(* (c) Copyright, John Harrison 1998-2014 *)
(* (c) Copyright, Valentina Bruno 2010 *)
(* Distributed under the same license as HOL Light *)
(* *)
(* Theorems taken directly from Multivariate/topology.ml which run after *)
(* loading Topology.ml. *)
needs "Library/card.ml";;
needs "Multivariate/determinants.ml";;
needs "RichterHilbertAxiomGeometry/Topology.ml";;
(* ------------------------------------------------------------------------- *)
(* Open and closed balls and spheres. *)
(* ------------------------------------------------------------------------- *)
add_translation_invariants
[BALL_TRANSLATION; CBALL_TRANSLATION; SPHERE_TRANSLATION];;
let BALL_LINEAR_IMAGE = prove
(`!f:real^M->real^N x r.
linear f /\ (!y. ?x. f x = y) /\ (!x. norm(f x) = norm x)
==> ball(f x,r) =
IMAGE f (ball(x,r))`,
REWRITE_TAC[ball] THEN GEOM_TRANSFORM_TAC[]);;
let CBALL_LINEAR_IMAGE = prove
(`!f:real^M->real^N x r.
linear f /\ (!y. ?x. f x = y) /\ (!x. norm(f x) = norm x)
==> cball(f x,r) =
IMAGE f (cball(x,r))`,
REWRITE_TAC[cball] THEN GEOM_TRANSFORM_TAC[]);;
let SPHERE_LINEAR_IMAGE = prove
(`!f:real^M->real^N x r.
linear f /\ (!y. ?x. f x = y) /\ (!x. norm(f x) = norm x)
==> sphere(f x,r) =
IMAGE f (sphere(x,r))`,
REWRITE_TAC[sphere] THEN GEOM_TRANSFORM_TAC[]);;
let CLOSED_POSITIVE_ORTHANT = prove
(`closed {x:real^N | !i. 1 <= i /\ i <= dimindex(:N)
==> &0 <= x$i}`,
REWRITE_TAC[
CLOSED_LIMPT;
LIMPT_APPROACHABLE] THEN
REWRITE_TAC[
IN_ELIM_THM] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
X_GEN_TAC `i:num` THEN STRIP_TAC THEN REWRITE_TAC[GSYM
REAL_NOT_LT] THEN
DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `--(x:real^N $ i)`) THEN
ASM_REWRITE_TAC[
REAL_LT_RNEG; REAL_ADD_LID;
NOT_EXISTS_THM] THEN
X_GEN_TAC `y:real^N` THEN
MATCH_MP_TAC(TAUT `(a ==> ~c) ==> ~(a /\ b /\ c)`) THEN DISCH_TAC THEN
MATCH_MP_TAC(REAL_ARITH `!b. abs x <= b /\ b <= a ==> ~(a + x < &0)`) THEN
EXISTS_TAC `abs((y - x :real^N)$i)` THEN
ASM_SIMP_TAC[dist;
COMPONENT_LE_NORM] THEN
ASM_SIMP_TAC[
VECTOR_SUB_COMPONENT; REAL_ARITH
`x < &0 /\ &0 <= y ==> abs(x) <= abs(y - x)`]);;
let FINITE_SET_AVOID = prove
(`!a:real^N s.
FINITE s
==> ?d. &0 < d /\ !x. x
IN s /\ ~(x = a) ==> d <= dist(a,x)`,
GEN_TAC THEN MATCH_MP_TAC
FINITE_INDUCT_STRONG THEN
REWRITE_TAC[
NOT_IN_EMPTY] THEN
CONJ_TAC THENL [MESON_TAC[
REAL_LT_01]; ALL_TAC] THEN
MAP_EVERY X_GEN_TAC [`x:real^N`; `s:real^N->bool`] THEN
DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC) THEN
FIRST_X_ASSUM(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN
ASM_CASES_TAC `x:real^N = a` THEN REWRITE_TAC[
IN_INSERT] THENL
[ASM_MESON_TAC[]; ALL_TAC] THEN
EXISTS_TAC `min d (dist(a:real^N,x))` THEN
ASM_REWRITE_TAC[
REAL_LT_MIN; GSYM
DIST_NZ;
REAL_MIN_LE] THEN
ASM_MESON_TAC[
REAL_LE_REFL]);;
let DISCRETE_IMP_CLOSED = prove
(`!s:real^N->bool e.
&0 < e /\
(!x y. x
IN s /\ y
IN s /\ norm(y - x) < e ==> y = x)
==> closed s`,
REPEAT STRIP_TAC THEN
SUBGOAL_THEN `!x:real^N. ~(x
limit_point_of s)`
(fun th -> MESON_TAC[th;
CLOSED_LIMPT]) THEN
GEN_TAC THEN REWRITE_TAC[
LIMPT_APPROACHABLE] THEN DISCH_TAC THEN
FIRST_ASSUM(MP_TAC o SPEC `e / &2`) THEN
REWRITE_TAC[
REAL_HALF; ASSUME `&0 < e`] THEN
DISCH_THEN(X_CHOOSE_THEN `y:real^N` STRIP_ASSUME_TAC) THEN
FIRST_X_ASSUM(MP_TAC o SPEC `min (e / &2) (dist(x:real^N,y))`) THEN
ASM_SIMP_TAC[
REAL_LT_MIN;
DIST_POS_LT;
REAL_HALF] THEN
DISCH_THEN(X_CHOOSE_THEN `z:real^N` STRIP_ASSUME_TAC) THEN
FIRST_X_ASSUM(MP_TAC o SPECL [`y:real^N`; `z:real^N`]) THEN
ASM_REWRITE_TAC[] THEN ASM_NORM_ARITH_TAC);;
let LIMPT_OF_OPEN_IN = prove
(`!s t x:real^N.
open_in (subtopology euclidean s) t /\ x
limit_point_of s /\ x
IN t
==> x
limit_point_of t`,
REWRITE_TAC[
open_in;
SUBSET;
LIMPT_APPROACHABLE] THEN
REPEAT GEN_TAC THEN STRIP_TAC THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN
FIRST_X_ASSUM(MP_TAC o SPEC `min d e / &2`) THEN
ANTS_TAC THENL [ASM_REAL_ARITH_TAC; MATCH_MP_TAC
MONO_EXISTS] THEN
GEN_TAC THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THEN
TRY(FIRST_X_ASSUM MATCH_MP_TAC) THEN ASM_REWRITE_TAC[] THEN
ASM_REAL_ARITH_TAC);;
let NET = prove
(`!n x y. (!z. netord n z x ==> netord n z y) \/
(!z. netord n z y ==> netord n z x)`,
REWRITE_TAC[net_tybij; ETA_AX]);;
let OLDNET = prove
(`!n x y. netord n x x /\ netord n y y
==> ?z. netord n z z /\
!w. netord n w z ==> netord n w x /\ netord n w y`,
let NET_DILEMMA = prove
(`!net. (?a. (?x. netord net x a) /\ (!x. netord net x a ==> P x)) /\
(?b. (?x. netord net x b) /\ (!x. netord net x b ==> Q x))
==> ?c. (?x. netord net x c) /\ (!x. netord net x c ==> P x /\ Q x)`,
let AT = prove
(`!a:real^N x y.
netord(at a) x y <=> &0 < dist(x,a) /\ dist(x,a) <= dist(y,a)`,
let WITHIN = prove
(`!n s x y. netord(n within s) x y <=> netord n x y /\ x
IN s`,
GEN_TAC THEN GEN_TAC THEN REWRITE_TAC[within; GSYM
FUN_EQ_THM] THEN
REWRITE_TAC[GSYM(CONJUNCT2 net_tybij); ETA_AX] THEN
MESON_TAC[
NET]);;
let EVENTUALLY_WITHIN_LE = prove
(`!s a:real^M p.
eventually p (at a within s) <=>
?d. &0 < d /\ !x. x
IN s /\ &0 < dist(x,a) /\ dist(x,a) <= d ==> p(x)`,
let EVENTUALLY_WITHIN = prove
(`!s a:real^M p.
eventually p (at a within s) <=>
?d. &0 < d /\ !x. x
IN s /\ &0 < dist(x,a) /\ dist(x,a) < d ==> p(x)`,
let EVENTUALLY_AT = prove
(`!a p. eventually p (at a) <=>
?d. &0 < d /\ !x. &0 < dist(x,a) /\ dist(x,a) < d ==> p(x)`,
let EVENTUALLY_AND = prove
(`!net:(A net) p q.
eventually (\x. p x /\ q x) net <=>
eventually p net /\ eventually q net`,
REPEAT GEN_TAC THEN REWRITE_TAC[eventually] THEN
ASM_CASES_TAC `
trivial_limit(net:(A net))` THEN ASM_REWRITE_TAC[] THEN
EQ_TAC THEN SIMP_TAC[
NET_DILEMMA] THEN MESON_TAC[]);;
let EVENTUALLY_MONO = prove
(`!net:(A net) p q.
(!x. p x ==> q x) /\ eventually p net
==> eventually q net`,
REWRITE_TAC[eventually] THEN MESON_TAC[]);;
let EVENTUALLY_MP = prove
(`!net:(A net) p q.
eventually (\x. p x ==> q x) net /\ eventually p net
==> eventually q net`,
REWRITE_TAC[GSYM
EVENTUALLY_AND] THEN
REWRITE_TAC[eventually] THEN MESON_TAC[]);;
let LIM = prove
(`(f --> l) net <=>
trivial_limit net \/
!e. &0 < e ==> ?y. (?x. netord(net) x y) /\
!x. netord(net) x y ==> dist(f(x),l) < e`,
REWRITE_TAC[tendsto; eventually] THEN MESON_TAC[]);;
let LIM_WITHIN_LE = prove
(`!f:real^M->real^N l a s.
(f --> l)(at a within s) <=>
!e. &0 < e ==> ?d. &0 < d /\
!x. x
IN s /\ &0 < dist(x,a) /\ dist(x,a) <= d
==> dist(f(x),l) < e`,
let LIM_WITHIN = prove
(`!f:real^M->real^N l a s.
(f --> l) (at a within s) <=>
!e. &0 < e
==> ?d. &0 < d /\
!x. x
IN s /\ &0 < dist(x,a) /\ dist(x,a) < d
==> dist(f(x),l) < e`,
let LIM_AT_LE = prove
(`!f l a. (f --> l) (at a) <=>
!e. &0 < e
==> ?d. &0 < d /\
!x. &0 < dist(x,a) /\ dist(x,a) <= d
==> dist (f x,l) < e`,
let LIM_AT = prove
(`!f l:real^N a:real^M.
(f --> l) (at a) <=>
!e. &0 < e
==> ?d. &0 < d /\ !x. &0 < dist(x,a) /\ dist(x,a) < d
==> dist(f(x),l) < e`,
let LIM_UNION = prove
(`!f x l s t.
(f --> l) (at x within s) /\ (f --> l) (at x within t)
==> (f --> l) (at x within (s
UNION t))`,
REPEAT GEN_TAC THEN REWRITE_TAC[
LIM_WITHIN;
IN_UNION] THEN
REWRITE_TAC[
AND_FORALL_THM] THEN MATCH_MP_TAC
MONO_FORALL THEN
X_GEN_TAC `e:real` THEN ASM_CASES_TAC `&0 < e` THEN ASM_SIMP_TAC[] THEN
DISCH_THEN(CONJUNCTS_THEN2
(X_CHOOSE_TAC `d1:real`) (X_CHOOSE_TAC `d2:real`)) THEN
EXISTS_TAC `min d1 d2` THEN ASM_MESON_TAC[
REAL_LT_MIN]);;
let LIM_UNION_UNIV = prove
(`!f x l s t.
(f --> l) (at x within s) /\ (f --> l) (at x within t) /\
s
UNION t = (:real^N)
==> (f --> l) (at x)`,
let LIM_COMPOSE_WITHIN = prove
(`!net f:real^M->real^N g:real^N->real^P s y z.
(f --> y) net /\
eventually (\w. f w
IN s /\ (f w = y ==> g y = z)) net /\
(g --> z) (at y within s)
==> ((g o f) --> z) net`,
let LIM_COMPOSE_AT = prove
(`!net f:real^M->real^N g:real^N->real^P y z.
(f --> y) net /\
eventually (\w. f w = y ==> g y = z) net /\
(g --> z) (at y)
==> ((g o f) --> z) net`,
REPEAT STRIP_TAC THEN
MP_TAC(ISPECL [`net:(real^M)net`; `f:real^M->real^N`; `g:real^N->real^P`;
`(:real^N)`; `y:real^N`; `z:real^P`]
LIM_COMPOSE_WITHIN) THEN
ASM_REWRITE_TAC[
IN_UNIV;
WITHIN_UNIV]);;
let LIM_WITHIN_OPEN = prove
(`!f l a:real^M s.
a
IN s /\ open s ==> ((f --> l)(at a within s) <=> (f --> l)(at a))`,
REPEAT STRIP_TAC THEN EQ_TAC THEN SIMP_TAC[
LIM_AT_WITHIN] THEN
REWRITE_TAC[
LIM_AT;
LIM_WITHIN] THEN
MATCH_MP_TAC
MONO_FORALL THEN X_GEN_TAC `e:real` THEN
ASM_CASES_TAC `&0 < e` THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_THEN `d1:real` STRIP_ASSUME_TAC) THEN
FIRST_X_ASSUM(MP_TAC o SPEC `a:real^M` o GEN_REWRITE_RULE I [
open_def]) THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_THEN `d2:real` STRIP_ASSUME_TAC) THEN
MP_TAC(SPECL [`d1:real`; `d2:real`]
REAL_DOWN2) THEN ASM_REWRITE_TAC[] THEN
ASM_MESON_TAC[
REAL_LT_TRANS]);;
let INFINITE_OPEN_IN = prove
(`!u s:real^N->bool.
open_in (subtopology euclidean u) s /\ (?x. x
IN s /\ x
limit_point_of u)
==>
INFINITE s`,
REPEAT STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
OPEN_IN_OPEN]) THEN
DISCH_THEN(X_CHOOSE_THEN `t:real^N->bool` STRIP_ASSUME_TAC) THEN
FIRST_X_ASSUM(MP_TAC o SPEC `t:real^N->bool` o
GEN_REWRITE_RULE I [LIMPT_INFINITE_OPEN]) THEN
FIRST_X_ASSUM SUBST_ALL_TAC THEN ASM SET_TAC[]);;
let LIM_LINEAR = prove
(`!net:(A)net h f l.
(f --> l) net /\ linear h ==> ((\x. h(f x)) --> h l) net`,
let LIM_CMUL = prove
(`!f l c. (f --> l) net ==> ((\x. c % f x) --> c % l) net`,
let LIM_CMUL_EQ = prove
(`!net f l c.
~(c = &0) ==> (((\x. c % f x) --> c % l) net <=> (f --> l) net)`,
let LIM_NEG = prove
(`!net f l:real^N. (f --> l) net ==> ((\x. --(f x)) --> --l) net`,
REPEAT GEN_TAC THEN REWRITE_TAC[
LIM; dist] THEN
REWRITE_TAC[VECTOR_ARITH `--x - --y = --(x - y:real^N)`;
NORM_NEG]);;
let LIM_NEG_EQ = prove
(`!net f l:real^N. ((\x. --(f x)) --> --l) net <=> (f --> l) net`,
REPEAT GEN_TAC THEN EQ_TAC THEN
DISCH_THEN(MP_TAC o MATCH_MP
LIM_NEG) THEN
REWRITE_TAC[
VECTOR_NEG_NEG; ETA_AX]);;
let LIM_ADD = prove
(`!net:(A)net f g l m.
(f --> l) net /\ (g --> m) net ==> ((\x. f(x) + g(x)) --> l + m) net`,
let LIM_ABS = prove
(`!net:(A)net f:A->real^N l.
(f --> l) net
==> ((\x. lambda i. (abs(f(x)$i))) --> (lambda i. abs(l$i)):real^N) net`,
REPEAT GEN_TAC THEN REWRITE_TAC[
LIM] THEN
ASM_CASES_TAC `
trivial_limit (net:(A)net)` THEN ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC
MONO_FORALL THEN GEN_TAC THEN
MATCH_MP_TAC MONO_IMP THEN REWRITE_TAC[] THEN
MATCH_MP_TAC
MONO_EXISTS THEN GEN_TAC THEN
MATCH_MP_TAC MONO_AND THEN REWRITE_TAC[] THEN
MATCH_MP_TAC
MONO_FORALL THEN GEN_TAC THEN
MATCH_MP_TAC MONO_IMP THEN REWRITE_TAC[] THEN
MATCH_MP_TAC(NORM_ARITH
`norm(x - y) <= norm(a - b) ==> dist(a,b) < e ==> dist(x,y) < e`) THEN
MATCH_MP_TAC
NORM_LE_COMPONENTWISE THEN
SIMP_TAC[
LAMBDA_BETA;
VECTOR_SUB_COMPONENT] THEN
REAL_ARITH_TAC);;
let LIM_SUB = prove
(`!net:(A)net f g l m.
(f --> l) net /\ (g --> m) net ==> ((\x. f(x) - g(x)) --> l - m) net`,
let LIM_MAX = prove
(`!net:(A)net f g l:real^N m:real^N.
(f --> l) net /\ (g --> m) net
==> ((\x. lambda i. max (f(x)$i) (g(x)$i))
--> (lambda i. max (l$i) (m$i)):real^N) net`,
let LIM_MIN = prove
(`!net:(A)net f g l:real^N m:real^N.
(f --> l) net /\ (g --> m) net
==> ((\x. lambda i. min (f(x)$i) (g(x)$i))
--> (lambda i. min (l$i) (m$i)):real^N) net`,
let LIM_NORM = prove
(`!net f:A->real^N l.
(f --> l) net ==> ((\x. lift(norm(f x))) --> lift(norm l)) net`,
REPEAT GEN_TAC THEN REWRITE_TAC[tendsto;
DIST_LIFT] THEN
MATCH_MP_TAC
MONO_FORALL THEN GEN_TAC THEN MATCH_MP_TAC MONO_IMP THEN
REWRITE_TAC[] THEN
MATCH_MP_TAC(REWRITE_RULE[
IMP_CONJ]
EVENTUALLY_MONO) THEN
REWRITE_TAC[] THEN NORM_ARITH_TAC);;
let LIM_NULL = prove
(`!net f l. (f --> l) net <=> ((\x. f(x) - l) --> vec 0) net`,
let LIM_NULL_COMPARISON = prove
(`!net f g. eventually (\x. norm(f x) <= g x) net /\
((\x. lift(g x)) --> vec 0) net
==> (f --> vec 0) net`,
let LIM_COMPONENT = prove
(`!net f i l:real^N. (f --> l) net /\ 1 <= i /\ i <= dimindex(:N)
==> ((\a. lift(f(a)$i)) --> lift(l$i)) net`,
let LIM_NULL_CMUL_BOUNDED = prove
(`!f g:A->real^N B.
eventually (\a. g a = vec 0 \/ abs(f a) <= B) net /\
(g --> vec 0) net
==> ((\n. f n % g n) --> vec 0) net`,
let LIM_NULL_VMUL_BOUNDED = prove
(`!f g:A->real^N B.
((lift o f) --> vec 0) net /\
eventually (\a. f a = &0 \/ norm(g a) <= B) net
==> ((\n. f n % g n) --> vec 0) net`,
let LIM_VSUM = prove
(`!f:A->B->real^N s.
FINITE s /\ (!i. i
IN s ==> ((f i) --> (l i)) net)
==> ((\x. vsum s (\i. f i x)) --> vsum s l) net`,
let LIM_IN_CLOSED_SET = prove
(`!net f:A->real^N s l.
closed s /\ eventually (\x. f(x)
IN s) net /\
~(
trivial_limit net) /\ (f --> l) net
==> l
IN s`,
REWRITE_TAC[closed] THEN REPEAT STRIP_TAC THEN
MATCH_MP_TAC(SET_RULE `~(x
IN (
UNIV DIFF s)) ==> x
IN s`) THEN
DISCH_TAC THEN
FIRST_ASSUM(MP_TAC o SPEC `l:real^N` o GEN_REWRITE_RULE I
[
OPEN_CONTAINS_BALL]) THEN
ASM_REWRITE_TAC[
SUBSET;
IN_BALL;
IN_DIFF;
IN_UNION] THEN
DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN
FIRST_X_ASSUM(MP_TAC o SPEC `e:real` o GEN_REWRITE_RULE I [tendsto]) THEN
UNDISCH_TAC `eventually (\x. (f:A->real^N) x
IN s) net` THEN
ASM_REWRITE_TAC[GSYM
EVENTUALLY_AND; TAUT `a ==> ~b <=> ~(a /\ b)`] THEN
MATCH_MP_TAC
NOT_EVENTUALLY THEN ASM_MESON_TAC[
DIST_SYM]);;
let LIM_NORM_UBOUND = prove
(`!net:(A)net f (l:real^N) b.
~(
trivial_limit net) /\
(f --> l) net /\
eventually (\x. norm(f x) <= b) net
==> norm(l) <= b`,
REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
ASM_REWRITE_TAC[
LIM] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
ASM_REWRITE_TAC[eventually] THEN
STRIP_TAC THEN REWRITE_TAC[GSYM
REAL_NOT_LT] THEN
ONCE_REWRITE_TAC[GSYM
REAL_SUB_LT] THEN DISCH_TAC THEN
SUBGOAL_THEN
`?x:A. dist(f(x):real^N,l) < norm(l:real^N) - b /\ norm(f x) <= b`
(CHOOSE_THEN MP_TAC) THENL [ASM_MESON_TAC[
NET]; ALL_TAC] THEN
REWRITE_TAC[
REAL_NOT_LT;
REAL_LE_SUB_RADD; DE_MORGAN_THM; dist] THEN
NORM_ARITH_TAC);;
let LIM_NORM_LBOUND = prove
(`!net:(A)net f (l:real^N) b.
~(
trivial_limit net) /\ (f --> l) net /\
eventually (\x. b <= norm(f x)) net
==> b <= norm(l)`,
REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
ASM_REWRITE_TAC[
LIM] THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
ASM_REWRITE_TAC[eventually] THEN
STRIP_TAC THEN REWRITE_TAC[GSYM
REAL_NOT_LT] THEN
ONCE_REWRITE_TAC[GSYM
REAL_SUB_LT] THEN DISCH_TAC THEN
SUBGOAL_THEN
`?x:A. dist(f(x):real^N,l) < b - norm(l:real^N) /\ b <= norm(f x)`
(CHOOSE_THEN MP_TAC) THENL [ASM_MESON_TAC[
NET]; ALL_TAC] THEN
REWRITE_TAC[
REAL_NOT_LT;
REAL_LE_SUB_RADD; DE_MORGAN_THM; dist] THEN
NORM_ARITH_TAC);;
let LIM_UNIQUE = prove
(`!net:(A)net f l:real^N l'.
~(
trivial_limit net) /\ (f --> l) net /\ (f --> l') net ==> (l = l')`,
REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
DISCH_THEN(ASSUME_TAC o REWRITE_RULE[
VECTOR_SUB_REFL] o MATCH_MP
LIM_SUB) THEN
SUBGOAL_THEN `!e. &0 < e ==> norm(l:real^N - l') <= e` MP_TAC THENL
[GEN_TAC THEN DISCH_TAC THEN MATCH_MP_TAC
LIM_NORM_UBOUND THEN
MAP_EVERY EXISTS_TAC [`net:(A)net`; `\x:A. vec 0 : real^N`] THEN
ASM_SIMP_TAC[
NORM_0;
REAL_LT_IMP_LE; eventually] THEN
ASM_MESON_TAC[
trivial_limit];
ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[
DIST_NZ; dist] THEN
DISCH_TAC THEN DISCH_THEN(MP_TAC o SPEC `norm(l - l':real^N) / &2`) THEN
ASM_SIMP_TAC[
REAL_LT_RDIV_EQ;
REAL_LE_RDIV_EQ;
REAL_OF_NUM_LT; ARITH] THEN
UNDISCH_TAC `&0 < norm(l - l':real^N)` THEN REAL_ARITH_TAC]);;
let LIM_CONST_EQ = prove
(`!net:(A net) c d:real^N.
((\x. c) --> d) net <=>
trivial_limit net \/ c = d`,
REPEAT GEN_TAC THEN
ASM_CASES_TAC `
trivial_limit (net:A net)` THEN ASM_REWRITE_TAC[] THENL
[ASM_REWRITE_TAC[
LIM]; ALL_TAC] THEN
EQ_TAC THEN SIMP_TAC[
LIM_CONST] THEN DISCH_TAC THEN
MATCH_MP_TAC(SPEC `net:A net`
LIM_UNIQUE) THEN
EXISTS_TAC `(\x. c):A->real^N` THEN ASM_REWRITE_TAC[
LIM_CONST]);;
let UNIFORM_LIM_ADD = prove
(`!net:(A)net P f g l m.
(!e. &0 < e
==> eventually (\x. !n:B. P n ==> norm(f n x - l n) < e) net) /\
(!e. &0 < e
==> eventually (\x. !n. P n ==> norm(g n x - m n) < e) net)
==> !e. &0 < e
==> eventually
(\x. !n. P n
==> norm((f n x + g n x) - (l n + m n)) < e)
net`,
REPEAT GEN_TAC THEN REWRITE_TAC[
AND_FORALL_THM] THEN DISCH_TAC THEN
X_GEN_TAC `e:real` THEN DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `e / &2`) THEN
ASM_REWRITE_TAC[
REAL_HALF; GSYM
EVENTUALLY_AND] THEN
MATCH_MP_TAC(REWRITE_RULE[
IMP_CONJ]
EVENTUALLY_MONO) THEN
GEN_TAC THEN REWRITE_TAC[
AND_FORALL_THM] THEN
MATCH_MP_TAC
MONO_FORALL THEN X_GEN_TAC `n:B` THEN
ASM_CASES_TAC `(P:B->bool) n` THEN ASM_REWRITE_TAC[] THEN
CONV_TAC NORM_ARITH);;
let UNIFORM_LIM_SUB = prove
(`!net:(A)net P f g l m.
(!e. &0 < e
==> eventually (\x. !n:B. P n ==> norm(f n x - l n) < e) net) /\
(!e. &0 < e
==> eventually (\x. !n. P n ==> norm(g n x - m n) < e) net)
==> !e. &0 < e
==> eventually
(\x. !n. P n
==> norm((f n x - g n x) - (l n - m n)) < e)
net`,
REPEAT GEN_TAC THEN REWRITE_TAC[
AND_FORALL_THM] THEN DISCH_TAC THEN
X_GEN_TAC `e:real` THEN DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `e / &2`) THEN
ASM_REWRITE_TAC[
REAL_HALF; GSYM
EVENTUALLY_AND] THEN
MATCH_MP_TAC(REWRITE_RULE[
IMP_CONJ]
EVENTUALLY_MONO) THEN
GEN_TAC THEN REWRITE_TAC[
AND_FORALL_THM] THEN
MATCH_MP_TAC
MONO_FORALL THEN X_GEN_TAC `n:B` THEN
ASM_CASES_TAC `(P:B->bool) n` THEN ASM_REWRITE_TAC[] THEN
CONV_TAC NORM_ARITH);;
let UNIFORM_LIM_BILINEAR = prove
(`!net:(A)net P (h:real^M->real^N->real^P) f g l m b1 b2.
bilinear h /\
eventually (\x. !n. P n ==> norm(l n) <= b1) net /\
eventually (\x. !n. P n ==> norm(m n) <= b2) net /\
(!e. &0 < e
==> eventually (\x. !n:B. P n ==> norm(f n x - l n) < e) net) /\
(!e. &0 < e
==> eventually (\x. !n. P n ==> norm(g n x - m n) < e) net)
==> !e. &0 < e
==> eventually
(\x. !n. P n
==> norm(h (f n x) (g n x) - h (l n) (m n)) < e)
net`,
REPEAT GEN_TAC THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
FIRST_ASSUM(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC o MATCH_MP
BILINEAR_BOUNDED_POS) THEN
REWRITE_TAC[
AND_FORALL_THM;
RIGHT_AND_FORALL_THM] THEN DISCH_TAC THEN
X_GEN_TAC `e:real` THEN DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC
`min (abs b2 + &1) (e / &2 / (B * (abs b1 + abs b2 + &2)))`) THEN
ASM_SIMP_TAC[
REAL_HALF;
REAL_LT_DIV;
REAL_LT_MUL;
REAL_LT_MIN;
REAL_ARITH `&0 < abs x + &1`;
REAL_ARITH `&0 < abs x + abs y + &2`] THEN
REWRITE_TAC[GSYM
EVENTUALLY_AND] THEN
MATCH_MP_TAC(REWRITE_RULE[
IMP_CONJ]
EVENTUALLY_MONO) THEN
X_GEN_TAC `x:A` THEN REWRITE_TAC[
AND_FORALL_THM] THEN
MATCH_MP_TAC
MONO_FORALL THEN X_GEN_TAC `n:B` THEN
ASM_CASES_TAC `(P:B->bool) n` THEN ASM_REWRITE_TAC[] THEN
STRIP_TAC THEN
ONCE_REWRITE_TAC[VECTOR_ARITH
`h a b - h c d :real^N = (h a b - h a d) + (h a d - h c d)`] THEN
ASM_SIMP_TAC[GSYM
BILINEAR_LSUB; GSYM
BILINEAR_RSUB] THEN
MATCH_MP_TAC
NORM_TRIANGLE_LT THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP
(MESON[
REAL_LE_ADD2;
REAL_LET_TRANS]
`(!x y. norm(h x y:real^P) <= B * norm x * norm y)
==> B * norm a * norm b + B * norm c * norm d < e
==> norm(h a b) + norm(h c d) < e`)) THEN
MATCH_MP_TAC(REAL_ARITH
`x * B < e / &2 /\ y * B < e / &2 ==> B * x + B * y < e`) THEN
CONJ_TAC THEN ASM_SIMP_TAC[GSYM
REAL_LT_RDIV_EQ] THENL
[ONCE_REWRITE_TAC[REAL_MUL_SYM]; ALL_TAC] THEN
MATCH_MP_TAC
REAL_LET_TRANS THEN
EXISTS_TAC `e / &2 / (B * (abs b1 + abs b2 + &2)) *
(abs b1 + abs b2 + &1)` THEN
(CONJ_TAC THENL
[MATCH_MP_TAC
REAL_LE_MUL2 THEN
ASM_SIMP_TAC[
NORM_POS_LE;
REAL_LT_IMP_LE] THEN
ASM_SIMP_TAC[REAL_ARITH `a <= b2 ==> a <= abs b1 + abs b2 + &1`] THEN
ASM_MESON_TAC[NORM_ARITH
`norm(f - l:real^P) < abs b2 + &1 /\ norm(l) <= b1
==> norm(f) <= abs b1 + abs b2 + &1`];
ONCE_REWRITE_TAC[
real_div] THEN
ASM_SIMP_TAC[
REAL_LT_LMUL_EQ;
REAL_HALF; GSYM REAL_MUL_ASSOC;
REAL_INV_MUL] THEN
REWRITE_TAC[REAL_ARITH `B * inv x * y < B <=> B * y / x < B * &1`] THEN
ASM_SIMP_TAC[
REAL_LT_INV_EQ;
REAL_LT_LMUL_EQ;
REAL_LT_LDIV_EQ;
REAL_ARITH `&0 < abs x + abs y + &2`] THEN
REAL_ARITH_TAC]));;
let LIM_BILINEAR = prove
(`!net:(A)net (h:real^M->real^N->real^P) f g l m.
(f --> l) net /\ (g --> m) net /\ bilinear h
==> ((\x. h (f x) (g x)) --> (h l m)) net`,
REPEAT STRIP_TAC THEN
MP_TAC(ISPECL
[`net:(A)net`; `\x:one. T`; `h:real^M->real^N->real^P`;
`\n:one. (f:A->real^M)`; `\n:one. (g:A->real^N)`;
`\n:one. (l:real^M)`; `\n:one. (m:real^N)`;
`norm(l:real^M)`; `norm(m:real^N)`]
UNIFORM_LIM_BILINEAR) THEN
ASM_REWRITE_TAC[
REAL_LE_REFL;
EVENTUALLY_TRUE] THEN
ASM_REWRITE_TAC[GSYM dist; GSYM tendsto]);;
let LIM_AT_ZERO = prove
(`!f:real^M->real^N l a.
(f --> l) (at a) <=> ((\x. f(a + x)) --> l) (at(vec 0))`,
REPEAT GEN_TAC THEN REWRITE_TAC[
LIM_AT] THEN
AP_TERM_TAC THEN ABS_TAC THEN
ASM_CASES_TAC `&0 < e` THEN ASM_REWRITE_TAC[] THEN
AP_TERM_TAC THEN ABS_TAC THEN
ASM_CASES_TAC `&0 < d` THEN ASM_REWRITE_TAC[] THEN
EQ_TAC THEN DISCH_TAC THEN X_GEN_TAC `x:real^M` THENL
[FIRST_X_ASSUM(MP_TAC o SPEC `a + x:real^M`) THEN
REWRITE_TAC[dist;
VECTOR_ADD_SUB;
VECTOR_SUB_RZERO];
FIRST_X_ASSUM(MP_TAC o SPEC `x - a:real^M`) THEN
REWRITE_TAC[dist;
VECTOR_SUB_RZERO;
VECTOR_SUB_ADD2]]);;
let LIM_TRANSFORM = prove
(`!net f g l.
((\x. f x - g x) --> vec 0) net /\ (f --> l) net ==> (g --> l) net`,
REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP
LIM_SUB) THEN
DISCH_THEN(MP_TAC o MATCH_MP
LIM_NEG) THEN MATCH_MP_TAC EQ_IMP THEN
AP_THM_TAC THEN BINOP_TAC THEN REWRITE_TAC[
FUN_EQ_THM] THEN
VECTOR_ARITH_TAC);;
let LIM_TRANSFORM_WITHIN = prove
(`!f g x s d.
&0 < d /\
(!x'. x'
IN s /\ &0 < dist(x',x) /\ dist(x',x) < d ==> f(x') = g(x')) /\
(f --> l) (at x within s)
==> (g --> l) (at x within s)`,
let LIM_TRANSFORM_AT = prove
(`!f g x d.
&0 < d /\
(!x'. &0 < dist(x',x) /\ dist(x',x) < d ==> f(x') = g(x')) /\
(f --> l) (at x)
==> (g --> l) (at x)`,
let LIM_TRANSFORM_EQ = prove
(`!net f:A->real^N g l.
((\x. f x - g x) --> vec 0) net ==> ((f --> l) net <=> (g --> l) net)`,
REPEAT STRIP_TAC THEN EQ_TAC THEN
DISCH_TAC THEN MATCH_MP_TAC
LIM_TRANSFORM THENL
[EXISTS_TAC `f:A->real^N` THEN ASM_REWRITE_TAC[];
EXISTS_TAC `g:A->real^N` THEN ASM_REWRITE_TAC[] THEN
ONCE_REWRITE_TAC[GSYM
LIM_NEG_EQ] THEN
ASM_REWRITE_TAC[
VECTOR_NEG_SUB;
VECTOR_NEG_0]]);;
let LIM_TRANSFORM_WITHIN_SET = prove
(`!f a s t.
eventually (\x. x
IN s <=> x
IN t) (at a)
==> ((f --> l) (at a within s) <=> (f --> l) (at a within t))`,
REPEAT GEN_TAC THEN REWRITE_TAC[
EVENTUALLY_AT;
LIM_WITHIN] THEN
DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN
EQ_TAC THEN DISCH_TAC THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `e:real`) THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_THEN `k:real` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `min d k:real` THEN ASM_REWRITE_TAC[
REAL_LT_MIN] THEN
ASM_MESON_TAC[]);;
let LIM_TRANSFORM_AWAY_WITHIN = prove
(`!f:real^M->real^N g a b s.
~(a = b) /\
(!x. x
IN s /\ ~(x = a) /\ ~(x = b) ==> f(x) = g(x)) /\
(f --> l) (at a within s)
==> (g --> l) (at a within s)`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC
LIM_TRANSFORM_WITHIN THEN
MAP_EVERY EXISTS_TAC [`f:real^M->real^N`; `dist(a:real^M,b)`] THEN
ASM_REWRITE_TAC[GSYM
DIST_NZ] THEN X_GEN_TAC `y:real^M` THEN
REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
ASM_MESON_TAC[
DIST_SYM;
REAL_LT_REFL]);;
let LIM_TRANSFORM_AWAY_AT = prove
(`!f:real^M->real^N g a b.
~(a = b) /\
(!x. ~(x = a) /\ ~(x = b) ==> f(x) = g(x)) /\
(f --> l) (at a)
==> (g --> l) (at a)`,
let LIM_CASES_SEQUENTIALLY = prove
(`!f g l m. (((\n. if m <= n then f n else g n) --> l) sequentially <=>
(f --> l) sequentially) /\
(((\n. if m < n then f n else g n) --> l) sequentially <=>
(f --> l) sequentially) /\
(((\n. if n <= m then f n else g n) --> l) sequentially <=>
(g --> l) sequentially) /\
(((\n. if n < m then f n else g n) --> l) sequentially <=>
(g --> l) sequentially)`,
let LIM_CONG_WITHIN = prove
(`(!x. ~(x = a) ==> f x = g x)
==> (((\x. f x) --> l) (at a within s) <=> ((g --> l) (at a within s)))`,
let LIM_CONG_AT = prove
(`(!x. ~(x = a) ==> f x = g x)
==> (((\x. f x) --> l) (at a) <=> ((g --> l) (at a)))`,
let CLOSURE_SEQUENTIAL = prove
(`!s l:real^N.
l
IN closure(s) <=> ?x. (!n. x(n)
IN s) /\ (x --> l) sequentially`,
REWRITE_TAC[closure;
IN_UNION;
LIMPT_SEQUENTIAL;
IN_ELIM_THM;
IN_DELETE] THEN
REPEAT GEN_TAC THEN MATCH_MP_TAC(TAUT
`((b ==> c) /\ (~a /\ c ==> b)) /\ (a ==> c) ==> (a \/ b <=> c)`) THEN
CONJ_TAC THENL [MESON_TAC[]; ALL_TAC] THEN DISCH_TAC THEN
EXISTS_TAC `\n:num. l:real^N` THEN
ASM_REWRITE_TAC[
LIM_CONST]);;
let SEQ_OFFSET = prove
(`!f l k. (f --> l) sequentially ==> ((\i. f(i + k)) --> l) sequentially`,
REWRITE_TAC[
LIM_SEQUENTIALLY] THEN
MESON_TAC[ARITH_RULE `N <= n ==> N <= n + k:num`]);;
let SEQ_OFFSET_NEG = prove
(`!f l k. (f --> l) sequentially ==> ((\i. f(i - k)) --> l) sequentially`,
REWRITE_TAC[
LIM_SEQUENTIALLY] THEN
MESON_TAC[ARITH_RULE `N + k <= n ==> N <= n - k:num`]);;
let SEQ_OFFSET_REV = prove
(`!f l k. ((\i. f(i + k)) --> l) sequentially ==> (f --> l) sequentially`,
REWRITE_TAC[
LIM_SEQUENTIALLY] THEN
MESON_TAC[ARITH_RULE `N + k <= n ==> N <= n - k /\ (n - k) + k = n:num`]);;
let SPHERE_EQ_SING = prove
(`!a:real^N r x. sphere(a,r) = {x} <=> x = a /\ r = &0`,
REPEAT GEN_TAC THEN EQ_TAC THEN SIMP_TAC[
SPHERE_SING] THEN
ASM_CASES_TAC `r < &0` THEN ASM_SIMP_TAC[
SPHERE_EMPTY;
NOT_INSERT_EMPTY] THEN
ASM_CASES_TAC `r = &0` THEN ASM_SIMP_TAC[
SPHERE_SING] THENL
[ASM SET_TAC[]; ALL_TAC] THEN
MATCH_MP_TAC(SET_RULE
`!y. (x
IN s ==> y
IN s /\ ~(y = x)) ==> ~(s = {x})`) THEN
EXISTS_TAC `a - (x - a):real^N` THEN REWRITE_TAC[
IN_SPHERE] THEN
REPEAT(POP_ASSUM MP_TAC) THEN CONV_TAC NORM_ARITH);;
let BOUNDED_CBALL = prove
(`!x:real^N e. bounded(cball(x,e))`,
REPEAT GEN_TAC THEN REWRITE_TAC[bounded] THEN
EXISTS_TAC `norm(x:real^N) + e` THEN REWRITE_TAC[
IN_CBALL; dist] THEN
NORM_ARITH_TAC);;
let FINITE_IMP_BOUNDED = prove
(`!s:real^N->bool.
FINITE s ==> bounded s`,
MATCH_MP_TAC
FINITE_INDUCT_STRONG THEN REWRITE_TAC[
BOUNDED_EMPTY] THEN
REWRITE_TAC[bounded;
IN_INSERT] THEN X_GEN_TAC `x:real^N` THEN GEN_TAC THEN
DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_TAC `B:real`) STRIP_ASSUME_TAC) THEN
EXISTS_TAC `norm(x:real^N) + abs B` THEN REPEAT STRIP_TAC THEN
ASM_MESON_TAC[
NORM_POS_LE; REAL_ARITH
`(y <= b /\ &0 <= x ==> y <= x + abs b) /\ x <= x + abs b`]);;
let BOUNDED_POS = prove
(`!s. bounded s <=> ?b. &0 < b /\ !x. x
IN s ==> norm(x) <= b`,
REWRITE_TAC[bounded] THEN
MESON_TAC[REAL_ARITH `&0 < &1 + abs(y) /\ (x <= y ==> x <= &1 + abs(y))`]);;
let BOUNDED_POS_LT = prove
(`!s. bounded s <=> ?b. &0 < b /\ !x. x
IN s ==> norm(x) < b`,
REWRITE_TAC[bounded] THEN
MESON_TAC[
REAL_LT_IMP_LE;
REAL_ARITH `&0 < &1 + abs(y) /\ (x <= y ==> x < &1 + abs(y))`]);;
let BOUNDED_TRANSLATION = prove
(`!a:real^N s. bounded s ==> bounded (
IMAGE (\x. a + x) s)`,
REPEAT GEN_TAC THEN REWRITE_TAC[
BOUNDED_POS;
FORALL_IN_IMAGE] THEN
DISCH_THEN(X_CHOOSE_TAC `B:real`) THEN
EXISTS_TAC `B + norm(a:real^N)` THEN POP_ASSUM MP_TAC THEN
MATCH_MP_TAC MONO_AND THEN CONJ_TAC THENL [NORM_ARITH_TAC; ALL_TAC] THEN
MATCH_MP_TAC
MONO_FORALL THEN GEN_TAC THEN MATCH_MP_TAC MONO_IMP THEN
REWRITE_TAC[] THEN NORM_ARITH_TAC);;
let BOUNDED_DIFFS = prove
(`!s t:real^N->bool.
bounded s /\ bounded t ==> bounded {x - y | x
IN s /\ y
IN t}`,
REPEAT GEN_TAC THEN REWRITE_TAC[
BOUNDED_POS] THEN
DISCH_THEN(CONJUNCTS_THEN2
(X_CHOOSE_TAC `B:real`) (X_CHOOSE_TAC `C:real`)) THEN
EXISTS_TAC `B + C:real` THEN REWRITE_TAC[
IN_ELIM_THM] THEN
CONJ_TAC THENL [ASM_REAL_ARITH_TAC; REPEAT STRIP_TAC] THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(NORM_ARITH
`norm x <= a /\ norm y <= b ==> norm(x - y) <= a + b`) THEN
ASM_SIMP_TAC[]);;
let BOUNDED_SUMS = prove
(`!s t:real^N->bool.
bounded s /\ bounded t ==> bounded {x + y | x
IN s /\ y
IN t}`,
REPEAT GEN_TAC THEN REWRITE_TAC[
BOUNDED_POS] THEN
DISCH_THEN(CONJUNCTS_THEN2
(X_CHOOSE_TAC `B:real`) (X_CHOOSE_TAC `C:real`)) THEN
EXISTS_TAC `B + C:real` THEN REWRITE_TAC[
IN_ELIM_THM] THEN
CONJ_TAC THENL [ASM_REAL_ARITH_TAC; REPEAT STRIP_TAC] THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(NORM_ARITH
`norm x <= a /\ norm y <= b ==> norm(x + y) <= a + b`) THEN
ASM_SIMP_TAC[]);;
let BOUNDED_SUBSET_BALL = prove
(`!s x:real^N. bounded(s) ==> ?r. &0 < r /\ s
SUBSET ball(x,r)`,
REPEAT GEN_TAC THEN REWRITE_TAC[
BOUNDED_POS] THEN
DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `&2 * B + norm(x:real^N)` THEN
ASM_SIMP_TAC[
NORM_POS_LE; REAL_ARITH
`&0 < B /\ &0 <= x ==> &0 < &2 * B + x`] THEN
REWRITE_TAC[
SUBSET] THEN X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `y:real^N`) THEN ASM_REWRITE_TAC[
IN_BALL] THEN
UNDISCH_TAC `&0 < B` THEN NORM_ARITH_TAC);;
let SUBSET_BALLS = prove
(`(!a a':real^N r r'.
ball(a,r)
SUBSET ball(a',r') <=> dist(a,a') + r <= r' \/ r <= &0) /\
(!a a':real^N r r'.
ball(a,r)
SUBSET cball(a',r') <=> dist(a,a') + r <= r' \/ r <= &0) /\
(!a a':real^N r r'.
cball(a,r)
SUBSET ball(a',r') <=> dist(a,a') + r < r' \/ r < &0) /\
(!a a':real^N r r'.
cball(a,r)
SUBSET cball(a',r') <=> dist(a,a') + r <= r' \/ r < &0)`,
let INTER_BALLS_EQ_EMPTY = prove
(`(!a b:real^N r s. ball(a,r)
INTER ball(b,s) = {} <=>
r <= &0 \/ s <= &0 \/ r + s <= dist(a,b)) /\
(!a b:real^N r s. ball(a,r)
INTER cball(b,s) = {} <=>
r <= &0 \/ s < &0 \/ r + s <= dist(a,b)) /\
(!a b:real^N r s. cball(a,r)
INTER ball(b,s) = {} <=>
r < &0 \/ s <= &0 \/ r + s <= dist(a,b)) /\
(!a b:real^N r s. cball(a,r)
INTER cball(b,s) = {} <=>
r < &0 \/ s < &0 \/ r + s < dist(a,b))`,
let compact = new_definition
`compact s <=>
!f:num->real^N.
(!n. f(n) IN s)
==> ?l r. l IN s /\ (!m n:num. m < n ==> r(m) < r(n)) /\
((f o r) --> l) sequentially`;;let MONOTONE_BIGGER = prove
(`!r. (!m n. m < n ==> r(m) < r(n)) ==> !n:num. n <= r(n)`,
GEN_TAC THEN DISCH_TAC THEN INDUCT_TAC THEN
ASM_MESON_TAC[
LE_0; ARITH_RULE `n <= m /\ m < p ==> SUC n <= p`;
LT]);;
let LIM_SUBSEQUENCE = prove
(`!s r l. (!m n. m < n ==> r(m) < r(n)) /\ (s --> l) sequentially
==> (s o r --> l) sequentially`,
let MONOTONE_SUBSEQUENCE = prove
(`!s:num->real. ?r:num->num.
(!m n. m < n ==> r(m) < r(n)) /\
((!m n. m <= n ==> s(r(m)) <= s(r(n))) \/
(!m n. m <= n ==> s(r(n)) <= s(r(m))))`,
GEN_TAC THEN
ASM_CASES_TAC `!n:num. ?p. n < p /\ !m. p <= m ==> s(m) <= s(p)` THEN
POP_ASSUM MP_TAC THEN
REWRITE_TAC[
NOT_FORALL_THM;
NOT_EXISTS_THM; NOT_IMP; DE_MORGAN_THM] THEN
REWRITE_TAC[
RIGHT_OR_EXISTS_THM;
SKOLEM_THM;
REAL_NOT_LE;
REAL_NOT_LT] THENL
[ABBREV_TAC `N = 0`; DISCH_THEN(X_CHOOSE_THEN `N:num` MP_TAC)] THEN
DISCH_THEN(X_CHOOSE_THEN `next:num->num` STRIP_ASSUME_TAC) THEN
(MP_TAC o prove_recursive_functions_exist num_RECURSION)
`(r 0 = next(SUC N)) /\ (!n. r(SUC n) = next(r n))` THEN
MATCH_MP_TAC
MONO_EXISTS THEN GEN_TAC THEN STRIP_TAC THENL
[SUBGOAL_THEN `!m:num n:num. r n <= m ==> s(m) <= s(r n):real`
ASSUME_TAC THEN TRY CONJ_TAC THEN TRY DISJ2_TAC THEN
GEN_TAC THEN INDUCT_TAC THEN ASM_REWRITE_TAC[
LT;
LE] THEN
ASM_MESON_TAC[
REAL_LE_TRANS;
REAL_LE_REFL;
LT_IMP_LE;
LT_TRANS];
SUBGOAL_THEN `!n. N < (r:num->num) n` ASSUME_TAC THEN
TRY(CONJ_TAC THENL [GEN_TAC; DISJ1_TAC THEN GEN_TAC]) THEN
INDUCT_TAC THEN ASM_REWRITE_TAC[
LT;
LE] THEN
TRY STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
ASM_MESON_TAC[
REAL_LT_REFL;
LT_LE;
LTE_TRANS;
REAL_LE_REFL;
REAL_LT_LE;
REAL_LE_TRANS;
LT]]);;
let CONVERGENT_BOUNDED_INCREASING = prove
(`!s:num->real b. (!m n. m <= n ==> s m <= s n) /\ (!n. abs(s n) <= b)
==> ?l. !e. &0 < e ==> ?N. !n. N <= n ==> abs(s n - l) < e`,
REPEAT STRIP_TAC THEN
MP_TAC(SPEC `\x. ?n. (s:num->real) n = x`
REAL_COMPLETE) THEN
REWRITE_TAC[] THEN ANTS_TAC THENL
[ASM_MESON_TAC[REAL_ARITH `abs(x) <= b ==> x <= b`]; ALL_TAC] THEN
MATCH_MP_TAC
MONO_EXISTS THEN X_GEN_TAC `l:real` THEN STRIP_TAC THEN
X_GEN_TAC `e:real` THEN STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `l - e`) THEN
ASM_MESON_TAC[REAL_ARITH `&0 < e ==> ~(l <= l - e)`;
REAL_ARITH `x <= y /\ y <= l /\ ~(x <= l - e) ==> abs(y - l) < e`]);;
let CONVERGENT_BOUNDED_MONOTONE = prove
(`!s:num->real b. (!n. abs(s n) <= b) /\
((!m n. m <= n ==> s m <= s n) \/
(!m n. m <= n ==> s n <= s m))
==> ?l. !e. &0 < e ==> ?N. !n. N <= n ==> abs(s n - l) < e`,
let COMPACT_REAL_LEMMA = prove
(`!s b. (!n:num. abs(s n) <= b)
==> ?l r. (!m n:num. m < n ==> r(m) < r(n)) /\
!e. &0 < e ==> ?N. !n. N <= n ==> abs(s(r n) - l) < e`,
let COMPACT_LEMMA = prove
(`!s. bounded s /\ (!n. (x:num->real^N) n
IN s)
==> !d. d <= dimindex(:N)
==> ?l:real^N r. (!m n. m < n ==> r m < (r:num->num) n) /\
!e. &0 < e
==> ?N. !n i. 1 <= i /\ i <= d
==> N <= n
==> abs(x(r n)$i - l$i) < e`,
GEN_TAC THEN REWRITE_TAC[bounded] THEN
DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_TAC `b:real`) ASSUME_TAC) THEN
INDUCT_TAC THENL
[REWRITE_TAC[ARITH_RULE `1 <= i /\ i <= 0 <=> F`;
CONJ_ASSOC] THEN
DISCH_TAC THEN EXISTS_TAC `\n:num. n` THEN REWRITE_TAC[];
ALL_TAC] THEN
DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o check (is_imp o concl)) THEN
ASM_SIMP_TAC[ARITH_RULE `SUC d <= n ==> d <= n`] THEN STRIP_TAC THEN
MP_TAC(SPECL [`\n:num. (x:num->real^N) (r n) $ (SUC d)`; `b:real`]
COMPACT_REAL_LEMMA) THEN
REWRITE_TAC[] THEN ANTS_TAC THENL
[ASM_MESON_TAC[
REAL_LE_TRANS;
COMPONENT_LE_NORM; ARITH_RULE `1 <= SUC n`];
ALL_TAC] THEN
DISCH_THEN(X_CHOOSE_THEN `y:real` (X_CHOOSE_THEN `s:num->num`
STRIP_ASSUME_TAC)) THEN
MAP_EVERY EXISTS_TAC
[`(lambda k. if k = SUC d then y else (l:real^N)$k):real^N`;
`(r:num->num) o (s:num->num)`] THEN
ASM_SIMP_TAC[
o_THM] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN
REPEAT(FIRST_ASSUM(C UNDISCH_THEN (MP_TAC o SPEC `e:real`) o concl)) THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_TAC `N1:num`) THEN
DISCH_THEN(X_CHOOSE_TAC `N2:num`) THEN EXISTS_TAC `N1 + N2:num` THEN
FIRST_ASSUM(fun th -> SIMP_TAC[
LAMBDA_BETA; MATCH_MP(ARITH_RULE
`SUC d <= n ==> !i. 1 <= i /\ i <= SUC d ==> 1 <= i /\ i <= n`) th]) THEN
REWRITE_TAC[
LE] THEN REPEAT STRIP_TAC THEN
ASM_REWRITE_TAC[] THEN TRY COND_CASES_TAC THEN
ASM_MESON_TAC[
MONOTONE_BIGGER;
LE_TRANS;
ARITH_RULE `N1 + N2 <= n ==> N2 <= n:num /\ N1 <= n`;
ARITH_RULE `1 <= i /\ i <= d /\ SUC d <= n
==> ~(i = SUC d) /\ 1 <= SUC d /\ d <= n /\ i <= n`]);;
let CAUCHY = prove
(`!s:num->real^N.
cauchy s <=> !e. &0 < e ==> ?N. !n. n >= N ==> dist(s n,s N) < e`,
REPEAT GEN_TAC THEN REWRITE_TAC[cauchy;
GE] THEN EQ_TAC THENL
[MESON_TAC[
LE_REFL]; DISCH_TAC] THEN
X_GEN_TAC `e:real` THEN DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `e / &2`) THEN ASM_REWRITE_TAC[
REAL_HALF] THEN
MESON_TAC[
DIST_TRIANGLE_HALF_L]);;
let COMPACT_IMP_COMPLETE = prove
(`!s:real^N->bool. compact s ==> complete s`,
GEN_TAC THEN REWRITE_TAC[complete; compact] THEN
MATCH_MP_TAC
MONO_FORALL THEN X_GEN_TAC `f:num->real^N` THEN
DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC
MONO_EXISTS THEN GEN_TAC THEN
DISCH_THEN(X_CHOOSE_THEN `r:num->num` STRIP_ASSUME_TAC) THEN
FIRST_X_ASSUM(MP_TAC o MATCH_MP (REWRITE_RULE[
IMP_CONJ]
LIM_ADD)) THEN
DISCH_THEN(MP_TAC o SPEC `\n. (f:num->real^N)(n) - f(r n)`) THEN
DISCH_THEN(MP_TAC o SPEC `vec 0: real^N`) THEN ASM_REWRITE_TAC[
o_THM] THEN
REWRITE_TAC[
VECTOR_ADD_RID;
VECTOR_SUB_ADD2; ETA_AX] THEN
DISCH_THEN MATCH_MP_TAC THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [cauchy]) THEN
REWRITE_TAC[
GE;
LIM;
SEQUENTIALLY; dist;
VECTOR_SUB_RZERO] THEN
SUBGOAL_THEN `!n:num. n <= r(n)` MP_TAC THENL [INDUCT_TAC; ALL_TAC] THEN
ASM_MESON_TAC[
LE_TRANS;
LE_REFL;
LT;
LET_TRANS;
LE_0;
LE_SUC_LT]);;
let COMPACT_IMP_TOTALLY_BOUNDED = prove
(`!s:real^N->bool.
compact s
==> !e. &0 < e ==> ?k.
FINITE k /\ k
SUBSET s /\
s
SUBSET (
UNIONS(
IMAGE (\x. ball(x,e)) k))`,
GEN_TAC THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN
REWRITE_TAC[
NOT_FORALL_THM; NOT_IMP;
NOT_EXISTS_THM] THEN
REWRITE_TAC[TAUT `~(a /\ b /\ c) <=> a /\ b ==> ~c`;
SUBSET] THEN
DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN
SUBGOAL_THEN
`?x:num->real^N. !n. x(n)
IN s /\ !m. m < n ==> ~(dist(x(m),x(n)) < e)`
MP_TAC THENL
[SUBGOAL_THEN
`?x:num->real^N.
!n. x(n) = @y. y
IN s /\ !m. m < n ==> ~(dist(x(m),y) < e)`
MP_TAC THENL
[MATCH_MP_TAC(MATCH_MP
WF_REC WF_num) THEN SIMP_TAC[]; ALL_TAC] THEN
MATCH_MP_TAC
MONO_EXISTS THEN X_GEN_TAC `x:num->real^N` THEN
DISCH_TAC THEN MATCH_MP_TAC
num_WF THEN X_GEN_TAC `n:num` THEN
FIRST_X_ASSUM(SUBST1_TAC o SPEC `n:num`) THEN STRIP_TAC THEN
CONV_TAC SELECT_CONV THEN
FIRST_X_ASSUM(MP_TAC o SPEC `
IMAGE (x:num->real^N) {m | m < n}`) THEN
SIMP_TAC[
FINITE_IMAGE;
FINITE_NUMSEG_LT;
NOT_FORALL_THM; NOT_IMP] THEN
REWRITE_TAC[
IN_UNIONS;
IN_IMAGE;
IN_ELIM_THM] THEN ASM_MESON_TAC[
IN_BALL];
ALL_TAC] THEN
REWRITE_TAC[compact;
NOT_FORALL_THM] THEN MATCH_MP_TAC
MONO_EXISTS THEN
X_GEN_TAC `x:num->real^N` THEN REWRITE_TAC[NOT_IMP;
FORALL_AND_THM] THEN
STRIP_TAC THEN ASM_REWRITE_TAC[
NOT_EXISTS_THM] THEN REPEAT STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o MATCH_MP
CONVERGENT_IMP_CAUCHY) THEN
REWRITE_TAC[cauchy] THEN DISCH_THEN(MP_TAC o SPEC `e:real`) THEN
ASM_REWRITE_TAC[
o_THM;
NOT_EXISTS_THM; NOT_IMP;
NOT_FORALL_THM; NOT_IMP] THEN
X_GEN_TAC `N:num` THEN MAP_EVERY EXISTS_TAC [`N:num`; `SUC N`] THEN
CONJ_TAC THENL [ARITH_TAC; ASM_MESON_TAC[
LT]]);;
let HEINE_BOREL_LEMMA = prove
(`!s:real^N->bool.
compact s
==> !t. s
SUBSET (
UNIONS t) /\ (!b. b
IN t ==> open b)
==> ?e. &0 < e /\
!x. x
IN s ==> ?b. b
IN t /\ ball(x,e)
SUBSET b`,
GEN_TAC THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN
REWRITE_TAC[
NOT_FORALL_THM; NOT_IMP;
NOT_EXISTS_THM] THEN
DISCH_THEN(CHOOSE_THEN (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
DISCH_THEN(MP_TAC o GEN `n:num` o SPEC `&1 / (&n + &1)`) THEN
SIMP_TAC[
REAL_LT_DIV;
REAL_LT_01; REAL_ARITH `x <= y ==> x < y + &1`;
FORALL_AND_THM;
REAL_POS;
NOT_FORALL_THM; NOT_IMP;
SKOLEM_THM; compact] THEN
MATCH_MP_TAC
MONO_EXISTS THEN GEN_TAC THEN REWRITE_TAC[
NOT_EXISTS_THM] THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ASM_REWRITE_TAC[] THEN
DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`l:real^N`; `r:num->num`] THEN
STRIP_TAC THEN
SUBGOAL_THEN `?b:real^N->bool. l
IN b /\ b
IN t` STRIP_ASSUME_TAC THENL
[ASM_MESON_TAC[
SUBSET;
IN_UNIONS]; ALL_TAC] THEN
SUBGOAL_THEN `?e. &0 < e /\ !z:real^N. dist(z,l) < e ==> z
IN b`
STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[
open_def]; ALL_TAC] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
LIM_SEQUENTIALLY]) THEN
DISCH_THEN(MP_TAC o SPEC `e / &2`) THEN
SUBGOAL_THEN `&0 < e / &2` (fun th ->
REWRITE_TAC[th;
o_THM] THEN MP_TAC(GEN_REWRITE_RULE I [
REAL_ARCH_INV] th))
THENL [ASM_REWRITE_TAC[
REAL_HALF]; ALL_TAC] THEN
DISCH_THEN(X_CHOOSE_THEN `N1:num` STRIP_ASSUME_TAC) THEN
DISCH_THEN(X_CHOOSE_THEN `N2:num` STRIP_ASSUME_TAC) THEN
FIRST_X_ASSUM(MP_TAC o SPECL
[`(r:num->num)(N1 + N2)`; `b:real^N->bool`]) THEN
ASM_REWRITE_TAC[
SUBSET] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN MATCH_MP_TAC
DIST_TRIANGLE_HALF_R THEN
EXISTS_TAC `(f:num->real^N)(r(N1 + N2:num))` THEN CONJ_TAC THENL
[ALL_TAC; FIRST_X_ASSUM MATCH_MP_TAC THEN ARITH_TAC] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
IN_BALL]) THEN
MATCH_MP_TAC(REAL_ARITH `a <= b ==> x < a ==> x < b`) THEN
MATCH_MP_TAC
REAL_LE_TRANS THEN EXISTS_TAC `inv(&N1)` THEN
ASM_SIMP_TAC[
REAL_LT_IMP_LE] THEN REWRITE_TAC[
real_div; REAL_MUL_LID] THEN
MATCH_MP_TAC
REAL_LE_INV2 THEN
REWRITE_TAC[REAL_OF_NUM_ADD; REAL_OF_NUM_LE;
REAL_OF_NUM_LT] THEN
ASM_MESON_TAC[ARITH_RULE `(~(n = 0) ==> 0 < n)`;
LE_ADD;
MONOTONE_BIGGER;
LT_IMP_LE;
LE_TRANS]);;
let BOUNDED_CLOSED_NEST = prove
(`!s. (!n. closed(s n)) /\ (!n. ~(s n = {})) /\
(!m n. m <= n ==> s(n)
SUBSET s(m)) /\
bounded(s 0)
==> ?a:real^N. !n:num. a
IN s(n)`,
GEN_TAC THEN REWRITE_TAC[GSYM
MEMBER_NOT_EMPTY;
SKOLEM_THM] THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
DISCH_THEN(CONJUNCTS_THEN2
(X_CHOOSE_TAC `a:num->real^N`) STRIP_ASSUME_TAC) THEN
SUBGOAL_THEN `compact(s 0:real^N->bool)` MP_TAC THENL
[ASM_MESON_TAC[
BOUNDED_CLOSED_IMP_COMPACT]; ALL_TAC] THEN
REWRITE_TAC[compact] THEN
DISCH_THEN(MP_TAC o SPEC `a:num->real^N`) THEN
ANTS_TAC THENL [ASM_MESON_TAC[
SUBSET;
LE_0]; ALL_TAC] THEN
MATCH_MP_TAC
MONO_EXISTS THEN X_GEN_TAC `l:real^N` THEN
REWRITE_TAC[
LIM_SEQUENTIALLY;
o_THM] THEN
DISCH_THEN(X_CHOOSE_THEN `r:num->num` STRIP_ASSUME_TAC) THEN
GEN_REWRITE_TAC I [TAUT `p <=> ~(~p)`] THEN
GEN_REWRITE_TAC RAND_CONV [
NOT_FORALL_THM] THEN
DISCH_THEN(X_CHOOSE_THEN `N:num` MP_TAC) THEN
MP_TAC(ISPECL [`l:real^N`; `(s:num->real^N->bool) N`]
CLOSED_APPROACHABLE) THEN
ASM_MESON_TAC[
SUBSET;
LE_REFL;
LE_TRANS;
LE_CASES;
MONOTONE_BIGGER]);;
let DECREASING_CLOSED_NEST = prove
(`!s. (!n. closed(s n)) /\ (!n. ~(s n = {})) /\
(!m n. m <= n ==> s(n)
SUBSET s(m)) /\
(!e. &0 < e ==> ?n. !x y. x
IN s(n) /\ y
IN s(n) ==> dist(x,y) < e)
==> ?a:real^N. !n:num. a
IN s(n)`,
let BOUNDED_CLOSED_CHAIN = prove
(`!f b:real^N->bool.
(!s. s
IN f ==> closed s /\ ~(s = {})) /\
(!s t. s
IN f /\ t
IN f ==> s
SUBSET t \/ t
SUBSET s) /\
b
IN f /\ bounded b
==> ~(
INTERS f = {})`,
REPEAT GEN_TAC THEN STRIP_TAC THEN
SUBGOAL_THEN `~(b
INTER (
INTERS f):real^N->bool = {})` MP_TAC THENL
[ALL_TAC; SET_TAC[]] THEN
MATCH_MP_TAC
COMPACT_IMP_FIP THEN
ASM_SIMP_TAC[
COMPACT_EQ_BOUNDED_CLOSED] THEN
X_GEN_TAC `u:(real^N->bool)->bool` THEN STRIP_TAC THEN
SUBGOAL_THEN `?s:real^N->bool. s
IN f /\ !t. t
IN u ==> s
SUBSET t`
MP_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN
UNDISCH_TAC `(u:(real^N->bool)->bool)
SUBSET f` THEN
UNDISCH_TAC `
FINITE(u:(real^N->bool)->bool)` THEN
SPEC_TAC(`u:(real^N->bool)->bool`,`u:(real^N->bool)->bool`) THEN
MATCH_MP_TAC
FINITE_INDUCT_STRONG THEN
CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
MAP_EVERY X_GEN_TAC [`t:real^N->bool`; `u:(real^N->bool)->bool`] THEN
REWRITE_TAC[
INSERT_SUBSET] THEN
DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN(CONJUNCTS_THEN2 MP_TAC STRIP_ASSUME_TAC) THEN
DISCH_THEN(X_CHOOSE_THEN `s:real^N->bool` STRIP_ASSUME_TAC) THEN
FIRST_X_ASSUM(MP_TAC o SPECL [`s:real^N->bool`; `t:real^N->bool`]) THEN
ASM SET_TAC[]);;
let UNIFORMLY_CONVERGENT_EQ_CAUCHY = prove
(`!P s:num->A->real^N.
(?l. !e. &0 < e
==> ?N. !n x. N <= n /\ P x ==> dist(s n x,l x) < e) <=>
(!e. &0 < e
==> ?N. !m n x. N <= m /\ N <= n /\ P x
==> dist(s m x,s n x) < e)`,
REPEAT GEN_TAC THEN EQ_TAC THENL
[DISCH_THEN(X_CHOOSE_TAC `l:A->real^N`) THEN X_GEN_TAC `e:real` THEN
DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `e / &2`) THEN
ASM_REWRITE_TAC[
REAL_HALF] THEN MESON_TAC[
DIST_TRIANGLE_HALF_L];
ALL_TAC] THEN
DISCH_TAC THEN
SUBGOAL_THEN `!x:A. P x ==> cauchy (\n. s n x :real^N)` MP_TAC THENL
[REWRITE_TAC[cauchy;
GE] THEN ASM_MESON_TAC[]; ALL_TAC] THEN
REWRITE_TAC[GSYM
CONVERGENT_EQ_CAUCHY;
LIM_SEQUENTIALLY] THEN
GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [
RIGHT_IMP_EXISTS_THM] THEN
REWRITE_TAC[
SKOLEM_THM] THEN MATCH_MP_TAC
MONO_EXISTS THEN
X_GEN_TAC `l:A->real^N` THEN DISCH_TAC THEN X_GEN_TAC `e:real` THEN
DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `e / &2`) THEN
ASM_REWRITE_TAC[
REAL_HALF] THEN MATCH_MP_TAC
MONO_EXISTS THEN
X_GEN_TAC `N:num` THEN STRIP_TAC THEN
MAP_EVERY X_GEN_TAC [`n:num`; `x:A`] THEN STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `x:A`) THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN(MP_TAC o SPEC `e / &2`) THEN ASM_REWRITE_TAC[
REAL_HALF] THEN
DISCH_THEN(X_CHOOSE_TAC `M:num`) THEN
FIRST_X_ASSUM(MP_TAC o SPECL [`n:num`; `N + M:num`; `x:A`]) THEN
ASM_REWRITE_TAC[
LE_ADD] THEN ONCE_REWRITE_TAC[
ADD_SYM] THEN
FIRST_X_ASSUM(MP_TAC o SPEC `M + N:num`) THEN REWRITE_TAC[
LE_ADD] THEN
ASM_MESON_TAC[
DIST_TRIANGLE_HALF_L;
DIST_SYM]);;
let UNIFORMLY_CAUCHY_IMP_UNIFORMLY_CONVERGENT = prove
(`!P (s:num->A->real^N) l.
(!e. &0 < e
==> ?N. !m n x. N <= m /\ N <= n /\ P x ==> dist(s m x,s n x) < e) /\
(!x. P x ==> !e. &0 < e ==> ?N. !n. N <= n ==> dist(s n x,l x) < e)
==> (!e. &0 < e ==> ?N. !n x. N <= n /\ P x ==> dist(s n x,l x) < e)`,
REPEAT GEN_TAC THEN REWRITE_TAC[GSYM
UNIFORMLY_CONVERGENT_EQ_CAUCHY] THEN
DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_TAC `l':A->real^N`) ASSUME_TAC) THEN
SUBGOAL_THEN `!x. P x ==> (l:A->real^N) x = l' x` MP_TAC THENL
[ALL_TAC; ASM_MESON_TAC[]] THEN
REPEAT STRIP_TAC THEN MATCH_MP_TAC(ISPEC `sequentially`
LIM_UNIQUE) THEN
EXISTS_TAC `\n. (s:num->A->real^N) n x` THEN
REWRITE_TAC[
LIM_SEQUENTIALLY;
TRIVIAL_LIMIT_SEQUENTIALLY] THEN
ASM_MESON_TAC[]);;
let CONTINUOUS_TRANSFORM_WITHIN = prove
(`!f g:real^M->real^N s x d.
&0 < d /\ x
IN s /\
(!x'. x'
IN s /\ dist(x',x) < d ==> f(x') = g(x')) /\
f continuous (at x within s)
==> g continuous (at x within s)`,
let CONTINUOUS_TRANSFORM_AT = prove
(`!f g:real^M->real^N x d.
&0 < d /\ (!x'. dist(x',x) < d ==> f(x') = g(x')) /\
f continuous (at x)
==> g continuous (at x)`,
let continuous_within = prove
(`f continuous (at x within s) <=>
!e. &0 < e
==> ?d. &0 < d /\
!x'. x'
IN s /\ dist(x',x) < d ==> dist(f(x'),f(x)) < e`,
let continuous_at = prove
(`f continuous (at x) <=>
!e. &0 < e ==> ?d. &0 < d /\
!x'. dist(x',x) < d ==> dist(f(x'),f(x)) < e`,
let CONTINUOUS_ADD = prove
(`!f g net. f continuous net /\ g continuous net
==> (\x. f(x) + g(x)) continuous net`,
REWRITE_TAC[continuous;
LIM_ADD]);;
let CONTINUOUS_SUB = prove
(`!f g net. f continuous net /\ g continuous net
==> (\x. f(x) - g(x)) continuous net`,
REWRITE_TAC[continuous;
LIM_SUB]);;
let CONTINUOUS_ABS = prove
(`!(f:A->real^N) net.
f continuous net
==> (\x. (lambda i. abs(f(x)$i)):real^N) continuous net`,
REWRITE_TAC[continuous;
LIM_ABS]);;
let CONTINUOUS_MAX = prove
(`!(f:A->real^N) (g:A->real^N) net.
f continuous net /\ g continuous net
==> (\x. (lambda i. max (f(x)$i) (g(x)$i)):real^N) continuous net`,
REWRITE_TAC[continuous;
LIM_MAX]);;
let CONTINUOUS_MIN = prove
(`!(f:A->real^N) (g:A->real^N) net.
f continuous net /\ g continuous net
==> (\x. (lambda i. min (f(x)$i) (g(x)$i)):real^N) continuous net`,
REWRITE_TAC[continuous;
LIM_MIN]);;
let CONTINUOUS_OPEN_PREIMAGE = prove
(`!f:real^M->real^N s t.
f
continuous_on s /\ open s /\ open t
==> open {x | x
IN s /\ f(x)
IN t}`,
REPEAT STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
CONTINUOUS_ON_OPEN]) THEN
REWRITE_TAC [
OPEN_IN_OPEN] THEN
DISCH_THEN(MP_TAC o SPEC `
IMAGE (f:real^M->real^N) s
INTER t`) THEN
ANTS_TAC THENL
[EXISTS_TAC `t:real^N->bool` THEN ASM_REWRITE_TAC [];
STRIP_TAC THEN
SUBGOAL_THEN `{x | x
IN s /\ (f:real^M->real^N) x
IN t} =
s
INTER t'` SUBST1_TAC THENL
[ASM SET_TAC []; ASM_MESON_TAC [
OPEN_INTER]]]);;
let LINEAR_CONTINUOUS_AT = prove
(`!f:real^M->real^N a. linear f ==> f continuous (at a)`,
REPEAT STRIP_TAC THEN
MP_TAC(ISPEC `\x. (f:real^M->real^N) (a + x) - f(a)`
LINEAR_LIM_0) THEN
ANTS_TAC THENL
[POP_ASSUM MP_TAC THEN SIMP_TAC[linear] THEN
REPEAT STRIP_TAC THEN VECTOR_ARITH_TAC;
ALL_TAC] THEN
REWRITE_TAC[GSYM
LIM_NULL;
CONTINUOUS_AT] THEN
GEN_REWRITE_TAC RAND_CONV [
LIM_AT_ZERO] THEN SIMP_TAC[]);;
let BILINEAR_CONTINUOUS_COMPOSE = prove
(`!net f:A->real^M g:A->real^N h:real^M->real^N->real^P.
f continuous net /\ g continuous net /\ bilinear h
==> (\x. h (f x) (g x)) continuous net`,
let COMPACT_CONTINUOUS_IMAGE = prove
(`!f:real^M->real^N s.
f
continuous_on s /\ compact s ==> compact(
IMAGE f s)`,
REPEAT GEN_TAC THEN REWRITE_TAC[
continuous_on; compact] THEN
STRIP_TAC THEN X_GEN_TAC `y:num->real^N` THEN
REWRITE_TAC[
IN_IMAGE;
SKOLEM_THM;
FORALL_AND_THM] THEN
DISCH_THEN(X_CHOOSE_THEN `x:num->real^M` STRIP_ASSUME_TAC) THEN
FIRST_X_ASSUM(MP_TAC o SPEC `x:num->real^M`) THEN ASM_REWRITE_TAC[] THEN
ONCE_REWRITE_TAC[
SWAP_EXISTS_THM] THEN MATCH_MP_TAC
MONO_EXISTS THEN
X_GEN_TAC `r:num->num` THEN
DISCH_THEN(X_CHOOSE_THEN `l:real^M` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `(f:real^M->real^N) l` THEN ASM_REWRITE_TAC[] THEN
CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
REWRITE_TAC[
LIM_SEQUENTIALLY] THEN
FIRST_X_ASSUM(MP_TAC o SPEC `l:real^M`) THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC
MONO_FORALL THEN X_GEN_TAC `e:real` THEN
DISCH_THEN(fun th -> DISCH_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
LIM_SEQUENTIALLY]) THEN
DISCH_THEN(MP_TAC o SPEC `d:real`) THEN ASM_REWRITE_TAC[
o_THM] THEN
ASM_MESON_TAC[]);;
let CONTINUOUS_ON_COMPOSE_QUOTIENT = prove
(`!f:real^M->real^N g:real^N->real^P s t u.
IMAGE f s
SUBSET t /\
IMAGE g t
SUBSET u /\
(!v. v
SUBSET t
==> (
open_in (subtopology euclidean s) {x | x
IN s /\ f x
IN v} <=>
open_in (subtopology euclidean t) v)) /\
(g o f)
continuous_on s
==> g
continuous_on t`,
REPEAT GEN_TAC THEN
REPLICATE_TAC 3 (DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
FIRST_ASSUM(fun th -> REWRITE_TAC[MATCH_MP
CONTINUOUS_ON_OPEN_GEN th]) THEN
SUBGOAL_THEN
`
IMAGE ((g:real^N->real^P) o (f:real^M->real^N)) s
SUBSET u`
(fun th -> REWRITE_TAC[MATCH_MP
CONTINUOUS_ON_OPEN_GEN th])
THENL [REWRITE_TAC[
IMAGE_o] THEN ASM SET_TAC[]; DISCH_TAC] THEN
X_GEN_TAC `v:real^P->bool` THEN DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `v:real^P->bool`) THEN
ASM_REWRITE_TAC[
o_THM] THEN DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `{x | x
IN t /\ (g:real^N->real^P) x
IN v}`) THEN
ASM_REWRITE_TAC[
SUBSET_RESTRICT] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (MESON[]
`
open_in top s ==> s = t ==>
open_in top t`)) THEN
ASM SET_TAC[]);;
let QUOTIENT_MAP_COMPOSE = prove
(`!f:real^M->real^N g:real^N->real^P s t u.
IMAGE f s
SUBSET t /\
(!v. v
SUBSET t
==> (
open_in (subtopology euclidean s) {x | x
IN s /\ f x
IN v} <=>
open_in (subtopology euclidean t) v)) /\
(!v. v
SUBSET u
==> (
open_in (subtopology euclidean t) {x | x
IN t /\ g x
IN v} <=>
open_in (subtopology euclidean u) v))
==> !v. v
SUBSET u
==> (
open_in (subtopology euclidean s)
{x | x
IN s /\ (g o f) x
IN v} <=>
open_in (subtopology euclidean u) v)`,
REPEAT STRIP_TAC THEN REWRITE_TAC[
o_THM] THEN
SUBGOAL_THEN
`{x | x
IN s /\ (g:real^N->real^P) ((f:real^M->real^N) x)
IN v} =
{x | x
IN s /\ f x
IN {x | x
IN t /\ g x
IN v}}`
SUBST1_TAC THENL [ASM SET_TAC[]; ASM_SIMP_TAC[
SUBSET_RESTRICT]]);;
let CONNECTED_MONOTONE_QUOTIENT_PREIMAGE = prove
(`!f:real^M->real^N s t.
f
continuous_on s /\
IMAGE f s = t /\
(!u. u
SUBSET t
==> (
open_in (subtopology euclidean s) {x | x
IN s /\ f x
IN u} <=>
open_in (subtopology euclidean t) u)) /\
(!y. y
IN t ==> connected {x | x
IN s /\ f x = y}) /\
connected t
==> connected s`,
REPEAT STRIP_TAC THEN REWRITE_TAC[connected;
NOT_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC [`u:real^M->bool`; `v:real^M->bool`] THEN STRIP_TAC THEN
UNDISCH_TAC `connected(t:real^N->bool)` THEN SIMP_TAC[
CONNECTED_OPEN_IN] THEN
MAP_EVERY EXISTS_TAC
[`
IMAGE (f:real^M->real^N) (s
INTER u)`;
`
IMAGE (f:real^M->real^N) (s
INTER v)`] THEN
ASM_REWRITE_TAC[
IMAGE_EQ_EMPTY] THEN
SUBGOAL_THEN
`
IMAGE (f:real^M->real^N) (s
INTER u)
INTER IMAGE f (s
INTER v) = {}`
ASSUME_TAC THENL
[REWRITE_TAC[
EXTENSION;
IN_INTER;
NOT_IN_EMPTY] THEN
X_GEN_TAC `y:real^N` THEN STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `y:real^N`) THEN
ANTS_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[connected]] THEN
MAP_EVERY EXISTS_TAC [`u:real^M->bool`; `v:real^M->bool`] THEN
ASM SET_TAC[];
ALL_TAC] THEN
ONCE_REWRITE_TAC[
CONJ_ASSOC] THEN
CONJ_TAC THENL [CONJ_TAC; ASM SET_TAC[]] THEN
FIRST_X_ASSUM(fun th ->
W(MP_TAC o PART_MATCH (rand o rand) th o snd)) THEN
(ANTS_TAC THENL [ASM SET_TAC[]; DISCH_THEN(SUBST1_TAC o SYM)]) THEN
MATCH_MP_TAC(MESON[]
`{x | x
IN s /\ f x
IN IMAGE f u} = u /\
open_in top u
==>
open_in top {x | x
IN s /\ f x
IN IMAGE f u}`) THEN
ASM_SIMP_TAC[
OPEN_IN_OPEN_INTER] THEN ASM SET_TAC[]);;
let PROPER_MAP_COMPOSE = prove
(`!f:real^M->real^N g:real^N->real^P s t u.
IMAGE f s
SUBSET t /\
(!k. k
SUBSET t /\ compact k ==> compact {x | x
IN s /\ f x
IN k}) /\
(!k. k
SUBSET u /\ compact k ==> compact {x | x
IN t /\ g x
IN k})
==> !k. k
SUBSET u /\ compact k
==> compact {x | x
IN s /\ (g o f) x
IN k}`,
REPEAT STRIP_TAC THEN REWRITE_TAC[
o_THM] THEN
FIRST_X_ASSUM(MP_TAC o SPEC `k:real^P->bool`) THEN
ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `{x | x
IN t /\ (g:real^N->real^P) x
IN k}`) THEN
ANTS_TAC THENL [ASM SET_TAC[]; MATCH_MP_TAC EQ_IMP] THEN
AP_TERM_TAC THEN ASM SET_TAC[]);;
let PASTING_LEMMA_EXISTS = prove
(`!f:A->real^M->real^N t s k.
s
SUBSET UNIONS {t i | i
IN k} /\
(!i. i
IN k
==>
open_in (subtopology euclidean s) (t i) /\
(f i)
continuous_on (t i)) /\
(!i j x. i
IN k /\ j
IN k /\ x
IN s
INTER t i
INTER t j
==> f i x = f j x)
==> ?g. g
continuous_on s /\
(!x i. i
IN k /\ x
IN s
INTER t i ==> g x = f i x)`,
REPEAT STRIP_TAC THEN
EXISTS_TAC `\x. (f:A->real^M->real^N)(@i. i
IN k /\ x
IN t i) x` THEN
CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC
PASTING_LEMMA THEN
MAP_EVERY EXISTS_TAC
[`f:A->real^M->real^N`; `t:A->real^M->bool`; `k:A->bool`] THEN
ASM SET_TAC[]);;
let LIM_LIFT_DOT = prove
(`!f:real^M->real^N a.
(f --> l) net ==> ((lift o (\y. a dot f(y))) --> lift(a dot l)) net`,
let CONTINUOUS_ON_CLOSURE = prove
(`!f:real^M->real^N s.
f
continuous_on closure s <=>
!x e. x
IN closure s /\ &0 < e
==> ?d. &0 < d /\
!y. y
IN s /\ dist(y,x) < d ==> dist(f y,f x) < e`,
REPEAT GEN_TAC THEN REWRITE_TAC[
continuous_on] THEN
EQ_TAC THENL [MESON_TAC[REWRITE_RULE[
SUBSET]
CLOSURE_SUBSET]; ALL_TAC] THEN
DISCH_TAC THEN X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN
X_GEN_TAC `e:real` THEN DISCH_TAC THEN
FIRST_ASSUM(MP_TAC o SPECL [`x:real^M`; `e / &2`]) THEN
ANTS_TAC THENL [ASM_REWRITE_TAC[
REAL_HALF]; ALL_TAC] THEN
DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `d / &2` THEN ASM_REWRITE_TAC[
REAL_HALF] THEN
X_GEN_TAC `y:real^M` THEN STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPECL [`y:real^M`; `e / &2`]) THEN
ASM_REWRITE_TAC[
REAL_HALF] THEN
DISCH_THEN(X_CHOOSE_THEN `k:real` STRIP_ASSUME_TAC) THEN
MP_TAC(ISPECL [`y:real^M`; `s:real^M->bool`]
CLOSURE_APPROACHABLE) THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o SPEC `min k (d / &2)`) THEN
ASM_REWRITE_TAC[
REAL_HALF;
REAL_LT_MIN] THEN
ASM_MESON_TAC[
DIST_SYM; NORM_ARITH
`dist(a,b) < e / &2 /\ dist(b,c) < e / &2 ==> dist(a,c) < e`]);;
let UNIFORMLY_CONTINUOUS_ON_CLOSURE = prove
(`!f:real^M->real^N s.
f
uniformly_continuous_on s /\ f
continuous_on closure s
==> f
uniformly_continuous_on closure s`,
REPEAT GEN_TAC THEN
REWRITE_TAC[
uniformly_continuous_on] THEN STRIP_TAC THEN
X_GEN_TAC `e:real` THEN DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `e / &3`) THEN
ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `d / &3` THEN CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
MAP_EVERY X_GEN_TAC [`x:real^M`; `y:real^M`] THEN STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
continuous_on]) THEN
DISCH_THEN(fun th ->
MP_TAC(SPEC `y:real^M` th) THEN MP_TAC(SPEC `x:real^M` th)) THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN(MP_TAC o SPEC `e / &3`) THEN
ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
DISCH_THEN(X_CHOOSE_THEN `d1:real` (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
MP_TAC(ISPECL [`x:real^M`; `s:real^M->bool`]
CLOSURE_APPROACHABLE) THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o SPEC `min d1 (d / &3)`) THEN
ANTS_TAC THENL [ASM_REAL_ARITH_TAC; REWRITE_TAC[
REAL_LT_MIN]] THEN
DISCH_THEN(X_CHOOSE_THEN `x':real^M` STRIP_ASSUME_TAC) THEN
DISCH_THEN(MP_TAC o SPEC `x':real^M`) THEN
ASM_SIMP_TAC[REWRITE_RULE[
SUBSET]
CLOSURE_SUBSET] THEN DISCH_TAC THEN
DISCH_THEN(MP_TAC o SPEC `e / &3`) THEN
ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
DISCH_THEN(X_CHOOSE_THEN `d2:real` (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
MP_TAC(ISPECL [`y:real^M`; `s:real^M->bool`]
CLOSURE_APPROACHABLE) THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o SPEC `min d2 (d / &3)`) THEN
ANTS_TAC THENL [ASM_REAL_ARITH_TAC; REWRITE_TAC[
REAL_LT_MIN]] THEN
DISCH_THEN(X_CHOOSE_THEN `y':real^M` STRIP_ASSUME_TAC) THEN
DISCH_THEN(MP_TAC o SPEC `y':real^M`) THEN
ASM_SIMP_TAC[REWRITE_RULE[
SUBSET]
CLOSURE_SUBSET] THEN DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPECL [`x':real^M`; `y':real^M`]) THEN
ASM_MESON_TAC[
DIST_SYM; NORM_ARITH
`dist(y,x) < d / &3 /\ dist(x',x) < d / &3 /\ dist(y',y) < d / &3
==> dist(y',x') < d`]);;
let CAUCHY_CONTINUOUS_UNIQUENESS_LEMMA = prove
(`!f:real^M->real^N s.
(!x. cauchy x /\ (!n. (x n)
IN s) ==> cauchy(f o x))
==> !a x. (!n. (x n)
IN s) /\ (x --> a) sequentially
==> ?l. ((f o x) --> l) sequentially /\
!y. (!n. (y n)
IN s) /\ (y --> a) sequentially
==> ((f o y) --> l) sequentially`,
REPEAT STRIP_TAC THEN
FIRST_ASSUM(MP_TAC o SPEC `x:num->real^M`) THEN
ANTS_TAC THENL [ASM_MESON_TAC[
CONVERGENT_IMP_CAUCHY]; ALL_TAC] THEN
REWRITE_TAC[GSYM
CONVERGENT_EQ_CAUCHY] THEN MATCH_MP_TAC
MONO_EXISTS THEN
X_GEN_TAC `l:real^N` THEN DISCH_TAC THEN ASM_REWRITE_TAC[] THEN
X_GEN_TAC `y:num->real^M` THEN STRIP_TAC THEN
FIRST_ASSUM(MP_TAC o SPEC `y:num->real^M`) THEN
ANTS_TAC THENL [ASM_MESON_TAC[
CONVERGENT_IMP_CAUCHY]; ALL_TAC] THEN
REWRITE_TAC[GSYM
CONVERGENT_EQ_CAUCHY] THEN
DISCH_THEN(X_CHOOSE_THEN `m:real^N` STRIP_ASSUME_TAC) THEN
SUBGOAL_THEN `l:real^N = m` (fun th -> ASM_REWRITE_TAC[th]) THEN
ONCE_REWRITE_TAC[GSYM
VECTOR_SUB_EQ] THEN
MATCH_MP_TAC(ISPEC `sequentially`
LIM_UNIQUE) THEN
EXISTS_TAC `\n:num. (f:real^M->real^N)(x n) - f(y n)` THEN
RULE_ASSUM_TAC(REWRITE_RULE[
o_DEF]) THEN
ASM_SIMP_TAC[
LIM_SUB;
TRIVIAL_LIMIT_SEQUENTIALLY] THEN
FIRST_X_ASSUM(MP_TAC o SPEC
`\n. if
EVEN n then x(n DIV 2):real^M else y(n DIV 2)`) THEN
REWRITE_TAC[cauchy;
o_THM;
LIM_SEQUENTIALLY] THEN ANTS_TAC THENL
[CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[]] THEN
X_GEN_TAC `e:real` THEN DISCH_TAC THEN MAP_EVERY UNDISCH_TAC
[`((y:num->real^M) --> a) sequentially`;
`((x:num->real^M) --> a) sequentially`] THEN
REPEAT(FIRST_X_ASSUM(K ALL_TAC o check (is_forall o concl))) THEN
REWRITE_TAC[
LIM_SEQUENTIALLY] THEN
DISCH_THEN(MP_TAC o SPEC `e / &2`) THEN ASM_REWRITE_TAC[
REAL_HALF] THEN
DISCH_THEN(X_CHOOSE_TAC `N1:num`) THEN
DISCH_THEN(MP_TAC o SPEC `e / &2`) THEN ASM_REWRITE_TAC[
REAL_HALF] THEN
DISCH_THEN(X_CHOOSE_TAC `N2:num`) THEN
EXISTS_TAC `2 * (N1 + N2)` THEN
MAP_EVERY X_GEN_TAC [`m:num`; `n:num`] THEN STRIP_TAC THEN
REPEAT(FIRST_X_ASSUM(fun th ->
MP_TAC(SPEC `m DIV 2` th) THEN MP_TAC(SPEC `n DIV 2` th))) THEN
REPEAT(ANTS_TAC THENL [ASM_ARITH_TAC; DISCH_TAC]) THEN
REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[]) THEN
REPEAT(FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM
REAL_NOT_LE])) THEN
CONV_TAC NORM_ARITH;
MATCH_MP_TAC
MONO_FORALL THEN X_GEN_TAC `e:real` THEN
ASM_CASES_TAC `&0 < e` THEN ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC
MONO_EXISTS THEN X_GEN_TAC `N:num` THEN DISCH_TAC THEN
X_GEN_TAC `n:num` THEN DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPECL [`2 * n`; `2 * n + 1`]) THEN
ANTS_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN
REWRITE_TAC[
EVEN_ADD;
EVEN_MULT;
ARITH_EVEN] THEN
REWRITE_TAC[ARITH_RULE `(2 * n) DIV 2 = n /\ (2 * n + 1) DIV 2 = n`] THEN
REWRITE_TAC[dist;
VECTOR_SUB_RZERO]]);;
let CONTINUOUS_AT_COMPOSE_EQ = prove
(`!f:real^M->real^N g:real^M->real^M h:real^M->real^M.
g continuous at x /\ h continuous at (g x) /\
(!y. g(h y) = y) /\ h(g x) = x
==> (f continuous at (g x) <=> (\x. f(g x)) continuous at x)`,
REPEAT STRIP_TAC THEN EQ_TAC THEN
ASM_SIMP_TAC[REWRITE_RULE[
o_DEF]
CONTINUOUS_AT_COMPOSE] THEN
DISCH_TAC THEN
SUBGOAL_THEN
`((f:real^M->real^N) o (g:real^M->real^M) o (h:real^M->real^M))
continuous at (g(x:real^M))`
MP_TAC THENL
[REWRITE_TAC[
o_ASSOC] THEN MATCH_MP_TAC
CONTINUOUS_AT_COMPOSE THEN
ASM_REWRITE_TAC[
o_DEF];
ASM_REWRITE_TAC[
o_DEF; ETA_AX]]);;
let CONTINUOUS_AT_LINEAR_IMAGE = prove
(`!h:real^M->real^M z f:real^M->real^N.
linear h /\ (!x. norm(h x) = norm x)
==> (f continuous at (h z) <=> (\x. f(h x)) continuous at z)`,
let CONTINUOUS_WITHIN_AVOID = prove
(`!f:real^M->real^N x s a.
f continuous (at x within s) /\ x
IN s /\ ~(f x = a)
==> ?e. &0 < e /\ !y. y
IN s /\ dist(x,y) < e ==> ~(f y = a)`,
REPEAT STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
continuous_within]) THEN
DISCH_THEN(MP_TAC o SPEC `norm((f:real^M->real^N) x - a)`) THEN
ASM_REWRITE_TAC[
NORM_POS_LT;
VECTOR_SUB_EQ] THEN
MATCH_MP_TAC
MONO_EXISTS THEN GEN_TAC THEN MATCH_MP_TAC MONO_AND THEN
REWRITE_TAC[] THEN MATCH_MP_TAC
MONO_FORALL THEN
GEN_TAC THEN MATCH_MP_TAC MONO_IMP THEN SIMP_TAC[] THEN NORM_ARITH_TAC);;
let CONTINUOUS_AT_AVOID = prove
(`!f:real^M->real^N x a.
f continuous (at x) /\ ~(f x = a)
==> ?e. &0 < e /\ !y. dist(x,y) < e ==> ~(f y = a)`,
let OPEN_SUMS = prove
(`!s t:real^N->bool.
open s \/ open t ==> open {x + y | x
IN s /\ y
IN t}`,
REPEAT GEN_TAC THEN REWRITE_TAC[
open_def] THEN STRIP_TAC THEN
REWRITE_TAC[
FORALL_IN_GSPEC] THEN
MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN STRIP_TAC THENL
[FIRST_X_ASSUM(MP_TAC o SPEC `x:real^N`);
FIRST_X_ASSUM(MP_TAC o SPEC `y:real^N`)] THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC
MONO_EXISTS THEN
X_GEN_TAC `e:real` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
X_GEN_TAC `z:real^N` THEN DISCH_TAC THEN REWRITE_TAC[
IN_ELIM_THM] THEN
ASM_MESON_TAC[
VECTOR_ADD_SYM; VECTOR_ARITH `(z - y) + y:real^N = z`;
NORM_ARITH `dist(z:real^N,x + y) < e ==> dist(z - y,x) < e`]);;
let UPPER_LOWER_HEMICONTINUOUS_EXPLICIT = prove
(`!f:real^M->real^N->bool t s.
(!x. x
IN s ==> f(x)
SUBSET t) /\
(!u.
open_in (subtopology euclidean t) u
==>
open_in (subtopology euclidean s)
{x | x
IN s /\ f(x)
SUBSET u}) /\
(!u.
closed_in (subtopology euclidean t) u
==>
closed_in (subtopology euclidean s)
{x | x
IN s /\ f(x)
SUBSET u})
==> !x e. x
IN s /\ &0 < e /\ bounded(f x) /\ ~(f x = {})
==> ?d. &0 < d /\
!x'. x'
IN s /\ dist(x,x') < d
==> (!y. y
IN f x
==> ?y'. y'
IN f x' /\ dist(y,y') < e) /\
(!y'. y'
IN f x'
==> ?y. y
IN f x /\ dist(y',y) < e)`,
let COMPONENTS_LINEAR_IMAGE = prove
(`!f s. linear f /\ (!x y. f x = f y ==> x = y) /\ (!y. ?x. f x = y)
==> components(
IMAGE f s) =
IMAGE (
IMAGE f) (components s)`,
REWRITE_TAC[components] THEN GEOM_TRANSFORM_TAC[] THEN SET_TAC[]);;
let IN_COMPONENTS = prove
(`!u:real^N->bool s. s
IN components u
<=> ?x. x
IN u /\ s =
connected_component u x`,
REPEAT GEN_TAC THEN REWRITE_TAC[components] THEN EQ_TAC
THENL [SET_TAC[];STRIP_TAC THEN ASM_SIMP_TAC[] THEN
UNDISCH_TAC `x:real^N
IN u` THEN SET_TAC[]]);;
let COMPONENTS_UNIQUE = prove
(`!s:real^N->bool k.
UNIONS k = s /\
(!c. c
IN k
==> connected c /\ ~(c = {}) /\
!c'. connected c' /\ c
SUBSET c' /\ c'
SUBSET s ==> c' = c)
==> components s = k`,
REPEAT STRIP_TAC THEN GEN_REWRITE_TAC I [
EXTENSION] THEN
X_GEN_TAC `c:real^N->bool` THEN REWRITE_TAC[
IN_COMPONENTS] THEN
EQ_TAC THENL
[DISCH_THEN(X_CHOOSE_THEN `x:real^N`
(CONJUNCTS_THEN2 ASSUME_TAC SUBST1_TAC)) THEN
FIRST_ASSUM(MP_TAC o SPEC `x:real^N` o GEN_REWRITE_RULE I [
EXTENSION]) THEN
REWRITE_TAC[
IN_UNIONS] THEN ASM_REWRITE_TAC[
LEFT_IMP_EXISTS_THM] THEN
X_GEN_TAC `c:real^N->bool` THEN STRIP_TAC THEN
SUBGOAL_THEN `
connected_component s (x:real^N) = c`
(fun th -> ASM_REWRITE_TAC[th]) THEN
MATCH_MP_TAC
CONNECTED_COMPONENT_UNIQUE THEN
FIRST_X_ASSUM(MP_TAC o SPEC `c:real^N->bool`) THEN
ASM_REWRITE_TAC[] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
X_GEN_TAC `c':real^N->bool` THEN STRIP_TAC THEN
REWRITE_TAC[SET_RULE `c'
SUBSET c <=> c'
UNION c = c`] THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN CONJ_TAC THENL
[MATCH_MP_TAC
CONNECTED_UNION; ASM SET_TAC[]] THEN
ASM SET_TAC[];
DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `c:real^N->bool`) THEN
ASM_REWRITE_TAC[] THEN STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM
MEMBER_NOT_EMPTY]) THEN
MATCH_MP_TAC
MONO_EXISTS THEN X_GEN_TAC `x:real^N` THEN STRIP_TAC THEN
CONJ_TAC THENL [ASM SET_TAC[]; CONV_TAC SYM_CONV] THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN
REWRITE_TAC[
CONNECTED_CONNECTED_COMPONENT;
CONNECTED_COMPONENT_SUBSET] THEN
MATCH_MP_TAC
CONNECTED_COMPONENT_MAXIMAL THEN
ASM_REWRITE_TAC[] THEN ASM SET_TAC[]]);;
let CLOPEN_IN_COMPONENTS = prove
(`!u s:real^N->bool.
closed_in (subtopology euclidean u) s /\
open_in (subtopology euclidean u) s /\
connected s /\ ~(s = {})
==> s
IN components u`,
REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[
CONJ_ASSOC] THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
FIRST_ASSUM(MP_TAC o MATCH_MP
CLOPEN_UNIONS_COMPONENTS) THEN
DISCH_THEN(X_CHOOSE_THEN `k:(real^N->bool)->bool` STRIP_ASSUME_TAC) THEN
ASM_CASES_TAC `k:(real^N->bool)->bool = {}` THEN
ASM_REWRITE_TAC[
UNIONS_0] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM
MEMBER_NOT_EMPTY]) THEN
DISCH_THEN(X_CHOOSE_TAC `c:real^N->bool`) THEN
ASM_CASES_TAC `k = {c:real^N->bool}` THENL
[ASM_MESON_TAC[
UNIONS_1; GSYM
SING_SUBSET]; ALL_TAC] THEN
MATCH_MP_TAC(TAUT `~p ==> p /\ q ==> r`) THEN
SUBGOAL_THEN `?c':real^N->bool. c'
IN k /\ ~(c = c')` STRIP_ASSUME_TAC THENL
[ASM_MESON_TAC[SET_RULE
`a
IN s /\ ~(s = {a}) ==> ?b. b
IN s /\ ~(b = a)`];
REWRITE_TAC[
CONNECTED_EQ_CONNECTED_COMPONENTS_EQ] THEN
DISCH_THEN(MP_TAC o SPECL [`c:real^N->bool`; `c':real^N->bool`]) THEN
ASM_REWRITE_TAC[NOT_IMP] THEN CONJ_TAC THEN
MATCH_MP_TAC
COMPONENTS_INTERMEDIATE_SUBSET THEN
EXISTS_TAC `u:real^N->bool` THEN
MP_TAC(ISPEC `u:real^N->bool`
UNIONS_COMPONENTS) THEN ASM SET_TAC[]]);;
let COMPACT_UNIFORMLY_EQUICONTINUOUS = prove
(`!(fs:(real^M->real^N)->bool) s.
(!x e. x
IN s /\ &0 < e
==> ?d. &0 < d /\
(!f x'. f
IN fs /\ x'
IN s /\ dist (x',x) < d
==> dist (f x',f x) < e)) /\
compact s
==> !e. &0 < e
==> ?d. &0 < d /\
!f x x'. f
IN fs /\ x
IN s /\ x'
IN s /\ dist (x',x) < d
==> dist(f x',f x) < e`,
REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN
GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV) [
RIGHT_IMP_EXISTS_THM] THEN
REWRITE_TAC[
SKOLEM_THM;
LEFT_IMP_EXISTS_THM] THEN
X_GEN_TAC `d:real^M->real->real` THEN DISCH_TAC THEN X_GEN_TAC `e:real` THEN
DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP
HEINE_BOREL_LEMMA) THEN
DISCH_THEN(MP_TAC o SPEC
`{ ball(x:real^M,d x (e / &2)) | x
IN s}`) THEN
SIMP_TAC[
FORALL_IN_GSPEC;
OPEN_BALL;
UNIONS_GSPEC;
SUBSET;
IN_ELIM_THM] THEN
ANTS_TAC THENL [ASM_MESON_TAC[
CENTRE_IN_BALL;
REAL_HALF]; ALL_TAC] THEN
MATCH_MP_TAC
MONO_EXISTS THEN X_GEN_TAC `k:real` THEN STRIP_TAC THEN
ASM_REWRITE_TAC[] THEN
MAP_EVERY X_GEN_TAC [`f:real^M->real^N`; `u:real^M`; `v:real^M`] THEN
STRIP_TAC THEN FIRST_X_ASSUM(fun th -> MP_TAC(SPEC `v:real^M` th) THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN(CHOOSE_THEN MP_TAC)) THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
DISCH_THEN(fun th ->
MP_TAC(SPEC `u:real^M` th) THEN MP_TAC(SPEC `v:real^M` th)) THEN
ASM_REWRITE_TAC[
DIST_REFL] THEN
FIRST_X_ASSUM(X_CHOOSE_THEN `w:real^M` (CONJUNCTS_THEN2 ASSUME_TAC
SUBST_ALL_TAC)) THEN
ASM_REWRITE_TAC[
CENTRE_IN_BALL] THEN ASM_REWRITE_TAC[
IN_BALL] THEN
ONCE_REWRITE_TAC[
DIST_SYM] THEN REPEAT STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPECL [`w:real^M`; `e / &2`]) THEN
ASM_REWRITE_TAC[
REAL_HALF] THEN
DISCH_THEN(MP_TAC o SPEC `f:real^M->real^N` o CONJUNCT2) THEN
DISCH_THEN(fun th -> MP_TAC(SPEC `u:real^M` th) THEN
MP_TAC(SPEC `v:real^M` th)) THEN
ASM_REWRITE_TAC[] THEN CONV_TAC NORM_ARITH);;
let CONTINUOUS_UNIFORM_LIMIT = prove
(`!net f:A->real^M->real^N g s.
~(
trivial_limit net) /\
eventually (\n. (f n)
continuous_on s) net /\
(!e. &0 < e
==> eventually (\n. !x. x
IN s ==> norm(f n x - g x) < e) net)
==> g
continuous_on s`,
REWRITE_TAC[
continuous_on] THEN REPEAT GEN_TAC THEN STRIP_TAC THEN
X_GEN_TAC `x:real^M` THEN STRIP_TAC THEN
X_GEN_TAC `e:real` THEN DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `e / &3`) THEN
ASM_SIMP_TAC[
REAL_LT_DIV;
REAL_OF_NUM_LT; ARITH] THEN
FIRST_X_ASSUM(fun th -> MP_TAC th THEN REWRITE_TAC[IMP_IMP] THEN
GEN_REWRITE_TAC LAND_CONV [GSYM
EVENTUALLY_AND]) THEN
DISCH_THEN(MP_TAC o MATCH_MP
EVENTUALLY_HAPPENS) THEN
ASM_REWRITE_TAC[
LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `a:A` THEN
DISCH_THEN(CONJUNCTS_THEN2 (MP_TAC o SPEC `x:real^M`) ASSUME_TAC) THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o SPEC `e / &3`) THEN
ASM_SIMP_TAC[
REAL_LT_DIV;
REAL_OF_NUM_LT; ARITH] THEN
MATCH_MP_TAC
MONO_EXISTS THEN X_GEN_TAC `d:real` THEN
MATCH_MP_TAC MONO_AND THEN REWRITE_TAC[] THEN
MATCH_MP_TAC
MONO_FORALL THEN X_GEN_TAC `y:real^M` THEN
DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[] THEN
FIRST_X_ASSUM(fun th ->
MP_TAC(SPEC `x:real^M` th) THEN MP_TAC(SPEC `y:real^M` th)) THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(REAL_ARITH
`w <= x + y + z
==> x < e / &3 ==> y < e / &3 ==> z < e / &3 ==> w < e`) THEN
REWRITE_TAC[dist] THEN
SUBST1_TAC(VECTOR_ARITH
`(g:real^M->real^N) y - g x =
--(f (a:A) y - g y) + (f a x - g x) + (f a y - f a x)`) THEN
MATCH_MP_TAC
NORM_TRIANGLE_LE THEN REWRITE_TAC[
NORM_NEG;
REAL_LE_LADD] THEN
MATCH_MP_TAC
NORM_TRIANGLE_LE THEN REWRITE_TAC[
NORM_NEG;
REAL_LE_REFL]);;
let COMPACT_ATTAINS_SUP = prove
(`!s. compact (
IMAGE lift s) /\ ~(s = {})
==> ?x. x
IN s /\ !y. y
IN s ==> y <= x`,
REWRITE_TAC[
COMPACT_EQ_BOUNDED_CLOSED] THEN REPEAT STRIP_TAC THEN
MP_TAC(SPEC `s:real->bool`
BOUNDED_HAS_SUP) THEN ASM_REWRITE_TAC[] THEN
STRIP_TAC THEN EXISTS_TAC `sup s` THEN ASM_REWRITE_TAC[] THEN
ASM_MESON_TAC[
CLOSED_LIFT; REAL_ARITH `s <= s - e <=> ~(&0 < e)`;
REAL_ARITH `x <= s /\ ~(x <= s - e) ==> abs(x - s) < e`]);;
let COMPACT_ATTAINS_INF = prove
(`!s. compact (
IMAGE lift s) /\ ~(s = {})
==> ?x. x
IN s /\ !y. y
IN s ==> x <= y`,
REWRITE_TAC[
COMPACT_EQ_BOUNDED_CLOSED] THEN REPEAT STRIP_TAC THEN
MP_TAC(SPEC `s:real->bool`
BOUNDED_HAS_INF) THEN ASM_REWRITE_TAC[] THEN
STRIP_TAC THEN EXISTS_TAC `inf s` THEN ASM_REWRITE_TAC[] THEN
ASM_MESON_TAC[
CLOSED_LIFT; REAL_ARITH `s + e <= s <=> ~(&0 < e)`;
REAL_ARITH `s <= x /\ ~(s + e <= x) ==> abs(x - s) < e`]);;
let LIM_MUL = prove
(`!net:(A)net f l:real^N c d.
((lift o c) --> lift d) net /\ (f --> l) net
==> ((\x. c(x) % f(x)) --> (d % l)) net`,
REPEAT STRIP_TAC THEN
MP_TAC(ISPECL [`net:(A)net`; `\x (y:real^N). drop x % y`;
`lift o (c:A->real)`; `f:A->real^N`; `lift d`; `l:real^N`]
LIM_BILINEAR) THEN
ASM_REWRITE_TAC[
LIFT_DROP;
o_THM] THEN DISCH_THEN MATCH_MP_TAC THEN
REWRITE_TAC[bilinear; linear;
DROP_ADD;
DROP_CMUL] THEN
REPEAT STRIP_TAC THEN VECTOR_ARITH_TAC);;
let LIM_VMUL = prove
(`!net:(A)net c d v:real^N.
((lift o c) --> lift d) net ==> ((\x. c(x) % v) --> d % v) net`,
let CONTINUOUS_MUL = prove
(`!net f c. (lift o c) continuous net /\ f continuous net
==> (\x. c(x) % f(x)) continuous net`,
let LIM_INV = prove
(`!net:(A)net f l.
((lift o f) --> lift l) net /\ ~(l = &0)
==> ((lift o inv o f) --> lift(inv l)) net`,
let CONTINUOUS_INV = prove
(`!net f. (lift o f) continuous net /\ ~(f(netlimit net) = &0)
==> (lift o inv o f) continuous net`,
let LIM_PASTECART = prove
(`!net f:A->real^M g:A->real^N.
(f --> a) net /\ (g --> b) net
==> ((\x. pastecart (f x) (g x)) --> pastecart a b) net`,
REPEAT GEN_TAC THEN REWRITE_TAC[
LIM] THEN
ASM_CASES_TAC `
trivial_limit(net:(A)net)` THEN ASM_REWRITE_TAC[] THEN
REWRITE_TAC[
AND_FORALL_THM] THEN DISCH_TAC THEN X_GEN_TAC `e:real` THEN
DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `e / &2`) THEN
ASM_REWRITE_TAC[
REAL_HALF] THEN
DISCH_THEN(MP_TAC o MATCH_MP
NET_DILEMMA) THEN
MATCH_MP_TAC
MONO_EXISTS THEN GEN_TAC THEN MATCH_MP_TAC MONO_AND THEN
REWRITE_TAC[] THEN MATCH_MP_TAC
MONO_FORALL THEN GEN_TAC THEN
MATCH_MP_TAC MONO_IMP THEN REWRITE_TAC[] THEN
REWRITE_TAC[dist;
PASTECART_SUB] THEN
MATCH_MP_TAC(REAL_ARITH
`z <= x + y ==> x < e / &2 /\ y < e / &2 ==> z < e`) THEN
REWRITE_TAC[
NORM_PASTECART_LE]);;
let LIM_PASTECART_EQ = prove
(`!net f:A->real^M g:A->real^N.
((\x. pastecart (f x) (g x)) --> pastecart a b) net <=>
(f --> a) net /\ (g --> b) net`,
let CONTINUOUS_PASTECART = prove
(`!net f:A->real^M g:A->real^N.
f continuous net /\ g continuous net
==> (\x. pastecart (f x) (g x)) continuous net`,
let DIAMETER_BOUNDED = prove
(`!s. bounded s
==> (!x:real^N y. x
IN s /\ y
IN s ==> norm(x - y) <= diameter s) /\
(!d. &0 <= d /\ d < diameter s
==> ?x y. x
IN s /\ y
IN s /\ norm(x - y) > d)`,
let DIAMETER_POS_LE = prove
(`!s:real^N->bool. bounded s ==> &0 <= diameter s`,
REPEAT STRIP_TAC THEN REWRITE_TAC[diameter] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[
REAL_LE_REFL] THEN
MP_TAC(SPEC `{norm(x - y:real^N) | x
IN s /\ y
IN s}`
SUP) THEN
REWRITE_TAC[
FORALL_IN_GSPEC] THEN ANTS_TAC THENL
[CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
FIRST_X_ASSUM(X_CHOOSE_TAC `B:real` o GEN_REWRITE_RULE I [
BOUNDED_POS]) THEN
EXISTS_TAC `&2 * B` THEN
ASM_SIMP_TAC[NORM_ARITH
`norm x <= B /\ norm y <= B ==> norm(x - y) <= &2 * B`];
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM
MEMBER_NOT_EMPTY]) THEN
DISCH_THEN(X_CHOOSE_TAC `a:real^N`) THEN
DISCH_THEN(MP_TAC o SPECL [`a:real^N`; `a:real^N`] o CONJUNCT1) THEN
ASM_REWRITE_TAC[
VECTOR_SUB_REFL;
NORM_0]]);;
let DIAMETER_SUBSET = prove
(`!s t:real^N->bool. s
SUBSET t /\ bounded t ==> diameter s <= diameter t`,
REPEAT STRIP_TAC THEN
ASM_CASES_TAC `s:real^N->bool = {}` THEN
ASM_SIMP_TAC[
DIAMETER_EMPTY;
DIAMETER_POS_LE] THEN
ASM_REWRITE_TAC[diameter] THEN
COND_CASES_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
MATCH_MP_TAC
REAL_SUP_LE_SUBSET THEN
REPEAT(CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC]) THEN
REWRITE_TAC[
FORALL_IN_GSPEC] THEN
FIRST_X_ASSUM(X_CHOOSE_TAC `B:real` o GEN_REWRITE_RULE I [
BOUNDED_POS]) THEN
EXISTS_TAC `&2 * B` THEN
ASM_SIMP_TAC[NORM_ARITH
`norm x <= B /\ norm y <= B ==> norm(x - y) <= &2 * B`]);;
let DIAMETER_CLOSURE = prove
(`!s:real^N->bool. bounded s ==> diameter(closure s) = diameter s`,
REWRITE_TAC[GSYM REAL_LE_ANTISYM] THEN REPEAT STRIP_TAC THEN
ASM_SIMP_TAC[
DIAMETER_SUBSET;
BOUNDED_CLOSURE;
CLOSURE_SUBSET] THEN
REWRITE_TAC[GSYM
REAL_NOT_LT] THEN ONCE_REWRITE_TAC[GSYM
REAL_SUB_LT] THEN
DISCH_TAC THEN MP_TAC(ISPEC `closure s:real^N->bool`
DIAMETER_BOUNDED) THEN
ABBREV_TAC `d = diameter(closure s) - diameter(s:real^N->bool)` THEN
ASM_SIMP_TAC[
BOUNDED_CLOSURE] THEN DISCH_THEN(MP_TAC o
SPEC `diameter(closure(s:real^N->bool)) - d / &2` o CONJUNCT2) THEN
REWRITE_TAC[NOT_IMP; GSYM
CONJ_ASSOC;
NOT_EXISTS_THM] THEN
FIRST_ASSUM(ASSUME_TAC o MATCH_MP
DIAMETER_POS_LE) THEN
REPEAT(CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC]) THEN
MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN
REWRITE_TAC[
CLOSURE_APPROACHABLE;
CONJ_ASSOC;
AND_FORALL_THM] THEN
DISCH_THEN(CONJUNCTS_THEN2 (MP_TAC o SPEC `d / &4`) ASSUME_TAC) THEN
ASM_REWRITE_TAC[REAL_ARITH `&0 < d / &4 <=> &0 < d`] THEN
DISCH_THEN(CONJUNCTS_THEN2
(X_CHOOSE_THEN `u:real^N` (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC))
(X_CHOOSE_THEN `v:real^N` (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC))) THEN
FIRST_ASSUM(MP_TAC o MATCH_MP
DIAMETER_BOUNDED) THEN
DISCH_THEN(MP_TAC o SPECL [`u:real^N`; `v:real^N`] o CONJUNCT1) THEN
ASM_REWRITE_TAC[] THEN REPEAT(POP_ASSUM MP_TAC) THEN NORM_ARITH_TAC);;
let DIAMETER_EQ_0 = prove
(`!s:real^N->bool.
bounded s ==> (diameter s = &0 <=> s = {} \/ ?a. s = {a})`,
REPEAT STRIP_TAC THEN EQ_TAC THEN STRIP_TAC THEN
ASM_REWRITE_TAC[
DIAMETER_EMPTY;
DIAMETER_SING] THEN
REWRITE_TAC[SET_RULE
`s = {} \/ (?a. s = {a}) <=> !a b. a
IN s /\ b
IN s ==> a = b`] THEN
MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`] THEN STRIP_TAC THEN
MP_TAC(ISPECL [`s:real^N->bool`; `a:real^N`; `b:real^N`]
DIAMETER_BOUNDED_BOUND) THEN
ASM_REWRITE_TAC[] THEN NORM_ARITH_TAC);;
let DIAMETER_LE = prove
(`!s:real^N->bool.
(~(s = {}) \/ &0 <= d) /\
(!x y. x
IN s /\ y
IN s ==> norm(x - y) <= d) ==> diameter s <= d`,
GEN_TAC THEN REWRITE_TAC[diameter] THEN
COND_CASES_TAC THEN ASM_SIMP_TAC[] THEN
STRIP_TAC THEN MATCH_MP_TAC
REAL_SUP_LE THEN
CONJ_TAC THENL [ASM SET_TAC[]; ASM_SIMP_TAC[
FORALL_IN_GSPEC]]);;
let DIAMETER_SUMS = prove
(`!s t:real^N->bool.
bounded s /\ bounded t
==> diameter {x + y | x
IN s /\ y
IN t} <= diameter s + diameter t`,
let LEBESGUE_COVERING_LEMMA = prove
(`!s:real^N->bool c.
compact s /\ ~(c = {}) /\ s
SUBSET UNIONS c /\ (!b. b
IN c ==> open b)
==> ?d. &0 < d /\
!t. t
SUBSET s /\ diameter t <= d
==> ?b. b
IN c /\ t
SUBSET b`,
REPEAT STRIP_TAC THEN
FIRST_ASSUM(MP_TAC o MATCH_MP
HEINE_BOREL_LEMMA) THEN
DISCH_THEN(MP_TAC o SPEC `c:(real^N->bool)->bool`) THEN ASM_SIMP_TAC[] THEN
ASM_REWRITE_TAC[
LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `e:real` THEN
STRIP_TAC THEN EXISTS_TAC `e / &2` THEN ASM_REWRITE_TAC[
REAL_HALF] THEN
X_GEN_TAC `t:real^N->bool` THEN STRIP_TAC THEN
ASM_CASES_TAC `t:real^N->bool = {}` THENL [ASM SET_TAC[]; ALL_TAC] THEN
MP_TAC(ISPEC `t:real^N->bool`
DIAMETER_SUBSET_CBALL_NONEMPTY) THEN
ANTS_TAC THENL
[ASM_MESON_TAC[
BOUNDED_SUBSET;
COMPACT_IMP_BOUNDED]; ALL_TAC] THEN
DISCH_THEN(X_CHOOSE_THEN `x:real^N` STRIP_ASSUME_TAC) THEN
FIRST_X_ASSUM(MP_TAC o SPEC `x:real^N`) THEN
ANTS_TAC THENL [ASM SET_TAC[]; MATCH_MP_TAC
MONO_EXISTS] THEN
X_GEN_TAC `b:real^N->bool` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC
SUBSET_TRANS THEN
EXISTS_TAC `cball(x:real^N,diameter(t:real^N->bool))` THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC
SUBSET_TRANS THEN
EXISTS_TAC `ball(x:real^N,e)` THEN ASM_REWRITE_TAC[] THEN
REWRITE_TAC[
SUBSET;
IN_CBALL;
IN_BALL] THEN
MAP_EVERY UNDISCH_TAC [`&0 < e`; `diameter(t:real^N->bool) <= e / &2`] THEN
NORM_ARITH_TAC);;
let CLOSED_NEGATIONS = prove
(`!s:real^N->bool. closed s ==> closed (
IMAGE (--) s)`,
REPEAT GEN_TAC THEN
SUBGOAL_THEN `
IMAGE (--) s =
IMAGE (\x:real^N. --(&1) % x) s`
SUBST1_TAC THEN SIMP_TAC[
CLOSED_SCALING] THEN
REWRITE_TAC[VECTOR_ARITH `--(&1) % x = --x`] THEN REWRITE_TAC[ETA_AX]);;
let COMPACT_CLOSED_SUMS = prove
(`!s:real^N->bool t.
compact s /\ closed t ==> closed {x + y | x
IN s /\ y
IN t}`,
REPEAT GEN_TAC THEN
REWRITE_TAC[compact;
IN_ELIM_THM;
CLOSED_SEQUENTIAL_LIMITS] THEN
STRIP_TAC THEN X_GEN_TAC `f:num->real^N` THEN X_GEN_TAC `l:real^N` THEN
REWRITE_TAC[
SKOLEM_THM;
FORALL_AND_THM] THEN
DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN
DISCH_THEN(X_CHOOSE_THEN `a:num->real^N` MP_TAC) THEN
DISCH_THEN(X_CHOOSE_THEN `b:num->real^N` STRIP_ASSUME_TAC) THEN
FIRST_X_ASSUM(MP_TAC o check(is_imp o concl) o SPEC `a:num->real^N`) THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_THEN `la:real^N` (X_CHOOSE_THEN `sub:num->num`
STRIP_ASSUME_TAC)) THEN
MAP_EVERY EXISTS_TAC [`la:real^N`; `l - la:real^N`] THEN
ASM_REWRITE_TAC[VECTOR_ARITH `a + (b - a) = b:real^N`] THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN
EXISTS_TAC `\n. (f o (sub:num->num)) n - (a o sub) n:real^N` THEN
CONJ_TAC THENL [ASM_REWRITE_TAC[
VECTOR_ADD_SUB;
o_THM]; ALL_TAC] THEN
MATCH_MP_TAC
LIM_SUB THEN ASM_SIMP_TAC[
LIM_SUBSEQUENCE; ETA_AX]);;
let CLOSURE_SUMS = prove
(`!s t:real^N->bool.
bounded s \/ bounded t
==> closure {x + y | x
IN s /\ y
IN t} =
{x + y | x
IN closure s /\ y
IN closure t}`,
REWRITE_TAC[TAUT `p \/ q ==> r <=> (p ==> r) /\ (q ==> r)`] THEN
REWRITE_TAC[
FORALL_AND_THM] THEN
GEN_REWRITE_TAC (RAND_CONV o ONCE_DEPTH_CONV) [
SUMS_SYM] THEN
MATCH_MP_TAC(TAUT `(p ==> q) /\ p ==> p /\ q`) THEN
SIMP_TAC[] THEN
REPEAT STRIP_TAC THEN REWRITE_TAC[
EXTENSION;
CLOSURE_SEQUENTIAL] THEN
X_GEN_TAC `z:real^N` THEN REWRITE_TAC[
IN_ELIM_THM] THEN EQ_TAC THENL
[REWRITE_TAC[
IN_ELIM_THM;
IN_DELETE;
SKOLEM_THM;
LEFT_AND_EXISTS_THM] THEN
REWRITE_TAC[
FORALL_AND_THM] THEN
ONCE_REWRITE_TAC[TAUT `(p /\ q) /\ r <=> q /\ p /\ r`] THEN
ONCE_REWRITE_TAC[MESON[] `(?f x y. P f x y) <=> (?x y f. P f x y)`] THEN
ONCE_REWRITE_TAC[GSYM
FUN_EQ_THM] THEN
REWRITE_TAC[ETA_AX;
UNWIND_THM2] THEN
REWRITE_TAC[
LEFT_IMP_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC [`a:num->real^N`; `b:num->real^N`] THEN
STRIP_TAC THEN
MP_TAC(ISPEC `closure s:real^N->bool` compact) THEN
ASM_REWRITE_TAC[
COMPACT_CLOSURE] THEN
DISCH_THEN(MP_TAC o SPEC `a:num->real^N`) THEN
ASM_SIMP_TAC[REWRITE_RULE[
SUBSET]
CLOSURE_SUBSET;
LEFT_IMP_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC [`u:real^N`; `r:num->num`] THEN STRIP_TAC THEN
EXISTS_TAC `z - u:real^N` THEN
EXISTS_TAC `(a:num->real^N) o (r:num->num)` THEN EXISTS_TAC `u:real^N` THEN
ASM_REWRITE_TAC[
o_THM] THEN
CONJ_TAC THENL [ALL_TAC; VECTOR_ARITH_TAC] THEN
EXISTS_TAC `(\n. ((\n. a n + b n) o (r:num->num)) n - (a o r) n)
:num->real^N` THEN
CONJ_TAC THENL
[ASM_REWRITE_TAC[
o_DEF; VECTOR_ARITH `(a + b) - a:real^N = b`];
MATCH_MP_TAC
LIM_SUB THEN ASM_REWRITE_TAC[ETA_AX] THEN
MATCH_MP_TAC
LIM_SUBSEQUENCE THEN ASM_REWRITE_TAC[]];
REWRITE_TAC[
LEFT_AND_EXISTS_THM] THEN
REWRITE_TAC[
LEFT_IMP_EXISTS_THM;
LEFT_AND_EXISTS_THM;
RIGHT_AND_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC
[`x:real^N`; `y:real^N`; `a:num->real^N`; `b:num->real^N`] THEN
STRIP_TAC THEN EXISTS_TAC `(\n. a n + b n):num->real^N` THEN
ASM_SIMP_TAC[
LIM_ADD] THEN ASM_MESON_TAC[]]);;
let open_interval = new_definition
`open_interval(a:real^N,b:real^N) =
{x:real^N | !i. 1 <= i /\ i <= dimindex(:N)
==> a$i < x$i /\ x$i < b$i}`;;let interval = prove
(`(interval (a,b) = {x:real^N | !i. 1 <= i /\ i <= dimindex(:N)
==> a$i < x$i /\ x$i < b$i}) /\
(interval [a,b] = {x:real^N | !i. 1 <= i /\ i <= dimindex(:N)
==> a$i <= x$i /\ x$i <= b$i})`,
let IN_INTERVAL = prove
(`(!x:real^N.
x
IN interval (a,b) <=>
!i. 1 <= i /\ i <= dimindex(:N)
==> a$i < x$i /\ x$i < b$i) /\
(!x:real^N.
x
IN interval [a,b] <=>
!i. 1 <= i /\ i <= dimindex(:N)
==> a$i <= x$i /\ x$i <= b$i)`,
let INTERVAL_EQ_EMPTY = prove
(`((interval [a:real^N,b] = {}) <=>
?i. 1 <= i /\ i <= dimindex(:N) /\ b$i < a$i) /\
((interval (a:real^N,b) = {}) <=>
?i. 1 <= i /\ i <= dimindex(:N) /\ b$i <= a$i)`,
let INTERVAL_NE_EMPTY = prove
(`(~(interval [a:real^N,b] = {}) <=>
!i. 1 <= i /\ i <= dimindex(:N) ==> a$i <= b$i) /\
(~(interval (a:real^N,b) = {}) <=>
!i. 1 <= i /\ i <= dimindex(:N) ==> a$i < b$i)`,
let SUBSET_INTERVAL_IMP = prove
(`((!i. 1 <= i /\ i <= dimindex(:N) ==> a$i <= c$i /\ d$i <= b$i)
==> interval[c,d]
SUBSET interval[a:real^N,b]) /\
((!i. 1 <= i /\ i <= dimindex(:N) ==> a$i < c$i /\ d$i < b$i)
==> interval[c,d]
SUBSET interval(a:real^N,b)) /\
((!i. 1 <= i /\ i <= dimindex(:N) ==> a$i <= c$i /\ d$i <= b$i)
==> interval(c,d)
SUBSET interval[a:real^N,b]) /\
((!i. 1 <= i /\ i <= dimindex(:N) ==> a$i <= c$i /\ d$i <= b$i)
==> interval(c,d)
SUBSET interval(a:real^N,b))`,
REWRITE_TAC[
SUBSET;
IN_INTERVAL] THEN REPEAT CONJ_TAC THEN
DISCH_TAC THEN GEN_TAC THEN POP_ASSUM MP_TAC THEN
REWRITE_TAC[IMP_IMP;
AND_FORALL_THM] THEN MATCH_MP_TAC
MONO_FORALL THEN
GEN_TAC THEN DISCH_THEN(fun th -> DISCH_TAC THEN MP_TAC th) THEN
ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC);;
let SUBSET_INTERVAL = prove
(`(interval[c,d]
SUBSET interval[a:real^N,b] <=>
(!i. 1 <= i /\ i <= dimindex(:N) ==> c$i <= d$i)
==> (!i. 1 <= i /\ i <= dimindex(:N) ==> a$i <= c$i /\ d$i <= b$i)) /\
(interval[c,d]
SUBSET interval(a:real^N,b) <=>
(!i. 1 <= i /\ i <= dimindex(:N) ==> c$i <= d$i)
==> (!i. 1 <= i /\ i <= dimindex(:N) ==> a$i < c$i /\ d$i < b$i)) /\
(interval(c,d)
SUBSET interval[a:real^N,b] <=>
(!i. 1 <= i /\ i <= dimindex(:N) ==> c$i < d$i)
==> (!i. 1 <= i /\ i <= dimindex(:N) ==> a$i <= c$i /\ d$i <= b$i)) /\
(interval(c,d)
SUBSET interval(a:real^N,b) <=>
(!i. 1 <= i /\ i <= dimindex(:N) ==> c$i < d$i)
==> (!i. 1 <= i /\ i <= dimindex(:N) ==> a$i <= c$i /\ d$i <= b$i))`,
let lemma = prove
(`(!x:real^N. (!i. 1 <= i /\ i <= dimindex(:N) ==> Q i (x$i))
==> (!i. 1 <= i /\ i <= dimindex(:N) ==> R i (x$i)))
==> (!i. 1 <= i /\ i <= dimindex(:N) ==> ?y. Q i y)
==> !i y. 1 <= i /\ i <= dimindex(:N) /\ Q i y ==> R i y`,
DISCH_TAC THEN REWRITE_TAC[RIGHT_IMP_EXISTS_THM; SKOLEM_THM] THEN
DISCH_THEN(X_CHOOSE_THEN `f:num->real` STRIP_ASSUME_TAC) THEN
REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o
SPEC `(lambda j. if j = i then y else f j):real^N`) THEN
SIMP_TAC[LAMBDA_BETA] THEN ASM_MESON_TAC[]) in
REPEAT STRIP_TAC THEN
(MATCH_MP_TAC(TAUT
`(~q ==> p) /\ (q ==> (p <=> r)) ==> (p <=> q ==> r)`) THEN
CONJ_TAC THENL
[DISCH_TAC THEN MATCH_MP_TAC(SET_RULE `s = {} ==> s SUBSET t`) THEN
REWRITE_TAC[INTERVAL_EQ_EMPTY] THEN ASM_MESON_TAC[REAL_NOT_LT];
ALL_TAC] THEN
DISCH_TAC THEN EQ_TAC THEN REWRITE_TAC[SUBSET_INTERVAL_IMP] THEN
REWRITE_TAC[SUBSET; IN_INTERVAL] THEN
DISCH_THEN(MP_TAC o MATCH_MP lemma) THEN ANTS_TAC THENL
[ASM_MESON_TAC[REAL_LT_BETWEEN; REAL_LE_BETWEEN]; ALL_TAC] THEN
MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `i:num` THEN
DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[] THEN
FIRST_X_ASSUM(MP_TAC o SPEC `i:num`) THEN
ASM_REWRITE_TAC[] THEN POP_ASSUM_LIST(K ALL_TAC) THEN STRIP_TAC)
THENL
[ASM_MESON_TAC[REAL_LE_TRANS; REAL_LE_REFL];
ASM_MESON_TAC[REAL_LE_TRANS; REAL_LE_REFL];
ALL_TAC; ALL_TAC] THEN
(REPEAT STRIP_TAC THENL
[FIRST_X_ASSUM(MP_TAC o SPEC
`((c:real^N)$i + min ((a:real^N)$i) ((d:real^N)$i)) / &2`) THEN
POP_ASSUM MP_TAC THEN REAL_ARITH_TAC;
FIRST_X_ASSUM(MP_TAC o SPEC
`(max ((b:real^N)$i) ((c:real^N)$i) + (d:real^N)$i) / &2`) THEN
POP_ASSUM MP_TAC THEN REAL_ARITH_TAC]));;
let DISJOINT_INTERVAL = prove
(`!a b c d:real^N.
(interval[a,b]
INTER interval[c,d] = {} <=>
?i. 1 <= i /\ i <= dimindex(:N) /\
(b$i < a$i \/ d$i < c$i \/ b$i < c$i \/ d$i < a$i)) /\
(interval[a,b]
INTER interval(c,d) = {} <=>
?i. 1 <= i /\ i <= dimindex(:N) /\
(b$i < a$i \/ d$i <= c$i \/ b$i <= c$i \/ d$i <= a$i)) /\
(interval(a,b)
INTER interval[c,d] = {} <=>
?i. 1 <= i /\ i <= dimindex(:N) /\
(b$i <= a$i \/ d$i < c$i \/ b$i <= c$i \/ d$i <= a$i)) /\
(interval(a,b)
INTER interval(c,d) = {} <=>
?i. 1 <= i /\ i <= dimindex(:N) /\
(b$i <= a$i \/ d$i <= c$i \/ b$i <= c$i \/ d$i <= a$i))`,
REWRITE_TAC[
EXTENSION;
IN_INTER;
IN_INTERVAL;
NOT_IN_EMPTY] THEN
REWRITE_TAC[
AND_FORALL_THM;
NOT_FORALL_THM] THEN
REWRITE_TAC[TAUT `~((p ==> q) /\ (p ==> r)) <=> p /\ (~q \/ ~r)`] THEN
REWRITE_TAC[DE_MORGAN_THM] THEN REPEAT STRIP_TAC THEN
(EQ_TAC THENL
[DISCH_THEN(MP_TAC o SPEC
`(lambda i. (max ((a:real^N)$i) ((c:real^N)$i) +
min ((b:real^N)$i) ((d:real^N)$i)) / &2):real^N`) THEN
MATCH_MP_TAC
MONO_EXISTS THEN GEN_TAC THEN
DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC) THEN
ASM_SIMP_TAC[
LAMBDA_BETA] THEN REAL_ARITH_TAC;
DISCH_THEN(fun th -> GEN_TAC THEN MP_TAC th) THEN
MATCH_MP_TAC
MONO_EXISTS THEN GEN_TAC THEN SIMP_TAC[] THEN
REAL_ARITH_TAC]));;
let ENDS_IN_INTERVAL = prove
(`(!a b. a
IN interval[a,b] <=> ~(interval[a,b] = {})) /\
(!a b. b
IN interval[a,b] <=> ~(interval[a,b] = {})) /\
(!a b. ~(a
IN interval(a,b))) /\
(!a b. ~(b
IN interval(a,b)))`,
let OPEN_INTERVAL_LEMMA = prove
(`!a b x. a < x /\ x < b
==> ?d. &0 < d /\ !x'. abs(x' - x) < d ==> a < x' /\ x' < b`,
REPEAT STRIP_TAC THEN
EXISTS_TAC `min (x - a) (b - x)` THEN REWRITE_TAC[
REAL_LT_MIN] THEN
ASM_REAL_ARITH_TAC);;
let CLOSED_INTERVAL = prove
(`!a:real^N b. closed(interval [a,b])`,
REWRITE_TAC[
CLOSED_LIMPT;
LIMPT_APPROACHABLE;
IN_INTERVAL] THEN
REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM
REAL_NOT_LT] THEN DISCH_TAC THENL
[FIRST_X_ASSUM(MP_TAC o SPEC `(a:real^N)$i - (x:real^N)$i`);
FIRST_X_ASSUM(MP_TAC o SPEC `(x:real^N)$i - (b:real^N)$i`)] THEN
ASM_REWRITE_TAC[
REAL_SUB_LT] THEN
DISCH_THEN(X_CHOOSE_THEN `z:real^N` MP_TAC) THEN
REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
REWRITE_TAC[dist;
REAL_NOT_LT] THEN
MATCH_MP_TAC
REAL_LE_TRANS THEN EXISTS_TAC `abs((z - x :real^N)$i)` THEN
ASM_SIMP_TAC[
COMPONENT_LE_NORM] THEN
ASM_SIMP_TAC[
VECTOR_SUB_COMPONENT] THEN
ASM_SIMP_TAC[REAL_ARITH `x < a /\ a <= z ==> a - x <= abs(z - x)`;
REAL_ARITH `z <= b /\ b < x ==> x - b <= abs(z - x)`]);;
let INTERIOR_CLOSED_INTERVAL = prove
(`!a:real^N b. interior(interval [a,b]) = interval (a,b)`,
REPEAT GEN_TAC THEN MATCH_MP_TAC
SUBSET_ANTISYM THEN CONJ_TAC THENL
[ALL_TAC;
MATCH_MP_TAC
INTERIOR_MAXIMAL THEN
REWRITE_TAC[
INTERVAL_OPEN_SUBSET_CLOSED;
OPEN_INTERVAL]] THEN
REWRITE_TAC[interior;
SUBSET;
IN_INTERVAL;
IN_ELIM_THM] THEN
X_GEN_TAC `x:real^N` THEN
DISCH_THEN(X_CHOOSE_THEN `s:real^N->bool` STRIP_ASSUME_TAC) THEN
X_GEN_TAC `i:num` THEN STRIP_TAC THEN
ASM_SIMP_TAC[
REAL_LT_LE] THEN REPEAT STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
open_def]) THEN
DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_THEN `e:real` (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THENL
[(let t = `x - (e / &2) % basis i :real^N` in
DISCH_THEN(MP_TAC o SPEC t) THEN FIRST_X_ASSUM(MP_TAC o SPEC t));
(let t = `x + (e / &2) % basis i :real^N` in
DISCH_THEN(MP_TAC o SPEC t) THEN FIRST_X_ASSUM(MP_TAC o SPEC t))] THEN
REWRITE_TAC[dist;
VECTOR_ADD_SUB; VECTOR_ARITH `x - y - x = --y:real^N`] THEN
ASM_SIMP_TAC[
NORM_MUL;
NORM_BASIS;
NORM_NEG;
REAL_MUL_RID;
REAL_ARITH `&0 < e ==> abs(e / &2) < e`] THEN
MATCH_MP_TAC(TAUT `~b ==> (a ==> b) ==> ~a`) THEN
REWRITE_TAC[
NOT_FORALL_THM] THEN EXISTS_TAC `i:num` THEN
ASM_SIMP_TAC[DE_MORGAN_THM;
VECTOR_SUB_COMPONENT;
VECTOR_ADD_COMPONENT] THENL
[DISJ1_TAC THEN REWRITE_TAC[REAL_ARITH `a <= a - b <=> ~(&0 < b)`];
DISJ2_TAC THEN REWRITE_TAC[REAL_ARITH `a + b <= a <=> ~(&0 < b)`]] THEN
ASM_SIMP_TAC[
VECTOR_MUL_COMPONENT; basis;
LAMBDA_BETA;
REAL_MUL_RID] THEN
ASM_REWRITE_TAC[
REAL_HALF]);;
let INTERIOR_INTERVAL = prove
(`(!a b. interior(interval[a,b]) = interval(a,b)) /\
(!a b. interior(interval(a,b)) = interval(a,b))`,
let BOUNDED_CLOSED_INTERVAL = prove
(`!a b:real^N. bounded (interval [a,b])`,
REPEAT STRIP_TAC THEN REWRITE_TAC[bounded; interval] THEN
EXISTS_TAC `sum(1..dimindex(:N))
(\i. abs((a:real^N)$i) + abs((b:real^N)$i))` THEN
X_GEN_TAC `x:real^N` THEN REWRITE_TAC[
IN_ELIM_THM] THEN
STRIP_TAC THEN MATCH_MP_TAC
REAL_LE_TRANS THEN
EXISTS_TAC `sum(1..dimindex(:N)) (\i. abs((x:real^N)$i))` THEN
REWRITE_TAC[
NORM_LE_L1] THEN MATCH_MP_TAC
SUM_LE THEN
ASM_SIMP_TAC[
FINITE_NUMSEG;
IN_NUMSEG; REAL_ARITH
`a <= x /\ x <= b ==> abs(x) <= abs(a) + abs(b)`]);;
let CLOSURE_INTERVAL = prove
(`(!a b. closure(interval[a,b]) = interval[a,b]) /\
(!a b. closure(interval(a,b)) =
if interval(a,b) = {} then {} else interval[a,b])`,
let IMAGE_STRETCH_INTERVAL = prove
(`!a b:real^N m.
IMAGE (\x. lambda k. m(k) * x$k) (interval[a,b]) =
if interval[a,b] = {} then {}
else interval[(lambda k. min (m(k) * a$k) (m(k) * b$k)):real^N,
(lambda k. max (m(k) * a$k) (m(k) * b$k))]`,
let SUMS_INTERVALS = prove
(`(!a b c d:real^N.
~(interval[a,b] = {}) /\ ~(interval[c,d] = {})
==> {x + y | x
IN interval[a,b] /\ y
IN interval[c,d]} =
interval[a+c,b+d]) /\
(!a b c d:real^N.
~(interval(a,b) = {}) /\ ~(interval(c,d) = {})
==> {x + y | x
IN interval(a,b) /\ y
IN interval(c,d)} =
interval(a+c,b+d))`,
CONJ_TAC THEN REPEAT GEN_TAC THEN REWRITE_TAC[
INTERVAL_NE_EMPTY] THEN
STRIP_TAC THEN REWRITE_TAC[
EXTENSION;
IN_INTERVAL;
IN_ELIM_THM] THEN
REWRITE_TAC[TAUT `(a /\ b) /\ c <=> c /\ a /\ b`] THEN
REWRITE_TAC[VECTOR_ARITH `x:real^N = y + z <=> z = x - y`] THEN
REWRITE_TAC[
UNWIND_THM2;
VECTOR_ADD_COMPONENT;
VECTOR_SUB_COMPONENT] THEN
(X_GEN_TAC `x:real^N` THEN EQ_TAC THENL
[DISCH_THEN(X_CHOOSE_THEN `y:real^N` STRIP_ASSUME_TAC);
DISCH_TAC THEN
REWRITE_TAC[
AND_FORALL_THM; GSYM
LAMBDA_SKOLEM;
TAUT `(p ==> q) /\ (p ==> r) <=> p ==> q /\ r`] THEN
REWRITE_TAC[REAL_ARITH
`((a <= y /\ y <= b) /\ c <= x - y /\ x - y <= d <=>
max a (x - d) <= y /\ y <= min b (x - c)) /\
((a < y /\ y < b) /\ c < x - y /\ x - y < d <=>
max a (x - d) < y /\ y < min b (x - c))`] THEN
REWRITE_TAC[GSYM
REAL_LE_BETWEEN; GSYM
REAL_LT_BETWEEN]] THEN
X_GEN_TAC `i:num` THEN STRIP_TAC THEN
REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `i:num`)) THEN
ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC));;
let PCROSS_INTERVAL = prove
(`!a b:real^M c d:real^N.
interval[a,b]
PCROSS interval[c,d] =
interval[pastecart a c,pastecart b d]`,
REPEAT GEN_TAC THEN REWRITE_TAC[
PCROSS] THEN
REWRITE_TAC[
EXTENSION;
FORALL_PASTECART;
IN_ELIM_PASTECART_THM] THEN
SIMP_TAC[
IN_INTERVAL; pastecart;
LAMBDA_BETA;
DIMINDEX_FINITE_SUM] THEN
MAP_EVERY X_GEN_TAC [`x:real^M`; `y:real^N`] THEN EQ_TAC THEN STRIP_TAC THENL
[X_GEN_TAC `i:num` THEN STRIP_TAC THEN
COND_CASES_TAC THEN ASM_SIMP_TAC[] THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC;
CONJ_TAC THEN X_GEN_TAC `i:num` THEN STRIP_TAC THENL
[FIRST_X_ASSUM(MP_TAC o SPEC `i:num`) THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN MATCH_MP_TAC THEN ASM_ARITH_TAC;
FIRST_X_ASSUM(MP_TAC o SPEC `i + dimindex(:M)`) THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[
ADD_SUB] THENL
[ASM_ARITH_TAC;
DISCH_THEN MATCH_MP_TAC THEN ASM_ARITH_TAC]]]);;
let DIAMETER_INTERVAL = prove
(`(!a b:real^N.
diameter(interval[a,b]) =
if interval[a,b] = {} then &0 else norm(b - a)) /\
(!a b:real^N.
diameter(interval(a,b)) =
if interval(a,b) = {} then &0 else norm(b - a))`,
let IN_INTERVAL_1 = prove
(`!a b x:real^1.
(x
IN interval[a,b] <=> drop a <= drop x /\ drop x <= drop b) /\
(x
IN interval(a,b) <=> drop a < drop x /\ drop x < drop b)`,
let INTERVAL_NE_EMPTY_1 = prove
(`(!a b:real^1. ~(interval[a,b] = {}) <=> drop a <= drop b) /\
(!a b:real^1. ~(interval(a,b) = {}) <=> drop a < drop b)`,
let SUBSET_INTERVAL_1 = prove
(`!a b c d.
(interval[a,b]
SUBSET interval[c,d] <=>
drop b < drop a \/
drop c <= drop a /\ drop a <= drop b /\ drop b <= drop d) /\
(interval[a,b]
SUBSET interval(c,d) <=>
drop b < drop a \/
drop c < drop a /\ drop a <= drop b /\ drop b < drop d) /\
(interval(a,b)
SUBSET interval[c,d] <=>
drop b <= drop a \/
drop c <= drop a /\ drop a < drop b /\ drop b <= drop d) /\
(interval(a,b)
SUBSET interval(c,d) <=>
drop b <= drop a \/
drop c <= drop a /\ drop a < drop b /\ drop b <= drop d)`,
let EQ_INTERVAL_1 = prove
(`!a b c d:real^1.
(interval[a,b] = interval[c,d] <=>
drop b < drop a /\ drop d < drop c \/
drop a = drop c /\ drop b = drop d)`,
let DISJOINT_INTERVAL_1 = prove
(`!a b c d:real^1.
(interval[a,b]
INTER interval[c,d] = {} <=>
drop b < drop a \/ drop d < drop c \/
drop b < drop c \/ drop d < drop a) /\
(interval[a,b]
INTER interval(c,d) = {} <=>
drop b < drop a \/ drop d <= drop c \/
drop b <= drop c \/ drop d <= drop a) /\
(interval(a,b)
INTER interval[c,d] = {} <=>
drop b <= drop a \/ drop d < drop c \/
drop b <= drop c \/ drop d <= drop a) /\
(interval(a,b)
INTER interval(c,d) = {} <=>
drop b <= drop a \/ drop d <= drop c \/
drop b <= drop c \/ drop d <= drop a)`,
let BALL_1 = prove
(`!x:real^1 r. cball(x,r) = interval[x - lift r,x + lift r] /\
ball(x,r) = interval(x - lift r,x + lift r)`,
let SPHERE_1 = prove
(`!a:real^1 r. sphere(a,r) = if r < &0 then {} else {a - lift r,a + lift r}`,
let BALL_INTERVAL_0 = prove
(`!e. ball(vec 0:real^1,e) = interval(--lift e,lift e)`,
GEN_TAC THEN REWRITE_TAC[
BALL_INTERVAL] THEN AP_TERM_TAC THEN
BINOP_TAC THEN VECTOR_ARITH_TAC);;
let CBALL_INTERVAL_0 = prove
(`!e. cball(vec 0:real^1,e) = interval[--lift e,lift e]`,
GEN_TAC THEN REWRITE_TAC[
CBALL_INTERVAL] THEN AP_TERM_TAC THEN
AP_THM_TAC THEN AP_TERM_TAC THEN BINOP_TAC THEN VECTOR_ARITH_TAC);;
let INTER_INTERVAL_1 = prove
(`!a b c d:real^1.
interval[a,b]
INTER interval[c,d] =
interval[lift(max (drop a) (drop c)),lift(min (drop b) (drop d))]`,
let is_interval = new_definition
`is_interval(s:real^N->bool) <=>
!a b x. a IN s /\ b IN s /\
(!i. 1 <= i /\ i <= dimindex(:N)
==> (a$i <= x$i /\ x$i <= b$i) \/
(b$i <= x$i /\ x$i <= a$i))
==> x IN s`;;let IS_INTERVAL_POINTWISE = prove
(`!s:real^N->bool x.
is_interval s /\
(!i. 1 <= i /\ i <= dimindex(:N) ==> ?a. a
IN s /\ a$i = x$i)
==> x
IN s`,
REWRITE_TAC[
is_interval] THEN REPEAT STRIP_TAC THEN
SUBGOAL_THEN
`!n. ?y:real^N. (!i. 1 <= i /\ i <= n ==> y$i = (x:real^N)$i) /\ y
IN s`
MP_TAC THENL
[INDUCT_TAC THEN REWRITE_TAC[ARITH_RULE `~(1 <= i /\ i <= 0)`] THENL
[ASM_MESON_TAC[
DIMINDEX_GE_1;
LE_REFL]; ALL_TAC] THEN
FIRST_X_ASSUM(X_CHOOSE_TAC `y:real^N`) THEN
ASM_CASES_TAC `SUC n <= dimindex(:N)` THENL
[FIRST_X_ASSUM(MP_TAC o SPEC `SUC n`) THEN
ANTS_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN
DISCH_THEN(X_CHOOSE_THEN `z:real^N` STRIP_ASSUME_TAC) THEN
EXISTS_TAC
`(lambda i. if i <= n then (y:real^N)$i else (z:real^N)$i):real^N` THEN
CONJ_TAC THENL
[X_GEN_TAC `i:num` THEN STRIP_TAC THEN
SUBGOAL_THEN `i <= dimindex(:N)` ASSUME_TAC THENL
[ASM_ARITH_TAC; ASM_SIMP_TAC[
LAMBDA_BETA]] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
SUBGOAL_THEN `i = SUC n` (fun th -> ASM_REWRITE_TAC[th]) THEN
ASM_ARITH_TAC;
FIRST_X_ASSUM(ASSUME_TAC o CONJUNCT2) THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN
MAP_EVERY EXISTS_TAC [`y:real^N`; `z:real^N`] THEN
ASM_SIMP_TAC[
LAMBDA_BETA] THEN REAL_ARITH_TAC];
EXISTS_TAC `y:real^N` THEN ASM_REWRITE_TAC[] THEN
SUBGOAL_THEN `y:real^N = x` (fun th -> REWRITE_TAC[th]) THEN
REWRITE_TAC[
CART_EQ] THEN
ASM_MESON_TAC[ARITH_RULE `i <= N /\ ~(SUC n <= N) ==> i <= n`]];
DISCH_THEN(MP_TAC o SPEC `dimindex(:N)`) THEN
REWRITE_TAC[GSYM
CART_EQ] THEN MESON_TAC[]]);;
let IS_INTERVAL_1_CASES = prove
(`!s:real^1->bool.
is_interval s <=>
s = {} \/
s = (:real^1) \/
(?a. s = {x | a < drop x}) \/
(?a. s = {x | a <= drop x}) \/
(?b. s = {x | drop x <= b}) \/
(?b. s = {x | drop x < b}) \/
(?a b. s = {x | a < drop x /\ drop x < b}) \/
(?a b. s = {x | a < drop x /\ drop x <= b}) \/
(?a b. s = {x | a <= drop x /\ drop x < b}) \/
(?a b. s = {x | a <= drop x /\ drop x <= b})`,
GEN_TAC THEN REWRITE_TAC[
IS_INTERVAL_1] THEN EQ_TAC THENL
[DISCH_TAC;
STRIP_TAC THEN ASM_REWRITE_TAC[
IN_ELIM_THM;
IN_UNIV;
NOT_IN_EMPTY] THEN
REAL_ARITH_TAC] THEN
ASM_CASES_TAC `s:real^1->bool = {}` THEN ASM_REWRITE_TAC[] THEN
MP_TAC(ISPEC `
IMAGE drop s`
SUP) THEN
MP_TAC(ISPEC `
IMAGE drop s`
INF) THEN
ASM_REWRITE_TAC[
IMAGE_EQ_EMPTY;
FORALL_IN_IMAGE] THEN
ASM_CASES_TAC `?a. !x. x
IN s ==> a <= drop x` THEN
ASM_CASES_TAC `?b. !x. x
IN s ==> drop x <= b` THEN
ASM_REWRITE_TAC[] THENL
[STRIP_TAC THEN STRIP_TAC THEN
MAP_EVERY ASM_CASES_TAC
[`inf(
IMAGE drop s)
IN IMAGE drop s`; `sup(
IMAGE drop s)
IN IMAGE drop s`]
THENL
[REPLICATE_TAC 8 DISJ2_TAC;
REPLICATE_TAC 7 DISJ2_TAC THEN DISJ1_TAC;
REPLICATE_TAC 6 DISJ2_TAC THEN DISJ1_TAC;
REPLICATE_TAC 5 DISJ2_TAC THEN DISJ1_TAC] THEN
MAP_EVERY EXISTS_TAC [`inf(
IMAGE drop s)`; `sup(
IMAGE drop s)`];
STRIP_TAC THEN ASM_CASES_TAC `inf(
IMAGE drop s)
IN IMAGE drop s` THENL
[REPLICATE_TAC 2 DISJ2_TAC THEN DISJ1_TAC;
DISJ2_TAC THEN DISJ1_TAC] THEN
EXISTS_TAC `inf(
IMAGE drop s)`;
STRIP_TAC THEN ASM_CASES_TAC `sup(
IMAGE drop s)
IN IMAGE drop s` THENL
[REPLICATE_TAC 3 DISJ2_TAC THEN DISJ1_TAC;
REPLICATE_TAC 4 DISJ2_TAC THEN DISJ1_TAC] THEN
EXISTS_TAC `sup(
IMAGE drop s)`;
DISJ1_TAC] THEN
REWRITE_TAC[
EXTENSION;
IN_ELIM_THM;
IN_UNIV] THEN
RULE_ASSUM_TAC(REWRITE_RULE[
IN_IMAGE]) THEN
REWRITE_TAC[GSYM
REAL_NOT_LE] THEN
ASM_MESON_TAC[
REAL_LE_TRANS;
REAL_LE_TOTAL; REAL_LE_ANTISYM]);;
let IS_INTERVAL_PCROSS_EQ = prove
(`!s:real^M->bool t:real^N->bool.
is_interval(s
PCROSS t) <=>
s = {} \/ t = {} \/
is_interval s /\
is_interval t`,
REPEAT GEN_TAC THEN
ASM_CASES_TAC `s:real^M->bool = {}` THEN
ASM_REWRITE_TAC[
PCROSS_EMPTY;
IS_INTERVAL_EMPTY] THEN
ASM_CASES_TAC `t:real^N->bool = {}` THEN
ASM_REWRITE_TAC[
PCROSS_EMPTY;
IS_INTERVAL_EMPTY] THEN
EQ_TAC THEN REWRITE_TAC[
IS_INTERVAL_PCROSS] THEN
REWRITE_TAC[
is_interval] THEN
REWRITE_TAC[
FORALL_PASTECART;
PASTECART_IN_PCROSS] THEN
STRIP_TAC THEN CONJ_TAC THENL
[MAP_EVERY X_GEN_TAC [`a:real^M`; `b:real^M`; `x:real^M`] THEN
STRIP_TAC THEN UNDISCH_TAC `~(t:real^N->bool = {})` THEN
REWRITE_TAC[GSYM
MEMBER_NOT_EMPTY] THEN
DISCH_THEN(X_CHOOSE_TAC `y:real^N`) THEN
FIRST_X_ASSUM(MP_TAC o SPECL
[`a:real^M`; `y:real^N`; `b:real^M`;
`y:real^N`; `x:real^M`; `y:real^N`]);
MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`; `x:real^N`] THEN
STRIP_TAC THEN UNDISCH_TAC `~(s:real^M->bool = {})` THEN
REWRITE_TAC[GSYM
MEMBER_NOT_EMPTY] THEN
DISCH_THEN(X_CHOOSE_TAC `w:real^M`) THEN
FIRST_X_ASSUM(MP_TAC o SPECL
[`w:real^M`; `a:real^N`; `w:real^M`;
`b:real^N`; `w:real^M`; `x:real^N`])] THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN MATCH_MP_TAC THEN
SIMP_TAC[pastecart;
LAMBDA_BETA] THEN
REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[
REAL_LE_REFL] THEN
ASM_MESON_TAC[
DIMINDEX_FINITE_SUM; ARITH_RULE
`1 <= i /\ i <= m + n /\ ~(i <= m) ==> 1 <= i - m /\ i - m <= n`]);;
let IS_INTERVAL_SUMS = prove
(`!s t:real^N->bool.
is_interval s /\
is_interval t
==>
is_interval {x + y | x
IN s /\ y
IN t}`,
REPEAT GEN_TAC THEN REWRITE_TAC[
is_interval] THEN
REWRITE_TAC[
IMP_CONJ;
RIGHT_FORALL_IMP_THM] THEN
REWRITE_TAC[
FORALL_IN_GSPEC] THEN
REWRITE_TAC[
RIGHT_IMP_FORALL_THM] THEN
REWRITE_TAC[IMP_IMP; GSYM
CONJ_ASSOC] THEN
MAP_EVERY X_GEN_TAC
[`a:real^N`; `a':real^N`; `b:real^N`; `b':real^N`; `y:real^N`] THEN
DISCH_THEN(CONJUNCTS_THEN2
(MP_TAC o SPECL [`a:real^N`; `b:real^N`]) MP_TAC) THEN
DISCH_THEN(CONJUNCTS_THEN2
(MP_TAC o SPECL [`a':real^N`; `b':real^N`]) STRIP_ASSUME_TAC) THEN
ASM_REWRITE_TAC[IMP_IMP;
IN_ELIM_THM] THEN ONCE_REWRITE_TAC[
CONJ_SYM] THEN
ONCE_REWRITE_TAC[VECTOR_ARITH `z:real^N = x + y <=> y = z - x`] THEN
REWRITE_TAC[
UNWIND_THM2] THEN MATCH_MP_TAC(MESON[]
`(?x. P x /\ Q(f x))
==> (!x. P x ==> x
IN s) /\ (!x. Q x ==> x
IN t)
==> ?x. x
IN s /\ f x
IN t`) THEN
REWRITE_TAC[
VECTOR_SUB_COMPONENT;
AND_FORALL_THM;
TAUT `(p ==> q) /\ (p ==> r) <=> p ==> q /\ r`] THEN
REWRITE_TAC[GSYM
LAMBDA_SKOLEM] THEN
X_GEN_TAC `i:num` THEN STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `i:num`) THEN
ASM_REWRITE_TAC[
VECTOR_ADD_COMPONENT] THEN
REWRITE_TAC[REAL_ARITH
`c <= y - x /\ y - x <= d <=> y - d <= x /\ x <= y - c`] THEN
REWRITE_TAC[REAL_ARITH
`a <= x /\ x <= b \/ b <= x /\ x <= a <=> min a b <= x /\ x <= max a b`] THEN
ONCE_REWRITE_TAC[TAUT `(p /\ q) /\ (r /\ s) <=> (p /\ r) /\ (q /\ s)`] THEN
REWRITE_TAC[GSYM
REAL_LE_MIN; GSYM
REAL_MAX_LE] THEN
REWRITE_TAC[GSYM
REAL_LE_BETWEEN] THEN REAL_ARITH_TAC);;
let segment = prove
(`segment[a,b] = {(&1 - u) % a + u % b | &0 <= u /\ u <= &1} /\
segment(a,b) = segment[a,b]
DIFF {a,b}`,
let SEGMENT_REFL = prove
(`(!a. segment[a,a] = {a}) /\
(!a. segment(a,a) = {})`,
REWRITE_TAC[segment; VECTOR_ARITH `(&1 - u) % a + u % a = a`] THEN
SET_TAC[
REAL_POS]);;
let IN_SEGMENT = prove
(`!a b x:real^N.
(x
IN segment[a,b] <=>
?u. &0 <= u /\ u <= &1 /\ x = (&1 - u) % a + u % b) /\
(x
IN segment(a,b) <=>
~(a = b) /\ ?u. &0 < u /\ u < &1 /\ x = (&1 - u) % a + u % b)`,
let SEGMENT_SYM = prove
(`(!a b:real^N. segment[a,b] = segment[b,a]) /\
(!a b:real^N. segment(a,b) = segment(b,a))`,
MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN
SIMP_TAC[
open_segment] THEN
CONJ_TAC THENL [ALL_TAC; SIMP_TAC[
INSERT_AC]] THEN
REWRITE_TAC[
EXTENSION;
IN_SEGMENT] THEN REPEAT GEN_TAC THEN EQ_TAC THEN
DISCH_THEN(X_CHOOSE_TAC `u:real`) THEN EXISTS_TAC `&1 - u` THEN
ASM_REWRITE_TAC[] THEN
REPEAT CONJ_TAC THEN TRY ASM_ARITH_TAC THEN VECTOR_ARITH_TAC);;
let ENDS_IN_SEGMENT = prove
(`!a b. a
IN segment[a,b] /\ b
IN segment[a,b]`,
REPEAT STRIP_TAC THEN REWRITE_TAC[segment;
IN_ELIM_THM] THENL
[EXISTS_TAC `&0`; EXISTS_TAC `&1`] THEN
(CONJ_TAC THENL [REAL_ARITH_TAC; VECTOR_ARITH_TAC]));;
let MIDPOINT_IN_SEGMENT = prove
(`(!a b:real^N. midpoint(a,b)
IN segment[a,b]) /\
(!a b:real^N. midpoint(a,b)
IN segment(a,b) <=> ~(a = b))`,
REWRITE_TAC[
IN_SEGMENT] THEN REPEAT STRIP_TAC THENL
[ALL_TAC; ASM_CASES_TAC `a:real^N = b` THEN ASM_REWRITE_TAC[]] THEN
EXISTS_TAC `&1 / &2` THEN REWRITE_TAC[midpoint] THEN
CONV_TAC REAL_RAT_REDUCE_CONV THEN VECTOR_ARITH_TAC);;
let BETWEEN_IN_SEGMENT = prove
(`!x a b:real^N. between x (a,b) <=> x
IN segment[a,b]`,
REPEAT GEN_TAC THEN REWRITE_TAC[between] THEN
ASM_CASES_TAC `a:real^N = b` THEN
ASM_REWRITE_TAC[
SEGMENT_REFL;
IN_SING] THENL [NORM_ARITH_TAC; ALL_TAC] THEN
REWRITE_TAC[segment;
IN_ELIM_THM] THEN EQ_TAC THENL
[DISCH_THEN(ASSUME_TAC o SYM) THEN
EXISTS_TAC `dist(a:real^N,x) / dist(a,b)` THEN
ASM_SIMP_TAC[
REAL_LE_LDIV_EQ;
REAL_LE_RDIV_EQ;
DIST_POS_LT] THEN CONJ_TAC
THENL [FIRST_ASSUM(SUBST1_TAC o SYM) THEN NORM_ARITH_TAC; ALL_TAC] THEN
MATCH_MP_TAC
VECTOR_MUL_LCANCEL_IMP THEN EXISTS_TAC `dist(a:real^N,b)` THEN
ASM_SIMP_TAC[
VECTOR_MUL_ASSOC;
VECTOR_ADD_LDISTRIB;
REAL_SUB_LDISTRIB;
REAL_DIV_LMUL;
DIST_EQ_0] THEN
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
DIST_TRIANGLE_EQ] o SYM) THEN
FIRST_ASSUM(SUBST1_TAC o SYM) THEN
REWRITE_TAC[dist; REAL_ARITH `(a + b) * &1 - a = b`] THEN
VECTOR_ARITH_TAC;
STRIP_TAC THEN ASM_REWRITE_TAC[dist] THEN
REWRITE_TAC[VECTOR_ARITH `a - ((&1 - u) % a + u % b) = u % (a - b)`;
VECTOR_ARITH `((&1 - u) % a + u % b) - b = (&1 - u) % (a - b)`;
NORM_MUL; GSYM REAL_ADD_LDISTRIB] THEN
REPEAT(POP_ASSUM MP_TAC) THEN CONV_TAC REAL_FIELD]);;
let IN_SEGMENT_COMPONENT = prove
(`!a b x:real^N i.
x
IN segment[a,b] /\ 1 <= i /\ i <= dimindex(:N)
==> min (a$i) (b$i) <= x$i /\ x$i <= max (a$i) (b$i)`,
REPEAT STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
IN_SEGMENT]) THEN
DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC) THEN
FIRST_X_ASSUM(X_CHOOSE_THEN `t:real` STRIP_ASSUME_TAC) THEN
ASM_REWRITE_TAC[
VECTOR_ADD_COMPONENT;
VECTOR_MUL_COMPONENT] THEN
SIMP_TAC[REAL_ARITH `c <= u * a + t * b <=> u * --a + t * --b <= --c`] THEN
MATCH_MP_TAC
REAL_CONVEX_BOUND_LE THEN ASM_REAL_ARITH_TAC);;
let SEGMENT_1 = prove
(`(!a b. segment[a,b] =
if drop a <= drop b then interval[a,b] else interval[b,a]) /\
(!a b. segment(a,b) =
if drop a <= drop b then interval(a,b) else interval(b,a))`,
let SEGMENT_TRANSLATION = prove
(`(!c a b. segment[c + a,c + b] =
IMAGE (\x. c + x) (segment[a,b])) /\
(!c a b. segment(c + a,c + b) =
IMAGE (\x. c + x) (segment(a,b)))`,
REWRITE_TAC[
EXTENSION;
IN_SEGMENT;
IN_IMAGE] THEN
REWRITE_TAC[VECTOR_ARITH `(&1 - u) % (c + a) + u % (c + b) =
c + (&1 - u) % a + u % b`] THEN
REWRITE_TAC[VECTOR_ARITH `c + a:real^N = c + b <=> a = b`] THEN
MESON_TAC[]);;
let SEGMENT_SCALAR_MULTIPLE = prove
(`(!a b v. segment[a % v,b % v] =
{x % v:real^N | a <= x /\ x <= b \/ b <= x /\ x <= a}) /\
(!a b v. ~(v = vec 0)
==> segment(a % v,b % v) =
{x % v:real^N | a < x /\ x < b \/ b < x /\ x < a})`,
let FINITE_INTER_COLLINEAR_OPEN_SEGMENTS = prove
(`!a b c d:real^N.
collinear{a,b,c}
==> (
FINITE(segment(a,b)
INTER segment(c,d)) <=>
segment(a,b)
INTER segment(c,d) = {})`,
REPEAT GEN_TAC THEN ABBREV_TAC `m:real^N = b - a` THEN POP_ASSUM MP_TAC THEN
GEOM_NORMALIZE_TAC `m:real^N` THEN
SIMP_TAC[
VECTOR_SUB_EQ;
SEGMENT_REFL;
INTER_EMPTY;
FINITE_EMPTY] THEN
X_GEN_TAC `m:real^N` THEN DISCH_TAC THEN REPEAT GEN_TAC THEN
DISCH_THEN(SUBST_ALL_TAC o SYM) THEN POP_ASSUM MP_TAC THEN
GEOM_ORIGIN_TAC `a:real^N` THEN GEOM_BASIS_MULTIPLE_TAC 1 `b:real^N` THEN
X_GEN_TAC `b:real` THEN DISCH_TAC THEN
MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN
SIMP_TAC[
VECTOR_SUB_RZERO;
NORM_MUL;
NORM_BASIS;
DIMINDEX_GE_1;
LE_REFL] THEN
ASM_REWRITE_TAC[
real_abs;
REAL_MUL_RID] THEN DISCH_THEN SUBST_ALL_TAC THEN
POP_ASSUM(K ALL_TAC) THEN
ASM_CASES_TAC `collinear{vec 0:real^N,&1 % basis 1,y}` THENL
[POP_ASSUM MP_TAC THEN
SIMP_TAC[
COLLINEAR_LEMMA_ALT;
BASIS_NONZERO;
DIMINDEX_GE_1;
LE_REFL] THEN
MATCH_MP_TAC(TAUT
`~a /\ (b ==> c ==> d) ==> a \/ b ==> a \/ c ==> d`) THEN
CONJ_TAC THENL
[SIMP_TAC[
VECTOR_MUL_LID;
BASIS_NONZERO;
DIMINDEX_GE_1;
LE_REFL];
REWRITE_TAC[
LEFT_IMP_EXISTS_THM]] THEN
X_GEN_TAC `b:real` THEN DISCH_THEN SUBST_ALL_TAC THEN
X_GEN_TAC `a:real` THEN DISCH_THEN SUBST_ALL_TAC THEN
REWRITE_TAC[
VECTOR_MUL_ASSOC;
REAL_MUL_RID] THEN
SUBST1_TAC(VECTOR_ARITH `vec 0:real^N = &0 % basis 1`) THEN
SIMP_TAC[
SEGMENT_SCALAR_MULTIPLE;
BASIS_NONZERO;
DIMINDEX_GE_1;
LE_REFL;
VECTOR_MUL_RCANCEL;
IMAGE_EQ_EMPTY;
FINITE_IMAGE_INJ_EQ; SET_RULE
`(!x y. x % v = y % v <=> x = y)
==> {x % v | P x}
INTER {x % v | Q x} =
IMAGE (\x. x % v) {x | P x /\ Q x}`] THEN
REWRITE_TAC[REAL_ARITH `(&0 < x /\ x < &1 \/ &1 < x /\ x < &0) /\
(b < x /\ x < a \/ a < x /\ x < b) <=>
max (&0) (min a b) < x /\ x < min (&1) (max a b)`] THEN
SIMP_TAC[
FINITE_REAL_INTERVAL;
EXTENSION;
NOT_IN_EMPTY;
IN_ELIM_THM] THEN
SIMP_TAC[GSYM
REAL_LT_BETWEEN; GSYM
NOT_EXISTS_THM] THEN REAL_ARITH_TAC;
DISCH_TAC THEN ASM_CASES_TAC
`segment(vec 0:real^N,&1 % basis 1)
INTER segment (x,y) = {}` THEN
ASM_REWRITE_TAC[
FINITE_EMPTY] THEN DISCH_THEN(K ALL_TAC) THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM
MEMBER_NOT_EMPTY]) THEN
REWRITE_TAC[
open_segment;
IN_DIFF;
NOT_IN_EMPTY;
DE_MORGAN_THM;
IN_INTER;
IN_INSERT] THEN
DISCH_THEN(X_CHOOSE_THEN `p:real^N` STRIP_ASSUME_TAC) THEN
UNDISCH_TAC `~collinear{vec 0:real^N,&1 % basis 1, y}` THEN
RULE_ASSUM_TAC(REWRITE_RULE[
VECTOR_MUL_LID]) THEN
REWRITE_TAC[
VECTOR_MUL_LID] THEN
MATCH_MP_TAC
COLLINEAR_SUBSET THEN
EXISTS_TAC `{p,x:real^N, y, vec 0, basis 1}` THEN
CONJ_TAC THENL [ALL_TAC; SET_TAC[]] THEN
MP_TAC(ISPECL [`{y:real^N,vec 0,basis 1}`; `p:real^N`; `x:real^N`]
COLLINEAR_TRIPLES) THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN SUBST1_TAC THEN
REWRITE_TAC[
FORALL_IN_INSERT;
NOT_IN_EMPTY] THEN CONJ_TAC THENL
[ONCE_REWRITE_TAC[SET_RULE `{p,x,y} = {x,p,y}`] THEN
MATCH_MP_TAC
BETWEEN_IMP_COLLINEAR THEN
ASM_REWRITE_TAC[
BETWEEN_IN_SEGMENT];
ALL_TAC] THEN
ASM_SIMP_TAC[GSYM
COLLINEAR_4_3] THEN
ONCE_REWRITE_TAC[SET_RULE `{p,x,z,w} = {w,z,p,x}`] THEN
SIMP_TAC[
COLLINEAR_4_3;
BASIS_NONZERO;
DIMINDEX_GE_1; ARITH] THEN
REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP
BETWEEN_IMP_COLLINEAR o
GEN_REWRITE_RULE I [GSYM
BETWEEN_IN_SEGMENT])) THEN
REPEAT(POP_ASSUM MP_TAC) THEN SIMP_TAC[
INSERT_AC]]);;
let LIM_COMPONENT_LE = prove
(`!net:(A)net f:A->real^N g:A->real^N k l m.
~(
trivial_limit net) /\ (f --> l) net /\ (g --> m) net /\
eventually (\x. (f x)$k <= (g x)$k) net /\
1 <= k /\ k <= dimindex(:N)
==> l$k <= m$k`,
let LIM_DROP_LE = prove
(`!net:(A)net f g l m.
~(
trivial_limit net) /\ (f --> l) net /\ (g --> m) net /\
eventually (\x. drop(f x) <= drop(g x)) net
==> drop l <= drop m`,
REWRITE_TAC[drop] THEN REPEAT STRIP_TAC THEN
MATCH_MP_TAC(ISPEC `net:(A)net`
LIM_COMPONENT_LE) THEN
MAP_EVERY EXISTS_TAC [`f:A->real^1`; `g:A->real^1`] THEN
ASM_REWRITE_TAC[DIMINDEX_1;
LE_REFL]);;
let LIM_WITHIN_UNION = prove
(`(f --> l) (at x within (s
UNION t)) <=>
(f --> l) (at x within s) /\ (f --> l) (at x within t)`,
REWRITE_TAC[
LIM_WITHIN;
IN_UNION;
AND_FORALL_THM] THEN
AP_TERM_TAC THEN REWRITE_TAC[
FUN_EQ_THM] THEN X_GEN_TAC `e:real` THEN
ASM_CASES_TAC `&0 < e` THEN ASM_REWRITE_TAC[] THEN
EQ_TAC THENL [MESON_TAC[]; ALL_TAC] THEN DISCH_THEN
(CONJUNCTS_THEN2 (X_CHOOSE_TAC `d:real`) (X_CHOOSE_TAC `k:real`)) THEN
EXISTS_TAC `min d k` THEN ASM_REWRITE_TAC[
REAL_LT_MIN] THEN
ASM_MESON_TAC[]);;
let LIM_COMPONENTWISE_LIFT = prove
(`!net f:A->real^N.
(f --> l) net <=>
!i. 1 <= i /\ i <= dimindex(:N)
==> ((\x. lift((f x)$i)) --> lift(l$i)) net`,
let MAPPING_CONNECTED_ONTO_SEGMENT = prove
(`!s:real^M->bool a b:real^N.
connected s /\ ~(?a. s
SUBSET {a})
==> ?f. f
continuous_on s /\
IMAGE f s = segment[a,b]`,
REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP (SET_RULE
`~(?a. s
SUBSET {a}) ==> ?a b. a
IN s /\ b
IN s /\ ~(a = b)`)) THEN
REWRITE_TAC[
LEFT_IMP_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC [`u:real^M`; `v:real^M`] THEN STRIP_TAC THEN EXISTS_TAC
`\x:real^M. a + dist(u,x) / (dist(u,x) + dist(v,x)) % (b - a:real^N)` THEN
CONJ_TAC THENL
[MATCH_MP_TAC
CONTINUOUS_ON_ADD THEN REWRITE_TAC[
CONTINUOUS_ON_CONST] THEN
MATCH_MP_TAC
CONTINUOUS_ON_MUL THEN SIMP_TAC[
o_DEF;
CONTINUOUS_ON_CONST];
REWRITE_TAC[segment; VECTOR_ARITH
`(&1 - u) % a + u % b:real^N = a + u % (b - a)`] THEN
MATCH_MP_TAC(SET_RULE
`
IMAGE f s = {x | P x}
==>
IMAGE (\x. a + f x % b) s = {a + u % b:real^N | P u}`) THEN
REWRITE_TAC[GSYM
SUBSET_ANTISYM_EQ;
SUBSET;
FORALL_IN_IMAGE] THEN
ASM_SIMP_TAC[
IN_ELIM_THM;
REAL_LE_RDIV_EQ;
REAL_LE_LDIV_EQ;
NORM_ARITH `~(u:real^N = v) ==> &0 < dist(u,x) + dist(v,x)`] THEN
CONJ_TAC THENL [CONV_TAC NORM_ARITH; REWRITE_TAC[
IN_IMAGE]] THEN
X_GEN_TAC `t:real` THEN STRIP_TAC THEN
MP_TAC(ISPECL
[`
IMAGE (\x:real^M. lift(dist(u,x) / (dist(u,x) + dist(v,x)))) s`;
`vec 0:real^1`; `vec 1:real^1`; `t:real`; `1`]
CONNECTED_IVT_COMPONENT) THEN
ASM_SIMP_TAC[
VEC_COMPONENT; DIMINDEX_1;
ARITH_LE] THEN
REWRITE_TAC[
EXISTS_IN_IMAGE; GSYM drop;
LIFT_DROP] THEN
ANTS_TAC THENL [REWRITE_TAC[
IN_IMAGE]; MESON_TAC[]] THEN
REPEAT CONJ_TAC THENL
[MATCH_MP_TAC
CONNECTED_CONTINUOUS_IMAGE THEN ASM_REWRITE_TAC[];
EXISTS_TAC `u:real^M` THEN ASM_REWRITE_TAC[
DIST_REFL;
real_div] THEN
REWRITE_TAC[GSYM
LIFT_NUM;
LIFT_EQ] THEN REAL_ARITH_TAC;
EXISTS_TAC `v:real^M` THEN ASM_REWRITE_TAC[
DIST_REFL] THEN
ASM_SIMP_TAC[
REAL_DIV_REFL;
DIST_EQ_0;
REAL_ADD_RID] THEN
REWRITE_TAC[GSYM
LIFT_NUM;
LIFT_EQ]]] THEN
REWRITE_TAC[
real_div;
LIFT_CMUL] THEN
MATCH_MP_TAC
CONTINUOUS_ON_MUL THEN
REWRITE_TAC[
CONTINUOUS_ON_LIFT_DIST] THEN
MATCH_MP_TAC(REWRITE_RULE[
o_DEF]
CONTINUOUS_ON_INV) THEN
ASM_SIMP_TAC[
LIFT_ADD; NORM_ARITH
`~(u:real^N = v) ==> ~(dist(u,x) + dist(v,x) = &0)`] THEN
MATCH_MP_TAC
CONTINUOUS_ON_ADD THEN
REWRITE_TAC[REWRITE_RULE[
o_DEF]
CONTINUOUS_ON_LIFT_DIST]);;
let HOMEOMORPHISM_SYM = prove
(`!f:real^M->real^N g s t.
homeomorphism (s,t) (f,g) <=> homeomorphism (t,s) (g,f)`,
REWRITE_TAC[homeomorphism] THEN MESON_TAC[]);;
let HOMEOMORPHIC_SYM = prove
(`!s t. s homeomorphic t <=> t homeomorphic s`,
REPEAT GEN_TAC THEN REWRITE_TAC[homeomorphic; homeomorphism] THEN
GEN_REWRITE_TAC RAND_CONV [
SWAP_EXISTS_THM] THEN
REPEAT(AP_TERM_TAC THEN ABS_TAC) THEN CONV_TAC TAUT);;
let HOMEOMORPHISM_COMPOSE = prove
(`!f:real^M->real^N g h:real^N->real^P k s t u.
homeomorphism (s,t) (f,g) /\ homeomorphism (t,u) (h,k)
==> homeomorphism (s,u) (h o f,g o k)`,
let HOMEOMORPHIC_TRANS = prove
(`!s:real^M->bool t:real^N->bool u:real^P->bool.
s homeomorphic t /\ t homeomorphic u ==> s homeomorphic u`,
let HOMEOMORPHIC_EMPTY = prove
(`(!s. (s:real^N->bool) homeomorphic ({}:real^M->bool) <=> s = {}) /\
(!s. ({}:real^M->bool) homeomorphic (s:real^N->bool) <=> s = {})`,
let CONTINUOUS_ON_INVERSE_OPEN_MAP = prove
(`!f:real^M->real^N g s t.
f
continuous_on s /\
IMAGE f s = t /\ (!x. x
IN s ==> g(f x) = x) /\
(!u.
open_in (subtopology euclidean s) u
==>
open_in (subtopology euclidean t) (
IMAGE f u))
==> g
continuous_on t`,
REPEAT STRIP_TAC THEN
MP_TAC(ISPECL [`g:real^N->real^M`; `t:real^N->bool`; `s:real^M->bool`]
CONTINUOUS_ON_OPEN_GEN) THEN
ANTS_TAC THENL [ASM SET_TAC[]; DISCH_THEN SUBST1_TAC] THEN
X_GEN_TAC `u:real^M->bool` THEN DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `u:real^M->bool`) THEN ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC EQ_IMP THEN AP_TERM_TAC THEN
FIRST_ASSUM(MP_TAC o CONJUNCT1 o GEN_REWRITE_RULE I [
open_in]) THEN
ASM SET_TAC[]);;
let HOMEOMORPHISM_IMP_OPEN_MAP = prove
(`!f:real^M->real^N g s t u.
homeomorphism (s,t) (f,g) /\
open_in (subtopology euclidean s) u
==>
open_in (subtopology euclidean t) (
IMAGE f u)`,
REWRITE_TAC[homeomorphism] THEN REPEAT STRIP_TAC THEN
SUBGOAL_THEN `
IMAGE (f:real^M->real^N) u =
{y | y
IN t /\ g(y)
IN u}`
SUBST1_TAC THENL
[FIRST_ASSUM(MP_TAC o CONJUNCT1 o GEN_REWRITE_RULE I [
open_in]) THEN
ASM SET_TAC[];
MATCH_MP_TAC
CONTINUOUS_ON_IMP_OPEN_IN THEN ASM_REWRITE_TAC[]]);;
let CARD_EQ_INTERVAL = prove
(`(!a b:real^N. ~(interval(a,b) = {}) ==> interval[a,b] =_c (:real)) /\
(!a b:real^N. ~(interval(a,b) = {}) ==> interval(a,b) =_c (:real))`,
let UNCOUNTABLE_INTERVAL = prove
(`(!a b. ~(interval(a,b) = {}) ==> ~COUNTABLE(interval[a,b])) /\
(!a b. ~(interval(a,b) = {}) ==> ~COUNTABLE(interval(a,b)))`,
let CAUCHY_ISOMETRIC = prove
(`!f s e x.
&0 < e /\ subspace s /\
linear f /\ (!x. x
IN s ==> norm(f x) >= e * norm(x)) /\
(!n. x(n)
IN s) /\ cauchy(f o x)
==> cauchy x`,
let INJECTIVE_IMP_ISOMETRIC = prove
(`!f:real^M->real^N s.
closed s /\ subspace s /\
linear f /\ (!x. x
IN s /\ (f x = vec 0) ==> (x = vec 0))
==> ?e. &0 < e /\ !x. x
IN s ==> norm(f x) >= e * norm(x)`,
let DIM_SUBSTANDARD = prove
(`!d. d <= dimindex(:N)
==> (dim {x:real^N | !i. d < i /\ i <= dimindex(:N)
==> x$i = &0} =
d)`,
let CLOSED_SUBSPACE = prove
(`!s:real^N->bool. subspace s ==> closed s`,
REPEAT STRIP_TAC THEN ABBREV_TAC `d = dim(s:real^N->bool)` THEN
MP_TAC(MATCH_MP
DIM_SUBSTANDARD
(ISPEC `s:real^N->bool`
DIM_SUBSET_UNIV)) THEN
ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
MP_TAC(ISPECL
[`{x:real^N | !i. d < i /\ i <= dimindex(:N)
==> x$i = &0}`;
`s:real^N->bool`]
SUBSPACE_ISOMORPHISM) THEN
ASM_REWRITE_TAC[
SUBSPACE_SUBSTANDARD] THEN
DISCH_THEN(X_CHOOSE_THEN `f:real^N->real^N` MP_TAC) THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
DISCH_THEN(CONJUNCTS_THEN2 (SUBST_ALL_TAC o SYM) STRIP_ASSUME_TAC) THEN
MATCH_MP_TAC(ISPEC `f:real^N->real^N`
CLOSED_INJECTIVE_IMAGE_SUBSPACE) THEN
ASM_REWRITE_TAC[
SUBSPACE_SUBSTANDARD;
CLOSED_SUBSTANDARD] THEN
X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
ASM_REWRITE_TAC[] THEN
CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[
LINEAR_0]] THEN
REWRITE_TAC[
IN_ELIM_THM] THEN
ASM_MESON_TAC[
VEC_COMPONENT; ARITH_RULE `d < i ==> 1 <= i`]);;
let BASIS_COORDINATES_LIPSCHITZ = prove
(`!b:real^N->bool.
independent b
==> ?B. &0 < B /\
!c v. v
IN b
==> abs(c v) <= B * norm(vsum b (\v. c(v) % v))`,
X_GEN_TAC `k:real^N->bool` THEN DISCH_TAC THEN
FIRST_ASSUM(STRIP_ASSUME_TAC o MATCH_MP
INDEPENDENT_BOUND) THEN
FIRST_ASSUM(X_CHOOSE_THEN `b:num->real^N` STRIP_ASSUME_TAC o
GEN_REWRITE_RULE I [
FINITE_INDEX_NUMSEG]) THEN
ABBREV_TAC `n =
CARD(k:real^N->bool)` THEN
MP_TAC(ISPECL
[`(\x. vsum(1..n) (\i. x$i % b i)):real^N->real^N`;
`span(
IMAGE basis (1..n)):real^N->bool`]
INJECTIVE_IMP_ISOMETRIC) THEN
REWRITE_TAC[
SUBSPACE_SPAN] THEN ANTS_TAC THENL
[CONJ_TAC THENL [SIMP_TAC[
CLOSED_SUBSPACE;
SUBSPACE_SPAN]; ALL_TAC] THEN
CONJ_TAC THENL
[MATCH_MP_TAC
LINEAR_COMPOSE_VSUM THEN
REWRITE_TAC[
FINITE_NUMSEG;
IN_NUMSEG] THEN REPEAT STRIP_TAC THEN
MATCH_MP_TAC
LINEAR_VMUL_COMPONENT THEN
SIMP_TAC[
LINEAR_ID] THEN ASM_ARITH_TAC;
ALL_TAC] THEN
X_GEN_TAC `x:real^N` THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
IN_SPAN_IMAGE_BASIS]) THEN
REWRITE_TAC[
IN_NUMSEG] THEN DISCH_TAC THEN
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
INJECTIVE_ON_LEFT_INVERSE]) THEN
DISCH_THEN(X_CHOOSE_TAC `c:real^N->num`) THEN
SUBGOAL_THEN
`vsum(1..n) (\i. (x:real^N)$i % b i:real^N) = vsum k (\v. x$(c v) % v)`
SUBST1_TAC THENL
[MATCH_MP_TAC
VSUM_EQ_GENERAL_INVERSES THEN
MAP_EVERY EXISTS_TAC [`b:num->real^N`; `c:real^N->num`] THEN
ASM SET_TAC[];
ALL_TAC] THEN
DISCH_TAC THEN
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
INDEPENDENT_EXPLICIT]) THEN
DISCH_THEN(MP_TAC o SPEC `\v:real^N. (x:real^N)$(c v)` o CONJUNCT2) THEN
ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
REWRITE_TAC[
CART_EQ;
FORALL_IN_IMAGE;
VEC_COMPONENT] THEN
ASM_MESON_TAC[
IN_NUMSEG];
ALL_TAC] THEN
DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `inv(B:real)` THEN ASM_REWRITE_TAC[
REAL_LT_INV_EQ] THEN
ASM_REWRITE_TAC[
FORALL_IN_IMAGE;
IN_NUMSEG] THEN
MAP_EVERY X_GEN_TAC [`c:real^N->real`; `j:num`] THEN STRIP_TAC THEN
ONCE_REWRITE_TAC[REAL_ARITH `inv B * x = x / B`] THEN
ASM_SIMP_TAC[
REAL_LE_RDIV_EQ] THEN
W(MP_TAC o PART_MATCH (lhs o rand)
VSUM_IMAGE o rand o rand o snd) THEN
ASM_REWRITE_TAC[
FINITE_NUMSEG] THEN DISCH_THEN SUBST1_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC
`(lambda i. if 1 <= i /\ i <= n then c(b i:real^N) else &0):real^N`) THEN
SIMP_TAC[
IN_SPAN_IMAGE_BASIS;
LAMBDA_BETA] THEN
ANTS_TAC THENL [MESON_TAC[
IN_NUMSEG]; ALL_TAC] THEN
MATCH_MP_TAC(REAL_ARITH `x = v /\ u <= y ==> x >= y ==> u <= v`) THEN
CONJ_TAC THENL
[AP_TERM_TAC THEN MATCH_MP_TAC
VSUM_EQ_NUMSEG THEN
SUBGOAL_THEN `!i. i <= n ==> i <= dimindex(:N)` MP_TAC THENL
[ASM_ARITH_TAC; SIMP_TAC[
LAMBDA_BETA] THEN DISCH_THEN(K ALL_TAC)] THEN
REWRITE_TAC[
o_THM];
GEN_REWRITE_TAC RAND_CONV [REAL_MUL_SYM] THEN
ASM_SIMP_TAC[
REAL_LE_RMUL_EQ] THEN
MP_TAC(ISPECL
[`(lambda i. if 1 <= i /\ i <= n then c(b i:real^N) else &0):real^N`;
`j:num`]
COMPONENT_LE_NORM) THEN
SUBGOAL_THEN `1 <= j /\ j <= dimindex(:N)` MP_TAC THENL
[ASM_ARITH_TAC; SIMP_TAC[
LAMBDA_BETA] THEN ASM_REWRITE_TAC[]]]);;
let BASIS_COORDINATES_CONTINUOUS = prove
(`!b:real^N->bool e.
independent b /\ &0 < e
==> ?d. &0 < d /\
!c. norm(vsum b (\v. c(v) % v)) < d
==> !v. v
IN b ==> abs(c v) < e`,
REPEAT STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o MATCH_MP
BASIS_COORDINATES_LIPSCHITZ) THEN
DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `e / B:real` THEN ASM_SIMP_TAC[
REAL_LT_DIV] THEN
X_GEN_TAC `c:real^N->real` THEN DISCH_TAC THEN
X_GEN_TAC `v:real^N` THEN DISCH_TAC THEN
MATCH_MP_TAC
REAL_LET_TRANS THEN
EXISTS_TAC `B * norm(vsum b (\v:real^N. c v % v))` THEN
ASM_SIMP_TAC[] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
ASM_SIMP_TAC[GSYM
REAL_LT_RDIV_EQ]);;
let AFFINITY_INVERSES = prove
(`!m c. ~(m = &0)
==> (\x. m % x + c) o (\x. inv(m) % x + (--(inv(m) % c))) = I /\
(\x. inv(m) % x + (--(inv(m) % c))) o (\x. m % x + c) = I`,
let REAL_AFFINITY_LE = prove
(`!m c x y. &0 < m ==> (m * x + c <= y <=> x <= inv(m) * y + --(c / m))`,
REWRITE_TAC[REAL_ARITH `m * x + c <= y <=> x * m <= y - c`] THEN
SIMP_TAC[GSYM
REAL_LE_RDIV_EQ] THEN REAL_ARITH_TAC);;
let REAL_LE_AFFINITY = prove
(`!m c x y. &0 < m ==> (y <= m * x + c <=> inv(m) * y + --(c / m) <= x)`,
REWRITE_TAC[REAL_ARITH `y <= m * x + c <=> y - c <= x * m`] THEN
SIMP_TAC[GSYM
REAL_LE_LDIV_EQ] THEN REAL_ARITH_TAC);;
let REAL_AFFINITY_EQ = prove
(`!m c x y. ~(m = &0) ==> (m * x + c = y <=> x = inv(m) * y + --(c / m))`,
CONV_TAC REAL_FIELD);;
let REAL_EQ_AFFINITY = prove
(`!m c x y. ~(m = &0) ==> (y = m * x + c <=> inv(m) * y + --(c / m) = x)`,
CONV_TAC REAL_FIELD);;
let IMAGE_AFFINITY_INTERVAL = prove
(`!a b:real^N m c.
IMAGE (\x. m % x + c) (interval[a,b]) =
if interval[a,b] = {} then {}
else if &0 <= m then interval[m % a + c,m % b + c]
else interval[m % b + c,m % a + c]`,
let CONTINUOUS_LIFT_DET = prove
(`!(A:A->real^N^N) net.
(!i j. 1 <= i /\ i <= dimindex(:N) /\
1 <= j /\ j <= dimindex(:N)
==> (\x. lift(A x$i$j)) continuous net)
==> (\x. lift(det(A x))) continuous net`,
let MATRIX_WLOG_INVERTIBLE = prove
(`!P. (!A:real^N^N. invertible A ==> P A) /\
(!A:real^N^N. ?d. &0 < d /\
closed {x | x
IN cball(vec 0,d) /\
P(A + drop x %% mat 1)})
==> !A:real^N^N. P A`,
let COFACTOR_COFACTOR = prove
(`!A:real^N^N.
2 <= dimindex(:N)
==> cofactor(cofactor A) = (det(A) pow (dimindex(:N) - 2)) %% A`,
let RANK_COFACTOR = prove
(`!A:real^N^N.
rank(cofactor A) = if rank(A) = dimindex(:N) then dimindex(:N)
else if rank(A) = dimindex(:N) - 1 then 1
else 0`,
let DET_OPEN_MAP = prove
(`!A:real^N^N e.
&0 < e
==> (?B:real^N^N. (!i j. abs(B$i$j - A$i$j) < e) /\ det B < det A) /\
(?C:real^N^N. (!i j. abs(C$i$j - A$i$j) < e) /\ det C > det A)`,
let SUMS_SUMMABLE = prove
(`!f l s. (f sums l) s ==> summable s f`,
REWRITE_TAC[summable] THEN MESON_TAC[]);;
let SUMS_INFSUM = prove
(`!f s. (f sums (infsum s f)) s <=> summable s f`,
REWRITE_TAC[infsum; summable] THEN MESON_TAC[]);;
let SUMS_LIM = prove
(`!f:num->real^N s.
(f sums lim sequentially (\n. vsum (s
INTER (0..n)) f)) s
<=> summable s f`,
GEN_TAC THEN GEN_TAC THEN EQ_TAC THENL [MESON_TAC[summable];
REWRITE_TAC[summable; sums] THEN STRIP_TAC THEN REWRITE_TAC[lim] THEN
ASM_MESON_TAC[]]);;
let SERIES_FROM = prove
(`!f l k. (f sums l) (from k) = ((\n. vsum(k..n) f) --> l) sequentially`,
REPEAT GEN_TAC THEN REWRITE_TAC[sums] THEN
AP_THM_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN ABS_TAC THEN
AP_THM_TAC THEN AP_TERM_TAC THEN
REWRITE_TAC[
EXTENSION; numseg; from;
IN_ELIM_THM;
IN_INTER] THEN ARITH_TAC);;
let SERIES_TERMS_TOZERO = prove
(`!f l n. (f sums l) (from n) ==> (f --> vec 0) sequentially`,
REPEAT GEN_TAC THEN SIMP_TAC[sums;
LIM_SEQUENTIALLY;
FROM_INTER_NUMSEG] THEN
DISCH_TAC THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `e / &2`) THEN
ASM_REWRITE_TAC[
REAL_HALF] THEN DISCH_THEN(X_CHOOSE_TAC `N:num`) THEN
EXISTS_TAC `N + n + 1` THEN X_GEN_TAC `m:num` THEN DISCH_TAC THEN
FIRST_X_ASSUM(fun th ->
MP_TAC(SPEC `m - 1` th) THEN MP_TAC(SPEC `m:num` th)) THEN
SUBGOAL_THEN `0 < m /\ n <= m` (fun th -> SIMP_TAC[
VSUM_CLAUSES_RIGHT; th])
THENL [ASM_ARITH_TAC; ALL_TAC] THEN
REPEAT(ANTS_TAC THENL [ASM_ARITH_TAC; DISCH_TAC]) THEN
REPEAT(POP_ASSUM MP_TAC) THEN NORM_ARITH_TAC);;
let SERIES_ADD = prove
(`!x x0 y y0 s.
(x sums x0) s /\ (y sums y0) s ==> ((\n. x n + y n) sums (x0 + y0)) s`,
let SERIES_SUB = prove
(`!x x0 y y0 s.
(x sums x0) s /\ (y sums y0) s ==> ((\n. x n - y n) sums (x0 - y0)) s`,
let SUMS_IFF = prove
(`!f g k. (!x. x
IN k ==> f x = g x) ==> ((f sums l) k <=> (g sums l) k)`,
REPEAT STRIP_TAC THEN REWRITE_TAC[sums] THEN
AP_THM_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN ABS_TAC THEN
MATCH_MP_TAC
VSUM_EQ THEN ASM_SIMP_TAC[
IN_INTER]);;
let SUMS_EQ = prove
(`!f g k. (!x. x
IN k ==> f x = g x) /\ (f sums l) k ==> (g sums l) k`,
let SUMS_0 = prove
(`!f:num->real^N s. (!n. n
IN s ==> f n = vec 0) ==> (f sums vec 0) s`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC
SUMS_EQ THEN
EXISTS_TAC `\n:num. vec 0:real^N` THEN ASM_SIMP_TAC[
SERIES_0]);;
let SERIES_COMPONENT = prove
(`!f s l:real^N k. (f sums l) s /\ 1 <= k /\ k <= dimindex(:N)
==> ((\i. lift(f(i)$k)) sums lift(l$k)) s`,
let SERIES_DIFFS = prove
(`!f:num->real^N k.
(f --> vec 0) sequentially
==> ((\n. f(n) - f(n + 1)) sums f(k)) (from k)`,
let SERIES_RESTRICT = prove
(`!f k l:real^N.
((\n. if n
IN k then f(n) else vec 0) sums l) (:num) <=>
(f sums l) k`,
REPEAT GEN_TAC THEN REWRITE_TAC[sums] THEN
AP_THM_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN
REWRITE_TAC[
FUN_EQ_THM;
INTER_UNIV] THEN GEN_TAC THEN
MATCH_MP_TAC(MESON[] `vsum s f = vsum t f /\ vsum t f = vsum t g
==> vsum s f = vsum t g`) THEN
CONJ_TAC THENL
[MATCH_MP_TAC
VSUM_SUPERSET THEN SET_TAC[];
MATCH_MP_TAC
VSUM_EQ THEN SIMP_TAC[
IN_INTER]]);;
let SUMS_REINDEX = prove
(`!k a l n. ((\x. a(x + k)) sums l) (from n) <=> (a sums l) (from(n + k))`,
REPEAT GEN_TAC THEN REWRITE_TAC[sums;
FROM_INTER_NUMSEG] THEN
REPEAT GEN_TAC THEN REWRITE_TAC[GSYM
VSUM_OFFSET] THEN
REWRITE_TAC[
LIM_SEQUENTIALLY] THEN
ASM_MESON_TAC[ARITH_RULE `N + k:num <= n ==> n = (n - k) + k /\ N <= n - k`;
ARITH_RULE `N + k:num <= n ==> N <= n + k`]);;
let SUMMABLE_EQ = prove
(`!f g k. (!x. x
IN k ==> f x = g x) /\ summable k f ==> summable k g`,
REWRITE_TAC[summable] THEN MESON_TAC[
SUMS_EQ]);;
let SUMMABLE_COMPONENT = prove
(`!f:num->real^N s k.
summable s f /\ 1 <= k /\ k <= dimindex(:N)
==> summable s (\i. lift(f(i)$k))`,
REPEAT STRIP_TAC THEN
FIRST_X_ASSUM(X_CHOOSE_TAC `l:real^N` o REWRITE_RULE[summable]) THEN
REWRITE_TAC[summable] THEN EXISTS_TAC `lift((l:real^N)$k)` THEN
ASM_SIMP_TAC[
SERIES_COMPONENT]);;
let SERIES_SUBSET = prove
(`!x s t l.
s
SUBSET t /\
((\i. if i
IN s then x i else vec 0) sums l) t
==> (x sums l) s`,
REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
REWRITE_TAC[sums] THEN MATCH_MP_TAC EQ_IMP THEN
AP_THM_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN ABS_TAC THEN
ASM_SIMP_TAC[GSYM
VSUM_RESTRICT_SET;
FINITE_INTER_NUMSEG] THEN
AP_THM_TAC THEN AP_TERM_TAC THEN POP_ASSUM MP_TAC THEN SET_TAC[]);;
let SUMS_FINITE_DIFF = prove
(`!f:num->real^N t s l.
t
SUBSET s /\
FINITE t /\ (f sums l) s
==> (f sums (l - vsum t f)) (s
DIFF t)`,
REPEAT GEN_TAC THEN
REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
FIRST_ASSUM(MP_TAC o ISPEC `f:num->real^N` o MATCH_MP
SERIES_FINITE) THEN
ONCE_REWRITE_TAC[GSYM
SERIES_RESTRICT] THEN
REWRITE_TAC[IMP_IMP] THEN ONCE_REWRITE_TAC[
CONJ_SYM] THEN
DISCH_THEN(MP_TAC o MATCH_MP
SERIES_SUB) THEN
MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN
REWRITE_TAC[
FUN_EQ_THM] THEN X_GEN_TAC `x:num` THEN REWRITE_TAC[
IN_DIFF] THEN
FIRST_ASSUM(MP_TAC o SPEC `x:num` o GEN_REWRITE_RULE I [
SUBSET]) THEN
MAP_EVERY ASM_CASES_TAC [`(x:num)
IN s`; `(x:num)
IN t`] THEN
ASM_REWRITE_TAC[] THEN VECTOR_ARITH_TAC);;
let SUMS_FINITE_UNION = prove
(`!f:num->real^N s t l.
FINITE t /\ (f sums l) s
==> (f sums (l + vsum (t
DIFF s) f)) (s
UNION t)`,
REPEAT GEN_TAC THEN
REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
FIRST_ASSUM(MP_TAC o SPEC `s:num->bool` o MATCH_MP
FINITE_DIFF) THEN
DISCH_THEN(MP_TAC o ISPEC `f:num->real^N` o MATCH_MP
SERIES_FINITE) THEN
ONCE_REWRITE_TAC[GSYM
SERIES_RESTRICT] THEN
REWRITE_TAC[IMP_IMP] THEN ONCE_REWRITE_TAC[
CONJ_SYM] THEN
DISCH_THEN(MP_TAC o MATCH_MP
SERIES_ADD) THEN
MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN
REWRITE_TAC[
FUN_EQ_THM] THEN X_GEN_TAC `x:num` THEN
REWRITE_TAC[
IN_DIFF;
IN_UNION] THEN
MAP_EVERY ASM_CASES_TAC [`(x:num)
IN s`; `(x:num)
IN t`] THEN
ASM_REWRITE_TAC[] THEN VECTOR_ARITH_TAC);;
let SUMS_OFFSET = prove
(`!f:num->real^N l m n.
(f sums l) (from m) /\ m < n
==> (f sums (l - vsum(m..(n-1)) f)) (from n)`,
let SUMS_OFFSET_REV = prove
(`!f:num->real^N l m n.
(f sums l) (from m) /\ n < m
==> (f sums (l + vsum(n..m-1) f)) (from n)`,
let SERIES_DROP_LE = prove
(`!f g s a b.
(f sums a) s /\ (g sums b) s /\
(!x. x
IN s ==> drop(f x) <= drop(g x))
==> drop a <= drop b`,
let SERIES_DROP_POS = prove
(`!f s a.
(f sums a) s /\ (!x. x
IN s ==> &0 <= drop(f x))
==> &0 <= drop a`,
REPEAT STRIP_TAC THEN
MP_TAC(ISPECL [`(\n. vec 0):num->real^1`; `f:num->real^1`; `s:num->bool`;
`vec 0:real^1`; `a:real^1`]
SERIES_DROP_LE) THEN
ASM_SIMP_TAC[
SUMS_0;
DROP_VEC]);;
let SERIES_BOUND = prove
(`!f:num->real^N g s a b.
(f sums a) s /\ ((lift o g) sums (lift b)) s /\
(!i. i
IN s ==> norm(f i) <= g i)
==> norm(a) <= b`,
let INFSUM_ADD = prove
(`!x y s. summable s x /\ summable s y
==> infsum s (\i. x i + y i) = infsum s x + infsum s y`,
let INFSUM_SUB = prove
(`!x y s. summable s x /\ summable s y
==> infsum s (\i. x i - y i) = infsum s x - infsum s y`,
let INFSUM_EQ = prove
(`!f g k. summable k f /\ summable k g /\ (!x. x
IN k ==> f x = g x)
==> infsum k f = infsum k g`,
REPEAT STRIP_TAC THEN REWRITE_TAC[infsum] THEN
AP_TERM_TAC THEN ABS_TAC THEN ASM_MESON_TAC[
SUMS_EQ;
SUMS_INFSUM]);;
let SEQUENCE_CAUCHY_WLOG = prove
(`!P s. (!m n:num. P m /\ P n ==> dist(s m,s n) < e) <=>
(!m n. P m /\ P n /\ m <= n ==> dist(s m,s n) < e)`,
let SERIES_CAUCHY = prove
(`!f s. (?l. (f sums l) s) =
!e. &0 < e
==> ?N. !m n. m >= N
==> norm(vsum(s
INTER (m..n)) f) < e`,
REPEAT GEN_TAC THEN REWRITE_TAC[sums;
CONVERGENT_EQ_CAUCHY; cauchy] THEN
REWRITE_TAC[
SEQUENCE_CAUCHY_WLOG] THEN ONCE_REWRITE_TAC[
DIST_SYM] THEN
SIMP_TAC[dist;
VSUM_DIFF_LEMMA;
NORM_VSUM_TRIVIAL_LEMMA] THEN
REWRITE_TAC[
GE; TAUT `a ==> b \/ c <=> a /\ ~b ==> c`] THEN
REWRITE_TAC[
NOT_LT; ARITH_RULE
`(N <= m /\ N <= n /\ m <= n) /\ m + 1 <= n <=>
N + 1 <= m + 1 /\ m + 1 <= n`] THEN
AP_TERM_TAC THEN REWRITE_TAC[
FUN_EQ_THM] THEN X_GEN_TAC `e:real` THEN
ASM_CASES_TAC `&0 < e` THEN ASM_REWRITE_TAC[] THEN
EQ_TAC THEN DISCH_THEN(X_CHOOSE_TAC `N:num`) THENL
[EXISTS_TAC `N + 1`; EXISTS_TAC `N:num`] THEN
REPEAT STRIP_TAC THEN
ASM_SIMP_TAC[ARITH_RULE `N + 1 <= m + 1 ==> N <= m + 1`] THEN
FIRST_X_ASSUM(MP_TAC o SPECL [`m - 1`; `n:num`]) THEN
SUBGOAL_THEN `m - 1 + 1 = m` SUBST_ALL_TAC THENL
[ALL_TAC; ANTS_TAC THEN SIMP_TAC[]] THEN
ASM_ARITH_TAC);;
let SUMMABLE_IFF_EVENTUALLY = prove
(`!f g k. (?N. !n. N <= n /\ n
IN k ==> f n = g n)
==> (summable k f <=> summable k g)`,
REWRITE_TAC[summable;
SERIES_CAUCHY] THEN REPEAT GEN_TAC THEN
DISCH_THEN(X_CHOOSE_THEN `N0:num` STRIP_ASSUME_TAC) THEN
AP_TERM_TAC THEN REWRITE_TAC[
FUN_EQ_THM] THEN X_GEN_TAC `e:real` THEN
AP_TERM_TAC THEN EQ_TAC THEN
DISCH_THEN(X_CHOOSE_THEN `N1:num`
(fun th -> EXISTS_TAC `N0 + N1:num` THEN MP_TAC th)) THEN
REPEAT(MATCH_MP_TAC
MONO_FORALL THEN GEN_TAC) THEN
DISCH_THEN(fun th -> DISCH_TAC THEN MP_TAC th) THEN
(ANTS_TAC THENL [ASM_ARITH_TAC; ALL_TAC]) THEN
MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN AP_TERM_TAC THEN
MATCH_MP_TAC
VSUM_EQ THEN ASM_SIMP_TAC[
IN_INTER;
IN_NUMSEG] THEN
REPEAT STRIP_TAC THENL [ALL_TAC; CONV_TAC SYM_CONV] THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN
ASM_ARITH_TAC);;
let SERIES_CAUCHY_UNIFORM = prove
(`!P f:A->num->real^N k.
(?l. !e. &0 < e
==> ?N. !n x. N <= n /\ P x
==> dist(vsum(k
INTER (0..n)) (f x),
l x) < e) <=>
(!e. &0 < e ==> ?N. !m n x. N <= m /\ P x
==> norm(vsum(k
INTER (m..n)) (f x)) < e)`,
REPEAT GEN_TAC THEN
REWRITE_TAC[sums;
UNIFORMLY_CONVERGENT_EQ_CAUCHY; cauchy] THEN
ONCE_REWRITE_TAC[MESON[]
`(!m n:num y. N <= m /\ N <= n /\ P y ==> Q m n y) <=>
(!y. P y ==> !m n. N <= m /\ N <= n ==> Q m n y)`] THEN
REWRITE_TAC[
SEQUENCE_CAUCHY_WLOG] THEN ONCE_REWRITE_TAC[
DIST_SYM] THEN
SIMP_TAC[dist;
VSUM_DIFF_LEMMA;
NORM_VSUM_TRIVIAL_LEMMA] THEN
REWRITE_TAC[
GE; TAUT `a ==> b \/ c <=> a /\ ~b ==> c`] THEN
REWRITE_TAC[
NOT_LT; ARITH_RULE
`(N <= m /\ N <= n /\ m <= n) /\ m + 1 <= n <=>
N + 1 <= m + 1 /\ m + 1 <= n`] THEN
AP_TERM_TAC THEN REWRITE_TAC[
FUN_EQ_THM] THEN X_GEN_TAC `e:real` THEN
ASM_CASES_TAC `&0 < e` THEN ASM_REWRITE_TAC[] THEN
EQ_TAC THEN DISCH_THEN(X_CHOOSE_TAC `N:num`) THENL
[EXISTS_TAC `N + 1`; EXISTS_TAC `N:num`] THEN
REPEAT STRIP_TAC THEN
ASM_SIMP_TAC[ARITH_RULE `N + 1 <= m + 1 ==> N <= m + 1`] THEN
FIRST_X_ASSUM(MP_TAC o SPEC `x:A`) THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN(MP_TAC o SPECL [`m - 1`; `n:num`]) THEN
SUBGOAL_THEN `m - 1 + 1 = m` SUBST_ALL_TAC THENL
[ALL_TAC; ANTS_TAC THEN SIMP_TAC[]] THEN
ASM_ARITH_TAC);;
let SERIES_GOESTOZERO = prove
(`!s x. summable s x
==> !e. &0 < e
==> eventually (\n. n
IN s ==> norm(x n) < e) sequentially`,
REPEAT GEN_TAC THEN REWRITE_TAC[summable;
SERIES_CAUCHY] THEN
MATCH_MP_TAC
MONO_FORALL THEN X_GEN_TAC `e:real` THEN
MATCH_MP_TAC MONO_IMP THEN REWRITE_TAC[
EVENTUALLY_SEQUENTIALLY] THEN
MATCH_MP_TAC
MONO_EXISTS THEN X_GEN_TAC `N:num` THEN DISCH_TAC THEN
X_GEN_TAC `n:num` THEN REPEAT STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPECL [`n:num`; `n:num`]) THEN
ASM_SIMP_TAC[
NUMSEG_SING;
GE; SET_RULE `n
IN s ==> s
INTER {n} = {n}`] THEN
REWRITE_TAC[
VSUM_SING]);;
let SUMMABLE_IMP_TOZERO = prove
(`!f:num->real^N k.
summable k f
==> ((\n. if n
IN k then f(n) else vec 0) --> vec 0) sequentially`,
REPEAT GEN_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM
SUMMABLE_RESTRICT] THEN
REWRITE_TAC[summable;
LIM_SEQUENTIALLY;
INTER_UNIV; sums] THEN
DISCH_THEN(X_CHOOSE_TAC `l:real^N`) THEN X_GEN_TAC `e:real` THEN
DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `e / &2`) THEN
ASM_REWRITE_TAC[
REAL_HALF;
LEFT_IMP_EXISTS_THM] THEN
X_GEN_TAC `N:num` THEN DISCH_TAC THEN EXISTS_TAC `N + 1` THEN
X_GEN_TAC `n:num` THEN DISCH_TAC THEN
FIRST_X_ASSUM(fun th ->
MP_TAC(SPEC `n - 1` th) THEN MP_TAC(SPEC `n:num` th)) THEN
ASM_SIMP_TAC[ARITH_RULE `N + 1 <= n ==> N <= n /\ N <= n - 1`] THEN
ABBREV_TAC `m = n - 1` THEN
SUBGOAL_THEN `n = SUC m` SUBST1_TAC THENL
[ASM_ARITH_TAC; ALL_TAC] THEN
REWRITE_TAC[
VSUM_CLAUSES_NUMSEG;
LE_0] THEN
REWRITE_TAC[NORM_ARITH `dist(x,vec 0) = norm x`] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[
NORM_0] THEN CONV_TAC NORM_ARITH);;
let SERIES_COMPARISON = prove
(`!f g s. (?l. ((lift o g) sums l) s) /\
(?N. !n. n >= N /\ n
IN s ==> norm(f n) <= g n)
==> ?l:real^N. (f sums l) s`,
REPEAT GEN_TAC THEN REWRITE_TAC[
SERIES_CAUCHY] THEN
DISCH_THEN(CONJUNCTS_THEN2 MP_TAC (X_CHOOSE_TAC `N1:num`)) 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 `N2:num`) THEN
EXISTS_TAC `N1 + N2:num` THEN
MAP_EVERY X_GEN_TAC [`m:num`; `n:num`] THEN DISCH_TAC THEN
MATCH_MP_TAC
REAL_LET_TRANS THEN
EXISTS_TAC `norm (vsum (s
INTER (m .. n)) (lift o g))` THEN CONJ_TAC THENL
[SIMP_TAC[GSYM
LIFT_SUM;
FINITE_INTER_NUMSEG;
NORM_LIFT] THEN
MATCH_MP_TAC(REAL_ARITH `x <= a ==> x <= abs(a)`) THEN
MATCH_MP_TAC
VSUM_NORM_LE THEN
REWRITE_TAC[
FINITE_INTER_NUMSEG;
IN_INTER;
IN_NUMSEG] THEN
ASM_MESON_TAC[ARITH_RULE `m >= N1 + N2:num /\ m <= x ==> x >= N1`];
ASM_MESON_TAC[ARITH_RULE `m >= N1 + N2:num ==> m >= N2`]]);;
let SERIES_COMPARISON_BOUND = prove
(`!f:num->real^N g s a.
(g sums a) s /\ (!i. i
IN s ==> norm(f i) <= drop(g i))
==> ?l. (f sums l) s /\ norm(l) <= drop a`,
REPEAT STRIP_TAC THEN
MP_TAC(ISPECL [`f:num->real^N`; `drop o (g:num->real^1)`; `s:num->bool`]
SUMMABLE_COMPARISON) THEN
REWRITE_TAC[
o_DEF;
LIFT_DROP;
GE; ETA_AX; summable] THEN
ANTS_TAC THENL [ASM_MESON_TAC[]; MATCH_MP_TAC
MONO_EXISTS] THEN
X_GEN_TAC `l:real^N` THEN DISCH_TAC THEN ASM_REWRITE_TAC[] THEN
RULE_ASSUM_TAC(REWRITE_RULE[
FROM_0;
INTER_UNIV; sums]) THEN
MATCH_MP_TAC
SERIES_BOUND THEN MAP_EVERY EXISTS_TAC
[`f:num->real^N`; `drop o (g:num->real^1)`; `s:num->bool`] THEN
ASM_REWRITE_TAC[sums;
o_DEF;
LIFT_DROP; ETA_AX]);;
let SERIES_COMPARISON_UNIFORM = prove
(`!f g P s. (?l. ((lift o g) sums l) s) /\
(?N. !n x. N <= n /\ n
IN s /\ P x ==> norm(f x n) <= g n)
==> ?l:A->real^N.
!e. &0 < e
==> ?N. !n x. N <= n /\ P x
==> dist(vsum(s
INTER (0..n)) (f x),
l x) < e`,
REPEAT GEN_TAC THEN SIMP_TAC[
GE;
SERIES_CAUCHY;
SERIES_CAUCHY_UNIFORM] THEN
DISCH_THEN(CONJUNCTS_THEN2 MP_TAC (X_CHOOSE_TAC `N1:num`)) 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 `N2:num`) THEN
EXISTS_TAC `N1 + N2:num` THEN
MAP_EVERY X_GEN_TAC [`m:num`; `n:num`; `x:A`] THEN DISCH_TAC THEN
MATCH_MP_TAC
REAL_LET_TRANS THEN
EXISTS_TAC `norm (vsum (s
INTER (m .. n)) (lift o g))` THEN CONJ_TAC THENL
[SIMP_TAC[GSYM
LIFT_SUM;
FINITE_INTER_NUMSEG;
NORM_LIFT] THEN
MATCH_MP_TAC(REAL_ARITH `x <= a ==> x <= abs(a)`) THEN
MATCH_MP_TAC
VSUM_NORM_LE THEN
REWRITE_TAC[
FINITE_INTER_NUMSEG;
IN_INTER;
IN_NUMSEG] THEN
ASM_MESON_TAC[ARITH_RULE `N1 + N2:num <= m /\ m <= x ==> N1 <= x`];
ASM_MESON_TAC[ARITH_RULE `N1 + N2:num <= m ==> N2 <= m`]]);;
let SERIES_RATIO = prove
(`!c a s N.
c < &1 /\
(!n. n >= N ==> norm(a(SUC n)) <= c * norm(a(n)))
==> ?l:real^N. (a sums l) s`,
let BOUNDED_PARTIAL_SUMS = prove
(`!f:num->real^N k.
bounded { vsum(k..n) f | n
IN (:num) }
==> bounded { vsum(m..n) f | m
IN (:num) /\ n
IN (:num) }`,
REPEAT STRIP_TAC THEN
SUBGOAL_THEN `bounded { vsum(0..n) f:real^N | n
IN (:num) }` MP_TAC THENL
[FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
BOUNDED_POS]) THEN
REWRITE_TAC[bounded] THEN
REWRITE_TAC[
SIMPLE_IMAGE;
FORALL_IN_IMAGE;
IN_UNIV] THEN
DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `sum { i:num | i < k} (\i. norm(f i:real^N)) + B` THEN
X_GEN_TAC `i:num` THEN ASM_CASES_TAC `i:num < k` THENL
[MATCH_MP_TAC(REAL_ARITH
`!y. x <= y /\ y <= a /\ &0 < b ==> x <= a + b`) THEN
EXISTS_TAC `sum (0..i) (\i. norm(f i:real^N))` THEN
ASM_SIMP_TAC[
VSUM_NORM;
FINITE_NUMSEG] THEN
MATCH_MP_TAC
SUM_SUBSET THEN
REWRITE_TAC[
FINITE_NUMSEG;
FINITE_NUMSEG_LT;
NORM_POS_LE] THEN
REWRITE_TAC[
IN_DIFF;
IN_NUMSEG;
IN_ELIM_THM] THEN ASM_ARITH_TAC;
ALL_TAC] THEN
ASM_CASES_TAC `k = 0` THENL
[FIRST_X_ASSUM SUBST_ALL_TAC THEN MATCH_MP_TAC(REAL_ARITH
`x <= B /\ &0 <= b ==> x <= b + B`) THEN
ASM_SIMP_TAC[
SUM_POS_LE;
FINITE_NUMSEG_LT;
NORM_POS_LE];
ALL_TAC] THEN
MP_TAC(ISPECL [`f:num->real^N`; `0`; `k:num`; `i:num`]
VSUM_COMBINE_L) THEN
ANTS_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN
DISCH_THEN(SUBST1_TAC o SYM) THEN ASM_REWRITE_TAC[
NUMSEG_LT] THEN
MATCH_MP_TAC(NORM_ARITH
`norm(x) <= a /\ norm(y) <= b ==> norm(x + y) <= a + b`) THEN
ASM_SIMP_TAC[
VSUM_NORM;
FINITE_NUMSEG];
ALL_TAC] THEN
DISCH_THEN(fun th ->
MP_TAC(MATCH_MP
BOUNDED_DIFFS (W CONJ th)) THEN MP_TAC th) THEN
REWRITE_TAC[IMP_IMP; GSYM
BOUNDED_UNION] THEN
MATCH_MP_TAC(REWRITE_RULE[TAUT `a /\ b ==> c <=> b ==> a ==> c`]
BOUNDED_SUBSET) THEN
REWRITE_TAC[
SUBSET;
IN_ELIM_THM;
IN_UNION;
LEFT_IMP_EXISTS_THM;
IN_UNIV] THEN
MAP_EVERY X_GEN_TAC [`x:real^N`; `m:num`; `n:num`] THEN
DISCH_THEN SUBST1_TAC THEN
ASM_CASES_TAC `m = 0` THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
ASM_CASES_TAC `n:num < m` THENL
[DISJ2_TAC THEN REPEAT(EXISTS_TAC `vsum(0..0) (f:num->real^N)`) THEN
ASM_SIMP_TAC[
VSUM_TRIV_NUMSEG;
VECTOR_SUB_REFL] THEN MESON_TAC[];
ALL_TAC] THEN
DISJ2_TAC THEN MAP_EVERY EXISTS_TAC
[`vsum(0..n) (f:num->real^N)`; `vsum(0..(m-1)) (f:num->real^N)`] THEN
CONJ_TAC THENL [MESON_TAC[]; ALL_TAC] THEN
MP_TAC(ISPECL [`f:num->real^N`; `0`; `m:num`; `n:num`]
VSUM_COMBINE_L) THEN
ANTS_TAC THENL [ASM_ARITH_TAC; VECTOR_ARITH_TAC]);;
let SUMMABLE_BILINEAR_PARTIAL_PRE = prove
(`!f g h:real^M->real^N->real^P l k.
bilinear h /\
((\n. h (f(n + 1)) (g(n))) --> l) sequentially /\
summable (from k) (\n. h (f(n + 1) - f(n)) (g(n)))
==> summable (from k) (\n. h (f n) (g(n) - g(n - 1)))`,
REPEAT GEN_TAC THEN
REWRITE_TAC[summable; sums;
FROM_INTER_NUMSEG] THEN
REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
FIRST_ASSUM(fun th ->
REWRITE_TAC[MATCH_MP
BILINEAR_VSUM_PARTIAL_PRE th]) THEN
DISCH_THEN(X_CHOOSE_TAC `l':real^P`) THEN
EXISTS_TAC `l - (h:real^M->real^N->real^P) (f k) (g(k - 1)) - l'` THEN
REWRITE_TAC[
LIM_CASES_SEQUENTIALLY] THEN
REPEAT(MATCH_MP_TAC
LIM_SUB THEN ASM_REWRITE_TAC[
LIM_CONST]));;
let SERIES_DIRICHLET_BILINEAR = prove
(`!f g h:real^M->real^N->real^P k m p l.
bilinear h /\
bounded { vsum (m..n) f | n
IN (:num)} /\
summable (from p) (\n. lift(norm(g(n + 1) - g(n)))) /\
((\n. h (g(n + 1)) (vsum(1..n) f)) --> l) sequentially
==> summable (from k) (\n. h (g n) (f n))`,
let SERIES_DIRICHLET = prove
(`!f:num->real^N g N k m.
bounded { vsum (m..n) f | n
IN (:num)} /\
(!n. N <= n ==> g(n + 1) <= g(n)) /\
((lift o g) --> vec 0) sequentially
==> summable (from k) (\n. g(n) % f(n))`,
let SERIES_REARRANGE_EQ = prove
(`!x:num->real^N s p l.
summable s (\n. lift(norm(x n))) /\ p permutes s
==> (((x o p) sums l) s <=> (x sums l) s)`,
let SERIES_REARRANGE = prove
(`!x:num->real^N s p l.
summable s (\n. lift(norm(x n))) /\ p permutes s /\ (x sums l) s
==> ((x o p) sums l) s`,
let BANACH_FIX = prove
(`!f s c. complete s /\ ~(s = {}) /\
&0 <= c /\ c < &1 /\
(
IMAGE f s)
SUBSET s /\
(!x y. x
IN s /\ y
IN s ==> dist(f(x),f(y)) <= c * dist(x,y))
==> ?!x:real^N. x
IN s /\ (f x = x)`,
REPEAT STRIP_TAC THEN REWRITE_TAC[
EXISTS_UNIQUE_THM] THEN CONJ_TAC THENL
[ALL_TAC;
MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN STRIP_TAC THEN
SUBGOAL_THEN `dist((f:real^N->real^N) x,f y) <= c * dist(x,y)` MP_TAC THENL
[ASM_MESON_TAC[]; ALL_TAC] THEN
ASM_REWRITE_TAC[REAL_ARITH `a <= c * a <=> &0 <= --a * (&1 - c)`] THEN
ASM_SIMP_TAC[GSYM
REAL_LE_LDIV_EQ;
REAL_SUB_LT;
real_div] THEN
REWRITE_TAC[
REAL_MUL_LZERO; REAL_ARITH `&0 <= --x <=> ~(&0 < x)`] THEN
MESON_TAC[
DIST_POS_LT]] THEN
STRIP_ASSUME_TAC(prove_recursive_functions_exist num_RECURSION
`(z 0 = @x:real^N. x
IN s) /\ (!n. z(SUC n) = f(z n))`) THEN
SUBGOAL_THEN `!n. (z:num->real^N) n
IN s` ASSUME_TAC THENL
[INDUCT_TAC THEN ASM_REWRITE_TAC[] THEN
ASM_MESON_TAC[
MEMBER_NOT_EMPTY;
SUBSET;
IN_IMAGE];
ALL_TAC] THEN
UNDISCH_THEN `z 0 = @x:real^N. x
IN s` (K ALL_TAC) THEN
SUBGOAL_THEN `?x:real^N. x
IN s /\ (z --> x) sequentially` MP_TAC THENL
[ALL_TAC;
MATCH_MP_TAC
MONO_EXISTS THEN X_GEN_TAC `a:real^N` THEN
STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
ABBREV_TAC `e = dist(f(a:real^N),a)` THEN
SUBGOAL_THEN `~(&0 < e)` (fun th -> ASM_MESON_TAC[th;
DIST_POS_LT]) THEN
DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
LIM_SEQUENTIALLY]) THEN
DISCH_THEN(MP_TAC o SPEC `e / &2`) THEN
ASM_REWRITE_TAC[
REAL_HALF] THEN DISCH_THEN(X_CHOOSE_TAC `N:num`) THEN
SUBGOAL_THEN
`dist(f(z N),a:real^N) < e / &2 /\ dist(f(z(N:num)),f(a)) < e / &2`
(fun th -> ASM_MESON_TAC[th;
DIST_TRIANGLE_HALF_R;
REAL_LT_REFL]) THEN
CONJ_TAC THENL [ASM_MESON_TAC[ARITH_RULE `N <= SUC N`]; ALL_TAC] THEN
MATCH_MP_TAC
REAL_LET_TRANS THEN
EXISTS_TAC `c * dist((z:num->real^N) N,a)` THEN ASM_SIMP_TAC[] THEN
MATCH_MP_TAC(REAL_ARITH `x < y /\ c * x <= &1 * x ==> c * x < y`) THEN
ASM_SIMP_TAC[
LE_REFL;
REAL_LE_RMUL;
DIST_POS_LE;
REAL_LT_IMP_LE]] THEN
FIRST_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [complete]) THEN
ASM_REWRITE_TAC[
CAUCHY] THEN
SUBGOAL_THEN `!n. dist(z(n):real^N,z(SUC n)) <= c pow n * dist(z(0),z(1))`
ASSUME_TAC THENL
[INDUCT_TAC THEN
REWRITE_TAC[
real_pow; ARITH; REAL_MUL_LID;
REAL_LE_REFL] THEN
MATCH_MP_TAC
REAL_LE_TRANS THEN
EXISTS_TAC `c * dist(z(n):real^N,z(SUC n))` THEN
CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
REWRITE_TAC[GSYM REAL_MUL_ASSOC] THEN ASM_SIMP_TAC[
REAL_LE_LMUL];
ALL_TAC] THEN
SUBGOAL_THEN
`!m n:num. (&1 - c) * dist(z(m):real^N,z(m+n))
<= c pow m * dist(z(0),z(1)) * (&1 - c pow n)`
ASSUME_TAC THENL
[GEN_TAC THEN INDUCT_TAC THENL
[REWRITE_TAC[
ADD_CLAUSES;
DIST_REFL;
REAL_MUL_RZERO] THEN
MATCH_MP_TAC
REAL_LE_MUL THEN
ASM_SIMP_TAC[
REAL_LE_MUL;
REAL_POW_LE;
DIST_POS_LE;
REAL_SUB_LE;
REAL_POW_1_LE;
REAL_LT_IMP_LE];
ALL_TAC] THEN
MATCH_MP_TAC
REAL_LE_TRANS THEN EXISTS_TAC
`(&1 - c) * (dist(z m:real^N,z(m + n)) + dist(z(m + n),z(m + SUC n)))` THEN
ASM_SIMP_TAC[
REAL_LE_LMUL;
REAL_SUB_LE;
REAL_LT_IMP_LE;
DIST_TRIANGLE] THEN
FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH
`c * x <= y ==> c * x' + y <= y' ==> c * (x + x') <= y'`)) THEN
REWRITE_TAC[REAL_ARITH
`q + a * b * (&1 - x) <= a * b * (&1 - y) <=> q <= a * b * (x - y)`] THEN
REWRITE_TAC[
ADD_CLAUSES;
real_pow] THEN
REWRITE_TAC[REAL_ARITH `a * b * (d - c * d) = (&1 - c) * a * d * b`] THEN
MATCH_MP_TAC
REAL_LE_LMUL THEN
ASM_SIMP_TAC[
REAL_SUB_LE;
REAL_LT_IMP_LE] THEN
REWRITE_TAC[GSYM
REAL_POW_ADD; REAL_MUL_ASSOC] THEN ASM_MESON_TAC[];
ALL_TAC] THEN
X_GEN_TAC `e:real` THEN DISCH_TAC THEN
ASM_CASES_TAC `(z:num->real^N) 0 = z 1` THENL
[FIRST_X_ASSUM SUBST_ALL_TAC THEN EXISTS_TAC `0` THEN
REWRITE_TAC[
GE;
LE_0] THEN X_GEN_TAC `n:num` THEN
FIRST_X_ASSUM(MP_TAC o SPECL [`0`; `n:num`]) THEN
REWRITE_TAC[
ADD_CLAUSES;
DIST_REFL;
REAL_MUL_LZERO;
REAL_MUL_RZERO] THEN
ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN
ASM_CASES_TAC `(z:num->real^N) 0 = z n` THEN
ASM_REWRITE_TAC[
DIST_REFL;
REAL_NOT_LE] THEN
ASM_SIMP_TAC[
REAL_LT_MUL;
DIST_POS_LT;
REAL_SUB_LT];
ALL_TAC] THEN
MP_TAC(SPECL [`c:real`; `e * (&1 - c) / dist((z:num->real^N) 0,z 1)`]
REAL_ARCH_POW_INV) THEN
ASM_SIMP_TAC[
REAL_LT_MUL;
REAL_LT_DIV;
REAL_SUB_LT;
DIST_POS_LT] THEN
MATCH_MP_TAC
MONO_EXISTS THEN X_GEN_TAC `N:num` THEN
REWRITE_TAC[
real_div;
GE; REAL_MUL_ASSOC] THEN
ASM_SIMP_TAC[
REAL_LT_RDIV_EQ; GSYM
real_div;
DIST_POS_LT] THEN
ASM_SIMP_TAC[GSYM
REAL_LT_LDIV_EQ;
REAL_SUB_LT] THEN DISCH_TAC THEN
REWRITE_TAC[
LE_EXISTS;
LEFT_IMP_EXISTS_THM] THEN
GEN_TAC THEN X_GEN_TAC `d:num` THEN DISCH_THEN SUBST_ALL_TAC THEN
ONCE_REWRITE_TAC[
DIST_SYM] THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP(REAL_ARITH
`d < e ==> x <= d ==> x < e`)) THEN
ASM_SIMP_TAC[
REAL_LE_RDIV_EQ;
REAL_SUB_LT] THEN
FIRST_X_ASSUM(MP_TAC o SPECL [`N:num`; `d:num`]) THEN
MATCH_MP_TAC(REAL_ARITH
`(c * d) * e <= (c * d) * &1 ==> x * y <= c * d * e ==> y * x <= c * d`) THEN
MATCH_MP_TAC
REAL_LE_LMUL THEN
ASM_SIMP_TAC[
REAL_LE_MUL;
REAL_POW_LE;
DIST_POS_LE; REAL_ARITH
`&0 <= x ==> &1 - x <= &1`]);;
let EDELSTEIN_FIX = prove
(`!f s. compact s /\ ~(s = {}) /\ (
IMAGE f s)
SUBSET s /\
(!x y. x
IN s /\ y
IN s /\ ~(x = y) ==> dist(f(x),f(y)) < dist(x,y))
==> ?!x:real^N. x
IN s /\ f x = x`,
MAP_EVERY X_GEN_TAC [`g:real^N->real^N`; `s:real^N->bool`] THEN
REPEAT STRIP_TAC THEN REWRITE_TAC[
EXISTS_UNIQUE_THM] THEN CONJ_TAC THENL
[ALL_TAC; ASM_MESON_TAC[
REAL_LT_REFL]] THEN
SUBGOAL_THEN
`!x y. x
IN s /\ y
IN s ==> dist((g:real^N->real^N)(x),g(y)) <= dist(x,y)`
ASSUME_TAC THENL
[REPEAT STRIP_TAC THEN ASM_CASES_TAC `x:real^N = y` THEN
ASM_SIMP_TAC[
DIST_REFL;
REAL_LE_LT];
ALL_TAC] THEN
ASM_CASES_TAC `?x:real^N. x
IN s /\ ~(g x = x)` THENL
[ALL_TAC; ASM SET_TAC[]] THEN
FIRST_X_ASSUM(X_CHOOSE_THEN `x:real^N` STRIP_ASSUME_TAC) THEN
ABBREV_TAC `y = (g:real^N->real^N) x` THEN
SUBGOAL_THEN `(y:real^N)
IN s` ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
FIRST_ASSUM(MP_TAC o MATCH_MP
COMPACT_PCROSS o W CONJ) THEN
REWRITE_TAC[compact;
PCROSS] THEN
(STRIP_ASSUME_TAC o prove_general_recursive_function_exists)
`?f:num->real^N->real^N.
(!z. f 0 z = z) /\ (!z n. f (SUC n) z = g(f n z))` THEN
SUBGOAL_THEN `!n z. z
IN s ==> (f:num->real^N->real^N) n z
IN s`
STRIP_ASSUME_TAC THENL [INDUCT_TAC THEN ASM SET_TAC[]; ALL_TAC] THEN
SUBGOAL_THEN
`!m n w z. m <= n /\ w
IN s /\ z
IN s
==> dist((f:num->real^N->real^N) n w,f n z) <= dist(f m w,f m z)`
ASSUME_TAC THENL
[REWRITE_TAC[
RIGHT_FORALL_IMP_THM;
IMP_CONJ] THEN
MATCH_MP_TAC
TRANSITIVE_STEPWISE_LE THEN
RULE_ASSUM_TAC(REWRITE_RULE[
SUBSET;
FORALL_IN_IMAGE]) THEN
ASM_SIMP_TAC[
REAL_LE_REFL] THEN MESON_TAC[
REAL_LE_TRANS];
ALL_TAC] THEN
DISCH_THEN(MP_TAC o SPEC
`\n:num. pastecart (f n (x:real^N)) (f n y:real^N)`) THEN
ANTS_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[
LEFT_IMP_EXISTS_THM]] THEN
MAP_EVERY X_GEN_TAC [`l:real^(N,N)finite_sum`; `s:num->num`] THEN
REWRITE_TAC[
o_DEF;
IN_ELIM_THM] THEN
DISCH_THEN(CONJUNCTS_THEN2 MP_TAC STRIP_ASSUME_TAC) THEN
REWRITE_TAC[
LEFT_IMP_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`] THEN
DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC SUBST_ALL_TAC) THEN
SUBGOAL_THEN
`(\x:real^(N,N)finite_sum. fstcart x)
continuous_on UNIV /\
(\x:real^(N,N)finite_sum. sndcart x)
continuous_on UNIV`
MP_TAC THENL
[CONJ_TAC THEN MATCH_MP_TAC
LINEAR_CONTINUOUS_ON THEN
REWRITE_TAC[ETA_AX;
LINEAR_FSTCART;
LINEAR_SNDCART];
ALL_TAC] THEN
REWRITE_TAC[
CONTINUOUS_ON_SEQUENTIALLY;
IN_UNIV] THEN
DISCH_THEN(CONJUNCTS_THEN(fun th -> FIRST_ASSUM(MP_TAC o MATCH_MP th))) THEN
REWRITE_TAC[
o_DEF;
FSTCART_PASTECART;
SNDCART_PASTECART; IMP_IMP] THEN
ONCE_REWRITE_TAC[
CONJ_SYM] THEN
DISCH_THEN(fun th -> CONJUNCTS_THEN2 (LABEL_TAC "A") (LABEL_TAC "B") th THEN
MP_TAC(MATCH_MP
LIM_SUB th)) THEN
REWRITE_TAC[] THEN DISCH_THEN(LABEL_TAC "AB") THEN
SUBGOAL_THEN
`!n. dist(a:real^N,b) <= dist((f:num->real^N->real^N) n x,f n y)`
STRIP_ASSUME_TAC THENL
[X_GEN_TAC `N:num` THEN REWRITE_TAC[GSYM
REAL_NOT_LT] THEN
ONCE_REWRITE_TAC[GSYM
REAL_SUB_LT] THEN DISCH_TAC THEN
USE_THEN "AB" (MP_TAC o REWRITE_RULE[
LIM_SEQUENTIALLY]) THEN
DISCH_THEN(fun th -> FIRST_X_ASSUM(MP_TAC o MATCH_MP th)) THEN
REWRITE_TAC[
NOT_EXISTS_THM] THEN X_GEN_TAC `M:num` THEN
DISCH_THEN(MP_TAC o SPEC `M + N:num`) THEN REWRITE_TAC[
LE_ADD] THEN
MATCH_MP_TAC(NORM_ARITH
`dist(fx,fy) <= dist(x,y)
==> ~(dist(fx - fy,a - b) < dist(a,b) - dist(x,y))`) THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN
FIRST_X_ASSUM(MP_TAC o SPEC `M + N:num` o MATCH_MP
MONOTONE_BIGGER) THEN
ARITH_TAC;
ALL_TAC] THEN
SUBGOAL_THEN `b:real^N = a` SUBST_ALL_TAC THENL
[MATCH_MP_TAC(TAUT `(~p ==> F) ==> p`) THEN DISCH_TAC THEN
ABBREV_TAC `e = dist(a,b) - dist((g:real^N->real^N) a,g b)` THEN
SUBGOAL_THEN `&0 < e` ASSUME_TAC THENL
[ASM_MESON_TAC[
REAL_SUB_LT]; ALL_TAC] THEN
SUBGOAL_THEN
`?n. dist((f:num->real^N->real^N) n x,a) < e / &2 /\
dist(f n y,b) < e / &2`
STRIP_ASSUME_TAC THENL
[MAP_EVERY (fun s -> USE_THEN s (MP_TAC o SPEC `e / &2` o
REWRITE_RULE[
LIM_SEQUENTIALLY])) ["A";
let DINI = prove
(`!f:num->real^N->real^1 g s.
compact s /\ (!n. (f n)
continuous_on s) /\ g
continuous_on s /\
(!x. x
IN s ==> ((\n. (f n x)) --> g x) sequentially) /\
(!n x. x
IN s ==> drop(f n x) <= drop(f (n + 1) x))
==> !e. &0 < e
==> eventually (\n. !x. x
IN s ==> norm(f n x - g x) < e)
sequentially`,
let SETDIST_EMPTY = prove
(`(!t. setdist({},t) = &0) /\ (!s. setdist(s,{}) = &0)`,
REWRITE_TAC[setdist]);;
let REAL_LE_SETDIST = prove
(`!s t:real^N->bool d.
~(s = {}) /\ ~(t = {}) /\
(!x y. x
IN s /\ y
IN t ==> d <= dist(x,y))
==> d <= setdist(s,t)`,
REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[setdist] THEN
MP_TAC(ISPEC `{dist(x:real^N,y) | x
IN s /\ y
IN t}`
INF) THEN
REWRITE_TAC[
FORALL_IN_GSPEC] THEN ANTS_TAC THENL
[CONJ_TAC THENL [ASM SET_TAC[]; MESON_TAC[
DIST_POS_LE]]; ALL_TAC] THEN
ASM_MESON_TAC[]);;
let SETDIST_LE_DIST = prove
(`!s t x y:real^N. x
IN s /\ y
IN t ==> setdist(s,t) <= dist(x,y)`,
REPEAT GEN_TAC THEN REWRITE_TAC[setdist] THEN
COND_CASES_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
MP_TAC(ISPEC `{dist(x:real^N,y) | x
IN s /\ y
IN t}`
INF) THEN
REWRITE_TAC[
FORALL_IN_GSPEC] THEN ANTS_TAC THENL
[CONJ_TAC THENL [ASM SET_TAC[]; MESON_TAC[
DIST_POS_LE]]; ALL_TAC] THEN
ASM_MESON_TAC[]);;
let REAL_LE_SETDIST_EQ = prove
(`!d s t:real^N->bool.
d <= setdist(s,t) <=>
(!x y. x
IN s /\ y
IN t ==> d <= dist(x,y)) /\
(s = {} \/ t = {} ==> d <= &0)`,
let SETDIST_CLOSURE = prove
(`(!s t:real^N->bool. setdist(closure s,t) = setdist(s,t)) /\
(!s t:real^N->bool. setdist(s,closure t) = setdist(s,t))`,
let SETDIST_TRANSLATION = prove
(`!a:real^N s t.
setdist(
IMAGE (\x. a + x) s,
IMAGE (\x. a + x) t) = setdist(s,t)`,
REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[
SETDIST_DIFFERENCES] THEN
AP_TERM_TAC THEN AP_TERM_TAC THEN
REWRITE_TAC[SET_RULE
`{f x y | x
IN IMAGE g s /\ y
IN IMAGE g t} =
{f (g x) (g y) | x
IN s /\ y
IN t}`] THEN
REWRITE_TAC[VECTOR_ARITH `(a + x) - (a + y):real^N = x - y`]);;
let SETDIST_LINEAR_IMAGE = prove
(`!f:real^M->real^N s t.
linear f /\ (!x. norm(f x) = norm x)
==> setdist(
IMAGE f s,
IMAGE f t) = setdist(s,t)`,
REPEAT STRIP_TAC THEN REWRITE_TAC[setdist;
IMAGE_EQ_EMPTY] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[dist] THEN AP_TERM_TAC THEN
REWRITE_TAC[SET_RULE
`{f x y | x
IN IMAGE g s /\ y
IN IMAGE g t} =
{f (g x) (g y) | x
IN s /\ y
IN t}`] THEN
FIRST_X_ASSUM(fun th -> REWRITE_TAC[GSYM(MATCH_MP
LINEAR_SUB th)]) THEN
ASM_REWRITE_TAC[]);;
let SETDIST_UNIQUE = prove
(`!s t a b:real^N d.
a
IN s /\ b
IN t /\ dist(a,b) = d /\
(!x y. x
IN s /\ y
IN t ==> dist(a,b) <= dist(x,y))
==> setdist(s,t) = d`,
REPEAT STRIP_TAC THEN REWRITE_TAC[GSYM REAL_LE_ANTISYM] THEN CONJ_TAC THENL
[ASM_MESON_TAC[
SETDIST_LE_DIST];
MATCH_MP_TAC
REAL_LE_SETDIST THEN ASM SET_TAC[]]);;
let SETDIST_EQ_0_SING = prove
(`(!s x:real^N. setdist({x},s) = &0 <=> s = {} \/ x
IN closure s) /\
(!s x:real^N. setdist(s,{x}) = &0 <=> s = {} \/ x
IN closure s)`,
let SETDIST_BALLS = prove
(`(!a b:real^N r s.
setdist(ball(a,r),ball(b,s)) =
if r <= &0 \/ s <= &0 then &0 else max (&0) (dist(a,b) - (r + s))) /\
(!a b:real^N r s.
setdist(ball(a,r),cball(b,s)) =
if r <= &0 \/ s < &0 then &0 else max (&0) (dist(a,b) - (r + s))) /\
(!a b:real^N r s.
setdist(cball(a,r),ball(b,s)) =
if r < &0 \/ s <= &0 then &0 else max (&0) (dist(a,b) - (r + s))) /\
(!a b:real^N r s.
setdist(cball(a,r),cball(b,s)) =
if r < &0 \/ s < &0 then &0 else max (&0) (dist(a,b) - (r + s)))`,
let hausdist = new_definition
`hausdist(s:real^N->bool,t:real^N->bool) =
let ds = {setdist({x},t) | x IN s} UNION {setdist({y},s) | y IN t} in
if ~(ds = {}) /\ (?b. !d. d IN ds ==> d <= b) then sup ds
else &0`;;let HAUSDIST_EMPTY = prove
(`(!t:real^N->bool. hausdist ({},t) = &0) /\
(!s:real^N->bool. hausdist (s,{}) = &0)`,
REWRITE_TAC[hausdist;
LET_DEF;
LET_END_DEF;
SETDIST_EMPTY] THEN
REWRITE_TAC[SET_RULE `{f x | x
IN {}} = {}`;
UNION_EMPTY] THEN
REWRITE_TAC[SET_RULE `{c |x| x
IN s} = {} <=> s = {}`] THEN
X_GEN_TAC `s:real^N->bool` THEN
ASM_CASES_TAC `s:real^N->bool = {}` THEN ASM_REWRITE_TAC[] THEN
ASM_SIMP_TAC[SET_RULE `~(s = {}) ==> {c |x| x
IN s} = {c}`] THEN
REWRITE_TAC[
SUP_SING;
COND_ID]);;
let HAUSDIST_EQ = prove
(`!s t:real^M->bool s' t':real^N->bool.
(!b. (!x. x
IN s ==> setdist({x},t) <= b) /\
(!y. y
IN t ==> setdist({y},s) <= b) <=>
(!x. x
IN s' ==> setdist({x},t') <= b) /\
(!y. y
IN t' ==> setdist({y},s') <= b))
==> hausdist(s,t) = hausdist(s',t')`,
REPEAT STRIP_TAC THEN REWRITE_TAC[hausdist;
LET_DEF;
LET_END_DEF] THEN
MATCH_MP_TAC(MESON[]
`(p <=> p') /\ s = s'
==> (if p then s else &0) = (if p' then s' else &0)`) THEN
CONJ_TAC THENL
[BINOP_TAC THENL
[PURE_REWRITE_TAC[SET_RULE `s = {} <=> !x. x
IN s ==> F`];
AP_TERM_TAC THEN ABS_TAC];
MATCH_MP_TAC
SUP_EQ] THEN
PURE_REWRITE_TAC[
FORALL_IN_UNION;
FORALL_IN_GSPEC] THEN
ASM_REWRITE_TAC[] THEN
REWRITE_TAC[DE_MORGAN_THM;
NOT_FORALL_THM;
MEMBER_NOT_EMPTY] THEN
REWRITE_TAC[GSYM DE_MORGAN_THM] THEN AP_TERM_TAC THEN EQ_TAC THEN
DISCH_THEN(fun th -> POP_ASSUM MP_TAC THEN ASSUME_TAC th) THEN
ASM_REWRITE_TAC[
NOT_IN_EMPTY] THEN
DISCH_THEN(MP_TAC o SPEC `--(&1):real`) THEN
SIMP_TAC[
SETDIST_POS_LE; REAL_ARITH `&0 <= x ==> ~(x <= --(&1))`] THEN
SET_TAC[]);;
let HAUSDIST_LINEAR_IMAGE = prove
(`!f:real^M->real^N s t.
linear f /\ (!x. norm(f x) = norm x)
==> hausdist(
IMAGE f s,
IMAGE f t) = hausdist(s,t)`,
REPEAT STRIP_TAC THEN
REPEAT GEN_TAC THEN REWRITE_TAC[hausdist] THEN
REWRITE_TAC[SET_RULE `{f x | x
IN IMAGE g s} = {f(g x) | x
IN s}`] THEN
ONCE_REWRITE_TAC[SET_RULE `{(f:real^M->real^N) x} =
IMAGE f {x}`] THEN
ASM_SIMP_TAC[
SETDIST_LINEAR_IMAGE]);;
let HAUSDIST_CLOSURE = prove
(`(!s t:real^N->bool. hausdist(closure s,t) = hausdist(s,t)) /\
(!s t:real^N->bool. hausdist(s,closure t) = hausdist(s,t))`,
let REAL_HAUSDIST_LE = prove
(`!s t:real^N->bool b.
~(s = {}) /\ ~(t = {}) /\
(!x. x
IN s ==> setdist({x},t) <= b) /\
(!y. y
IN t ==> setdist({y},s) <= b)
==> hausdist(s,t) <= b`,
let REAL_LE_HAUSDIST = prove
(`!s t:real^N->bool a b c z.
~(s = {}) /\ ~(t = {}) /\
(!x. x
IN s ==> setdist({x},t) <= b) /\
(!y. y
IN t ==> setdist({y},s) <= c) /\
(z
IN s /\ a <= setdist({z},t) \/ z
IN t /\ a <= setdist({z},s))
==> a <= hausdist(s,t)`,
let UPPER_LOWER_HEMICONTINUOUS = prove
(`!f:real^M->real^N->bool t s.
(!x. x
IN s ==> f(x)
SUBSET t) /\
(!u.
open_in (subtopology euclidean t) u
==>
open_in (subtopology euclidean s)
{x | x
IN s /\ f(x)
SUBSET u}) /\
(!u.
closed_in (subtopology euclidean t) u
==>
closed_in (subtopology euclidean s)
{x | x
IN s /\ f(x)
SUBSET u})
==> !x e. x
IN s /\ &0 < e /\ bounded(f x)
==> ?d. &0 < d /\
!x'. x'
IN s /\ dist(x,x') < d
==> hausdist(f x,f x') < e`,
REPEAT GEN_TAC THEN DISCH_TAC THEN REPEAT STRIP_TAC THEN
ASM_CASES_TAC `(f:real^M->real^N->bool) x = {}` THENL
[ASM_REWRITE_TAC[
HAUSDIST_EMPTY] THEN MESON_TAC[
REAL_LT_01]; ALL_TAC] THEN
FIRST_ASSUM(MP_TAC o SPECL [`x:real^M`; `e / &2`] o MATCH_MP
UPPER_LOWER_HEMICONTINUOUS_EXPLICIT) THEN
ASM_REWRITE_TAC[
REAL_HALF] THEN
DISCH_THEN(X_CHOOSE_THEN `d1:real` STRIP_ASSUME_TAC) THEN
FIRST_ASSUM(MP_TAC o SPEC `vec 0:real^N` o MATCH_MP
BOUNDED_SUBSET_BALL) THEN
DISCH_THEN(X_CHOOSE_THEN `r:real` STRIP_ASSUME_TAC) THEN
FIRST_ASSUM(MP_TAC o SPEC `t
INTER ball(vec 0:real^N,r)` o
CONJUNCT1 o CONJUNCT2) THEN
SIMP_TAC[
OPEN_IN_OPEN_INTER;
OPEN_BALL] THEN REWRITE_TAC[
open_in] THEN
DISCH_THEN(MP_TAC o SPEC `x:real^M` o CONJUNCT2) THEN
ASM_SIMP_TAC[
SUBSET_INTER;
IN_ELIM_THM] THEN
DISCH_THEN(X_CHOOSE_THEN `d2:real` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `min d1 d2:real` THEN ASM_REWRITE_TAC[
REAL_LT_MIN] THEN
X_GEN_TAC `x':real^M` THEN STRIP_TAC THEN
REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `x':real^M`)) THEN
ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[
DIST_SYM] THEN ASM_REWRITE_TAC[] THEN
STRIP_TAC THEN STRIP_TAC THEN
ASM_CASES_TAC `(f:real^M->real^N->bool) x' = {}` THEN
ASM_REWRITE_TAC[
HAUSDIST_EMPTY] THEN
MATCH_MP_TAC(REAL_ARITH `&0 < e /\ x <= e / &2 ==> x < e`) THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC
REAL_HAUSDIST_LE THEN
ASM_MESON_TAC[
SETDIST_LE_DIST;
DIST_SYM;
REAL_LE_TRANS;
IN_SING;
REAL_LT_IMP_LE]);;
let HAUSDIST_NONTRIVIAL = prove
(`!s t:real^N->bool.
bounded s /\ bounded t /\ ~(s = {}) /\ ~(t = {})
==> hausdist(s,t) =
sup({setdist ({x},t) | x
IN s}
UNION {setdist ({y},s) | y
IN t})`,
let HAUSDIST_NONTRIVIAL_ALT = prove
(`!s t:real^N->bool.
bounded s /\ bounded t /\ ~(s = {}) /\ ~(t = {})
==> hausdist(s,t) = max (sup {setdist ({x},t) | x
IN s})
(sup {setdist ({y},s) | y
IN t})`,
let REAL_HAUSDIST_LE_EQ = prove
(`!s t:real^N->bool b.
~(s = {}) /\ ~(t = {}) /\ bounded s /\ bounded t
==> (hausdist(s,t) <= b <=>
(!x. x
IN s ==> setdist({x},t) <= b) /\
(!y. y
IN t ==> setdist({y},s) <= b))`,
let HAUSDIST_TRANS = prove
(`!s t u:real^N->bool.
bounded s /\ bounded t /\ bounded u /\ ~(t = {})
==> hausdist(s,u) <= hausdist(s,t) + hausdist(t,u)`,
let HAUSDIST_EQ_0 = prove
(`!s t:real^N->bool.
bounded s /\ bounded t
==> (hausdist(s,t) = &0 <=> s = {} \/ t = {} \/ closure s = closure t)`,
let HAUSDIST_BALLS = prove
(`(!a b:real^N r s.
hausdist(ball(a,r),ball(b,s)) =
if r <= &0 \/ s <= &0 then &0 else dist(a,b) + abs(r - s)) /\
(!a b:real^N r s.
hausdist(ball(a,r),cball(b,s)) =
if r <= &0 \/ s < &0 then &0 else dist(a,b) + abs(r - s)) /\
(!a b:real^N r s.
hausdist(cball(a,r),ball(b,s)) =
if r < &0 \/ s <= &0 then &0 else dist(a,b) + abs(r - s)) /\
(!a b:real^N r s.
hausdist(cball(a,r),cball(b,s)) =
if r < &0 \/ s < &0 then &0 else dist(a,b) + abs(r - s))`,
let HAUSDIST_ALT = prove
(`!s t:real^N->bool.
bounded s /\ bounded t /\ ~(s = {}) /\ ~(t = {})
==> hausdist(s,t) =
sup {abs(setdist({x},s) - setdist({x},t)) | x
IN (:real^N)}`,
let CONTINUOUS_DIAMETER = prove
(`!s:real^N->bool e.
bounded s /\ ~(s = {}) /\ &0 < e
==> ?d. &0 < d /\
!t. bounded t /\ ~(t = {}) /\ hausdist(s,t) < d
==> abs(diameter s - diameter t) < e`,
let ISOMETRY_IMP_OPEN_MAP = prove
(`!f:real^M->real^N s t u.
IMAGE f s = t /\
(!x y. x
IN s /\ y
IN s ==> dist(f x,f y) = dist(x,y)) /\
open_in (subtopology euclidean s) u
==>
open_in (subtopology euclidean t) (
IMAGE f u)`,
REWRITE_TAC[
open_in;
FORALL_IN_IMAGE] THEN REPEAT GEN_TAC THEN STRIP_TAC THEN
CONJ_TAC THENL [ASM SET_TAC[]; X_GEN_TAC `x:real^M` THEN DISCH_TAC] THEN
FIRST_X_ASSUM(MP_TAC o SPEC `x:real^M`) THEN ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC
MONO_EXISTS THEN X_GEN_TAC `e:real` THEN
STRIP_TAC THEN ASM_REWRITE_TAC[
IMP_CONJ] THEN
EXPAND_TAC "t" THEN REWRITE_TAC[
FORALL_IN_IMAGE] THEN
RULE_ASSUM_TAC(REWRITE_RULE[
SUBSET]) THEN
ASM_SIMP_TAC[
IN_IMAGE] THEN ASM_MESON_TAC[]);;
let ISOMETRY_IMP_HOMEOMORPHISM_COMPACT = prove
(`!f s:real^N->bool.
compact s /\
IMAGE f s
SUBSET s /\
(!x y. x
IN s /\ y
IN s ==> dist(f x,f y) = dist(x,y))
==> ?g. homeomorphism (s,s) (f,g)`,
REPEAT STRIP_TAC THEN
SUBGOAL_THEN `
IMAGE (f:real^N->real^N) s = s`
(fun th -> ASM_MESON_TAC[th;
ISOMETRY_IMP_EMBEDDING]) THEN
FIRST_ASSUM(ASSUME_TAC o MATCH_MP
ISOMETRY_ON_IMP_CONTINUOUS_ON) THEN
ASM_REWRITE_TAC[GSYM
SUBSET_ANTISYM_EQ] THEN REWRITE_TAC[
SUBSET] THEN
X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
SUBGOAL_THEN `setdist({x},
IMAGE (f:real^N->real^N) s) = &0` MP_TAC THENL
[MATCH_MP_TAC(REAL_ARITH `&0 <= x /\ ~(&0 < x) ==> x = &0`) THEN
REWRITE_TAC[
SETDIST_POS_LE] THEN DISCH_TAC THEN
(X_CHOOSE_THEN `z:num->real^N` STRIP_ASSUME_TAC o
prove_recursive_functions_exist num_RECURSION)
`z 0 = (x:real^N) /\ !n. z(SUC n) = f(z n)` THEN
SUBGOAL_THEN `!n. (z:num->real^N) n
IN s` ASSUME_TAC THENL
[INDUCT_TAC THEN ASM SET_TAC[]; ALL_TAC] THEN
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [compact]) THEN
DISCH_THEN(MP_TAC o SPEC `z:num->real^N`) THEN
ASM_REWRITE_TAC[
NOT_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC [`l:real^N`; `r:num->num`] THEN STRIP_TAC THEN
FIRST_ASSUM(MP_TAC o MATCH_MP
CONVERGENT_IMP_CAUCHY) THEN
REWRITE_TAC[cauchy] THEN
DISCH_THEN(MP_TAC o SPEC `setdist({x},
IMAGE (f:real^N->real^N) s)`) THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `N:num`
(MP_TAC o SPECL [`N:num`; `N + 1`])) THEN
ANTS_TAC THENL [ARITH_TAC; REWRITE_TAC[
REAL_NOT_LT;
o_THM]] THEN
SUBGOAL_THEN `(r:num->num) N < r (N + 1)` MP_TAC THENL
[FIRST_X_ASSUM MATCH_MP_TAC THEN ARITH_TAC;
REWRITE_TAC[
LT_EXISTS;
LEFT_IMP_EXISTS_THM]] THEN
X_GEN_TAC `d:num` THEN DISCH_THEN SUBST1_TAC THEN
TRANS_TAC
REAL_LE_TRANS `dist(x:real^N,z(SUC d))` THEN CONJ_TAC THENL
[MATCH_MP_TAC
SETDIST_LE_DIST THEN ASM SET_TAC[]; ALL_TAC] THEN
MATCH_MP_TAC
REAL_EQ_IMP_LE THEN
SPEC_TAC(`(r:num->num) N`,`m:num`) THEN
INDUCT_TAC THEN ASM_MESON_TAC[
ADD_CLAUSES];
REWRITE_TAC[
SETDIST_EQ_0_SING;
IMAGE_EQ_EMPTY] THEN
ASM_MESON_TAC[
COMPACT_IMP_CLOSED;
NOT_IN_EMPTY;
COMPACT_CONTINUOUS_IMAGE;
CLOSURE_CLOSED]]);;
let URYSOHN_LOCAL_STRONG = prove
(`!s t u a b.
closed_in (subtopology euclidean u) s /\
closed_in (subtopology euclidean u) t /\
s
INTER t = {} /\ ~(a = b)
==> ?f:real^N->real^M.
f
continuous_on u /\
(!x. x
IN u ==> f(x)
IN segment[a,b]) /\
(!x. x
IN u ==> (f x = a <=> x
IN s)) /\
(!x. x
IN u ==> (f x = b <=> x
IN t))`,
let lemma = prove
(`!s t u a b.
closed_in (subtopology euclidean u) s /\
closed_in (subtopology euclidean u) t /\
s INTER t = {} /\ ~(s = {}) /\ ~(t = {}) /\ ~(a = b)
==> ?f:real^N->real^M.
f continuous_on u /\
(!x. x IN u ==> f(x) IN segment[a,b]) /\
(!x. x IN u ==> (f x = a <=> x IN s)) /\
(!x. x IN u ==> (f x = b <=> x IN t))`,
REPEAT STRIP_TAC THEN EXISTS_TAC
`\x:real^N. a + setdist({x},s) / (setdist({x},s) + setdist({x},t)) %
(b - a:real^M)` THEN REWRITE_TAC[] THEN
SUBGOAL_THEN
`(!x:real^N. x IN u ==> (setdist({x},s) = &0 <=> x IN s)) /\
(!x:real^N. x IN u ==> (setdist({x},t) = &0 <=> x IN t))`
STRIP_ASSUME_TAC THENL
[ASM_REWRITE_TAC[SETDIST_EQ_0_SING] THEN CONJ_TAC THENL
[MP_TAC(ISPEC `s:real^N->bool` CLOSED_IN_CLOSED);
MP_TAC(ISPEC `t:real^N->bool` CLOSED_IN_CLOSED)] THEN
DISCH_THEN(MP_TAC o SPEC `u:real^N->bool`) THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN(X_CHOOSE_THEN `v:real^N->bool`
(CONJUNCTS_THEN2 ASSUME_TAC SUBST_ALL_TAC)) THEN
ASM_MESON_TAC[CLOSURE_CLOSED; INTER_SUBSET; SUBSET_CLOSURE; SUBSET;
IN_INTER; CLOSURE_SUBSET];
ALL_TAC] THEN
SUBGOAL_THEN `!x:real^N. x IN u ==> &0 < setdist({x},s) + setdist({x},t)`
ASSUME_TAC THENL
[REPEAT STRIP_TAC THEN MATCH_MP_TAC(REAL_ARITH
`&0 <= x /\ &0 <= y /\ ~(x = &0 /\ y = &0) ==> &0 < x + y`) THEN
REWRITE_TAC[SETDIST_POS_LE] THEN ASM SET_TAC[];
ALL_TAC] THEN
REPEAT CONJ_TAC THENL
[MATCH_MP_TAC CONTINUOUS_ON_ADD THEN REWRITE_TAC[CONTINUOUS_ON_CONST] THEN
REWRITE_TAC[real_div; GSYM VECTOR_MUL_ASSOC] THEN
REPEAT(MATCH_MP_TAC CONTINUOUS_ON_MUL THEN CONJ_TAC) THEN
REWRITE_TAC[CONTINUOUS_ON_CONST; o_DEF] THEN
REWRITE_TAC[CONTINUOUS_ON_LIFT_SETDIST] THEN
MATCH_MP_TAC(REWRITE_RULE[o_DEF] CONTINUOUS_ON_INV) THEN
ASM_SIMP_TAC[REAL_LT_IMP_NZ] THEN
REWRITE_TAC[LIFT_ADD] THEN MATCH_MP_TAC CONTINUOUS_ON_ADD THEN
REWRITE_TAC[CONTINUOUS_ON_LIFT_SETDIST];
X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
REWRITE_TAC[segment; IN_ELIM_THM] THEN
REWRITE_TAC[VECTOR_MUL_EQ_0; LEFT_OR_DISTRIB; VECTOR_ARITH
`a + x % (b - a):real^N = (&1 - u) % a + u % b <=>
(x - u) % (b - a) = vec 0`;
EXISTS_OR_THM] THEN
DISJ1_TAC THEN ONCE_REWRITE_TAC[CONJ_SYM] THEN
REWRITE_TAC[REAL_SUB_0; UNWIND_THM1] THEN
ASM_SIMP_TAC[REAL_LE_DIV; REAL_LE_ADD; SETDIST_POS_LE; REAL_LE_LDIV_EQ;
REAL_ARITH `a <= &1 * (a + b) <=> &0 <= b`];
REWRITE_TAC[VECTOR_ARITH `a + x:real^N = a <=> x = vec 0`];
REWRITE_TAC[VECTOR_ARITH `a + x % (b - a):real^N = b <=>
(x - &1) % (b - a) = vec 0`]] THEN
ASM_REWRITE_TAC[VECTOR_MUL_EQ_0; VECTOR_SUB_EQ] THEN
ASM_SIMP_TAC[REAL_SUB_0; REAL_EQ_LDIV_EQ;
REAL_MUL_LZERO; REAL_MUL_LID] THEN
REWRITE_TAC[REAL_ARITH `x:real = x + y <=> y = &0`] THEN
ASM_REWRITE_TAC[]) in
MATCH_MP_TAC(MESON[]
`(!s t. P s t <=> P t s) /\
(!s t. ~(s = {}) /\ ~(t = {}) ==> P s t) /\
P {} {} /\ (!t. ~(t = {}) ==> P {} t)
==> !s t. P s t`) THEN
REPEAT CONJ_TAC THENL
[REPEAT GEN_TAC THEN
GEN_REWRITE_TAC (RAND_CONV o BINDER_CONV) [SWAP_FORALL_THM] THEN
REPEAT(AP_TERM_TAC THEN ABS_TAC) THEN
REWRITE_TAC[SEGMENT_SYM; INTER_COMM; CONJ_ACI; EQ_SYM_EQ];
SIMP_TAC[lemma];
REPEAT STRIP_TAC THEN EXISTS_TAC `(\x. midpoint(a,b)):real^N->real^M` THEN
ASM_SIMP_TAC[NOT_IN_EMPTY; CONTINUOUS_ON_CONST; MIDPOINT_IN_SEGMENT] THEN
REWRITE_TAC[midpoint] THEN CONJ_TAC THEN GEN_TAC THEN DISCH_TAC THEN
UNDISCH_TAC `~(a:real^M = b)` THEN REWRITE_TAC[CONTRAPOS_THM] THEN
VECTOR_ARITH_TAC;
REPEAT STRIP_TAC THEN ASM_CASES_TAC `t:real^N->bool = u` THENL
[EXISTS_TAC `(\x. b):real^N->real^M` THEN
ASM_REWRITE_TAC[NOT_IN_EMPTY; ENDS_IN_SEGMENT; IN_UNIV;
CONTINUOUS_ON_CONST];
SUBGOAL_THEN `?c:real^N. c IN u /\ ~(c IN t)` STRIP_ASSUME_TAC THENL
[REPEAT(FIRST_X_ASSUM(MP_TAC o MATCH_MP CLOSED_IN_SUBSET)) THEN
REWRITE_TAC[TOPSPACE_EUCLIDEAN_SUBTOPOLOGY] THEN ASM SET_TAC[];
ALL_TAC] THEN
MP_TAC(ISPECL [`{c:real^N}`; `t:real^N->bool`; `u:real^N->bool`;
`midpoint(a,b):real^M`; `b:real^M`] lemma) THEN
ASM_REWRITE_TAC[CLOSED_IN_SING; MIDPOINT_EQ_ENDPOINT] THEN
ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
MATCH_MP_TAC MONO_EXISTS THEN SIMP_TAC[NOT_IN_EMPTY] THEN
X_GEN_TAC `f:real^N->real^M` THEN STRIP_TAC THEN CONJ_TAC THENL
[SUBGOAL_THEN
`segment[midpoint(a,b):real^M,b] SUBSET segment[a,b]` MP_TAC
THENL
[REWRITE_TAC[SUBSET; IN_SEGMENT; midpoint] THEN GEN_TAC THEN
DISCH_THEN(X_CHOOSE_THEN `u:real` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `(&1 + u) / &2` THEN ASM_REWRITE_TAC[] THEN
REPEAT(CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC]) THEN
VECTOR_ARITH_TAC;
ASM SET_TAC[]];
SUBGOAL_THEN `~(a IN segment[midpoint(a,b):real^M,b])` MP_TAC THENL
[ALL_TAC; ASM_MESON_TAC[]] THEN
DISCH_THEN(MP_TAC o CONJUNCT2 o MATCH_MP DIST_IN_CLOSED_SEGMENT) THEN
REWRITE_TAC[DIST_MIDPOINT] THEN
UNDISCH_TAC `~(a:real^M = b)` THEN NORM_ARITH_TAC]]]);;
let URYSOHN = prove
(`!s t a b.
closed s /\ closed t /\ s
INTER t = {}
==> ?f:real^N->real^M.
f
continuous_on (:real^N) /\ (!x. f(x)
IN segment[a,b]) /\
(!x. x
IN s ==> f x = a) /\ (!x. x
IN t ==> f x = b)`,
let SUBSET_SECOND_COUNTABLE = prove
(`!s:real^N->bool.
?b.
COUNTABLE b /\
(!c. c
IN b ==> ~(c = {}) /\
open_in(subtopology euclidean s) c) /\
!t.
open_in(subtopology euclidean s) t
==> ?u. u
SUBSET b /\ t =
UNIONS u`,
GEN_TAC THEN
SUBGOAL_THEN
`?b.
COUNTABLE b /\
(!c:real^N->bool. c
IN b ==>
open_in(subtopology euclidean s) c) /\
!t.
open_in(subtopology euclidean s) t
==> ?u. u
SUBSET b /\ t =
UNIONS u`
STRIP_ASSUME_TAC THENL
[X_CHOOSE_THEN `B:(real^N->bool)->bool` STRIP_ASSUME_TAC
UNIV_SECOND_COUNTABLE THEN
EXISTS_TAC `{s
INTER c :real^N->bool | c
IN B}` THEN
ASM_SIMP_TAC[
SIMPLE_IMAGE;
COUNTABLE_IMAGE] THEN
ASM_SIMP_TAC[
FORALL_IN_IMAGE;
EXISTS_SUBSET_IMAGE;
OPEN_IN_OPEN_INTER] THEN
REWRITE_TAC[
OPEN_IN_OPEN] THEN
X_GEN_TAC `t:real^N->bool` THEN
DISCH_THEN(X_CHOOSE_THEN `u:real^N->bool` STRIP_ASSUME_TAC) THEN
FIRST_X_ASSUM SUBST_ALL_TAC THEN
SUBGOAL_THEN `?b. b
SUBSET B /\ u:real^N->bool =
UNIONS b`
STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
FIRST_X_ASSUM SUBST_ALL_TAC THEN
EXISTS_TAC `b:(real^N->bool)->bool` THEN ASM_REWRITE_TAC[] THEN
REWRITE_TAC[
INTER_UNIONS] THEN AP_TERM_TAC THEN SET_TAC[];
EXISTS_TAC `b
DELETE ({}:real^N->bool)` THEN
ASM_SIMP_TAC[
COUNTABLE_DELETE;
IN_DELETE;
SUBSET_DELETE] THEN
X_GEN_TAC `t:real^N->bool` THEN DISCH_THEN(ANTE_RES_THEN MP_TAC) THEN
DISCH_THEN(X_CHOOSE_THEN `u:(real^N->bool)->bool` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `u
DELETE ({}:real^N->bool)` THEN
REPEAT(CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC]) THEN
FIRST_X_ASSUM SUBST_ALL_TAC THEN
REWRITE_TAC[
EXTENSION;
IN_UNIONS] THEN
GEN_TAC THEN AP_TERM_TAC THEN ABS_TAC THEN
REWRITE_TAC[
IN_DELETE] THEN SET_TAC[]]);;
let DISCRETE_IMP_COUNTABLE = prove
(`!s:real^N->bool.
(!x. x
IN s ==> ?e. &0 < e /\
!y. y
IN s /\ ~(y = x) ==> e <= norm(y - x))
==>
COUNTABLE s`,
REPEAT STRIP_TAC THEN
SUBGOAL_THEN
`!x. x
IN s
==> ?q. (!i. 1 <= i /\ i <= dimindex(:N) ==> rational(q$i)) /\
!y:real^N. y
IN s /\ ~(y = x) ==> norm(x - q) < norm(y - q)`
MP_TAC THENL
[X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN
MP_TAC(SET_RULE `x
IN (:real^N)`) THEN
REWRITE_TAC[GSYM
CLOSURE_RATIONAL_COORDINATES] THEN
REWRITE_TAC[
CLOSURE_APPROACHABLE;
IN_ELIM_THM] THEN
DISCH_THEN(MP_TAC o SPEC `e / &2`) THEN ASM_REWRITE_TAC[
REAL_HALF] THEN
MATCH_MP_TAC
MONO_EXISTS THEN X_GEN_TAC `q:real^N` THEN
STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
X_GEN_TAC `y:real^N` THEN STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `y:real^N`) THEN ASM_REWRITE_TAC[] THEN
REPEAT(POP_ASSUM MP_TAC) THEN NORM_ARITH_TAC;
POP_ASSUM(K ALL_TAC) THEN
REWRITE_TAC[
RIGHT_IMP_EXISTS_THM;
SKOLEM_THM;
LEFT_IMP_EXISTS_THM] THEN
X_GEN_TAC `q:real^N->real^N` THEN DISCH_TAC THEN
MP_TAC(ISPECL
[`s:real^N->bool`;
`{ x:real^N | !i. 1 <= i /\ i <= dimindex(:N) ==> rational(x$i) }`;
`(:num)`]
CARD_LE_TRANS) THEN
REWRITE_TAC[
COUNTABLE;
ge_c] THEN DISCH_THEN MATCH_MP_TAC THEN
SIMP_TAC[REWRITE_RULE[
COUNTABLE;
ge_c]
COUNTABLE_RATIONAL_COORDINATES] THEN
REWRITE_TAC[
le_c] THEN EXISTS_TAC `q:real^N->real^N` THEN
ASM_SIMP_TAC[
IN_ELIM_THM] THEN ASM_MESON_TAC[
REAL_LT_ANTISYM]]);;
let BROUWER_REDUCTION_THEOREM_GEN = prove
(`!P s:real^N->bool.
(!f. (!n. closed(f n) /\ P(f n)) /\ (!n. f(SUC n)
SUBSET f(n))
==> P(
INTERS {f n | n
IN (:num)})) /\
closed s /\ P s
==> ?t. t
SUBSET s /\ closed t /\ P t /\
(!u. u
SUBSET s /\ closed u /\ P u ==> ~(u
PSUBSET t))`,
REPEAT STRIP_TAC THEN
SUBGOAL_THEN
`?b:num->real^N->bool.
(!m n. b m = b n <=> m = n) /\
(!n. open (b n)) /\
(!s. open s ==> (?k. s =
UNIONS {b n | n
IN k}))`
STRIP_ASSUME_TAC THENL
[REWRITE_TAC[
UNIV_SECOND_COUNTABLE_SEQUENCE]; ALL_TAC] THEN
X_CHOOSE_THEN `a:num->real^N->bool` MP_TAC
(prove_recursive_functions_exist num_RECURSION
`a 0 = (s:real^N->bool) /\
(!n. a(SUC n) =
if ?u. u
SUBSET a(n) /\ closed u /\ P u /\ u
INTER (b n) = {}
then @u. u
SUBSET a(n) /\ closed u /\ P u /\ u
INTER (b n) = {}
else a(n))`) THEN
DISCH_THEN(CONJUNCTS_THEN2 (LABEL_TAC "base") (LABEL_TAC "step")) THEN
EXISTS_TAC `
INTERS {a n :real^N->bool | n
IN (:num)}` THEN
SUBGOAL_THEN `!n. (a:num->real^N->bool)(SUC n)
SUBSET a(n)` ASSUME_TAC THENL
[GEN_TAC THEN ASM_REWRITE_TAC[] THEN
COND_CASES_TAC THEN REWRITE_TAC[
SUBSET_REFL] THEN
FIRST_X_ASSUM(MP_TAC o SELECT_RULE) THEN MESON_TAC[];
ALL_TAC] THEN
SUBGOAL_THEN `!n. (a:num->real^N->bool) n
SUBSET s` ASSUME_TAC THENL
[INDUCT_TAC THEN ASM_MESON_TAC[
SUBSET_REFL;
SUBSET_TRANS]; ALL_TAC] THEN
SUBGOAL_THEN `!n. closed((a:num->real^N->bool) n) /\ P(a n)` ASSUME_TAC THENL
[INDUCT_TAC THEN ASM_REWRITE_TAC[] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
FIRST_X_ASSUM(MP_TAC o SELECT_RULE) THEN MESON_TAC[];
ALL_TAC] THEN
REPEAT CONJ_TAC THENL
[ASM SET_TAC[];
MATCH_MP_TAC
CLOSED_INTERS THEN
ASM_REWRITE_TAC[
FORALL_IN_GSPEC;
IN_UNIV] THEN SET_TAC[];
FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[];
X_GEN_TAC `t:real^N->bool` THEN STRIP_TAC THEN
REWRITE_TAC[
PSUBSET_ALT] THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
REWRITE_TAC[
INTERS_GSPEC;
EXISTS_IN_GSPEC;
IN_UNIV] THEN
DISCH_THEN(X_CHOOSE_THEN `x:real^N` STRIP_ASSUME_TAC) THEN
SUBGOAL_THEN
`?n. x
IN (b:num->real^N->bool)(n) /\ t
INTER b n = {}`
STRIP_ASSUME_TAC THENL
[MP_TAC(ISPEC `(:real^N)
DIFF t`
OPEN_CONTAINS_BALL) THEN
ASM_REWRITE_TAC[GSYM closed] THEN
DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN
ASM_REWRITE_TAC[
IN_DIFF;
IN_UNIV;
LEFT_IMP_EXISTS_THM] THEN
REWRITE_TAC[SET_RULE `s
SUBSET UNIV DIFF t <=> t
INTER s = {}`] THEN
X_GEN_TAC `e:real` THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
MP_TAC(ISPECL [`x:real^N`; `e:real`]
CENTRE_IN_BALL) THEN
FIRST_X_ASSUM(MP_TAC o SPEC `ball(x:real^N,e)`) THEN
ASM_REWRITE_TAC[
OPEN_BALL;
LEFT_IMP_EXISTS_THM] THEN
X_GEN_TAC `k:num->bool` THEN DISCH_THEN SUBST1_TAC THEN
REWRITE_TAC[
IN_UNIONS;
INTER_UNIONS;
EMPTY_UNIONS;
FORALL_IN_GSPEC] THEN
SET_TAC[];
REMOVE_THEN "step" (MP_TAC o SPEC `n:num`) THEN
COND_CASES_TAC THENL
[DISCH_THEN(ASSUME_TAC o SYM) THEN
FIRST_X_ASSUM(MP_TAC o SELECT_RULE) THEN ASM_REWRITE_TAC[] THEN
ASM SET_TAC[];
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
NOT_EXISTS_THM]) THEN
DISCH_THEN(MP_TAC o SPEC `t:real^N->bool`) THEN ASM_REWRITE_TAC[] THEN
ASM SET_TAC[]]]]);;
let BROUWER_REDUCTION_THEOREM = prove
(`!P s:real^N->bool.
(!f. (!n. compact(f n) /\ ~(f n = {}) /\ P(f n)) /\
(!n. f(SUC n)
SUBSET f(n))
==> P(
INTERS {f n | n
IN (:num)})) /\
compact s /\ ~(s = {}) /\ P s
==> ?t. t
SUBSET s /\ compact t /\ ~(t = {}) /\ P t /\
(!u. u
SUBSET s /\ closed u /\ ~(u = {}) /\ P u
==> ~(u
PSUBSET t))`,
let SUBSEQUENCE_DIAGONALIZATION_LEMMA = prove
(`!P:num->(num->A)->bool.
(!i r:num->A. ?k. (!m n. m < n ==> k m < k n) /\ P i (r o k)) /\
(!i r:num->A k1 k2 N.
P i (r o k1) /\ (!j. N <= j ==> ?j'. j <= j' /\ k2 j = k1 j')
==> P i (r o k2))
==> !r:num->A. ?k. (!m n. m < n ==> k m < k n) /\ (!i. P i (r o k))`,
REPEAT GEN_TAC THEN
DISCH_THEN(CONJUNCTS_THEN2 MP_TAC STRIP_ASSUME_TAC) THEN
GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV) [
SKOLEM_THM] THEN
REWRITE_TAC[
FORALL_AND_THM; TAUT
`(p ==> q /\ r) <=> (p ==> q) /\ (p ==> r)`] THEN
DISCH_THEN(X_CHOOSE_THEN
`kk:num->(num->A)->num->num` STRIP_ASSUME_TAC) THEN
X_GEN_TAC `r:num->A` THEN
(STRIP_ASSUME_TAC o prove_recursive_functions_exist num_RECURSION)
`(rr 0 = (kk:num->(num->A)->num->num) 0 r) /\
(!n. rr(SUC n) = rr n o kk (SUC n) (r o rr n))` THEN
EXISTS_TAC `\n. (rr:num->num->num) n n` THEN REWRITE_TAC[ETA_AX] THEN
SUBGOAL_THEN
`(!i. (!m n. m < n ==> (rr:num->num->num) i m < rr i n)) /\
(!i. (P:num->(num->A)->bool) i (r o rr i))`
STRIP_ASSUME_TAC THENL
[REWRITE_TAC[
AND_FORALL_THM] THEN
INDUCT_TAC THEN ASM_REWRITE_TAC[
o_ASSOC] THEN
REWRITE_TAC[
o_THM] THEN ASM_MESON_TAC[];
ALL_TAC] THEN
SUBGOAL_THEN `!i j n. i <= j ==> (rr:num->num->num) i n <= rr j n`
ASSUME_TAC THENL
[REPEAT GEN_TAC THEN GEN_REWRITE_TAC LAND_CONV [
LE_EXISTS] THEN
SIMP_TAC[
LEFT_IMP_EXISTS_THM] THEN SPEC_TAC(`j:num`,`j:num`) THEN
ONCE_REWRITE_TAC[
SWAP_FORALL_THM] THEN SIMP_TAC[
FORALL_UNWIND_THM2] THEN
INDUCT_TAC THEN REWRITE_TAC[
ADD_CLAUSES;
LE_REFL] THEN
ASM_REWRITE_TAC[] THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP
(REWRITE_RULE[
IMP_CONJ]
LE_TRANS)) THEN REWRITE_TAC[
o_THM] THEN
FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP
(MESON[
LE_LT]
`!f:num->num.
(!m n. m < n ==> f m < f n) ==> (!m n. m <= n ==> f m <= f n)`) o
SPEC `i + d:num`) THEN
SPEC_TAC(`n:num`,`n:num`) THEN MATCH_MP_TAC
MONOTONE_BIGGER THEN
ASM_SIMP_TAC[];
ALL_TAC] THEN
CONJ_TAC THENL
[MAP_EVERY X_GEN_TAC [`m:num`; `n:num`] THEN DISCH_TAC THEN
MATCH_MP_TAC
LET_TRANS THEN
EXISTS_TAC `(rr:num->num->num) n m` THEN
ASM_MESON_TAC[
LT_IMP_LE];
ALL_TAC] THEN
SUBGOAL_THEN
`!m n i. n <= m ==> ?j. i <= j /\ (rr:num->num->num) m i = rr n j`
ASSUME_TAC THENL
[ALL_TAC;
X_GEN_TAC `i:num` THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
EXISTS_TAC `(rr:num->num->num) i` THEN ASM_REWRITE_TAC[] THEN
EXISTS_TAC `i:num` THEN ASM_MESON_TAC[]] THEN
SUBGOAL_THEN
`!p d i. ?j. i <= j /\ (rr:num->num->num) (p + d) i = rr p j`
(fun th -> MESON_TAC[
LE_EXISTS; th]) THEN
X_GEN_TAC `p:num` THEN MATCH_MP_TAC
num_INDUCTION THEN
ASM_REWRITE_TAC[
ADD_CLAUSES] THEN CONJ_TAC THENL
[MESON_TAC[
LE_REFL]; ALL_TAC] THEN
X_GEN_TAC `d:num` THEN DISCH_THEN(LABEL_TAC "+") THEN
X_GEN_TAC `i:num` THEN ASM_REWRITE_TAC[
o_THM] THEN
REMOVE_THEN "+" (MP_TAC o SPEC
`(kk:num->(num->A)->num->num) (SUC(p + d))
((r:num->A) o (rr:num->num->num) (p + d)) i`) THEN
MATCH_MP_TAC
MONO_EXISTS THEN X_GEN_TAC `j:num` THEN
MATCH_MP_TAC MONO_AND THEN REWRITE_TAC[] THEN
MATCH_MP_TAC(REWRITE_RULE[
IMP_CONJ]
LE_TRANS) THEN
SPEC_TAC(`i:num`,`i:num`) THEN MATCH_MP_TAC
MONOTONE_BIGGER THEN
ASM_REWRITE_TAC[
o_THM] THEN ASM_MESON_TAC[]);;
let FUNCTION_CONVERGENT_SUBSEQUENCE = prove
(`!f:num->real^M->real^N s M.
COUNTABLE s /\ (!n x. x
IN s ==> norm(f n x) <= M)
==> ?k. (!m n:num. m < n ==> k m < k n) /\
!x. x
IN s ==> ?l. ((\n. f (k n) x) --> l) sequentially`,
REPEAT STRIP_TAC THEN
ASM_CASES_TAC `s:real^M->bool = {}` THENL
[EXISTS_TAC `\n:num. n` THEN ASM_REWRITE_TAC[
NOT_IN_EMPTY];
ALL_TAC] THEN
MP_TAC(ISPEC `s:real^M->bool`
COUNTABLE_AS_IMAGE) THEN
ASM_REWRITE_TAC[
LEFT_IMP_EXISTS_THM] THEN
X_GEN_TAC `X:num->real^M` THEN DISCH_THEN SUBST_ALL_TAC THEN
MP_TAC(ISPEC
`\i r. ?l. ((\n. ((f:num->real^M->real^N) o (r:num->num)) n
((X:num->real^M) i)) --> l) sequentially`
SUBSEQUENCE_DIAGONALIZATION_LEMMA) THEN
REWRITE_TAC[
FORALL_IN_IMAGE;
o_THM;
IN_UNIV] THEN
ANTS_TAC THENL [ALL_TAC; DISCH_THEN MATCH_ACCEPT_TAC] THEN CONJ_TAC THENL
[RULE_ASSUM_TAC(REWRITE_RULE[
FORALL_IN_IMAGE;
IN_UNIV]) THEN
MAP_EVERY X_GEN_TAC [`i:num`; `r:num->num`] THEN
MP_TAC(ISPEC `cball(vec 0:real^N,M)` compact) THEN
REWRITE_TAC[
COMPACT_CBALL] THEN DISCH_THEN(MP_TAC o SPEC
`\n. (f:num->real^M->real^N) ((r:num->num) n) (X(i:num))`) THEN
ASM_REWRITE_TAC[
IN_CBALL_0;
o_DEF] THEN MESON_TAC[];
REPEAT GEN_TAC THEN REWRITE_TAC[
LIM_SEQUENTIALLY;
GE] THEN
DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN
MATCH_MP_TAC
MONO_EXISTS THEN GEN_TAC THEN
MATCH_MP_TAC
MONO_FORALL THEN GEN_TAC THEN
MATCH_MP_TAC MONO_IMP THEN REWRITE_TAC[] THEN
ASM_MESON_TAC[
LE_TRANS; ARITH_RULE `
MAX a b <= c <=> a <= c /\ b <= c`]]);;
let ARZELA_ASCOLI = prove
(`!f:num->real^M->real^N s M.
compact s /\
(!n x. x
IN s ==> norm(f n x) <= M) /\
(!x e. x
IN s /\ &0 < e
==> ?d. &0 < d /\
!n y. y
IN s /\ norm(x - y) < d
==> norm(f n x - f n y) < e)
==> ?g. g
continuous_on s /\
?r. (!m n:num. m < n ==> r m < r n) /\
!e. &0 < e
==> ?N. !n x. n >= N /\ x
IN s
==> norm(f(r n) x - g x) < e`,
REPEAT STRIP_TAC THEN REWRITE_TAC[
GE] THEN
MATCH_MP_TAC(MESON[]
`(!k g. V k g ==> N g) /\ (?k. M k /\ ?g. V k g)
==> ?g. N g /\ ?k. M k /\ V k g`) THEN
CONJ_TAC THENL
[MAP_EVERY X_GEN_TAC [`k:num->num`; `g:real^M->real^N`] THEN
STRIP_TAC THEN MATCH_MP_TAC(ISPEC `sequentially`
CONTINUOUS_UNIFORM_LIMIT) THEN
EXISTS_TAC `(f:num->real^M->real^N) o (k:num->num)` THEN
ASM_SIMP_TAC[
EVENTUALLY_SEQUENTIALLY;
o_THM;
TRIVIAL_LIMIT_SEQUENTIALLY;
RIGHT_IMP_FORALL_THM; IMP_IMP] THEN
EXISTS_TAC `0` THEN REWRITE_TAC[
continuous_on; dist] THEN
ASM_MESON_TAC[
NORM_SUB];
ALL_TAC] THEN
MP_TAC(ISPECL
[`
IMAGE (f:num->real^M->real^N) (:num)`;
`s:real^M->bool`]
COMPACT_UNIFORMLY_EQUICONTINUOUS) THEN
REWRITE_TAC[
IMP_CONJ;
RIGHT_FORALL_IMP_THM;
FORALL_IN_IMAGE;
IN_UNIV] THEN
ANTS_TAC THENL
[REWRITE_TAC[dist] THEN ONCE_REWRITE_TAC[
NORM_SUB] THEN ASM_MESON_TAC[];
ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(K ALL_TAC o SPEC `x:real^M`)] THEN
REWRITE_TAC[
RIGHT_IMP_FORALL_THM] THEN
REWRITE_TAC[IMP_IMP; GSYM
CONJ_ASSOC; dist] THEN
DISCH_THEN(ASSUME_TAC o ONCE_REWRITE_RULE[
NORM_SUB]) THEN
REWRITE_TAC[GSYM dist;
UNIFORMLY_CONVERGENT_EQ_CAUCHY] THEN
X_CHOOSE_THEN `r:real^M->bool` STRIP_ASSUME_TAC
(ISPEC `s:real^M->bool`
SEPARABLE) THEN
MP_TAC(ISPECL [`f:num->real^M->real^N`; `r:real^M->bool`; `M:real`]
FUNCTION_CONVERGENT_SUBSEQUENCE) THEN
ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
MATCH_MP_TAC
MONO_EXISTS THEN X_GEN_TAC `k:num->num` THEN
REWRITE_TAC[
CONVERGENT_EQ_CAUCHY; cauchy] THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (LABEL_TAC "*")) THEN
ASM_REWRITE_TAC[] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `e / &3`) THEN
ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
COMPACT_EQ_HEINE_BOREL]) THEN
DISCH_THEN(MP_TAC o SPEC `
IMAGE (\x:real^M. ball(x,d)) r`) THEN
REWRITE_TAC[
FORALL_IN_IMAGE;
OPEN_BALL] THEN
ONCE_REWRITE_TAC[TAUT `a /\ b /\ c <=> b /\ a /\ c`] THEN
REWRITE_TAC[
EXISTS_FINITE_SUBSET_IMAGE] THEN ANTS_TAC THENL
[MATCH_MP_TAC
SUBSET_TRANS THEN EXISTS_TAC `closure r:real^M->bool` THEN
ASM_REWRITE_TAC[] THEN REWRITE_TAC[
SUBSET;
CLOSURE_APPROACHABLE] THEN
X_GEN_TAC `x:real^M` THEN DISCH_THEN(MP_TAC o SPEC `d:real`) THEN
ASM_REWRITE_TAC[
UNIONS_IMAGE;
IN_ELIM_THM;
IN_BALL];
DISCH_THEN(X_CHOOSE_THEN `t:real^M->bool` STRIP_ASSUME_TAC)] THEN
REMOVE_THEN "*" MP_TAC THEN REWRITE_TAC[
RIGHT_IMP_FORALL_THM] THEN
GEN_REWRITE_TAC LAND_CONV [
SWAP_FORALL_THM] THEN
DISCH_THEN(MP_TAC o SPEC `e / &3`) THEN
ASM_REWRITE_TAC[REAL_ARITH `&0 < e / &3 <=> &0 < e`] THEN
GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [
RIGHT_IMP_EXISTS_THM] THEN
REWRITE_TAC[
SKOLEM_THM;
LEFT_IMP_EXISTS_THM] THEN
X_GEN_TAC `M:real^M->num` THEN DISCH_THEN(LABEL_TAC "*") THEN
MP_TAC(ISPECL [`M:real^M->num`; `t:real^M->bool`]
UPPER_BOUND_FINITE_SET) THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC
MONO_EXISTS THEN X_GEN_TAC `N:num` THEN
DISCH_TAC THEN
MAP_EVERY X_GEN_TAC [`m:num`; `n:num`; `x:real^M`] THEN STRIP_TAC THEN
UNDISCH_TAC `s
SUBSET UNIONS (
IMAGE (\x:real^M. ball (x,d)) t)` THEN
REWRITE_TAC[
SUBSET;
UNIONS_IMAGE;
IN_ELIM_THM] THEN
DISCH_THEN(MP_TAC o SPEC `x:real^M`) THEN
ASM_REWRITE_TAC[
IN_BALL;
LEFT_IMP_EXISTS_THM; dist] THEN
X_GEN_TAC `y:real^M` THEN STRIP_TAC THEN
MATCH_MP_TAC(NORM_ARITH
`norm(f (k(m:num)) y - f (k m) x) < e / &3 /\
norm(f (k n) y - f (k n) x) < e / &3 /\
norm(f (k m) y - f (k n) y) < e / &3
==> norm(f (k m) x - f (k n) x :real^M) < e`) THEN
ASM_SIMP_TAC[] THEN REMOVE_THEN "*" (MP_TAC o SPEC `y:real^M`) THEN
ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
DISCH_THEN(MP_TAC o SPECL [`m:num`; `n:num`]) THEN
ASM_REWRITE_TAC[dist;
GE] THEN ASM_MESON_TAC[
SUBSET;
LE_TRANS]);;
let PARACOMPACT = prove
(`!s c. (!t:real^N->bool. t
IN c ==> open t) /\ s
SUBSET UNIONS c
==> ?c'. s
SUBSET UNIONS c' /\
(!u. u
IN c'
==> open u /\ ?t. t
IN c /\ u
SUBSET t) /\
(!x. x
IN s
==> ?v. open v /\ x
IN v /\
FINITE {u | u
IN c' /\ ~(u
INTER v = {})})`,
REPEAT STRIP_TAC THEN
ASM_CASES_TAC `s:real^N->bool = {}` THENL
[EXISTS_TAC `{}:(real^N->bool)->bool` THEN
ASM_REWRITE_TAC[
EMPTY_SUBSET;
NOT_IN_EMPTY];
ALL_TAC] THEN
SUBGOAL_THEN
`!x:real^N. x
IN s
==> ?t u. x
IN u /\ open u /\ closure u
SUBSET t /\ t
IN c`
MP_TAC THENL
[X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `x:real^N` o GEN_REWRITE_RULE I [
SUBSET]) THEN
ASM_REWRITE_TAC[
IN_UNIONS] THEN MATCH_MP_TAC
MONO_EXISTS THEN
X_GEN_TAC `t:real^N->bool` THEN STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `t:real^N->bool`) THEN
ASM_REWRITE_TAC[] THEN
GEN_REWRITE_TAC LAND_CONV [
OPEN_CONTAINS_CBALL] THEN
DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `ball(x:real^N,e)` THEN
ASM_SIMP_TAC[
OPEN_BALL;
CENTRE_IN_BALL;
CLOSURE_BALL];
GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV) [
RIGHT_IMP_EXISTS_THM] THEN
REWRITE_TAC[
LEFT_IMP_EXISTS_THM;
SKOLEM_THM] THEN
MAP_EVERY X_GEN_TAC
[`f:real^N->real^N->bool`; `e:real^N->real^N->bool`] THEN
STRIP_TAC] THEN
MP_TAC(ISPEC `
IMAGE (e:real^N->real^N->bool) s`
LINDELOF) THEN
ASM_SIMP_TAC[
FORALL_IN_IMAGE] THEN
ONCE_REWRITE_TAC[TAUT `p /\ q /\ r <=> q /\ p /\ r`] THEN
REWRITE_TAC[
EXISTS_COUNTABLE_SUBSET_IMAGE] THEN
DISCH_THEN(X_CHOOSE_THEN `k:real^N->bool`
(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
ASM_CASES_TAC `k:real^N->bool = {}` THENL
[ASM_REWRITE_TAC[] THEN ASM SET_TAC[]; ALL_TAC] THEN
MP_TAC(ISPEC `k:real^N->bool`
COUNTABLE_AS_IMAGE) THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_THEN `a:num->real^N` SUBST_ALL_TAC) THEN
STRIP_TAC THEN EXISTS_TAC
`{ f(a n:real^N)
DIFF UNIONS {closure(e(a m)):real^N->bool | m < n} |
n
IN (:num)}` THEN
REWRITE_TAC[
FORALL_IN_GSPEC;
IN_UNIV] THEN REPEAT CONJ_TAC THENL
[X_GEN_TAC `n:num` THEN CONJ_TAC THENL
[MATCH_MP_TAC
OPEN_DIFF THEN
CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
MATCH_MP_TAC
CLOSED_UNIONS THEN
REWRITE_TAC[
FORALL_IN_GSPEC;
CLOSED_CLOSURE] THEN
ONCE_REWRITE_TAC[
SIMPLE_IMAGE_GEN] THEN
SIMP_TAC[
FINITE_IMAGE;
FINITE_NUMSEG_LT];
EXISTS_TAC `f((a:num->real^N) n):real^N->bool` THEN ASM SET_TAC[]];
REWRITE_TAC[
SUBSET;
UNIONS_GSPEC;
IN_ELIM_THM;
IN_DIFF] THEN
X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
SUBGOAL_THEN `?n. x
IN (f((a:num->real^N) n):real^N->bool)` MP_TAC THENL
[RULE_ASSUM_TAC(REWRITE_RULE[
UNIONS_IMAGE;
EXISTS_IN_IMAGE]) THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
EXTENSION]) THEN
DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN
ASM_REWRITE_TAC[
IN_ELIM_THM;
IN_UNIV] THEN
DISCH_THEN(MP_TAC o snd o EQ_IMP_RULE) THEN
ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
MATCH_MP_TAC
MONO_EXISTS THEN X_GEN_TAC `n:num` THEN
STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `(a:num->real^N) n`) THEN
ANTS_TAC THENL [ASM SET_TAC[]; ASM_MESON_TAC[
CLOSURE_SUBSET;
SUBSET]];
GEN_REWRITE_TAC LAND_CONV [
num_WOP] THEN
MATCH_MP_TAC
MONO_EXISTS THEN ASM SET_TAC[]];
X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
RULE_ASSUM_TAC(REWRITE_RULE[
UNIONS_IMAGE;
EXISTS_IN_IMAGE]) THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
EXTENSION]) THEN
DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN
ASM_REWRITE_TAC[
IN_ELIM_THM;
IN_UNIV] THEN
DISCH_THEN(MP_TAC o snd o EQ_IMP_RULE) THEN
ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
DISCH_THEN(X_CHOOSE_TAC `n:num`) THEN
EXISTS_TAC `e((a:num->real^N) n):real^N->bool` THEN
ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
REWRITE_TAC[SET_RULE
`{u | (?n. u = f n) /\ P u} =
IMAGE f {n |n| P(f n) /\ n
IN (:num)}`] THEN
MATCH_MP_TAC
FINITE_IMAGE THEN MATCH_MP_TAC
FINITE_SUBSET THEN
EXISTS_TAC `{m:num | m <= n}` THEN REWRITE_TAC[
FINITE_NUMSEG_LE] THEN
REWRITE_TAC[
SUBSET;
IN_ELIM_THM;
IN_UNIV] THEN
X_GEN_TAC `m:num` THEN ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN
REWRITE_TAC[
NOT_LE] THEN DISCH_TAC THEN
MATCH_MP_TAC(SET_RULE `u
SUBSET t ==> (s
DIFF t)
INTER u = {}`) THEN
REWRITE_TAC[
SUBSET;
IN_UNIONS;
EXISTS_IN_GSPEC] THEN
ASM_MESON_TAC[
CLOSURE_SUBSET;
SUBSET]]);;
let PARACOMPACT_CLOSED_IN = prove
(`!u:real^N->bool s c.
closed_in (subtopology euclidean u) s /\
(!t:real^N->bool. t
IN c ==>
open_in (subtopology euclidean u) t) /\
s
SUBSET UNIONS c
==> ?c'. s
SUBSET UNIONS c' /\
(!v. v
IN c'
==>
open_in (subtopology euclidean u) v /\
?t. t
IN c /\ v
SUBSET t) /\
(!x. x
IN u
==> ?v.
open_in (subtopology euclidean u) v /\ x
IN v /\
FINITE {n | n
IN c' /\ ~(n
INTER v = {})})`,
REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC
(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC)) THEN
REWRITE_TAC[
OPEN_IN_OPEN] THEN
GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [
RIGHT_IMP_EXISTS_THM] THEN
REWRITE_TAC[
SKOLEM_THM;
LEFT_IMP_EXISTS_THM] THEN
X_GEN_TAC `uu:(real^N->bool)->(real^N->bool)` THEN
DISCH_THEN(ASSUME_TAC o GSYM) THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
CLOSED_IN_CLOSED]) THEN
DISCH_THEN(X_CHOOSE_THEN `k:real^N->bool`
(CONJUNCTS_THEN2 ASSUME_TAC SUBST_ALL_TAC)) THEN
MP_TAC(ISPECL
[`u:real^N->bool`;
`((:real^N)
DIFF k)
INSERT IMAGE (uu:(real^N->bool)->(real^N->bool)) c`]
PARACOMPACT) THEN
ASM_SIMP_TAC[
FORALL_IN_IMAGE;
UNIONS_IMAGE;
UNIONS_INSERT;
FORALL_IN_INSERT;
EXISTS_IN_IMAGE;
EXISTS_IN_INSERT; GSYM closed] THEN
ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
DISCH_THEN(X_CHOOSE_THEN `d:(real^N->bool)->bool` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `{u
INTER v:real^N->bool | v
IN d /\ ~(v
INTER k = {})}` THEN
REPEAT CONJ_TAC THENL
[REWRITE_TAC[
UNIONS_GSPEC] THEN ASM SET_TAC[];
REWRITE_TAC[
FORALL_IN_GSPEC] THEN ASM SET_TAC[];
X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `x:real^N`) THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_THEN `v:real^N->bool` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `u
INTER v:real^N->bool` THEN ASM_REWRITE_TAC[
IN_INTER] THEN
CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
ONCE_REWRITE_TAC[SET_RULE
`{y | y
IN {f x | P x} /\ Q y} =
IMAGE f {x | P x /\ Q(f x)}`] THEN
MATCH_MP_TAC
FINITE_IMAGE THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP
(REWRITE_RULE[
IMP_CONJ]
FINITE_SUBSET)) THEN SET_TAC[]]);;