC THEN ASM_SIMP_TAC[CEXP_N; CEXP_CLOG]);;
let CPOW_1 = 
prove (`!z. Cx(&1) cpow z = Cx(&1)`,
REWRITE_TAC[cpow; CX_INJ; REAL_OF_NUM_EQ; ARITH_EQ; CLOG_1] THEN REWRITE_TAC[CEXP_0; COMPLEX_MUL_RZERO]);;
let CPOW_ADD = 
prove (`!w z1 z2. w cpow (z1 + z2) = w cpow z1 * w cpow z2`,
REPEAT GEN_TAC THEN REWRITE_TAC[cpow] THEN ASM_CASES_TAC `w = Cx(&0)` THEN ASM_REWRITE_TAC[COMPLEX_MUL_RZERO] THEN REWRITE_TAC[COMPLEX_ADD_RDISTRIB; CEXP_ADD]);;
let CPOW_NEG = 
prove (`!w z. w cpow (--z) = inv(w cpow z)`,
REPEAT GEN_TAC THEN REWRITE_TAC[cpow] THEN ASM_CASES_TAC `w = Cx(&0)` THEN ASM_REWRITE_TAC[COMPLEX_MUL_RZERO; COMPLEX_INV_0] THEN REWRITE_TAC[COMPLEX_MUL_LNEG; CEXP_NEG]);;
let CPOW_SUB = 
prove (`!w z1 z2. w cpow (z1 - z2) = w cpow z1 / w cpow z2`,
let CPOW_EQ_0 = 
prove (`!w z. w cpow z = Cx(&0) <=> w = Cx(&0)`,
REPEAT GEN_TAC THEN REWRITE_TAC[cpow] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[CEXP_NZ]);;
let NORM_CPOW_REAL = 
prove (`!w z. real w /\ &0 < Re w ==> norm(w cpow z) = exp(Re z * log(Re w))`,
REPEAT STRIP_TAC THEN FIRST_X_ASSUM(SUBST_ALL_TAC o SYM o GEN_REWRITE_RULE I [REAL]) THEN RULE_ASSUM_TAC(REWRITE_RULE[RE_CX]) THEN ASM_SIMP_TAC[cpow; CX_INJ; REAL_LT_IMP_NZ] THEN ASM_SIMP_TAC[NORM_CEXP; GSYM CX_LOG; RE_MUL_CX; RE_CX]);;
let CPOW_REAL_REAL = 
prove (`!w z. real w /\ real z /\ &0 < Re w ==> w cpow z = Cx(exp(Re z * log(Re w)))`,
REPEAT STRIP_TAC THEN REPEAT(FIRST_X_ASSUM(SUBST_ALL_TAC o SYM o GEN_REWRITE_RULE I [REAL])) THEN RULE_ASSUM_TAC(REWRITE_RULE[RE_CX]) THEN ASM_SIMP_TAC[cpow; CX_INJ; REAL_LT_IMP_NZ] THEN ASM_SIMP_TAC[NORM_CEXP; GSYM CX_LOG; RE_MUL_CX; RE_CX; CX_EXP; CX_MUL]);;
let NORM_CPOW_REAL_MONO = 
prove (`!w z1 z2. real w /\ &1 < Re w ==> (norm(w cpow z1) <= norm(w cpow z2) <=> Re(z1) <= Re(z2))`,
SIMP_TAC[NORM_CPOW_REAL; REAL_ARITH `&1 < x ==> &0 < x`] THEN SIMP_TAC[REAL_EXP_MONO_LE; REAL_LE_RMUL_EQ; LOG_POS_LT]);;
let CPOW_MUL_REAL = 
prove (`!x y z. real x /\ real y /\ &0 <= Re x /\ &0 <= Re y ==> (x * y) cpow z = x cpow z * y cpow z`,
REPEAT STRIP_TAC THEN REPEAT(FIRST_X_ASSUM(SUBST_ALL_TAC o SYM o GEN_REWRITE_RULE I [REAL])) THEN REPEAT(POP_ASSUM MP_TAC) THEN REWRITE_TAC[RE_CX; IM_CX] THEN REWRITE_TAC[REAL_ARITH `&0 <= x <=> x = &0 \/ &0 < x`] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[COMPLEX_MUL_LZERO; COMPLEX_MUL_RZERO; CPOW_0] THEN ASM_SIMP_TAC[cpow; COMPLEX_ENTIRE; CX_INJ; REAL_LT_IMP_NZ] THEN REWRITE_TAC[GSYM CEXP_ADD; GSYM COMPLEX_ADD_LDISTRIB] THEN ASM_SIMP_TAC[GSYM CX_LOG; GSYM CX_ADD; GSYM CX_MUL; REAL_LT_MUL] THEN ASM_SIMP_TAC[LOG_MUL]);;
let HAS_COMPLEX_DERIVATIVE_CPOW = 
prove (`!s z. (Im z = &0 ==> &0 < Re z) ==> ((\z. z cpow s) has_complex_derivative (s * z cpow (s - Cx(&1)))) (at z)`,
REPEAT GEN_TAC THEN ASM_CASES_TAC `z = Cx(&0)` THEN ASM_REWRITE_TAC[IM_CX; RE_CX; REAL_LT_REFL] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[cpow] THEN MATCH_MP_TAC HAS_COMPLEX_DERIVATIVE_TRANSFORM_AT THEN MAP_EVERY EXISTS_TAC [`\z. cexp (s * clog z)`; `norm(z:complex)`] THEN ASM_REWRITE_TAC[COMPLEX_NORM_NZ] THEN CONJ_TAC THENL [GEN_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[dist] THEN REWRITE_TAC[COMPLEX_SUB_LZERO; NORM_NEG; REAL_LT_REFL]; COMPLEX_DIFF_TAC THEN ASM_REWRITE_TAC[CEXP_SUB; COMPLEX_SUB_RDISTRIB] THEN ASM_SIMP_TAC[CEXP_CLOG; COMPLEX_MUL_LID] THEN REPEAT(POP_ASSUM MP_TAC) THEN CONV_TAC COMPLEX_FIELD]);;
add_complex_differentiation_theorems (CONJUNCTS(REWRITE_RULE[FORALL_AND_THM] (GEN `s:complex` (MATCH_MP HAS_COMPLEX_DERIVATIVE_CHAIN (SPEC `s:complex` HAS_COMPLEX_DERIVATIVE_CPOW)))));;
let HAS_COMPLEX_DERIVATIVE_CPOW_RIGHT = 
prove (`!w z. ~(w = Cx(&0)) ==> ((\z. w cpow z) has_complex_derivative clog(w) * w cpow z) (at z)`,
REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[cpow] THEN COMPLEX_DIFF_TAC THEN REWRITE_TAC[COMPLEX_MUL_LID]);;
add_complex_differentiation_theorems (CONJUNCTS(REWRITE_RULE[FORALL_AND_THM] (GEN `s:complex` (MATCH_MP HAS_COMPLEX_DERIVATIVE_CHAIN (SPEC `s:complex` HAS_COMPLEX_DERIVATIVE_CPOW_RIGHT)))));;
let COMPLEX_DIFFERENTIABLE_CPOW_RIGHT = 
prove (`!w z. ~(w = Cx(&0)) ==> (\z. w cpow z) complex_differentiable (at z)`,
let HOLOMORPHIC_ON_CPOW_RIGHT = 
prove (`!w f s. ~(w = Cx(&0)) /\ f holomorphic_on s ==> (\z. w cpow (f z)) holomorphic_on s`,
REPEAT STRIP_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN MATCH_MP_TAC HOLOMORPHIC_ON_COMPOSE THEN ASM_REWRITE_TAC[] THEN GEN_REWRITE_TAC LAND_CONV [GSYM ETA_AX] THEN REWRITE_TAC[holomorphic_on; GSYM complex_differentiable] THEN ASM_SIMP_TAC[COMPLEX_DIFFERENTIABLE_CPOW_RIGHT; COMPLEX_DIFFERENTIABLE_AT_WITHIN]);;
(* ------------------------------------------------------------------------- *) (* Product rule. *) (* ------------------------------------------------------------------------- *)
let CLOG_MUL = 
prove (`!w z. ~(w = Cx(&0)) /\ ~(z = Cx(&0)) ==> clog(w * z) = if Im(clog w + clog z) <= --pi then (clog(w) + clog(z)) + ii * Cx(&2 * pi) else if Im(clog w + clog z) > pi then (clog(w) + clog(z)) - ii * Cx(&2 * pi) else clog(w) + clog(z)`,
REPEAT STRIP_TAC THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[]) THEN MATCH_MP_TAC CLOG_UNIQUE THEN ASM_SIMP_TAC[CEXP_ADD; CEXP_SUB; CEXP_EULER; CEXP_CLOG; CONJ_ASSOC; GSYM CX_SIN; GSYM CX_COS; COS_NPI; SIN_NPI] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN TRY(CONJ_TAC THENL [ALL_TAC; CONV_TAC COMPLEX_FIELD]) THEN REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP CLOG_WORKS)) THEN REPEAT(POP_ASSUM MP_TAC) THEN REWRITE_TAC[IM_ADD; IM_SUB; IM_MUL_II; RE_CX] THEN REAL_ARITH_TAC);;
let CLOG_MUL_SIMPLE = 
prove (`!w z. ~(w = Cx(&0)) /\ ~(z = Cx(&0)) /\ --pi < Im(clog(w)) + Im(clog(z)) /\ Im(clog(w)) + Im(clog(z)) <= pi ==> clog(w * z) = clog(w) + clog(z)`,
SIMP_TAC[CLOG_MUL; IM_ADD] THEN REAL_ARITH_TAC);;
let CLOG_NEG = 
prove (`!z. ~(z = Cx(&0)) ==> clog(--z) = if Im(z) <= &0 /\ ~(Re(z) < &0 /\ Im(z) = &0) then clog(z) + ii * Cx(pi) else clog(z) - ii * Cx(pi)`,
REPEAT STRIP_TAC THEN SUBST1_TAC(SIMPLE_COMPLEX_ARITH `--z = --Cx(&1) * z`) THEN ASM_SIMP_TAC[CLOG_MUL; COMPLEX_RING `~(--Cx(&1) = Cx(&0))`] THEN REWRITE_TAC[CLOG_NEG_1; IM_ADD; IM_MUL_II; RE_CX] THEN ASM_SIMP_TAC[CLOG_WORKS; REAL_ARITH `--p < x /\ x <= p ==> ~(p + x <= --p)`] THEN REWRITE_TAC[REAL_ARITH `p + x > p <=> &0 < x`] THEN ASM_SIMP_TAC[GSYM IM_CLOG_EQ_PI] THEN ONCE_REWRITE_TAC[REAL_ARITH `Im z <= &0 <=> ~(&0 < Im z)`] THEN ASM_SIMP_TAC[GSYM IM_CLOG_POS_LT] THEN ASM_SIMP_TAC[CLOG_WORKS; REAL_ARITH `x <= p ==> (x < p <=> ~(x = p))`] THEN REWRITE_TAC[TAUT `~(a /\ ~b) /\ ~b <=> ~a /\ ~b`] THEN ASM_CASES_TAC `Im(clog z) = pi` THEN ASM_REWRITE_TAC[PI_POS] THEN ASM_CASES_TAC `&0 < Im(clog z)` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[CX_MUL] THEN CONV_TAC COMPLEX_RING);;
let CLOG_MUL_II = 
prove (`!z. ~(z = Cx(&0)) ==> clog(ii * z) = if &0 <= Re(z) \/ Im(z) < &0 then clog(z) + ii * Cx(pi / &2) else clog(z) - ii * Cx(&3 * pi / &2)`,
REPEAT STRIP_TAC THEN ASM_SIMP_TAC[CLOG_MUL; II_NZ; CLOG_II] THEN REWRITE_TAC[IM_ADD; IM_MUL_II; RE_CX] THEN ASM_SIMP_TAC[CLOG_WORKS; REAL_ARITH `--p < x /\ x <= p ==> ~(p / &2 + x <= --p)`] THEN REWRITE_TAC[REAL_ARITH `p / &2 + x > p <=> p / &2 < x`] THEN REWRITE_TAC[REAL_ARITH `Im z < &0 <=> ~(&0 <= Im z)`] THEN ASM_SIMP_TAC[GSYM RE_CLOG_POS_LE; GSYM IM_CLOG_POS_LE] THEN MATCH_MP_TAC(MESON[] `(p <=> ~q) /\ x = a /\ y = b ==> ((if p then x else y) = (if q then b else a))`) THEN CONJ_TAC THENL [MP_TAC PI_POS THEN REAL_ARITH_TAC; REWRITE_TAC[CX_MUL; CX_DIV] THEN CONV_TAC COMPLEX_RING]);;
(* ------------------------------------------------------------------------- *) (* Unwinding number gives another version of log-product formula. *) (* Note that in this special case the unwinding number is -1, 0 or 1. *) (* ------------------------------------------------------------------------- *)
let unwinding = new_definition
 `unwinding(z) = (z - clog(cexp z)) / (Cx(&2 * pi) * ii)`;;
let UNWINDING_2PI = 
prove (`Cx(&2 * pi) * ii * unwinding(z) = z - clog(cexp z)`,
REWRITE_TAC[unwinding; COMPLEX_MUL_ASSOC] THEN MATCH_MP_TAC COMPLEX_DIV_LMUL THEN REWRITE_TAC[COMPLEX_ENTIRE; CX_INJ; II_NZ] THEN MP_TAC PI_POS THEN REAL_ARITH_TAC);;
let CLOG_MUL_UNWINDING = 
prove (`!w z. ~(w = Cx(&0)) /\ ~(z = Cx(&0)) ==> clog(w * z) = clog(w) + clog(z) - Cx(&2 * pi) * ii * unwinding(clog w + clog z)`,
REWRITE_TAC[UNWINDING_2PI; COMPLEX_RING `w + z - ((w + z) - c) = c:complex`] THEN ASM_SIMP_TAC[CEXP_ADD; CEXP_CLOG]);;
(* ------------------------------------------------------------------------- *) (* Complex arctangent (branch cut gives standard bounds in real case). *) (* ------------------------------------------------------------------------- *)
let catn = new_definition
 `catn z = (ii / Cx(&2)) * clog((Cx(&1) - ii * z) / (Cx(&1) + ii * z))`;;
let IM_COMPLEX_DIV_LEMMA = 
prove (`!z. Im((Cx(&1) - ii * z) / (Cx(&1) + ii * z)) = &0 <=> Re z = &0`,
REWRITE_TAC[IM_COMPLEX_DIV_EQ_0] THEN REWRITE_TAC[complex_mul; IM; RE; IM_CNJ; RE_CNJ; RE_CX; IM_CX; RE_II; IM_II; RE_SUB; RE_ADD; IM_SUB; IM_ADD] THEN REAL_ARITH_TAC);;
let RE_COMPLEX_DIV_LEMMA = 
prove (`!z. &0 < Re((Cx(&1) - ii * z) / (Cx(&1) + ii * z)) <=> norm(z) < &1`,
REWRITE_TAC[RE_COMPLEX_DIV_GT_0; NORM_LT_SQUARE; REAL_LT_01] THEN REWRITE_TAC[GSYM NORM_POW_2; COMPLEX_SQNORM] THEN REWRITE_TAC[complex_mul; IM; RE; IM_CNJ; RE_CNJ; RE_CX; IM_CX; RE_II; IM_II; RE_SUB; RE_ADD; IM_SUB; IM_ADD] THEN REAL_ARITH_TAC);;
let CTAN_CATN = 
prove (`!z. ~(z pow 2 = --Cx(&1)) ==> ctan(catn z) = z`,
REPEAT STRIP_TAC THEN REWRITE_TAC[catn; ctan; csin; ccos; COMPLEX_RING `--i * i / Cx(&2) * z = --(i * i) / Cx(&2) * z`; COMPLEX_RING `i * i / Cx(&2) * z = (i * i) / Cx(&2) * z`] THEN REWRITE_TAC[COMPLEX_POW_II_2; GSYM COMPLEX_POW_2] THEN REWRITE_TAC[COMPLEX_RING `--Cx(&1) / Cx(&2) * x = --(Cx(&1) / Cx(&2) * x)`; CEXP_NEG] THEN SUBGOAL_THEN `~(cexp(Cx(&1) / Cx(&2) * (clog((Cx(&1) - ii * z) / (Cx(&1) + ii * z)))) pow 2 = --Cx(&1))` ASSUME_TAC THENL [REWRITE_TAC[GSYM CEXP_N; CEXP_SUB; COMPLEX_RING `Cx(&2) * Cx(&1) / Cx(&2) * z = z`] THEN ASM_SIMP_TAC[CEXP_CLOG; COMPLEX_POW_II_2; COMPLEX_FIELD `~(w = Cx(&0)) /\ ~(z = Cx(&0)) ==> ~(w / z = Cx(&0))`; COMPLEX_FIELD `~(w = Cx(&0)) ==> (x / w = y <=> x = y * w)`; COMPLEX_FIELD `ii pow 2 = --Cx(&1) /\ ~(z pow 2 = --Cx(&1)) ==> ~(Cx(&1) - ii * z = Cx(&0)) /\ ~(Cx(&1) + ii * z = Cx(&0))`] THEN POP_ASSUM MP_TAC THEN CONV_TAC COMPLEX_FIELD; ALL_TAC] THEN REWRITE_TAC[COMPLEX_RING `-- --Cx (&1) / Cx (&2) = Cx(&1) / Cx(&2)`] THEN ASM_SIMP_TAC[CEXP_NZ; COMPLEX_FIELD `~(z = Cx(&0)) /\ ~(z pow 2 = --Cx(&1)) ==> ((inv(z) - z) / (Cx(&2) * ii)) / ((inv(z) + z) / Cx(&2)) = inv ii * ((Cx(&1) - z pow 2) / (Cx(&1) + z pow 2))`] THEN ASM_SIMP_TAC[GSYM CEXP_N; CEXP_SUB; COMPLEX_RING `Cx(&2) * Cx(&1) / Cx(&2) * z = z`] THEN ASM_SIMP_TAC[CEXP_CLOG; COMPLEX_FIELD `~(z pow 2 = --Cx(&1)) ==> ~((Cx(&1) - ii * z) / (Cx(&1) + ii * z) = Cx(&0))`] THEN UNDISCH_TAC `~(z pow 2 = --Cx(&1))` THEN CONV_TAC COMPLEX_FIELD);;
let CATN_CTAN = 
prove (`!z. abs(Re z) < pi / &2 ==> catn(ctan z) = z`,
REPEAT STRIP_TAC THEN REWRITE_TAC[catn; ctan; csin; ccos] THEN ASM_SIMP_TAC[COMPLEX_FIELD `ii * (a / (Cx(&2) * ii)) / (b / Cx(&2)) = a / b`] THEN SIMP_TAC[COMPLEX_FIELD `ii / Cx(&2) * x = y <=> x = Cx(&2) * --(ii * y)`] THEN SUBGOAL_THEN `~(cexp(ii * z) pow 2 = --Cx(&1))` ASSUME_TAC THENL [SUBGOAL_THEN `--Cx(&1) = cexp(ii * Cx pi)` SUBST1_TAC THENL [REWRITE_TAC[CEXP_EULER; GSYM CX_SIN; GSYM CX_COS; SIN_PI; COS_PI] THEN CONV_TAC COMPLEX_RING; ALL_TAC] THEN REWRITE_TAC[GSYM CEXP_N; CEXP_EQ] THEN DISCH_THEN(X_CHOOSE_THEN `n:real` STRIP_ASSUME_TAC) THEN FIRST_X_ASSUM(MP_TAC o AP_TERM `Im`) THEN REWRITE_TAC[IM_MUL_CX; IM_MUL_II; IM_ADD; RE_CX; IM_II; REAL_MUL_RID] THEN MATCH_MP_TAC(REAL_ARITH `abs(z) < p / &2 /\ (w = &0 \/ abs(w) >= &2 * p) ==> ~(&2 * z = p + w)`) THEN ASM_REWRITE_TAC[REAL_ABS_MUL; REAL_ABS_PI; REAL_ABS_NUM] THEN SIMP_TAC[real_ge; REAL_MUL_ASSOC; REAL_LE_RMUL_EQ; PI_POS] THEN REWRITE_TAC[REAL_ENTIRE; PI_NZ] THEN MP_TAC(SPEC `n:real` REAL_ABS_INTEGER_LEMMA) THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC; ASM_SIMP_TAC[CEXP_NEG; CEXP_NZ; COMPLEX_MUL_LNEG; COMPLEX_FIELD `~(w = Cx(&0)) /\ ~(w pow 2 = --Cx(&1)) ==> (Cx(&1) - (w - inv w) / (w + inv w)) / (Cx(&1) + (w - inv w) / (w + inv w)) = inv(w) pow 2`] THEN REWRITE_TAC[GSYM CEXP_N; GSYM CEXP_NEG] THEN MATCH_MP_TAC CLOG_CEXP THEN REWRITE_TAC[IM_MUL_CX; IM_NEG; IM_MUL_II] THEN ASM_REAL_ARITH_TAC]);;
let RE_CATN_BOUNDS = 
prove (`!z. (Re z = &0 ==> abs(Im z) < &1) ==> abs(Re(catn z)) < pi / &2`,
REWRITE_TAC[catn; complex_div; GSYM CX_INV; GSYM COMPLEX_MUL_ASSOC] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[RE_MUL_II; IM_MUL_CX] THEN MATCH_MP_TAC(REAL_ARITH `abs x < p ==> abs(--(inv(&2) * x)) < p / &2`) THEN MATCH_MP_TAC(REAL_ARITH `(--p < x /\ x <= p) /\ ~(x = p) ==> abs x < p`) THEN SUBGOAL_THEN `~(z = ii) /\ ~(z = --ii)` STRIP_ASSUME_TAC THENL [CONJ_TAC THEN DISCH_THEN(fun th -> POP_ASSUM MP_TAC THEN SUBST1_TAC th) THEN REWRITE_TAC[RE_II; IM_II; RE_NEG; IM_NEG] THEN REAL_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[GSYM complex_div] THEN CONJ_TAC THENL [SUBGOAL_THEN `~((Cx(&1) - ii * z) / (Cx(&1) + ii * z) = Cx(&0))` (fun th -> MESON_TAC[th; CLOG_WORKS]) THEN REPEAT(POP_ASSUM MP_TAC) THEN CONV_TAC COMPLEX_FIELD; ALL_TAC] THEN DISCH_TAC THEN MP_TAC(ISPEC `clog((Cx(&1) - ii * z) / (Cx(&1) + ii * z))` EULER) THEN ASM_REWRITE_TAC[SIN_PI; COS_PI; CX_NEG] THEN REWRITE_TAC[COMPLEX_RING `x = y * (--Cx(&1) + z * Cx(&0)) <=> x + y = Cx(&0)`] THEN REWRITE_TAC[CX_EXP] THEN ASM_SIMP_TAC[CEXP_CLOG; COMPLEX_FIELD `~(z = ii) /\ ~(z = --ii) ==> ~((Cx(&1) - ii * z) / (Cx(&1) + ii * z) = Cx(&0))`] THEN REWRITE_TAC[GSYM CX_EXP] THEN DISCH_THEN(MP_TAC o AP_TERM `Im`) THEN REWRITE_TAC[IM_ADD; IM_CX; REAL_ADD_RID; IM_COMPLEX_DIV_LEMMA] THEN DISCH_TAC THEN UNDISCH_TAC `Re z = &0 ==> abs (Im z) < &1` THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN SUBGOAL_THEN `ii * z = --Cx(Im z)` SUBST_ALL_TAC THENL [ASM_REWRITE_TAC[COMPLEX_EQ; RE_NEG; IM_NEG; RE_MUL_II; IM_MUL_II; RE_CX; IM_CX; REAL_NEG_0]; ALL_TAC] THEN UNDISCH_TAC `Im(clog((Cx(&1) - --Cx(Im z)) / (Cx(&1) + --Cx(Im z)))) = pi` THEN REWRITE_TAC[COMPLEX_SUB_RNEG; GSYM complex_sub] THEN REWRITE_TAC[GSYM CX_ADD; GSYM CX_SUB; GSYM CX_DIV] THEN SUBGOAL_THEN `&0 < (&1 + Im z) / (&1 - Im z)` ASSUME_TAC THENL [MATCH_MP_TAC REAL_LT_DIV THEN ASM_REAL_ARITH_TAC; ASM_SIMP_TAC[GSYM CX_LOG; IM_CX; PI_NZ]]);;
let HAS_COMPLEX_DERIVATIVE_CATN = 
prove (`!z. (Re z = &0 ==> abs(Im z) < &1) ==> (catn has_complex_derivative inv(Cx(&1) + z pow 2)) (at z)`,
REPEAT STRIP_TAC THEN SUBGOAL_THEN `~(z = ii) /\ ~(z = --ii)` STRIP_ASSUME_TAC THENL [CONJ_TAC THEN DISCH_THEN(fun th -> POP_ASSUM MP_TAC THEN SUBST1_TAC th) THEN REWRITE_TAC[RE_II; IM_II; RE_NEG; IM_NEG] THEN REAL_ARITH_TAC; ALL_TAC] THEN GEN_REWRITE_TAC (RATOR_CONV o LAND_CONV) [GSYM ETA_AX] THEN REWRITE_TAC[catn] THEN COMPLEX_DIFF_TAC THEN REWRITE_TAC[RE_SUB; RE_ADD; IM_SUB; IM_ADD; RE_CX; RE_MUL_II; IM_CX; IM_MUL_II] THEN REWRITE_TAC[GSYM CONJ_ASSOC] THEN CONJ_TAC THENL [ASM_REWRITE_TAC[IM_COMPLEX_DIV_LEMMA; RE_COMPLEX_DIV_LEMMA] THEN SIMP_TAC[complex_norm] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_REWRITE_TAC[REAL_ADD_LID; POW_2_SQRT_ABS]; REPEAT(POP_ASSUM MP_TAC) THEN CONV_TAC COMPLEX_FIELD]);;
let COMPLEX_DIFFERENTIABLE_AT_CATN = 
prove (`!z. (Re z = &0 ==> abs(Im z) < &1) ==> catn complex_differentiable at z`,
REWRITE_TAC[complex_differentiable] THEN MESON_TAC[HAS_COMPLEX_DERIVATIVE_CATN]);;
let COMPLEX_DIFFERENTIABLE_WITHIN_CATN = 
prove (`!s z. (Re z = &0 ==> abs(Im z) < &1) ==> catn complex_differentiable (at z within s)`,
add_complex_differentiation_theorems (CONJUNCTS(REWRITE_RULE[FORALL_AND_THM] (MATCH_MP HAS_COMPLEX_DERIVATIVE_CHAIN HAS_COMPLEX_DERIVATIVE_CATN)));;
let CONTINUOUS_AT_CATN = 
prove (`!z. (Re z = &0 ==> abs(Im z) < &1) ==> catn continuous at z`,
let CONTINUOUS_WITHIN_CATN = 
prove (`!s z. (Re z = &0 ==> abs(Im z) < &1) ==> catn continuous (at z within s)`,
let CONTINUOUS_ON_CATN = 
prove (`!s. (!z. z IN s /\ Re z = &0 ==> abs(Im z) < &1) ==> catn continuous_on s`,
let HOLOMORPHIC_ON_CATN = 
prove (`!s. (!z. z IN s /\ Re z = &0 ==> abs(Im z) < &1) ==> catn holomorphic_on s`,
(* ------------------------------------------------------------------------- *) (* Real arctangent. *) (* ------------------------------------------------------------------------- *)
let atn = new_definition
 `atn(x) = Re(catn(Cx x))`;;
let CX_ATN = 
prove (`!x. Cx(atn x) = catn(Cx x)`,
GEN_TAC THEN REWRITE_TAC[atn; catn; GSYM REAL; real] THEN REWRITE_TAC[complex_div; IM_MUL_II; GSYM CX_INV; GSYM COMPLEX_MUL_ASSOC] THEN REWRITE_TAC[RE_MUL_CX; REAL_ARITH `inv(&2) * x = &0 <=> x = &0`] THEN MATCH_MP_TAC NORM_CEXP_IMAGINARY THEN SUBGOAL_THEN `~(Cx(&1) - ii * Cx(x) = Cx(&0)) /\ ~(Cx(&1) + ii * Cx(x) = Cx(&0))` STRIP_ASSUME_TAC THENL [CONJ_TAC THEN DISCH_THEN(MP_TAC o AP_TERM `Re`) THEN REWRITE_TAC[RE_ADD; RE_SUB; RE_MUL_II; IM_CX; RE_CX] THEN CONV_TAC REAL_RAT_REDUCE_CONV; ALL_TAC] THEN ASM_SIMP_TAC[CEXP_SUB; CEXP_CLOG; COMPLEX_FIELD `~(a = Cx(&0)) /\ ~(b = Cx(&0)) ==> ~(a * inv b = Cx(&0))`] THEN REWRITE_TAC[GSYM complex_div; COMPLEX_NORM_DIV] THEN MATCH_MP_TAC(REAL_FIELD `~(b = &0) /\ a = b ==> a / b = &1`) THEN ASM_REWRITE_TAC[COMPLEX_NORM_ZERO] THEN MATCH_MP_TAC(MESON[COMPLEX_NORM_CNJ] `cnj a = b ==> norm a = norm b`) THEN REWRITE_TAC[CNJ_SUB; CNJ_MUL; CNJ_MUL; CNJ_II; CNJ_CX] THEN CONV_TAC COMPLEX_RING);;
let ATN_TAN = 
prove (`!y. tan(atn y) = y`,
GEN_TAC THEN REWRITE_TAC[tan_def; atn] THEN MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `Re(ctan(catn(Cx y)))` THEN CONJ_TAC THENL [REWRITE_TAC[GSYM CX_ATN; RE_CX]; ALL_TAC] THEN GEN_REWRITE_TAC RAND_CONV [GSYM RE_CX] THEN AP_TERM_TAC THEN MATCH_MP_TAC CTAN_CATN THEN MATCH_MP_TAC(COMPLEX_RING `~(z = ii) /\ ~(z = --ii) ==> ~(z pow 2 = --Cx(&1))`) THEN CONJ_TAC THEN DISCH_THEN(MP_TAC o AP_TERM `Im`) THEN REWRITE_TAC[IM_II; IM_CX; IM_NEG] THEN REAL_ARITH_TAC);;
let ATN_BOUND = 
prove (`!y. abs(atn y) < pi / &2`,
GEN_TAC THEN REWRITE_TAC[atn] THEN MATCH_MP_TAC RE_CATN_BOUNDS THEN REWRITE_TAC[IM_CX] THEN CONV_TAC REAL_RAT_REDUCE_CONV);;
let ATN_BOUNDS = 
prove (`!y. --(pi / &2) < atn(y) /\ atn(y) < (pi / &2)`,
MP_TAC ATN_BOUND THEN MATCH_MP_TAC MONO_FORALL THEN REAL_ARITH_TAC);;
let TAN_ATN = 
prove (`!x. --(pi / &2) < x /\ x < pi / &2 ==> atn(tan(x)) = x`,
REPEAT STRIP_TAC THEN REWRITE_TAC[tan_def; atn] THEN MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `Re(catn(ctan(Cx x)))` THEN CONJ_TAC THENL [REWRITE_TAC[GSYM CX_TAN; RE_CX]; ALL_TAC] THEN GEN_REWRITE_TAC RAND_CONV [GSYM RE_CX] THEN AP_TERM_TAC THEN MATCH_MP_TAC CATN_CTAN THEN REWRITE_TAC[RE_CX] THEN ASM_REAL_ARITH_TAC);;
let ATN_0 = 
prove (`atn(&0) = &0`,
GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [SYM TAN_0] THEN MATCH_MP_TAC TAN_ATN THEN MP_TAC PI_POS THEN REAL_ARITH_TAC);;
let ATN_1 = 
prove (`atn(&1) = pi / &4`,
MP_TAC(AP_TERM `atn` TAN_PI4) THEN DISCH_THEN(SUBST1_TAC o SYM) THEN MATCH_MP_TAC TAN_ATN THEN MP_TAC PI_POS THEN REAL_ARITH_TAC);;
let ATN_NEG = 
prove (`!x. atn(--x) = --(atn x)`,
GEN_TAC THEN MP_TAC(SPEC `atn(x)` TAN_NEG) THEN REWRITE_TAC[ATN_TAN] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN MATCH_MP_TAC TAN_ATN THEN MP_TAC(SPEC `x:real` ATN_BOUNDS) THEN REAL_ARITH_TAC);;
let ATN_MONO_LT = 
prove (`!x y. x < y ==> atn(x) < atn(y)`,
REPEAT GEN_TAC THEN GEN_REWRITE_TAC (LAND_CONV o BINOP_CONV) [GSYM ATN_TAN] THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[REAL_NOT_LT] THEN SIMP_TAC[TAN_MONO_LE; ATN_BOUNDS]);;
let ATN_MONO_LT_EQ = 
prove (`!x y. atn(x) < atn(y) <=> x < y`,
let ATN_MONO_LE_EQ = 
prove (`!x y. atn(x) <= atn(y) <=> x <= y`,
REWRITE_TAC[GSYM REAL_NOT_LT; ATN_MONO_LT_EQ]);;
let ATN_INJ = 
prove (`!x y. (atn x = atn y) <=> (x = y)`,
REWRITE_TAC[GSYM REAL_LE_ANTISYM; ATN_MONO_LE_EQ]);;
let ATN_POS_LT = 
prove (`&0 < atn(x) <=> &0 < x`,
MESON_TAC[ATN_0; ATN_MONO_LT_EQ]);;
let ATN_POS_LE = 
prove (`&0 <= atn(x) <=> &0 <= x`,
MESON_TAC[ATN_0; ATN_MONO_LE_EQ]);;
let ATN_LT_PI4_POS = 
prove (`!x. x < &1 ==> atn(x) < pi / &4`,
SIMP_TAC[GSYM ATN_1; ATN_MONO_LT]);;
let ATN_LT_PI4_NEG = 
prove (`!x. --(&1) < x ==> --(pi / &4) < atn(x)`,
SIMP_TAC[GSYM ATN_1; GSYM ATN_NEG; ATN_MONO_LT]);;
let ATN_LT_PI4 = 
prove (`!x. abs(x) < &1 ==> abs(atn x) < pi / &4`,
GEN_TAC THEN MATCH_MP_TAC(REAL_ARITH `(&0 < x ==> &0 < y) /\ (x < &0 ==> y < &0) /\ ((x = &0) ==> (y = &0)) /\ (x < a ==> y < b) /\ (--a < x ==> --b < y) ==> abs(x) < a ==> abs(y) < b`) THEN SIMP_TAC[ATN_LT_PI4_POS; ATN_LT_PI4_NEG; ATN_0] THEN CONJ_TAC THEN GEN_REWRITE_TAC (RAND_CONV o ONCE_DEPTH_CONV) [GSYM ATN_0] THEN SIMP_TAC[ATN_MONO_LT]);;
let ATN_LE_PI4 = 
prove (`!x. abs(x) <= &1 ==> abs(atn x) <= pi / &4`,
REWRITE_TAC[REAL_LE_LT] THEN REPEAT STRIP_TAC THEN ASM_SIMP_TAC[ATN_LT_PI4] THEN DISJ2_TAC THEN FIRST_ASSUM(DISJ_CASES_THEN SUBST1_TAC o MATCH_MP (REAL_ARITH `(abs(x) = a) ==> (x = a) \/ (x = --a)`)) THEN ASM_REWRITE_TAC[ATN_1; ATN_NEG] THEN REWRITE_TAC[REAL_ABS_DIV; REAL_ABS_NUM; REAL_ABS_NEG] THEN SIMP_TAC[real_abs; REAL_LT_IMP_LE; PI_POS]);;
let COS_ATN_NZ = 
prove (`!x. ~(cos(atn(x)) = &0)`,
GEN_TAC THEN MATCH_MP_TAC REAL_LT_IMP_NZ THEN MATCH_MP_TAC COS_POS_PI THEN REWRITE_TAC[ATN_BOUNDS]);;
let TAN_SEC = 
prove (`!x. ~(cos(x) = &0) ==> (&1 + (tan(x) pow 2) = inv(cos x) pow 2)`,
MP_TAC SIN_CIRCLE THEN MATCH_MP_TAC MONO_FORALL THEN REWRITE_TAC[tan] THEN CONV_TAC REAL_FIELD);;
let COS_ATN = 
prove (`!x. cos(atn x) = &1 / sqrt(&1 + x pow 2)`,
SIMP_TAC[COS_TAN; ATN_BOUND; ATN_TAN]);;
let SIN_ATN = 
prove (`!x. sin(atn x) = x / sqrt(&1 + x pow 2)`,
SIMP_TAC[SIN_TAN; ATN_BOUND; ATN_TAN]);;
(* ------------------------------------------------------------------------- *) (* Some bound theorems where a bit of simple calculus is handy. *) (* ------------------------------------------------------------------------- *)
let ATN_ABS_LE_X = 
prove (`!x. abs(atn x) <= abs x`,
REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`catn`; `\z. inv(Cx(&1) + z pow 2)`; `real`; `&1`] COMPLEX_MVT) THEN REWRITE_TAC[CONVEX_REAL; IN] THEN ANTS_TAC THENL [CONJ_TAC THENL [REWRITE_TAC[real] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC HAS_COMPLEX_DERIVATIVE_AT_WITHIN THEN MATCH_MP_TAC HAS_COMPLEX_DERIVATIVE_CATN THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC; GEN_TAC THEN REWRITE_TAC[REAL] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN REWRITE_TAC[GSYM CX_POW; GSYM CX_ADD; GSYM CX_INV; COMPLEX_NORM_CX] THEN REWRITE_TAC[REAL_ABS_INV] THEN MATCH_MP_TAC REAL_INV_LE_1 THEN MP_TAC(SPEC `Re z` REAL_LE_SQUARE) THEN REAL_ARITH_TAC]; DISCH_THEN(MP_TAC o SPECL [`Cx(&0)`; `Cx(x)`]) THEN REWRITE_TAC[GSYM CX_ATN; COMPLEX_SUB_RZERO; REAL_CX; ATN_0] THEN REWRITE_TAC[COMPLEX_NORM_CX; REAL_MUL_LID]]);;
let ATN_LE_X = 
prove (`!x. &0 <= x ==> atn(x) <= x`,
MP_TAC ATN_ABS_LE_X THEN MATCH_MP_TAC MONO_FORALL THEN REAL_ARITH_TAC);;
let TAN_ABS_GE_X = 
prove (`!x. abs(x) < pi / &2 ==> abs(x) <= abs(tan x)`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `abs(atn(tan x))` THEN REWRITE_TAC[ATN_ABS_LE_X] THEN MATCH_MP_TAC REAL_EQ_IMP_LE THEN AP_TERM_TAC THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC TAN_ATN THEN POP_ASSUM MP_TAC THEN REAL_ARITH_TAC);;
(* ------------------------------------------------------------------------- *) (* Probably not very useful, but for compatibility with old analysis theory. *) (* ------------------------------------------------------------------------- *)
let TAN_TOTAL = 
prove (`!y. ?!x. --(pi / &2) < x /\ x < (pi / &2) /\ tan(x) = y`,
MESON_TAC[TAN_ATN; ATN_TAN; ATN_BOUNDS]);;
let TAN_TOTAL_POS = 
prove (`!y. &0 <= y ==> ?x. &0 <= x /\ x < pi / &2 /\ tan(x) = y`,
MESON_TAC[ATN_TAN; ATN_BOUNDS; ATN_POS_LE]);;
let TAN_TOTAL_LEMMA = 
prove (`!y. &0 < y ==> ?x. &0 < x /\ x < pi / &2 /\ y < tan(x)`,
REPEAT STRIP_TAC THEN EXISTS_TAC `atn(y + &1)` THEN REWRITE_TAC[ATN_TAN; ATN_BOUNDS; ATN_POS_LT] THEN POP_ASSUM MP_TAC THEN REAL_ARITH_TAC);;
(* ------------------------------------------------------------------------- *) (* Some slightly ad hoc lemmas useful here. *) (* ------------------------------------------------------------------------- *)
let RE_POW_2 = 
prove (`Re(z pow 2) = Re(z) pow 2 - Im(z) pow 2`,
REWRITE_TAC[COMPLEX_POW_2; complex_mul; RE] THEN REAL_ARITH_TAC);;
let IM_POW_2 = 
prove (`Im(z pow 2) = &2 * Re(z) * Im(z)`,
REWRITE_TAC[COMPLEX_POW_2; complex_mul; IM] THEN REAL_ARITH_TAC);;
let ABS_SQUARE_LT_1 = 
prove (`!x. x pow 2 < &1 <=> abs(x) < &1`,
ONCE_REWRITE_TAC[GSYM REAL_ABS_NUM] THEN REWRITE_TAC[REAL_LT_SQUARE_ABS] THEN REAL_ARITH_TAC);;
let ABS_SQUARE_LE_1 = 
prove (`!x. x pow 2 <= &1 <=> abs(x) <= &1`,
ONCE_REWRITE_TAC[GSYM REAL_ABS_NUM] THEN REWRITE_TAC[REAL_LT_SQUARE_ABS; GSYM REAL_NOT_LT] THEN REAL_ARITH_TAC);;
let ABS_SQUARE_EQ_1 = 
prove (`!x. x pow 2 = &1 <=> abs(x) = &1`,
REWRITE_TAC[REAL_RING `x pow 2 = &1 <=> x = &1 \/ x = -- &1`] THEN REAL_ARITH_TAC);;
(* ------------------------------------------------------------------------- *) (* Inverse sine. *) (* ------------------------------------------------------------------------- *)
let casn = new_definition
 `casn z = --ii * clog(ii * z + csqrt(Cx(&1) - z pow 2))`;;
let CASN_BODY_LEMMA = 
prove (`!z. ~(ii * z + csqrt(Cx(&1) - z pow 2) = Cx(&0))`,
GEN_TAC THEN MP_TAC(SPEC `Cx(&1) - z pow 2` CSQRT) THEN CONV_TAC COMPLEX_FIELD);;
let CSIN_CASN = 
prove (`!z. csin(casn z) = z`,
GEN_TAC THEN REWRITE_TAC[csin; casn; COMPLEX_MUL_LNEG; COMPLEX_MUL_RNEG] THEN REWRITE_TAC[COMPLEX_MUL_ASSOC; COMPLEX_NEG_NEG] THEN REWRITE_TAC[COMPLEX_POW_II_2; GSYM COMPLEX_POW_2] THEN REWRITE_TAC[COMPLEX_NEG_NEG; COMPLEX_MUL_LNEG; COMPLEX_MUL_LID] THEN REWRITE_TAC[CEXP_NEG] THEN ASM_SIMP_TAC[CASN_BODY_LEMMA; CEXP_CLOG; COMPLEX_FIELD `~(z = Cx(&0)) ==> ((z - inv z) / (Cx(&2) * ii) = c <=> z pow 2 - Cx(&1) = Cx(&2) * ii * c * z)`] THEN MP_TAC(SPEC `Cx(&1) - z pow 2` CSQRT) THEN CONV_TAC COMPLEX_FIELD);;
let CASN_CSIN = 
prove (`!z. abs(Re z) < pi / &2 \/ (abs(Re z) = pi / &2 /\ Im z = &0) ==> casn(csin z) = z`,
GEN_TAC THEN DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC) THEN REWRITE_TAC[csin; casn; COMPLEX_MUL_LNEG; CEXP_NEG] THEN SIMP_TAC[CEXP_NZ; COMPLEX_FIELD `~(z = Cx(&0)) ==> Cx(&1) - ((z - inv z) / (Cx(&2) * ii)) pow 2 = ((z + inv z) / Cx(&2)) pow 2`] THEN SUBGOAL_THEN `csqrt(((cexp(ii * z) + inv(cexp(ii * z))) / Cx(&2)) pow 2) = (cexp(ii * z) + inv(cexp(ii * z))) / Cx(&2)` SUBST1_TAC THENL [MATCH_MP_TAC POW_2_CSQRT THEN REWRITE_TAC[GSYM CEXP_NEG] THEN REWRITE_TAC[complex_div; GSYM CX_INV; RE_MUL_CX; IM_MUL_CX] THEN REWRITE_TAC[REAL_ARITH `&0 < r * inv(&2) \/ r * inv(&2) = &0 /\ &0 <= i * inv(&2) <=> &0 < r \/ r = &0 /\ &0 <= i`] THEN REWRITE_TAC[RE_ADD; IM_ADD; RE_CEXP; IM_CEXP] THEN REWRITE_TAC[RE_MUL_II; RE_NEG; IM_MUL_II; IM_NEG] THEN REWRITE_TAC[SIN_NEG; COS_NEG; REAL_NEG_NEG] THEN REWRITE_TAC[REAL_MUL_RNEG; GSYM real_sub] THEN REWRITE_TAC[GSYM REAL_ADD_RDISTRIB; GSYM REAL_SUB_RDISTRIB] THEN FIRST_X_ASSUM(DISJ_CASES_THEN STRIP_ASSUME_TAC) THENL [DISJ1_TAC THEN MATCH_MP_TAC REAL_LT_MUL THEN ASM_SIMP_TAC[REAL_LT_ADD; REAL_EXP_POS_LT] THEN MATCH_MP_TAC COS_POS_PI THEN POP_ASSUM MP_TAC THEN REAL_ARITH_TAC; DISJ2_TAC THEN ASM_REWRITE_TAC[SIN_PI2; COS_PI2] THEN REWRITE_TAC[REAL_EXP_NEG; REAL_EXP_0; REAL_INV_1; REAL_SUB_REFL] THEN REWRITE_TAC[REAL_MUL_LZERO; REAL_LE_REFL; REAL_ENTIRE] THEN FIRST_X_ASSUM(DISJ_CASES_THEN SUBST1_TAC o MATCH_MP (REAL_ARITH `abs(x) = p ==> x = p \/ x = --p`)) THEN REWRITE_TAC[COS_PI2; COS_NEG] THEN REAL_ARITH_TAC]; ALL_TAC] THEN SIMP_TAC[COMPLEX_FIELD `ii * (a - b) / (Cx(&2) * ii) + (a + b) / Cx(&2) = a`] THEN SIMP_TAC[COMPLEX_FIELD `--(ii * w) = z <=> w = ii * z`] THEN MATCH_MP_TAC CLOG_CEXP THEN REWRITE_TAC[IM_MUL_II] THEN MP_TAC PI_POS THEN POP_ASSUM MP_TAC THEN REAL_ARITH_TAC);;
let CASN_UNIQUE = 
prove (`!w z. csin(z) = w /\ (abs(Re z) < pi / &2 \/ (abs(Re z) = pi / &2 /\ Im z = &0)) ==> casn w = z`,
MESON_TAC[CASN_CSIN]);;
let CASN_0 = 
prove (`casn(Cx(&0)) = Cx(&0)`,
let CASN_1 = 
prove (`casn(Cx(&1)) = Cx(pi / &2)`,
REWRITE_TAC[casn; GSYM CX_POW; GSYM CX_SUB] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[CSQRT_0; COMPLEX_MUL_RID; COMPLEX_ADD_RID] THEN REWRITE_TAC[CLOG_II] THEN CONV_TAC COMPLEX_RING);;
let CASN_NEG_1 = 
prove (`casn(--Cx(&1)) = --Cx(pi / &2)`,
REWRITE_TAC[casn; GSYM CX_NEG; GSYM CX_POW; GSYM CX_SUB] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[CSQRT_0; COMPLEX_MUL_RID; COMPLEX_ADD_RID] THEN REWRITE_TAC[CX_NEG; COMPLEX_MUL_RID; COMPLEX_MUL_RNEG] THEN REWRITE_TAC[CLOG_NEG_II] THEN CONV_TAC COMPLEX_RING);;
let HAS_COMPLEX_DERIVATIVE_CASN = 
prove (`!z. (Im z = &0 ==> abs(Re z) < &1) ==> (casn has_complex_derivative inv(ccos(casn z))) (at z)`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC HAS_COMPLEX_DERIVATIVE_INVERSE_BASIC THEN EXISTS_TAC `csin` THEN REWRITE_TAC[CSIN_CASN; HAS_COMPLEX_DERIVATIVE_CSIN; CONTINUOUS_AT_CSIN] THEN EXISTS_TAC `ball(z:complex,&1)` THEN REWRITE_TAC[OPEN_BALL; CENTRE_IN_BALL; REAL_LT_01] THEN CONJ_TAC THENL [DISCH_THEN(MP_TAC o MATCH_MP (COMPLEX_RING `ccos z = Cx(&0) ==> csin(z) pow 2 + ccos(z) pow 2 = Cx(&1) ==> csin(z) pow 2 = Cx(&1)`)) THEN REWRITE_TAC[CSIN_CASN; CSIN_CIRCLE] THEN REWRITE_TAC[COMPLEX_RING `z pow 2 = Cx(&1) <=> z = Cx(&1) \/ z = --Cx(&1)`] THEN DISCH_THEN(DISJ_CASES_THEN SUBST_ALL_TAC) THEN POP_ASSUM MP_TAC THEN REWRITE_TAC[RE_CX; IM_CX; RE_NEG; IM_NEG] THEN REAL_ARITH_TAC; ALL_TAC] THEN GEN_REWRITE_TAC LAND_CONV [GSYM ETA_AX] THEN REWRITE_TAC[casn] THEN MATCH_MP_TAC CONTINUOUS_COMPLEX_MUL THEN REWRITE_TAC[CONTINUOUS_CONST] THEN ONCE_REWRITE_TAC[GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_AT_COMPOSE THEN CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ADD THEN SIMP_TAC[CONTINUOUS_COMPLEX_MUL; CONTINUOUS_CONST; CONTINUOUS_AT_ID] THEN ONCE_REWRITE_TAC[GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_AT_COMPOSE THEN SIMP_TAC[CONTINUOUS_COMPLEX_POW; CONTINUOUS_SUB; CONTINUOUS_CONST; CONTINUOUS_AT_ID] THEN MATCH_MP_TAC CONTINUOUS_AT_CSQRT THEN REWRITE_TAC[RE_SUB; IM_SUB; RE_CX; IM_CX; RE_POW_2; IM_POW_2] THEN REWRITE_TAC[REAL_RING `&0 - &2 * x * y = &0 <=> x = &0 \/ y = &0`] THEN STRIP_TAC THEN ASM_REWRITE_TAC[REAL_POW_2; REAL_MUL_LZERO; REAL_SUB_RZERO; REAL_ARITH `&1 - (&0 - x) = &1 + x`] THEN ASM_SIMP_TAC[REAL_LE_SQUARE; REAL_ARITH `&0 <= x ==> &0 < &1 + x`] THEN REWRITE_TAC[REAL_ARITH `&0 < &1 - x * x <=> x pow 2 < &1 pow 2`] THEN ONCE_REWRITE_TAC[GSYM REAL_POW2_ABS] THEN MATCH_MP_TAC REAL_POW_LT2 THEN ASM_SIMP_TAC[REAL_ABS_POS; REAL_ABS_NUM; ARITH]; ALL_TAC] THEN MATCH_MP_TAC CONTINUOUS_AT_CLOG THEN REWRITE_TAC[IM_ADD; IM_MUL_II; RE_ADD; RE_MUL_II] THEN ASM_CASES_TAC `Im z = &0` THENL [DISCH_THEN(K ALL_TAC) THEN ASM_REWRITE_TAC[csqrt] THEN ASM_REWRITE_TAC[IM_SUB; RE_SUB; IM_CX; RE_CX; IM_POW_2; RE_POW_2; REAL_MUL_RZERO; REAL_SUB_REFL] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[REAL_ARITH `&0 <= &1 - (z pow 2 - &0) <=> z pow 2 <= &1 pow 2`; GSYM REAL_LE_SQUARE_ABS] THEN ASM_SIMP_TAC[REAL_LT_IMP_LE; REAL_ABS_NUM; RE; REAL_ADD_LID] THEN MATCH_MP_TAC SQRT_POS_LT THEN REWRITE_TAC[REAL_ARITH `&0 < &1 - (z pow 2 - &0) <=> z pow 2 < &1 pow 2`; GSYM REAL_LT_SQUARE_ABS] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[csqrt; IM_SUB; RE_SUB; IM_CX; RE_CX; IM_POW_2; RE_POW_2] THEN REWRITE_TAC[REAL_RING `&0 - &2 * x * y = &0 <=> x = &0 \/ y = &0`] THEN ASM_CASES_TAC `Re z = &0` THEN ASM_REWRITE_TAC[RE; IM] THENL [CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[REAL_ARITH `&1 - (&0 - x) = &1 + x`] THEN SIMP_TAC[REAL_POW_2; REAL_LE_ADD; REAL_LE_SQUARE; REAL_POS] THEN REWRITE_TAC[RE; IM; REAL_ADD_LID; REAL_ARITH `&0 < --x + y <=> x < y`] THEN MATCH_MP_TAC REAL_LT_RSQRT THEN REAL_ARITH_TAC; ALL_TAC] THEN DISCH_TAC THEN REWRITE_TAC[REAL_ARITH `&0 < --x + y <=> x < y`] THEN MATCH_MP_TAC REAL_LT_RSQRT THEN REWRITE_TAC[REAL_POW_2; REAL_ARITH `a < (n + &1 - (b - a)) / &2 <=> (a + b) - &1 < n`] THEN REWRITE_TAC[complex_norm] THEN MATCH_MP_TAC REAL_LT_RSQRT THEN REWRITE_TAC[RE_SUB; IM_SUB; RE_CX; IM_CX; RE_POW_2; IM_POW_2] THEN REPEAT(FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM REAL_LT_SQUARE])) THEN REAL_ARITH_TAC);;
let COMPLEX_DIFFERENTIABLE_AT_CASN = 
prove (`!z. (Im z = &0 ==> abs(Re z) < &1) ==> casn complex_differentiable at z`,
REWRITE_TAC[complex_differentiable] THEN MESON_TAC[HAS_COMPLEX_DERIVATIVE_CASN]);;
let COMPLEX_DIFFERENTIABLE_WITHIN_CASN = 
prove (`!s z. (Im z = &0 ==> abs(Re z) < &1) ==> casn complex_differentiable (at z within s)`,
add_complex_differentiation_theorems (CONJUNCTS(REWRITE_RULE[FORALL_AND_THM] (MATCH_MP HAS_COMPLEX_DERIVATIVE_CHAIN HAS_COMPLEX_DERIVATIVE_CASN)));;
let CONTINUOUS_AT_CASN = 
prove (`!z. (Im z = &0 ==> abs(Re z) < &1) ==> casn continuous at z`,
let CONTINUOUS_WITHIN_CASN = 
prove (`!s z. (Im z = &0 ==> abs(Re z) < &1) ==> casn continuous (at z within s)`,
let CONTINUOUS_ON_CASN = 
prove (`!s. (!z. z IN s /\ Im z = &0 ==> abs(Re z) < &1) ==> casn continuous_on s`,
let HOLOMORPHIC_ON_CASN = 
prove (`!s. (!z. z IN s /\ Im z = &0 ==> abs(Re z) < &1) ==> casn holomorphic_on s`,
(* ------------------------------------------------------------------------- *) (* Inverse cosine. *) (* ------------------------------------------------------------------------- *)
let cacs = new_definition
 `cacs z = --ii * clog(z + ii * csqrt(Cx(&1) - z pow 2))`;;
let CACS_BODY_LEMMA = 
prove (`!z. ~(z + ii * csqrt(Cx(&1) - z pow 2) = Cx(&0))`,
GEN_TAC THEN MP_TAC(SPEC `Cx(&1) - z pow 2` CSQRT) THEN CONV_TAC COMPLEX_FIELD);;
let CCOS_CACS = 
prove (`!z. ccos(cacs z) = z`,
GEN_TAC THEN REWRITE_TAC[ccos; cacs; COMPLEX_MUL_LNEG; COMPLEX_MUL_RNEG] THEN REWRITE_TAC[COMPLEX_MUL_ASSOC; COMPLEX_NEG_NEG] THEN REWRITE_TAC[COMPLEX_POW_II_2; GSYM COMPLEX_POW_2] THEN REWRITE_TAC[COMPLEX_NEG_NEG; COMPLEX_MUL_LNEG; COMPLEX_MUL_LID] THEN REWRITE_TAC[CEXP_NEG] THEN ASM_SIMP_TAC[CACS_BODY_LEMMA; CEXP_CLOG; COMPLEX_POW_II_2; COMPLEX_FIELD `~(z = Cx(&0)) ==> ((z + inv z) / Cx(&2) = c <=> z pow 2 + Cx(&1) = Cx(&2) * c * z)`] THEN MP_TAC(SPEC `Cx(&1) - z pow 2` CSQRT) THEN CONV_TAC COMPLEX_FIELD);;
let CACS_CCOS = 
prove (`!z. &0 < Re z /\ Re z < pi \/ Re(z) = &0 /\ &0 <= Im(z) \/ Re(z) = pi /\ Im(z) <= &0 ==> cacs(ccos z) = z`,
GEN_TAC THEN DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC) THEN REWRITE_TAC[ccos; cacs; COMPLEX_MUL_LNEG; CEXP_NEG] THEN SIMP_TAC[CEXP_NZ; COMPLEX_FIELD `~(z = Cx(&0)) ==> Cx(&1) - ((z + inv z) / Cx(&2)) pow 2 = --(((z - inv z) / Cx(&2)) pow 2)`] THEN SUBGOAL_THEN `csqrt(--(((cexp(ii * z) - inv(cexp(ii * z))) / Cx(&2)) pow 2)) = --ii * (cexp(ii * z) - inv(cexp(ii * z))) / Cx(&2)` SUBST1_TAC THENL [SIMP_TAC[COMPLEX_FIELD `--(x pow 2) = (--ii * x) pow 2`] THEN MATCH_MP_TAC POW_2_CSQRT THEN REWRITE_TAC[GSYM CEXP_NEG] THEN REWRITE_TAC[complex_div; GSYM CX_INV; RE_MUL_CX; IM_MUL_CX; RE_NEG; IM_NEG; COMPLEX_MUL_LNEG; RE_MUL_II; IM_MUL_II; RE_SUB; IM_SUB] THEN REWRITE_TAC[REAL_NEG_NEG; REAL_NEG_EQ_0] THEN REWRITE_TAC[REAL_ARITH `&0 < r * inv(&2) \/ r * inv(&2) = &0 /\ &0 <= --(i * inv(&2)) <=> &0 < r \/ r = &0 /\ &0 <= --i`] THEN REWRITE_TAC[RE_ADD; IM_ADD; RE_CEXP; IM_CEXP] THEN REWRITE_TAC[RE_MUL_II; RE_NEG; IM_MUL_II; IM_NEG] THEN REWRITE_TAC[SIN_NEG; COS_NEG; REAL_NEG_NEG] THEN REWRITE_TAC[REAL_MUL_RNEG; GSYM real_sub; REAL_SUB_RNEG; REAL_NEG_SUB] THEN REWRITE_TAC[GSYM REAL_ADD_RDISTRIB; GSYM REAL_SUB_RDISTRIB] THEN ASM_SIMP_TAC[REAL_LT_ADD; REAL_EXP_POS_LT; REAL_LT_MUL_EQ] THEN POP_ASSUM(REPEAT_TCL DISJ_CASES_THEN STRIP_ASSUME_TAC) THEN ASM_SIMP_TAC[SIN_POS_PI] THEN DISJ2_TAC THEN REWRITE_TAC[SIN_PI; REAL_MUL_RZERO; COS_PI; SIN_0; COS_0] THEN REWRITE_TAC[REAL_MUL_RID; REAL_MUL_RNEG] THEN REWRITE_TAC[REAL_NEG_SUB; REAL_SUB_LE; REAL_EXP_MONO_LE] THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN SIMP_TAC[COMPLEX_FIELD `(e + e') / Cx(&2) + ii * --ii * (e - e') / Cx(&2) = e`] THEN SIMP_TAC[COMPLEX_FIELD `--(ii * w) = z <=> w = ii * z`] THEN MATCH_MP_TAC CLOG_CEXP THEN REWRITE_TAC[IM_MUL_II] THEN MP_TAC PI_POS THEN ASM_REAL_ARITH_TAC);;
let CACS_UNIQUE = 
prove (`!w z. ccos z = w /\ (&0 < Re z /\ Re z < pi \/ Re(z) = &0 /\ &0 <= Im(z) \/ Re(z) = pi /\ Im(z) <= &0) ==> cacs(w) = z`,
MESON_TAC[CACS_CCOS]);;
let CACS_0 = 
prove (`cacs(Cx(&0)) = Cx(pi / &2)`,
MATCH_MP_TAC CACS_UNIQUE THEN REWRITE_TAC[RE_CX; IM_CX; GSYM CX_COS; COS_PI2] THEN MP_TAC PI_POS THEN REAL_ARITH_TAC);;
let CACS_1 = 
prove (`cacs(Cx(&1)) = Cx(&0)`,
MATCH_MP_TAC CACS_UNIQUE THEN REWRITE_TAC[RE_CX; IM_CX; GSYM CX_COS; COS_0; REAL_LE_REFL]);;
let CACS_NEG_1 = 
prove (`cacs(--Cx(&1)) = Cx pi`,
MATCH_MP_TAC CACS_UNIQUE THEN REWRITE_TAC[RE_CX; IM_CX; GSYM CX_COS; COS_PI; CX_NEG; REAL_LE_REFL]);;
let HAS_COMPLEX_DERIVATIVE_CACS = 
prove (`!z. (Im z = &0 ==> abs(Re z) < &1) ==> (cacs has_complex_derivative --inv(csin(cacs z))) (at z)`,
REPEAT STRIP_TAC THEN REWRITE_TAC[COMPLEX_NEG_INV] THEN MATCH_MP_TAC HAS_COMPLEX_DERIVATIVE_INVERSE_BASIC THEN EXISTS_TAC `ccos` THEN REWRITE_TAC[CCOS_CACS; HAS_COMPLEX_DERIVATIVE_CCOS; CONTINUOUS_AT_CCOS] THEN EXISTS_TAC `ball(z:complex,&1)` THEN REWRITE_TAC[OPEN_BALL; CENTRE_IN_BALL; REAL_LT_01] THEN CONJ_TAC THENL [DISCH_THEN(MP_TAC o MATCH_MP (COMPLEX_RING `--(csin z) = Cx(&0) ==> csin(z) pow 2 + ccos(z) pow 2 = Cx(&1) ==> ccos(z) pow 2 = Cx(&1)`)) THEN REWRITE_TAC[CCOS_CACS; CSIN_CIRCLE] THEN REWRITE_TAC[COMPLEX_RING `z pow 2 = Cx(&1) <=> z = Cx(&1) \/ z = --Cx(&1)`] THEN DISCH_THEN(DISJ_CASES_THEN SUBST_ALL_TAC) THEN POP_ASSUM MP_TAC THEN REWRITE_TAC[RE_CX; IM_CX; RE_NEG; IM_NEG] THEN REAL_ARITH_TAC; ALL_TAC] THEN GEN_REWRITE_TAC LAND_CONV [GSYM ETA_AX] THEN REWRITE_TAC[cacs] THEN MATCH_MP_TAC CONTINUOUS_COMPLEX_MUL THEN REWRITE_TAC[CONTINUOUS_CONST] THEN ONCE_REWRITE_TAC[GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_AT_COMPOSE THEN CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ADD THEN REWRITE_TAC[CONTINUOUS_AT_ID] THEN MATCH_MP_TAC CONTINUOUS_COMPLEX_MUL THEN REWRITE_TAC[CONTINUOUS_CONST] THEN ONCE_REWRITE_TAC[GSYM o_DEF] THEN MATCH_MP_TAC CONTINUOUS_AT_COMPOSE THEN SIMP_TAC[CONTINUOUS_COMPLEX_POW; CONTINUOUS_SUB; CONTINUOUS_CONST; CONTINUOUS_AT_ID] THEN MATCH_MP_TAC CONTINUOUS_AT_CSQRT THEN REWRITE_TAC[RE_SUB; IM_SUB; RE_CX; IM_CX; RE_POW_2; IM_POW_2] THEN REWRITE_TAC[REAL_RING `&0 - &2 * x * y = &0 <=> x = &0 \/ y = &0`] THEN STRIP_TAC THEN ASM_REWRITE_TAC[REAL_POW_2; REAL_MUL_LZERO; REAL_SUB_RZERO; REAL_ARITH `&1 - (&0 - x) = &1 + x`] THEN ASM_SIMP_TAC[REAL_LE_SQUARE; REAL_ARITH `&0 <= x ==> &0 < &1 + x`] THEN REWRITE_TAC[REAL_ARITH `&0 < &1 - x * x <=> x pow 2 < &1 pow 2`] THEN ONCE_REWRITE_TAC[GSYM REAL_POW2_ABS] THEN MATCH_MP_TAC REAL_POW_LT2 THEN ASM_SIMP_TAC[REAL_ABS_POS; REAL_ABS_NUM; ARITH]; ALL_TAC] THEN MATCH_MP_TAC CONTINUOUS_AT_CLOG THEN REWRITE_TAC[IM_ADD; IM_MUL_II; RE_ADD; RE_MUL_II] THEN ASM_CASES_TAC `Im z = &0` THENL [ASM_REWRITE_TAC[csqrt] THEN ASM_REWRITE_TAC[IM_SUB; RE_SUB; IM_CX; RE_CX; IM_POW_2; RE_POW_2; REAL_MUL_RZERO; REAL_SUB_REFL] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[REAL_ARITH `&0 <= &1 - (z pow 2 - &0) <=> z pow 2 <= &1 pow 2`; GSYM REAL_LE_SQUARE_ABS] THEN ASM_SIMP_TAC[REAL_LT_IMP_LE; REAL_ABS_NUM; RE; REAL_ADD_LID] THEN REWRITE_TAC[GSYM real_sub; IM; REAL_SUB_LT; REAL_SUB_RZERO] THEN MATCH_MP_TAC(REAL_ARITH `&0 < x ==> x = &0 ==> &0 < y`) THEN MATCH_MP_TAC SQRT_POS_LT THEN ASM_SIMP_TAC[REAL_SUB_LT; ABS_SQUARE_LT_1]; ALL_TAC] THEN REWRITE_TAC[csqrt; IM_SUB; RE_SUB; IM_CX; RE_CX; IM_POW_2; RE_POW_2] THEN REWRITE_TAC[REAL_RING `&0 - &2 * x * y = &0 <=> x = &0 \/ y = &0`] THEN ASM_CASES_TAC `Re z = &0` THEN ASM_REWRITE_TAC[RE; IM] THENL [CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[REAL_ARITH `&1 - (&0 - x) = &1 + x`] THEN SIMP_TAC[REAL_POW_2; REAL_LE_ADD; REAL_LE_SQUARE; REAL_POS] THEN REWRITE_TAC[RE; IM; REAL_ADD_LID] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN DISCH_THEN(MP_TAC o MATCH_MP (REAL_ARITH `a + b = &0 ==> a = --b`)) THEN DISCH_THEN(MP_TAC o AP_TERM `\x:real. x pow 2`) THEN SIMP_TAC[SQRT_POW_2; REAL_POW_NEG; ARITH; REAL_LE_SQUARE; REAL_LE_ADD; REAL_POS] THEN REAL_ARITH_TAC; ALL_TAC] THEN DISCH_THEN(MP_TAC o MATCH_MP (REAL_ARITH `a + b = &0 ==> a = --b`)) THEN DISCH_THEN(MP_TAC o AP_TERM `\x:real. x pow 2`) THEN SUBGOAL_THEN `&0 < (norm(Cx (&1) - z pow 2) + &1 - (Re z pow 2 - Im z pow 2)) / &2` ASSUME_TAC THENL [REWRITE_TAC[REAL_ARITH `&0 < (x + y - z) / &2 <=> z - y < x`] THEN REWRITE_TAC[complex_norm] THEN MATCH_MP_TAC REAL_LT_RSQRT THEN REWRITE_TAC[RE_SUB; IM_SUB; RE_CX; IM_CX; RE_POW_2; IM_POW_2] THEN REPEAT(FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM REAL_LT_SQUARE])) THEN REWRITE_TAC[IMP_IMP] THEN DISCH_THEN(MP_TAC o MATCH_MP REAL_LT_MUL) THEN REAL_ARITH_TAC; ALL_TAC] THEN ASM_SIMP_TAC[SQRT_POW_2; REAL_POW_NEG; ARITH; REAL_LT_IMP_LE] THEN REWRITE_TAC[REAL_POW_2; REAL_ARITH `a = (n + &1 - (b - a)) / &2 <=> (a + b) - &1 = n`] THEN REWRITE_TAC[complex_norm] THEN DISCH_THEN(MP_TAC o AP_TERM `\x:real. x pow 2`) THEN SIMP_TAC[SQRT_POW_2; REWRITE_RULE[GSYM REAL_POW_2] REAL_LE_SQUARE; REAL_LE_ADD] THEN REWRITE_TAC[RE_SUB; RE_CX; RE_POW_2; IM_SUB; IM_CX; IM_POW_2] THEN REPEAT(FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM REAL_LT_SQUARE])) THEN REAL_ARITH_TAC);;
let COMPLEX_DIFFERENTIABLE_AT_CACS = 
prove (`!z. (Im z = &0 ==> abs(Re z) < &1) ==> cacs complex_differentiable at z`,
REWRITE_TAC[complex_differentiable] THEN MESON_TAC[HAS_COMPLEX_DERIVATIVE_CACS]);;
let COMPLEX_DIFFERENTIABLE_WITHIN_CACS = 
prove (`!s z. (Im z = &0 ==> abs(Re z) < &1) ==> cacs complex_differentiable (at z within s)`,
add_complex_differentiation_theorems (CONJUNCTS(REWRITE_RULE[FORALL_AND_THM] (MATCH_MP HAS_COMPLEX_DERIVATIVE_CHAIN HAS_COMPLEX_DERIVATIVE_CACS)));;
let CONTINUOUS_AT_CACS = 
prove (`!z. (Im z = &0 ==> abs(Re z) < &1) ==> cacs continuous at z`,
let CONTINUOUS_WITHIN_CACS = 
prove (`!s z. (Im z = &0 ==> abs(Re z) < &1) ==> cacs continuous (at z within s)`,
let CONTINUOUS_ON_CACS = 
prove (`!s. (!z. z IN s /\ Im z = &0 ==> abs(Re z) < &1) ==> cacs continuous_on s`,
let HOLOMORPHIC_ON_CACS = 
prove (`!s. (!z. z IN s /\ Im z = &0 ==> abs(Re z) < &1) ==> cacs holomorphic_on s`,
(* ------------------------------------------------------------------------- *) (* Some crude range theorems (could be sharpened). *) (* ------------------------------------------------------------------------- *)
let CASN_RANGE_LEMMA = 
prove (`!z. abs (Re z) < &1 ==> &0 < Re(ii * z + csqrt(Cx(&1) - z pow 2))`,
REPEAT STRIP_TAC THEN REWRITE_TAC[RE_ADD; RE_MUL_II] THEN REWRITE_TAC[REAL_ARITH `&0 < --i + r <=> i < r`] THEN REWRITE_TAC[csqrt; IM_SUB; RE_SUB; COMPLEX_POW_2; RE_CX; IM_CX] THEN REWRITE_TAC[complex_mul; RE; IM] THEN REWRITE_TAC[GSYM complex_mul] THEN REWRITE_TAC[REAL_ARITH `r * i + i * r = &2 * r * i`] THEN REWRITE_TAC[REAL_SUB_LZERO; REAL_NEG_EQ_0; REAL_ABS_NEG] THEN REWRITE_TAC[REAL_NEG_SUB; REAL_ENTIRE; REAL_OF_NUM_EQ; ARITH] THEN MAP_EVERY ASM_CASES_TAC [`Re z = &0`; `Im z = &0`] THEN ASM_REWRITE_TAC[REAL_SUB_LZERO; REAL_SUB_RZERO] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[RE; SQRT_1] THEN CONV_TAC REAL_RAT_REDUCE_CONV THENL [REWRITE_TAC[REAL_ARITH `&1 - (&0 - z) = &1 + z`] THEN SIMP_TAC[REAL_LE_ADD; REAL_POS; REAL_LE_SQUARE; RE] THEN MATCH_MP_TAC REAL_LT_RSQRT THEN REAL_ARITH_TAC; SUBGOAL_THEN `Re(z) pow 2 < &1 pow 2` MP_TAC THENL [ONCE_REWRITE_TAC[GSYM REAL_POW2_ABS] THEN MATCH_MP_TAC REAL_POW_LT2 THEN ASM_REWRITE_TAC[REAL_ABS_POS; REAL_ABS_NUM; ARITH]; REWRITE_TAC[REAL_POW_ONE] THEN STRIP_TAC] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[RE] THEN TRY(MATCH_MP_TAC SQRT_POS_LT) THEN ASM_REAL_ARITH_TAC; MATCH_MP_TAC REAL_LT_RSQRT THEN REWRITE_TAC[REAL_POW_2; REAL_ARITH `a < (n + &1 - (b - a)) / &2 <=> (a + b) - &1 < n`] THEN REWRITE_TAC[complex_norm] THEN MATCH_MP_TAC REAL_LT_RSQRT THEN REWRITE_TAC[RE_SUB; IM_SUB; RE_CX; IM_CX] THEN REWRITE_TAC[complex_mul; RE; IM] THEN REPEAT(FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM REAL_LT_SQUARE])) THEN REAL_ARITH_TAC]);;
let CACS_RANGE_LEMMA = 
prove (`!z. abs(Re z) < &1 ==> &0 < Im(z + ii * csqrt(Cx(&1) - z pow 2))`,
REPEAT STRIP_TAC THEN MP_TAC(SPEC `--z:complex` CASN_RANGE_LEMMA) THEN ASM_SIMP_TAC[IM_NEG; RE_NEG; IM_ADD; RE_ADD; IM_MUL_II; RE_MUL_II; COMPLEX_POW_NEG; ARITH; REAL_ABS_NEG] THEN REAL_ARITH_TAC);;
let RE_CASN = 
prove (`!z. Re(casn z) = Im(clog(ii * z + csqrt(Cx(&1) - z pow 2)))`,
let RE_CACS = 
prove (`!z. Re(cacs z) = Im(clog(z + ii * csqrt(Cx(&1) - z pow 2)))`,
let CASN_BOUNDS = 
prove (`!z. abs(Re z) < &1 ==> abs(Re(casn z)) < pi / &2`,
GEN_TAC THEN DISCH_TAC THEN REWRITE_TAC[RE_CASN] THEN MATCH_MP_TAC RE_CLOG_POS_LT_IMP THEN ASM_SIMP_TAC[CASN_RANGE_LEMMA]);;
let CACS_BOUNDS = 
prove (`!z. abs(Re z) < &1 ==> &0 < Re(cacs z) /\ Re(cacs z) < pi`,
GEN_TAC THEN DISCH_TAC THEN REWRITE_TAC[RE_CACS] THEN MATCH_MP_TAC IM_CLOG_POS_LT_IMP THEN ASM_SIMP_TAC[CACS_RANGE_LEMMA]);;
let RE_CACS_BOUNDS = 
prove (`!z. --pi < Re(cacs z) /\ Re(cacs z) <= pi`,
REWRITE_TAC[RE_CACS] THEN SIMP_TAC[CLOG_WORKS; CACS_BODY_LEMMA]);;
let RE_CACS_BOUND = 
prove (`!z. abs(Re(cacs z)) <= pi`,
MP_TAC RE_CACS_BOUNDS THEN MATCH_MP_TAC MONO_FORALL THEN REAL_ARITH_TAC);;
let RE_CASN_BOUNDS = 
prove (`!z. --pi < Re(casn z) /\ Re(casn z) <= pi`,
REWRITE_TAC[RE_CASN] THEN SIMP_TAC[CLOG_WORKS; CASN_BODY_LEMMA]);;
let RE_CASN_BOUND = 
prove (`!z. abs(Re(casn z)) <= pi`,
MP_TAC RE_CASN_BOUNDS THEN MATCH_MP_TAC MONO_FORALL THEN REAL_ARITH_TAC);;
(* ------------------------------------------------------------------------- *) (* Interrelations between the two functions. *) (* ------------------------------------------------------------------------- *)
let CCOS_CASN_NZ = 
prove (`!z. ~(z pow 2 = Cx(&1)) ==> ~(ccos(casn z) = Cx(&0))`,
REWRITE_TAC[ccos; casn; CEXP_NEG; COMPLEX_RING `ii * --ii * z = z`; COMPLEX_RING `--ii * --ii * z = --z`] THEN SIMP_TAC[CEXP_CLOG; CASN_BODY_LEMMA; COMPLEX_FIELD `~(x = Cx(&0)) ==> ((x + inv(x)) / Cx(&2) = Cx(&0) <=> x pow 2 = --Cx(&1))`] THEN SIMP_TAC[CSQRT; COMPLEX_FIELD `s pow 2 = Cx(&1) - z pow 2 ==> ((ii * z + s) pow 2 = --Cx(&1) <=> ii * s * z = Cx(&1) - z pow 2)`] THEN GEN_TAC THEN STRIP_TAC THEN MATCH_MP_TAC(COMPLEX_RING `~(x pow 2 + y pow 2 = Cx(&0)) ==> ~(ii * x = y)`) THEN REPEAT(POP_ASSUM MP_TAC) THEN MP_TAC(SPEC `Cx(&1) - z pow 2` CSQRT) THEN CONV_TAC COMPLEX_RING);;
let CSIN_CACS_NZ = 
prove (`!z. ~(z pow 2 = Cx(&1)) ==> ~(csin(cacs z) = Cx(&0))`,
REWRITE_TAC[csin; cacs; CEXP_NEG; COMPLEX_RING `ii * --ii * z = z`; COMPLEX_RING `--ii * --ii * z = --z`] THEN SIMP_TAC[CEXP_CLOG; CACS_BODY_LEMMA; COMPLEX_FIELD `~(x = Cx(&0)) ==> ((x - inv(x)) / (Cx(&2) * ii) = Cx(&0) <=> x pow 2 = Cx(&1))`] THEN SIMP_TAC[CSQRT; COMPLEX_FIELD `s pow 2 = Cx(&1) - z pow 2 ==> ((z + ii * s) pow 2 = Cx(&1) <=> ii * s * z = Cx(&1) - z pow 2)`] THEN GEN_TAC THEN STRIP_TAC THEN MATCH_MP_TAC(COMPLEX_RING `~(x pow 2 + y pow 2 = Cx(&0)) ==> ~(ii * x = y)`) THEN REPEAT(POP_ASSUM MP_TAC) THEN MP_TAC(SPEC `Cx(&1) - z pow 2` CSQRT) THEN CONV_TAC COMPLEX_RING);;
let CCOS_CSIN_CSQRT = 
prove (`!z. &0 < cos(Re z) \/ cos(Re z) = &0 /\ Im(z) * sin(Re z) <= &0 ==> ccos(z) = csqrt(Cx(&1) - csin(z) pow 2)`,
REPEAT STRIP_TAC THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC CSQRT_UNIQUE THEN REWRITE_TAC[COMPLEX_EQ_SUB_LADD] THEN ONCE_REWRITE_TAC[COMPLEX_ADD_SYM] THEN REWRITE_TAC[CSIN_CIRCLE] THEN REWRITE_TAC[RE_CCOS; IM_CCOS] THEN ASM_SIMP_TAC[REAL_LT_MUL_EQ; REAL_HALF; REAL_LT_ADD; REAL_EXP_POS_LT] THEN DISJ2_TAC THEN REWRITE_TAC[REAL_MUL_RZERO] THEN FIRST_ASSUM(MP_TAC o MATCH_MP(REAL_ARITH `x * y <= &0 ==> &0 <= --x * y`)) THEN REWRITE_TAC[REAL_MUL_POS_LE] THEN SIMP_TAC[REAL_ARITH `x / &2 = &0 <=> x = &0`; REAL_LT_RDIV_EQ; REAL_ADD_LID; REAL_SUB_LT; REAL_LT_LDIV_EQ; REAL_OF_NUM_LT; ARITH; REAL_MUL_LZERO; REAL_SUB_0; REAL_EXP_MONO_LT; REAL_LT_SUB_RADD; REAL_EXP_INJ] THEN REAL_ARITH_TAC);;
let CSIN_CCOS_CSQRT = 
prove (`!z. &0 < sin(Re z) \/ sin(Re z) = &0 /\ &0 <= Im(z) * cos(Re z) ==> csin(z) = csqrt(Cx(&1) - ccos(z) pow 2)`,
REPEAT STRIP_TAC THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC CSQRT_UNIQUE THEN REWRITE_TAC[COMPLEX_EQ_SUB_LADD] THEN ONCE_REWRITE_TAC[COMPLEX_ADD_SYM] THEN REWRITE_TAC[ONCE_REWRITE_RULE[COMPLEX_ADD_SYM] CSIN_CIRCLE] THEN REWRITE_TAC[RE_CSIN; IM_CSIN] THEN ASM_SIMP_TAC[REAL_LT_MUL_EQ; REAL_HALF; REAL_LT_ADD; REAL_EXP_POS_LT] THEN DISJ2_TAC THEN REWRITE_TAC[REAL_MUL_RZERO] THEN REPEAT(POP_ASSUM MP_TAC) THEN REWRITE_TAC[REAL_MUL_POS_LE] THEN SIMP_TAC[REAL_ARITH `x / &2 = &0 <=> x = &0`; REAL_LT_RDIV_EQ; REAL_ADD_LID; REAL_SUB_LT; REAL_LT_LDIV_EQ; REAL_OF_NUM_LT; ARITH; REAL_MUL_LZERO; REAL_SUB_0; REAL_EXP_MONO_LT; REAL_LT_SUB_RADD; REAL_EXP_INJ] THEN REAL_ARITH_TAC);;
let CASN_CACS_SQRT_POS = 
prove (`!z. (&0 < Re z \/ Re z = &0 /\ &0 <= Im z) ==> casn(z) = cacs(csqrt(Cx(&1) - z pow 2))`,
GEN_TAC THEN DISCH_TAC THEN REWRITE_TAC[casn; cacs] THEN AP_TERM_TAC THEN AP_TERM_TAC THEN MATCH_MP_TAC(COMPLEX_RING `w = z ==> ii * z + s = s + ii * w`) THEN MATCH_MP_TAC CSQRT_UNIQUE THEN ASM_REWRITE_TAC[CSQRT] THEN CONV_TAC COMPLEX_RING);;
let CACS_CASN_SQRT_POS = 
prove (`!z. (&0 < Re z \/ Re z = &0 /\ &0 <= Im z) ==> cacs(z) = casn(csqrt(Cx(&1) - z pow 2))`,
GEN_TAC THEN DISCH_TAC THEN REWRITE_TAC[casn; cacs] THEN AP_TERM_TAC THEN AP_TERM_TAC THEN MATCH_MP_TAC(COMPLEX_RING `w = z ==> z + ii * s = ii * s + w`) THEN MATCH_MP_TAC CSQRT_UNIQUE THEN ASM_REWRITE_TAC[CSQRT] THEN CONV_TAC COMPLEX_RING);;
let CSIN_CACS = 
prove (`!z. &0 < Re z \/ Re(z) = &0 /\ &0 <= Im z ==> csin(cacs z) = csqrt(Cx(&1) - z pow 2)`,
GEN_TAC THEN DISCH_TAC THEN GEN_REWRITE_TAC RAND_CONV [GSYM CSIN_CASN] THEN AP_TERM_TAC THEN MATCH_MP_TAC CACS_CASN_SQRT_POS THEN ASM_REWRITE_TAC[]);;
let CCOS_CASN = 
prove (`!z. &0 < Re z \/ Re(z) = &0 /\ &0 <= Im z ==> ccos(casn z) = csqrt(Cx(&1) - z pow 2)`,
GEN_TAC THEN DISCH_TAC THEN GEN_REWRITE_TAC RAND_CONV [GSYM CCOS_CACS] THEN AP_TERM_TAC THEN MATCH_MP_TAC CASN_CACS_SQRT_POS THEN ASM_REWRITE_TAC[]);;
(* ------------------------------------------------------------------------- *) (* Real arcsin. *) (* ------------------------------------------------------------------------- *)
let asn = new_definition `asn(x) = Re(casn(Cx x))`;;
let REAL_ASN = 
prove (`!z. real z /\ abs(Re z) <= &1 ==> real(casn z)`,
GEN_TAC THEN REWRITE_TAC[IMP_CONJ] THEN GEN_REWRITE_TAC LAND_CONV [REAL] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN SPEC_TAC(`Re z`,`x:real`) THEN REWRITE_TAC[real; casn; COMPLEX_MUL_LNEG; IM_NEG; IM_MUL_II] THEN GEN_TAC THEN REWRITE_TAC[RE_CX; REAL_NEG_EQ_0] THEN DISCH_TAC THEN MATCH_MP_TAC NORM_CEXP_IMAGINARY THEN SIMP_TAC[CEXP_CLOG; CASN_BODY_LEMMA; NORM_EQ_SQUARE] THEN REWRITE_TAC[DOT_SQUARE_NORM; COMPLEX_SQNORM] THEN REWRITE_TAC[RE_ADD; IM_ADD; RE_MUL_II; IM_MUL_II; RE_CX; IM_CX] THEN ASM_SIMP_TAC[GSYM CX_POW; GSYM CX_SUB; GSYM CX_SQRT; REAL_SUB_LE; ABS_SQUARE_LE_1; RE_CX; IM_CX; REAL_NEG_0; REAL_ADD_LID; SQRT_POW_2] THEN REAL_ARITH_TAC);;
let CX_ASN = 
prove (`!x. abs(x) <= &1 ==> Cx(asn x) = casn(Cx x)`,
REWRITE_TAC[asn] THEN MESON_TAC[REAL; RE_CX; REAL_CX; REAL_ASN]);;
let SIN_ASN = 
prove (`!y. --(&1) <= y /\ y <= &1 ==> sin(asn(y)) = y`,
REWRITE_TAC[REAL_ARITH `--(&1) <= y /\ y <= &1 <=> abs(y) <= &1`] THEN ONCE_REWRITE_TAC[GSYM CX_INJ] THEN SIMP_TAC[CX_ASN; CX_SIN; CSIN_CASN]);;
let ASN_SIN = 
prove (`!x. --(pi / &2) <= x /\ x <= pi / &2 ==> asn(sin(x)) = x`,
ONCE_REWRITE_TAC[GSYM CX_INJ] THEN SIMP_TAC[CX_ASN; SIN_BOUND; CX_SIN] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC CASN_CSIN THEN REWRITE_TAC[IM_CX; RE_CX] THEN REPEAT(POP_ASSUM MP_TAC) THEN REAL_ARITH_TAC);;
let ASN_BOUNDS_LT = 
prove (`!y. --(&1) < y /\ y < &1 ==> --(pi / &2) < asn(y) /\ asn(y) < pi / &2`,
GEN_TAC THEN REWRITE_TAC[asn] THEN MP_TAC(SPEC `Cx y` CASN_BOUNDS) THEN REWRITE_TAC[RE_CX] THEN REAL_ARITH_TAC);;
let ASN_0 = 
prove (`asn(&0) = &0`,
REWRITE_TAC[asn; CASN_0; RE_CX]);;
let ASN_1 = 
prove (`asn(&1) = pi / &2`,
REWRITE_TAC[asn; CASN_1; RE_CX]);;
let ASN_NEG_1 = 
prove (`asn(-- &1) = --(pi / &2)`,
REWRITE_TAC[asn; CX_NEG; CASN_NEG_1; RE_CX; RE_NEG]);;
let ASN_BOUNDS = 
prove (`!y. --(&1) <= y /\ y <= &1 ==> --(pi / &2) <= asn(y) /\ asn(y) <= pi / &2`,
REWRITE_TAC[REAL_LE_LT] THEN REPEAT STRIP_TAC THEN MAP_EVERY MP_TAC [ASN_1; ASN_NEG_1; SPEC `y:real` ASN_BOUNDS_LT] THEN ASM_REWRITE_TAC[] THEN REPEAT(POP_ASSUM MP_TAC) THEN MP_TAC PI_POS THEN REAL_ARITH_TAC);;
let ASN_NEG = 
prove (`!x. -- &1 <= x /\ x <= &1 ==> asn(--x) = --asn(x)`,
GEN_TAC THEN DISCH_TAC THEN FIRST_ASSUM(fun th -> GEN_REWRITE_TAC (LAND_CONV o RAND_CONV o RAND_CONV) [GSYM(MATCH_MP SIN_ASN th)]) THEN REWRITE_TAC[GSYM SIN_NEG] THEN MATCH_MP_TAC ASN_SIN THEN REWRITE_TAC[REAL_ARITH `--a <= --x /\ --x <= a <=> --a <= x /\ x <= a`] THEN ASM_SIMP_TAC[ASN_BOUNDS]);;
let COS_ASN_NZ = 
prove (`!x. --(&1) < x /\ x < &1 ==> ~(cos(asn(x)) = &0)`,
ONCE_REWRITE_TAC[GSYM CX_INJ] THEN SIMP_TAC[CX_ASN; CX_COS; REAL_ARITH `--(&1) < x /\ x < &1 ==> abs(x) <= &1`] THEN GEN_TAC THEN STRIP_TAC THEN MATCH_MP_TAC CCOS_CASN_NZ THEN SIMP_TAC[COMPLEX_RING `x pow 2 = Cx(&1) <=> x = Cx(&1) \/ x = --Cx(&1)`] THEN REWRITE_TAC[GSYM CX_NEG; CX_INJ] THEN ASM_REAL_ARITH_TAC);;
let ASN_MONO_LT_EQ = 
prove (`!x y. abs(x) <= &1 /\ abs(y) <= &1 ==> (asn(x) < asn(y) <=> x < y)`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `sin(asn(x)) < sin(asn(y))` THEN CONJ_TAC THENL [CONV_TAC SYM_CONV THEN MATCH_MP_TAC SIN_MONO_LT_EQ THEN ONCE_REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THEN MATCH_MP_TAC ASN_BOUNDS; BINOP_TAC THEN MATCH_MP_TAC SIN_ASN] THEN ASM_REAL_ARITH_TAC);;
let ASN_MONO_LE_EQ = 
prove (`!x y. abs(x) <= &1 /\ abs(y) <= &1 ==> (asn(x) <= asn(y) <=> x <= y)`,
REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM REAL_NOT_LT] THEN ASM_SIMP_TAC[ASN_MONO_LT_EQ]);;
let ASN_MONO_LT = 
prove (`!x y. --(&1) <= x /\ x < y /\ y <= &1 ==> asn(x) < asn(y)`,
MP_TAC ASN_MONO_LT_EQ THEN REPEAT(MATCH_MP_TAC MONO_FORALL THEN GEN_TAC) THEN REAL_ARITH_TAC);;
let ASN_MONO_LE = 
prove (`!x y. --(&1) <= x /\ x <= y /\ y <= &1 ==> asn(x) <= asn(y)`,
MP_TAC ASN_MONO_LE_EQ THEN REPEAT(MATCH_MP_TAC MONO_FORALL THEN GEN_TAC) THEN REAL_ARITH_TAC);;
let COS_ASN = 
prove (`!x. --(&1) <= x /\ x <= &1 ==> cos(asn x) = sqrt(&1 - x pow 2)`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC(GSYM SQRT_UNIQUE) THEN ASM_SIMP_TAC[ASN_BOUNDS; COS_POS_PI_LE; REAL_EQ_SUB_RADD] THEN ASM_MESON_TAC[SIN_ASN; SIN_CIRCLE; REAL_ADD_SYM]);;
(* ------------------------------------------------------------------------- *) (* Real arccosine. *) (* ------------------------------------------------------------------------- *)
let acs = new_definition `acs(x) = Re(cacs(Cx x))`;;
let REAL_ACS = 
prove (`!z. real z /\ abs(Re z) <= &1 ==> real(cacs z)`,
GEN_TAC THEN REWRITE_TAC[IMP_CONJ] THEN GEN_REWRITE_TAC LAND_CONV [REAL] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN SPEC_TAC(`Re z`,`x:real`) THEN REWRITE_TAC[real; cacs; COMPLEX_MUL_LNEG; IM_NEG; IM_MUL_II] THEN GEN_TAC THEN REWRITE_TAC[RE_CX; REAL_NEG_EQ_0] THEN DISCH_TAC THEN MATCH_MP_TAC NORM_CEXP_IMAGINARY THEN SIMP_TAC[CEXP_CLOG; CACS_BODY_LEMMA; NORM_EQ_SQUARE] THEN REWRITE_TAC[DOT_SQUARE_NORM; COMPLEX_SQNORM] THEN REWRITE_TAC[RE_ADD; IM_ADD; RE_MUL_II; IM_MUL_II; RE_CX; IM_CX] THEN ASM_SIMP_TAC[GSYM CX_POW; GSYM CX_SUB; GSYM CX_SQRT; REAL_SUB_LE; ABS_SQUARE_LE_1; RE_CX; IM_CX; REAL_NEG_0; REAL_ADD_LID; SQRT_POW_2] THEN REAL_ARITH_TAC);;
let CX_ACS = 
prove (`!x. abs(x) <= &1 ==> Cx(acs x) = cacs(Cx x)`,
REWRITE_TAC[acs] THEN MESON_TAC[REAL; RE_CX; REAL_CX; REAL_ACS]);;
let COS_ACS = 
prove (`!y. --(&1) <= y /\ y <= &1 ==> cos(acs(y)) = y`,
REWRITE_TAC[REAL_ARITH `--(&1) <= y /\ y <= &1 <=> abs(y) <= &1`] THEN ONCE_REWRITE_TAC[GSYM CX_INJ] THEN SIMP_TAC[CX_ACS; CX_COS; CCOS_CACS]);;
let ACS_COS = 
prove (`!x. &0 <= x /\ x <= pi ==> acs(cos(x)) = x`,
ONCE_REWRITE_TAC[GSYM CX_INJ] THEN SIMP_TAC[CX_ACS; COS_BOUND; CX_COS] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC CACS_CCOS THEN REWRITE_TAC[IM_CX; RE_CX] THEN ASM_REAL_ARITH_TAC);;
let ACS_BOUNDS_LT = 
prove (`!y. --(&1) < y /\ y < &1 ==> &0 < acs(y) /\ acs(y) < pi`,
GEN_TAC THEN REWRITE_TAC[acs] THEN MP_TAC(SPEC `Cx y` CACS_BOUNDS) THEN REWRITE_TAC[RE_CX] THEN REAL_ARITH_TAC);;
let ACS_0 = 
prove (`acs(&0) = pi / &2`,
REWRITE_TAC[acs; CACS_0; RE_CX]);;
let ACS_1 = 
prove (`acs(&1) = &0`,
REWRITE_TAC[acs; CACS_1; RE_CX]);;
let ACS_NEG_1 = 
prove (`acs(-- &1) = pi`,
REWRITE_TAC[acs; CX_NEG; CACS_NEG_1; RE_CX; RE_NEG]);;
let ACS_BOUNDS = 
prove (`!y. --(&1) <= y /\ y <= &1 ==> &0 <= acs(y) /\ acs(y) <= pi`,
REWRITE_TAC[REAL_LE_LT] THEN REPEAT STRIP_TAC THEN MAP_EVERY MP_TAC [ACS_1; ACS_NEG_1; SPEC `y:real` ACS_BOUNDS_LT] THEN ASM_REWRITE_TAC[] THEN REPEAT(POP_ASSUM MP_TAC) THEN MP_TAC PI_POS THEN REAL_ARITH_TAC);;
let ACS_NEG = 
prove (`!x. -- &1 <= x /\ x <= &1 ==> acs(--x) = pi - acs(x)`,
GEN_TAC THEN DISCH_TAC THEN FIRST_ASSUM(fun th -> GEN_REWRITE_TAC (LAND_CONV o RAND_CONV o RAND_CONV) [GSYM(MATCH_MP COS_ACS th)]) THEN ONCE_REWRITE_TAC[GSYM COS_NEG] THEN REWRITE_TAC[GSYM COS_PERIODIC_PI] THEN REWRITE_TAC[REAL_ARITH `--x + y:real = y - x`] THEN MATCH_MP_TAC ACS_COS THEN SIMP_TAC[REAL_ARITH `&0 <= p - x /\ p - x <= p <=> &0 <= x /\ x <= p`] THEN ASM_SIMP_TAC[ACS_BOUNDS]);;
let SIN_ACS_NZ = 
prove (`!x. --(&1) < x /\ x < &1 ==> ~(sin(acs(x)) = &0)`,
ONCE_REWRITE_TAC[GSYM CX_INJ] THEN SIMP_TAC[CX_ACS; CX_SIN; REAL_ARITH `--(&1) < x /\ x < &1 ==> abs(x) <= &1`] THEN GEN_TAC THEN STRIP_TAC THEN MATCH_MP_TAC CSIN_CACS_NZ THEN SIMP_TAC[COMPLEX_RING `x pow 2 = Cx(&1) <=> x = Cx(&1) \/ x = --Cx(&1)`] THEN REWRITE_TAC[GSYM CX_NEG; CX_INJ] THEN ASM_REAL_ARITH_TAC);;
let ACS_MONO_LT_EQ = 
prove (`!x y. abs(x) <= &1 /\ abs(y) <= &1 ==> (acs(x) < acs(y) <=> y < x)`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `cos(acs(y)) < cos(acs(x))` THEN CONJ_TAC THENL [CONV_TAC SYM_CONV THEN MATCH_MP_TAC COS_MONO_LT_EQ THEN ONCE_REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THEN MATCH_MP_TAC ACS_BOUNDS; BINOP_TAC THEN MATCH_MP_TAC COS_ACS] THEN ASM_REAL_ARITH_TAC);;
let ACS_MONO_LE_EQ = 
prove (`!x y. abs(x) <= &1 /\ abs(y) <= &1 ==> (acs(x) <= acs(y) <=> y <= x)`,
REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM REAL_NOT_LT] THEN ASM_SIMP_TAC[ACS_MONO_LT_EQ]);;
let ACS_MONO_LT = 
prove (`!x y. --(&1) <= x /\ x < y /\ y <= &1 ==> acs(y) < acs(x)`,
REPEAT GEN_TAC THEN MP_TAC(SPECL [`y:real`; `x:real`] ACS_MONO_LT_EQ) THEN REAL_ARITH_TAC);;
let ACS_MONO_LE = 
prove (`!x y. --(&1) <= x /\ x <= y /\ y <= &1 ==> acs(y) <= acs(x)`,
REPEAT GEN_TAC THEN MP_TAC(SPECL [`y:real`; `x:real`] ACS_MONO_LE_EQ) THEN REAL_ARITH_TAC);;
let SIN_ACS = 
prove (`!x. --(&1) <= x /\ x <= &1 ==> sin(acs x) = sqrt(&1 - x pow 2)`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC(GSYM SQRT_UNIQUE) THEN ASM_SIMP_TAC[ACS_BOUNDS; SIN_POS_PI_LE; REAL_EQ_SUB_RADD] THEN ASM_MESON_TAC[COS_ACS; SIN_CIRCLE]);;
let ACS_INJ = 
prove (`!x y. abs(x) <= &1 /\ abs(y) <= &1 ==> (acs x = acs y <=> x = y)`,
REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM REAL_LE_ANTISYM] THEN ASM_SIMP_TAC[ACS_MONO_LE_EQ] THEN REAL_ARITH_TAC);;
(* ------------------------------------------------------------------------- *) (* Some interrelationships among the real inverse trig functions. *) (* ------------------------------------------------------------------------- *)
let ACS_ATN = 
prove (`!x. -- &1 < x /\ x < &1 ==> acs(x) = pi / &2 - atn(x / sqrt(&1 - x pow 2))`,
REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[REAL_ARITH `x:real = p - y <=> y - (p - x) = &0`] THEN MATCH_MP_TAC SIN_EQ_0_PI THEN ASM_SIMP_TAC[ATN_BOUND; ACS_BOUNDS; REAL_LT_IMP_LE; REAL_ARITH `abs(x) < pi / &2 /\ &0 <= y /\ y <= pi ==> --pi < x - (pi / &2 - y) /\ x - (pi / &2 - y) < pi`] THEN SUBGOAL_THEN `tan(atn(x / sqrt(&1 - x pow 2))) = tan(pi / &2 - acs x)` MP_TAC THENL [REWRITE_TAC[TAN_COT; ATN_TAN] THEN REWRITE_TAC[tan] THEN ASM_SIMP_TAC[SIN_ACS; COS_ACS; REAL_LT_IMP_LE; REAL_INV_DIV]; ALL_TAC] THEN GEN_REWRITE_TAC LAND_CONV [GSYM REAL_SUB_0] THEN ASM_SIMP_TAC[SIN_ACS_NZ; GSYM SIN_COS; COS_ATN_NZ; REAL_SUB_TAN; REAL_FIELD `~(y = &0) /\ ~(z = &0) ==> (x / (y * z) = &0 <=> x = &0)`]);;
let ASN_PLUS_ACS = 
prove (`!x. -- &1 <= x /\ x <= &1 ==> asn(x) + acs(x) = pi / &2`,
REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[REAL_ARITH `x + y:real = p <=> x = p - y`] THEN MATCH_MP_TAC SIN_INJ_PI THEN ASM_SIMP_TAC[SIN_PI2; COS_PI2; SIN_SUB; REAL_MUL_LZERO; REAL_SUB_RZERO] THEN ASM_SIMP_TAC[SIN_ASN; COS_ACS; REAL_MUL_LID] THEN REWRITE_TAC[REAL_ARITH `--p <= p - x <=> x <= &2 * p`; REAL_ARITH `p - x <= p <=> &0 <= x`] THEN ASM_SIMP_TAC[ASN_BOUNDS; ACS_BOUNDS; REAL_ARITH `&2 * x / &2 = x`]);;
let ASN_ACS = 
prove (`!x. -- &1 <= x /\ x <= &1 ==> asn(x) = pi / &2 - acs(x)`,
let ACS_ASN = 
prove (`!x. -- &1 <= x /\ x <= &1 ==> acs(x) = pi / &2 - asn(x)`,
SIMP_TAC[ASN_ACS] THEN REAL_ARITH_TAC);;
let ASN_ATN = 
prove (`!x. -- &1 < x /\ x < &1 ==> asn(x) = atn(x / sqrt(&1 - x pow 2))`,
SIMP_TAC[ASN_ACS; REAL_LT_IMP_LE; ACS_ATN] THEN REAL_ARITH_TAC);;
let ASN_ACS_SQRT_POS = 
prove (`!x. &0 <= x /\ x <= &1 ==> asn(x) = acs(sqrt(&1 - x pow 2))`,
REPEAT STRIP_TAC THEN REWRITE_TAC[asn; acs] THEN ASM_SIMP_TAC[CX_SQRT; REAL_SUB_LE; REAL_POW_1_LE; CX_SUB; CX_POW] THEN AP_TERM_TAC THEN MATCH_MP_TAC CASN_CACS_SQRT_POS THEN ASM_REWRITE_TAC[RE_CX; IM_CX] THEN ASM_REAL_ARITH_TAC);;
let ASN_ACS_SQRT_NEG = 
prove (`!x. -- &1 <= x /\ x <= &0 ==> asn(x) = --acs(sqrt(&1 - x pow 2))`,
REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[REAL_ARITH `x = --y <=> (--x:real) = y`] THEN ASM_SIMP_TAC[GSYM ASN_NEG; REAL_ARITH `x <= &0 ==> x <= &1`] THEN ONCE_REWRITE_TAC[REAL_ARITH `(x:real) pow 2 = (--x) pow 2`] THEN MATCH_MP_TAC ASN_ACS_SQRT_POS THEN ASM_REAL_ARITH_TAC);;
let ACS_ASN_SQRT_POS = 
prove (`!x. &0 <= x /\ x <= &1 ==> acs(x) = asn(sqrt(&1 - x pow 2))`,
REPEAT STRIP_TAC THEN REWRITE_TAC[asn; acs] THEN ASM_SIMP_TAC[CX_SQRT; REAL_SUB_LE; REAL_POW_1_LE; CX_SUB; CX_POW] THEN AP_TERM_TAC THEN MATCH_MP_TAC CACS_CASN_SQRT_POS THEN ASM_REWRITE_TAC[RE_CX; IM_CX] THEN ASM_REAL_ARITH_TAC);;
let ACS_ASN_SQRT_NEG = 
prove (`!x. -- &1 <= x /\ x <= &0 ==> acs(x) = pi - asn(sqrt(&1 - x pow 2))`,
REPEAT STRIP_TAC THEN MP_TAC(SPEC `--x:real` ACS_ASN_SQRT_POS) THEN ANTS_TAC THENL [ASM_REAL_ARITH_TAC; SIMP_TAC[REAL_POW_NEG; ARITH]] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [GSYM REAL_NEG_NEG] THEN MATCH_MP_TAC ACS_NEG THEN ASM_REAL_ARITH_TAC);;
(* ------------------------------------------------------------------------- *) (* More delicate continuity results for arcsin and arccos. *) (* ------------------------------------------------------------------------- *)
let CONTINUOUS_ON_CASN_REAL = 
prove (`casn continuous_on {w | real w /\ abs(Re w) <= &1}`,
MATCH_MP_TAC CONTINUOUS_ON_SUBSET THEN EXISTS_TAC `IMAGE csin {z | real z /\ abs(Re z) <= pi / &2}` THEN CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_INVERSE THEN REWRITE_TAC[CONTINUOUS_ON_CSIN] THEN CONJ_TAC THENL [REWRITE_TAC[COMPACT_EQ_BOUNDED_CLOSED] THEN CONJ_TAC THENL [MATCH_MP_TAC BOUNDED_SUBSET THEN EXISTS_TAC `cball(Cx(&0),pi / &2)` THEN REWRITE_TAC[BOUNDED_CBALL; SUBSET; IN_ELIM_THM; IN_CBALL] THEN REWRITE_TAC[dist; COMPLEX_SUB_LZERO; NORM_NEG; real] THEN X_GEN_TAC `z:complex` THEN MP_TAC(SPEC `z:complex` COMPLEX_NORM_LE_RE_IM) THEN REAL_ARITH_TAC; SIMP_TAC[SET_RULE `{x | P x /\ Q x} = {x | P x} INTER {x | Q x}`; GSYM REAL_BOUNDS_LE] THEN SIMP_TAC[CLOSED_INTER; CLOSED_REAL_SET; CLOSED_HALFSPACE_RE_LE; REWRITE_RULE[real_ge] CLOSED_HALFSPACE_RE_GE]]; SIMP_TAC[SUBSET; IMP_CONJ; FORALL_REAL; IN_ELIM_THM; RE_CX] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC CASN_CSIN THEN REWRITE_TAC[RE_CX; IM_CX] THEN ASM_REAL_ARITH_TAC]; SIMP_TAC[SUBSET; IMP_CONJ; FORALL_REAL; IN_ELIM_THM; RE_CX; IN_IMAGE] THEN X_GEN_TAC `x:real` THEN DISCH_TAC THEN EXISTS_TAC `Cx(asn x)` THEN ASM_SIMP_TAC[RE_CX; ASN_BOUNDS; REAL_BOUNDS_LE; REAL_CX; SIN_ASN; GSYM CX_SIN] THEN ASM_MESON_TAC[REAL_BOUNDS_LE; ASN_BOUNDS]]);;
let CONTINUOUS_WITHIN_CASN_REAL = 
prove (`!z. casn continuous (at z within {w | real w /\ abs(Re w) <= &1})`,
GEN_TAC THEN ASM_CASES_TAC `z IN {w | real w /\ abs(Re w) <= &1}` THENL [ASM_SIMP_TAC[REWRITE_RULE[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN] CONTINUOUS_ON_CASN_REAL]; MATCH_MP_TAC CONTINUOUS_WITHIN_CLOSED_NONTRIVIAL THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[GSYM REAL_BOUNDS_LE] THEN ASM_SIMP_TAC[SET_RULE `{x | P x /\ Q x} = {x | P x} INTER {x | Q x}`] THEN SIMP_TAC[CLOSED_INTER; CLOSED_REAL_SET; CLOSED_HALFSPACE_RE_LE; REWRITE_RULE[real_ge] CLOSED_HALFSPACE_RE_GE]]);;
let CONTINUOUS_ON_CACS_REAL = 
prove (`cacs continuous_on {w | real w /\ abs(Re w) <= &1}`,
MATCH_MP_TAC CONTINUOUS_ON_SUBSET THEN EXISTS_TAC `IMAGE ccos {z | real z /\ &0 <= Re z /\ Re z <= pi}` THEN CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_INVERSE THEN REWRITE_TAC[CONTINUOUS_ON_CCOS] THEN CONJ_TAC THENL [REWRITE_TAC[COMPACT_EQ_BOUNDED_CLOSED] THEN CONJ_TAC THENL [MATCH_MP_TAC BOUNDED_SUBSET THEN EXISTS_TAC `cball(Cx(&0),&2 * pi)` THEN REWRITE_TAC[BOUNDED_CBALL; SUBSET; IN_ELIM_THM; IN_CBALL] THEN REWRITE_TAC[dist; COMPLEX_SUB_LZERO; NORM_NEG; real] THEN X_GEN_TAC `z:complex` THEN MP_TAC(SPEC `z:complex` COMPLEX_NORM_LE_RE_IM) THEN REAL_ARITH_TAC; SIMP_TAC[SET_RULE `{x | P x /\ Q x} = {x | P x} INTER {x | Q x}`] THEN SIMP_TAC[CLOSED_INTER; CLOSED_REAL_SET; CLOSED_HALFSPACE_RE_LE; REWRITE_RULE[real_ge] CLOSED_HALFSPACE_RE_GE]]; SIMP_TAC[SUBSET; IMP_CONJ; FORALL_REAL; IN_ELIM_THM; RE_CX] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC CACS_CCOS THEN REWRITE_TAC[RE_CX; IM_CX] THEN ASM_REAL_ARITH_TAC]; SIMP_TAC[SUBSET; IMP_CONJ; FORALL_REAL; IN_ELIM_THM; RE_CX; IN_IMAGE] THEN X_GEN_TAC `x:real` THEN DISCH_TAC THEN EXISTS_TAC `Cx(acs x)` THEN ASM_SIMP_TAC[RE_CX; ACS_BOUNDS; REAL_BOUNDS_LE; REAL_CX; COS_ACS; GSYM CX_COS]]);;
let CONTINUOUS_WITHIN_CACS_REAL = 
prove (`!z. cacs continuous (at z within {w | real w /\ abs(Re w) <= &1})`,
GEN_TAC THEN ASM_CASES_TAC `z IN {w | real w /\ abs(Re w) <= &1}` THENL [ASM_SIMP_TAC[REWRITE_RULE[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN] CONTINUOUS_ON_CACS_REAL]; MATCH_MP_TAC CONTINUOUS_WITHIN_CLOSED_NONTRIVIAL THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[GSYM REAL_BOUNDS_LE] THEN ASM_SIMP_TAC[SET_RULE `{x | P x /\ Q x} = {x | P x} INTER {x | Q x}`] THEN SIMP_TAC[CLOSED_INTER; CLOSED_REAL_SET; CLOSED_HALFSPACE_RE_LE; REWRITE_RULE[real_ge] CLOSED_HALFSPACE_RE_GE]]);;
(* ------------------------------------------------------------------------- *) (* Some limits, most involving sequences of transcendentals. *) (* ------------------------------------------------------------------------- *)
let LIM_LOG_OVER_POWER = 
prove (`!s. &0 < Re s ==> ((\n. clog(Cx(&n)) / Cx(&n) cpow s) --> Cx(&0)) sequentially`,
REPEAT STRIP_TAC THEN REWRITE_TAC[LIM_SEQUENTIALLY] THEN FIRST_ASSUM(MP_TAC o SPEC `&2` o MATCH_MP REAL_ARCH) THEN DISCH_THEN(X_CHOOSE_THEN `N:num` MP_TAC) THEN ASM_CASES_TAC `N = 0` THEN ASM_REWRITE_TAC[REAL_MUL_LZERO; REAL_OF_NUM_LT; ARITH] THEN DISCH_TAC THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN MP_TAC(ISPEC `(&N / e) pow N + &1` REAL_ARCH_SIMPLE) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `M:num` THEN ASM_CASES_TAC `M = 0` THENL [ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(TAUT `~a ==> a ==> b`) THEN MATCH_MP_TAC(REAL_ARITH `&0 <= x ==> ~(x + &1 <= &0)`) THEN ASM_SIMP_TAC[REAL_POW_LE; REAL_POS; REAL_LT_IMP_LE; REAL_LE_DIV]; ALL_TAC] THEN STRIP_TAC THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN SUBGOAL_THEN `~(n = 0)` ASSUME_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[dist; COMPLEX_SUB_RZERO; COMPLEX_NORM_DIV] THEN ASM_SIMP_TAC[NORM_CPOW_REAL; REAL_CX; RE_CX; REAL_OF_NUM_LT; LT_NZ] THEN ASM_SIMP_TAC[REAL_LT_LDIV_EQ; REAL_EXP_POS_LT] THEN ASM_SIMP_TAC[GSYM CX_LOG; REAL_OF_NUM_LT; LT_NZ; COMPLEX_NORM_CX] THEN ASM_SIMP_TAC[real_abs; LOG_POS; REAL_OF_NUM_LE; ARITH_RULE `~(n = 0) ==> 1 <= n`] THEN MATCH_MP_TAC REAL_LET_TRANS THEN EXISTS_TAC `&N * log(exp(log(&n) / &N))` THEN CONJ_TAC THENL [ASM_SIMP_TAC[LOG_EXP; REAL_LE_REFL; REAL_DIV_LMUL; REAL_OF_NUM_EQ]; ALL_TAC] THEN MATCH_MP_TAC REAL_LET_TRANS THEN EXISTS_TAC `&N * exp(log(&n) / &N)` THEN CONJ_TAC THENL [MATCH_MP_TAC REAL_LE_LMUL THEN REWRITE_TAC[REAL_POS] THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `log(&1 + exp(log(&n) / &N))` THEN CONJ_TAC THENL [MATCH_MP_TAC LOG_MONO_LE_IMP THEN REWRITE_TAC[REAL_EXP_POS_LT] THEN REAL_ARITH_TAC; MATCH_MP_TAC LOG_LE THEN REWRITE_TAC[REAL_EXP_POS_LE]]; ALL_TAC] THEN SIMP_TAC[GSYM REAL_LT_RDIV_EQ; REAL_EXP_POS_LT] THEN REWRITE_TAC[real_div; GSYM REAL_MUL_ASSOC; GSYM REAL_EXP_NEG; GSYM REAL_EXP_ADD] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN ASM_SIMP_TAC[GSYM REAL_LT_LDIV_EQ] THEN MATCH_MP_TAC REAL_LTE_TRANS THEN EXISTS_TAC `exp(log(&n) / &N)` THEN CONJ_TAC THENL [ALL_TAC; REWRITE_TAC[REAL_EXP_MONO_LE] THEN REWRITE_TAC[REAL_ARITH `l / n <= r * l + --(l * inv n) <=> &0 <= l * (r - &2 / n)`] THEN MATCH_MP_TAC REAL_LE_MUL THEN ASM_SIMP_TAC[LOG_POS; REAL_OF_NUM_LE; ARITH_RULE `1 <= n <=> ~(n = 0)`; REAL_SUB_LE; REAL_LE_LDIV_EQ; REAL_OF_NUM_LT; LT_NZ] THEN ASM_REAL_ARITH_TAC] THEN W(MP_TAC o PART_MATCH (rand o rand) LOG_MONO_LT o snd) THEN ASM_SIMP_TAC[REAL_LT_DIV; REAL_OF_NUM_LT; LT_NZ; REAL_EXP_POS_LT] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN REWRITE_TAC[LOG_EXP] THEN ASM_SIMP_TAC[REAL_LT_RDIV_EQ; REAL_OF_NUM_LT; LT_NZ] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN ASM_SIMP_TAC[GSYM LOG_POW; REAL_LT_DIV; REAL_OF_NUM_LT; LT_NZ] THEN MATCH_MP_TAC LOG_MONO_LT_IMP THEN ASM_SIMP_TAC[REAL_POW_LT; REAL_LT_DIV; REAL_OF_NUM_LT; LT_NZ] THEN MATCH_MP_TAC REAL_LTE_TRANS THEN EXISTS_TAC `&M` THEN ASM_REWRITE_TAC[REAL_OF_NUM_LE] THEN ASM_REAL_ARITH_TAC);;
let LIM_LOG_OVER_N = 
prove (`((\n. clog(Cx(&n)) / Cx(&n)) --> Cx(&0)) sequentially`,
MP_TAC(SPEC `Cx(&1)` LIM_LOG_OVER_POWER) THEN SIMP_TAC[RE_CX; REAL_LT_01] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] LIM_TRANSFORM_EVENTUALLY) THEN REWRITE_TAC[EVENTUALLY_SEQUENTIALLY; CPOW_N; CX_INJ] THEN EXISTS_TAC `1` THEN SIMP_TAC[COMPLEX_POW_1; REAL_OF_NUM_EQ; ARITH_RULE `1 <= n <=> ~(n = 0)`]);;
let LIM_1_OVER_POWER = 
prove (`!s. &0 < Re s ==> ((\n. Cx(&1) / Cx(&n) cpow s) --> Cx(&0)) sequentially`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC LIM_NULL_COMPLEX_BOUND THEN EXISTS_TAC `\n. clog(Cx(&n)) / Cx(&n) cpow s` THEN ASM_SIMP_TAC[LIM_LOG_OVER_POWER] THEN REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN MP_TAC(ISPEC `exp(&1)` REAL_ARCH_SIMPLE) THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `N:num` THEN ASM_CASES_TAC `N = 0` THENL [ASM_SIMP_TAC[GSYM REAL_NOT_LT; REAL_EXP_POS_LT]; ALL_TAC] THEN DISCH_TAC THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN REWRITE_TAC[complex_div; COMPLEX_NORM_MUL] THEN MATCH_MP_TAC REAL_LE_RMUL THEN REWRITE_TAC[NORM_POS_LE] THEN SUBGOAL_THEN `~(n = 0)` ASSUME_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN ASM_SIMP_TAC[GSYM CX_LOG; REAL_OF_NUM_LT; LT_NZ; COMPLEX_NORM_CX; REAL_ABS_NUM] THEN MATCH_MP_TAC(REAL_ARITH `x <= y ==> x <= abs y`) THEN ONCE_REWRITE_TAC[GSYM REAL_EXP_MONO_LE] THEN ASM_SIMP_TAC[EXP_LOG; REAL_OF_NUM_LT; LT_NZ] THEN RULE_ASSUM_TAC(REWRITE_RULE[GSYM REAL_OF_NUM_LE]) THEN ASM_REAL_ARITH_TAC);;
let LIM_1_OVER_N = 
prove (`((\n. Cx(&1) / Cx(&n)) --> Cx(&0)) sequentially`,
MP_TAC(SPEC `Cx(&1)` LIM_1_OVER_POWER) THEN SIMP_TAC[RE_CX; REAL_LT_01] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] LIM_TRANSFORM_EVENTUALLY) THEN REWRITE_TAC[EVENTUALLY_SEQUENTIALLY; CPOW_N; CX_INJ] THEN EXISTS_TAC `1` THEN SIMP_TAC[COMPLEX_POW_1; REAL_OF_NUM_EQ; ARITH_RULE `1 <= n <=> ~(n = 0)`]);;
let LIM_INV_N = 
prove (`((\n. inv(Cx(&n))) --> Cx(&0)) sequentially`,
MP_TAC LIM_1_OVER_N THEN REWRITE_TAC[complex_div; COMPLEX_MUL_LID]);;
let LIM_1_OVER_LOG = 
prove (`((\n. Cx(&1) / clog(Cx(&n))) --> Cx(&0)) sequentially`,
REWRITE_TAC[LIM_SEQUENTIALLY] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN X_CHOOSE_TAC `N:num` (SPEC `exp(inv e)` REAL_ARCH_SIMPLE) THEN EXISTS_TAC `N + 1` THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN REWRITE_TAC[dist; COMPLEX_SUB_RZERO; COMPLEX_MUL_LID; complex_div] THEN SUBGOAL_THEN `0 < n` ASSUME_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN RULE_ASSUM_TAC(REWRITE_RULE [GSYM REAL_OF_NUM_LT; GSYM REAL_OF_NUM_LE; GSYM REAL_OF_NUM_ADD]) THEN ASM_SIMP_TAC[GSYM CX_LOG; COMPLEX_NORM_CX; COMPLEX_NORM_INV] THEN GEN_REWRITE_TAC RAND_CONV [GSYM REAL_INV_INV] THEN MATCH_MP_TAC REAL_LT_INV2 THEN ASM_REWRITE_TAC[REAL_LT_INV_EQ] THEN MATCH_MP_TAC(REAL_ARITH `a < x ==> a < abs x`) THEN ONCE_REWRITE_TAC[GSYM REAL_EXP_MONO_LT] THEN ASM_SIMP_TAC[EXP_LOG] THEN ASM_REAL_ARITH_TAC);;
let LIM_N_TIMES_POWN = 
prove (`!z. norm(z) < &1 ==> ((\n. Cx(&n) * z pow n) --> Cx(&0)) sequentially`,
REPEAT STRIP_TAC THEN ASM_CASES_TAC `z = Cx(&0)` THEN ASM_SIMP_TAC[COMPLEX_POW_ZERO; LIM_CASES_FINITE_SEQUENTIALLY; LIM_CONST; COND_RAND; FINITE_SING; SING_GSPEC; COMPLEX_MUL_RZERO] THEN MP_TAC LIM_LOG_OVER_N THEN REWRITE_TAC[LIM_SEQUENTIALLY; dist; COMPLEX_SUB_RZERO] THEN DISCH_THEN(MP_TAC o SPEC `log(inv(norm(z:complex))) / &2`) THEN ASM_SIMP_TAC[LOG_POS_LT; REAL_INV_1_LT; COMPLEX_NORM_NZ; REAL_HALF] THEN DISCH_THEN(X_CHOOSE_THEN `N1:num` (LABEL_TAC "+")) THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN FIRST_ASSUM(X_CHOOSE_THEN `N2:num` STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [REAL_ARCH_INV]) THEN EXISTS_TAC `MAX 1 (MAX N1 N2)` THEN REWRITE_TAC[ARITH_RULE `MAX a b <= c <=> a <= c /\ b <= c`] THEN X_GEN_TAC `n:num` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `n:num`) THEN ASM_SIMP_TAC[GSYM CX_LOG; REAL_OF_NUM_LT; LE_1; GSYM CX_DIV; COMPLEX_NORM_CX; REAL_ABS_DIV; REAL_ABS_NUM] THEN ASM_SIMP_TAC[REAL_LT_RDIV_EQ; REAL_OF_NUM_LT; ARITH; real_abs; LOG_POS; REAL_OF_NUM_LE] THEN ONCE_REWRITE_TAC[REAL_ARITH `a / b * &2 = (&2 * a) / b`] THEN ASM_SIMP_TAC[REAL_LT_LDIV_EQ; REAL_OF_NUM_LT; LE_1] THEN GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [REAL_MUL_SYM] THEN GEN_REWRITE_TAC LAND_CONV [GSYM REAL_EXP_MONO_LT] THEN ASM_SIMP_TAC[REAL_EXP_N; EXP_LOG; REAL_OF_NUM_LT; LE_1; REAL_LT_INV_EQ; COMPLEX_NORM_NZ] THEN REWRITE_TAC[REAL_POW_INV] THEN GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [GSYM REAL_MUL_LID] THEN ASM_SIMP_TAC[GSYM real_div; REAL_LT_RDIV_EQ; REAL_POW_LT; COMPLEX_NORM_NZ; COMPLEX_NORM_MUL; COMPLEX_NORM_NUM; COMPLEX_NORM_POW] THEN DISCH_TAC THEN MATCH_MP_TAC REAL_LET_TRANS THEN EXISTS_TAC `inv(&N2)` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `inv(&n)` THEN ASM_SIMP_TAC[REAL_LE_INV2; REAL_OF_NUM_LE; REAL_OF_NUM_LT; LE_1] THEN MATCH_MP_TAC REAL_LE_LCANCEL_IMP THEN EXISTS_TAC `&n` THEN ASM_SIMP_TAC[REAL_MUL_RINV; REAL_LT_IMP_NZ; REAL_OF_NUM_LT; LE_1] THEN ASM_REAL_ARITH_TAC);;
let LIM_N_OVER_POWN = 
prove (`!z. &1 < norm(z) ==> ((\n. Cx(&n) / z pow n) --> Cx(&0)) sequentially`,
let LIM_POWN = 
prove (`!z. norm(z) < &1 ==> ((\n. z pow n) --> Cx(&0)) sequentially`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC LIM_NULL_COMPARISON_COMPLEX THEN EXISTS_TAC `\n. Cx(&n) * z pow n` THEN ASM_SIMP_TAC[LIM_N_TIMES_POWN] THEN REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN EXISTS_TAC `1` THEN REWRITE_TAC[COMPLEX_NORM_MUL; COMPLEX_NORM_CX; REAL_ABS_NUM] THEN REWRITE_TAC[REAL_ARITH `a <= n * a <=> &0 <= (n - &1) * a`] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_LE_MUL THEN ASM_REWRITE_TAC[NORM_POS_LE; REAL_SUB_LE; REAL_OF_NUM_LE]);;
(* ------------------------------------------------------------------------- *) (* Roots of unity. *) (* ------------------------------------------------------------------------- *)
let COMPLEX_ROOT_POLYFUN = 
prove (`!n z a. 1 <= n ==> (z pow n = a <=> vsum(0..n) (\i. (if i = 0 then --a else if i = n then Cx(&1) else Cx(&0)) * z pow i) = Cx(&0))`,
ASM_SIMP_TAC[VSUM_CLAUSES_RIGHT; LE_1; LE_0] THEN SIMP_TAC[VSUM_CLAUSES_LEFT; LE_0; ADD_CLAUSES] THEN ASM_SIMP_TAC[LE_1; ARITH_RULE `1 <= n /\ 1 <= i /\ i <= n - 1 ==> ~(i = n)`] THEN REWRITE_TAC[COMPLEX_MUL_LZERO; complex_pow; COMPLEX_MUL_RID] THEN REWRITE_TAC[GSYM COMPLEX_VEC_0; VSUM_0; VECTOR_ADD_RID] THEN REWRITE_TAC[COMPLEX_VEC_0] THEN CONV_TAC COMPLEX_RING);;
let COMPLEX_ROOT_UNITY = 
prove (`!n j. ~(n = 0) ==> cexp(Cx(&2) * Cx pi * ii * Cx(&j / &n)) pow n = Cx(&1)`,
REWRITE_TAC[GSYM CEXP_N; CX_DIV] THEN ASM_SIMP_TAC[CX_INJ; complex_div; REAL_OF_NUM_EQ; COMPLEX_FIELD `~(n = Cx(&0)) ==> n * t * p * ii * j * inv(n) = j * (ii * t * p)`] THEN REWRITE_TAC[CEXP_N; GSYM CX_MUL] THEN REWRITE_TAC[CEXP_EULER; GSYM CX_MUL; GSYM CX_SIN; GSYM CX_COS] THEN REWRITE_TAC[COS_NPI; SIN_NPI; REAL_POW_NEG; COMPLEX_MUL_RZERO; REAL_POW_ONE; ARITH_EVEN; COMPLEX_ADD_RID; COMPLEX_POW_ONE]);;
let COMPLEX_ROOT_UNITY_EQ = 
prove (`!n j k. ~(n = 0) ==> (cexp(Cx(&2) * Cx pi * ii * Cx(&j / &n)) = cexp(Cx(&2) * Cx pi * ii * Cx(&k / &n)) <=> (j == k) (mod n))`,
REPEAT STRIP_TAC THEN REWRITE_TAC[CEXP_EQ; num_congruent; CX_MUL] THEN REWRITE_TAC[COMPLEX_RING `t * p * ii * j = t * p * ii * k + (t * n * p) * ii <=> (t * p * ii = Cx(&0)) \/ j - k = n`] THEN SIMP_TAC[COMPLEX_ENTIRE; II_NZ; CX_INJ; PI_NZ; REAL_OF_NUM_EQ; ARITH] THEN REWRITE_TAC[GSYM CX_SUB; CX_INJ] THEN ASM_SIMP_TAC[REAL_OF_NUM_EQ; REAL_FIELD `~(n = &0) ==> (j / n - k / n = m <=> j - k = n * m)`] THEN REWRITE_TAC[int_congruent] THEN REWRITE_TAC[int_eq; int_sub_th; int_mul_th; int_of_num_th] THEN MESON_TAC[int_abstr; int_rep]);;
let COMPLEX_ROOT_UNITY_EQ_1 = 
prove (`!n j. ~(n = 0) ==> (cexp(Cx(&2) * Cx pi * ii * Cx(&j / &n)) = Cx(&1) <=> n divides j)`,
REPEAT STRIP_TAC THEN SUBGOAL_THEN `Cx(&1) = cexp(Cx(&2) * Cx pi * ii * Cx(&n / &n))` SUBST1_TAC THENL [ASM_SIMP_TAC[REAL_DIV_REFL; REAL_OF_NUM_EQ; COMPLEX_MUL_RID] THEN ONCE_REWRITE_TAC[COMPLEX_RING `t * p * ii = ii * t * p`] THEN REWRITE_TAC[CEXP_EULER; GSYM CX_MUL; GSYM CX_SIN; GSYM CX_COS] THEN REWRITE_TAC[COS_NPI; SIN_NPI] THEN SIMPLE_COMPLEX_ARITH_TAC; ASM_SIMP_TAC[COMPLEX_ROOT_UNITY_EQ] THEN CONV_TAC NUMBER_RULE]);;
let FINITE_CARD_COMPLEX_ROOTS_UNITY = 
prove (`!n. 1 <= n ==> FINITE {z | z pow n = Cx(&1)} /\ CARD {z | z pow n = Cx(&1)} <= n`,
REPEAT GEN_TAC THEN DISCH_TAC THEN ASM_SIMP_TAC[COMPLEX_ROOT_POLYFUN] THEN MATCH_MP_TAC COMPLEX_POLYFUN_ROOTBOUND THEN DISCH_THEN(MP_TAC o SPEC `n:num`) THEN ASM_SIMP_TAC[IN_NUMSEG; LE_1; LE_0; LE_REFL] THEN CONV_TAC COMPLEX_RING);;
let FINITE_COMPLEX_ROOTS_UNITY = 
prove (`!n. ~(n = 0) ==> FINITE {z | z pow n = Cx(&1)}`,
let FINITE_CARD_COMPLEX_ROOTS_UNITY_EXPLICIT = 
prove (`!n. 1 <= n ==> FINITE {cexp(Cx(&2) * Cx pi * ii * Cx(&j / &n)) | j | j < n} /\ CARD {cexp(Cx(&2) * Cx pi * ii * Cx(&j / &n)) | j | j < n} = n`,
let lemma = prove (* So we don't need to load number theories yet *)
   (`!x y n:num. (x == y) (mod n) /\ x < y + n /\ y < x + n ==> x = y`,
    REWRITE_TAC[num_congruent; GSYM INT_OF_NUM_EQ; GSYM INT_OF_NUM_LT] THEN
    REWRITE_TAC[GSYM INT_OF_NUM_ADD] THEN
    REWRITE_TAC[INT_ARITH `x < y + n /\ y < x + n <=> abs(x - y:int) < n`] THEN
    REPEAT GEN_TAC THEN REWRITE_TAC[int_congruent] THEN
    DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_TAC `d:int`) MP_TAC) THEN
    ONCE_REWRITE_TAC[GSYM INT_SUB_0] THEN
    ASM_SIMP_TAC[INT_ABS_MUL; INT_ENTIRE; INT_ABS_NUM;
                 INT_ARITH `n * x:int < n <=> n * x < n * &1`] THEN
    DISJ_CASES_TAC(INT_ARITH `&n:int = &0 \/ &0:int < &n`) THEN
    ASM_SIMP_TAC[INT_LT_LMUL_EQ] THEN INT_ARITH_TAC) in
  REWRITE_TAC[GSYM HAS_SIZE] THEN
  REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC [SIMPLE_IMAGE_GEN] THEN
  MATCH_MP_TAC HAS_SIZE_IMAGE_INJ THEN REWRITE_TAC[IN_ELIM_THM] THEN
  ASM_SIMP_TAC[HAS_SIZE_NUMSEG_LT; COMPLEX_ROOT_UNITY_EQ; LE_1] THEN
  REPEAT STRIP_TAC THEN MATCH_MP_TAC lemma THEN EXISTS_TAC `n:num` THEN
  ASM_REWRITE_TAC[] THEN ASM_ARITH_TAC);;
let COMPLEX_ROOTS_UNITY = 
prove (`!n. 1 <= n ==> {z | z pow n = Cx(&1)} = {cexp(Cx(&2) * Cx pi * ii * Cx(&j / &n)) | j | j < n}`,
REPEAT STRIP_TAC THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC CARD_SUBSET_LE THEN ASM_SIMP_TAC[FINITE_CARD_COMPLEX_ROOTS_UNITY; FINITE_CARD_COMPLEX_ROOTS_UNITY_EXPLICIT] THEN GEN_REWRITE_TAC LAND_CONV [SIMPLE_IMAGE_GEN] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_ELIM_THM] THEN ASM_SIMP_TAC[COMPLEX_ROOT_UNITY; LE_1]);;
let CARD_COMPLEX_ROOTS_UNITY = 
prove (`!n. 1 <= n ==> CARD {z | z pow n = Cx(&1)} = n`,
let HAS_SIZE_COMPLEX_ROOTS_UNITY = 
prove (`!n. 1 <= n ==> {z | z pow n = Cx(&1)} HAS_SIZE n`,
let COMPLEX_NOT_ROOT_UNITY = 
prove (`!n. 1 <= n ==> ?u. norm u = &1 /\ ~(u pow n = Cx(&1))`,
GEN_TAC THEN DISCH_TAC THEN ABBREV_TAC `u = cexp (Cx pi * ii * Cx (&1 / &n))` THEN EXISTS_TAC `u : complex` THEN CONJ_TAC THEN EXPAND_TAC "u" THEN REWRITE_TAC [NORM_CEXP; RE_MUL_CX; RE_II; REAL_MUL_LZERO; REAL_MUL_RZERO; REAL_EXP_0] THEN EXPAND_TAC "u" THEN REWRITE_TAC[GSYM CEXP_N] THEN ASM_SIMP_TAC[CX_DIV; LE_1; CX_INJ; REAL_OF_NUM_EQ; COMPLEX_FIELD `~(n = Cx(&0)) ==> n * p * i * Cx(&1) / n = i * p`] THEN REWRITE_TAC[CEXP_EULER; RE_CX; IM_CX; GSYM CX_COS; GSYM CX_SIN] THEN REWRITE_TAC[COS_PI; SIN_PI] THEN CONV_TAC COMPLEX_RING);;
(* ------------------------------------------------------------------------- *) (* Relation between clog and Arg, and hence continuity of Arg. *) (* ------------------------------------------------------------------------- *)
let ARG_CLOG = 
prove (`!z. &0 < Arg z ==> Arg z = Im(clog(--z)) + pi`,
GEN_TAC THEN ASM_CASES_TAC `z = Cx(&0)` THENL [ASM_REWRITE_TAC[Arg_DEF; REAL_LT_REFL]; ALL_TAC] THEN DISCH_TAC THEN MP_TAC(last(CONJUNCTS(SPEC `z:complex` ARG))) THEN ASM_SIMP_TAC[CX_INJ; COMPLEX_NORM_ZERO; COMPLEX_FIELD `~(z = Cx(&0)) ==> (w = z * a <=> a = w / z)`] THEN DISCH_THEN(MP_TAC o AP_TERM `( * ) (cexp(--(ii * Cx pi)))`) THEN REWRITE_TAC[GSYM CEXP_ADD] THEN DISCH_THEN(MP_TAC o AP_TERM `clog`) THEN W(MP_TAC o PART_MATCH (lhs o rand) CLOG_CEXP o lhand o lhand o snd) THEN REWRITE_TAC[IM_ADD; IM_MUL_II; RE_CX; IM_NEG] THEN ASM_SIMP_TAC[REAL_LT_ADDR; ARG; REAL_ARITH `z < &2 * pi ==> --pi + z <= pi`] THEN DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[CEXP_NEG; CEXP_EULER] THEN REWRITE_TAC[GSYM CX_SIN; GSYM CX_COS; SIN_PI; COS_PI] THEN REWRITE_TAC[CX_NEG; COMPLEX_MUL_RZERO; COMPLEX_ADD_RID; SIMPLE_COMPLEX_ARITH `inv(--Cx(&1)) * z / w = --z / w`] THEN DISCH_THEN(MP_TAC o AP_TERM `Im`) THEN REWRITE_TAC[IM_ADD; IM_NEG; IM_MUL_II; RE_CX] THEN MATCH_MP_TAC(REAL_RING `w = z ==> --pi + x = w ==> x = z + pi`) THEN REWRITE_TAC[complex_div] THEN W(MP_TAC o PART_MATCH (lhs o rand) CLOG_MUL_SIMPLE o rand o lhand o snd) THEN ASM_SIMP_TAC[CX_INJ; REAL_INV_EQ_0; COMPLEX_NORM_ZERO; COMPLEX_NEG_EQ_0; GSYM CX_INV; GSYM CX_LOG; REAL_LT_INV_EQ; COMPLEX_NORM_NZ; IM_CX] THEN ASM_SIMP_TAC[REAL_ADD_RID; CLOG_WORKS; COMPLEX_NEG_EQ_0] THEN DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[IM_ADD; IM_CX; REAL_ADD_RID]);;
let CONTINUOUS_AT_ARG = 
prove (`!z. ~(real z /\ &0 <= Re z) ==> (Cx o Arg) continuous (at z)`,
let lemma = prove
   (`(\z. Cx(Im(f z) + pi)) = (Cx o Im) o (\z. f z + ii * Cx pi)`,
    REWRITE_TAC[FUN_EQ_THM; o_DEF; IM_ADD; IM_CX; IM_MUL_II; RE_CX]) in
  REPEAT STRIP_TAC THEN REWRITE_TAC[CONTINUOUS_AT] THEN
  MATCH_MP_TAC LIM_TRANSFORM_WITHIN_OPEN THEN
  EXISTS_TAC `\z. Cx(Im(clog(--z)) + pi)` THEN
  EXISTS_TAC `(:complex) DIFF {z | real z /\ &0 <= Re z}` THEN
  ASM_REWRITE_TAC[IN_DIFF; IN_UNIV; IN_ELIM_THM; GSYM closed] THEN
  ASM_SIMP_TAC[o_THM; ARG_CLOG; ARG_LT_NZ; ARG_EQ_0] THEN CONJ_TAC THENL
   [REWRITE_TAC[SET_RULE `{z | P z /\ Q z} = P INTER {z | Q z}`] THEN
    MATCH_MP_TAC CLOSED_INTER THEN
    REWRITE_TAC[CLOSED_REAL; GSYM real_ge; CLOSED_HALFSPACE_RE_GE];
    REWRITE_TAC[GSYM CONTINUOUS_AT; lemma] THEN
    MATCH_MP_TAC CONTINUOUS_AT_COMPOSE THEN
    REWRITE_TAC[CONTINUOUS_AT_CX_IM] THEN
    MATCH_MP_TAC CONTINUOUS_ADD THEN REWRITE_TAC[CONTINUOUS_CONST] THEN
    MATCH_MP_TAC(REWRITE_RULE[o_DEF] CONTINUOUS_AT_COMPOSE) THEN
    GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [GSYM ETA_AX] THEN
    SIMP_TAC[CONTINUOUS_NEG; CONTINUOUS_AT_ID] THEN
    MATCH_MP_TAC CONTINUOUS_AT_CLOG THEN POP_ASSUM MP_TAC THEN
    REWRITE_TAC[real; IM_NEG; RE_NEG] THEN REAL_ARITH_TAC]);;
let CONTINUOUS_WITHIN_UPPERHALF_ARG = 
prove (`!z. ~(z = Cx(&0)) ==> (Cx o Arg) continuous (at z) within {z | &0 <= Im z}`,
REPEAT STRIP_TAC THEN ASM_CASES_TAC `real z /\ &0 <= Re z` THEN ASM_SIMP_TAC[CONTINUOUS_AT_ARG; CONTINUOUS_AT_WITHIN] THEN FIRST_X_ASSUM(CONJUNCTS_THEN2 (ASSUME_TAC o GEN_REWRITE_RULE I [real]) MP_TAC) THEN SUBGOAL_THEN `~(Re z = &0)` ASSUME_TAC THENL [DISCH_TAC THEN UNDISCH_TAC `~(z = Cx(&0))` THEN ASM_REWRITE_TAC[COMPLEX_EQ; RE_CX; IM_CX]; GEN_REWRITE_TAC LAND_CONV [REAL_LE_LT]] THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN MP_TAC(ISPEC `rotate2d (pi / &2) z` CONTINUOUS_AT_ARG) THEN ANTS_TAC THENL [ASM_REWRITE_TAC[ROTATE2D_PI2; real; IM_MUL_II]; ALL_TAC] THEN REWRITE_TAC[continuous_at; continuous_within] THEN MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `e:real` THEN ASM_CASES_TAC `&0 < e` THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:real` THEN REWRITE_TAC[o_THM; dist; GSYM CX_SUB; COMPLEX_NORM_CX] THEN STRIP_TAC THEN ASM_REWRITE_TAC[IN_ELIM_THM] THEN X_GEN_TAC `w:complex` THEN STRIP_TAC THEN SUBGOAL_THEN `Arg z = &0` ASSUME_TAC THENL [ASM_SIMP_TAC[ARG_EQ_0; real; REAL_LT_IMP_LE]; ALL_TAC] THEN ASM_CASES_TAC `Arg w = &0` THEN ASM_REWRITE_TAC[REAL_SUB_REFL; REAL_ABS_NUM] THEN SUBGOAL_THEN `&0 < Arg w` ASSUME_TAC THENL [ASM_REWRITE_TAC[ARG; REAL_LT_LE]; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o SPEC `rotate2d (pi / &2) w`) THEN ASM_REWRITE_TAC[GSYM ROTATE2D_SUB; NORM_ROTATE2D] THEN MP_TAC(ISPECL [`pi / &2`; `z:complex`] ARG_ROTATE2D) THEN ANTS_TAC THENL [ASM_REWRITE_TAC[] THEN MP_TAC PI_POS THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN DISCH_THEN SUBST1_TAC THEN ASM_REWRITE_TAC[REAL_ADD_RID] THEN MATCH_MP_TAC(REAL_ARITH `w' = p + w ==> abs(w' - p) < e ==> abs(w - &0) < e`) THEN MATCH_MP_TAC ARG_ROTATE2D THEN CONJ_TAC THENL [DISCH_TAC THEN UNDISCH_TAC `&0 < Arg w` THEN ASM_REWRITE_TAC[Arg_DEF; REAL_LT_REFL]; FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM ARG_LE_PI]) THEN MP_TAC(SPEC `w:complex` ARG) THEN REAL_ARITH_TAC]);;
let CONTINUOUS_ON_UPPERHALF_ARG = 
prove (`(Cx o Arg) continuous_on ({z | &0 <= Im z} DIFF {Cx(&0)})`,
REWRITE_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN] THEN X_GEN_TAC `z:complex` THEN REWRITE_TAC[IN_DIFF; IN_SING; IN_ELIM_THM] THEN STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP CONTINUOUS_WITHIN_UPPERHALF_ARG) THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] CONTINUOUS_WITHIN_SUBSET) THEN SET_TAC[]);;
let CONTINUOUS_ON_COMPOSE_ARG = 
prove (`!s p:real->real^N. (p o drop) continuous_on interval[vec 0,lift(&2 * pi)] /\ p(&2 * pi) = p(&0) /\ ~(Cx(&0) IN s) ==> (\z. p(Arg z)) continuous_on s`,
let ulemma = prove
   (`!s. s INTER {z | &0 <= Im z} UNION s INTER {z | Im z <= &0} = s`,
    SET_TAC[REAL_LE_TOTAL]) in
  REPEAT STRIP_TAC THEN MATCH_MP_TAC CONTINUOUS_ON_EQ THEN
  EXISTS_TAC
   `\z. if &0 <= Im z then p(Arg z)
        else p(&2 * pi - Arg(cnj z)):real^N` THEN
  REWRITE_TAC[IN_UNIV; IN_SING; IN_DIFF] THEN CONJ_TAC THENL
   [X_GEN_TAC `z:complex` THEN DISCH_TAC THEN
    COND_CASES_TAC THEN ASM_REWRITE_TAC[ARG_CNJ] THEN
    COND_CASES_TAC THEN ASM_REWRITE_TAC[REAL_SUB_SUB2] THEN
    SUBGOAL_THEN `Arg z = &0`
     (fun th -> ASM_REWRITE_TAC[REAL_SUB_RZERO; th]) THEN
    ASM_REWRITE_TAC[ARG_EQ_0];
    GEN_REWRITE_TAC RAND_CONV [GSYM ulemma] THEN
    MATCH_MP_TAC CONTINUOUS_ON_CASES_LOCAL THEN REWRITE_TAC[ulemma] THEN
    SIMP_TAC[CLOSED_IN_CLOSED_INTER; CLOSED_HALFSPACE_IM_LE;
             REWRITE_RULE[real_ge] CLOSED_HALFSPACE_IM_GE] THEN
    REWRITE_TAC[IN_INTER; IN_DIFF; IN_UNIV; IN_SING; IN_ELIM_THM] THEN
    SIMP_TAC[GSYM CONJ_ASSOC; REAL_LE_ANTISYM; TAUT `~(p /\ ~p)`] THEN
    REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL
     [GEN_REWRITE_TAC (BINOP_CONV o LAND_CONV) [GSYM o_DEF] THEN
      SUBGOAL_THEN `(p:real->real^N) = (p o drop) o lift` SUBST1_TAC THENL
       [REWRITE_TAC[o_DEF; LIFT_DROP; ETA_AX]; ALL_TAC] THEN
      ONCE_REWRITE_TAC[GSYM o_ASSOC] THEN
      CONJ_TAC THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN CONJ_TAC THENL
       [REWRITE_TAC[o_DEF; GSYM CONTINUOUS_ON_CX_LIFT] THEN
        MP_TAC CONTINUOUS_ON_UPPERHALF_ARG THEN REWRITE_TAC[o_DEF] THEN
        MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] CONTINUOUS_ON_SUBSET) THEN
        ASM SET_TAC[];
        FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ]
          CONTINUOUS_ON_SUBSET)) THEN
        REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_INTER; IN_ELIM_THM] THEN
        REWRITE_TAC[IN_INTERVAL_1; LIFT_DROP; o_THM; DROP_VEC] THEN
        SIMP_TAC[ARG; REAL_LT_IMP_LE];
        REWRITE_TAC[o_DEF; LIFT_SUB] THEN MATCH_MP_TAC CONTINUOUS_ON_SUB THEN
        REWRITE_TAC[CONTINUOUS_ON_CONST] THEN
        GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV) [GSYM o_DEF] THEN
        REWRITE_TAC[o_ASSOC] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN
        REWRITE_TAC[CONTINUOUS_ON_CNJ; o_DEF; GSYM CONTINUOUS_ON_CX_LIFT] THEN
        MP_TAC CONTINUOUS_ON_UPPERHALF_ARG THEN REWRITE_TAC[o_DEF] THEN
        MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] CONTINUOUS_ON_SUBSET) THEN
        SIMP_TAC[SUBSET; FORALL_IN_IMAGE; IN_INTER; IN_ELIM_THM; IN_DIFF] THEN
        SIMP_TAC[IN_SING; CNJ_EQ_0; IM_CNJ; REAL_NEG_GE0] THEN ASM SET_TAC[];
        FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ]
          CONTINUOUS_ON_SUBSET)) THEN
        REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_INTER; IN_ELIM_THM] THEN
        REWRITE_TAC[IN_INTERVAL_1; LIFT_DROP; o_THM; DROP_VEC] THEN
        X_GEN_TAC `z:complex` THEN STRIP_TAC THEN
        MP_TAC(SPEC `cnj z` ARG) THEN REAL_ARITH_TAC];
      REWRITE_TAC[GSYM ARG_EQ_0_PI; GSYM real; ARG_CNJ] THEN
      REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
      COND_CASES_TAC THEN ASM_REWRITE_TAC[REAL_SUB_SUB2; REAL_SUB_RZERO] THEN
      ASM_REWRITE_TAC[REAL_ARITH `&2 * x - x = x`]]]);;
let OPEN_ARG_LTT = 
prove (`!s t. &0 <= s /\ t <= &2 * pi ==> open {z | s < Arg z /\ Arg z < t}`,
REPEAT STRIP_TAC THEN MP_TAC(ISPECL [`Cx o Arg`; `(:complex) DIFF {z | real z /\ &0 <= Re z}`; `{z | Re(z) > s} INTER {z | Re(z) < t}`] CONTINUOUS_OPEN_PREIMAGE) THEN ASM_SIMP_TAC[OPEN_INTER; OPEN_HALFSPACE_RE_GT; OPEN_HALFSPACE_RE_LT] THEN ANTS_TAC THENL [CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_AT_IMP_CONTINUOUS_ON THEN REWRITE_TAC[IN_DIFF; IN_UNIV; IN_ELIM_THM; CONTINUOUS_AT_ARG]; REWRITE_TAC[GSYM closed] THEN REWRITE_TAC[SET_RULE `{z | P z /\ Q z} = P INTER {z | Q z}`] THEN MATCH_MP_TAC CLOSED_INTER THEN REWRITE_TAC[CLOSED_REAL; GSYM real_ge; CLOSED_HALFSPACE_RE_GE]]; MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN REWRITE_TAC[EXTENSION] THEN ASM_SIMP_TAC[IN_DIFF; IN_INTER; IN_UNIV; IN_ELIM_THM; o_THM; RE_CX; GSYM ARG_EQ_0] THEN ASM_REAL_ARITH_TAC]);;
let OPEN_ARG_GT = 
prove (`!t. open {z | t < Arg z}`,
GEN_TAC THEN DISJ_CASES_TAC(REAL_ARITH `t < &0 \/ &0 <= t`) THENL [SUBGOAL_THEN `{z | t < Arg z} = (:complex)` (fun th -> SIMP_TAC[th; OPEN_UNIV]) THEN REWRITE_TAC[EXTENSION; IN_UNIV; IN_ELIM_THM] THEN MP_TAC ARG THEN MATCH_MP_TAC MONO_FORALL THEN ASM_REAL_ARITH_TAC; MP_TAC(ISPECL [`t:real`; `&2 * pi`] OPEN_ARG_LTT) THEN ASM_REWRITE_TAC[ARG; REAL_LE_REFL]]);;
let CLOSED_ARG_LE = 
prove (`!t. closed {z | Arg z <= t}`,
REWRITE_TAC[closed; DIFF; IN_UNIV; IN_ELIM_THM] THEN REWRITE_TAC[REAL_NOT_LE; OPEN_ARG_GT]);;
(* ------------------------------------------------------------------------- *) (* Relation between Arg and arctangent in upper halfplane. *) (* ------------------------------------------------------------------------- *)
let ARG_ATAN_UPPERHALF = 
prove (`!z. &0 < Im z ==> Arg(z) = pi / &2 - atn(Re z / Im z)`,
GEN_TAC THEN ASM_CASES_TAC `z = Cx(&0)` THEN ASM_REWRITE_TAC[IM_CX; REAL_LT_REFL] THEN DISCH_TAC THEN MATCH_MP_TAC ARG_UNIQUE THEN EXISTS_TAC `norm(z:complex)` THEN ASM_REWRITE_TAC[COMPLEX_NORM_NZ] THEN CONJ_TAC THENL [ALL_TAC; MP_TAC(ISPEC `Re z / Im z` ATN_BOUNDS) THEN REAL_ARITH_TAC] THEN REWRITE_TAC[CEXP_EULER; GSYM CX_SIN; GSYM CX_COS] THEN REWRITE_TAC[SIN_SUB; COS_SUB; SIN_PI2; COS_PI2] THEN REWRITE_TAC[REAL_MUL_LZERO; REAL_MUL_LID; SIN_ATN; COS_ATN] THEN SUBGOAL_THEN `sqrt(&1 + (Re z / Im z) pow 2) = norm(z) / Im z` SUBST1_TAC THENL [MATCH_MP_TAC SQRT_UNIQUE THEN ASM_SIMP_TAC[REAL_LE_DIV; NORM_POS_LE; REAL_LT_IMP_LE] THEN REWRITE_TAC[REAL_POW_DIV; COMPLEX_SQNORM] THEN UNDISCH_TAC `&0 < Im z` THEN CONV_TAC REAL_FIELD; REWRITE_TAC[REAL_ADD_LID; REAL_SUB_RZERO; real_div] THEN REWRITE_TAC[COMPLEX_EQ; RE_MUL_CX; IM_MUL_CX; RE_MUL_II; IM_MUL_II; RE_ADD; IM_ADD; RE_CX; IM_CX] THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM COMPLEX_NORM_NZ]) THEN POP_ASSUM MP_TAC THEN CONV_TAC REAL_FIELD]);;
(* ------------------------------------------------------------------------- *) (* Real n'th roots. *) (* ------------------------------------------------------------------------- *)
let root = new_definition
 `root(n) x = @u:real. (&0 <= x ==> &0 <= u) /\ u pow n = x`;;
let ROOT_0 = 
prove (`!n. ~(n = 0) ==> root n (&0) = &0`,
SIMP_TAC[root; REAL_LT_REFL; REAL_POW_EQ_0; REAL_LE_REFL; SELECT_REFL; REAL_ARITH `&0 <= u /\ u = &0 <=> u = &0`]);;
let ROOT_1 = 
prove (`!n. ~(n = 0) ==> root n (&1) = &1`,
SIMP_TAC[root; REAL_POS; REAL_POW_EQ_1; CONJ_ASSOC] THEN REWRITE_TAC[REAL_ARITH `&0 <= u /\ abs u = &1 <=> u = &1`] THEN SIMP_TAC[TAUT `a /\ b <=> ~(a ==> ~b)`] THEN CONV_TAC REAL_RAT_REDUCE_CONV);;
let ROOT_2 = 
prove (`!x. root 2 x = sqrt x`,
GEN_TAC THEN REWRITE_TAC[sqrt; root] THEN AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN X_GEN_TAC `y:real` THEN ASM_CASES_TAC `x:real = y pow 2` THEN ASM_REWRITE_TAC[REAL_POW_2; REAL_LE_SQUARE] THEN REAL_ARITH_TAC);;
let ROOT_WORKS = 
prove (`!n x. ODD n \/ ~(n = 0) /\ &0 <= x ==> (&0 <= x ==> &0 <= root n x) /\ (root n x) pow n = x`,
REPEAT GEN_TAC THEN ASM_CASES_TAC `n = 0` THEN ASM_REWRITE_TAC[ARITH] THEN DISCH_TAC THEN REWRITE_TAC[root] THEN CONV_TAC SELECT_CONV THEN REPEAT_TCL DISJ_CASES_THEN ASSUME_TAC (REAL_ARITH `x = &0 \/ &0 < x \/ &0 < --x`) THENL [EXISTS_TAC `&0` THEN ASM_REWRITE_TAC[REAL_POW_ZERO]; EXISTS_TAC `exp(log x / &n)` THEN REWRITE_TAC[REAL_EXP_POS_LE] THEN ASM_SIMP_TAC[GSYM REAL_EXP_N; REAL_DIV_LMUL; REAL_OF_NUM_EQ; EXP_LOG]; FIRST_X_ASSUM(DISJ_CASES_THEN ASSUME_TAC) THENL [ALL_TAC; ASM_REAL_ARITH_TAC] THEN EXISTS_TAC `--exp(log(--x) / &n)` THEN REWRITE_TAC[REAL_POW_NEG] THEN ASM_SIMP_TAC[GSYM NOT_ODD; REAL_ARITH `&0 < --x ==> ~(&0 <= x)`] THEN ASM_SIMP_TAC[GSYM REAL_EXP_N; REAL_DIV_LMUL; REAL_OF_NUM_EQ; EXP_LOG; REAL_NEG_NEG]]);;
let REAL_POW_ROOT = 
prove (`!n x. ODD n \/ ~(n = 0) /\ &0 <= x ==> (root n x) pow n = x`,
SIMP_TAC[ROOT_WORKS]);;
let ROOT_POS_LE = 
prove (`!n x. ~(n = 0) /\ &0 <= x ==> &0 <= root n x`,
SIMP_TAC[ROOT_WORKS]);;
let ROOT_POS_LT = 
prove (`!n x. ~(n = 0) /\ &0 < x ==> &0 < root n x`,
REPEAT GEN_TAC THEN SIMP_TAC[REAL_LT_LE; ROOT_POS_LE] THEN STRIP_TAC THEN DISCH_THEN(ASSUME_TAC o SYM) THEN MP_TAC(SPECL [`n:num`; `x:real`] REAL_POW_ROOT) THEN ASM_REWRITE_TAC[REAL_POW_ZERO]);;
let REAL_ROOT_POW = 
prove (`!n x. ODD n \/ ~(n = 0) /\ &0 <= x ==> root n (x pow n) = x`,
REPEAT GEN_TAC THEN DISCH_TAC THEN REWRITE_TAC[root] THEN GEN_REWRITE_TAC RAND_CONV [GSYM SELECT_REFL] THEN AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN X_GEN_TAC `y:real` THEN MP_TAC(SPECL [`n:num`; `&0`; `x:real`] REAL_POW_LE2_ODD_EQ) THEN POP_ASSUM MP_TAC THEN REWRITE_TAC[REAL_POW_EQ_EQ; GSYM NOT_EVEN] THEN ASM_CASES_TAC `n = 0` THEN ASM_REWRITE_TAC[ARITH] THEN COND_CASES_TAC THEN ASM_SIMP_TAC[REAL_POW_LE; REAL_POW_ZERO] THEN ASM_REAL_ARITH_TAC);;
let ROOT_UNIQUE = 
prove (`!n x y. y pow n = x /\ (ODD n \/ ~(n = 0) /\ &0 <= y) ==> root n x = y`,
REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN ASSUME_TAC) THEN UNDISCH_THEN `(y:real) pow n = x` (SUBST_ALL_TAC o SYM) THEN MATCH_MP_TAC REAL_ROOT_POW THEN ASM_REWRITE_TAC[]);;
let REAL_ROOT_MUL = 
prove (`!n x y. ~(n = 0) /\ &0 <= x /\ &0 <= y ==> root n (x * y) = root n x * root n y`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC ROOT_UNIQUE THEN ASM_SIMP_TAC[REAL_POW_MUL; REAL_POW_ROOT; REAL_LE_MUL; ROOT_POS_LE]);;
let REAL_ROOT_INV = 
prove (`!n x. ~(n = 0) /\ &0 <= x ==> root n (inv x) = inv(root n x)`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC ROOT_UNIQUE THEN ASM_SIMP_TAC[REAL_POW_INV; REAL_POW_ROOT; REAL_LE_INV_EQ; ROOT_POS_LE]);;
let REAL_ROOT_DIV = 
prove (`!n x y. ~(n = 0) /\ &0 <= x /\ &0 <= y ==> root n (x / y) = root n x / root n y`,
let ROOT_MONO_LT = 
prove (`!n x y. ~(n = 0) /\ &0 <= x /\ x < y ==> root n x < root n y`,
REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN MATCH_MP_TAC(REAL_ARITH `(&0 <= x /\ &0 <= y ==> (v <= u ==> y <= x)) ==> &0 <= x /\ x < y ==> u < v`) THEN REPEAT STRIP_TAC THEN SUBGOAL_THEN `(root n y) pow n <= (root n x) pow n` MP_TAC THENL [MATCH_MP_TAC REAL_POW_LE2 THEN ASM_SIMP_TAC[ROOT_POS_LE]; ASM_SIMP_TAC[REAL_POW_ROOT]]);;
let ROOT_MONO_LE = 
prove (`!n x y. ~(n = 0) /\ &0 <= x /\ x <= y ==> root n x <= root n y`,
MESON_TAC[ROOT_MONO_LT; REAL_LE_LT]);;
let ROOT_MONO_LT_EQ = 
prove (`!n x y. ~(n = 0) /\ &0 <= x /\ &0 <= y ==> (root n x < root n y <=> x < y)`,
let ROOT_MONO_LE_EQ = 
prove (`!n x y. ~(n = 0) /\ &0 <= x /\ &0 <= y ==> (root n x <= root n y <=> x <= y)`,
let ROOT_INJ = 
prove (`!n x y. ~(n = 0) /\ &0 <= x /\ &0 <= y ==> (root n x = root n y <=> x = y)`,
SIMP_TAC[GSYM REAL_LE_ANTISYM; ROOT_MONO_LE_EQ]);;
let REAL_ROOT_LE = 
prove (`!n x y. ~(n = 0) /\ &0 <= x /\ &0 <= y ==> (root n x <= y <=> x <= y pow n)`,
let REAL_LE_ROOT = 
prove (`!n x y. ~(n = 0) /\ &0 <= x /\ &0 <= y ==> (x <= root n y <=> x pow n <= y)`,
let LOG_ROOT = 
prove (`!n x. ~(n = 0) /\ &0 < x ==> log(root n x) = log x / &n`,
SIMP_TAC[REAL_EQ_RDIV_EQ; REAL_OF_NUM_LT; LE_1] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN SIMP_TAC[GSYM LOG_POW; ROOT_POS_LT; REAL_POW_ROOT; REAL_LT_IMP_LE]);;
let ROOT_EXP_LOG = 
prove (`!n x. ~(n = 0) /\ &0 < x ==> root n x = exp(log x / &n)`,
REPEAT GEN_TAC THEN DISCH_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP ROOT_POS_LT) THEN REWRITE_TAC[GSYM REAL_EXP_LOG] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN AP_TERM_TAC THEN ASM_SIMP_TAC[LOG_ROOT]);;
(* ------------------------------------------------------------------------- *) (* Real power function. This involves a few arbitrary choices. *) (* *) (* The value of x^y is unarguable when x > 0. *) (* *) (* We make 0^0 = 1 to agree with "pow", but otherwise 0^y = 0. *) (* *) (* There is a sensible real value for (-x)^(p/q) where q is odd and either *) (* p is even [(-x)^y = x^y] or odd [(-x)^y = -x^y]. *) (* *) (* In all other cases, we return (-x)^y = -x^y. This is meaningless but at *) (* least it covers half the cases above without another case split. *) (* *) (* As for laws of indices, we do have x^-y = 1/x^y. Of course we can't have *) (* x^(yz) = x^y^z or x^(y+z) = x^y x^z since then (-1)^(1/2)^2 = -1. *) (* ------------------------------------------------------------------------- *) parse_as_infix("rpow",(24,"left"));;
let rpow = new_definition
  `x rpow y = if &0 < x then exp(y * log x)
               else if x = &0 then if y = &0 then &1 else &0
               else if ?m n. ODD(m) /\ ODD(n) /\ (abs y = &m / &n)
                    then --(exp(y * log(--x)))
                    else exp(y * log(--x))`;;
let RPOW_POW = 
prove (`!x n. x rpow &n = x pow n`,
REPEAT GEN_TAC THEN REWRITE_TAC[rpow] THEN COND_CASES_TAC THEN ASM_SIMP_TAC[REAL_EXP_N; EXP_LOG] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[REAL_POW_ZERO; REAL_OF_NUM_EQ] THEN ASM_SIMP_TAC[EXP_LOG; REAL_ARITH `~(&0 < x) /\ ~(x = &0) ==> &0 < --x`] THEN REWRITE_TAC[REAL_POW_NEG; REAL_ABS_NUM] THEN SUBGOAL_THEN `(?p q. ODD(p) /\ ODD(q) /\ &n = &p / &q) <=> ODD n` (fun th -> SIMP_TAC[th; GSYM NOT_ODD; REAL_NEG_NEG; COND_ID]) THEN EQ_TAC THEN REWRITE_TAC[LEFT_IMP_EXISTS_THM] THENL [REPEAT GEN_TAC THEN ASM_CASES_TAC `q = 0` THEN ASM_REWRITE_TAC[ARITH_ODD] THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN ASM_SIMP_TAC[REAL_OF_NUM_EQ; REAL_FIELD `~(q = &0) ==> (n = p / q <=> q * n = p)`] THEN REWRITE_TAC[REAL_OF_NUM_MUL; REAL_OF_NUM_EQ] THEN ASM_MESON_TAC[ODD_MULT]; DISCH_TAC THEN MAP_EVERY EXISTS_TAC [`n:num`; `1`] THEN ASM_REWRITE_TAC[REAL_DIV_1; ARITH_ODD]]);;
let RPOW_NEG = 
prove (`!x y. x rpow (--y) = inv(x rpow y)`,
REPEAT GEN_TAC THEN REWRITE_TAC[rpow] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[REAL_MUL_LNEG; REAL_EXP_NEG] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[REAL_NEG_EQ_0] THENL [COND_CASES_TAC THEN ASM_REWRITE_TAC[REAL_INV_0; REAL_INV_1]; REWRITE_TAC[REAL_ABS_NEG] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[REAL_INV_NEG]]);;
let RPOW_ZERO = 
prove (`!y. &0 rpow y = if y = &0 then &1 else &0`,
REWRITE_TAC[rpow; REAL_LT_REFL]);;
let RPOW_POS_LT = 
prove (`!x y. &0 < x ==> &0 < x rpow y`,
SIMP_TAC[rpow; REAL_EXP_POS_LT]);;
let RPOW_POS_LE = 
prove (`!x y. &0 <= x ==> &0 <= x rpow y`,
REPEAT GEN_TAC THEN ASM_CASES_TAC `x = &0` THENL [ASM_REWRITE_TAC[RPOW_ZERO] THEN MESON_TAC[REAL_POS]; ASM_SIMP_TAC[RPOW_POS_LT; REAL_LE_LT]]);;
let RPOW_LT2 = 
prove (`!x y z. &0 <= x /\ x < y /\ &0 < z ==> x rpow z < y rpow z`,
REPEAT GEN_TAC THEN ASM_CASES_TAC `x = &0` THEN ASM_SIMP_TAC[RPOW_ZERO; REAL_LT_IMP_NZ; RPOW_POS_LT] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[rpow] THEN ASM_CASES_TAC `&0 < x /\ &0 < y` THENL [ALL_TAC; MATCH_MP_TAC(TAUT `F ==> p`) THEN ASM_REAL_ARITH_TAC] THEN ASM_SIMP_TAC[REAL_EXP_MONO_LT; REAL_LT_LMUL_EQ] THEN MATCH_MP_TAC LOG_MONO_LT_IMP THEN ASM_REAL_ARITH_TAC);;
let RPOW_LE2 = 
prove (`!x y z. &0 <= x /\ x <= y /\ &0 <= z ==> x rpow z <= y rpow z`,
REPEAT GEN_TAC THEN ASM_CASES_TAC `z = &0` THEN ASM_REWRITE_TAC[RPOW_POW; real_pow; REAL_LE_REFL] THEN ASM_CASES_TAC `x:real = y` THEN ASM_REWRITE_TAC[REAL_LE_REFL] THEN ASM_MESON_TAC[RPOW_LT2; REAL_LE_LT]);;
let REAL_ABS_RPOW = 
prove (`!x y. abs(x rpow y) = abs(x) rpow y`,
REPEAT GEN_TAC THEN REWRITE_TAC[rpow] THEN ASM_CASES_TAC `x = &0` THEN ASM_REWRITE_TAC[REAL_ABS_NUM; REAL_LT_REFL] THENL [REAL_ARITH_TAC; ALL_TAC] THEN ASM_REWRITE_TAC[GSYM REAL_ABS_NZ; REAL_ABS_ZERO] THEN COND_CASES_TAC THEN ASM_SIMP_TAC[REAL_ABS_EXP; REAL_ARITH `&0 < x ==> abs x = x`] THEN COND_CASES_TAC THEN REWRITE_TAC[REAL_ABS_NEG; REAL_ABS_EXP] THEN AP_TERM_TAC THEN AP_TERM_TAC THEN AP_TERM_TAC THEN ASM_REAL_ARITH_TAC);;
let RPOW_ONE = 
prove (`!z. &1 rpow z = &1`,
let RPOW_RPOW = 
prove (`!x y z. &0 <= x ==> x rpow y rpow z = x rpow (y * z)`,
REPEAT GEN_TAC THEN REWRITE_TAC[REAL_LE_LT] THEN ASM_CASES_TAC `x = &0` THEN ASM_REWRITE_TAC[] THENL [ASM_REWRITE_TAC[RPOW_ZERO; REAL_ENTIRE] THEN ASM_CASES_TAC `y = &0` THEN ASM_REWRITE_TAC[RPOW_ZERO; RPOW_ONE]; SIMP_TAC[rpow; REAL_EXP_POS_LT; LOG_EXP] THEN REWRITE_TAC[REAL_MUL_AC]]);;
let RPOW_LNEG = 
prove (`!x y. --x rpow y = if ?m n. ODD m /\ ODD n /\ abs y = &m / &n then --(x rpow y) else x rpow y`,
REPEAT GEN_TAC THEN REWRITE_TAC[rpow] THEN ASM_CASES_TAC `x = &0` THEN ASM_REWRITE_TAC[REAL_NEG_0; REAL_ABS_NUM; REAL_LT_REFL] THENL [ASM_CASES_TAC `y = &0` THEN ASM_REWRITE_TAC[REAL_NEG_0; COND_ID] THEN REWRITE_TAC[REAL_ARITH `abs(&0) = m / n <=> m * inv n = &0`] THEN SIMP_TAC[REAL_ENTIRE; REAL_INV_EQ_0; REAL_OF_NUM_EQ] THEN MESON_TAC[ODD]; ASM_SIMP_TAC[REAL_ARITH `~(x = &0) ==> (&0 < --x <=> ~(&0 < x))`] THEN ASM_REWRITE_TAC[REAL_NEG_EQ_0] THEN ASM_CASES_TAC `&0 < x` THEN ASM_REWRITE_TAC[REAL_NEG_NEG; COND_ID]]);;
let RPOW_EQ_0 = 
prove (`!x y. x rpow y = &0 <=> x = &0 /\ ~(y = &0)`,
REPEAT GEN_TAC THEN REWRITE_TAC[rpow] THEN ASM_CASES_TAC `x = &0` THEN ASM_REWRITE_TAC[REAL_LT_REFL] THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[REAL_NEG_EQ_0; REAL_EXP_NZ]) THEN REAL_ARITH_TAC);;
let RPOW_MUL = 
prove (`!x y z. (x * y) rpow z = x rpow z * y rpow z`,
SUBGOAL_THEN `!x y z. &0 <= x /\ &0 <= y ==> (x * y) rpow z = x rpow z * y rpow z` ASSUME_TAC THENL [REPEAT GEN_TAC THEN REWRITE_TAC[REAL_LE_LT] THEN ASM_CASES_TAC `z = &0` THEN ASM_REWRITE_TAC[RPOW_POW; real_pow; REAL_MUL_LID] THEN ASM_CASES_TAC `x = &0` THEN ASM_REWRITE_TAC[REAL_MUL_LZERO; RPOW_ZERO] THEN ASM_CASES_TAC `y = &0` THEN ASM_REWRITE_TAC[REAL_MUL_RZERO; RPOW_ZERO] THEN SIMP_TAC[rpow; REAL_LT_MUL; LOG_MUL; REAL_ADD_LDISTRIB; REAL_EXP_ADD]; REPEAT GEN_TAC THEN REPEAT_TCL DISJ_CASES_THEN (ANTE_RES_THEN (MP_TAC o SPEC `z:real`)) (REAL_ARITH `&0 <= x /\ &0 <= y \/ &0 <= x /\ &0 <= --y \/ &0 <= --x /\ &0 <= y \/ &0 <= --x /\ &0 <= --y`) THEN REWRITE_TAC[RPOW_LNEG; REAL_MUL_RNEG; REAL_MUL_LNEG] THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[REAL_MUL_RNEG; REAL_MUL_LNEG; REAL_EQ_NEG2]]);;
let RPOW_INV = 
prove (`!x y. inv(x) rpow y = inv(x rpow y)`,
REPEAT GEN_TAC THEN REWRITE_TAC[rpow; REAL_LT_INV_EQ] THEN SIMP_TAC[LOG_INV; REAL_MUL_RNEG; REAL_EXP_NEG] THEN COND_CASES_TAC THEN REWRITE_TAC[] THEN REWRITE_TAC[REAL_INV_EQ_0] THEN REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[REAL_INV_1; REAL_INV_0]) THEN ASM_SIMP_TAC[GSYM REAL_INV_NEG; LOG_INV; REAL_ARITH `~(&0 < x) /\ ~(x = &0) ==> &0 < --x`] THEN REWRITE_TAC[REAL_MUL_RNEG; REAL_EXP_NEG; REAL_INV_NEG]);;
let REAL_INV_RPOW = 
prove (`!x y. inv(x rpow y) = inv(x) rpow y`,
REWRITE_TAC[RPOW_INV]);;
let RPOW_ADD = 
prove (`!x y z. &0 < x ==> x rpow (y + z) = x rpow y * x rpow z`,
REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[rpow; REAL_ADD_RDISTRIB; REAL_EXP_ADD]);;
let RPOW_ADD_ALT = 
prove (`!x y z. &0 <= x /\ (x = &0 /\ y + z = &0 ==> y = &0 \/ z = &0) ==> x rpow (y + z) = x rpow y * x rpow z`,
REPEAT GEN_TAC THEN ASM_CASES_TAC `x = &0` THEN ASM_SIMP_TAC[REAL_LE_LT; RPOW_ADD] THEN REWRITE_TAC[RPOW_ZERO] THEN ASM_CASES_TAC `y = &0` THEN ASM_REWRITE_TAC[REAL_MUL_LID; REAL_ADD_LID] THEN ASM_CASES_TAC `y + z = &0` THEN ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC);;
let RPOW_SQRT = 
prove (`!x. &0 <= x ==> x rpow (&1 / &2) = sqrt x`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC(REAL_RING `x pow 2 = y pow 2 /\ (x + y = &0 ==> x = &0 /\ y = &0) ==> x = y`) THEN CONJ_TAC THENL [ASM_SIMP_TAC[SQRT_POW_2] THEN ASM_SIMP_TAC[GSYM RPOW_POW; RPOW_RPOW] THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[RPOW_POW; REAL_POW_1]; MATCH_MP_TAC(REAL_ARITH `&0 <= x /\ &0 <= y ==> x + y = &0 ==> x = &0 /\ y = &0`) THEN ASM_SIMP_TAC[SQRT_POS_LE; RPOW_POS_LE]]);;
let RPOW_MONO = 
prove (`!a b x. &1 <= x /\ a <= b ==> x rpow a <= x rpow b`,
SIMP_TAC[rpow; REAL_ARITH `&1 <= x ==> &0 < x`] THEN SIMP_TAC[REAL_EXP_MONO_LE; LOG_POS; REAL_LE_RMUL]);;
let RPOW_MONO_INV = 
prove (`!a b x. &0 < x /\ x <= &1 /\ b <= a ==> x rpow a <= x rpow b`,
REPEAT STRIP_TAC THEN GEN_REWRITE_TAC BINOP_CONV [GSYM REAL_INV_INV] THEN MATCH_MP_TAC REAL_LE_INV2 THEN ASM_SIMP_TAC[REAL_LT_INV_EQ; RPOW_POS_LT; GSYM RPOW_INV] THEN MATCH_MP_TAC RPOW_MONO THEN ASM_SIMP_TAC[REAL_INV_1_LE]);;
let RPOW_1_LE = 
prove (`!a x. &0 <= x /\ x <= &1 /\ &0 <= a ==> x rpow a <= &1`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `&1 rpow a` THEN CONJ_TAC THENL [MATCH_MP_TAC RPOW_LE2 THEN ASM_REAL_ARITH_TAC; REWRITE_TAC[RPOW_ONE; REAL_LE_REFL]]);;
let REAL_ROOT_RPOW = 
prove (`!n x. ~(n = 0) /\ (&0 <= x \/ ODD n) ==> root n x = x rpow (inv(&n))`,
REPEAT GEN_TAC THEN ASM_CASES_TAC `x = &0` THEN ASM_SIMP_TAC[ROOT_0; RPOW_ZERO; REAL_INV_EQ_0; REAL_OF_NUM_EQ] THEN ASM_CASES_TAC `&0 <= x` THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THENL [ASM_SIMP_TAC[ROOT_EXP_LOG; rpow; REAL_LT_LE] THEN AP_TERM_TAC THEN REAL_ARITH_TAC; ASM_REWRITE_TAC[rpow] THEN COND_CASES_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[REAL_ABS_INV; REAL_ABS_NUM] THEN REWRITE_TAC[REAL_ARITH `inv x = &1 / x`] THEN COND_CASES_TAC THENL [ALL_TAC; ASM_MESON_TAC[ARITH]] THEN MATCH_MP_TAC ROOT_UNIQUE THEN ASM_REWRITE_TAC[REAL_POW_NEG; GSYM REAL_EXP_N; GSYM NOT_ODD] THEN ASM_SIMP_TAC[REAL_OF_NUM_EQ; REAL_FIELD `~(n = &0) ==> n * &1 / n * x = x`] THEN ONCE_REWRITE_TAC[REAL_ARITH `--x:real = y <=> x = --y`] THEN MATCH_MP_TAC EXP_LOG THEN ASM_REAL_ARITH_TAC]);;
(* ------------------------------------------------------------------------- *) (* Formulation of loop homotopy in terms of maps out of S^1 *) (* ------------------------------------------------------------------------- *)
let HOMOTOPIC_CIRCLEMAPS_IMP_HOMOTOPIC_LOOPS = 
prove (`!f:complex->real^N g s. homotopic_with (\h. T) (sphere(vec 0,&1),s) f g ==> homotopic_loops s (f o cexp o (\t. Cx(&2 * pi * drop t) * ii)) (g o cexp o (\t. Cx(&2 * pi * drop t) * ii))`,
REWRITE_TAC[homotopic_loops; sphere; DIST_0] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC HOMOTOPIC_WITH_COMPOSE_CONTINUOUS_RIGHT THEN EXISTS_TAC `{z:complex | norm z = &1}` THEN REWRITE_TAC[pathstart; pathfinish; o_THM; DROP_VEC] THEN ONCE_REWRITE_TAC[REAL_ARITH `&2 * pi * n = &2 * n * pi`] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; o_THM; IN_ELIM_THM] THEN ASM_SIMP_TAC[CEXP_INTEGER_2PI; INTEGER_CLOSED] THEN REWRITE_TAC[ONCE_REWRITE_RULE[COMPLEX_MUL_SYM] NORM_CEXP_II] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN REWRITE_TAC[CONTINUOUS_ON_CEXP] THEN REWRITE_TAC[CX_MUL] THEN REPEAT(MATCH_MP_TAC CONTINUOUS_ON_COMPLEX_MUL THEN REWRITE_TAC[CONTINUOUS_ON_CONST]) THEN SIMP_TAC[CONTINUOUS_ON_CX_DROP; CONTINUOUS_ON_ID]);;
let HOMOTOPIC_LOOPS_IMP_HOMOTOPIC_CIRCLEMAPS = 
prove (`!p q s:real^N->bool. homotopic_loops s p q ==> homotopic_with (\h. T) (sphere(vec 0,&1),s) (p o (\z. lift(Arg z / (&2 * pi)))) (q o (\z. lift(Arg z / (&2 * pi))))`,
let ulemma = prove
   (`!s. s INTER (UNIV PCROSS {z | &0 <= Im z}) UNION
         s INTER (UNIV PCROSS {z | Im z <= &0}) = s`,
    REWRITE_TAC[EXTENSION; FORALL_PASTECART; IN_INTER; IN_UNION;
                 PASTECART_IN_PCROSS] THEN
    SET_TAC[REAL_LE_TOTAL]) in
  REPEAT GEN_TAC THEN REWRITE_TAC[homotopic_loops; sphere; DIST_0] THEN
  GEN_REWRITE_TAC LAND_CONV [homotopic_with] THEN
  SIMP_TAC[pathstart; pathfinish; LEFT_IMP_EXISTS_THM; HOMOTOPIC_WITH] THEN
  X_GEN_TAC `h:real^(1,1)finite_sum->real^N` THEN STRIP_TAC THEN
  EXISTS_TAC `\w. (h:real^(1,1)finite_sum->real^N)
                  (pastecart (fstcart w)
                             (lift(Arg(sndcart w) / (&2 * pi))))` THEN
  ASM_REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART; o_THM] THEN
  CONJ_TAC THENL
   [MATCH_MP_TAC CONTINUOUS_ON_EQ THEN
    EXISTS_TAC
     `(\z. if &0 <= Im(sndcart z)
           then h (pastecart (fstcart z) (lift(Arg(sndcart z) / (&2 * pi))))
           else h (pastecart (fstcart z)
                             (vec 1 - lift(Arg(cnj(sndcart z)) / (&2 * pi)))))
      :real^(1,2)finite_sum->real^N` THEN
    REWRITE_TAC[FORALL_IN_PCROSS; FSTCART_PASTECART; SNDCART_PASTECART] THEN
    REWRITE_TAC[IN_ELIM_THM] THEN CONJ_TAC THENL
     [MAP_EVERY X_GEN_TAC [`t:real^1`; `z:complex`] THEN STRIP_TAC THEN
      COND_CASES_TAC THEN ASM_REWRITE_TAC[ARG_CNJ] THEN
      COND_CASES_TAC THENL [ASM_MESON_TAC[real; REAL_LE_REFL]; ALL_TAC] THEN
      SIMP_TAC[PI_POS; LIFT_SUB; LIFT_NUM; REAL_FIELD
        `&0 < pi ==> (&2 * pi - z) / (&2 * pi) = &1 - z / (&2 * pi)`] THEN
      REWRITE_TAC[VECTOR_ARITH `a - (a - b):real^N = b`];
      GEN_REWRITE_TAC RAND_CONV [GSYM ulemma] THEN
      MATCH_MP_TAC CONTINUOUS_ON_CASES_LOCAL THEN REWRITE_TAC[ulemma] THEN
      SIMP_TAC[CLOSED_IN_CLOSED_INTER; CLOSED_HALFSPACE_IM_LE; CLOSED_UNIV;
        CLOSED_PCROSS; REWRITE_RULE[real_ge] CLOSED_HALFSPACE_IM_GE] THEN
      REWRITE_TAC[FORALL_PASTECART; PASTECART_IN_PCROSS; IN_INTER; IN_DIFF;
       FSTCART_PASTECART; SNDCART_PASTECART; IN_UNIV; IN_SING; IN_ELIM_THM;
       GSYM CONJ_ASSOC; REAL_LE_ANTISYM; TAUT `~(p /\ ~p)`] THEN
      REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL
       [ALL_TAC;
        REWRITE_TAC[GSYM ARG_EQ_0_PI; GSYM real; ARG_CNJ] THEN
        REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
        SIMP_TAC[REAL_ARITH `&2 * x - x = x`; COND_ID; GSYM LIFT_NUM; PI_POS;
         GSYM LIFT_SUB; REAL_FIELD
          `&0 < pi ==> &1 - pi / (&2 * pi) = pi / (&2 * pi)`] THEN
        COND_CASES_TAC THEN
        SIMP_TAC[REAL_SUB_RZERO; REAL_DIV_REFL; REAL_ENTIRE; REAL_OF_NUM_EQ;
                 ARITH_EQ; PI_NZ] THEN
        SIMP_TAC[real_div; REAL_MUL_LZERO; REAL_SUB_REFL; REAL_SUB_RZERO] THEN
        ASM_SIMP_TAC[LIFT_NUM]] THEN
      GEN_REWRITE_TAC (BINOP_CONV o LAND_CONV) [GSYM o_DEF] THEN
      CONJ_TAC THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN CONJ_TAC THENL
       [MATCH_MP_TAC CONTINUOUS_ON_PASTECART THEN
        SIMP_TAC[LINEAR_CONTINUOUS_ON; LINEAR_FSTCART] THEN
        REWRITE_TAC[real_div; REWRITE_RULE[REAL_MUL_SYM] LIFT_CMUL] THEN
        MATCH_MP_TAC CONTINUOUS_ON_VMUL THEN
        GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [GSYM o_DEF] THEN
        REWRITE_TAC[o_ASSOC] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN
        SIMP_TAC[LINEAR_CONTINUOUS_ON; LINEAR_SNDCART] THEN
        REWRITE_TAC[o_DEF; GSYM CONTINUOUS_ON_CX_LIFT] THEN
        MP_TAC CONTINUOUS_ON_UPPERHALF_ARG THEN REWRITE_TAC[o_DEF] THEN
        MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] CONTINUOUS_ON_SUBSET) THEN
        REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_PASTECART; IN_INTER;
                    PASTECART_IN_PCROSS; IN_ELIM_THM; SNDCART_PASTECART] THEN
        MAP_EVERY X_GEN_TAC [`t:real^1`; `z:complex`] THEN
        SIMP_TAC[IN_DIFF; IN_ELIM_THM; IN_SING] THEN
        ASM_CASES_TAC `z = Cx(&0)` THEN ASM_REWRITE_TAC[COMPLEX_NORM_0] THEN
        REWRITE_TAC[REAL_OF_NUM_EQ; ARITH_EQ];
        FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ]
          CONTINUOUS_ON_SUBSET)) THEN
        REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_PASTECART; IN_INTER;
                    PASTECART_IN_PCROSS; IN_ELIM_THM; SNDCART_PASTECART;
                    FSTCART_PASTECART] THEN
        SIMP_TAC[IN_INTERVAL_1; LIFT_DROP; DROP_VEC] THEN
        SIMP_TAC[REAL_LE_LDIV_EQ; REAL_LE_RDIV_EQ; PI_POS; REAL_MUL_LZERO;
             REAL_MUL_LID; REAL_ARITH `&0 < &2 * x <=> &0 < x`] THEN
        SIMP_TAC[ARG; REAL_LT_IMP_LE];
        MATCH_MP_TAC CONTINUOUS_ON_PASTECART THEN
        SIMP_TAC[LINEAR_CONTINUOUS_ON; LINEAR_FSTCART] THEN
        MATCH_MP_TAC CONTINUOUS_ON_SUB THEN
        REWRITE_TAC[CONTINUOUS_ON_CONST] THEN
        REWRITE_TAC[real_div; REWRITE_RULE[REAL_MUL_SYM] LIFT_CMUL] THEN
        MATCH_MP_TAC CONTINUOUS_ON_VMUL THEN
        GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [GSYM o_DEF] THEN
        REWRITE_TAC[o_ASSOC] THEN MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN
        GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [GSYM o_DEF] THEN
        SIMP_TAC[LINEAR_CONTINUOUS_ON; LINEAR_SNDCART; CONTINUOUS_ON_COMPOSE;
                 CONTINUOUS_ON_CNJ] THEN
        REWRITE_TAC[o_DEF; GSYM CONTINUOUS_ON_CX_LIFT] THEN
        MP_TAC CONTINUOUS_ON_UPPERHALF_ARG THEN REWRITE_TAC[o_DEF] THEN
        MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] CONTINUOUS_ON_SUBSET) THEN
        REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_PASTECART; IN_INTER;
                    PASTECART_IN_PCROSS; IN_ELIM_THM; SNDCART_PASTECART] THEN
        MAP_EVERY X_GEN_TAC [`t:real^1`; `z:complex`] THEN
        SIMP_TAC[IN_DIFF; IN_ELIM_THM; IN_SING] THEN
        SIMP_TAC[IM_CNJ; REAL_NEG_GE0; CNJ_EQ_0] THEN
        ASM_CASES_TAC `z = Cx(&0)` THEN ASM_REWRITE_TAC[COMPLEX_NORM_0] THEN
        REWRITE_TAC[REAL_OF_NUM_EQ; ARITH_EQ];
        FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ]
          CONTINUOUS_ON_SUBSET)) THEN
        REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_PASTECART; IN_INTER;
                    PASTECART_IN_PCROSS; IN_ELIM_THM; SNDCART_PASTECART;
                    FSTCART_PASTECART] THEN
        SIMP_TAC[IN_INTERVAL_1; DROP_SUB; DROP_VEC; LIFT_DROP] THEN
        REWRITE_TAC[REAL_ARITH `&0 <= &1 - x /\ &1 - x <= &1 <=>
                                &0 <= x /\ x <= &1`] THEN
        SIMP_TAC[REAL_LE_LDIV_EQ; REAL_LE_RDIV_EQ; PI_POS; REAL_MUL_LZERO;
             REAL_MUL_LID; REAL_ARITH `&0 < &2 * x <=> &0 < x`] THEN
        SIMP_TAC[ARG; REAL_LT_IMP_LE]]];
    REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; FORALL_IN_PCROSS; IN_ELIM_THM] THEN
    REWRITE_TAC[FSTCART_PASTECART; SNDCART_PASTECART] THEN
    REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
     `IMAGE h s SUBSET t ==> y IN s ==> h y IN t`)) THEN
    ASM_REWRITE_TAC[PASTECART_IN_PCROSS; IN_INTERVAL_1; LIFT_DROP] THEN
    SIMP_TAC[DROP_VEC; REAL_LE_LDIV_EQ; REAL_LE_RDIV_EQ; PI_POS;
             REAL_ARITH `&0 < &2 * x <=> &0 < x`] THEN
    SIMP_TAC[REAL_MUL_LZERO; REAL_MUL_LID; ARG; REAL_LT_IMP_LE]]);;
let SIMPLY_CONNECTED_EQ_HOMOTOPIC_CIRCLEMAPS, SIMPLY_CONNECTED_EQ_CONTRACTIBLE_CIRCLEMAP = (CONJ_PAIR o prove) (`(!s:real^N->bool. simply_connected s <=> !f g:complex->real^N. f continuous_on sphere(vec 0,&1) /\ IMAGE f (sphere(vec 0,&1)) SUBSET s /\ g continuous_on sphere(vec 0,&1) /\ IMAGE g (sphere(vec 0,&1)) SUBSET s ==> homotopic_with (\h. T) (sphere(vec 0,&1),s) f g) /\ (!s:real^N->bool. simply_connected s <=> path_connected s /\ !f:real^2->real^N. f continuous_on sphere(vec 0,&1) /\ IMAGE f (sphere(vec 0,&1)) SUBSET s ==> ?a. homotopic_with (\h. T) (sphere(vec 0,&1),s) f (\x. a))`, REWRITE_TAC[AND_FORALL_THM] THEN GEN_TAC THEN MATCH_MP_TAC(TAUT `(p ==> q) /\ (q ==> r) /\ (r ==> p) ==> (p <=> q) /\ (p <=> r)`) THEN REPEAT CONJ_TAC THENL [REWRITE_TAC[simply_connected] THEN DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`f:complex->real^N`; `g:complex->real^N`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`(f:complex->real^N) o cexp o (\t. Cx(&2 * pi * drop t) * ii)`; `(g:complex->real^N) o cexp o (\t. Cx(&2 * pi * drop t) * ii)`]) THEN ONCE_REWRITE_TAC[TAUT `p1 /\ q1 /\ r1 /\ p2 /\ q2 /\ r2 <=> (p1 /\ r1 /\ q1) /\ (p2 /\ r2 /\ q2)`] THEN REWRITE_TAC[GSYM HOMOTOPIC_LOOPS_REFL] THEN ASM_SIMP_TAC[HOMOTOPIC_CIRCLEMAPS_IMP_HOMOTOPIC_LOOPS; HOMOTOPIC_WITH_REFL] THEN DISCH_THEN(MP_TAC o MATCH_MP HOMOTOPIC_LOOPS_IMP_HOMOTOPIC_CIRCLEMAPS) THEN MATCH_MP_TAC(ONCE_REWRITE_RULE[IMP_CONJ_ALT] HOMOTOPIC_WITH_EQ) THEN REWRITE_TAC[IN_SPHERE_0; LIFT_DROP; o_DEF] THEN X_GEN_TAC `z:complex` THEN REPEAT STRIP_TAC THEN AP_TERM_TAC THEN MP_TAC(SPEC `z:complex` ARG) THEN ASM_REWRITE_TAC[COMPLEX_MUL_LID] THEN DISCH_THEN(STRIP_ASSUME_TAC o GSYM) THEN SIMP_TAC[PI_POS; REAL_FIELD `&0 < pi ==> &2 * pi * x / (&2 * pi) = x`] THEN ASM_MESON_TAC[COMPLEX_MUL_SYM]; DISCH_TAC THEN CONJ_TAC THENL [REWRITE_TAC[PATH_CONNECTED_EQ_HOMOTOPIC_POINTS] THEN MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`(\x. a):complex->real^N`; `(\x. b):complex->real^N`]) THEN REWRITE_TAC[CONTINUOUS_ON_CONST] THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN DISCH_THEN (MP_TAC o MATCH_MP HOMOTOPIC_CIRCLEMAPS_IMP_HOMOTOPIC_LOOPS) THEN REWRITE_TAC[o_DEF; LINEPATH_REFL]; X_GEN_TAC `f:complex->real^N` THEN STRIP_TAC THEN EXISTS_TAC `f(Cx(&1)):real^N` THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[CONTINUOUS_ON_CONST] THEN RULE_ASSUM_TAC(REWRITE_RULE[SUBSET; FORALL_IN_IMAGE; IN_SPHERE_0]) THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_SPHERE_0] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[COMPLEX_NORM_CX] THEN REAL_ARITH_TAC]; STRIP_TAC THEN ASM_REWRITE_TAC[SIMPLY_CONNECTED_EQ_CONTRACTIBLE_LOOP_SOME] THEN X_GEN_TAC `p:real^1->real^N` THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `(p:real^1->real^N) o (\z. lift(Arg z / (&2 * pi)))`) THEN ANTS_TAC THENL [MP_TAC(ISPECL [`s:real^N->bool`; `p:real^1->real^N`] HOMOTOPIC_LOOPS_REFL) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o MATCH_MP HOMOTOPIC_LOOPS_IMP_HOMOTOPIC_CIRCLEMAPS) THEN SIMP_TAC[HOMOTOPIC_WITH_REFL]; MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `a:real^N` THEN STRIP_TAC THEN FIRST_ASSUM (MP_TAC o MATCH_MP HOMOTOPIC_CIRCLEMAPS_IMP_HOMOTOPIC_LOOPS) THEN FIRST_ASSUM(MP_TAC o MATCH_MP HOMOTOPIC_WITH_IMP_SUBSET) THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_SPHERE_0; o_DEF] THEN DISCH_THEN(MP_TAC o SPEC `Cx(&1)` o CONJUNCT2) THEN REWRITE_TAC[COMPLEX_NORM_CX; REAL_ABS_NUM] THEN STRIP_TAC THEN ASM_REWRITE_TAC[LINEPATH_REFL] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] HOMOTOPIC_LOOPS_TRANS) THEN MATCH_MP_TAC HOMOTOPIC_LOOPS_EQ THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[IN_INTERVAL_1; FORALL_LIFT; LIFT_DROP; DROP_VEC] THEN X_GEN_TAC `t:real` THEN STRIP_TAC THEN ASM_CASES_TAC `t = &1` THENL [ASM_REWRITE_TAC[REAL_ARITH `&2 * pi * &1 = &2 * &1 * pi`] THEN SIMP_TAC[CEXP_INTEGER_2PI; INTEGER_CLOSED; ARG_NUM] THEN REWRITE_TAC[real_div; REAL_MUL_LZERO; LIFT_NUM] THEN ASM_MESON_TAC[pathstart; pathfinish]; AP_TERM_TAC THEN AP_TERM_TAC THEN SIMP_TAC[PI_POS; REAL_FIELD `&0 < pi ==> (t = x / (&2 * pi) <=> x = &2 * pi * t)`] THEN MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `Im(Cx (&2 * pi * t) * ii)` THEN CONJ_TAC THENL [MATCH_MP_TAC ARG_CEXP; ALL_TAC] THEN SIMP_TAC[IM_MUL_II; RE_CX; REAL_ARITH `a < &2 * pi <=> a < &2 * pi * &1`] THEN ASM_SIMP_TAC[REAL_LE_MUL; REAL_LT_LMUL_EQ; REAL_OF_NUM_LT; ARITH; PI_POS; REAL_LT_IMP_LE; REAL_POS; REAL_LE_MUL] THEN ASM_REWRITE_TAC[REAL_LT_LE]]]]);;
let HOMOTOPY_EQUIVALENT_SIMPLE_CONNECTEDNESS = 
prove (`!s:real^M->bool t:real^N->bool. s homotopy_equivalent t ==> (simply_connected s <=> simply_connected t)`,
REWRITE_TAC[SIMPLY_CONNECTED_EQ_HOMOTOPIC_CIRCLEMAPS] THEN REWRITE_TAC[HOMOTOPY_EQUIVALENT_HOMOTOPIC_TRIVIALITY]);;
(* ------------------------------------------------------------------------- *) (* Homeomorphism of simple closed curves to circles. *) (* ------------------------------------------------------------------------- *)
let HOMEOMORPHIC_SIMPLE_PATH_IMAGE_CIRCLE = 
prove (`!g:real^1->real^N a:real^2 r. simple_path g /\ pathfinish g = pathstart g /\ &0 < r ==> (path_image g) homeomorphic sphere(a,r)`,
REPEAT STRIP_TAC THEN TRANS_TAC HOMEOMORPHIC_TRANS `sphere(vec 0:real^2,&1)` THEN ASM_SIMP_TAC[HOMEOMORPHIC_SPHERES; REAL_LT_01] THEN MP_TAC(ISPECL [`g:real^1->real^N`; `g:real^1->real^N`; `path_image(g:real^1->real^N)`] HOMOTOPIC_LOOPS_IMP_HOMOTOPIC_CIRCLEMAPS) THEN REWRITE_TAC[HOMOTOPIC_LOOPS_REFL; HOMOTOPIC_WITH_REFL; SUBSET_REFL] THEN ASM_SIMP_TAC[SIMPLE_PATH_IMP_PATH] THEN STRIP_TAC THEN ONCE_REWRITE_TAC[HOMEOMORPHIC_SYM] THEN REWRITE_TAC[homeomorphic] THEN EXISTS_TAC `(g:real^1->real^N) o (\z. lift(Arg z / (&2 * pi)))` THEN MATCH_MP_TAC HOMEOMORPHISM_COMPACT THEN ASM_REWRITE_TAC[COMPACT_SPHERE] THEN CONJ_TAC THENL [MATCH_MP_TAC SUBSET_ANTISYM THEN ASM_REWRITE_TAC[] THEN REWRITE_T