(* ========================================================================= *)
(* Complex path integrals and Cauchy's theorem. *)
(* *)
(* (c) Copyright, John Harrison 1998-2008 *)
(* (c) Copyright, Gianni Ciolli, Graziano Gentili, Marco Maggesi 2008-2009. *)
(* (c) Copyright, Valentina Bruno 2010 *)
(* ========================================================================= *)
needs "Library/binomial.ml";;
needs "Library/iter.ml";;
needs "Multivariate/moretop.ml";;
prioritize_complex();;
(* ------------------------------------------------------------------------- *)
(* A couple of extra tactics used in some proofs below. *)
(* ------------------------------------------------------------------------- *)
let ASSERT_TAC tm =
SUBGOAL_THEN tm STRIP_ASSUME_TAC;;
let EQ_TRANS_TAC tm =
MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC tm THEN CONJ_TAC;;
(* ------------------------------------------------------------------------- *)
(* Piecewise differentiability on a 1-D interval. The definition doesn't *)
(* tie it to real^1 but it's not obviously that useful elsewhere. *)
(* ------------------------------------------------------------------------- *)
parse_as_infix("piecewise_differentiable_on",(12,"right"));;
(* ------------------------------------------------------------------------- *)
(* Valid paths, and their start and finish. *)
(* ------------------------------------------------------------------------- *)
(* ------------------------------------------------------------------------- *)
(* In particular, all results for paths apply. *)
(* ------------------------------------------------------------------------- *)
(* ------------------------------------------------------------------------- *)
(* Theorems about rectifiable valid paths. *)
(* ------------------------------------------------------------------------- *)
(* ------------------------------------------------------------------------- *)
(* Negligibility of valid_path image *)
(* ------------------------------------------------------------------------- *)
(* ------------------------------------------------------------------------- *)
(* Integrals along a path (= piecewise differentiable function on [0,1]). *)
(* ------------------------------------------------------------------------- *)
parse_as_infix("has_path_integral",(12,"right"));;
parse_as_infix("path_integrable_on",(12,"right"));;
(* ------------------------------------------------------------------------- *)
(* Show that we can forget about the localized derivative. *)
(* ------------------------------------------------------------------------- *)
(* ------------------------------------------------------------------------- *)
(* Reversing a path. *)
(* ------------------------------------------------------------------------- *)
(* ------------------------------------------------------------------------- *)
(* Joining two paths together. *)
(* ------------------------------------------------------------------------- *)
(* ------------------------------------------------------------------------- *)
(* Reparametrizing to shift the starting point of a (closed) path. *)
(* ------------------------------------------------------------------------- *)
(* ------------------------------------------------------------------------- *)
(* More about straight-line paths. *)
(* ------------------------------------------------------------------------- *)
(* ------------------------------------------------------------------------- *)
(* Relation to subpath construction. *)
(* ------------------------------------------------------------------------- *)
(* ------------------------------------------------------------------------- *)
(* Easier to reason about segments via convex hulls. *)
(* ------------------------------------------------------------------------- *)
let SEGMENTS_SUBSET_CONVEX_HULL = prove
(`!a b c. segment[a,b]
SUBSET (convex hull {a,b,c}) /\
segment[a,c]
SUBSET (convex hull {a,b,c}) /\
segment[b,c]
SUBSET (convex hull {a,b,c}) /\
segment[b,a]
SUBSET (convex hull {a,b,c}) /\
segment[c,a]
SUBSET (convex hull {a,b,c}) /\
segment[c,b]
SUBSET (convex hull {a,b,c})`,
let NORM_SUM_LEMMA = prove
(`norm(a + b + c + d:complex) >= e
==> norm(a) >= e / &4 \/
norm(b) >= e / &4 \/
norm(c) >= e / &4 \/
norm(d) >= e / &4`,
NORM_ARITH_TAC);;
let CAUCHY_THEOREM_QUADRISECTION = prove
(`!f a b c e K.
f
continuous_on (convex hull {a,b,c}) /\
dist (a,b) <= K /\
dist (b,c) <= K /\
dist (c,a) <= K /\
norm(
path_integral(linepath(a,b)) f +
path_integral(linepath(b,c)) f +
path_integral(linepath(c,a)) f) >= e * K pow 2
==> ?a' b' c'. a'
IN convex hull {a,b,c} /\
b'
IN convex hull {a,b,c} /\
c'
IN convex hull {a,b,c} /\
dist(a',b') <= K / &2 /\
dist(b',c') <= K / &2 /\
dist(c',a') <= K / &2 /\
norm(
path_integral(linepath(a',b')) f +
path_integral(linepath(b',c')) f +
path_integral(linepath(c',a')) f)
>= e * (K / &2) pow 2`,
REPEAT STRIP_TAC THEN MAP_EVERY ABBREV_TAC
[`a':complex = midpoint(b,c)`;
`b':complex = midpoint(c,a)`;
`c':complex = midpoint(a,b)`] THEN
SUBGOAL_THEN
`
path_integral(linepath(a,b)) f +
path_integral(linepath(b,c)) f +
path_integral(linepath(c,a)) f =
(
path_integral(linepath(a,c')) f +
path_integral(linepath(c',b')) f +
path_integral(linepath(b',a)) f) +
(
path_integral(linepath(a',c')) f +
path_integral(linepath(c',b)) f +
path_integral(linepath(b,a')) f) +
(
path_integral(linepath(a',c)) f +
path_integral(linepath(c,b')) f +
path_integral(linepath(b',a')) f) +
(
path_integral(linepath(a',b')) f +
path_integral(linepath(b',c')) f +
path_integral(linepath(c',a')) f)`
SUBST_ALL_TAC THENL
[MP_TAC(SPEC `f:complex->complex`
PATH_INTEGRAL_MIDPOINT) THEN DISCH_THEN
(fun th -> MP_TAC(SPECL [`a:complex`; `b:complex`] th) THEN
MP_TAC(SPECL [`b:complex`; `c:complex`] th) THEN
MP_TAC(SPECL [`c:complex`; `a:complex`] th)) THEN
MP_TAC(SPEC `f:complex->complex`
PATH_INTEGRAL_REVERSE_LINEPATH) THEN DISCH_THEN
(fun th -> MP_TAC(SPECL [`a':complex`; `b':complex`] th) THEN
MP_TAC(SPECL [`b':complex`; `c':complex`] th) THEN
MP_TAC(SPECL [`c':complex`; `a':complex`] th)) THEN
ASM_REWRITE_TAC[] THEN
REPEAT(MATCH_MP_TAC(TAUT
`((a /\ c ==> b /\ d) ==> e) ==> (a ==> b) ==> (c ==> d) ==> e`)) THEN
ANTS_TAC THENL [ALL_TAC; CONV_TAC COMPLEX_RING] THEN
REWRITE_TAC[
SEGMENT_CONVEX_HULL] THEN
REPEAT CONJ_TAC THEN MATCH_MP_TAC
CONTINUOUS_ON_SUBSET THEN
EXISTS_TAC `convex hull {a:complex,b,c}` THEN ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC
CONVEX_HULL_SUBSET THEN
SIMP_TAC[
IN_INSERT;
NOT_IN_EMPTY;
TAUT `(a \/ b ==> c) <=> (a ==> c) /\ (b ==> c)`] THEN
MAP_EVERY EXPAND_TAC ["a'";
let TRIANGLE_POINTS_CLOSER = prove
(`!a b c x y:real^N.
x
IN convex hull {a,b,c} /\
y
IN convex hull {a,b,c}
==> norm(x - y) <= norm(a - b) \/
norm(x - y) <= norm(b - c) \/
norm(x - y) <= norm(c - a)`,
let HOLOMORPHIC_POINT_SMALL_TRIANGLE = prove
(`!f s x e.
x
IN s /\ f
continuous_on s /\
f
complex_differentiable (at x within s) /\
&0 < e
==> ?k. &0 < k /\
!a b c. dist(a,b) <= k /\ dist(b,c) <= k /\ dist(c,a) <= k /\
x
IN convex hull {a,b,c} /\ convex hull {a,b,c}
SUBSET s
==> norm(
path_integral(linepath(a,b)) f +
path_integral(linepath(b,c)) f +
path_integral(linepath(c,a)) f)
<= e * (dist(a,b) + dist(b,c) + dist(c,a)) pow 2`,
REPEAT STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
complex_differentiable]) THEN
DISCH_THEN(X_CHOOSE_THEN `f':complex` MP_TAC) THEN
GEN_REWRITE_TAC LAND_CONV [
has_complex_derivative] THEN
REWRITE_TAC[
HAS_DERIVATIVE_WITHIN_ALT] THEN
DISCH_THEN(MP_TAC o SPEC `e:real` o CONJUNCT2) THEN
ASM_SIMP_TAC[
REAL_LT_DIV;
REAL_OF_NUM_LT; ARITH] THEN
GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV)
[TAUT `a /\ b ==> c <=> b ==> a ==> c`] THEN
REWRITE_TAC[
APPROACHABLE_LT_LE] THEN
ONCE_REWRITE_TAC[TAUT `b ==> a ==> c <=> a /\ b ==> c`] THEN
MATCH_MP_TAC
MONO_EXISTS THEN X_GEN_TAC `d:real` THEN STRIP_TAC THEN
ASM_REWRITE_TAC[dist] THEN
MAP_EVERY X_GEN_TAC [`a:complex`; `b:complex`; `c:complex`] THEN
STRIP_TAC THEN
SUBGOAL_THEN
`
path_integral (linepath(a,b)) f +
path_integral (linepath(b,c)) f +
path_integral (linepath(c,a)) f =
path_integral (linepath(a,b)) (\y. f y - f x - f' * (y - x)) +
path_integral (linepath(b,c)) (\y. f y - f x - f' * (y - x)) +
path_integral (linepath(c,a)) (\y. f y - f x - f' * (y - x))`
SUBST1_TAC THENL
[SUBGOAL_THEN
`
path_integral (linepath(a,b)) (\y. f y - f x - f' * (y - x)) =
path_integral (linepath(a,b)) f -
path_integral (linepath(a,b)) (\y. f x + f' * (y - x)) /\
path_integral (linepath(b,c)) (\y. f y - f x - f' * (y - x)) =
path_integral (linepath(b,c)) f -
path_integral (linepath(b,c)) (\y. f x + f' * (y - x)) /\
path_integral (linepath(c,a)) (\y. f y - f x - f' * (y - x)) =
path_integral (linepath(c,a)) f -
path_integral (linepath(c,a)) (\y. f x + f' * (y - x))`
(REPEAT_TCL CONJUNCTS_THEN SUBST1_TAC) THENL
[REWRITE_TAC[SIMPLE_COMPLEX_ARITH `a - b - c = a - (b + c)`] THEN
REPEAT CONJ_TAC THEN MATCH_MP_TAC
PATH_INTEGRAL_UNIQUE THEN
MATCH_MP_TAC
HAS_PATH_INTEGRAL_SUB THEN
CONJ_TAC THEN MATCH_MP_TAC
HAS_PATH_INTEGRAL_INTEGRAL THEN
MATCH_MP_TAC
PATH_INTEGRABLE_CONTINUOUS_LINEPATH THEN
MATCH_MP_TAC
CONTINUOUS_ON_SUBSET THEN
EXISTS_TAC `s:complex->bool` THEN
ASM_SIMP_TAC[
CONTINUOUS_ON_ID;
CONTINUOUS_ON_ADD;
CONTINUOUS_ON_CONST;
CONTINUOUS_ON_COMPLEX_MUL;
CONTINUOUS_ON_SUB] THEN
MATCH_MP_TAC
SUBSET_TRANS THEN
EXISTS_TAC `convex hull {a:complex,b,c}` THEN
ASM_REWRITE_TAC[
SEGMENT_CONVEX_HULL] THEN
MATCH_MP_TAC
CONVEX_HULL_SUBSET THEN
REWRITE_TAC[
IN_INSERT;
NOT_IN_EMPTY] THEN REPEAT STRIP_TAC THEN
ASM_SIMP_TAC[
MIDPOINTS_IN_CONVEX_HULL;
POINTS_IN_CONVEX_HULL;
IN_INSERT];
ALL_TAC] THEN
REWRITE_TAC[COMPLEX_RING
`x + y + z = (x - x') + (y - y') + (z - z') <=>
x' + y' + z' = Cx(&0)`] THEN
MP_TAC(ISPECL [`a:complex`; `b:complex`; `c:complex`;
`f':complex`; `f x - f' * x`]
TRIANGLE_LINEAR_HAS_CHAIN_INTEGRAL) THEN
REWRITE_TAC[COMPLEX_RING
`f' * x' + f x - f' * x = f x + f' * (x' - x)`] THEN
DISCH_THEN(MP_TAC o MATCH_MP
HAS_CHAIN_INTEGRAL_CHAIN_INTEGRAL) THEN
REWRITE_TAC[];
ALL_TAC] THEN
ONCE_REWRITE_TAC[
NORM_SUB] THEN MATCH_MP_TAC(REAL_ARITH
`&0 <= x * y /\ &0 <= x * z /\ &0 <= y * z /\
a <= (e * (x + y + z)) * x +
(e * (x + y + z)) * y +
(e * (x + y + z)) * z
==> a <= e * (x + y + z) pow 2`) THEN
SIMP_TAC[
REAL_LE_MUL;
NORM_POS_LE] THEN
REPEAT(MATCH_MP_TAC
NORM_TRIANGLE_LE THEN
MATCH_MP_TAC
REAL_LE_ADD2 THEN CONJ_TAC) THEN
(MATCH_MP_TAC
HAS_PATH_INTEGRAL_BOUND_LINEPATH THEN
EXISTS_TAC `\y:complex. f y - f x - f' * (y - x)` THEN
ASM_SIMP_TAC[
REAL_LE_MUL;
REAL_LE_ADD;
REAL_LT_IMP_LE;
NORM_POS_LE] THEN
CONJ_TAC THENL
[MATCH_MP_TAC
HAS_PATH_INTEGRAL_INTEGRAL THEN
MATCH_MP_TAC
PATH_INTEGRABLE_CONTINUOUS_LINEPATH THEN
MATCH_MP_TAC
CONTINUOUS_ON_SUBSET THEN EXISTS_TAC `s:complex->bool` THEN
ASM_SIMP_TAC[
CONTINUOUS_ON_SUB; ETA_AX;
CONTINUOUS_ON_COMPLEX_MUL;
CONTINUOUS_ON_CONST;
CONTINUOUS_ON_ID] THEN
MATCH_MP_TAC
SUBSET_TRANS THEN
EXISTS_TAC `convex hull {a:complex,b,c}` THEN
ASM_REWRITE_TAC[
SEGMENT_CONVEX_HULL] THEN
MATCH_MP_TAC
CONVEX_HULL_SUBSET THEN
REWRITE_TAC[
IN_INSERT;
NOT_IN_EMPTY] THEN REPEAT STRIP_TAC THEN
ASM_SIMP_TAC[
MIDPOINTS_IN_CONVEX_HULL;
POINTS_IN_CONVEX_HULL;
IN_INSERT];
ALL_TAC] THEN
X_GEN_TAC `y:complex` THEN STRIP_TAC THEN
MATCH_MP_TAC
REAL_LE_TRANS THEN
EXISTS_TAC `e * norm(y - x:complex)` THEN CONJ_TAC THENL
[FIRST_X_ASSUM MATCH_MP_TAC THEN CONJ_TAC THENL
[MATCH_MP_TAC(SET_RULE `!t. y
IN t /\ t
SUBSET s ==> y
IN s`) THEN
EXISTS_TAC `convex hull {a:complex,b,c}` THEN ASM_REWRITE_TAC[];
MATCH_MP_TAC(REAL_ARITH
`!n1 n2 n3. n1 <= d /\ n2 <= d /\ n3 <= d /\
(n <= n1 \/ n <= n2 \/ n <= n3)
==> n <= d`) THEN
MAP_EVERY EXISTS_TAC
[`norm(a - b:complex)`; `norm(b - c:complex)`;
`norm(c - a:complex)`] THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC
TRIANGLE_POINTS_CLOSER];
ASM_SIMP_TAC[
REAL_LE_LMUL_EQ] THEN
ONCE_REWRITE_TAC[
NORM_SUB] THEN
MATCH_MP_TAC(REAL_ARITH
`(x <= a \/ x <= b \/ x <= c) /\ (&0 <= a /\ &0 <= b /\ &0 <= c)
==> x <= a + b + c`) THEN
REWRITE_TAC[
NORM_POS_LE] THEN MATCH_MP_TAC
TRIANGLE_POINTS_CLOSER THEN
ASM_REWRITE_TAC[]] THEN
REPEAT CONJ_TAC THEN
FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
`x
IN s ==> s
SUBSET t ==> x
IN t`)) THEN
REWRITE_TAC[
SEGMENT_CONVEX_HULL] THEN MATCH_MP_TAC
CONVEX_HULL_SUBSET THEN
REWRITE_TAC[
IN_INSERT;
NOT_IN_EMPTY] THEN REPEAT STRIP_TAC THEN
ASM_SIMP_TAC[
MIDPOINTS_IN_CONVEX_HULL;
POINTS_IN_CONVEX_HULL;
IN_INSERT]));;
let CAUCHY_THEOREM_TRIANGLE = prove
(`!f a b c.
f
holomorphic_on (convex hull {a,b,c})
==> (f
has_path_integral Cx(&0))
(linepath(a,b) ++ linepath(b,c) ++ linepath(c,a))`,
let lemma1 = prove
(`!P Q abc.
P abc 0 /\
(!abc:A n. P abc n ==> ?abc'. P abc' (SUC n) /\ Q abc' abc)
==> ?ABC. ABC 0 = abc /\ !n. P (ABC n) n /\ Q (ABC(SUC n)) (ABC n)`,
REPEAT STRIP_TAC THEN
(MP_TAC o prove_recursive_functions_exist num_RECURSION)
`ABC 0 = abc:A /\
!n. ABC(SUC n) = @abc. P abc (SUC n) /\ Q abc (ABC n)` THEN
MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN
STRIP_TAC THEN CONJ_TAC THENL [ASM_REWRITE_TAC[]; ALL_TAC] THEN
REWRITE_TAC[FORALL_AND_THM] THEN
MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN CONJ_TAC THENL
[INDUCT_TAC THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[]; ALL_TAC] THEN
DISCH_TAC THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[]) in
let lemma3 = prove
(`!P Q a:A b:A c:A.
P a b c 0 /\
(!a b c n. P a b c n
==> ?a' b' c'. P a' b' c' (SUC n) /\ Q a' b' c' a b c)
==> ?A B C. A 0 = a /\ B 0 = b /\ C 0 = c /\
!n. P (A n) (B n) (C n) n /\
Q (A(SUC n)) (B(SUC n)) (C(SUC n)) (A n) (B n) (C n)`,
REPEAT STRIP_TAC THEN
MP_TAC(ISPECL
[`\(a,b,c). (P:A->A->A->num->bool) a b c`;
`\(a,b,c) (a',b',c'). (Q:A->A->A->A->A->A->bool) a b c a' b' c'`;
`(a:A,b:A,c:A)`]
lemma1) THEN
REWRITE_TAC[FORALL_PAIR_THM; EXISTS_PAIR_THM] THEN
ANTS_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
DISCH_THEN(X_CHOOSE_THEN `ABC:num->A#A#A` STRIP_ASSUME_TAC) THEN
MAP_EVERY EXISTS_TAC
[`(\(a,b,c). a) o (ABC:num->A#A#A)`;
`(\(a,b,c). b) o (ABC:num->A#A#A)`;
`(\(a,b,c). c) o (ABC:num->A#A#A)`] THEN
REWRITE_TAC[o_THM] THEN
REPEAT(CONJ_TAC THENL [ASM_REWRITE_TAC[]; ALL_TAC]) THEN
X_GEN_TAC `n:num` THEN
FIRST_X_ASSUM(MP_TAC o SPEC `n:num`) THEN
SPEC_TAC(`(ABC:num->A#A#A) (SUC n)`,`y:A#A#A`) THEN
SPEC_TAC(`(ABC:num->A#A#A) n`,`x:A#A#A`) THEN
REWRITE_TAC[FORALL_PAIR_THM]) in
REPEAT STRIP_TAC THEN
STRIP_ASSUME_TAC(ISPECL [`a:complex`; `b:complex`; `c:complex`]
SEGMENTS_SUBSET_CONVEX_HULL) THEN
FIRST_ASSUM(ASSUME_TAC o MATCH_MP HOLOMORPHIC_ON_IMP_CONTINUOUS_ON) THEN
SUBGOAL_THEN
`f path_integrable_on (linepath(a,b) ++ linepath(b,c) ++ linepath(c,a))`
MP_TAC THENL
[SIMP_TAC[PATH_INTEGRABLE_JOIN; VALID_PATH_JOIN; VALID_PATH_LINEPATH;
PATHSTART_JOIN; PATHFINISH_JOIN; PATHSTART_LINEPATH;
PATHFINISH_LINEPATH] THEN
ASM_MESON_TAC[PATH_INTEGRABLE_CONTINUOUS_LINEPATH; CONTINUOUS_ON_SUBSET];
ALL_TAC] THEN
SIMP_TAC[path_integrable_on] THEN DISCH_THEN(X_CHOOSE_TAC `y:complex`) THEN
ASM_CASES_TAC `y = Cx(&0)` THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
ABBREV_TAC
`K = &1 + max (dist(a:complex,b)) (max (dist(b,c)) (dist(c,a)))` THEN
SUBGOAL_THEN `&0 < K` ASSUME_TAC THENL
[EXPAND_TAC "K" THEN MATCH_MP_TAC(REAL_ARITH `&0 <= x ==> &0 < &1 + x`) THEN
REWRITE_TAC[REAL_LE_MAX; DIST_POS_LE];
ALL_TAC] THEN
ABBREV_TAC `e = norm(y:complex) / K pow 2` THEN
SUBGOAL_THEN `&0 < e` ASSUME_TAC THENL
[EXPAND_TAC "e" THEN
ASM_SIMP_TAC[REAL_LT_DIV; REAL_POW_LT; COMPLEX_NORM_NZ];
ALL_TAC] THEN
SUBGOAL_THEN
`?A B C. A 0 = a /\ B 0 = b /\ C 0 = c /\
!n. (convex hull {A n,B n,C n} SUBSET convex hull {a,b,c} /\
dist(A n,B n) <= K / &2 pow n /\
dist(B n,C n) <= K / &2 pow n /\
dist(C n,A n) <= K / &2 pow n /\
norm(path_integral(linepath (A n,B n)) f +
path_integral(linepath (B n,C n)) f +
path_integral(linepath (C n,A n)) f) >=
e * (K / &2 pow n) pow 2) /\
convex hull {A(SUC n),B(SUC n),C(SUC n)} SUBSET
convex hull {A n,B n,C n}`
MP_TAC THENL
[MATCH_MP_TAC lemma3 THEN CONJ_TAC THENL
[ASM_REWRITE_TAC[real_pow; REAL_DIV_1; CONJ_ASSOC; SUBSET_REFL] THEN
CONJ_TAC THENL [EXPAND_TAC "K" THEN REAL_ARITH_TAC; ALL_TAC] THEN
EXPAND_TAC "e" THEN
ASM_SIMP_TAC[REAL_DIV_RMUL; REAL_LT_IMP_NZ; REAL_POW_LT] THEN
MATCH_MP_TAC(REAL_ARITH `x = y ==> x >= y`) THEN AP_TERM_TAC THEN
FIRST_ASSUM(SUBST1_TAC o SYM o
MATCH_MP HAS_CHAIN_INTEGRAL_CHAIN_INTEGRAL) THEN
REWRITE_TAC[];
ALL_TAC] THEN
MAP_EVERY X_GEN_TAC
[`a':complex`; `b':complex`; `c':complex`; `n:num`] THEN
REPEAT STRIP_TAC THEN
MP_TAC(SPECL [`f:complex->complex`; `a':complex`; `b':complex`;
`c':complex`; `e:real`; `K / &2 pow n`]
CAUCHY_THEOREM_QUADRISECTION) THEN
ASM_REWRITE_TAC[] THEN
ANTS_TAC THENL [ASM_MESON_TAC[CONTINUOUS_ON_SUBSET]; ALL_TAC] THEN
REPEAT(MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC) THEN STRIP_TAC THEN
ASM_REWRITE_TAC[real_pow; REAL_FIELD `x / (&2 * y) = x / y / &2`] THEN
MATCH_MP_TAC(SET_RULE
`s SUBSET t /\ t SUBSET u ==> s SUBSET u /\ s SUBSET t`) THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC CONVEX_HULL_SUBSET THEN
ASM SET_TAC[];
ALL_TAC] THEN
REWRITE_TAC[FORALL_AND_THM] THEN STRIP_TAC THEN
SUBGOAL_THEN
`?x:complex. !n:num. x IN convex hull {A n,B n,C n}`
STRIP_ASSUME_TAC THENL
[MATCH_MP_TAC BOUNDED_CLOSED_NEST THEN REPEAT CONJ_TAC THENL
[GEN_TAC THEN MATCH_MP_TAC COMPACT_IMP_CLOSED;
REWRITE_TAC[CONVEX_HULL_EQ_EMPTY; NOT_INSERT_EMPTY];
MATCH_MP_TAC TRANSITIVE_STEPWISE_LE THEN ASM_REWRITE_TAC[] THEN
MESON_TAC[SUBSET_REFL; SUBSET_TRANS];
MATCH_MP_TAC COMPACT_IMP_BOUNDED] THEN
MATCH_MP_TAC FINITE_IMP_COMPACT_CONVEX_HULL THEN
REWRITE_TAC[FINITE_INSERT; FINITE_RULES];
ALL_TAC] THEN
MP_TAC(ISPECL [`f:complex->complex`; `convex hull {a:complex,b,c}`;
`x:complex`; `e / &10`] HOLOMORPHIC_POINT_SMALL_TRIANGLE) THEN
ANTS_TAC THENL
[ASM_SIMP_TAC[HOLOMORPHIC_ON_IMP_CONTINUOUS_ON; complex_differentiable] THEN
ASM_SIMP_TAC[REAL_LT_DIV; REAL_OF_NUM_LT; ARITH] THEN
ASM_MESON_TAC[holomorphic_on; SUBSET];
ALL_TAC] THEN
DISCH_THEN(X_CHOOSE_THEN `k:real` STRIP_ASSUME_TAC) THEN
MP_TAC(SPEC `K:real / k` REAL_ARCH_POW2) THEN
ASM_SIMP_TAC[REAL_LT_LDIV_EQ] THEN
ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
SIMP_TAC[GSYM REAL_LT_LDIV_EQ; REAL_POW_LT; REAL_OF_NUM_LT; ARITH] THEN
DISCH_THEN(X_CHOOSE_TAC `n:num`) THEN FIRST_X_ASSUM(MP_TAC o SPECL
[`(A:num->complex) n`; `(B:num->complex) n`; `(C:num->complex) n`]) THEN
ASM_REWRITE_TAC[] THEN ANTS_TAC THENL
[ASM_MESON_TAC[REAL_LE_TRANS; REAL_LT_IMP_LE]; ALL_TAC] THEN
ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN DISCH_THEN(K ALL_TAC) THEN
REWRITE_TAC[REAL_NOT_LE] THEN
MATCH_MP_TAC REAL_LTE_TRANS THEN
EXISTS_TAC `e * (K / &2 pow n) pow 2` THEN
CONJ_TAC THENL [ALL_TAC; ASM_REWRITE_TAC[GSYM real_ge]] THEN
ASM_SIMP_TAC[real_div; GSYM REAL_MUL_ASSOC; REAL_LT_LMUL_EQ] THEN
MATCH_MP_TAC(REAL_ARITH
`&0 < x /\ y <= &9 * x ==> inv(&10) * y < x`) THEN
ASM_SIMP_TAC[REAL_POW_LT; REAL_LT_MUL; REAL_LT_INV_EQ;
REAL_OF_NUM_LT; ARITH] THEN
REWRITE_TAC[REAL_ARITH `&9 * x pow 2 = (&3 * x) pow 2`] THEN
MATCH_MP_TAC REAL_POW_LE2 THEN
SIMP_TAC[REAL_LE_ADD; DIST_POS_LE; GSYM real_div] THEN
MATCH_MP_TAC(REAL_ARITH
`x <= a /\ y <= a /\ z <= a ==> x + y + z <= &3 * a`) THEN
ASM_REWRITE_TAC[]);;
let CAUCHY_THEOREM_TRIANGLE_INTERIOR = prove
(`!f a b c.
f
continuous_on (convex hull {a,b,c}) /\
f
holomorphic_on interior (convex hull {a,b,c})
==> (f
has_path_integral Cx(&0))
(linepath(a,b) ++ linepath(b,c) ++ linepath(c,a))`,
REPEAT STRIP_TAC THEN
STRIP_ASSUME_TAC(ISPECL [`a:complex`; `b:complex`; `c:complex`]
SEGMENTS_SUBSET_CONVEX_HULL) THEN
SUBGOAL_THEN
`?B. &0 < B /\
!y. y
IN IMAGE (f:complex->complex) (convex hull {a,b,c})
==> norm(y) <= B`
MP_TAC THENL
[REWRITE_TAC[GSYM
BOUNDED_POS] THEN MATCH_MP_TAC
COMPACT_IMP_BOUNDED THEN
MATCH_MP_TAC
COMPACT_CONTINUOUS_IMAGE THEN
ASM_SIMP_TAC[
FINITE_IMP_COMPACT_CONVEX_HULL;
FINITE_INSERT;
FINITE_RULES];
REWRITE_TAC[
FORALL_IN_IMAGE] THEN STRIP_TAC] THEN
SUBGOAL_THEN
`?C. &0 < C /\ !x:complex. x
IN convex hull {a,b,c} ==> norm(x) <= C`
MP_TAC THENL
[REWRITE_TAC[GSYM
BOUNDED_POS] THEN
MATCH_MP_TAC
COMPACT_IMP_BOUNDED THEN
ASM_SIMP_TAC[
FINITE_IMP_COMPACT_CONVEX_HULL;
FINITE_INSERT;
FINITE_RULES];
STRIP_TAC] THEN
SUBGOAL_THEN
`(f:complex->complex)
uniformly_continuous_on (convex hull {a,b,c})`
MP_TAC THENL
[MATCH_MP_TAC
COMPACT_UNIFORMLY_CONTINUOUS THEN
ASM_SIMP_TAC[
FINITE_IMP_COMPACT_CONVEX_HULL;
FINITE_RULES;
FINITE_INSERT];
ALL_TAC] THEN
REWRITE_TAC[
uniformly_continuous_on] THEN DISCH_TAC THEN
SUBGOAL_THEN
`f
path_integrable_on
(linepath (a,b) ++ linepath(b,c) ++ linepath(c,a))`
MP_TAC THENL
[SIMP_TAC[
PATH_INTEGRABLE_JOIN;
VALID_PATH_JOIN;
VALID_PATH_LINEPATH;
PATHSTART_JOIN;
PATHFINISH_JOIN;
PATHSTART_LINEPATH;
PATHFINISH_LINEPATH] THEN
ASM_MESON_TAC[
PATH_INTEGRABLE_CONTINUOUS_LINEPATH;
CONTINUOUS_ON_SUBSET];
ALL_TAC] THEN
SIMP_TAC[
path_integrable_on] THEN DISCH_THEN(X_CHOOSE_TAC `y:complex`) THEN
ASM_CASES_TAC `y = Cx(&0)` THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
UNDISCH_TAC `~(y = Cx(&0))` THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN
DISCH_THEN(K ALL_TAC) THEN REWRITE_TAC[] THEN
FIRST_ASSUM(ASSUME_TAC o SYM o MATCH_MP
HAS_CHAIN_INTEGRAL_CHAIN_INTEGRAL) THEN
ASM_REWRITE_TAC[] THEN
ASM_CASES_TAC `c:complex = a` THENL
[MATCH_MP_TAC
CAUCHY_THEOREM_FLAT THEN
EXISTS_TAC `&0` THEN ASM_REWRITE_TAC[
VECTOR_MUL_LZERO;
VECTOR_SUB_EQ];
ALL_TAC] THEN
ASM_CASES_TAC `b:complex = c` THENL
[ONCE_REWRITE_TAC[COMPLEX_RING `a + b + c:complex = c + a + b`] THEN
MATCH_MP_TAC
CAUCHY_THEOREM_FLAT THEN
EXISTS_TAC `&0` THEN ASM_REWRITE_TAC[
VECTOR_MUL_LZERO;
VECTOR_SUB_EQ] THEN
ASM_MESON_TAC[
INSERT_AC];
ALL_TAC] THEN
ASM_CASES_TAC `a:complex = b` THENL
[ONCE_REWRITE_TAC[COMPLEX_RING `a + b + c:complex = b + c + a`] THEN
MATCH_MP_TAC
CAUCHY_THEOREM_FLAT THEN
EXISTS_TAC `&0` THEN ASM_REWRITE_TAC[
VECTOR_MUL_LZERO;
VECTOR_SUB_EQ] THEN
ASM_MESON_TAC[
INSERT_AC];
ALL_TAC] THEN
ASM_CASES_TAC `interior(convex hull {a:complex,b,c}) = {}` THENL
[MATCH_MP_TAC
CAUCHY_THEOREM_FLAT THEN
SUBGOAL_THEN `{a:complex,b,c}
HAS_SIZE (dimindex(:2) + 1)`
MP_TAC THENL
[ASM_SIMP_TAC[
HAS_SIZE;
CARD_CLAUSES;
FINITE_INSERT;
FINITE_EMPTY] THEN
ASM_REWRITE_TAC[DIMINDEX_2; ARITH;
IN_INSERT;
NOT_IN_EMPTY];
ALL_TAC] THEN
DISCH_THEN(MP_TAC o MATCH_MP
INTERIOR_CONVEX_HULL_EQ_EMPTY) THEN
ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
SUBGOAL_THEN `collinear{a:complex,b,c}` MP_TAC THENL
[ASM_REWRITE_TAC[
COLLINEAR_3_EQ_AFFINE_DEPENDENT]; ALL_TAC] THEN
ONCE_REWRITE_TAC[SET_RULE `{a,b,c} = {b,a,c}`] THEN
ONCE_REWRITE_TAC[
COLLINEAR_3] THEN
ASM_REWRITE_TAC[
COLLINEAR_LEMMA;
VECTOR_SUB_EQ];
ALL_TAC] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM
MEMBER_NOT_EMPTY]) THEN
DISCH_THEN(X_CHOOSE_TAC `d:complex`) THEN FIRST_X_ASSUM(MP_TAC o SYM) THEN
DISCH_TAC THEN
ASM_REWRITE_TAC[] THEN ASM_CASES_TAC `y = Cx(&0)` THEN ASM_REWRITE_TAC[] THEN
FIRST_X_ASSUM(MP_TAC o SPEC `norm(y:complex) / &24 / C`) THEN
SUBGOAL_THEN `&0 < norm(y:complex) / &24 / C` ASSUME_TAC THENL
[ASM_SIMP_TAC[
REAL_LT_DIV;
REAL_OF_NUM_LT; ARITH;
NORM_POS_LE;
REAL_LTE_ADD;
COMPLEX_NORM_NZ;
COMPLEX_SUB_0];
ASM_REWRITE_TAC[dist]] THEN
DISCH_THEN(X_CHOOSE_THEN `d1:real` STRIP_ASSUME_TAC) THEN ABBREV_TAC
`e = min (&1)
(min (d1 / (&4 * C))
((norm(y:complex) / &24 / C) / B))` THEN
SUBGOAL_THEN `&0 < e` ASSUME_TAC THENL
[EXPAND_TAC "e" THEN
ASM_SIMP_TAC[
REAL_HALF;
REAL_LT_MIN;
REAL_LT_DIV;
COMPLEX_NORM_NZ;
REAL_LT_MUL;
REAL_OF_NUM_LT; ARITH];
ALL_TAC] THEN
ABBREV_TAC `shrink = \x:complex. x - e % (x - d)` THEN
SUBGOAL_THEN `shrink (a:complex)
IN interior(convex hull {a,b,c}) /\
shrink b
IN interior(convex hull {a,b,c}) /\
shrink c
IN interior(convex hull {a,b,c})`
STRIP_ASSUME_TAC THENL
[REPEAT CONJ_TAC THEN EXPAND_TAC "shrink" THEN
MATCH_MP_TAC
IN_INTERIOR_CONVEX_SHRINK THEN
ASM_REWRITE_TAC[
CONVEX_CONVEX_HULL] THEN
(CONJ_TAC THENL [ALL_TAC; EXPAND_TAC "e" THEN REAL_ARITH_TAC]) THEN
MATCH_MP_TAC(REWRITE_RULE[
SUBSET]
HULL_SUBSET) THEN
REWRITE_TAC[
IN_INSERT];
ALL_TAC] THEN
SUBGOAL_THEN
`norm((
path_integral(linepath(shrink a,shrink b)) f -
path_integral(linepath(a,b)) f) +
(
path_integral(linepath(shrink b,shrink c)) f -
path_integral(linepath(b,c)) f) +
(
path_integral(linepath(shrink c,shrink a)) f -
path_integral(linepath(c,a)) f)) <= norm(y:complex) / &2`
MP_TAC THENL
[ALL_TAC;
ASM_REWRITE_TAC[COMPLEX_RING
`(ab' - ab) + (bc' - bc) + (ca' - ca) =
(ab' + bc' + ca') - (ab + bc + ca)`] THEN
SUBGOAL_THEN
`(f
has_path_integral (Cx(&0)))
(linepath (shrink a,shrink b) ++
linepath (shrink b,shrink c) ++
linepath (shrink c,shrink (a:complex)))`
MP_TAC THENL
[MATCH_MP_TAC
CAUCHY_THEOREM_TRIANGLE THEN
MATCH_MP_TAC
HOLOMORPHIC_ON_SUBSET THEN
EXISTS_TAC `interior(convex hull {a:complex,b,c})` THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC
HULL_MINIMAL THEN
SIMP_TAC[
CONVEX_INTERIOR;
CONVEX_CONVEX_HULL] THEN
ASM SET_TAC[];
ALL_TAC] THEN
DISCH_THEN(MP_TAC o MATCH_MP
HAS_CHAIN_INTEGRAL_CHAIN_INTEGRAL) THEN
SIMP_TAC[] THEN DISCH_THEN(K ALL_TAC) THEN
REWRITE_TAC[
COMPLEX_SUB_LZERO;
NORM_NEG] THEN
MATCH_MP_TAC(REAL_ARITH `&0 <= y /\ ~(y = &0) ==> ~(y <= y / &2)`) THEN
ASM_REWRITE_TAC[
COMPLEX_NORM_ZERO;
NORM_POS_LE]] THEN
SUBGOAL_THEN
`!x y. x
IN convex hull {a,b,c} /\ y
IN convex hull {a,b,c}
==> norm(x - y) <= &2 * C`
ASSUME_TAC THENL
[REPEAT STRIP_TAC THEN REWRITE_TAC[
REAL_MUL_2;
VECTOR_SUB] THEN
MATCH_MP_TAC
NORM_TRIANGLE_LE THEN REWRITE_TAC[
NORM_NEG] THEN
MATCH_MP_TAC
REAL_LE_ADD2 THEN ASM_SIMP_TAC[];
ALL_TAC] THEN
REWRITE_TAC[REAL_ARITH `x / &2 = x / &6 + x / &6 + x / &6`] THEN
REPEAT(MATCH_MP_TAC
NORM_TRIANGLE_LE THEN
MATCH_MP_TAC
REAL_LE_ADD2 THEN CONJ_TAC) THEN
GEN_REWRITE_TAC RAND_CONV [GSYM
REAL_MUL_RID] THEN
GEN_REWRITE_TAC (RAND_CONV o RAND_CONV) [GSYM
CONTENT_UNIT_1] THEN
MATCH_MP_TAC
HAS_INTEGRAL_BOUND THENL
[EXISTS_TAC `\x. f(linepath(shrink a,shrink b) x) *
(shrink b - shrink a) -
f(linepath(a,b) x) * (b - a)`;
EXISTS_TAC `\x. f(linepath(shrink b,shrink c) x) *
(shrink c - shrink b) -
f(linepath(b,c) x) * (c - b)`;
EXISTS_TAC `\x. f(linepath(shrink c,shrink a) x) *
(shrink a - shrink c) -
f(linepath(c,a) x) * (a - c)`] THEN
ASM_SIMP_TAC[
COMPLEX_NORM_NZ; REAL_ARITH `&0 < x ==> &0 <= x / &6`] THEN
(CONJ_TAC THENL
[MATCH_MP_TAC
HAS_INTEGRAL_SUB THEN
REWRITE_TAC[GSYM
HAS_PATH_INTEGRAL_LINEPATH] THEN
CONJ_TAC THEN MATCH_MP_TAC
HAS_PATH_INTEGRAL_INTEGRAL THEN
MATCH_MP_TAC
PATH_INTEGRABLE_CONTINUOUS_LINEPATH THEN
MATCH_MP_TAC
CONTINUOUS_ON_SUBSET THEN
EXISTS_TAC `convex hull {a:complex,b,c}` THEN
ASM_REWRITE_TAC[
SEGMENT_CONVEX_HULL] THEN
MATCH_MP_TAC
HULL_MINIMAL THEN
REWRITE_TAC[
CONVEX_CONVEX_HULL;
SUBSET;
IN_INSERT;
NOT_IN_EMPTY] THEN
ASM_MESON_TAC[
SUBSET;
INTERIOR_SUBSET];
ALL_TAC] THEN
REPEAT STRIP_TAC THEN
ONCE_REWRITE_TAC[COMPLEX_RING
`f' * x' - f * x = f' * (x' - x) + x * (f' - f):complex`] THEN
MATCH_MP_TAC
NORM_TRIANGLE_LE THEN REWRITE_TAC[
COMPLEX_NORM_MUL] THEN
MATCH_MP_TAC
REAL_LE_TRANS THEN
EXISTS_TAC `B * (norm(y:complex) / &24 / C / B) * &2 * C +
(&2 * C) * (norm y / &24 / C)` THEN
CONJ_TAC THENL
[ALL_TAC;
MATCH_MP_TAC
REAL_EQ_IMP_LE THEN
MAP_EVERY UNDISCH_TAC [`&0 < B`; `&0 < C`] THEN CONV_TAC REAL_FIELD] THEN
MATCH_MP_TAC
REAL_LE_ADD2 THEN CONJ_TAC THEN
MATCH_MP_TAC
REAL_LE_MUL2 THEN REWRITE_TAC[
NORM_POS_LE] THENL
[CONJ_TAC THENL
[FIRST_X_ASSUM MATCH_MP_TAC THEN
W(fun (asl,w) ->
MP_TAC(PART_MATCH (lhand o rand)
LINEPATH_IN_PATH (lhand w))) THEN
ASM_REWRITE_TAC[] THEN
W(fun (asl,w) -> SPEC_TAC(lhand(rand w),`x:complex`)) THEN
REWRITE_TAC[GSYM
SUBSET;
SEGMENT_CONVEX_HULL] THEN
MATCH_MP_TAC
HULL_MINIMAL THEN
REWRITE_TAC[
CONVEX_CONVEX_HULL;
SUBSET;
IN_INSERT;
NOT_IN_EMPTY] THEN
ASM_MESON_TAC[
SUBSET;
INTERIOR_SUBSET];
ALL_TAC] THEN
EXPAND_TAC "shrink" THEN
REWRITE_TAC[VECTOR_ARITH `(b - e % (b - d)) - (a - e % (a - d)) -
(b - a) = e % (a - b)`] THEN
REWRITE_TAC[
NORM_MUL] THEN MATCH_MP_TAC
REAL_LE_MUL2 THEN
ASM_SIMP_TAC[
NORM_POS_LE; REAL_ARITH `&0 < x ==> abs x = x`;
REAL_ABS_POS] THEN
CONJ_TAC THENL [EXPAND_TAC "e" THEN REAL_ARITH_TAC; ALL_TAC] THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN CONJ_TAC THEN
MATCH_MP_TAC(REWRITE_RULE[
SUBSET]
HULL_SUBSET) THEN
REWRITE_TAC[
IN_INSERT];
ALL_TAC] THEN
CONJ_TAC THENL
[FIRST_X_ASSUM MATCH_MP_TAC THEN CONJ_TAC THEN
MATCH_MP_TAC(REWRITE_RULE[
SUBSET]
HULL_SUBSET) THEN
REWRITE_TAC[
IN_INSERT];
ALL_TAC] THEN
MATCH_MP_TAC
REAL_LT_IMP_LE THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
CONJ_TAC THENL
[W(fun (asl,w) ->
MP_TAC(PART_MATCH (lhand o rand)
LINEPATH_IN_PATH (lhand w))) THEN
ASM_MESON_TAC[
SUBSET];
ALL_TAC] THEN
CONJ_TAC THENL
[W(fun (asl,w) ->
MP_TAC(PART_MATCH (lhand o rand)
LINEPATH_IN_PATH (lhand w))) THEN
ASM_REWRITE_TAC[] THEN
W(fun (asl,w) -> SPEC_TAC(lhand(rand w),`x:complex`)) THEN
REWRITE_TAC[GSYM
SUBSET;
SEGMENT_CONVEX_HULL] THEN
MATCH_MP_TAC
HULL_MINIMAL THEN
REWRITE_TAC[
CONVEX_CONVEX_HULL;
SUBSET;
IN_INSERT;
NOT_IN_EMPTY] THEN
ASM_MESON_TAC[
SUBSET;
INTERIOR_SUBSET];
ALL_TAC] THEN
REWRITE_TAC[linepath] THEN REWRITE_TAC[VECTOR_ARITH
`((&1 - x) % a' + x % b') - ((&1 - x) % a + x % b) =
(&1 - x) % (a' - a) + x % (b' - b)`] THEN
EXPAND_TAC "shrink" THEN REWRITE_TAC[VECTOR_ARITH `a - b - a = --b`] THEN
MATCH_MP_TAC
NORM_TRIANGLE_LT THEN REWRITE_TAC[
NORM_MUL;
NORM_NEG] THEN
MATCH_MP_TAC
REAL_CONVEX_BOUND_LT THEN ONCE_REWRITE_TAC[TAUT
`a /\ b /\ c /\ d /\ e <=> (c /\ d /\ e) /\ a /\ b`] THEN
CONJ_TAC THENL
[FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
IN_INTERVAL_1]) THEN
REWRITE_TAC[
DROP_VEC] THEN REAL_ARITH_TAC;
ALL_TAC] THEN
CONJ_TAC THEN
MATCH_MP_TAC
REAL_LET_TRANS THEN
EXISTS_TAC `e * &2 * C` THEN
ASM_SIMP_TAC[
REAL_LE_LMUL_EQ; REAL_ARITH `&0 < x ==> abs x = x`] THEN
(CONJ_TAC THENL
[FIRST_X_ASSUM MATCH_MP_TAC THEN
ASM_MESON_TAC[
SUBSET;
INTERIOR_SUBSET;
HULL_SUBSET;
IN_INSERT];
ALL_TAC]) THEN
ASM_SIMP_TAC[GSYM
REAL_LT_RDIV_EQ;
REAL_LT_MUL;
REAL_OF_NUM_LT; ARITH] THEN
EXPAND_TAC "e" THEN REWRITE_TAC[
REAL_MIN_LT] THEN
DISJ2_TAC THEN DISJ1_TAC THEN
REWRITE_TAC[REAL_FIELD `d / (a * b) = inv(a:real) * d / b`] THEN
REWRITE_TAC[REAL_ARITH `inv(&4) * x < inv(&2) * x <=> &0 < x`] THEN
ASM_SIMP_TAC[
REAL_LT_DIV]));;
let CAUCHY_THEOREM_TRIANGLE_COFINITE = prove
(`!f s a b c.
f
continuous_on (convex hull {a,b,c}) /\
FINITE s /\
(!x. x
IN interior(convex hull {a,b,c})
DIFF s
==> f
complex_differentiable (at x))
==> (f
has_path_integral Cx(&0))
(linepath (a,b) ++ linepath(b,c) ++ linepath(c,a))`,
GEN_TAC THEN GEN_TAC THEN WF_INDUCT_TAC `
CARD(s:complex->bool)` THEN
REPEAT STRIP_TAC THEN ASM_CASES_TAC `s:complex->bool = {}` THENL
[MATCH_MP_TAC
CAUCHY_THEOREM_TRIANGLE_INTERIOR THEN
ASM_REWRITE_TAC[
holomorphic_on] THEN X_GEN_TAC `z:complex` THEN
DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `z:complex`) THEN
ASM_REWRITE_TAC[
complex_differentiable;
IN_DIFF;
NOT_IN_EMPTY] THEN
MESON_TAC[
HAS_COMPLEX_DERIVATIVE_AT_WITHIN];
ALL_TAC] THEN
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM
MEMBER_NOT_EMPTY]) THEN
DISCH_THEN(X_CHOOSE_TAC `d:complex`) THEN
FIRST_X_ASSUM(MP_TAC o SPEC `s
DELETE (d:complex)`) THEN
ASM_SIMP_TAC[
CARD_DELETE;
CARD_EQ_0;
ARITH_RULE `n - 1 < n <=> ~(n = 0)`] THEN
ASM_CASES_TAC `(d:complex)
IN convex hull {a,b,c}` THENL
[ALL_TAC;
DISCH_THEN MATCH_MP_TAC THEN
ASM_REWRITE_TAC[
FINITE_DELETE;
IN_DIFF;
IN_DELETE] THEN
REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
ASM_REWRITE_TAC[
IN_DIFF] THEN ASM_MESON_TAC[
INTERIOR_SUBSET;
SUBSET]] THEN
DISCH_TAC THEN SUBGOAL_THEN
`(f
has_path_integral Cx(&0))
(linepath(a,b) ++ linepath(b,d) ++ linepath(d,a)) /\
(f
has_path_integral Cx(&0))
(linepath(b,c) ++ linepath(c,d) ++ linepath(d,b)) /\
(f
has_path_integral Cx(&0))
(linepath(c,a) ++ linepath(a,d) ++ linepath(d,c))`
MP_TAC THENL
[RULE_ASSUM_TAC(REWRITE_RULE[IMP_IMP;
RIGHT_IMP_FORALL_THM]) THEN
REPEAT CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
ASM_REWRITE_TAC[
FINITE_DELETE] THEN
(CONJ_TAC THENL
[MATCH_MP_TAC
CONTINUOUS_ON_SUBSET THEN
EXISTS_TAC `convex hull {a:complex,b,c}` THEN ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC
CONVEX_HULL_SUBSET THEN
REWRITE_TAC[
IN_INSERT;
NOT_IN_EMPTY] THEN
REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC(REWRITE_RULE[
SUBSET]
HULL_SUBSET) THEN
REWRITE_TAC[
IN_INSERT];
ALL_TAC]) THEN
ASM_REWRITE_TAC[
FINITE_DELETE;
IN_DIFF;
IN_DELETE] THEN
REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [DE_MORGAN_THM]) THEN
(ASM_CASES_TAC `x:complex = d` THEN ASM_REWRITE_TAC[] THENL
[ASM_MESON_TAC[
NOT_IN_INTERIOR_CONVEX_HULL_3]; ALL_TAC]) THEN
DISCH_TAC THEN ASM_REWRITE_TAC[
IN_DIFF] THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
`x
IN interior s
==> interior s
SUBSET interior t ==> x
IN interior t`)) THEN
MATCH_MP_TAC
SUBSET_INTERIOR THEN
MATCH_MP_TAC
CONVEX_HULL_SUBSET THEN
SIMP_TAC[
IN_INSERT;
NOT_IN_EMPTY] THEN
REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC(REWRITE_RULE[
SUBSET]
HULL_SUBSET) THEN REWRITE_TAC[
IN_INSERT];
ALL_TAC] THEN
SUBGOAL_THEN
`f
path_integrable_on
(linepath (a,b) ++ linepath(b,c) ++ linepath(c,a))`
MP_TAC THENL
[SIMP_TAC[
PATH_INTEGRABLE_JOIN;
VALID_PATH_JOIN;
VALID_PATH_LINEPATH;
PATHSTART_JOIN;
PATHFINISH_JOIN;
PATHSTART_LINEPATH;
PATHFINISH_LINEPATH] THEN
STRIP_ASSUME_TAC(ISPECL [`a:complex`; `b:complex`; `c:complex`]
SEGMENTS_SUBSET_CONVEX_HULL) THEN
ASM_MESON_TAC[
PATH_INTEGRABLE_CONTINUOUS_LINEPATH;
CONTINUOUS_ON_SUBSET];
ALL_TAC] THEN
REWRITE_TAC[
path_integrable_on;
LEFT_IMP_EXISTS_THM] THEN
X_GEN_TAC `y:complex` THEN
DISCH_THEN(fun th -> ASSUME_TAC th THEN MP_TAC th) THEN
REWRITE_TAC[IMP_IMP] THEN
DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN
(MP_TAC o MATCH_MP
HAS_CHAIN_INTEGRAL_CHAIN_INTEGRAL)) THEN
ASM_CASES_TAC `y = Cx(&0)` THEN ASM_REWRITE_TAC[] THENL
[ASM_MESON_TAC[]; UNDISCH_TAC `~(y = Cx(&0))`] THEN
REWRITE_TAC[] THEN
SUBGOAL_THEN `(f:complex->complex)
continuous_on segment[a,d] /\
f
continuous_on segment[b,d] /\
f
continuous_on segment[c,d]`
MP_TAC THENL
[ALL_TAC;
DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN (MP_TAC o MATCH_MP
PATH_INTEGRAL_REVERSE_LINEPATH)) THEN
CONV_TAC COMPLEX_RING] THEN
REPEAT CONJ_TAC THEN MATCH_MP_TAC
CONTINUOUS_ON_SUBSET THEN
EXISTS_TAC `convex hull {a:complex,b,c}` THEN
ASM_REWRITE_TAC[
SEGMENT_CONVEX_HULL] THEN
MATCH_MP_TAC
CONVEX_HULL_SUBSET THEN
SIMP_TAC[
IN_INSERT;
NOT_IN_EMPTY] THEN
REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC(REWRITE_RULE[
SUBSET]
HULL_SUBSET) THEN REWRITE_TAC[
IN_INSERT]);;
let WINDING_NUMBER_UNIQUE = prove
(`!g z e n.
path g /\ ~(z
IN path_image g) /\
(!e. &0 < e
==> ?p.
valid_path p /\ ~(z
IN path_image p) /\
pathstart p = pathstart g /\
pathfinish p = pathfinish g /\
(!t. t
IN interval[vec 0,vec 1]
==> norm(g t - p t) < e) /\
path_integral p (\w. Cx(&1) / (w - z)) =
Cx(&2) * Cx(pi) * ii * n)
==>
winding_number(g,z) = n`,
REPEAT STRIP_TAC THEN
MP_TAC(ISPECL [`(:complex)
DELETE z`; `g:real^1->complex`]
PATH_INTEGRAL_NEARBY_ENDS) THEN
ASM_SIMP_TAC[
OPEN_DELETE;
OPEN_UNIV] THEN
ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN
FIRST_X_ASSUM(MP_TAC o SPEC `e:real`) THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_THEN `p:real^1->complex` STRIP_ASSUME_TAC) THEN
MP_TAC(ISPECL [`g:real^1->complex`; `z:complex`]
WINDING_NUMBER) THEN
DISCH_THEN(MP_TAC o SPEC `e:real`) THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_THEN `q:real^1->complex` STRIP_ASSUME_TAC) THEN
FIRST_X_ASSUM(MP_TAC o SPECL [`p:real^1->complex`; `q:real^1->complex`]) THEN
ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[
NORM_SUB] THEN ASM_SIMP_TAC[] THEN
STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `\w. Cx(&1) / (w - z)`) THEN
ANTS_TAC THENL
[ASM_SIMP_TAC[
OPEN_DELETE;
OPEN_UNIV;
HOLOMORPHIC_ON_OPEN] THEN
REWRITE_TAC[
IN_DELETE;
IN_UNIV; GSYM
complex_differentiable] THEN
REPEAT STRIP_TAC THEN COMPLEX_DIFFERENTIABLE_TAC THEN
ASM_REWRITE_TAC[
COMPLEX_SUB_0];
ASM_REWRITE_TAC[] THEN MP_TAC
CX_2PII_NZ THEN
CONV_TAC COMPLEX_RING]);;
let WINDING_NUMBER_UNIQUE_LOOP = prove
(`!g z e n.
path g /\ ~(z
IN path_image g) /\ pathfinish g = pathstart g /\
(!e. &0 < e
==> ?p.
valid_path p /\ ~(z
IN path_image p) /\
pathfinish p = pathstart p /\
(!t. t
IN interval[vec 0,vec 1]
==> norm(g t - p t) < e) /\
path_integral p (\w. Cx(&1) / (w - z)) =
Cx(&2) * Cx(pi) * ii * n)
==>
winding_number(g,z) = n`,
REPEAT STRIP_TAC THEN
MP_TAC(ISPECL [`(:complex)
DELETE z`; `g:real^1->complex`]
PATH_INTEGRAL_NEARBY_LOOP) THEN
ASM_SIMP_TAC[
OPEN_DELETE;
OPEN_UNIV] THEN
ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN
FIRST_X_ASSUM(MP_TAC o SPEC `e:real`) THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_THEN `p:real^1->complex` STRIP_ASSUME_TAC) THEN
MP_TAC(ISPECL [`g:real^1->complex`; `z:complex`]
WINDING_NUMBER) THEN
DISCH_THEN(MP_TAC o SPEC `e:real`) THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_THEN `q:real^1->complex` STRIP_ASSUME_TAC) THEN
FIRST_X_ASSUM(MP_TAC o SPECL [`p:real^1->complex`; `q:real^1->complex`]) THEN
ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[
NORM_SUB] THEN ASM_SIMP_TAC[] THEN
STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `\w. Cx(&1) / (w - z)`) THEN
ANTS_TAC THENL
[ASM_SIMP_TAC[
OPEN_DELETE;
OPEN_UNIV;
HOLOMORPHIC_ON_OPEN] THEN
REWRITE_TAC[
IN_DELETE;
IN_UNIV; GSYM
complex_differentiable] THEN
REPEAT STRIP_TAC THEN COMPLEX_DIFFERENTIABLE_TAC THEN
ASM_REWRITE_TAC[
COMPLEX_SUB_0];
ASM_REWRITE_TAC[] THEN MP_TAC
CX_2PII_NZ THEN
CONV_TAC COMPLEX_RING]);;
let WINDING_NUMBER_AHLFORS_LEMMA = prove
(`!g a b.
g
piecewise_differentiable_on interval [a,b] /\
drop a <= drop b /\ (!x. x
IN interval [a,b] ==> ~(g x = z))
==> (\x.
vector_derivative g (at x within interval[a,b]) / (g(x) - z))
integrable_on interval[a,b] /\
cexp(--(integral (interval[a,b])
(\x.
vector_derivative g (at x within interval[a,b]) /
(g(x) - z)))) *
(g(b) - z) = g(a) - z`,
let lemma = prove
(`!f g g' s x z.
(g has_vector_derivative g') (at x within s) /\
(f has_vector_derivative (g' / (g x - z))) (at x within s) /\
~(g x = z)
==> ((\x. cexp(--f x) * (g x - z)) has_vector_derivative Cx(&0))
(at x within s)`,
REPEAT STRIP_TAC THEN
SUBGOAL_THEN
`cexp(--f x) * (g' - Cx(&0)) +
(cexp(--f x) * --(g' / ((g:real^1->complex) x - z))) * (g x - z) = Cx(&0)`
(SUBST1_TAC o SYM)
THENL
[FIRST_X_ASSUM(MP_TAC o check (is_neg o concl)) THEN
CONV_TAC COMPLEX_FIELD;
ALL_TAC] THEN
MATCH_MP_TAC(ISPEC `( * ):complex->complex->complex`
HAS_VECTOR_DERIVATIVE_BILINEAR_WITHIN) THEN
REWRITE_TAC[BILINEAR_COMPLEX_MUL; GSYM COMPLEX_VEC_0] THEN
ASM_SIMP_TAC[HAS_VECTOR_DERIVATIVE_SUB; ETA_AX;
HAS_VECTOR_DERIVATIVE_CONST] THEN
GEN_REWRITE_TAC (RATOR_CONV o LAND_CONV) [GSYM o_DEF] THEN
REWRITE_TAC[has_vector_derivative] THEN
SUBGOAL_THEN `!x y. (\z. drop z % (x * y :complex)) =
(\w. x * w) o (\z. drop z % y)`
(fun th -> REWRITE_TAC[th])
THENL
[REWRITE_TAC[FUN_EQ_THM; o_THM; COMPLEX_CMUL] THEN
SIMPLE_COMPLEX_ARITH_TAC;
ALL_TAC] THEN
MATCH_MP_TAC DIFF_CHAIN_WITHIN THEN
REWRITE_TAC[GSYM has_complex_derivative; GSYM has_vector_derivative] THEN
SIMP_TAC[HAS_COMPLEX_DERIVATIVE_CEXP; HAS_COMPLEX_DERIVATIVE_AT_WITHIN] THEN
ASM_SIMP_TAC[HAS_VECTOR_DERIVATIVE_NEG]) in
REPEAT GEN_TAC THEN STRIP_TAC THEN
SUBGOAL_THEN
`!w. ~(w = z)
==> ?h. !y. norm(y - w) < norm(w - z)
==> (h has_complex_derivative inv(y - z)) (at y)`
(LABEL_TAC "P")
THENL
[REPEAT STRIP_TAC THEN
MP_TAC(ISPECL [`\w:complex. inv(w - z)`;
`ball(w:complex,norm(w - z))`;
`{}:complex->bool`]
HOLOMORPHIC_CONVEX_PRIMITIVE) THEN
SIMP_TAC[HAS_COMPLEX_DERIVATIVE_WITHIN_OPEN; OPEN_BALL; INTERIOR_OPEN] THEN
REWRITE_TAC[CONVEX_BALL; FINITE_RULES; DIFF_EMPTY] THEN ANTS_TAC THENL
[SUBGOAL_THEN `(\w. inv(w - z)) holomorphic_on ball(w:complex,norm(w - z))`
(fun th ->
MESON_TAC[HOLOMORPHIC_ON_OPEN; HOLOMORPHIC_ON_IMP_CONTINUOUS_ON;
OPEN_BALL; complex_differentiable; th]) THEN
SIMP_TAC[HOLOMORPHIC_ON_OPEN; OPEN_BALL; IN_BALL] THEN
X_GEN_TAC `u:complex` THEN DISCH_TAC THEN
EXISTS_TAC `--Cx(&1) / (u - z) pow 2` THEN COMPLEX_DIFF_TAC THEN
REWRITE_TAC[COMPLEX_SUB_RZERO; COMPLEX_SUB_0] THEN
ASM_MESON_TAC[REAL_LT_REFL; dist];
ALL_TAC] THEN
REWRITE_TAC[IN_BALL; dist] THEN MESON_TAC[NORM_SUB];
ALL_TAC] THEN
SUBGOAL_THEN
`!t. t IN interval[a,b]
==> (\x. vector_derivative g (at x within interval[a,b]) / (g(x) - z))
integrable_on interval[a,t] /\
cexp(--(integral (interval[a,t])
(\x. vector_derivative g (at x within interval[a,b]) /
(g(x) - z)))) *
(g(t) - z) = g(a) - z`
(fun th -> MATCH_MP_TAC th THEN
ASM_REWRITE_TAC[IN_INTERVAL_1; REAL_LE_REFL]) THEN
REWRITE_TAC[TAUT `a ==> b /\ c <=> (a ==> b) /\ (a ==> c)`] THEN
REWRITE_TAC[FORALL_AND_THM] THEN
MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN CONJ_TAC THENL
[REPEAT STRIP_TAC THEN MATCH_MP_TAC INTEGRABLE_SUBINTERVAL THEN
MAP_EVERY EXISTS_TAC [`a:real^1`; `b:real^1`] THEN
ASM_REWRITE_TAC[SUBSET_INTERVAL_1; REAL_LE_REFL] THEN
CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[IN_INTERVAL_1]] THEN
REWRITE_TAC[integrable_on; complex_div] THEN
ONCE_REWRITE_TAC[COMPLEX_MUL_SYM] THEN
REWRITE_TAC[HAS_INTEGRAL_LOCALIZED_VECTOR_DERIVATIVE] THEN
REWRITE_TAC[GSYM integrable_on] THEN
MATCH_MP_TAC PATH_INTEGRAL_LOCAL_PRIMITIVE_ANY THEN
EXISTS_TAC `(:complex) DELETE z` THEN
ASM_SIMP_TAC[IN_DELETE; IN_UNIV;
DIFFERENTIABLE_ON_IMP_PIECEWISE_DIFFERENTIABLE] THEN
X_GEN_TAC `w:complex` THEN DISCH_TAC THEN
EXISTS_TAC `norm(w - z:complex)` THEN
ASM_REWRITE_TAC[COMPLEX_NORM_NZ; COMPLEX_SUB_0] THEN
ASM_MESON_TAC[HAS_COMPLEX_DERIVATIVE_AT_WITHIN];
ALL_TAC] THEN
DISCH_TAC THEN MATCH_MP_TAC HAS_DERIVATIVE_ZERO_UNIQUE_STRONG_INTERVAL THEN
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [piecewise_differentiable_on]) THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
MATCH_MP_TAC MONO_EXISTS THEN SIMP_TAC[IN_DIFF; FINITE_IMP_COUNTABLE] THEN
X_GEN_TAC `k:real^1->bool` THEN STRIP_TAC THEN
ASM_SIMP_TAC[CONVEX_INTERVAL; INTEGRAL_REFL] THEN
REWRITE_TAC[COMPLEX_VEC_0; COMPLEX_NEG_0; CEXP_0; COMPLEX_MUL_LID] THEN
CONJ_TAC THENL
[MATCH_MP_TAC CONTINUOUS_ON_COMPLEX_MUL THEN
ASM_SIMP_TAC[CONTINUOUS_ON_SUB; CONTINUOUS_ON_CONST; ETA_AX;
PIECEWISE_DIFFERENTIABLE_ON_IMP_CONTINUOUS_ON] THEN
GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN
MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN
REWRITE_TAC[CONTINUOUS_ON_CEXP] THEN
MATCH_MP_TAC CONTINUOUS_ON_NEG THEN
MATCH_MP_TAC INDEFINITE_INTEGRAL_CONTINUOUS_RIGHT THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN
ASM_REWRITE_TAC[IN_INTERVAL_1; REAL_LE_REFL];
ALL_TAC] THEN
X_GEN_TAC `t:real^1` THEN DISCH_TAC THEN
MP_TAC(ISPECL [`\w:complex. inv(w - z)`;
`ball((g:real^1->complex) t,dist(g t,z))`;
`{}:complex->bool`]
HOLOMORPHIC_CONVEX_PRIMITIVE) THEN
SIMP_TAC[HAS_COMPLEX_DERIVATIVE_WITHIN_OPEN; OPEN_BALL; INTERIOR_OPEN] THEN
REWRITE_TAC[CONVEX_BALL; FINITE_RULES; DIFF_EMPTY] THEN ANTS_TAC THENL
[SUBGOAL_THEN `(\w. inv(w - z)) holomorphic_on ball(g(t:real^1),dist(g t,z))`
(fun th ->
MESON_TAC[HOLOMORPHIC_ON_OPEN; HOLOMORPHIC_ON_IMP_CONTINUOUS_ON;
OPEN_BALL; complex_differentiable; th]) THEN
SIMP_TAC[HOLOMORPHIC_ON_OPEN; OPEN_BALL; IN_BALL] THEN
X_GEN_TAC `w:complex` THEN DISCH_TAC THEN
EXISTS_TAC `--Cx(&1) / (w - z) pow 2` THEN COMPLEX_DIFF_TAC THEN
REWRITE_TAC[COMPLEX_SUB_RZERO; COMPLEX_SUB_0] THEN
ASM_MESON_TAC[REAL_LT_REFL];
ALL_TAC] THEN
REWRITE_TAC[IN_BALL; dist] THEN
DISCH_THEN(X_CHOOSE_TAC `h:complex->complex`) THEN
SUBGOAL_THEN `(\h. Cx(&0)) = (\h. drop h % Cx(&0))` SUBST1_TAC THENL
[REWRITE_TAC[FUN_EQ_THM; GSYM COMPLEX_VEC_0; VECTOR_MUL_RZERO];
ALL_TAC] THEN
REWRITE_TAC[GSYM has_vector_derivative] THEN MATCH_MP_TAC lemma THEN
EXISTS_TAC `vector_derivative g (at t within interval[a,b]):complex` THEN
REPEAT CONJ_TAC THENL
[REWRITE_TAC[GSYM VECTOR_DERIVATIVE_WORKS] THEN
ASM_MESON_TAC[DIFFERENTIABLE_AT_WITHIN];
ALL_TAC;
ASM_MESON_TAC[]] THEN
REWRITE_TAC[has_vector_derivative] THEN
MATCH_MP_TAC HAS_DERIVATIVE_TRANSFORM_WITHIN THEN
ASM_REWRITE_TAC[GSYM has_vector_derivative] THEN
EXISTS_TAC `\u. integral (interval [a,t])
(\x. vector_derivative g (at x within interval [a,b]) /
((g:real^1->complex) x - z)) + (h(g(u)) - h(g(t)))` THEN
REWRITE_TAC[LEFT_EXISTS_AND_THM; CONJ_ASSOC] THEN
REWRITE_TAC[GSYM CONJ_ASSOC] THEN CONJ_TAC THENL
[ALL_TAC;
ONCE_REWRITE_TAC[COMPLEX_RING `a + (b - c) = b + (a - c):complex`] THEN
GEN_REWRITE_TAC LAND_CONV [GSYM VECTOR_ADD_RID] THEN
MATCH_MP_TAC HAS_VECTOR_DERIVATIVE_ADD THEN
REWRITE_TAC[HAS_VECTOR_DERIVATIVE_CONST] THEN
REWRITE_TAC[has_vector_derivative] THEN
SUBGOAL_THEN `!x y. (\h. drop h % x / y) =
(\x. inv(y) * x) o (\h. drop h % x)`
(fun th -> REWRITE_TAC[th])
THENL
[REWRITE_TAC[FUN_EQ_THM; o_THM; COMPLEX_CMUL] THEN
SIMPLE_COMPLEX_ARITH_TAC;
ALL_TAC] THEN
GEN_REWRITE_TAC (RATOR_CONV o LAND_CONV) [GSYM o_DEF] THEN
MATCH_MP_TAC DIFF_CHAIN_WITHIN THEN
REWRITE_TAC[GSYM has_complex_derivative; GSYM has_vector_derivative] THEN
REWRITE_TAC[GSYM VECTOR_DERIVATIVE_WORKS] THEN
CONJ_TAC THENL [ASM_MESON_TAC[DIFFERENTIABLE_AT_WITHIN]; ALL_TAC] THEN
MATCH_MP_TAC HAS_COMPLEX_DERIVATIVE_AT_WITHIN THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN
REWRITE_TAC[COMPLEX_SUB_REFL; COMPLEX_NORM_0; COMPLEX_NORM_NZ] THEN
ASM_SIMP_TAC[COMPLEX_SUB_0]] THEN
SUBGOAL_THEN
`?d. &0 < d /\
!y:real^1. y IN interval[a,b] /\ dist(y,t) < d
==> dist(g y:complex,g t) < norm(g t - z) /\ ~(y IN k)`
MP_TAC THENL
[SUBGOAL_THEN `(g:real^1->complex) continuous (at t within interval[a,b])`
MP_TAC THENL
[ASM_MESON_TAC[PIECEWISE_DIFFERENTIABLE_ON_IMP_CONTINUOUS_ON;
CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN];
ALL_TAC] THEN
REWRITE_TAC[continuous_within] THEN
DISCH_THEN(MP_TAC o SPEC `norm((g:real^1->complex) t - z)`) THEN
ASM_SIMP_TAC[COMPLEX_NORM_NZ; COMPLEX_SUB_0] THEN
DISCH_THEN(X_CHOOSE_THEN `d1:real` STRIP_ASSUME_TAC) THEN
FIRST_X_ASSUM(X_CHOOSE_THEN `d2:real` STRIP_ASSUME_TAC o
SPEC `t:real^1` o MATCH_MP FINITE_SET_AVOID) THEN
EXISTS_TAC `min d1 d2` THEN ASM_SIMP_TAC[REAL_LT_MIN] THEN
ASM_MESON_TAC[DIST_SYM; REAL_NOT_LE];
ALL_TAC] THEN
REWRITE_TAC[TAUT `a ==> b /\ c <=> (a ==> b) /\ (a ==> c)`] THEN
REWRITE_TAC[FORALL_AND_THM] THEN
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:real` THEN STRIP_TAC THEN
ASM_REWRITE_TAC[] THEN X_GEN_TAC `u:real^1` THEN REWRITE_TAC[dist] THEN
STRIP_TAC THEN
DISJ_CASES_TAC(REAL_ARITH `drop t <= drop u \/ drop u <= drop t`) THENL
[SUBGOAL_THEN
`integral (interval [a,u])
(\x. vector_derivative g (at x within interval [a,b]) / (g x - z)) =
integral (interval [a,t])
(\x. vector_derivative g (at x within interval [a,b]) / (g x - z)) +
integral (interval [t,u])
(\x. vector_derivative g (at x within interval [a,b]) / (g x - z))`
SUBST1_TAC THENL
[CONV_TAC SYM_CONV THEN MATCH_MP_TAC INTEGRAL_COMBINE THEN
ASM_MESON_TAC[IN_INTERVAL_1];
ALL_TAC] THEN
SIMP_TAC[COMPLEX_RING `a + x = a + y <=> y:complex = x`];
SUBGOAL_THEN
`integral (interval [a,t])
(\x. vector_derivative g (at x within interval [a,b]) / (g x - z)) =
integral (interval [a,u])
(\x. vector_derivative g (at x within interval [a,b]) / (g x - z)) +
integral (interval [u,t])
(\x. vector_derivative g (at x within interval [a,b]) / (g x - z))`
SUBST1_TAC THENL
[CONV_TAC SYM_CONV THEN MATCH_MP_TAC INTEGRAL_COMBINE THEN
ASM_MESON_TAC[IN_INTERVAL_1];
ALL_TAC] THEN
SIMP_TAC[COMPLEX_RING `(a + x) + (w - z) = a <=> x:complex = z - w`]] THEN
(MATCH_MP_TAC INTEGRAL_UNIQUE THEN
MATCH_MP_TAC FUNDAMENTAL_THEOREM_OF_CALCULUS THEN
ASM_REWRITE_TAC[GSYM o_DEF] THEN X_GEN_TAC `x:real^1` THEN
REPEAT STRIP_TAC THEN REWRITE_TAC[has_vector_derivative; COMPLEX_CMUL] THEN
SUBGOAL_THEN `!x y. (\h. Cx(drop h) * x / y) =
(\x. inv(y) * x) o (\h. drop h % x)`
(fun th -> REWRITE_TAC[th])
THENL
[REWRITE_TAC[FUN_EQ_THM; o_THM; COMPLEX_CMUL] THEN
SIMPLE_COMPLEX_ARITH_TAC;
ALL_TAC] THEN
MATCH_MP_TAC DIFF_CHAIN_WITHIN THEN
REWRITE_TAC[GSYM has_complex_derivative; GSYM has_vector_derivative] THEN
CONJ_TAC THENL
[MATCH_MP_TAC HAS_VECTOR_DERIVATIVE_WITHIN_SUBSET THEN
EXISTS_TAC `interval[a:real^1,b]` THEN
REWRITE_TAC[GSYM VECTOR_DERIVATIVE_WORKS] THEN CONJ_TAC THENL
[MATCH_MP_TAC DIFFERENTIABLE_AT_WITHIN THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN CONJ_TAC THENL
[ALL_TAC; FIRST_X_ASSUM MATCH_MP_TAC];
ALL_TAC] THEN
REPEAT(FIRST_X_ASSUM(MP_TAC o
check (fun t -> not(is_forall (concl t))))) THEN
REWRITE_TAC[dist; NORM_REAL; GSYM drop; DROP_SUB] THEN
REWRITE_TAC[SUBSET_INTERVAL_1; IN_INTERVAL_1; REAL_LE_REFL] THEN
REAL_ARITH_TAC;
ALL_TAC] THEN
MATCH_MP_TAC HAS_COMPLEX_DERIVATIVE_AT_WITHIN THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM dist] THEN
ONCE_REWRITE_TAC[DIST_SYM] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
CONJ_TAC THENL [ASM_MESON_TAC[IN_INTERVAL_1; REAL_LE_TRANS]; ALL_TAC] THEN
REPEAT(FIRST_X_ASSUM(MP_TAC o
check (fun t -> not(is_forall (concl t))))) THEN
REWRITE_TAC[dist; NORM_REAL; GSYM drop; DROP_SUB] THEN
REWRITE_TAC[SUBSET_INTERVAL_1; IN_INTERVAL_1; REAL_LE_REFL] THEN
REAL_ARITH_TAC));;
let WINDING_NUMBER_TRIANGLE = prove
(`!a b c z.
z
IN interior(convex hull {a,b,c})
==>
winding_number(linepath(a,b) ++ linepath(b,c) ++ linepath(c,a),z) =
if &0 < Im((b - a) * cnj (b - z)) then Cx(&1) else --Cx(&1)`,
let lemma1 = prove
(`!a b c. vec 0 IN interior(convex hull {a,b,c})
==> ~(Im(a / b) <= &0 /\ &0 <= Im(b / c))`,
REPEAT GEN_TAC THEN
ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN SIMP_TAC[] THEN
GEN_REWRITE_TAC (LAND_CONV o RAND_CONV o ONCE_DEPTH_CONV)
[GSYM COMPLEX_INV_DIV] THEN
REWRITE_TAC[IM_COMPLEX_INV_GE_0] THEN
GEOM_BASIS_MULTIPLE_TAC 1 `b:complex` THEN
REWRITE_TAC[COMPLEX_CMUL; COMPLEX_BASIS; GSYM CX_MUL; REAL_MUL_RID] THEN
X_GEN_TAC `x:real` THEN GEN_REWRITE_TAC LAND_CONV [REAL_LE_LT] THEN
REWRITE_TAC[IM_DIV_CX] THEN ASM_CASES_TAC `x = &0` THEN
ASM_REWRITE_TAC[NOT_IN_INTERIOR_CONVEX_HULL_3; COMPLEX_VEC_0] THEN
DISCH_TAC THEN REPEAT GEN_TAC THEN
ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_MUL_LZERO] THEN STRIP_TAC THEN
MATCH_MP_TAC(SET_RULE
`!s. ~(x IN s) /\ t SUBSET s ==> ~(x IN t)`) THEN
EXISTS_TAC `interior {z | Im z <= &0}` THEN CONJ_TAC THENL
[REWRITE_TAC[IM_DEF; INTERIOR_HALFSPACE_COMPONENT_LE] THEN
REWRITE_TAC[GSYM COMPLEX_VEC_0; IN_ELIM_THM; VEC_COMPONENT] THEN
REAL_ARITH_TAC;
MATCH_MP_TAC SUBSET_INTERIOR THEN MATCH_MP_TAC HULL_MINIMAL THEN
REWRITE_TAC[CONVEX_HALFSPACE_IM_LE] THEN
ASM_REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET; IN_ELIM_THM] THEN
REWRITE_TAC[IM_CX; REAL_LE_REFL]]) in
let lemma2 = prove
(`z IN interior(convex hull {a,b,c})
==> &0 < Im((b - a) * cnj (b - z)) /\
&0 < Im((c - b) * cnj (c - z)) /\
&0 < Im((a - c) * cnj (a - z)) \/
Im((b - a) * cnj (b - z)) < &0 /\
&0 < Im((b - c) * cnj (b - z)) /\
&0 < Im((a - b) * cnj (a - z)) /\
&0 < Im((c - a) * cnj (c - z))`,
GEOM_ORIGIN_TAC `z:complex` THEN
REWRITE_TAC[VECTOR_SUB_RZERO; COMPLEX_SUB_RDISTRIB] THEN
REWRITE_TAC[COMPLEX_MUL_CNJ; IM_SUB; GSYM CX_POW; IM_CX] THEN
REWRITE_TAC[REAL_ARITH `&0 < &0 - x <=> x < &0`;
REAL_ARITH `&0 - x < &0 <=> &0 < x`] THEN
REWRITE_TAC[GSYM IM_COMPLEX_DIV_GT_0; GSYM IM_COMPLEX_DIV_LT_0] THEN
REPEAT STRIP_TAC THEN
GEN_REWRITE_TAC (RAND_CONV o ONCE_DEPTH_CONV) [GSYM COMPLEX_INV_DIV] THEN
REWRITE_TAC[IM_COMPLEX_INV_LT_0; IM_COMPLEX_INV_GT_0] THEN
GEN_REWRITE_TAC (RAND_CONV o LAND_CONV o LAND_CONV o RAND_CONV)
[GSYM COMPLEX_INV_DIV] THEN
REWRITE_TAC[IM_COMPLEX_INV_LT_0] THEN
MP_TAC(ISPECL [`a:complex`; `b:complex`; `c:complex`] lemma1) THEN
MP_TAC(ISPECL [`b:complex`; `c:complex`; `a:complex`] lemma1) THEN
MP_TAC(ISPECL [`c:complex`; `a:complex`; `b:complex`] lemma1) THEN
POP_ASSUM MP_TAC THEN SIMP_TAC[INSERT_AC] THEN REAL_ARITH_TAC) in
let lemma3 = prove
(`!a b c z.
z IN interior(convex hull {a,b,c}) /\
&0 < Im((b - a) * cnj (b - z)) /\
&0 < Im((c - b) * cnj (c - z)) /\
&0 < Im((a - c) * cnj (a - z))
==> winding_number
(linepath(a,b) ++ linepath(b,c) ++ linepath(c,a),z) = Cx(&1)`,
REPEAT STRIP_TAC THEN
MATCH_MP_TAC WINDING_NUMBER_EQ_1 THEN
REWRITE_TAC[PATHSTART_JOIN; PATHFINISH_JOIN; CONJ_ASSOC;
PATHSTART_LINEPATH; PATHFINISH_LINEPATH] THEN
CONJ_TAC THENL
[REWRITE_TAC[GSYM CONJ_ASSOC] THEN
REPEAT(MATCH_MP_TAC WINDING_NUMBER_JOIN_POS_COMBINED THEN
REWRITE_TAC[PATHSTART_JOIN; PATHFINISH_JOIN;
PATHSTART_LINEPATH; PATHFINISH_LINEPATH] THEN
CONJ_TAC) THEN
ASM_SIMP_TAC[WINDING_NUMBER_LINEPATH_POS_LT; VALID_PATH_LINEPATH] THEN
RULE_ASSUM_TAC(REWRITE_RULE
[INTERIOR_OF_TRIANGLE; IN_DIFF; IN_UNION; DE_MORGAN_THM]) THEN
ASM_REWRITE_TAC[PATH_IMAGE_LINEPATH];
RULE_ASSUM_TAC(REWRITE_RULE
[INTERIOR_OF_TRIANGLE; IN_DIFF; IN_UNION; DE_MORGAN_THM]) THEN
ASM_SIMP_TAC[WINDING_NUMBER_JOIN; PATH_IMAGE_JOIN; PATH_JOIN; IN_UNION;
PATH_LINEPATH; PATHSTART_JOIN; PATHFINISH_JOIN; RE_ADD;
PATHSTART_LINEPATH; PATHFINISH_LINEPATH; PATH_IMAGE_LINEPATH] THEN
MATCH_MP_TAC(REAL_ARITH
`abs a < &1 / &2 /\ abs b < &1 / &2 /\ abs c < &1 / &2
==> a + b + c < &2`) THEN
REPEAT CONJ_TAC THEN MATCH_MP_TAC WINDING_NUMBER_LT_HALF_LINEPATH THEN
ASM_REWRITE_TAC[]]) in
REPEAT STRIP_TAC THEN
FIRST_ASSUM(STRIP_ASSUME_TAC o MATCH_MP lemma2) THEN
ASM_SIMP_TAC[lemma3; COMPLEX_NORM_CX; REAL_ABS_NUM] THEN
SUBGOAL_THEN
`winding_number
(linepath(c,b) ++ linepath(b,a) ++ linepath(a,c),z) = Cx(&1)`
MP_TAC THENL
[MATCH_MP_TAC lemma3 THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[INSERT_AC];
COND_CASES_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC]] THEN
DISCH_THEN(SUBST1_TAC o SYM) THEN CONV_TAC SYM_CONV THEN
REWRITE_TAC[COMPLEX_NORM_CX; REAL_ABS_NUM] THEN
RULE_ASSUM_TAC(REWRITE_RULE
[INTERIOR_OF_TRIANGLE; IN_DIFF; IN_UNION; DE_MORGAN_THM]) THEN
FIRST_ASSUM(ASSUME_TAC o ONCE_REWRITE_RULE[SEGMENT_SYM] o CONJUNCT2) THEN
ASM_SIMP_TAC[WINDING_NUMBER_JOIN; PATH_IMAGE_JOIN; PATH_JOIN; IN_UNION;
PATH_LINEPATH; PATHSTART_JOIN; PATHFINISH_JOIN; RE_ADD;
PATHSTART_LINEPATH; PATHFINISH_LINEPATH; PATH_IMAGE_LINEPATH] THEN
ASM_SIMP_TAC[COMPLEX_NEG_ADD; GSYM WINDING_NUMBER_REVERSEPATH;
PATH_IMAGE_LINEPATH; PATH_LINEPATH; REVERSEPATH_LINEPATH] THEN
CONV_TAC COMPLEX_RING);;
let CAUCHY_THEOREM_HOMOTOPIC_PATHS = prove
(`!f g h s.
open s /\ f
holomorphic_on s /\
valid_path g /\
valid_path h /\
homotopic_paths s g h
==>
path_integral g f =
path_integral h f`,
REPEAT STRIP_TAC THEN
FIRST_ASSUM(ASSUME_TAC o SYM o MATCH_MP
HOMOTOPIC_PATHS_IMP_PATHSTART) THEN
FIRST_ASSUM(ASSUME_TAC o SYM o MATCH_MP
HOMOTOPIC_PATHS_IMP_PATHFINISH) THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
homotopic_paths]) THEN
REWRITE_TAC[
homotopic_with;
LEFT_IMP_EXISTS_THM;
PCROSS] THEN
X_GEN_TAC `k:real^(1,1)finite_sum->complex` THEN STRIP_TAC THEN
SUBGOAL_THEN
`!t. t
IN interval[vec 0:real^1,vec 1]
==> ?e. &0 < e /\
!t1 t2. t1
IN interval[vec 0:real^1,vec 1] /\
t2
IN interval[vec 0,vec 1] /\
norm(t1 - t) < e /\ norm(t2 - t) < e
==> ?d. &0 < d /\
!g1 g2.
valid_path g1 /\
valid_path g2 /\
(!u. u
IN interval[vec 0,vec 1]
==> norm(g1 u - k(pastecart t1 u)) < d /\
norm(g2 u - k(pastecart t2 u)) < d) /\
pathstart g1 = pathstart g /\
pathfinish g1 = pathfinish g /\
pathstart g2 = pathstart g /\
pathfinish g2 = pathfinish g
==>
path_image g1
SUBSET s /\
path_image g2
SUBSET s /\
path_integral g2 f =
path_integral g1 f`
MP_TAC THENL
[X_GEN_TAC `t:real^1` THEN STRIP_TAC THEN
MP_TAC(ISPECL
[`s:complex->bool`; `\u. (k:real^(1,1)finite_sum->complex)(pastecart t u)`]
PATH_INTEGRAL_NEARBY_ENDS) THEN
REWRITE_TAC[] THEN ANTS_TAC THENL
[ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[GSYM
o_DEF] THEN
REWRITE_TAC[
path_image; path;
IMAGE_o] THEN CONJ_TAC THENL
[ALL_TAC; ASM SET_TAC[]] THEN
MATCH_MP_TAC
CONTINUOUS_ON_COMPOSE THEN
SIMP_TAC[
CONTINUOUS_ON_PASTECART;
CONTINUOUS_ON_ID;
CONTINUOUS_ON_CONST] THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[
IMP_CONJ]
CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[];
DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC)] THEN
FIRST_ASSUM(MP_TAC o MATCH_MP (REWRITE_RULE[
IMP_CONJ]
COMPACT_UNIFORMLY_CONTINUOUS)) THEN
SIMP_TAC[REWRITE_RULE[
PCROSS]
COMPACT_PCROSS;
COMPACT_INTERVAL] THEN
REWRITE_TAC[
uniformly_continuous_on] THEN
DISCH_THEN(MP_TAC o SPEC `e / &4`) THEN
ASM_REWRITE_TAC[REAL_ARITH `&0 < e / &4 <=> &0 < e`] THEN
MATCH_MP_TAC
MONO_EXISTS THEN X_GEN_TAC `d:real` THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
REWRITE_TAC[
IMP_CONJ;
RIGHT_FORALL_IMP_THM] THEN
REWRITE_TAC[
FORALL_IN_GSPEC] THEN
REWRITE_TAC[
RIGHT_IMP_FORALL_THM; IMP_IMP; GSYM
CONJ_ASSOC] THEN
DISCH_THEN(MP_TAC o MATCH_MP (MESON[]
`(!t x t' x'. P t x t' x') ==> (!t t' u. P t u t' u)`)) THEN
REWRITE_TAC[dist;
NORM_PASTECART;
PASTECART_SUB] THEN
REWRITE_TAC[
VECTOR_SUB_REFL;
NORM_0] THEN
CONV_TAC REAL_RAT_REDUCE_CONV THEN
REWRITE_TAC[TAUT `a /\ b /\ c /\ b /\ d <=> a /\ c /\ b /\ d`] THEN
SIMP_TAC[
REAL_ADD_RID;
POW_2_SQRT;
NORM_POS_LE] THEN DISCH_TAC THEN
ASM_REWRITE_TAC[] THEN
MAP_EVERY X_GEN_TAC [`t1:real^1`; `t2:real^1`] THEN
STRIP_TAC THEN EXISTS_TAC `e / &4` THEN
ASM_REWRITE_TAC[REAL_ARITH `&0 < e / &4 <=> &0 < e`] THEN
MAP_EVERY X_GEN_TAC [`g1:real^1->complex`; `g2:real^1->complex`] THEN
STRIP_TAC THEN FIRST_X_ASSUM
(MP_TAC o SPECL [`g1:real^1->complex`; `g2:real^1->complex`]) THEN
ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ALL_TAC; ASM_MESON_TAC[]] THEN
X_GEN_TAC `u:real^1` THEN STRIP_TAC THEN
ASM_MESON_TAC[NORM_ARITH
`norm(g1 - k1) < e / &4 /\ norm(g2 - k2) < e / &4 /\
norm(k1 - kt) < e / &4 /\ norm(k2 - kt) < e / &4
==> norm(g1 - kt) < e /\ norm(g2 - kt) < e`];
GEN_REWRITE_TAC (LAND_CONV o BINDER_CONV) [
RIGHT_IMP_EXISTS_THM] THEN
REWRITE_TAC[
SKOLEM_THM;
LEFT_IMP_EXISTS_THM]] THEN
X_GEN_TAC `ee:real^1->real` THEN DISCH_THEN(LABEL_TAC "*") THEN
MP_TAC(ISPEC `interval[vec 0:real^1,vec 1]`
COMPACT_IMP_HEINE_BOREL) THEN
REWRITE_TAC[
COMPACT_INTERVAL] THEN
DISCH_THEN(MP_TAC o SPEC
`
IMAGE (\t:real^1. ball(t,ee t / &3)) (interval[vec 0,vec 1])`) THEN
ANTS_TAC THENL
[REWRITE_TAC[
FORALL_IN_IMAGE;
OPEN_BALL;
SUBSET] THEN
X_GEN_TAC `t:real^1` THEN DISCH_TAC THEN
REWRITE_TAC[
UNIONS_IMAGE;
IN_ELIM_THM] THEN EXISTS_TAC `t:real^1` THEN
ASM_SIMP_TAC[
CENTRE_IN_BALL; REAL_ARITH `&0 < e / &3 <=> &0 < e`];
ALL_TAC] THEN
ONCE_REWRITE_TAC[TAUT `a /\ b /\ c <=> b /\ a /\ c`] THEN
REWRITE_TAC[
CONJ_ASSOC;
FINITE_SUBSET_IMAGE] THEN
REWRITE_TAC[
LEFT_AND_EXISTS_THM; MESON[]
`(?f s. (P s /\ f = g s) /\ Q f) <=> ?s. P s /\ Q(g s)`] THEN
REWRITE_TAC[
UNIONS_IMAGE;
LEFT_IMP_EXISTS_THM] THEN
X_GEN_TAC `k:real^1->bool` THEN
DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC) THEN
GEN_REWRITE_TAC LAND_CONV [
SUBSET] THEN REWRITE_TAC[
IN_ELIM_THM] THEN
ASM_CASES_TAC `k:real^1->bool = {}` THENL
[ASM_REWRITE_TAC[
NOT_IN_EMPTY; GSYM
NOT_EXISTS_THM;
MEMBER_NOT_EMPTY] THEN
REWRITE_TAC[
INTERVAL_EQ_EMPTY_1;
DROP_VEC] THEN REAL_ARITH_TAC;
DISCH_THEN(LABEL_TAC "+")] THEN
SUBGOAL_THEN `!i:real^1. i
IN k ==> &0 < ee(i)`
ASSUME_TAC THENL [ASM_MESON_TAC[
SUBSET]; ALL_TAC] THEN
ABBREV_TAC `e = inf(
IMAGE (ee:real^1->real) k)` THEN
MP_TAC(ISPEC `
IMAGE (ee:real^1->real) k`
INF_FINITE) THEN
MP_TAC(ISPECL [`
IMAGE (ee:real^1->real) k`; `&0`]
REAL_LT_INF_FINITE) THEN
ASM_SIMP_TAC[
FINITE_IMAGE;
IMAGE_EQ_EMPTY;
FORALL_IN_IMAGE] THEN
DISCH_TAC THEN DISCH_THEN(ASSUME_TAC o CONJUNCT2) THEN
MP_TAC(ISPEC `e / &3`
REAL_ARCH_INV) THEN
ASM_REWRITE_TAC[REAL_ARITH `&0 < e / &3 <=> &0 < e`] THEN
DISCH_THEN(X_CHOOSE_THEN `N:num` STRIP_ASSUME_TAC) THEN
SUBGOAL_THEN
`!n. n <= N
==> ?d. &0 < d /\
!j.
valid_path j /\
(!u. u
IN interval [vec 0,vec 1]
==> norm(j u - k(pastecart (lift(&n / &N)) u)) < d) /\
pathstart j = pathstart g /\
pathfinish j = pathfinish g
==>
path_image j
SUBSET s /\
path_integral j f =
path_integral g f`
(MP_TAC o SPEC `N:num`) THENL
[ALL_TAC;
REWRITE_TAC[
LE_REFL;
LEFT_IMP_EXISTS_THM] THEN
GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
DISCH_THEN(MP_TAC o SPEC `h:real^1->complex`) THEN
ASM_SIMP_TAC[
REAL_DIV_REFL; REAL_OF_NUM_EQ;
LIFT_NUM] THEN
ASM_REWRITE_TAC[
VECTOR_SUB_REFL;
NORM_0] THEN MESON_TAC[]] THEN
INDUCT_TAC THENL
[REMOVE_THEN "*" (MP_TAC o SPEC `vec 0:real^1`) THEN
ASM_REWRITE_TAC[
real_div;
REAL_MUL_LZERO;
LE_0;
LIFT_NUM] THEN
REWRITE_TAC[
IN_INTERVAL_1;
DROP_VEC;
REAL_POS;
REAL_LE_REFL] THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
REPEAT(DISCH_THEN(MP_TAC o SPEC `vec 0:real^1`) THEN
REWRITE_TAC[
IN_INTERVAL_1;
DROP_VEC;
REAL_POS;
REAL_LE_REFL]) THEN
ASM_REWRITE_TAC[
VECTOR_SUB_REFL;
NORM_0] THEN
MATCH_MP_TAC
MONO_EXISTS THEN GEN_TAC THEN STRIP_TAC THEN
ASM_REWRITE_TAC[] THEN X_GEN_TAC `j:real^1->complex` THEN
STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL
[`g:real^1->complex`; `j:real^1->complex`]) THEN
ASM_REWRITE_TAC[
VECTOR_SUB_REFL;
NORM_0] THEN MESON_TAC[];
DISCH_TAC] THEN
SUBGOAL_THEN `lift(&n / &N)
IN interval[vec 0,vec 1] /\
lift(&(SUC n) / &N)
IN interval[vec 0,vec 1]`
STRIP_ASSUME_TAC THENL
[REWRITE_TAC[
IN_INTERVAL_1;
DROP_VEC;
LIFT_DROP] THEN
ASM_SIMP_TAC[
REAL_LE_RDIV_EQ;
REAL_LE_LDIV_EQ;
REAL_OF_NUM_LT;
LE_1] THEN
REWRITE_TAC[
REAL_MUL_LZERO; REAL_MUL_LID; REAL_OF_NUM_LE] THEN
ASM_ARITH_TAC;
ALL_TAC] THEN
REMOVE_THEN "+" (MP_TAC o SPEC `lift(&n / &N)`) THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_THEN `t:real^1` STRIP_ASSUME_TAC) THEN
FIRST_X_ASSUM(MP_TAC o check (is_imp o concl)) THEN
ANTS_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN
DISCH_THEN(X_CHOOSE_THEN `d1:real`
(CONJUNCTS_THEN2 ASSUME_TAC (LABEL_TAC "1"))) THEN
REMOVE_THEN "*" (MP_TAC o SPEC `t:real^1`) THEN
ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
DISCH_THEN(MP_TAC o SPECL [`lift(&n / &N)`; `lift(&(SUC n) / &N)`]) THEN
ASM_REWRITE_TAC[] THEN ANTS_TAC THENL
[FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
IN_BALL]) THEN
MATCH_MP_TAC(NORM_ARITH
`!e. norm(n' - n:real^N) < e / &3 /\ e <= ee
==> dist(t,n) < ee / &3 ==> norm(n - t) < ee /\ norm(n' - t) < ee`) THEN
EXISTS_TAC `e:real` THEN
REWRITE_TAC[
NORM_REAL; GSYM drop;
DROP_SUB;
LIFT_DROP] THEN
REWRITE_TAC[
real_div; GSYM
REAL_SUB_RDISTRIB] THEN
SIMP_TAC[
REAL_OF_NUM_SUB; ARITH_RULE `n <= SUC n`] THEN
REWRITE_TAC[ARITH_RULE `SUC n - n = 1`; REAL_MUL_LID] THEN
REWRITE_TAC[
REAL_ABS_INV;
REAL_ABS_NUM] THEN
ASM_SIMP_TAC[GSYM
real_div];
ALL_TAC] THEN
MATCH_MP_TAC
MONO_EXISTS THEN X_GEN_TAC `d2:real` THEN
STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
X_GEN_TAC `j:real^1->complex` THEN STRIP_TAC THEN
MP_TAC(ISPECL
[`\u:real^1. (k(pastecart (lift (&n / &N)) u):complex)`;
`min d1 d2`]
PATH_APPROX_VECTOR_POLYNOMIAL_FUNCTION) THEN
ASM_SIMP_TAC[
REAL_LT_MIN; REAL_ARITH `&0 < e / &4 <=> &0 < e`] THEN
ANTS_TAC THENL
[REWRITE_TAC[path] THEN GEN_REWRITE_TAC LAND_CONV [GSYM
o_DEF] THEN
MATCH_MP_TAC
CONTINUOUS_ON_COMPOSE THEN
SIMP_TAC[
CONTINUOUS_ON_PASTECART;
CONTINUOUS_ON_ID;
CONTINUOUS_ON_CONST] THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[
IMP_CONJ]
CONTINUOUS_ON_SUBSET)) THEN
ASM SET_TAC[];
DISCH_THEN(X_CHOOSE_THEN `p:real^1->complex` STRIP_ASSUME_TAC)] THEN
REMOVE_THEN "1" (MP_TAC o SPEC `p:real^1->complex`) THEN
ASM_SIMP_TAC[
VALID_PATH_VECTOR_POLYNOMIAL_FUNCTION] THEN STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPECL [`p:real^1->complex`; `j:real^1->complex`]) THEN
ASM_SIMP_TAC[
VALID_PATH_VECTOR_POLYNOMIAL_FUNCTION]);;
let CAUCHY_THEOREM_HOMOTOPIC_LOOPS = prove
(`!f g h s.
open s /\ f
holomorphic_on s /\
valid_path g /\
valid_path h /\
homotopic_loops s g h
==>
path_integral g f =
path_integral h f`,
REPEAT STRIP_TAC THEN
FIRST_ASSUM(STRIP_ASSUME_TAC o MATCH_MP
HOMOTOPIC_LOOPS_IMP_LOOP) THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
homotopic_loops]) THEN
REWRITE_TAC[
homotopic_with;
PCROSS;
LEFT_IMP_EXISTS_THM] THEN
X_GEN_TAC `k:real^(1,1)finite_sum->complex` THEN STRIP_TAC THEN
SUBGOAL_THEN
`!t. t
IN interval[vec 0:real^1,vec 1]
==> ?e. &0 < e /\
!t1 t2. t1
IN interval[vec 0:real^1,vec 1] /\
t2
IN interval[vec 0,vec 1] /\
norm(t1 - t) < e /\ norm(t2 - t) < e
==> ?d. &0 < d /\
!g1 g2.
valid_path g1 /\
valid_path g2 /\
(!u. u
IN interval[vec 0,vec 1]
==> norm(g1 u - k(pastecart t1 u)) < d /\
norm(g2 u - k(pastecart t2 u)) < d) /\
pathfinish g1 = pathstart g1 /\
pathfinish g2 = pathstart g2
==>
path_image g1
SUBSET s /\
path_image g2
SUBSET s /\
path_integral g2 f =
path_integral g1 f`
MP_TAC THENL
[X_GEN_TAC `t:real^1` THEN STRIP_TAC THEN
MP_TAC(ISPECL
[`s:complex->bool`; `\u. (k:real^(1,1)finite_sum->complex)(pastecart t u)`]
PATH_INTEGRAL_NEARBY_LOOP) THEN
REWRITE_TAC[] THEN ANTS_TAC THENL
[ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[GSYM
o_DEF] THEN
REWRITE_TAC[
path_image; path;
IMAGE_o] THEN CONJ_TAC THENL
[ALL_TAC; ASM SET_TAC[]] THEN
MATCH_MP_TAC
CONTINUOUS_ON_COMPOSE THEN
SIMP_TAC[
CONTINUOUS_ON_PASTECART;
CONTINUOUS_ON_ID;
CONTINUOUS_ON_CONST] THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[
IMP_CONJ]
CONTINUOUS_ON_SUBSET)) THEN ASM SET_TAC[];
DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC)] THEN
FIRST_ASSUM(MP_TAC o MATCH_MP (REWRITE_RULE[
IMP_CONJ]
COMPACT_UNIFORMLY_CONTINUOUS)) THEN
SIMP_TAC[REWRITE_RULE[
PCROSS]
COMPACT_PCROSS;
COMPACT_INTERVAL] THEN
REWRITE_TAC[
uniformly_continuous_on] THEN
DISCH_THEN(MP_TAC o SPEC `e / &4`) THEN
ASM_REWRITE_TAC[REAL_ARITH `&0 < e / &4 <=> &0 < e`] THEN
MATCH_MP_TAC
MONO_EXISTS THEN X_GEN_TAC `d:real` THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
REWRITE_TAC[
IMP_CONJ;
RIGHT_FORALL_IMP_THM] THEN
REWRITE_TAC[
FORALL_IN_GSPEC] THEN
REWRITE_TAC[
RIGHT_IMP_FORALL_THM; IMP_IMP; GSYM
CONJ_ASSOC] THEN
DISCH_THEN(MP_TAC o MATCH_MP (MESON[]
`(!t x t' x'. P t x t' x') ==> (!t t' u. P t u t' u)`)) THEN
REWRITE_TAC[dist;
NORM_PASTECART;
PASTECART_SUB] THEN
REWRITE_TAC[
VECTOR_SUB_REFL;
NORM_0] THEN
CONV_TAC REAL_RAT_REDUCE_CONV THEN
REWRITE_TAC[TAUT `a /\ b /\ c /\ b /\ d <=> a /\ c /\ b /\ d`] THEN
SIMP_TAC[
REAL_ADD_RID;
POW_2_SQRT;
NORM_POS_LE] THEN DISCH_TAC THEN
ASM_REWRITE_TAC[] THEN
MAP_EVERY X_GEN_TAC [`t1:real^1`; `t2:real^1`] THEN
STRIP_TAC THEN EXISTS_TAC `e / &4` THEN
ASM_REWRITE_TAC[REAL_ARITH `&0 < e / &4 <=> &0 < e`] THEN
MAP_EVERY X_GEN_TAC [`g1:real^1->complex`; `g2:real^1->complex`] THEN
STRIP_TAC THEN FIRST_X_ASSUM
(MP_TAC o SPECL [`g1:real^1->complex`; `g2:real^1->complex`]) THEN
ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ALL_TAC; ASM_MESON_TAC[]] THEN
X_GEN_TAC `u:real^1` THEN STRIP_TAC THEN
ASM_MESON_TAC[NORM_ARITH
`norm(g1 - k1) < e / &4 /\ norm(g2 - k2) < e / &4 /\
norm(k1 - kt) < e / &4 /\ norm(k2 - kt) < e / &4
==> norm(g1 - kt) < e /\ norm(g2 - kt) < e`];
GEN_REWRITE_TAC (LAND_CONV o BINDER_CONV) [
RIGHT_IMP_EXISTS_THM] THEN
REWRITE_TAC[
SKOLEM_THM;
LEFT_IMP_EXISTS_THM]] THEN
X_GEN_TAC `ee:real^1->real` THEN DISCH_THEN(LABEL_TAC "*") THEN
MP_TAC(ISPEC `interval[vec 0:real^1,vec 1]`
COMPACT_IMP_HEINE_BOREL) THEN
REWRITE_TAC[
COMPACT_INTERVAL] THEN
DISCH_THEN(MP_TAC o SPEC
`
IMAGE (\t:real^1. ball(t,ee t / &3)) (interval[vec 0,vec 1])`) THEN
ANTS_TAC THENL
[REWRITE_TAC[
FORALL_IN_IMAGE;
OPEN_BALL;
SUBSET] THEN
X_GEN_TAC `t:real^1` THEN DISCH_TAC THEN
REWRITE_TAC[
UNIONS_IMAGE;
IN_ELIM_THM] THEN EXISTS_TAC `t:real^1` THEN
ASM_SIMP_TAC[
CENTRE_IN_BALL; REAL_ARITH `&0 < e / &3 <=> &0 < e`];
ALL_TAC] THEN
ONCE_REWRITE_TAC[TAUT `a /\ b /\ c <=> b /\ a /\ c`] THEN
REWRITE_TAC[
CONJ_ASSOC;
FINITE_SUBSET_IMAGE] THEN
REWRITE_TAC[
LEFT_AND_EXISTS_THM; MESON[]
`(?f s. (P s /\ f = g s) /\ Q f) <=> ?s. P s /\ Q(g s)`] THEN
REWRITE_TAC[
UNIONS_IMAGE;
LEFT_IMP_EXISTS_THM] THEN
X_GEN_TAC `k:real^1->bool` THEN
DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC) THEN
GEN_REWRITE_TAC LAND_CONV [
SUBSET] THEN REWRITE_TAC[
IN_ELIM_THM] THEN
ASM_CASES_TAC `k:real^1->bool = {}` THENL
[ASM_REWRITE_TAC[
NOT_IN_EMPTY; GSYM
NOT_EXISTS_THM;
MEMBER_NOT_EMPTY] THEN
REWRITE_TAC[
INTERVAL_EQ_EMPTY_1;
DROP_VEC] THEN REAL_ARITH_TAC;
DISCH_THEN(LABEL_TAC "+")] THEN
SUBGOAL_THEN `!i:real^1. i
IN k ==> &0 < ee(i)`
ASSUME_TAC THENL [ASM_MESON_TAC[
SUBSET]; ALL_TAC] THEN
ABBREV_TAC `e = inf(
IMAGE (ee:real^1->real) k)` THEN
MP_TAC(ISPEC `
IMAGE (ee:real^1->real) k`
INF_FINITE) THEN
MP_TAC(ISPECL [`
IMAGE (ee:real^1->real) k`; `&0`]
REAL_LT_INF_FINITE) THEN
ASM_SIMP_TAC[
FINITE_IMAGE;
IMAGE_EQ_EMPTY;
FORALL_IN_IMAGE] THEN
DISCH_TAC THEN DISCH_THEN(ASSUME_TAC o CONJUNCT2) THEN
MP_TAC(ISPEC `e / &3`
REAL_ARCH_INV) THEN
ASM_REWRITE_TAC[REAL_ARITH `&0 < e / &3 <=> &0 < e`] THEN
DISCH_THEN(X_CHOOSE_THEN `N:num` STRIP_ASSUME_TAC) THEN
SUBGOAL_THEN
`!n. n <= N
==> ?d. &0 < d /\
!j.
valid_path j /\
(!u. u
IN interval [vec 0,vec 1]
==> norm(j u - k(pastecart (lift(&n / &N)) u)) < d) /\
pathfinish j = pathstart j
==>
path_image j
SUBSET s /\
path_integral j f =
path_integral g f`
(MP_TAC o SPEC `N:num`) THENL
[ALL_TAC;
REWRITE_TAC[
LE_REFL;
LEFT_IMP_EXISTS_THM] THEN
GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
DISCH_THEN(MP_TAC o SPEC `h:real^1->complex`) THEN
ASM_SIMP_TAC[
REAL_DIV_REFL; REAL_OF_NUM_EQ;
LIFT_NUM] THEN
ASM_REWRITE_TAC[
VECTOR_SUB_REFL;
NORM_0] THEN MESON_TAC[]] THEN
INDUCT_TAC THENL
[REMOVE_THEN "*" (MP_TAC o SPEC `vec 0:real^1`) THEN
ASM_REWRITE_TAC[
real_div;
REAL_MUL_LZERO;
LE_0;
LIFT_NUM] THEN
REWRITE_TAC[
IN_INTERVAL_1;
DROP_VEC;
REAL_POS;
REAL_LE_REFL] THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
REPEAT(DISCH_THEN(MP_TAC o SPEC `vec 0:real^1`) THEN
REWRITE_TAC[
IN_INTERVAL_1;
DROP_VEC;
REAL_POS;
REAL_LE_REFL]) THEN
ASM_REWRITE_TAC[
VECTOR_SUB_REFL;
NORM_0] THEN
MATCH_MP_TAC
MONO_EXISTS THEN GEN_TAC THEN STRIP_TAC THEN
ASM_REWRITE_TAC[] THEN X_GEN_TAC `j:real^1->complex` THEN
STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL
[`g:real^1->complex`; `j:real^1->complex`]) THEN
ASM_REWRITE_TAC[
VECTOR_SUB_REFL;
NORM_0] THEN MESON_TAC[];
DISCH_TAC] THEN
SUBGOAL_THEN `lift(&n / &N)
IN interval[vec 0,vec 1] /\
lift(&(SUC n) / &N)
IN interval[vec 0,vec 1]`
STRIP_ASSUME_TAC THENL
[REWRITE_TAC[
IN_INTERVAL_1;
DROP_VEC;
LIFT_DROP] THEN
ASM_SIMP_TAC[
REAL_LE_RDIV_EQ;
REAL_LE_LDIV_EQ;
REAL_OF_NUM_LT;
LE_1] THEN
REWRITE_TAC[
REAL_MUL_LZERO; REAL_MUL_LID; REAL_OF_NUM_LE] THEN
ASM_ARITH_TAC;
ALL_TAC] THEN
REMOVE_THEN "+" (MP_TAC o SPEC `lift(&n / &N)`) THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_THEN `t:real^1` STRIP_ASSUME_TAC) THEN
FIRST_X_ASSUM(MP_TAC o check (is_imp o concl)) THEN
ANTS_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN
DISCH_THEN(X_CHOOSE_THEN `d1:real`
(CONJUNCTS_THEN2 ASSUME_TAC (LABEL_TAC "1"))) THEN
REMOVE_THEN "*" (MP_TAC o SPEC `t:real^1`) THEN
ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
DISCH_THEN(MP_TAC o SPECL [`lift(&n / &N)`; `lift(&(SUC n) / &N)`]) THEN
ASM_REWRITE_TAC[] THEN ANTS_TAC THENL
[FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
IN_BALL]) THEN
MATCH_MP_TAC(NORM_ARITH
`!e. norm(n' - n:real^N) < e / &3 /\ e <= ee
==> dist(t,n) < ee / &3 ==> norm(n - t) < ee /\ norm(n' - t) < ee`) THEN
EXISTS_TAC `e:real` THEN
REWRITE_TAC[
NORM_REAL; GSYM drop;
DROP_SUB;
LIFT_DROP] THEN
REWRITE_TAC[
real_div; GSYM
REAL_SUB_RDISTRIB] THEN
SIMP_TAC[
REAL_OF_NUM_SUB; ARITH_RULE `n <= SUC n`] THEN
REWRITE_TAC[ARITH_RULE `SUC n - n = 1`; REAL_MUL_LID] THEN
REWRITE_TAC[
REAL_ABS_INV;
REAL_ABS_NUM] THEN
ASM_SIMP_TAC[GSYM
real_div];
ALL_TAC] THEN
MATCH_MP_TAC
MONO_EXISTS THEN X_GEN_TAC `d2:real` THEN
STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
X_GEN_TAC `j:real^1->complex` THEN STRIP_TAC THEN
MP_TAC(ISPECL
[`\u:real^1. (k(pastecart (lift (&n / &N)) u):complex)`;
`min d1 d2`]
PATH_APPROX_VECTOR_POLYNOMIAL_FUNCTION) THEN
ASM_SIMP_TAC[
REAL_LT_MIN; REAL_ARITH `&0 < e / &4 <=> &0 < e`] THEN
ANTS_TAC THENL
[REWRITE_TAC[path] THEN GEN_REWRITE_TAC LAND_CONV [GSYM
o_DEF] THEN
MATCH_MP_TAC
CONTINUOUS_ON_COMPOSE THEN
SIMP_TAC[
CONTINUOUS_ON_PASTECART;
CONTINUOUS_ON_ID;
CONTINUOUS_ON_CONST] THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[
IMP_CONJ]
CONTINUOUS_ON_SUBSET)) THEN
ASM SET_TAC[];
DISCH_THEN(X_CHOOSE_THEN `p:real^1->complex` STRIP_ASSUME_TAC)] THEN
REMOVE_THEN "1" (MP_TAC o SPEC `p:real^1->complex`) THEN
ASM_SIMP_TAC[
VALID_PATH_VECTOR_POLYNOMIAL_FUNCTION] THEN STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPECL [`p:real^1->complex`; `j:real^1->complex`]) THEN
ASM_SIMP_TAC[
VALID_PATH_VECTOR_POLYNOMIAL_FUNCTION]);;
let SIMPLE_CLOSED_PATH_WINDING_NUMBER_INSIDE = prove
(`!g.
simple_path g
==> (!z. z
IN inside(
path_image g) ==>
winding_number(g,z) = Cx(&1)) \/
(!z. z
IN inside(
path_image g) ==>
winding_number(g,z) = --Cx(&1))`,
let lemma1 = prove
(`!p a e.
&0 < e /\
simple_path(p ++ linepath(a - e % basis 1,a + e % basis 1)) /\
pathstart p = a + e % basis 1 /\ pathfinish p = a - e % basis 1 /\
ball(a,e) INTER path_image p = {}
==> ?z. z IN inside(path_image
(p ++ linepath(a - e % basis 1,a + e % basis 1))) /\
norm(winding_number
(p ++ linepath(a - e % basis 1,a + e % basis 1),z)) = &1`,
REPEAT STRIP_TAC THEN
MP_TAC(ISPECL
[`p:real^1->complex`; `linepath(a - e % basis 1,a + e % basis 1)`]
SIMPLE_PATH_JOIN_LOOP_EQ) THEN
ASM_REWRITE_TAC[PATHFINISH_LINEPATH; PATHSTART_LINEPATH] THEN
STRIP_TAC THEN
SUBGOAL_THEN
`(a:complex) IN frontier(inside
(path_image(p ++ linepath(a - e % basis 1,a + e % basis 1))))`
MP_TAC THENL
[FIRST_ASSUM
(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] JORDAN_INSIDE_OUTSIDE)) THEN
ASM_REWRITE_TAC[PATHSTART_JOIN; PATHFINISH_JOIN;
PATHFINISH_LINEPATH] THEN
STRIP_TAC THEN ASM_SIMP_TAC[PATH_IMAGE_JOIN; PATHSTART_LINEPATH] THEN
REWRITE_TAC[IN_UNION; PATH_IMAGE_LINEPATH] THEN DISJ2_TAC THEN
REWRITE_TAC[IN_SEGMENT] THEN EXISTS_TAC `&1 / &2` THEN
CONV_TAC REAL_RAT_REDUCE_CONV THEN VECTOR_ARITH_TAC;
ALL_TAC] THEN
REWRITE_TAC[FRONTIER_STRADDLE] THEN
DISCH_THEN(MP_TAC o SPEC `e:real`) THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_THEN `c:complex` STRIP_ASSUME_TAC o CONJUNCT1) THEN
MP_TAC(ISPEC
`path_image (p ++ linepath(a - e % basis 1:complex,a + e % basis 1))`
INSIDE_NO_OVERLAP) THEN
REWRITE_TAC[EXTENSION] THEN DISCH_THEN(MP_TAC o SPEC `c:complex`) THEN
ASM_REWRITE_TAC[IN_INTER; NOT_IN_EMPTY] THEN
ASM_SIMP_TAC[PATH_IMAGE_JOIN; PATHSTART_LINEPATH; PATH_IMAGE_LINEPATH] THEN
REWRITE_TAC[IN_UNION; DE_MORGAN_THM] THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [SEGMENT_AS_BALL] THEN
ASM_REWRITE_TAC[IN_INTER;
VECTOR_ARITH `inv(&2) % ((a - e) + (a + e)):complex = a`;
VECTOR_ARITH `(a + e) - (a - e):complex = &2 % e`] THEN
ASM_SIMP_TAC[NORM_MUL; NORM_BASIS; DIMINDEX_2; ARITH] THEN
ASM_SIMP_TAC[REAL_ARITH `&0 < e ==> (abs(&2) * abs e * &1) / &2 = e`] THEN
ASM_SIMP_TAC[IN_CBALL; REAL_LT_IMP_LE] THEN STRIP_TAC THEN
SUBGOAL_THEN
`~collinear{a - e % basis 1,c:complex,a + e % basis 1}`
ASSUME_TAC THENL
[MP_TAC(ISPECL
[`a - e % basis 1:complex`; `a + e % basis 1:complex`; `c:complex`]
COLLINEAR_3_AFFINE_HULL) THEN
ASM_SIMP_TAC[VECTOR_ARITH `a - x:complex = a + x <=> x = vec 0`;
BASIS_NONZERO; DIMINDEX_2; ARITH; VECTOR_MUL_EQ_0;
REAL_LT_IMP_NZ] THEN
REWRITE_TAC[INSERT_AC];
ALL_TAC] THEN
SUBGOAL_THEN
`~(interior(convex hull {a - e % basis 1,c:complex,a + e % basis 1}) = {})`
MP_TAC THENL
[ASM_SIMP_TAC[INTERIOR_CONVEX_HULL_3_MINIMAL] THEN
REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_ELIM_THM] THEN
REPEAT(ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN EXISTS_TAC `&1 / &3`) THEN
CONV_TAC REAL_RAT_REDUCE_CONV THEN MESON_TAC[];
ALL_TAC] THEN
REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `z:complex` THEN DISCH_TAC THEN
FIRST_ASSUM(MP_TAC o AP_TERM `norm:complex->real` o
MATCH_MP WINDING_NUMBER_TRIANGLE) THEN
REWRITE_TAC[] THEN ONCE_REWRITE_TAC[COND_RAND] THEN
REWRITE_TAC[NORM_NEG; COND_ID; COMPLEX_NORM_CX; REAL_ABS_NUM] THEN
DISCH_TAC THEN
MP_TAC(ISPECL
[`linepath(a + e % basis 1:complex,a - e % basis 1)`;
`p:real^1->complex`;
`linepath(a + e % basis 1:complex,c) ++ linepath(c,a - e % basis 1)`;
`a + e % basis 1:complex`; `a - e % basis 1:complex`;
`z:complex`;
`winding_number
(linepath(a - e % basis 1,c) ++
linepath(c,a + e % basis 1) ++
linepath(a + e % basis 1,a - e % basis 1),
z)`] WINDING_NUMBER_FROM_INNERPATH) THEN
ASM_SIMP_TAC[SIMPLE_PATH_LINEPATH; PATHSTART_JOIN; PATHFINISH_JOIN;
VECTOR_ARITH `a + x:complex = a - x <=> x = vec 0`;
BASIS_NONZERO; DIMINDEX_2; ARITH; VECTOR_MUL_EQ_0;
REAL_LT_IMP_NZ; PATHSTART_LINEPATH; PATHFINISH_LINEPATH;
ARC_IMP_SIMPLE_PATH; PATH_IMAGE_JOIN; PATH_IMAGE_LINEPATH] THEN
ANTS_TAC THENL
[ALL_TAC;
MATCH_MP_TAC(TAUT
`(p ==> p') /\ (p /\ q ==> q') ==> p /\ q ==> p' /\ q'`) THEN
CONJ_TAC THENL [MESON_TAC[UNION_COMM; SEGMENT_SYM]; ALL_TAC] THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (SUBST_ALL_TAC o SYM)) THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (NORM_ARITH
`norm(z:complex) = &1 ==> u = --z ==> norm u = &1`)) THEN
GEN_REWRITE_TAC (RAND_CONV o ONCE_DEPTH_CONV)
[GSYM REVERSEPATH_LINEPATH] THEN
ASM_SIMP_TAC[GSYM REVERSEPATH_JOINPATHS; PATHSTART_LINEPATH] THEN
ONCE_REWRITE_TAC[COMPLEX_RING `a:complex = --b <=> b = --a`] THEN
MATCH_MP_TAC WINDING_NUMBER_REVERSEPATH THEN
ASM_SIMP_TAC[PATH_JOIN; PATHSTART_LINEPATH; PATH_IMAGE_JOIN;
PATH_LINEPATH; ARC_IMP_PATH; PATH_IMAGE_LINEPATH] THEN
ONCE_REWRITE_TAC[SEGMENT_SYM] THEN ONCE_REWRITE_TAC[UNION_COMM] THEN
ASM_MESON_TAC[INSIDE_NO_OVERLAP; IN_INTER; NOT_IN_EMPTY]] THEN
REPEAT CONJ_TAC THENL
[MATCH_MP_TAC ARC_IMP_SIMPLE_PATH THEN MATCH_MP_TAC ARC_JOIN THEN
REWRITE_TAC[ARC_LINEPATH_EQ; PATHSTART_LINEPATH;
PATHFINISH_LINEPATH] THEN
REPEAT(CONJ_TAC THENL
[DISCH_THEN SUBST_ALL_TAC THEN
RULE_ASSUM_TAC(REWRITE_RULE[INSERT_AC; COLLINEAR_2]) THEN
FIRST_X_ASSUM CONTR_TAC;
ALL_TAC]) THEN
REWRITE_TAC[PATH_IMAGE_LINEPATH] THEN
MATCH_MP_TAC(SET_RULE `s = t ==> s SUBSET t`) THEN
MATCH_MP_TAC INTER_SEGMENT THEN ASM_MESON_TAC[INSERT_AC];
REWRITE_TAC[SEGMENT_CLOSED_OPEN] THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
`b INTER p = {}
==> s SUBSET b /\ k SUBSET p
==> (s UNION k) INTER p = k`)) THEN
CONJ_TAC THENL
[REWRITE_TAC[SUBSET; IN_SEGMENT; IN_BALL] THEN
REWRITE_TAC[VECTOR_ARITH
`(&1 - u) % (a + e) + u % (a - e):complex =
a + (&1 - &2 * u) % e`] THEN
REPEAT STRIP_TAC THEN
ASM_REWRITE_TAC[NORM_ARITH `dist(a:complex,a + e) = norm e`] THEN
SIMP_TAC[NORM_MUL; NORM_BASIS; DIMINDEX_2; ARITH] THEN
MATCH_MP_TAC(REAL_ARITH
`x * e < &1 * e /\ &0 < e ==> x * abs e * &1 < e`) THEN
ASM_SIMP_TAC[REAL_LT_RMUL_EQ] THEN ASM_REAL_ARITH_TAC;
REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET] THEN
ASM_MESON_TAC[PATHSTART_IN_PATH_IMAGE; PATHFINISH_IN_PATH_IMAGE]];
MATCH_MP_TAC(SET_RULE
`s INTER t1 = {a} /\ s INTER t2 = {b}
==> s INTER (t1 UNION t2) = {a,b}`) THEN
CONJ_TAC THENL
[GEN_REWRITE_TAC (LAND_CONV o LAND_CONV) [SEGMENT_SYM];
GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [SEGMENT_SYM]] THEN
MATCH_MP_TAC INTER_SEGMENT THEN DISJ2_TAC THEN
ASM_MESON_TAC[INSERT_AC];
MATCH_MP_TAC(SET_RULE
`s INTER t1 = {a} /\ s INTER t2 = {b}
==> s INTER (t1 UNION t2) = {a,b}`) THEN
CONJ_TAC THENL [ONCE_REWRITE_TAC[SEGMENT_SYM]; ALL_TAC] THEN
REWRITE_TAC[SEGMENT_CLOSED_OPEN] THEN
MATCH_MP_TAC(SET_RULE
`b IN p /\ ~(c IN p) /\ p INTER s = {}
==> p INTER (s UNION {c,b}) = {b}`) THEN
(CONJ_TAC THENL
[ASM_MESON_TAC[PATHSTART_IN_PATH_IMAGE; PATHFINISH_IN_PATH_IMAGE];
ASM_REWRITE_TAC[]]) THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
`b INTER p = {} ==> s SUBSET b ==> p INTER s = {}`)) THEN
REWRITE_TAC[GSYM INTERIOR_CBALL] THEN
MATCH_MP_TAC IN_INTERIOR_CLOSURE_CONVEX_SEGMENT THEN
ASM_REWRITE_TAC[CONVEX_CBALL; INTERIOR_CBALL; IN_BALL] THEN
MATCH_MP_TAC(REWRITE_RULE[SUBSET] CLOSURE_SUBSET) THEN
REWRITE_TAC[IN_CBALL;
NORM_ARITH `dist(a:complex,a - e) = norm e`;
NORM_ARITH `dist(a:complex,a + e) = norm e`] THEN
ASM_SIMP_TAC[NORM_MUL; NORM_BASIS; DIMINDEX_2; ARITH] THEN
ASM_REAL_ARITH_TAC;
REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN EXISTS_TAC `c:complex` THEN
REWRITE_TAC[IN_INTER; ENDS_IN_SEGMENT; IN_UNION] THEN
FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
`c IN s ==> s = t ==> c IN t`)) THEN
ASM_SIMP_TAC[PATH_IMAGE_JOIN; PATHSTART_LINEPATH] THEN
REWRITE_TAC[UNION_COMM; PATH_IMAGE_LINEPATH; SEGMENT_SYM];
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [GSYM
INSIDE_OF_TRIANGLE]) THEN
REWRITE_TAC[UNION_ACI; SEGMENT_SYM];
ASM_SIMP_TAC[REVERSEPATH_JOINPATHS; PATHSTART_JOIN; PATHFINISH_JOIN;
PATHSTART_LINEPATH; PATHFINISH_LINEPATH; REVERSEPATH_LINEPATH] THEN
RULE_ASSUM_TAC(REWRITE_RULE
[INTERIOR_OF_TRIANGLE; IN_DIFF; IN_UNION; DE_MORGAN_THM]) THEN
ASM_SIMP_TAC[WINDING_NUMBER_JOIN; PATH_JOIN; PATH_LINEPATH;
PATH_IMAGE_JOIN; IN_UNION; PATHSTART_JOIN; PATHFINISH_JOIN;
PATHSTART_LINEPATH; PATHFINISH_LINEPATH; PATH_IMAGE_LINEPATH] THEN
CONV_TAC COMPLEX_RING;
DISCH_THEN SUBST_ALL_TAC THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE LAND_CONV [COMPLEX_NORM_CX]) THEN
REAL_ARITH_TAC]) in
let lemma2 = prove
(`!p a d e.
&0 < d /\ &0 < e /\
simple_path(p ++ linepath(a - d % basis 1,a + e % basis 1)) /\
pathstart p = a + e % basis 1 /\ pathfinish p = a - d % basis 1
==> ?z. z IN inside(path_image
(p ++ linepath(a - d % basis 1,a + e % basis 1))) /\
norm(winding_number
(p ++ linepath(a - d % basis 1,a + e % basis 1),z)) = &1`,
REPEAT STRIP_TAC THEN
MP_TAC(ISPECL
[`p:real^1->complex`; `linepath(a - d % basis 1,a + e % basis 1)`]
SIMPLE_PATH_JOIN_LOOP_EQ) THEN
ASM_REWRITE_TAC[PATHFINISH_LINEPATH; PATHSTART_LINEPATH] THEN
REWRITE_TAC[ARC_LINEPATH_EQ; PATH_IMAGE_LINEPATH] THEN STRIP_TAC THEN
SUBGOAL_THEN `~((a:complex) IN path_image p)` ASSUME_TAC THENL
[FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
`p INTER s SUBSET {d,e}
==> a IN s /\ ~(d = a) /\ ~(e = a) ==> ~(a IN p)`)) THEN
REWRITE_TAC[GSYM BETWEEN_IN_SEGMENT; between] THEN
REWRITE_TAC[NORM_ARITH `dist(a - d:complex,a + e) = norm(d + e)`;
NORM_ARITH `dist(a - d:complex,a) + dist(a,a + e) = norm(d) + norm(e)`;
VECTOR_ARITH `a + e:complex = a <=> e = vec 0`;
VECTOR_ARITH `a - d:complex = a <=> d = vec 0`] THEN
SIMP_TAC[GSYM VECTOR_ADD_RDISTRIB; NORM_MUL; VECTOR_MUL_EQ_0] THEN
ASM_SIMP_TAC[BASIS_NONZERO; NORM_BASIS; DIMINDEX_2; ARITH] THEN
ASM_REAL_ARITH_TAC;
ALL_TAC] THEN
MP_TAC(ISPEC `(:complex) DIFF path_image p` OPEN_CONTAINS_BALL) THEN
ASM_SIMP_TAC[GSYM closed; CLOSED_ARC_IMAGE; IN_UNIV; IN_DIFF] THEN
DISCH_THEN(MP_TAC o SPEC `a:complex`) THEN ASM_REWRITE_TAC[] THEN
REWRITE_TAC[SET_RULE `s SUBSET UNIV DIFF t <=> s INTER t = {}`] THEN
DISCH_THEN(X_CHOOSE_THEN `k:real` STRIP_ASSUME_TAC) THEN
ABBREV_TAC `kde:real = min k (min d e) / &2` THEN
SUBGOAL_THEN `&0 < kde /\ kde < k /\ kde < d /\ kde < e`
STRIP_ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
MP_TAC(ISPECL
[`linepath(a + kde % basis 1,a + e % basis 1) ++ p ++
linepath(a - d % basis 1,a - kde % basis 1)`;
`a:complex`; `kde:real`] lemma1) THEN
ASM_SIMP_TAC[PATHSTART_JOIN; PATHFINISH_JOIN; PATH_IMAGE_JOIN;
PATHSTART_LINEPATH; PATHFINISH_LINEPATH; PATH_IMAGE_LINEPATH;
SIMPLE_PATH_JOIN_LOOP_EQ] THEN
ANTS_TAC THENL
[REPEAT CONJ_TAC THENL
[MATCH_MP_TAC ARC_JOIN THEN
ASM_SIMP_TAC[ARC_JOIN_EQ; PATHSTART_LINEPATH; PATHFINISH_LINEPATH;
PATHSTART_JOIN; PATHFINISH_JOIN; PATH_IMAGE_LINEPATH;
ARC_LINEPATH_EQ; PATH_IMAGE_JOIN] THEN
REWRITE_TAC[VECTOR_ARITH `a + e:complex = a + d <=> e - d = vec 0`;
VECTOR_ARITH `a - d:complex = a - e <=> e - d = vec 0`] THEN
REWRITE_TAC[GSYM VECTOR_SUB_RDISTRIB; VECTOR_MUL_EQ_0; REAL_SUB_0] THEN
ASM_SIMP_TAC[BASIS_NONZERO; NORM_BASIS; DIMINDEX_2; ARITH] THEN
ASM_SIMP_TAC[REAL_LT_IMP_NE] THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
`p INTER de SUBSET {e,d}
==> dk SUBSET de /\ ke SUBSET de /\ ~(e IN dk) /\ ~(d IN ke) /\
ke INTER dk = {}
==> p INTER dk SUBSET {d} /\ ke INTER (p UNION dk) SUBSET {e}`)) THEN
REWRITE_TAC[SUBSET_SEGMENT; ENDS_IN_SEGMENT] THEN
REWRITE_TAC[GSYM BETWEEN_IN_SEGMENT; between] THEN
REWRITE_TAC[NORM_ARITH `dist(a - d:complex,a + e) = norm(d + e) /\
dist(a + d,a - e) = norm(d + e) /\
dist(a - d,a - e) = norm(d - e) /\
dist(a + d,a + e) = norm(d - e)`] THEN
REWRITE_TAC[GSYM VECTOR_ADD_RDISTRIB; GSYM VECTOR_SUB_RDISTRIB] THEN
ASM_SIMP_TAC[NORM_MUL; NORM_BASIS; DIMINDEX_2; ARITH] THEN
REPEAT(CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC]) THEN
REWRITE_TAC[EXTENSION; IN_ELIM_THM; IN_INTER; NOT_IN_EMPTY] THEN
MATCH_MP_TAC(MESON[REAL_LT_ANTISYM]
`!a:complex. (!x. x IN t ==> x$1 < a$1) /\ (!x. x IN s ==> a$1 < x$1)
==> !x. ~(x IN s /\ x IN t)`) THEN
EXISTS_TAC `a:complex` THEN
SIMP_TAC[IN_SEGMENT; LEFT_IMP_EXISTS_THM] THEN
SIMP_TAC[VECTOR_SUB_COMPONENT; VECTOR_ADD_COMPONENT;
VECTOR_MUL_COMPONENT; BASIS_COMPONENT; DIMINDEX_2; ARITH] THEN
REWRITE_TAC[REAL_ARITH
`(a < (&1 - u) * (a + x) + u * (a + y) <=>
&0 < (&1 - u) * x + u * y) /\
((&1 - u) * (a - x) + u * (a - y) < a <=>
&0 < (&1 - u) * x + u * y)`] THEN
REPEAT STRIP_TAC THEN REWRITE_TAC[REAL_MUL_RID] THEN
REWRITE_TAC[REAL_ARITH `&0 < (&1 - u) * x + u * y <=>
(&1 - u) * --x + u * --y < &0`] THEN
MATCH_MP_TAC REAL_CONVEX_BOUND_LT THEN ASM_REAL_ARITH_TAC;
REWRITE_TAC[ARC_LINEPATH_EQ; VECTOR_MUL_EQ_0;
VECTOR_ARITH `a - k:complex = a + k <=> k = vec 0`] THEN
ASM_SIMP_TAC[REAL_LT_IMP_NZ; BASIS_NONZERO; DIMINDEX_2; ARITH];
MATCH_MP_TAC(SET_RULE
`kk INTER p = {} /\ kk INTER ke = {kp} /\ dk INTER kk = {kn}
==> (ke UNION p UNION dk) INTER kk SUBSET {kp,kn}`) THEN
CONJ_TAC THENL
[FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
`b INTER p = {} ==> s SUBSET b ==> s INTER p = {}`)) THEN
SIMP_TAC[SUBSET; IN_SEGMENT; IN_BALL; LEFT_IMP_EXISTS_THM] THEN
REWRITE_TAC[VECTOR_ARITH
`(&1 - u) % (a - d) + u % (a + d):complex = a - (&1 - &2 * u) % d`;
NORM_ARITH `dist(a:complex,a - d) = norm d`] THEN
REPEAT STRIP_TAC THEN
SIMP_TAC[NORM_MUL; NORM_BASIS; DIMINDEX_2; ARITH] THEN
MATCH_MP_TAC(REAL_ARITH
`&0 < kd /\ a * kd <= &1 * kd /\ kd < k
==> a * abs kd * &1 < k`) THEN
ASM_SIMP_TAC[REAL_LE_RMUL_EQ] THEN ASM_REAL_ARITH_TAC;
CONJ_TAC THEN MATCH_MP_TAC INTER_SEGMENT THEN DISJ1_TAC THEN
REWRITE_TAC[GSYM BETWEEN_IN_SEGMENT; between] THEN
REWRITE_TAC[NORM_ARITH `dist(a - d:complex,a + e) = norm(d + e) /\
dist(a + d,a - e) = norm(d + e) /\
dist(a - d,a - e) = norm(d - e) /\
dist(a + d,a + e) = norm(d - e)`] THEN
REWRITE_TAC[GSYM VECTOR_ADD_RDISTRIB; GSYM VECTOR_SUB_RDISTRIB] THEN
ASM_SIMP_TAC[NORM_MUL; NORM_BASIS; DIMINDEX_2; ARITH] THEN
ASM_REAL_ARITH_TAC];
REWRITE_TAC[UNION_OVER_INTER; EMPTY_UNION] THEN
ONCE_REWRITE_TAC[TAUT `a /\ b /\ c <=> b /\ a /\ c`] THEN CONJ_TAC THENL
[FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
`b INTER p = {} ==> c SUBSET b ==> c INTER p = {}`)) THEN
MATCH_MP_TAC SUBSET_BALL THEN ASM_SIMP_TAC[REAL_LT_IMP_LE];
ALL_TAC] THEN
REWRITE_TAC[SET_RULE `s INTER t = {} <=>
!x. x IN t ==> ~(x IN s)`] THEN
SIMP_TAC[IN_SEGMENT; LEFT_IMP_EXISTS_THM; IN_BALL] THEN
REWRITE_TAC[VECTOR_ARITH
`(&1 - u) % (a - d) + u % (a - e):complex =
a - ((&1 - u) % d + u % e) /\
(&1 - u) % (a + d) + u % (a + e):complex =
a + ((&1 - u) % d + u % e)`;
NORM_ARITH
`dist(a:complex,a + d) = norm d /\ dist(a,a - e) = norm e`] THEN
REWRITE_TAC[VECTOR_MUL_ASSOC; GSYM VECTOR_ADD_RDISTRIB] THEN
SIMP_TAC[NORM_MUL; NORM_BASIS; DIMINDEX_2; ARITH] THEN
REWRITE_TAC[REAL_NOT_LT; REAL_MUL_RID] THEN REPEAT STRIP_TAC THEN
MATCH_MP_TAC(REAL_ARITH `x <= y ==> x <= abs y`) THEN
REWRITE_TAC[REAL_ARITH
`(k <= (&1 - u) * k + u * e <=> &0 <= u * (e - k)) /\
(k <= (&1 - u) * d + u * k <=> &0 <= (&1 - u) * (d - k))`] THEN
MATCH_MP_TAC REAL_LE_MUL THEN ASM_REAL_ARITH_TAC];
ALL_TAC] THEN
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `z:complex` THEN
MATCH_MP_TAC(TAUT
`(p <=> p') /\ (p /\ p' ==> (q <=> q')) ==> p /\ q ==> p' /\ q'`) THEN
CONJ_TAC THENL
[AP_TERM_TAC THEN AP_TERM_TAC THEN
ONCE_REWRITE_TAC[SET_RULE
`(c UNION p UNION a) UNION b = p UNION (a UNION b UNION c)`] THEN
AP_TERM_TAC THEN
W(MP_TAC o PART_MATCH (lhand o rand) UNION_SEGMENT o
rand o lhand o snd) THEN
REWRITE_TAC[GSYM BETWEEN_IN_SEGMENT; between;
NORM_ARITH `dist(a - d:complex,a + e) = norm(d + e)`;
NORM_ARITH `dist(a + d:complex,a + e) = norm(d - e)`] THEN
ASM_SIMP_TAC[GSYM VECTOR_ADD_RDISTRIB; GSYM VECTOR_SUB_RDISTRIB;
NORM_MUL; NORM_BASIS; DIMINDEX_2; ARITH] THEN
ANTS_TAC THENL [ASM_REAL_ARITH_TAC; DISCH_THEN SUBST1_TAC] THEN
MATCH_MP_TAC UNION_SEGMENT THEN
REWRITE_TAC[GSYM BETWEEN_IN_SEGMENT; between;
NORM_ARITH `dist(a - d:complex,a + e) = norm(d + e)`;
NORM_ARITH `dist(a - d:complex,a - e) = norm(d - e)`] THEN
ASM_SIMP_TAC[GSYM VECTOR_ADD_RDISTRIB; GSYM VECTOR_SUB_RDISTRIB;
NORM_MUL; NORM_BASIS; DIMINDEX_2; ARITH] THEN
ASM_REAL_ARITH_TAC;
ALL_TAC] THEN
DISCH_THEN(CONJUNCTS_THEN (MP_TAC o MATCH_MP
(MESON[INSIDE_NO_OVERLAP; IN_INTER; NOT_IN_EMPTY]
`z IN inside s ==> ~(z IN s)`))) THEN
REWRITE_TAC[IN_UNION; DE_MORGAN_THM] THEN REPEAT STRIP_TAC THEN
AP_THM_TAC THEN AP_TERM_TAC THEN AP_TERM_TAC THEN
ASM_SIMP_TAC[WINDING_NUMBER_JOIN; PATH_JOIN; ARC_IMP_PATH; PATH_LINEPATH;
PATH_IMAGE_JOIN; IN_UNION; PATH_IMAGE_LINEPATH; PATHSTART_JOIN;
PATHFINISH_JOIN; PATHSTART_LINEPATH; PATHFINISH_LINEPATH] THEN
MATCH_MP_TAC(COMPLEX_RING
`d + k + e:complex = z ==> (e + p + d) + k = p + z`) THEN
MATCH_MP_TAC EQ_TRANS THEN
EXISTS_TAC
`winding_number(linepath (a - d % basis 1:complex,a - kde % basis 1),z) +
winding_number(linepath (a - kde % basis 1,a + e % basis 1),z)` THEN
CONJ_TAC THENL [AP_TERM_TAC; ALL_TAC] THEN CONV_TAC SYM_CONV THEN
MATCH_MP_TAC WINDING_NUMBER_SPLIT_LINEPATH THEN
ASM_REWRITE_TAC[] THENL
[CONJ_TAC THENL
[ALL_TAC;
SUBGOAL_THEN
`~(z IN segment[a - kde % basis 1:complex,a + kde % basis 1]) /\
~(z IN segment[a + kde % basis 1,a + e % basis 1])`
MP_TAC THENL [ASM_REWRITE_TAC[]; ALL_TAC] THEN
MATCH_MP_TAC(SET_RULE
`s UNION t = u ==> ~(z IN s) /\ ~(z IN t) ==> ~(z IN u)`) THEN
MATCH_MP_TAC UNION_SEGMENT];
ALL_TAC] THEN
REWRITE_TAC[GSYM BETWEEN_IN_SEGMENT; between] THEN
REWRITE_TAC[NORM_ARITH `dist(a - d:complex,a + e) = norm(d + e)`;
NORM_ARITH `dist(a - d:complex,a - e) = norm(d - e)`;
NORM_ARITH `dist(a + d:complex,a + e) = norm(d - e)`] THEN
ASM_SIMP_TAC[GSYM VECTOR_ADD_RDISTRIB; GSYM VECTOR_SUB_RDISTRIB; NORM_MUL;
NORM_BASIS; DIMINDEX_2; ARITH] THEN
ASM_REAL_ARITH_TAC) in
let lemma3 = prove
(`!p:real^1->complex.
simple_path p /\ pathfinish p = pathstart p
==> ?z. z IN inside(path_image p) /\ norm(winding_number(p,z)) = &1`,
GEN_TAC THEN STRIP_TAC THEN
MP_TAC(ISPEC `p:real^1->complex` JORDAN_INSIDE_OUTSIDE) THEN
ASM_REWRITE_TAC[] THEN STRIP_TAC THEN
UNDISCH_TAC `~(inside(path_image p):complex->bool = {})` THEN
REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN
DISCH_THEN(X_CHOOSE_TAC `a:complex`) THEN
MP_TAC(ISPECL [`inside(path_image p):complex->bool`;
`a:complex`; `basis 1:complex`]
RAY_TO_FRONTIER) THEN
MP_TAC(ISPECL [`inside(path_image p):complex->bool`;
`a:complex`; `--basis 1:complex`]
RAY_TO_FRONTIER) THEN
ASM_SIMP_TAC[INTERIOR_OPEN; VECTOR_NEG_EQ_0; BASIS_NONZERO;
DIMINDEX_2; ARITH] THEN
REWRITE_TAC[VECTOR_ARITH `a + d % --b:complex = a - d % b`] THEN
DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN
DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN
SUBGOAL_THEN
`?t. t IN interval[vec 0,vec 1] /\
(p:real^1->complex) t = a - d % basis 1`
STRIP_ASSUME_TAC THENL
[RULE_ASSUM_TAC(REWRITE_RULE[path_image; IN_IMAGE]) THEN
ASM_MESON_TAC[];
ALL_TAC] THEN
SUBGOAL_THEN
`?q. simple_path q /\
pathstart q:complex = a - d % basis 1 /\
pathfinish q = a - d % basis 1 /\
path_image q = path_image p /\
(!z. z IN inside(path_image p)
==> winding_number(q,z) = winding_number(p,z))`
MP_TAC THENL
[EXISTS_TAC `shiftpath t (p:real^1->complex)` THEN
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_INTERVAL_1]) THEN
ASM_SIMP_TAC[PATHSTART_SHIFTPATH; PATHFINISH_SHIFTPATH; DROP_VEC;
SIMPLE_PATH_SHIFTPATH; PATH_IMAGE_SHIFTPATH] THEN
REPEAT STRIP_TAC THEN MATCH_MP_TAC WINDING_NUMBER_SHIFTPATH THEN
ASM_SIMP_TAC[SIMPLE_PATH_IMP_PATH] THEN
ASM_MESON_TAC[INSIDE_NO_OVERLAP; IN_INTER; NOT_IN_EMPTY];
DISCH_THEN(X_CHOOSE_THEN `q:real^1->complex` MP_TAC) THEN
REPLICATE_TAC 3 (DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
REWRITE_TAC[IMP_CONJ] THEN DISCH_THEN(SUBST_ALL_TAC o SYM) THEN
SUBGOAL_THEN
`?z. z IN inside(path_image q) /\ norm(winding_number(q,z)) = &1`
(fun th -> MESON_TAC[th]) THEN
POP_ASSUM_LIST(MP_TAC o end_itlist CONJ o rev o
filter (fun tm -> not(free_in `t:real^1` (concl tm) or
free_in `p:real^1->complex` (concl tm)))) THEN
STRIP_TAC] THEN
SUBGOAL_THEN
`?t. t IN interval[vec 0,vec 1] /\
(q:real^1->complex) t = a + e % basis 1`
STRIP_ASSUME_TAC THENL
[RULE_ASSUM_TAC(REWRITE_RULE[path_image; IN_IMAGE]) THEN
ASM_MESON_TAC[];
ALL_TAC] THEN
SUBGOAL_THEN `~(a - d % basis 1:complex = a + e % basis 1)`
ASSUME_TAC THENL
[REWRITE_TAC[VECTOR_ARITH
`a - d % l:complex = a + e % l <=> (e + d) % l = vec 0`] THEN
SIMP_TAC[VECTOR_MUL_EQ_0; BASIS_NONZERO; DIMINDEX_2; ARITH] THEN
ASM_REAL_ARITH_TAC;
ALL_TAC] THEN
SUBGOAL_THEN
`path_image q INTER segment[a - d % basis 1,a + e % basis 1] =
{a - d % basis 1:complex,a + e % basis 1}`
ASSUME_TAC THENL
[REWRITE_TAC[SEGMENT_CLOSED_OPEN] THEN
MATCH_MP_TAC(SET_RULE
`a IN p /\ b IN p /\ p INTER s = {}
==> p INTER (s UNION {a,b}) = {a,b}`) THEN
CONJ_TAC THENL [ASM_MESON_TAC[PATHSTART_IN_PATH_IMAGE]; ALL_TAC] THEN
CONJ_TAC THENL [ASM_MESON_TAC[path_image; IN_IMAGE]; ALL_TAC] THEN
ASM_SIMP_TAC[PATH_IMAGE_SUBPATH_SUBSET; SIMPLE_PATH_IMP_PATH;
ENDS_IN_UNIT_INTERVAL] THEN
REWRITE_TAC[SET_RULE `s INTER t = {} <=> !x. x IN t ==> ~(x IN s)`] THEN
REWRITE_TAC[IN_SEGMENT; VECTOR_ARITH
`(&1 - u) % (a - d % l) + u % (a + e % l):complex =
a + (u * e - (&1 - u) * d) % l`] THEN
X_GEN_TAC `y:complex` THEN
DISCH_THEN(X_CHOOSE_THEN `k:real` STRIP_ASSUME_TAC o CONJUNCT2) THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(MESON
[INSIDE_NO_OVERLAP; IN_INTER; NOT_IN_EMPTY]
`x IN inside s ==> ~(x IN s)`) THEN
ASM_CASES_TAC `&0 <= k * e - (&1 - k) * d` THENL
[ALL_TAC;
ONCE_REWRITE_TAC[VECTOR_ARITH
`a + (s - t) % l:complex = a - (t - s) % l`]] THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN
ASM_SIMP_TAC[REAL_ARITH `~(&0 <= a - b) ==> &0 <= b - a`] THEN
REWRITE_TAC[REAL_ARITH `k * e - (&1 - k) * d < e <=>
&0 < (&1 - k) * (d + e)`] THEN
REWRITE_TAC[REAL_ARITH `(&1 - k) * d - k * e < d <=>
&0 < k * (d + e)`] THEN
MATCH_MP_TAC REAL_LT_MUL THEN ASM_REAL_ARITH_TAC;
ALL_TAC] THEN
MP_TAC(ISPECL
[`subpath t (vec 0) (q:real^1->complex)`;
`a:complex`; `d:real`; `e:real`] lemma2) THEN
ASM_SIMP_TAC[PATHSTART_SUBPATH; PATHFINISH_SUBPATH;
PATH_IMAGE_JOIN; PATHSTART_LINEPATH] THEN
ANTS_TAC THENL
[CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[pathstart]] THEN
MATCH_MP_TAC SIMPLE_PATH_JOIN_LOOP THEN
ASM_REWRITE_TAC[PATHSTART_SUBPATH; PATHFINISH_SUBPATH;
PATHSTART_LINEPATH; PATHFINISH_LINEPATH] THEN
ASM_REWRITE_TAC[ARC_LINEPATH_EQ] THEN REPEAT CONJ_TAC THENL
[MATCH_MP_TAC ARC_SIMPLE_PATH_SUBPATH THEN
RULE_ASSUM_TAC(REWRITE_RULE[pathstart]) THEN
ASM_REWRITE_TAC[ENDS_IN_UNIT_INTERVAL];
RULE_ASSUM_TAC(REWRITE_RULE[pathstart]) THEN ASM_REWRITE_TAC[];
REWRITE_TAC[PATH_IMAGE_LINEPATH] THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
`p INTER s = {a,b} ==> p' SUBSET p ==> p' INTER s SUBSET {b,a}`)) THEN
ASM_SIMP_TAC[PATH_IMAGE_SUBPATH_SUBSET; SIMPLE_PATH_IMP_PATH;
ENDS_IN_UNIT_INTERVAL]];
DISCH_THEN(X_CHOOSE_THEN `z:complex` STRIP_ASSUME_TAC)] THEN
MP_TAC(ISPECL
[`subpath (vec 0) t (q:real^1->complex)`;
`subpath (vec 1) t (q:real^1->complex)`;
`linepath(a - d % basis 1:complex,a + e % basis 1)`;
`a - d % basis 1:complex`; `a + e % basis 1:complex`;
`z:complex`;
`--winding_number
(subpath t (vec 0) q ++
linepath (a - d % basis 1,a + e % basis 1),z)`]
WINDING_NUMBER_FROM_INNERPATH) THEN
ASM_REWRITE_TAC[PATHSTART_SUBPATH; PATHFINISH_SUBPATH;
PATHSTART_LINEPATH; PATHFINISH_LINEPATH] THEN
REWRITE_TAC[REVERSEPATH_SUBPATH; REVERSEPATH_LINEPATH] THEN
SUBGOAL_THEN
`path_image (subpath (vec 0) t q) UNION
path_image (subpath (vec 1) t q) :complex->bool =
path_image q`
SUBST1_TAC THENL
[FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_INTERVAL_1]) THEN
SIMP_TAC[DROP_VEC; PATH_IMAGE_SUBPATH] THEN
ONCE_REWRITE_TAC[GSYM PATH_IMAGE_REVERSEPATH] THEN
REWRITE_TAC[REVERSEPATH_SUBPATH] THEN
SIMP_TAC[DROP_VEC; PATH_IMAGE_SUBPATH] THEN STRIP_TAC THEN
REWRITE_TAC[GSYM IMAGE_UNION; PATH_IMAGE_REVERSEPATH] THEN
SUBGOAL_THEN `interval[vec 0:real^1,t] UNION interval[t,vec 1] =
interval[vec 0,vec 1]`
(fun th -> ASM_REWRITE_TAC[th; GSYM path_image]) THEN
REWRITE_TAC[EXTENSION; IN_UNION; IN_INTERVAL_1; DROP_VEC] THEN
ASM_REAL_ARITH_TAC;
ALL_TAC] THEN
ANTS_TAC THENL
[RULE_ASSUM_TAC(REWRITE_RULE[pathstart; pathfinish]) THEN
REPLICATE_TAC 2 (ASM_REWRITE_TAC[] THEN CONJ_TAC THENL
[MATCH_MP_TAC SIMPLE_PATH_SUBPATH THEN
ASM_REWRITE_TAC[ENDS_IN_UNIT_INTERVAL] THEN ASM_MESON_TAC[];
ALL_TAC]) THEN
ASM_REWRITE_TAC[SIMPLE_PATH_LINEPATH_EQ; PATH_IMAGE_LINEPATH] THEN
REPEAT CONJ_TAC THENL
[FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_INTERVAL_1]) THEN
SIMP_TAC[DROP_VEC; PATH_IMAGE_SUBPATH] THEN
ONCE_REWRITE_TAC[GSYM PATH_IMAGE_REVERSEPATH] THEN
REWRITE_TAC[REVERSEPATH_SUBPATH] THEN
SIMP_TAC[DROP_VEC; PATH_IMAGE_SUBPATH] THEN STRIP_TAC THEN
MATCH_MP_TAC(SET_RULE
`a IN s /\ a IN t /\ b IN s /\ b IN t /\
(!x. x IN s ==> !y. y IN t ==> x = y ==> x = a \/ x = b)
==> s INTER t = {a,b}`) THEN
REPEAT CONJ_TAC THENL
[REWRITE_TAC[IN_IMAGE] THEN EXISTS_TAC `vec 0:real^1` THEN
ASM_REWRITE_TAC[ENDS_IN_INTERVAL; INTERVAL_EQ_EMPTY_1; DROP_VEC] THEN
ASM_REAL_ARITH_TAC;
REWRITE_TAC[IN_IMAGE] THEN EXISTS_TAC `vec 1:real^1` THEN
ASM_REWRITE_TAC[ENDS_IN_INTERVAL; INTERVAL_EQ_EMPTY_1; DROP_VEC] THEN
ASM_REAL_ARITH_TAC;
REWRITE_TAC[IN_IMAGE] THEN EXISTS_TAC `t:real^1` THEN
ASM_REWRITE_TAC[ENDS_IN_INTERVAL; INTERVAL_EQ_EMPTY_1; DROP_VEC] THEN
ASM_REAL_ARITH_TAC;
REWRITE_TAC[IN_IMAGE] THEN EXISTS_TAC `t:real^1` THEN
ASM_REWRITE_TAC[ENDS_IN_INTERVAL; INTERVAL_EQ_EMPTY_1; DROP_VEC] THEN
ASM_REAL_ARITH_TAC;
ALL_TAC] THEN
REWRITE_TAC[FORALL_IN_IMAGE; IN_INTERVAL_1; DROP_VEC] THEN
X_GEN_TAC `s:real^1` THEN STRIP_TAC THEN
X_GEN_TAC `u:real^1` THEN STRIP_TAC THEN DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [simple_path]) THEN
DISCH_THEN(MP_TAC o SPECL [`s:real^1`; `u:real^1`] o CONJUNCT2) THEN
ASM_REWRITE_TAC[IN_INTERVAL_1; DROP_VEC] THEN
ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
DISCH_THEN(REPEAT_TCL DISJ_CASES_THEN
(REPEAT_TCL CONJUNCTS_THEN SUBST_ALL_TAC)) THEN
ASM_REWRITE_TAC[] THEN SUBGOAL_THEN `drop u = drop t` MP_TAC THENL
[ASM_REAL_ARITH_TAC; ASM_MESON_TAC[DROP_EQ]];
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
`p INTER s = {a,b}
==> a IN q /\ b IN q /\ q SUBSET p ==> q INTER s = {a,b}`)) THEN
ASM_SIMP_TAC[PATH_IMAGE_SUBPATH_SUBSET; SIMPLE_PATH_IMP_PATH;
ENDS_IN_UNIT_INTERVAL] THEN
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_INTERVAL_1]) THEN
SIMP_TAC[DROP_VEC; PATH_IMAGE_SUBPATH] THEN STRIP_TAC THEN
REWRITE_TAC[IN_IMAGE] THEN CONJ_TAC THENL
[EXISTS_TAC `vec 0:real^1`; EXISTS_TAC `t:real^1`] THEN
ASM_REWRITE_TAC[ENDS_IN_INTERVAL; INTERVAL_EQ_EMPTY_1; DROP_VEC] THEN
ASM_REAL_ARITH_TAC;
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
`p INTER s = {a,b}
==> a IN q /\ b IN q /\ q SUBSET p ==> q INTER s = {a,b}`)) THEN
ASM_SIMP_TAC[PATH_IMAGE_SUBPATH_SUBSET; SIMPLE_PATH_IMP_PATH;
ENDS_IN_UNIT_INTERVAL] THEN
ONCE_REWRITE_TAC[GSYM PATH_IMAGE_REVERSEPATH] THEN
REWRITE_TAC[REVERSEPATH_SUBPATH] THEN
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_INTERVAL_1]) THEN
SIMP_TAC[DROP_VEC; PATH_IMAGE_SUBPATH] THEN STRIP_TAC THEN
REWRITE_TAC[IN_IMAGE] THEN CONJ_TAC THENL
[EXISTS_TAC `vec 1:real^1`; EXISTS_TAC `t:real^1`] THEN
ASM_REWRITE_TAC[ENDS_IN_INTERVAL; INTERVAL_EQ_EMPTY_1; DROP_VEC] THEN
ASM_REAL_ARITH_TAC;
REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTER] THEN
EXISTS_TAC `a:complex` THEN
ASM_REWRITE_TAC[GSYM BETWEEN_IN_SEGMENT; between] THEN
REWRITE_TAC[NORM_ARITH `dist(a - d:complex,a + e) = norm(d + e)`;
NORM_ARITH `dist(a - d:complex,a) = norm(d)`;
NORM_ARITH `dist(a:complex,a + e) = norm e`] THEN
ASM_SIMP_TAC[GSYM VECTOR_ADD_RDISTRIB; NORM_MUL;
NORM_BASIS; DIMINDEX_2; ARITH] THEN
ASM_REAL_ARITH_TAC;
ONCE_REWRITE_TAC[GSYM PATH_IMAGE_REVERSEPATH] THEN
RULE_ASSUM_TAC(REWRITE_RULE[PATH_IMAGE_LINEPATH]) THEN
ASM_REWRITE_TAC[REVERSEPATH_SUBPATH];
W(MP_TAC o PART_MATCH (rand o rand) WINDING_NUMBER_REVERSEPATH o
rand o snd) THEN
ANTS_TAC THENL
[ASM_SIMP_TAC[PATH_JOIN_EQ; PATH_IMAGE_JOIN; PATH_LINEPATH;
SIMPLE_PATH_IMP_PATH; PATHSTART_LINEPATH; PATHFINISH_SUBPATH;
PATH_SUBPATH; ENDS_IN_UNIT_INTERVAL] THEN
ASM_MESON_TAC[INSIDE_NO_OVERLAP; IN_INTER; NOT_IN_EMPTY];
DISCH_THEN(SUBST1_TAC o SYM) THEN
ASM_SIMP_TAC[REVERSEPATH_JOINPATHS; REVERSEPATH_LINEPATH;
REVERSEPATH_SUBPATH; PATHFINISH_SUBPATH;
PATHSTART_LINEPATH] THEN
MATCH_MP_TAC(MESON[COMPLEX_ADD_SYM]
`winding_number(g ++ h,z) =
winding_number(g,z) + winding_number(h,z) /\
winding_number(h ++ g,z) =
winding_number(h,z) + winding_number(g,z)
==> winding_number(g ++ h,z) =winding_number(h ++ g,z)`) THEN
CONJ_TAC THEN MATCH_MP_TAC WINDING_NUMBER_JOIN THEN
ASM_SIMP_TAC[PATH_LINEPATH; PATH_SUBPATH; PATH_SUBPATH;
SIMPLE_PATH_IMP_PATH; ENDS_IN_UNIT_INTERVAL;
PATHSTART_LINEPATH; PATHFINISH_LINEPATH;
PATHSTART_SUBPATH; PATHFINISH_SUBPATH]
THENL [ALL_TAC; ONCE_REWRITE_TAC[CONJ_SYM]] THEN
REWRITE_TAC[SET_RULE
`~(z IN s) /\ ~(z IN t) <=> ~(z IN s UNION t)`] THEN
ONCE_REWRITE_TAC[GSYM PATH_IMAGE_REVERSEPATH] THEN
REWRITE_TAC[REVERSEPATH_LINEPATH; REVERSEPATH_SUBPATH] THEN
ASM_MESON_TAC[INSIDE_NO_OVERLAP; IN_INTER; NOT_IN_EMPTY]];
REWRITE_TAC[COMPLEX_NEG_EQ_0] THEN DISCH_THEN SUBST_ALL_TAC THEN
RULE_ASSUM_TAC(REWRITE_RULE
[COMPLEX_NORM_CX; REAL_OF_NUM_EQ; REAL_ABS_NUM; ARITH]) THEN
FIRST_X_ASSUM CONTR_TAC];
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
ONCE_REWRITE_TAC[COMPLEX_RING `a:complex = --b <=> --a = b`] THEN
DISCH_THEN(SUBST_ALL_TAC o SYM) THEN
RULE_ASSUM_TAC(REWRITE_RULE[NORM_NEG])] THEN
EXISTS_TAC `z:complex` THEN ASM_REWRITE_TAC[] THEN
SUBGOAL_THEN
`winding_number(subpath (vec 0) t q ++ subpath t (vec 1) q,z) =
winding_number(subpath (vec 0) (vec 1) q,z)`
(fun th -> ASM_MESON_TAC[th; SUBPATH_TRIVIAL]) THEN
MATCH_MP_TAC EQ_TRANS THEN
EXISTS_TAC `winding_number(subpath (vec 0) t q,z) +
winding_number(subpath t (vec 1) q,z)` THEN
CONJ_TAC THENL
[MATCH_MP_TAC WINDING_NUMBER_JOIN THEN
ASM_SIMP_TAC[PATH_SUBPATH; ENDS_IN_UNIT_INTERVAL; SIMPLE_PATH_IMP_PATH;
PATHSTART_SUBPATH; PATHFINISH_SUBPATH] THEN
SUBGOAL_THEN `~((z:complex) IN path_image q)` MP_TAC THENL
[ASM_MESON_TAC[INSIDE_NO_OVERLAP; IN_INTER; NOT_IN_EMPTY];
MATCH_MP_TAC(SET_RULE
`s1 SUBSET s /\ s2 SUBSET s
==> ~(z IN s) ==> ~(z IN s1) /\ ~(z IN s2)`) THEN
ASM_SIMP_TAC[PATH_IMAGE_SUBPATH_SUBSET; ENDS_IN_UNIT_INTERVAL;
SIMPLE_PATH_IMP_PATH]];
MATCH_MP_TAC WINDING_NUMBER_SUBPATH_COMBINE THEN
ASM_REWRITE_TAC[ENDS_IN_INTERVAL; GSYM IN_INTERVAL_1] THEN
ASM_SIMP_TAC[UNIT_INTERVAL_NONEMPTY; SIMPLE_PATH_IMP_PATH] THEN
ASM_MESON_TAC[INSIDE_NO_OVERLAP; IN_INTER; NOT_IN_EMPTY]]) in
GEN_TAC THEN DISCH_TAC THEN
ASM_CASES_TAC `pathfinish g:complex = pathstart g` THENL
[ALL_TAC; ASM_MESON_TAC[INSIDE_SIMPLE_CURVE_IMP_CLOSED]] THEN
MATCH_MP_TAC(MESON[]
`(?k. !z. z IN s ==> f z = k) /\
(?z. z IN s /\ (f z = a \/ f z = b))
==> (!z. z IN s ==> f z = a) \/ (!z. z IN s ==> f z = b)`) THEN
CONJ_TAC THENL
[MATCH_MP_TAC WINDING_NUMBER_CONSTANT THEN
ASM_SIMP_TAC[INSIDE_NO_OVERLAP; SIMPLE_PATH_IMP_PATH] THEN
ASM_SIMP_TAC[JORDAN_INSIDE_OUTSIDE];
MP_TAC(SPEC `g:real^1->complex` lemma3) THEN ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `z:complex` THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ASM_REWRITE_TAC[] THEN
MP_TAC(ISPECL [`g:real^1->complex`; `z:complex`]
INTEGER_WINDING_NUMBER) THEN
ANTS_TAC THENL
[ASM_SIMP_TAC[SIMPLE_PATH_IMP_PATH] THEN
ASM_MESON_TAC[INSIDE_NO_OVERLAP; IN_INTER; NOT_IN_EMPTY];
SIMP_TAC[complex_integer; COMPLEX_EQ; IM_NEG; IM_CX] THEN
SIMP_TAC[GSYM real; REAL_NORM; RE_NEG; RE_CX] THEN REAL_ARITH_TAC]]);;
let HAS_PATH_INTEGRAL_BOUND_PARTCIRCLEPATH_STRONG = prove
(`!f i z r s t B k.
FINITE k /\
(f
has_path_integral i) (partcirclepath(z,r,s,t)) /\
&0 <= B /\ &0 < r /\ s <= t /\
(!x. x
IN path_image(partcirclepath(z,r,s,t))
DIFF k
==> norm(f x) <= B)
==> norm(i) <= B * r * (t - s)`,
let lemma1 = prove
(`!b w. FINITE {z | norm(z) <= b /\ cexp(z) = w}`,
REPEAT GEN_TAC THEN ASM_CASES_TAC `w = Cx(&0)` THEN
ASM_REWRITE_TAC[CEXP_NZ; SET_RULE `{x | F} = {}`; FINITE_RULES] THEN
FIRST_ASSUM(SUBST1_TAC o SYM o MATCH_MP CEXP_CLOG) THEN
REWRITE_TAC[CEXP_EQ] THEN
REWRITE_TAC[SET_RULE
`{z | P z /\ ?n. Q n /\ z = f n} = IMAGE f {n | Q n /\ P(f n)}`] THEN
MATCH_MP_TAC FINITE_IMAGE THEN
MATCH_MP_TAC FINITE_SUBSET THEN
EXISTS_TAC `{n | integer n /\
norm(Cx(&2 * n * pi) * ii) <= b + norm(clog w)}` THEN
CONJ_TAC THENL
[ALL_TAC; SIMP_TAC[SUBSET; IN_ELIM_THM] THEN NORM_ARITH_TAC] THEN
REWRITE_TAC[COMPLEX_NORM_MUL; COMPLEX_NORM_CX; COMPLEX_NORM_II] THEN
REWRITE_TAC[REAL_MUL_RID; REAL_ABS_MUL; REAL_ABS_NUM; REAL_ABS_PI] THEN
ASM_SIMP_TAC[REAL_MUL_ASSOC; GSYM REAL_LE_RDIV_EQ; PI_POS] THEN
REWRITE_TAC[REAL_ARITH `&2 * x <= a <=> x <= a / &2`] THEN
REWRITE_TAC[GSYM REAL_BOUNDS_LE; FINITE_INTSEG]) in
let lemma2 = prove
(`!a b. ~(a = Cx(&0)) ==> FINITE {z | norm(z) <= b /\ cexp(a * z) = w}`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC FINITE_SUBSET THEN
EXISTS_TAC
`IMAGE (\z. z / a) {z | norm(z) <= b * norm(a) /\ cexp(z) = w}` THEN
SIMP_TAC[lemma1; FINITE_IMAGE] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
REWRITE_TAC[SUBSET; IN_IMAGE; IN_ELIM_THM] THEN
ASM_SIMP_TAC[COMPLEX_FIELD `~(a = Cx(&0)) ==> (x = y / a <=> a * x = y)`;
UNWIND_THM1; COMPLEX_NORM_MUL; REAL_LE_LMUL; NORM_POS_LE]) in
REPEAT GEN_TAC THEN REWRITE_TAC[HAS_PATH_INTEGRAL] THEN STRIP_TAC THEN
MP_TAC(ASSUME `s <= t`) THEN GEN_REWRITE_TAC LAND_CONV [REAL_LE_LT] THEN
STRIP_TAC THENL
[ALL_TAC;
FIRST_X_ASSUM SUBST_ALL_TAC THEN
REPEAT(POP_ASSUM MP_TAC) THEN
REWRITE_TAC[VECTOR_DERIVATIVE_PARTCIRCLEPATH] THEN
REWRITE_TAC[COMPLEX_SUB_REFL; COMPLEX_MUL_LZERO; COMPLEX_MUL_RZERO] THEN
SIMP_TAC[GSYM COMPLEX_VEC_0; HAS_INTEGRAL_0_EQ; NORM_0] THEN
REAL_ARITH_TAC] THEN
GEN_REWRITE_TAC RAND_CONV [GSYM REAL_MUL_RID] THEN
REWRITE_TAC[GSYM CONTENT_UNIT_1] THEN MATCH_MP_TAC HAS_INTEGRAL_BOUND THEN
EXISTS_TAC `\x. if (partcirclepath(z,r,s,t) x) IN k then Cx(&0)
else f(partcirclepath(z,r,s,t) x) *
vector_derivative (partcirclepath(z,r,s,t)) (at x)` THEN
ASM_SIMP_TAC[] THEN CONJ_TAC THENL
[ASM_MESON_TAC[REAL_LE_MUL; REAL_POS; REAL_LT_IMP_LE; REAL_SUB_LE];
ALL_TAC] THEN
CONJ_TAC THENL
[MATCH_MP_TAC HAS_INTEGRAL_SPIKE THEN
EXISTS_TAC `\x. f(partcirclepath(z,r,s,t) x) *
vector_derivative (partcirclepath(z,r,s,t)) (at x)` THEN
EXISTS_TAC `{x | x IN interval[vec 0,vec 1] /\
(partcirclepath(z,r,s,t) x) IN k}` THEN
ASM_SIMP_TAC[IN_DIFF; IN_ELIM_THM; IMP_CONJ] THEN
MATCH_MP_TAC NEGLIGIBLE_FINITE THEN
MATCH_MP_TAC FINITE_FINITE_PREIMAGE_GENERAL THEN ASM_REWRITE_TAC[] THEN
X_GEN_TAC `y:complex` THEN DISCH_TAC THEN
REWRITE_TAC[partcirclepath] THEN
ASM_SIMP_TAC[CX_INJ; REAL_LT_IMP_NZ; COMPLEX_FIELD
`~(r = Cx(&0)) ==> (z + r * e = y <=> e = (y - z) / r)`] THEN
REWRITE_TAC[linepath; COMPLEX_CMUL] THEN
REWRITE_TAC[GSYM CX_MUL; GSYM CX_ADD] THEN
REWRITE_TAC[REAL_ARITH `(&1 - t) * x + t * y = x + t * (y - x)`] THEN
REWRITE_TAC[CX_ADD; COMPLEX_ADD_LDISTRIB; CEXP_ADD] THEN
SIMP_TAC[CEXP_NZ; COMPLEX_FIELD
`~(e = Cx(&0)) ==> (e * x = y <=> x = y / e)`] THEN
ABBREV_TAC `w = (y - z) / Cx r / cexp(ii * Cx s)` THEN
REWRITE_TAC[CX_MUL; COMPLEX_RING
`ii * Cx x * Cx(t - s) = (ii * Cx(t - s)) * Cx x`] THEN
MATCH_MP_TAC FINITE_SUBSET THEN
EXISTS_TAC
`{x | Cx(drop x) IN
{z | norm(z) <= &1 /\ cexp((ii * Cx(t - s)) * z) = w}}` THEN
CONJ_TAC THENL
[MATCH_MP_TAC FINITE_IMAGE_INJ THEN REWRITE_TAC[CX_INJ; DROP_EQ] THEN
MATCH_MP_TAC lemma2 THEN
REWRITE_TAC[COMPLEX_RING `ii * x = Cx(&0) <=> x = Cx(&0)`] THEN
ASM_SIMP_TAC[CX_INJ; REAL_SUB_0; REAL_LT_IMP_NE];
SIMP_TAC[SUBSET; IN_ELIM_THM; IN_INTERVAL_1; DROP_VEC] THEN
SIMP_TAC[COMPLEX_NORM_CX] THEN REAL_ARITH_TAC];
ALL_TAC] THEN
X_GEN_TAC `x:real^1` THEN DISCH_TAC THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[COMPLEX_NORM_0] THEN
ASM_SIMP_TAC[REAL_LE_MUL; REAL_LT_IMP_LE; REAL_SUB_LE] THEN
REWRITE_TAC[VECTOR_DERIVATIVE_PARTCIRCLEPATH] THEN
REWRITE_TAC[COMPLEX_NORM_MUL; COMPLEX_NORM_CX; COMPLEX_NORM_II] THEN
REWRITE_TAC[REAL_ABS_NUM; REAL_MUL_LID] THEN
REWRITE_TAC[NORM_CEXP; RE_MUL_II; IM_LINEPATH_CX] THEN
REWRITE_TAC[REAL_EXP_0; REAL_NEG_0; REAL_MUL_RID] THEN
MATCH_MP_TAC REAL_LE_MUL2 THEN REWRITE_TAC[NORM_POS_LE] THEN CONJ_TAC THENL
[FIRST_X_ASSUM MATCH_MP_TAC THEN
REWRITE_TAC[path_image] THEN ASM SET_TAC[];
ALL_TAC] THEN
SIMP_TAC[REAL_LE_MUL; NORM_POS_LE; REAL_ABS_POS] THEN
MATCH_MP_TAC REAL_LE_MUL2 THEN
ASM_SIMP_TAC[NORM_POS_LE; GSYM CX_SUB; COMPLEX_NORM_CX] THEN
ASM_REAL_ARITH_TAC);;
let ARC_PARTCIRCLEPATH = prove
(`!z r s t. ~(r = &0) /\ ~(s = t) /\ abs(s - t) < &2 * pi
==> arc(partcirclepath(z,r,s,t))`,
REPEAT STRIP_TAC THEN REWRITE_TAC[arc;
PATH_PARTCIRCLEPATH] THEN
REWRITE_TAC[partcirclepath] THEN
SIMP_TAC[COMPLEX_RING `z + r * x = z + r * y <=> r * (x - y) = Cx(&0)`] THEN
ASM_REWRITE_TAC[
COMPLEX_ENTIRE;
CX_INJ] THEN
REWRITE_TAC[
COMPLEX_SUB_0;
CEXP_EQ] THEN
REWRITE_TAC[COMPLEX_RING
`ii * x = ii * y + z * ii <=> ii * (x - (y + z)) = Cx(&0)`] THEN
REWRITE_TAC[
COMPLEX_ENTIRE;
II_NZ;
LINEPATH_CX] THEN
REWRITE_TAC[GSYM
CX_SUB; GSYM
CX_ADD;
CX_INJ] THEN
REWRITE_TAC[REAL_ARITH
`((&1 - x) * s + x * t) - (((&1 - y) * s + y * t) + z) = &0 <=>
(x - y) * (t - s) = z`] THEN
REPEAT GEN_TAC THEN DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC) THEN
FIRST_X_ASSUM(X_CHOOSE_THEN `n:real` MP_TAC) THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
ASM_CASES_TAC `n = &0` THEN
ASM_REWRITE_TAC[
REAL_MUL_LZERO;
REAL_MUL_RZERO;
REAL_ENTIRE;
REAL_SUB_0;
DROP_EQ] THEN
MP_TAC(SPEC `n:real`
REAL_ABS_INTEGER_LEMMA) THEN
ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN DISCH_THEN(K ALL_TAC) THEN
MATCH_MP_TAC(REAL_ARITH `abs x < abs y ==> ~(x = y)`) THEN
REWRITE_TAC[
REAL_ABS_MUL;
REAL_ABS_NUM;
REAL_ABS_PI] THEN
MATCH_MP_TAC
REAL_LTE_TRANS THEN EXISTS_TAC `&1 * &2 * pi` THEN
CONJ_TAC THENL
[ALL_TAC;
ONCE_REWRITE_TAC[REAL_ARITH `&2 * n * pi = n * &2 * pi`] THEN
MATCH_MP_TAC
REAL_LE_RMUL THEN ASM_REWRITE_TAC[] THEN
MP_TAC
PI_POS THEN REAL_ARITH_TAC] THEN
MATCH_MP_TAC
REAL_LET_TRANS THEN EXISTS_TAC `&1 * abs(t - s)` THEN
CONJ_TAC THENL
[ALL_TAC;
ASM_REWRITE_TAC[REAL_MUL_LID] THEN ASM_MESON_TAC[
REAL_ABS_SUB]] THEN
MATCH_MP_TAC
REAL_LE_RMUL THEN REWRITE_TAC[
REAL_ABS_POS] THEN
REPEAT(FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
IN_INTERVAL_1])) THEN
REWRITE_TAC[
DROP_VEC] THEN REAL_ARITH_TAC);;
let CIRCLEPATH = prove
(`circlepath(z,r) = \x. z + Cx(r) * cexp(Cx(&2) * Cx pi * ii * Cx(drop x))`,
let HOLOMORPHIC_POWER_SERIES = prove
(`!f z w r.
f
holomorphic_on ball(z,r) /\ w
IN ball(z,r)
==> ((\n.
higher_complex_derivative n f z / Cx(&(
FACT n)) * (w - z) pow n)
sums f(w)) (from 0)`,
REPEAT STRIP_TAC THEN
SUBGOAL_THEN
`?r. &0 < r /\ f
holomorphic_on cball(z,r) /\ w
IN ball(z,r)`
MP_TAC THENL
[EXISTS_TAC `(r + dist(w:complex,z)) / &2` THEN REPEAT CONJ_TAC THENL
[ALL_TAC;
MATCH_MP_TAC
HOLOMORPHIC_ON_SUBSET THEN
EXISTS_TAC `ball(z:complex,r)` THEN ASM_REWRITE_TAC[
SUBSET];
ALL_TAC] THEN
UNDISCH_TAC `(w:complex)
IN ball(z,r)` THEN
REWRITE_TAC[
IN_BALL;
IN_CBALL] THEN NORM_ARITH_TAC;
ALL_TAC] THEN
POP_ASSUM_LIST(K ALL_TAC) THEN
DISCH_THEN(X_CHOOSE_THEN `r:real` STRIP_ASSUME_TAC) THEN
SUBGOAL_THEN `f
holomorphic_on ball(z,r) /\ f
continuous_on cball(z,r)`
STRIP_ASSUME_TAC THENL
[ASM_SIMP_TAC[
HOLOMORPHIC_ON_IMP_CONTINUOUS_ON] THEN
MATCH_MP_TAC
HOLOMORPHIC_ON_SUBSET THEN ASM_MESON_TAC[
BALL_SUBSET_CBALL];
ALL_TAC] THEN
SUBGOAL_THEN
`((\k.
path_integral (circlepath(z,r)) (\u. f u / (u - z) pow (k + 1)) *
(w - z) pow k)
sums
path_integral (circlepath(z,r)) (\u. f u / (u - w))) (from 0)`
MP_TAC THENL
[ALL_TAC;
DISCH_THEN(MP_TAC o SPEC `inv(Cx(&2) * Cx(pi) * ii)` o
MATCH_MP
SERIES_COMPLEX_LMUL) THEN
MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN BINOP_TAC THENL
[REWRITE_TAC[
FUN_EQ_THM] THEN X_GEN_TAC `k:num` THEN
MP_TAC(SPECL [`f:complex->complex`; `z:complex`; `r:real`; `k:num`;
`z:complex`]
CAUCHY_HAS_PATH_INTEGRAL_HIGHER_DERIVATIVE_CIRCLEPATH) THEN
ASM_REWRITE_TAC[
CENTRE_IN_BALL] THEN
DISCH_THEN(SUBST1_TAC o MATCH_MP
PATH_INTEGRAL_UNIQUE) THEN
MATCH_MP_TAC(COMPLEX_FIELD
`~(pit = Cx(&0)) /\ ~(fact = Cx(&0))
==> inv(pit) * ((pit / fact) * d) * wz = d / fact * wz`) THEN
REWRITE_TAC[
CX_2PII_NZ;
CX_INJ; REAL_OF_NUM_EQ;
FACT_NZ];
MP_TAC(SPECL [`f:complex->complex`; `z:complex`; `r:real`; `w:complex`]
CAUCHY_INTEGRAL_CIRCLEPATH_SIMPLE) THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN(SUBST1_TAC o MATCH_MP
PATH_INTEGRAL_UNIQUE) THEN
MATCH_MP_TAC(COMPLEX_FIELD
`~(x * y * z = Cx(&0)) ==> inv(x * y * z) * x * y * z * w = w`) THEN
REWRITE_TAC[
CX_2PII_NZ]]] THEN
REWRITE_TAC[sums;
FROM_0;
INTER_UNIV] THEN
MATCH_MP_TAC
LIM_TRANSFORM_EVENTUALLY THEN
EXISTS_TAC `\n.
path_integral (circlepath(z,r))
(\u. vsum (0..n)
(\k. f u * (w - z) pow k / (u - z) pow (k + 1)))` THEN
CONJ_TAC THENL
[MATCH_MP_TAC
ALWAYS_EVENTUALLY THEN
X_GEN_TAC `k:num` THEN REWRITE_TAC[] THEN
W(MP_TAC o PART_MATCH (lhs o rand)
PATH_INTEGRAL_VSUM o lhand o snd) THEN
ANTS_TAC THENL
[REWRITE_TAC[
FINITE_NUMSEG] THEN X_GEN_TAC `m:num` THEN DISCH_TAC THEN
ONCE_REWRITE_TAC[SIMPLE_COMPLEX_ARITH
`a * b / c:complex = b * a / c`] THEN
MATCH_MP_TAC
PATH_INTEGRABLE_COMPLEX_LMUL THEN
ASM_SIMP_TAC[
CAUCHY_HIGHER_DERIVATIVE_INTEGRAL_CIRCLEPATH;
CENTRE_IN_BALL];
ALL_TAC] THEN
DISCH_THEN SUBST1_TAC THEN MATCH_MP_TAC
VSUM_EQ THEN
X_GEN_TAC `m:num` THEN DISCH_TAC THEN REWRITE_TAC[] THEN
ONCE_REWRITE_TAC[SIMPLE_COMPLEX_ARITH `a * b / c:complex = a / c * b`] THEN
MATCH_MP_TAC
PATH_INTEGRAL_COMPLEX_RMUL THEN
ASM_SIMP_TAC[
CAUCHY_HIGHER_DERIVATIVE_INTEGRAL_CIRCLEPATH;
CENTRE_IN_BALL];
ALL_TAC] THEN
MATCH_MP_TAC(CONJUNCT2
(REWRITE_RULE[
FORALL_AND_THM; TAUT `a ==> b /\ c <=> (a ==> b) /\ (a ==> c)`]
PATH_INTEGRAL_UNIFORM_LIMIT_CIRCLEPATH)) THEN
ASM_REWRITE_TAC[
TRIVIAL_LIMIT_SEQUENTIALLY] THEN CONJ_TAC THENL
[MATCH_MP_TAC
ALWAYS_EVENTUALLY THEN
X_GEN_TAC `k:num` THEN REWRITE_TAC[] THEN
MATCH_MP_TAC
PATH_INTEGRABLE_VSUM THEN
REWRITE_TAC[
FINITE_NUMSEG] THEN X_GEN_TAC `m:num` THEN DISCH_TAC THEN
ONCE_REWRITE_TAC[SIMPLE_COMPLEX_ARITH `a * b / c:complex = b * a / c`] THEN
MATCH_MP_TAC
PATH_INTEGRABLE_COMPLEX_LMUL THEN
ASM_SIMP_TAC[
CAUCHY_HIGHER_DERIVATIVE_INTEGRAL_CIRCLEPATH;
CENTRE_IN_BALL];
ALL_TAC] THEN
X_GEN_TAC `e:real` THEN STRIP_TAC THEN
ASM_SIMP_TAC[
PATH_IMAGE_CIRCLEPATH;
REAL_LT_IMP_LE;
IN_ELIM_THM] THEN
SIMP_TAC[
VSUM_COMPLEX_LMUL;
FINITE_NUMSEG;
complex_div] THEN
REWRITE_TAC[GSYM
COMPLEX_SUB_LDISTRIB;
COMPLEX_NORM_MUL] THEN
REWRITE_TAC[
COMPLEX_POW_ADD;
COMPLEX_INV_MUL;
COMPLEX_POW_1] THEN
SIMP_TAC[
COMPLEX_MUL_ASSOC;
VSUM_COMPLEX_RMUL;
FINITE_NUMSEG] THEN
REWRITE_TAC[GSYM
complex_div; GSYM
COMPLEX_POW_DIV] THEN
REWRITE_TAC[
VSUM_GP; CONJUNCT1
LT; CONJUNCT1
complex_pow] THEN
REWRITE_TAC[
EVENTUALLY_SEQUENTIALLY] THEN
SUBGOAL_THEN
`?B. &0 < B /\
!u:complex. u
IN cball(z,r) ==> norm(f u:complex) <= B`
STRIP_ASSUME_TAC THENL
[MP_TAC(ISPEC `
IMAGE (f:complex->complex) (cball(z,r))`
COMPACT_IMP_BOUNDED) THEN
ASM_SIMP_TAC[
COMPACT_CONTINUOUS_IMAGE;
COMPACT_CBALL] THEN
REWRITE_TAC[
BOUNDED_POS;
FORALL_IN_IMAGE];
ALL_TAC] THEN
SUBGOAL_THEN `?k. &0 < k /\ k <= r /\ norm(w - z) <= r - k /\
!u. norm(u - z) = r ==> k <= norm(u - w)`
STRIP_ASSUME_TAC THENL
[EXISTS_TAC `r - dist(z:complex,w)` THEN
REPEAT(POP_ASSUM MP_TAC) THEN REWRITE_TAC[
IN_BALL] THEN
NORM_ARITH_TAC;
ALL_TAC] THEN
REWRITE_TAC[
IN_SPHERE; NORM_ARITH `dist(z,x) = r <=> norm(x - z) = r`] THEN
MP_TAC(SPECL [`(r - k) / r:real`; `e / B * k:real`]
REAL_ARCH_POW_INV) THEN
ASM_SIMP_TAC[
REAL_LT_LDIV_EQ;
REAL_LT_DIV;
REAL_HALF;
REAL_LT_MUL] THEN
ASM_REWRITE_TAC[REAL_ARITH `r - k < &1 * r <=> &0 < k`] THEN
MATCH_MP_TAC
MONO_EXISTS THEN X_GEN_TAC `N:num` THEN DISCH_TAC THEN
X_GEN_TAC `n:num` THEN DISCH_TAC THEN
X_GEN_TAC `u:complex` THEN DISCH_TAC THEN
SUBGOAL_THEN `~(u:complex = z) /\ ~(u = w)` STRIP_ASSUME_TAC THENL
[FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
IN_BALL]) THEN
MAP_EVERY UNDISCH_TAC [`&0 < r`; `norm(u - z:complex) = r`] THEN
NORM_ARITH_TAC;
ALL_TAC] THEN
ASM_SIMP_TAC[COMPLEX_FIELD
`~(u = z) /\ ~(u = w) ==> ~((w - z) / (u - z) = Cx(&1))`] THEN
ASM_SIMP_TAC[COMPLEX_FIELD
`~(u = z) /\ ~(u = w)
==> x / (Cx(&1) - (w - z) / (u - z)) / (u - z) = x / (u - w)`] THEN
ASM_SIMP_TAC[COMPLEX_FIELD
`~(u = w)
==> (Cx(&1) - e) / (u - w) - inv(u - w) = --(e / (u - w))`] THEN
REWRITE_TAC[
COMPLEX_NORM_DIV;
NORM_NEG;
COMPLEX_NORM_POW] THEN
MATCH_MP_TAC
REAL_LET_TRANS THEN
EXISTS_TAC `B * ((r - k) / r) pow N / k:real` THEN CONJ_TAC THENL
[ALL_TAC;
ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
ASM_SIMP_TAC[GSYM
REAL_LT_RDIV_EQ;
REAL_LT_LDIV_EQ]] THEN
MATCH_MP_TAC
REAL_LE_MUL2 THEN ASM_SIMP_TAC[
NORM_POS_LE] THEN
REPEAT CONJ_TAC THENL
[FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[
IN_CBALL] THEN
ONCE_REWRITE_TAC[
DIST_SYM] THEN ASM_REWRITE_TAC[dist;
REAL_LE_REFL];
MATCH_MP_TAC
REAL_LE_DIV THEN REWRITE_TAC[
NORM_POS_LE] THEN
MATCH_MP_TAC
REAL_POW_LE THEN MATCH_MP_TAC
REAL_LE_DIV THEN
ASM_SIMP_TAC[
REAL_LT_IMP_LE;
NORM_POS_LE];
ALL_TAC] THEN
REWRITE_TAC[
real_div] THEN MATCH_MP_TAC
REAL_LE_MUL2 THEN
REWRITE_TAC[GSYM
real_div] THEN REPEAT CONJ_TAC THENL
[MATCH_MP_TAC
REAL_POW_LE THEN MATCH_MP_TAC
REAL_LE_DIV THEN
ASM_SIMP_TAC[
REAL_LT_IMP_LE;
NORM_POS_LE];
ALL_TAC;
REWRITE_TAC[
REAL_LE_INV_EQ;
NORM_POS_LE];
MATCH_MP_TAC
REAL_LE_INV2 THEN ASM_SIMP_TAC[]] THEN
MATCH_MP_TAC
REAL_LE_TRANS THEN
EXISTS_TAC `((r - k) / r:real) pow (SUC n)` THEN CONJ_TAC THENL
[MATCH_MP_TAC
REAL_POW_LE2 THEN
ASM_SIMP_TAC[
REAL_LE_DIV2_EQ;
REAL_LE_DIV;
NORM_POS_LE;
REAL_LT_IMP_LE];
MATCH_MP_TAC
REAL_POW_MONO_INV THEN
ASM_SIMP_TAC[
REAL_LE_RDIV_EQ;
REAL_LE_LDIV_EQ] THEN
ASM_SIMP_TAC[ARITH_RULE `N <= n ==> N <= SUC n`] THEN
ASM_REAL_ARITH_TAC]);;
let FTA = prove
(`!a n. a(0) = Cx(&0) \/ ~(!k. k
IN 1..n ==> a(k) = Cx(&0))
==> ?z. vsum(0..n) (\i. a(i) * z pow i) = Cx(&0)`,
let SERIES_AND_DERIVATIVE_COMPARISON_COMPLEX = prove
(`!f f' s k.
open s /\
(!n x. n
IN k /\ x
IN s ==> (f n
has_complex_derivative f' n x) (at x)) /\
(!x. x
IN s
==> ?d h N. &0 < d /\ summable k h /\
(!n. n
IN k ==> real(h n) /\ &0 <= Re(h n)) /\
(!n y. N <= n /\ n
IN k /\ y
IN ball(x,d)
==> norm(f n y) <= norm(h n)))
==> ?g g'. !x. x
IN s
==> ((\n. f n x) sums g x) k /\
((\n. f' n x) sums g' x) k /\
(g
has_complex_derivative g' x) (at x)`,
REPEAT STRIP_TAC THEN
MATCH_MP_TAC
SERIES_AND_DERIVATIVE_COMPARISON_LOCAL THEN
ASM_REWRITE_TAC[] THEN X_GEN_TAC `z:complex` THEN STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `z:complex`) THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC
MONO_EXISTS THEN
X_GEN_TAC `d:real` THEN ONCE_REWRITE_TAC[
SWAP_EXISTS_THM] THEN
MATCH_MP_TAC
MONO_EXISTS THEN X_GEN_TAC `N:num` THEN
DISCH_THEN(X_CHOOSE_THEN `h:num->complex` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `\n. norm((h:num->complex) n)` THEN ASM_REWRITE_TAC[] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [summable]) THEN
DISCH_THEN(X_CHOOSE_THEN `l:complex` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `lift(Re l)` THEN MATCH_MP_TAC
SUMS_EQ THEN
EXISTS_TAC `\i:num. lift(Re(h i))` THEN
ASM_SIMP_TAC[
REAL_NORM_POS;
o_DEF] THEN
REWRITE_TAC[
RE_DEF] THEN MATCH_MP_TAC
SERIES_COMPONENT THEN
ASM_REWRITE_TAC[DIMINDEX_2; ARITH]);;
let POWER_SERIES_AND_DERIVATIVE = prove
(`!k a r w.
summable k (\n. a(n) * Cx(r) pow n)
==> ?g g'.
!z. z
IN ball(w,r)
==> ((\n. a(n) * (z - w) pow n) sums g(z)) k /\
((\n. Cx(&n) * a(n) * (z - w) pow (n - 1)) sums g'(z)) k /\
(g
has_complex_derivative g' z) (at z)`,
REPEAT STRIP_TAC THEN
FIRST_ASSUM(MP_TAC o MATCH_MP
POWER_SERIES_AND_DERIVATIVE_0) THEN
REWRITE_TAC[
LEFT_IMP_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC [`g:complex->complex`; `g':complex->complex`] THEN
DISCH_TAC THEN
EXISTS_TAC `(\z. g(z - w)):complex->complex` THEN
EXISTS_TAC `(\z. g'(z - w)):complex->complex` THEN
REWRITE_TAC[
IN_BALL; dist] THEN X_GEN_TAC `z:complex` THEN
DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `z - w:complex`) THEN
ANTS_TAC THENL [ASM_MESON_TAC[
NORM_SUB]; ALL_TAC] THEN
STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
GEN_REWRITE_TAC (RATOR_CONV o LAND_CONV) [GSYM
o_DEF] THEN
GEN_REWRITE_TAC (RATOR_CONV o RAND_CONV) [GSYM
COMPLEX_MUL_RID] THEN
MATCH_MP_TAC
COMPLEX_DIFF_CHAIN_AT THEN ASM_REWRITE_TAC[] THEN
COMPLEX_DIFF_TAC THEN REWRITE_TAC[
COMPLEX_SUB_RZERO]);;
let POWER_SERIES_HOLOMORPHIC = prove
(`!a k f z r. (!w. w
IN ball(z,r) ==> ((\n. a(n) * (w - z) pow n) sums f w) k)
==> f
holomorphic_on ball(z,r)`,
REPEAT STRIP_TAC THEN SIMP_TAC[
HOLOMORPHIC_ON_OPEN;
OPEN_BALL] THEN
X_GEN_TAC `w:complex` THEN REWRITE_TAC[
IN_BALL; dist] THEN DISCH_TAC THEN
MP_TAC(ISPECL [`k:num->bool`; `a:num->complex`;
`(norm(z - w:complex) + r) / &2`; `z:complex`]
POWER_SERIES_AND_DERIVATIVE) THEN
ANTS_TAC THENL
[FIRST_X_ASSUM(MP_TAC o SPEC `z + Cx((norm(z - w) + r) / &2)`) THEN
REWRITE_TAC[
IN_BALL; dist; COMPLEX_RING `(z + w) - z:complex = w`;
NORM_ARITH `norm(z - (z + w)) = norm w`; summable] THEN
ANTS_TAC THENL [REWRITE_TAC[
COMPLEX_NORM_CX]; MESON_TAC[]] THEN
POP_ASSUM MP_TAC THEN NORM_ARITH_TAC;
ALL_TAC] THEN
DISCH_THEN(X_CHOOSE_THEN `g:complex->complex` MP_TAC) THEN
DISCH_THEN(X_CHOOSE_THEN `g':complex->complex` (LABEL_TAC "*")) THEN
EXISTS_TAC `(g':complex->complex) w` THEN
MATCH_MP_TAC
HAS_COMPLEX_DERIVATIVE_TRANSFORM_AT THEN
MAP_EVERY EXISTS_TAC
[`g:complex->complex`; `(r - norm(z - w:complex)) / &2`] THEN
REPEAT CONJ_TAC THENL
[UNDISCH_TAC `norm(z - w:complex) < r` THEN NORM_ARITH_TAC;
ALL_TAC;
REMOVE_THEN "*" (MP_TAC o SPEC `w:complex`) THEN ANTS_TAC THENL
[ALL_TAC; SIMP_TAC[]] THEN REWRITE_TAC[
IN_BALL] THEN
UNDISCH_TAC `norm(z - w:complex) < r` THEN NORM_ARITH_TAC] THEN
X_GEN_TAC `u:complex` THEN REWRITE_TAC[dist] THEN DISCH_TAC THEN
MATCH_MP_TAC
SERIES_UNIQUE THEN
EXISTS_TAC `(\n. a(n) * (u - z) pow n):num->complex` THEN
EXISTS_TAC `k:num->bool` THEN CONJ_TAC THENL
[REMOVE_THEN "*" (MP_TAC o SPEC `u:complex`) THEN
ANTS_TAC THENL [ALL_TAC; SIMP_TAC[]];
FIRST_X_ASSUM MATCH_MP_TAC] THEN
REWRITE_TAC[
IN_BALL] THEN ASM_NORM_ARITH_TAC);;
let CATAN_CONVERGS = prove
(`!z. norm(z) < &1
==> ((\n. --(Cx(&1)) pow n / Cx(&(2 * n + 1)) * z pow (2 * n + 1))
sums catn(z)) (from 0)`,
let TAYLOR_CATN = prove
(`!n z. norm(z) < &1
==> norm(catn z -
vsum(0..n) (\k. --(Cx(&1)) pow k / Cx(&(2 * k + 1)) *
z pow (2 * k + 1)))
<= norm(z) pow (2 * n + 3) /
((&2 * &n + &3) * (&1 - norm z pow 2))`,
let CLOG_CONVERGES = prove
(`!z. norm(z) < &1
==> ((\n. --Cx(&1) pow (n + 1) * z pow n / Cx(&n)) sums clog(Cx(&1) + z))
(from 1)`,
let TAYLOR_CLOG = prove
(`!n z. norm(z) < &1
==> norm(clog(Cx(&1) + z) -
vsum(1..n) (\k. --Cx(&1) pow (k + 1) * z pow k / Cx(&k)))
<= norm z pow (n + 1) / (&1 - norm z)`,
let TAYLOR_CLOG_NEG = prove
(`!n z. norm(z) < &1
==> norm(clog(Cx(&1) - z) + vsum(1..n) (\k. z pow k / Cx(&k)))
<= norm z pow (n + 1) / (&1 - norm z)`,
let CEXP_LIMIT = prove
(`!z. ((\n. (Cx(&1) + z / Cx(&n)) pow n) --> cexp(z)) sequentially`,
let LIM_N_MUL_SUB_CLOG = prove
(`!w z. ((\n. Cx(&n) * (clog(Cx(&n) + w) - clog(Cx(&n) + z))) --> w - z)
sequentially`,
REPEAT GEN_TAC THEN ASM_CASES_TAC `w:complex = z` THEN
ASM_REWRITE_TAC[
COMPLEX_SUB_REFL;
LIM_CONST;
COMPLEX_MUL_RZERO] THEN
MATCH_MP_TAC
LIM_TRANSFORM_EVENTUALLY THEN
EXISTS_TAC `\n. (Cx(&n) + z) / (Cx(&1) + z / Cx(&n)) *
clog(Cx(&1) + (w - z) / (Cx(&n) + z))` THEN
REWRITE_TAC[] THEN CONJ_TAC THENL
[REWRITE_TAC[
EVENTUALLY_SEQUENTIALLY] THEN
MP_TAC(SPEC `max (norm(w:complex)) (norm(z:complex)) + &1`
REAL_ARCH_SIMPLE) THEN
MATCH_MP_TAC
MONO_EXISTS THEN X_GEN_TAC `N:num` THEN DISCH_TAC THEN
X_GEN_TAC `n:num` THEN DISCH_TAC THEN
SUBGOAL_THEN `&0 < Re(Cx(&n) + w) /\ &0 < Re(Cx(&n) + z)` MP_TAC THENL
[REWRITE_TAC[
RE_ADD;
RE_CX] THEN
RULE_ASSUM_TAC(REWRITE_RULE[GSYM REAL_OF_NUM_LE]) THEN CONJ_TAC THEN
MATCH_MP_TAC(REAL_ARITH
`norm z < n /\ abs(Re z) <= norm z ==> &0 < n + Re z`) THEN
REWRITE_TAC[
COMPLEX_NORM_GE_RE_IM] THEN ASM_REAL_ARITH_TAC;
MAP_EVERY ASM_CASES_TAC
[`Cx(&n) + w = Cx(&0)`; `Cx(&n) + z = Cx(&0)`] THEN
ASM_REWRITE_TAC[
RE_CX;
REAL_LT_REFL] THEN STRIP_TAC] THEN
SUBGOAL_THEN `~(Cx(&n) = Cx(&0))` ASSUME_TAC THENL
[REWRITE_TAC[
CX_INJ; REAL_OF_NUM_EQ] THEN DISCH_THEN SUBST_ALL_TAC THEN
UNDISCH_TAC `max (norm(w:complex)) (norm (z:complex)) + &1 <= &N` THEN
RULE_ASSUM_TAC(REWRITE_RULE[CONJUNCT1
LE]) THEN
ASM_REWRITE_TAC[] THEN CONV_TAC NORM_ARITH;
ASM_SIMP_TAC[COMPLEX_FIELD
`~(n + z = Cx(&0)) /\ ~(n = Cx(&0))
==> (n + z) / (Cx(&1) + z / n) = n`] THEN
AP_TERM_TAC THEN ASM_SIMP_TAC[COMPLEX_FIELD
`~(n + z = Cx(&0))
==> Cx(&1) + (w - z) / (n + z) = (n + w) / (n + z)`] THEN
REWRITE_TAC[
complex_div] THEN IMP_REWRITE_TAC[
CLOG_MUL_SIMPLE] THEN
ASM_REWRITE_TAC[
COMPLEX_INV_EQ_0] THEN ASM_SIMP_TAC[
CLOG_INV] THEN
CONJ_TAC THENL [CONV_TAC COMPLEX_RING; REWRITE_TAC[
IM_NEG]] THEN
MATCH_MP_TAC(REAL_ARITH
`abs(x) < pi / &2 /\ abs(y) < pi / &2
==> --pi < x + --y /\ x + --y <= pi`) THEN
ASM_SIMP_TAC[
RE_CLOG_POS_LT]];
REWRITE_TAC[SIMPLE_COMPLEX_ARITH `a / b * c:complex = inv b * a * c`] THEN
GEN_REWRITE_TAC LAND_CONV [GSYM
COMPLEX_MUL_LID] THEN
MATCH_MP_TAC
LIM_COMPLEX_MUL THEN CONJ_TAC THENL
[GEN_REWRITE_TAC LAND_CONV [GSYM
COMPLEX_INV_1] THEN
MATCH_MP_TAC
LIM_COMPLEX_INV THEN
CONJ_TAC THENL [ALL_TAC; CONV_TAC COMPLEX_RING] THEN
GEN_REWRITE_TAC LAND_CONV [GSYM
COMPLEX_ADD_RID] THEN
MATCH_MP_TAC
LIM_ADD THEN REWRITE_TAC[
LIM_CONST;
complex_div] THEN
SIMP_TAC[
LIM_NULL_COMPLEX_LMUL;
LIM_INV_N];
ALL_TAC] THEN
SUBGOAL_THEN
`(\n. (Cx(&n) + z) * clog (Cx(&1) + (w - z) / (Cx(&n) + z))) =
(\x. (w - z) * clog(Cx(&1) + x) / x) o (\n. (w - z) / (Cx(&n) + z))`
SUBST1_TAC THENL
[REWRITE_TAC[
FUN_EQ_THM;
o_THM;
complex_div] THEN
REWRITE_TAC[
COMPLEX_INV_MUL;
COMPLEX_INV_INV] THEN
POP_ASSUM MP_TAC THEN CONV_TAC COMPLEX_FIELD;
ALL_TAC] THEN
MATCH_MP_TAC
LIM_COMPOSE_AT THEN EXISTS_TAC `Cx(&0)` THEN
REPEAT CONJ_TAC THENL
[REWRITE_TAC[
complex_div] THEN
SIMP_TAC[
LIM_INV_N_OFFSET;
LIM_NULL_COMPLEX_LMUL];
REWRITE_TAC[
EVENTUALLY_SEQUENTIALLY] THEN
MP_TAC(SPEC `norm(z:complex) + &1`
REAL_ARCH_SIMPLE) THEN
MATCH_MP_TAC
MONO_EXISTS THEN X_GEN_TAC `N:num` THEN DISCH_TAC THEN
REWRITE_TAC[GSYM REAL_OF_NUM_LE] THEN X_GEN_TAC `n:num` THEN
STRIP_TAC THEN MATCH_MP_TAC(TAUT `~p ==> p ==> q`) THEN
ASM_REWRITE_TAC[
COMPLEX_DIV_EQ_0;
COMPLEX_SUB_0] THEN
REWRITE_TAC[COMPLEX_RING `n + z = Cx(&0) <=> z = --n`] THEN
DISCH_TAC THEN UNDISCH_TAC `norm(z:complex) + &1 <= &N` THEN
ASM_REWRITE_TAC[
NORM_NEG;
COMPLEX_NORM_CX] THEN ASM_REAL_ARITH_TAC;
GEN_REWRITE_TAC LAND_CONV [GSYM
COMPLEX_MUL_RID] THEN
SIMP_TAC[
LIM_COMPLEX_LMUL;
LIM_LOGPLUS1_OVER_X]]]);;
let LIM_SUB_CLOG = prove
(`!w z. ((\n. clog(Cx(&n) + w) - clog(Cx(&n) + z)) --> Cx(&0)) sequentially`,
REPEAT GEN_TAC THEN MATCH_MP_TAC
LIM_TRANSFORM_EVENTUALLY THEN
SUBST1_TAC(COMPLEX_RING `Cx(&0) = Cx(&0) * (w - z)`) THEN EXISTS_TAC
`\n. inv(Cx(&n)) * Cx(&n) * (clog(Cx(&n) + w) - clog(Cx(&n) + z))` THEN
CONJ_TAC THENL
[REWRITE_TAC[
EVENTUALLY_SEQUENTIALLY] THEN EXISTS_TAC `1` THEN
REWRITE_TAC[ARITH_RULE `1 <= n <=> ~(n = 0)`; GSYM REAL_OF_NUM_EQ] THEN
REWRITE_TAC[GSYM
CX_INJ] THEN CONV_TAC COMPLEX_FIELD;
MATCH_MP_TAC
LIM_COMPLEX_MUL THEN
REWRITE_TAC[
LIM_INV_N;
LIM_N_MUL_SUB_CLOG]]);;
let POLE_THEOREM = prove
(`!f g s a.
g
holomorphic_on s /\ a
IN interior(s) /\
(!z. z
IN s /\ ~(z = a) ==> g(z) = (z - a) * f(z))
==> (\z. if z = a then
complex_derivative g a
else f(z) - g(a) / (z - a))
holomorphic_on s`,
REPEAT GEN_TAC THEN REWRITE_TAC[
CONJ_ASSOC] THEN
DISCH_THEN(CONJUNCTS_THEN ASSUME_TAC) THEN
FIRST_ASSUM(MP_TAC o MATCH_MP
POLE_LEMMA) THEN
MATCH_MP_TAC(REWRITE_RULE[
IMP_CONJ]
HOLOMORPHIC_TRANSFORM) THEN
X_GEN_TAC `z:complex` THEN DISCH_TAC THEN REWRITE_TAC[] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
FIRST_X_ASSUM(MP_TAC o SPEC `z:complex` o last o CONJUNCTS) THEN
ASM_REWRITE_TAC[] THEN REPEAT(POP_ASSUM MP_TAC) THEN
CONV_TAC COMPLEX_FIELD);;
let POLE_THEOREM_OPEN = prove
(`!f g s a.
open s /\ g
holomorphic_on s /\
(!z. z
IN s /\ ~(z = a) ==> g(z) = (z - a) * f(z))
==> (\z. if z = a then
complex_derivative g a
else f(z) - g(a) / (z - a))
holomorphic_on s`,
REPEAT GEN_TAC THEN REWRITE_TAC[
CONJ_ASSOC] THEN
DISCH_THEN(CONJUNCTS_THEN ASSUME_TAC) THEN
FIRST_ASSUM(MP_TAC o SPEC `a:complex` o MATCH_MP
POLE_LEMMA_OPEN) THEN
MATCH_MP_TAC(REWRITE_RULE[
IMP_CONJ]
HOLOMORPHIC_TRANSFORM) THEN
X_GEN_TAC `z:complex` THEN DISCH_TAC THEN REWRITE_TAC[] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
FIRST_X_ASSUM(MP_TAC o SPEC `z:complex` o last o CONJUNCTS) THEN
ASM_REWRITE_TAC[] THEN REPEAT(POP_ASSUM MP_TAC) THEN
CONV_TAC COMPLEX_FIELD);;
let CAUCHY_INTEGRAL_FORMULA_GLOBAL = prove
(`!f s g z.
open s /\ f
holomorphic_on s /\ z
IN s /\
valid_path g /\ pathfinish g = pathstart g /\
path_image g
SUBSET s
DELETE z /\
(!w. ~(w
IN s) ==>
winding_number(g,w) = Cx(&0))
==> ((\w. f(w) / (w - z))
has_path_integral
(Cx(&2) * Cx(pi) * ii *
winding_number(g,z) * f(z))) g`,
MATCH_MP_TAC(MESON[]
`((!f s g.
vector_polynomial_function g ==> P f s g) ==> !f s g. P f s g) /\
(!f s g.
vector_polynomial_function g ==> P f s g)
==> !f s g. P f s g`) THEN
CONJ_TAC THENL
[REPEAT STRIP_TAC THEN
MP_TAC(ISPECL [`s
DELETE (z:complex)`; `g:real^1->complex`]
PATH_INTEGRAL_NEARBY_ENDS) THEN
ASM_SIMP_TAC[
VALID_PATH_IMP_PATH;
OPEN_DELETE] THEN
DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN
MP_TAC(ISPECL [`g:real^1->complex`; `d:real`]
PATH_APPROX_VECTOR_POLYNOMIAL_FUNCTION) THEN
ASM_SIMP_TAC[
VALID_PATH_IMP_PATH] THEN
DISCH_THEN(X_CHOOSE_THEN `p:real^1->complex` STRIP_ASSUME_TAC) THEN
FIRST_X_ASSUM(MP_TAC o SPECL
[`g:real^1->complex`; `p:real^1->complex`]) THEN
ASM_SIMP_TAC[
VECTOR_SUB_REFL;
NORM_0;
VALID_PATH_VECTOR_POLYNOMIAL_FUNCTION] THEN
STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL
[`f:complex->complex`; `s:complex->bool`; `p:real^1->complex`]) THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o SPEC `z:complex`) THEN
SUBGOAL_THEN
`
winding_number(p,z) =
winding_number(g,z) /\
!w. ~(w
IN s) ==>
winding_number(p,w) =
winding_number(g,w)`
(fun th -> SIMP_TAC[th])
THENL
[FIRST_X_ASSUM(K ALL_TAC o SPEC `z:complex`) THEN
REPEAT(FIRST_X_ASSUM(STRIP_ASSUME_TAC o MATCH_MP (SET_RULE
`g
SUBSET s
DELETE z
==> ~(z
IN g) /\ (!y. ~(y
IN s) ==> ~(y
IN g))`))) THEN
ASM_SIMP_TAC[
WINDING_NUMBER_VALID_PATH;
VALID_PATH_VECTOR_POLYNOMIAL_FUNCTION] THEN
REPEAT STRIP_TAC THEN AP_TERM_TAC THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN
REWRITE_TAC[
complex_div;
COMPLEX_MUL_LID] THEN
MATCH_MP_TAC
HOLOMORPHIC_ON_INV THEN
SIMP_TAC[
HOLOMORPHIC_ON_SUB;
HOLOMORPHIC_ON_ID;
HOLOMORPHIC_ON_CONST;
IN_DELETE;
COMPLEX_SUB_0] THEN ASM SET_TAC[];
ALL_TAC] THEN
ASM_SIMP_TAC[
VALID_PATH_VECTOR_POLYNOMIAL_FUNCTION] THEN
MATCH_MP_TAC(MESON[
HAS_PATH_INTEGRAL_INTEGRAL;
path_integrable_on;
PATH_INTEGRAL_UNIQUE]
`f
path_integrable_on g /\
path_integral p f =
path_integral g f
==> (f
has_path_integral y) p ==> (f
has_path_integral y) g`) THEN
CONJ_TAC THENL
[MATCH_MP_TAC
PATH_INTEGRABLE_HOLOMORPHIC_SIMPLE THEN
EXISTS_TAC `s
DELETE (z:complex)` THEN ASM_SIMP_TAC[
OPEN_DELETE];
FIRST_X_ASSUM MATCH_MP_TAC] THEN
MATCH_MP_TAC
HOLOMORPHIC_ON_DIV THEN
SIMP_TAC[
HOLOMORPHIC_ON_SUB;
HOLOMORPHIC_ON_ID;
HOLOMORPHIC_ON_CONST;
IN_DELETE;
COMPLEX_SUB_0] THEN
ASM_MESON_TAC[
HOLOMORPHIC_ON_SUBSET;
DELETE_SUBSET];
ALL_TAC] THEN
MAP_EVERY X_GEN_TAC
[`f:complex->complex`; `u:complex->bool`; `g:real^1->complex`] THEN
DISCH_TAC THEN X_GEN_TAC `z:complex` THEN STRIP_TAC THEN
FIRST_ASSUM(X_CHOOSE_THEN `g':real^1->complex` STRIP_ASSUME_TAC o
MATCH_MP
HAS_VECTOR_DERIVATIVE_VECTOR_POLYNOMIAL_FUNCTION) THEN
SUBGOAL_THEN
`bounded(
IMAGE (g':real^1->complex) (interval[vec 0,vec 1]))`
MP_TAC THENL
[MATCH_MP_TAC
COMPACT_IMP_BOUNDED THEN
MATCH_MP_TAC
COMPACT_CONTINUOUS_IMAGE THEN
REWRITE_TAC[
COMPACT_INTERVAL] THEN
ASM_MESON_TAC[
CONTINUOUS_VECTOR_POLYNOMIAL_FUNCTION;
CONTINUOUS_AT_IMP_CONTINUOUS_ON];
REWRITE_TAC[
BOUNDED_POS;
FORALL_IN_IMAGE] THEN
DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC)] THEN
FIRST_ASSUM(ASSUME_TAC o MATCH_MP
VALID_PATH_IMP_PATH) THEN
MAP_EVERY ABBREV_TAC
[`d = \z w. if w = z then
complex_derivative f z
else (f(w) - f(z)) / (w - z)`;
`v = {w | ~(w
IN path_image g) /\
winding_number(g,w) = Cx(&0)}`] THEN
SUBGOAL_THEN `open(v:complex->bool)` ASSUME_TAC THENL
[EXPAND_TAC "v" THEN MATCH_MP_TAC
OPEN_WINDING_NUMBER_LEVELSETS THEN
ASM_REWRITE_TAC[];
ALL_TAC] THEN
SUBGOAL_THEN `u
UNION v = (:complex)` ASSUME_TAC THENL
[ASM SET_TAC[]; ALL_TAC] THEN
SUBGOAL_THEN `!y:complex. y
IN u ==> (d y)
holomorphic_on u` ASSUME_TAC THENL
[X_GEN_TAC `y:complex` THEN STRIP_TAC THEN EXPAND_TAC "d" THEN
MATCH_MP_TAC
NO_ISOLATED_SINGULARITY THEN EXISTS_TAC `{y:complex}` THEN
ASM_REWRITE_TAC[
FINITE_SING] THEN CONJ_TAC THENL
[ASM_SIMP_TAC[
CONTINUOUS_ON_EQ_CONTINUOUS_AT] THEN
X_GEN_TAC `w:complex` THEN DISCH_TAC THEN
ASM_CASES_TAC `w:complex = y` THENL
[UNDISCH_THEN `w:complex = y` SUBST_ALL_TAC THEN
REWRITE_TAC[
CONTINUOUS_AT] THEN
MATCH_MP_TAC
LIM_TRANSFORM_AWAY_AT THEN
EXISTS_TAC `\w:complex. (f w - f y) / (w - y)` THEN SIMP_TAC[] THEN
EXISTS_TAC `y + Cx(&1)` THEN
CONJ_TAC THENL [CONV_TAC COMPLEX_RING; ALL_TAC] THEN
REWRITE_TAC[GSYM
HAS_COMPLEX_DERIVATIVE_AT] THEN
REWRITE_TAC[
HAS_COMPLEX_DERIVATIVE_DIFFERENTIABLE] THEN
ASM_MESON_TAC[
HOLOMORPHIC_ON_IMP_DIFFERENTIABLE_AT];
MATCH_MP_TAC
COMPLEX_DIFFERENTIABLE_IMP_CONTINUOUS_AT];
ASM_SIMP_TAC[
HOLOMORPHIC_ON_OPEN;
OPEN_DELETE;
IN_DELETE;
SET_RULE `s
DIFF {x} = s
DELETE x`; GSYM
complex_differentiable] THEN
X_GEN_TAC `w:complex` THEN STRIP_TAC] THEN
MATCH_MP_TAC
COMPLEX_DIFFERENTIABLE_TRANSFORM_AT THEN
EXISTS_TAC `\w:complex. (f w - f y) / (w - y)` THEN
EXISTS_TAC `dist(w:complex,y)` THEN ASM_SIMP_TAC[
DIST_POS_LT] THEN
(CONJ_TAC THENL [MESON_TAC[
DIST_SYM;
REAL_LT_REFL]; ALL_TAC]) THEN
MATCH_MP_TAC
COMPLEX_DIFFERENTIABLE_DIV_AT THEN
ASM_REWRITE_TAC[
COMPLEX_SUB_0] THEN CONJ_TAC THEN
MATCH_MP_TAC
COMPLEX_DIFFERENTIABLE_SUB THEN
ASM_SIMP_TAC[ETA_AX;
COMPLEX_DIFFERENTIABLE_CONST;
COMPLEX_DIFFERENTIABLE_ID] THEN
ASM_MESON_TAC[
HOLOMORPHIC_ON_IMP_DIFFERENTIABLE_AT];
ALL_TAC] THEN
SUBGOAL_THEN
`!y. ~(y
IN path_image g)
==> (\x. (f x - f y) / (x - y))
path_integrable_on g`
ASSUME_TAC THENL
[X_GEN_TAC `y:complex` THEN DISCH_TAC THEN
MATCH_MP_TAC
PATH_INTEGRABLE_HOLOMORPHIC_SIMPLE THEN
EXISTS_TAC `u
DELETE (y:complex)` THEN ASM_SIMP_TAC[
OPEN_DELETE] THEN
CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN
MATCH_MP_TAC
HOLOMORPHIC_ON_DIV THEN
SIMP_TAC[
IN_DELETE;
COMPLEX_SUB_0] THEN
CONJ_TAC THEN MATCH_MP_TAC
HOLOMORPHIC_ON_SUB THEN
ASM_REWRITE_TAC[
HOLOMORPHIC_ON_CONST;
HOLOMORPHIC_ON_ID] THEN
MATCH_MP_TAC
HOLOMORPHIC_ON_SUBSET THEN
EXISTS_TAC `u:complex->bool` THEN ASM_REWRITE_TAC[
DELETE_SUBSET];
ALL_TAC] THEN
SUBGOAL_THEN
`!y:complex. d y
path_integrable_on g`
ASSUME_TAC THENL
[X_GEN_TAC `y:complex` THEN
ASM_CASES_TAC `(y:complex)
IN path_image g` THENL
[MATCH_MP_TAC
PATH_INTEGRABLE_HOLOMORPHIC_SIMPLE THEN
EXISTS_TAC `u:complex->bool` THEN ASM_SIMP_TAC[] THEN ASM SET_TAC[];
MATCH_MP_TAC
PATH_INTEGRABLE_EQ THEN
EXISTS_TAC `\x:complex. (f x - f y) / (x - y)` THEN
ASM_SIMP_TAC[] THEN EXPAND_TAC "d" THEN ASM_MESON_TAC[]];
ALL_TAC] THEN
SUBGOAL_THEN
`?h. (!z. z
IN u ==> ((d z)
has_path_integral h(z)) g) /\
(!z. z
IN v ==> ((\w. f(w) / (w - z))
has_path_integral h(z)) g)`
(CHOOSE_THEN (CONJUNCTS_THEN2 (LABEL_TAC "u") (LABEL_TAC "v")))
THENL
[EXISTS_TAC `\z. if z
IN u then
path_integral g (d z)
else
path_integral g (\w. f(w) / (w - z))` THEN
SIMP_TAC[] THEN CONJ_TAC THEN X_GEN_TAC `w:complex` THEN DISCH_TAC THENL
[ASM_MESON_TAC[
HAS_PATH_INTEGRAL_INTEGRAL]; ALL_TAC] THEN
ASM_CASES_TAC `(w:complex)
IN u` THEN ASM_REWRITE_TAC[] THENL
[ALL_TAC;
MATCH_MP_TAC
HAS_PATH_INTEGRAL_INTEGRAL THEN
MATCH_MP_TAC
PATH_INTEGRABLE_HOLOMORPHIC_SIMPLE THEN
EXISTS_TAC `u:complex->bool` THEN ASM_SIMP_TAC[] THEN CONJ_TAC THENL
[MATCH_MP_TAC
HOLOMORPHIC_ON_DIV THEN
ASM_SIMP_TAC[
COMPLEX_SUB_0;
HOLOMORPHIC_ON_SUB;
HOLOMORPHIC_ON_CONST;
HOLOMORPHIC_ON_ID] THEN
ASM_MESON_TAC[];
ASM SET_TAC[]]] THEN
MATCH_MP_TAC
HAS_PATH_INTEGRAL_EQ THEN
EXISTS_TAC `\x:complex. (f x - f w) / (x - w) + f(w) / (x - w)` THEN
CONJ_TAC THENL
[X_GEN_TAC `x:complex` THEN DISCH_TAC THEN REWRITE_TAC[] THEN
SIMPLE_COMPLEX_ARITH_TAC;
ALL_TAC] THEN
GEN_REWRITE_TAC LAND_CONV [GSYM
COMPLEX_ADD_RID] THEN
MATCH_MP_TAC
HAS_PATH_INTEGRAL_ADD THEN
UNDISCH_TAC `(w:complex)
IN v` THEN EXPAND_TAC "v" THEN
REWRITE_TAC[
IN_ELIM_THM] THEN REPEAT STRIP_TAC THENL
[MATCH_MP_TAC(MESON[
PATH_INTEGRAL_UNIQUE;
HAS_PATH_INTEGRAL_INTEGRAL;
path_integrable_on;
PATH_INTEGRAL_EQ;
PATH_INTEGRABLE_EQ]
`g
path_integrable_on p /\
(!x. x
IN path_image p ==> f x = g x)
==> (f
has_path_integral path_integral p g) p`) THEN
ASM_REWRITE_TAC[] THEN EXPAND_TAC "d" THEN ASM_MESON_TAC[];
SUBGOAL_THEN
`Cx(&0) = (f w) * Cx(&2) * Cx pi * ii *
winding_number(g,w)`
SUBST1_TAC THENL [ASM_REWRITE_TAC[
COMPLEX_MUL_RZERO]; ALL_TAC] THEN
ONCE_REWRITE_TAC[SIMPLE_COMPLEX_ARITH `x / y = x * Cx(&1) / y`] THEN
MATCH_MP_TAC
HAS_PATH_INTEGRAL_COMPLEX_LMUL THEN
MATCH_MP_TAC
HAS_PATH_INTEGRAL_WINDING_NUMBER THEN
ASM_REWRITE_TAC[]];
ALL_TAC] THEN
SUBGOAL_THEN `!z. (h:complex->complex) z = Cx(&0)` ASSUME_TAC THENL
[ALL_TAC;
REMOVE_THEN "u" (MP_TAC o SPEC `z:complex`) THEN ASM_REWRITE_TAC[] THEN
EXPAND_TAC "d" THEN REWRITE_TAC[] THEN
DISCH_THEN(MP_TAC o SPEC `\w. (f w - f z) / (w - z)` o
MATCH_MP (REWRITE_RULE[
IMP_CONJ_ALT]
HAS_PATH_INTEGRAL_EQ)) THEN
REWRITE_TAC[] THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
MP_TAC(SPECL [`g:real^1->complex`; `z:complex`]
HAS_PATH_INTEGRAL_WINDING_NUMBER) THEN ASM_REWRITE_TAC[] THEN
ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
DISCH_THEN(MP_TAC o MATCH_MP
HAS_PATH_INTEGRAL_COMPLEX_RMUL) THEN
DISCH_THEN(MP_TAC o SPEC `(f:complex->complex) z`) THEN
REWRITE_TAC[IMP_IMP] THEN
DISCH_THEN(MP_TAC o MATCH_MP
HAS_PATH_INTEGRAL_ADD) THEN
REWRITE_TAC[
complex_div;
COMPLEX_ADD_RID; COMPLEX_RING
`(Cx(&1) * i) * fz + (fx - fz) * i = fx * i`] THEN
REWRITE_TAC[GSYM
COMPLEX_MUL_ASSOC]] THEN
UNDISCH_THEN `(z:complex)
IN u` (K ALL_TAC) THEN
FIRST_X_ASSUM(ASSUME_TAC o MATCH_MP (SET_RULE
`p
SUBSET u
DELETE z ==> p
SUBSET u`)) THEN
POP_ASSUM_LIST(MP_TAC o end_itlist CONJ o rev) THEN STRIP_TAC THEN
MATCH_MP_TAC
LIOUVILLE_WEAK THEN
MATCH_MP_TAC(TAUT `b /\ (b ==> a) ==> a /\ b`) THEN CONJ_TAC THENL
[SUBGOAL_THEN
`?t:complex->bool.
compact t /\
path_image g
SUBSET interior t /\ t
SUBSET u`
STRIP_ASSUME_TAC THENL
[SUBGOAL_THEN
`?dd. &0 < dd /\
{y + k | y
IN path_image g /\ k
IN ball(vec 0,dd)}
SUBSET u`
STRIP_ASSUME_TAC THENL
[ASM_CASES_TAC `u = (:complex)` THENL
[EXISTS_TAC `&1` THEN ASM_REWRITE_TAC[
REAL_LT_01;
SUBSET_UNIV];
ALL_TAC] THEN
MP_TAC(ISPECL [`
path_image g:complex->bool`; `(:complex)
DIFF u`]
SEPARATE_COMPACT_CLOSED) THEN
ASM_SIMP_TAC[
COMPACT_PATH_IMAGE; GSYM
OPEN_CLOSED] THEN
ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
DISCH_THEN(X_CHOOSE_THEN `dd:real` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `dd / &2` THEN ASM_REWRITE_TAC[
REAL_HALF] THEN
REWRITE_TAC[
SUBSET;
FORALL_IN_GSPEC] THEN
MAP_EVERY X_GEN_TAC [`y:complex`; `k:complex`] THEN
MATCH_MP_TAC(TAUT `(a /\ ~c ==> ~b) ==> a /\ b ==> c`) THEN
STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPECL [`y:complex`; `y + k:complex`]) THEN
ASM_REWRITE_TAC[
IN_DIFF;
IN_UNIV;
IN_BALL] THEN CONV_TAC NORM_ARITH;
ALL_TAC] THEN
EXISTS_TAC `{y + k:complex |
y
IN path_image g /\ k
IN cball(vec 0,dd / &2)}` THEN
ASM_SIMP_TAC[
COMPACT_SUMS;
COMPACT_PATH_IMAGE;
COMPACT_CBALL] THEN
CONJ_TAC THENL
[REWRITE_TAC[
SUBSET;
IN_INTERIOR;
IN_ELIM_THM] THEN
X_GEN_TAC `y:complex` THEN DISCH_TAC THEN
EXISTS_TAC `dd / &2` THEN ASM_REWRITE_TAC[
REAL_HALF] THEN
X_GEN_TAC `x:complex` THEN REWRITE_TAC[
IN_BALL] THEN DISCH_TAC THEN
MAP_EVERY EXISTS_TAC [`y:complex`; `x - y:complex`] THEN
ASM_REWRITE_TAC[
IN_CBALL] THEN
UNDISCH_TAC `dist(y:complex,x) < dd / &2` THEN CONV_TAC NORM_ARITH;
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
`{x + y:real^N | x
IN s /\ y
IN t}
SUBSET u
==> t'
SUBSET t ==> {x + y | x
IN s /\ y
IN t'}
SUBSET u`)) THEN
REWRITE_TAC[
SUBSET;
IN_BALL;
IN_CBALL] THEN
UNDISCH_TAC `&0 < dd` THEN CONV_TAC NORM_ARITH];
ALL_TAC] THEN
MP_TAC(ISPECL [`interior t:complex->bool`; `g:real^1->complex`]
PATH_INTEGRAL_BOUND_EXISTS) THEN
ASM_REWRITE_TAC[
OPEN_INTERIOR] THEN
DISCH_THEN(X_CHOOSE_THEN `L:real` STRIP_ASSUME_TAC) THEN
SUBGOAL_THEN `bounded(
IMAGE (f:complex->complex) t)` MP_TAC THENL
[MATCH_MP_TAC
COMPACT_IMP_BOUNDED THEN
MATCH_MP_TAC
COMPACT_CONTINUOUS_IMAGE THEN
ASM_MESON_TAC[
HOLOMORPHIC_ON_IMP_CONTINUOUS_ON;
CONTINUOUS_ON_SUBSET];
REWRITE_TAC[
BOUNDED_POS;
FORALL_IN_IMAGE] THEN
DISCH_THEN(X_CHOOSE_THEN `D:real` STRIP_ASSUME_TAC)] THEN
FIRST_ASSUM(MP_TAC o MATCH_MP
COMPACT_IMP_BOUNDED) THEN
REWRITE_TAC[
BOUNDED_POS] THEN
DISCH_THEN(X_CHOOSE_THEN `C:real` STRIP_ASSUME_TAC) THEN
REWRITE_TAC[
LIM_AT_INFINITY] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN
EXISTS_TAC `(D * L) / (e / &2) + C:real` THEN REWRITE_TAC[
real_ge] THEN
X_GEN_TAC `y:complex` THEN DISCH_TAC THEN
REWRITE_TAC[dist;
COMPLEX_SUB_RZERO] THEN
SUBGOAL_THEN `h y =
path_integral g (\w. f w / (w - y))` SUBST1_TAC THENL
[CONV_TAC SYM_CONV THEN MATCH_MP_TAC
PATH_INTEGRAL_UNIQUE THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN EXPAND_TAC "v" THEN
REWRITE_TAC[
IN_ELIM_THM] THEN CONJ_TAC THENL
[DISCH_TAC THEN
UNDISCH_TAC `(D * L) / (e / &2) + C <= norm(y:complex)` THEN
MATCH_MP_TAC(REAL_ARITH `&0 < d /\ x <= c ==> d + c <= x ==> F`) THEN
ASM_SIMP_TAC[
REAL_LT_MUL;
REAL_LT_DIV;
REAL_HALF] THEN
ASM_MESON_TAC[
INTERIOR_SUBSET;
SUBSET];
MATCH_MP_TAC
WINDING_NUMBER_ZERO_OUTSIDE THEN
EXISTS_TAC `cball(Cx(&0),C)` THEN
ASM_REWRITE_TAC[
CONVEX_CBALL;
SUBSET;
IN_CBALL; dist;
COMPLEX_SUB_LZERO;
NORM_NEG] THEN
CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[
INTERIOR_SUBSET;
SUBSET]] THEN
UNDISCH_TAC `(D * L) / (e / &2) + C <= norm(y:complex)` THEN
MATCH_MP_TAC(REAL_ARITH `&0 < d ==> d + c <= x ==> ~(x <= c)`) THEN
ASM_SIMP_TAC[
REAL_LT_MUL;
REAL_LT_DIV;
REAL_HALF]];
ALL_TAC] THEN
MATCH_MP_TAC
REAL_LET_TRANS THEN EXISTS_TAC `L * (e / &2 / L)` THEN
CONJ_TAC THENL
[ALL_TAC;
ASM_SIMP_TAC[
REAL_DIV_LMUL;
REAL_LT_IMP_NZ;
REAL_HALF] THEN
ASM_REAL_ARITH_TAC] THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[] THEN CONJ_TAC THENL
[MATCH_MP_TAC
HOLOMORPHIC_ON_DIV THEN CONJ_TAC THENL
[ASM_MESON_TAC[
HOLOMORPHIC_ON_SUBSET;
SUBSET_TRANS;
INTERIOR_SUBSET];
SIMP_TAC[
HOLOMORPHIC_ON_SUB;
HOLOMORPHIC_ON_ID;
HOLOMORPHIC_ON_CONST;
COMPLEX_SUB_0]] THEN
X_GEN_TAC `w:complex` THEN DISCH_TAC THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (NORM_ARITH
`d + c <= norm y ==> &0 < d /\ norm w <= c ==> ~(w = y)`)) THEN
ASM_SIMP_TAC[
REAL_LT_MUL;
REAL_LT_DIV;
REAL_HALF] THEN
ASM_MESON_TAC[
SUBSET;
INTERIOR_SUBSET];
ALL_TAC] THEN
X_GEN_TAC `w:complex` THEN DISCH_TAC THEN SIMP_TAC[
COMPLEX_NORM_DIV] THEN
SUBGOAL_THEN `&0 < norm(w - y)` ASSUME_TAC THENL
[FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (NORM_ARITH
`d + c <= norm y ==> &0 < d /\ norm w <= c ==> &0 < norm(w - y)`)) THEN
ASM_SIMP_TAC[
REAL_LT_MUL;
REAL_LT_DIV;
REAL_HALF] THEN
ASM_MESON_TAC[
SUBSET;
INTERIOR_SUBSET];
ASM_SIMP_TAC[
REAL_LE_LDIV_EQ]] THEN
MATCH_MP_TAC
REAL_LE_TRANS THEN EXISTS_TAC `D:real` THEN CONJ_TAC THENL
[ASM_MESON_TAC[
SUBSET;
INTERIOR_SUBSET]; ALL_TAC] THEN
REWRITE_TAC[REAL_ARITH `e / &2 / L * x = (x * (e / &2)) / L`] THEN
ASM_SIMP_TAC[
REAL_LE_RDIV_EQ; GSYM
REAL_LE_LDIV_EQ;
REAL_HALF] THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (NORM_ARITH
`d + c <= norm y ==> norm w <= c ==> d <= norm(w - y)`)) THEN
ASM_MESON_TAC[
SUBSET;
INTERIOR_SUBSET];
DISCH_TAC] THEN
SUBGOAL_THEN
`(\y. (d:complex->complex->complex) (fstcart y) (sndcart y))
continuous_on
{pastecart x z | x
IN u /\ z
IN u}`
ASSUME_TAC THENL
[REWRITE_TAC[
CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN] THEN EXPAND_TAC "d" THEN
REWRITE_TAC[
FORALL_IN_GSPEC;
continuous_within;
IMP_CONJ] THEN
MAP_EVERY X_GEN_TAC [`x:complex`; `z:complex`] THEN REPEAT DISCH_TAC THEN
X_GEN_TAC `e:real` THEN DISCH_TAC THEN
REWRITE_TAC[
FSTCART_PASTECART;
SNDCART_PASTECART;
FORALL_PASTECART] THEN
REWRITE_TAC[dist; IMP_IMP; GSYM
CONJ_ASSOC;
PASTECART_SUB] THEN
ASM_CASES_TAC `z:complex = x` THEN ASM_REWRITE_TAC[] THENL
[UNDISCH_THEN `z:complex = x` (SUBST_ALL_TAC o SYM);
SUBGOAL_THEN
`(\y. (f(sndcart y) - f(fstcart y)) / (sndcart y - fstcart y))
continuous at (pastecart x z)`
MP_TAC THENL
[MATCH_MP_TAC
CONTINUOUS_COMPLEX_DIV_AT THEN
ASM_SIMP_TAC[
FSTCART_PASTECART;
SNDCART_PASTECART;
COMPLEX_SUB_0] THEN
CONJ_TAC THEN MATCH_MP_TAC
CONTINUOUS_SUB THEN
SIMP_TAC[
LINEAR_CONTINUOUS_AT;
LINEAR_FSTCART;
LINEAR_SNDCART] THEN
CONJ_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM
o_DEF] THEN
MATCH_MP_TAC
CONTINUOUS_AT_COMPOSE THEN
SIMP_TAC[
LINEAR_CONTINUOUS_AT;
LINEAR_FSTCART;
LINEAR_SNDCART] THEN
REWRITE_TAC[
FSTCART_PASTECART;
SNDCART_PASTECART] THEN
ASM_MESON_TAC[
HOLOMORPHIC_ON_IMP_CONTINUOUS_ON;
CONTINUOUS_ON_EQ_CONTINUOUS_AT];
ALL_TAC] THEN
REWRITE_TAC[
continuous_at; dist;
FORALL_PASTECART] THEN
REWRITE_TAC[
FSTCART_PASTECART;
SNDCART_PASTECART;
PASTECART_SUB] THEN
DISCH_THEN(MP_TAC o SPEC `e:real`) THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_THEN `k1:real` STRIP_ASSUME_TAC) THEN
SUBGOAL_THEN
`open({pastecart x z | x
IN u /\ z
IN u}
DIFF
{y | y
IN UNIV /\ fstcart y - sndcart y = Cx(&0)})`
MP_TAC THENL
[MATCH_MP_TAC
OPEN_DIFF THEN
ASM_SIMP_TAC[REWRITE_RULE[
PCROSS]
OPEN_PCROSS] THEN
MATCH_MP_TAC
CONTINUOUS_CLOSED_PREIMAGE_CONSTANT THEN
REWRITE_TAC[
CLOSED_UNIV] THEN MATCH_MP_TAC
CONTINUOUS_ON_SUB THEN
SIMP_TAC[
LINEAR_CONTINUOUS_ON;
LINEAR_FSTCART;
LINEAR_SNDCART];
SIMP_TAC[
OPEN_CONTAINS_BALL;
IN_DIFF;
IMP_CONJ;
FORALL_IN_GSPEC] THEN
DISCH_THEN(MP_TAC o SPECL [`x:complex`; `z:complex`]) THEN
ASM_REWRITE_TAC[
IN_ELIM_THM;
IN_UNIV;
COMPLEX_SUB_0] THEN
ASM_REWRITE_TAC[
SUBSET;
IN_BALL;
FORALL_PASTECART;
IN_DIFF;
IN_ELIM_PASTECART_THM;
FSTCART_PASTECART;
SNDCART_PASTECART] THEN
REWRITE_TAC[
IN_ELIM_THM; IMP_IMP; GSYM
CONJ_ASSOC] THEN
REWRITE_TAC[ONCE_REWRITE_RULE[
NORM_SUB] dist;
PASTECART_SUB;
FSTCART_PASTECART;
SNDCART_PASTECART] THEN
DISCH_THEN(X_CHOOSE_THEN `k2:real` STRIP_ASSUME_TAC)] THEN
EXISTS_TAC `min k1 k2:real` THEN
ASM_SIMP_TAC[
REAL_LT_MIN;
COMPLEX_NORM_NZ;
COMPLEX_SUB_0]] THEN
SUBGOAL_THEN `(
complex_derivative f) continuous at z` MP_TAC THENL
[MATCH_MP_TAC
CONTINUOUS_ON_INTERIOR THEN
EXISTS_TAC `u:complex->bool` THEN ASM_SIMP_TAC[
INTERIOR_OPEN] THEN
MATCH_MP_TAC
HOLOMORPHIC_ON_IMP_CONTINUOUS_ON THEN
MATCH_MP_TAC
HOLOMORPHIC_COMPLEX_DERIVATIVE THEN
ASM_REWRITE_TAC[];
REWRITE_TAC[
continuous_at] THEN DISCH_THEN(MP_TAC o SPEC `e / &2`) THEN
ASM_REWRITE_TAC[dist;
REAL_HALF]] THEN
DISCH_THEN(X_CHOOSE_THEN `k1:real` STRIP_ASSUME_TAC) THEN
MP_TAC(ISPEC `u:complex->bool`
OPEN_CONTAINS_BALL) THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o SPEC `z:complex`) THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_THEN `k2:real` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `min k1 k2:real` THEN ASM_REWRITE_TAC[
REAL_LT_MIN] THEN
MAP_EVERY X_GEN_TAC [`x':complex`; `z':complex`] THEN STRIP_TAC THEN
MATCH_MP_TAC(REAL_ARITH `&0 < e /\ x <= e / &2 ==> x < e`) THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[] THENL
[ASM_MESON_TAC[
NORM_LE_PASTECART;
REAL_LET_TRANS;
REAL_LT_IMP_LE];
ALL_TAC] THEN
SUBGOAL_THEN `e / &2 = e / &2 / norm(z' - x') * norm(z' - x':complex)`
SUBST1_TAC THENL
[ASM_SIMP_TAC[
REAL_DIV_RMUL;
NORM_EQ_0;
VECTOR_SUB_EQ]; ALL_TAC] THEN
MATCH_MP_TAC
HAS_PATH_INTEGRAL_BOUND_LINEPATH THEN
EXISTS_TAC `\u. (
complex_derivative f u -
complex_derivative f z) /
(z' - x')` THEN
ASM_SIMP_TAC[
REAL_LE_DIV;
NORM_POS_LE;
REAL_LT_IMP_LE;
REAL_HALF] THEN
CONJ_TAC THENL
[ASM_SIMP_TAC[COMPLEX_FIELD
`~(z:complex = x)
==> a / (z - x) - b = (a - b * (z - x)) / (z - x)`] THEN
MATCH_MP_TAC
HAS_PATH_INTEGRAL_COMPLEX_DIV THEN
MATCH_MP_TAC
HAS_PATH_INTEGRAL_SUB THEN
REWRITE_TAC[
HAS_PATH_INTEGRAL_CONST_LINEPATH] THEN
MP_TAC(ISPECL [`f:complex->complex`; `
complex_derivative f`;
`linepath(x':complex,z')`; `u:complex->bool`]
PATH_INTEGRAL_PRIMITIVE) THEN
REWRITE_TAC[ETA_AX;
PATHSTART_LINEPATH;
PATHFINISH_LINEPATH] THEN
DISCH_THEN MATCH_MP_TAC THEN
REWRITE_TAC[
VALID_PATH_LINEPATH] THEN CONJ_TAC THENL
[ASM_MESON_TAC[
HAS_COMPLEX_DERIVATIVE_DIFFERENTIABLE;
GSYM
HOLOMORPHIC_ON_DIFFERENTIABLE;
HAS_COMPLEX_DERIVATIVE_AT_WITHIN;
HOLOMORPHIC_ON_OPEN;
complex_differentiable];
MATCH_MP_TAC
SUBSET_TRANS THEN EXISTS_TAC `ball(z:complex,k2)`];
X_GEN_TAC `w:complex` THEN DISCH_TAC THEN
REWRITE_TAC[
COMPLEX_NORM_DIV;
real_div] THEN
MATCH_MP_TAC
REAL_LE_RMUL THEN
REWRITE_TAC[
REAL_LE_INV_EQ;
NORM_POS_LE] THEN
MATCH_MP_TAC(REAL_ARITH `x < e / &2 ==> x <= e * inv(&2)`) THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN
REWRITE_TAC[REWRITE_RULE[ONCE_REWRITE_RULE[
NORM_SUB] dist]
(GSYM
IN_BALL)] THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
`w
IN s ==> s
SUBSET t ==> w
IN t`))] THEN
ASM_REWRITE_TAC[
PATH_IMAGE_LINEPATH;
SEGMENT_CONVEX_HULL] THEN
MATCH_MP_TAC
HULL_MINIMAL THEN REWRITE_TAC[
CONVEX_BALL] THEN
REWRITE_TAC[
INSERT_SUBSET;
EMPTY_SUBSET;
IN_BALL; dist] THEN
ONCE_REWRITE_TAC[
NORM_SUB] THEN
ASM_MESON_TAC[
NORM_LE_PASTECART;
REAL_LET_TRANS];
ALL_TAC] THEN
SIMP_TAC[
HOLOMORPHIC_ON_OPEN;
OPEN_UNIV;
IN_UNIV;
GSYM
complex_differentiable] THEN
X_GEN_TAC `z0:complex` THEN ASM_CASES_TAC `(z0:complex)
IN v` THENL
[MP_TAC(ISPECL
[`f:complex->complex`; `h:complex->complex`; `g:real^1->complex`;
`v:complex->bool`; `1`; `B:real`]
CAUCHY_NEXT_DERIVATIVE) THEN
ASM_SIMP_TAC[
IN_DIFF;
ARITH_EQ;
COMPLEX_POW_1] THEN ANTS_TAC THENL
[CONJ_TAC THENL
[ASM_MESON_TAC[
HAS_VECTOR_DERIVATIVE_UNIQUE_AT]; ALL_TAC] THEN
MATCH_MP_TAC
HOLOMORPHIC_ON_IMP_CONTINUOUS_ON THEN
MATCH_MP_TAC
HOLOMORPHIC_ON_SUBSET THEN
EXISTS_TAC `u:complex->bool` THEN ASM SET_TAC[];
DISCH_THEN(MP_TAC o SPEC `z0:complex`) THEN
UNDISCH_TAC `(z0:complex)
IN v` THEN EXPAND_TAC "v" THEN
SIMP_TAC[
IN_ELIM_THM;
complex_differentiable] THEN MESON_TAC[]];
ALL_TAC] THEN
SUBGOAL_THEN `(z0:complex)
IN u` ASSUME_TAC THENL
[ASM SET_TAC[]; ALL_TAC] THEN
MP_TAC(ISPEC `u:complex->bool`
OPEN_CONTAINS_BALL) THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o SPEC `z0:complex`) THEN
ASM_SIMP_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN
MATCH_MP_TAC
HOLOMORPHIC_ON_IMP_DIFFERENTIABLE_AT THEN
EXISTS_TAC `ball(z0:complex,e)` THEN
ASM_REWRITE_TAC[
OPEN_BALL;
CENTRE_IN_BALL] THEN
MATCH_MP_TAC
ANALYTIC_IMP_HOLOMORPHIC THEN MATCH_MP_TAC
MORERA_TRIANGLE THEN
REWRITE_TAC[
OPEN_BALL] THEN
SUBGOAL_THEN `(h:complex->complex)
continuous_on u` ASSUME_TAC THENL
[REWRITE_TAC[
CONTINUOUS_ON_SEQUENTIALLY] THEN
MAP_EVERY X_GEN_TAC [`a:num->complex`; `x:complex`] THEN STRIP_TAC THEN
MP_TAC(ISPECL
[`sequentially`; `\n:num x. (d:complex->complex->complex) (a n) x`;
`B:real`; `g:real^1->complex`; `(d:complex->complex->complex) x`]
PATH_INTEGRAL_UNIFORM_LIMIT) THEN
ASM_REWRITE_TAC[
TRIVIAL_LIMIT_SEQUENTIALLY; ETA_AX;
EVENTUALLY_TRUE] THEN
ANTS_TAC THENL
[ALL_TAC;
MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN BINOP_TAC THEN
REWRITE_TAC[
FUN_EQ_THM;
o_THM] THEN REPEAT GEN_TAC THEN
MATCH_MP_TAC
PATH_INTEGRAL_UNIQUE THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[]] THEN
CONJ_TAC THENL
[ASM_MESON_TAC[
HAS_VECTOR_DERIVATIVE_UNIQUE_AT]; ALL_TAC] THEN
X_GEN_TAC `ee:real` THEN DISCH_TAC THEN
MP_TAC(ISPEC `u:complex->bool`
OPEN_CONTAINS_CBALL) THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o SPEC `x:complex`) THEN
ASM_SIMP_TAC[] THEN
DISCH_THEN(X_CHOOSE_THEN `dd:real` STRIP_ASSUME_TAC) THEN
SUBGOAL_THEN
`(\y. (d:complex->complex->complex) (fstcart y) (sndcart y))
uniformly_continuous_on
{pastecart w z | w
IN cball(x,dd) /\ z
IN path_image g}`
MP_TAC THENL
[MATCH_MP_TAC
COMPACT_UNIFORMLY_CONTINUOUS THEN
ASM_SIMP_TAC[REWRITE_RULE[
PCROSS]
COMPACT_PCROSS;
COMPACT_CBALL;
COMPACT_VALID_PATH_IMAGE] THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[
IMP_CONJ]
CONTINUOUS_ON_SUBSET)) THEN
REWRITE_TAC[
SUBSET;
FORALL_IN_GSPEC;
IN_ELIM_PASTECART_THM] THEN
ASM SET_TAC[];
ALL_TAC] THEN
REWRITE_TAC[
uniformly_continuous_on] THEN
DISCH_THEN(MP_TAC o SPEC `ee:real`) THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_THEN `kk:real`
(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
DISCH_THEN(MP_TAC o GENL [`w:complex`; `z:complex`] o
SPECL [`pastecart (x:complex) (z:complex)`;
`pastecart (w:complex) (z:complex)`]) THEN
SIMP_TAC[
IN_ELIM_PASTECART_THM;
FSTCART_PASTECART;
SNDCART_PASTECART] THEN
ASM_SIMP_TAC[
CENTRE_IN_CBALL;
REAL_LT_IMP_LE; dist;
PASTECART_SUB] THEN
REWRITE_TAC[
VECTOR_SUB_REFL;
NORM_0;
NORM_PASTECART] THEN
CONV_TAC REAL_RAT_REDUCE_CONV THEN
REWRITE_TAC[TAUT `b /\ (a /\ b) /\ c ==> d <=> a /\ b /\ c ==> d`] THEN
SIMP_TAC[
REAL_ADD_RID;
POW_2_SQRT;
NORM_POS_LE] THEN DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
LIM_SEQUENTIALLY]) THEN
DISCH_THEN(MP_TAC o SPEC `min dd kk:real`) THEN
ASM_REWRITE_TAC[
EVENTUALLY_SEQUENTIALLY;
REAL_LT_MIN] THEN
MATCH_MP_TAC
MONO_EXISTS THEN X_GEN_TAC `N:num` THEN
REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
ASM_SIMP_TAC[ONCE_REWRITE_RULE[
DIST_SYM]
IN_CBALL; GSYM dist;
REAL_LT_IMP_LE];
ALL_TAC] THEN
CONJ_TAC THENL [ASM_MESON_TAC[
CONTINUOUS_ON_SUBSET]; ALL_TAC] THEN
SUBGOAL_THEN
`!w. w
IN u ==> (\z. d z w)
holomorphic_on u`
ASSUME_TAC THENL
[EXPAND_TAC "d" THEN X_GEN_TAC `y:complex` THEN STRIP_TAC THEN
MATCH_MP_TAC
NO_ISOLATED_SINGULARITY THEN EXISTS_TAC `{y:complex}` THEN
ASM_REWRITE_TAC[
FINITE_SING] THEN CONJ_TAC THENL
[SUBGOAL_THEN
`((\y. (d:complex->complex->complex) (fstcart y) (sndcart y)) o
(\z. pastecart y z))
continuous_on u`
MP_TAC THENL
[MATCH_MP_TAC
CONTINUOUS_ON_COMPOSE THEN
SIMP_TAC[
CONTINUOUS_ON_PASTECART;
CONTINUOUS_ON_ID;
CONTINUOUS_ON_CONST] THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[
IMP_CONJ]
CONTINUOUS_ON_SUBSET)) THEN
ASM_SIMP_TAC[
SUBSET;
FORALL_IN_IMAGE;
IN_ELIM_PASTECART_THM];
EXPAND_TAC "d" THEN
REWRITE_TAC[
o_DEF;
FSTCART_PASTECART;
SNDCART_PASTECART] THEN
MATCH_MP_TAC(REWRITE_RULE[
IMP_CONJ]
CONTINUOUS_ON_EQ) THEN
GEN_TAC THEN REWRITE_TAC[] THEN COND_CASES_TAC THEN
ASM_REWRITE_TAC[] THEN
DISCH_TAC THEN REWRITE_TAC[
complex_div] THEN MATCH_MP_TAC(COMPLEX_RING
`x':complex = --x /\ y' = --y ==> x * y = x' * y'`) THEN
REWRITE_TAC[GSYM
COMPLEX_INV_NEG;
COMPLEX_NEG_SUB]];
ASM_SIMP_TAC[
HOLOMORPHIC_ON_OPEN;
OPEN_DELETE;
IN_DELETE;
SET_RULE `s
DIFF {x} = s
DELETE x`; GSYM
complex_differentiable] THEN
X_GEN_TAC `w:complex` THEN STRIP_TAC THEN
MATCH_MP_TAC
COMPLEX_DIFFERENTIABLE_TRANSFORM_AT THEN
EXISTS_TAC `\w:complex. (f y - f w) / (y - w)` THEN
EXISTS_TAC `dist(w:complex,y)` THEN ASM_SIMP_TAC[
DIST_POS_LT] THEN
(CONJ_TAC THENL [MESON_TAC[
DIST_SYM;
REAL_LT_REFL]; ALL_TAC]) THEN
MATCH_MP_TAC
COMPLEX_DIFFERENTIABLE_DIV_AT THEN
ASM_REWRITE_TAC[
COMPLEX_SUB_0] THEN CONJ_TAC THEN
MATCH_MP_TAC
COMPLEX_DIFFERENTIABLE_SUB THEN
ASM_SIMP_TAC[ETA_AX;
COMPLEX_DIFFERENTIABLE_CONST;
COMPLEX_DIFFERENTIABLE_ID] THEN
ASM_MESON_TAC[
HOLOMORPHIC_ON_IMP_DIFFERENTIABLE_AT]];
ALL_TAC] THEN
SUBGOAL_THEN
`!w a b:complex. w
IN u /\ segment[a,b]
SUBSET u
==> (\z. d z w)
path_integrable_on (linepath(a,b))`
ASSUME_TAC THENL
[REPEAT STRIP_TAC THEN
MATCH_MP_TAC
PATH_INTEGRABLE_CONTINUOUS_LINEPATH THEN
ASM_MESON_TAC[
CONTINUOUS_ON_SUBSET;
HOLOMORPHIC_ON_IMP_CONTINUOUS_ON];
ALL_TAC] THEN
SUBGOAL_THEN
`!a b:complex.
segment[a,b]
SUBSET u
==> (\w.
path_integral (linepath(a,b)) (\z. d z w))
continuous_on u`
ASSUME_TAC THENL
[REPEAT STRIP_TAC THEN ASM_CASES_TAC `a:complex = b` THENL
[ASM_SIMP_TAC[
PATH_INTEGRAL_TRIVIAL;
CONTINUOUS_ON_CONST]; ALL_TAC] THEN
REWRITE_TAC[
continuous_on] THEN X_GEN_TAC `w:complex` THEN DISCH_TAC THEN
X_GEN_TAC `ee:real` THEN DISCH_TAC THEN
ASM_SIMP_TAC[dist; GSYM
PATH_INTEGRAL_SUB] THEN
MP_TAC(ISPEC `u:complex->bool`
OPEN_CONTAINS_CBALL) THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o SPEC `w:complex`) THEN
ASM_SIMP_TAC[] THEN
DISCH_THEN(X_CHOOSE_THEN `dd:real` STRIP_ASSUME_TAC) THEN
SUBGOAL_THEN
`(\y. (d:complex->complex->complex) (fstcart y) (sndcart y))
uniformly_continuous_on
{pastecart z t | z
IN segment[a,b] /\ t
IN cball(w,dd)}`
MP_TAC THENL
[MATCH_MP_TAC
COMPACT_UNIFORMLY_CONTINUOUS THEN
ASM_SIMP_TAC[REWRITE_RULE[
PCROSS]
COMPACT_PCROSS;
COMPACT_CBALL;
COMPACT_SEGMENT] THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[
IMP_CONJ]
CONTINUOUS_ON_SUBSET)) THEN
REWRITE_TAC[
SUBSET;
FORALL_IN_GSPEC;
IN_ELIM_PASTECART_THM] THEN
ASM SET_TAC[];
ALL_TAC] THEN
REWRITE_TAC[
uniformly_continuous_on] THEN
DISCH_THEN(MP_TAC o SPEC `ee / &2 / norm(b - a:complex)`) THEN
ASM_SIMP_TAC[
REAL_HALF;
REAL_LT_DIV;
COMPLEX_NORM_NZ;
COMPLEX_SUB_0] THEN
DISCH_THEN(X_CHOOSE_THEN `kk:real`
(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
DISCH_THEN(MP_TAC o GENL [`z:complex`; `r:complex`] o
SPECL [`pastecart (r:complex) (z:complex)`;
`pastecart (r:complex) (w:complex)`]) THEN
SIMP_TAC[
IN_ELIM_PASTECART_THM;
FSTCART_PASTECART;
SNDCART_PASTECART] THEN
ASM_SIMP_TAC[
CENTRE_IN_CBALL;
REAL_LT_IMP_LE; dist;
PASTECART_SUB] THEN
REWRITE_TAC[
VECTOR_SUB_REFL;
NORM_0;
NORM_PASTECART] THEN
CONV_TAC REAL_RAT_REDUCE_CONV THEN
REWRITE_TAC[TAUT `(a /\ b) /\ a /\ c ==> d <=> a /\ b /\ c ==> d`] THEN
SIMP_TAC[REAL_ADD_LID;
POW_2_SQRT;
NORM_POS_LE] THEN DISCH_TAC THEN
EXISTS_TAC `min dd kk:real` THEN ASM_REWRITE_TAC[
REAL_LT_MIN] THEN
X_GEN_TAC `x:complex` THEN REPEAT STRIP_TAC THEN
MATCH_MP_TAC(REAL_ARITH `&0 < e /\ x <= e / &2 ==> x < e`) THEN
ASM_REWRITE_TAC[] THEN
SUBGOAL_THEN `ee / &2 = ee / &2 / norm(b - a) * norm(b - a:complex)`
SUBST1_TAC THENL
[ASM_SIMP_TAC[
REAL_DIV_RMUL;
NORM_EQ_0;
VECTOR_SUB_EQ]; ALL_TAC] THEN
MATCH_MP_TAC
HAS_PATH_INTEGRAL_BOUND_LINEPATH THEN
EXISTS_TAC `\r. (d:complex->complex->complex) r x - d r w` THEN
ASM_SIMP_TAC[
REAL_LE_DIV;
NORM_POS_LE;
REAL_LT_IMP_LE;
REAL_HALF] THEN
CONJ_TAC THENL
[MATCH_MP_TAC
HAS_PATH_INTEGRAL_INTEGRAL THEN
MATCH_MP_TAC
PATH_INTEGRABLE_SUB THEN ASM_SIMP_TAC[];
REPEAT STRIP_TAC THEN GEN_REWRITE_TAC LAND_CONV [
NORM_SUB] THEN
MATCH_MP_TAC
REAL_LT_IMP_LE THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
ASM_REWRITE_TAC[
IN_CBALL; dist] THEN
ASM_MESON_TAC[
NORM_SUB;
REAL_LT_IMP_LE]];
ALL_TAC] THEN
SUBGOAL_THEN
`!a b. segment[a,b]
SUBSET u
==> (\w.
path_integral (linepath (a,b)) (\z. d z w))
path_integrable_on g`
ASSUME_TAC THENL
[REPEAT STRIP_TAC THEN REWRITE_TAC[
PATH_INTEGRABLE_ON] THEN
MATCH_MP_TAC
INTEGRABLE_CONTINUOUS THEN
MATCH_MP_TAC
CONTINUOUS_ON_COMPLEX_MUL THEN CONJ_TAC THENL
[SUBGOAL_THEN
`((\w.
path_integral (linepath(a,b)) (\z. d z w)) o (g:real^1->complex))
continuous_on interval[vec 0,vec 1]`
MP_TAC THENL
[MATCH_MP_TAC
CONTINUOUS_ON_COMPOSE THEN
ASM_SIMP_TAC[GSYM path;
VALID_PATH_IMP_PATH] THEN
MATCH_MP_TAC
CONTINUOUS_ON_SUBSET THEN
EXISTS_TAC `u:complex->bool` THEN ASM_SIMP_TAC[GSYM
path_image];
REWRITE_TAC[
o_DEF]];
FIRST_ASSUM(fun th -> REWRITE_TAC
[MATCH_MP
HAS_VECTOR_DERIVATIVE_UNIQUE_AT (SPEC_ALL th)]) THEN
ASM_SIMP_TAC[ETA_AX; GSYM path;
VALID_PATH_IMP_PATH;
VALID_PATH_VECTOR_POLYNOMIAL_FUNCTION]];
ALL_TAC] THEN
SUBGOAL_THEN
`!a b. segment[a,b]
SUBSET u
==>
path_integral (linepath(a,b)) h =
path_integral g (\w.
path_integral (linepath (a,b)) (\z. d z w))`
ASSUME_TAC THENL
[ALL_TAC;
MAP_EVERY X_GEN_TAC [`a:complex`; `b:complex`; `c:complex`] THEN
DISCH_TAC THEN
SUBGOAL_THEN
`segment[a:complex,b]
SUBSET u /\
segment[b,c]
SUBSET u /\ segment[c,a]
SUBSET u`
STRIP_ASSUME_TAC THENL
[ASM_MESON_TAC[
SEGMENTS_SUBSET_CONVEX_HULL;
SUBSET_TRANS]; ALL_TAC] THEN
ASM_SIMP_TAC[] THEN
ASM_SIMP_TAC[GSYM
PATH_INTEGRAL_ADD;
PATH_INTEGRABLE_ADD] THEN
MATCH_MP_TAC
PATH_INTEGRAL_EQ_0 THEN
X_GEN_TAC `w:complex` THEN REWRITE_TAC[] THEN DISCH_TAC THEN
SUBGOAL_THEN `(w:complex)
IN u` ASSUME_TAC THENL
[ASM SET_TAC[]; ALL_TAC] THEN
ASM_SIMP_TAC[GSYM
PATH_INTEGRAL_JOIN;
VALID_PATH_LINEPATH;
VALID_PATH_JOIN;
PATHSTART_JOIN;
PATH_INTEGRABLE_JOIN;
PATHSTART_LINEPATH;
PATHFINISH_LINEPATH] THEN
MATCH_MP_TAC
PATH_INTEGRAL_UNIQUE THEN
MATCH_MP_TAC
CAUCHY_THEOREM_TRIANGLE THEN
MATCH_MP_TAC
HOLOMORPHIC_ON_SUBSET THEN EXISTS_TAC `u:complex->bool` THEN
ASM_SIMP_TAC[] THEN ASM SET_TAC[]] THEN
MAP_EVERY X_GEN_TAC [`a:complex`; `b:complex`] THEN DISCH_TAC THEN
MATCH_MP_TAC
EQ_TRANS THEN
EXISTS_TAC `
path_integral (linepath(a,b)) (\z.
path_integral g (d z))` THEN
CONJ_TAC THENL
[MATCH_MP_TAC
PATH_INTEGRAL_EQ THEN
REWRITE_TAC[
PATH_IMAGE_LINEPATH] THEN
REPEAT STRIP_TAC THEN CONV_TAC SYM_CONV THEN
MATCH_MP_TAC
PATH_INTEGRAL_UNIQUE THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_MESON_TAC[
SUBSET];
MATCH_MP_TAC(REWRITE_RULE[
PCROSS]
PATH_INTEGRAL_SWAP) THEN
REWRITE_TAC[
VALID_PATH_LINEPATH;
VECTOR_DERIVATIVE_LINEPATH_AT;
CONTINUOUS_ON_CONST] THEN
FIRST_ASSUM(fun th -> REWRITE_TAC
[MATCH_MP
HAS_VECTOR_DERIVATIVE_UNIQUE_AT (SPEC_ALL th)]) THEN
ASM_SIMP_TAC[ETA_AX;
CONTINUOUS_VECTOR_POLYNOMIAL_FUNCTION;
CONTINUOUS_AT_IMP_CONTINUOUS_ON] THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[
IMP_CONJ]
CONTINUOUS_ON_SUBSET)) THEN
REWRITE_TAC[
SUBSET;
FORALL_IN_GSPEC;
IN_ELIM_PASTECART_THM] THEN
REWRITE_TAC[
PATH_IMAGE_LINEPATH] THEN ASM SET_TAC[]]);;
let SECOND_CARTAN_THM_DIM_1 = prove
(`!g f r.
&0 < r /\
g
holomorphic_on ball(Cx(&0),r) /\
(!z. z
IN ball(Cx(&0),r) ==> g z
IN ball(Cx(&0),r)) /\
g(Cx(&0)) = Cx(&0) /\
f
holomorphic_on ball(Cx(&0),r) /\
(!z. z
IN ball(Cx(&0),r) ==> f z
IN ball(Cx(&0),r)) /\
f (Cx(&0)) = Cx(&0) /\
(!z. z
IN ball(Cx(&0),r) ==> g (f z) = z) /\
(!z. z
IN ball(Cx(&0),r) ==> f (g z) = z)
==> ?t. !z. z
IN ball(Cx(&0),r) ==> g z = cexp(ii * Cx t) * z`,
let COMPLEX_DERIVATIVE_LEFT_INVERSE = prove
(`!s t f g w.
open s /\ open t /\
(!z. z IN s ==> f z IN t) /\ f holomorphic_on s /\
(!z. z IN t ==> g z IN s) /\ g holomorphic_on t /\
(!z. z IN s ==> g (f z) = z) /\ w IN s
==> complex_derivative f w * complex_derivative g (f w) = Cx(&1)`,
REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC [COMPLEX_MUL_SYM] THEN
SUBGOAL_THEN `complex_derivative g (f w) * complex_derivative f w =
complex_derivative (g o f) w ` SUBST1_TAC THENL
[ASM_MESON_TAC [HOLOMORPHIC_ON_IMP_DIFFERENTIABLE_AT;
COMPLEX_DERIVATIVE_CHAIN];
EQ_TRANS_TAC `complex_derivative (\u. u) w` THENL
[MATCH_MP_TAC COMPLEX_DERIVATIVE_TRANSFORM_WITHIN_OPEN THEN
EXISTS_TAC `s:complex->bool` THEN
ASM_SIMP_TAC[HOLOMORPHIC_ON_ID;o_THM] THEN
ASM_MESON_TAC [HOLOMORPHIC_ON_COMPOSE_GEN];
ASM_SIMP_TAC[COMPLEX_DERIVATIVE_ID]]]) in
let LEMMA_1 = prove
(`!s f.
open s /\ connected s /\ f holomorphic_on s /\ Cx(&0) IN s /\
(!u z. norm u = &1 /\ z IN s ==> u * z IN s) /\
(!u z. norm u = &1 /\ z IN s ==> f (u * z) = u * f z)
==> ?c. !z. z IN s ==> f z = c * z`,
REPEAT STRIP_TAC THEN ABBREV_TAC `c = complex_derivative f (Cx(&0))` THEN
EXISTS_TAC `c : complex` THEN
SUBGOAL_THEN `f(Cx(&0)) = Cx(&0)` ASSUME_TAC THENL
[FIRST_X_ASSUM (MP_TAC o SPECL [`--Cx(&1)`;`Cx(&0)`]) THEN
ASM_REWRITE_TAC [NORM_NEG; COMPLEX_NORM_NUM; COMPLEX_MUL_RZERO] THEN
CONV_TAC COMPLEX_RING; ALL_TAC] THEN
SUBGOAL_THEN
`!n u z.
norm u = &1 /\ z IN s ==>
u pow n * higher_complex_derivative n f (u * z) =
u * higher_complex_derivative n f z`
ASSUME_TAC THENL
[REPEAT STRIP_TAC THEN
EQ_TRANS_TAC `higher_complex_derivative n (\w. f (u * w)) z` THENL
[MATCH_MP_TAC EQ_SYM THEN
MATCH_MP_TAC HIGHER_COMPLEX_DERIVATIVE_COMPOSE_LINEAR THEN
EXISTS_TAC `s:complex->bool` THEN EXISTS_TAC `s:complex->bool` THEN
ASM_SIMP_TAC[]; ALL_TAC] THEN
EQ_TRANS_TAC `higher_complex_derivative n (\w. u * f w) z` THENL
[MATCH_MP_TAC HIGHER_COMPLEX_DERIVATIVE_TRANSFORM_WITHIN_OPEN THEN
EXISTS_TAC `s:complex->bool` THEN ASM_SIMP_TAC[] THEN CONJ_TAC THENL
[MATCH_MP_TAC
(REWRITE_RULE [o_DEF]
(SPECL [`\w:complex. u*w`; `f:complex->complex`]
HOLOMORPHIC_ON_COMPOSE_GEN)) THEN
EXISTS_TAC `s:complex->bool` THEN
ASM_SIMP_TAC [HOLOMORPHIC_ON_LINEAR];
MATCH_MP_TAC
(REWRITE_RULE [o_DEF]
(SPECL [`f:complex->complex`; `\w:complex. u*w`]
HOLOMORPHIC_ON_COMPOSE_GEN)) THEN
EXISTS_TAC `(:complex)` THEN
ASM_REWRITE_TAC [HOLOMORPHIC_ON_LINEAR; IN_UNIV]];
POP_ASSUM MP_TAC THEN SPEC_TAC (`z:complex`,`z:complex`) THEN
SPEC_TAC (`n:num`,`n:num`) THEN INDUCT_TAC THEN
REWRITE_TAC [higher_complex_derivative] THEN GEN_TAC THEN
DISCH_TAC THEN EQ_TRANS_TAC
`complex_derivative (\w. u * higher_complex_derivative n f w) z`
THENL
[MATCH_MP_TAC COMPLEX_DERIVATIVE_TRANSFORM_WITHIN_OPEN THEN
EXISTS_TAC `s:complex->bool` THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL
[MATCH_MP_TAC HOLOMORPHIC_HIGHER_COMPLEX_DERIVATIVE THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC HOLOMORPHIC_ON_MUL THEN
ASM_REWRITE_TAC [HOLOMORPHIC_ON_CONST];
MATCH_MP_TAC HOLOMORPHIC_ON_MUL THEN
ASM_REWRITE_TAC [HOLOMORPHIC_ON_CONST; ETA_AX] THEN
MATCH_MP_TAC HOLOMORPHIC_HIGHER_COMPLEX_DERIVATIVE THEN
ASM_REWRITE_TAC[]];
MATCH_MP_TAC COMPLEX_DERIVATIVE_LMUL THEN
MATCH_MP_TAC HOLOMORPHIC_ON_IMP_DIFFERENTIABLE_AT THEN
ASM_MESON_TAC [HOLOMORPHIC_HIGHER_COMPLEX_DERIVATIVE]]];
SUBGOAL_THEN
`!n. 2 <= n ==> higher_complex_derivative n f (Cx(&0)) = Cx(&0)`
ASSUME_TAC THENL
[GEN_TAC THEN DISCH_TAC THEN SUBGOAL_THEN
`!n z. 2 <= n /\
(!u. norm u = &1 ==> u pow n * z = u * z) ==> z = Cx(&0)`
MATCH_MP_TAC THENL
[REPEAT STRIP_TAC THEN MATCH_MP_TAC
(COMPLEX_RING
`!u. ~(u pow n' = u) /\ u pow n' * z = u * z ==> z = Cx(&0)`) THEN
SUBGOAL_THEN `2 <= n' ==> ?u. norm u = &1 /\ ~(u pow n' = u)`
(fun th -> ASM_MESON_TAC [th]) THEN
STRUCT_CASES_TAC (SPEC `n':num` num_CASES) THEN
REWRITE_TAC
[ARITH_LE; ARITH_RULE `2 <= SUC n'' <=> 1 <= n''`; complex_pow] THEN
DISCH_TAC THEN MP_TAC (SPEC `n'':num` COMPLEX_NOT_ROOT_UNITY) THEN
ASM_REWRITE_TAC [] THEN STRIP_TAC THEN EXISTS_TAC `u:complex` THEN
ASM_REWRITE_TAC [] THEN POP_ASSUM MP_TAC THEN
REWRITE_TAC [CONTRAPOS_THM] THEN
SUBGOAL_THEN `~(u = Cx(&0))` MP_TAC THENL
[ASM_REWRITE_TAC [GSYM COMPLEX_NORM_ZERO; REAL_OF_NUM_EQ; ARITH_EQ];
CONV_TAC COMPLEX_FIELD];
EXISTS_TAC `n:num` THEN ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN
FIRST_X_ASSUM (MP_TAC o SPECL [`n:num`;`u:complex`;`Cx(&0)`]) THEN
ASM_REWRITE_TAC[COMPLEX_MUL_RZERO]];
REPEAT STRIP_TAC THEN MATCH_MP_TAC
(REWRITE_RULE []
(SPECL [`f:complex->complex`; `\z. c*z`; `Cx(&0)`;
`s:complex->bool`]
HOLOMORPHIC_FUN_EQ_ON_CONNECTED)) THEN
ASM_REWRITE_TAC [COMPLEX_MUL_RZERO; HOLOMORPHIC_ON_LINEAR;
HIGHER_COMPLEX_DERIVATIVE_LINEAR] THEN
GEN_TAC THEN FIRST_X_ASSUM (MP_TAC o SPEC `n:num`) THEN
STRUCT_CASES_TAC (ARITH_RULE `n = 0 \/ n = 1 \/ 2 <= n`) THEN
ASM_SIMP_TAC [higher_complex_derivative; ARITH_EQ; ARITH_LE; ONE] THEN
ASM_SIMP_TAC [ARITH_RULE `2 <= n ==> ~(n=0)`] THEN
ASM_SIMP_TAC [ARITH_RULE `2 <= n ==> ~(n=SUC 0)`]]]) in
let LEMMA_2 = prove
(`!r c. &0 < r /\ &0 <= c /\
(!x. &0 <= x /\ x < r ==> c * x < r)
==> c <= &1`,
REPEAT STRIP_TAC THEN REWRITE_TAC [GSYM REAL_NOT_LT] THEN STRIP_TAC THEN
FIRST_X_ASSUM (MP_TAC o SPEC `r * (c + &1) / (&2 * c)`) THEN
REWRITE_TAC [MESON [] `((a ==> b) ==> F) <=> (a /\ ~b)`] THEN
CONJ_TAC THENL
[CONJ_TAC THENL
[MATCH_MP_TAC REAL_LE_MUL THEN CONJ_TAC THENL
[ASM_REAL_ARITH_TAC; MATCH_MP_TAC REAL_LE_DIV THEN ASM_REAL_ARITH_TAC];
ALL_TAC] THEN
MATCH_MP_TAC REAL_LTE_TRANS THEN EXISTS_TAC `r * &1` THEN
CONJ_TAC THENL [ALL_TAC; REWRITE_TAC [REAL_MUL_RID; REAL_LE_REFL]] THEN
MATCH_MP_TAC REAL_LT_LMUL THEN ASM_REWRITE_TAC[] THEN
SUBGOAL_THEN `&0 < &2 * c` ASSUME_TAC THENL
[ASM_REAL_ARITH_TAC;
ASM_SIMP_TAC [REAL_LT_LDIV_EQ] THEN ASM_REAL_ARITH_TAC];
REWRITE_TAC [REAL_NOT_LT] THEN
ONCE_REWRITE_TAC [REAL_RING `!a b c:real. a * b * c = b * a * c`] THEN
MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `r * &1` THEN CONJ_TAC THENL
[REWRITE_TAC [REAL_MUL_RID; REAL_LE_REFL]; ALL_TAC] THEN
MATCH_MP_TAC REAL_LE_LMUL THEN CONJ_TAC THENL
[ASM_REAL_ARITH_TAC; ALL_TAC] THEN
SUBGOAL_THEN `&0 < &2 * c` ASSUME_TAC THENL
[ASM_REAL_ARITH_TAC; ALL_TAC] THEN
ASM_SIMP_TAC [REAL_ARITH `&0 < c ==> a * b / c = (a * b) / c`] THEN
SUBGOAL_THEN `(c * (c + &1)) / (&2 * c) = (c + &1) / &2`
SUBST1_TAC THENL
[ASM_SIMP_TAC [RAT_LEMMA5; REAL_ARITH `&0 < &2`] THEN
ASM_REAL_ARITH_TAC;
ASM_REAL_ARITH_TAC]]) in
REPEAT STRIP_TAC THEN SUBGOAL_THEN
`!u z. norm u = &1 /\ z IN ball(Cx(&0),r) ==> u * g z = g (u * z)`
ASSUME_TAC THENL
[REPEAT STRIP_TAC THEN SUBGOAL_THEN `~(u = Cx(&0))` ASSUME_TAC THENL
[ASM_REWRITE_TAC[GSYM COMPLEX_NORM_NZ] THEN REAL_ARITH_TAC; ALL_TAC] THEN
SUBGOAL_THEN `!w. w IN ball(Cx(&0),r) ==> f (u * g w) / u = w`
ASSUME_TAC THENL
[REPEAT STRIP_TAC THEN MATCH_MP_TAC FIRST_CARTAN_THM_DIM_1 THEN
EXISTS_TAC `ball(Cx(&0),r)` THEN EXISTS_TAC `Cx(&0)` THEN
ASM_REWRITE_TAC [OPEN_BALL;CONNECTED_BALL;BOUNDED_BALL;
COMPLEX_MUL_RZERO; CENTRE_IN_BALL] THEN
ASSERT_TAC `!z. norm (u * z) = norm z` THENL
[ASM_REWRITE_TAC [COMPLEX_NORM_MUL; REAL_MUL_LID]; ALL_TAC] THEN
ASSERT_TAC `!z. z IN ball(Cx(&0),r) ==> u * z IN ball(Cx(&0),r)` THENL
[ASM_REWRITE_TAC [COMPLEX_IN_BALL_0]; ALL_TAC] THEN
ASSERT_TAC `!z. z IN ball(Cx(&0),r) ==> z / u IN ball(Cx(&0),r)` THENL
[ASM_REWRITE_TAC [COMPLEX_IN_BALL_0; COMPLEX_NORM_DIV; REAL_DIV_1];
ALL_TAC] THEN
CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN CONJ_TAC THENL
[MATCH_MP_TAC HOLOMORPHIC_ON_DIV THEN CONJ_TAC THENL
[ALL_TAC; ASM_REWRITE_TAC[HOLOMORPHIC_ON_CONST]] THEN
SUBGOAL_THEN `(\w:complex. f (u * g w) : complex) = f o (\w. u * g w)`
SUBST1_TAC THENL
[REWRITE_TAC [o_DEF]; MATCH_MP_TAC HOLOMORPHIC_ON_COMPOSE_GEN] THEN
EXISTS_TAC `ball(Cx(&0),r)` THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL
[MATCH_MP_TAC HOLOMORPHIC_ON_MUL THEN
ASM_REWRITE_TAC[HOLOMORPHIC_ON_CONST];
ASM_SIMP_TAC[]];
ALL_TAC] THEN
CONJ_TAC THENL
[REWRITE_TAC [complex_div; COMPLEX_MUL_LZERO]; ALL_TAC] THEN
SUBGOAL_THEN `Cx(&1) = u / u` SUBST1_TAC THENL
[ASM_SIMP_TAC [COMPLEX_DIV_REFL]; ALL_TAC] THEN
MATCH_MP_TAC HAS_COMPLEX_DERIVATIVE_DERIVATIVE THEN
MATCH_MP_TAC HAS_COMPLEX_DERIVATIVE_CDIV_AT THEN
SUBGOAL_THEN `(\w:complex. f (u * g w) : complex) = f o (\w. u * g w)`
SUBST1_TAC THENL [REWRITE_TAC [o_DEF]; ALL_TAC] THEN
SUBGOAL_THEN
`((\w. f (u * g w)) has_complex_derivative
complex_derivative f (u * g(Cx(&0))) *
(u * complex_derivative g (Cx(&0))))
(at (Cx(&0)))` MP_TAC THENL
[MATCH_MP_TAC (REWRITE_RULE [o_DEF]
(SPECL [`\w:complex. u * g(w):complex`; `f:complex->complex`]
COMPLEX_DIFF_CHAIN_AT)) THEN CONJ_TAC THENL
[MATCH_MP_TAC HAS_COMPLEX_DERIVATIVE_LMUL_AT THEN
REWRITE_TAC [HAS_COMPLEX_DERIVATIVE_DIFFERENTIABLE] THEN
MATCH_MP_TAC HOLOMORPHIC_ON_IMP_DIFFERENTIABLE_AT THEN
EXISTS_TAC `ball(Cx(&0),r)` THEN
ASM_REWRITE_TAC[OPEN_BALL; CENTRE_IN_BALL];
REWRITE_TAC [HAS_COMPLEX_DERIVATIVE_DIFFERENTIABLE] THEN
MATCH_MP_TAC HOLOMORPHIC_ON_IMP_DIFFERENTIABLE_AT THEN
EXISTS_TAC `ball(Cx(&0),r)` THEN
ASM_REWRITE_TAC[OPEN_BALL; CENTRE_IN_BALL; COMPLEX_MUL_RZERO]];
SUBGOAL_THEN
`complex_derivative f (u * g (Cx(&0))) *
(u * complex_derivative g (Cx(&0))) = u`
SUBST1_TAC THENL
[ALL_TAC; REWRITE_TAC[o_DEF]] THEN
ABBREV_TAC `g' = complex_derivative g (Cx(&0))` THEN
ABBREV_TAC `f' = complex_derivative f (Cx(&0))` THEN
SUBGOAL_THEN `f' * g' = Cx(&1)` ASSUME_TAC THENL
[EXPAND_TAC "g'" THEN EXPAND_TAC "f'" THEN
SUBGOAL_THEN `complex_derivative g (Cx(&0)) =
complex_derivative g (f (Cx(&0)))` SUBST1_TAC THENL
[ASM_REWRITE_TAC [];
MATCH_MP_TAC COMPLEX_DERIVATIVE_LEFT_INVERSE THEN
EXISTS_TAC `ball(Cx(&0),r)` THEN EXISTS_TAC `ball(Cx(&0),r)` THEN
ASM_REWRITE_TAC [OPEN_BALL; CENTRE_IN_BALL]];
ASM_REWRITE_TAC [COMPLEX_MUL_RZERO] THEN
POP_ASSUM MP_TAC THEN CONV_TAC COMPLEX_RING]];
SUBGOAL_THEN `f(u*g(z)) = f (g (u * z)) : complex` MP_TAC THENL
[MATCH_MP_TAC EQ_TRANS THEN EXISTS_TAC `u * z:complex` THEN CONJ_TAC THENL
[SUBGOAL_THEN `!x y:complex. x / u = y ==> x = u * y` MATCH_MP_TAC THENL
[REWRITE_TAC [complex_div] THEN GEN_TAC THEN GEN_TAC THEN
DISCH_THEN (SUBST1_TAC o GSYM) THEN
SUBGOAL_THEN `x = (inv u * u) * x` MP_TAC THENL
[ASM_SIMP_TAC [COMPLEX_MUL_LINV; COMPLEX_MUL_LID];
REWRITE_TAC [COMPLEX_MUL_AC]];
POP_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC []];
MATCH_MP_TAC EQ_SYM THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
ASM_REWRITE_TAC [COMPLEX_IN_BALL_0; COMPLEX_NORM_MUL; REAL_MUL_LID] THEN
ASM_REWRITE_TAC [GSYM COMPLEX_IN_BALL_0]];
DISCH_TAC THEN SUBGOAL_THEN
`g (f (u * g z)) = g (f (g (u * z : complex))) : complex` MP_TAC THENL
[POP_ASSUM SUBST1_TAC THEN REWRITE_TAC [];
SUBGOAL_THEN `u * g z IN ball (Cx(&0),r) /\ u * z IN ball(Cx(&0),r)`
MP_TAC THENL
[ASM_REWRITE_TAC [COMPLEX_IN_BALL_0; COMPLEX_NORM_MUL; REAL_MUL_LID] THEN
REWRITE_TAC [GSYM COMPLEX_IN_BALL_0] THEN ASM_SIMP_TAC[];
ASM_SIMP_TAC[]]]]];
SUBGOAL_THEN `?c. !z. z IN ball(Cx(&0),r) ==> g z = c * z`
STRIP_ASSUME_TAC THENL
[MATCH_MP_TAC LEMMA_1 THEN
ASM_SIMP_TAC [OPEN_BALL; CONNECTED_BALL; CENTRE_IN_BALL] THEN
SIMP_TAC [COMPLEX_IN_BALL_0; COMPLEX_NORM_MUL; REAL_MUL_LID];
ALL_TAC] THEN
SUBGOAL_THEN `norm (c:complex) = &1` ASSUME_TAC THENL
[ALL_TAC; ASM_MESON_TAC [COMPLEX_NORM_EQ_1_CEXP]] THEN
SUBGOAL_THEN `~(norm (c:complex) = &0)` ASSUME_TAC THENL
[REWRITE_TAC [COMPLEX_NORM_ZERO] THEN STRIP_TAC THEN
SUBGOAL_THEN `Cx(&0) = Cx(r / &2)` MP_TAC THENL
[ALL_TAC; REWRITE_TAC [CX_INJ] THEN ASM_REAL_ARITH_TAC] THEN
SUBGOAL_THEN `Cx(r / &2) IN ball(Cx(&0),r)` ASSUME_TAC THENL
[REWRITE_TAC [COMPLEX_IN_BALL_0; CX_DIV; COMPLEX_NORM_DIV;
COMPLEX_NORM_NUM] THEN
REWRITE_TAC [COMPLEX_NORM_CX] THEN ASM_REAL_ARITH_TAC;
EQ_TRANS_TAC `g (f (Cx(r / &2)):complex):complex` THENL
[EQ_TRANS_TAC `c * (f (Cx(r / &2)):complex)` THENL
[ASM_REWRITE_TAC [COMPLEX_MUL_LZERO]; ASM_MESON_TAC[]];
ASM_MESON_TAC[]]];
ALL_TAC] THEN SUBGOAL_THEN `&0 < norm (c:complex)` ASSUME_TAC THENL
[POP_ASSUM MP_TAC THEN CONV_TAC NORM_ARITH; ALL_TAC] THEN
REWRITE_TAC [GSYM REAL_LE_ANTISYM] THEN CONJ_TAC THENL
[MATCH_MP_TAC LEMMA_2 THEN EXISTS_TAC `r : real` THEN
ASM_REWRITE_TAC [NORM_POS_LE] THEN GEN_TAC THEN STRIP_TAC THEN
ABBREV_TAC `p = Cx x` THEN
SUBGOAL_THEN `x = norm (p:complex)` SUBST_ALL_TAC THENL
[EXPAND_TAC "p" THEN REWRITE_TAC [COMPLEX_NORM_CX] THEN
ASM_REAL_ARITH_TAC;
REWRITE_TAC [GSYM COMPLEX_NORM_MUL] THEN
SUBGOAL_THEN `c * p = g p` SUBST1_TAC THENL
[ALL_TAC; ASM_MESON_TAC [COMPLEX_IN_BALL_0]] THEN
FIRST_X_ASSUM (MATCH_MP_TAC o GSYM) THEN
ASM_MESON_TAC [COMPLEX_IN_BALL_0]];
ALL_TAC] THEN
SUBST1_TAC (GSYM (SPEC `norm (c:complex)` REAL_INV_INV)) THEN
MATCH_MP_TAC REAL_INV_1_LE THEN CONJ_TAC THENL
[ASM_MESON_TAC [REAL_LT_INV]; ALL_TAC] THEN
MATCH_MP_TAC LEMMA_2 THEN EXISTS_TAC `r:real` THEN ASM_REWRITE_TAC [] THEN
CONJ_TAC THENL
[MATCH_MP_TAC REAL_LE_INV THEN ASM_REAL_ARITH_TAC; ALL_TAC] THEN
GEN_TAC THEN STRIP_TAC THEN
SUBGOAL_THEN `x = norm (g (f (Cx x):complex):complex)` SUBST1_TAC THENL
[SUBGOAL_THEN `g (f (Cx x):complex) = Cx x` SUBST1_TAC THENL
[FIRST_X_ASSUM MATCH_MP_TAC THEN
REWRITE_TAC [COMPLEX_IN_BALL_0; COMPLEX_NORM_CX] THEN
ASM_REAL_ARITH_TAC;
REWRITE_TAC [COMPLEX_NORM_CX] THEN ASM_REAL_ARITH_TAC];
SUBGOAL_THEN `g (f (Cx x):complex) = c * f (Cx x) : complex`
SUBST1_TAC THENL
[FIRST_X_ASSUM MATCH_MP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
REWRITE_TAC [COMPLEX_IN_BALL_0; COMPLEX_NORM_CX] THEN
ASM_REAL_ARITH_TAC;
REWRITE_TAC [COMPLEX_NORM_MUL; REAL_MUL_ASSOC] THEN
ASM_SIMP_TAC [REAL_MUL_LINV; REAL_MUL_LID; GSYM COMPLEX_IN_BALL_0] THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN
REWRITE_TAC [COMPLEX_IN_BALL_0; COMPLEX_NORM_CX] THEN
ASM_REAL_ARITH_TAC]]]);;
let ISOLATED_ZEROS = prove
(`!f a z w.
open a /\ connected a /\ f
holomorphic_on a /\ z
IN a /\ f z = Cx(&0) /\
w
IN a /\ ~(f w = Cx(&0))
==> (?r. &0 < r /\ ball(z,r)
SUBSET a /\
(!w. w
IN ball(z,r) /\ ~(w=z) ==> ~(f w = Cx(&0))))`,
REPEAT STRIP_TAC THEN ASSERT_TAC `?k.
~(
higher_complex_derivative k f z = Cx(&0)) /\
(!n. n < k ==>
higher_complex_derivative n f z = Cx(&0))` THENL
[EXISTS_TAC `minimal n. (~(
higher_complex_derivative n f z = Cx(&0)))`
THEN SUBGOAL_THEN `?k'. ~(
higher_complex_derivative k' f z = Cx(&0))`
(fun th-> ASM_MESON_TAC[th;
MINIMAL]) THEN REWRITE_TAC[GSYM
NOT_FORALL_THM]
THEN STRIP_TAC THEN ASM_MESON_TAC[
HOLOMORPHIC_FUN_EQ_0_ON_CONNECTED];
ALL_TAC] THEN SUBGOAL_THEN `~(k = 0)`ASSUME_TAC THENL
[STRIP_TAC THEN MP_TAC(ASSUME `~(
higher_complex_derivative k f z = Cx(&0))`)
THEN ASM_MESON_TAC[
higher_complex_derivative];
STRIP_ASSUME_TAC (MESON [
OPEN_CONTAINS_BALL;ASSUME `open (a:complex->bool)`;
ASSUME `z:complex
IN a`] `?s. &0 < s /\ ball (z:complex,s)
SUBSET a`)
THEN ASSUME_TAC (MESON [
HOLOMORPHIC_POWER_SERIES;
ASSUME `f
holomorphic_on a`;ASSUME `ball (z:complex,s)
SUBSET a`;
HOLOMORPHIC_ON_SUBSET] `!w:complex. w
IN ball(z,s) ==>
((\n.
higher_complex_derivative n f z / Cx(&(
FACT n))*(w -z) pow n) sums f w)
(from 0)`) THEN ASSERT_TAC `?g:complex->complex. !x:complex.
x
IN ball(z,s) ==>
(((\n.
higher_complex_derivative n f z / Cx(&(
FACT n)) *
(x - z) pow (n-k))) sums g x) (from k)` THENL
[EXISTS_TAC `\x:complex. lim sequentially
(\m. vsum (k..m) (\n.
higher_complex_derivative n f z / Cx(&(
FACT n)) *
(x - z) pow (n-k)))` THEN GEN_TAC THEN DISCH_TAC THEN
SUBGOAL_THEN `!m. k..m = (0..m)
INTER from k` ASSUME_TAC THENL
[REWRITE_TAC[
EXTENSION;
IN_FROM;
IN_INTER;
IN_ELIM_THM;
IN_NUMSEG] THEN
ARITH_TAC;ASM_REWRITE_TAC[] THEN REWRITE_TAC
[SET_RULE `!m. (0..m)
INTER from k = from k
INTER (0..m)`;
SUMS_LIM]] THEN
ASM_CASES_TAC `x:complex = z` THENL
[ASM_REWRITE_TAC[
COMPLEX_SUB_REFL;summable] THEN
EXISTS_TAC `
higher_complex_derivative k f z / Cx(&(
FACT k))` THEN
MATCH_MP_TAC
SUMS_EQ THEN EXISTS_TAC `\n. if n = k then
higher_complex_derivative k f z / Cx(&(
FACT k)) else Cx(&0)`
THEN CONJ_TAC THENL [REWRITE_TAC [
IN_FROM] THEN GEN_TAC THEN DISCH_TAC
THEN COND_CASES_TAC THENL
[ASM_REWRITE_TAC[
COMPLEX_POW_ZERO;
SUB_REFL;
COMPLEX_MUL_RID];
ASM_SIMP_TAC[
COMPLEX_POW_ZERO; ARITH_RULE `k <= x' /\ ~(x' = k) ==>
~(x' - k = 0)`;
COMPLEX_MUL_RZERO]]; MATCH_MP_TAC
SERIES_VSUM THEN
EXISTS_TAC `{k:num}` THEN SIMP_TAC [
FINITE_SING;from;
IN_SING;
COMPLEX_VEC_0;
VSUM_SING] THEN SET_TAC[
LE_REFL]];
MATCH_MP_TAC
SUMMABLE_EQ THEN EXISTS_TAC
`\n.
higher_complex_derivative n f z / Cx(&(
FACT n)) *
(x - z) pow n / (x-z) pow k` THEN CONJ_TAC THENL [REWRITE_TAC [
IN_FROM]
THEN GEN_TAC THEN DISCH_TAC THEN SUBGOAL_THEN `(x:complex - z) pow (x' - k)
= (x - z) pow x' / (x - z) pow k` (fun th->
REWRITE_TAC[th;
COMPLEX_EQ_MUL_LCANCEL]) THEN MATCH_MP_TAC
COMPLEX_DIV_POW THEN ASM_SIMP_TAC [
COMPLEX_SUB_0];
SUBGOAL_THEN `(\n.
higher_complex_derivative n f z / Cx(&(
FACT n)) *
(x - z) pow n / (x - z) pow k) = (\n. (
higher_complex_derivative n f z /
Cx(&(
FACT n)) *(x - z) pow n) / (x - z) pow k) ` SUBST1_TAC
THENL [REWRITE_TAC [
FUN_EQ_THM] THEN GEN_TAC THEN CONV_TAC COMPLEX_FIELD;
MATCH_MP_TAC
SUMMABLE_COMPLEX_DIV THEN MATCH_MP_TAC
SUMMABLE_FROM_ELSEWHERE
THEN EXISTS_TAC `0` THEN ASM_MESON_TAC[summable]]]];ALL_TAC] THEN
ASSERT_TAC `~(g (z:complex) = Cx(&0)) /\
(!x. x
IN ball(z,s) ==> f x = (x - z) pow k * g(x))` THENL
[CONJ_TAC THENL [MATCH_MP_TAC
(COMPLEX_FIELD `!x y:complex. x = y /\ ~(y= Cx(&0)) ==> ~(x=Cx(&0))`) THEN
EXISTS_TAC `
higher_complex_derivative k f z / Cx(&(
FACT k))` THEN
CONJ_TAC THENL [ONCE_REWRITE_TAC [GSYM
COMPLEX_SUB_0] THEN
MATCH_MP_TAC
SERIES_UNIQUE THEN EXISTS_TAC
`(\n.
higher_complex_derivative n f z / Cx(&(
FACT n)) *
Cx(&0) pow (n-k))` THEN EXISTS_TAC `from (k +1)` THEN
CONJ_TAC THENL [SUBST1_TAC (MESON [
VSUM_SING_NUMSEG]
`
higher_complex_derivative k f z / Cx(&(
FACT k)) =
vsum (k..k) (\n.
higher_complex_derivative n f z / Cx(&(
FACT n))) `)
THEN SUBGOAL_THEN `vsum (k..k) (\n.
higher_complex_derivative n f z
/ Cx(&(
FACT n))) = vsum (k..((k+1)-1)) (\n.
higher_complex_derivative n f z
/ Cx(&(
FACT n)) * Cx(&0) pow (n - k))` SUBST1_TAC THENL [
REWRITE_TAC[
VSUM_SING_NUMSEG;
COMPLEX_POW_ZERO;
SUB_REFL;
COMPLEX_MUL_RID;
ARITH_RULE `((k:num) + 1) -1 = k`];
MATCH_MP_TAC
SUMS_OFFSET THEN
ASM_REWRITE_TAC[ARITH_RULE `k:num <= k+1 /\ 0 < k+1`]
THEN POP_ASSUM (MP_TAC o SPEC `z:complex`) THEN
ASM_REWRITE_TAC[
CENTRE_IN_BALL;
COMPLEX_SUB_REFL]];MATCH_MP_TAC
SUMS_COMPLEX_0 THEN GEN_TAC THEN SIMP_TAC [
IN_FROM;
COMPLEX_POW_ZERO;
ARITH_RULE `k + 1 <= n <=> ~(n-k= 0)`;
COMPLEX_MUL_RZERO]];
MATCH_MP_TAC (COMPLEX_FIELD `!x y. ~(x = Cx(&0)) /\ ~(y = Cx(&0))
==> ~(x / y = Cx(&0))`) THEN ASM_REWRITE_TAC[GSYM
COMPLEX_NORM_ZERO] THEN
SUBST1_TAC (MESON [
COMPLEX_NORM_CX]
`norm (Cx(&(
FACT k))) = abs ((&(
FACT k)))`) THEN
SIMP_TAC [
REAL_ABS_ZERO;
FACT_LT;
REAL_OF_NUM_LT;
REAL_LT_IMP_NZ]]; ALL_TAC]
THEN GEN_TAC THEN DISCH_TAC THEN MATCH_MP_TAC
SERIES_UNIQUE THEN
EXISTS_TAC `(\n.
higher_complex_derivative n f z / Cx(&(
FACT n)) *
(x - z) pow n)`THEN EXISTS_TAC `(from 0)` THEN
CONJ_TAC THENL [FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[];
ASM_CASES_TAC `x:complex = z` THENL [
ASM_REWRITE_TAC[
COMPLEX_SUB_REFL] THEN MATCH_MP_TAC
SUMS_EQ THEN
EXISTS_TAC `\n:num. Cx(&0)` THEN CONJ_TAC THENL
[REWRITE_TAC[
IN_FROM;
COMPLEX_POW_ZERO] THEN X_GEN_TAC `n:num` THEN
DISCH_TAC THEN COND_CASES_TAC THENL [
ASM_REWRITE_TAC[
higher_complex_derivative] THEN CONV_TAC COMPLEX_FIELD;
REWRITE_TAC[
COMPLEX_MUL_RZERO]];
ASM_REWRITE_TAC[
COMPLEX_POW_ZERO;
COMPLEX_MUL_LZERO] THEN
ASM_REWRITE_TAC[
SERIES_0;GSYM
COMPLEX_VEC_0]];ALL_TAC] THEN
MATCH_MP_TAC
SUMS_EQ THEN EXISTS_TAC `\n.(x-z) pow k *
higher_complex_derivative n f z / Cx(&(
FACT n)) *(x - z) pow (n - k)`
THEN CONJ_TAC THENL [REWRITE_TAC[
IN_FROM] THEN X_GEN_TAC `n:num`
THEN DISCH_TAC THEN ASM_CASES_TAC `n:num < k` THENL [ASM_SIMP_TAC[]
THEN CONV_TAC COMPLEX_FIELD;
SUBGOAL_THEN `(x:complex-z) pow (n-k) = (x-z) pow n / (x-z) pow k`
SUBST1_TAC THENL [MATCH_MP_TAC
COMPLEX_DIV_POW THEN
ASM_SIMP_TAC[
COMPLEX_SUB_0; ARITH_RULE `~(n:num < k) ==> k <= n`];
SUBST1_TAC (COMPLEX_FIELD `(x - z) pow k *
higher_complex_derivative n f z / Cx(&(
FACT n)) *
(x - z) pow n / (x - z) pow k =
higher_complex_derivative n f z / Cx(&(
FACT n)) * (x-z) pow k *
(x - z) pow n / (x - z) pow k`) THEN MESON_TAC [ASSUME `~(x:complex = z)`;
COMPLEX_DIV_LMUL;
COMPLEX_SUB_0;
COMPLEX_POW_EQ_0]]];
MATCH_MP_TAC
SERIES_COMPLEX_LMUL THEN SUBST1_TAC
(MESON [
COMPLEX_ADD_RID] `(g:complex->complex) x = g x + Cx(&0)`) THEN
SUBGOAL_THEN `Cx(&0) = vsum (0.. (k-1))
(\n.
higher_complex_derivative n f z / Cx(&(
FACT n)) * (x - z) pow (n - k))`
SUBST1_TAC THENL [ONCE_REWRITE_TAC[
EQ_SYM_EQ] THEN
REWRITE_TAC [GSYM
COMPLEX_VEC_0] THEN MATCH_MP_TAC
VSUM_EQ_0 THEN
REWRITE_TAC [
IN_NUMSEG] THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN
ASM_SIMP_TAC[ARITH_RULE ` ~(k = 0) /\ n <= k - 1 ==> n < k`] THEN
REWRITE_TAC[
COMPLEX_VEC_0] THEN CONV_TAC COMPLEX_FIELD;
MATCH_MP_TAC
SUMS_OFFSET_REV THEN
ASM_SIMP_TAC[ARITH_RULE `0 <= k /\ ~(k = 0) ==> 0 < k`;
LE_0]]]];ALL_TAC] THEN
ASSERT_TAC `?r. &0 < r /\ (!x:complex. dist (z,x) < r ==>
~((g:complex->complex) x = Cx(&0)))` THENL [
MATCH_MP_TAC
CONTINUOUS_ON_OPEN_AVOID THEN
EXISTS_TAC `ball(z:complex, s)` THEN
ASM_REWRITE_TAC[
OPEN_BALL;
CENTRE_IN_BALL]
THEN MATCH_MP_TAC
HOLOMORPHIC_ON_IMP_CONTINUOUS_ON THEN
MATCH_MP_TAC
ANALYTIC_IMP_HOLOMORPHIC THEN MATCH_MP_TAC
POWER_SERIES_ANALYTIC
THEN EXISTS_TAC `\n.
higher_complex_derivative (n+k) f z / Cx(&(
FACT (n+k)))`
THEN EXISTS_TAC `from 0` THEN REWRITE_TAC[] THEN GEN_TAC THEN DISCH_TAC
THEN REWRITE_TAC[
SERIES_FROM] THEN MATCH_MP_TAC
LIM_TRANSFORM THEN
EXISTS_TAC `(\n.vsum (k..(k+n))
(\n.
higher_complex_derivative n f z / Cx(&(
FACT n)) *(w' - z) pow (n-k)))`
THEN CONJ_TAC THENL [SIMP_TAC [
VSUM_OFFSET_0;ARITH_RULE
`!k n :num.(k + n) - k = n`; ARITH_RULE `!k n:num. k <= k + n`;
ADD_ASSOC;
ARITH_RULE `!k n :num.(n + k) - k = n`] THEN
SUBGOAL_THEN `(\x. vsum (0..x) (\i.
higher_complex_derivative (i + k)
f z / Cx(&(
FACT (i + k))) * (w' - z) pow i)
- vsum (0..x) (\n.
higher_complex_derivative (n + k) f z
/ Cx(&(
FACT (n + k))) * (w' - z) pow n)) = (\x. Cx(&0))`
(fun th-> SIMP_TAC[th;
COMPLEX_VEC_0;
LIM_CONST]) THEN
REWRITE_TAC[
FUN_EQ_THM] THEN GEN_TAC THEN REWRITE_TAC[
COMPLEX_SUB_0];
SUBGOAL_THEN `(\n. vsum (k..k + n)
(\n.
higher_complex_derivative n f z / Cx(&(
FACT n)) *(w' - z) pow (n - k)))
= (\n. vsum (k..n+k)(\n.
higher_complex_derivative n f z / Cx(&(
FACT n)) *
(w' - z) pow (n - k)))` SUBST1_TAC THENL [
REWRITE_TAC[
FUN_EQ_THM] THEN GEN_TAC THEN REWRITE_TAC[
ADD_SYM];
MP_TAC (ISPECL [`(\n. vsum (k..n)
(\n.
higher_complex_derivative n f z / Cx(&(
FACT n)) *
(w' - z) pow (n - k)))`;`(g:complex->complex) w'`;`k:num`]
SEQ_OFFSET) THEN ONCE_REWRITE_TAC[GSYM
SERIES_FROM] THEN ASM_SIMP_TAC[]]];
ALL_TAC] THEN EXISTS_TAC `min r s` THEN CONJ_TAC THENL
[MP_TAC (CONJ (ASSUME `&0 < r`) (ASSUME `&0 < s`)) THEN REAL_ARITH_TAC;
CONJ_TAC THENL [REWRITE_TAC[
real_min] THEN COND_CASES_TAC
THENL [MATCH_MP_TAC
SUBSET_TRANS THEN EXISTS_TAC `ball(z:complex,s)`
THEN ASM_REWRITE_TAC[ball] THEN SET_TAC[ASSUME `r:real <= s`;
REAL_LTE_TRANS];
ASM_REWRITE_TAC[]];GEN_TAC THEN STRIP_TAC THEN
SUBGOAL_THEN `(f:complex->complex) w' =
(w' - z) pow k * (g:complex->complex) w'` SUBST1_TAC
THENL [FIRST_ASSUM MATCH_MP_TAC THEN
MP_TAC (ASSUME `w':complex
IN ball (z,min r s)`) THEN REWRITE_TAC [
real_min]
THEN COND_CASES_TAC THENL [ASM_MESON_TAC[
IN_BALL;
REAL_LTE_TRANS];
REWRITE_TAC[]];SIMP_TAC [
COMPLEX_ENTIRE;DE_MORGAN_THM] THEN
CONJ_TAC THENL [REWRITE_TAC[
COMPLEX_POW_EQ_0;DE_MORGAN_THM]
THEN DISJ1_TAC THEN ASM_REWRITE_TAC [
COMPLEX_SUB_0];
FIRST_X_ASSUM MATCH_MP_TAC THEN MP_TAC (ASSUME `w':complex
IN
ball (z,min r s)`) THEN REWRITE_TAC [
real_min] THEN COND_CASES_TAC
THENL [REWRITE_TAC[
IN_BALL];
ASM_MESON_TAC[
REAL_NOT_LE;
IN_BALL;
REAL_LT_TRANS]]]]]]]);;
let OPEN_MAPPING_THM = prove
(`!a f.
open a /\ connected a /\ f
holomorphic_on a /\
~(?c:complex. !z:complex. z
IN a ==> f z = c)
==> (!u. open u /\ u
SUBSET a ==> open(
IMAGE f u))`,
let LEMMA_ZERO = prove
(`!f z r. f continuous_on cball(z,r) /\ f holomorphic_on ball(z,r) /\
&0 < r /\ (!w. norm(z-w) =r ==> norm(f z) < norm(f w))
==> (?w. w IN ball(z,r) /\ f w = Cx(&0))`,
REPEAT STRIP_TAC THEN SUBGOAL_THEN ` ((!x:complex. x IN ball(z,r) ==>
~((f:complex->complex) x = Cx(&0))) ==> F ) ==> ( ?w:complex. w IN ball(z,r)
/\ f w = Cx(&0))` MATCH_MP_TAC THENL [MESON_TAC[];
STRIP_TAC THEN SUBGOAL_THEN `&0 < norm ((f:complex->complex) z)` ASSUME_TAC
THENL [ASM_SIMP_TAC[COMPLEX_NORM_NZ; CENTRE_IN_BALL; SPEC `z:complex`
(ASSUME`!x:complex. x IN ball(z,r) ==> ~((f:complex->complex) x = Cx(&0))`)];
ALL_TAC] THEN SUBGOAL_THEN
`(!x:complex. x IN cball(z,r) ==> ~((f:complex->complex) x = Cx(&0)))`
ASSUME_TAC THENL [GEN_TAC THEN REWRITE_TAC [IN_CBALL;dist]
THEN REWRITE_TAC[REAL_ARITH `a <= b <=> a < b \/ a = b`] THEN
REWRITE_TAC [TAUT `((p \/ q) ==> r ) <=> ((p ==> r ) /\ (q ==> r))`] THEN
CONJ_TAC THENL [ASM_MESON_TAC[IN_BALL;dist];
DISCH_TAC THEN REWRITE_TAC[GSYM COMPLEX_NORM_ZERO] THEN MATCH_MP_TAC
REAL_LT_IMP_NZ THEN MATCH_MP_TAC REAL_LT_TRANS THEN EXISTS_TAC
`norm ((f:complex->complex) z)` THEN
ASM_SIMP_TAC [SPEC `z':complex` (ASSUME `!w:complex. norm (w - z) = r
==> norm ((f:complex->complex) z) < norm (f w)`)]];
ALL_TAC] THEN SUBGOAL_THEN `~(frontier(cball(z:complex,r))={})` ASSUME_TAC
THENL [REWRITE_TAC[FRONTIER_CBALL;sphere;dist] THEN SUBGOAL_THEN `?x:complex.
norm(z-x) = r` (fun th-> SET_TAC [MEMBER_NOT_EMPTY;th]) THEN EXISTS_TAC
`z + Cx r` THEN ASM_SIMP_TAC[COMPLEX_ADD_SUB2;NORM_NEG;COMPLEX_NORM_CX;
REAL_ABS_REFL;REAL_LT_IMP_LE];ALL_TAC] THEN
ABBREV_TAC `g = \z. inv ((f:complex->complex) z)` THEN ASSERT_TAC
`(g:complex->complex) continuous_on cball(z,r) /\ g holomorphic_on ball(z,r)`
THENL [CONJ_TAC THENL [EXPAND_TAC "g" THEN
REWRITE_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN] THEN GEN_TAC THEN DISCH_TAC
THEN MATCH_MP_TAC CONTINUOUS_COMPLEX_INV_WITHIN THEN ASM_MESON_TAC
[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN];EXPAND_TAC "g" THEN MATCH_MP_TAC
HOLOMORPHIC_ON_INV THEN ASM_REWRITE_TAC[]];ALL_TAC] THEN
SUBGOAL_THEN `?w:complex. w IN frontier(cball(z,r)) /\
(!x:complex. x IN frontier(cball(z,r)) ==>
norm ((f:complex->complex) w) <= norm (f x))`
STRIP_ASSUME_TAC THENL [MATCH_MP_TAC CONTINUOUS_ATTAINS_INF
THEN ASM_SIMP_TAC[COMPACT_FRONTIER;COMPACT_CBALL;CBALL_EQ_EMPTY;
REAL_ARITH `!r:real. &0 < r ==> ~(r < &0)` ] THEN
SUBGOAL_THEN `lift o (\x. norm ((f:complex->complex) x)) =
(lift o norm) o (\x. f x) ` SUBST1_TAC THENL
[REWRITE_TAC[o_DEF]; MATCH_MP_TAC CONTINUOUS_ON_COMPOSE
THEN CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_SUBSET THEN EXISTS_TAC
`cball(z:complex,r)` THEN ASM_REWRITE_TAC[ETA_AX] THEN
ASM_SIMP_TAC[SUBSET_TRANS;CLOSED_CBALL;FRONTIER_SUBSET_CLOSED];
ASM_MESON_TAC [CONTINUOUS_ON_LIFT_NORM; HOLOMORPHIC_ON_SUBSET;
HOLOMORPHIC_ON_IMP_CONTINUOUS_ON;SUBSET_TRANS;CLOSED_CBALL;
FRONTIER_SUBSET_CLOSED]]];ALL_TAC] THEN
SUBGOAL_THEN `?w:complex. norm (z-w) = r /\
norm ((f:complex->complex) w) <= norm (f z)`
(fun thm -> ASM_MESON_TAC[thm;REAL_NOT_LE])
THEN EXISTS_TAC `w:complex` THEN CONJ_TAC
THENL [MP_TAC (ASSUME `w:complex IN frontier (cball (z,r))`) THEN
REWRITE_TAC[FRONTIER_CBALL;sphere;dist] THEN SET_TAC[];ALL_TAC] THEN
SUBGOAL_THEN `&0 < norm ((f:complex->complex) w)` ASSUME_TAC THENL
[REWRITE_TAC[NORM_POS_LT;COMPLEX_VEC_0] THEN MATCH_MP_TAC (ASSUME `!x.
x:complex IN cball (z,r) ==> ~(f x = Cx(&0))`) THEN MATCH_MP_TAC
(SET_RULE `!x:complex u s. x IN u /\ u SUBSET s ==> x IN s `) THEN
EXISTS_TAC `frontier(cball(z:complex,r))` THEN
ASM_SIMP_TAC[CLOSED_CBALL;FRONTIER_SUBSET_CLOSED];ALL_TAC] THEN
SUBGOAL_THEN `inv (norm ((f:complex-> complex) w)) = &1/ (norm (f w))`
ASSUME_TAC THENL [MATCH_MP_TAC REAL_MUL_RINV_UNIQ THEN MATCH_MP_TAC
REAL_DIV_LMUL THEN ASM_REWRITE_TAC[COMPLEX_NORM_ZERO;GSYM COMPLEX_NORM_NZ];
ASSERT_TAC `?x:complex. x IN frontier(cball(z,r)) /\ (!y. y IN
frontier(cball(z,r)) ==> norm ((g:complex->complex) y) <= norm (g x))`
THENL [MATCH_MP_TAC CONTINUOUS_ATTAINS_SUP THEN
ASM_SIMP_TAC[COMPACT_FRONTIER;
COMPACT_CBALL;CBALL_EQ_EMPTY; REAL_ARITH `!r:real. &0 < r ==> ~(r < &0)`]
THEN SUBGOAL_THEN `lift o (\x. norm ((g:complex->complex) x)) =
(lift o norm) o (\x. g x) ` SUBST1_TAC
THENL [REWRITE_TAC[o_DEF]; MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN
CONJ_TAC THENL [MATCH_MP_TAC CONTINUOUS_ON_SUBSET THEN
EXISTS_TAC `cball(z:complex,r)` THEN ASM_REWRITE_TAC[ETA_AX]
THEN ASM_SIMP_TAC[SUBSET_TRANS;CLOSED_CBALL;
FRONTIER_SUBSET_CLOSED]; ASM_MESON_TAC [CONTINUOUS_ON_LIFT_NORM;
HOLOMORPHIC_ON_SUBSET; HOLOMORPHIC_ON_IMP_CONTINUOUS_ON;SUBSET_TRANS;
CLOSED_CBALL; FRONTIER_SUBSET_CLOSED]]];ALL_TAC] THEN SUBGOAL_THEN
`&0 < norm ((f:complex->complex) x)` ASSUME_TAC THENL
[REWRITE_TAC[NORM_POS_LT;COMPLEX_VEC_0] THEN MATCH_MP_TAC
(ASSUME `!x. x:complex IN cball (z,r) ==> ~(f x = Cx(&0))`)
THEN MATCH_MP_TAC (SET_RULE `!x:complex u s. x IN u /\ u SUBSET s
==> x IN s `) THEN EXISTS_TAC `frontier(cball(z:complex,r))`
THEN ASM_SIMP_TAC[CLOSED_CBALL;FRONTIER_SUBSET_CLOSED];
ABBREV_TAC `B = norm ((g:complex->complex) x)`
THEN SUBGOAL_THEN `norm (higher_complex_derivative 0 g z) <=
(&(FACT 0)) * B / (r pow 0) `
MP_TAC THENL[MATCH_MP_TAC CAUCHY_INEQUALITY THEN
ASM_REWRITE_TAC[] THEN MP_TAC
(ASSUME `!y:complex. y IN frontier (cball (z,r)) ==>
norm ((g:complex ->complex) y) <= B`)
THEN SIMP_TAC [FRONTIER_CBALL;sphere;dist] THEN SET_TAC[];
REWRITE_TAC [higher_complex_derivative;FACT;real_pow;
REAL_MUL_LID;REAL_DIV_1] THEN DISCH_TAC THEN SUBGOAL_THEN
`inv (norm ((f:complex->complex) z)) <=
inv (norm (f w)) ==> norm (f w) <= norm (f z)` MATCH_MP_TAC
THENL [SUBGOAL_THEN `inv (norm ((f:complex-> complex) z)) =
&1/ (norm (f z))` SUBST1_TAC
THENL [MATCH_MP_TAC REAL_MUL_RINV_UNIQ THEN MATCH_MP_TAC REAL_DIV_LMUL THEN
ASM_SIMP_TAC[REAL_ARITH `&0 < norm ((f:complex->complex) z) ==>
~(norm (f z) = &0) `]; ASM_REWRITE_TAC[] THEN DISCH_TAC THEN SUBST1_TAC
(REAL_ARITH `norm ((f:complex->complex) w)= &1 * norm (f w)`) THEN
SUBST1_TAC(REAL_ARITH `norm ((f:complex->complex) z)=
&1 * norm (f z)`) THEN POP_ASSUM
MP_TAC THEN MATCH_MP_TAC (TAUT `(p <=> q ) ==> ( p ==> q)`)
THEN MATCH_MP_TAC RAT_LEMMA4 THEN ASM_REWRITE_TAC[]];
REWRITE_TAC[GSYM COMPLEX_NORM_INV] THEN
SUBGOAL_THEN `inv ((f:complex->complex) z) = g z /\ inv (f w) = g w`
(fun thm -> REWRITE_TAC[thm])
THENL [ASM_MESON_TAC[];MATCH_MP_TAC (REAL_ARITH
`!x y z:real. x <= y /\ y = z ==> x <= z`) THEN EXISTS_TAC `B:real` THEN
ASM_REWRITE_TAC[GSYM REAL_LE_ANTISYM] THEN CONJ_TAC THENL [EXPAND_TAC "B"
THEN REWRITE_TAC[SYM (ASSUME`(\z. inv ((f:complex->complex) z)) =
g`);COMPLEX_NORM_INV] THEN SUBGOAL_THEN `inv (norm ((f:complex->complex) x))
= &1 / norm (f x)` (fun thm -> REWRITE_TAC[thm]) THENL [MATCH_MP_TAC
REAL_MUL_RINV_UNIQ THEN MATCH_MP_TAC REAL_DIV_LMUL THEN
ASM_SIMP_TAC[REAL_LT_IMP_NZ]; ASM_REWRITE_TAC[] THEN
MP_TAC (SPEC `x:complex`(ASSUME`!x:complex. x IN frontier (cball (z,r))
==> norm ((f:complex->complex) w) <= norm (f x)`))
THEN REWRITE_TAC [ASSUME`x:complex IN frontier
(cball (z,r))`] THEN SUBST1_TAC
(REAL_ARITH `norm ((f:complex->complex) w)= &1* norm (f w)`) THEN
SUBST1_TAC (REAL_ARITH `norm ((f:complex->complex) x)= &1 * norm (f x)`)
THEN DISCH_TAC THEN REWRITE_TAC[REAL_MUL_LID] THEN POP_ASSUM
MP_TAC THEN MATCH_MP_TAC (TAUT `(q <=> p ) ==> ( p ==> q)`) THEN MATCH_MP_TAC
(RAT_LEMMA4) THEN ASM_REWRITE_TAC[]];ASM_MESON_TAC[]]]]]]]]) in
REPEAT STRIP_TAC THEN ASSUME_TAC (MESON [HOLOMORPHIC_ON_SUBSET;
ASSUME `(u:complex->bool) SUBSET a`;ASSUME `f holomorphic_on a`]
`f holomorphic_on u`) THEN ASM_CASES_TAC `(u:complex->bool)={}` THENL [
ASM_MESON_TAC[SUBSET_EMPTY;IMAGE_EQ_EMPTY;OPEN_EMPTY];ALL_TAC] THEN
SUBGOAL_THEN `!f u. ~(u={}) /\ open u /\ connected u /\
f holomorphic_on u /\
~(?c:complex. !z:complex. z IN u ==> f z=c) ==>
open (IMAGE f u)` ASSUME_TAC
THENL [REPEAT STRIP_TAC THEN REWRITE_TAC[OPEN_CONTAINS_BALL;IN_IMAGE]
THEN GEN_TAC THEN STRIP_TAC THEN
ASSERT_TAC `(\z:complex.(f':complex->complex)z - f' x') holomorphic_on
(u':complex->bool) /\ (\z:complex. f' z - f' x')x' = Cx(&0)` THENL [
ASM_SIMP_TAC[HOLOMORPHIC_ON_CONST;HOLOMORPHIC_ON_SUB;
BETA_THM;COMPLEX_SUB_REFL];ALL_TAC] THEN
ASSERT_TAC `?s:real. &0 < s /\ ball(x',s) SUBSET u' /\
(!z:complex. z IN ball(x',s) /\ ~(z = x') ==>
~((\z:complex.(f':complex->complex)z - f' x') z = Cx(&0)))` THENL [
MATCH_MP_TAC ISOLATED_ZEROS THEN ASM_REWRITE_TAC[] THEN
ASM_MESON_TAC[COMPLEX_SUB_0];
ASSERT_TAC `?r. &0 < r /\ cball(x':complex,r) SUBSET ball(x',s)` THENL[
EXISTS_TAC `s:real / &2` THEN ASM_SIMP_TAC [REAL_ARITH `&0 < s
==> &0 < s/ &2`;SUBSET;IN_CBALL;IN_BALL] THEN MP_TAC (ASSUME `&0 < s`)
THEN REAL_ARITH_TAC;ALL_TAC] THEN
ASSERT_TAC `cball(x',r) SUBSET u' /\
(!z:complex. z IN cball(x',r) /\
~(z=x')==> ~((\z:complex.(f':complex->complex)z - f' x') z = Cx(&0)))`
THENL [CONJ_TAC THENL [ASM_MESON_TAC[SUBSET_TRANS];
MESON_TAC[ASSUME `!z:complex. z IN ball (x',s) /\ ~(z = x')
==> ~((\z. (f':complex->complex) z - f' x') z = Cx(&0))`;
ASSUME `cball (x':complex,r) SUBSET ball (x',s)`;SUBSET]];ALL_TAC]
THEN SUBGOAL_THEN `frontier (cball (x':complex,r)) SUBSET u'` ASSUME_TAC
THENL [MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC `cball(x':complex,r)`
THEN ASM_MESON_TAC[CLOSED_CBALL;FRONTIER_SUBSET_CLOSED];ALL_TAC] THEN
ASSERT_TAC `?w. w IN frontier(cball(x':complex,r)) /\
(!z. z IN frontier(cball(x',r)) ==>
norm ((f':complex->complex)w - f' x') <= norm(f' z - f' x'))`
THENL [MATCH_MP_TAC CONTINUOUS_ATTAINS_INF THEN
ASM_SIMP_TAC[COMPACT_FRONTIER;COMPACT_CBALL;CBALL_EQ_EMPTY;
REAL_ARITH `!r:real. &0 < r ==> ~(r < &0)` ] THEN
CONJ_TAC THENL [REWRITE_TAC[REWRITE_RULE[sphere] FRONTIER_CBALL;dist] THEN
SUBGOAL_THEN `?x:complex. norm(x'-x) = r` (fun th-> SET_TAC
[MEMBER_NOT_EMPTY;th]) THEN EXISTS_TAC `x' + Cx r` THEN
ASM_SIMP_TAC[COMPLEX_ADD_SUB2;NORM_NEG;COMPLEX_NORM_CX;
REAL_ABS_REFL;REAL_LT_IMP_LE];
SUBGOAL_THEN `lift o (\z. norm ((f':complex->complex) z - f' x')) =
(lift o norm) o (\z. f' z - f' x') ` SUBST1_TAC THENL [
REWRITE_TAC[o_DEF]; MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN
ASM_MESON_TAC [CONTINUOUS_ON_LIFT_NORM; HOLOMORPHIC_ON_SUBSET;
HOLOMORPHIC_ON_IMP_CONTINUOUS_ON]]];ALL_TAC] THEN
ABBREV_TAC `e = (norm ((f':complex->complex) w - f' x'))*(&1/ &3)`
THEN SUBGOAL_THEN `&0<e` ASSUME_TAC THENL [
EXPAND_TAC "e" THEN MATCH_MP_TAC REAL_LT_MUL THEN
REWRITE_TAC [REAL_ARITH `&0 < &1 / &3`; COMPLEX_NORM_NZ] THEN
SUBST1_TAC (MESON [BETA_THM] `(f':complex->complex) w - f' x' =
(\w. f' w - f' x')w `) THEN FIRST_ASSUM MATCH_MP_TAC THEN
CONJ_TAC THENL[MESON_TAC[ASSUME `w:complex IN frontier (cball (x',r))`;
FRONTIER_SUBSET_CLOSED; CLOSED_CBALL;SET_RULE `!x:complex s t. x IN s /\
s SUBSET t ==> x IN t` ];ONCE_REWRITE_TAC[GSYM COMPLEX_SUB_0] THEN
REWRITE_TAC[GSYM COMPLEX_NORM_ZERO] THEN MATCH_MP_TAC REAL_LT_IMP_NZ
THEN MATCH_MP_TAC (REAL_ARITH `&0 < r /\ r = norm (w:complex - x') ==>
&0 < norm (w - x')`) THEN ASM_REWRITE_TAC[] THEN
MP_TAC (ASSUME `w:complex IN frontier (cball (x',r))`) THEN
SIMP_TAC[FRONTIER_CBALL; sphere; dist; IN_ELIM_THM; NORM_SUB]];
ALL_TAC]
THEN EXISTS_TAC `e:real` THEN REWRITE_TAC[ASSUME `&0<e`] THEN
REWRITE_TAC[SUBSET;IN_IMAGE] THEN GEN_TAC THEN DISCH_TAC THEN
ONCE_REWRITE_TAC [GSYM COMPLEX_SUB_0] THEN
SUBGOAL_THEN `(?x:complex. x IN ball(x',r) /\
x'' - (f':complex->complex) x = Cx(&0)) ==>
?x. x'' - f' x = Cx(&0) /\ x IN u'` MATCH_MP_TAC THENL [
STRIP_TAC THEN EXISTS_TAC `x''':complex` THEN
ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC (SET_RULE `!x:complex u s. x IN u /\ u SUBSET s ==> x IN s`)
THEN EXISTS_TAC `ball(x':complex,r)` THEN ASM_REWRITE_TAC[]
THEN ASM_MESON_TAC[BALL_SUBSET_CBALL;SUBSET_TRANS];
MATCH_MP_TAC LEMMA_ZERO THEN CONJ_TAC THENL
[MATCH_MP_TAC HOLOMORPHIC_ON_IMP_CONTINUOUS_ON THEN
MATCH_MP_TAC HOLOMORPHIC_ON_SUB THEN ASM_MESON_TAC
[HOLOMORPHIC_ON_CONST; HOLOMORPHIC_ON_SUBSET];
CONJ_TAC THENL [MATCH_MP_TAC HOLOMORPHIC_ON_SUB THEN ASM_MESON_TAC[
HOLOMORPHIC_ON_CONST;HOLOMORPHIC_ON_SUBSET;BALL_SUBSET_CBALL];
ASM_REWRITE_TAC[] THEN X_GEN_TAC `w':complex` THEN DISCH_TAC THEN
MATCH_MP_TAC REAL_LTE_TRANS THEN
EXISTS_TAC `e:real` THEN CONJ_TAC THENL
[MESON_TAC [NORM_SUB;dist;IN_BALL; ASSUME`x'':complex IN ball (x,e)`;
ASSUME `x:complex = (f':complex->complex) x'`];
MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `&2*e` THEN
ASM_SIMP_TAC[REAL_ARITH `&0<e ==> e <= &2 * e`;NORM_SUB] THEN
SUBST1_TAC (COMPLEX_RING `(f':complex->complex) w' - x'' =
f' w' -x + x - x''`) THEN
MATCH_MP_TAC REAL_LE_TRANS THEN
EXISTS_TAC `norm ((f':complex->complex) w' - x) - norm (x-x'')` THEN
CONJ_TAC THENL [SUBST1_TAC (REAL_ARITH `&2 * e = &3 *e - e`) THEN
MATCH_MP_TAC (REAL_ARITH `!x y z w:real. x<=y /\ z<w ==> x-w <= y-z`)
THEN CONJ_TAC THENL [EXPAND_TAC "e" THEN
ASM_REWRITE_TAC[REAL_ARITH `&3 * norm ((f':complex->complex) w - f' x') *
&1 / &3 = norm (f' w - f' x')`] THEN FIRST_ASSUM MATCH_MP_TAC THEN
POP_ASSUM MP_TAC THEN
REWRITE_TAC[FRONTIER_CBALL; sphere; NORM_SUB; IN_ELIM_THM; dist];
UNDISCH_TAC `x'':complex IN ball (x,e)` THEN
REWRITE_TAC [IN_BALL;dist;ASSUME`x:complex = (f':complex->complex) x'`]];
MATCH_MP_TAC (REAL_ARITH `!x y z:real. x<=y+z ==> x-z<=y`) THEN
REWRITE_TAC[COMPLEX_NORM_TRIANGLE_SUB]]]]]]];ALL_TAC] THEN
ASM_CASES_TAC `connected (u:complex->bool)` THENL [
SUBGOAL_THEN `~(?c:complex. !z:complex. z IN u ==> f z=c)`
(fun th-> ASM_MESON_TAC [th]) THEN
ONCE_REWRITE_TAC[GSYM COMPLEX_SUB_0]
THEN STRIP_TAC THEN ABBREV_TAC `w:complex= CHOICE u` THEN
ASSUME_TAC (MESON [CHOICE_DEF;GSYM (ASSUME `CHOICE u = w:complex`);
ASSUME `~(u:complex->bool = {})`] `w:complex IN u`) THEN
ASSERT_TAC `w:complex limit_point_of u` THENL
[MATCH_MP_TAC INTERIOR_LIMIT_POINT THEN ASM_SIMP_TAC [INTERIOR_OPEN];
SUBGOAL_THEN `(\z. (f:complex->complex) z - c) holomorphic_on a` ASSUME_TAC
THENL [ASM_SIMP_TAC [HOLOMORPHIC_ON_SUB; HOLOMORPHIC_ON_CONST];
ASSUME_TAC (MESON [ASSUME `w:complex IN u`;ASSUME `u:complex->bool SUBSET a`;
SET_RULE `w:complex IN u /\ u SUBSET a ==> w IN a`] `w:complex IN a`) THEN
MP_TAC(SPECL [`\z:complex.(f:complex->complex)z - c`;
`a:complex->bool`; `u:complex->bool`; `w:complex`]
ANALYTIC_CONTINUATION) THEN
ASM_REWRITE_TAC [] THEN MP_TAC (ASSUME `~(?c:complex. !z. z IN a ==>
(f:complex->complex) z = c)`) THEN ONCE_REWRITE_TAC [GSYM COMPLEX_SUB_0;
GSYM COMPLEX_SUB_RZERO] THEN ONCE_REWRITE_TAC [COMPLEX_SUB_RZERO] THEN
MESON_TAC[]]];ALL_TAC] THEN SUBST1_TAC (MESON [UNIONS_COMPONENTS]
`u:complex->bool = UNIONS ( components u)`) THEN
REWRITE_TAC [IMAGE_UNIONS] THEN MATCH_MP_TAC OPEN_UNIONS THEN
REWRITE_TAC[IN_IMAGE] THEN GEN_TAC THEN STRIP_TAC THEN
FIRST_X_ASSUM SUBST1_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
STRIP_ASSUME_TAC(MESON [IN_COMPONENTS;
ASSUME `(x:complex->bool) IN components u`]
`?w:complex. w IN u /\ x = connected_component u w`) THEN
ASM_SIMP_TAC[CONNECTED_COMPONENT_EQ_EMPTY;OPEN_CONNECTED_COMPONENT;
CONNECTED_CONNECTED_COMPONENT] THEN CONJ_TAC THENL
[ASM_MESON_TAC [CONNECTED_COMPONENT_SUBSET;
HOLOMORPHIC_ON_SUBSET]; ONCE_REWRITE_TAC[GSYM COMPLEX_SUB_0] THEN
STRIP_TAC THEN ABBREV_TAC `y = CHOICE (x:complex->bool)` THEN
SUBGOAL_THEN `y:complex IN x` ASSUME_TAC THENL
[EXPAND_TAC "y" THEN MATCH_MP_TAC CHOICE_DEF THEN
ASM_MESON_TAC [CONNECTED_COMPONENT_EQ_EMPTY];
ASSUME_TAC (MESON [OPEN_COMPONENTS;ASSUME `open (u:complex->bool)`;
ASSUME` x:complex->bool IN components u`] `open (x:complex->bool)`) THEN
ASSERT_TAC `y:complex limit_point_of x` THENL [
MATCH_MP_TAC INTERIOR_LIMIT_POINT THEN ASSUME_TAC
(MESON [OPEN_COMPONENTS;ASSUME `open (u:complex->bool)`;
ASSUME` x:complex->bool IN components u`] `open (x:complex->bool)`) THEN
SIMP_TAC [INTERIOR_OPEN;ASSUME `open (x:complex->bool)`;
ASSUME `y:complex IN x`]; SUBGOAL_THEN `(\z. (f:complex->complex) z - c)
holomorphic_on a` ASSUME_TAC THENL [
ASM_SIMP_TAC [HOLOMORPHIC_ON_SUB; HOLOMORPHIC_ON_CONST];
SUBGOAL_THEN `x:complex->bool SUBSET a` ASSUME_TAC THENL [
MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC `u:complex->bool` THEN
ASM_REWRITE_TAC[CONNECTED_COMPONENT_SUBSET];
SUBGOAL_THEN `y:complex IN a` ASSUME_TAC THENL [
MATCH_MP_TAC (SET_RULE `y:complex IN x /\ x SUBSET a ==> y IN a`)
THEN ASM_REWRITE_TAC[]; MP_TAC(SPECL [`\z:complex.(f:complex->complex)z - c`;
`a:complex->bool`; `x:complex->bool`; `y:complex`] ANALYTIC_CONTINUATION)
THEN ASM_REWRITE_TAC [] THEN MP_TAC (ASSUME `~(?c:complex. !z. z IN a ==>
(f:complex->complex) z = c)`) THEN
ONCE_REWRITE_TAC [GSYM COMPLEX_SUB_0;GSYM COMPLEX_SUB_RZERO] THEN
ONCE_REWRITE_TAC [COMPLEX_SUB_RZERO] THEN MESON_TAC[]]]]]]]);;
let MAXIMUM_MODULUS_PRINCIPLE = prove
(`!f a u w.
open a /\ connected a /\ f
holomorphic_on a /\
open u /\ u
SUBSET a /\ w
IN u /\
(!z. z
IN u ==> norm(f z) <= norm(f w))
==> (?c. !z. z
IN a ==> f z = c)`,
REPEAT STRIP_TAC THEN SUBGOAL_THEN
`~(open (
IMAGE (f:complex->complex) u))`
(fun th -> ASM_MESON_TAC[th;
OPEN_MAPPING_THM]) THEN
REWRITE_TAC[
OPEN_CONTAINS_BALL;
NOT_FORALL_THM] THEN
EXISTS_TAC `(f:complex->complex) w` THEN
MATCH_MP_TAC (TAUT `!p q. (p /\ ~ q) ==> ~(p ==> q)`) THEN CONJ_TAC THENL
[ASM_MESON_TAC[
IN_IMAGE]; ALL_TAC] THEN
REWRITE_TAC[
NOT_EXISTS_THM;DE_MORGAN_THM;
SUBSET] THEN
GEN_TAC THEN ASM_CASES_TAC `~(&0 < e)` THENL
[ASM_REWRITE_TAC[]; ALL_TAC] THEN POP_ASSUM MP_TAC THEN REWRITE_TAC[] THEN
DISCH_TAC THEN DISJ2_TAC THEN REWRITE_TAC[
NOT_FORALL_THM] THEN
EXISTS_TAC `if &0 < Re((f:complex->complex) w)
then f w + Cx(e / &2)
else f w - Cx(e/ &2) ` THEN
ABBREV_TAC `x = if &0<Re((f:complex->complex) w)
then f w + Cx(e / &2)
else f w - Cx(e / &2)` THEN
MATCH_MP_TAC (TAUT `!p q. (p /\ ~ q) ==> ~(p ==> q)`) THEN CONJ_TAC THENL
[REWRITE_TAC[
IN_BALL;dist] THEN
MATCH_MP_TAC (REAL_ARITH `!x y z:real. x = y /\ y < z ==> x < z `) THEN
EXISTS_TAC `e / &2` THEN EXPAND_TAC "x" THEN COND_CASES_TAC THENL
[ASM_SIMP_TAC [
NORM_NEG;
COMPLEX_ADD_SUB2;REAL_ARITH `&0 < e ==> e / &2 <e`;
COMPLEX_NORM_CX;
REAL_ABS_REFL;
REAL_ARITH `&0 < e ==> &0 <= e / &2`];
ASM_SIMP_TAC [
COMPLEX_SUB_SUB2; REAL_ARITH `&0 < e ==> e / &2 <e`;
COMPLEX_NORM_CX;
REAL_ABS_REFL;
REAL_ARITH `&0 < e ==> &0 <= e / &2`]]; ALL_TAC] THEN
REWRITE_TAC[
IN_IMAGE;
NOT_EXISTS_THM; DE_MORGAN_THM] THEN
GEN_TAC THEN ASM_CASES_TAC `~(x':complex
IN u)` THENL
[ASM_REWRITE_TAC[]; ALL_TAC] THEN DISJ1_TAC THEN POP_ASSUM MP_TAC THEN
REWRITE_TAC[] THEN DISCH_TAC THEN
MATCH_MP_TAC (NORM_ARITH `!x y:complex. ~(norm x=norm y) ==> ~(x=y)`) THEN
REWRITE_TAC[
REAL_NOT_EQ] THEN DISJ2_TAC THEN
MATCH_MP_TAC
REAL_LET_TRANS THEN
EXISTS_TAC `norm ((f:complex->complex) w)` THEN ASM_SIMP_TAC[] THEN
EXPAND_TAC "x" THEN COND_CASES_TAC THEN
REWRITE_TAC [
complex_norm;
RE_ADD;
IM_ADD;
IM_CX;
RE_CX;
REAL_ADD_RID] THENL
[MATCH_MP_TAC
SQRT_MONO_LT THEN
MATCH_MP_TAC (REAL_ARITH `!x:real y z. x < y ==> x + z < y + z`) THEN
REWRITE_TAC[GSYM
REAL_LT_SQUARE_ABS] THEN
ASM_SIMP_TAC [REAL_ARITH `!x y. &0 < x /\ &0 < y
==> abs (x+y) = abs x + abs y`;
REAL_ARITH `!x:real. &0 < x ==> &0 < x / &2`] THEN
ASM_REAL_ARITH_TAC; ALL_TAC] THEN
REWRITE_TAC [
complex_norm;
RE_SUB;
IM_SUB;
IM_CX;
RE_CX;
REAL_SUB_RZERO] THEN
MATCH_MP_TAC
SQRT_MONO_LT THEN
MATCH_MP_TAC (REAL_ARITH `!x:real y z. x < y ==> x + z < y + z`) THEN
REWRITE_TAC[GSYM
REAL_LT_SQUARE_ABS] THEN POP_ASSUM MP_TAC THEN
REWRITE_TAC[
REAL_NOT_LT] THEN DISCH_TAC THEN
ASM_SIMP_TAC [REAL_ARITH `!x y. x <= &0 /\ &0 < y
==> abs (x - y) = abs x + abs y`;
REAL_ARITH `!x. &0 < x ==> &0 < x/ &2`] THEN
ASM_REAL_ARITH_TAC);;
let SCHWARZ_LEMMA = prove
(`!f. f
holomorphic_on ball(Cx(&0),&1) /\
(!z:complex. norm z < &1 ==> norm (f z) < &1) /\
f(Cx(&0)) = Cx(&0)
==> (!z. norm z < &1 ==> norm(f z) <= norm z) /\
norm(
complex_derivative f(Cx(&0))) <= &1 /\
((?z. norm z < &1 /\ ~(z= Cx(&0)) /\ norm(f z) = norm z) \/
norm(
complex_derivative f (Cx(&0))) = &1
==> ?c. (!z. norm z < &1 ==> f z = c*z) /\ norm c = &1)`,
let LEMMA1 = prove
(`!f a. open a /\ connected a /\ bounded a /\ ~(a = {}) /\
f holomorphic_on a /\ f continuous_on (closure a)
==> (?w. w IN (frontier a) /\
(!z. z IN (closure a) ==> norm (f z) <= norm (f w)))`,
REPEAT STRIP_TAC THEN ASSERT_TAC
`?x. x IN closure a /\
(!z. z IN closure a ==>
norm((f:complex->complex) z) <= norm(f x))` THENL
[MATCH_MP_TAC CONTINUOUS_ATTAINS_SUP THEN
ASM_SIMP_TAC [COMPACT_CLOSURE;CLOSURE_EQ_EMPTY] THEN
SUBGOAL_THEN `lift o (\x. norm((f:complex->complex) x)) =
(lift o norm) o (\x. f x) ` SUBST1_TAC THENL
[REWRITE_TAC[o_DEF]; MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN
ASM_REWRITE_TAC [CONTINUOUS_ON_LIFT_NORM;ETA_AX]]; ALL_TAC] THEN
ASM_CASES_TAC `x:complex IN frontier a` THENL
[EXISTS_TAC `x:complex` THEN ASM_REWRITE_TAC[]; ALL_TAC] THEN
SUBGOAL_THEN `x:complex IN interior a` MP_TAC THENL
[POP_ASSUM MP_TAC THEN REWRITE_TAC[frontier;DIFF] THEN
SET_TAC[ASSUME `x:complex IN closure a`]; ALL_TAC] THEN
ASM_SIMP_TAC[INTERIOR_OPEN] THEN DISCH_TAC THEN
SUBGOAL_THEN `?c. !z. z IN a ==> (f:complex->complex) z = c`
STRIP_ASSUME_TAC THENL
[MATCH_MP_TAC MAXIMUM_MODULUS_PRINCIPLE THEN
EXISTS_TAC `a:complex->bool` THEN
EXISTS_TAC `x:complex` THEN ASM_REWRITE_TAC[SUBSET_REFL] THEN GEN_TAC THEN
DISCH_TAC THEN FIRST_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[closure;UNION] THEN
SET_TAC[ASSUME `z:complex IN a`]; ALL_TAC] THEN
SUBGOAL_THEN `CHOICE(frontier(a:complex->bool)) IN frontier a`
ASSUME_TAC THENL
[MATCH_MP_TAC CHOICE_DEF THEN MATCH_MP_TAC FRONTIER_NOT_EMPTY THEN
CONJ_TAC THENL [ASM SET_TAC[]; ASM_MESON_TAC[NOT_BOUNDED_UNIV]];
ALL_TAC] THEN
EXISTS_TAC `CHOICE(frontier(a:complex->bool))` THEN ASM_REWRITE_TAC[] THEN
REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_EQ_IMP_LE THEN
SUBGOAL_THEN `!z. z IN closure a ==> (f:complex->complex) z = c`
ASSUME_TAC THENL
[MP_TAC (ISPECL [`f:complex->complex`; `closure (a:complex->bool)`;
`{c:complex}`] CONTINUOUS_CLOSED_PREIMAGE) THEN
ASM_REWRITE_TAC [CLOSED_CLOSURE; CLOSED_SING] THEN
ABBREV_TAC `s = {x | x IN closure(a:complex->bool) /\
(f:complex->complex) x IN {c}}` THEN DISCH_TAC THEN
SUBGOAL_THEN `closure a SUBSET (s:complex->bool)` ASSUME_TAC THENL
[MATCH_MP_TAC CLOSURE_MINIMAL THEN CONJ_TAC THENL
[REWRITE_TAC[SUBSET] THEN EXPAND_TAC "s" THEN
ASSUME_TAC (MESON [CLOSURE_SUBSET;GSYM SUBSET]
`!x:complex. x IN a ==> x IN closure a`) THEN
SET_TAC [ASSUME `!x:complex. x IN a ==> x IN closure a`;
ASSUME `!z:complex. z IN a ==> f z = c:complex`];
ASM_REWRITE_TAC[]];
POP_ASSUM MP_TAC THEN EXPAND_TAC "s" THEN SET_TAC[]];
EQ_TRANS_TAC `norm(c:complex)` THENL
[ASM_SIMP_TAC[]; ONCE_REWRITE_TAC [EQ_SYM_EQ] THEN
MATCH_MP_TAC (NORM_ARITH `!x y:complex. x = y ==> norm x = norm y`) THEN
FIRST_ASSUM MATCH_MP_TAC THEN ASM_MESON_TAC[frontier;IN_DIFF]]])
in
let LEMMA2 = prove
(`!(f:complex->complex) r w s.
&0 < r /\ f holomorphic_on ball(Cx(&0),r) /\
&0 < s /\ ball(w,s) SUBSET ball(Cx(&0),r) /\
(!z. norm (w-z) < s ==> norm(f z) <= norm(f w))
==> (?c. !z. norm z < r ==> f z = c)`,
REPEAT STRIP_TAC THEN
MP_TAC (SPECL[`f:complex->complex`;`ball (Cx(&0),r)`; `ball (w:complex,s)`;
`w:complex`] MAXIMUM_MODULUS_PRINCIPLE) THEN
ASM_REWRITE_TAC[OPEN_BALL; CONNECTED_BALL; IN_BALL;DIST_REFL] THEN
ASM_REWRITE_TAC[dist;COMPLEX_SUB_LZERO;NORM_NEG])
in
let LEMMA3 = prove
(`!r:real f. f holomorphic_on (ball(Cx(&0),r)) /\ f (Cx(&0))=Cx(&0)
==> (?h. h holomorphic_on (ball(Cx(&0),r)) /\
((!z. norm z < r ==> f z=z*(h z)) /\
(complex_derivative f (Cx(&0)))= h (Cx(&0))))`,
REPEAT STRIP_TAC THEN ABBREV_TAC `h = \z. if z = Cx(&0) then
complex_derivative f (Cx(&0)) else f z/z` THEN EXISTS_TAC
`h:complex->complex` THEN ASSERT_TAC `(!z:complex. norm z < r ==>
(f:complex->complex) z = z * h z) /\ complex_derivative f (Cx(&0))
= h (Cx(&0))` THENL [CONJ_TAC THENL
[GEN_TAC THEN DISCH_TAC THEN EXPAND_TAC "h" THEN
COND_CASES_TAC THENL [ASM_REWRITE_TAC[COMPLEX_MUL_LZERO];
POP_ASSUM MP_TAC THEN CONV_TAC COMPLEX_FIELD];
EXPAND_TAC "h" THEN ASM_REWRITE_TAC[]];ALL_TAC] THEN ASM_REWRITE_TAC[]
THEN MATCH_MP_TAC POLE_THEOREM_OPEN_0 THEN EXISTS_TAC `(f:complex->complex)`
THEN EXISTS_TAC `Cx(&0)` THEN
ASM_SIMP_TAC[OPEN_BALL;IN_BALL;COMPLEX_SUB_RZERO;
dist;COMPLEX_SUB_LZERO;NORM_NEG])
in
GEN_TAC THEN STRIP_TAC THEN
MP_TAC (SPECL [`&1`;`f:complex->complex`] LEMMA3) THEN ASM_REWRITE_TAC[] THEN
STRIP_TAC THEN SUBGOAL_THEN
`!z. norm z < &1 ==> norm ((h:complex->complex) z) <= &1`
ASSUME_TAC THENL
[GEN_TAC THEN DISCH_TAC THEN MATCH_MP_TAC
(prove
(`!x y:real. (!a. y<a ==> x<a) ==> x <= y`,
REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM REAL_NOT_LT] THEN
ONCE_REWRITE_TAC[REAL_LT_BETWEEN] THEN REWRITE_TAC[NOT_EXISTS_THM;
DE_MORGAN_THM] THEN X_GEN_TAC `z:real` THEN
POP_ASSUM (MP_TAC o SPEC `z:real`) THEN REAL_ARITH_TAC)) THEN
X_GEN_TAC `a:real` THEN
DISCH_TAC THEN SUBGOAL_THEN
`?r. norm (z:complex) < r /\ inv r < a /\ r < &1` MP_TAC THENL
[SUBGOAL_THEN `max (inv a) (norm(z:complex)) < &1` MP_TAC THENL
[ASM_SIMP_TAC[REAL_MAX_LT; REAL_INV_LT_1];
GEN_REWRITE_TAC LAND_CONV [REAL_LT_BETWEEN] THEN
DISCH_THEN (X_CHOOSE_TAC `r:real`) THEN EXISTS_TAC `r:real` THEN
POP_ASSUM MP_TAC THEN ASM_REWRITE_TAC[REAL_MAX_LT] THEN STRIP_TAC THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC REAL_LT_LINV THEN
ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC]; ALL_TAC] THEN
STRIP_TAC THEN
SUBGOAL_THEN `&0 < r` ASSUME_TAC THENL
[ASM_MESON_TAC[REAL_LET_TRANS; NORM_POS_LE]; ALL_TAC] THEN
SUBGOAL_THEN `inv (r:real) = &1/r` ASSUME_TAC THENL
[MATCH_MP_TAC REAL_MUL_RINV_UNIQ THEN MATCH_MP_TAC REAL_DIV_LMUL THEN
ASM_SIMP_TAC[REAL_LT_IMP_NZ]; ALL_TAC] THEN
SUBGOAL_THEN `?w. norm w = r /\ (!z. norm z < r
==> norm((h:complex->complex) z) <= norm(h w))`
STRIP_ASSUME_TAC THENL
[MATCH_MP_TAC(prove (`!f r. &0 < r /\ f holomorphic_on ball(Cx(&0),r) /\
f continuous_on cball(Cx(&0),r)
==> (?w. norm w = r /\ (!z. norm z < r ==> norm(f z) <= norm(f w)))`,
REPEAT GEN_TAC THEN STRIP_TAC THEN
MP_TAC(SPECL[`f:complex->complex`; `ball(Cx(&0),r)`] LEMMA1) THEN
ASM_REWRITE_TAC[OPEN_BALL; CONNECTED_BALL; BOUNDED_BALL; BALL_EQ_EMPTY;
REAL_ARITH `!r:real. ~(r <= &0) <=> &0 < r`] THEN
ASM_SIMP_TAC[CLOSURE_BALL] THEN STRIP_TAC THEN EXISTS_TAC `w:complex` THEN
CONJ_TAC THENL
[UNDISCH_TAC `w:complex IN frontier(ball(Cx(&0),r))` THEN
ASM_SIMP_TAC[FRONTIER_BALL;sphere;dist;COMPLEX_SUB_LZERO;NORM_NEG] THEN
SET_TAC[];
POP_ASSUM MP_TAC THEN
REWRITE_TAC[IN_CBALL;dist;COMPLEX_SUB_LZERO;NORM_NEG] THEN
MESON_TAC [REAL_LT_IMP_LE]])) THEN ASM_REWRITE_TAC[] THEN
CONJ_TAC THENL [MATCH_MP_TAC HOLOMORPHIC_ON_SUBSET
THEN EXISTS_TAC `ball(Cx(&0),&1)` THEN
ASM_SIMP_TAC [SUBSET_BALL;REAL_LT_IMP_LE];
MATCH_MP_TAC HOLOMORPHIC_ON_IMP_CONTINUOUS_ON THEN
MATCH_MP_TAC HOLOMORPHIC_ON_SUBSET THEN EXISTS_TAC `ball(Cx(&0),&1)` THEN
ASM_REWRITE_TAC[SUBSET; IN_CBALL; IN_BALL] THEN
ASM_MESON_TAC[REAL_LET_TRANS]]; ALL_TAC] THEN
MATCH_MP_TAC REAL_LET_TRANS THEN
EXISTS_TAC `norm(h(w:complex):complex)` THEN CONJ_TAC THENL
[ASM_SIMP_TAC[]; ALL_TAC] THEN
SUBGOAL_THEN `h w:complex = f w / w` SUBST1_TAC THENL
[ASM_SIMP_TAC[] THEN
MP_TAC (MESON [GSYM COMPLEX_NORM_ZERO;REAL_NOT_EQ;
ASSUME `norm(w:complex) =r`;
ASSUME `&0 < r`] `~(w=Cx(&0))`) THEN
CONV_TAC(COMPLEX_FIELD);
ASM_REWRITE_TAC[COMPLEX_NORM_DIV] THEN MATCH_MP_TAC REAL_LT_TRANS THEN
EXISTS_TAC `&1/(r:real)` THEN ASM_SIMP_TAC [REAL_LT_DIV2_EQ] THEN
MATCH_MP_TAC REAL_LET_TRANS THEN EXISTS_TAC `inv (r:real)` THEN
ASM_REWRITE_TAC[REAL_LE_REFL]]; ALL_TAC] THEN
CONJ_TAC THENL
[GEN_TAC THEN DISCH_TAC THEN ASM_CASES_TAC `z = Cx(&0)` THENL
[ASM_SIMP_TAC[COMPLEX_MUL_LZERO;REAL_LE_REFL];
SUBST1_TAC (REAL_ARITH `norm (z:complex) = norm z * &1`) THEN
ASM_SIMP_TAC[COMPLEX_NORM_MUL] THEN MATCH_MP_TAC REAL_LE_LMUL THEN
ASM_SIMP_TAC[NORM_POS_LE]]; ALL_TAC] THEN CONJ_TAC THENL
[ASM_MESON_TAC [COMPLEX_NORM_ZERO;REAL_LT_01]; ALL_TAC] THEN
REWRITE_TAC[TAUT `((p \/ q) ==> r) <=> ((p ==> r) /\ (q ==> r))`] THEN
CONJ_TAC THENL [STRIP_TAC THEN SUBGOAL_THEN
`norm ((h:complex->complex) z) = &1` ASSUME_TAC THENL
[SUBGOAL_THEN `(h:complex->complex) z = f z/z` SUBST1_TAC THENL
[UNDISCH_THEN `!z:complex. norm z < &1 ==> f z = z * h z`
(MP_TAC o SPEC `z:complex`) THEN ASM_REWRITE_TAC[] THEN
UNDISCH_TAC `~(z = Cx(&0))` THEN CONV_TAC(COMPLEX_FIELD);
ASM_SIMP_TAC[COMPLEX_NORM_ZERO;REAL_DIV_REFL;COMPLEX_NORM_DIV]];
SUBGOAL_THEN `?c. (!z. norm z < &1 ==> (h:complex->complex) z = c)`
STRIP_ASSUME_TAC THENL [MATCH_MP_TAC LEMMA2
THEN EXISTS_TAC `z:complex` THEN EXISTS_TAC `&1 - norm(z:complex)`
THEN ASM_REWRITE_TAC[REAL_LT_01] THEN CONJ_TAC THENL
[ASM_MESON_TAC[REAL_SUB_LT]; CONJ_TAC THENL
[REWRITE_TAC[SUBSET;IN_BALL] THEN GEN_TAC THEN DISCH_TAC THEN
MATCH_MP_TAC REAL_LET_TRANS THEN
EXISTS_TAC `dist(Cx(&0), z) + dist(z,x)` THEN
REWRITE_TAC[DIST_TRIANGLE] THEN POP_ASSUM MP_TAC THEN
REWRITE_TAC[dist;COMPLEX_SUB_LZERO;NORM_NEG] THEN REAL_ARITH_TAC;
GEN_TAC THEN DISCH_TAC THEN FIRST_ASSUM MATCH_MP_TAC THEN
MATCH_MP_TAC REAL_LET_TRANS THEN
EXISTS_TAC `norm(z:complex) + norm(z' - z)` THEN
REWRITE_TAC[NORM_TRIANGLE_SUB] THEN REWRITE_TAC[NORM_SUB] THEN
POP_ASSUM MP_TAC THEN REWRITE_TAC[NORM_SUB] THEN REAL_ARITH_TAC]];
EXISTS_TAC `c:complex` THEN CONJ_TAC THENL
[ASM_SIMP_TAC[COMPLEX_MUL_SYM];
POP_ASSUM (MP_TAC o SPEC `z:complex`) THEN ASM_MESON_TAC[]]]];
ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
SUBGOAL_THEN `?c. (!z. norm z < &1 ==> (h:complex->complex) z = c)`
STRIP_ASSUME_TAC THENL[MATCH_MP_TAC LEMMA2 THEN EXISTS_TAC `Cx(&0)`
THEN EXISTS_TAC `&1` THEN ASM_REWRITE_TAC[REAL_ARITH `&0 < &1`;
SUBSET_REFL; COMPLEX_SUB_LZERO; NORM_NEG];
EXISTS_TAC `c:complex` THEN CONJ_TAC THENL
[ASM_SIMP_TAC[COMPLEX_MUL_SYM];POP_ASSUM (MP_TAC o SPEC `Cx(&0)`) THEN
ASM_MESON_TAC[COMPLEX_NORM_0; REAL_LT_01]]]]);;
let HOLOMORPHIC_ON_PASTE_ACROSS_LINE = prove
(`!f s a k.
open s /\ ~(a = vec 0) /\
f
holomorphic_on {z | z
IN s /\ k < a dot z} /\
f
holomorphic_on {z | z
IN s /\ a dot z < k} /\
f
continuous_on s
==> f
holomorphic_on s`,
let lemma0 = prove
(`!d a b:real^N k.
d dot a <= k /\ k <= d dot b
==> ?c. c IN segment[a,b] /\ d dot c = k /\
(!z. z IN segment[a,c] ==> d dot z <= k) /\
(!z. z IN segment[c,b] ==> k <= d dot z)`,
REPEAT STRIP_TAC THEN
MP_TAC(ISPECL [`segment[a:real^N,b]`; `a:real^N`; `b:real^N`;
`d:real^N`; `k:real`] CONNECTED_IVT_HYPERPLANE) THEN
ASM_REWRITE_TAC[CONNECTED_SEGMENT; ENDS_IN_SEGMENT] THEN
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `c:real^N` THEN STRIP_TAC THEN
ASM_REWRITE_TAC[SET_RULE
`(!z. z IN s ==> P z) <=> s SUBSET {x | P x}`] THEN
REWRITE_TAC[SEGMENT_CONVEX_HULL] THEN CONJ_TAC THEN
MATCH_MP_TAC HULL_MINIMAL THEN
REWRITE_TAC[CONVEX_HALFSPACE_LE; REWRITE_RULE[real_ge] CONVEX_HALFSPACE_GE;
SUBSET; FORALL_IN_INSERT; NOT_IN_EMPTY] THEN
ASM_REWRITE_TAC[IN_ELIM_THM; REAL_LE_REFL]) in
let lemma1 = prove
(`!f s d k a b c.
convex s /\ open s /\ a IN s /\ b IN s /\ c IN s /\ ~(d = vec 0) /\
d dot a <= k /\ d dot b <= k /\ d dot c <= k /\
f holomorphic_on {z | z IN s /\ d dot z < k} /\
f holomorphic_on {z | z IN s /\ k < d dot z} /\
f continuous_on s
==> path_integral (linepath (a,b)) f +
path_integral (linepath (b,c)) f +
path_integral (linepath (c,a)) f = Cx(&0)`,
REPEAT STRIP_TAC THEN
MP_TAC(SPECL [`f:complex->complex`; `a:complex`; `b:complex`; `c:complex`]
CAUCHY_THEOREM_TRIANGLE_INTERIOR) THEN
ANTS_TAC THENL
[CONJ_TAC THENL
[MATCH_MP_TAC CONTINUOUS_ON_SUBSET THEN
EXISTS_TAC `s:complex->bool` THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC HULL_MINIMAL THEN ASM SET_TAC[];
MATCH_MP_TAC HOLOMORPHIC_ON_SUBSET THEN
EXISTS_TAC `{z:complex | z IN s /\ d dot z < k}` THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC SUBSET_TRANS THEN
EXISTS_TAC `interior(s INTER {x:complex | d dot x <= k})` THEN
CONJ_TAC THENL
[MATCH_MP_TAC SUBSET_INTERIOR THEN MATCH_MP_TAC HULL_MINIMAL THEN
ASM_SIMP_TAC[CONVEX_INTER; CONVEX_HALFSPACE_LE] THEN ASM SET_TAC[];
ASM_SIMP_TAC[INTERIOR_INTER; INTERIOR_HALFSPACE_LE;
INTERIOR_OPEN] THEN
SET_TAC[]]];
REWRITE_TAC[HAS_CHAIN_INTEGRAL_CHAIN_INTEGRAL]]) in
let lemma2 = prove
(`!f s d k a b c.
convex s /\ open s /\ a IN s /\ b IN s /\ c IN s /\ ~(d = vec 0) /\
d dot a <= k /\ d dot b <= k /\
f holomorphic_on {z | z IN s /\ d dot z < k} /\
f holomorphic_on {z | z IN s /\ k < d dot z} /\
f continuous_on s
==> path_integral (linepath (a,b)) f +
path_integral (linepath (b,c)) f +
path_integral (linepath (c,a)) f = Cx(&0)`,
REPEAT STRIP_TAC THEN ASM_CASES_TAC `(d:complex) dot c <= k` THENL
[MATCH_MP_TAC lemma1 THEN ASM_MESON_TAC[]; ALL_TAC] THEN
RULE_ASSUM_TAC(REWRITE_RULE[REAL_NOT_LE]) THEN
MP_TAC(ISPECL [`d:complex`; `b:complex`; `c:complex`; `k:real`]
lemma0) THEN
MP_TAC(ISPECL [`d:complex`; `a:complex`; `c:complex`; `k:real`]
lemma0) THEN
ASM_SIMP_TAC[REAL_LT_IMP_LE] THEN
DISCH_THEN(X_CHOOSE_THEN `a':complex` STRIP_ASSUME_TAC) THEN
DISCH_THEN(X_CHOOSE_THEN `b':complex` STRIP_ASSUME_TAC) THEN
SUBGOAL_THEN `(a':complex) IN s /\ b' IN s` STRIP_ASSUME_TAC THENL
[ASM_MESON_TAC[CONVEX_CONTAINS_SEGMENT; SEGMENT_SYM; SUBSET];
ALL_TAC] THEN
MP_TAC(SPECL
[`f:complex->complex`; `c:complex`; `a:complex`; `a':complex`]
PATH_INTEGRAL_SPLIT_LINEPATH) THEN
MP_TAC(SPECL
[`f:complex->complex`; `b:complex`; `c:complex`; `b':complex`]
PATH_INTEGRAL_SPLIT_LINEPATH) THEN
ASM_REWRITE_TAC[] THEN REPEAT(ANTS_TAC THENL
[ASM_MESON_TAC[CONVEX_CONTAINS_SEGMENT; SEGMENT_SYM;
CONTINUOUS_ON_SUBSET];
ONCE_REWRITE_TAC[TAUT `p ==> q ==> r <=> q ==> p ==> r`]]) THEN
MP_TAC(ISPECL [`f:complex->complex`; `linepath(a':complex,b')`]
PATH_INTEGRAL_REVERSEPATH) THEN
REWRITE_TAC[REVERSEPATH_LINEPATH; VALID_PATH_LINEPATH] THEN ANTS_TAC THENL
[MATCH_MP_TAC PATH_INTEGRABLE_CONTINUOUS_LINEPATH THEN
ASM_MESON_TAC[CONVEX_CONTAINS_SEGMENT; CONTINUOUS_ON_SUBSET];
ALL_TAC] THEN
MP_TAC(ISPECL [`f:complex->complex`; `s INTER {x:complex | d dot x <= k}`;
`{}:complex->bool`;
`linepath(a:complex,b) ++ linepath(b,b') ++
linepath(b',a') ++ linepath(a',a)`]
CAUCHY_THEOREM_CONVEX) THEN
MP_TAC(ISPECL [`f:complex->complex`; `s INTER {x:complex | k <= d dot x}`;
`{}:complex->bool`;
`linepath(b':complex,c) ++ linepath(c,a') ++
linepath(a',b')`]
CAUCHY_THEOREM_CONVEX) THEN
MATCH_MP_TAC(TAUT
`(q /\ q' ==> r) /\ (p /\ p') ==> (p ==> q) ==> (p' ==> q') ==> r`) THEN
CONJ_TAC THENL
[DISCH_THEN(CONJUNCTS_THEN
(fun th -> MP_TAC(MATCH_MP PATH_INTEGRAL_UNIQUE th) THEN
MP_TAC(MATCH_MP HAS_PATH_INTEGRAL_INTEGRABLE th)));
ASM_SIMP_TAC[DIFF_EMPTY; INTERIOR_INTER; INTERIOR_HALFSPACE_LE;
REWRITE_RULE[real_ge] INTERIOR_HALFSPACE_GE] THEN
ASM_SIMP_TAC[CONVEX_INTER; CONVEX_HALFSPACE_LE; FINITE_EMPTY;
REWRITE_RULE[real_ge] CONVEX_HALFSPACE_GE]] THEN
SIMP_TAC[PATH_INTEGRABLE_JOIN; VALID_PATH_JOIN_EQ;
PATHSTART_JOIN; PATHFINISH_JOIN; PATH_IMAGE_JOIN;
VALID_PATH_LINEPATH; PATHSTART_LINEPATH; PATHFINISH_LINEPATH;
PATH_INTEGRAL_JOIN]
THENL [CONV_TAC COMPLEX_RING; ALL_TAC] THEN
REWRITE_TAC[PATH_IMAGE_LINEPATH; UNION_SUBSET; SUBSET_INTER] THEN
ASM_SIMP_TAC[fst(EQ_IMP_RULE(SPEC_ALL CONVEX_CONTAINS_SEGMENT_EQ));
CONVEX_HALFSPACE_LE; REWRITE_RULE[real_ge] CONVEX_HALFSPACE_GE;
IN_ELIM_THM; REAL_LT_IMP_LE; REAL_LE_REFL] THEN
ASM_SIMP_TAC[complex_differentiable; GSYM HOLOMORPHIC_ON_OPEN;
OPEN_INTER; INTERIOR_OPEN; OPEN_HALFSPACE_LT;
OPEN_HALFSPACE_GT] THEN
RULE_ASSUM_TAC(REWRITE_RULE[SET_RULE
`{x | x IN s /\ P x} = s INTER {x | P x}`]) THEN
ASM_REWRITE_TAC[real_gt] THEN
ASM_MESON_TAC[INTER_SUBSET; CONTINUOUS_ON_SUBSET]) in
let lemma3 = prove
(`!f s d k a b c.
convex s /\ open s /\ a IN s /\ b IN s /\ c IN s /\ ~(d = vec 0) /\
d dot a <= k /\
f holomorphic_on {z | z IN s /\ d dot z < k} /\
f holomorphic_on {z | z IN s /\ k < d dot z} /\
f continuous_on s
==> path_integral (linepath (a,b)) f +
path_integral (linepath (b,c)) f +
path_integral (linepath (c,a)) f = Cx(&0)`,
REPEAT STRIP_TAC THEN ASM_CASES_TAC `(d:complex) dot b <= k` THENL
[MATCH_MP_TAC lemma2 THEN ASM_MESON_TAC[]; ALL_TAC] THEN
ASM_CASES_TAC `(d:complex) dot c <= k` THENL
[ONCE_REWRITE_TAC[COMPLEX_RING `a + b + c:complex = c + a + b`] THEN
MATCH_MP_TAC(GEN_ALL lemma2) THEN ASM_MESON_TAC[];
ALL_TAC] THEN
ONCE_REWRITE_TAC[COMPLEX_RING `a + b + c:complex = b + c + a`] THEN
MATCH_MP_TAC(GEN_ALL lemma2) THEN
MAP_EVERY EXISTS_TAC
[`s:complex->bool`; `--d:real^2`; `--k:real`] THEN
ASM_REWRITE_TAC[DOT_LNEG; REAL_LE_NEG2; REAL_LT_NEG2; VECTOR_NEG_EQ_0] THEN
ASM_REAL_ARITH_TAC) in
let lemma4 = prove
(`!f s d k a b c.
convex s /\ open s /\ a IN s /\ b IN s /\ c IN s /\ ~(d = vec 0) /\
f holomorphic_on {z | z IN s /\ d dot z < k} /\
f holomorphic_on {z | z IN s /\ k < d dot z} /\
f continuous_on s
==> path_integral (linepath (a,b)) f +
path_integral (linepath (b,c)) f +
path_integral (linepath (c,a)) f = Cx(&0)`,
REPEAT STRIP_TAC THEN ASM_CASES_TAC `(d:complex) dot a <= k` THENL
[MATCH_MP_TAC lemma3 THEN ASM_MESON_TAC[]; ALL_TAC] THEN
MATCH_MP_TAC lemma3 THEN
MAP_EVERY EXISTS_TAC
[`s:complex->bool`; `--d:real^2`; `--k:real`] THEN
ASM_REWRITE_TAC[DOT_LNEG; REAL_LE_NEG2; REAL_LT_NEG2; VECTOR_NEG_EQ_0] THEN
ASM_REAL_ARITH_TAC) in
REPEAT STRIP_TAC THEN MATCH_MP_TAC ANALYTIC_IMP_HOLOMORPHIC THEN
MATCH_MP_TAC MORERA_LOCAL_TRIANGLE THEN
X_GEN_TAC `p:complex` THEN DISCH_TAC THEN
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_CONTAINS_BALL]) THEN
DISCH_THEN(MP_TAC o SPEC `p:complex`) THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `ball(p:complex,e)` THEN
ASM_REWRITE_TAC[OPEN_BALL; CENTRE_IN_BALL] THEN
CONJ_TAC THENL [ASM_MESON_TAC[CONTINUOUS_ON_SUBSET]; ALL_TAC] THEN
MAP_EVERY X_GEN_TAC [`u:complex`; `v:complex`; `w:complex`] THEN
SIMP_TAC[SUBSET_HULL; CONVEX_BALL; INSERT_SUBSET; EMPTY_SUBSET] THEN
STRIP_TAC THEN MATCH_MP_TAC lemma4 THEN
MAP_EVERY EXISTS_TAC [`ball(p:complex,e)`; `a:complex`; `k:real`] THEN
ASM_REWRITE_TAC[CONVEX_BALL; OPEN_BALL] THEN REPEAT CONJ_TAC THENL
[MATCH_MP_TAC HOLOMORPHIC_ON_SUBSET THEN
EXISTS_TAC `{z:complex | z IN s /\ a dot z < k}`;
MATCH_MP_TAC HOLOMORPHIC_ON_SUBSET THEN
EXISTS_TAC `{z:complex | z IN s /\ k < a dot z}`;
MATCH_MP_TAC CONTINUOUS_ON_SUBSET THEN
EXISTS_TAC `s:complex->bool`] THEN
ASM_REWRITE_TAC[] THEN ASM SET_TAC[]);;
let BLOCH_LEMMA = prove
(`!f a r.
&0 < r /\ f
holomorphic_on cball(a,r) /\
(!z. z
IN ball(a,r)
==> norm(
complex_derivative f z) <= &2 * norm(
complex_derivative f a))
==> ball(f(a),(&3 - &2 * sqrt(&2)) * r * norm(
complex_derivative f a))
SUBSET IMAGE f (ball(a,r))`,
SUBGOAL_THEN
`!f r.
&0 < r /\ f
holomorphic_on cball(Cx(&0),r) /\ f(Cx(&0)) = Cx(&0) /\
(!z. z
IN ball(Cx(&0),r)
==> norm(
complex_derivative f z)
<= &2 * norm(
complex_derivative f (Cx(&0))))
==> ball(Cx(&0),
(&3 - &2 * sqrt(&2)) *
r * norm(
complex_derivative f (Cx(&0))))
SUBSET IMAGE f (ball(Cx(&0),r))`
ASSUME_TAC THENL
[ALL_TAC;
REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL
[`\z. (f:complex->complex)(a + z) - f(a)`; `r:real`]) THEN
ASM_REWRITE_TAC[
COMPLEX_ADD_RID;
COMPLEX_SUB_REFL] THEN
SUBGOAL_THEN
`!z. z
IN ball(Cx(&0),r)
==>
complex_derivative (\w. f (a + w) - f a) z =
complex_derivative f (a + z)`
(fun th -> ASM_SIMP_TAC[
CENTRE_IN_BALL;
COMPLEX_ADD_RID; th])
THENL
[REWRITE_TAC[
COMPLEX_IN_BALL_0] THEN REPEAT STRIP_TAC THEN
MATCH_MP_TAC
HAS_COMPLEX_DERIVATIVE_DERIVATIVE THEN
ONCE_REWRITE_TAC [COMPLEX_RING
`
complex_derivative f z =
complex_derivative f z * (Cx(&0) + Cx(&1)) - Cx(&0)`] THEN
MATCH_MP_TAC
HAS_COMPLEX_DERIVATIVE_SUB THEN
REWRITE_TAC[
HAS_COMPLEX_DERIVATIVE_CONST] THEN
GEN_REWRITE_TAC (RATOR_CONV o LAND_CONV) [GSYM
o_DEF] THEN
MATCH_MP_TAC
COMPLEX_DIFF_CHAIN_AT THEN
SIMP_TAC[
HAS_COMPLEX_DERIVATIVE_ADD;
HAS_COMPLEX_DERIVATIVE_CONST;
HAS_COMPLEX_DERIVATIVE_ID;
HAS_COMPLEX_DERIVATIVE_DIFFERENTIABLE] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
holomorphic_on]) THEN
DISCH_THEN(MP_TAC o SPEC `a + z:complex`) THEN
ASM_SIMP_TAC[
IN_CBALL; NORM_ARITH `norm z < r ==> dist(a,a+z) <= r`] THEN
REWRITE_TAC[GSYM
complex_differentiable] THEN
DISCH_THEN(MP_TAC o SPEC `ball(a:complex,r)` o
MATCH_MP (REWRITE_RULE[
IMP_CONJ]
COMPLEX_DIFFERENTIABLE_WITHIN_SUBSET)) THEN
ASM_REWRITE_TAC[
BALL_SUBSET_CBALL] THEN MATCH_MP_TAC EQ_IMP THEN
MATCH_MP_TAC
COMPLEX_DIFFERENTIABLE_WITHIN_OPEN THEN
ASM_REWRITE_TAC[
IN_BALL;
OPEN_BALL; NORM_ARITH `dist(a,a + z) = norm z`];
ANTS_TAC THENL
[CONJ_TAC THENL
[MATCH_MP_TAC
HOLOMORPHIC_ON_SUB THEN
REWRITE_TAC[
HOLOMORPHIC_ON_CONST] THEN
GEN_REWRITE_TAC LAND_CONV [GSYM
o_DEF] THEN
MATCH_MP_TAC
HOLOMORPHIC_ON_COMPOSE_GEN THEN
EXISTS_TAC `cball(a:complex,r)` THEN
ASM_SIMP_TAC[
HOLOMORPHIC_ON_ADD;
HOLOMORPHIC_ON_ID;
HOLOMORPHIC_ON_CONST;
COMPLEX_IN_CBALL_0] THEN
REWRITE_TAC[
IN_CBALL] THEN NORM_ARITH_TAC;
X_GEN_TAC `z:complex` THEN REWRITE_TAC[
COMPLEX_IN_BALL_0] THEN
STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
ASM_REWRITE_TAC[
IN_BALL; NORM_ARITH `dist(a,a + z) = norm z`]];
REWRITE_TAC[
SUBSET;
COMPLEX_IN_BALL_0;
IN_IMAGE] THEN
REWRITE_TAC[
IN_BALL; ONCE_REWRITE_RULE[
DIST_SYM] dist] THEN
DISCH_THEN(fun th ->
X_GEN_TAC `z:complex` THEN DISCH_TAC THEN
MP_TAC(SPEC `z - (f:complex->complex) a` th)) THEN
ASM_REWRITE_TAC[COMPLEX_RING `z - a:complex = w - a <=> z = w`] THEN
DISCH_THEN(X_CHOOSE_TAC `x:complex`) THEN
EXISTS_TAC `a + x:complex` THEN
ASM_REWRITE_TAC[
COMPLEX_ADD_SUB]]]] THEN
REPEAT GEN_TAC THEN
SUBGOAL_THEN `&0 < &3 - &2 * sqrt(&2)` ASSUME_TAC THENL
[REWRITE_TAC[REAL_ARITH `&0 < a - &2 * b <=> b < a / &2`] THEN
MATCH_MP_TAC
REAL_LT_LSQRT THEN REAL_ARITH_TAC;
ALL_TAC] THEN
ASM_CASES_TAC `&0 < r` THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THEN
ASM_CASES_TAC `
complex_derivative f (Cx(&0)) = Cx(&0)` THEN
ASM_SIMP_TAC[
COMPLEX_NORM_0;
REAL_MUL_RZERO;
BALL_TRIVIAL;
EMPTY_SUBSET] THEN
ABBREV_TAC `C = &2 * norm(
complex_derivative f (Cx(&0)))` THEN
SUBGOAL_THEN `&0 < C` ASSUME_TAC THENL
[ASM_MESON_TAC[
COMPLEX_NORM_NZ; REAL_ARITH `&0 < &2 * x <=> &0 < x`];
ALL_TAC] THEN
SUBGOAL_THEN
`!z. z
IN ball(Cx(&0),r)
==> norm(
complex_derivative f z -
complex_derivative f (Cx(&0)))
<= norm(z) / (r - norm(z)) * C`
(LABEL_TAC "+") THENL
[REPEAT STRIP_TAC THEN
SUBGOAL_THEN
`!R. norm z < R /\ R < r
==> norm(
complex_derivative f z -
complex_derivative f (Cx(&0)))
<= norm(z) / (R - norm(z)) * C`
MP_TAC THENL
[REPEAT STRIP_TAC THEN
MP_TAC(ISPECL
[`
complex_derivative f`;
`cball(Cx(&0),R)`;
`circlepath(Cx(&0),R)`]
CAUCHY_INTEGRAL_FORMULA_CONVEX_SIMPLE) THEN
REWRITE_TAC[
CONVEX_CBALL;
VALID_PATH_CIRCLEPATH;
INTERIOR_CBALL;
PATHSTART_CIRCLEPATH;
PATHFINISH_CIRCLEPATH] THEN
SUBGOAL_THEN `&0 < R` ASSUME_TAC THENL
[ASM_MESON_TAC[
REAL_LET_TRANS;
NORM_POS_LE]; ALL_TAC] THEN
ASM_SIMP_TAC[
PATH_IMAGE_CIRCLEPATH;
REAL_LT_IMP_LE] THEN
REWRITE_TAC[sphere; NORM_ARITH `dist(z,w) = norm(w - z)`] THEN
REWRITE_TAC[
SUBSET;
FORALL_IN_GSPEC;
IN_CBALL;
IN_BALL;
IN_DELETE] THEN
SIMP_TAC[
WINDING_NUMBER_CIRCLEPATH;
COMPLEX_SUB_RZERO;
COMPLEX_SUB_LZERO;
dist;
NORM_NEG;
REAL_LE_REFL; MESON[
REAL_LT_REFL]
`norm z < R /\ (!w. norm w = R ==> ~(w = z)) <=> norm z < R`] THEN
REWRITE_TAC[
RIGHT_FORALL_IMP_THM;
IMP_CONJ] THEN ANTS_TAC THENL
[MATCH_MP_TAC
HOLOMORPHIC_ON_SUBSET THEN
EXISTS_TAC `ball(Cx(&0),r)` THEN CONJ_TAC THENL
[MATCH_MP_TAC
HOLOMORPHIC_COMPLEX_DERIVATIVE THEN
REWRITE_TAC[
OPEN_BALL] THEN
MATCH_MP_TAC
HOLOMORPHIC_ON_SUBSET THEN
EXISTS_TAC `cball(Cx(&0),r)` THEN ASM_REWRITE_TAC[
BALL_SUBSET_CBALL];
ASM_REWRITE_TAC[
SUBSET_BALLS;
DIST_REFL; REAL_ADD_LID]];
REWRITE_TAC[
COMPLEX_MUL_LID]] THEN
DISCH_THEN(fun th ->
MP_TAC (CONJ (SPEC `z:complex` th) (SPEC `Cx(&0)` th))) THEN
ASM_REWRITE_TAC[
COMPLEX_NORM_0;
COMPLEX_SUB_RZERO] THEN
DISCH_THEN(MP_TAC o MATCH_MP
HAS_PATH_INTEGRAL_SUB) THEN
DISCH_THEN(MP_TAC o SPEC `C * norm(z) / (R * (R - norm(z:complex)))` o
MATCH_MP (ONCE_REWRITE_RULE[
IMP_CONJ]
HAS_PATH_INTEGRAL_BOUND_CIRCLEPATH)) THEN
ASM_REWRITE_TAC[GSYM
COMPLEX_SUB_LDISTRIB] THEN
REWRITE_TAC[
COMPLEX_NORM_MUL;
COMPLEX_NORM_CX;
COMPLEX_NORM_II] THEN
REWRITE_TAC[
REAL_ABS_NUM; REAL_MUL_LID;
REAL_ABS_PI] THEN
ASM_SIMP_TAC[REAL_FIELD
`&0 < R /\ z < R
==> (C * z / (R * (R - z))) * &2 * pi * R =
&2 * pi * z / (R - z) * C`] THEN
ASM_SIMP_TAC[
REAL_LE_LMUL_EQ; REAL_ARITH `&0 < &2`;
PI_POS] THEN
DISCH_THEN MATCH_MP_TAC THEN
ASM_SIMP_TAC[
REAL_LE_MUL_EQ;
REAL_LE_DIV;
REAL_LE_MUL;
REAL_SUB_LE;
REAL_LT_IMP_LE;
NORM_POS_LE;
COMPLEX_SUB_RZERO] THEN
X_GEN_TAC `x:complex` THEN DISCH_TAC THEN
SUBGOAL_THEN `~(x = Cx(&0)) /\ ~(x = z)` STRIP_ASSUME_TAC THENL
[ASM_MESON_TAC[
REAL_LT_REFL;
COMPLEX_NORM_0]; ALL_TAC] THEN
ASM_SIMP_TAC[COMPLEX_FIELD
`~(x = Cx(&0)) /\ ~(x = z)
==> d / (x - z) - d / x = d * z / (x * (x - z))`] THEN
REWRITE_TAC[
COMPLEX_NORM_MUL] THEN MATCH_MP_TAC
REAL_LE_MUL2 THEN
ASM_SIMP_TAC[
NORM_POS_LE;
REAL_LT_IMP_LE;
IN_BALL; dist;
NORM_NEG;
COMPLEX_SUB_LZERO] THEN
REWRITE_TAC[
COMPLEX_NORM_DIV;
real_div] THEN
MATCH_MP_TAC
REAL_LE_LMUL THEN REWRITE_TAC[
NORM_POS_LE] THEN
MATCH_MP_TAC
REAL_LE_INV2 THEN
ASM_SIMP_TAC[
REAL_LT_MUL;
REAL_SUB_LT;
COMPLEX_NORM_MUL] THEN
ASM_SIMP_TAC[
REAL_LE_LMUL_EQ] THEN
UNDISCH_TAC `norm(x:complex) = R` THEN CONV_TAC NORM_ARITH;
DISCH_TAC THEN MP_TAC(ISPECL
[`\x. lift(norm(z:complex) / (drop x - norm z) * C)`;
`interval(lift((norm(z:complex) + r) / &2),lift r)`; `lift r`;
`norm(
complex_derivative f z -
complex_derivative f (Cx(&0)))`;
`1`]
CONTINUOUS_ON_CLOSURE_COMPONENT_GE) THEN
REWRITE_TAC[GSYM drop;
LIFT_DROP;
CLOSURE_INTERVAL] THEN
DISCH_THEN MATCH_MP_TAC THEN REWRITE_TAC[
INTERVAL_EQ_EMPTY_1] THEN
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
IN_BALL]) THEN
REWRITE_TAC[dist;
COMPLEX_SUB_LZERO;
NORM_NEG] THEN DISCH_TAC THEN
ASM_SIMP_TAC[
ENDS_IN_INTERVAL;
INTERVAL_EQ_EMPTY_1;
LIFT_DROP; REAL_ARITH
`z < r ==> ~(r <= (z + r) / &2) /\ ~(r < (z + r) / &2)`] THEN
REWRITE_TAC[
FORALL_LIFT;
LIFT_DROP;
IN_INTERVAL_1] THEN
CONJ_TAC THENL
[ALL_TAC;
ASM_MESON_TAC[REAL_ARITH `(z + r) / &2 < R /\ R < r ==> z < R`]] THEN
REWRITE_TAC[
LIFT_CMUL;
real_div] THEN
MATCH_MP_TAC
CONTINUOUS_ON_MUL THEN
REWRITE_TAC[
CONTINUOUS_ON_CONST;
o_DEF;
LIFT_CMUL] THEN
MATCH_MP_TAC
CONTINUOUS_ON_MUL THEN
REWRITE_TAC[
CONTINUOUS_ON_CONST;
o_DEF;
LIFT_CMUL] THEN
MATCH_MP_TAC(REWRITE_RULE[
o_DEF]
CONTINUOUS_ON_INV) THEN
SIMP_TAC[
LIFT_SUB;
CONTINUOUS_ON_SUB;
CONTINUOUS_ON_CONST;
LIFT_DROP;
CONTINUOUS_ON_LIFT_NORM_COMPOSE;
CONTINUOUS_ON_ID] THEN
REWRITE_TAC[
IN_INTERVAL_1;
DROP_CMUL;
LIFT_DROP] THEN
ASM_REAL_ARITH_TAC];
ALL_TAC] THEN
SUBGOAL_THEN
`!z. z
IN ball(Cx(&0),r)
==> (norm(z) - norm(z) pow 2 / (r - norm(z))) *
norm(
complex_derivative f (Cx(&0)))
<= norm(f z)`
(LABEL_TAC "*") THENL
[REPEAT STRIP_TAC THEN
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
IN_BALL]) THEN
REWRITE_TAC[dist;
COMPLEX_SUB_LZERO;
NORM_NEG] THEN DISCH_TAC THEN
MP_TAC(ISPECL[`\z. f(z) -
complex_derivative f (Cx(&0)) * z`;
`\z.
complex_derivative f z -
complex_derivative f (Cx(&0))`;
`linepath(Cx(&0),z)`; `ball(Cx(&0),r)`]
PATH_INTEGRAL_PRIMITIVE) THEN
REWRITE_TAC[
PATHSTART_LINEPATH;
PATHFINISH_LINEPATH] THEN ANTS_TAC THENL
[REWRITE_TAC[
VALID_PATH_LINEPATH;
PATH_IMAGE_LINEPATH] THEN
ONCE_REWRITE_TAC[COMPLEX_RING
`a -
complex_derivative f b = a -
complex_derivative f b * Cx(&1)`] THEN
CONJ_TAC THENL
[X_GEN_TAC `x:complex` THEN STRIP_TAC THEN
MATCH_MP_TAC
HAS_COMPLEX_DERIVATIVE_SUB THEN
SIMP_TAC[
HAS_COMPLEX_DERIVATIVE_LMUL_WITHIN;
HAS_COMPLEX_DERIVATIVE_ID] THEN
MATCH_MP_TAC
HAS_COMPLEX_DERIVATIVE_AT_WITHIN THEN
REWRITE_TAC[
HAS_COMPLEX_DERIVATIVE_DIFFERENTIABLE] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
holomorphic_on]) THEN
DISCH_THEN(MP_TAC o SPEC `x:complex`) THEN
ASM_SIMP_TAC[REWRITE_RULE[
SUBSET]
BALL_SUBSET_CBALL] THEN
REWRITE_TAC[GSYM
complex_differentiable] THEN
DISCH_THEN(MP_TAC o SPEC `ball(Cx(&0),r)` o
MATCH_MP (REWRITE_RULE[
IMP_CONJ]
COMPLEX_DIFFERENTIABLE_WITHIN_SUBSET)) THEN
ASM_SIMP_TAC[
COMPLEX_DIFFERENTIABLE_WITHIN_OPEN;
OPEN_BALL] THEN
REWRITE_TAC[
BALL_SUBSET_CBALL];
MATCH_MP_TAC(REWRITE_RULE[
CONVEX_CONTAINS_SEGMENT]
CONVEX_BALL) THEN
ASM_REWRITE_TAC[
CENTRE_IN_BALL]];
ALL_TAC] THEN
SIMP_TAC[
HAS_INTEGRAL_INTEGRABLE_INTEGRAL;
HAS_PATH_INTEGRAL_LINEPATH] THEN
REWRITE_TAC[
COMPLEX_SUB_RZERO;
COMPLEX_MUL_RZERO] THEN
REWRITE_TAC[linepath;
COMPLEX_CMUL;
COMPLEX_MUL_RZERO;
LIFT_DROP] THEN
REWRITE_TAC[
COMPLEX_ADD_LID;
FORALL_LIFT;
IN_INTERVAL_1;
LIFT_DROP] THEN
STRIP_TAC THEN FIRST_ASSUM(MP_TAC o
SPEC `\t. lift(norm(z:complex) pow 2 * drop t / (r - norm(z)) * C)` o
MATCH_MP (REWRITE_RULE[
IMP_CONJ]
INTEGRAL_NORM_BOUND_INTEGRAL)) THEN
REWRITE_TAC[linepath;
COMPLEX_CMUL;
COMPLEX_MUL_RZERO;
LIFT_DROP] THEN
REWRITE_TAC[
COMPLEX_ADD_LID;
FORALL_LIFT;
IN_INTERVAL_1;
LIFT_DROP] THEN
REWRITE_TAC[REAL_ARITH `a * b / c * d:real = (a / c * d) * b`] THEN
REWRITE_TAC[
LIFT_CMUL;
LIFT_DROP;
DROP_VEC] THEN
MP_TAC(ISPECL
[`\x. inv(&2) * x pow 2`; `\x:real. x`; `&0`; `&1`]
REAL_FUNDAMENTAL_THEOREM_OF_CALCULUS) THEN
REWRITE_TAC[
REAL_POS] THEN ANTS_TAC THENL
[REPEAT STRIP_TAC THEN REAL_DIFF_TAC THEN CONV_TAC NUM_REDUCE_CONV THEN
REAL_ARITH_TAC;
REWRITE_TAC[
has_real_integral;
o_DEF;
IMAGE_LIFT_REAL_INTERVAL] THEN
CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[
LIFT_DROP;
LIFT_NUM] THEN
DISCH_THEN(MP_TAC o SPEC `norm(z:complex) pow 2 / (r - norm z) * C` o
MATCH_MP
HAS_INTEGRAL_CMUL) THEN
REWRITE_TAC[
HAS_INTEGRAL_INTEGRABLE_INTEGRAL] THEN
STRIP_TAC THEN ASM_REWRITE_TAC[]] THEN
ANTS_TAC THENL
[X_GEN_TAC `t:real` THEN STRIP_TAC THEN
REWRITE_TAC[REAL_ARITH
`(z pow 2 / y * c) * t:real = (z / y * t * c) * z`] THEN
REWRITE_TAC[
COMPLEX_NORM_MUL] THEN MATCH_MP_TAC
REAL_LE_RMUL THEN
REWRITE_TAC[
NORM_POS_LE] THEN
REMOVE_THEN "+" (MP_TAC o SPEC `Cx(t) * z`) THEN
REWRITE_TAC[
IN_BALL; dist;
COMPLEX_SUB_LZERO;
NORM_NEG] THEN
SUBGOAL_THEN `norm(Cx t * z) <= norm z` ASSUME_TAC THENL
[GEN_REWRITE_TAC RAND_CONV [GSYM REAL_MUL_LID] THEN
REWRITE_TAC[
COMPLEX_NORM_MUL] THEN MATCH_MP_TAC
REAL_LE_RMUL THEN
REWRITE_TAC[
NORM_POS_LE;
COMPLEX_NORM_CX] THEN ASM_REAL_ARITH_TAC;
ALL_TAC] THEN
ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
MATCH_MP_TAC(REWRITE_RULE[
IMP_CONJ_ALT]
REAL_LE_TRANS) THEN
ASM_SIMP_TAC[
REAL_LE_RMUL_EQ; REAL_MUL_ASSOC;
real_div] THEN
ASM_REWRITE_TAC[
COMPLEX_NORM_MUL;
COMPLEX_NORM_CX;
real_abs] THEN
GEN_REWRITE_TAC LAND_CONV
[REAL_ARITH `(t * z) * w:real = (z * w) * t`] THEN
MATCH_MP_TAC
REAL_LE_RMUL THEN ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC
REAL_LE_MUL2 THEN
REWRITE_TAC[
NORM_POS_LE;
REAL_LE_INV_EQ;
REAL_SUB_LE] THEN
REWRITE_TAC[
REAL_LE_REFL] THEN CONJ_TAC THENL
[ALL_TAC; MATCH_MP_TAC
REAL_LE_INV2] THEN
ASM_REWRITE_TAC[
REAL_SUB_LT] THEN FIRST_X_ASSUM(MP_TAC o
GEN_REWRITE_RULE LAND_CONV [
COMPLEX_NORM_MUL]) THEN
REWRITE_TAC[
COMPLEX_NORM_CX] THEN ASM_REAL_ARITH_TAC;
REWRITE_TAC[
COMPLEX_SUB_RZERO]] THEN
MATCH_MP_TAC(NORM_ARITH
`abc <= norm d - e ==> norm(f - d) <= e ==> abc <= norm f`) THEN
REWRITE_TAC[
REAL_SUB_RDISTRIB;
ONCE_REWRITE_RULE[
COMPLEX_MUL_SYM]
COMPLEX_NORM_MUL] THEN
MATCH_MP_TAC(REAL_ARITH `y <= x ==> a - x <= a - y`) THEN
REWRITE_TAC[
DROP_CMUL; GSYM REAL_MUL_ASSOC;
LIFT_DROP] THEN
MATCH_MP_TAC
REAL_LE_LMUL THEN
ASM_SIMP_TAC[
REAL_LE_DIV;
REAL_SUB_LE;
REAL_LT_IMP_LE;
REAL_LE_POW_2] THEN
EXPAND_TAC "C" THEN REAL_ARITH_TAC;
ALL_TAC] THEN
MATCH_MP_TAC
SUBSET_TRANS THEN
EXISTS_TAC `
IMAGE (f:complex->complex)
(ball(Cx(&0),(&1 - sqrt(&2) / &2) * r))` THEN
SUBGOAL_THEN `&0 < &1 - sqrt(&2) / &2 /\ &1 - sqrt(&2) / &2 < &1`
STRIP_ASSUME_TAC THENL
[REWRITE_TAC[REAL_ARITH
`&0 < &1 - s / &2 /\ &1 - s / &2 < &1 <=> &0 < s /\ s < &2`] THEN
CONJ_TAC THENL
[MATCH_MP_TAC
REAL_LT_RSQRT; MATCH_MP_TAC
REAL_LT_LSQRT] THEN
REAL_ARITH_TAC;
ALL_TAC] THEN
CONJ_TAC THENL
[ALL_TAC;
MATCH_MP_TAC
IMAGE_SUBSET THEN MATCH_MP_TAC
SUBSET_BALL THEN
REWRITE_TAC[REAL_ARITH `x * r <= r <=> &0 <= r * (&1 - x)`] THEN
MATCH_MP_TAC
REAL_LE_MUL THEN ASM_REAL_ARITH_TAC] THEN
FIRST_ASSUM(fun th ->
GEN_REWRITE_TAC (LAND_CONV o RAND_CONV o LAND_CONV) [SYM th]) THEN
MATCH_MP_TAC
BALL_SUBSET_OPEN_MAP_IMAGE THEN
ASM_SIMP_TAC[
REAL_LT_MUL;
BOUNDED_BALL;
CLOSURE_BALL;
CENTRE_IN_BALL] THEN
REPEAT CONJ_TAC THENL
[MATCH_MP_TAC
CONTINUOUS_ON_SUBSET THEN
EXISTS_TAC `cball(Cx(&0),r)` THEN
ASM_SIMP_TAC[
HOLOMORPHIC_ON_IMP_CONTINUOUS_ON] THEN
MATCH_MP_TAC
SUBSET_CBALL THEN
REWRITE_TAC[REAL_ARITH `x * r <= r <=> &0 <= r * (&1 - x)`] THEN
MATCH_MP_TAC
REAL_LE_MUL THEN ASM_REAL_ARITH_TAC;
MATCH_MP_TAC(REWRITE_RULE[
RIGHT_IMP_FORALL_THM; IMP_IMP]
OPEN_MAPPING_THM) THEN
EXISTS_TAC `ball(Cx(&0),r)` THEN
ASM_SIMP_TAC[
OPEN_BALL;
CONNECTED_BALL;
INTERIOR_OPEN;
SUBSET_REFL] THEN
REPEAT CONJ_TAC THENL
[ASM_MESON_TAC[
HOLOMORPHIC_ON_SUBSET;
BALL_SUBSET_CBALL];
ALL_TAC;
MATCH_MP_TAC
SUBSET_BALL THEN
REWRITE_TAC[REAL_ARITH `x * r <= r <=> &0 <= r * (&1 - x)`] THEN
MATCH_MP_TAC
REAL_LE_MUL THEN ASM_REAL_ARITH_TAC] THEN
DISCH_THEN(X_CHOOSE_TAC `y:complex`) THEN
MP_TAC(ISPECL
[`f:complex->complex`; `(\x. y):complex->complex`;
`ball(Cx(&0),r)`; `Cx(&0)`]
COMPLEX_DERIVATIVE_TRANSFORM_WITHIN_OPEN) THEN
ASM_REWRITE_TAC[
OPEN_BALL;
HOLOMORPHIC_ON_CONST;
COMPLEX_DERIVATIVE_CONST;
CENTRE_IN_BALL] THEN
ASM_MESON_TAC[
HOLOMORPHIC_ON_SUBSET;
BALL_SUBSET_CBALL];
REPEAT(MATCH_MP_TAC
REAL_LT_MUL THEN CONJ_TAC) THEN
ASM_REWRITE_TAC[REAL_ARITH `&0 < &3 - &2 * s <=> s < &3 / &2`] THEN
ASM_REAL_ARITH_TAC;
ALL_TAC] THEN
ASM_SIMP_TAC[
FRONTIER_BALL; sphere;
REAL_LT_MUL; dist;
IN_ELIM_THM] THEN
X_GEN_TAC `z:complex` THEN REWRITE_TAC[
COMPLEX_SUB_LZERO;
NORM_NEG] THEN
DISCH_TAC THEN REMOVE_THEN "*" (MP_TAC o SPEC `z:complex`) THEN
ASM_REWRITE_TAC[
IN_BALL; dist;
COMPLEX_SUB_LZERO;
COMPLEX_SUB_RZERO] THEN
ASM_REWRITE_TAC[
NORM_NEG] THEN ANTS_TAC THENL
[REWRITE_TAC[REAL_ARITH `x * r < r <=> &0 < r * (&1 - x)`] THEN
MATCH_MP_TAC
REAL_LT_MUL THEN ASM_REAL_ARITH_TAC;
MATCH_MP_TAC(REWRITE_RULE[
IMP_CONJ]
REAL_LE_TRANS)] THEN
REWRITE_TAC[REAL_MUL_ASSOC] THEN MATCH_MP_TAC
REAL_LE_RMUL THEN
REWRITE_TAC[
NORM_POS_LE; REAL_ARITH `r - (&1 - s) * r = s * r`] THEN
REWRITE_TAC[
real_div;
REAL_INV_MUL;
REAL_INV_INV] THEN
ASM_SIMP_TAC[
REAL_LE_RMUL_EQ; REAL_FIELD
`&0 < r
==> a * r - (b * r) pow 2 * x * inv r = (a - b pow 2 * x) * r`] THEN
MATCH_MP_TAC
REAL_EQ_IMP_LE THEN
MP_TAC(SPEC `&2`
SQRT_WORKS) THEN CONV_TAC REAL_FIELD);;
let SCHOTTKY = prove
(`!f r. f
holomorphic_on cball(Cx(&0),&1) /\ norm(f(Cx(&0))) <= r /\
(!z. z
IN cball(Cx(&0),&1) ==> ~(f z = Cx(&0) \/ f z = Cx(&1)))
==> !t z. &0 < t /\ t < &1 /\ norm(z) <= t
==> norm(f z)
<= exp(pi * exp(pi *
(&2 + &2 * r + &12 * t / (&1 - t))))`,
let lemma0 = prove
(`!f s a.
f holomorphic_on s /\
contractible s /\
a IN s /\
(!z. z IN s ==> ~(f z = Cx(&1)) /\ ~(f z = --Cx(&1)))
==> (?g. g holomorphic_on s /\
norm(g a) <= &1 + norm(f a) / &3 /\
(!z. z IN s ==> f z = ccos(Cx pi * g z)))`,
REPEAT GEN_TAC THEN
DISCH_THEN(X_CHOOSE_THEN `g:complex->complex` STRIP_ASSUME_TAC o MATCH_MP
CONTRACTIBLE_IMP_HOLOMORPHIC_ACS_BOUNDED) THEN
EXISTS_TAC `\z:complex. g z / Cx pi` THEN
ASM_SIMP_TAC[COMPLEX_DIV_LMUL; CX_INJ; PI_NZ; COMPLEX_NORM_DIV;
HOLOMORPHIC_ON_DIV; HOLOMORPHIC_ON_CONST; REAL_LE_LDIV_EQ;
COMPLEX_NORM_CX; REAL_ABS_PI; PI_POS] THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH
`x <= pi + a ==> a * &3 <= n * pi ==> x <= (&1 + n / &3) * pi`)) THEN
MATCH_MP_TAC REAL_LE_LMUL THEN REWRITE_TAC[NORM_POS_LE] THEN
MP_TAC PI_APPROX_32 THEN REAL_ARITH_TAC) in
let lemma1 = prove
(`!n. 0 < n ==> &0 < &n + sqrt(&n pow 2 - &1)`,
MESON_TAC[REAL_LTE_ADD; REAL_OF_NUM_LT; SQRT_POS_LE; REAL_POW_LE_1;
REAL_SUB_LE; REAL_OF_NUM_LE; LE_1]) in
let lemma2 = prove
(`!x. &0 <= x
==> ?n. 0 < n /\
abs(x - log(&n + sqrt(&n pow 2 - &1)) / pi) < &1 / &2`,
REPEAT STRIP_TAC THEN MP_TAC(SPEC
`\n. 0 < n /\ log(&n + sqrt(&n pow 2 - &1)) / pi <= x` num_MAX) THEN
SIMP_TAC[] THEN
MATCH_MP_TAC(TAUT `p /\ (q ==> r) ==> (p <=> q) ==> r`) THEN
REPEAT CONJ_TAC THENL
[EXISTS_TAC `1` THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN
ASM_REWRITE_TAC[ARITH; SQRT_0; REAL_ADD_RID; LOG_1] THEN
REWRITE_TAC[real_div; REAL_MUL_LZERO] THEN ASM_REAL_ARITH_TAC;
MP_TAC(ISPEC `exp(x * pi)` REAL_ARCH_SIMPLE) THEN
MATCH_MP_TAC MONO_EXISTS THEN
X_GEN_TAC `n:num` THEN DISCH_TAC THEN X_GEN_TAC `m:num` THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
SIMP_TAC[REAL_LE_LDIV_EQ; PI_POS] THEN
GEN_REWRITE_TAC LAND_CONV [GSYM REAL_EXP_MONO_LE] THEN
ASM_SIMP_TAC[lemma1; EXP_LOG] THEN
REWRITE_TAC[GSYM REAL_OF_NUM_LE] THEN MATCH_MP_TAC(REAL_ARITH
`e <= n /\ &0 <= x ==> m + x <= e ==> m <= n`) THEN
ASM_SIMP_TAC[SQRT_POS_LE; REAL_POW_LE_1; REAL_SUB_LE;
REAL_OF_NUM_LE; LE_1];
DISCH_THEN(X_CHOOSE_THEN `n:num`
(CONJUNCTS_THEN2 (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)
(MP_TAC o SPEC `n + 1`))) THEN
REWRITE_TAC[ARITH_RULE `~(n + 1 <= n) /\ 0 < n + 1`] THEN
REWRITE_TAC[REAL_NOT_LE; IMP_IMP] THEN
DISCH_THEN(MP_TAC o MATCH_MP (REAL_ARITH
`x < b /\ a <= x ==> b - a < &1
==> abs(x - a) < &1 / &2 \/ abs(x - b) < &1 / &2`)) THEN
ANTS_TAC THENL [ALL_TAC; ASM_MESON_TAC[ARITH_RULE `0 < n + 1`]] THEN
REWRITE_TAC[REAL_ARITH `x / pi - y / pi = (x - y) / pi`] THEN
SIMP_TAC[PI_POS; REAL_LT_LDIV_EQ; REAL_MUL_LID] THEN
MATCH_MP_TAC REAL_LET_TRANS THEN EXISTS_TAC `&3` THEN
CONJ_TAC THENL [ALL_TAC; MP_TAC PI_APPROX_32 THEN REAL_ARITH_TAC] THEN
ASM_SIMP_TAC[lemma1; GSYM LOG_DIV; ARITH_RULE `0 < n + 1`] THEN
FIRST_ASSUM(DISJ_CASES_TAC o MATCH_MP (ARITH_RULE
`0 < n ==> n = 1 \/ 2 <= n`))
THENL
[ASM_REWRITE_TAC[] THEN CONV_TAC NUM_REDUCE_CONV THEN
CONV_TAC REAL_RAT_REDUCE_CONV THEN
REWRITE_TAC[SQRT_0; REAL_ADD_RID; REAL_DIV_1] THEN
ONCE_REWRITE_TAC[GSYM REAL_EXP_MONO_LE] THEN
SIMP_TAC[EXP_LOG; REAL_LTE_ADD; SQRT_POS_LE; REAL_POS; REAL_OF_NUM_LT;
ARITH] THEN
MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `&1 + &3` THEN
SIMP_TAC[REAL_EXP_LE_X; REAL_POS] THEN
REWRITE_TAC[REAL_ARITH `&2 + s <= a <=> s <= a - &2`] THEN
MATCH_MP_TAC REAL_LE_LSQRT THEN CONV_TAC REAL_RAT_REDUCE_CONV;
MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `log(&2)` THEN
CONJ_TAC THENL
[MATCH_MP_TAC LOG_MONO_LE_IMP THEN
ASM_SIMP_TAC[lemma1; ARITH_RULE `0 < n + 1`; REAL_LT_DIV;
REAL_LE_LDIV_EQ] THEN
REWRITE_TAC[GSYM REAL_OF_NUM_ADD] THEN MATCH_MP_TAC(REAL_ARITH
`&1 <= n /\ s <= &2 * t ==> (n + &1) + s <= &2 * (n + t)`) THEN
ASM_SIMP_TAC[REAL_OF_NUM_LE; LE_1] THEN
MATCH_MP_TAC REAL_LE_LSQRT THEN
ASM_SIMP_TAC[REAL_SUB_LE; REAL_POW_LE_1; REAL_ARITH `&1 <= &n + &1`;
REAL_ARITH `&0 <= &2 * x <=> &0 <= x`; REAL_POW_MUL; SQRT_POW_2;
REAL_LE_MUL; REAL_POS; SQRT_POS_LE; REAL_OF_NUM_LE; LE_1] THEN
MATCH_MP_TAC(REAL_ARITH
`&2 <= n /\ &2 * n <= n * n
==> (n + &1) pow 2 - &1 <= &2 pow 2 * (n pow 2 - &1)`) THEN
ASM_SIMP_TAC[REAL_LE_RMUL; REAL_OF_NUM_LE; LE_0];
ONCE_REWRITE_TAC[GSYM REAL_EXP_MONO_LE] THEN
SIMP_TAC[EXP_LOG; REAL_OF_NUM_LT; ARITH] THEN
MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `&1 + &3` THEN
SIMP_TAC[REAL_EXP_LE_X; REAL_POS] THEN REAL_ARITH_TAC]]]) in
let lemma3 = prove
(`!z.
z IN
({complex(m,log(&n + sqrt(&n pow 2 - &1)) / pi) | integer m /\ 0 < n}
UNION
{complex(m,--log(&n + sqrt(&n pow 2 - &1)) / pi) | integer m /\ 0 < n})
==> ccos(Cx(pi) * ccos(Cx pi * z)) = Cx(&1) \/
ccos(Cx(pi) * ccos(Cx pi * z)) = --Cx(&1)`,
REWRITE_TAC[COMPLEX_RING
`x = Cx(&1) \/ x = --Cx(&1) <=> Cx(&1) - x pow 2 = Cx(&0)`] THEN
REWRITE_TAC[COMPLEX_POW_EQ_0; ARITH_EQ; CSIN_EQ_0;
REWRITE_RULE[COMPLEX_RING
`s pow 2 + c pow 2 = Cx(&1) <=>
Cx(&1) - c pow 2 = s pow 2`] CSIN_CIRCLE] THEN
ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN REWRITE_TAC[CX_MUL] THEN
REWRITE_TAC[COMPLEX_EQ_MUL_LCANCEL; CX_INJ; PI_NZ] THEN
REWRITE_TAC[IN_UNION; TAUT `p \/ q ==> r <=> (p ==> r) /\ (q ==> r)`] THEN
REWRITE_TAC[FORALL_AND_THM; FORALL_IN_GSPEC] THEN
REWRITE_TAC[complex_mul; RE; IM; RE_CX; IM_CX; REAL_MUL_LZERO] THEN
ASM_SIMP_TAC[REAL_DIV_LMUL; PI_NZ; REAL_ADD_RID; REAL_SUB_RZERO] THEN
ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
REWRITE_TAC[ccos; COMPLEX_MUL_LNEG; CEXP_NEG] THEN CONJ_TAC THENL
[ASM_SIMP_TAC[CEXP_NZ; COMPLEX_FIELD
`~(e = Cx(&0))
==> ((e + inv e) / Cx(&2) = n <=>
inv e pow 2 - Cx(&2) * n * inv e + Cx(&1) = Cx(&0))`];
ASM_SIMP_TAC[CEXP_NZ; COMPLEX_FIELD
`~(e = Cx(&0))
==> ((e + inv e) / Cx(&2) = n <=>
e pow 2 - Cx(&2) * n * e + Cx(&1) = Cx(&0))`]] THEN
SIMP_TAC[COMPLEX_TRAD; COMPLEX_RING
`ii * (a + ii * b) = --b + ii * a`] THEN
REWRITE_TAC[GSYM COMPLEX_TRAD; GSYM CX_NEG; CEXP_COMPLEX] THEN
SIMP_TAC[REAL_EXP_NEG; EXP_LOG; lemma1] THEN
SIMP_TAC[SIN_INTEGER_PI; REAL_INV_INV] THEN
REWRITE_TAC[COMPLEX_TRAD; COMPLEX_MUL_RZERO; COMPLEX_ADD_RID] THEN
REWRITE_TAC[GSYM CX_POW; GSYM CX_MUL; GSYM CX_ADD; GSYM CX_ADD;
GSYM CX_SUB; GSYM CX_INV; CX_INJ] THEN
REWRITE_TAC[REAL_INV_MUL; REAL_INV_INV; REAL_POW_MUL] THEN
ONCE_REWRITE_TAC[GSYM COS_ABS] THEN REWRITE_TAC[REAL_ABS_MUL] THEN
MAP_EVERY X_GEN_TAC [`i:real`; `n:num`] THEN REWRITE_TAC[integer] THEN
DISCH_THEN(CONJUNCTS_THEN2
(X_CHOOSE_THEN `m:num` SUBST_ALL_TAC) ASSUME_TAC) THEN
REWRITE_TAC[GSYM integer] THEN REWRITE_TAC[real_abs; PI_POS_LE] THEN
REWRITE_TAC[COS_NPI; REAL_POW_INV; REAL_POW_POW] THEN
REWRITE_TAC[REAL_POW_NEG; EVEN_MULT; ARITH; REAL_POW_ONE] THEN
(ASM_CASES_TAC `EVEN m` THEN
ASM_REWRITE_TAC[REAL_INV_NEG; REAL_INV_1; REAL_MUL_RID] THEN
REWRITE_TAC[REAL_ARITH `a - &2 * n * x * --(&1) = a - &2 * --n * x`] THENL
[EXISTS_TAC `&n:real`; EXISTS_TAC `--(&n):real`] THEN
REWRITE_TAC[REAL_NEG_NEG; REAL_RING
`(n + s) pow 2 - &2 * n * (n + s) + &1 = &0 <=>
s pow 2 = n pow 2 - &1`] THEN
SIMP_TAC[INTEGER_CLOSED] THEN MATCH_MP_TAC SQRT_POW_2 THEN
ASM_SIMP_TAC[REAL_SUB_LE; REAL_POW_LE_1; REAL_OF_NUM_LE; LE_1])) in
REPEAT GEN_TAC THEN STRIP_TAC THEN
MP_TAC(ISPECL
[`\z:complex. Cx(&2) * f z - Cx(&1)`; `cball(Cx(&0),&1)`; `Cx(&0)`]
lemma0) THEN
ASM_SIMP_TAC[HOLOMORPHIC_ON_SUB; HOLOMORPHIC_ON_MUL;
HOLOMORPHIC_ON_CONST; CENTRE_IN_CBALL; REAL_POS;
COMPLEX_RING `Cx(&2) * z - Cx(&1) = Cx(&1) <=> z = Cx(&1)`;
COMPLEX_RING `Cx(&2) * z - Cx(&1) = --Cx(&1) <=> z = Cx(&0)`;
CONVEX_IMP_CONTRACTIBLE; CONVEX_CBALL] THEN
ANTS_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
DISCH_THEN(X_CHOOSE_THEN `h:complex->complex` STRIP_ASSUME_TAC) THEN
MP_TAC(ISPECL
[`h:complex->complex`; `cball(Cx(&0),&1)`; `Cx(&0)`]
lemma0) THEN
ASM_SIMP_TAC[CENTRE_IN_CBALL; REAL_POS; CONVEX_IMP_CONTRACTIBLE;
CONVEX_CBALL] THEN
ANTS_TAC THENL
[X_GEN_TAC `z:complex` THEN REPEAT STRIP_TAC THEN
REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `z:complex`)) THEN
ASM_REWRITE_TAC[COMPLEX_MUL_RID; COMPLEX_MUL_RNEG; CCOS_NEG;
GSYM CX_COS; COS_PI; CX_NEG] THEN
CONV_TAC COMPLEX_RING;
DISCH_THEN(X_CHOOSE_THEN `g:complex->complex` STRIP_ASSUME_TAC)] THEN
MAP_EVERY UNDISCH_TAC
[`!z. z IN cball (Cx(&0),&1)
==> Cx(&2) * f z - Cx(&1) = ccos(Cx pi * h z)`;
`!z. z IN cball(Cx(&0),&1) ==> h z = ccos(Cx pi * g z)`] THEN
SIMP_TAC[] THEN DISCH_THEN(K ALL_TAC) THEN DISCH_TAC THEN
SUBGOAL_THEN
`norm(g(Cx(&0)):complex) <= &2 + norm(f(Cx(&0)):complex)`
ASSUME_TAC THENL
[FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ]
REAL_LE_TRANS)) THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH
`h <= p ==> p / &3 <= &1 + f ==> &1 + h / &3 <= &2 + f`)) THEN
MP_TAC(ISPEC `&1` COMPLEX_NORM_CX) THEN
REWRITE_TAC[GSYM COMPLEX_CMUL] THEN CONV_TAC NORM_ARITH;
MAP_EVERY (C UNDISCH_THEN (K ALL_TAC))
[`h holomorphic_on cball(Cx(&0),&1)`;
`norm(g(Cx(&0)):complex) <= &1 + norm(h(Cx(&0)):complex) / &3`;
`norm(h(Cx(&0)):complex) <=
&1 + norm(Cx(&2) * f(Cx(&0)) - Cx(&1)) / &3`]] THEN
MAP_EVERY X_GEN_TAC [`t:real`; `z:complex`] THEN STRIP_TAC THEN
SUBGOAL_THEN `z IN ball(Cx(&0),&1)` ASSUME_TAC THENL
[REWRITE_TAC[COMPLEX_IN_BALL_0] THEN ASM_REAL_ARITH_TAC;
FIRST_ASSUM(ASSUME_TAC o MATCH_MP
(REWRITE_RULE[SUBSET] BALL_SUBSET_CBALL))] THEN
SUBGOAL_THEN
`norm(g(z) - g(Cx(&0))) <= &12 * t / (&1 - t)`
ASSUME_TAC THENL
[FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [holomorphic_on]) THEN
REWRITE_TAC[RIGHT_IMP_EXISTS_THM; SKOLEM_THM] THEN
DISCH_THEN(X_CHOOSE_TAC `g':complex->complex`) THEN
MP_TAC(ISPECL [`g:complex->complex`; `g':complex->complex`;
`linepath(Cx(&0),z)`; `cball(Cx(&0),&1)`]
PATH_INTEGRAL_PRIMITIVE) THEN
ASM_REWRITE_TAC[VALID_PATH_LINEPATH; PATH_IMAGE_LINEPATH;
PATHSTART_LINEPATH; PATHFINISH_LINEPATH] THEN
ASM_SIMP_TAC[CONVEX_CONTAINS_SEGMENT_IMP; CONVEX_CBALL] THEN
REWRITE_TAC[CENTRE_IN_CBALL; REAL_POS] THEN
DISCH_THEN(MP_TAC o SPEC `&12 / (&1 - t)` o MATCH_MP
(ONCE_REWRITE_RULE[IMP_CONJ] HAS_PATH_INTEGRAL_BOUND_LINEPATH)) THEN
ANTS_TAC THENL
[ASM_SIMP_TAC[REAL_LE_DIV; REAL_POS; REAL_SUB_LT; REAL_LT_IMP_LE] THEN
X_GEN_TAC `w:complex` THEN DISCH_TAC THEN
MP_TAC(ISPECL [`Cx(&0)`; `z:complex`; `w:complex`] SEGMENT_BOUND) THEN
ASM_REWRITE_TAC[COMPLEX_SUB_RZERO] THEN STRIP_TAC THEN
MP_TAC(ISPECL
[`g:complex->complex`; `cball(Cx(&0),&1)`; `w:complex`;
`&1 - t`; `&1`] BLOCH_COROLLARY) THEN
ASM_REWRITE_TAC[FRONTIER_CBALL; COMPLEX_IN_CBALL_0;
COMPLEX_IN_SPHERE_0] THEN
MATCH_MP_TAC(TAUT
`p /\ q /\ ~s /\ (~r ==> t) ==> (p /\ q /\ r ==> s) ==> t`) THEN
REWRITE_TAC[REAL_NOT_LE] THEN REPEAT CONJ_TAC THENL
[ASM_REAL_ARITH_TAC;
MAP_EVERY UNDISCH_TAC
[`norm(w:complex) <= norm(z:complex)`; `norm(z:complex) <= t`] THEN
CONV_TAC NORM_ARITH;
MATCH_MP_TAC(SET_RULE
`!t u. (!b. (?w. w IN t /\ w IN ball(b,&1)) \/
(?w. w IN u /\ w IN ball(b,&1))) /\
(!x. x IN d ==> ~(g x IN t UNION u))
==> ~(?b. ball(b,&1) SUBSET IMAGE g d)`) THEN
MAP_EVERY EXISTS_TAC
[`{ complex(m,log(&n + sqrt(&n pow 2 - &1)) / pi) |
integer m /\ 0 < n}`;
`{ complex(m,--log(&n + sqrt(&n pow 2 - &1)) / pi) |
integer m /\ 0 < n}`] THEN
REWRITE_TAC[EXISTS_IN_GSPEC] THEN CONJ_TAC THENL
[X_GEN_TAC `b:complex` THEN REWRITE_TAC[OR_EXISTS_THM] THEN
MP_TAC(ISPEC `Re b` INTEGER_ROUND) THEN
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `m:real` THEN
STRIP_TAC THEN ASM_REWRITE_TAC[IN_BALL] THEN
DISJ_CASES_TAC(REAL_ARITH `&0 <= Im b \/ &0 <= --(Im b)`) THENL
[MP_TAC(SPEC `Im b` lemma2); MP_TAC(SPEC `--(Im b)` lemma2)] THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN
X_GEN_TAC `n:num` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THENL
[DISJ1_TAC; DISJ2_TAC] THEN
REWRITE_TAC[dist] THEN
W(MP_TAC o PART_MATCH lhand COMPLEX_NORM_LE_RE_IM o lhand o snd) THEN
MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LET_TRANS) THEN
MATCH_MP_TAC(REAL_ARITH
`x <= &1 / &2 /\ y < &1 / &2 ==> x + y < &1`) THEN
ASM_REWRITE_TAC[RE_SUB; IM_SUB; RE; IM] THEN ASM_REAL_ARITH_TAC;
X_GEN_TAC `v:complex` THEN DISCH_TAC THEN
DISCH_THEN(DISJ_CASES_TAC o MATCH_MP lemma3) THEN
REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `v:complex`)) THEN
ASM_REWRITE_TAC[] THEN CONV_TAC COMPLEX_RING];
REWRITE_TAC[REAL_ARITH `a * c / &12 < &1 <=> c * a < &12`] THEN
ASM_SIMP_TAC[GSYM REAL_LT_RDIV_EQ; REAL_SUB_LT] THEN MATCH_MP_TAC
(NORM_ARITH `x = y ==> norm(x) < d ==> norm(y) <= d`) THEN
MATCH_MP_TAC COMPLEX_DERIVATIVE_UNIQUE_AT THEN
MAP_EVERY EXISTS_TAC [`g:complex->complex`; `w:complex`] THEN
REWRITE_TAC[HAS_COMPLEX_DERIVATIVE_DIFFERENTIABLE] THEN
MATCH_MP_TAC(TAUT `(q ==> p) /\ q ==> p /\ q`) THEN
CONJ_TAC THENL [MESON_TAC[complex_differentiable]; ALL_TAC] THEN
MATCH_MP_TAC(MESON[]
`!s. (g has_complex_derivative g') (at x within s) /\
((g has_complex_derivative g') (at x within s) <=>
(g has_complex_derivative g') (at x))
==> (g has_complex_derivative g') (at x)`) THEN
EXISTS_TAC `cball(Cx(&0),&1)` THEN CONJ_TAC THENL
[FIRST_X_ASSUM MATCH_MP_TAC THEN
REWRITE_TAC[COMPLEX_IN_CBALL_0] THEN ASM_REAL_ARITH_TAC;
REWRITE_TAC[HAS_COMPLEX_DERIVATIVE_WITHIN;
HAS_COMPLEX_DERIVATIVE_AT] THEN
MATCH_MP_TAC LIM_WITHIN_INTERIOR THEN
REWRITE_TAC[INTERIOR_CBALL; COMPLEX_IN_BALL_0] THEN
ASM_REAL_ARITH_TAC]];
MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN
ONCE_REWRITE_TAC[REAL_ARITH `&12 * t / s = &12 / s * t`] THEN
MATCH_MP_TAC REAL_LE_LMUL THEN
ASM_SIMP_TAC[REAL_LE_DIV; REAL_POS; REAL_SUB_LT; REAL_LT_IMP_LE] THEN
ASM_REWRITE_TAC[COMPLEX_SUB_RZERO]];
GEN_REWRITE_TAC (LAND_CONV o RAND_CONV)
[COMPLEX_RING `y = (Cx(&1) + (Cx(&2) * y - Cx(&1))) / Cx(&2)`] THEN
ASM_SIMP_TAC[COMPLEX_NORM_DIV; COMPLEX_NORM_CX; REAL_ABS_NUM] THEN
ONCE_REWRITE_TAC[REAL_ARITH `x / &2 <= y <=> x <= &2 * y`] THEN
W(MP_TAC o PART_MATCH lhand NORM_CCOS_PLUS1_LE o lhand o snd) THEN
MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN
MATCH_MP_TAC REAL_LE_LMUL THEN
REWRITE_TAC[REAL_POS; REAL_EXP_MONO_LE; COMPLEX_NORM_MUL] THEN
REWRITE_TAC[COMPLEX_NORM_CX; REAL_ABS_PI] THEN
MATCH_MP_TAC REAL_LE_LMUL THEN REWRITE_TAC[PI_POS_LE] THEN
W(MP_TAC o PART_MATCH lhand NORM_CCOS_LE o lhand o snd) THEN
MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN
REWRITE_TAC[REAL_EXP_MONO_LE; COMPLEX_NORM_MUL] THEN
REWRITE_TAC[COMPLEX_NORM_CX; REAL_ABS_PI] THEN
MATCH_MP_TAC REAL_LE_LMUL THEN REWRITE_TAC[PI_POS_LE] THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (NORM_ARITH
`norm(z - w) <= c ==> norm w <= a + b ==> norm z <= a + b + c`)) THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ]
REAL_LE_TRANS)) THEN
UNDISCH_TAC `norm(f(Cx(&0)):complex) <= r` THEN
CONV_TAC NORM_ARITH]);;
let HOLOMORPHIC_PERIODIC_FIXPOINT = prove
(`!f p. f
holomorphic_on (:complex) /\ ~(p = Cx(&0)) /\ (!z. f(z + p) = f(z))
==> ?x. f(x) = x`,
REWRITE_TAC[MESON[] `(?x. P x) <=> ~(!x. ~P x)`] THEN REPEAT STRIP_TAC THEN
MP_TAC(ISPECL
[`\z:complex. f(z) - z`; `Cx(&0)`; `p:complex`]
LITTLE_PICARD) THEN
ASM_SIMP_TAC[
HOLOMORPHIC_ON_SUB;
HOLOMORPHIC_ON_ID; NOT_IMP] THEN
REWRITE_TAC[SET_RULE `
IMAGE f
UNIV INTER {a,b} = {} <=>
!x. ~(f x = a) /\ ~(f x = b)`] THEN
CONJ_TAC THENL
[REWRITE_TAC[COMPLEX_RING `a - b:complex = c <=> a = b + c`;
COMPLEX_ADD_RID] THEN
ASM_MESON_TAC[];
REWRITE_TAC[
NOT_EXISTS_THM;
FUN_EQ_THM] THEN GEN_TAC THEN
DISCH_THEN(fun th ->
MP_TAC(SPEC `p + p:complex` th) THEN
MP_TAC(SPEC `p:complex` th)) THEN
ASM_REWRITE_TAC[] THEN
UNDISCH_TAC `~(p = Cx(&0))` THEN CONV_TAC COMPLEX_RING]);;
let MONTEL = prove
(`!(f:num->complex->complex) p s.
open s /\ (!h. h
IN p ==> h
holomorphic_on s) /\
(!k. compact k /\ k
SUBSET s
==> ?b. !h z. h
IN p /\ z
IN k ==> norm(h z) <= b) /\
(!n. (f n)
IN p)
==> ?g r. g
holomorphic_on s /\
(!m n:num. m < n ==> r m < r n) /\
(!x. x
IN s ==> ((\n. f (r n) x) --> g(x)) sequentially) /\
(!k e. compact k /\ k
SUBSET s /\ &0 < e
==> ?N. !n x. n >= N /\ x
IN k
==> norm(f (r n) x - g x) < e)`,
REPEAT GEN_TAC THEN
REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
SPEC_TAC(`f:num->complex->complex`,`f:num->complex->complex`) THEN
REWRITE_TAC[
LIM_SEQUENTIALLY] THEN
REWRITE_TAC[
RIGHT_IMP_FORALL_THM; IMP_IMP; GSYM
GE; dist] THEN
FIRST_ASSUM(MP_TAC o MATCH_MP
OPEN_UNION_COMPACT_SUBSETS) THEN
DISCH_THEN(X_CHOOSE_THEN `k:num->complex->bool`
(fun th -> FIRST_X_ASSUM(MP_TAC o GEN `i:num `o
SPEC `(k:num->complex->bool) i`) THEN
STRIP_ASSUME_TAC th)) THEN
ASM_REWRITE_TAC[
SKOLEM_THM;
LEFT_IMP_EXISTS_THM] THEN
X_GEN_TAC `B:num->real` THEN DISCH_TAC THEN
SUBGOAL_THEN
`!(f:num->complex->complex) (i:num).
(!n. f n
IN p)
==> ?r g. (!m n:num. m < n ==> r m < r n) /\
(!e. &0 < e ==> ?N. !n x. n >= N /\ x
IN k i
==> norm((f o r) n x - g x) < e)`
MP_TAC THENL
[REPEAT STRIP_TAC THEN REWRITE_TAC[
o_THM] THEN
MP_TAC(ISPECL [`f:num->complex->complex`; `(k:num->complex->bool) i`;
`(B:num->real) i`]
ARZELA_ASCOLI) THEN
ANTS_TAC THENL [ASM_SIMP_TAC[]; MESON_TAC[]] THEN
MAP_EVERY X_GEN_TAC [`z:complex`; `e:real`] THEN STRIP_TAC THEN
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
OPEN_CONTAINS_CBALL]) THEN
DISCH_THEN(MP_TAC o SPEC `z:complex`) THEN
ANTS_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[
SUBSET;
IN_CBALL]] THEN
DISCH_THEN(X_CHOOSE_THEN `r:real` STRIP_ASSUME_TAC) THEN
SUBGOAL_THEN
`?M. &0 < M /\
!n w. dist(z,w) <= &2 / &3 * r
==> norm((f:num->complex->complex) n w) <= M`
STRIP_ASSUME_TAC THENL
[FIRST_X_ASSUM(MP_TAC o SPEC `cball(z:complex,&2 / &3 * r)`) THEN
ASM_SIMP_TAC[
SUBSET;
IN_CBALL;
COMPACT_CBALL;
NORM_ARITH `dist(a,b) <= &2 / &3 * r ==> dist(a,b) <= r`] THEN
DISCH_THEN(X_CHOOSE_THEN `N:num` (MP_TAC o SPEC `N:num`)) THEN
REWRITE_TAC[
GE;
LE_REFL] THEN DISCH_TAC THEN
EXISTS_TAC `abs(B(N:num)) + &1` THEN
REWRITE_TAC[REAL_ARITH `&0 < abs x + &1`] THEN
ASM_MESON_TAC[
SUBSET; REAL_ARITH `x <= b ==> x <= abs b + &1`];
ALL_TAC] THEN
EXISTS_TAC `min (r / &3) ((e * r) / (&6 * M))` THEN
ASM_SIMP_TAC[
REAL_LT_MIN;
REAL_LT_DIV;
REAL_LT_MUL;
REAL_OF_NUM_LT; ARITH] THEN
MAP_EVERY X_GEN_TAC [`n:num`; `y:complex`] THEN STRIP_TAC THEN
MP_TAC
(ISPECL [`(f:num->complex->complex) n`; `cball(z:complex,&2 / &3 * r)`;
`circlepath(z:complex,&2 / &3 * r)`]
CAUCHY_INTEGRAL_FORMULA_CONVEX_SIMPLE) THEN
REWRITE_TAC[
CONVEX_CBALL;
VALID_PATH_CIRCLEPATH] THEN
REWRITE_TAC[
PATHSTART_CIRCLEPATH;
PATHFINISH_CIRCLEPATH] THEN
SIMP_TAC[
INTERIOR_CBALL;
IN_BALL;
WINDING_NUMBER_CIRCLEPATH;
NORM_ARITH `dist(z,w) = norm(w - z)`] THEN
ASM_SIMP_TAC[
PATH_IMAGE_CIRCLEPATH;
REAL_ARITH `&0 < r ==> &0 <= &2 / &3 * r`] THEN
REWRITE_TAC[sphere; NORM_ARITH `dist(z,w) = norm(w - z)`] THEN
SIMP_TAC[
SUBSET;
IN_CBALL;
IN_DELETE;
IN_ELIM_THM;
REAL_LE_REFL;
NORM_ARITH `dist(z,w) = norm(w - z)`] THEN
ONCE_REWRITE_TAC[TAUT `p ==> ~q <=> q ==> ~p`] THEN
SIMP_TAC[
FORALL_UNWIND_THM2;
IMP_CONJ;
REAL_LT_IMP_NE] THEN
REWRITE_TAC[
RIGHT_FORALL_IMP_THM;
COMPLEX_MUL_LID] THEN ANTS_TAC THENL
[MATCH_MP_TAC
HOLOMORPHIC_ON_SUBSET THEN
EXISTS_TAC `s:complex->bool` THEN ASM_SIMP_TAC[
SUBSET;
IN_CBALL] THEN
ASM_SIMP_TAC[NORM_ARITH `dist(a,b) <= &2 / &3 * r ==> dist(a,b) <= r`];
ALL_TAC] THEN
DISCH_THEN(fun th ->
MP_TAC(SPEC `y:complex` th) THEN MP_TAC(SPEC `z:complex` th)) THEN
ASM_SIMP_TAC[
VECTOR_SUB_REFL;
NORM_0;
REAL_LT_MUL;
REAL_LT_DIV;
REAL_OF_NUM_LT; ARITH; NORM_ARITH
`norm(z - y) < r / &3 ==> norm(y - z) < &2 / &3 * r`] THEN
REWRITE_TAC[IMP_IMP] THEN
DISCH_THEN(MP_TAC o MATCH_MP
HAS_PATH_INTEGRAL_SUB) THEN
DISCH_THEN(MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[
IMP_CONJ]
HAS_PATH_INTEGRAL_BOUND_CIRCLEPATH)) THEN
REWRITE_TAC[GSYM
COMPLEX_SUB_LDISTRIB;
COMPLEX_NORM_MUL] THEN
REWRITE_TAC[
COMPLEX_NORM_II;
COMPLEX_NORM_CX;
REAL_ABS_PI;
REAL_ABS_NUM; REAL_MUL_LID] THEN
DISCH_THEN(MP_TAC o SPEC `e / r:real`) THEN
ASM_SIMP_TAC[REAL_FIELD
`&0 < r ==> e / r * &2 * pi * c * r = &2 * pi * e * c`] THEN
SIMP_TAC[
REAL_LE_LMUL_EQ;
REAL_OF_NUM_LT; ARITH;
PI_POS] THEN
ANTS_TAC THENL [ALL_TAC; ASM_REAL_ARITH_TAC] THEN
ASM_SIMP_TAC[
REAL_LT_IMP_LE;
REAL_LT_DIV;
REAL_OF_NUM_LT; ARITH;
REAL_LT_MUL] THEN
X_GEN_TAC `w:complex` THEN STRIP_TAC THEN
SUBGOAL_THEN `~(w:complex = z) /\ ~(w = y)` STRIP_ASSUME_TAC THENL
[CONJ_TAC THEN DISCH_THEN SUBST_ALL_TAC THEN
RULE_ASSUM_TAC(REWRITE_RULE[
NORM_0;
VECTOR_SUB_REFL]) THEN
RULE_ASSUM_TAC(REWRITE_RULE[
NORM_SUB]) THEN ASM_REAL_ARITH_TAC;
ALL_TAC] THEN
ASM_SIMP_TAC[COMPLEX_FIELD
`~(w:complex = z) /\ ~(w = y)
==> (a / (w - z) - a / (w - y) =
(a * (z - y)) / ((w - z) * (w - y)))`] THEN
REWRITE_TAC[
COMPLEX_NORM_MUL;
COMPLEX_NORM_DIV] THEN
ASM_SIMP_TAC[
REAL_LE_LDIV_EQ;
REAL_LT_MUL;
NORM_POS_LT;
VECTOR_SUB_EQ;
REAL_FIELD `&0 < r ==> e / r * (&2 / &3 * r) * x = &2 / &3 * e * x`] THEN
MATCH_MP_TAC
REAL_LE_TRANS THEN
EXISTS_TAC `M * (e * r) / (&6 * M)` THEN CONJ_TAC THENL
[MATCH_MP_TAC
REAL_LE_MUL2 THEN
ASM_SIMP_TAC[
REAL_LT_IMP_LE;
NORM_POS_LE] THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN
ASM_REWRITE_TAC[NORM_ARITH `dist(x,y) = norm(y - x)`;
REAL_LE_REFL];
ASM_SIMP_TAC[REAL_FIELD `&0 < M ==> M * e / (&6 * M) = e / &6`] THEN
MATCH_MP_TAC(REAL_ARITH
`&0 < x /\ x <= y * &3 ==> x / &6 <= &2 / &3 * y`) THEN
ASM_SIMP_TAC[
REAL_LT_MUL; GSYM REAL_MUL_ASSOC;
REAL_LE_LMUL_EQ] THEN
MAP_EVERY UNDISCH_TAC
[`norm(w - z:complex) = &2 / &3 * r`;
`norm(z - y:complex) < r / &3`] THEN
CONV_TAC NORM_ARITH];
ALL_TAC] THEN
REWRITE_TAC[
RIGHT_EXISTS_AND_THM] THEN
DISCH_THEN(fun th -> X_GEN_TAC `f:num->complex->complex` THEN
DISCH_TAC THEN MP_TAC th) THEN
DISCH_THEN(MP_TAC o GENL [`i:num`; `r:num->num`] o
SPECL [`(f:num->complex->complex) o (r:num->num)`; `i:num`]) THEN
GEN_REWRITE_TAC
(LAND_CONV o funpow 2 BINDER_CONV o LAND_CONV o ONCE_DEPTH_CONV)
[
o_THM] THEN ASM_REWRITE_TAC[GSYM
o_ASSOC] THEN
DISCH_THEN(MP_TAC o MATCH_MP (ONCE_REWRITE_RULE[
IMP_CONJ]
SUBSEQUENCE_DIAGONALIZATION_LEMMA)) THEN
ANTS_TAC THENL
[SIMP_TAC[
o_THM;
GE] THEN REPEAT GEN_TAC THEN
DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN
MATCH_MP_TAC
MONO_EXISTS THEN X_GEN_TAC `g:complex->complex` THEN
MATCH_MP_TAC
MONO_FORALL THEN X_GEN_TAC `e:real` THEN
MATCH_MP_TAC MONO_IMP THEN REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_TAC `M:num`) THEN
EXISTS_TAC `
MAX M N` THEN
REWRITE_TAC[ARITH_RULE `
MAX m n <= x <=> m <= x /\ n <= x`] THEN
ASM_MESON_TAC[
LE_TRANS];
ALL_TAC] THEN
DISCH_THEN(MP_TAC o SPEC `I:num->num`) THEN
REWRITE_TAC[
I_O_ID;
RIGHT_AND_EXISTS_THM] THEN
ONCE_REWRITE_TAC[
SWAP_EXISTS_THM] THEN
MATCH_MP_TAC
MONO_EXISTS THEN X_GEN_TAC `r:num->num` THEN
REWRITE_TAC[
o_THM] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
SUBGOAL_THEN
`!x. x
IN s
==> ?l. !e. &0 < e
==> ?N:num. !n. n >= N
==> norm((f:num->complex->complex) (r n) x - l)
< e`
MP_TAC THENL
[X_GEN_TAC `z:complex` THEN DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `{z:complex}`) THEN
ASM_REWRITE_TAC[
COMPACT_SING;
SING_SUBSET] THEN
DISCH_THEN(X_CHOOSE_THEN `N:num` STRIP_ASSUME_TAC) THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
SKOLEM_THM]) THEN
DISCH_THEN(X_CHOOSE_THEN `G:num->complex->complex` MP_TAC) THEN
DISCH_THEN(LABEL_TAC "*" o SPEC `N:num`) THEN
EXISTS_TAC `(G:num->complex->complex) N z` THEN
X_GEN_TAC `e:real` THEN DISCH_TAC THEN
REMOVE_THEN "*" (MP_TAC o SPEC `e:real`) THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_THEN `M:num` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `
MAX M N` THEN
REWRITE_TAC[ARITH_RULE `a >=
MAX m n <=> a >= m /\ a >= n`] THEN
ASM_MESON_TAC[
GE_REFL];
ALL_TAC] THEN
GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [
RIGHT_IMP_EXISTS_THM] THEN
REWRITE_TAC[
SKOLEM_THM;
RIGHT_IMP_FORALL_THM; IMP_IMP] THEN
MATCH_MP_TAC
MONO_EXISTS THEN X_GEN_TAC `g:complex->complex` THEN
DISCH_TAC THEN ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC(TAUT `b /\ (b ==> a) ==> a /\ b`) THEN CONJ_TAC THENL
[MAP_EVERY X_GEN_TAC [`t:complex->bool`; `e:real`] THEN STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `t:complex->bool`) THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_TAC `N:num`) THEN
FIRST_X_ASSUM(X_CHOOSE_THEN `h:complex->complex` (LABEL_TAC "*") o
SPEC `N:num`) THEN
SUBGOAL_THEN
`!w. w
IN t ==> g w = (h:complex->complex) w`
(fun th -> ASM_MESON_TAC[
GE_REFL;
SUBSET; th]) THEN
X_GEN_TAC `w:complex` THEN DISCH_TAC THEN
MATCH_MP_TAC(ISPEC `sequentially`
LIM_UNIQUE) THEN
EXISTS_TAC `\n:num. (f:num->complex->complex)(r n) w` THEN
ASM_REWRITE_TAC[
TRIVIAL_LIMIT_SEQUENTIALLY;
LIM_SEQUENTIALLY] THEN
REWRITE_TAC[GSYM
GE; dist;
o_THM] THEN
ASM_MESON_TAC[
SUBSET;
GE_REFL];
DISCH_THEN(LABEL_TAC "*")] THEN
MATCH_MP_TAC
HOLOMORPHIC_UNIFORM_SEQUENCE THEN
EXISTS_TAC `(f:num->complex->complex) o (r:num->num)` THEN
ASM_SIMP_TAC[
o_THM] THEN X_GEN_TAC `z:complex` THEN DISCH_TAC THEN
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
OPEN_CONTAINS_CBALL]) THEN
DISCH_THEN(MP_TAC o SPEC `z:complex`) THEN ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC
MONO_EXISTS THEN X_GEN_TAC `d:real` THEN
STRIP_TAC THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN
REWRITE_TAC[
EVENTUALLY_SEQUENTIALLY] THEN ASM_MESON_TAC[
COMPACT_CBALL;
GE]);;
let BALL_BIHOLOMORPHISM_EXISTS = prove
(`!a. a
IN ball(Cx(&0),&1)
==> ?f g. f(a) = Cx(&0) /\
f
holomorphic_on ball (Cx(&0),&1) /\
(!z. z
IN ball (Cx(&0),&1) ==> f z
IN ball (Cx(&0),&1)) /\
g
holomorphic_on ball (Cx(&0),&1) /\
(!z. z
IN ball (Cx(&0),&1) ==> g z
IN ball (Cx(&0),&1)) /\
(!z. z
IN ball (Cx(&0),&1) ==> f (g z) = z) /\
(!z. z
IN ball (Cx(&0),&1) ==> g (f z) = z)`,
let BALL_BIHOLOMORPHISM_MOEBIUS_FUNCTION = prove
(`!f g.
f
holomorphic_on ball (Cx(&0),&1) /\
(!z. z
IN ball (Cx(&0),&1) ==> f z
IN ball (Cx(&0),&1)) /\
g
holomorphic_on ball (Cx(&0),&1) /\
(!z. z
IN ball (Cx(&0),&1) ==> g z
IN ball (Cx(&0),&1)) /\
(!z. z
IN ball (Cx(&0),&1) ==> f (g z) = z) /\
(!z. z
IN ball (Cx(&0),&1) ==> g (f z) = z)
==> ?t w. w
IN ball (Cx(&0),&1) /\
(!z. z
IN ball (Cx(&0),&1) ==> f z =
moebius_function t w z)`,
let HURWITZ_NO_ZEROS = prove
(`!f:num->complex->complex g s.
open s /\ connected s /\
(!n. (f n)
holomorphic_on s) /\ g
holomorphic_on s /\
(!k e. compact k /\ k
SUBSET s /\ &0 < e
==> ?N. !n x. n >= N /\ x
IN k ==> norm(f n x - g x) < e) /\
~(?c. !z. z
IN s ==> g z = c) /\
(!n z. z
IN s ==> ~(f n z = Cx(&0)))
==> (!z. z
IN s ==> ~(g z = Cx(&0)))`,
REPEAT GEN_TAC THEN STRIP_TAC THEN X_GEN_TAC `z0:complex` THEN
REPEAT DISCH_TAC THEN
MP_TAC(ISPECL [`g:complex->complex`; `s:complex->bool`; `z0:complex`]
HOLOMORPHIC_FACTOR_ZERO_NONCONSTANT) THEN ASM_REWRITE_TAC[] THEN
REWRITE_TAC[
NOT_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC [`h:complex->complex`; `r:real`; `m:num`] THEN
STRIP_TAC THEN
MP_TAC(ISPECL
[`sequentially`; `\n:num z.
complex_derivative (f n) z / f n z`;
`\z.
complex_derivative g z / g z`; `z0:complex`; `r / &2`]
PATH_INTEGRAL_UNIFORM_LIMIT_CIRCLEPATH) THEN
ASM_REWRITE_TAC[
REAL_HALF;
TRIVIAL_LIMIT_SEQUENTIALLY; NOT_IMP] THEN
SUBGOAL_THEN
`!n:num. ((\z.
complex_derivative (f n) z / f n z)
has_path_integral (Cx(&0))) (circlepath(z0,r / &2))`
ASSUME_TAC THENL
[X_GEN_TAC `n:num` THEN MATCH_MP_TAC
CAUCHY_THEOREM_DISC_SIMPLE THEN
MAP_EVERY EXISTS_TAC [`z0:complex`; `r:real`] THEN
ASM_SIMP_TAC[
VALID_PATH_CIRCLEPATH;
PATHSTART_CIRCLEPATH;
PATHFINISH_CIRCLEPATH;
PATH_IMAGE_CIRCLEPATH;
REAL_HALF;
REAL_LT_IMP_LE] THEN
REWRITE_TAC[sphere; NORM_ARITH `dist(z,w) = norm(w - z)`] THEN
REWRITE_TAC[
SUBSET;
IN_BALL; ONCE_REWRITE_RULE[
DIST_SYM] dist] THEN
ASM_SIMP_TAC[
IN_ELIM_THM; REAL_ARITH `&0 < r ==> r / &2 < r`] THEN
MATCH_MP_TAC
HOLOMORPHIC_ON_DIV THEN REPEAT CONJ_TAC THENL
[REWRITE_TAC[ETA_AX] THEN MATCH_MP_TAC
HOLOMORPHIC_COMPLEX_DERIVATIVE THEN
REWRITE_TAC[
OPEN_BALL];
REWRITE_TAC[ETA_AX];
ASM_MESON_TAC[
SUBSET]] THEN
ASM_MESON_TAC[
HOLOMORPHIC_ON_SUBSET];
ALL_TAC] THEN
REPEAT CONJ_TAC THENL
[MATCH_MP_TAC
ALWAYS_EVENTUALLY THEN
REWRITE_TAC[
path_integrable_on] THEN ASM_MESON_TAC[];
MATCH_MP_TAC
UNIFORM_LIM_COMPLEX_DIV THEN
REWRITE_TAC[
LEFT_EXISTS_AND_THM;
CONJ_ASSOC] THEN
REWRITE_TAC[
RIGHT_EXISTS_AND_THM; GSYM
CONJ_ASSOC] THEN
REWRITE_TAC[
LEFT_EXISTS_AND_THM] THEN
ASM_SIMP_TAC[
PATH_IMAGE_CIRCLEPATH;
REAL_HALF;
REAL_LT_IMP_LE] THEN
REWRITE_TAC[sphere; NORM_ARITH `dist(z,w) = norm(w - z)`] THEN
REWRITE_TAC[
IN_ELIM_THM] THEN REPEAT CONJ_TAC THENL
[MP_TAC(ISPEC `
IMAGE (
complex_derivative g) {w | norm(w - z0) = r / &2}`
COMPACT_IMP_BOUNDED) THEN
ANTS_TAC THENL
[MATCH_MP_TAC
COMPACT_CONTINUOUS_IMAGE THEN
REWRITE_TAC[
o_DEF;
REWRITE_RULE[sphere; NORM_ARITH `dist(w:real^N,z) = norm(z - w)`]
COMPACT_SPHERE] THEN
MATCH_MP_TAC
HOLOMORPHIC_ON_IMP_CONTINUOUS_ON THEN
MATCH_MP_TAC
HOLOMORPHIC_ON_SUBSET THEN
EXISTS_TAC `s:complex->bool` THEN
ASM_SIMP_TAC[
HOLOMORPHIC_COMPLEX_DERIVATIVE] THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[
IMP_CONJ_ALT]
SUBSET_TRANS)) THEN
REWRITE_TAC[
SUBSET;
IN_BALL;
IN_ELIM_THM] THEN
UNDISCH_TAC `&0 < r` THEN CONV_TAC NORM_ARITH;
REWRITE_TAC[bounded;
FORALL_IN_IMAGE;
IN_ELIM_THM] THEN
MATCH_MP_TAC
MONO_EXISTS THEN GEN_TAC THEN DISCH_TAC THEN
MATCH_MP_TAC
ALWAYS_EVENTUALLY THEN ASM_SIMP_TAC[]];
MP_TAC(ISPEC `
IMAGE (norm o (g:complex->complex))
{w | norm(w - z0) = r / &2}`
COMPACT_ATTAINS_INF) THEN
REWRITE_TAC[
EXISTS_IN_IMAGE;
FORALL_IN_IMAGE;
IMAGE_EQ_EMPTY] THEN
REWRITE_TAC[GSYM
IMAGE_o;
FORALL_IN_GSPEC;
EXISTS_IN_GSPEC;
o_THM] THEN
ANTS_TAC THENL
[CONJ_TAC THENL
[MATCH_MP_TAC
COMPACT_CONTINUOUS_IMAGE THEN
REWRITE_TAC[
o_DEF;
REWRITE_RULE[sphere; NORM_ARITH `dist(w:real^N,z) = norm(z - w)`]
COMPACT_SPHERE] THEN
MATCH_MP_TAC
CONTINUOUS_ON_LIFT_NORM_COMPOSE THEN
MATCH_MP_TAC
HOLOMORPHIC_ON_IMP_CONTINUOUS_ON THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[
IMP_CONJ]
HOLOMORPHIC_ON_SUBSET)) THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[
IMP_CONJ_ALT]
SUBSET_TRANS)) THEN
REWRITE_TAC[
SUBSET;
IN_BALL;
IN_ELIM_THM] THEN
UNDISCH_TAC `&0 < r` THEN CONV_TAC NORM_ARITH;
REWRITE_TAC[GSYM
MEMBER_NOT_EMPTY;
IN_ELIM_THM] THEN
EXISTS_TAC `z0 + Cx(r / &2)` THEN
REWRITE_TAC[VECTOR_ARITH `(a + b) - a:real^N = b`] THEN
REWRITE_TAC[
COMPLEX_NORM_CX] THEN ASM_REAL_ARITH_TAC];
DISCH_THEN(X_CHOOSE_THEN `ww:complex` MP_TAC) THEN
STRIP_TAC THEN EXISTS_TAC `norm((g:complex->complex) ww)` THEN
ASM_SIMP_TAC[
ALWAYS_EVENTUALLY;
COMPLEX_NORM_NZ] THEN
DISCH_THEN(ASSUME_TAC o REWRITE_RULE[
COMPLEX_NORM_ZERO]) THEN
UNDISCH_TAC `!w. w
IN ball(z0,r) ==> g w = (w - z0) pow m * h w` THEN
DISCH_THEN(MP_TAC o SPEC `ww:complex`) THEN
CONV_TAC(ONCE_DEPTH_CONV SYM_CONV) THEN
ASM_SIMP_TAC[
COMPLEX_ENTIRE;
COMPLEX_POW_EQ_0] THEN
REWRITE_TAC[
IN_BALL; GSYM
COMPLEX_NORM_ZERO] THEN
ASM_REWRITE_TAC[ONCE_REWRITE_RULE[
DIST_SYM] dist] THEN
ASM_REAL_ARITH_TAC];
X_GEN_TAC `e:real` THEN DISCH_TAC THEN
FIRST_ASSUM(MP_TAC o SPECL
[`cball(z0:complex,&3 * r / &4)`; `r / &4 * e / &2`]) THEN
REWRITE_TAC[
COMPACT_CBALL] THEN ANTS_TAC THENL
[ASM_SIMP_TAC[
REAL_LT_MUL;
REAL_LT_DIV;
REAL_OF_NUM_LT; ARITH] THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[
IMP_CONJ_ALT]
SUBSET_TRANS)) THEN
REWRITE_TAC[
SUBSET;
IN_BALL;
IN_CBALL;
IN_ELIM_THM] THEN
UNDISCH_TAC `&0 < r` THEN CONV_TAC NORM_ARITH;
REWRITE_TAC[
GE;
EVENTUALLY_SEQUENTIALLY;
IN_CBALL; dist] THEN
MATCH_MP_TAC
MONO_EXISTS THEN X_GEN_TAC `N:num` THEN
MATCH_MP_TAC
MONO_FORALL THEN X_GEN_TAC `n:num` THEN
DISCH_THEN(fun th -> DISCH_TAC THEN MP_TAC th) THEN
ASM_REWRITE_TAC[] THEN DISCH_TAC] THEN
X_GEN_TAC `w:complex` THEN DISCH_TAC THEN
MP_TAC(ISPECL
[`\z. (f:num->complex->complex) n z - g z`;
`w:complex`; `Cx(&0)`; `r / &4`; `r / &4 * e / &2`; `1`]
CAUCHY_HIGHER_COMPLEX_DERIVATIVE_BOUND) THEN
REWRITE_TAC[
HIGHER_COMPLEX_DERIVATIVE_1;
COMPLEX_IN_BALL_0] THEN
ANTS_TAC THENL
[CONJ_TAC THENL [ASM_REAL_ARITH_TAC; CONV_TAC NUM_REDUCE_CONV] THEN
REPEAT CONJ_TAC THENL
[ALL_TAC;
MATCH_MP_TAC
HOLOMORPHIC_ON_IMP_CONTINUOUS_ON;
X_GEN_TAC `y:complex` THEN REWRITE_TAC[
IN_BALL] THEN
DISCH_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
MAP_EVERY UNDISCH_TAC
[`norm(w - z0:complex) = r / &2`; `dist(w:complex,y) < r / &4`] THEN
CONV_TAC NORM_ARITH] THEN
(MATCH_MP_TAC
HOLOMORPHIC_ON_SUB THEN
CONJ_TAC THEN MATCH_MP_TAC
HOLOMORPHIC_ON_SUBSET THEN
EXISTS_TAC `s:complex->bool` THEN ASM_REWRITE_TAC[ETA_AX] THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[
IMP_CONJ_ALT]
SUBSET_TRANS)) THEN
REWRITE_TAC[
SUBSET;
IN_BALL;
IN_CBALL;
IN_ELIM_THM] THEN
UNDISCH_TAC `norm(w - z0:complex) = r / &2` THEN
UNDISCH_TAC `&0 < r` THEN CONV_TAC NORM_ARITH);
CONV_TAC NUM_REDUCE_CONV THEN
ASM_SIMP_TAC[REAL_FIELD
`&0 < r /\ &0 < e
==> &1 * (r / &4 * e / &2) / (r / &4) pow 1 = e / &2`] THEN
MATCH_MP_TAC(NORM_ARITH
`x = y /\ &0 < e ==> norm(x) <= e / &2 ==> norm(y) < e`) THEN
ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC
COMPLEX_DERIVATIVE_SUB THEN CONJ_TAC THEN
MATCH_MP_TAC
HOLOMORPHIC_ON_IMP_DIFFERENTIABLE_AT THEN
EXISTS_TAC `s:complex->bool` THEN ASM_REWRITE_TAC[] THEN
FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [
SUBSET]) THEN
ASM_REWRITE_TAC[
IN_BALL; ONCE_REWRITE_RULE[
DIST_SYM] dist] THEN
ASM_REAL_ARITH_TAC];
X_GEN_TAC `e:real` THEN DISCH_TAC THEN
REWRITE_TAC[
EVENTUALLY_SEQUENTIALLY] THEN
FIRST_X_ASSUM(MP_TAC o SPECL
[`{w:complex | norm(w - z0) = r / &2}`; `e:real`]) THEN
ASM_REWRITE_TAC[
GE;
IN_ELIM_THM;
REWRITE_RULE[sphere; NORM_ARITH `dist(w:real^N,z) = norm(z - w)`]
COMPACT_SPHERE] THEN
ANTS_TAC THENL [ALL_TAC; MESON_TAC[]] THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[
IMP_CONJ_ALT]
SUBSET_TRANS)) THEN
REWRITE_TAC[
SUBSET;
IN_BALL;
IN_ELIM_THM] THEN
UNDISCH_TAC `&0 < r` THEN CONV_TAC NORM_ARITH];
FIRST_ASSUM(ASSUME_TAC o GEN `n:num` o MATCH_MP
PATH_INTEGRAL_UNIQUE o
SPEC `n:num`) THEN
DISCH_THEN(MP_TAC o CONJUNCT2) THEN
ASM_REWRITE_TAC[
LIM_CONST_EQ;
TRIVIAL_LIMIT_SEQUENTIALLY] THEN
MATCH_MP_TAC(COMPLEX_RING
`!q r. p = q /\ q = r /\ ~(r = Cx(&0)) ==> ~(Cx(&0) = p)`) THEN
MAP_EVERY EXISTS_TAC
[`
path_integral (circlepath(z0,r / &2))
(\z. Cx(&m) / (z - z0) +
complex_derivative h z / h z)`;
`Cx(&2) * Cx pi * ii * Cx(&m)`] THEN
REPEAT CONJ_TAC THENL
[MATCH_MP_TAC
PATH_INTEGRAL_EQ THEN X_GEN_TAC `w:complex` THEN
ASM_SIMP_TAC[
PATH_IMAGE_CIRCLEPATH;
IN_ELIM_THM;
REAL_HALF;
REAL_LT_IMP_LE; sphere; NORM_ARITH `dist(z,w) = norm(w - z)`] THEN
ASM_CASES_TAC `w:complex = z0` THEN
ASM_REWRITE_TAC[
VECTOR_SUB_REFL;
NORM_0] THENL
[ASM_REAL_ARITH_TAC; DISCH_TAC] THEN
SUBGOAL_THEN `w
IN ball(z0:complex,r)` ASSUME_TAC THENL
[REWRITE_TAC[
IN_BALL] THEN
MAP_EVERY UNDISCH_TAC [`norm (w - z0) = r / &2`; `&0 < r`] THEN
CONV_TAC NORM_ARITH;
ALL_TAC] THEN
ASM_SIMP_TAC[] THEN
ASM_SIMP_TAC[
COMPLEX_ENTIRE;
COMPLEX_POW_EQ_0;
COMPLEX_SUB_0;
COMPLEX_FIELD `~(y = Cx(&0)) ==> (x / y = w <=> x = y * w)`] THEN
ASM_SIMP_TAC[COMPLEX_FIELD
`~(h = Cx(&0)) ==> (m * h) * (x + y / h) = m * y + m * h * x`] THEN
MATCH_MP_TAC
HAS_COMPLEX_DERIVATIVE_DERIVATIVE THEN
MATCH_MP_TAC
HAS_COMPLEX_DERIVATIVE_TRANSFORM_WITHIN_OPEN THEN
EXISTS_TAC `\w:complex. (w - z0) pow m * h w` THEN
EXISTS_TAC `ball(z0:complex,r)` THEN ASM_SIMP_TAC[
OPEN_BALL] THEN
SUBGOAL_THEN
`(w - z0) pow m * h w * Cx(&m) / (w - z0) =
(Cx(&m) * (w - z0) pow (m - 1)) * h w`
SUBST1_TAC THENL
[MATCH_MP_TAC(COMPLEX_FIELD
`w * mm = z /\ ~(w = Cx(&0))
==> z * h * m / w = (m * mm) * h`) THEN
ASM_REWRITE_TAC[
COMPLEX_SUB_0; GSYM(CONJUNCT2
complex_pow)] THEN
AP_TERM_TAC THEN ASM_ARITH_TAC;
MATCH_MP_TAC
HAS_COMPLEX_DERIVATIVE_MUL_AT THEN CONJ_TAC THENL
[COMPLEX_DIFF_TAC THEN CONV_TAC COMPLEX_RING;
REWRITE_TAC[
HAS_COMPLEX_DERIVATIVE_DIFFERENTIABLE] THEN
ASM_MESON_TAC[
HOLOMORPHIC_ON_IMP_DIFFERENTIABLE_AT;
OPEN_BALL]]];
GEN_REWRITE_TAC RAND_CONV [GSYM
COMPLEX_ADD_RID] THEN
MATCH_MP_TAC
PATH_INTEGRAL_UNIQUE THEN
MATCH_MP_TAC
HAS_PATH_INTEGRAL_ADD THEN CONJ_TAC THENL
[MATCH_MP_TAC
CAUCHY_INTEGRAL_CIRCLEPATH_SIMPLE THEN
ASM_REWRITE_TAC[
CENTRE_IN_BALL;
REAL_HALF;
HOLOMORPHIC_ON_CONST];
MATCH_MP_TAC
CAUCHY_THEOREM_DISC_SIMPLE THEN
MAP_EVERY EXISTS_TAC [`z0:complex`; `r:real`] THEN
ASM_SIMP_TAC[
VALID_PATH_CIRCLEPATH;
PATHSTART_CIRCLEPATH;
PATHFINISH_CIRCLEPATH;
PATH_IMAGE_CIRCLEPATH;
REAL_HALF;
REAL_LT_IMP_LE] THEN
REWRITE_TAC[sphere; NORM_ARITH `dist(z,w) = norm(w - z)`] THEN
REWRITE_TAC[
SUBSET;
IN_BALL; ONCE_REWRITE_RULE[
DIST_SYM] dist] THEN
ASM_SIMP_TAC[
IN_ELIM_THM; REAL_ARITH `&0 < r ==> r / &2 < r`] THEN
MATCH_MP_TAC
HOLOMORPHIC_ON_DIV THEN ASM_REWRITE_TAC[ETA_AX] THEN
MATCH_MP_TAC
HOLOMORPHIC_COMPLEX_DERIVATIVE THEN
ASM_REWRITE_TAC[
OPEN_BALL]];
REWRITE_TAC[
COMPLEX_ENTIRE;
CX_INJ;
PI_NZ;
II_NZ; REAL_OF_NUM_EQ] THEN
ASM_SIMP_TAC[
LE_1;
ARITH_EQ] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
ASM_REWRITE_TAC[
CENTRE_IN_BALL]]]);;
let HURWITZ_INJECTIVE = prove
(`!f:num->complex->complex g s.
open s /\ connected s /\
(!n. (f n)
holomorphic_on s) /\ g
holomorphic_on s /\
(!k e. compact k /\ k
SUBSET s /\ &0 < e
==> ?N. !n x. n >= N /\ x
IN k ==> norm(f n x - g x) < e) /\
~(?c. !z. z
IN s ==> g z = c) /\
(!n w z. w
IN s /\ z
IN s /\ f n w = f n z ==> w = z)
==> (!w z. w
IN s /\ z
IN s /\ g w = g z ==> w = z)`,
REPEAT GEN_TAC THEN STRIP_TAC THEN
MAP_EVERY X_GEN_TAC [`z1:complex`; `z2:complex`] THEN
REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REPEAT DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
NOT_EXISTS_THM]) THEN
DISCH_THEN(MP_TAC o SPEC `(g:complex->complex) z2`) THEN
REWRITE_TAC[] THEN X_GEN_TAC `z0:complex` THEN
ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REPEAT DISCH_TAC THEN
RULE_ASSUM_TAC(REWRITE_RULE[MESON[]
`(!x y. x
IN s /\ y
IN s /\ g x = g y ==> x = y) <=>
(!x y. x
IN s /\ y
IN s ==> (g x = g y <=> x = y))`]) THEN
MP_TAC(ISPECL
[`\z. (g:complex->complex) z - g z1`; `s:complex->bool`;
`z2:complex`; `z0:complex`]
ISOLATED_ZEROS) THEN
ASM_SIMP_TAC[
COMPLEX_SUB_0;
HOLOMORPHIC_ON_SUB;
HOLOMORPHIC_ON_ID;
HOLOMORPHIC_ON_CONST] THEN
DISCH_THEN(X_CHOOSE_THEN `r:real` STRIP_ASSUME_TAC) THEN
MP_TAC(ISPECL
[`\n z. (f:num->complex->complex) n z - f n z1`;
`\z. (g:complex->complex) z - g z1`; `s
DELETE (z1:complex)`]
HURWITZ_NO_ZEROS) THEN
REWRITE_TAC[NOT_IMP;
COMPLEX_SUB_0] THEN REPEAT CONJ_TAC THENL
[ASM_SIMP_TAC[
OPEN_DELETE];
ASM_SIMP_TAC[
CONNECTED_OPEN_DELETE; DIMINDEX_2;
LE_REFL];
GEN_TAC THEN MATCH_MP_TAC
HOLOMORPHIC_ON_SUBSET THEN
EXISTS_TAC `s:complex->bool` THEN
ASM_SIMP_TAC[
HOLOMORPHIC_ON_SUB; ETA_AX;
HOLOMORPHIC_ON_CONST] THEN
SET_TAC[];
MATCH_MP_TAC
HOLOMORPHIC_ON_SUBSET THEN
EXISTS_TAC `s:complex->bool` THEN
ASM_SIMP_TAC[
HOLOMORPHIC_ON_SUB; ETA_AX;
HOLOMORPHIC_ON_CONST] THEN
SET_TAC[];
MAP_EVERY X_GEN_TAC [`k:complex->bool`; `e:real`] THEN STRIP_TAC THEN
FIRST_ASSUM(ASSUME_TAC o MATCH_MP (SET_RULE
`k
SUBSET s
DELETE z ==> k
SUBSET s`)) THEN
FIRST_X_ASSUM(fun th ->
MP_TAC(SPECL [`k:complex->bool`; `e / &2`] th) THEN
MP_TAC(SPECL [`{z1:complex}`; `e / &2`] th)) THEN
ASM_REWRITE_TAC[
COMPACT_SING;
SING_SUBSET;
REAL_HALF] THEN
SIMP_TAC[
IMP_CONJ;
RIGHT_FORALL_IMP_THM;
IN_SING;
FORALL_UNWIND_THM2] THEN
REWRITE_TAC[IMP_IMP;
RIGHT_IMP_FORALL_THM] THEN DISCH_THEN(CONJUNCTS_THEN2
(X_CHOOSE_TAC `N1:num`) (X_CHOOSE_TAC `N2:num`)) THEN
EXISTS_TAC `
MAX N1 N2` THEN REPEAT STRIP_TAC THEN
UNDISCH_THEN `(g:complex->complex) z1 = g z2` (SUBST1_TAC o SYM) THEN
MATCH_MP_TAC(NORM_ARITH
`norm(x1 - x2) < e / &2 /\ norm(y1 - y2) < e / &2
==> norm(x1 - y1 - (x2 - y2)) < e`) THEN
ASM_MESON_TAC[ARITH_RULE `x >=
MAX m n <=> x >= m /\ x >= n`];
REWRITE_TAC[
IN_DELETE;
COMPLEX_EQ_SUB_RADD] THEN DISCH_THEN(CHOOSE_THEN
(fun th -> MAP_EVERY (MP_TAC o C SPEC th)
[`z0:complex`; `z1:complex`; `z2:complex`])) THEN
ASM_MESON_TAC[];
REWRITE_TAC[
IN_DELETE] THEN ASM_MESON_TAC[];
REWRITE_TAC[
IN_DELETE] THEN ASM_MESON_TAC[]]);;
let GREAT_PICARD = prove
(`!f n a b z.
open n /\ z
IN n /\ ~(a = b) /\ f
holomorphic_on (n
DELETE z) /\
(!w. w
IN n
DELETE z ==> ~(f w = a) /\ ~(f w = b))
==> ?l. (f --> l) (at z) \/ ((inv o f) --> l) (at z)`,
let lemma1 = prove
(`!p q r s w.
open s /\ connected s /\ w IN s /\ &0 < r /\
(!h. h IN p
==> h holomorphic_on s /\
!z. z IN s ==> ~(h z = Cx(&0)) /\ ~(h z = Cx(&1))) /\
(!h. h IN q ==> h IN p /\ norm(h w) <= r)
==> ?B n. &0 < B /\ open n /\ w IN n /\ n SUBSET s /\
!h z. h IN q /\ z IN n ==> norm(h z) <= B`,
REPEAT STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_CONTAINS_CBALL]) THEN
DISCH_THEN(MP_TAC o SPEC `w:complex`) THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN
MAP_EVERY EXISTS_TAC
[`exp(pi * exp(pi * (&2 + &2 * r + &12)))`;
`ball(w:complex,e / &2)`] THEN
ASM_REWRITE_TAC[OPEN_BALL; CENTRE_IN_BALL; REAL_HALF] THEN
REWRITE_TAC[REAL_EXP_POS_LT] THEN CONJ_TAC THENL
[FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT]
SUBSET_TRANS)) THEN
REWRITE_TAC[SUBSET_BALLS; DIST_REFL] THEN ASM_REAL_ARITH_TAC;
ALL_TAC] THEN
MAP_EVERY X_GEN_TAC [`f:complex->complex`; `z:complex`] THEN
STRIP_TAC THEN
REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `f:complex->complex`)) THEN
ASM_CASES_TAC `(f:complex->complex) IN p` THEN ASM_REWRITE_TAC[] THEN
REPEAT STRIP_TAC THEN
MP_TAC(ISPECL
[`\z. (f:complex->complex) (w + Cx e * z)`; `r:real`]
SCHOTTKY) THEN
ASM_REWRITE_TAC[DE_MORGAN_THM; COMPLEX_MUL_RZERO; COMPLEX_ADD_RID] THEN
ANTS_TAC THENL
[CONJ_TAC THENL
[GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN
MATCH_MP_TAC HOLOMORPHIC_ON_COMPOSE THEN
SIMP_TAC[HOLOMORPHIC_ON_ADD; HOLOMORPHIC_ON_MUL;
HOLOMORPHIC_ON_CONST; HOLOMORPHIC_ON_ID] THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ]
HOLOMORPHIC_ON_SUBSET)) THEN
REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN
X_GEN_TAC `u:complex` THEN DISCH_TAC;
X_GEN_TAC `u:complex` THEN DISCH_TAC THEN
FIRST_X_ASSUM MATCH_MP_TAC] THEN
FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [SUBSET]) THEN
REWRITE_TAC[IN_CBALL; NORM_ARITH `dist(w,w + z) = norm z`] THEN
REWRITE_TAC[COMPLEX_NORM_MUL; COMPLEX_NORM_CX] THEN
ASM_SIMP_TAC[REAL_LE_LMUL_EQ; REAL_ARITH
`&0 < e ==> (abs e * u <= e <=> e * u <= e * &1)`] THEN
ASM_MESON_TAC[COMPLEX_IN_CBALL_0];
DISCH_THEN(MP_TAC o SPECL [`&1 / &2`; `Cx(inv e) * (z - w)`]) THEN
REWRITE_TAC[COMPLEX_MUL_ASSOC; GSYM CX_MUL] THEN
ASM_SIMP_TAC[REAL_MUL_RINV; COMPLEX_NORM_MUL; REAL_LT_IMP_NZ] THEN
REWRITE_TAC[COMPLEX_RING `w + Cx(&1) * (z - w) = z`] THEN
CONV_TAC REAL_RAT_REDUCE_CONV THEN DISCH_THEN MATCH_MP_TAC THEN
REWRITE_TAC[COMPLEX_NORM_CX; REAL_ABS_INV] THEN
ASM_SIMP_TAC[real_abs; REAL_LT_IMP_LE] THEN
ONCE_REWRITE_TAC[REAL_ARITH `inv e * x:real = x / e`] THEN
ASM_SIMP_TAC[REAL_LE_LDIV_EQ] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_BALL]) THEN
CONV_TAC NORM_ARITH]) in
let lemma2 = prove
(`!s t:real^N->bool.
connected t /\ ~(s = {}) /\ s SUBSET t /\ open s /\
(!x. x limit_point_of s /\ x IN t ==> x IN s)
==> s = t`,
REPEAT STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [CONNECTED_CLOPEN]) THEN
DISCH_THEN(MP_TAC o SPEC `s:real^N->bool`) THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN MATCH_MP_TAC THEN
ASM_REWRITE_TAC[CLOSED_IN_LIMPT] THEN ASM_SIMP_TAC[OPEN_SUBSET]) in
let lemma3 = prove
(`!p s w q.
open s /\ connected s /\ w IN s /\
(!h. h IN p
==> h holomorphic_on s /\
!z. z IN s ==> ~(h z = Cx(&0)) /\ ~(h z = Cx(&1))) /\
(!h. h IN q ==> h IN p /\ norm(h w) <= &1)
==> !k. compact k /\ k SUBSET s
==> ?b. !h z. h IN q /\ z IN k ==> norm(h z) <= b`,
REPEAT GEN_TAC THEN STRIP_TAC THEN
ABBREV_TAC
`u = {z | z IN s /\
?B n. &0 < B /\ open n /\ z IN n /\ n SUBSET s /\
!h:complex->complex z'.
h IN q /\ z' IN n ==> norm(h z') <= B}` THEN
SUBGOAL_THEN `(u:complex->bool) SUBSET s` ASSUME_TAC THENL
[EXPAND_TAC "u" THEN REWRITE_TAC[SUBSET_RESTRICT]; ALL_TAC] THEN
SUBGOAL_THEN `u:complex->bool = s` ASSUME_TAC THENL
[MATCH_MP_TAC lemma2 THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL
[REWRITE_TAC[GSYM MEMBER_NOT_EMPTY] THEN EXISTS_TAC `w:complex` THEN
EXPAND_TAC "u" THEN REWRITE_TAC[IN_ELIM_THM] THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC lemma1 THEN
MAP_EVERY EXISTS_TAC [`p:(complex->complex)->bool`; `&1`] THEN
ASM_REWRITE_TAC[REAL_LT_01];
ALL_TAC] THEN
CONJ_TAC THENL
[ONCE_REWRITE_TAC[OPEN_SUBOPEN] THEN X_GEN_TAC `z:complex` THEN
EXPAND_TAC "u" THEN REWRITE_TAC[IN_ELIM_THM] THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
DISCH_THEN(X_CHOOSE_THEN `B:real` MP_TAC) THEN
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `n:complex->bool` THEN
STRIP_TAC THEN ASM_REWRITE_TAC[] THEN EXPAND_TAC "u" THEN
ONCE_REWRITE_TAC[SUBSET] THEN REWRITE_TAC[IN_ELIM_THM] THEN
X_GEN_TAC `v:complex` THEN DISCH_TAC THEN
CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
MAP_EVERY EXISTS_TAC [`B:real`; `n:complex->bool`] THEN
ASM_REWRITE_TAC[];
ALL_TAC] THEN
REWRITE_TAC[SUBSET] THEN X_GEN_TAC `v:complex` THEN STRIP_TAC THEN
EXPAND_TAC "u" THEN REWRITE_TAC[IN_ELIM_THM] THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC lemma1 THEN
EXISTS_TAC `p:(complex->complex)->bool` THEN
ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[MESON[]
`(?r. P r /\ Q r) <=> ~(!r. P r ==> ~Q r)`] THEN
DISCH_THEN(MP_TAC o GEN `n:num` o SPEC `&n + &1:real`) THEN
REWRITE_TAC[REAL_ARITH `&0 < &n + &1`] THEN
GEN_REWRITE_TAC (RAND_CONV o ONCE_DEPTH_CONV) [NOT_FORALL_THM] THEN
ASM_SIMP_TAC[SKOLEM_THM] THEN
REWRITE_TAC[NOT_IMP; GSYM CONJ_ASSOC; REAL_NOT_LE] THEN
REWRITE_TAC[FORALL_AND_THM] THEN
DISCH_THEN(X_CHOOSE_THEN `f:num->complex->complex` STRIP_ASSUME_TAC) THEN
ABBREV_TAC `g:num->complex->complex = \n z. inv(f n z)` THEN
SUBGOAL_THEN `!n:num. (g n) holomorphic_on s` ASSUME_TAC THENL
[GEN_TAC THEN EXPAND_TAC "g" THEN MATCH_MP_TAC HOLOMORPHIC_ON_INV THEN
REWRITE_TAC[ETA_AX] THEN ASM SET_TAC[];
ALL_TAC] THEN
SUBGOAL_THEN
`!n:num z:complex. z IN s ==> ~(g n z = Cx(&0)) /\ ~(g n z = Cx(&1))`
STRIP_ASSUME_TAC THENL
[X_GEN_TAC `n:num` THEN EXPAND_TAC "g" THEN
REWRITE_TAC[COMPLEX_INV_EQ_0; COMPLEX_INV_EQ_1] THEN
ASM SET_TAC[];
ALL_TAC] THEN
SUBGOAL_THEN
`?B n. &0 < B /\
open n /\
v IN n /\
n SUBSET s /\
!h z. h IN {(g:num->complex->complex) n | n IN (:num)} /\ z IN n
==> norm(h z) <= B`
MP_TAC THENL
[MATCH_MP_TAC lemma1 THEN
EXISTS_TAC `{h | h holomorphic_on s /\
!z. z IN s ==> ~(h z = Cx(&0)) /\ ~(h z = Cx(&1))}` THEN
ASM_REWRITE_TAC[FORALL_IN_GSPEC] THEN EXISTS_TAC `&1` THEN
ASM_REWRITE_TAC[IN_ELIM_THM; IN_UNIV; REAL_LT_01] THEN
X_GEN_TAC `n:num` THEN EXPAND_TAC "g" THEN
REWRITE_TAC[COMPLEX_NORM_INV] THEN MATCH_MP_TAC REAL_INV_LE_1 THEN
ASM_MESON_TAC[REAL_ARITH `&n + &1 < f ==> &1 <= f`];
REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_GSPEC] THEN
REWRITE_TAC[IN_UNIV] THEN STRIP_TAC] THEN
UNDISCH_TAC `open(n:complex->bool)` THEN
REWRITE_TAC[OPEN_CONTAINS_BALL] THEN
DISCH_THEN(MP_TAC o SPEC `v:complex`) THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN
MP_TAC(ISPECL [`g:num->complex->complex`;
`{(g:num->complex->complex) n | n IN (:num)}`;
`ball(v:complex,e)`] MONTEL) THEN
ASM_REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_GSPEC] THEN
REWRITE_TAC[IN_UNIV; IMP_IMP; OPEN_BALL; GSYM CONJ_ASSOC] THEN
REWRITE_TAC[NOT_IMP] THEN CONJ_TAC THENL
[CONJ_TAC THENL [ASM_MESON_TAC[HOLOMORPHIC_ON_SUBSET]; ASM SET_TAC[]];
ALL_TAC] THEN
REWRITE_TAC[NOT_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC [`h:complex->complex`; `j:num->num`] THEN
STRIP_TAC THEN
SUBGOAL_THEN `h(v:complex) = Cx(&0)` ASSUME_TAC THENL
[MATCH_MP_TAC(ISPEC `sequentially` LIM_UNIQUE) THEN
EXISTS_TAC `\n:num. (g:num->complex->complex) (j n) v` THEN
ASM_SIMP_TAC[CENTRE_IN_BALL; TRIVIAL_LIMIT_SEQUENTIALLY] THEN
REWRITE_TAC[GSYM COMPLEX_VEC_0] THEN
MATCH_MP_TAC LIM_NULL_COMPARISON THEN EXISTS_TAC `\n. inv(&n)` THEN
REWRITE_TAC[SEQ_HARMONIC] THEN
REWRITE_TAC[EVENTUALLY_SEQUENTIALLY] THEN EXISTS_TAC `1` THEN
X_GEN_TAC `i:num` THEN DISCH_TAC THEN EXPAND_TAC "g" THEN
REWRITE_TAC[COMPLEX_NORM_INV] THEN MATCH_MP_TAC REAL_LE_INV2 THEN
ASM_SIMP_TAC[REAL_OF_NUM_LT; LE_1] THEN
TRANS_TAC REAL_LE_TRANS `&i + &1` THEN
CONJ_TAC THENL [REAL_ARITH_TAC; ALL_TAC] THEN
TRANS_TAC REAL_LE_TRANS `&((j:num->num) i) + &1` THEN
ASM_SIMP_TAC[REAL_LT_IMP_LE; REAL_LE_RADD; REAL_OF_NUM_LE] THEN
ASM_MESON_TAC[MONOTONE_BIGGER];
ALL_TAC] THEN
MP_TAC(ISPECL
[`(g:num->complex->complex) o (j:num->num)`;
`h:complex->complex`; `ball(v:complex,e)`]
HURWITZ_NO_ZEROS) THEN
ASM_REWRITE_TAC[OPEN_BALL; CONNECTED_BALL] THEN
ASM_REWRITE_TAC[NOT_IMP; o_THM] THEN REPEAT CONJ_TAC THENL
[ASM_MESON_TAC[HOLOMORPHIC_ON_SUBSET; SUBSET_TRANS];
ASM_MESON_TAC[];
ALL_TAC;
ASM SET_TAC[];
DISCH_THEN(MP_TAC o SPEC `v:complex`) THEN
ASM_REWRITE_TAC[CENTRE_IN_BALL]] THEN
DISCH_THEN(X_CHOOSE_THEN `c:complex` (fun th ->
MP_TAC th THEN MP_TAC(SPEC `v:complex` th))) THEN
ASM_REWRITE_TAC[CENTRE_IN_BALL] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN
DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [LIMPT_APPROACHABLE]) THEN
DISCH_THEN(MP_TAC o SPEC `e:real`) THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_THEN `y:complex` STRIP_ASSUME_TAC) THEN
SUBGOAL_THEN `y IN ball(v:complex,e)` ASSUME_TAC THENL
[REWRITE_TAC[IN_BALL] THEN ASM_MESON_TAC[DIST_SYM]; ALL_TAC] THEN
UNDISCH_TAC `(y:complex) IN u` THEN EXPAND_TAC "u" THEN
REWRITE_TAC[IN_ELIM_THM] THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC
(X_CHOOSE_THEN `C:real` MP_TAC)) THEN
DISCH_THEN(X_CHOOSE_THEN `nn:complex->bool` MP_TAC) THEN
REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
DISCH_THEN(MP_TAC o GEN `n:num` o
SPECL [`(f:num->complex->complex) n`; `y:complex`]) THEN
ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `{y:complex}`) THEN
ASM_REWRITE_TAC[COMPACT_SING; SING_SUBSET] THEN
DISCH_THEN(MP_TAC o SPEC `inv(C:real)`) THEN
ASM_REWRITE_TAC[REAL_LT_INV_EQ; FORALL_IN_INSERT; NOT_IN_EMPTY] THEN
DISCH_THEN(X_CHOOSE_THEN `n:num` (MP_TAC o SPEC `n:num`)) THEN
ASM_SIMP_TAC[GE; LE_REFL; COMPLEX_SUB_RZERO; REAL_NOT_LT] THEN
EXPAND_TAC "g" THEN REWRITE_TAC[COMPLEX_NORM_INV] THEN
MATCH_MP_TAC REAL_LE_INV2 THEN ASM_REWRITE_TAC[COMPLEX_NORM_NZ] THEN
ASM SET_TAC[];
X_GEN_TAC `k:complex->bool` THEN STRIP_TAC THEN
SUBGOAL_THEN
`!x:complex. x IN k ==> x IN u` MP_TAC
THENL [ASM SET_TAC[]; ALL_TAC] THEN
EXPAND_TAC "u" THEN REWRITE_TAC[IN_ELIM_THM] THEN
DISCH_THEN(MP_TAC o MATCH_MP (MESON[]
`(!x. P x ==> Q x /\ ?y z. R x y z) ==> !x. ?y z. P x ==> R x y z`)) THEN
REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC [`b:complex->real`; `n:complex->complex->bool`] THEN
DISCH_TAC THEN FIRST_ASSUM(MP_TAC o
MATCH_MP COMPACT_IMP_HEINE_BOREL) THEN
DISCH_THEN(MP_TAC o SPEC `IMAGE (n:complex->complex->bool) k`) THEN
ASM_SIMP_TAC[FORALL_IN_IMAGE; UNIONS_IMAGE] THEN
ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
ONCE_REWRITE_TAC[TAUT `p /\ q /\ r <=> q /\ p /\ r`] THEN
REWRITE_TAC[EXISTS_FINITE_SUBSET_IMAGE] THEN
DISCH_THEN(X_CHOOSE_THEN `j:complex->bool` MP_TAC) THEN
ASM_CASES_TAC `j:complex->bool = {}` THEN
ASM_REWRITE_TAC[IMAGE_CLAUSES; UNIONS_0] THENL
[SET_TAC[]; STRIP_TAC] THEN
EXISTS_TAC `sup(IMAGE (b:complex->real) j)` THEN
ASM_SIMP_TAC[REAL_LE_SUP_FINITE; IMAGE_EQ_EMPTY; FINITE_IMAGE] THEN
REWRITE_TAC[EXISTS_IN_IMAGE] THEN ASM SET_TAC[]]) in
let lemma4 = prove
(`!f k B.
&0 < k /\ f holomorphic_on ball(Cx(&0),k) DELETE Cx(&0) /\
(!e. &0 < e /\ e < k
==> ?d. &0 < d /\ d < e /\
!z. z IN sphere(Cx(&0),d) ==> norm(f z) <= B)
==> ?e. &0 < e /\ e < k /\
!z. z IN ball(Cx(&0),e) DELETE Cx(&0) ==> norm(f z) <= B`,
REPEAT STRIP_TAC THEN
FIRST_ASSUM(MP_TAC o SPEC `k / &2`) THEN
ANTS_TAC THENL [ASM_REAL_ARITH_TAC; MATCH_MP_TAC MONO_EXISTS] THEN
X_GEN_TAC `e:real` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
REWRITE_TAC[IN_DELETE; COMPLEX_IN_BALL_0] THEN
X_GEN_TAC `z:complex` THEN STRIP_TAC THEN
FIRST_ASSUM(MP_TAC o SPEC `norm(z:complex)`) THEN
REWRITE_TAC[COMPLEX_NORM_NZ] THEN ASM_REWRITE_TAC[] THEN
ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN
SUBGOAL_THEN
`!w. w IN cball(Cx(&0),e) DIFF ball(Cx(&0),d) ==> norm(f w:complex) <= B`
MATCH_MP_TAC THENL
[MATCH_MP_TAC MAXIMUM_MODULUS_FRONTIER;
ASM_REWRITE_TAC[IN_DIFF; COMPLEX_IN_BALL_0; COMPLEX_IN_CBALL_0] THEN
ASM_REAL_ARITH_TAC] THEN
SIMP_TAC[BOUNDED_CBALL; BOUNDED_DIFF; CONJ_ASSOC] THEN CONJ_TAC THENL
[SIMP_TAC[CLOSURE_CLOSED; CLOSED_DIFF; CLOSED_CBALL; OPEN_BALL] THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP
(MESON[INTERIOR_SUBSET; HOLOMORPHIC_ON_SUBSET; SUBSET_TRANS;
HOLOMORPHIC_ON_IMP_CONTINUOUS_ON]
`f holomorphic_on t ==> s SUBSET t
==> f holomorphic_on interior s /\ f continuous_on s`)) THEN
MATCH_MP_TAC(SET_RULE
`s SUBSET t /\ a IN u ==> s DIFF u SUBSET t DELETE a`) THEN
ASM_REWRITE_TAC[CENTRE_IN_BALL; SUBSET_BALLS; DIST_REFL] THEN
ASM_REAL_ARITH_TAC;
X_GEN_TAC `w:complex` THEN
ONCE_REWRITE_TAC[GSYM FRONTIER_COMPLEMENT] THEN
REWRITE_TAC[SET_RULE `UNIV DIFF (s DIFF t) = (UNIV DIFF s) UNION t`] THEN
DISCH_THEN(MP_TAC o MATCH_MP
(REWRITE_RULE[SUBSET] FRONTIER_UNION_SUBSET)) THEN
ASM_SIMP_TAC[FRONTIER_COMPLEMENT; FRONTIER_BALL; FRONTIER_CBALL] THEN
ASM SET_TAC[]]) in
let lemma5 = prove
(`!f. f holomorphic_on (ball(Cx(&0),&1) DELETE (Cx(&0))) /\
(!z. z IN ball(Cx(&0),&1) DELETE Cx(&0)
==> ~(f z = Cx(&0)) /\ ~(f z = Cx(&1)))
==> ?e b. &0 < e /\ e < &1 /\ &0 < b /\
((!z. z IN ball(Cx(&0),e) DELETE Cx(&0)
==> norm(f z) <= b) \/
(!z. z IN ball(Cx(&0),e) DELETE Cx(&0)
==> norm(f z) >= b))`,
REPEAT STRIP_TAC THEN
ABBREV_TAC `h = \n z. (f:complex->complex) (z / Cx(&n + &1))` THEN
SUBGOAL_THEN
`(!n:num. (h n) holomorphic_on ball(Cx(&0),&1) DELETE Cx(&0)) /\
(!n z. z IN ball(Cx(&0),&1) DELETE Cx(&0)
==> ~(h n z = Cx(&0)) /\ ~(h n z = Cx(&1)))`
STRIP_ASSUME_TAC THENL
[CONJ_TAC THEN X_GEN_TAC `n:num` THEN EXPAND_TAC "h" THEN SIMP_TAC[] THENL
[ONCE_REWRITE_TAC[GSYM o_DEF] THEN
MATCH_MP_TAC HOLOMORPHIC_ON_COMPOSE THEN CONJ_TAC THENL
[MATCH_MP_TAC HOLOMORPHIC_ON_DIV THEN
REWRITE_TAC[HOLOMORPHIC_ON_ID; HOLOMORPHIC_ON_CONST; CX_INJ] THEN
REAL_ARITH_TAC;
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ]
HOLOMORPHIC_ON_SUBSET))];
SUBGOAL_THEN
`!z. z IN ball (Cx(&0),&1) DELETE Cx(&0)
==> z / Cx(&n + &1) IN ball (Cx(&0),&1) DELETE Cx(&0)`
(fun th -> ASM_MESON_TAC[th])] THEN
REWRITE_TAC[IN_DELETE; FORALL_IN_IMAGE; SUBSET; COMPLEX_IN_BALL_0] THEN
SIMP_TAC[COMPLEX_DIV_EQ_0; CX_INJ; REAL_ARITH `~(&n + &1 = &0)`] THEN
SIMP_TAC[COMPLEX_NORM_DIV; COMPLEX_NORM_CX; REAL_ARITH `&0 < &n + &1`;
REAL_ARITH `abs(&n + &1) = &n + &1`; REAL_LT_LDIV_EQ] THEN
REAL_ARITH_TAC;
ALL_TAC] THEN
SUBGOAL_THEN `?w. w IN ball(Cx(&0),&1) DELETE Cx(&0)`
STRIP_ASSUME_TAC THENL
[EXISTS_TAC `Cx(&1 / &2)` THEN
REWRITE_TAC[IN_DELETE; COMPLEX_IN_BALL_0; COMPLEX_NORM_CX; CX_INJ] THEN
CONV_TAC REAL_RAT_REDUCE_CONV;
ALL_TAC] THEN
MP_TAC(ISPECL
[`{g | g holomorphic_on ball(Cx(&0),&1) DELETE Cx(&0) /\
!z. z IN ball(Cx(&0),&1) DELETE Cx(&0)
==> ~(g z = Cx(&0)) /\ ~(g z = Cx(&1))}`;
`ball(Cx(&0),&1) DELETE Cx(&0)`; `w:complex`] lemma3) THEN
ASM_REWRITE_TAC[FORALL_IN_GSPEC] THEN
SIMP_TAC[OPEN_BALL; OPEN_DELETE; CONNECTED_BALL; DIMINDEX_2; LE_REFL;
CONNECTED_OPEN_DELETE; IN_ELIM_THM] THEN
SUBGOAL_THEN
`INFINITE {n | norm((h:num->complex->complex) n w) <= &1} \/
INFINITE {n | &1 <= norm((h:num->complex->complex) n w)}`
MP_TAC THENL
[MP_TAC num_INFINITE THEN
REWRITE_TAC[INFINITE; GSYM DE_MORGAN_THM; GSYM FINITE_UNION] THEN
REWRITE_TAC[CONTRAPOS_THM] THEN MATCH_MP_TAC EQ_IMP THEN
AP_TERM_TAC THEN SIMP_TAC[EXTENSION; IN_UNIV; IN_UNION; IN_ELIM_THM] THEN
REAL_ARITH_TAC;
ALL_TAC] THEN
STRIP_TAC THEN FIRST_ASSUM(MP_TAC o MATCH_MP INFINITE_ENUMERATE_WEAK) THEN
REWRITE_TAC[LEFT_IMP_EXISTS_THM; IN_ELIM_THM] THEN
X_GEN_TAC `r:num->num` THEN STRIP_TAC THENL
[DISCH_THEN(MP_TAC o SPEC
`{(h:num->complex->complex) (r n) | n IN (:num)}`);
DISCH_THEN(MP_TAC o SPEC
`{inv o (h:num->complex->complex) (r n) | n IN (:num)}`)] THEN
ASM_REWRITE_TAC[FORALL_IN_GSPEC; IN_UNIV] THEN
ASM_SIMP_TAC[o_DEF; COMPLEX_INV_EQ_0; COMPLEX_INV_EQ_1] THEN
ASM_SIMP_TAC[COMPLEX_NORM_INV; REAL_INV_LE_1] THEN
ASM_SIMP_TAC[HOLOMORPHIC_ON_INV; ETA_AX] THEN
DISCH_THEN(MP_TAC o SPEC `sphere(Cx(&0),&1 / &2)`) THEN
(ANTS_TAC THENL
[REWRITE_TAC[SUBSET; COMPLEX_IN_SPHERE_0; IN_DELETE; COMPLEX_IN_BALL_0;
COMPACT_SPHERE; GSYM COMPLEX_NORM_NZ] THEN
SIMP_TAC[] THEN CONV_TAC REAL_RAT_REDUCE_CONV;
ALL_TAC]) THEN
REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_GSPEC] THEN
EXPAND_TAC "h" THEN REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_TAC `b:real`) THEN
ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THENL
[EXISTS_TAC `abs b + &1`; EXISTS_TAC `inv(abs b + &1)`] THEN
REWRITE_TAC[REAL_LT_INV_EQ; REAL_ARITH `&0 < abs b + &1`] THEN
REWRITE_TAC[LEFT_OR_DISTRIB; EXISTS_OR_THM] THENL
[DISJ1_TAC THEN MATCH_MP_TAC lemma4 THEN
ASM_REWRITE_TAC[REAL_LT_01];
DISJ2_TAC THEN
MP_TAC(ISPECL [`inv o (f:complex->complex)`; `&1`; `abs b + &1`]
lemma4) THEN
ASM_SIMP_TAC[HOLOMORPHIC_ON_INV; ETA_AX; o_DEF; REAL_LT_01] THEN
ANTS_TAC THENL
[ALL_TAC;
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `e:real` THEN
REWRITE_TAC[COMPLEX_NORM_INV; real_ge;
IN_DELETE; COMPLEX_IN_BALL_0] THEN
REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
GEN_REWRITE_TAC RAND_CONV [GSYM REAL_INV_INV] THEN
MATCH_MP_TAC REAL_LE_INV2 THEN ASM_SIMP_TAC[REAL_LT_INV_EQ] THEN
REWRITE_TAC[COMPLEX_NORM_NZ] THEN
MATCH_MP_TAC(TAUT `!q. ~p /\ ~q ==> ~p`) THEN
EXISTS_TAC `f(z:complex) = Cx(&1)` THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN
ASM_REWRITE_TAC[IN_DELETE; COMPLEX_IN_BALL_0] THEN
ASM_REAL_ARITH_TAC]] THEN
(X_GEN_TAC `e:real` THEN STRIP_TAC THEN
MP_TAC(ISPEC `e:real` REAL_ARCH_INV) THEN
ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
X_GEN_TAC `n:num` THEN STRIP_TAC THEN
EXISTS_TAC `inv(&2 * (&(r(n:num)) + &1))` THEN
REWRITE_TAC[REAL_LT_INV_EQ] THEN
CONJ_TAC THENL [REAL_ARITH_TAC; ALL_TAC] THEN CONJ_TAC THENL
[TRANS_TAC REAL_LET_TRANS `inv(&n)` THEN ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC REAL_LE_INV2 THEN
ASM_SIMP_TAC[REAL_OF_NUM_LT; LE_1; REAL_OF_NUM_ADD] THEN
REWRITE_TAC[REAL_OF_NUM_MUL; REAL_OF_NUM_LE] THEN
MATCH_MP_TAC(ARITH_RULE `m <= n ==> m <= 2 * (n + 1)`) THEN
ASM_MESON_TAC[MONOTONE_BIGGER];
ALL_TAC] THEN
X_GEN_TAC `z:complex` THEN REWRITE_TAC[COMPLEX_IN_SPHERE_0] THEN
DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL
[`n:num`; `Cx(&(r(n:num)) + &1) * z`]) THEN
ASM_REWRITE_TAC[COMPLEX_IN_SPHERE_0; COMPLEX_NORM_MUL; COMPLEX_NORM_CX;
REAL_ARITH `abs(&n + &1) = &n + &1`] THEN
ANTS_TAC THENL [CONV_TAC REAL_FIELD; ALL_TAC] THEN
MATCH_MP_TAC(NORM_ARITH
`x = y ==> norm x <= b ==> norm y <= abs b + &1`) THEN
REPEAT AP_TERM_TAC THEN MATCH_MP_TAC(COMPLEX_FIELD
`~(z = Cx(&0)) ==> (z * w) / z = w`) THEN
REWRITE_TAC[CX_INJ] THEN REAL_ARITH_TAC)) in
let lemma6 = prove
(`!f n a z.
open n /\ z IN n /\ ~(a = Cx(&0)) /\ f holomorphic_on (n DELETE z) /\
(!w. w IN n DELETE z ==> ~(f w = Cx(&0)) /\ ~(f w = a))
==> ?r. &0 < r /\ ball(z,r) SUBSET n /\
(bounded(IMAGE f (ball (z,r) DELETE z)) \/
bounded(IMAGE (inv o f) (ball (z,r) DELETE z)))`,
REPEAT STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_CONTAINS_BALL]) THEN
DISCH_THEN(MP_TAC o SPEC `z:complex`) THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_THEN `r:real` STRIP_ASSUME_TAC) THEN
MP_TAC(ISPEC
`\w. (f:complex->complex) (z + Cx r * w) / a` lemma5) THEN
REWRITE_TAC[] THEN ANTS_TAC THENL
[CONJ_TAC THENL
[MATCH_MP_TAC HOLOMORPHIC_ON_DIV THEN
ASM_REWRITE_TAC[HOLOMORPHIC_ON_CONST] THEN
GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN
MATCH_MP_TAC HOLOMORPHIC_ON_COMPOSE THEN
SIMP_TAC[HOLOMORPHIC_ON_ADD; HOLOMORPHIC_ON_MUL;
HOLOMORPHIC_ON_ID; HOLOMORPHIC_ON_CONST] THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ]
HOLOMORPHIC_ON_SUBSET)) THEN
REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; COMPLEX_IN_BALL_0; IN_DELETE] THEN
GEN_TAC THEN STRIP_TAC;
ASM_SIMP_TAC[COMPLEX_FIELD
`~(a = Cx(&0)) ==> (x / a = z <=> x = a * z)`] THEN
ASM_REWRITE_TAC[COMPLEX_MUL_RZERO; COMPLEX_MUL_RID] THEN
REWRITE_TAC[COMPLEX_IN_BALL_0; IN_DELETE] THEN GEN_TAC THEN
STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
REWRITE_TAC[IN_DELETE]] THEN
ASM_SIMP_TAC[CX_INJ; REAL_LT_IMP_NZ; COMPLEX_RING
`z + a * b = z <=> a = Cx(&0) \/ b = Cx(&0)`] THEN
FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [SUBSET]) THEN
REWRITE_TAC[IN_BALL; NORM_ARITH `dist(a,a + b) = norm b`] THEN
ASM_SIMP_TAC[COMPLEX_NORM_MUL; COMPLEX_NORM_CX; real_abs; REAL_LT_IMP_LE;
REAL_ARITH `r * x < r <=> &0 < r * (&1 - x)`] THEN
MATCH_MP_TAC REAL_LT_MUL THEN ASM_REAL_ARITH_TAC;
REWRITE_TAC[LEFT_IMP_EXISTS_THM; bounded; FORALL_IN_IMAGE; o_THM]] THEN
MAP_EVERY X_GEN_TAC [`e:real`; `b:real`] THEN
DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC) THEN
EXISTS_TAC `e * r:real` THEN ASM_SIMP_TAC[REAL_LT_MUL] THEN
MATCH_MP_TAC(TAUT `p /\ (p ==> q) ==> p /\ q`) THEN CONJ_TAC THENL
[FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT]
SUBSET_TRANS)) THEN
ASM_SIMP_TAC[SUBSET_BALLS; REAL_ADD_LID; DIST_REFL; REAL_LT_MUL;
REAL_SUB_LT; REAL_ARITH `&0 < r * (&1 - e) ==> e * r <= r`];
DISCH_TAC] THEN
FIRST_X_ASSUM(DISJ_CASES_THEN (LABEL_TAC "*")) THENL
[DISJ1_TAC THEN EXISTS_TAC `norm(a:complex) * b`;
DISJ2_TAC THEN EXISTS_TAC `inv(norm(a:complex) * b)`] THEN
X_GEN_TAC `w:complex` THEN REWRITE_TAC[IN_BALL; IN_DELETE] THEN
STRIP_TAC THEN
REMOVE_THEN "*" (MP_TAC o SPEC `(w - z) / Cx r`) THEN
ASM_SIMP_TAC[IN_DELETE; COMPLEX_IN_BALL_0; COMPLEX_DIV_EQ_0;
COMPLEX_SUB_0; CX_INJ; REAL_LT_IMP_NZ; COMPLEX_NORM_DIV;
COMPLEX_NORM_CX; real_abs; REAL_LT_IMP_LE; REAL_LT_LDIV_EQ;
NORM_ARITH `norm(w - z) = dist(z,w)`; COMPLEX_DIV_LMUL] THEN
REWRITE_TAC[real_ge; COMPLEX_RING `z + w - z:complex = w`] THEN
ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_LE_RDIV_EQ; COMPLEX_NORM_NZ] THEN
DISCH_TAC THEN REWRITE_TAC[COMPLEX_NORM_INV] THEN
MATCH_MP_TAC REAL_LE_INV2 THEN
ASM_SIMP_TAC[REAL_LT_MUL; COMPLEX_NORM_NZ]) in
REPEAT STRIP_TAC THEN
MP_TAC(ISPECL [`\z. (f:complex->complex) z - a`; `n:complex->bool`;
`b - a:complex`; `z:complex`]
lemma6) THEN
ASM_SIMP_TAC[COMPLEX_SUB_0; HOLOMORPHIC_ON_SUB; ETA_AX; HOLOMORPHIC_ON_CONST;
COMPLEX_RING `x - a:complex = y - a <=> x = y`] THEN
DISCH_THEN(X_CHOOSE_THEN `r:real` MP_TAC) THEN
REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
REWRITE_TAC[EXISTS_OR_THM] THEN MATCH_MP_TAC(TAUT
`(p ==> r) /\ (~r /\ q ==> s) ==> p \/ q ==> r \/ s`) THEN
REPEAT STRIP_TAC THENL
[MP_TAC(ISPECL [`f:complex->complex`; `z:complex`; `ball(z:complex,r)`]
HOLOMORPHIC_ON_EXTEND_BOUNDED) THEN
ASM_SIMP_TAC[INTERIOR_OPEN; OPEN_BALL; CENTRE_IN_BALL] THEN ANTS_TAC THENL
[FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ]
HOLOMORPHIC_ON_SUBSET)) THEN
ASM SET_TAC[];
MATCH_MP_TAC(TAUT `q /\ (p ==> r) ==> (p <=> q) ==> r`)] THEN
CONJ_TAC THENL
[REWRITE_TAC[EVENTUALLY_AT; FORALL_IN_IMAGE; IN_BALL; IN_DELETE] THEN
ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN EXISTS_TAC `r:real` THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [bounded]) THEN
REWRITE_TAC[FORALL_IN_IMAGE; IN_BALL; DIST_NZ; IN_DELETE] THEN
ASM_MESON_TAC[NORM_ARITH `norm(x - y) <= B ==> norm(x) <= norm(y) + B`;
DIST_SYM];
DISCH_THEN(X_CHOOSE_THEN `g:complex->complex` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `(g:complex->complex) z` THEN
MATCH_MP_TAC LIM_TRANSFORM_WITHIN_OPEN THEN
MAP_EVERY EXISTS_TAC [`g:complex->complex`; `ball(z:complex,r)`] THEN
ASM_SIMP_TAC[OPEN_BALL; CENTRE_IN_BALL; IN_DELETE] THEN
ASM_SIMP_TAC[GSYM CONTINUOUS_AT] THEN
ASM_MESON_TAC[HOLOMORPHIC_ON_IMP_CONTINUOUS_ON; OPEN_BALL;
CONTINUOUS_ON_EQ_CONTINUOUS_AT; CENTRE_IN_BALL]];
MP_TAC(ISPECL [`\z. inv((f:complex->complex) z - a)`;
`z:complex`; `ball(z:complex,r)`]
HOLOMORPHIC_ON_EXTEND_BOUNDED) THEN
ASM_SIMP_TAC[INTERIOR_OPEN; OPEN_BALL; CENTRE_IN_BALL] THEN ANTS_TAC THENL
[MATCH_MP_TAC HOLOMORPHIC_ON_INV THEN REWRITE_TAC[COMPLEX_SUB_0] THEN
CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN
MATCH_MP_TAC HOLOMORPHIC_ON_SUB THEN
REWRITE_TAC[HOLOMORPHIC_ON_CONST; ETA_AX] THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ]
HOLOMORPHIC_ON_SUBSET)) THEN
ASM SET_TAC[];
MATCH_MP_TAC(TAUT `q /\ (p ==> r) ==> (p <=> q) ==> r`)] THEN
CONJ_TAC THENL
[FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [bounded]) THEN
SIMP_TAC[EVENTUALLY_AT; o_DEF; FORALL_IN_IMAGE; IN_BALL; IN_DELETE] THEN
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `B:real` THEN STRIP_TAC THEN
EXISTS_TAC `r:real` THEN ASM_MESON_TAC[DIST_NZ; DIST_SYM];
DISCH_THEN(X_CHOOSE_THEN `g:complex->complex` STRIP_ASSUME_TAC)] THEN
SUBGOAL_THEN `((g:complex->complex) --> g z) (at z)` ASSUME_TAC THENL
[ASM_SIMP_TAC[GSYM CONTINUOUS_AT] THEN
ASM_MESON_TAC[HOLOMORPHIC_ON_IMP_CONTINUOUS_ON; OPEN_BALL;
CONTINUOUS_ON_EQ_CONTINUOUS_AT; CENTRE_IN_BALL];
ALL_TAC] THEN
ASM_CASES_TAC `(g:complex->complex) z = Cx(&0)` THENL
[EXISTS_TAC `Cx(&0)` THEN
MATCH_MP_TAC LIM_TRANSFORM_WITHIN_OPEN THEN
EXISTS_TAC `\w:complex. g(w) / (Cx(&1) + a * g w)` THEN
EXISTS_TAC `ball(z:complex,r)` THEN
ASM_REWRITE_TAC[OPEN_BALL; CENTRE_IN_BALL; o_DEF] THEN CONJ_TAC THENL
[X_GEN_TAC `w:complex` THEN
DISCH_TAC THEN ONCE_REWRITE_TAC[GSYM COMPLEX_INV_DIV] THEN
AP_TERM_TAC THEN MATCH_MP_TAC(COMPLEX_FIELD
`~(g = Cx(&0)) /\ inv(g) = f - a
==> (Cx(&1) + a * g) / g = f`) THEN
ASM_SIMP_TAC[IN_DELETE; COMPLEX_INV_INV; COMPLEX_INV_EQ_0] THEN
REWRITE_TAC[COMPLEX_SUB_0] THEN ASM SET_TAC[];
SUBST1_TAC(COMPLEX_FIELD
`Cx(&0) = Cx(&0) / (Cx(&1) + a * Cx(&0))`) THEN
MATCH_MP_TAC LIM_COMPLEX_DIV THEN REPEAT CONJ_TAC THENL
[ASM_MESON_TAC[]; ALL_TAC; CONV_TAC COMPLEX_RING] THEN
MATCH_MP_TAC LIM_ADD THEN REWRITE_TAC[LIM_CONST] THEN
MATCH_MP_TAC LIM_COMPLEX_MUL THEN REWRITE_TAC[LIM_CONST] THEN
ASM_MESON_TAC[]];
EXISTS_TAC `g(z:complex) / (Cx(&1) + a * g z)` THEN
MATCH_MP_TAC LIM_TRANSFORM_WITHIN_OPEN THEN
EXISTS_TAC `\w:complex. g(w) / (Cx(&1) + a * g w)` THEN
EXISTS_TAC `ball(z:complex,r)` THEN
ASM_REWRITE_TAC[OPEN_BALL; CENTRE_IN_BALL; o_DEF] THEN CONJ_TAC THENL
[X_GEN_TAC `w:complex` THEN
DISCH_TAC THEN ONCE_REWRITE_TAC[GSYM COMPLEX_INV_DIV] THEN
AP_TERM_TAC THEN MATCH_MP_TAC(COMPLEX_FIELD
`~(g = Cx(&0)) /\ inv(g) = f - a
==> (Cx(&1) + a * g) / g = f`) THEN
ASM_SIMP_TAC[IN_DELETE; COMPLEX_INV_INV; COMPLEX_INV_EQ_0] THEN
REWRITE_TAC[COMPLEX_SUB_0] THEN ASM SET_TAC[];
ALL_TAC] THEN
MATCH_MP_TAC LIM_COMPLEX_DIV THEN
ASM_SIMP_TAC[LIM_ADD; LIM_COMPLEX_MUL; LIM_CONST] THEN DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [NOT_EXISTS_THM]) THEN
DISCH_THEN(MP_TAC o SPEC `Cx(&0) / g(z:complex)`) THEN REWRITE_TAC[] THEN
MATCH_MP_TAC LIM_TRANSFORM_WITHIN_OPEN THEN
EXISTS_TAC `\w:complex. (Cx(&1) + a * g w) / g w` THEN
EXISTS_TAC `ball(z:complex,r)` THEN
ASM_REWRITE_TAC[OPEN_BALL; CENTRE_IN_BALL; o_DEF] THEN CONJ_TAC THENL
[X_GEN_TAC `w:complex` THEN
DISCH_TAC THEN MATCH_MP_TAC(COMPLEX_FIELD
`~(g = Cx(&0)) /\ inv(g) = f - a
==> (Cx(&1) + a * g) / g = f`) THEN
ASM_SIMP_TAC[IN_DELETE; COMPLEX_INV_INV; COMPLEX_INV_EQ_0] THEN
REWRITE_TAC[COMPLEX_SUB_0] THEN ASM SET_TAC[];
ALL_TAC] THEN
MATCH_MP_TAC LIM_COMPLEX_DIV THEN ASM_REWRITE_TAC[] THEN
FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN
ASM_SIMP_TAC[LIM_ADD; LIM_CONST; LIM_COMPLEX_MUL]]]);;
let MONODROMY_CONTINUOUS_LOG = prove
(`!f:complex->complex s.
open s /\ f
continuous_on s /\
(!z. z
IN s ==> ~(f z = Cx(&0)))
==> ((!p. path p /\
path_image p
SUBSET s /\
pathfinish p = pathstart p
==>
winding_number(f o p,Cx(&0)) = Cx(&0)) <=>
(?g. g
continuous_on s /\ !z. z
IN s ==> f(z) = cexp(g z)))`,
let lemma = prove
(`!f g s p.
f continuous_on s /\ g continuous_on s /\
(!z:complex. z IN s ==> f(z) = cexp(g z)) /\
path p /\ path_image p SUBSET s
==> winding_number(f o p,Cx(&0)) =
Cx(&1) / (Cx(&2) * Cx pi * ii) *
(pathfinish(g o p) - pathstart(g o p))`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC EQ_TRANS THEN
EXISTS_TAC `winding_number(cexp o g o (p:real^1->complex),Cx(&0))` THEN
CONJ_TAC THENL
[MATCH_MP_TAC WINDING_NUMBER_NEARBY_PATHS_EQ THEN
REPEAT CONJ_TAC THENL
[MATCH_MP_TAC PATH_CONTINUOUS_IMAGE THEN
REWRITE_TAC[CONTINUOUS_ON_CEXP] THEN
MATCH_MP_TAC PATH_CONTINUOUS_IMAGE THEN
ASM_MESON_TAC[CONTINUOUS_ON_SUBSET];
MATCH_MP_TAC PATH_CONTINUOUS_IMAGE THEN
ASM_MESON_TAC[CONTINUOUS_ON_SUBSET];
REWRITE_TAC[PATHSTART_COMPOSE] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
ASM_MESON_TAC[SUBSET; PATHSTART_IN_PATH_IMAGE];
REWRITE_TAC[PATHFINISH_COMPOSE] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
ASM_MESON_TAC[SUBSET; PATHFINISH_IN_PATH_IMAGE];
GEN_TAC THEN DISCH_TAC THEN REWRITE_TAC[o_THM; COMPLEX_SUB_RZERO] THEN
MATCH_MP_TAC(NORM_ARITH
`x = y /\ ~(z = vec 0) ==> norm(x - y) < norm z`) THEN
REWRITE_TAC[COMPLEX_VEC_0; CEXP_NZ] THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN
ASM_MESON_TAC[SUBSET; path_image; IN_IMAGE]];
MATCH_MP_TAC WINDING_NUMBER_COMPOSE_CEXP THEN
ASM_REWRITE_TAC[PATHSTART_COMPOSE; PATHFINISH_COMPOSE] THEN
MATCH_MP_TAC PATH_CONTINUOUS_IMAGE THEN
ASM_MESON_TAC[CONTINUOUS_ON_SUBSET]]) in
REPEAT STRIP_TAC THEN EQ_TAC THENL
[ALL_TAC;
DISCH_THEN(X_CHOOSE_THEN `g:complex->complex` STRIP_ASSUME_TAC) THEN
X_GEN_TAC `p:real^1->complex` THEN STRIP_TAC THEN
MP_TAC(ISPECL [`f:complex->complex`; `g:complex->complex`;
`s:complex->bool`; `p:real^1->complex`]
lemma) THEN
ASM_REWRITE_TAC[PATHSTART_COMPOSE; PATHFINISH_COMPOSE] THEN
REWRITE_TAC[COMPLEX_SUB_REFL; COMPLEX_MUL_RZERO]] THEN
DISCH_TAC THEN
EXISTS_TAC `\z. let c = connected_component s (z:complex) in
let z0 = (@) c in
let p = @p. path p /\ path_image p SUBSET c /\
pathstart p = z0 /\ pathfinish p = z in
Cx(&2) * Cx(pi) * ii * winding_number(f o p,Cx(&0)) +
clog(f z0)` THEN
CONJ_TAC THENL
[ALL_TAC;
X_GEN_TAC `z:complex` THEN DISCH_TAC THEN REWRITE_TAC[] THEN
REPEAT LET_TAC THEN
SUBGOAL_THEN `(z:complex) IN c` ASSUME_TAC THENL
[ASM_MESON_TAC[CONNECTED_COMPONENT_REFL; IN]; ALL_TAC] THEN
SUBGOAL_THEN `(z0:complex) IN c` ASSUME_TAC THENL
[EXPAND_TAC "z0" THEN REWRITE_TAC[IN] THEN MATCH_MP_TAC SELECT_AX THEN
ASM_MESON_TAC[IN];
ALL_TAC] THEN
SUBGOAL_THEN `(c:complex->bool) SUBSET s` ASSUME_TAC THENL
[ASM_MESON_TAC[CONNECTED_COMPONENT_SUBSET]; ALL_TAC] THEN
SUBGOAL_THEN `connected(c:complex->bool)` ASSUME_TAC THENL
[ASM_MESON_TAC[CONNECTED_CONNECTED_COMPONENT]; ALL_TAC] THEN
SUBGOAL_THEN `open(c:complex->bool)` ASSUME_TAC THENL
[ASM_MESON_TAC[OPEN_CONNECTED_COMPONENT]; ALL_TAC] THEN
SUBGOAL_THEN `path_connected(c:complex->bool)` ASSUME_TAC THENL
[ASM_MESON_TAC[CONNECTED_OPEN_PATH_CONNECTED]; ALL_TAC] THEN
SUBGOAL_THEN
`path p /\ path_image p SUBSET c /\
pathstart p = z0 /\ pathfinish p = (z:complex)`
STRIP_ASSUME_TAC THENL
[EXPAND_TAC "p" THEN CONV_TAC SELECT_CONV THEN
FIRST_X_ASSUM(MATCH_MP_TAC o REWRITE_RULE[path_connected]) THEN
ASM_REWRITE_TAC[];
ALL_TAC] THEN
MP_TAC(ISPECL [`(f:complex->complex) o (p:real^1->complex)`; `Cx(&0)`]
WINDING_NUMBER_AHLFORS_FULL) THEN
REWRITE_TAC[CEXP_ADD] THEN ANTS_TAC THENL
[CONJ_TAC THENL
[MATCH_MP_TAC PATH_CONTINUOUS_IMAGE THEN
ASM_MESON_TAC[CONTINUOUS_ON_SUBSET];
REWRITE_TAC[path_image; IMAGE_o] THEN
REWRITE_TAC[GSYM path_image] THEN ASM SET_TAC[]];
ASM_REWRITE_TAC[PATHSTART_COMPOSE; PATHFINISH_COMPOSE] THEN
REWRITE_TAC[COMPLEX_SUB_RZERO] THEN DISCH_THEN SUBST1_TAC THEN
AP_TERM_TAC THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC CEXP_CLOG THEN
ASM SET_TAC[]]] THEN
MATCH_MP_TAC CONTINUOUS_ON_COMPONENTS_OPEN THEN
ASM_REWRITE_TAC[] THEN X_GEN_TAC `c:complex->bool` THEN DISCH_TAC THEN
ABBREV_TAC `z0:complex = (@) c` THEN
MATCH_MP_TAC CONTINUOUS_ON_EQ THEN
ABBREV_TAC
`g = \z. let p = @p. path p /\ path_image p SUBSET c /\
pathstart p = z0 /\ pathfinish p = z in
Cx(&2) * Cx(pi) * ii * winding_number(f o p,Cx(&0)) +
clog(f(z0:complex))` THEN
EXISTS_TAC `g:complex->complex` THEN REWRITE_TAC[] THEN CONJ_TAC THENL
[X_GEN_TAC `z:complex` THEN DISCH_TAC THEN EXPAND_TAC "g" THEN
CONV_TAC(TOP_DEPTH_CONV let_CONV) THEN EXPAND_TAC "z0" THEN
SUBGOAL_THEN `connected_component s (z:complex) = c`
(fun th -> REWRITE_TAC[th]) THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_COMPONENTS]) THEN
ASM_MESON_TAC[CONNECTED_COMPONENT_EQ];
ALL_TAC] THEN
SUBGOAL_THEN `(z0:complex) IN c` ASSUME_TAC THENL
[EXPAND_TAC "z0" THEN REWRITE_TAC[IN] THEN MATCH_MP_TAC SELECT_AX THEN
FIRST_ASSUM(MP_TAC o MATCH_MP IN_COMPONENTS_NONEMPTY) THEN SET_TAC[];
ALL_TAC] THEN
SUBGOAL_THEN `(c:complex->bool) SUBSET s` ASSUME_TAC THENL
[ASM_MESON_TAC[IN_COMPONENTS_SUBSET]; ALL_TAC] THEN
SUBGOAL_THEN `connected(c:complex->bool)` ASSUME_TAC THENL
[ASM_MESON_TAC[IN_COMPONENTS_CONNECTED]; ALL_TAC] THEN
SUBGOAL_THEN `open(c:complex->bool)` ASSUME_TAC THENL
[ASM_MESON_TAC[OPEN_COMPONENTS]; ALL_TAC] THEN
SUBGOAL_THEN `path_connected(c:complex->bool)` ASSUME_TAC THENL
[ASM_MESON_TAC[CONNECTED_OPEN_PATH_CONNECTED]; ALL_TAC] THEN
SUBGOAL_THEN
`!x. x IN c
==> ?p. path (p:real^1->complex) /\ path_image p SUBSET c /\
pathstart p = z0 /\ pathfinish p = x /\
g(x) = Cx(&2) * Cx pi * ii * winding_number(f o p,Cx(&0)) +
clog (f z0)`
(LABEL_TAC "*")
THENL
[X_GEN_TAC `z:complex` THEN DISCH_TAC THEN EXPAND_TAC "g" THEN
ABBREV_TAC `p = @p. path p /\ path_image p SUBSET c /\
pathstart p = z0 /\ pathfinish p = (z:complex)` THEN
EXISTS_TAC `p:real^1->complex` THEN
CONV_TAC(TOP_DEPTH_CONV let_CONV) THEN REWRITE_TAC[] THEN
EXPAND_TAC "p" THEN CONV_TAC SELECT_CONV THEN
FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [path_connected]) THEN
ASM_REWRITE_TAC[];
ALL_TAC] THEN
ASM_SIMP_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_AT] THEN
X_GEN_TAC `z:complex` THEN DISCH_TAC THEN
FIRST_ASSUM(MP_TAC o SPEC `z:complex` o GEN_REWRITE_RULE I
[OPEN_CONTAINS_BALL]) THEN
ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `e:real` THEN
STRIP_TAC THEN
MP_TAC(SPEC `ball(z:complex,e)` SIMPLY_CONNECTED_EQ_CONTINUOUS_LOG) THEN
SIMP_TAC[OPEN_BALL; CONVEX_BALL; CONVEX_IMP_SIMPLY_CONNECTED] THEN
DISCH_THEN(MP_TAC o SPEC `f:complex->complex` o CONJUNCT2) THEN
ANTS_TAC THENL
[ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; SUBSET];
DISCH_THEN(X_CHOOSE_THEN `l:complex->complex` STRIP_ASSUME_TAC)] THEN
REWRITE_TAC[CONTINUOUS_AT] THEN ONCE_REWRITE_TAC[LIM_NULL] THEN
MATCH_MP_TAC LIM_TRANSFORM_AT THEN
ONCE_REWRITE_TAC[DIST_SYM] THEN EXISTS_TAC
`\w. Cx(&2) * Cx pi * ii *
winding_number((f:complex->complex) o linepath(z,w),Cx(&0))` THEN
EXISTS_TAC `e:real` THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL
[X_GEN_TAC `w:complex` THEN STRIP_TAC THEN REMOVE_THEN "*"
(fun th -> MP_TAC(SPEC `w:complex` th) THEN
MP_TAC(SPEC `z:complex` th)) THEN
ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
X_GEN_TAC `p:real^1->complex` THEN STRIP_TAC THEN
ANTS_TAC THENL [ASM_MESON_TAC[SUBSET; IN_BALL; DIST_SYM]; ALL_TAC] THEN
DISCH_THEN(X_CHOOSE_THEN `q:real^1->complex` STRIP_ASSUME_TAC) THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(COMPLEX_RING
`(z + x) - y = Cx(&0)
==> a * b * c * x = (a * b * c * y + l) - (a * b * c * z + l)`) THEN
FIRST_X_ASSUM(MP_TAC o SPEC
`p ++ linepath(z:complex,w) ++ reversepath q`) THEN
ASM_SIMP_TAC[PATHSTART_JOIN; PATHFINISH_JOIN;
PATHSTART_REVERSEPATH; PATHFINISH_REVERSEPATH;
PATH_JOIN_EQ; PATH_LINEPATH; PATH_REVERSEPATH;
PATHSTART_LINEPATH; PATHFINISH_LINEPATH; PATH_IMAGE_JOIN] THEN
ASM_REWRITE_TAC[UNION_SUBSET; PATH_IMAGE_REVERSEPATH] THEN ANTS_TAC THENL
[REPEAT STRIP_TAC THEN
MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC `c:complex->bool` THEN
ASM_REWRITE_TAC[PATH_IMAGE_LINEPATH] THEN
MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC `ball(z:complex,e)` THEN
ASM_REWRITE_TAC[SEGMENT_CONVEX_HULL] THEN MATCH_MP_TAC HULL_MINIMAL THEN
ASM_REWRITE_TAC[INSERT_SUBSET; CENTRE_IN_BALL; EMPTY_SUBSET] THEN
ASM_REWRITE_TAC[IN_BALL; CONVEX_BALL];
DISCH_THEN(fun th -> GEN_REWRITE_TAC RAND_CONV [GSYM th]) THEN
REWRITE_TAC[PATH_COMPOSE_JOIN; PATH_COMPOSE_REVERSEPATH] THEN
W(MP_TAC o PART_MATCH (lhand o rand) WINDING_NUMBER_JOIN o
rand o snd) THEN
ANTS_TAC THENL
[ALL_TAC;
DISCH_THEN SUBST1_TAC THEN
REWRITE_TAC[VECTOR_SUB; GSYM VECTOR_ADD_ASSOC] THEN
AP_TERM_TAC THEN
W(MP_TAC o PART_MATCH (lhand o rand) WINDING_NUMBER_JOIN o
rand o snd) THEN
ANTS_TAC THENL
[ALL_TAC;
DISCH_THEN SUBST1_TAC THEN AP_TERM_TAC THEN
MATCH_MP_TAC(GSYM WINDING_NUMBER_REVERSEPATH)]] THEN
ASM_SIMP_TAC[PATHSTART_REVERSEPATH; PATHFINISH_REVERSEPATH;
PATHSTART_COMPOSE; PATHFINISH_COMPOSE; PATH_IMAGE_REVERSEPATH;
PATHSTART_JOIN; PATHFINISH_JOIN; PATH_REVERSEPATH;
PATHSTART_LINEPATH; PATHFINISH_LINEPATH; PATH_JOIN;
PATH_IMAGE_JOIN; IN_UNION; DE_MORGAN_THM] THEN
REWRITE_TAC[PATH_IMAGE_COMPOSE; SET_RULE
`~(z IN IMAGE f s) <=> !x. x IN s ==> ~(f x = z)`] THEN
REPEAT CONJ_TAC THEN
((MATCH_MP_TAC PATH_CONTINUOUS_IMAGE)
ORELSE
(X_GEN_TAC `x:complex` THEN DISCH_TAC THEN
FIRST_X_ASSUM MATCH_MP_TAC)) THEN
ASM_REWRITE_TAC[PATH_LINEPATH] THEN
TRY(FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ]
CONTINUOUS_ON_SUBSET)) THEN
REWRITE_TAC[SUBSET] THEN X_GEN_TAC `x:complex` THEN STRIP_TAC) THEN
FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [SUBSET]) THEN
TRY(FIRST_X_ASSUM(fun th ->
MATCH_MP_TAC(GEN_REWRITE_RULE I [SUBSET] th) THEN
FIRST_X_ASSUM ACCEPT_TAC)) THEN
UNDISCH_TAC `(x:complex) IN path_image(linepath(z,w))` THEN
SPEC_TAC(`x:complex`,`x:complex`) THEN
REWRITE_TAC[GSYM SUBSET; PATH_IMAGE_LINEPATH; SEGMENT_CONVEX_HULL] THEN
MATCH_MP_TAC SUBSET_TRANS THEN EXISTS_TAC `ball(z:complex,e)` THEN
ASM_REWRITE_TAC[SEGMENT_CONVEX_HULL] THEN MATCH_MP_TAC HULL_MINIMAL THEN
ASM_REWRITE_TAC[INSERT_SUBSET; CENTRE_IN_BALL; EMPTY_SUBSET] THEN
ASM_REWRITE_TAC[IN_BALL; CONVEX_BALL]];
MATCH_MP_TAC LIM_TRANSFORM THEN
EXISTS_TAC `\w. Cx(&2) * Cx pi * ii *
Cx(&1) / (Cx(&2) * Cx pi * ii) *
(pathfinish(l o linepath(z:complex,w)) -
pathstart (l o linepath(z,w)))` THEN
REWRITE_TAC[] THEN CONJ_TAC THENL
[MATCH_MP_TAC LIM_EVENTUALLY THEN REWRITE_TAC[EVENTUALLY_AT] THEN
EXISTS_TAC `e:real` THEN ONCE_REWRITE_TAC[DIST_SYM] THEN
ASM_REWRITE_TAC[] THEN X_GEN_TAC `w:complex` THEN STRIP_TAC THEN
REWRITE_TAC[VECTOR_ARITH `x - y = vec 0 <=> y = x`] THEN
REPLICATE_TAC 3 AP_TERM_TAC THEN MATCH_MP_TAC lemma THEN
EXISTS_TAC `ball(z:complex,e)` THEN ASM_REWRITE_TAC[PATH_LINEPATH] THEN
CONJ_TAC THENL[ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; SUBSET]; ALL_TAC] THEN
REWRITE_TAC[PATH_IMAGE_LINEPATH; SEGMENT_CONVEX_HULL] THEN
MATCH_MP_TAC HULL_MINIMAL THEN REWRITE_TAC[CONVEX_BALL] THEN
ASM_REWRITE_TAC[INSERT_SUBSET; CENTRE_IN_BALL; EMPTY_SUBSET] THEN
ASM_REWRITE_TAC[IN_BALL];
REWRITE_TAC[COMPLEX_VEC_0] THEN
REPEAT(MATCH_MP_TAC LIM_NULL_COMPLEX_LMUL) THEN
REWRITE_TAC[PATHSTART_COMPOSE; PATHSTART_LINEPATH;
PATHFINISH_COMPOSE; PATHFINISH_LINEPATH] THEN
REWRITE_TAC[GSYM COMPLEX_VEC_0; GSYM LIM_NULL; GSYM CONTINUOUS_AT] THEN
ASM_MESON_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_AT; OPEN_BALL;
CENTRE_IN_BALL]]]);;