(* ========================================================================= *)
(* Kurzweil-Henstock gauge integration in many dimensions. *)
(* *)
(* (c) Copyright, John Harrison 1998-2008 *)
(* ========================================================================= *)
needs "Library/products.ml";;
needs "Library/floor.ml";;
needs "Multivariate/derivatives.ml";;
prioritize_real();;
(* ------------------------------------------------------------------------- *)
(* Some useful lemmas about intervals. *)
(* ------------------------------------------------------------------------- *)
(* ------------------------------------------------------------------------- *)
(* This lemma about iterations comes up in a few places. *)
(* ------------------------------------------------------------------------- *)
let ITERATE_NONZERO_IMAGE_LEMMA = prove
(`!op s f g a.
monoidal op /\
FINITE s /\
g(a) = neutral op /\
(!x y. x
IN s /\ y
IN s /\ f x = f y /\ ~(x = y) ==> g(f x) = neutral op)
==> iterate op {f x | x | x
IN s /\ ~(f x = a)} g =
iterate op s (g o f)`,
let interval_upperbound = new_definition
`(interval_upperbound:(real^M->bool)->real^M) s =
lambda i. sup {a | ?x. x IN s /\ (x$i = a)}`;;let interval_lowerbound = new_definition
`(interval_lowerbound:(real^M->bool)->real^M) s =
lambda i. inf {a | ?x. x IN s /\ (x$i = a)}`;;let CONTENT_CLOSED_INTERVAL = prove
(`!a b:real^N. (!i. 1 <= i /\ i <= dimindex(:N) ==> a$i <= b$i)
==> content(interval[a,b]) =
product(1..dimindex(:N)) (\i. b$i - a$i)`,
let CONTENT_1 = prove
(`!a b. drop a <= drop b ==> content(interval[a,b]) = drop b - drop a`,
let CONTENT_POS_LT = prove
(`!a b:real^N.
(!i. 1 <= i /\ i <= dimindex(:N) ==> a$i < b$i)
==> &0 < content(interval[a,b])`,
let CONTENT_EQ_0_GEN = prove
(`!s:real^N->bool.
bounded s
==> (content s = &0 <=>
?i a. 1 <= i /\ i <= dimindex(:N) /\ !x. x
IN s ==> x$i = a)`,
let CONTENT_EQ_0 = prove
(`!a b:real^N.
content(interval[a,b]) = &0 <=>
?i. 1 <= i /\ i <= dimindex(:N) /\ b$i <= a$i`,
let CONTENT_0_SUBSET_GEN = prove
(`!s t:real^N->bool.
s
SUBSET t /\ bounded t /\ content t = &0 ==> content s = &0`,
REPEAT GEN_TAC THEN
REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
SUBGOAL_THEN `bounded(s:real^N->bool)` ASSUME_TAC THENL
[ASM_MESON_TAC[
BOUNDED_SUBSET]; ALL_TAC] THEN
ASM_SIMP_TAC[
CONTENT_EQ_0_GEN] THEN ASM SET_TAC[]);;
let CONTENT_CLOSED_INTERVAL_CASES = prove
(`!a b:real^N.
content(interval[a,b]) =
if !i. 1 <= i /\ i <= dimindex(:N) ==> a$i <= b$i
then product(1..dimindex(:N)) (\i. b$i - a$i)
else &0`,
let CONTENT_SUBSET = prove
(`!a b c d:real^N.
interval[a,b]
SUBSET interval[c,d]
==> content(interval[a,b]) <= content(interval[c,d])`,
REPEAT STRIP_TAC THEN GEN_REWRITE_TAC LAND_CONV [content] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[
CONTENT_POS_LE] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
SUBSET]) THEN
RULE_ASSUM_TAC(REWRITE_RULE[
INTERVAL_NE_EMPTY]) THEN
REWRITE_TAC[
IN_INTERVAL] THEN DISCH_THEN(fun th ->
MP_TAC(SPEC `a:real^N` th) THEN MP_TAC(SPEC `b:real^N` th)) THEN
ASM_SIMP_TAC[
REAL_LE_REFL; content] THEN REPEAT STRIP_TAC THEN
ONCE_REWRITE_TAC[TAUT `(if b then c else d) = (if ~b then d else c)`] THEN
REWRITE_TAC[
INTERVAL_NE_EMPTY] THEN COND_CASES_TAC THENL
[ALL_TAC; ASM_MESON_TAC[
REAL_LE_TRANS]] THEN
MATCH_MP_TAC
PRODUCT_LE_NUMSEG THEN
ASM_SIMP_TAC[
INTERVAL_LOWERBOUND;
INTERVAL_UPPERBOUND] THEN
REPEAT(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 -> STRIP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[] THEN
REAL_ARITH_TAC);;
let CONTENT_PASTECART = prove
(`!a b:real^M c d:real^N.
content(interval[pastecart a c,pastecart b d]) =
content(interval[a,b]) * content(interval[c,d])`,
REPEAT GEN_TAC THEN
SIMP_TAC[
CONTENT_CLOSED_INTERVAL_CASES;
LAMBDA_BETA] THEN
MATCH_MP_TAC(MESON[
REAL_MUL_LZERO;
REAL_MUL_RZERO]
`(p <=> p1 /\ p2) /\ z = x * y
==> (if p then z else &0) =
(if p1 then x else &0) * (if p2 then y else &0)`) THEN
CONJ_TAC THENL
[EQ_TAC THEN DISCH_TAC THEN TRY CONJ_TAC THEN X_GEN_TAC `i:num` THEN
STRIP_TAC THENL
[FIRST_X_ASSUM(MP_TAC o SPEC `i:num`) THEN
ASM_SIMP_TAC[pastecart;
LAMBDA_BETA;
DIMINDEX_FINITE_SUM] THEN
DISCH_THEN MATCH_MP_TAC THEN ASM_ARITH_TAC;
FIRST_X_ASSUM(MP_TAC o SPEC `i + dimindex(:M)`) THEN
ASM_SIMP_TAC[pastecart;
LAMBDA_BETA;
DIMINDEX_FINITE_SUM] THEN
ANTS_TAC THENL [ASM_ARITH_TAC; REWRITE_TAC[
ADD_SUB]] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN ASM_ARITH_TAC;
RULE_ASSUM_TAC(REWRITE_RULE[
DIMINDEX_FINITE_SUM]) THEN
ASM_CASES_TAC `i <= dimindex(:M)` THENL
[FIRST_X_ASSUM(MP_TAC o SPEC `i:num` o CONJUNCT1);
FIRST_X_ASSUM(MP_TAC o SPEC `i - dimindex(:M)` o CONJUNCT2)] THEN
ASM_SIMP_TAC[pastecart;
LAMBDA_BETA;
DIMINDEX_FINITE_SUM;
ARITH_RULE `i:num <= m ==> i <= m + n`] THEN
DISCH_THEN MATCH_MP_TAC THEN ASM_ARITH_TAC];
SIMP_TAC[
DIMINDEX_FINITE_SUM; ARITH_RULE `1 <= n + 1`;
PRODUCT_ADD_SPLIT] THEN
BINOP_TAC THENL
[ALL_TAC;
ONCE_REWRITE_TAC[
ADD_SYM] THEN REWRITE_TAC[
PRODUCT_OFFSET]] THEN
MATCH_MP_TAC
PRODUCT_EQ_NUMSEG THEN
SIMP_TAC[pastecart;
LAMBDA_BETA;
DIMINDEX_FINITE_SUM;
ADD_SUB;
ARITH_RULE `i:num <= m ==> i <= m + n`;
ARITH_RULE `i:num <= n ==> i + m <= m + n`] THEN
REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
ASM_ARITH_TAC]);;
let DIVISION_UNION_INTERVALS_EXISTS = prove
(`!a b c d:real^N.
~(interval[a,b] = {})
==> ?p. (interval[a,b]
INSERT p)
division_of
(interval[a,b]
UNION interval[c,d])`,
REPEAT STRIP_TAC THEN
ASM_CASES_TAC `interval[c:real^N,d] = {}` THENL
[ASM_REWRITE_TAC[
UNION_EMPTY] THEN ASM_MESON_TAC[
DIVISION_OF_SELF];
ALL_TAC] THEN
ASM_CASES_TAC `interval[a:real^N,b]
INTER interval[c,d] = {}` THENL
[EXISTS_TAC `{interval[c:real^N,d]}` THEN
ONCE_REWRITE_TAC[SET_RULE `{a,b} = {a}
UNION {b}`] THEN
MATCH_MP_TAC
DIVISION_DISJOINT_UNION THEN
ASM_SIMP_TAC[
DIVISION_OF_SELF] THEN
MATCH_MP_TAC(SET_RULE
`interior s
SUBSET s /\ interior t
SUBSET t /\ s
INTER t = {}
==> interior s
INTER interior t = {}`) THEN
ASM_REWRITE_TAC[
INTERIOR_SUBSET];
ALL_TAC] THEN
SUBGOAL_THEN
`?u v:real^N. interval[a,b]
INTER interval[c,d] = interval[u,v]`
STRIP_ASSUME_TAC THENL [MESON_TAC[
INTER_INTERVAL]; ALL_TAC] THEN
MP_TAC(ISPECL [`c:real^N`; `d:real^N`; `u:real^N`; `v:real^N`]
PARTIAL_DIVISION_EXTEND_1) THEN
ANTS_TAC THENL [ASM_MESON_TAC[
INTER_SUBSET]; ALL_TAC] THEN
DISCH_THEN(X_CHOOSE_THEN `p:(real^N->bool)->bool` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `p
DELETE interval[u:real^N,v]` THEN
SUBGOAL_THEN `interval[a:real^N,b]
UNION interval[c,d] =
interval[a,b]
UNION UNIONS(p
DELETE interval[u,v])`
SUBST1_TAC THENL
[FIRST_ASSUM(SUBST1_TAC o SYM o last o CONJUNCTS o
GEN_REWRITE_RULE I [
division_of]) THEN
ASM SET_TAC[];
ALL_TAC] THEN
ONCE_REWRITE_TAC[SET_RULE `x
INSERT s = {x}
UNION s`] THEN
MATCH_MP_TAC
DIVISION_DISJOINT_UNION THEN
ASM_SIMP_TAC[
DIVISION_OF_SELF] THEN CONJ_TAC THENL
[MATCH_MP_TAC
DIVISION_OF_SUBSET THEN
EXISTS_TAC `p:(real^N->bool)->bool` THEN
ASM_MESON_TAC[
DIVISION_OF_UNION_SELF;
DELETE_SUBSET];
ALL_TAC] THEN
REWRITE_TAC[GSYM
INTERIOR_INTER] THEN
MATCH_MP_TAC
EQ_TRANS THEN
EXISTS_TAC `interior(interval[u:real^N,v]
INTER
UNIONS (p
DELETE interval[u,v]))` THEN
CONJ_TAC THENL
[AP_TERM_TAC THEN MATCH_MP_TAC(SET_RULE
`!cd. p
SUBSET cd /\ uv = ab
INTER cd
==> (ab
INTER p = uv
INTER p)`) THEN
EXISTS_TAC `interval[c:real^N,d]` THEN
ASM_REWRITE_TAC[
UNIONS_SUBSET;
IN_DELETE] THEN
ASM_MESON_TAC[
division_of];
REWRITE_TAC[
INTERIOR_INTER] THEN
MATCH_MP_TAC
INTER_INTERIOR_UNIONS_INTERVALS THEN
REWRITE_TAC[
IN_DELETE;
OPEN_INTERIOR;
FINITE_DELETE] THEN
ASM_MESON_TAC[
division_of]]);;
let ELEMENTARY_UNION_INTERVAL_STRONG = prove
(`!p a b:real^N.
p
division_of (
UNIONS p)
==> ?q. p
SUBSET q /\ q
division_of (interval[a,b]
UNION UNIONS p)`,
REPEAT STRIP_TAC THEN ASM_CASES_TAC `p:(real^N->bool)->bool = {}` THENL
[ASM_REWRITE_TAC[
UNIONS_0;
UNION_EMPTY;
EMPTY_SUBSET] THEN
MESON_TAC[
ELEMENTARY_INTERVAL];
ALL_TAC] THEN
ASM_CASES_TAC `interval[a:real^N,b] = {}` THEN
ASM_REWRITE_TAC[
UNION_EMPTY] THENL [ASM_MESON_TAC[
SUBSET_REFL]; ALL_TAC] THEN
ASM_CASES_TAC `interior(interval[a:real^N,b]) = {}` THENL
[EXISTS_TAC `interval[a:real^N,b]
INSERT p` THEN
REWRITE_TAC[
division_of] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
division_of]) THEN
SIMP_TAC[
FINITE_INSERT;
UNIONS_INSERT] THEN ASM SET_TAC[];
ALL_TAC] THEN
ASM_CASES_TAC `interval[a:real^N,b]
SUBSET UNIONS p` THENL
[ASM_SIMP_TAC[SET_RULE `s
SUBSET t ==> s
UNION t = t`] THEN
ASM_MESON_TAC[
SUBSET_REFL];
ALL_TAC] THEN
SUBGOAL_THEN
`!k:real^N->bool. k
IN p
==> ?q. ~(k
IN q) /\ ~(q = {}) /\
(k
INSERT q)
division_of (interval[a,b]
UNION k)`
MP_TAC THENL
[X_GEN_TAC `k:real^N->bool` THEN DISCH_TAC THEN
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
division_of]) THEN
DISCH_THEN(MP_TAC o SPEC `k:real^N->bool` o CONJUNCT1 o CONJUNCT2) THEN
ASM_REWRITE_TAC[] THEN
REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
REWRITE_TAC[
LEFT_IMP_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC [`c:real^N`; `d:real^N`] THEN
DISCH_THEN SUBST_ALL_TAC THEN
ONCE_REWRITE_TAC[
UNION_COMM] THEN
MP_TAC(ISPECL [`c:real^N`; `d:real^N`; `a:real^N`; `b:real^N`]
DIVISION_UNION_INTERVALS_EXISTS) THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_TAC `q:(real^N->bool)->bool`) THEN
EXISTS_TAC `q
DELETE interval[c:real^N,d]` THEN
ASM_REWRITE_TAC[
IN_DELETE; SET_RULE
`x
INSERT (q
DELETE x) = x
INSERT q`] THEN
DISCH_TAC THEN
UNDISCH_TAC `(interval[c:real^N,d]
INSERT q)
division_of
(interval [c,d]
UNION interval [a,b])` THEN
ASM_SIMP_TAC[SET_RULE `s
DELETE x = {} ==> x
INSERT s = {x}`] THEN
REWRITE_TAC[
division_of;
UNIONS_1] THEN ASM SET_TAC[];
ALL_TAC] THEN
GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [
RIGHT_IMP_EXISTS_THM] THEN
REWRITE_TAC[
SKOLEM_THM] THEN
DISCH_THEN(X_CHOOSE_TAC `q:(real^N->bool)->(real^N->bool)->bool`) THEN
MP_TAC(ISPEC `
IMAGE (
UNIONS o (q:(real^N->bool)->(real^N->bool)->bool)) p`
ELEMENTARY_INTERS) THEN
FIRST_ASSUM(ASSUME_TAC o MATCH_MP
DIVISION_OF_FINITE) THEN
ASM_SIMP_TAC[
FINITE_IMAGE;
IMAGE_EQ_EMPTY;
FORALL_IN_IMAGE] THEN
ANTS_TAC THENL
[X_GEN_TAC `k:real^N->bool` THEN DISCH_TAC THEN
EXISTS_TAC `(q:(real^N->bool)->(real^N->bool)->bool) k` THEN
REWRITE_TAC[
o_THM] THEN MATCH_MP_TAC
DIVISION_OF_SUBSET THEN
EXISTS_TAC `(k:real^N->bool)
INSERT q k` THEN
CONJ_TAC THENL [ASM_MESON_TAC[
DIVISION_OF_UNION_SELF]; SET_TAC[]];
DISCH_THEN(X_CHOOSE_TAC `r:(real^N->bool)->bool`)] THEN
EXISTS_TAC `p
UNION r:(real^N->bool)->bool` THEN SIMP_TAC[
SUBSET_UNION] THEN
SUBGOAL_THEN
`interval[a:real^N,b]
UNION UNIONS p =
UNIONS p
UNION INTERS(
IMAGE (
UNIONS o q) p)`
SUBST1_TAC THENL
[GEN_REWRITE_TAC I [
EXTENSION] THEN X_GEN_TAC `y:real^N` THEN
REWRITE_TAC[
IN_UNION] THEN
ASM_CASES_TAC `(y:real^N)
IN UNIONS p` THEN ASM_REWRITE_TAC[
IN_INTERS] THEN
REWRITE_TAC[
FORALL_IN_UNIONS;
IMP_CONJ;
FORALL_IN_IMAGE;
RIGHT_FORALL_IMP_THM] THEN
SUBGOAL_THEN
`!k. k
IN p ==>
UNIONS(k
INSERT q k) = interval[a:real^N,b]
UNION k`
MP_TAC THENL [ASM_MESON_TAC[
division_of]; ALL_TAC] THEN
REWRITE_TAC[
UNIONS_INSERT;
o_THM] THEN
GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [
EXTENSION] THEN
REWRITE_TAC[
RIGHT_IMP_FORALL_THM;
IN_UNION] THEN
ONCE_REWRITE_TAC[
SWAP_FORALL_THM] THEN
DISCH_THEN(MP_TAC o SPEC `y:real^N`) THEN
UNDISCH_TAC `~((y:real^N)
IN UNIONS p)` THEN
SIMP_TAC[
IN_UNIONS;
NOT_EXISTS_THM; TAUT `~(a /\ b) <=> a ==> ~b`] THEN
ASM_CASES_TAC `(y:real^N)
IN interval[a,b]` THEN
ASM_REWRITE_TAC[] THEN ASM SET_TAC[];
ALL_TAC] THEN
MATCH_MP_TAC
DIVISION_DISJOINT_UNION THEN
ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[
INTER_COMM] THEN
MATCH_MP_TAC
INTER_INTERIOR_UNIONS_INTERVALS THEN
ASM_REWRITE_TAC[
OPEN_INTERIOR] THEN
CONJ_TAC THENL [ASM_MESON_TAC[
division_of]; ALL_TAC] THEN
X_GEN_TAC `k:real^N->bool` THEN DISCH_TAC THEN
ASM_SIMP_TAC[
INTERIOR_FINITE_INTERS;
FINITE_IMAGE] THEN
MATCH_MP_TAC(SET_RULE `(?x. x
IN p /\ f x
INTER s = {})
==>
INTERS (
IMAGE f p)
INTER s = {}`) THEN
REWRITE_TAC[
EXISTS_IN_IMAGE;
o_THM] THEN EXISTS_TAC `k:real^N->bool` THEN
ASM_REWRITE_TAC[] THEN
ONCE_REWRITE_TAC[
INTER_COMM] THEN
MATCH_MP_TAC
INTER_INTERIOR_UNIONS_INTERVALS THEN
ASM_REWRITE_TAC[
OPEN_INTERIOR] THEN REPEAT CONJ_TAC THENL
[ASM_MESON_TAC[
division_of;
FINITE_INSERT;
IN_INSERT];
ASM_MESON_TAC[
division_of;
FINITE_INSERT;
IN_INSERT];
ALL_TAC] THEN
FIRST_X_ASSUM(MP_TAC o SPEC `k:real^N->bool`) THEN
ASM_REWRITE_TAC[
division_of;
IN_INSERT] THEN
REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_MESON_TAC[]);;
let DIVISION_OF_NONTRIVIAL = prove
(`!s a b:real^N.
s
division_of interval[a,b] /\ ~(content(interval[a,b]) = &0)
==> {k | k
IN s /\ ~(content k = &0)}
division_of interval[a,b]`,
REPEAT GEN_TAC THEN WF_INDUCT_TAC `
CARD(s:(real^N->bool)->bool)` THEN
REPEAT STRIP_TAC THEN
ASM_CASES_TAC `{k:real^N->bool | k
IN s /\ ~(content k = &0)} = s` THEN
ASM_REWRITE_TAC[] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [
EXTENSION]) THEN
REWRITE_TAC[
IN_ELIM_THM;
NOT_FORALL_THM;
LEFT_IMP_EXISTS_THM] THEN
REWRITE_TAC[TAUT `~(a /\ ~b <=> a) <=> a /\ b`] THEN
X_GEN_TAC `k:real^N->bool` THEN STRIP_TAC THEN
FIRST_ASSUM(ASSUME_TAC o MATCH_MP
DIVISION_OF_FINITE) THEN
FIRST_X_ASSUM(MP_TAC o SPEC `s
DELETE (k:real^N->bool)`) THEN
ASM_SIMP_TAC[
CARD_DELETE; ARITH_RULE `n - 1 < n <=> ~(n = 0)`] THEN
ASM_SIMP_TAC[
CARD_EQ_0] THEN ANTS_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
ANTS_TAC THENL
[ALL_TAC;
MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN
ASM SET_TAC[]] THEN
REWRITE_TAC[
DIVISION_OF] THEN
FIRST_X_ASSUM(STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [
division_of]) THEN
ASM_SIMP_TAC[
FINITE_DELETE;
IN_DELETE] THEN
FIRST_ASSUM(MP_TAC o C MATCH_MP (ASSUME `(k:real^N->bool)
IN s`)) THEN
REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
REWRITE_TAC[
LEFT_IMP_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC [`c:real^N`; `d:real^N`] THEN
DISCH_THEN SUBST_ALL_TAC THEN
MATCH_MP_TAC(SET_RULE
`
UNIONS s = i /\ k
SUBSET UNIONS(s
DELETE k)
==>
UNIONS(s
DELETE k) = i`) THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(MESON[
CLOSED_LIMPT;
SUBSET]
`closed s /\ (!x. x
IN k ==> x
limit_point_of s) ==> k
SUBSET s`) THEN
CONJ_TAC THENL
[MATCH_MP_TAC
CLOSED_UNIONS THEN
ASM_REWRITE_TAC[
FINITE_DELETE;
IN_DELETE] THEN
ASM_MESON_TAC[
CLOSED_INTERVAL];
ALL_TAC] THEN
X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN REWRITE_TAC[
LIMPT_APPROACHABLE] THEN
X_GEN_TAC `e:real` THEN DISCH_TAC THEN REWRITE_TAC[dist] THEN
SUBGOAL_THEN `?y:real^N. y
IN UNIONS s /\ ~(y
IN interval[c,d]) /\
~(y = x) /\ norm(y - x) < e`
MP_TAC THENL [ALL_TAC; SET_TAC[]] THEN ASM_REWRITE_TAC[] THEN
MAP_EVERY UNDISCH_TAC
[`~(content(interval[a:real^N,b]) = &0)`;
`content(interval[c:real^N,d]) = &0`] THEN
REWRITE_TAC[
CONTENT_EQ_0;
NOT_EXISTS_THM] THEN
DISCH_THEN(X_CHOOSE_THEN `i:num` STRIP_ASSUME_TAC) THEN
DISCH_THEN(MP_TAC o SPEC `i:num`) THEN ASM_REWRITE_TAC[
REAL_NOT_LE] THEN
DISCH_TAC THEN UNDISCH_TAC `~(interval[c:real^N,d] = {})` THEN
REWRITE_TAC[
INTERVAL_EQ_EMPTY;
NOT_EXISTS_THM] THEN
DISCH_THEN(MP_TAC o SPEC `i:num`) THEN ASM_REWRITE_TAC[
REAL_NOT_LT] THEN
ASM_SIMP_TAC[REAL_ARITH `a <= b ==> (b <= a <=> a = b)`] THEN
DISCH_THEN(fun th -> SUBST_ALL_TAC th THEN ASSUME_TAC th) THEN
UNDISCH_TAC `interval[c:real^N,d]
SUBSET interval[a,b]` THEN
REWRITE_TAC[
SUBSET] THEN DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN
ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
MP_TAC(ASSUME `(x:real^N)
IN interval[c,d]`) THEN
GEN_REWRITE_TAC LAND_CONV [
IN_INTERVAL] THEN
DISCH_THEN(MP_TAC o SPEC `i:num`) THEN ASM_REWRITE_TAC[] THEN
ASM_SIMP_TAC[REAL_ARITH `d = c ==> (c <= x /\ x <= d <=> x = c)`] THEN
DISCH_TAC THEN
MP_TAC(ASSUME `(x:real^N)
IN interval[a,b]`) THEN
GEN_REWRITE_TAC LAND_CONV [
IN_INTERVAL] THEN
DISCH_THEN(MP_TAC o SPEC `i:num`) THEN ASM_REWRITE_TAC[] THEN
STRIP_TAC THEN EXISTS_TAC
`(lambda j. if j = i then
if (c:real^N)$i <= ((a:real^N)$i + (b:real^N)$i) / &2
then c$i + min e (b$i - c$i) / &2
else c$i - min e (c$i - a$i) / &2
else (x:real^N)$j):real^N` THEN
SIMP_TAC[
IN_INTERVAL;
LAMBDA_BETA;
CART_EQ] THEN REPEAT CONJ_TAC THENL
[X_GEN_TAC `j:num` THEN STRIP_TAC THEN
UNDISCH_TAC `(x:real^N)
IN interval[a,b]` THEN
REWRITE_TAC[
IN_INTERVAL] THEN DISCH_THEN(MP_TAC o SPEC `j:num`) THEN
ASM_REWRITE_TAC[] THEN COND_CASES_TAC THEN REWRITE_TAC[] THEN
FIRST_X_ASSUM SUBST_ALL_TAC THEN
ASM_REAL_ARITH_TAC;
DISCH_THEN(MP_TAC o SPEC `i:num`) THEN ASM_REWRITE_TAC[] THEN
ASM_REAL_ARITH_TAC;
DISCH_THEN(MP_TAC o SPEC `i:num`) THEN ASM_REWRITE_TAC[] THEN
ASM_REAL_ARITH_TAC;
REWRITE_TAC[
vector_norm; dot] THEN
SIMP_TAC[
LAMBDA_BETA;
VECTOR_SUB_COMPONENT; GSYM REAL_POW_2] THEN
REWRITE_TAC[REAL_ARITH
`((if p then x else y) - y) pow 2 = if p then (x - y) pow 2 else &0`] THEN
ASM_SIMP_TAC[
SUM_DELTA;
IN_NUMSEG;
POW_2_SQRT_ABS] THEN
ASM_REAL_ARITH_TAC]);;
let VSUM_CONTENT_NULL = prove
(`!f:real^M->real^N a b p.
content(interval[a,b]) = &0 /\
p
tagged_division_of interval[a,b]
==> vsum p (\(x,k). content k % f x) = vec 0`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC
VSUM_EQ_0 THEN
REWRITE_TAC[
FORALL_PAIR_THM] THEN
MAP_EVERY X_GEN_TAC [`p:real^M`; `k:real^M->bool`] THEN
DISCH_TAC THEN REWRITE_TAC[
VECTOR_MUL_EQ_0] THEN DISJ1_TAC THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
TAGGED_DIVISION_OF]) THEN
DISCH_THEN(MP_TAC o CONJUNCT1 o CONJUNCT2) THEN
DISCH_THEN(MP_TAC o SPECL [`p:real^M`; `k:real^M->bool`]) THEN
ASM_MESON_TAC[
CONTENT_SUBSET;
CONTENT_POS_LE; REAL_ARITH
`&0 <= x /\ x <= y /\ y = &0 ==> x = &0`]);;
let INTERVAL_BISECTION_STEP = prove
(`!P. P {} /\
(!s t. P s /\ P t /\ interior(s)
INTER interior(t) = {}
==> P(s
UNION t))
==> !a b:real^N.
~(P(interval[a,b]))
==> ?c d. ~(P(interval[c,d])) /\
!i. 1 <= i /\ i <= dimindex(:N)
==> a$i <= c$i /\ c$i <= d$i /\ d$i <= b$i /\
&2 * (d$i - c$i) <= b$i - a$i`,
REPEAT GEN_TAC THEN STRIP_TAC THEN REPEAT GEN_TAC THEN
ASM_CASES_TAC `!i. 1 <= i /\ i <= dimindex(:N)
==> (a:real^N)$i <= (b:real^N)$i` THENL
[ALL_TAC;
RULE_ASSUM_TAC(REWRITE_RULE[GSYM
INTERVAL_NE_EMPTY]) THEN
ASM_REWRITE_TAC[]] THEN
SUBGOAL_THEN
`!f.
FINITE f /\
(!s:real^N->bool. s
IN f ==> P s) /\
(!s:real^N->bool. s
IN f ==> ?a b. s = interval[a,b]) /\
(!s t. s
IN f /\ t
IN f /\ ~(s = t)
==> interior(s)
INTER interior(t) = {})
==> P(
UNIONS f)`
ASSUME_TAC THENL
[ONCE_REWRITE_TAC[
IMP_CONJ] THEN MATCH_MP_TAC
FINITE_INDUCT_STRONG THEN
ASM_SIMP_TAC[
UNIONS_0;
UNIONS_INSERT;
NOT_IN_EMPTY;
FORALL_IN_INSERT] THEN
REWRITE_TAC[IMP_IMP] THEN REPEAT GEN_TAC THEN DISCH_THEN(fun th ->
FIRST_X_ASSUM MATCH_MP_TAC THEN STRIP_ASSUME_TAC th) THEN
ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
MATCH_MP_TAC
INTER_INTERIOR_UNIONS_INTERVALS THEN
ASM_REWRITE_TAC[
OPEN_INTERIOR] THEN REPEAT STRIP_TAC THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[
IN_INSERT] THEN
ASM_MESON_TAC[];
ALL_TAC] THEN
DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC
`{ interval[c,d] |
!i. 1 <= i /\ i <= dimindex(:N)
==> ((c:real^N)$i = (a:real^N)$i) /\ (d$i = (a$i + b$i) / &2) \/
(c$i = (a$i + b$i) / &2) /\ ((d:real^N)$i = (b:real^N)$i)}`) THEN
ONCE_REWRITE_TAC[
IMP_CONJ] THEN ANTS_TAC THENL
[MATCH_MP_TAC
FINITE_SUBSET THEN
EXISTS_TAC
`
IMAGE (\s.
closed_interval
[(lambda i. if i
IN s then (a:real^N)$i else (a$i + b$i) / &2):real^N,
(lambda i. if i
IN s then (a$i + b$i) / &2 else (b:real^N)$i)])
{s | s
SUBSET (1..dimindex(:N))}` THEN
CONJ_TAC THENL
[SIMP_TAC[
FINITE_POWERSET;
FINITE_IMAGE;
FINITE_NUMSEG]; ALL_TAC] THEN
REWRITE_TAC[
SUBSET;
IN_ELIM_THM;
IN_IMAGE] THEN
X_GEN_TAC `k:real^N->bool` THEN
DISCH_THEN(X_CHOOSE_THEN `c:real^N` (X_CHOOSE_THEN `d:real^N`
(CONJUNCTS_THEN2 ASSUME_TAC SUBST1_TAC))) THEN
EXISTS_TAC `{i | 1 <= i /\ i <= dimindex(:N) /\
((c:real^N)$i = (a:real^N)$i)}` THEN
CONJ_TAC THENL [ALL_TAC; SIMP_TAC[
IN_ELIM_THM;
IN_NUMSEG]] THEN
AP_TERM_TAC THEN REWRITE_TAC[
CONS_11;
PAIR_EQ] THEN
SIMP_TAC[
CART_EQ;
LAMBDA_BETA;
IN_ELIM_THM] THEN
REPEAT(FIRST_X_ASSUM(MP_TAC o GEN `i:num` o SPEC `i:num`)) THEN
REWRITE_TAC[IMP_IMP;
AND_FORALL_THM] THEN
MATCH_MP_TAC
MONO_FORALL THEN GEN_TAC THEN
REWRITE_TAC[TAUT `(a ==> b) /\ (a ==> c) <=> (a ==> b /\ c)`] THEN
MATCH_MP_TAC MONO_IMP THEN REWRITE_TAC[] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
SIMP_TAC[
REAL_EQ_RDIV_EQ;
REAL_OF_NUM_LT; ARITH] THEN
REAL_ARITH_TAC;
ALL_TAC] THEN
GEN_REWRITE_TAC LAND_CONV [GSYM CONTRAPOS_THM] THEN ANTS_TAC THENL
[UNDISCH_TAC `~P(interval[a:real^N,b])` THEN MATCH_MP_TAC EQ_IMP THEN
AP_TERM_TAC THEN AP_TERM_TAC THEN CONV_TAC SYM_CONV THEN
GEN_REWRITE_TAC I [
EXTENSION] THEN
REWRITE_TAC[
IN_UNIONS;
IN_ELIM_THM] THEN X_GEN_TAC `x:real^N` THEN
REWRITE_TAC[
LEFT_AND_EXISTS_THM] THEN
ONCE_REWRITE_TAC[TAUT `(a /\ b) /\ c <=> b /\ a /\ c`] THEN
GEN_REWRITE_TAC LAND_CONV [
SWAP_EXISTS_THM] THEN
GEN_REWRITE_TAC (LAND_CONV o BINDER_CONV) [
SWAP_EXISTS_THM] THEN
REWRITE_TAC[
UNWIND_THM2;
IN_INTERVAL] THEN
REWRITE_TAC[
AND_FORALL_THM] THEN
REWRITE_TAC[TAUT `(a ==> b) /\ (a ==> c) <=> (a ==> b /\ c)`] THEN
REWRITE_TAC[GSYM
LAMBDA_SKOLEM] THEN AP_TERM_TAC THEN
GEN_REWRITE_TAC I [
FUN_EQ_THM] THEN X_GEN_TAC `i:num` THEN
REWRITE_TAC[] THEN
MATCH_MP_TAC(TAUT `(a ==> (b <=> c)) ==> ((a ==> b) <=> (a ==> c))`) THEN
STRIP_TAC THEN
ONCE_REWRITE_TAC[TAUT `(a \/ b) /\ c <=> ~(a ==> ~c) \/ ~(b ==> ~c)`] THEN
SIMP_TAC[] THEN
REWRITE_TAC[TAUT `~(a ==> ~b) <=> a /\ b`; GSYM
CONJ_ASSOC] THEN
REWRITE_TAC[
EXISTS_OR_THM;
RIGHT_EXISTS_AND_THM] THEN
REWRITE_TAC[
LEFT_EXISTS_AND_THM;
EXISTS_REFL] THEN
SIMP_TAC[
REAL_LE_LDIV_EQ;
REAL_LE_RDIV_EQ;
REAL_OF_NUM_LT; ARITH] THEN
REAL_ARITH_TAC;
ALL_TAC] THEN
REWRITE_TAC[
FORALL_IN_GSPEC] THEN
MATCH_MP_TAC(TAUT `b /\ (~a ==> e) /\ c ==> ~(a /\ b /\ c) ==> e`) THEN
CONJ_TAC THENL [MESON_TAC[]; ALL_TAC] THEN CONJ_TAC THENL
[REWRITE_TAC[
NOT_FORALL_THM; NOT_IMP] THEN
REPEAT(MATCH_MP_TAC
MONO_EXISTS THEN GEN_TAC) THEN
DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC
MONO_FORALL THEN X_GEN_TAC `i:num` THEN
DISCH_THEN(fun th -> REPEAT DISCH_TAC THEN MP_TAC th) THEN
FIRST_X_ASSUM(MP_TAC o SPEC `i:num`) THEN ASM_REWRITE_TAC[] THEN
REAL_ARITH_TAC;
ALL_TAC] THEN
REWRITE_TAC[
IMP_CONJ;
RIGHT_FORALL_IMP_THM;
FORALL_IN_GSPEC] THEN
REWRITE_TAC[IMP_IMP;
INTERIOR_CLOSED_INTERVAL] THEN
REWRITE_TAC[
RIGHT_IMP_FORALL_THM] THEN
MAP_EVERY X_GEN_TAC
[`c1:real^N`; `d1:real^N`; `c2:real^N`; `d2:real^N`] THEN
ASM_CASES_TAC `(c1 = c2:real^N) /\ (d1 = d2:real^N)` THENL
[ASM_REWRITE_TAC[]; ALL_TAC] THEN
DISCH_THEN(fun th ->
DISCH_THEN(CONJUNCTS_THEN2 MP_TAC (K ALL_TAC)) THEN MP_TAC th) THEN
REWRITE_TAC[IMP_IMP] THEN
UNDISCH_TAC `~((c1 = c2:real^N) /\ (d1 = d2:real^N))` THEN
REWRITE_TAC[
CART_EQ;
INTERIOR_CLOSED_INTERVAL] THEN
REWRITE_TAC[
AND_FORALL_THM] THEN
REWRITE_TAC[TAUT `(a ==> b) /\ (a ==> c) <=> (a ==> b /\ c)`] THEN
REWRITE_TAC[
NOT_FORALL_THM] THEN
DISCH_THEN(X_CHOOSE_THEN `j:num` (fun th ->
DISCH_THEN(MP_TAC o SPEC `j:num`) THEN MP_TAC th)) THEN
REWRITE_TAC[NOT_IMP] THEN
DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC) THEN
ASM_REWRITE_TAC[IMP_IMP] THEN
DISCH_THEN(CONJUNCTS_THEN2 MP_TAC STRIP_ASSUME_TAC) THEN
ASM_REWRITE_TAC[
EXTENSION;
IN_INTERVAL;
NOT_IN_EMPTY;
IN_INTER] THEN
SIMP_TAC[
REAL_EQ_RDIV_EQ;
REAL_EQ_LDIV_EQ;
REAL_OF_NUM_LT; ARITH] THEN
REWRITE_TAC[
REAL_ARITH `~((a * &2 = a + b) /\ (a + b = b * &2)) <=> ~(a = b)`;
REAL_ARITH `~((a + b = a * &2) /\ (b * &2 = a + b)) <=> ~(a = b)`] THEN
DISCH_THEN(fun th -> X_GEN_TAC `x:real^N` THEN MP_TAC th) THEN
REWRITE_TAC[
AND_FORALL_THM] THEN
REWRITE_TAC[TAUT `(a ==> b) /\ (a ==> c) <=> (a ==> b /\ c)`] THEN
ASM_REWRITE_TAC[CONTRAPOS_THM] THEN
DISCH_THEN(MP_TAC o SPEC `j:num`) THEN ASM_REWRITE_TAC[] THEN
REAL_ARITH_TAC);;
let INTERVAL_BISECTION = prove
(`!P. P {} /\
(!s t. P s /\ P t /\ interior(s)
INTER interior(t) = {}
==> P(s
UNION t))
==> !a b:real^N.
~(P(interval[a,b]))
==> ?x. x
IN interval[a,b] /\
!e. &0 < e
==> ?c d. x
IN interval[c,d] /\
interval[c,d]
SUBSET ball(x,e) /\
interval[c,d]
SUBSET interval[a,b] /\
~P(interval[c,d])`,
REPEAT STRIP_TAC THEN
SUBGOAL_THEN
`?A B. (A(0) = a:real^N) /\ (B(0) = b) /\
!n. ~(P(interval[A(SUC n),B(SUC n)])) /\
!i. 1 <= i /\ i <= dimindex(:N)
==> A(n)$i <= A(SUC n)$i /\
A(SUC n)$i <= B(SUC n)$i /\
B(SUC n)$i <= B(n)$i /\
&2 * (B(SUC n)$i - A(SUC n)$i) <= B(n)$i - A(n)$i`
STRIP_ASSUME_TAC THENL
[MP_TAC(ISPEC `P:(real^N->bool)->bool`
INTERVAL_BISECTION_STEP) THEN
ASM_REWRITE_TAC[] THEN
GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV) [
RIGHT_IMP_EXISTS_THM] THEN
REWRITE_TAC[
SKOLEM_THM] THEN
DISCH_THEN(X_CHOOSE_THEN `C:real^N->real^N->real^N`
(X_CHOOSE_THEN `D:real^N->real^N->real^N` ASSUME_TAC)) THEN
MP_TAC(prove_recursive_functions_exist num_RECURSION
`(E 0 = a:real^N,b:real^N) /\
(!n. E(SUC n) = C (
FST(E n)) (
SND(E n)),
D (
FST(E n)) (
SND(E n)))`) THEN
DISCH_THEN(X_CHOOSE_THEN `E:num->real^N#real^N` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `\n.
FST((E:num->real^N#real^N) n)` THEN
EXISTS_TAC `\n.
SND((E:num->real^N#real^N) n)` THEN
ASM_REWRITE_TAC[] THEN INDUCT_TAC THEN ASM_SIMP_TAC[];
ALL_TAC] THEN
SUBGOAL_THEN
`!e. &0 < e
==> ?n:num. !x y. x
IN interval[A(n),B(n)] /\ y
IN interval[A(n),B(n)]
==> dist(x,y:real^N) < e`
ASSUME_TAC THENL
[X_GEN_TAC `e:real` THEN DISCH_TAC THEN MP_TAC(SPEC
`sum(1..dimindex(:N)) (\i. (b:real^N)$i - (a:real^N)$i) / e`
REAL_ARCH_POW2) THEN
MATCH_MP_TAC
MONO_EXISTS THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN
MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN STRIP_TAC THEN
MATCH_MP_TAC
REAL_LET_TRANS THEN
EXISTS_TAC `sum(1..dimindex(:N))(\i. abs((x - y:real^N)$i))` THEN
REWRITE_TAC[dist;
NORM_LE_L1] THEN
MATCH_MP_TAC
REAL_LET_TRANS THEN
EXISTS_TAC `sum(1..dimindex(:N))
(\i. (B:num->real^N)(n)$i - (A:num->real^N)(n)$i)` THEN
CONJ_TAC THENL
[MATCH_MP_TAC
SUM_LE_NUMSEG THEN SIMP_TAC[
VECTOR_SUB_COMPONENT] THEN
REPEAT STRIP_TAC THEN
MATCH_MP_TAC(REAL_ARITH `a <= x /\ x <= b /\ a <= y /\ y <= b
==> abs(x - y) <= b - a`) THEN
UNDISCH_TAC `x
IN interval[(A:num->real^N) n,B n]` THEN
UNDISCH_TAC `y
IN interval[(A:num->real^N) n,B n]` THEN
REWRITE_TAC[
IN_INTERVAL] THEN ASM_SIMP_TAC[];
ALL_TAC] THEN
MATCH_MP_TAC
REAL_LET_TRANS THEN EXISTS_TAC
`sum(1..dimindex(:N)) (\i. (b:real^N)$i - (a:real^N)$i) /
&2 pow n` THEN
CONJ_TAC THENL
[ALL_TAC;
SIMP_TAC[
REAL_LT_LDIV_EQ;
REAL_POW_LT;
REAL_OF_NUM_LT; ARITH] THEN
ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
ASM_SIMP_TAC[GSYM
REAL_LT_LDIV_EQ]] THEN
REWRITE_TAC[
real_div] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
REWRITE_TAC[GSYM
SUM_LMUL] THEN MATCH_MP_TAC
SUM_LE_NUMSEG THEN
X_GEN_TAC `j:num` THEN STRIP_TAC THEN REWRITE_TAC[] THEN
ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN REWRITE_TAC[GSYM
real_div] THEN
SPEC_TAC(`n:num`,`m:num`) THEN INDUCT_TAC THEN
ASM_REWRITE_TAC[
real_pow;
REAL_DIV_1;
REAL_LE_REFL] THEN
ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
REWRITE_TAC[
real_div;
REAL_INV_MUL; REAL_MUL_ASSOC] THEN
SIMP_TAC[GSYM
real_div;
REAL_LE_RDIV_EQ;
REAL_OF_NUM_LT; ARITH] THEN
ASM_MESON_TAC[
REAL_LE_TRANS; REAL_MUL_SYM];
ALL_TAC] THEN
SUBGOAL_THEN `?a:real^N. !n:num. a
IN interval[A(n),B(n)]` MP_TAC THENL
[MATCH_MP_TAC
DECREASING_CLOSED_NEST THEN
ASM_REWRITE_TAC[
CLOSED_INTERVAL] THEN CONJ_TAC THENL
[REWRITE_TAC[
INTERVAL_EQ_EMPTY] THEN
ASM_MESON_TAC[
REAL_NOT_LT;
REAL_LE_TRANS];
ALL_TAC] THEN
REWRITE_TAC[
LE_EXISTS] THEN SIMP_TAC[
LEFT_IMP_EXISTS_THM] THEN
X_GEN_TAC `m:num` THEN ONCE_REWRITE_TAC[
SWAP_FORALL_THM] THEN
REWRITE_TAC[GSYM
LEFT_IMP_EXISTS_THM;
EXISTS_REFL] THEN
INDUCT_TAC THEN REWRITE_TAC[
ADD_CLAUSES;
SUBSET_REFL] THEN
MATCH_MP_TAC
SUBSET_TRANS THEN
EXISTS_TAC `interval[A(m + d:num):real^N,B(m + d)]` THEN
ASM_REWRITE_TAC[] THEN
REWRITE_TAC[
SUBSET;
IN_INTERVAL] THEN ASM_MESON_TAC[
REAL_LE_TRANS];
ALL_TAC] THEN
MATCH_MP_TAC
MONO_EXISTS THEN X_GEN_TAC `x0:real^N` THEN
DISCH_TAC THEN CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_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_TAC `n:num`) THEN
MAP_EVERY EXISTS_TAC [`(A:num->real^N) n`; `(B:num->real^N) n`] THEN
ASM_REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL
[REWRITE_TAC[
SUBSET;
IN_BALL] THEN ASM_MESON_TAC[];
ALL_TAC;
SPEC_TAC(`n:num`,`p:num`) THEN INDUCT_TAC THEN ASM_REWRITE_TAC[]] THEN
SUBGOAL_THEN
`!m n. m <= n ==> interval[(A:num->real^N) n,B n]
SUBSET interval[A m,B m]`
(fun th -> ASM_MESON_TAC[
SUBSET;
LE_0; th]) THEN
MATCH_MP_TAC
TRANSITIVE_STEPWISE_LE THEN
REPEAT(CONJ_TAC THENL [SET_TAC[]; ALL_TAC]) THEN
REWRITE_TAC[
SUBSET_INTERVAL] THEN ASM_MESON_TAC[]);;
let HAS_INTEGRAL_UNIQUE = prove
(`!f:real^M->real^N i k1 k2.
(f
has_integral k1) i /\ (f
has_integral k2) i ==> k1 = k2`,
REPEAT GEN_TAC THEN
SUBGOAL_THEN
`!f:real^M->real^N a b k1 k2.
(f
has_integral k1) (interval[a,b]) /\
(f
has_integral k2) (interval[a,b])
==> k1 = k2`
MP_TAC THENL
[REPEAT GEN_TAC THEN REWRITE_TAC[
has_integral] THEN
REWRITE_TAC[
AND_FORALL_THM] THEN
REWRITE_TAC[TAUT `(a ==> b) /\ (a ==> c) <=> a ==> b /\ c`] THEN
ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN
ONCE_REWRITE_TAC[GSYM
VECTOR_SUB_EQ] THEN
REWRITE_TAC[GSYM
NORM_POS_LT] THEN DISCH_TAC THEN
DISCH_THEN(MP_TAC o SPEC `norm(k1 - k2 :real^N) / &2`) THEN
ASM_REWRITE_TAC[
REAL_HALF] THEN
DISCH_THEN(CONJUNCTS_THEN2
(X_CHOOSE_THEN `d1:real^M->real^M->bool` STRIP_ASSUME_TAC)
(X_CHOOSE_THEN `d2:real^M->real^M->bool` STRIP_ASSUME_TAC)) THEN
MP_TAC(ISPEC `\x. ((d1:real^M->real^M->bool) x)
INTER (d2 x)`
FINE_DIVISION_EXISTS) THEN
ASM_SIMP_TAC[
GAUGE_INTER] THEN
DISCH_THEN(MP_TAC o SPECL [`a:real^M`; `b:real^M`]) THEN
REPEAT(FIRST_X_ASSUM(MP_TAC o check (is_forall o concl))) THEN
REWRITE_TAC[] THEN REWRITE_TAC[IMP_IMP;
NOT_EXISTS_THM] THEN
REWRITE_TAC[
AND_FORALL_THM] THEN MATCH_MP_TAC
MONO_FORALL THEN
GEN_TAC THEN
MATCH_MP_TAC(TAUT
`(f0 ==> f1 /\ f2) /\ ~(n1 /\ n2)
==> (t /\ f1 ==> n1) /\ (t /\ f2 ==> n2) ==> ~(t /\ f0)`) THEN
CONJ_TAC THENL [SIMP_TAC[fine;
SUBSET_INTER]; ALL_TAC] THEN
MATCH_MP_TAC(REAL_ARITH `c <= a + b ==> ~(a < c / &2 /\ b < c / &2)`) THEN
MESON_TAC[
NORM_SUB;
NORM_TRIANGLE; VECTOR_ARITH
`k1 - k2:real^N = (k1 - x) + (x - k2)`];
ALL_TAC] THEN
DISCH_TAC THEN ONCE_REWRITE_TAC[
has_integral_alt] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[] THENL
[ASM_MESON_TAC[]; ALL_TAC] THEN
DISCH_TAC THEN MATCH_MP_TAC(NORM_ARITH
`~(&0 < norm(x - y)) ==> x = y`) THEN
DISCH_TAC THEN
FIRST_X_ASSUM(CONJUNCTS_THEN (MP_TAC o SPEC `norm(k1 - k2:real^N) / &2`)) THEN
ASM_REWRITE_TAC[
REAL_HALF] THEN
DISCH_THEN(X_CHOOSE_THEN `B1:real` STRIP_ASSUME_TAC) THEN
DISCH_THEN(X_CHOOSE_THEN `B2:real` STRIP_ASSUME_TAC) THEN
MP_TAC(ISPEC
`ball(vec 0,B1)
UNION ball(vec 0:real^M,B2)`
BOUNDED_SUBSET_CLOSED_INTERVAL) THEN
REWRITE_TAC[
BOUNDED_UNION;
BOUNDED_BALL;
UNION_SUBSET;
NOT_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC [`a:real^M`; `b:real^M`] THEN
DISCH_THEN(CONJUNCTS_THEN(ANTE_RES_THEN MP_TAC)) THEN
DISCH_THEN(X_CHOOSE_THEN `w:real^N` STRIP_ASSUME_TAC) THEN
DISCH_THEN(X_CHOOSE_THEN `z:real^N` STRIP_ASSUME_TAC) THEN
SUBGOAL_THEN `w:real^N = z:real^N` SUBST_ALL_TAC THEN
ASM_MESON_TAC[NORM_ARITH
`~(norm(z - k1) < norm(k1 - k2) / &2 /\
norm(z - k2) < norm(k1 - k2) / &2)`]);;
let HAS_INTEGRAL_LINEAR = prove
(`!f:real^M->real^N y s h:real^N->real^P.
(f
has_integral y) s /\ linear h ==> ((h o f)
has_integral h(y)) s`,
SUBGOAL_THEN
`!f:real^M->real^N y a b h:real^N->real^P.
(f
has_integral y) (interval[a,b]) /\ linear h
==> ((h o f)
has_integral h(y)) (interval[a,b])`
MP_TAC THENL
[REPEAT GEN_TAC THEN REWRITE_TAC[
has_integral] THEN STRIP_TAC THEN
FIRST_ASSUM(MP_TAC o MATCH_MP
LINEAR_BOUNDED_POS) THEN
DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN
X_GEN_TAC `e:real` THEN DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `e:real / B`) THEN
ASM_SIMP_TAC[
REAL_LT_DIV] THEN
MATCH_MP_TAC
MONO_EXISTS THEN GEN_TAC THEN
STRIP_TAC THEN ASM_SIMP_TAC[] THEN
X_GEN_TAC `p:real^M#(real^M->bool)->bool` THEN STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `p:real^M#(real^M->bool)->bool`) THEN
ASM_SIMP_TAC[
REAL_LT_RDIV_EQ] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
MATCH_MP_TAC(REAL_ARITH `x <= y ==> y < e ==> x < e`) THEN
FIRST_ASSUM(fun th -> W(fun (asl,w) ->
MP_TAC(PART_MATCH rand th (rand w)))) THEN
MATCH_MP_TAC(REAL_ARITH `x <= y ==> y <= e ==> x <= e`) THEN
FIRST_ASSUM(ASSUME_TAC o MATCH_MP
TAGGED_DIVISION_OF_FINITE) THEN
ASM_SIMP_TAC[
LINEAR_SUB;
LINEAR_VSUM;
o_DEF;
LAMBDA_PAIR_THM;
LINEAR_CMUL;
REAL_LE_REFL];
ALL_TAC] THEN
DISCH_TAC THEN REPEAT GEN_TAC THEN
DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN
ONCE_REWRITE_TAC[
has_integral_alt] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[] THENL
[ASM_MESON_TAC[]; ALL_TAC] THEN
DISCH_TAC THEN
FIRST_ASSUM(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC o MATCH_MP
LINEAR_BOUNDED_POS) THEN
X_GEN_TAC `e:real` THEN DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `e / B:real`) THEN
ASM_SIMP_TAC[
REAL_LT_DIV] THEN
MATCH_MP_TAC
MONO_EXISTS THEN X_GEN_TAC `M:real` THEN
MATCH_MP_TAC MONO_AND THEN REWRITE_TAC[] THEN
REPEAT(MATCH_MP_TAC
MONO_FORALL THEN GEN_TAC) THEN
MATCH_MP_TAC MONO_IMP THEN REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_THEN `z:real^N` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `(h:real^N->real^P) z` THEN
SUBGOAL_THEN
`(\x. if x
IN s then ((h:real^N->real^P) o (f:real^M->real^N)) x else vec 0)
= h o (\x. if x
IN s then f x else vec 0)`
SUBST1_TAC THENL
[REWRITE_TAC[
FUN_EQ_THM;
o_THM] THEN ASM_MESON_TAC[
LINEAR_0]; ALL_TAC] THEN
ASM_SIMP_TAC[GSYM
LINEAR_SUB] THEN
MATCH_MP_TAC
REAL_LET_TRANS THEN EXISTS_TAC `B * norm(z - y:real^N)` THEN
ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
ASM_SIMP_TAC[GSYM
REAL_LT_RDIV_EQ]);;
let HAS_INTEGRAL_ADD = prove
(`!f:real^M->real^N g s.
(f
has_integral k) s /\ (g
has_integral l) s
==> ((\x. f(x) + g(x))
has_integral (k + l)) s`,
SUBGOAL_THEN
`!f:real^M->real^N g k l a b.
(f
has_integral k) (interval[a,b]) /\
(g
has_integral l) (interval[a,b])
==> ((\x. f(x) + g(x))
has_integral (k + l)) (interval[a,b])`
ASSUME_TAC THENL
[REPEAT GEN_TAC THEN REWRITE_TAC[
has_integral;
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(CONJUNCTS_THEN2
(X_CHOOSE_THEN `d1:real^M->real^M->bool` STRIP_ASSUME_TAC)
(X_CHOOSE_THEN `d2:real^M->real^M->bool` STRIP_ASSUME_TAC)) THEN
EXISTS_TAC `\x. ((d1:real^M->real^M->bool) x)
INTER (d2 x)` THEN
ASM_SIMP_TAC[
GAUGE_INTER] THEN
REWRITE_TAC[
tagged_division_of;
tagged_partial_division_of] THEN
SIMP_TAC[
VSUM_ADD;
VECTOR_ADD_LDISTRIB;
LAMBDA_PAIR] THEN
REWRITE_TAC[GSYM
LAMBDA_PAIR] THEN
REWRITE_TAC[GSYM
tagged_partial_division_of] THEN
REWRITE_TAC[GSYM
tagged_division_of;
FINE_INTER] THEN
SIMP_TAC[VECTOR_ARITH `(a + b) - (c + d) = (a - c) + (b - d):real^N`] THEN
REPEAT STRIP_TAC THEN MATCH_MP_TAC
NORM_TRIANGLE_LT THEN
MATCH_MP_TAC(REAL_ARITH `x < e / &2 /\ y < e / &2 ==> x + y < e`) THEN
ASM_SIMP_TAC[];
ALL_TAC] THEN
REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[
has_integral_alt] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[] THENL
[ASM_MESON_TAC[]; ALL_TAC] THEN
DISCH_TAC THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN
FIRST_X_ASSUM(CONJUNCTS_THEN (MP_TAC o SPEC `e / &2`)) THEN
ASM_REWRITE_TAC[
REAL_HALF] THEN
DISCH_THEN(X_CHOOSE_THEN `B1:real` STRIP_ASSUME_TAC) THEN
DISCH_THEN(X_CHOOSE_THEN `B2:real` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `max B1 B2:real` THEN ASM_REWRITE_TAC[
REAL_LT_MAX] THEN
REWRITE_TAC[
BALL_MAX_UNION;
UNION_SUBSET] THEN
MAP_EVERY X_GEN_TAC [`a:real^M`; `b:real^M`] THEN
DISCH_THEN(CONJUNCTS_THEN(ANTE_RES_THEN MP_TAC)) THEN
DISCH_THEN(X_CHOOSE_THEN `w:real^N` STRIP_ASSUME_TAC) THEN
DISCH_THEN(X_CHOOSE_THEN `z:real^N` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `w + z:real^N` THEN
SUBGOAL_THEN
`(\x. if x
IN s then (f:real^M->real^N) x + g x else vec 0) =
(\x. (if x
IN s then f x else vec 0) + (if x
IN s then g x else vec 0))`
SUBST1_TAC THENL
[REWRITE_TAC[
FUN_EQ_THM] THEN GEN_TAC THEN COND_CASES_TAC THEN
ASM_REWRITE_TAC[] THEN VECTOR_ARITH_TAC;
ALL_TAC] THEN
ASM_SIMP_TAC[] THEN
REPEAT(FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM
REAL_NOT_LE])) THEN
NORM_ARITH_TAC);;
let INTEGRABLE_CAUCHY = prove
(`!f:real^M->real^N a b.
f
integrable_on interval[a,b] <=>
!e. &0 < e
==> ?d. gauge d /\
!p1 p2. p1
tagged_division_of interval[a,b] /\ d fine p1 /\
p2
tagged_division_of interval[a,b] /\ d fine p2
==> norm(vsum p1 (\(x,k). content k % f x) -
vsum p2 (\(x,k). content k % f x)) < e`,
REPEAT GEN_TAC THEN REWRITE_TAC[
integrable_on;
has_integral] THEN
EQ_TAC THEN DISCH_TAC THENL
[X_GEN_TAC `e:real` THEN DISCH_TAC THEN
FIRST_X_ASSUM(X_CHOOSE_THEN `y:real^N` (MP_TAC o SPEC `e / &2`)) THEN
ASM_REWRITE_TAC[
REAL_HALF] THEN MATCH_MP_TAC
MONO_EXISTS THEN
X_GEN_TAC `d:real^M->real^M->bool` THEN
REWRITE_TAC[GSYM dist] THEN MESON_TAC[
DIST_TRIANGLE_HALF_L];
ALL_TAC] THEN
FIRST_X_ASSUM(MP_TAC o GEN `n:num` o SPEC `inv(&n + &1)`) THEN
REWRITE_TAC[
REAL_LT_INV_EQ; REAL_ARITH `&0 < &n + &1`;
SKOLEM_THM] THEN
DISCH_THEN(X_CHOOSE_THEN `d:num->real^M->real^M->bool` MP_TAC) THEN
REWRITE_TAC[
FORALL_AND_THM] THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (LABEL_TAC "*")) THEN
MP_TAC(GEN `n:num`
(ISPECL [`\x.
INTERS {(d:num->real^M->real^M->bool) i x | i
IN 0..n}`;
`a:real^M`; `b:real^M`]
FINE_DIVISION_EXISTS)) THEN
ASM_SIMP_TAC[
GAUGE_INTERS;
FINE_INTERS;
FINITE_NUMSEG;
SKOLEM_THM] THEN
REWRITE_TAC[
IN_NUMSEG;
LE_0;
FORALL_AND_THM] THEN
DISCH_THEN(X_CHOOSE_THEN `p:num->(real^M#(real^M->bool))->bool`
STRIP_ASSUME_TAC) THEN
SUBGOAL_THEN
`cauchy (\n. vsum (p n)
(\(x,k:real^M->bool). content k % (f:real^M->real^N) x))`
MP_TAC THENL
[REWRITE_TAC[cauchy] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
REAL_ARCH_INV]) THEN
MATCH_MP_TAC
MONO_EXISTS THEN X_GEN_TAC `N:num` THEN STRIP_TAC THEN
MATCH_MP_TAC
WLOG_LE THEN CONJ_TAC THENL
[MESON_TAC[
DIST_SYM]; ALL_TAC] THEN
MAP_EVERY X_GEN_TAC [`m:num`; `n:num`] THEN REWRITE_TAC[
GE] THEN
REPEAT STRIP_TAC THEN
MATCH_MP_TAC
REAL_LTE_TRANS THEN EXISTS_TAC `inv(&m + &1)` THEN
CONJ_TAC THENL
[REWRITE_TAC[dist] THEN ASM_MESON_TAC[
LE_REFL]; ALL_TAC] THEN
MATCH_MP_TAC
REAL_LE_TRANS THEN EXISTS_TAC `inv(&N)` THEN
ASM_SIMP_TAC[
REAL_LT_IMP_LE] THEN MATCH_MP_TAC
REAL_LE_INV2 THEN
REWRITE_TAC[REAL_OF_NUM_ADD; REAL_OF_NUM_LE;
REAL_OF_NUM_LT] THEN
ASM_ARITH_TAC;
ALL_TAC] THEN
REWRITE_TAC[GSYM
CONVERGENT_EQ_CAUCHY;
LIM_SEQUENTIALLY] THEN
MATCH_MP_TAC
MONO_EXISTS THEN X_GEN_TAC `y:real^N` THEN
REWRITE_TAC[dist] THEN STRIP_TAC THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN
MP_TAC(SPEC `e / &2`
REAL_ARCH_INV) THEN ASM_REWRITE_TAC[
REAL_HALF] THEN
DISCH_THEN(X_CHOOSE_THEN `N1:num` STRIP_ASSUME_TAC) THEN
FIRST_X_ASSUM(MP_TAC o SPEC `e / &2`) THEN ASM_REWRITE_TAC[
REAL_HALF] THEN
DISCH_THEN(X_CHOOSE_TAC `N2:num`) THEN EXISTS_TAC
`(d:num->real^M->real^M->bool) (N1 + N2)` THEN
ASM_REWRITE_TAC[] THEN
X_GEN_TAC `q:(real^M#(real^M->bool))->bool` THEN STRIP_TAC THEN
REWRITE_TAC[GSYM dist] THEN MATCH_MP_TAC
DIST_TRIANGLE_HALF_L THEN
EXISTS_TAC `vsum (p(N1+N2:num))
(\(x,k:real^M->bool). content k % (f:real^M->real^N) x)` THEN
CONJ_TAC THENL
[REWRITE_TAC[dist] THEN MATCH_MP_TAC
REAL_LTE_TRANS THEN
EXISTS_TAC `inv(&(N1 + N2) + &1)` THEN CONJ_TAC THENL
[FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_MESON_TAC[
LE_REFL]; ALL_TAC] THEN
MATCH_MP_TAC
REAL_LE_TRANS THEN EXISTS_TAC `inv(&N1)` THEN
ASM_SIMP_TAC[
REAL_LT_IMP_LE] THEN MATCH_MP_TAC
REAL_LE_INV2 THEN
REWRITE_TAC[REAL_OF_NUM_ADD; REAL_OF_NUM_LE;
REAL_OF_NUM_LT] THEN
ASM_ARITH_TAC;
ONCE_REWRITE_TAC[
DIST_SYM] THEN REWRITE_TAC[dist] THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN ARITH_TAC]);;
let INTERVAL_SPLIT = prove
(`!a b:real^N c k.
1 <= k /\ k <= dimindex(:N)
==> interval[a,b]
INTER {x | x$k <= c} =
interval[a,(lambda i. if i = k then min (b$k) c else b$i)] /\
interval[a,b]
INTER {x | x$k >= c} =
interval[(lambda i. if i = k then max (a$k) c else a$i),b]`,
let CONTENT_SPLIT = prove
(`!a b:real^N k.
1 <= k /\ k <= dimindex(:N)
==> content(interval[a,b]) =
content(interval[a,b]
INTER {x | x$k <= c}) +
content(interval[a,b]
INTER {x | x$k >= c})`,
SIMP_TAC[
INTERVAL_SPLIT;
CONTENT_CLOSED_INTERVAL_CASES;
LAMBDA_BETA] THEN
REPEAT GEN_TAC THEN REWRITE_TAC[REAL_ARITH
`((a <= if p then b else c) <=> (p ==> a <= b) /\ (~p ==> a <= c)) /\
((if p then b else c) <= a <=> (p ==> b <= a) /\ (~p ==> c <= a))`] THEN
REWRITE_TAC[
REAL_LE_MIN;
REAL_MAX_LE] THEN
REWRITE_TAC[MESON[] `(i = k ==> p i k) <=> (i = k ==> p i i)`] THEN
REWRITE_TAC[TAUT `(p ==> a /\ b) /\ (~p ==> a) <=> a /\ (p ==> b)`] THEN
REWRITE_TAC[TAUT `a ==> b /\ c <=> (a ==> b) /\ (a ==> c)`] THEN
REWRITE_TAC[
FORALL_AND_THM] THEN STRIP_TAC THEN
ASM_CASES_TAC
`!i. 1 <= i /\ i <= dimindex(:N) ==> (a:real^N)$i <= (b:real^N)$i` THEN
ASM_REWRITE_TAC[
REAL_ADD_RID] THEN
REWRITE_TAC[MESON[] `(!i. P i ==> i = k ==> Q i) <=> (P k ==> Q k)`] THEN
ASM_REWRITE_TAC[] THEN
REWRITE_TAC[REAL_ARITH `min b c = if c <= b then c else b`;
REAL_ARITH `max a c = if a <= c then c else a`] THEN
REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[REAL_ADD_LID;
REAL_ADD_RID]) THEN
REWRITE_TAC[MESON[] `(if i = k then a k else a i) = a i`] THENL
[ALL_TAC; ASM_MESON_TAC[
REAL_LE_TRANS;
REAL_LE_TOTAL]] THEN
SUBGOAL_THEN `1..dimindex(:N) = k
INSERT ((1..dimindex(:N))
DELETE k)`
SUBST1_TAC THENL
[REWRITE_TAC[
EXTENSION;
IN_INSERT;
IN_DELETE;
IN_NUMSEG] THEN
ASM_MESON_TAC[];
ALL_TAC] THEN
SIMP_TAC[
PRODUCT_CLAUSES;
FINITE_DELETE;
FINITE_NUMSEG;
IN_DELETE] THEN
MATCH_MP_TAC(REAL_RING
`p'' = p /\ p':real = p
==> (b - a) * p = (c - a) * p' + (b - c) * p''`) THEN
CONJ_TAC THEN MATCH_MP_TAC
PRODUCT_EQ THEN SIMP_TAC[
IN_DELETE]);;
let lemma = prove
(`!a b:real^N c k.
1 <= k /\ k <= dimindex(:N)
==> (content(interval[a,b]
INTER {x | x$k <= c}) = &0 <=>
interior(interval[a,b]
INTER {x | x$k <= c}) = {}) /\
(content(interval[a,b]
INTER {x | x$k >= c}) = &0 <=>
interior(interval[a,b]
INTER {x | x$k >= c}) = {})`,
SIMP_TAC[
INTERVAL_SPLIT;
CONTENT_EQ_0_INTERIOR]) in
REPEAT STRIP_TAC THEN
REWRITE_TAC[
CONTENT_EQ_0_INTERIOR] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
division_of]) THEN
DISCH_THEN(CONJUNCTS_THEN2 MP_TAC (MP_TAC o CONJUNCT1) o CONJUNCT2) THEN
DISCH_THEN(MP_TAC o SPECL
[`k1:real^N->bool`; `k2:real^N->bool`]) THEN
ASM_REWRITE_TAC[
PAIR_EQ] THEN DISCH_TAC THEN
DISCH_THEN(MP_TAC o SPEC `k2:real^N->bool`) THEN
ASM_REWRITE_TAC[] THEN
REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
DISCH_THEN(X_CHOOSE_THEN `u:real^N` (X_CHOOSE_THEN `v:real^N`
SUBST_ALL_TAC)) THEN
ASM_SIMP_TAC[lemma] THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
`s
INTER t = {}
==> u
SUBSET s /\ u
SUBSET t ==> u = {}`)) THEN
CONJ_TAC THEN MATCH_MP_TAC
SUBSET_INTERIOR THEN ASM SET_TAC[]);;
let DIVISION_SPLIT = prove
(`!p a b:real^N k c.
p
division_of interval[a,b] /\ 1 <= k /\ k <= dimindex(:N)
==> {l
INTER {x | x$k <= c} |l| l
IN p /\ ~(l
INTER {x | x$k <= c} = {})}
division_of (interval[a,b]
INTER {x | x$k <= c}) /\
{l
INTER {x | x$k >= c} |l| l
IN p /\ ~(l
INTER {x | x$k >= c} = {})}
division_of (interval[a,b]
INTER {x | x$k >= c})`,
REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC STRIP_ASSUME_TAC) THEN
SIMP_TAC[
division_of;
FINITE_IMAGE] THEN
SIMP_TAC[SET_RULE `(!x. x
IN {f x | P x} ==> Q x) <=> (!x. P x ==> Q (f x))`;
MESON[] `(!x y. x
IN s /\ y
IN t /\ Q x y ==> P x y) <=>
(!x. x
IN s ==> !y. y
IN t ==> Q x y ==> P x y)`;
RIGHT_FORALL_IMP_THM] THEN
REPEAT(MATCH_MP_TAC(TAUT
`(a ==> a' /\ a'') /\ (b ==> b' /\ b'')
==> a /\ b ==> (a' /\ b') /\ (a'' /\ b'')`) THEN CONJ_TAC)
THENL
[ONCE_REWRITE_TAC[SET_RULE
`{f x |x| x
IN s /\ ~(f x = {})} = {y | y
IN IMAGE f s /\ ~(y = {})}`] THEN
SIMP_TAC[
FINITE_RESTRICT;
FINITE_IMAGE];
REWRITE_TAC[
AND_FORALL_THM] THEN
MATCH_MP_TAC
MONO_FORALL THEN X_GEN_TAC `l:real^N->bool` THEN
DISCH_THEN(fun th -> CONJ_TAC THEN STRIP_TAC THEN MP_TAC th) THEN
(ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_AND THEN
CONJ_TAC THENL [SET_TAC[]; ALL_TAC] THEN
STRIP_TAC THEN ASM_MESON_TAC[
INTERVAL_SPLIT]);
DISCH_THEN(fun th -> CONJ_TAC THEN MP_TAC th) THEN
(REPEAT(MATCH_MP_TAC
MONO_FORALL THEN GEN_TAC) THEN
DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_SIMP_TAC[] THEN
REPEAT(MATCH_MP_TAC
MONO_FORALL THEN GEN_TAC) THEN
DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_SIMP_TAC[] THEN
DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_SIMP_TAC[] THEN
ANTS_TAC THENL [ASM_MESON_TAC[
PAIR_EQ]; ALL_TAC] THEN
MATCH_MP_TAC(SET_RULE
`s
SUBSET s' /\ t
SUBSET t'
==> s'
INTER t' = {} ==> s
INTER t = {}`) THEN
CONJ_TAC THEN MATCH_MP_TAC
SUBSET_INTERIOR THEN SET_TAC[]);
DISCH_THEN(SUBST1_TAC o SYM) THEN REWRITE_TAC[
INTER_UNIONS] THEN
ONCE_REWRITE_TAC[
EXTENSION] THEN REWRITE_TAC[
IN_UNIONS] THEN
CONJ_TAC THEN GEN_TAC THEN AP_TERM_TAC THEN
GEN_REWRITE_TAC I [
FUN_EQ_THM] THEN GEN_TAC THEN
REWRITE_TAC[
IN_ELIM_THM;
PAIR_EQ] THEN MESON_TAC[
NOT_IN_EMPTY]]);;
let HAS_INTEGRAL_SPLIT = prove
(`!f:real^M->real^N k a b c.
(f
has_integral i) (interval[a,b]
INTER {x | x$k <= c}) /\
(f
has_integral j) (interval[a,b]
INTER {x | x$k >= c}) /\
1 <= k /\ k <= dimindex(:M)
==> (f
has_integral (i + j)) (interval[a,b])`,
let lemma1 = prove
(`(!x k. (x,k) IN {x,f k | P x k} ==> Q x k) <=>
(!x k. P x k ==> Q x (f k))`,
REWRITE_TAC[IN_ELIM_THM; PAIR_EQ] THEN
SET_TAC[]) in
let lemma2 = prove
(`!f:B->B s:(A#B)->bool.
FINITE s ==> FINITE {x,f k | (x,k) IN s /\ P x k}`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC FINITE_SUBSET THEN
EXISTS_TAC `IMAGE (\(x:A,k:B). x,(f k:B)) s` THEN
ASM_SIMP_TAC[FINITE_IMAGE] THEN
REWRITE_TAC[SUBSET; FORALL_PAIR_THM; lemma1; IN_IMAGE] THEN
REWRITE_TAC[EXISTS_PAIR_THM; PAIR_EQ] THEN MESON_TAC[]) in
let lemma3 = prove
(`!f:real^M->real^N g:(real^M->bool)->(real^M->bool) p.
FINITE p
==> vsum {x,g k |x,k| (x,k) IN p /\ ~(g k = {})}
(\(x,k). content k % f x) =
vsum (IMAGE (\(x,k). x,g k) p) (\(x,k). content k % f x)`,
REPEAT STRIP_TAC THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC VSUM_SUPERSET THEN
ASM_SIMP_TAC[FINITE_IMAGE; lemma2] THEN
REWRITE_TAC[IMP_CONJ; FORALL_IN_IMAGE] THEN
REWRITE_TAC[FORALL_PAIR_THM; SUBSET; IN_IMAGE; EXISTS_PAIR_THM] THEN
REWRITE_TAC[IN_ELIM_THM; PAIR_EQ; VECTOR_MUL_EQ_0] THEN
MESON_TAC[CONTENT_EMPTY]) in
let lemma4 = prove
(`(\(x,l). content (g l) % f x) =
(\(x,l). content l % f x) o (\(x,l). x,g l)`,
REWRITE_TAC[FUN_EQ_THM; o_THM; FORALL_PAIR_THM]) in
REPEAT GEN_TAC THEN
ASM_CASES_TAC `1 <= k /\ k <= dimindex(:M)` THEN ASM_REWRITE_TAC[] THEN
ASM_SIMP_TAC[INTERVAL_SPLIT] THEN REWRITE_TAC[has_integral] THEN
ASM_SIMP_TAC[GSYM INTERVAL_SPLIT] THEN FIRST_X_ASSUM STRIP_ASSUME_TAC THEN
DISCH_TAC THEN X_GEN_TAC `e:real` THEN STRIP_TAC THEN
FIRST_X_ASSUM(CONJUNCTS_THEN2 (MP_TAC o SPEC `e / &2`) STRIP_ASSUME_TAC) THEN
FIRST_X_ASSUM(MP_TAC o SPEC `e / &2`) THEN ASM_REWRITE_TAC[REAL_HALF] THEN
DISCH_THEN(X_CHOOSE_THEN `d2:real^M->real^M->bool`
(CONJUNCTS_THEN2 ASSUME_TAC (LABEL_TAC "I2"))) THEN
DISCH_THEN(X_CHOOSE_THEN `d1:real^M->real^M->bool`
(CONJUNCTS_THEN2 ASSUME_TAC (LABEL_TAC "I1"))) THEN
EXISTS_TAC `\x. if x$k = c then (d1(x:real^M) INTER d2(x)):real^M->bool
else ball(x,abs(x$k - c)) INTER d1(x) INTER d2(x)` THEN
CONJ_TAC THENL
[REWRITE_TAC[gauge] THEN GEN_TAC THEN
RULE_ASSUM_TAC(REWRITE_RULE[gauge]) THEN COND_CASES_TAC THEN
ASM_SIMP_TAC[OPEN_INTER; IN_INTER; OPEN_BALL; IN_BALL] THEN
ASM_REWRITE_TAC[DIST_REFL; GSYM REAL_ABS_NZ; REAL_SUB_0];
ALL_TAC] THEN
X_GEN_TAC `p:(real^M#(real^M->bool))->bool` THEN STRIP_TAC THEN
SUBGOAL_THEN
`(!x:real^M kk. (x,kk) IN p /\ ~(kk INTER {x:real^M | x$k <= c} = {})
==> x$k <= c) /\
(!x:real^M kk. (x,kk) IN p /\ ~(kk INTER {x:real^M | x$k >= c} = {})
==> x$k >= c)`
STRIP_ASSUME_TAC THENL
[CONJ_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [fine]) THEN
MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `x:real^M` THEN
MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `kk:real^M->bool` THEN
DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[REAL_LE_REFL; real_ge] THEN DISCH_THEN
(MP_TAC o MATCH_MP (SET_RULE `k SUBSET (a INTER b) ==> k SUBSET a`)) THEN
REWRITE_TAC[SUBSET; IN_BALL; dist] THEN DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
DISCH_THEN(X_CHOOSE_THEN `u:real^M` MP_TAC) THEN
REWRITE_TAC[IN_INTER; IN_ELIM_THM] THEN REPEAT STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `u:real^M`) THEN ASM_REWRITE_TAC[] THEN
ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN
REWRITE_TAC[REAL_NOT_LE; REAL_NOT_LT] THEN STRIP_TAC THEN
MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `abs((x - u:real^M)$k)` THEN
ASM_SIMP_TAC[COMPONENT_LE_NORM] THEN
ASM_SIMP_TAC[VECTOR_SUB_COMPONENT] THEN
ASM_REAL_ARITH_TAC;
ALL_TAC] THEN
REMOVE_THEN "I2" (MP_TAC o SPEC
`{(x:real^M,kk INTER {x:real^M | x$k >= c}) |x,kk|
(x,kk) IN p /\ ~(kk INTER {x:real^M | x$k >= c} = {})}`) THEN
REMOVE_THEN "I1" (MP_TAC o SPEC
`{(x:real^M,kk INTER {x:real^M | x$k <= c}) |x,kk|
(x,kk) IN p /\ ~(kk INTER {x:real^M | x$k <= c} = {})}`) THEN
MATCH_MP_TAC(TAUT
`(a /\ b) /\ (a' /\ b' ==> c) ==> (a ==> a') ==> (b ==> b') ==> c`) THEN
CONJ_TAC THENL
[FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [TAGGED_DIVISION_OF]) THEN
REWRITE_TAC[TAGGED_DIVISION_OF] THEN
REPEAT(MATCH_MP_TAC(TAUT
`(a ==> (a' /\ a'')) /\ (b ==> (b' /\ d) /\ (b'' /\ e))
==> a /\ b ==> ((a' /\ b') /\ d) /\ ((a'' /\ b'') /\ e)`) THEN
CONJ_TAC) THEN
REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
REWRITE_TAC[lemma1] THEN REWRITE_TAC[IMP_IMP] THENL
[SIMP_TAC[lemma2];
REWRITE_TAC[AND_FORALL_THM] THEN
MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `x:real^M` THEN
MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `kk:real^M->bool` THEN
DISCH_THEN(fun th -> CONJ_TAC THEN STRIP_TAC THEN MP_TAC th) THEN
(ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_AND THEN CONJ_TAC THENL
[SIMP_TAC[IN_INTER; IN_ELIM_THM] THEN ASM_MESON_TAC[]; ALL_TAC]) THEN
(MATCH_MP_TAC MONO_AND THEN CONJ_TAC THENL [SET_TAC[]; ALL_TAC]) THEN
ASM_MESON_TAC[INTERVAL_SPLIT];
DISCH_THEN(fun th -> CONJ_TAC THEN MP_TAC th) THEN
(REPEAT(MATCH_MP_TAC MONO_FORALL THEN GEN_TAC) THEN
DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_SIMP_TAC[] THEN
REPEAT(MATCH_MP_TAC MONO_FORALL THEN GEN_TAC) THEN
DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_SIMP_TAC[] THEN
ANTS_TAC THENL [ASM_MESON_TAC[PAIR_EQ]; ALL_TAC] THEN
MATCH_MP_TAC(SET_RULE
`s SUBSET s' /\ t SUBSET t'
==> s' INTER t' = {} ==> s INTER t = {}`) THEN
CONJ_TAC THEN MATCH_MP_TAC SUBSET_INTERIOR THEN SET_TAC[]);
ALL_TAC] THEN
MATCH_MP_TAC(TAUT `(a ==> b /\ c) /\ d /\ e
==> (a ==> (b /\ d) /\ (c /\ e))`) THEN
CONJ_TAC THENL
[DISCH_THEN(fun th -> CONJ_TAC THEN MP_TAC th) THEN
DISCH_THEN(SUBST1_TAC o SYM) THEN REWRITE_TAC[INTER_UNIONS] THEN
ONCE_REWRITE_TAC[EXTENSION] THEN REWRITE_TAC[IN_UNIONS] THEN
X_GEN_TAC `x:real^M` THEN AP_TERM_TAC THEN
GEN_REWRITE_TAC I [FUN_EQ_THM] THEN X_GEN_TAC `kk:real^M->bool` THEN
REWRITE_TAC[IN_ELIM_THM; PAIR_EQ] THEN MESON_TAC[NOT_IN_EMPTY];
ALL_TAC] THEN
CONJ_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [fine]) THEN
REWRITE_TAC[fine; lemma1] THEN
REPEAT(MATCH_MP_TAC MONO_FORALL THEN GEN_TAC) THEN
DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN
ASM_SIMP_TAC[] THEN SET_TAC[];
ALL_TAC] THEN
DISCH_THEN(MP_TAC o MATCH_MP (REAL_ARITH
`x < e / &2 /\ y < e / &2 ==> x + y < e`)) THEN
DISCH_THEN(MP_TAC o MATCH_MP NORM_TRIANGLE_LT) THEN
MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN AP_TERM_TAC THEN
REWRITE_TAC[VECTOR_ARITH
`(a - i) + (b - j) = c - (i + j) <=> a + b = c:real^N`] THEN
FIRST_ASSUM(ASSUME_TAC o MATCH_MP TAGGED_DIVISION_OF_FINITE) THEN
MATCH_MP_TAC EQ_TRANS THEN
EXISTS_TAC
`vsum p (\(x,l). content (l INTER {x:real^M | x$k <= c}) %
(f:real^M->real^N) x) +
vsum p (\(x,l). content (l INTER {x:real^M | x$k >= c}) %
(f:real^M->real^N) x)` THEN
CONJ_TAC THENL
[ALL_TAC;
ASM_SIMP_TAC[GSYM VSUM_ADD] THEN MATCH_MP_TAC VSUM_EQ THEN
REWRITE_TAC[FORALL_PAIR_THM; GSYM VECTOR_ADD_RDISTRIB] THEN
MAP_EVERY X_GEN_TAC [`x:real^M`; `l:real^M->bool`] THEN
DISCH_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [TAGGED_DIVISION_OF]) THEN
DISCH_THEN(MP_TAC o SPECL [`x:real^M`; `l:real^M->bool`] o
el 1 o CONJUNCTS) THEN
ASM_REWRITE_TAC[] THEN STRIP_TAC THEN
ASM_SIMP_TAC[GSYM CONTENT_SPLIT]] THEN
ASM_SIMP_TAC[lemma3] THEN BINOP_TAC THEN
(GEN_REWRITE_TAC (RAND_CONV o RAND_CONV) [lemma4] THEN
MATCH_MP_TAC VSUM_IMAGE_NONZERO THEN ASM_REWRITE_TAC[FORALL_PAIR_THM] THEN
REWRITE_TAC[PAIR_EQ] THEN
ASM_MESON_TAC[TAGGED_DIVISION_SPLIT_LEFT_INJ; VECTOR_MUL_LZERO;
TAGGED_DIVISION_SPLIT_RIGHT_INJ]));;
let INTEGRABLE_SPLIT = prove
(`!f:real^M->real^N a b.
f
integrable_on (interval[a,b]) /\ 1 <= k /\ k <= dimindex(:M)
==> f
integrable_on (interval[a,b]
INTER {x | x$k <= c}) /\
f
integrable_on (interval[a,b]
INTER {x | x$k >= c})`,
let lemma = prove
(`b - a = c
==> norm(a:real^N) < e / &2 ==> norm(b) < e / &2 ==> norm(c) < e`,
DISCH_THEN(SUBST1_TAC o SYM) THEN REWRITE_TAC[GSYM dist] THEN
REPEAT STRIP_TAC THEN MATCH_MP_TAC DIST_TRIANGLE_HALF_L THEN
EXISTS_TAC `vec 0:real^N` THEN
ASM_REWRITE_TAC[dist; VECTOR_SUB_LZERO; VECTOR_SUB_RZERO; NORM_NEG]) in
REPEAT GEN_TAC THEN
GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [integrable_on] THEN
REWRITE_TAC[LEFT_IMP_EXISTS_THM; LEFT_AND_EXISTS_THM] THEN
X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN CONJ_TAC THEN
ASM_SIMP_TAC[INTERVAL_SPLIT; INTEGRABLE_CAUCHY] THEN
X_GEN_TAC `e:real` THEN DISCH_TAC THEN
FIRST_ASSUM(MP_TAC o SPEC `e / &2` o
MATCH_MP HAS_INTEGRAL_SEPARATE_SIDES) THEN
MAP_EVERY ABBREV_TAC
[`b' = (lambda i. if i = k then min ((b:real^M)$k) c else b$i):real^M`;
`a' = (lambda i. if i = k then max ((a:real^M)$k) c else a$i):real^M`] THEN
ASM_SIMP_TAC[REAL_HALF; INTERVAL_SPLIT] THEN
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:real^M->real^M->bool` THEN
STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
FIRST_ASSUM(MP_TAC o MATCH_MP FINE_DIVISION_EXISTS) THENL
[DISCH_THEN(MP_TAC o SPECL [`a':real^M`; `b:real^M`]) THEN
RULE_ASSUM_TAC(ONCE_REWRITE_RULE[SWAP_FORALL_THM]);
DISCH_THEN(MP_TAC o SPECL [`a:real^M`; `b':real^M`])] THEN
DISCH_THEN(X_CHOOSE_THEN `p:(real^M#(real^M->bool))->bool`
STRIP_ASSUME_TAC) THEN
REPEAT STRIP_TAC THEN FIRST_X_ASSUM(fun th ->
MP_TAC(SPECL [`p:(real^M#(real^M->bool))->bool`;
`p1:(real^M#(real^M->bool))->bool`] th) THEN
MP_TAC(SPECL [`p:(real^M#(real^M->bool))->bool`;
`p2:(real^M#(real^M->bool))->bool`] th)) THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC lemma THEN VECTOR_ARITH_TAC);;
let operative = new_definition
`operative op (f:(real^N->bool)->A) <=>
(!a b. content(interval[a,b]) = &0 ==> f(interval[a,b]) = neutral(op)) /\
(!a b c k. 1 <= k /\ k <= dimindex(:N)
==> f(interval[a,b]) =
op (f(interval[a,b] INTER {x | x$k <= c}))
(f(interval[a,b] INTER {x | x$k >= c})))`;;let OPERATIVE_TRIVIAL = prove
(`!op f a b.
operative op f /\ content(interval[a,b]) = &0
==> f(interval[a,b]) = neutral op`,
REWRITE_TAC[operative] THEN MESON_TAC[]);;
let lifted = define
`(lifted op NONE _ = NONE) /\
(lifted op _ NONE = NONE) /\
(lifted op (SOME x) (SOME y) = SOME(op x y))`;;
let OPERATIVE_DIVISION = prove
(`!op d a b f:(real^N->bool)->A.
monoidal op /\ operative op f /\ d
division_of interval[a,b]
==> iterate(op) d f = f(interval[a,b])`,
REPEAT GEN_TAC THEN CONV_TAC(RAND_CONV SYM_CONV) THEN WF_INDUCT_TAC
`
CARD (
division_points (interval[a,b]:real^N->bool) d)` THEN
POP_ASSUM(fun th -> REPEAT STRIP_TAC THEN MP_TAC th) THEN
ASM_REWRITE_TAC[] THEN
ASM_CASES_TAC `content(interval[a:real^N,b]) = &0` THENL
[SUBGOAL_THEN `iterate op d (f:(real^N->bool)->A) = neutral op`
(fun th -> ASM_MESON_TAC[th; operative]) THEN
MATCH_MP_TAC(REWRITE_RULE[
RIGHT_IMP_FORALL_THM; IMP_IMP]
ITERATE_EQ_NEUTRAL) THEN
FIRST_ASSUM(fun th -> REWRITE_TAC[MATCH_MP
FORALL_IN_DIVISION th]) THEN
ASM_MESON_TAC[operative;
DIVISION_OF_CONTENT_0];
ALL_TAC] THEN
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM
CONTENT_LT_NZ]) THEN
REWRITE_TAC[
CONTENT_POS_LT_EQ] THEN STRIP_TAC THEN
FIRST_ASSUM(ASSUME_TAC o MATCH_MP
DIVISION_OF_FINITE) THEN
ASM_CASES_TAC `
division_points (interval[a,b]:real^N->bool) d = {}` THENL
[DISCH_THEN(K ALL_TAC) THEN
SUBGOAL_THEN
`!i. i
IN d
==> ?u v:real^N. i = interval[u,v] /\
!j. 1 <= j /\ j <= dimindex(:N)
==> u$j = (a:real^N)$j /\ v$j = a$j \/
u$j = (b:real^N)$j /\ v$j = b$j \/
u$j = a$j /\ v$j = b$j`
(LABEL_TAC "*") THENL
[FIRST_ASSUM(fun th -> REWRITE_TAC[MATCH_MP
FORALL_IN_DIVISION th]) THEN
MAP_EVERY X_GEN_TAC [`u:real^N`; `v:real^N`] THEN DISCH_TAC THEN
MAP_EVERY EXISTS_TAC [`u:real^N`; `v:real^N`] THEN REWRITE_TAC[] THEN
REPEAT STRIP_TAC THEN
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
division_of]) THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN(MP_TAC o SPEC `interval[u:real^N,v]` o CONJUNCT1) THEN
ASM_REWRITE_TAC[
INTERVAL_NE_EMPTY] THEN
DISCH_THEN(CONJUNCTS_THEN2 MP_TAC (ASSUME_TAC o CONJUNCT1)) THEN
ASM_REWRITE_TAC[
SUBSET_INTERVAL] THEN STRIP_TAC THEN
MATCH_MP_TAC(REAL_ARITH
`a <= u /\ u <= v /\ v <= b /\ ~(a < u /\ u < b \/ a < v /\ v < b)
==> u = a /\ v = a \/ u = b /\ v = b \/ u = a /\ v = b`) THEN
ASM_SIMP_TAC[] THEN DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
EXTENSION]) THEN
REWRITE_TAC[
division_points;
NOT_IN_EMPTY;
FORALL_PAIR_THM] THEN
REWRITE_TAC[
IN_ELIM_PAIR_THM] THEN DISCH_THEN(MP_TAC o SPEC `j:num`) THEN
ASM_REWRITE_TAC[] THEN REWRITE_TAC[
RIGHT_AND_EXISTS_THM] THEN
REWRITE_TAC[
NOT_EXISTS_THM] THEN ONCE_REWRITE_TAC[
SWAP_FORALL_THM] THEN
DISCH_THEN(MP_TAC o SPEC `interval[u:real^N,v]`) THEN
ASM_SIMP_TAC[
INTERVAL_UPPERBOUND;
INTERVAL_LOWERBOUND;
REAL_LT_IMP_LE] THEN
DISCH_THEN(fun th ->
MP_TAC(SPEC `(u:real^N)$j` th) THEN
MP_TAC(SPEC `(v:real^N)$j` th)) THEN
FIRST_X_ASSUM(DISJ_CASES_THEN MP_TAC) THEN REAL_ARITH_TAC;
ALL_TAC] THEN
SUBGOAL_THEN `interval[a:real^N,b]
IN d` MP_TAC THENL
[FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
division_of]) THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o last o CONJUNCTS) THEN
REWRITE_TAC[
EXTENSION;
IN_INTERVAL;
IN_UNIONS] THEN
DISCH_THEN(MP_TAC o SPEC `inv(&2) % (a + b:real^N)`) THEN
MATCH_MP_TAC(TAUT `b /\ (a ==> c) ==> (a <=> b) ==> c`) THEN
CONJ_TAC THENL
[SIMP_TAC[
VECTOR_ADD_COMPONENT;
VECTOR_MUL_COMPONENT] THEN
X_GEN_TAC `j:num` THEN STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `j:num`) THEN ASM_REWRITE_TAC[] THEN
REAL_ARITH_TAC;
ALL_TAC] THEN
DISCH_THEN(X_CHOOSE_THEN `i:real^N->bool`
(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
REMOVE_THEN "*" (MP_TAC o SPEC `i:real^N->bool`) THEN
ASM_REWRITE_TAC[] THEN REWRITE_TAC[
LEFT_IMP_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC [`u:real^N`; `v:real^N`] THEN
DISCH_THEN(CONJUNCTS_THEN2 SUBST_ALL_TAC MP_TAC) THEN
SIMP_TAC[
IN_INTERVAL;
VECTOR_ADD_COMPONENT;
VECTOR_MUL_COMPONENT] THEN
REWRITE_TAC[IMP_IMP;
AND_FORALL_THM] THEN
REWRITE_TAC[TAUT `(a ==> b) /\ (a ==> c) <=> a ==> b /\ c`] THEN
ASM_SIMP_TAC[REAL_ARITH
`a < b
==> ((u = a /\ v = a \/ u = b /\ v = b \/ u = a /\ v = b) /\
u <= inv(&2) * (a + b) /\ inv(&2) * (a + b) <= v <=>
u = a /\ v = b)`] THEN
ASM_MESON_TAC[
CART_EQ];
ALL_TAC] THEN
DISCH_THEN(fun th -> ASSUME_TAC th THEN MP_TAC th) THEN
DISCH_THEN(SUBST1_TAC o MATCH_MP (SET_RULE
`a
IN d ==> d = a
INSERT (d
DELETE a)`)) THEN
ASM_SIMP_TAC[
ITERATE_CLAUSES;
FINITE_DELETE;
IN_DELETE] THEN
SUBGOAL_THEN
`iterate op (d
DELETE interval[a,b]) (f:(real^N->bool)->A) = neutral op`
(fun th -> ASM_MESON_TAC[th; monoidal]) THEN
MATCH_MP_TAC(REWRITE_RULE[
RIGHT_IMP_FORALL_THM; IMP_IMP]
ITERATE_EQ_NEUTRAL) THEN
ASM_REWRITE_TAC[] THEN X_GEN_TAC `l:real^N->bool` THEN
REWRITE_TAC[
IN_DELETE] THEN STRIP_TAC THEN
SUBGOAL_THEN `content(l:real^N->bool) = &0`
(fun th -> ASM_MESON_TAC[th; operative]) THEN
REMOVE_THEN "*" (MP_TAC o SPEC `l:real^N->bool`) THEN
ASM_REWRITE_TAC[
LEFT_IMP_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC [`u:real^N`; `v:real^N`] THEN
DISCH_THEN(CONJUNCTS_THEN2 SUBST_ALL_TAC MP_TAC) THEN
UNDISCH_TAC `~(interval[u:real^N,v] = interval[a,b])` THEN
ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN
REWRITE_TAC[] THEN DISCH_THEN(fun th -> AP_TERM_TAC THEN MP_TAC th) THEN
REWRITE_TAC[
CONS_11;
PAIR_EQ;
CART_EQ;
CONTENT_EQ_0] THEN
GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV)
[TAUT `a ==> b <=> ~a \/ b`] THEN
REWRITE_TAC[
NOT_FORALL_THM;
OR_EXISTS_THM] THEN
REWRITE_TAC[
NOT_EXISTS_THM;
AND_FORALL_THM] THEN
MATCH_MP_TAC
MONO_FORALL THEN X_GEN_TAC `j:num` THEN
ASM_CASES_TAC `1 <= j /\ j <= dimindex(:N)` THEN ASM_REWRITE_TAC[] THEN
FIRST_X_ASSUM(MP_TAC o SPEC `j:num`) THEN ASM_REWRITE_TAC[] THEN
REAL_ARITH_TAC;
ALL_TAC] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM
MEMBER_NOT_EMPTY]) THEN
GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [
division_points] THEN
REWRITE_TAC[
IN_ELIM_THM;
LEFT_IMP_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC [`whatever:num#real`; `k:num`; `c:real`] THEN
ASM_SIMP_TAC[
INTERVAL_LOWERBOUND;
INTERVAL_UPPERBOUND;
REAL_LT_IMP_LE] THEN
DISCH_THEN(CONJUNCTS_THEN2 MP_TAC (K ALL_TAC)) THEN
DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC) THEN
MP_TAC(ISPECL [`a:real^N`; `b:real^N`; `c:real`; `d:(real^N->bool)->bool`;
`k:num`]
DIVISION_POINTS_PSUBSET) THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN(CONJUNCTS_THEN
(MP_TAC o MATCH_MP (REWRITE_RULE [
IMP_CONJ]
CARD_PSUBSET))) THEN
MP_TAC(ISPECL [`d:(real^N->bool)->bool`; `a:real^N`; `b:real^N`; `k:num`;
`c:real`]
DIVISION_SPLIT) THEN
ASM_SIMP_TAC[
DIVISION_POINTS_FINITE] THEN
ASM_SIMP_TAC[
INTERVAL_SPLIT] THEN
ASM_SIMP_TAC[REAL_ARITH `a < c ==> max a c = c`;
REAL_ARITH `c < b ==> min b c = c`] THEN
MAP_EVERY ABBREV_TAC
[`d1:(real^N->bool)->bool =
{l
INTER {x | x$k <= c} | l | l
IN d /\ ~(l
INTER {x | x$k <= c} = {})}`;
`d2:(real^N->bool)->bool =
{l
INTER {x | x$k >= c} | l | l
IN d /\ ~(l
INTER {x | x$k >= c} = {})}`;
`cb:real^N = (lambda i. if i = k then c else (b:real^N)$i)`;
`ca:real^N = (lambda i. if i = k then c else (a:real^N)$i)`] THEN
STRIP_TAC THEN STRIP_TAC THEN STRIP_TAC THEN DISCH_THEN(fun th ->
MP_TAC(SPECL [`a:real^N`; `cb:real^N`; `d1:(real^N->bool)->bool`] th) THEN
MP_TAC(SPECL [`ca:real^N`; `b:real^N`; `d2:(real^N->bool)->bool`] th)) THEN
ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN
MATCH_MP_TAC
EQ_TRANS THEN
EXISTS_TAC `op (iterate op d1 (f:(real^N->bool)->A))
(iterate op d2 (f:(real^N->bool)->A))` THEN
CONJ_TAC THENL
[FIRST_ASSUM(MP_TAC o CONJUNCT2 o GEN_REWRITE_RULE I [operative]) THEN
DISCH_THEN(MP_TAC o SPECL [`a:real^N`; `b:real^N`; `c:real`; `k:num`]) THEN
ASM_SIMP_TAC[
INTERVAL_SPLIT] THEN
ASM_SIMP_TAC[REAL_ARITH `a < c ==> max a c = c`;
REAL_ARITH `c < b ==> min b c = c`];
ALL_TAC] THEN
MATCH_MP_TAC
EQ_TRANS THEN
EXISTS_TAC
`op (iterate op d (\l. f(l
INTER {x | x$k <= c}):A))
(iterate op d (\l. f(l
INTER {x:real^N | x$k >= c})))` THEN
CONJ_TAC THENL
[ALL_TAC;
ASM_SIMP_TAC[GSYM
ITERATE_OP] THEN
MATCH_MP_TAC(REWRITE_RULE[
RIGHT_IMP_FORALL_THM; IMP_IMP]
ITERATE_EQ) THEN
ASM_REWRITE_TAC[MATCH_MP
FORALL_IN_DIVISION
(ASSUME `d
division_of interval[a:real^N,b]`)] THEN
ASM_MESON_TAC[operative]] THEN
MAP_EVERY EXPAND_TAC ["d1";
let INTEGRAL_PASTECART_CONST = prove
(`!a b:real^M c d:real^N k:real^P.
integral (interval[pastecart a c,pastecart b d]) (\x. k) =
integral (interval[a,b])
(\x. integral (interval[c,d]) (\y. k))`,
let DSUM_BOUND = prove
(`!p a b:real^M c:real^N e.
p
division_of interval[a,b] /\ norm(c) <= e
==> norm(vsum p (\l. content l % c)) <= e * content(interval[a,b])`,
let RSUM_BOUND = prove
(`!p a b f:real^M->real^N e.
p
tagged_division_of interval[a,b] /\
(!x. x
IN interval[a,b] ==> norm(f x) <= e)
==> norm(vsum p (\(x,k). content k % f x))
<= e * content(interval[a,b])`,
let RSUM_DIFF_BOUND = prove
(`!p a b f g:real^M->real^N.
p
tagged_division_of interval[a,b] /\
(!x. x
IN interval[a,b] ==> norm(f x - g x) <= e)
==> norm(vsum p (\(x,k). content k % f x) -
vsum p (\(x,k). content k % g x))
<= e * content(interval[a,b])`,
let HAS_INTEGRAL_BOUND = prove
(`!f:real^M->real^N a b i B.
&0 <= B /\
(f
has_integral i) (interval[a,b]) /\
(!x. x
IN interval[a,b] ==> norm(f x) <= B)
==> norm i <= B * content(interval[a,b])`,
let lemma = prove
(`norm(s) <= B ==> ~(norm(s - i) < norm(i) - B)`,
MATCH_MP_TAC(REAL_ARITH `n1 <= n + n2 ==> n <= B ==> ~(n2 < n1 - B)`) THEN
ONCE_REWRITE_TAC[NORM_SUB] THEN REWRITE_TAC[NORM_TRIANGLE_SUB]) in
REPEAT STRIP_TAC THEN
ASM_CASES_TAC `&0 < content(interval[a:real^M,b])` THENL
[ALL_TAC;
SUBGOAL_THEN `i:real^N = vec 0` SUBST1_TAC THEN
ASM_SIMP_TAC[REAL_LE_MUL; NORM_0; CONTENT_POS_LE] THEN
ASM_MESON_TAC[HAS_INTEGRAL_NULL_EQ; CONTENT_LT_NZ]] THEN
ONCE_REWRITE_TAC[GSYM REAL_NOT_LT] THEN DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [has_integral]) THEN
DISCH_THEN(MP_TAC o SPEC
`norm(i:real^N) - B * content(interval[a:real^M,b])`) THEN
ASM_REWRITE_TAC[REAL_SUB_LT] THEN
DISCH_THEN(X_CHOOSE_THEN `d:real^M->real^M->bool` STRIP_ASSUME_TAC) THEN
MP_TAC(SPECL [`d:real^M->real^M->bool`; `a:real^M`; `b:real^M`]
FINE_DIVISION_EXISTS) THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN
(X_CHOOSE_THEN `p:(real^M#(real^M->bool)->bool)` STRIP_ASSUME_TAC) THEN
FIRST_X_ASSUM(MP_TAC o SPEC `p:(real^M#(real^M->bool)->bool)`) THEN
ASM_MESON_TAC[lemma; RSUM_BOUND]);;
let RSUM_COMPONENT_LE = prove
(`!p a b f:real^M->real^N g:real^M->real^N.
p
tagged_division_of interval[a,b] /\ 1 <= i /\ i <= dimindex(:N) /\
(!x. x
IN interval[a,b] ==> (f x)$i <= (g x)$i)
==> vsum p (\(x,k). content k % f x)$i <=
vsum p (\(x,k). content k % g x)$i`,
let HAS_INTEGRAL_COMPONENT_LE = prove
(`!f:real^M->real^N g:real^M->real^N s i j k.
1 <= k /\ k <= dimindex(:N) /\
(f
has_integral i) s /\ (g
has_integral j) s /\
(!x. x
IN s ==> (f x)$k <= (g x)$k)
==> i$k <= j$k`,
SUBGOAL_THEN
`!f:real^M->real^N g:real^M->real^N a b i j k.
1 <= k /\ k <= dimindex(:N) /\
(f
has_integral i) (interval[a,b]) /\
(g
has_integral j) (interval[a,b]) /\
(!x. x
IN interval[a,b] ==> (f x)$k <= (g x)$k)
==> i$k <= j$k`
ASSUME_TAC THENL
[REPEAT STRIP_TAC THEN
MATCH_MP_TAC(REAL_ARITH `~(&0 < i - j) ==> i <= j`) THEN DISCH_TAC THEN
REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `((i:real^N)$k - (j:real^N)$k) / &3` o
GEN_REWRITE_RULE I [
has_integral])) THEN
ASM_SIMP_TAC[
REAL_LT_DIV;
REAL_OF_NUM_LT; ARITH] THEN
DISCH_THEN(X_CHOOSE_THEN `d1:real^M->real^M->bool` STRIP_ASSUME_TAC) THEN
DISCH_THEN(X_CHOOSE_THEN `d2:real^M->real^M->bool` STRIP_ASSUME_TAC) THEN
SUBGOAL_THEN `?p. p
tagged_division_of interval[a:real^M,b] /\
d1 fine p /\ d2 fine p`
STRIP_ASSUME_TAC THENL
[REWRITE_TAC[GSYM
FINE_INTER] THEN MATCH_MP_TAC
FINE_DIVISION_EXISTS THEN
ASM_SIMP_TAC[
GAUGE_INTER];
ALL_TAC] THEN
REPEAT
(FIRST_X_ASSUM(MP_TAC o SPEC `p:real^M#(real^M->bool)->bool`) THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN(MP_TAC o MATCH_MP
REAL_LT_IMP_LE) THEN
DISCH_THEN(MP_TAC o SPEC `k:num` o MATCH_MP
NORM_BOUND_COMPONENT_LE) THEN
ASM_SIMP_TAC[
VECTOR_SUB_COMPONENT]) THEN
SUBGOAL_THEN
`vsum p (\(x,l:real^M->bool). content l % (f:real^M->real^N) x)$k <=
vsum p (\(x,l). content l % (g:real^M->real^N) x)$k`
MP_TAC THENL
[MATCH_MP_TAC
RSUM_COMPONENT_LE THEN ASM_MESON_TAC[];
UNDISCH_TAC `&0 < (i:real^N)$k - (j:real^N)$k` THEN
SPEC_TAC(`vsum p (\(x:real^M,l:real^M->bool).
content l % (f x):real^N)$k`,
`fs:real`) THEN
SPEC_TAC(`vsum p (\(x:real^M,l:real^M->bool).
content l % (g x):real^N)$k`,
`gs:real`) THEN
REAL_ARITH_TAC];
ALL_TAC] THEN
REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[
has_integral_alt] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[] THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
STRIP_TAC THEN REWRITE_TAC[GSYM
REAL_NOT_LT] THEN DISCH_TAC THEN
REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC
`((i:real^N)$k - (j:real^N)$k) / &2`)) THEN
ASM_REWRITE_TAC[
REAL_HALF;
REAL_SUB_LT] THEN
DISCH_THEN(X_CHOOSE_THEN `B1:real` STRIP_ASSUME_TAC) THEN
DISCH_THEN(X_CHOOSE_THEN `B2:real` STRIP_ASSUME_TAC) THEN
MP_TAC(ISPEC
`ball(vec 0,B1)
UNION ball(vec 0:real^M,B2)`
BOUNDED_SUBSET_CLOSED_INTERVAL) THEN
REWRITE_TAC[
BOUNDED_UNION;
BOUNDED_BALL;
UNION_SUBSET;
NOT_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC [`a:real^M`; `b:real^M`] THEN
DISCH_THEN(CONJUNCTS_THEN(ANTE_RES_THEN MP_TAC)) THEN
DISCH_THEN(X_CHOOSE_THEN `w:real^N` STRIP_ASSUME_TAC) THEN
DISCH_THEN(X_CHOOSE_THEN `z:real^N` STRIP_ASSUME_TAC) THEN
SUBGOAL_THEN `(z:real^N)$k <= (w:real^N)$k` MP_TAC THENL
[FIRST_X_ASSUM MATCH_MP_TAC THEN
MAP_EVERY EXISTS_TAC
[`(\x. if x
IN s then f x else vec 0):real^M->real^N`;
`(\x. if x
IN s then g x else vec 0):real^M->real^N`;
`a:real^M`; `b:real^M`] THEN
ASM_MESON_TAC[
REAL_LE_REFL];
MP_TAC(ISPECL [`w - j:real^N`; `k:num`]
COMPONENT_LE_NORM) THEN
MP_TAC(ISPECL [`z - i:real^N`; `k:num`]
COMPONENT_LE_NORM) THEN
ASM_REWRITE_TAC[] THEN
SIMP_TAC[
VECTOR_SUB_COMPONENT; ASSUME `1 <= k`;
ASSUME `k <= dimindex(:N)`] THEN
REPEAT(FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM
REAL_NOT_LE])) THEN
NORM_ARITH_TAC]);;
let INTEGRABLE_UNIFORM_LIMIT = prove
(`!f a b. (!e. &0 < e
==> ?g. (!x. x
IN interval[a,b] ==> norm(f x - g x) <= e) /\
g
integrable_on interval[a,b] )
==> (f:real^M->real^N)
integrable_on interval[a,b]`,
let lemma = prove
(`x <= norm(a + b) + c ==> x <= norm(a) + norm(b) + c`,
MESON_TAC[REAL_ADD_AC; NORM_TRIANGLE; REAL_LE_TRANS; REAL_LE_RADD]) in
let (lemma1,lemma2) = (CONJ_PAIR o prove)
(`(norm(s2 - s1) <= e / &2 /\
norm(s1 - i1) < e / &4 /\ norm(s2 - i2) < e / &4
==> norm(i1 - i2) < e) /\
(norm(sf - sg) <= e / &3
==> norm(i - s) < e / &3 ==> norm(sg - i) < e / &3 ==> norm(sf - s) < e)`,
CONJ_TAC THENL
[REWRITE_TAC[CONJ_ASSOC] THEN
GEN_REWRITE_TAC (LAND_CONV o LAND_CONV o ONCE_DEPTH_CONV) [NORM_SUB] THEN
MATCH_MP_TAC(REAL_ARITH
`w <= x + y + z + &0
==> (x <= e / &2 /\ y < e / &4) /\ z < e / &4 ==> w < e`);
MATCH_MP_TAC(REAL_ARITH
`w <= x + y + z + &0
==> x <= e / &3 ==> y < e / &3 ==> z < e / &3 ==> w < e`)] THEN
REPEAT(MATCH_MP_TAC lemma) THEN REWRITE_TAC[REAL_ADD_RID] THEN
MATCH_MP_TAC REAL_EQ_IMP_LE THEN AP_TERM_TAC THEN VECTOR_ARITH_TAC) in
REPEAT STRIP_TAC THEN
ASM_CASES_TAC `&0 < content(interval[a:real^M,b])` THENL
[ALL_TAC;
ASM_MESON_TAC[HAS_INTEGRAL_NULL; CONTENT_LT_NZ; integrable_on]] THEN
FIRST_X_ASSUM(MP_TAC o GEN `n:num` o SPEC `inv(&n + &1)`) THEN
REWRITE_TAC[REAL_LT_INV_EQ; REAL_ARITH `&0 < &n + &1`] THEN
REWRITE_TAC[FORALL_AND_THM; SKOLEM_THM; integrable_on] THEN
DISCH_THEN(X_CHOOSE_THEN `g:num->real^M->real^N` (CONJUNCTS_THEN2
ASSUME_TAC (X_CHOOSE_TAC `i:num->real^N`))) THEN
SUBGOAL_THEN `cauchy(i:num->real^N)` MP_TAC THENL
[REWRITE_TAC[cauchy] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN
MP_TAC(SPEC `e / &4 / content(interval[a:real^M,b])`
REAL_ARCH_INV) THEN
ASM_SIMP_TAC[REAL_LT_DIV; REAL_OF_NUM_LT; ARITH] THEN
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `N:num` THEN STRIP_TAC THEN
MAP_EVERY X_GEN_TAC [`m:num`; `n:num`] THEN REWRITE_TAC[GE] THEN
STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE BINDER_CONV [has_integral]) THEN
ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN
DISCH_THEN(MP_TAC o SPEC `e / &4`) THEN
ASM_SIMP_TAC[REAL_LT_DIV; REAL_OF_NUM_LT; ARITH] THEN
DISCH_THEN(fun th -> MP_TAC(SPEC `m:num` th) THEN
MP_TAC(SPEC `n:num` th)) THEN
DISCH_THEN(X_CHOOSE_THEN `gn:real^M->real^M->bool` STRIP_ASSUME_TAC) THEN
DISCH_THEN(X_CHOOSE_THEN `gm:real^M->real^M->bool` STRIP_ASSUME_TAC) THEN
MP_TAC(ISPECL [`(\x. gm(x) INTER gn(x)):real^M->real^M->bool`;
`a:real^M`; `b:real^M`] FINE_DIVISION_EXISTS) THEN
ASM_SIMP_TAC[GAUGE_INTER; LEFT_IMP_EXISTS_THM] THEN
X_GEN_TAC `p:(real^M#(real^M->bool))->bool` THEN STRIP_TAC THEN
REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `p:(real^M#(real^M->bool))->bool`)) THEN
FIRST_ASSUM(fun th -> REWRITE_TAC[CONV_RULE(REWR_CONV FINE_INTER) th]) THEN
SUBGOAL_THEN `norm(vsum p (\(x,k:real^M->bool). content k % g (n:num) x) -
vsum p (\(x:real^M,k). content k % g m x :real^N))
<= e / &2`
MP_TAC THENL [ALL_TAC; ASM_REWRITE_TAC[dist] THEN MESON_TAC[lemma1]] THEN
MATCH_MP_TAC REAL_LE_TRANS THEN
EXISTS_TAC `&2 / &N * content(interval[a:real^M,b])` THEN CONJ_TAC THENL
[MATCH_MP_TAC RSUM_DIFF_BOUND;
ASM_SIMP_TAC[GSYM REAL_LE_RDIV_EQ] THEN
ASM_REAL_ARITH_TAC] THEN
ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN
FIRST_X_ASSUM(fun th -> MP_TAC(SPECL [`n:num`; `x:real^M`] th) THEN
MP_TAC(SPECL [`m:num`; `x:real^M`] th)) THEN
ASM_REWRITE_TAC[IMP_IMP] THEN
GEN_REWRITE_TAC (LAND_CONV o RAND_CONV o LAND_CONV) [NORM_SUB] THEN
DISCH_THEN(MP_TAC o MATCH_MP REAL_LE_ADD2) THEN
DISCH_THEN(MP_TAC o MATCH_MP NORM_TRIANGLE_LE) THEN
MATCH_MP_TAC(REAL_ARITH `u = v /\ a <= inv(x) /\ b <= inv(x) ==>
u <= a + b ==> v <= &2 / x`) THEN
CONJ_TAC THENL [AP_TERM_TAC THEN VECTOR_ARITH_TAC; ALL_TAC] THEN
CONJ_TAC THEN MATCH_MP_TAC REAL_LE_INV2 THEN
REWRITE_TAC[REAL_OF_NUM_ADD; REAL_OF_NUM_LE; REAL_OF_NUM_LT] THEN
ASM_ARITH_TAC;
ALL_TAC] THEN
REWRITE_TAC[GSYM CONVERGENT_EQ_CAUCHY] THEN
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `s:real^N` THEN DISCH_TAC THEN
REWRITE_TAC[has_integral] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `e / &3` o GEN_REWRITE_RULE I
[LIM_SEQUENTIALLY]) THEN
ASM_SIMP_TAC[dist; REAL_LT_DIV; REAL_OF_NUM_LT; ARITH] THEN
DISCH_THEN(X_CHOOSE_TAC `N1:num`) THEN
MP_TAC(SPEC `e / &3 / content(interval[a:real^M,b])` REAL_ARCH_INV) THEN
ASM_SIMP_TAC[REAL_LT_DIV; REAL_OF_NUM_LT; ARITH] THEN
DISCH_THEN(X_CHOOSE_THEN `N2:num` STRIP_ASSUME_TAC) THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE BINDER_CONV [has_integral]) THEN
DISCH_THEN(MP_TAC o SPECL [`N1 + N2:num`; `e / &3`]) THEN
ASM_SIMP_TAC[REAL_LT_DIV; REAL_OF_NUM_LT; ARITH] THEN
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `g:real^M->real^M->bool` THEN
STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
X_GEN_TAC `p:real^M#(real^M->bool)->bool` THEN STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `p:real^M#(real^M->bool)->bool`) THEN
ASM_REWRITE_TAC[] THEN
FIRST_X_ASSUM(MP_TAC o C MATCH_MP (ARITH_RULE `N1:num <= N1 + N2`)) THEN
MATCH_MP_TAC lemma2 THEN MATCH_MP_TAC REAL_LE_TRANS THEN
EXISTS_TAC `inv(&(N1 + N2) + &1) * content(interval[a:real^M,b])` THEN
CONJ_TAC THENL
[MATCH_MP_TAC RSUM_DIFF_BOUND THEN ASM_REWRITE_TAC[]; ALL_TAC] THEN
ASM_SIMP_TAC[GSYM REAL_LE_RDIV_EQ] THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH
`x < a ==> y <= x ==> y <= a`)) THEN
MATCH_MP_TAC REAL_LE_INV2 THEN
REWRITE_TAC[REAL_OF_NUM_ADD; REAL_OF_NUM_LE; REAL_OF_NUM_LT] THEN
ASM_ARITH_TAC);;
let DROP_INDICATOR = prove
(`!s x. drop(indicator s x) = if x
IN s then &1 else &0`,
REPEAT GEN_TAC THEN REWRITE_TAC[indicator] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[
DROP_VEC]);;
let VSUM_NONZERO_IMAGE_LEMMA = prove
(`!s f:A->B g:B->real^N a.
FINITE s /\ g(a) = vec 0 /\
(!x y. x
IN s /\ y
IN s /\ f x = f y /\ ~(x = y) ==> g(f x) = vec 0)
==> vsum {f x |x| x
IN s /\ ~(f x = a)} g =
vsum s (g o f)`,
let INTERVAL_DOUBLESPLIT = prove
(`1 <= k /\ k <= dimindex(:N)
==> interval[a,b]
INTER {x:real^N | abs(x$k - c) <= e} =
interval[(lambda i. if i = k then max (a$k) (c - e) else a$i),
(lambda i. if i = k then min (b$k) (c + e) else b$i)]`,
REWRITE_TAC[REAL_ARITH `abs(x - c) <= e <=> x >= c - e /\ x <= c + e`] THEN
REWRITE_TAC[SET_RULE `s
INTER {x | P x /\ Q x} =
(s
INTER {x | Q x})
INTER {x | P x}`] THEN
SIMP_TAC[
INTERVAL_SPLIT]);;
let DIVISION_DOUBLESPLIT = prove
(`!p a b:real^N k c e.
p
division_of interval[a,b] /\ 1 <= k /\ k <= dimindex(:N)
==> {l
INTER {x | abs(x$k - c) <= e} |l|
l
IN p /\ ~(l
INTER {x | abs(x$k - c) <= e} = {})}
division_of (interval[a,b]
INTER {x | abs(x$k - c) <= e})`,
REPEAT GEN_TAC THEN DISCH_TAC THEN
FIRST_ASSUM(MP_TAC o SPEC `c + e:real` o MATCH_MP
DIVISION_SPLIT) THEN
DISCH_THEN(MP_TAC o CONJUNCT1) THEN
ASM_SIMP_TAC[
INTERVAL_SPLIT] THEN
FIRST_ASSUM MP_TAC THEN REWRITE_TAC[IMP_IMP] THEN
DISCH_THEN(MP_TAC o MATCH_MP (TAUT
`(a /\ b /\ c) /\ d ==> d /\ b /\ c`)) THEN
DISCH_THEN(MP_TAC o CONJUNCT2 o SPEC `c - e:real` o
MATCH_MP
DIVISION_SPLIT) THEN
ASM_SIMP_TAC[
INTERVAL_DOUBLESPLIT;
INTERVAL_SPLIT] THEN
MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN
REWRITE_TAC[REAL_ARITH `abs(x - c) <= e <=> x >= c - e /\ x <= c + e`] THEN
GEN_REWRITE_TAC I [
EXTENSION] THEN REWRITE_TAC[
IN_INTER;
IN_ELIM_THM] THEN
GEN_TAC THEN REWRITE_TAC[
LEFT_AND_EXISTS_THM] THEN
ONCE_REWRITE_TAC[
SWAP_EXISTS_THM] THEN REWRITE_TAC[GSYM
CONJ_ASSOC] THEN
ONCE_REWRITE_TAC[TAUT `a /\ b /\ c /\ d <=> c /\ a /\ b /\ d`] THEN
REWRITE_TAC[
UNWIND_THM2] THEN AP_TERM_TAC THEN ABS_TAC THEN SET_TAC[]);;
let CONTENT_DOUBLESPLIT = prove
(`!a b:real^N k c e.
&0 < e /\ 1 <= k /\ k <= dimindex(:N)
==> ?d. &0 < d /\
content(interval[a,b]
INTER {x | abs(x$k - c) <= d}) < e`,
let TAGGED_DIVISION_FINER = prove
(`!p a b:real^N d. p
tagged_division_of interval[a,b] /\ gauge d
==> ?q. q
tagged_division_of interval[a,b] /\ d fine q /\
!x k. (x,k)
IN p /\ k
SUBSET d(x) ==> (x,k)
IN q`,
let lemma1 = prove
(`{k | ?x. (x,k) IN p} = IMAGE SND p`,
REWRITE_TAC[EXTENSION; EXISTS_PAIR_THM; IN_IMAGE; IN_ELIM_THM] THEN
MESON_TAC[]) in
SUBGOAL_THEN
`!a b:real^N d p.
FINITE p
==> p tagged_partial_division_of interval[a,b] /\ gauge d
==> ?q. q tagged_division_of (UNIONS {k | ?x. x,k IN p}) /\
d fine q /\
!x k. (x,k) IN p /\ k SUBSET d(x) ==> (x,k) IN q`
ASSUME_TAC THENL
[ALL_TAC;
REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN
GEN_REWRITE_TAC LAND_CONV [tagged_division_of] THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (SUBST1_TAC o SYM)) THEN
FIRST_ASSUM(MATCH_MP_TAC o REWRITE_RULE[IMP_IMP]) THEN
ASM_MESON_TAC[tagged_partial_division_of]] THEN
GEN_TAC THEN GEN_TAC THEN GEN_TAC THEN
MATCH_MP_TAC FINITE_INDUCT_STRONG THEN CONJ_TAC THENL
[DISCH_THEN(K ALL_TAC) THEN
REWRITE_TAC[SET_RULE `UNIONS {k | ?x. x,k IN {}} = {}`] THEN
EXISTS_TAC `{}:real^N#(real^N->bool)->bool` THEN
REWRITE_TAC[fine; NOT_IN_EMPTY; TAGGED_DIVISION_OF_EMPTY];
ALL_TAC] THEN
GEN_REWRITE_TAC I [FORALL_PAIR_THM] THEN MAP_EVERY X_GEN_TAC
[`x:real^N`; `k:real^N->bool`; `p:real^N#(real^N->bool)->bool`] THEN
DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN
DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ANTS_TAC THENL
[ASM_REWRITE_TAC[] THEN MATCH_MP_TAC TAGGED_PARTIAL_DIVISION_SUBSET THEN
EXISTS_TAC `(x:real^N,k:real^N->bool) INSERT p` THEN ASM SET_TAC[];
ALL_TAC] THEN
DISCH_THEN(X_CHOOSE_THEN `q1:real^N#(real^N->bool)->bool`
STRIP_ASSUME_TAC) THEN
SUBGOAL_THEN
`UNIONS {l:real^N->bool | ?y:real^N. (y,l) IN (x,k) INSERT p} =
k UNION UNIONS {l | ?y. (y,l) IN p}`
SUBST1_TAC THENL
[GEN_REWRITE_TAC I [EXTENSION] THEN REWRITE_TAC[IN_UNION; IN_UNIONS] THEN
REWRITE_TAC[IN_ELIM_THM; IN_INSERT; PAIR_EQ] THEN MESON_TAC[];
ALL_TAC] THEN
SUBGOAL_THEN `?u v:real^N. k = interval[u,v]` MP_TAC THENL
[ASM_MESON_TAC[IN_INSERT; tagged_partial_division_of]; ALL_TAC] THEN
DISCH_THEN(REPEAT_TCL CHOOSE_THEN SUBST_ALL_TAC) THEN
ASM_CASES_TAC `interval[u,v] SUBSET ((d:real^N->real^N->bool) x)` THENL
[EXISTS_TAC `{(x:real^N,interval[u:real^N,v])} UNION q1` THEN CONJ_TAC THENL
[MATCH_MP_TAC TAGGED_DIVISION_UNION THEN ASM_REWRITE_TAC[] THEN
CONJ_TAC THENL
[MATCH_MP_TAC TAGGED_DIVISION_OF_SELF THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I
[tagged_partial_division_of]) THEN
REWRITE_TAC[IN_INSERT; PAIR_EQ] THEN MESON_TAC[];
ALL_TAC];
CONJ_TAC THENL
[MATCH_MP_TAC FINE_UNION THEN ASM_REWRITE_TAC[] THEN
REWRITE_TAC[fine; IN_SING; PAIR_EQ] THEN ASM_MESON_TAC[];
ALL_TAC] THEN
ASM_REWRITE_TAC[IN_INSERT; PAIR_EQ; IN_UNION; IN_SING] THEN
ASM_MESON_TAC[]];
FIRST_ASSUM(MP_TAC o SPECL [`u:real^N`; `v:real^N`] o MATCH_MP
FINE_DIVISION_EXISTS) THEN
DISCH_THEN(X_CHOOSE_THEN `q2:real^N#(real^N->bool)->bool`
STRIP_ASSUME_TAC) THEN
EXISTS_TAC `q2 UNION q1:real^N#(real^N->bool)->bool` THEN CONJ_TAC THENL
[MATCH_MP_TAC TAGGED_DIVISION_UNION THEN ASM_REWRITE_TAC[];
ASM_SIMP_TAC[FINE_UNION] THEN
ASM_REWRITE_TAC[IN_INSERT; PAIR_EQ; IN_UNION; IN_SING] THEN
ASM_MESON_TAC[]]] THEN
(MATCH_MP_TAC INTER_INTERIOR_UNIONS_INTERVALS THEN
REWRITE_TAC[lemma1; IN_ELIM_THM; LEFT_IMP_EXISTS_THM] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I
[tagged_partial_division_of]) THEN
REWRITE_TAC[IN_INSERT; FINITE_INSERT; PAIR_EQ] THEN
STRIP_TAC THEN ASM_SIMP_TAC[FINITE_IMAGE] THEN CONJ_TAC THENL
[REWRITE_TAC[INTERIOR_CLOSED_INTERVAL; OPEN_INTERVAL]; ALL_TAC] THEN
CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
ASM_MESON_TAC[]));;
let HAS_INTEGRAL_SPIKE = prove
(`!f:real^M->real^N g s t.
negligible s /\ (!x. x
IN (t
DIFF s) ==> g x = f x) /\
(f
has_integral y) t
==> (g
has_integral y) t`,
SUBGOAL_THEN
`!f:real^M->real^N g s a b y.
negligible s /\ (!x. x
IN (interval[a,b]
DIFF s) ==> g x = f x)
==> (f
has_integral y) (interval[a,b])
==> (g
has_integral y) (interval[a,b])`
ASSUME_TAC THENL
[REPEAT STRIP_TAC THEN
SUBGOAL_THEN
`((\x. (f:real^M->real^N)(x) + (g(x) - f(x)))
has_integral (y + vec 0))
(interval[a,b])`
MP_TAC THENL
[ALL_TAC;
REWRITE_TAC[VECTOR_ARITH `f + g - f = g /\ f + vec 0 = f`; ETA_AX]] THEN
MATCH_MP_TAC
HAS_INTEGRAL_ADD THEN ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC
HAS_INTEGRAL_NEGLIGIBLE THEN
EXISTS_TAC `s:real^M->bool` THEN ASM_REWRITE_TAC[
VECTOR_SUB_EQ] THEN
ASM_MESON_TAC[];
ALL_TAC] THEN
REPEAT GEN_TAC THEN
REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
ONCE_REWRITE_TAC[
has_integral_alt] THEN COND_CASES_TAC THEN
ASM_REWRITE_TAC[] THENL
[FIRST_X_ASSUM(CHOOSE_THEN(CHOOSE_THEN SUBST_ALL_TAC)) THEN ASM_MESON_TAC[];
ALL_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
REPEAT(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
FIRST_X_ASSUM MATCH_MP_TAC THEN EXISTS_TAC `s:real^M->bool` THEN
ASM_REWRITE_TAC[] THEN ASM SET_TAC[]);;
let INTEGRAL_SPIKE = prove
(`!f:real^M->real^N g s t y.
negligible s /\ (!x. x
IN (t
DIFF s) ==> g x = f x)
==> integral t f = integral t g`,
REPEAT STRIP_TAC THEN REWRITE_TAC[integral] THEN
AP_TERM_TAC THEN ABS_TAC THEN MATCH_MP_TAC
HAS_INTEGRAL_SPIKE_EQ THEN
ASM_MESON_TAC[]);;
let NEGLIGIBLE_SUBSET = prove
(`!s:real^N->bool t:real^N->bool.
negligible s /\ t
SUBSET s ==> negligible t`,
REPEAT STRIP_TAC THEN REWRITE_TAC[negligible] THEN
MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`] THEN
MATCH_MP_TAC
HAS_INTEGRAL_SPIKE THEN
MAP_EVERY EXISTS_TAC [`(\x. vec 0):real^N->real^1`; `s:real^N->bool`] THEN
ASM_REWRITE_TAC[
HAS_INTEGRAL_0] THEN
REWRITE_TAC[indicator] THEN ASM SET_TAC[]);;
let NEGLIGIBLE = prove
(`!s:real^N->bool. negligible s <=> !t. (indicator s
has_integral vec 0) t`,
GEN_TAC THEN EQ_TAC THENL
[ALL_TAC; REWRITE_TAC[negligible] THEN SIMP_TAC[]] THEN
DISCH_TAC THEN GEN_TAC THEN ONCE_REWRITE_TAC[
has_integral_alt] THEN
COND_CASES_TAC THENL [ASM_MESON_TAC[negligible]; ALL_TAC] THEN
GEN_TAC THEN DISCH_TAC THEN EXISTS_TAC `&1` THEN REWRITE_TAC[
REAL_LT_01] THEN
REPEAT STRIP_TAC THEN EXISTS_TAC `vec 0:real^1` THEN
MP_TAC(ISPECL [`s:real^N->bool`; `s
INTER t:real^N->bool`]
NEGLIGIBLE_SUBSET) THEN
ASM_REWRITE_TAC[
INTER_SUBSET; negligible;
VECTOR_SUB_REFL;
NORM_0] THEN
REWRITE_TAC[indicator;
IN_INTER] THEN
SIMP_TAC[TAUT `(if p /\ q then r else s) =
(if q then if p then r else s else s)`]);;
let INTEGRAL_EQ = prove
(`!f:real^M->real^N g s.
(!x. x
IN s ==> f x = g x) ==> integral s f = integral s g`,
let INTEGRAL_EQ_0 = prove
(`!f:real^M->real^N s. (!x. x
IN s ==> f x = vec 0) ==> integral s f = vec 0`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC
EQ_TRANS THEN
EXISTS_TAC `integral s ((\x. vec 0):real^M->real^N)` THEN
CONJ_TAC THENL
[MATCH_MP_TAC
INTEGRAL_EQ THEN ASM_REWRITE_TAC[];
REWRITE_TAC[
INTEGRAL_0]]);;
let OPERATIVE_APPROXIMABLE = prove
(`!f:real^M->real^N e.
&0 <= e
==> operative(/\)
(\i. ?g. (!x. x
IN i ==> norm (f x - g x) <= e) /\
g
integrable_on i)`,
REPEAT STRIP_TAC THEN REWRITE_TAC[operative;
NEUTRAL_AND] THEN CONJ_TAC THENL
[REPEAT STRIP_TAC THEN EXISTS_TAC `f:real^M->real^N` THEN
ASM_REWRITE_TAC[
VECTOR_SUB_REFL;
NORM_0;
integrable_on] THEN
ASM_MESON_TAC[
HAS_INTEGRAL_NULL];
ALL_TAC] THEN
MAP_EVERY X_GEN_TAC [`u:real^M`; `v:real^M`; `c:real`; `k:num`] THEN
STRIP_TAC THEN EQ_TAC THENL
[ASM_MESON_TAC[
INTEGRABLE_SPLIT;
IN_INTER]; ALL_TAC] THEN
DISCH_THEN(CONJUNCTS_THEN2
(X_CHOOSE_THEN `g1:real^M->real^N` STRIP_ASSUME_TAC)
(X_CHOOSE_THEN `g2:real^M->real^N` STRIP_ASSUME_TAC)) THEN
EXISTS_TAC `\x. if x$k = c then (f:real^M->real^N)(x) else
if x$k <= c then g1(x) else g2(x)` THEN
CONJ_TAC THENL
[GEN_TAC THEN STRIP_TAC THEN REWRITE_TAC[] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[
VECTOR_SUB_REFL;
NORM_0] THEN
RULE_ASSUM_TAC(REWRITE_RULE[
IN_INTER;
IN_ELIM_THM]) THEN
ASM_MESON_TAC[REAL_ARITH `x <= c \/ x >= c`];
ALL_TAC] THEN
SUBGOAL_THEN
`(\x:real^M. if x$k = c then f x else if x$k <= c then g1 x else g2 x)
integrable_on (interval[u,v]
INTER {x | x$k <= c}) /\
(\x. if x$k = c then f x :real^N else if x$k <= c then g1 x else g2 x)
integrable_on (interval[u,v]
INTER {x | x$k >= c})`
MP_TAC THENL
[ALL_TAC;
REWRITE_TAC[
integrable_on] THEN ASM_MESON_TAC[
HAS_INTEGRAL_SPLIT]] THEN
CONJ_TAC THENL
[UNDISCH_TAC
`(g1:real^M->real^N)
integrable_on (interval[u,v]
INTER {x | x$k <= c})`;
UNDISCH_TAC
`(g2:real^M->real^N)
integrable_on (interval[u,v]
INTER {x | x$k >= c})`
] THEN
ASM_SIMP_TAC[
INTERVAL_SPLIT] THEN MATCH_MP_TAC
INTEGRABLE_SPIKE THEN
ASM_SIMP_TAC[GSYM
INTERVAL_SPLIT] THEN
EXISTS_TAC `{x:real^M | x$k = c}` THEN
ASM_SIMP_TAC[
NEGLIGIBLE_STANDARD_HYPERPLANE;
IN_DIFF;
IN_INTER;
IN_ELIM_THM;
REAL_ARITH `x >= c /\ ~(x = c) ==> ~(x <= c)`] THEN
EXISTS_TAC `e:real` THEN REPEAT STRIP_TAC THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[
IN_INTER;
IN_ELIM_THM]);;
let OPERATIVE_1_LT = prove
(`!op. monoidal op
==> !f. operative op f <=>
(!a b. drop b <= drop a ==> f(interval[a,b]) = neutral op) /\
(!a b c. drop a < drop c /\ drop c < drop b
==> op (f(interval[a,c])) (f(interval[c,b])) =
f(interval[a,b]))`,
REPEAT STRIP_TAC THEN REWRITE_TAC[operative;
CONTENT_EQ_0_1] THEN
MATCH_MP_TAC(TAUT `(a ==> (b <=> c)) ==> (a /\ b <=> a /\ c)`) THEN
DISCH_TAC THEN REWRITE_TAC[
FORALL_1; DIMINDEX_1] THEN
AP_TERM_TAC THEN REWRITE_TAC[
FUN_EQ_THM] THEN X_GEN_TAC `a:real^1` THEN
AP_TERM_TAC THEN REWRITE_TAC[
FUN_EQ_THM] THEN X_GEN_TAC `b:real^1` THEN
EQ_TAC THEN DISCH_TAC THENL
[X_GEN_TAC `c:real^1` THEN FIRST_ASSUM(SUBST1_TAC o SPEC `drop c`) THEN
DISCH_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP
REAL_LT_TRANS) THEN
ASM_SIMP_TAC[
INTERVAL_SPLIT; DIMINDEX_1;
LE_REFL;
REAL_LT_IMP_LE] THEN
BINOP_TAC THEN AP_TERM_TAC THEN AP_TERM_TAC THEN
REWRITE_TAC[
CONS_11;
PAIR_EQ] THEN
SIMP_TAC[
FORALL_1;
CART_EQ; DIMINDEX_1;
LAMBDA_BETA;
LE_REFL] THEN
REWRITE_TAC[GSYM drop] THEN ASM_REAL_ARITH_TAC;
ALL_TAC] THEN
X_GEN_TAC `d:real` THEN ABBREV_TAC `c = lift d` THEN
SUBGOAL_THEN `d = drop c` SUBST1_TAC THENL
[ASM_MESON_TAC[
LIFT_DROP]; ALL_TAC] THEN
SIMP_TAC[
INTERVAL_SPLIT;
LE_REFL; drop; DIMINDEX_1] THEN
REWRITE_TAC[GSYM drop] THEN
DISJ_CASES_TAC(REAL_ARITH `drop c <= drop a \/ drop a < drop c`) THENL
[SUBGOAL_THEN
`content(interval[a:real^1,
(lambda i. if i = 1 then min (drop b) (drop c) else b$i)]) = &0 /\
interval[(lambda i. if i = 1 then max (drop a) (drop c) else a$i),b] =
interval[a,b]`
(CONJUNCTS_THEN2 MP_TAC SUBST1_TAC) THENL
[CONJ_TAC THENL
[SIMP_TAC[
CONTENT_EQ_0_1];
AP_TERM_TAC THEN REWRITE_TAC[
CONS_11;
PAIR_EQ]] THEN
SIMP_TAC[drop;
CART_EQ;
FORALL_1;
LAMBDA_BETA; DIMINDEX_1;
LE_REFL] THEN
UNDISCH_TAC `drop c <= drop a` THEN REWRITE_TAC[drop] THEN
REAL_ARITH_TAC;
REWRITE_TAC[
CONTENT_EQ_0_1] THEN
DISCH_THEN(ANTE_RES_THEN SUBST1_TAC) THEN ASM_MESON_TAC[monoidal]];
ALL_TAC] THEN
DISJ_CASES_TAC(REAL_ARITH `drop b <= drop c \/ drop c < drop b`) THENL
[SUBGOAL_THEN
`interval[a,(lambda i. if i = 1 then min (drop b) (drop c) else b$i)] =
interval[a,b] /\
content(interval
[(lambda i. if i = 1 then max (drop a) (drop c) else a$i),b]) = &0`
(CONJUNCTS_THEN2 SUBST1_TAC MP_TAC) THENL
[CONJ_TAC THENL
[AP_TERM_TAC THEN REWRITE_TAC[
CONS_11;
PAIR_EQ];
SIMP_TAC[
CONTENT_EQ_0_1]] THEN
SIMP_TAC[drop;
CART_EQ;
FORALL_1;
LAMBDA_BETA; DIMINDEX_1;
LE_REFL] THEN
UNDISCH_TAC `drop b <= drop c` THEN REWRITE_TAC[drop] THEN
REAL_ARITH_TAC;
REWRITE_TAC[
CONTENT_EQ_0_1] THEN
DISCH_THEN(ANTE_RES_THEN SUBST1_TAC) THEN ASM_MESON_TAC[monoidal]];
ALL_TAC] THEN
SUBGOAL_THEN
`(lambda i. if i = 1 then min (drop b) (drop c) else b$i) = c /\
(lambda i. if i = 1 then max (drop a) (drop c) else a$i) = c`
(fun th -> REWRITE_TAC[th] THEN ASM_MESON_TAC[]) THEN
SIMP_TAC[
CART_EQ;
FORALL_1; DIMINDEX_1;
LE_REFL;
LAMBDA_BETA] THEN
REWRITE_TAC[GSYM drop] THEN ASM_REAL_ARITH_TAC);;
let OPERATIVE_1_LE = prove
(`!op. monoidal op
==> !f. operative op f <=>
(!a b. drop b <= drop a ==> f(interval[a,b]) = neutral op) /\
(!a b c. drop a <= drop c /\ drop c <= drop b
==> op (f(interval[a,c])) (f(interval[c,b])) =
f(interval[a,b]))`,
GEN_TAC THEN DISCH_TAC THEN GEN_TAC THEN EQ_TAC THENL
[ALL_TAC; ASM_SIMP_TAC[
OPERATIVE_1_LT] THEN MESON_TAC[
REAL_LT_IMP_LE]] THEN
REWRITE_TAC[operative;
CONTENT_EQ_0_1] THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ASM_REWRITE_TAC[] THEN
REWRITE_TAC[
FORALL_1; DIMINDEX_1] THEN
MAP_EVERY (fun t -> MATCH_MP_TAC
MONO_FORALL THEN X_GEN_TAC t)
[`a:real^1`; `b:real^1`] THEN DISCH_TAC THEN
X_GEN_TAC `c:real^1` THEN FIRST_ASSUM(SUBST1_TAC o SPEC `drop c`) THEN
DISCH_TAC THEN FIRST_ASSUM(ASSUME_TAC o MATCH_MP
REAL_LE_TRANS) THEN
ASM_SIMP_TAC[
INTERVAL_SPLIT; DIMINDEX_1;
LE_REFL] THEN
BINOP_TAC THEN AP_TERM_TAC THEN AP_TERM_TAC THEN
REWRITE_TAC[
CONS_11;
PAIR_EQ] THEN
SIMP_TAC[
FORALL_1;
CART_EQ; DIMINDEX_1;
LAMBDA_BETA;
LE_REFL] THEN
REWRITE_TAC[GSYM drop] THEN ASM_REAL_ARITH_TAC);;
let GAUGE_MODIFY = prove
(`!f:real^M->real^N.
(!s. open s ==> open {x | f(x)
IN s})
==> !d. gauge d ==> gauge (\x y. d (f x) (f y))`,
GEN_TAC THEN DISCH_TAC THEN GEN_TAC THEN
SIMP_TAC[gauge;
IN] THEN DISCH_TAC THEN
X_GEN_TAC `x:real^M` THEN
FIRST_X_ASSUM(MP_TAC o SPEC `(f:real^M->real^N) x`) THEN
DISCH_THEN(ANTE_RES_THEN MP_TAC o CONJUNCT2) THEN
MATCH_MP_TAC EQ_IMP THEN
AP_TERM_TAC THEN REWRITE_TAC[
EXTENSION;
IN_ELIM_THM] THEN
REWRITE_TAC[
IN]);;
let INTEGRAL_COMBINE = prove
(`!f:real^1->real^N a b c.
drop a <= drop c /\ drop c <= drop b /\ f
integrable_on (interval[a,b])
==> integral(interval[a,c]) f + integral(interval[c,b]) f =
integral(interval[a,b]) f`,
let INTEGRAL_HAS_VECTOR_DERIVATIVE = prove
(`!f:real^1->real^N a b.
(f
continuous_on interval[a,b])
==> !x. x
IN interval[a,b]
==> ((\u. integral (interval[a,u]) f)
has_vector_derivative f(x))
(at x within interval[a,b])`,
REWRITE_TAC[
IN_INTERVAL_1] THEN REPEAT STRIP_TAC THEN
REWRITE_TAC[
has_vector_derivative;
HAS_DERIVATIVE_WITHIN_ALT] THEN
CONJ_TAC THENL
[REWRITE_TAC[linear;
DROP_ADD;
DROP_CMUL] THEN
CONJ_TAC THEN VECTOR_ARITH_TAC;
ALL_TAC] THEN
X_GEN_TAC `e:real` THEN DISCH_TAC THEN REWRITE_TAC[
IN_INTERVAL_1] THEN
FIRST_ASSUM(MP_TAC o MATCH_MP (REWRITE_RULE[
IMP_CONJ]
COMPACT_UNIFORMLY_CONTINUOUS)) THEN
REWRITE_TAC[
COMPACT_INTERVAL;
uniformly_continuous_on] THEN
DISCH_THEN(MP_TAC o SPEC `e:real`) THEN ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC
MONO_EXISTS THEN REWRITE_TAC[dist] THEN
X_GEN_TAC `d:real` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
X_GEN_TAC `y:real^1` THEN STRIP_TAC THEN
REWRITE_TAC[
NORM_REAL; GSYM drop;
DROP_SUB] THEN
DISJ_CASES_TAC(REAL_ARITH `drop x <= drop y \/ drop y <= drop x`) THENL
[ASM_SIMP_TAC[REAL_ARITH `x <= y ==> abs(y - x) = y - x`];
ONCE_REWRITE_TAC[VECTOR_ARITH
`fy - fx - (x - y) % c = --(fx - fy - (y - x) % c)`] THEN
ASM_SIMP_TAC[
NORM_NEG; REAL_ARITH `x <= y ==> abs(x - y) = y - x`]] THEN
ASM_SIMP_TAC[GSYM
CONTENT_1] THEN MATCH_MP_TAC
HAS_INTEGRAL_BOUND THEN
EXISTS_TAC `(\u. f(u) - f(x)):real^1->real^N` THEN
(ASM_SIMP_TAC[
REAL_LT_IMP_LE] THEN CONJ_TAC THENL
[ALL_TAC;
REPEAT STRIP_TAC THEN
MATCH_MP_TAC
REAL_LT_IMP_LE THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
REPEAT(POP_ASSUM MP_TAC) THEN
REWRITE_TAC[
IN_INTERVAL_1;
NORM_REAL;
DROP_SUB; GSYM drop] THEN
REAL_ARITH_TAC] THEN
MATCH_MP_TAC
HAS_INTEGRAL_SUB THEN REWRITE_TAC[
HAS_INTEGRAL_CONST]) THENL
[SUBGOAL_THEN
`integral(interval[a,x]) f + integral(interval[x,y]) f =
integral(interval[a,y]) f /\
((f:real^1->real^N)
has_integral integral(interval[x,y]) f)
(interval[x,y])`
(fun th -> MESON_TAC[th;
VECTOR_ARITH `a + b = c:real^N ==> c - a = b`]);
SUBGOAL_THEN
`integral(interval[a,y]) f + integral(interval[y,x]) f =
integral(interval[a,x]) f /\
((f:real^1->real^N)
has_integral integral(interval[y,x]) f)
(interval[y,x])`
(fun th -> MESON_TAC[th;
VECTOR_ARITH `a + b = c:real^N ==> c - a = b`])] THEN
(CONJ_TAC THENL
[MATCH_MP_TAC
INTEGRAL_COMBINE;
MATCH_MP_TAC
INTEGRABLE_INTEGRAL] THEN
ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC
INTEGRABLE_SUBINTERVAL THEN
MAP_EVERY EXISTS_TAC [`a:real^1`; `b:real^1`] THEN
ASM_SIMP_TAC[
INTEGRABLE_CONTINUOUS;
SUBSET_INTERVAL_1] THEN
ASM_REAL_ARITH_TAC));;
let HAS_INTEGRAL_TWIDDLE = prove
(`!f:real^N->real^P (g:real^M->real^N) h r i a b.
&0 < r /\
(!x. h(g x) = x) /\ (!x. g(h x) = x) /\ (!x. g continuous at x) /\
(!u v. ?w z.
IMAGE g (interval[u,v]) = interval[w,z]) /\
(!u v. ?w z.
IMAGE h (interval[u,v]) = interval[w,z]) /\
(!u v. content(
IMAGE g (interval[u,v])) = r * content(interval[u,v])) /\
(f
has_integral i) (interval[a,b])
==> ((\x. f(g x))
has_integral (inv r) % i) (
IMAGE h (interval[a,b]))`,
let lemma0 = prove
(`(!x k. (x,k) IN IMAGE (\(x,k). f x,g k) p ==> P x k) <=>
(!x k. (x,k) IN p ==> P (f x) (g k))`,
REWRITE_TAC[IN_IMAGE; EXISTS_PAIR_THM; PAIR_EQ] THEN MESON_TAC[])
and lemma1 = prove
(`{k | ?x. (x,k) IN p} = IMAGE SND p`,
REWRITE_TAC[EXTENSION; EXISTS_PAIR_THM; IN_IMAGE; IN_ELIM_THM] THEN
MESON_TAC[])
and lemma2 = prove
(`SND o (\(x,k). f x,g k) = g o SND`,
REWRITE_TAC[FUN_EQ_THM; FORALL_PAIR_THM; o_DEF]) in
REPEAT GEN_TAC THEN ASM_CASES_TAC `interval[a:real^N,b] = {}` THEN
ASM_SIMP_TAC[IMAGE_CLAUSES; HAS_INTEGRAL_EMPTY_EQ; VECTOR_MUL_RZERO] THEN
REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
REWRITE_TAC[has_integral] THEN
ASM_REWRITE_TAC[has_integral_def; has_integral_compact_interval] THEN
DISCH_TAC THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `e * r:real`) THEN
ASM_SIMP_TAC[REAL_LT_MUL] THEN
DISCH_THEN(X_CHOOSE_THEN `d:real^N->real^N->bool` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `\x y:real^M. (d:real^N->real^N->bool) (g x) (g y)` THEN
CONJ_TAC THENL
[FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [gauge]) THEN
SIMP_TAC[gauge; IN; FORALL_AND_THM] THEN
STRIP_TAC THEN X_GEN_TAC `x:real^M` THEN
SUBGOAL_THEN `(\y:real^M. (d:real^N->real^N->bool) (g x) (g y)) =
{y | g y IN (d (g x))}` SUBST1_TAC
THENL [SET_TAC[]; ASM_SIMP_TAC[CONTINUOUS_OPEN_PREIMAGE_UNIV]];
ALL_TAC] THEN
X_GEN_TAC `p:real^M#(real^M->bool)->bool` THEN STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC
`IMAGE (\(x,k). (g:real^M->real^N) x, IMAGE g k) p`) THEN
ANTS_TAC THENL
[CONJ_TAC THENL
[ALL_TAC;
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [fine]) THEN
REWRITE_TAC[fine; lemma0] THEN
STRIP_TAC THEN REPEAT GEN_TAC THEN DISCH_THEN(ANTE_RES_THEN MP_TAC) THEN
ASM SET_TAC[]] THEN
SUBGOAL_THEN
`interval[a,b] = IMAGE ((g:real^M->real^N) o h) (interval[a,b])`
SUBST1_TAC THENL [SIMP_TAC[o_DEF] THEN ASM SET_TAC[]; ALL_TAC] THEN
SUBGOAL_THEN `?u v. IMAGE (h:real^N->real^M) (interval[a,b]) =
interval[u,v]`
(REPEAT_TCL CHOOSE_THEN
(fun th -> SUBST_ALL_TAC th THEN ASSUME_TAC th)) THENL
[ASM_MESON_TAC[]; ALL_TAC] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [TAGGED_DIVISION_OF]) THEN
REWRITE_TAC[TAGGED_DIVISION_OF; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
REWRITE_TAC[lemma0] THEN REWRITE_TAC[IMP_IMP; GSYM CONJ_ASSOC] THEN
REPEAT GEN_TAC THEN STRIP_TAC THEN CONJ_TAC THENL
[ASM_SIMP_TAC[FINITE_IMAGE]; ALL_TAC] THEN
CONJ_TAC THENL
[MAP_EVERY X_GEN_TAC [`x:real^M`; `k:real^M->bool`] THEN
DISCH_TAC THEN
UNDISCH_TAC
`!x:real^M k.
x,k IN p
==> x IN k /\
k SUBSET interval[u,v] /\
?w z. k = interval[w,z]` THEN
DISCH_THEN(MP_TAC o SPECL [`x:real^M`; `k:real^M->bool`]) THEN
ASM_REWRITE_TAC[] THEN
REPEAT(MATCH_MP_TAC MONO_AND THEN CONJ_TAC) THENL
[SET_TAC[];
REWRITE_TAC[IMAGE_o] THEN ASM SET_TAC[];
STRIP_TAC THEN ASM_REWRITE_TAC[]];
ALL_TAC] THEN
CONJ_TAC THENL
[ALL_TAC;
ASM_REWRITE_TAC[IMAGE_o] THEN FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN
REWRITE_TAC[lemma1; GSYM IMAGE_o; lemma2] THEN
REWRITE_TAC[IMAGE_o; GSYM IMAGE_UNIONS; ETA_AX]] THEN
MAP_EVERY X_GEN_TAC [`x1:real^M`; `k1:real^M->bool`] THEN DISCH_TAC THEN
MAP_EVERY X_GEN_TAC [`x2:real^M`; `k2:real^M->bool`] THEN STRIP_TAC THEN
UNDISCH_TAC
`!x1:real^M k1:real^M->bool.
x1,k1 IN p
==> (!x2 k2.
x2,k2 IN p /\ ~(x1,k1 = x2,k2)
==> interior k1 INTER interior k2 = {})` THEN
DISCH_THEN(MP_TAC o SPECL [`x1:real^M`; `k1:real^M->bool`]) THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN(MP_TAC o SPECL [`x2:real^M`; `k2:real^M->bool`]) THEN
ASM_REWRITE_TAC[] THEN ANTS_TAC THENL
[ASM_MESON_TAC[PAIR_EQ]; ALL_TAC] THEN
MATCH_MP_TAC(SET_RULE
`interior(IMAGE f s) SUBSET IMAGE f (interior s) /\
interior(IMAGE f t) SUBSET IMAGE f (interior t) /\
(!x y. f x = f y ==> x = y)
==> interior s INTER interior t = {}
==> interior(IMAGE f s) INTER interior(IMAGE f t) = {}`) THEN
REPEAT CONJ_TAC THEN TRY(MATCH_MP_TAC INTERIOR_IMAGE_SUBSET) THEN
ASM_MESON_TAC[];
ALL_TAC] THEN
W(fun (asl,w) -> MP_TAC(PART_MATCH (lhand o rand) VSUM_IMAGE
(lhand(rand(lhand(lhand w)))))) THEN
ANTS_TAC THENL
[FIRST_ASSUM(ASSUME_TAC o MATCH_MP TAGGED_DIVISION_OF_FINITE) THEN
ASM_REWRITE_TAC[FORALL_PAIR_THM; PAIR_EQ] THEN
REPEAT GEN_TAC THEN DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
MATCH_MP_TAC MONO_AND THEN CONJ_TAC THEN ASM SET_TAC[];
ALL_TAC] THEN
DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[o_DEF; LAMBDA_PAIR_THM] THEN
DISCH_TAC THEN MATCH_MP_TAC REAL_LT_LCANCEL_IMP THEN
EXISTS_TAC `abs r` THEN ASM_SIMP_TAC[REAL_ARITH `&0 < x ==> &0 < abs x`] THEN
REWRITE_TAC[GSYM NORM_MUL] THEN ASM_SIMP_TAC[real_abs; REAL_LT_IMP_LE] THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH
`x < a * b ==> x = y ==> y < b * a`)) THEN
AP_TERM_TAC THEN REWRITE_TAC[VECTOR_SUB_LDISTRIB] THEN
ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_RINV; REAL_LT_IMP_NZ] THEN
REWRITE_TAC[VECTOR_MUL_LID; GSYM VSUM_LMUL] THEN
AP_THM_TAC THEN AP_TERM_TAC THEN MATCH_MP_TAC VSUM_EQ THEN
REWRITE_TAC[FORALL_PAIR_THM; VECTOR_MUL_ASSOC] THEN
REPEAT STRIP_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN
ASM_MESON_TAC[TAGGED_DIVISION_OF]);;
let HAS_INTEGRAL_STRETCH = prove
(`!f:real^M->real^N i m a b.
(f
has_integral i) (interval[a,b]) /\
(!k. 1 <= k /\ k <= dimindex(:M) ==> ~(m k = &0))
==> ((\x:real^M. f(lambda k. m k * x$k))
has_integral
(inv(abs(product(1..dimindex(:M)) m)) % i))
(
IMAGE (\x. lambda k. inv(m k) * x$k) (interval[a,b]))`,
let INTEGRAL_REFLECT = prove
(`!f:real^M->real^N a b.
integral (interval[--b,--a]) (\x. f(--x)) =
integral (interval[a,b]) f`,
let HAS_INTEGRAL = prove
(`!f:real^M->real^N i s.
(f
has_integral i) s <=>
!e. &0 < e
==> ?B. &0 < B /\
!a b. ball(vec 0,B)
SUBSET interval[a,b]
==> ?z. ((\x. if x
IN s then f(x) else vec 0)
has_integral z) (interval[a,b]) /\
norm(z - i) < e`,
REPEAT GEN_TAC THEN GEN_REWRITE_TAC LAND_CONV [
has_integral_alt] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
POP_ASSUM(X_CHOOSE_THEN `a:real^M` (X_CHOOSE_THEN `b:real^M`
SUBST_ALL_TAC)) THEN
MP_TAC(ISPECL [`a:real^M`; `b:real^M`] (CONJUNCT1
BOUNDED_INTERVAL)) THEN
REWRITE_TAC[
BOUNDED_POS] THEN
DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN EQ_TAC THENL
[DISCH_TAC THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN
EXISTS_TAC `B + &1` THEN ASM_SIMP_TAC[
REAL_LT_ADD;
REAL_LT_01] THEN
MAP_EVERY X_GEN_TAC [`c:real^M`; `d:real^M`] THEN
REWRITE_TAC[
SUBSET;
IN_BALL; NORM_ARITH `dist(vec 0,x) = norm x`] THEN
DISCH_TAC THEN EXISTS_TAC `i:real^N` THEN
ASM_REWRITE_TAC[
VECTOR_SUB_REFL;
NORM_0] THEN
MATCH_MP_TAC
HAS_INTEGRAL_RESTRICT_CLOSED_SUBINTERVAL THEN
ASM_MESON_TAC[
SUBSET; REAL_ARITH `n <= B ==> n < B + &1`];
ALL_TAC] THEN
DISCH_TAC THEN
SUBGOAL_THEN `?y. ((f:real^M->real^N)
has_integral y) (interval[a,b])`
MP_TAC THENL
[SUBGOAL_THEN
`?c d. interval[a,b]
SUBSET interval[c,d] /\
(\x. if x
IN interval[a,b] then (f:real^M->real^N) x
else vec 0)
integrable_on interval[c,d]`
STRIP_ASSUME_TAC THENL
[FIRST_X_ASSUM(MP_TAC o C MATCH_MP
REAL_LT_01) THEN
DISCH_THEN(X_CHOOSE_THEN `C:real` STRIP_ASSUME_TAC) THEN
ABBREV_TAC `c:real^M = lambda i. --(max B C)` THEN
ABBREV_TAC `d:real^M = lambda i. max B C` THEN
MAP_EVERY EXISTS_TAC [`c:real^M`; `d:real^M`] THEN CONJ_TAC THENL
[REWRITE_TAC[
SUBSET] THEN X_GEN_TAC `x:real^M` THEN
DISCH_TAC THEN REWRITE_TAC[
IN_INTERVAL] THEN
X_GEN_TAC `k:num` THEN MAP_EVERY EXPAND_TAC ["c";
let NEGLIGIBLE_ON_INTERVALS = prove
(`!s. negligible s <=> !a b:real^N. negligible(s
INTER interval[a,b])`,
GEN_TAC THEN EQ_TAC THEN REPEAT STRIP_TAC THENL
[MATCH_MP_TAC
NEGLIGIBLE_SUBSET THEN EXISTS_TAC `s:real^N->bool` THEN
ASM_REWRITE_TAC[] THEN SET_TAC[];
ALL_TAC] THEN
REWRITE_TAC[negligible] THEN
MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`] THEN
FIRST_ASSUM(ASSUME_TAC o SPECL [`a:real^N`; `b:real^N`]) THEN
MATCH_MP_TAC
HAS_INTEGRAL_NEGLIGIBLE THEN
EXISTS_TAC `s
INTER interval[a:real^N,b]` THEN
ASM_REWRITE_TAC[] THEN SIMP_TAC[indicator;
IN_DIFF;
IN_INTER] THEN
MESON_TAC[]);;
let HAS_INTEGRAL_ALT = prove
(`!f:real^M->real^N s i.
(f
has_integral i) s <=>
(!a b. (\x. if x
IN s then f x else vec 0)
integrable_on interval[a,b]) /\
(!e. &0 < e
==> ?B. &0 < B /\
!a b. ball (vec 0,B)
SUBSET interval[a,b]
==> norm(integral(interval[a,b])
(\x. if x
IN s then f x else vec 0) -
i) < e)`,
REPEAT GEN_TAC THEN GEN_REWRITE_TAC LAND_CONV [
HAS_INTEGRAL] THEN
SPEC_TAC(`\x. if x
IN s then (f:real^M->real^N) x else vec 0`,
`f:real^M->real^N`) THEN
GEN_TAC THEN EQ_TAC THENL
[ALL_TAC; MESON_TAC[
INTEGRAL_UNIQUE;
integrable_on]] THEN
DISCH_TAC THEN CONJ_TAC THENL
[ALL_TAC; ASM_MESON_TAC[
INTEGRAL_UNIQUE]] THEN
MAP_EVERY X_GEN_TAC [`a:real^M`; `b:real^M`] THEN
POP_ASSUM(MP_TAC o C MATCH_MP
REAL_LT_01) THEN
DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN
MATCH_MP_TAC
INTEGRABLE_SUBINTERVAL THEN
EXISTS_TAC `(lambda i. min ((a:real^M)$i) (--B)):real^M` THEN
EXISTS_TAC `(lambda i. max ((b:real^M)$i) B):real^M` THEN CONJ_TAC THENL
[FIRST_X_ASSUM(MP_TAC o SPECL
[`(lambda i. min ((a:real^M)$i) (--B)):real^M`;
`(lambda i. max ((b:real^M)$i) B):real^M`]) THEN
ANTS_TAC THENL [ALL_TAC; MESON_TAC[
integrable_on]];
SIMP_TAC[
SUBSET;
IN_INTERVAL;
IN_BALL;
LAMBDA_BETA;
REAL_MIN_LE;
REAL_LE_MAX]] THEN
SIMP_TAC[
SUBSET;
IN_BALL;
IN_INTERVAL;
LAMBDA_BETA] THEN
GEN_TAC THEN REWRITE_TAC[NORM_ARITH `dist(vec 0,x) = norm x`] THEN
DISCH_TAC THEN X_GEN_TAC `k:num` THEN STRIP_TAC THEN
MATCH_MP_TAC(REAL_ARITH
`abs(x) <= B ==> min a (--B) <= x /\ x <= max b B`) THEN
MATCH_MP_TAC
REAL_LE_TRANS THEN EXISTS_TAC `norm(x:real^M)` THEN
ASM_SIMP_TAC[
REAL_LT_IMP_LE;
COMPONENT_LE_NORM]);;
let INTEGRABLE_ALT = prove
(`!f:real^M->real^N s.
f
integrable_on s <=>
(!a b. (\x. if x
IN s then f x else vec 0)
integrable_on
interval[a,b]) /\
(!e. &0 < e
==> ?B. &0 < B /\
!a b c d.
ball(vec 0,B)
SUBSET interval[a,b] /\
ball(vec 0,B)
SUBSET interval[c,d]
==> norm(integral (interval[a,b])
(\x. if x
IN s then f x else vec 0) -
integral (interval[c,d])
(\x. if x
IN s then f x else vec 0)) < e)`,
REPEAT GEN_TAC THEN
GEN_REWRITE_TAC LAND_CONV [
integrable_on] THEN
ONCE_REWRITE_TAC[
HAS_INTEGRAL_ALT] THEN
REWRITE_TAC[
RIGHT_EXISTS_AND_THM] THEN
MATCH_MP_TAC(TAUT `(a ==> (b <=> c)) ==> (a /\ b <=> a /\ c)`) THEN
DISCH_TAC THEN EQ_TAC THENL
[DISCH_THEN(X_CHOOSE_THEN `y:real^N` STRIP_ASSUME_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 `B:real` THEN
MESON_TAC[NORM_ARITH `norm(a - y) < e / &2 /\ norm(b - y) < e / &2
==> norm(a - b) < e`];
ALL_TAC] THEN
DISCH_TAC THEN
SUBGOAL_THEN
`cauchy (\n. integral (interval[(lambda i. --(&n)),(lambda i. &n)])
(\x. if x
IN s then (f:real^M->real^N) x else vec 0))`
MP_TAC THENL
[REWRITE_TAC[cauchy] 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 `B:real` STRIP_ASSUME_TAC) THEN
MP_TAC(SPEC `B:real`
REAL_ARCH_SIMPLE) THEN
MATCH_MP_TAC
MONO_EXISTS THEN X_GEN_TAC `N:num` THEN DISCH_TAC THEN
REPEAT STRIP_TAC THEN REWRITE_TAC[dist] THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN
REWRITE_TAC[
SUBSET;
IN_BALL; NORM_ARITH `dist(vec 0,x) = norm x`] THEN
CONJ_TAC;
REWRITE_TAC[GSYM
CONVERGENT_EQ_CAUCHY] THEN
MATCH_MP_TAC
MONO_EXISTS THEN X_GEN_TAC `i:real^N` THEN
REWRITE_TAC[
LIM_SEQUENTIALLY] THEN DISCH_THEN(LABEL_TAC "C") THEN
X_GEN_TAC `e:real` THEN DISCH_TAC THEN
REMOVE_THEN "C" (MP_TAC o SPEC `e / &2`) THEN
FIRST_X_ASSUM(MP_TAC o SPEC `e / &2`) THEN ASM_REWRITE_TAC[
REAL_HALF] THEN
DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN
DISCH_THEN(X_CHOOSE_THEN `N:num` ASSUME_TAC) THEN
MP_TAC(SPEC `max (&N) B`
REAL_ARCH_SIMPLE) THEN
REWRITE_TAC[
REAL_MAX_LE; REAL_OF_NUM_LE] THEN
DISCH_THEN(X_CHOOSE_THEN `n:num` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `&n` THEN CONJ_TAC THENL
[ASM_REAL_ARITH_TAC; ALL_TAC] THEN
MAP_EVERY X_GEN_TAC [`a:real^M`; `b:real^M`] THEN STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `n:num`) THEN ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC(NORM_ARITH
`norm(i1 - i2) < e / &2 ==> dist(i1,i) < e / &2 ==> norm(i2 - i) < e`) THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN CONJ_TAC THEN
MATCH_MP_TAC
SUBSET_TRANS THEN EXISTS_TAC `ball(vec 0:real^M,&n)` THEN
ASM_SIMP_TAC[
SUBSET_BALL] THEN
REWRITE_TAC[
SUBSET;
IN_BALL; NORM_ARITH `dist(vec 0,x) = norm x`]] THEN
X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN
SIMP_TAC[
IN_INTERVAL;
LAMBDA_BETA] THEN REPEAT GEN_TAC THEN STRIP_TAC THEN
REWRITE_TAC[
REAL_BOUNDS_LE] THEN MATCH_MP_TAC
REAL_LE_TRANS THEN
EXISTS_TAC `norm(x:real^M)` THEN ASM_SIMP_TAC[
COMPONENT_LE_NORM] THEN
REPEAT(POP_ASSUM MP_TAC) THEN REWRITE_TAC[GSYM
REAL_OF_NUM_GE] THEN
REAL_ARITH_TAC);;
let INTEGRABLE_ALT_SUBSET = prove
(`!f:real^M->real^N s.
f
integrable_on s <=>
(!a b. (\x. if x
IN s then f x else vec 0)
integrable_on
interval[a,b]) /\
(!e. &0 < e
==> ?B. &0 < B /\
!a b c d.
ball(vec 0,B)
SUBSET interval[a,b] /\
interval[a,b]
SUBSET interval[c,d]
==> norm(integral (interval[a,b])
(\x. if x
IN s then f x else vec 0) -
integral (interval[c,d])
(\x. if x
IN s then f x else vec 0)) < e)`,
REPEAT GEN_TAC THEN GEN_REWRITE_TAC LAND_CONV [
INTEGRABLE_ALT] THEN
ABBREV_TAC `g:real^M->real^N = \x. if x
IN s then f x else vec 0` THEN
POP_ASSUM(K ALL_TAC) THEN
MATCH_MP_TAC(TAUT `(a ==> (b <=> c)) ==> (a /\ b <=> a /\ c)`) THEN
DISCH_TAC THEN EQ_TAC THENL [MESON_TAC[
SUBSET_TRANS]; ALL_TAC] 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 `B:real` THEN
STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
MAP_EVERY X_GEN_TAC [`a:real^M`; `b:real^M`; `c:real^M`; `d:real^M`] THEN
STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPECL
[`(lambda i. max ((a:real^M)$i) ((c:real^M)$i)):real^M`;
`(lambda i. min ((b:real^M)$i) ((d:real^M)$i)):real^M`]) THEN
ASM_REWRITE_TAC[GSYM
INTER_INTERVAL;
SUBSET_INTER] THEN
DISCH_THEN(fun th ->
MP_TAC(ISPECL [`a:real^M`; `b:real^M`] th) THEN
MP_TAC(ISPECL [`c:real^M`; `d:real^M`] th)) THEN
ASM_REWRITE_TAC[
INTER_SUBSET] THEN NORM_ARITH_TAC);;
let INTEGRAL_SPLIT = prove
(`!f:real^M->real^N a b t k.
f
integrable_on interval[a,b] /\ 1 <= k /\ k <= dimindex(:M)
==> integral (interval[a,b]) f =
integral(interval
[a,(lambda i. if i = k then min (b$k) t else b$i)]) f +
integral(interval
[(lambda i. if i = k then max (a$k) t else a$i),b]) f`,
let INTEGRAL_SPLIT_SIGNED = prove
(`!f:real^M->real^N a b t k.
1 <= k /\ k <= dimindex(:M) /\ a$k <= t /\ a$k <= b$k /\
f
integrable_on
interval[a,(lambda i. if i = k then max (b$k) t else b$i)]
==> integral (interval[a,b]) f =
integral(interval
[a,(lambda i. if i = k then t else b$i)]) f +
(if b$k < t then -- &1 else &1) %
integral(interval
[(lambda i. if i = k then min (b$k) t else a$i),
(lambda i. if i = k then max (b$k) t else b$i)]) f`,
REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THENL
[MP_TAC(ISPECL
[`f:real^M->real^N`;
`a:real^M`;
`(lambda i. if i = k then t else (b:real^M)$i):real^M`;
`(b:real^M)$k`; `k:num`]
INTEGRAL_SPLIT) THEN
ASM_REWRITE_TAC[] THEN ANTS_TAC THENL
[FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP
(REWRITE_RULE[
IMP_CONJ]
INTEGRABLE_ON_SUBINTERVAL)) THEN
ASM_SIMP_TAC[
SUBSET_INTERVAL;
LAMBDA_BETA] THEN
REPEAT STRIP_TAC THEN TRY COND_CASES_TAC THEN
ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC;
DISCH_THEN SUBST1_TAC THEN MATCH_MP_TAC(VECTOR_ARITH
`x = y /\ w = z
==> x:real^N = (y + z) + --(&1) % w`) THEN
CONJ_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN AP_TERM_TAC THEN
REWRITE_TAC[
CONS_11;
PAIR_EQ;
CART_EQ] THEN TRY CONJ_TAC THEN
ASM_SIMP_TAC[
LAMBDA_BETA] THEN
GEN_TAC THEN STRIP_TAC THEN COND_CASES_TAC THEN ASM_SIMP_TAC[] THEN
ASM_REAL_ARITH_TAC];
MP_TAC(ISPECL
[`f:real^M->real^N`;
`a:real^M`;
`b:real^M`;
`t:real`; `k:num`]
INTEGRAL_SPLIT) THEN
ASM_REWRITE_TAC[] THEN ANTS_TAC THENL
[FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP
(REWRITE_RULE[
IMP_CONJ]
INTEGRABLE_ON_SUBINTERVAL)) THEN
ASM_SIMP_TAC[
SUBSET_INTERVAL;
LAMBDA_BETA] THEN
REPEAT STRIP_TAC THEN TRY COND_CASES_TAC THEN
ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC;
DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[
VECTOR_MUL_LID] THEN
BINOP_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN AP_TERM_TAC THEN
REWRITE_TAC[
CONS_11;
PAIR_EQ;
CART_EQ] THEN TRY CONJ_TAC THEN
ASM_SIMP_TAC[
LAMBDA_BETA] THEN
GEN_TAC THEN STRIP_TAC THEN COND_CASES_TAC THEN ASM_SIMP_TAC[] THEN
ASM_REAL_ARITH_TAC]]);;
let INTEGRAL_INTERVALS_INCLUSION_EXCLUSION = prove
(`!f:real^M->real^N a b c d.
f
integrable_on interval[a,b] /\
c
IN interval[a,b] /\ d
IN interval[a,b]
==> integral(interval[a,d]) f =
vsum {s | s
SUBSET 1..dimindex(:M)}
(\s. --(&1) pow
CARD {i | i
IN s /\ d$i < c$i} %
integral
(interval[(lambda i. if i
IN s
then min (c$i) (d$i)
else (a:real^M)$i),
(lambda i. if i
IN s
then max (c$i) (d$i)
else c$i)]) f)`,
let lemma1 = prove
(`!f:(num->bool)->real^N n.
vsum {s | s SUBSET 1..SUC n} f =
vsum {s | s SUBSET 1..n} f +
vsum {s | s SUBSET 1..n} (\s. f(SUC n INSERT s))`,
REPEAT STRIP_TAC THEN
REWRITE_TAC[NUMSEG_CLAUSES; ARITH_RULE `1 <= SUC n`; POWERSET_CLAUSES] THEN
W(MP_TAC o PART_MATCH (lhs o rand) VSUM_UNION o lhs o snd) THEN
ANTS_TAC THENL
[ASM_SIMP_TAC[FINITE_IMAGE; FINITE_POWERSET; FINITE_NUMSEG] THEN
REWRITE_TAC[SET_RULE
`DISJOINT s (IMAGE f t) <=> !x. x IN t ==> ~(f x IN s)`] THEN
GEN_TAC THEN DISCH_TAC THEN REWRITE_TAC[IN_ELIM_THM; SUBSET] THEN
DISCH_THEN(MP_TAC o SPEC `SUC n`) THEN
REWRITE_TAC[IN_INSERT; IN_NUMSEG] THEN ARITH_TAC;
DISCH_THEN SUBST1_TAC THEN AP_TERM_TAC THEN
MATCH_MP_TAC(REWRITE_RULE[o_DEF] VSUM_IMAGE) THEN
SIMP_TAC[FINITE_POWERSET; FINITE_NUMSEG] THEN
MAP_EVERY X_GEN_TAC [`s:num->bool`; `t:num->bool`] THEN
REWRITE_TAC[IN_ELIM_THM] THEN MATCH_MP_TAC(SET_RULE
`~(a IN i)
==> s SUBSET i /\ t SUBSET i /\ a INSERT s = a INSERT t
==> s = t`) THEN
REWRITE_TAC[IN_NUMSEG] THEN ARITH_TAC]) in
let lemma2 = prove
(`!f:real^M->real^N m a:real^M c:real^M d:real^M.
f integrable_on (:real^M) /\ m <= dimindex(:M) /\
(!i. m < i /\ i <= dimindex(:M) ==> a$i = c$i \/ d$i = c$i) /\
(!i. m < i /\ i <= dimindex(:M) ==> a$i = c$i ==> a$i = d$i) /\
(!i. 1 <= i /\ i <= dimindex(:M) ==> a$i <= c$i /\ a$i <= d$i)
==> integral(interval[a,d]) f =
vsum {s | s SUBSET 1..m}
(\s. --(&1) pow CARD {i | i IN s /\ d$i < c$i} %
integral
(interval[(lambda i. if i IN s
then min (c$i) (d$i)
else (a:real^M)$i),
(lambda i. if i IN s
then max (c$i) (d$i)
else c$i)]) f)`,
GEN_TAC THEN INDUCT_TAC THENL
[REWRITE_TAC[NUMSEG_CLAUSES; ARITH; SUBSET_EMPTY; SING_GSPEC] THEN
REWRITE_TAC[VSUM_SING; NOT_IN_EMPTY; EMPTY_GSPEC; CARD_CLAUSES] THEN
REWRITE_TAC[real_pow; LAMBDA_ETA; VECTOR_MUL_LID] THEN
REWRITE_TAC[ARITH_RULE `0 < i <=> 1 <= i`] THEN REPEAT STRIP_TAC THEN
ASM_CASES_TAC
`?k. 1 <= k /\ k <= dimindex(:M) /\ (a:real^M)$k = (c:real^M)$k`
THENL
[MATCH_MP_TAC(MESON[] `i = vec 0 /\ j = vec 0 ==> i:real^N = j`) THEN
CONJ_TAC THEN MATCH_MP_TAC INTEGRAL_NULL THEN
REWRITE_TAC[CONTENT_EQ_0] THEN ASM_MESON_TAC[];
SUBGOAL_THEN `d:real^M = c:real^M` (fun th -> REWRITE_TAC[th]) THEN
REWRITE_TAC[CART_EQ] THEN ASM_MESON_TAC[]];
ALL_TAC] THEN
REPEAT STRIP_TAC THEN REWRITE_TAC[lemma1] THEN
SUBGOAL_THEN
`!s. s SUBSET 1..m
==> --(&1) pow CARD {i | i IN SUC m INSERT s /\ d$i < c$i} =
(if (d:real^M)$(SUC m) < (c:real^M)$(SUC m) then -- &1 else &1) *
--(&1) pow CARD {i | i IN s /\ d$i < c$i}`
(fun th -> SIMP_TAC[th; IN_ELIM_THM]) THENL
[X_GEN_TAC `s:num->bool` THEN DISCH_TAC THEN
SUBGOAL_THEN `FINITE(s:num->bool)` ASSUME_TAC THENL
[ASM_MESON_TAC[FINITE_NUMSEG; FINITE_SUBSET]; ALL_TAC] THEN
COND_CASES_TAC THENL
[ASM_SIMP_TAC[CARD_CLAUSES; FINITE_RESTRICT; SET_RULE
`P a ==> {x | x IN a INSERT s /\ P x} =
a INSERT {x | x IN s /\ P x}`] THEN
REWRITE_TAC[IN_ELIM_THM] THEN COND_CASES_TAC THEN
ASM_REWRITE_TAC[real_pow] THEN
SUBGOAL_THEN `~(SUC m IN 1..m)` (fun th -> ASM SET_TAC[th]) THEN
REWRITE_TAC[IN_NUMSEG] THEN ARITH_TAC;
ASM_SIMP_TAC[REAL_MUL_LID; SET_RULE
`~(P a) ==> {x | x IN a INSERT s /\ P x} = {x | x IN s /\ P x}`]];
ALL_TAC] THEN
MP_TAC(ISPECL
[`f:real^M->real^N`; `a:real^M`; `d:real^M`; `(c:real^M)$SUC m`; `SUC m`]
INTEGRAL_SPLIT_SIGNED) THEN
ANTS_TAC THENL
[ASM_MESON_TAC[ARITH_RULE `1 <= SUC n`; INTEGRABLE_ON_SUBINTERVAL;
SUBSET_UNIV];
DISCH_THEN SUBST1_TAC] THEN
REWRITE_TAC[VSUM_LMUL; GSYM VECTOR_MUL_ASSOC] THEN
BINOP_TAC THENL [ALL_TAC; AP_TERM_TAC] THENL
[FIRST_X_ASSUM(MP_TAC o SPECL
[`a:real^M`;
`c:real^M`;
`(lambda i. if i = SUC m then (c:real^M)$SUC m
else (d:real^M)$i):real^M`]);
FIRST_X_ASSUM(MP_TAC o SPECL
[`(lambda i. if i = SUC m
then min ((d:real^M)$SUC m) ((c:real^M)$SUC m)
else (a:real^M)$i):real^M`;
`(lambda i. if i = SUC m
then max ((d:real^M)$SUC m) ((c:real^M)$SUC m)
else (c:real^M)$i):real^M`;
`(lambda i. if i = SUC m
then max ((d:real^M)$SUC m) ((c:real^M)$SUC m)
else d$i):real^M`])] THEN
(ANTS_TAC THENL
[ASM_REWRITE_TAC[] THEN
CONJ_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN
CONJ_TAC THENL
[X_GEN_TAC `i:num` THEN STRIP_TAC THEN
SUBGOAL_THEN `1 <= i` ASSUME_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN
ASM_SIMP_TAC[LAMBDA_BETA] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_ARITH_TAC;
ALL_TAC] THEN
CONJ_TAC THENL
[X_GEN_TAC `i:num` THEN STRIP_TAC THEN
SUBGOAL_THEN `1 <= i` ASSUME_TAC THENL [ASM_ARITH_TAC; ALL_TAC] THEN
ASM_SIMP_TAC[LAMBDA_BETA] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
ASM_MESON_TAC[ARITH_RULE `m < i <=> i = SUC m \/ SUC m < i`];
X_GEN_TAC `i:num` THEN STRIP_TAC THEN ASM_SIMP_TAC[LAMBDA_BETA] THEN
COND_CASES_TAC THEN ASM_SIMP_TAC[] THEN TRY REAL_ARITH_TAC THEN
FIRST_X_ASSUM SUBST_ALL_TAC THEN ASM_REWRITE_TAC[] THEN
ASM_MESON_TAC[]];
DISCH_THEN SUBST1_TAC THEN MATCH_MP_TAC VSUM_EQ THEN
X_GEN_TAC `s:num->bool` THEN REWRITE_TAC[IN_ELIM_THM] THEN
DISCH_TAC THEN BINOP_TAC THENL
[AP_TERM_TAC THEN AP_TERM_TAC THEN
REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN
X_GEN_TAC `i:num` THEN
ASM_CASES_TAC `(i:num) IN s` THEN ASM_REWRITE_TAC[] THEN
SUBGOAL_THEN `i IN 1..m` MP_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
REWRITE_TAC[IN_NUMSEG] THEN STRIP_TAC THEN
SUBGOAL_THEN `i <= dimindex(:M)` ASSUME_TAC THENL
[ASM_ARITH_TAC; ALL_TAC] THEN
ASM_SIMP_TAC[LAMBDA_BETA] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN ASM_ARITH_TAC;
AP_THM_TAC THEN AP_TERM_TAC THEN AP_TERM_TAC THEN
REWRITE_TAC[CONS_11; PAIR_EQ] THEN
SIMP_TAC[CART_EQ; LAMBDA_BETA; AND_FORALL_THM] THEN
X_GEN_TAC `i:num` THEN
ASM_CASES_TAC `1 <= i /\ i <= dimindex(:M)` THEN
ASM_REWRITE_TAC[] THEN
ASM_CASES_TAC `(i:num) IN s` THEN ASM_REWRITE_TAC[IN_INSERT] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN TRY REAL_ARITH_TAC THEN
SUBGOAL_THEN `~(SUC m IN 1..m)` (fun th -> ASM SET_TAC[th]) THEN
REWRITE_TAC[IN_NUMSEG] THEN ARITH_TAC]])) in
REWRITE_TAC[IN_INTERVAL] THEN REPEAT STRIP_TAC THEN
MP_TAC(ISPECL
[`\x. if x IN interval[a,b] then (f:real^M->real^N) x else vec 0`;
`dimindex(:M)`; `a:real^M`; `c:real^M`; `d:real^M`]
lemma2) THEN
ASM_SIMP_TAC[LTE_ANTISYM; INTEGRABLE_RESTRICT_UNIV; LE_REFL] THEN
MATCH_MP_TAC EQ_IMP THEN BINOP_TAC THENL
[ALL_TAC;
MATCH_MP_TAC VSUM_EQ THEN X_GEN_TAC `s:num->bool` THEN
REWRITE_TAC[IN_ELIM_THM] THEN DISCH_TAC THEN AP_TERM_TAC] THEN
MATCH_MP_TAC INTEGRAL_EQ THEN ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC(SET_RULE
`i SUBSET j ==> !x. x IN i ==> (if x IN j then f x else b) = f x`) THEN
ASM_SIMP_TAC[SUBSET_INTERVAL; REAL_LE_REFL; LAMBDA_BETA] THEN
DISCH_TAC 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 REAL_ARITH_TAC);;
let INTEGRAL_INTERVALS_DIFF_INCLUSION_EXCLUSION = prove
(`!f:real^M->real^N a b c d.
f
integrable_on interval[a,b] /\
c
IN interval[a,b] /\ d
IN interval[a,b]
==> integral(interval[a,d]) f - integral(interval[a,c]) f =
vsum {s | ~(s = {}) /\ s
SUBSET 1..dimindex(:M)}
(\s. --(&1) pow
CARD {i | i
IN s /\ d$i < c$i} %
integral
(interval[(lambda i. if i
IN s
then min (c$i) (d$i)
else (a:real^M)$i),
(lambda i. if i
IN s
then max (c$i) (d$i)
else c$i)]) f)`,
let INTEGRAL_INTERVALS_INCLUSION_EXCLUSION_RIGHT = prove
(`!f:real^M->real^N a b c.
f
integrable_on interval[a,b] /\ c
IN interval[a,b]
==> integral(interval[a,c]) f =
vsum {s | s
SUBSET 1..dimindex (:M)}
(\s. --(&1) pow
CARD s %
integral
(interval[(lambda i. if i
IN s then c$i else a$i),
b])
f)`,
REPEAT STRIP_TAC THEN MP_TAC(ISPECL
[`f:real^M->real^N`; `a:real^M`; `b:real^M`; `b:real^M`; `c:real^M`]
INTEGRAL_INTERVALS_INCLUSION_EXCLUSION) THEN
ASM_REWRITE_TAC[
ENDS_IN_INTERVAL] THEN ANTS_TAC THENL
[ASM_MESON_TAC[
ENDS_IN_INTERVAL;
MEMBER_NOT_EMPTY]; ALL_TAC] THEN
DISCH_THEN SUBST1_TAC THEN MATCH_MP_TAC
VSUM_EQ THEN
X_GEN_TAC `s:num->bool` THEN REWRITE_TAC[
IN_ELIM_THM] THEN STRIP_TAC THEN
ASM_CASES_TAC `?k. k
IN s /\ (c:real^M)$k = (b:real^M)$k` THENL
[FIRST_X_ASSUM(X_CHOOSE_THEN `k:num` STRIP_ASSUME_TAC) THEN
SUBGOAL_THEN `1 <= k /\ k <= dimindex(:M)` STRIP_ASSUME_TAC THENL
[ASM_MESON_TAC[
IN_NUMSEG;
SUBSET]; ALL_TAC] THEN
MATCH_MP_TAC(MESON[] `a:real^N = vec 0 /\ b = vec 0 ==> a = b`) THEN
CONJ_TAC THEN REWRITE_TAC[
VECTOR_MUL_EQ_0] THEN DISJ2_TAC THEN
MATCH_MP_TAC
INTEGRAL_NULL THEN REWRITE_TAC[
CONTENT_EQ_0] THEN
EXISTS_TAC `k:num` THEN ASM_SIMP_TAC[
LAMBDA_BETA] THEN REAL_ARITH_TAC;
ALL_TAC] THEN
RULE_ASSUM_TAC(REWRITE_RULE[
IN_INTERVAL]) THEN BINOP_TAC THENL
[AP_TERM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[
EXTENSION;
IN_ELIM_THM] THEN
ASM_MESON_TAC[
REAL_LT_LE;
SUBSET;
IN_NUMSEG];
AP_THM_TAC THEN AP_TERM_TAC THEN AP_TERM_TAC THEN
AP_THM_TAC THEN AP_TERM_TAC THEN
ASM_SIMP_TAC[
CART_EQ;
PAIR_EQ;
LAMBDA_BETA] THEN
CONJ_TAC THEN X_GEN_TAC `i:num` THEN STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `i:num`) THEN ASM_REWRITE_TAC[] THEN
REAL_ARITH_TAC]);;
let INTEGRABLE_STRADDLE_INTERVAL = prove
(`!f:real^N->real^1 a b.
(!e. &0 < e
==> ?g h i j. (g
has_integral i) (interval[a,b]) /\
(h
has_integral j) (interval[a,b]) /\
norm(i - j) < e /\
!x. x
IN interval[a,b]
==> drop(g x) <= drop(f x) /\
drop(f x) <= drop(h x))
==> f
integrable_on interval[a,b]`,
REPEAT STRIP_TAC THEN REWRITE_TAC[
INTEGRABLE_CAUCHY] 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;
LEFT_IMP_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC
[`g:real^N->real^1`; `h:real^N->real^1`; `i:real^1`; `j:real^1`] THEN
REWRITE_TAC[
has_integral] THEN REWRITE_TAC[
IMP_CONJ] THEN
DISCH_THEN(MP_TAC o SPEC `e / &3`) THEN
ASM_SIMP_TAC[
REAL_LT_DIV;
REAL_OF_NUM_LT; ARITH] THEN
DISCH_THEN(X_CHOOSE_THEN `d1:real^N->real^N->bool` STRIP_ASSUME_TAC) THEN
DISCH_THEN(MP_TAC o SPEC `e / &3`) THEN
ASM_SIMP_TAC[
REAL_LT_DIV;
REAL_OF_NUM_LT; ARITH] THEN
DISCH_THEN(X_CHOOSE_THEN `d2:real^N->real^N->bool` STRIP_ASSUME_TAC) THEN
DISCH_TAC THEN DISCH_TAC THEN
EXISTS_TAC `(\x. d1 x
INTER d2 x):real^N->real^N->bool` THEN
ASM_SIMP_TAC[
GAUGE_INTER;
FINE_INTER] THEN
MAP_EVERY X_GEN_TAC
[`p1:(real^N#(real^N->bool))->bool`;
`p2:(real^N#(real^N->bool))->bool`] THEN
REPEAT STRIP_TAC THEN
REPEAT(FIRST_X_ASSUM(fun th ->
MP_TAC(SPEC `p1:(real^N#(real^N->bool))->bool` th) THEN
MP_TAC(SPEC `p2:(real^N#(real^N->bool))->bool` th))) THEN
EVERY_ASSUM(fun th -> try ASSUME_TAC(MATCH_MP
TAGGED_DIVISION_OF_FINITE th)
with Failure _ -> ALL_TAC) THEN
ASM_SIMP_TAC[
VSUM_REAL] THEN REWRITE_TAC[
o_DEF;
LAMBDA_PAIR_THM] THEN
SUBST1_TAC(SYM(ISPEC `i:real^1` (CONJUNCT1
LIFT_DROP))) THEN
SUBST1_TAC(SYM(ISPEC `j:real^1` (CONJUNCT1
LIFT_DROP))) THEN
REWRITE_TAC[GSYM
LIFT_SUB;
DROP_CMUL;
NORM_LIFT] THEN
MATCH_MP_TAC(REAL_ARITH
`g1 - h2 <= f1 - f2 /\ f1 - f2 <= h1 - g2 /\
abs(i - j) < e / &3
==> abs(g2 - i) < e / &3
==> abs(g1 - i) < e / &3
==> abs(h2 - j) < e / &3
==> abs(h1 - j) < e / &3
==> abs(f1 - f2) < e`) THEN
ASM_REWRITE_TAC[GSYM
DROP_SUB; GSYM
NORM_LIFT;
LIFT_DROP] THEN CONJ_TAC THEN
MATCH_MP_TAC(REAL_ARITH `x <= x' /\ y' <= y ==> x - y <= x' - y'`) THEN
CONJ_TAC THEN MATCH_MP_TAC
SUM_LE THEN
REWRITE_TAC[
FORALL_PAIR_THM] THEN REPEAT STRIP_TAC THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC
REAL_LE_LMUL THEN
ASM_MESON_TAC[
TAGGED_DIVISION_OF;
CONTENT_POS_LE;
SUBSET]);;
let INTEGRABLE_STRADDLE = prove
(`!f:real^N->real^1 s.
(!e. &0 < e
==> ?g h i j. (g
has_integral i) s /\
(h
has_integral j) s /\
norm(i - j) < e /\
!x. x
IN s
==> drop(g x) <= drop(f x) /\
drop(f x) <= drop(h x))
==> f
integrable_on s`,
let lemma = prove
(`&0 <= drop x /\ drop x <= drop y ==> norm x <= norm y`,
REWRITE_TAC[NORM_REAL; GSYM drop] THEN REAL_ARITH_TAC) in
REPEAT STRIP_TAC THEN
SUBGOAL_THEN
`!a b. (\x. if x IN s then (f:real^N->real^1) x else vec 0)
integrable_on interval[a,b]`
ASSUME_TAC THENL
[RULE_ASSUM_TAC(REWRITE_RULE[HAS_INTEGRAL_ALT]) THEN
MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`] THEN
MATCH_MP_TAC INTEGRABLE_STRADDLE_INTERVAL THEN
X_GEN_TAC `e:real` THEN DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `e / &4`) THEN
ASM_SIMP_TAC[REAL_LT_DIV; REAL_OF_NUM_LT; ARITH] THEN
REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC
[`g:real^N->real^1`; `h:real^N->real^1`; `i:real^1`; `j:real^1`] THEN
REWRITE_TAC[GSYM CONJ_ASSOC] THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
DISCH_THEN(CONJUNCTS_THEN2 (MP_TAC o SPEC `e / &4`) MP_TAC) THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
DISCH_THEN(CONJUNCTS_THEN2 (MP_TAC o SPEC `e / &4`) STRIP_ASSUME_TAC) THEN
ASM_SIMP_TAC[REAL_LT_DIV; REAL_OF_NUM_LT; ARITH] THEN
DISCH_THEN(X_CHOOSE_THEN `B2:real`
(CONJUNCTS_THEN2 ASSUME_TAC (LABEL_TAC "H"))) THEN
DISCH_THEN(X_CHOOSE_THEN `B1:real`
(CONJUNCTS_THEN2 ASSUME_TAC (LABEL_TAC "G"))) THEN
MAP_EVERY EXISTS_TAC
[`\x. if x IN s then (g:real^N->real^1) x else vec 0`;
`\x. if x IN s then (h:real^N->real^1) x else vec 0`;
`integral(interval[a:real^N,b])
(\x. if x IN s then (g:real^N->real^1) x else vec 0:real^1)`;
`integral(interval[a:real^N,b])
(\x. if x IN s then (h:real^N->real^1) x else vec 0:real^1)`] THEN
ASM_SIMP_TAC[INTEGRABLE_INTEGRAL] THEN
CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[REAL_LE_REFL]] THEN
ABBREV_TAC `c:real^N = lambda i. min ((a:real^N)$i) (--(max B1 B2))` THEN
ABBREV_TAC `d:real^N = lambda i. max ((b:real^N)$i) (max B1 B2)` THEN
REMOVE_THEN "H" (MP_TAC o SPECL [`c:real^N`; `d:real^N`]) THEN
REMOVE_THEN "G" (MP_TAC o SPECL [`c:real^N`; `d:real^N`]) THEN
MATCH_MP_TAC(TAUT
`(a /\ c) /\ (b /\ d ==> e) ==> (a ==> b) ==> (c ==> d) ==> e`) THEN
CONJ_TAC THENL
[CONJ_TAC THEN MAP_EVERY EXPAND_TAC ["c";
let HENSTOCK_LEMMA_PART1 = prove
(`!f:real^M->real^N a b d e.
f
integrable_on interval[a,b] /\
&0 < e /\ gauge d /\
(!p. p
tagged_division_of interval[a,b] /\ d fine p
==> norm (vsum p (\(x,k). content k % f x) -
integral(interval[a,b]) f) < e)
==> !p. p
tagged_partial_division_of interval[a,b] /\ d fine p
==> norm(vsum p (\(x,k). content k % f x -
integral k f)) <= e`,
let lemma = prove
(`(!k. &0 < k ==> x <= e + k) ==> x <= e`,
DISCH_THEN(MP_TAC o SPEC `(x - e) / &2`) THEN REAL_ARITH_TAC) in
REPEAT GEN_TAC THEN STRIP_TAC THEN GEN_TAC THEN STRIP_TAC THEN
MATCH_MP_TAC lemma THEN X_GEN_TAC `k:real` THEN DISCH_TAC THEN
MP_TAC(ISPECL
[`IMAGE SND (p:(real^M#(real^M->bool))->bool)`; `a:real^M`; `b:real^M`]
PARTIAL_DIVISION_EXTEND_INTERVAL) THEN
ANTS_TAC THENL
[CONJ_TAC THENL
[ASM_MESON_TAC[PARTIAL_DIVISION_OF_TAGGED_DIVISION]; ALL_TAC] THEN
REWRITE_TAC[SUBSET; FORALL_IN_UNIONS] THEN
REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_IMAGE] THEN
REWRITE_TAC[FORALL_PAIR_THM] THEN
ASM_MESON_TAC[tagged_partial_division_of; SUBSET];
ALL_TAC] THEN
SUBGOAL_THEN `FINITE(p:(real^M#(real^M->bool))->bool)` ASSUME_TAC THENL
[ASM_MESON_TAC[tagged_partial_division_of]; ALL_TAC] THEN
DISCH_THEN(X_CHOOSE_THEN `q:(real^M->bool)->bool` STRIP_ASSUME_TAC) THEN
FIRST_X_ASSUM(MP_TAC o MATCH_MP(SET_RULE
`s SUBSET t ==> t = s UNION (t DIFF s)`)) THEN
ABBREV_TAC `r = q DIFF IMAGE SND (p:(real^M#(real^M->bool))->bool)` THEN
SUBGOAL_THEN `IMAGE SND (p:(real^M#(real^M->bool))->bool) INTER r = {}`
ASSUME_TAC THENL [EXPAND_TAC "r" THEN SET_TAC[]; ALL_TAC] THEN
DISCH_THEN SUBST_ALL_TAC THEN
SUBGOAL_THEN `FINITE(r:(real^M->bool)->bool)` ASSUME_TAC THENL
[ASM_MESON_TAC[division_of; FINITE_UNION]; ALL_TAC] THEN
SUBGOAL_THEN
`!i. i IN r
==> ?p. p tagged_division_of i /\ d fine p /\
norm(vsum p (\(x,j). content j % f x) -
integral i (f:real^M->real^N))
< k / (&(CARD(r:(real^M->bool)->bool)) + &1)`
MP_TAC THENL
[X_GEN_TAC `i:real^M->bool` THEN DISCH_TAC THEN
SUBGOAL_THEN `(i:real^M->bool) SUBSET interval[a,b]` ASSUME_TAC THENL
[ASM_MESON_TAC[division_of; IN_UNION]; ALL_TAC] THEN
SUBGOAL_THEN `?u v:real^M. i = interval[u,v]`
(REPEAT_TCL CHOOSE_THEN SUBST_ALL_TAC)
THENL [ASM_MESON_TAC[division_of; IN_UNION]; ALL_TAC] THEN
SUBGOAL_THEN `(f:real^M->real^N) integrable_on interval[u,v]` MP_TAC THENL
[ASM_MESON_TAC[INTEGRABLE_SUBINTERVAL]; ALL_TAC] THEN
DISCH_THEN(MP_TAC o MATCH_MP INTEGRABLE_INTEGRAL) THEN
REWRITE_TAC[has_integral] THEN
DISCH_THEN(MP_TAC o SPEC `k / (&(CARD(r:(real^M->bool)->bool)) + &1)`) THEN
ASM_SIMP_TAC[REAL_LT_DIV; REAL_ARITH `&0 < &n + &1`] THEN
DISCH_THEN(X_CHOOSE_THEN `dd:real^M->real^M->bool` MP_TAC) THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
MP_TAC(ISPECL [`d:real^M->real^M->bool`; `dd:real^M->real^M->bool`]
GAUGE_INTER) THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN(MP_TAC o MATCH_MP FINE_DIVISION_EXISTS) THEN
DISCH_THEN(MP_TAC o SPECL [`u:real^M`; `v:real^M`]) THEN
REWRITE_TAC[FINE_INTER] THEN MESON_TAC[];
ALL_TAC] THEN
REWRITE_TAC[RIGHT_IMP_EXISTS_THM; SKOLEM_THM] THEN
REWRITE_TAC[TAUT `(a ==> b /\ c) <=> (a ==> b) /\ (a ==> c)`] THEN
REWRITE_TAC[FORALL_AND_THM] THEN
DISCH_THEN(X_CHOOSE_THEN `q:(real^M->bool)->(real^M#(real^M->bool))->bool`
STRIP_ASSUME_TAC) THEN
FIRST_X_ASSUM(MP_TAC o SPEC
`p UNION UNIONS {q (i:real^M->bool) | i IN r}
:(real^M#(real^M->bool))->bool`) THEN
ANTS_TAC THENL
[CONJ_TAC THENL
[ALL_TAC;
MATCH_MP_TAC FINE_UNION THEN ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC FINE_UNIONS THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE] THEN
ASM_REWRITE_TAC[FORALL_IN_IMAGE]] THEN
FIRST_ASSUM(SUBST1_TAC o SYM o last o CONJUNCTS o
GEN_REWRITE_RULE I [division_of]) THEN
REWRITE_TAC[UNIONS_UNION] THEN
MATCH_MP_TAC TAGGED_DIVISION_UNION THEN CONJ_TAC THENL
[ASM_MESON_TAC[TAGGED_PARTIAL_DIVISION_OF_UNION_SELF]; ALL_TAC] THEN
CONJ_TAC THENL
[ONCE_REWRITE_TAC[SIMPLE_IMAGE] THEN
MATCH_MP_TAC TAGGED_DIVISION_UNIONS THEN
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [division_of]) THEN
SIMP_TAC[FINITE_UNION; IN_UNION] THEN ASM_MESON_TAC[];
ALL_TAC] THEN
MATCH_MP_TAC INTER_INTERIOR_UNIONS_INTERVALS THEN
REWRITE_TAC[OPEN_INTERIOR] THEN
REPEAT(CONJ_TAC THENL
[ASM_MESON_TAC[division_of; FINITE_UNION; IN_UNION]; ALL_TAC]) THEN
X_GEN_TAC `k:real^M->bool` THEN DISCH_TAC THEN
ONCE_REWRITE_TAC[INTER_COMM] THEN
MATCH_MP_TAC INTER_INTERIOR_UNIONS_INTERVALS THEN
REWRITE_TAC[FORALL_IN_IMAGE; FORALL_PAIR_THM; OPEN_INTERIOR] THEN
REPEAT(CONJ_TAC THENL
[ASM_MESON_TAC[tagged_partial_division_of; FINITE_IMAGE]; ALL_TAC]) THEN
REPEAT STRIP_TAC THEN
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [division_of]) THEN
DISCH_THEN(MATCH_MP_TAC o el 2 o CONJUNCTS) THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [EXTENSION]) THEN
REWRITE_TAC[NOT_IN_EMPTY; GSYM NOT_EXISTS_THM] THEN
ASM_REWRITE_TAC[EXISTS_PAIR_THM; IN_IMAGE; IN_INTER; IN_UNION] THEN
ASM_MESON_TAC[];
ALL_TAC] THEN
SUBGOAL_THEN
`vsum (p UNION UNIONS {q i | i IN r}) (\(x,k). content k % f x) =
vsum p (\(x:real^M,k:real^M->bool). content k % f x:real^N) +
vsum (UNIONS {q i | (i:real^M->bool) IN r}) (\(x,k). content k % f x)`
SUBST1_TAC THENL
[MATCH_MP_TAC VSUM_UNION_NONZERO THEN ASM_REWRITE_TAC[] THEN
ONCE_REWRITE_TAC[SIMPLE_IMAGE] THEN
ASM_SIMP_TAC[FINITE_UNIONS; FINITE_IMAGE; FORALL_IN_IMAGE] THEN
CONJ_TAC THENL [ASM_MESON_TAC[TAGGED_DIVISION_OF_FINITE]; ALL_TAC] THEN
REWRITE_TAC[IN_INTER] THEN ONCE_REWRITE_TAC[CONJ_SYM] THEN
REWRITE_TAC[IMP_CONJ; FORALL_IN_UNIONS; FORALL_IN_IMAGE] THEN
REWRITE_TAC[FORALL_PAIR_THM; FORALL_IN_IMAGE; RIGHT_FORALL_IMP_THM] THEN
X_GEN_TAC `k:real^M->bool` THEN DISCH_TAC THEN
MAP_EVERY X_GEN_TAC [`x:real^M`; `l:real^M->bool`] THEN
DISCH_TAC THEN
SUBGOAL_THEN `(l:real^M->bool) SUBSET k` ASSUME_TAC THENL
[ASM_MESON_TAC[TAGGED_DIVISION_OF]; ALL_TAC] THEN DISCH_TAC THEN
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [division_of]) THEN
DISCH_THEN(MP_TAC o SPECL [`k:real^M->bool`; `l:real^M->bool`] o
el 2 o CONJUNCTS) THEN
ANTS_TAC THENL
[ASM_REWRITE_TAC[IN_UNION; IN_IMAGE; EXISTS_PAIR_THM] THEN
CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
DISCH_THEN(SUBST_ALL_TAC o SYM) THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [EXTENSION]) THEN
REWRITE_TAC[NOT_IN_EMPTY; GSYM NOT_EXISTS_THM] THEN
ASM_REWRITE_TAC[EXISTS_PAIR_THM; IN_IMAGE; IN_INTER; IN_UNION] THEN
ASM_MESON_TAC[];
ALL_TAC] THEN
ASM_SIMP_TAC[SUBSET_INTERIOR; SET_RULE `s SUBSET t ==> t INTER s = s`] THEN
SUBGOAL_THEN `?u v:real^M. l = interval[u,v]`
(fun th -> REPEAT_TCL CHOOSE_THEN SUBST1_TAC th THEN
SIMP_TAC[VECTOR_MUL_LZERO; GSYM CONTENT_EQ_0_INTERIOR]) THEN
ASM_MESON_TAC[tagged_partial_division_of];
ALL_TAC] THEN
W(MP_TAC o PART_MATCH (lhand o rand) VSUM_UNIONS_NONZERO o
rand o lhand o rand o lhand o lhand o snd) THEN
ANTS_TAC THENL
[ONCE_REWRITE_TAC[SIMPLE_IMAGE] THEN ASM_SIMP_TAC[FINITE_IMAGE] THEN
REWRITE_TAC[IMP_CONJ; FORALL_IN_IMAGE; RIGHT_FORALL_IMP_THM] THEN
CONJ_TAC THENL [ASM_MESON_TAC[TAGGED_DIVISION_OF; IN_UNION]; ALL_TAC] THEN
X_GEN_TAC `k:real^M->bool` THEN DISCH_TAC THEN
X_GEN_TAC `l:real^M->bool` THEN DISCH_TAC THEN
DISCH_TAC THEN REWRITE_TAC[FORALL_PAIR_THM] THEN
MAP_EVERY X_GEN_TAC [`x:real^M`; `m:real^M->bool`] THEN
DISCH_TAC THEN DISCH_TAC THEN
REWRITE_TAC[VECTOR_MUL_EQ_0] THEN DISJ1_TAC THEN
SUBGOAL_THEN `?u v:real^M. m = interval[u,v]`
(REPEAT_TCL CHOOSE_THEN SUBST_ALL_TAC)
THENL [ASM_MESON_TAC[TAGGED_DIVISION_OF; IN_UNION]; ALL_TAC] THEN
REWRITE_TAC[CONTENT_EQ_0_INTERIOR] THEN
MATCH_MP_TAC(SET_RULE `!t. s SUBSET t /\ t = {} ==> s = {}`) THEN
EXISTS_TAC `interior(k INTER l:real^M->bool)` THEN CONJ_TAC THENL
[MATCH_MP_TAC SUBSET_INTERIOR THEN REWRITE_TAC[SUBSET_INTER] THEN
ASM_MESON_TAC[TAGGED_DIVISION_OF];
ALL_TAC] THEN
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [division_of]) THEN
REWRITE_TAC[INTERIOR_INTER] THEN
DISCH_THEN(MATCH_MP_TAC o SPECL [`k:real^M->bool`; `l:real^M->bool`] o
el 2 o CONJUNCTS) THEN
REWRITE_TAC[IN_IMAGE; EXISTS_PAIR_THM; IN_UNION] THEN ASM_MESON_TAC[];
ALL_TAC] THEN
DISCH_THEN SUBST1_TAC THEN ONCE_REWRITE_TAC[SIMPLE_IMAGE] THEN
W(MP_TAC o PART_MATCH (lhand o rand) VSUM_IMAGE_NONZERO o
rand o lhand o rand o lhand o lhand o snd) THEN
ASM_REWRITE_TAC[o_DEF] THEN ANTS_TAC THENL
[MAP_EVERY X_GEN_TAC [`k:real^M->bool`; `l:real^M->bool`] THEN
STRIP_TAC THEN MATCH_MP_TAC VSUM_EQ_0 THEN
REWRITE_TAC[FORALL_PAIR_THM] THEN
MAP_EVERY X_GEN_TAC [`x:real^M`; `m:real^M->bool`] THEN DISCH_TAC THEN
MP_TAC(ASSUME `!i:real^M->bool. i IN r ==> q i tagged_division_of i`) THEN
DISCH_THEN(fun th -> MP_TAC(SPEC `l:real^M->bool` th) THEN
ANTS_TAC THENL [ASM_REWRITE_TAC[]; ALL_TAC] THEN
MP_TAC(SPEC `k:real^M->bool` th) THEN
ANTS_TAC THENL [ASM_REWRITE_TAC[]; ALL_TAC]) THEN
ASM_REWRITE_TAC[tagged_division_of] THEN ASM_MESON_TAC[];
ALL_TAC] THEN
DISCH_THEN SUBST1_TAC THEN
SUBGOAL_THEN
`vsum p (\(x,k). content k % (f:real^M->real^N) x - integral k f) =
vsum p (\(x,k). content k % f x) - vsum p (\(x,k). integral k f)`
SUBST1_TAC THENL [ASM_SIMP_TAC[GSYM VSUM_SUB; LAMBDA_PAIR_THM]; ALL_TAC] THEN
MATCH_MP_TAC(NORM_ARITH
`!ir. ip + ir = i /\
norm(cr - ir) < k
==> norm((cp + cr) - i) < e ==> norm(cp - ip) <= e + k`) THEN
EXISTS_TAC `vsum r (\k. integral k (f:real^M->real^N))` THEN CONJ_TAC THENL
[MATCH_MP_TAC EQ_TRANS THEN
EXISTS_TAC `vsum (IMAGE SND (p:(real^M#(real^M->bool))->bool) UNION r)
(\k. integral k (f:real^M->real^N))` THEN
CONJ_TAC THENL
[ALL_TAC; ASM_MESON_TAC[INTEGRAL_COMBINE_DIVISION_TOPDOWN]] THEN
MATCH_MP_TAC EQ_TRANS THEN
EXISTS_TAC `vsum (IMAGE SND (p:(real^M#(real^M->bool))->bool))
(\k. integral k (f:real^M->real^N)) +
vsum r (\k. integral k f)` THEN
CONJ_TAC THENL
[ALL_TAC;
CONV_TAC SYM_CONV THEN MATCH_MP_TAC VSUM_UNION_NONZERO THEN
ASM_SIMP_TAC[FINITE_IMAGE; NOT_IN_EMPTY]] THEN
AP_THM_TAC THEN AP_TERM_TAC THEN
SUBGOAL_THEN `(\(x:real^M,k). integral k (f:real^M->real^N)) =
(\k. integral k f) o SND`
SUBST1_TAC THENL
[SIMP_TAC[o_THM; FUN_EQ_THM; FORALL_PAIR_THM]; ALL_TAC] THEN
CONV_TAC SYM_CONV THEN MATCH_MP_TAC VSUM_IMAGE_NONZERO THEN
ASM_REWRITE_TAC[FORALL_PAIR_THM] THEN
MAP_EVERY X_GEN_TAC
[`x:real^M`; `l:real^M->bool`; `y:real^M`; `m:real^M->bool`] THEN
REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
DISCH_THEN(SUBST_ALL_TAC o SYM) THEN
FIRST_X_ASSUM(MP_TAC o
GEN_REWRITE_RULE I [tagged_partial_division_of]) THEN
DISCH_THEN(CONJUNCTS_THEN MP_TAC o CONJUNCT2) THEN
DISCH_THEN(MP_TAC o SPECL
[`x:real^M`; `l:real^M->bool`; `y:real^M`; `l:real^M->bool`]) THEN
ASM_REWRITE_TAC[INTER_IDEMPOT] THEN DISCH_TAC THEN
DISCH_THEN(MP_TAC o SPECL [`x:real^M`; `l:real^M->bool`]) THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN(REPEAT_TCL CHOOSE_THEN SUBST_ALL_TAC o last o CONJUNCTS) THEN
MATCH_MP_TAC INTEGRAL_UNIQUE THEN MATCH_MP_TAC HAS_INTEGRAL_NULL THEN
ASM_REWRITE_TAC[CONTENT_EQ_0_INTERIOR];
ALL_TAC] THEN
ASM_SIMP_TAC[GSYM VSUM_SUB] THEN MATCH_MP_TAC REAL_LET_TRANS THEN
EXISTS_TAC `sum (r:(real^M->bool)->bool) (\x. k / (&(CARD r) + &1))` THEN
CONJ_TAC THENL
[MATCH_MP_TAC VSUM_NORM_LE THEN ASM_SIMP_TAC[REAL_LT_IMP_LE];
ASM_SIMP_TAC[SUM_CONST] THEN
REWRITE_TAC[real_div; REAL_MUL_ASSOC] THEN
SIMP_TAC[GSYM real_div; REAL_LT_LDIV_EQ; REAL_ARITH `&0 < &x + &1`] THEN
REWRITE_TAC[REAL_ARITH `a * k < k * b <=> &0 < k * (b - a)`] THEN
MATCH_MP_TAC REAL_LT_MUL THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC]);;
let HENSTOCK_LEMMA = prove
(`!f:real^M->real^N a b.
f
integrable_on interval[a,b]
==> !e. &0 < e
==> ?d. gauge d /\
!p. p
tagged_partial_division_of interval[a,b] /\
d fine p
==> sum p (\(x,k). norm(content k % f x -
integral k f)) < e`,
MP_TAC
HENSTOCK_LEMMA_PART2 THEN
REPEAT(MATCH_MP_TAC
MONO_FORALL THEN GEN_TAC) THEN
DISCH_THEN(fun th -> STRIP_TAC THEN X_GEN_TAC `e:real` THEN
STRIP_TAC THEN MP_TAC th) THEN
FIRST_ASSUM(MP_TAC o MATCH_MP
INTEGRABLE_INTEGRAL) THEN
GEN_REWRITE_TAC LAND_CONV [
has_integral] THEN
DISCH_THEN(MP_TAC o SPEC `e / (&2 * (&(dimindex(:N)) + &1))`) THEN
ASM_SIMP_TAC[
REAL_LT_DIV; REAL_ARITH `&0 < &2 * (&n + &1)`] THEN
DISCH_THEN(X_CHOOSE_THEN `d:real^M->real^M->bool` STRIP_ASSUME_TAC) THEN
DISCH_THEN(MP_TAC o SPECL
[`d:real^M->real^M->bool`; `e / (&2 * (&(dimindex(:N)) + &1))`]) THEN
ASM_SIMP_TAC[
REAL_LT_DIV; REAL_ARITH `&0 < &2 * (&n + &1)`] THEN
DISCH_THEN(fun th -> EXISTS_TAC `d:real^M->real^M->bool` THEN MP_TAC th) 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(REAL_ARITH `d < e ==> x <= d ==> x < e`) THEN
REWRITE_TAC[
real_div;
REAL_INV_MUL;
REAL_INV_INV; REAL_MUL_ASSOC] THEN
SIMP_TAC[GSYM
real_div;
REAL_LT_LDIV_EQ; REAL_ARITH `&0 < &n + &1`] THEN
UNDISCH_TAC `&0 < e` THEN REAL_ARITH_TAC);;
let MONOTONE_CONVERGENCE_INTERVAL = prove
(`!f:num->real^N->real^1 g a b.
(!k. (f k)
integrable_on interval[a,b]) /\
(!k x. x
IN interval[a,b] ==> drop(f k x) <= drop(f (SUC k) x)) /\
(!x. x
IN interval[a,b] ==> ((\k. f k x) --> g x) sequentially) /\
bounded {integral (interval[a,b]) (f k) | k
IN (:num)}
==> g
integrable_on interval[a,b] /\
((\k. integral (interval[a,b]) (f k))
--> integral (interval[a,b]) g) sequentially`,
let lemma = prove
(`{(x,y) | P x y} = {p | P (FST p) (SND p)}`,
REWRITE_TAC[EXTENSION; FORALL_PAIR_THM; IN_ELIM_PAIR_THM; IN_ELIM_THM]) in
REPEAT GEN_TAC THEN STRIP_TAC THEN
ASM_CASES_TAC `content(interval[a:real^N,b]) = &0` THENL
[ASM_SIMP_TAC[INTEGRAL_NULL; INTEGRABLE_ON_NULL; LIM_CONST];
RULE_ASSUM_TAC(REWRITE_RULE[GSYM CONTENT_LT_NZ])] THEN
SUBGOAL_THEN
`!x:real^N k:num. x IN interval[a,b] ==> drop(f k x) <= drop(g x)`
ASSUME_TAC THENL
[REPEAT STRIP_TAC THEN
MATCH_MP_TAC(ISPEC `sequentially` LIM_DROP_LBOUND) THEN
EXISTS_TAC `\k. (f:num->real^N->real^1) k x` THEN
ASM_SIMP_TAC[TRIVIAL_LIMIT_SEQUENTIALLY; EVENTUALLY_SEQUENTIALLY] THEN
EXISTS_TAC `k:num` THEN SPEC_TAC(`k:num`,`k:num`) THEN
MATCH_MP_TAC TRANSITIVE_STEPWISE_LE THEN REWRITE_TAC[REAL_LE_TRANS] THEN
ASM_SIMP_TAC[REAL_LE_REFL];
ALL_TAC] THEN
SUBGOAL_THEN
`?i. ((\k. integral (interval[a,b]) (f k:real^N->real^1)) --> i)
sequentially`
CHOOSE_TAC THENL
[MATCH_MP_TAC BOUNDED_INCREASING_CONVERGENT THEN ASM_REWRITE_TAC[] THEN
GEN_TAC THEN MATCH_MP_TAC INTEGRAL_DROP_LE THEN ASM_REWRITE_TAC[];
ALL_TAC] THEN
SUBGOAL_THEN
`!k. drop(integral(interval[a,b]) ((f:num->real^N->real^1) k)) <= drop i`
ASSUME_TAC THENL
[GEN_TAC THEN MATCH_MP_TAC(ISPEC `sequentially` LIM_DROP_LBOUND) THEN
EXISTS_TAC `\k. integral(interval[a,b]) ((f:num->real^N->real^1) k)` THEN
ASM_REWRITE_TAC[TRIVIAL_LIMIT_SEQUENTIALLY; EVENTUALLY_SEQUENTIALLY] THEN
EXISTS_TAC `k:num` THEN SPEC_TAC(`k:num`,`k:num`) THEN
MATCH_MP_TAC TRANSITIVE_STEPWISE_LE THEN
ASM_REWRITE_TAC[REAL_LE_REFL; REAL_LE_TRANS] THEN
GEN_TAC THEN MATCH_MP_TAC INTEGRAL_DROP_LE THEN ASM_REWRITE_TAC[];
ALL_TAC] THEN
SUBGOAL_THEN
`((g:real^N->real^1) has_integral i) (interval[a,b])`
ASSUME_TAC THENL
[REWRITE_TAC[has_integral] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE BINDER_CONV
[HAS_INTEGRAL_INTEGRAL]) THEN
REWRITE_TAC[has_integral] THEN
DISCH_THEN(MP_TAC o GEN `k:num` o
SPECL [`k:num`; `e / &2 pow (k + 2)`]) THEN
ASM_SIMP_TAC[REAL_LT_DIV; REAL_POW_LT; REAL_OF_NUM_LT; ARITH] THEN
GEN_REWRITE_TAC LAND_CONV [SKOLEM_THM] THEN
REWRITE_TAC[LEFT_IMP_EXISTS_THM; FORALL_AND_THM] THEN
X_GEN_TAC `b:num->real^N->real^N->bool` THEN STRIP_TAC THEN
SUBGOAL_THEN
`?r. !k. r:num <= k
==> &0 <= drop i - drop(integral(interval[a:real^N,b]) (f k)) /\
drop i - drop(integral(interval[a,b]) (f k)) < e / &4`
STRIP_ASSUME_TAC THENL
[FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [LIM_SEQUENTIALLY]) THEN
DISCH_THEN(MP_TAC o SPEC `e / &4`) THEN
ASM_SIMP_TAC[REAL_LT_DIV; REAL_OF_NUM_LT; ARITH] 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[ABS_DROP; dist; DROP_SUB] THEN
MATCH_MP_TAC(REAL_ARITH
`x <= y ==> abs(x - y) < e ==> &0 <= y - x /\ y - x < e`) THEN
ASM_REWRITE_TAC[];
ALL_TAC] THEN
SUBGOAL_THEN
`!x. x IN interval[a:real^N,b]
==> ?n. r:num <= n /\
!k. n <= k ==> &0 <= drop(g x) - drop(f k x) /\
drop(g x) - drop(f k x) <
e / (&4 * content(interval[a,b]))`
MP_TAC THENL
[X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE (BINDER_CONV o RAND_CONV)
[LIM_SEQUENTIALLY]) THEN
DISCH_THEN(MP_TAC o SPEC `x:real^N`) THEN ASM_SIMP_TAC[REAL_SUB_LE] THEN
DISCH_THEN(MP_TAC o SPEC `e / (&4 * content(interval[a:real^N,b]))`) THEN
ASM_SIMP_TAC[REAL_LT_DIV; REAL_LT_MUL; REAL_OF_NUM_LT; ARITH] THEN
REWRITE_TAC[dist; ABS_DROP; DROP_SUB] THEN
ASM_SIMP_TAC[REAL_ARITH `f <= g ==> abs(f - g) = g - f`] THEN
DISCH_THEN(X_CHOOSE_TAC `N:num`) THEN
EXISTS_TAC `N + r:num` THEN CONJ_TAC THENL [ARITH_TAC; ALL_TAC] THEN
ASM_MESON_TAC[ARITH_RULE `N + r:num <= k ==> N <= k`];
ALL_TAC] THEN
GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [RIGHT_IMP_EXISTS_THM] THEN
REWRITE_TAC[SKOLEM_THM] THEN
REWRITE_TAC[FORALL_AND_THM; TAUT
`a ==> b /\ c <=> (a ==> b) /\ (a ==> c)`] THEN
REWRITE_TAC[RIGHT_IMP_FORALL_THM; IMP_IMP] THEN
DISCH_THEN(X_CHOOSE_THEN `m:real^N->num` STRIP_ASSUME_TAC) THEN
ABBREV_TAC `d:real^N->real^N->bool = \x. b(m x:num) x` THEN
EXISTS_TAC `d:real^N->real^N->bool` THEN CONJ_TAC THENL
[EXPAND_TAC "d" THEN REWRITE_TAC[gauge] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE BINDER_CONV [gauge]) THEN
SIMP_TAC[];
ALL_TAC] THEN
X_GEN_TAC `p:(real^N#(real^N->bool))->bool` THEN STRIP_TAC THEN
MATCH_MP_TAC(NORM_ARITH
`!b c. norm(a - b) <= e / &4 /\
norm(b - c) < e / &2 /\
norm(c - d) < e / &4
==> norm(a - d) < e`) THEN
EXISTS_TAC `vsum p (\(x:real^N,k:real^N->bool).
content k % (f:num->real^N->real^1) (m x) x)` THEN
EXISTS_TAC `vsum p (\(x:real^N,k:real^N->bool).
integral k ((f:num->real^N->real^1) (m x)))` THEN
FIRST_ASSUM(ASSUME_TAC o MATCH_MP TAGGED_DIVISION_OF_FINITE) THEN
SUBGOAL_THEN `?s:num. !t:real^N#(real^N->bool). t IN p ==> m(FST t) <= s`
MP_TAC THENL [ASM_SIMP_TAC[UPPER_BOUND_FINITE_SET]; ALL_TAC] THEN
REWRITE_TAC[FORALL_PAIR_THM] THEN DISCH_THEN(X_CHOOSE_TAC `s:num`) THEN
REPEAT CONJ_TAC THENL
[ASM_SIMP_TAC[GSYM VSUM_SUB] THEN REWRITE_TAC[LAMBDA_PAIR_THM] THEN
REWRITE_TAC[GSYM VECTOR_SUB_LDISTRIB] THEN
W(MP_TAC o PART_MATCH (lhand o rand) VSUM_NORM o lhand o snd) THEN
ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC(REAL_ARITH `y <= e ==> x <= y ==> x <= e`) THEN
REWRITE_TAC[LAMBDA_PAIR_THM] THEN MATCH_MP_TAC REAL_LE_TRANS THEN
EXISTS_TAC
`sum p (\(x:real^N,k:real^N->bool).
content k * e / (&4 * content (interval[a:real^N,b])))` THEN
CONJ_TAC THENL
[MATCH_MP_TAC SUM_LE THEN ASM_REWRITE_TAC[FORALL_PAIR_THM] THEN
MAP_EVERY X_GEN_TAC [`x:real^N`; `k:real^N->bool`] THEN
DISCH_TAC THEN REWRITE_TAC[NORM_MUL; GSYM VECTOR_SUB_LDISTRIB] THEN
MATCH_MP_TAC REAL_LE_MUL2 THEN
REWRITE_TAC[REAL_ABS_POS; NORM_POS_LE] THEN
REWRITE_TAC[ABS_DROP; DROP_SUB] THEN
REWRITE_TAC[REAL_ARITH `abs(x) <= x <=> &0 <= x`] THEN CONJ_TAC THENL
[ASM_MESON_TAC[CONTENT_POS_LE; TAGGED_DIVISION_OF]; ALL_TAC] THEN
MATCH_MP_TAC(REAL_ARITH
`&0 <= g - f /\ g - f < e ==> abs(g - f) <= e`) THEN
CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
REWRITE_TAC[LE_REFL] THEN ASM_MESON_TAC[TAGGED_DIVISION_OF; SUBSET];
ALL_TAC] THEN
REWRITE_TAC[LAMBDA_PAIR; SUM_RMUL] THEN REWRITE_TAC[LAMBDA_PAIR_THM] THEN
FIRST_ASSUM(fun th -> REWRITE_TAC[MATCH_MP
ADDITIVE_CONTENT_TAGGED_DIVISION th]) THEN
MATCH_MP_TAC REAL_EQ_IMP_LE THEN
UNDISCH_TAC `&0 < content(interval[a:real^N,b])` THEN
CONV_TAC REAL_FIELD;
ASM_SIMP_TAC[GSYM VSUM_SUB] THEN REWRITE_TAC[LAMBDA_PAIR_THM] THEN
MATCH_MP_TAC REAL_LET_TRANS THEN EXISTS_TAC
`norm(vsum (0..s)
(\j. vsum {(x:real^N,k:real^N->bool) | (x,k) IN p /\ m(x) = j}
(\(x,k). content k % f (m x) x :real^1 -
integral k (f (m x)))))` THEN
CONJ_TAC THENL
[MATCH_MP_TAC REAL_EQ_IMP_LE THEN REWRITE_TAC[lemma] THEN
AP_TERM_TAC THEN MATCH_MP_TAC(GSYM VSUM_GROUP) THEN
ASM_REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_NUMSEG; LE_0] THEN
ASM_REWRITE_TAC[FORALL_PAIR_THM];
ALL_TAC] THEN
MATCH_MP_TAC REAL_LET_TRANS THEN
EXISTS_TAC `sum (0..s) (\i. e / &2 pow (i + 2))` THEN CONJ_TAC THENL
[ALL_TAC;
REWRITE_TAC[real_div; GSYM REAL_POW_INV; SUM_LMUL] THEN
REWRITE_TAC[REAL_POW_ADD; SUM_RMUL] THEN REWRITE_TAC[SUM_GP] THEN
CONV_TAC REAL_RAT_REDUCE_CONV THEN
ASM_SIMP_TAC[REAL_LT_LMUL_EQ; CONJUNCT1 LT] THEN
REWRITE_TAC[real_div; GSYM REAL_MUL_ASSOC] THEN
CONV_TAC REAL_RAT_REDUCE_CONV THEN
MATCH_MP_TAC(REAL_ARITH `&0 < x * y ==> (&1 - x) * y < y`) THEN
MATCH_MP_TAC REAL_LT_MUL THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN
MATCH_MP_TAC REAL_POW_LT THEN CONV_TAC REAL_RAT_REDUCE_CONV] THEN
MATCH_MP_TAC VSUM_NORM_LE THEN REWRITE_TAC[FINITE_NUMSEG] THEN
X_GEN_TAC `t:num` THEN REWRITE_TAC[IN_NUMSEG; LE_0] THEN DISCH_TAC THEN
MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC
`norm(vsum {x:real^N,k:real^N->bool | x,k IN p /\ m x:num = t}
(\(x,k). content k % f t x - integral k (f t)):real^1)` THEN
CONJ_TAC THENL
[MATCH_MP_TAC REAL_EQ_IMP_LE THEN AP_TERM_TAC THEN
MATCH_MP_TAC VSUM_EQ THEN SIMP_TAC[FORALL_PAIR_THM; IN_ELIM_PAIR_THM];
ALL_TAC] THEN
MATCH_MP_TAC(REWRITE_RULE[RIGHT_IMP_FORALL_THM; IMP_IMP]
HENSTOCK_LEMMA_PART1) THEN
MAP_EVERY EXISTS_TAC
[`a:real^N`; `b:real^N`; `(b(t:num)):real^N->real^N->bool`] THEN
ASM_REWRITE_TAC[] THEN
ASM_SIMP_TAC[REAL_LT_DIV; REAL_POW_LT; REAL_OF_NUM_LT; ARITH] THEN
CONJ_TAC THENL
[MATCH_MP_TAC TAGGED_PARTIAL_DIVISION_SUBSET THEN
EXISTS_TAC `p:(real^N#(real^N->bool))->bool` THEN
SIMP_TAC[SUBSET; FORALL_PAIR_THM; IN_ELIM_PAIR_THM] THEN
ASM_MESON_TAC[tagged_division_of];
ALL_TAC] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [fine]) THEN
EXPAND_TAC "d" THEN REWRITE_TAC[fine; IN_ELIM_PAIR_THM] THEN MESON_TAC[];
MP_TAC(ISPECL [`(f:num->real^N->real^1) s`; `a:real^N`; `b:real^N`;
`p:(real^N#(real^N->bool))->bool`]
INTEGRAL_COMBINE_TAGGED_DIVISION_TOPDOWN) THEN
MP_TAC(ISPECL [`(f:num->real^N->real^1) r`; `a:real^N`; `b:real^N`;
`p:(real^N#(real^N->bool))->bool`]
INTEGRAL_COMBINE_TAGGED_DIVISION_TOPDOWN) THEN
ASM_SIMP_TAC[ABS_DROP; DROP_SUB; DROP_VSUM; GSYM DROP_EQ] THEN
REWRITE_TAC[o_DEF; LAMBDA_PAIR_THM] THEN MATCH_MP_TAC(REAL_ARITH
`sr <= sx /\ sx <= ss /\ ks <= i /\ &0 <= i - kr /\ i - kr < e
==> kr = sr ==> ks = ss ==> abs(sx - i) < e`) THEN
ASM_SIMP_TAC[LE_REFL] THEN CONJ_TAC THEN MATCH_MP_TAC SUM_LE THEN
ASM_REWRITE_TAC[FORALL_PAIR_THM] THEN
MAP_EVERY X_GEN_TAC [`x:real^N`; `i:real^N->bool`] THEN DISCH_TAC THEN
(SUBGOAL_THEN `i SUBSET interval[a:real^N,b]` ASSUME_TAC THENL
[ASM_MESON_TAC[TAGGED_DIVISION_OF]; ALL_TAC] THEN
SUBGOAL_THEN `?u v:real^N. i = interval[u,v]`
(REPEAT_TCL CHOOSE_THEN SUBST_ALL_TAC)
THENL [ASM_MESON_TAC[TAGGED_DIVISION_OF]; ALL_TAC]) THEN
MATCH_MP_TAC INTEGRAL_DROP_LE THEN
REPEAT(CONJ_TAC THENL
[ASM_MESON_TAC[INTEGRABLE_SUBINTERVAL]; ALL_TAC]) THEN
X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN
MP_TAC(ISPEC
`\m n:num. drop (f m (y:real^N)) <= drop (f n y)`
TRANSITIVE_STEPWISE_LE) THEN
REWRITE_TAC[REAL_LE_TRANS; REAL_LE_REFL] THEN
(ANTS_TAC THENL [ASM_MESON_TAC[SUBSET]; ALL_TAC]) THEN
DISCH_THEN MATCH_MP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
ASM_MESON_TAC[TAGGED_DIVISION_OF; SUBSET]];
ALL_TAC] THEN
CONJ_TAC THENL [ASM_MESON_TAC[integrable_on]; ALL_TAC] THEN
FIRST_ASSUM(SUBST1_TAC o MATCH_MP INTEGRAL_UNIQUE) THEN
ASM_REWRITE_TAC[]);;
let MONOTONE_CONVERGENCE_INCREASING = prove
(`!f:num->real^N->real^1 g s.
(!k. (f k)
integrable_on s) /\
(!k x. x
IN s ==> drop(f k x) <= drop(f (SUC k) x)) /\
(!x. x
IN s ==> ((\k. f k x) --> g x) sequentially) /\
bounded {integral s (f k) | k
IN (:num)}
==> g
integrable_on s /\
((\k. integral s (f k)) --> integral s g) sequentially`,
SUBGOAL_THEN
`!f:num->real^N->real^1 g s.
(!k x. x
IN s ==> &0 <= drop(f k x)) /\
(!k. (f k)
integrable_on s) /\
(!k x. x
IN s ==> drop(f k x) <= drop(f (SUC k) x)) /\
(!x. x
IN s ==> ((\k. f k x) --> g x) sequentially) /\
bounded {integral s (f k) | k
IN (:num)}
==> g
integrable_on s /\
((\k. integral s (f k)) --> integral s g) sequentially`
ASSUME_TAC THENL
[ALL_TAC;
REPEAT GEN_TAC THEN STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o ISPECL
[`\n x:real^N. f(SUC n) x - f 0 x:real^1`;
`\x. (g:real^N->real^1) x - f 0 x`; `s:real^N->bool`]) THEN
REWRITE_TAC[] THEN ANTS_TAC THEN REPEAT CONJ_TAC THENL
[REPEAT STRIP_TAC THEN REWRITE_TAC[
DROP_SUB;
REAL_SUB_LE] THEN
MP_TAC(ISPEC
`\m n:num. drop (f m (x:real^N)) <= drop (f n x)`
TRANSITIVE_STEPWISE_LE) THEN
REWRITE_TAC[
REAL_LE_TRANS;
REAL_LE_REFL] THEN ASM_MESON_TAC[
LE_0];
GEN_TAC THEN MATCH_MP_TAC
INTEGRABLE_SUB THEN ASM_REWRITE_TAC[ETA_AX];
REPEAT STRIP_TAC THEN REWRITE_TAC[
DROP_SUB;
REAL_SUB_LE] THEN
ASM_SIMP_TAC[REAL_ARITH `x - a <= y - a <=> x <= y`];
REPEAT STRIP_TAC THEN MATCH_MP_TAC
LIM_SUB THEN SIMP_TAC[
LIM_CONST] THEN
REWRITE_TAC[
ADD1] THEN
MATCH_MP_TAC(ISPECL[`f:num->real^1`; `l:real^1`; `1`]
SEQ_OFFSET) THEN
ASM_SIMP_TAC[];
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [bounded]) THEN
ASM_SIMP_TAC[
INTEGRAL_SUB; ETA_AX; bounded] THEN
ONCE_REWRITE_TAC[
SIMPLE_IMAGE] THEN
REWRITE_TAC[
FORALL_IN_IMAGE;
IN_UNIV] THEN
DISCH_THEN(X_CHOOSE_THEN `B:real`
(fun th -> EXISTS_TAC `B + norm(integral s (f 0:real^N->real^1))` THEN
X_GEN_TAC `k:num` THEN MP_TAC(SPEC `SUC k` th))) THEN
NORM_ARITH_TAC;
ASM_SIMP_TAC[
INTEGRAL_SUB; ETA_AX;
IMP_CONJ] THEN
SUBGOAL_THEN `(f 0:real^N->real^1)
integrable_on s` MP_TAC THENL
[ASM_REWRITE_TAC[]; ONCE_REWRITE_TAC[IMP_IMP]] THEN
DISCH_THEN(MP_TAC o MATCH_MP
INTEGRABLE_ADD) THEN
REWRITE_TAC[ETA_AX; VECTOR_ARITH `f + (g - f):real^N = g`] THEN
DISCH_TAC THEN ASM_SIMP_TAC[
INTEGRAL_SUB; ETA_AX] THEN
MP_TAC(ISPECL [`sequentially`; `integral s (f 0:real^N->real^1)`]
LIM_CONST) THEN
REWRITE_TAC[IMP_IMP] THEN DISCH_THEN(MP_TAC o MATCH_MP
LIM_ADD) THEN
REWRITE_TAC[ETA_AX; VECTOR_ARITH `f + (g - f):real^N = g`] THEN
REWRITE_TAC[
ADD1] THEN
SIMP_TAC[ISPECL[`f:num->real^1`; `l:real^1`; `1`]
SEQ_OFFSET_REV]]] THEN
REPEAT GEN_TAC THEN STRIP_TAC THEN
SUBGOAL_THEN
`!x:real^N k:num. x
IN s ==> drop(f k x) <= drop(g x)`
ASSUME_TAC THENL
[REPEAT STRIP_TAC THEN
MATCH_MP_TAC(ISPEC `sequentially`
LIM_DROP_LBOUND) THEN
EXISTS_TAC `\k. (f:num->real^N->real^1) k x` THEN
ASM_SIMP_TAC[
TRIVIAL_LIMIT_SEQUENTIALLY;
EVENTUALLY_SEQUENTIALLY] THEN
EXISTS_TAC `k:num` THEN SPEC_TAC(`k:num`,`k:num`) THEN
MATCH_MP_TAC
TRANSITIVE_STEPWISE_LE THEN REWRITE_TAC[
REAL_LE_TRANS] THEN
ASM_SIMP_TAC[
REAL_LE_REFL];
ALL_TAC] THEN
SUBGOAL_THEN
`?i. ((\k. integral s (f k:real^N->real^1)) --> i)
sequentially`
CHOOSE_TAC THENL
[MATCH_MP_TAC
BOUNDED_INCREASING_CONVERGENT THEN ASM_REWRITE_TAC[] THEN
GEN_TAC THEN MATCH_MP_TAC
INTEGRAL_DROP_LE THEN ASM_REWRITE_TAC[];
ALL_TAC] THEN
SUBGOAL_THEN
`!k. drop(integral s ((f:num->real^N->real^1) k)) <= drop i`
ASSUME_TAC THENL
[GEN_TAC THEN MATCH_MP_TAC(ISPEC `sequentially`
LIM_DROP_LBOUND) THEN
EXISTS_TAC `\k. integral(s) ((f:num->real^N->real^1) k)` THEN
ASM_REWRITE_TAC[
TRIVIAL_LIMIT_SEQUENTIALLY;
EVENTUALLY_SEQUENTIALLY] THEN
EXISTS_TAC `k:num` THEN SPEC_TAC(`k:num`,`k:num`) THEN
MATCH_MP_TAC
TRANSITIVE_STEPWISE_LE THEN
ASM_REWRITE_TAC[
REAL_LE_REFL;
REAL_LE_TRANS] THEN
GEN_TAC THEN MATCH_MP_TAC
INTEGRAL_DROP_LE THEN ASM_REWRITE_TAC[];
ALL_TAC] THEN
SUBGOAL_THEN `((g:real^N->real^1)
has_integral i) s` ASSUME_TAC THENL
[ALL_TAC;
CONJ_TAC THENL [ASM_MESON_TAC[
integrable_on]; ALL_TAC] THEN
FIRST_ASSUM(SUBST1_TAC o MATCH_MP
INTEGRAL_UNIQUE) THEN
ASM_REWRITE_TAC[]] THEN
REWRITE_TAC[
HAS_INTEGRAL_ALT] THEN
MP_TAC(ISPECL
[`\k x. if x
IN s then (f:num->real^N->real^1) k x else vec 0`;
`\x. if x
IN s then (g:real^N->real^1) x else vec 0`]
(MATCH_MP(MESON[] `(!a b c d. P a b c d ==> Q a b c d)
==> !a b. (!c d. P a b c d) ==> (!c d. Q a b c d)`)
MONOTONE_CONVERGENCE_INTERVAL)) THEN
ANTS_TAC THENL
[REPEAT GEN_TAC THEN REWRITE_TAC[] THEN
MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN CONJ_TAC THENL
[FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE BINDER_CONV [
INTEGRABLE_ALT]) THEN
SIMP_TAC[];
DISCH_TAC] THEN
CONJ_TAC THENL
[REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_SIMP_TAC[
REAL_LE_REFL];
ALL_TAC] THEN
CONJ_TAC THENL
[REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_SIMP_TAC[
LIM_CONST];
ALL_TAC] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [bounded]) THEN
ONCE_REWRITE_TAC[
SIMPLE_IMAGE] THEN
REWRITE_TAC[bounded;
FORALL_IN_IMAGE;
IN_UNIV] THEN
MATCH_MP_TAC
MONO_EXISTS THEN GEN_TAC THEN
MATCH_MP_TAC
MONO_FORALL THEN X_GEN_TAC `k:num` THEN
REWRITE_TAC[
ABS_DROP] THEN MATCH_MP_TAC(REAL_ARITH
`&0 <= y /\ y <= x ==> abs(x) <= a ==> abs(y) <= a`) THEN
CONJ_TAC THENL
[MATCH_MP_TAC
INTEGRAL_DROP_POS THEN ASM_REWRITE_TAC[] THEN
REPEAT STRIP_TAC THEN COND_CASES_TAC THEN
ASM_SIMP_TAC[
REAL_LE_REFL;
DROP_VEC];
ALL_TAC] THEN
GEN_REWRITE_TAC (RAND_CONV o RAND_CONV) [GSYM
INTEGRAL_RESTRICT_UNIV] THEN
MATCH_MP_TAC
INTEGRAL_SUBSET_DROP_LE THEN
ASM_REWRITE_TAC[
SUBSET_UNIV;
IN_UNIV] THEN
ASM_REWRITE_TAC[
INTEGRABLE_RESTRICT_UNIV; ETA_AX] THEN
GEN_TAC THEN COND_CASES_TAC THEN
ASM_SIMP_TAC[
REAL_LE_REFL;
DROP_VEC;
REAL_LE_REFL];
ALL_TAC] THEN
REWRITE_TAC[
FORALL_AND_THM] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
X_GEN_TAC `e:real` THEN DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
LIM_SEQUENTIALLY]) THEN
DISCH_THEN(MP_TAC o SPEC `e / &4`) THEN
ASM_SIMP_TAC[dist;
REAL_LT_DIV;
REAL_OF_NUM_LT; ARITH] THEN
DISCH_THEN(X_CHOOSE_THEN `N:num` STRIP_ASSUME_TAC) THEN
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE BINDER_CONV
[
HAS_INTEGRAL_INTEGRAL]) THEN
GEN_REWRITE_TAC (LAND_CONV o BINDER_CONV) [
HAS_INTEGRAL_ALT] THEN
REWRITE_TAC[
FORALL_AND_THM] THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
DISCH_THEN(MP_TAC o SPECL [`N:num`; `e / &4`]) THEN
ASM_SIMP_TAC[dist;
REAL_LT_DIV;
REAL_OF_NUM_LT; ARITH] THEN
MATCH_MP_TAC
MONO_EXISTS THEN X_GEN_TAC `B:real` THEN
STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`] THEN DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPECL [`a:real^N`; `b:real^N`]) THEN
ASM_REWRITE_TAC[] THEN
FIRST_ASSUM(MP_TAC o C MATCH_MP (ARITH_RULE `N:num <= N`)) THEN
REWRITE_TAC[IMP_IMP] THEN
DISCH_THEN(MP_TAC o MATCH_MP (NORM_ARITH
`norm(x - y) < e / &4 /\ norm(z - x) < e / &4
==> norm(z - y) < e / &2`)) THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE (BINDER_CONV o BINDER_CONV)
[
LIM_SEQUENTIALLY]) THEN
DISCH_THEN(MP_TAC o SPECL [`a:real^N`; `b:real^N`; `e / &2`]) THEN
ASM_REWRITE_TAC[dist;
REAL_HALF] THEN
DISCH_THEN(X_CHOOSE_THEN `M:num` (MP_TAC o SPEC `M + N:num`)) THEN
REWRITE_TAC[
LE_ADD;
ABS_DROP;
DROP_SUB] THEN
MATCH_MP_TAC(REAL_ARITH
`f1 <= f2 /\ f2 <= i
==> abs(f2 - g) < e / &2 ==> abs(f1 - i) < e / &2 ==> abs(g - i) < e`) THEN
CONJ_TAC THENL
[MATCH_MP_TAC
INTEGRAL_DROP_LE THEN ASM_REWRITE_TAC[] THEN
X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[
REAL_LE_REFL] THEN
MP_TAC(ISPEC
`\m n:num. drop (f m (x:real^N)) <= drop (f n x)`
TRANSITIVE_STEPWISE_LE) THEN
REWRITE_TAC[
REAL_LE_REFL;
REAL_LE_TRANS] THEN
ANTS_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
DISCH_THEN MATCH_MP_TAC THEN ARITH_TAC;
ALL_TAC] THEN
MATCH_MP_TAC
REAL_LE_TRANS THEN
EXISTS_TAC `drop(integral s ((f:num->real^N->real^1) (M + N)))` THEN
ASM_REWRITE_TAC[] THEN
GEN_REWRITE_TAC (RAND_CONV o RAND_CONV) [GSYM
INTEGRAL_RESTRICT_UNIV] THEN
MATCH_MP_TAC
INTEGRAL_SUBSET_DROP_LE THEN
ASM_REWRITE_TAC[
SUBSET_UNIV;
IN_UNIV] THEN
ASM_REWRITE_TAC[
INTEGRABLE_RESTRICT_UNIV; ETA_AX] THEN
GEN_TAC THEN COND_CASES_TAC THEN
ASM_SIMP_TAC[
REAL_LE_REFL;
DROP_VEC;
REAL_LE_REFL]);;
let MONOTONE_CONVERGENCE_DECREASING = prove
(`!f:num->real^N->real^1 g s.
(!k. (f k)
integrable_on s) /\
(!k x. x
IN s ==> drop(f (SUC k) x) <= drop(f k x)) /\
(!x. x
IN s ==> ((\k. f k x) --> g x) sequentially) /\
bounded {integral s (f k) | k
IN (:num)}
==> g
integrable_on s /\
((\k. integral s (f k)) --> integral s g) sequentially`,
REPEAT GEN_TAC THEN DISCH_TAC THEN
MP_TAC(ISPECL
[`(\k x. --(f k x)):num->real^N->real^1`;
`(\x. --(g x)):real^N->real^1`; `s:real^N->bool`]
MONOTONE_CONVERGENCE_INCREASING) THEN
FIRST_ASSUM MP_TAC THEN
MATCH_MP_TAC(TAUT `(a ==> b) /\ (c ==> d) ==> a ==> (b ==> c) ==> d`) THEN
REWRITE_TAC[] THEN CONJ_TAC THENL
[REPEAT(MATCH_MP_TAC MONO_AND THEN CONJ_TAC) THENL
[MATCH_MP_TAC
MONO_FORALL THEN GEN_TAC THEN
DISCH_THEN(MP_TAC o MATCH_MP
INTEGRABLE_NEG) THEN REWRITE_TAC[];
SIMP_TAC[
DROP_NEG;
REAL_LE_NEG2];
REPEAT STRIP_TAC THEN MATCH_MP_TAC
LIM_NEG THEN ASM_SIMP_TAC[];
ALL_TAC] THEN
DISCH_TAC THEN MATCH_MP_TAC
BOUNDED_SUBSET THEN
EXISTS_TAC `
IMAGE (\x. --x)
{integral s (f k:real^N->real^1) | k
IN (:num)}` THEN
CONJ_TAC THENL
[MATCH_MP_TAC
BOUNDED_LINEAR_IMAGE THEN
ASM_SIMP_TAC[
LINEAR_COMPOSE_NEG;
LINEAR_ID];
ONCE_REWRITE_TAC[
SIMPLE_IMAGE] THEN REWRITE_TAC[GSYM
IMAGE_o] THEN
REWRITE_TAC[
SUBSET;
IN_IMAGE] THEN
GEN_TAC THEN MATCH_MP_TAC
MONO_EXISTS THEN
REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[
o_THM] THEN
MATCH_MP_TAC
INTEGRAL_NEG THEN ASM_REWRITE_TAC[]];
ALL_TAC] THEN
DISCH_THEN(CONJUNCTS_THEN2
(MP_TAC o MATCH_MP
INTEGRABLE_NEG) (MP_TAC o MATCH_MP
LIM_NEG)) THEN
REWRITE_TAC[
VECTOR_NEG_NEG; ETA_AX] THEN
DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN
BINOP_TAC THEN REWRITE_TAC[
FUN_EQ_THM] THEN TRY GEN_TAC THEN
MATCH_MP_TAC(VECTOR_ARITH `x:real^N = --y ==> --x = y`) THEN
MATCH_MP_TAC
INTEGRAL_NEG THEN ASM_REWRITE_TAC[]);;
let INTEGRAL_NORM_BOUND_INTEGRAL = prove
(`!f:real^M->real^N g s.
f
integrable_on s /\ g
integrable_on s /\
(!x. x
IN s ==> norm(f x) <= drop(g x))
==> norm(integral s f) <= drop(integral s g)`,
let lemma = prove
(`(!e. &0 < e ==> x < y + e) ==> x <= y`,
DISCH_THEN(MP_TAC o SPEC `x - y:real`) THEN REAL_ARITH_TAC) in
SUBGOAL_THEN
`!f:real^M->real^N g a b.
f integrable_on interval[a,b] /\ g integrable_on interval[a,b] /\
(!x. x IN interval[a,b] ==> norm(f x) <= drop(g x))
==> norm(integral(interval[a,b]) f) <= drop(integral(interval[a,b]) g)`
ASSUME_TAC THENL
[REPEAT STRIP_TAC THEN MATCH_MP_TAC lemma THEN
X_GEN_TAC `e:real` THEN DISCH_TAC THEN
UNDISCH_TAC `(f:real^M->real^N) integrable_on interval[a,b]` THEN
DISCH_THEN(MP_TAC o MATCH_MP INTEGRABLE_INTEGRAL) THEN
FIRST_X_ASSUM(MP_TAC o MATCH_MP INTEGRABLE_INTEGRAL) THEN
REWRITE_TAC[has_integral] THEN DISCH_THEN(MP_TAC o SPEC `e / &2`) THEN
ASM_REWRITE_TAC[REAL_HALF; LEFT_IMP_EXISTS_THM] THEN
X_GEN_TAC `d1:real^M->real^M->bool` THEN STRIP_TAC THEN
DISCH_THEN(MP_TAC o SPEC `e / &2`) THEN
ASM_REWRITE_TAC[REAL_HALF; LEFT_IMP_EXISTS_THM] THEN
X_GEN_TAC `d2:real^M->real^M->bool` THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
MP_TAC(ISPECL [`d1:real^M->real^M->bool`; `d2:real^M->real^M->bool`]
GAUGE_INTER) THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN(MP_TAC o MATCH_MP FINE_DIVISION_EXISTS) THEN
DISCH_THEN(MP_TAC o SPECL [`a:real^M`; `b:real^M`]) THEN
REWRITE_TAC[FINE_INTER; LEFT_IMP_EXISTS_THM] THEN
X_GEN_TAC `p:(real^M#(real^M->bool))->bool` THEN STRIP_TAC THEN
DISCH_THEN(MP_TAC o SPEC `p:(real^M#(real^M->bool))->bool`) THEN
FIRST_X_ASSUM(MP_TAC o SPEC `p:(real^M#(real^M->bool))->bool`) THEN
ASM_REWRITE_TAC[ABS_DROP; DROP_SUB] THEN MATCH_MP_TAC(NORM_ARITH
`norm(sg) <= dsa
==> abs(dsa - dia) < e / &2 ==> norm(sg - ig) < e / &2
==> norm(ig) < dia + e`) THEN
FIRST_ASSUM(ASSUME_TAC o MATCH_MP TAGGED_DIVISION_OF_FINITE) THEN
ASM_SIMP_TAC[DROP_VSUM] THEN MATCH_MP_TAC VSUM_NORM_LE THEN
ASM_REWRITE_TAC[o_DEF; FORALL_PAIR_THM] THEN
MAP_EVERY X_GEN_TAC [`x:real^M`; `k:real^M->bool`] THEN DISCH_TAC THEN
REWRITE_TAC[NORM_MUL; DROP_CMUL] THEN
MATCH_MP_TAC REAL_LE_MUL2 THEN REWRITE_TAC[REAL_ABS_POS; NORM_POS_LE] THEN
REWRITE_TAC[REAL_ARITH `abs x <= x <=> &0 <= x`] THEN
ASM_MESON_TAC[CONTENT_POS_LE; TAGGED_DIVISION_OF; SUBSET];
ALL_TAC] THEN
REPEAT GEN_TAC THEN REWRITE_TAC[CONJ_ASSOC] THEN
DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN
DISCH_THEN(CONJUNCTS_THEN (fun th ->
ASSUME_TAC(CONJUNCT1(GEN_REWRITE_RULE I [INTEGRABLE_ALT] th)) THEN
MP_TAC(MATCH_MP INTEGRABLE_INTEGRAL th))) THEN
ONCE_REWRITE_TAC[HAS_INTEGRAL] THEN
DISCH_THEN(LABEL_TAC "A") THEN DISCH_TAC THEN MATCH_MP_TAC lemma THEN
X_GEN_TAC `e:real` THEN DISCH_TAC THEN
REMOVE_THEN "A" (MP_TAC o SPEC `e / &2`) THEN
FIRST_X_ASSUM(MP_TAC o SPEC `e / &2`) THEN ASM_REWRITE_TAC[REAL_HALF] THEN
DISCH_THEN(X_CHOOSE_THEN `B1:real`
(CONJUNCTS_THEN2 ASSUME_TAC (LABEL_TAC "F"))) THEN
DISCH_THEN(X_CHOOSE_THEN `B2:real`
(CONJUNCTS_THEN2 ASSUME_TAC (LABEL_TAC "A"))) THEN
MP_TAC(ISPEC `ball(vec 0,max B1 B2):real^M->bool`
BOUNDED_SUBSET_CLOSED_INTERVAL) THEN
REWRITE_TAC[BOUNDED_BALL; LEFT_IMP_EXISTS_THM] THEN
REWRITE_TAC[BALL_MAX_UNION; UNION_SUBSET] THEN
MAP_EVERY X_GEN_TAC [`a:real^M`; `b:real^M`] THEN
DISCH_THEN(CONJUNCTS_THEN(ANTE_RES_THEN MP_TAC)) THEN
DISCH_THEN(X_CHOOSE_THEN `z:real^1` (CONJUNCTS_THEN2 ASSUME_TAC
(fun th -> DISCH_THEN(X_CHOOSE_THEN `w:real^N`
(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN MP_TAC th))) THEN
ASM_REWRITE_TAC[ABS_DROP; DROP_SUB] THEN MATCH_MP_TAC(NORM_ARITH
`norm(sg) <= dsa
==> abs(dsa - dia) < e / &2 ==> norm(sg - ig) < e / &2
==> norm(ig) < dia + e`) THEN
REPEAT(FIRST_X_ASSUM(SUBST1_TAC o SYM o MATCH_MP INTEGRAL_UNIQUE)) THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN
REPEAT STRIP_TAC THEN
COND_CASES_TAC THEN ASM_SIMP_TAC[NORM_0; DROP_VEC; REAL_LE_REFL]);;
let SET_VARIATION_EQ = prove
(`!f g:(real^M->bool)->real^N s.
(!a b. ~(interval[a,b] = {}) /\ interval[a,b]
SUBSET s
==> f(interval[a,b]) = g(interval[a,b]))
==>
set_variation s f =
set_variation s g`,
REPEAT STRIP_TAC THEN REWRITE_TAC[
set_variation] THEN AP_TERM_TAC THEN
MATCH_MP_TAC(SET_RULE
`(!x. P x ==> f x = g x) ==> {f x | P x} = {g x | P x}`) THEN
X_GEN_TAC `d:(real^M->bool)->bool` THEN
DISCH_THEN(X_CHOOSE_THEN `t:real^M->bool` STRIP_ASSUME_TAC) THEN
MATCH_MP_TAC
SUM_EQ THEN FIRST_ASSUM(fun th ->
GEN_REWRITE_TAC I [MATCH_MP
FORALL_IN_DIVISION_NONEMPTY th]) THEN
REPEAT STRIP_TAC THEN REWRITE_TAC[] THEN AP_TERM_TAC THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN
ASM_MESON_TAC[
division_of;
SUBSET_TRANS]);;
let SETVARIATION_EQUAL_LEMMA = prove
(`!mf:((real^M->bool)->real^N)->((real^M->bool)->real^N) ms ms'.
(!s. ms'(ms s) = s /\ ms(ms' s) = s) /\
(!f a b. ~(interval[a,b] = {})
==> mf f (ms (interval[a,b])) = f (interval[a,b]) /\
?a' b'. ~(interval[a',b'] = {}) /\
ms' (interval[a,b]) = interval[a',b']) /\
(!t u. t
SUBSET u ==> ms t
SUBSET ms u /\ ms' t
SUBSET ms' u) /\
(!d t. d
division_of t
==> (
IMAGE ms d)
division_of ms t /\
(
IMAGE ms' d)
division_of ms' t)
==> (!f s. (mf f)
has_bounded_setvariation_on (ms s) <=>
f
has_bounded_setvariation_on s) /\
(!f s.
set_variation (ms s) (mf f) =
set_variation s f)`,
REPEAT GEN_TAC THEN STRIP_TAC THEN
REWRITE_TAC[
has_bounded_setvariation_on;
set_variation] THEN
MATCH_MP_TAC(MESON[]
`((!f s. s1 f s = s2 f s) ==> P) /\
(!f s. s1 f s = s2 f s)
==> P /\ (!f s. sup (s1 f s) = sup (s2 f s))`) THEN
CONJ_TAC THENL
[REWRITE_TAC[
EXTENSION;
IN_ELIM_THM] THEN
REPEAT(MATCH_MP_TAC
MONO_FORALL THEN GEN_TAC) THEN MESON_TAC[];
ALL_TAC] THEN
REWRITE_TAC[
EXTENSION;
IN_ELIM_THM] THEN REPEAT GEN_TAC THEN EQ_TAC THEN
STRIP_TAC THEN ASM_REWRITE_TAC[] THENL
[EXISTS_TAC `
IMAGE (ms':(real^M->bool)->real^M->bool) d`;
EXISTS_TAC `
IMAGE (ms:(real^M->bool)->real^M->bool) d`] THEN
(CONJ_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
W(MP_TAC o PART_MATCH (lhand o rand)
SUM_IMAGE o rand o snd) THEN
ANTS_TAC THENL [ASM_MESON_TAC[]; DISCH_THEN SUBST1_TAC]) THEN
MATCH_MP_TAC
SUM_EQ THEN REWRITE_TAC[
o_THM] THEN FIRST_ASSUM
(fun th -> REWRITE_TAC[MATCH_MP
FORALL_IN_DIVISION_NONEMPTY th]) THEN
MAP_EVERY X_GEN_TAC [`a:real^M`; `b:real^M`] THEN STRIP_TAC THEN
AP_TERM_TAC THEN ASM_SIMP_TAC[] THEN
SUBGOAL_THEN `?a' b':real^M. ~(interval[a',b'] = {}) /\
ms' (interval[a:real^M,b]) = interval[a',b']`
STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[]);;
let SET_VARIATION_ELEMENTARY_LEMMA = prove
(`!f:(real^M->bool)->real^N s.
(?d. d
division_of s)
==> ((!d t. d
division_of t /\ t
SUBSET s
==> sum d (\k. norm(f k)) <= b) <=>
(!d. d
division_of s ==> sum d (\k. norm(f k)) <= b))`,
REPEAT GEN_TAC THEN DISCH_THEN(X_CHOOSE_TAC `d1:(real^M->bool)->bool`) THEN
EQ_TAC THENL [MESON_TAC[
SUBSET_REFL]; ALL_TAC] THEN
DISCH_TAC THEN X_GEN_TAC `d2:(real^M->bool)->bool` THEN
X_GEN_TAC `t:real^M->bool` THEN STRIP_TAC THEN MP_TAC(ISPECL
[`d2:(real^M->bool)->bool`; `d1:(real^M->bool)->bool`;
`t:real^M->bool`; `s:real^M->bool`]
PARTIAL_DIVISION_EXTEND) THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_TAC `d3:(real^M->bool)->bool`) THEN
MATCH_MP_TAC
REAL_LE_TRANS THEN
EXISTS_TAC `sum d3 (\k:real^M->bool. norm(f k:real^N))` THEN
ASM_SIMP_TAC[] THEN MATCH_MP_TAC
SUM_SUBSET_SIMPLE THEN
ASM_REWRITE_TAC[
NORM_POS_LE] THEN ASM_MESON_TAC[
DIVISION_OF_FINITE]);;
let OPERATIVE_LIFTED_SETVARIATION = prove
(`!f:(real^M->bool)->real^N.
operative(+) f
==> operative (lifted(+))
(\i. if f
has_bounded_setvariation_on i
then SOME(
set_variation i f) else NONE)`,
let lemma1 = prove
(`!f:(real^M->bool)->real B1 B2 k a b.
1 <= k /\ k <= dimindex(:M) /\
(!a b. content(interval[a,b]) = &0 ==> f(interval[a,b]) = &0) /\
(!a b c. f(interval[a,b]) <=
f(interval[a,b] INTER {x | x$k <= c}) +
f(interval[a,b] INTER {x | x$k >= c})) /\
(!d. d division_of (interval[a,b] INTER {x | x$k <= c})
==> sum d f <= B1) /\
(!d. d division_of (interval[a,b] INTER {x | x$k >= c})
==> sum d f <= B2)
==> !d. d division_of interval[a,b] ==> sum d f <= B1 + B2`,
REPEAT GEN_TAC THEN
REPLICATE_TAC 4 (DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
DISCH_THEN(CONJUNCTS_THEN2 (LABEL_TAC "L") (LABEL_TAC "R")) THEN
GEN_TAC THEN DISCH_TAC THEN MATCH_MP_TAC REAL_LE_TRANS THEN
EXISTS_TAC
`sum {l INTER {x:real^M | x$k <= c} | l | l IN d /\
~(l INTER {x | x$k <= c} = {})} f +
sum {l INTER {x | x$k >= c} | l | l IN d /\
~(l INTER {x | x$k >= c} = {})} f` THEN
CONJ_TAC THENL
[ALL_TAC;
MATCH_MP_TAC REAL_LE_ADD2 THEN CONJ_TAC THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_SIMP_TAC[DIVISION_SPLIT]] THEN
ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN
FIRST_ASSUM(ASSUME_TAC o MATCH_MP DIVISION_OF_FINITE) THEN
W(fun (asl,w) ->
MP_TAC(PART_MATCH (lhs o rand) SUM_IMAGE_NONZERO (lhand(rand w))) THEN
MP_TAC(PART_MATCH (lhs o rand) SUM_IMAGE_NONZERO (rand(rand w)))) THEN
MATCH_MP_TAC(TAUT
`(a1 /\ a2) /\ (b1 /\ b2 ==> c)
==> (a1 ==> b1) ==> (a2 ==> b2) ==> c`) THEN
CONJ_TAC THENL
[ASM_SIMP_TAC[FINITE_RESTRICT; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
REWRITE_TAC[FORALL_IN_GSPEC; IMP_CONJ] THEN
FIRST_ASSUM(fun th -> REWRITE_TAC[MATCH_MP FORALL_IN_DIVISION th]) THEN
REPEAT STRIP_TAC THEN ASM_SIMP_TAC[INTERVAL_SPLIT] THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_SIMP_TAC[GSYM INTERVAL_SPLIT] THENL
[MATCH_MP_TAC DIVISION_SPLIT_RIGHT_INJ;
MATCH_MP_TAC DIVISION_SPLIT_LEFT_INJ] THEN
ASM_MESON_TAC[];
DISCH_THEN(CONJUNCTS_THEN SUBST1_TAC)] THEN
MATCH_MP_TAC REAL_LE_TRANS THEN
EXISTS_TAC
`sum d (f o (\l. l INTER {x | x$k <= c})) +
sum d (f o (\l. l INTER {x:real^M | x$k >= c}))` THEN
CONJ_TAC THENL
[ASM_SIMP_TAC[GSYM SUM_ADD] THEN MATCH_MP_TAC SUM_LE THEN
ASM_REWRITE_TAC[o_THM] THEN
FIRST_ASSUM(fun th -> ASM_REWRITE_TAC[MATCH_MP FORALL_IN_DIVISION th]);
MATCH_MP_TAC(REAL_ARITH `x = y /\ w = z ==> x + w <= y + z`) THEN
CONJ_TAC THEN MATCH_MP_TAC SUM_SUPERSET THEN
REWRITE_TAC[SET_RULE `{x | x IN s /\ P x} SUBSET s`] THEN
REWRITE_TAC[SET_RULE `(x IN s /\ ~(x IN {x | x IN s /\ ~P x}) ==> Q x) <=>
(x IN s ==> P x ==> Q x)`] THEN
SIMP_TAC[o_THM] THEN ASM_MESON_TAC[EMPTY_AS_INTERVAL; CONTENT_EMPTY]])
and lemma2 = prove
(`!f:(real^M->bool)->real B k.
1 <= k /\ k <= dimindex(:M) /\
(!a b. content(interval[a,b]) = &0 ==> f(interval[a,b]) = &0) /\
(!d. d division_of interval[a,b] ==> sum d f <= B)
==> !d1 d2. d1 division_of (interval[a,b] INTER {x | x$k <= c}) /\
d2 division_of (interval[a,b] INTER {x | x$k >= c})
==> sum d1 f + sum d2 f <= B`,
REPEAT STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `d1 UNION d2:(real^M->bool)->bool`) THEN
ANTS_TAC THENL
[SUBGOAL_THEN
`interval[a,b] = (interval[a,b] INTER {x:real^M | x$k <= c}) UNION
(interval[a,b] INTER {x:real^M | x$k >= c})`
SUBST1_TAC THENL
[MATCH_MP_TAC(SET_RULE
`(!x. x IN t \/ x IN u) ==> (s = s INTER t UNION s INTER u)`) THEN
REWRITE_TAC[IN_ELIM_THM] THEN REAL_ARITH_TAC;
MATCH_MP_TAC DIVISION_DISJOINT_UNION THEN ASM_REWRITE_TAC[] THEN
REWRITE_TAC[GSYM INTERIOR_INTER] THEN
MATCH_MP_TAC(SET_RULE
`!t. interior s SUBSET interior t /\ interior t = {}
==> interior s = {}`) THEN
EXISTS_TAC `{x:real^M | x$k = c}` THEN CONJ_TAC THENL
[ALL_TAC; REWRITE_TAC[INTERIOR_STANDARD_HYPERPLANE]] THEN
MATCH_MP_TAC SUBSET_INTERIOR THEN
REWRITE_TAC[SUBSET; IN_INTER; IN_ELIM_THM] THEN REAL_ARITH_TAC];
MATCH_MP_TAC(REAL_ARITH `x = y ==> x <= b ==> y <= b`) THEN
MATCH_MP_TAC SUM_UNION_NONZERO THEN
REPEAT(CONJ_TAC THENL [ASM_MESON_TAC[DIVISION_OF_FINITE]; ALL_TAC]) THEN
X_GEN_TAC `k:real^M->bool` THEN REWRITE_TAC[IN_INTER] THEN STRIP_TAC THEN
SUBGOAL_THEN `?u v:real^M. k = interval[u,v]`
(REPEAT_TCL CHOOSE_THEN SUBST_ALL_TAC)
THENL [ASM_MESON_TAC[division_of]; ALL_TAC] THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN MATCH_MP_TAC CONTENT_0_SUBSET_GEN THEN
EXISTS_TAC `interval[a,b] INTER {x:real^M | x$k = c}` THEN CONJ_TAC THENL
[MATCH_MP_TAC SUBSET_TRANS THEN
EXISTS_TAC `(interval[a,b] INTER {x:real^M | x$k <= c}) INTER
(interval[a,b] INTER {x:real^M | x$k >= c})` THEN
CONJ_TAC THENL
[ONCE_REWRITE_TAC[SUBSET_INTER] THEN ASM_MESON_TAC[division_of];
REWRITE_TAC[SET_RULE
`(s INTER t) INTER (s INTER u) = s INTER t INTER u`] THEN
SIMP_TAC[SUBSET; IN_INTER; IN_ELIM_THM] THEN REAL_ARITH_TAC];
SIMP_TAC[BOUNDED_INTER; BOUNDED_INTERVAL] THEN
GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV)
[REAL_ARITH `x = y <=> x <= y /\ x >= y`] THEN
REWRITE_TAC[SET_RULE
`{x | P x /\ Q x} = {x | P x} INTER {x | Q x}`] THEN
ASM_SIMP_TAC[GSYM INTER_ASSOC; INTERVAL_SPLIT] THEN
REWRITE_TAC[CONTENT_EQ_0] THEN EXISTS_TAC `k:num` THEN
ASM_SIMP_TAC[LAMBDA_BETA] THEN REAL_ARITH_TAC]]) in
REWRITE_TAC[operative; NEUTRAL_VECTOR_ADD] THEN REPEAT GEN_TAC THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (ASSUME_TAC o GSYM)) THEN
ASM_SIMP_TAC[HAS_BOUNDED_SETVARIATION_ON_NULL; BOUNDED_INTERVAL;
MONOIDAL_REAL_ADD; SET_VARIATION_ON_NULL; NEUTRAL_LIFTED;
NEUTRAL_REAL_ADD] THEN
MAP_EVERY X_GEN_TAC [`a:real^M`; `b:real^M`; `c:real`; `k:num`] THEN
STRIP_TAC THEN ASM_CASES_TAC
`(f:(real^M->bool)->real^N) has_bounded_setvariation_on interval[a,b]` THEN
ASM_REWRITE_TAC[] THENL
[SUBGOAL_THEN
`(f:(real^M->bool)->real^N) has_bounded_setvariation_on
interval[a,b] INTER {x | x$k <= c} /\
(f:(real^M->bool)->real^N) has_bounded_setvariation_on
interval[a,b] INTER {x | x$k >= c}`
ASSUME_TAC THENL
[CONJ_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP
(REWRITE_RULE[IMP_CONJ] HAS_BOUNDED_SETVARIATION_ON_SUBSET)) THEN
REWRITE_TAC[INTER_SUBSET];
ALL_TAC] THEN
ASM_REWRITE_TAC[lifted] THEN AP_TERM_TAC THEN
REWRITE_TAC[GSYM REAL_LE_ANTISYM] THEN CONJ_TAC THENL
[MATCH_MP_TAC SET_VARIATION_UBOUND_ON_INTERVAL THEN ASM_REWRITE_TAC[] THEN
REPEAT STRIP_TAC THEN MATCH_MP_TAC
(REWRITE_RULE[IMP_IMP; RIGHT_IMP_FORALL_THM] lemma1) THEN
MAP_EVERY EXISTS_TAC [`k:num`; `a:real^M`; `b:real^M`] THEN
ASM_SIMP_TAC[NORM_0] THEN CONJ_TAC THENL
[REPEAT GEN_TAC THEN
MATCH_MP_TAC(NORM_ARITH
`x:real^N = y + z ==> norm(x) <= norm y + norm z`) THEN
ASM_SIMP_TAC[];
FIRST_X_ASSUM(fun th -> MP_TAC th THEN MATCH_MP_TAC MONO_AND) THEN
ASM_SIMP_TAC[INTERVAL_SPLIT; SET_VARIATION_WORKS_ON_INTERVAL]];
ONCE_REWRITE_TAC[REAL_ARITH `x + y <= z <=> x <= z - y`] THEN
ASM_SIMP_TAC[INTERVAL_SPLIT] THEN
MATCH_MP_TAC SET_VARIATION_UBOUND_ON_INTERVAL THEN
ASM_SIMP_TAC[GSYM INTERVAL_SPLIT] THEN
X_GEN_TAC `d1:(real^M->bool)->bool` THEN STRIP_TAC THEN
ONCE_REWRITE_TAC[REAL_ARITH `x <= y - z <=> z <= y - x`] THEN
ASM_SIMP_TAC[INTERVAL_SPLIT] THEN
MATCH_MP_TAC SET_VARIATION_UBOUND_ON_INTERVAL THEN
ASM_SIMP_TAC[GSYM INTERVAL_SPLIT] THEN
X_GEN_TAC `d2:(real^M->bool)->bool` THEN STRIP_TAC THEN
REWRITE_TAC[REAL_ARITH `x <= y - z <=> z + x <= y`] THEN
REPEAT STRIP_TAC THEN MATCH_MP_TAC
(REWRITE_RULE[IMP_IMP; RIGHT_IMP_FORALL_THM] lemma2) THEN
EXISTS_TAC `k:num` THEN
ASM_SIMP_TAC[NORM_0; SET_VARIATION_WORKS_ON_INTERVAL]];
REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[lifted]) THEN
FIRST_X_ASSUM(MP_TAC o check (is_neg o concl)) THEN
MATCH_MP_TAC(TAUT `p ==> ~p ==> q`) THEN
REWRITE_TAC[HAS_BOUNDED_SETVARIATION_ON_INTERVAL] THEN
EXISTS_TAC `set_variation (interval[a,b] INTER {x | x$k <= c})
(f:(real^M->bool)->real^N) +
set_variation (interval[a,b] INTER {x | x$k >= c}) f` THEN
REPEAT STRIP_TAC THEN MATCH_MP_TAC
(REWRITE_RULE[IMP_IMP; RIGHT_IMP_FORALL_THM] lemma1) THEN
MAP_EVERY EXISTS_TAC [`k:num`; `a:real^M`; `b:real^M`] THEN
ASM_SIMP_TAC[NORM_0] THEN REPEAT CONJ_TAC THENL
[REPEAT GEN_TAC THEN
MATCH_MP_TAC(NORM_ARITH
`x:real^N = y + z ==> norm(x) <= norm y + norm z`) THEN
ASM_SIMP_TAC[];
UNDISCH_TAC
`(f:(real^M->bool)->real^N) has_bounded_setvariation_on
(interval[a,b] INTER {x | x$k <= c})` THEN
ASM_SIMP_TAC[INTERVAL_SPLIT; SET_VARIATION_WORKS_ON_INTERVAL];
UNDISCH_TAC
`(f:(real^M->bool)->real^N) has_bounded_setvariation_on
(interval[a,b] INTER {x | x$k >= c})` THEN
ASM_SIMP_TAC[INTERVAL_SPLIT; SET_VARIATION_WORKS_ON_INTERVAL]]]);;
let lemma = prove
(`!P op f s z.
monoidal op /\
FINITE s /\
iterate(lifted op) s (\i. if P i then SOME(f i) else NONE) = SOME z
==> iterate op s f = z`,
let lemma = prove
(`!f:A->real^N g s e.
sum s (\x. norm(f x - g x)) < e
==>
FINITE s
==> abs(sum s (\x. norm(f x)) - sum s (\x. norm(g x))) < e`,
REPEAT GEN_TAC THEN SIMP_TAC[GSYM
SUM_SUB] THEN
DISCH_THEN(fun th -> DISCH_TAC THEN MP_TAC th) THEN
MATCH_MP_TAC(REAL_ARITH `x <= y ==> y < e ==> x < e`) THEN
W(MP_TAC o PART_MATCH (lhand o rand)
SUM_ABS o lhand o snd) THEN
ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC(REAL_ARITH `y <= z ==> x <= y ==> x <= z`) THEN
MATCH_MP_TAC
SUM_LE THEN ASM_REWRITE_TAC[] THEN
REPEAT STRIP_TAC THEN NORM_ARITH_TAC);;
let BOUNDED_SETVARIATION_ABSOLUTELY_INTEGRABLE_INTERVAL = prove
(`!f:real^M->real^N a b.
f
integrable_on interval[a,b] /\
(\k. integral k f)
has_bounded_setvariation_on interval[a,b]
==> f
absolutely_integrable_on interval[a,b]`,
REWRITE_TAC[
HAS_BOUNDED_SETVARIATION_ON_INTERVAL] THEN
REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[
absolutely_integrable_on] THEN
MP_TAC(ISPEC `
IMAGE (\d. sum d (\k. norm(integral k (f:real^M->real^N))))
{d | d
division_of interval[a,b] }`
SUP) THEN
REWRITE_TAC[
FORALL_IN_IMAGE;
IMAGE_EQ_EMPTY] THEN
REWRITE_TAC[GSYM
MEMBER_NOT_EMPTY;
IN_ELIM_THM] THEN
ABBREV_TAC
`i = sup (
IMAGE (\d. sum d (\k. norm(integral k (f:real^M->real^N))))
{d | d
division_of interval[a,b] })` THEN
ANTS_TAC THENL
[REWRITE_TAC[
ELEMENTARY_INTERVAL] THEN ASM_MESON_TAC[]; ALL_TAC] THEN
STRIP_TAC THEN REWRITE_TAC[
integrable_on] THEN EXISTS_TAC `lift i` THEN
REWRITE_TAC[
has_integral] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `i - e / &2`) THEN
ASM_SIMP_TAC[REAL_ARITH `&0 < e ==> ~(i <= i - e / &2)`] THEN
REWRITE_TAC[
NOT_FORALL_THM; NOT_IMP;
REAL_NOT_LE;
LEFT_IMP_EXISTS_THM] THEN
X_GEN_TAC `d:(real^M->bool)->bool` THEN STRIP_TAC THEN
FIRST_ASSUM(ASSUME_TAC o MATCH_MP
DIVISION_OF_FINITE) THEN
SUBGOAL_THEN
`!x. ?e. &0 < e /\
!i. i
IN d /\ ~(x
IN i) ==> ball(x:real^M,e)
INTER i = {}`
MP_TAC THENL
[X_GEN_TAC `x:real^M` THEN MP_TAC(ISPECL
[`
UNIONS {i:real^M->bool | i
IN d /\ ~(x
IN i)}`; `x:real^M`]
SEPARATE_POINT_CLOSED) THEN
ANTS_TAC THENL
[CONJ_TAC THENL [ALL_TAC; SET_TAC[]] THEN
MATCH_MP_TAC
CLOSED_UNIONS THEN
ASM_SIMP_TAC[
FINITE_RESTRICT;
IN_ELIM_THM;
IMP_CONJ] THEN
FIRST_ASSUM(fun t -> REWRITE_TAC[MATCH_MP
FORALL_IN_DIVISION t]) THEN
REWRITE_TAC[
CLOSED_INTERVAL];
ALL_TAC] THEN
MATCH_MP_TAC
MONO_EXISTS THEN X_GEN_TAC `k:real` THEN
SIMP_TAC[
FORALL_IN_UNIONS;
EXTENSION;
IN_INTER;
NOT_IN_EMPTY;
IN_BALL] THEN
REWRITE_TAC[
IN_ELIM_THM; DE_MORGAN_THM;
REAL_NOT_LT] THEN MESON_TAC[];
ALL_TAC] THEN
REWRITE_TAC[
SKOLEM_THM;
LEFT_IMP_EXISTS_THM;
FORALL_AND_THM] THEN
X_GEN_TAC `k:real^M->real` THEN STRIP_TAC THEN
FIRST_ASSUM(MP_TAC o SPEC `e / &2` o MATCH_MP
HENSTOCK_LEMMA) THEN
ASM_REWRITE_TAC[
REAL_HALF] THEN
DISCH_THEN(X_CHOOSE_THEN `g:real^M->real^M->bool` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `\x:real^M. g(x)
INTER ball(x,k x)` THEN CONJ_TAC THENL
[MATCH_MP_TAC
GAUGE_INTER THEN ASM_REWRITE_TAC[] THEN
ASM_REWRITE_TAC[gauge;
CENTRE_IN_BALL;
OPEN_BALL];
ALL_TAC] THEN
REWRITE_TAC[
FINE_INTER] THEN X_GEN_TAC `p:(real^M#(real^M->bool))->bool` THEN
STRIP_TAC THEN
FIRST_ASSUM(ASSUME_TAC o MATCH_MP
TAGGED_DIVISION_OF_FINITE) THEN
ABBREV_TAC
`p' = {(x:real^M,k:real^M->bool) |
?i l. x
IN i /\ i
IN d /\ (x,l)
IN p /\ k = i
INTER l}` THEN
SUBGOAL_THEN `g fine (p':(real^M#(real^M->bool))->bool)` ASSUME_TAC THENL
[EXPAND_TAC "p'" THEN
MP_TAC(ASSUME `g fine (p:(real^M#(real^M->bool))->bool)`) THEN
REWRITE_TAC[fine;
IN_ELIM_PAIR_THM] THEN
MESON_TAC[SET_RULE `k
SUBSET l ==> (i
INTER k)
SUBSET l`];
ALL_TAC] THEN
SUBGOAL_THEN `p'
tagged_division_of interval[a:real^M,b]` ASSUME_TAC THENL
[REWRITE_TAC[
TAGGED_DIVISION_OF] THEN EXPAND_TAC "p'" THEN
REWRITE_TAC[
IN_ELIM_PAIR_THM] THEN
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
TAGGED_DIVISION_OF]) THEN
MATCH_MP_TAC MONO_AND THEN CONJ_TAC THENL
[DISCH_TAC THEN MATCH_MP_TAC
FINITE_SUBSET THEN
EXISTS_TAC
`
IMAGE (\(k,(x,l)). x,k
INTER l)
{k,xl | k
IN (d:(real^M->bool)->bool) /\
xl
IN (p:(real^M#(real^M->bool))->bool)}` THEN
ASM_SIMP_TAC[
FINITE_IMAGE;
FINITE_PRODUCT] THEN
EXPAND_TAC "p'" THEN REWRITE_TAC[
SUBSET;
FORALL_PAIR_THM] THEN
REWRITE_TAC[
IN_ELIM_PAIR_THM;
IN_IMAGE;
EXISTS_PAIR_THM;
PAIR_EQ] THEN
MESON_TAC[];
ALL_TAC] THEN
MATCH_MP_TAC MONO_AND THEN CONJ_TAC THENL
[DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`x:real^M`; `k:real^M->bool`] THEN
REWRITE_TAC[
LEFT_IMP_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC [`i:real^M->bool`; `l:real^M->bool`] THEN
STRIP_TAC THEN FIRST_X_ASSUM SUBST_ALL_TAC THEN
ASM_SIMP_TAC[
IN_INTER] THEN CONJ_TAC THENL
[MATCH_MP_TAC(SET_RULE `l
SUBSET s ==> (k
INTER l)
SUBSET s`) THEN
ASM_MESON_TAC[];
ALL_TAC] THEN
FIRST_X_ASSUM(MP_TAC o SPECL [`x:real^M`; `l:real^M->bool`]) THEN
ASM_REWRITE_TAC[] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
division_of]) THEN
DISCH_THEN(MP_TAC o SPEC `i:real^M->bool` o el 1 o CONJUNCTS) THEN
ASM_REWRITE_TAC[] THEN STRIP_TAC THEN
ASM_REWRITE_TAC[
INTER_INTERVAL] THEN MESON_TAC[];
ALL_TAC] THEN
MATCH_MP_TAC MONO_AND THEN CONJ_TAC THENL
[DISCH_TAC THEN MAP_EVERY X_GEN_TAC
[`x1:real^M`; `k1:real^M->bool`; `x2:real^M`; `k2:real^M->bool`] THEN
DISCH_THEN(CONJUNCTS_THEN2
(X_CHOOSE_THEN `i1:real^M->bool` (X_CHOOSE_THEN `l1:real^M->bool`
STRIP_ASSUME_TAC)) MP_TAC) THEN
FIRST_X_ASSUM SUBST_ALL_TAC THEN
DISCH_THEN(CONJUNCTS_THEN2
(X_CHOOSE_THEN `i2:real^M->bool` (X_CHOOSE_THEN `l2:real^M->bool`
STRIP_ASSUME_TAC)) ASSUME_TAC) THEN
FIRST_X_ASSUM SUBST_ALL_TAC THEN
MATCH_MP_TAC(SET_RULE
`(interior(i1)
INTER interior(i2) = {} \/
interior(l1)
INTER interior(l2) = {}) /\
interior(i1
INTER l1)
SUBSET interior(i1) /\
interior(i2
INTER l2)
SUBSET interior(i2) /\
interior(i1
INTER l1)
SUBSET interior(l1) /\
interior(i2
INTER l2)
SUBSET interior(l2)
==> interior(i1
INTER l1)
INTER interior(i2
INTER l2) = {}`) THEN
SIMP_TAC[
SUBSET_INTERIOR;
INTER_SUBSET] THEN
FIRST_X_ASSUM(MP_TAC o SPECL
[`x1:real^M`; `l1:real^M->bool`; `x2:real^M`; `l2:real^M->bool`]) THEN
ASM_REWRITE_TAC[] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
division_of]) THEN
DISCH_THEN(MP_TAC o el 2 o CONJUNCTS) THEN
DISCH_THEN(MP_TAC o SPECL [`i1:real^M->bool`; `i2:real^M->bool`]) THEN
ASM_REWRITE_TAC[] THEN
FIRST_X_ASSUM(MP_TAC o check(is_neg o concl)) THEN
ASM_REWRITE_TAC[
PAIR_EQ] THEN MESON_TAC[];
ALL_TAC] THEN
DISCH_TAC THEN MATCH_MP_TAC
SUBSET_ANTISYM THEN CONJ_TAC THENL
[REWRITE_TAC[
UNIONS_SUBSET;
IN_ELIM_THM] THEN
REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC(SET_RULE `i
SUBSET s ==> (i
INTER k)
SUBSET s`) THEN
ASM_MESON_TAC[
division_of];
ALL_TAC] THEN
REWRITE_TAC[
SUBSET] THEN X_GEN_TAC `y:real^M` THEN DISCH_TAC THEN
REWRITE_TAC[
IN_UNIONS;
IN_ELIM_THM] THEN
REWRITE_TAC[
LEFT_AND_EXISTS_THM; GSYM
CONJ_ASSOC] THEN
REWRITE_TAC[MESON[]
`p /\ q /\ r /\ x = t /\ P x <=> x = t /\ p /\ q /\ r /\ P t`] THEN
ONCE_REWRITE_TAC[MESON[]
`(?a b c d. P a b c d) <=> (?d b c a. P a b c d)`] THEN
REWRITE_TAC[
IN_INTER;
UNWIND_THM2] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
EXTENSION]) THEN
DISCH_THEN(MP_TAC o SPEC `y:real^M`) THEN ASM_REWRITE_TAC[] THEN
REWRITE_TAC[
IN_UNIONS;
IN_ELIM_THM;
LEFT_AND_EXISTS_THM] THEN
MATCH_MP_TAC
MONO_EXISTS THEN X_GEN_TAC `l:real^M->bool` THEN
MATCH_MP_TAC
MONO_EXISTS THEN X_GEN_TAC `x:real^M` THEN
STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
FIRST_X_ASSUM(MP_TAC o last o CONJUNCTS o
GEN_REWRITE_RULE I [
division_of]) THEN
REWRITE_TAC[
EXTENSION] THEN DISCH_THEN(MP_TAC o SPEC `y:real^M`) THEN
ASM_REWRITE_TAC[
IN_UNIONS] THEN
MATCH_MP_TAC
MONO_EXISTS THEN X_GEN_TAC `k:real^M->bool` THEN
STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
FIRST_X_ASSUM(MP_TAC o SPECL [`x:real^M`; `k:real^M->bool`]) THEN
GEN_REWRITE_TAC LAND_CONV [GSYM CONTRAPOS_THM] THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN MATCH_MP_TAC THEN
REWRITE_TAC[GSYM
MEMBER_NOT_EMPTY] THEN
EXISTS_TAC `y:real^M` THEN ASM_REWRITE_TAC[
INTER;
IN_ELIM_THM] THEN
UNDISCH_TAC `(\x:real^M. ball (x,k x)) fine p` THEN
REWRITE_TAC[fine;
SUBSET] THEN ASM_MESON_TAC[];
ALL_TAC] THEN
FIRST_X_ASSUM(MP_TAC o SPEC `p':(real^M#(real^M->bool))->bool`) THEN
ASM_REWRITE_TAC[] THEN
ANTS_TAC THENL [ASM_MESON_TAC[
tagged_division_of]; ALL_TAC] THEN
FIRST_ASSUM(ASSUME_TAC o MATCH_MP
TAGGED_DIVISION_OF_FINITE) THEN
REWRITE_TAC[
LAMBDA_PAIR] THEN DISCH_THEN(MP_TAC o MATCH_MP lemma) THEN
ASM_SIMP_TAC[
DROP_VSUM;
o_DEF;
SUM_SUB;
DROP_SUB;
ABS_DROP] THEN
REWRITE_TAC[
LAMBDA_PAIR_THM;
DROP_CMUL;
NORM_MUL;
LIFT_DROP] THEN
MATCH_MP_TAC(REAL_ARITH
`!sni. i - e / &2 < sni /\
sni' <= i /\ sni <= sni' /\ sf' = sf
==> abs(sf' - sni') < e / &2
==> abs(sf - i) < e`) THEN
EXISTS_TAC `sum d (\k. norm (integral k (f:real^M->real^N)))` THEN
ASM_REWRITE_TAC[] THEN CONJ_TAC THENL
[MP_TAC(ISPECL [`\k. norm(integral k (f:real^M->real^N))`;
`p':(real^M#(real^M->bool))->bool`;
`interval[a:real^M,b]`]
SUM_OVER_TAGGED_DIVISION_LEMMA) THEN
ASM_SIMP_TAC[
INTEGRAL_NULL;
NORM_0] THEN DISCH_THEN SUBST1_TAC THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_MESON_TAC[
DIVISION_OF_TAGGED_DIVISION];
ALL_TAC] THEN
SUBGOAL_THEN
`p' = {x:real^M,(i
INTER l:real^M->bool) |
(x,l)
IN p /\ i
IN d /\ ~(i
INTER l = {})}`
(LABEL_TAC "p'") THENL
[EXPAND_TAC "p'" THEN GEN_REWRITE_TAC I [
EXTENSION] THEN
REWRITE_TAC[
FORALL_PAIR_THM;
IN_ELIM_PAIR_THM] THEN
REWRITE_TAC[
IN_ELIM_THM] THEN
MAP_EVERY X_GEN_TAC [`x:real^M`; `k:real^M->bool`] THEN
REWRITE_TAC[
PAIR_EQ; GSYM
CONJ_ASSOC] THEN
GEN_REWRITE_TAC RAND_CONV [
SWAP_EXISTS_THM] THEN
AP_TERM_TAC THEN GEN_REWRITE_TAC I [
FUN_EQ_THM] THEN
X_GEN_TAC `i:real^M->bool` THEN REWRITE_TAC[] THEN
CONV_TAC(RAND_CONV UNWIND_CONV) THEN
AP_TERM_TAC THEN GEN_REWRITE_TAC I [
FUN_EQ_THM] THEN
X_GEN_TAC `l:real^M->bool` THEN REWRITE_TAC[] THEN
ASM_CASES_TAC `k:real^M->bool = i
INTER l` THEN ASM_REWRITE_TAC[] THEN
ASM_REWRITE_TAC[
IN_INTER; GSYM
MEMBER_NOT_EMPTY] THEN
EQ_TAC THENL [ASM_MESON_TAC[
TAGGED_DIVISION_OF]; ALL_TAC] THEN
REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
DISCH_THEN(X_CHOOSE_THEN `y:real^M` STRIP_ASSUME_TAC) THEN
ASM_REWRITE_TAC[] THEN
FIRST_X_ASSUM(MP_TAC o SPECL [`x:real^M`; `i:real^M->bool`]) THEN
GEN_REWRITE_TAC LAND_CONV [GSYM CONTRAPOS_THM] THEN
ASM_REWRITE_TAC[] THEN DISCH_THEN MATCH_MP_TAC THEN
REWRITE_TAC[GSYM
MEMBER_NOT_EMPTY] THEN
EXISTS_TAC `y:real^M` THEN ASM_REWRITE_TAC[
INTER;
IN_ELIM_THM] THEN
UNDISCH_TAC `(\x:real^M. ball (x,k x)) fine p` THEN
REWRITE_TAC[fine;
SUBSET] THEN ASM_MESON_TAC[];
ALL_TAC] THEN
CONJ_TAC THENL
[MP_TAC(ISPECL
[`\y. norm(integral y (f:real^M->real^N))`;
`p':(real^M#(real^M->bool))->bool`;
`interval[a:real^M,b]`]
SUM_OVER_TAGGED_DIVISION_LEMMA) THEN
ASM_SIMP_TAC[
INTEGRAL_NULL;
NORM_0] THEN DISCH_THEN SUBST1_TAC THEN
MATCH_MP_TAC
REAL_LE_TRANS THEN
EXISTS_TAC `sum {i
INTER l | i
IN d /\
(l
IN IMAGE SND (p:(real^M#(real^M->bool))->bool))}
(\k. norm(integral k (f:real^M->real^N)))` THEN
CONJ_TAC THENL
[ALL_TAC;
MATCH_MP_TAC
REAL_EQ_IMP_LE THEN MATCH_MP_TAC
SUM_SUPERSET THEN
CONJ_TAC THENL
[REWRITE_TAC[
SUBSET;
FORALL_IN_IMAGE] THEN
REWRITE_TAC[
FORALL_PAIR_THM] THEN
REWRITE_TAC[
IN_ELIM_THM;
IN_IMAGE;
PAIR_EQ;
EXISTS_PAIR_THM] THEN
MESON_TAC[];
ALL_TAC] THEN
REWRITE_TAC[
IN_ELIM_THM;
LEFT_AND_EXISTS_THM;
LEFT_IMP_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC
[`kk:real^M->bool`; `i:real^M->bool`; `l:real^M->bool`] THEN
ASM_CASES_TAC `kk:real^M->bool = i
INTER l` THEN ASM_REWRITE_TAC[] THEN
REWRITE_TAC[
IN_IMAGE;
EXISTS_PAIR_THM;
UNWIND_THM1] THEN
DISCH_THEN(CONJUNCTS_THEN2
(CONJUNCTS_THEN2 ASSUME_TAC (X_CHOOSE_TAC `x:real^M`)) MP_TAC) THEN
REWRITE_TAC[
IN_ELIM_THM;
PAIR_EQ;
NOT_EXISTS_THM] THEN
DISCH_THEN(MP_TAC o SPECL
[`x:real^M`; `x:real^M`; `i:real^M->bool`; `l:real^M->bool`]) THEN
ASM_SIMP_TAC[
INTEGRAL_EMPTY;
NORM_0]] THEN
SUBGOAL_THEN
`{k
INTER l | k
IN d /\ l
IN IMAGE SND (p:(real^M#(real^M->bool))->bool)} =
IMAGE (\(k,l). k
INTER l) {k,l | k
IN d /\ l
IN IMAGE SND p}`
SUBST1_TAC THENL
[GEN_REWRITE_TAC I [
EXTENSION] THEN
REWRITE_TAC[
IN_ELIM_THM;
IN_IMAGE;
EXISTS_PAIR_THM;
FORALL_PAIR_THM] THEN
REWRITE_TAC[
PAIR_EQ] THEN
CONV_TAC(REDEPTH_CONV UNWIND_CONV) THEN MESON_TAC[];
ALL_TAC] THEN
W(MP_TAC o PART_MATCH (lhand o rand)
SUM_IMAGE_NONZERO o rand o snd) THEN
ANTS_TAC THENL
[ASSUME_TAC(MATCH_MP
DIVISION_OF_TAGGED_DIVISION
(ASSUME `p
tagged_division_of interval[a:real^M,b]`)) THEN
ASM_SIMP_TAC[
FINITE_PRODUCT;
FINITE_IMAGE] THEN
REWRITE_TAC[
FORALL_PAIR_THM;
IN_ELIM_PAIR_THM] THEN
MAP_EVERY X_GEN_TAC
[`l1:real^M->bool`; `k1:real^M->bool`;
`l2:real^M->bool`; `k2:real^M->bool`] THEN
REWRITE_TAC[
PAIR_EQ] THEN STRIP_TAC THEN
SUBGOAL_THEN `interior(l2
INTER k2:real^M->bool) = {}` MP_TAC THENL
[ALL_TAC;
MP_TAC(ASSUME `d
division_of interval[a:real^M,b]`) THEN
REWRITE_TAC[
division_of] THEN
DISCH_THEN(MP_TAC o SPEC `l2:real^M->bool` o el 1 o CONJUNCTS) THEN
ASM_REWRITE_TAC[] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
MP_TAC(ASSUME
`(
IMAGE SND (p:(real^M#(real^M->bool))->bool))
division_of interval[a:real^M,b]`) THEN
REWRITE_TAC[
division_of] THEN
DISCH_THEN(MP_TAC o SPEC `k2:real^M->bool` o el 1 o CONJUNCTS) THEN
ASM_REWRITE_TAC[] THEN STRIP_TAC THEN
ASM_REWRITE_TAC[
INTER_INTERVAL] THEN DISCH_TAC THEN
REWRITE_TAC[
NORM_EQ_0] THEN
MATCH_MP_TAC
INTEGRAL_NULL THEN
ASM_REWRITE_TAC[
CONTENT_EQ_0_INTERIOR]] THEN
MATCH_MP_TAC(SET_RULE
`(interior(k1)
INTER interior(k2) = {} \/
interior(l1)
INTER interior(l2) = {}) /\
interior(l1
INTER k1)
SUBSET interior(k1) /\
interior(l2
INTER k2)
SUBSET interior(k2) /\
interior(l1
INTER k1)
SUBSET interior(l1) /\
interior(l2
INTER k2)
SUBSET interior(l2) /\
interior(l1
INTER k1) = interior(l2
INTER k2)
==> interior(l2
INTER k2) = {}`) THEN
SIMP_TAC[
SUBSET_INTERIOR;
INTER_SUBSET] THEN ASM_REWRITE_TAC[] THEN
MP_TAC(ASSUME `d
division_of interval[a:real^M,b]`) THEN
REWRITE_TAC[
division_of] THEN DISCH_THEN(MP_TAC o el 2 o CONJUNCTS) THEN
DISCH_THEN(MP_TAC o SPECL [`l1:real^M->bool`; `l2:real^M->bool`]) THEN
ASM_REWRITE_TAC[] THEN
MP_TAC(ASSUME
`(
IMAGE SND (p:(real^M#(real^M->bool))->bool))
division_of interval[a:real^M,b]`) THEN
REWRITE_TAC[
division_of] THEN DISCH_THEN(MP_TAC o el 2 o CONJUNCTS) THEN
DISCH_THEN(MP_TAC o SPECL [`k1:real^M->bool`; `k2:real^M->bool`]) THEN
ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[];
ALL_TAC] THEN
DISCH_THEN SUBST1_TAC THEN
GEN_REWRITE_TAC (RAND_CONV o RAND_CONV) [GSYM ETA_AX] THEN
GEN_REWRITE_TAC (RAND_CONV o RAND_CONV) [
LAMBDA_PAIR_THM] THEN
ASM_SIMP_TAC[GSYM
SUM_SUM_PRODUCT;
FINITE_IMAGE] THEN
MATCH_MP_TAC
SUM_LE THEN ASM_REWRITE_TAC[] THEN
X_GEN_TAC `k:real^M->bool` THEN DISCH_TAC THEN REWRITE_TAC[
o_DEF] THEN
MATCH_MP_TAC
REAL_LE_TRANS THEN
EXISTS_TAC
`sum { k
INTER l |
l
IN IMAGE SND (p:(real^M#(real^M->bool))->bool)}
(\k. norm(integral k (f:real^M->real^N)))` THEN
CONJ_TAC THENL
[ALL_TAC;
ONCE_REWRITE_TAC[
SIMPLE_IMAGE] THEN
W(MP_TAC o PART_MATCH (lhs o rand)
SUM_IMAGE_NONZERO o lhand o snd) THEN
ANTS_TAC THENL [ALL_TAC; SIMP_TAC[
o_DEF;
REAL_LE_REFL]] THEN
ASM_SIMP_TAC[
FINITE_IMAGE] THEN
MAP_EVERY X_GEN_TAC [`k1:real^M->bool`; `k2:real^M->bool`] THEN
STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
SUBGOAL_THEN `interior(k
INTER k2:real^M->bool) = {}` MP_TAC THENL
[ALL_TAC;
MP_TAC(MATCH_MP
DIVISION_OF_TAGGED_DIVISION
(ASSUME `p
tagged_division_of interval[a:real^M,b]`)) THEN
MP_TAC(ASSUME `d
division_of interval[a:real^M,b]`) THEN
REWRITE_TAC[
division_of] THEN
DISCH_THEN(MP_TAC o SPEC `k:real^M->bool` o el 1 o CONJUNCTS) THEN
ASM_REWRITE_TAC[] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN(MP_TAC o SPEC `k2:real^M->bool` o el 1 o CONJUNCTS) THEN
ASM_REWRITE_TAC[
INTER_INTERVAL; GSYM
CONTENT_EQ_0_INTERIOR] THEN
STRIP_TAC THEN ASM_REWRITE_TAC[
INTER_INTERVAL] THEN
SIMP_TAC[GSYM
CONTENT_EQ_0_INTERIOR;
INTEGRAL_NULL;
NORM_0]] THEN
MATCH_MP_TAC(SET_RULE
`interior(k
INTER k2)
SUBSET interior(k1
INTER k2) /\
interior(k1
INTER k2) = {}
==> interior(k
INTER k2) = {}`) THEN
CONJ_TAC THENL
[MATCH_MP_TAC
SUBSET_INTERIOR THEN ASM SET_TAC[]; ALL_TAC] THEN
MP_TAC(MATCH_MP
DIVISION_OF_TAGGED_DIVISION
(ASSUME `p
tagged_division_of interval[a:real^M,b]`)) THEN
REWRITE_TAC[
division_of] THEN
DISCH_THEN(MP_TAC o el 2 o CONJUNCTS) THEN
REWRITE_TAC[
INTERIOR_INTER] THEN DISCH_THEN MATCH_MP_TAC THEN
ASM_REWRITE_TAC[]] THEN
SUBGOAL_THEN `?u v:real^M. k = interval[u,v]`
(REPEAT_TCL CHOOSE_THEN SUBST_ALL_TAC)
THENL [ASM_MESON_TAC[
division_of]; ALL_TAC] THEN
SUBGOAL_THEN `interval[u:real^M,v]
SUBSET interval[a,b]` ASSUME_TAC THENL
[ASM_MESON_TAC[
division_of]; ALL_TAC] THEN
ABBREV_TAC `d' =
{interval[u,v]
INTER l |l|
l
IN IMAGE SND (p:(real^M#(real^M->bool))->bool) /\
~(interval[u,v]
INTER l = {})}` THEN
MATCH_MP_TAC
REAL_LE_TRANS THEN
EXISTS_TAC
`sum d' (\k. norm (integral k (f:real^M->real^N)))` THEN
CONJ_TAC THENL
[ALL_TAC;
MATCH_MP_TAC
REAL_EQ_IMP_LE THEN CONV_TAC SYM_CONV THEN
MATCH_MP_TAC
SUM_SUPERSET THEN
EXPAND_TAC "d'" THEN REWRITE_TAC[
SUBSET; SET_RULE
`a
IN {f x |x| x
IN s /\ ~(f x = b)} <=>
a
IN {f x | x
IN s} /\ ~(a = b)`] THEN
SIMP_TAC[
IMP_CONJ;
INTEGRAL_EMPTY;
NORM_0]] THEN
SUBGOAL_THEN `d'
division_of interval[u:real^M,v]` ASSUME_TAC THENL
[EXPAND_TAC "d'" THEN MATCH_MP_TAC
DIVISION_INTER_1 THEN
EXISTS_TAC `interval[a:real^M,b]` THEN
ASM_SIMP_TAC[
DIVISION_OF_TAGGED_DIVISION];
ALL_TAC] THEN
MATCH_MP_TAC
REAL_LE_TRANS THEN
EXISTS_TAC `norm(vsum d' (\i. integral i (f:real^M->real^N)))` THEN
CONJ_TAC THENL
[MATCH_MP_TAC
REAL_EQ_IMP_LE THEN AP_TERM_TAC THEN
MATCH_MP_TAC
INTEGRAL_COMBINE_DIVISION_TOPDOWN THEN
ASM_MESON_TAC[
INTEGRABLE_ON_SUBINTERVAL];
ALL_TAC] THEN
MATCH_MP_TAC
VSUM_NORM_LE THEN
REWRITE_TAC[
REAL_LE_REFL] THEN ASM_MESON_TAC[
division_of];
ALL_TAC] THEN
REMOVE_THEN "p'" SUBST_ALL_TAC THEN
MATCH_MP_TAC
EQ_TRANS THEN
EXISTS_TAC `sum {x,i
INTER l | (x,l)
IN p /\ i
IN d}
(\(x,k:real^M->bool).
abs(content k) * norm((f:real^M->real^N) x))` THEN
CONJ_TAC THENL
[CONV_TAC SYM_CONV THEN MATCH_MP_TAC
SUM_SUPERSET THEN
CONJ_TAC THENL [SET_TAC[]; ALL_TAC] THEN
REWRITE_TAC[
FORALL_PAIR_THM] THEN
MAP_EVERY X_GEN_TAC [`x:real^M`; `i:real^M->bool`] THEN
ASM_CASES_TAC `i:real^M->bool = {}` THEN
ASM_SIMP_TAC[
CONTENT_EMPTY;
REAL_ABS_NUM;
REAL_MUL_LZERO] THEN
MATCH_MP_TAC(TAUT `(a <=> b) ==> a /\ ~b ==> c`) THEN
REWRITE_TAC[
IN_ELIM_THM] THEN
REPEAT(AP_TERM_TAC THEN ABS_TAC) THEN
REWRITE_TAC[
PAIR_EQ] THEN ASM_MESON_TAC[];
ALL_TAC] THEN
SUBGOAL_THEN
`{x,i
INTER l | x,l
IN (p:(real^M#(real^M->bool))->bool) /\ i
IN d} =
IMAGE (\((x,l),k). x,k
INTER l) {m,k | m
IN p /\ k
IN d}`
SUBST1_TAC THENL
[GEN_REWRITE_TAC I [
EXTENSION] THEN
REWRITE_TAC[
IN_ELIM_THM;
IN_IMAGE;
EXISTS_PAIR_THM;
FORALL_PAIR_THM] THEN
REWRITE_TAC[
PAIR_EQ] THEN
CONV_TAC(REDEPTH_CONV UNWIND_CONV) THEN MESON_TAC[];
ALL_TAC] THEN
W(MP_TAC o PART_MATCH (lhand o rand)
SUM_IMAGE_NONZERO o lhand o snd) THEN
ANTS_TAC THENL
[ASM_SIMP_TAC[
FINITE_PRODUCT] THEN
REWRITE_TAC[
FORALL_PAIR_THM;
IN_ELIM_PAIR_THM] THEN
MAP_EVERY X_GEN_TAC
[`x1:real^M`; `l1:real^M->bool`; `k1:real^M->bool`;
`x2:real^M`; `l2:real^M->bool`; `k2:real^M->bool`] THEN
REWRITE_TAC[
PAIR_EQ] THEN
ASM_CASES_TAC `x1:real^M = x2` THEN ASM_REWRITE_TAC[] THEN
STRIP_TAC THEN
REWRITE_TAC[
REAL_ENTIRE] THEN DISJ1_TAC THEN
REWRITE_TAC[
REAL_ABS_ZERO] THEN
SUBGOAL_THEN `interior(k2
INTER l2:real^M->bool) = {}` MP_TAC THENL
[ALL_TAC;
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
division_of]) THEN
DISCH_THEN(MP_TAC o SPEC `k2:real^M->bool` o el 1 o CONJUNCTS) THEN
ASM_REWRITE_TAC[] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
MP_TAC(ASSUME `p
tagged_division_of interval[a:real^M,b]`) THEN
REWRITE_TAC[
TAGGED_DIVISION_OF] THEN
DISCH_THEN(MP_TAC o el 1 o CONJUNCTS) THEN
DISCH_THEN(MP_TAC o SPECL [`x2:real^M`; `l2:real^M->bool`]) THEN
ASM_REWRITE_TAC[] THEN STRIP_TAC THEN
ASM_REWRITE_TAC[
INTER_INTERVAL;
CONTENT_EQ_0_INTERIOR]] THEN
MATCH_MP_TAC(SET_RULE
`(interior(k1)
INTER interior(k2) = {} \/
interior(l1)
INTER interior(l2) = {}) /\
interior(k1
INTER l1)
SUBSET interior(k1) /\
interior(k2
INTER l2)
SUBSET interior(k2) /\
interior(k1
INTER l1)
SUBSET interior(l1) /\
interior(k2
INTER l2)
SUBSET interior(l2) /\
interior(k1
INTER l1) = interior(k2
INTER l2)
==> interior(k2
INTER l2) = {}`) THEN
SIMP_TAC[
SUBSET_INTERIOR;
INTER_SUBSET] THEN ASM_REWRITE_TAC[] THEN
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
division_of]) THEN
DISCH_THEN(MP_TAC o el 2 o CONJUNCTS) THEN
DISCH_THEN(MP_TAC o SPECL [`k1:real^M->bool`; `k2:real^M->bool`]) THEN
MP_TAC(ASSUME `p
tagged_division_of interval[a:real^M,b]`) THEN
REWRITE_TAC[
TAGGED_DIVISION_OF] THEN
DISCH_THEN(MP_TAC o el 2 o CONJUNCTS) THEN
DISCH_THEN(MP_TAC o SPECL
[`x2:real^M`; `l1:real^M->bool`; `x2:real^M`; `l2:real^M->bool`]) THEN
ASM_REWRITE_TAC[
PAIR_EQ] THEN ASM_MESON_TAC[];
ALL_TAC] THEN
DISCH_THEN SUBST1_TAC THEN
GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [GSYM ETA_AX] THEN
GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [
LAMBDA_PAIR_THM] THEN
ASM_SIMP_TAC[GSYM
SUM_SUM_PRODUCT] THEN
MATCH_MP_TAC
SUM_EQ THEN REWRITE_TAC[
FORALL_PAIR_THM] THEN
MAP_EVERY X_GEN_TAC [`x:real^M`; `l:real^M->bool`] THEN
DISCH_TAC THEN REWRITE_TAC[
o_THM;
SUM_RMUL] THEN
AP_THM_TAC THEN AP_TERM_TAC THEN
SUBGOAL_THEN `?u v:real^M. l = interval[u,v]`
(REPEAT_TCL CHOOSE_THEN SUBST_ALL_TAC)
THENL [ASM_MESON_TAC[
TAGGED_DIVISION_OF]; ALL_TAC] THEN
MATCH_MP_TAC
EQ_TRANS THEN
EXISTS_TAC `sum d (\k. content(k
INTER interval[u:real^M,v]))` THEN
CONJ_TAC THENL
[MATCH_MP_TAC
SUM_EQ THEN REWRITE_TAC[
real_abs] THEN
X_GEN_TAC `k:real^M->bool` THEN DISCH_TAC THEN
SUBGOAL_THEN `?w z:real^M. k = interval[w,z]`
(REPEAT_TCL CHOOSE_THEN SUBST_ALL_TAC)
THENL [ASM_MESON_TAC[
division_of]; ALL_TAC] THEN
REWRITE_TAC[
INTER_INTERVAL;
CONTENT_POS_LE];
ALL_TAC] THEN
MATCH_MP_TAC
EQ_TRANS THEN
EXISTS_TAC `sum {k
INTER interval[u:real^M,v] | k
IN d} content` THEN
CONJ_TAC THENL
[ONCE_REWRITE_TAC[
SIMPLE_IMAGE] THEN ONCE_REWRITE_TAC[GSYM
o_DEF] THEN
CONV_TAC SYM_CONV THEN MATCH_MP_TAC
SUM_IMAGE_NONZERO THEN
ASM_REWRITE_TAC[] THEN
MAP_EVERY X_GEN_TAC [`k1:real^M->bool`; `k2:real^M->bool`] THEN
STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
SUBGOAL_THEN `interior(k2
INTER interval[u:real^M,v]) = {}` MP_TAC THENL
[ALL_TAC;
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
division_of]) THEN
DISCH_THEN(MP_TAC o SPEC `k2:real^M->bool` o el 1 o CONJUNCTS) THEN
ASM_REWRITE_TAC[] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
ASM_REWRITE_TAC[
INTER_INTERVAL;
CONTENT_EQ_0_INTERIOR]] THEN
MATCH_MP_TAC(SET_RULE
`interior(k2
INTER i)
SUBSET interior(k1
INTER k2) /\
interior(k1
INTER k2) = {}
==> interior(k2
INTER i) = {}`) THEN
CONJ_TAC THENL
[MATCH_MP_TAC
SUBSET_INTERIOR THEN ASM SET_TAC[]; ALL_TAC] THEN
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
division_of]) THEN
DISCH_THEN(MP_TAC o el 2 o CONJUNCTS) THEN
REWRITE_TAC[
INTERIOR_INTER] THEN DISCH_THEN MATCH_MP_TAC THEN
ASM_REWRITE_TAC[];
ALL_TAC] THEN
SUBGOAL_THEN `interval[u:real^M,v]
SUBSET interval[a,b]` ASSUME_TAC THENL
[ASM_MESON_TAC[
TAGGED_DIVISION_OF]; ALL_TAC] THEN
MATCH_MP_TAC
EQ_TRANS THEN
EXISTS_TAC `sum {k
INTER interval[u:real^M,v] |k|
k
IN d /\ ~(k
INTER interval[u,v] = {})} content` THEN
CONJ_TAC THENL
[MATCH_MP_TAC
SUM_SUPERSET THEN
REWRITE_TAC[
SUBSET; SET_RULE
`a
IN {f x |x| x
IN s /\ ~(f x = b)} <=>
a
IN {f x | x
IN s} /\ ~(a = b)`] THEN
SIMP_TAC[
IMP_CONJ;
CONTENT_EMPTY];
ALL_TAC] THEN
MATCH_MP_TAC
ADDITIVE_CONTENT_DIVISION THEN
ONCE_REWRITE_TAC[
INTER_COMM] THEN MATCH_MP_TAC
DIVISION_INTER_1 THEN
EXISTS_TAC `interval[a:real^M,b]` THEN ASM_REWRITE_TAC[]);;
let BEPPO_LEVI_INCREASING = prove
(`!f:num->real^N->real^1 s.
(!k. (f k)
integrable_on s) /\
(!k x. x
IN s ==> drop(f k x) <= drop(f (SUC k) x)) /\
bounded {integral s (f k) | k
IN (:num)}
==> ?g k. negligible k /\
!x. x
IN (s
DIFF k) ==> ((\k. f k x) --> g x) sequentially`,
SUBGOAL_THEN
`!f:num->real^N->real^1 s.
(!k x. x
IN s ==> &0 <= drop(f k x)) /\
(!k. (f k)
integrable_on s) /\
(!k x. x
IN s ==> drop(f k x) <= drop(f (SUC k) x)) /\
bounded {integral s (f k) | k
IN (:num)}
==> ?g k. negligible k /\
!x. x
IN (s
DIFF k) ==> ((\k. f k x) --> g x) sequentially`
ASSUME_TAC THENL
[ALL_TAC;
REPEAT GEN_TAC THEN STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o ISPECL
[`\n x:real^N. f(n:num) x - f 0 x:real^1`; `s:real^N->bool`]) THEN
REWRITE_TAC[] THEN ANTS_TAC THEN REPEAT CONJ_TAC THENL
[REPEAT STRIP_TAC THEN REWRITE_TAC[
DROP_SUB;
REAL_SUB_LE] THEN
MP_TAC(ISPEC
`\m n:num. drop (f m (x:real^N)) <= drop (f n x)`
TRANSITIVE_STEPWISE_LE) THEN
REWRITE_TAC[
REAL_LE_TRANS;
REAL_LE_REFL] THEN ASM_MESON_TAC[
LE_0];
GEN_TAC THEN MATCH_MP_TAC
INTEGRABLE_SUB THEN ASM_REWRITE_TAC[ETA_AX];
REPEAT STRIP_TAC THEN REWRITE_TAC[
DROP_SUB;
REAL_SUB_LE] THEN
ASM_SIMP_TAC[REAL_ARITH `x - a <= y - a <=> x <= y`];
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [bounded]) THEN
ASM_SIMP_TAC[
INTEGRAL_SUB; ETA_AX; bounded] THEN
ONCE_REWRITE_TAC[
SIMPLE_IMAGE] THEN
REWRITE_TAC[
FORALL_IN_IMAGE;
IN_UNIV] THEN
DISCH_THEN(X_CHOOSE_THEN `B:real`
(fun th -> EXISTS_TAC `B + norm(integral s (f 0:real^N->real^1))` THEN
X_GEN_TAC `k:num` THEN MP_TAC(SPEC `k:num` th))) THEN
NORM_ARITH_TAC;
ONCE_REWRITE_TAC[
SWAP_EXISTS_THM] THEN MATCH_MP_TAC
MONO_EXISTS THEN
GEN_TAC THEN
DISCH_THEN(X_CHOOSE_THEN `g:real^N->real^1` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `(\x. g x + f 0 x):real^N->real^1` THEN
ASM_REWRITE_TAC[] THEN X_GEN_TAC `x:real^N` THEN
DISCH_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `x:real^N`) THEN
ASM_REWRITE_TAC[
LIM_SEQUENTIALLY; dist] THEN
REWRITE_TAC[VECTOR_ARITH `a - b - c:real^1 = a - (c + b)`]]] THEN
REPEAT STRIP_TAC THEN
ABBREV_TAC
`g = \i n:num x:real^N. lift(min (drop(f n x) / (&i + &1)) (&1))` THEN
SUBGOAL_THEN
`!i n. ((g:num->num->real^N->real^1) i n)
integrable_on s`
ASSUME_TAC THENL
[REPEAT GEN_TAC THEN EXPAND_TAC "g" THEN
MATCH_MP_TAC
INTEGRABLE_MIN_CONST_1 THEN
ASM_SIMP_TAC[
REAL_POS;
REAL_LE_DIV;
REAL_LE_ADD] THEN
REWRITE_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM]
real_div] THEN
ASM_SIMP_TAC[
LIFT_CMUL;
LIFT_DROP;
INTEGRABLE_CMUL; ETA_AX];
ALL_TAC] THEN
SUBGOAL_THEN
`!i:num k:num x:real^N. x
IN s ==> drop(g i k x) <= drop(g i (SUC k) x)`
ASSUME_TAC THENL
[REPEAT STRIP_TAC THEN EXPAND_TAC "g" THEN REWRITE_TAC[
LIFT_DROP] THEN
MATCH_MP_TAC(REAL_ARITH `x <= y ==> min x a <= min y a`) THEN
ASM_SIMP_TAC[
REAL_LE_DIV2_EQ; REAL_ARITH `&0 < &n + &1`];
ALL_TAC] THEN
SUBGOAL_THEN `!i:num k:num x:real^N. x
IN s ==> norm(g i k x:real^1) <= &1`
ASSUME_TAC THENL
[REPEAT STRIP_TAC THEN EXPAND_TAC "g" THEN
REWRITE_TAC[
LIFT_DROP;
NORM_REAL; GSYM drop] THEN
MATCH_MP_TAC(REAL_ARITH `&0 <= x ==> abs(min x (&1)) <= &1`) THEN
ASM_SIMP_TAC[
REAL_POS;
REAL_LE_DIV;
REAL_LE_ADD];
ALL_TAC] THEN
SUBGOAL_THEN
`!i:num x:real^N. x
IN s ==> ?h:real^1. ((\n. g i n x) --> h) sequentially`
MP_TAC THENL
[REPEAT STRIP_TAC THEN
MP_TAC(ISPECL
[`\n. drop(g (i:num) (n:num) (x:real^N))`; `&1`]
CONVERGENT_BOUNDED_MONOTONE) THEN
REWRITE_TAC[] THEN ANTS_TAC THENL
[ASM_SIMP_TAC[GSYM
ABS_DROP] THEN DISJ1_TAC THEN
MATCH_MP_TAC
TRANSITIVE_STEPWISE_LE THEN
ASM_SIMP_TAC[
REAL_LE_REFL;
REAL_LE_TRANS] THEN REAL_ARITH_TAC;
DISCH_THEN(X_CHOOSE_THEN `l:real` (fun th ->
EXISTS_TAC `lift l` THEN MP_TAC th)) THEN
REWRITE_TAC[
LIM_SEQUENTIALLY;
DIST_REAL; GSYM drop;
LIFT_DROP]];
GEN_REWRITE_TAC (LAND_CONV o REDEPTH_CONV) [
RIGHT_IMP_EXISTS_THM] THEN
REWRITE_TAC[
SKOLEM_THM;
LEFT_IMP_EXISTS_THM]] THEN
X_GEN_TAC `h:num->real^N->real^1` THEN STRIP_TAC THEN
MP_TAC(GEN `i:num `(ISPECL
[`g(i:num):num->real^N->real^1`; `h(i:num):real^N->real^1`;
`s:real^N->bool`]
MONOTONE_CONVERGENCE_INCREASING)) THEN
DISCH_THEN(MP_TAC o MATCH_MP
MONO_FORALL) THEN
ASM_REWRITE_TAC[] THEN ANTS_TAC THENL
[GEN_TAC THEN REWRITE_TAC[bounded] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [bounded]) THEN
MATCH_MP_TAC
MONO_EXISTS THEN X_GEN_TAC `K:real` THEN
REWRITE_TAC[
FORALL_IN_GSPEC] THEN REWRITE_TAC[
IN_UNIV] THEN
MATCH_MP_TAC
MONO_FORALL THEN X_GEN_TAC `k:num` THEN
MATCH_MP_TAC(REWRITE_RULE[
IMP_CONJ]
REAL_LE_TRANS) THEN
MATCH_MP_TAC(REAL_ARITH
`norm a = drop a /\ x <= drop a ==> x <= norm a`) THEN
CONJ_TAC THENL
[REWRITE_TAC[
NORM_REAL; GSYM drop;
REAL_ABS_REFL] THEN
MATCH_MP_TAC
INTEGRAL_DROP_POS THEN ASM_SIMP_TAC[];
MATCH_MP_TAC
INTEGRAL_NORM_BOUND_INTEGRAL THEN
ASM_REWRITE_TAC[] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
EXPAND_TAC "g" THEN REWRITE_TAC[
NORM_REAL; GSYM drop;
LIFT_DROP] THEN
MATCH_MP_TAC(REAL_ARITH
`&0 <= x /\ x <= y ==> abs(min x (&1)) <= y`) THEN
ASM_SIMP_TAC[
REAL_LE_ADD;
REAL_POS;
REAL_LE_DIV] THEN
SIMP_TAC[
REAL_LE_LDIV_EQ; REAL_ARITH `&0 < &i + &1`] THEN
REWRITE_TAC[REAL_ARITH `a <= a * (x + &1) <=> &0 <= a * x`] THEN
ASM_SIMP_TAC[
REAL_LE_MUL;
REAL_POS]];
REWRITE_TAC[
FORALL_AND_THM] THEN STRIP_TAC] THEN
ABBREV_TAC
`Z =
{x:real^N | x
IN s /\ ~(?l:real^1. ((\k. f k x) --> l) sequentially)}` THEN
ONCE_REWRITE_TAC[
SWAP_EXISTS_THM] THEN EXISTS_TAC `Z:real^N->bool` THEN
REWRITE_TAC[
RIGHT_EXISTS_AND_THM; GSYM
SKOLEM_THM;
RIGHT_EXISTS_IMP_THM] THEN
CONJ_TAC THENL
[ALL_TAC; EXPAND_TAC "Z" THEN REWRITE_TAC[
IN_ELIM_THM] THEN SET_TAC[]] THEN
MP_TAC(ISPECL
[`h:num->real^N->real^1`;
`(\x. if x
IN Z then vec 1 else vec 0):real^N->real^1`;
`s:real^N->bool`]
MONOTONE_CONVERGENCE_DECREASING) THEN
ASM_REWRITE_TAC[] THEN
SUBGOAL_THEN
`!i x:real^N. x
IN s ==> drop(h (SUC i) x) <= drop(h i x)`
ASSUME_TAC THENL
[MAP_EVERY X_GEN_TAC [`i:num`; `x:real^N`] THEN DISCH_TAC THEN
MATCH_MP_TAC(ISPEC `sequentially`
LIM_DROP_LE) THEN
EXISTS_TAC `\n. (g:num->num->real^N->real^1) (SUC i) n x` THEN
EXISTS_TAC `\n. (g:num->num->real^N->real^1) i n x` THEN
ASM_SIMP_TAC[
TRIVIAL_LIMIT_SEQUENTIALLY] THEN
MATCH_MP_TAC
ALWAYS_EVENTUALLY THEN X_GEN_TAC `n:num` THEN
EXPAND_TAC "g" THEN REWRITE_TAC[
LIFT_DROP] THEN
MATCH_MP_TAC(REAL_ARITH `x <= y ==> min x a <= min y a`) THEN
REWRITE_TAC[
real_div] THEN MATCH_MP_TAC
REAL_LE_LMUL THEN
ASM_SIMP_TAC[] THEN MATCH_MP_TAC
REAL_LE_INV2 THEN
REWRITE_TAC[GSYM
REAL_OF_NUM_SUC] THEN REAL_ARITH_TAC;
ASM_REWRITE_TAC[]] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
BOUNDED_POS]) THEN
REWRITE_TAC[
FORALL_IN_GSPEC;
IN_UNIV] THEN
DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN
SUBGOAL_THEN
`!i. norm(integral s ((h:num->real^N->real^1) i)) <= B / (&i + &1)`
ASSUME_TAC THENL
[X_GEN_TAC `i:num` THEN
MATCH_MP_TAC(ISPEC `sequentially`
LIM_NORM_UBOUND) THEN
EXISTS_TAC `\k. integral s ((g:num->num->real^N->real^1) i k)` THEN
ASM_REWRITE_TAC[
TRIVIAL_LIMIT_SEQUENTIALLY] THEN
MATCH_MP_TAC
ALWAYS_EVENTUALLY THEN X_GEN_TAC `n:num` THEN
REWRITE_TAC[] THEN MATCH_MP_TAC
REAL_LE_TRANS THEN
EXISTS_TAC
`drop(integral s (\x. inv(&i + &1) % (f:num->real^N->real^1) n x))` THEN
CONJ_TAC THENL
[MATCH_MP_TAC
INTEGRAL_NORM_BOUND_INTEGRAL THEN
ASM_SIMP_TAC[
INTEGRABLE_CMUL; ETA_AX] THEN
X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN EXPAND_TAC "g" THEN
REWRITE_TAC[
NORM_REAL; GSYM drop;
LIFT_DROP;
DROP_CMUL] THEN
MATCH_MP_TAC(REAL_ARITH
`&0 <= x /\ x <= y ==> abs(min x (&1)) <= y`) THEN
ASM_SIMP_TAC[
REAL_LE_ADD;
REAL_POS;
REAL_LE_DIV] THEN
REAL_ARITH_TAC;
ASM_SIMP_TAC[
INTEGRAL_CMUL; ETA_AX;
DROP_CMUL] THEN
ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
ASM_SIMP_TAC[GSYM
real_div;
REAL_LE_DIV2_EQ;
REAL_ARITH `&0 < &n + &1`] THEN
MATCH_MP_TAC(REAL_ARITH `abs x <= a ==> x <= a`) THEN
ASM_REWRITE_TAC[GSYM
ABS_DROP]];
ALL_TAC] THEN
ANTS_TAC THENL
[REWRITE_TAC[bounded;
FORALL_IN_GSPEC] THEN CONJ_TAC THENL
[ALL_TAC;
EXISTS_TAC `B:real` THEN X_GEN_TAC `i:num` THEN
MATCH_MP_TAC
REAL_LE_TRANS THEN
EXISTS_TAC `B / (&i + &1)` THEN ASM_REWRITE_TAC[] THEN
ASM_SIMP_TAC[
REAL_LE_LDIV_EQ; REAL_ARITH `&0 < &i + &1`] THEN
REWRITE_TAC[REAL_ARITH `B <= B * (i + &1) <=> &0 <= i * B`] THEN
ASM_SIMP_TAC[
REAL_LE_MUL;
REAL_POS;
REAL_LT_IMP_LE]] THEN
X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
ASM_CASES_TAC `(x:real^N)
IN Z` THEN ASM_REWRITE_TAC[] THENL
[MATCH_MP_TAC
LIM_EVENTUALLY THEN
UNDISCH_TAC `(x:real^N)
IN Z` THEN EXPAND_TAC "Z" THEN
REWRITE_TAC[
IN_ELIM_THM] THEN ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
MP_TAC(GEN `B:real` (ISPECL
[`\n. drop(f (n:num) (x:real^N))`; `B:real`]
CONVERGENT_BOUNDED_MONOTONE)) THEN
REWRITE_TAC[
LEFT_FORALL_IMP_THM;
LEFT_EXISTS_AND_THM] THEN
MATCH_MP_TAC(TAUT
`q /\ ~r /\ (q ==> ~p ==> s)
==> (p /\ (q \/ q') ==> r) ==> s`) THEN
CONJ_TAC THENL
[MATCH_MP_TAC
TRANSITIVE_STEPWISE_LE THEN
ASM_SIMP_TAC[
REAL_LE_REFL;
REAL_LE_TRANS] THEN REAL_ARITH_TAC;
ALL_TAC] THEN
CONJ_TAC THENL
[FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
NOT_EXISTS_THM]) THEN
ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_THEN `l:real` STRIP_ASSUME_TAC) THEN
DISCH_THEN(MP_TAC o SPEC `lift l`) THEN
REWRITE_TAC[
LIM_SEQUENTIALLY] THEN
X_GEN_TAC `e:real` THEN DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `e:real`) THEN ASM_REWRITE_TAC[] THEN
REWRITE_TAC[
DIST_REAL; GSYM drop;
DROP_SUB;
LIFT_DROP];
ALL_TAC] THEN
DISCH_TAC THEN REWRITE_TAC[
NOT_FORALL_THM;
EVENTUALLY_SEQUENTIALLY] THEN
REWRITE_TAC[
NOT_EXISTS_THM;
NOT_FORALL_THM;
REAL_NOT_LE] THEN
DISCH_TAC THEN
EXISTS_TAC `0` THEN X_GEN_TAC `i:num` THEN DISCH_TAC THEN
MATCH_MP_TAC(ISPEC `sequentially`
LIM_UNIQUE) THEN
EXISTS_TAC `(\n. (g:num->num->real^N->real^1) i n x)` THEN
ASM_SIMP_TAC[
TRIVIAL_LIMIT_SEQUENTIALLY] THEN
MATCH_MP_TAC
LIM_EVENTUALLY THEN
EXPAND_TAC "g" THEN REWRITE_TAC[
EVENTUALLY_SEQUENTIALLY] THEN
FIRST_X_ASSUM(MP_TAC o SPEC `&i + &1`) 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
REWRITE_TAC[GSYM
DROP_EQ;
DROP_VEC;
LIFT_DROP] THEN
REWRITE_TAC[REAL_ARITH `min a b = b <=> b <= a`] THEN
SIMP_TAC[
REAL_LE_RDIV_EQ; REAL_ARITH `&0 < &i + &1`; REAL_MUL_LID] THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH
`a < abs N ==> &0 <= N /\ N <= n ==> a <= n`)) THEN
ASM_SIMP_TAC[];
UNDISCH_TAC `~((x:real^N)
IN Z)` THEN EXPAND_TAC "Z" THEN
REWRITE_TAC[
IN_ELIM_THM] THEN ASM_REWRITE_TAC[
LEFT_IMP_EXISTS_THM] THEN
X_GEN_TAC `l:real^1` THEN
DISCH_THEN(MP_TAC o MATCH_MP
CONVERGENT_IMP_BOUNDED) THEN
REWRITE_TAC[
BOUNDED_POS;
FORALL_IN_IMAGE;
IN_UNIV] THEN
DISCH_THEN(X_CHOOSE_THEN `C:real` STRIP_ASSUME_TAC) THEN
REWRITE_TAC[
LIM_SEQUENTIALLY;
DIST_0] THEN
X_GEN_TAC `e:real` THEN DISCH_TAC THEN
MP_TAC(ISPEC `e / C:real`
REAL_ARCH_INV) THEN
ASM_SIMP_TAC[
REAL_LT_DIV] THEN MATCH_MP_TAC
MONO_EXISTS THEN
X_GEN_TAC `N:num` THEN ASM_SIMP_TAC[
REAL_LT_RDIV_EQ] THEN STRIP_TAC THEN
X_GEN_TAC `i:num` THEN DISCH_TAC THEN
MATCH_MP_TAC
REAL_LET_TRANS THEN EXISTS_TAC `inv(&N) * C` THEN
ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC
REAL_LE_TRANS THEN EXISTS_TAC `C / (&i + &1)` THEN
CONJ_TAC THENL
[ALL_TAC;
REWRITE_TAC[ONCE_REWRITE_RULE[REAL_MUL_SYM]
real_div] THEN
ASM_SIMP_TAC[
REAL_LE_RMUL_EQ] THEN MATCH_MP_TAC
REAL_LE_INV2 THEN
ASM_REWRITE_TAC[
REAL_OF_NUM_LT; REAL_OF_NUM_LE; REAL_OF_NUM_ADD] THEN
ASM_ARITH_TAC] THEN
MATCH_MP_TAC(ISPEC `sequentially`
LIM_NORM_UBOUND) THEN
EXISTS_TAC `\n. (g:num->num->real^N->real^1) i n x` THEN
ASM_SIMP_TAC[
TRIVIAL_LIMIT_SEQUENTIALLY] THEN
MATCH_MP_TAC
ALWAYS_EVENTUALLY THEN X_GEN_TAC `n:num` THEN
EXPAND_TAC "g" THEN REWRITE_TAC[
NORM_REAL; GSYM drop;
LIFT_DROP] THEN
MATCH_MP_TAC(REAL_ARITH
`&0 <= x /\ x <= a ==> abs(min x (&1)) <= a`) THEN
ASM_SIMP_TAC[
REAL_LE_DIV;
REAL_LE_ADD;
REAL_POS] THEN
ASM_SIMP_TAC[
REAL_LE_DIV2_EQ; REAL_ARITH `&0 < &i + &1`] THEN
MATCH_MP_TAC(REAL_ARITH `abs x <= a ==> x <= a`) THEN
ASM_REWRITE_TAC[GSYM
NORM_LIFT;
LIFT_DROP]];
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
MATCH_MP_TAC(MESON[
LIM_UNIQUE;
TRIVIAL_LIMIT_SEQUENTIALLY]
`(f --> vec 0) sequentially /\ (i = vec 0 ==> p)
==> (f --> i) sequentially ==> p`) THEN
CONJ_TAC THENL
[MATCH_MP_TAC
LIM_NULL_COMPARISON THEN
EXISTS_TAC `\i. B / (&i + &1)` THEN
ASM_SIMP_TAC[
ALWAYS_EVENTUALLY] THEN
REWRITE_TAC[
real_div;
LIFT_CMUL] THEN
SUBST1_TAC(VECTOR_ARITH `vec 0:real^1 = B % vec 0`) THEN
MATCH_MP_TAC
LIM_CMUL THEN
REWRITE_TAC[
LIM_SEQUENTIALLY;
DIST_0] THEN
X_GEN_TAC `e:real` THEN GEN_REWRITE_TAC LAND_CONV [
REAL_ARCH_INV] THEN
MATCH_MP_TAC
MONO_EXISTS THEN X_GEN_TAC `N:num` THEN STRIP_TAC THEN
X_GEN_TAC `n:num` THEN DISCH_TAC THEN
REWRITE_TAC[
NORM_LIFT; GSYM drop;
LIFT_DROP;
REAL_ABS_INV] THEN
MATCH_MP_TAC
REAL_LET_TRANS THEN EXISTS_TAC `inv(&N)` THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC
REAL_LE_INV2 THEN
REWRITE_TAC[REAL_ARITH `abs(&n + &1) = &n + &1`] THEN
REWRITE_TAC[REAL_OF_NUM_ADD; REAL_OF_NUM_LE;
REAL_OF_NUM_LT] THEN
ASM_ARITH_TAC;
ASM_SIMP_TAC[
INTEGRAL_EQ_HAS_INTEGRAL] THEN
W(MP_TAC o PART_MATCH (lhs o rand)
HAS_INTEGRAL_NEGLIGIBLE_EQ o
lhand o snd) THEN
ANTS_TAC THENL
[REWRITE_TAC[
IMP_CONJ;
RIGHT_FORALL_IMP_THM] THEN
REWRITE_TAC[IMP_IMP; DIMINDEX_1;
FORALL_1; GSYM drop] THEN
REPEAT STRIP_TAC THEN COND_CASES_TAC THEN
REWRITE_TAC[
DROP_VEC;
REAL_POS];
DISCH_THEN SUBST1_TAC THEN
MATCH_MP_TAC(REWRITE_RULE[
IMP_CONJ_ALT]
NEGLIGIBLE_SUBSET) THEN
SIMP_TAC[
SUBSET;
IN_ELIM_THM;
VEC_EQ;
ARITH_EQ] THEN
EXPAND_TAC "Z" THEN SIMP_TAC[
IN_ELIM_THM]]]]);;
let BEPPO_LEVI_DECREASING = prove
(`!f:num->real^N->real^1 s.
(!k. (f k)
integrable_on s) /\
(!k x. x
IN s ==> drop(f (SUC k) x) <= drop(f k x)) /\
bounded {integral s (f k) | k
IN (:num)}
==> ?g k. negligible k /\
!x. x
IN (s
DIFF k) ==> ((\k. f k x) --> g x) sequentially`,
REPEAT STRIP_TAC THEN
MP_TAC(ISPECL [`\n x. --((f:num->real^N->real^1) n x)`; `s:real^N->bool`]
BEPPO_LEVI_INCREASING) THEN
ASM_SIMP_TAC[
INTEGRABLE_NEG;
DROP_NEG; ETA_AX;
REAL_LE_NEG2] THEN
ANTS_TAC THENL
[REWRITE_TAC[bounded] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [bounded]) THEN
REWRITE_TAC[
FORALL_IN_GSPEC] THEN
ASM_SIMP_TAC[
INTEGRAL_NEG; ETA_AX;
NORM_NEG];
ONCE_REWRITE_TAC[
SWAP_EXISTS_THM] THEN MATCH_MP_TAC
MONO_EXISTS THEN
X_GEN_TAC `k:real^N->bool` THEN
DISCH_THEN(X_CHOOSE_THEN `g:real^N->real^1` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `\x. --((g:real^N->real^1) x)` THEN
ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN
GEN_REWRITE_TAC (RATOR_CONV o LAND_CONV o ABS_CONV)
[GSYM
VECTOR_NEG_NEG] THEN
ASM_SIMP_TAC[
LIM_NEG_EQ]]);;
let FUNDAMENTAL_THEOREM_OF_CALCULUS_STRONG = prove
(`!f:real^1->real^N f' s a b.
COUNTABLE s /\
drop a <= drop b /\ f
continuous_on interval[a,b] /\
(!x. x
IN interval[a,b]
DIFF s
==> (f
has_vector_derivative f'(x)) (at x within interval[a,b]))
==> (f'
has_integral (f(b) - f(a))) (interval[a,b])`,
REPEAT STRIP_TAC THEN
MATCH_MP_TAC
HAS_INTEGRAL_SPIKE THEN
EXISTS_TAC `(\x. if x
IN s then vec 0 else f' x):real^1->real^N` THEN
EXISTS_TAC `s:real^1->bool` THEN
ASM_SIMP_TAC[
NEGLIGIBLE_COUNTABLE;
IN_DIFF] THEN
SUBGOAL_THEN
`?f t. s =
IMAGE (f:num->real^1) t /\
(!m n. m
IN t /\ n
IN t /\ f m = f n ==> m = n)`
MP_TAC THENL
[ASM_CASES_TAC `
FINITE(s:real^1->bool)` THENL
[FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
FINITE_INDEX_NUMSEG]) THEN
ASM_MESON_TAC[];
MP_TAC(ISPEC `s:real^1->bool`
COUNTABLE_AS_INJECTIVE_IMAGE) THEN
ASM_REWRITE_TAC[
INFINITE] THEN MESON_TAC[
IN_UNIV]];
REWRITE_TAC[
LEFT_IMP_EXISTS_THM;
INJECTIVE_ON_LEFT_INVERSE] THEN
MAP_EVERY X_GEN_TAC [`r:num->real^1`; `t:num->bool`] THEN
DISCH_THEN(CONJUNCTS_THEN2 SUBST_ALL_TAC MP_TAC) THEN
DISCH_THEN(X_CHOOSE_TAC `n:real^1->num`)] THEN
REWRITE_TAC[
HAS_INTEGRAL_FACTOR_CONTENT] THEN
X_GEN_TAC `e:real` THEN DISCH_TAC THEN
SUBGOAL_THEN
`!x. ?d. &0 < d /\
(x
IN interval[a,b]
==> (x
IN IMAGE (r:num->real^1) t
==> !y. norm(y - x) < d /\ y
IN interval[a,b]
==> norm(f y - f x)
<= e / &2 pow (4 + n x) * norm(b - a)) /\
(~(x
IN IMAGE r t)
==> !y. norm(y - x) < d /\ y
IN interval[a,b]
==> norm(f y - f x - drop(y - x) % f' x:real^N)
<= e / &2 * norm(y - x)))`
MP_TAC THENL
[X_GEN_TAC `x:real^1` THEN
ASM_CASES_TAC `(x:real^1)
IN interval[a,b]` THENL
[ALL_TAC; EXISTS_TAC `&1` THEN ASM_REWRITE_TAC[
REAL_LT_01]] THEN
ASM_CASES_TAC `x
IN IMAGE (r:num->real^1) t` THEN ASM_REWRITE_TAC[] THENL
[FIRST_ASSUM(MP_TAC o MATCH_MP (REAL_ARITH
`a <= b ==> a = b \/ a < b`)) THEN
REWRITE_TAC[
DROP_EQ] THEN STRIP_TAC THENL
[EXISTS_TAC `&1` THEN REWRITE_TAC[
REAL_LT_01] THEN
UNDISCH_TAC `(x:real^1)
IN interval[a,b]` THEN
ASM_SIMP_TAC[
INTERVAL_SING;
IN_SING;
VECTOR_SUB_REFL;
NORM_0] THEN
REAL_ARITH_TAC;
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
continuous_on]) THEN
DISCH_THEN(MP_TAC o SPEC `x:real^1`) THEN ASM_REWRITE_TAC[dist] THEN
DISCH_THEN(MP_TAC o SPEC
`e / &2 pow (4 + n(x:real^1)) * norm(b - a:real^1)`) THEN
ASM_SIMP_TAC[
REAL_LT_DIV;
REAL_LT_MUL;
NORM_POS_LT;
VECTOR_SUB_EQ;
REAL_LT_POW2; GSYM
DROP_EQ;
REAL_LT_IMP_NE] THEN
MESON_TAC[
REAL_LT_IMP_LE]];
FIRST_X_ASSUM(MP_TAC o SPEC `x:real^1`) THEN
ASM_REWRITE_TAC[
IN_DIFF;
has_vector_derivative;
HAS_DERIVATIVE_WITHIN_ALT] THEN
DISCH_THEN(MP_TAC o SPEC `e / &2` o CONJUNCT2) THEN
ASM_REWRITE_TAC[
REAL_HALF] THEN MESON_TAC[]];
GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [
RIGHT_IMP_EXISTS_THM] THEN
REWRITE_TAC[
SKOLEM_THM;
LEFT_IMP_EXISTS_THM;
FORALL_AND_THM; IMP_IMP;
TAUT `p ==> q /\ r <=> (p ==> q) /\ (p ==> r)`] THEN
X_GEN_TAC `d:real^1->real` THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
DISCH_THEN(CONJUNCTS_THEN2 (LABEL_TAC "E") (LABEL_TAC "U"))] THEN
EXISTS_TAC `\x. ball(x:real^1,d(x))` THEN
ASM_SIMP_TAC[
GAUGE_BALL_DEPENDENT] THEN
X_GEN_TAC `p:(real^1#(real^1->bool))->bool` THEN STRIP_TAC THEN
MP_TAC(ISPECL [`f:real^1->real^N`; `p:(real^1#(real^1->bool))->bool`;
`a:real^1`; `b:real^1`]
ADDITIVE_TAGGED_DIVISION_1) THEN
ASM_SIMP_TAC[
CONTENT_1] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN
FIRST_ASSUM(ASSUME_TAC o MATCH_MP
TAGGED_DIVISION_OF_FINITE) THEN
ASM_SIMP_TAC[GSYM
VSUM_SUB;
LAMBDA_PAIR_THM] THEN
SUBGOAL_THEN
`p:(real^1#(real^1->bool))->bool =
{(x,k) | (x,k)
IN p /\ x
IN IMAGE r (t:num->bool)}
UNION
{(x,k) | (x,k)
IN p /\ ~(x
IN IMAGE r (t:num->bool))}`
SUBST1_TAC THENL
[REWRITE_TAC[
EXTENSION;
FORALL_PAIR_THM;
IN_ELIM_PAIR_THM;
IN_UNION] THEN
MESON_TAC[];
ALL_TAC] THEN
W(MP_TAC o PART_MATCH (lhs o rand)
VSUM_UNION o rand o lhand o snd) THEN
ANTS_TAC THENL
[REWRITE_TAC[SET_RULE `
DISJOINT s t <=> !x. x
IN s ==> ~(x
IN t)`] THEN
SIMP_TAC[
FORALL_IN_GSPEC;
IN_ELIM_PAIR_THM] THEN CONJ_TAC THEN
MATCH_MP_TAC
FINITE_SUBSET THEN
EXISTS_TAC `p:(real^1#(real^1->bool))->bool` THEN
ASM_SIMP_TAC[
SUBSET;
FORALL_IN_GSPEC;
IN_ELIM_PAIR_THM];
DISCH_THEN SUBST1_TAC] THEN
SUBGOAL_THEN
`(!P.
FINITE {(x:real^1,k:real^1->bool) | (x,k)
IN p /\ P x k}) /\
(!P x.
FINITE {(x:real^1,k:real^1->bool) |k| (x,k)
IN p /\ P x k})`
STRIP_ASSUME_TAC THENL
[REPEAT STRIP_TAC THEN MATCH_MP_TAC
FINITE_SUBSET THEN
EXISTS_TAC `p:real^1#(real^1->bool)->bool` THEN
ASM_SIMP_TAC[
SUBSET;
FORALL_IN_GSPEC];
ALL_TAC] THEN
MATCH_MP_TAC(NORM_ARITH
`norm(x:real^N) <= e / &2 * a /\ norm(y) <= e / &2 * a
==> norm(x + y) <= e * a`) THEN
CONJ_TAC THENL
[MATCH_MP_TAC
REAL_LE_TRANS THEN EXISTS_TAC
`norm(vsum {(x,k) | (x,k)
IN p /\ x
IN IMAGE (r:num->real^1) t /\
~(content k = &0)}
(\(x,k). --(f(
interval_upperbound k) -
(f:real^1->real^N)(
interval_lowerbound k))))` THEN
CONJ_TAC THENL
[MATCH_MP_TAC
REAL_EQ_IMP_LE THEN AP_TERM_TAC THEN
MATCH_MP_TAC
VSUM_EQ_SUPERSET THEN
ASM_REWRITE_TAC[
FORALL_IN_GSPEC;
IMP_CONJ] THEN
CONJ_TAC THENL [SET_TAC[]; ALL_TAC] THEN
SIMP_TAC[VECTOR_ARITH `a % vec 0 - x:real^N = --x`] THEN
REWRITE_TAC[
IN_ELIM_PAIR_THM] THEN
MAP_EVERY X_GEN_TAC [`x:real^1`; `k:real^1->bool`] THEN DISCH_TAC THEN
SUBGOAL_THEN `?u v:real^1. k = interval[u,v] /\ x
IN interval[u,v]`
STRIP_ASSUME_TAC THENL
[ASM_MESON_TAC[
TAGGED_DIVISION_OF]; ALL_TAC] THEN
ASM_REWRITE_TAC[
CONTENT_EQ_0_1] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
IN_INTERVAL_1]) THEN
DISCH_THEN(MP_TAC o MATCH_MP
REAL_LE_TRANS) THEN
SIMP_TAC[
INTERVAL_LOWERBOUND_1;
INTERVAL_UPPERBOUND_1;
INTERVAL_EQ_EMPTY;
REAL_NOT_LE;
REAL_NOT_LT] THEN
REPEAT STRIP_TAC THEN MATCH_MP_TAC(VECTOR_ARITH
`x:real^N = y ==> --(x - y) = vec 0`) THEN
AP_TERM_TAC THEN ASM_REWRITE_TAC[GSYM
DROP_EQ; GSYM REAL_LE_ANTISYM];
ALL_TAC] THEN
MATCH_MP_TAC
REAL_LE_TRANS THEN EXISTS_TAC
`sum {(x,k:real^1->bool) | (x,k)
IN p /\ x
IN IMAGE (r:num->real^1) t /\
~(content k = &0)}
((\(x,k). e / &2 pow (3 + n x) * norm (b - a:real^1)))` THEN
CONJ_TAC THENL
[MATCH_MP_TAC
VSUM_NORM_LE THEN
ASM_REWRITE_TAC[
FORALL_IN_GSPEC;
IMP_CONJ] THEN
MAP_EVERY X_GEN_TAC [`x:real^1`; `k:real^1->bool`] THEN DISCH_TAC THEN
SUBGOAL_THEN `?u v:real^1. k = interval[u,v] /\ x
IN interval[u,v]`
MP_TAC THENL [ASM_MESON_TAC[
TAGGED_DIVISION_OF]; ALL_TAC] THEN
DISCH_THEN(REPEAT_TCL CHOOSE_THEN
(CONJUNCTS_THEN2 SUBST_ALL_TAC MP_TAC)) THEN
SIMP_TAC[
CONTENT_EQ_0_1;
REAL_NOT_LE;
REAL_LT_IMP_LE;
IN_INTERVAL_1;
INTERVAL_LOWERBOUND_1;
INTERVAL_UPPERBOUND_1] THEN
REPEAT STRIP_TAC THEN
REMOVE_THEN "E" (MP_TAC o SPEC `x:real^1`) THEN ANTS_TAC THENL
[ASM_MESON_TAC[
TAGGED_DIVISION_OF;
SUBSET]; ALL_TAC] THEN
DISCH_THEN(fun th ->
MP_TAC(ISPEC `u:real^1` th) THEN MP_TAC(ISPEC `v:real^1` th)) THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [fine]) THEN
DISCH_THEN(MP_TAC o SPECL [`x:real^1`; `interval[u:real^1,v]`]) THEN
ASM_REWRITE_TAC[
SUBSET;
IN_BALL] THEN
DISCH_THEN(fun th ->
MP_TAC(ISPEC `u:real^1` th) THEN MP_TAC(ISPEC `v:real^1` th)) THEN
ASM_REWRITE_TAC[dist;
ENDS_IN_INTERVAL;
INTERVAL_NE_EMPTY_1] THEN
ASM_SIMP_TAC[
REAL_LT_IMP_LE;
NORM_SUB] THEN DISCH_TAC THEN DISCH_TAC THEN
SUBGOAL_THEN `interval[u:real^1,v]
SUBSET interval[a,b]` ASSUME_TAC THENL
[ASM_MESON_TAC[
TAGGED_DIVISION_OF]; ALL_TAC] THEN
REPEAT(ANTS_TAC THENL
[ASM_MESON_TAC[
ENDS_IN_INTERVAL;
SUBSET;
INTERVAL_NE_EMPTY_1;
REAL_LT_IMP_LE];
ONCE_REWRITE_TAC[TAUT `p ==> q ==> r <=> q ==> p ==> r`]]) THEN
REWRITE_TAC[
REAL_POW_ADD;
real_div;
REAL_INV_MUL] THEN NORM_ARITH_TAC;
ALL_TAC] THEN
MP_TAC(ISPECL
[`FST:real^1#(real^1->bool)->real^1`;
`\(x:real^1,k:real^1->bool). e / &2 pow (3 + n x) * norm (b - a:real^1)`;
`{(x,k:real^1->bool) | (x,k)
IN p /\ x
IN IMAGE (r:num->real^1) t /\
~(content k = &0)}`;
`
IMAGE (r:num->real^1) t`
]
SUM_GROUP) THEN
ANTS_TAC THENL
[ASM_REWRITE_TAC[] THEN
SIMP_TAC[
SUBSET;
FORALL_IN_IMAGE;
FORALL_IN_GSPEC];
DISCH_THEN(SUBST1_TAC o SYM)] THEN
MATCH_MP_TAC
REAL_LE_TRANS THEN EXISTS_TAC
`sum (
IMAGE (r:num->real^1) t)
(\x. sum {(x,k:real^1->bool) |k|
(x,k)
IN p /\ ~(content k = &0)}
(\yk. e / &2 pow (3 + n x) * norm(b - a:real^1)))` THEN
CONJ_TAC THENL
[MATCH_MP_TAC
REAL_EQ_IMP_LE THEN MATCH_MP_TAC
SUM_EQ THEN
X_GEN_TAC `x:real^1` THEN DISCH_TAC THEN REWRITE_TAC[] THEN
MATCH_MP_TAC
SUM_EQ_SUPERSET THEN
ASM_REWRITE_TAC[
SUBSET;
FORALL_IN_GSPEC;
IMP_CONJ] THEN
REWRITE_TAC[
IN_ELIM_THM;
PAIR_EQ] THEN ASM_MESON_TAC[];
ALL_TAC] THEN
ASM_SIMP_TAC[
SUM_CONST] THEN
REWRITE_TAC[
SUM_RMUL;
NORM_1;
DROP_SUB; REAL_MUL_ASSOC] THEN
ASM_REWRITE_TAC[
real_abs;
REAL_SUB_LE] THEN MATCH_MP_TAC
REAL_LE_RMUL THEN
ASM_REWRITE_TAC[
REAL_SUB_LE;
REAL_POW_ADD;
real_div;
REAL_INV_MUL] THEN
ONCE_REWRITE_TAC[REAL_ARITH
`p * e * inv(&2 pow 3) * n = e / &8 * (p * n)`] THEN
ASM_SIMP_TAC[
REAL_LE_LMUL_EQ;
SUM_LMUL; REAL_ARITH
`e / &8 * x <= e * inv(&2) <=> e * x <= e * &4`] THEN
MATCH_MP_TAC
REAL_LE_TRANS THEN EXISTS_TAC
`sum (
IMAGE (r:num->real^1) t
INTER
IMAGE (FST:real^1#(real^1->bool)->real^1) p)
(\x. &(
CARD {(x,k:real^1->bool) | k |
(x,k)
IN p /\ ~(content k = &0)}) *
inv(&2 pow n x))` THEN
CONJ_TAC THENL
[MATCH_MP_TAC
REAL_EQ_IMP_LE THEN MATCH_MP_TAC
SUM_SUPERSET THEN
REWRITE_TAC[
INTER_SUBSET;
IMP_CONJ;
FORALL_IN_IMAGE] THEN
SIMP_TAC[
IN_INTER;
FUN_IN_IMAGE] THEN
REWRITE_TAC[
IN_IMAGE;
EXISTS_PAIR_THM] THEN
REPEAT STRIP_TAC THEN REWRITE_TAC[
REAL_ENTIRE] THEN
DISJ1_TAC THEN AP_TERM_TAC THEN
MATCH_MP_TAC(MESON[
CARD_CLAUSES] `s = {} ==>
CARD s = 0`) THEN
ASM SET_TAC[];
ALL_TAC] THEN
MATCH_MP_TAC
REAL_LE_TRANS THEN EXISTS_TAC
`sum (
IMAGE (r:num->real^1) t
INTER
IMAGE (FST:real^1#(real^1->bool)->real^1) p)
(\x. &2 / &2 pow (n x))` THEN
CONJ_TAC THENL
[MATCH_MP_TAC
SUM_LE THEN
ASM_SIMP_TAC[
FINITE_IMAGE;
FINITE_INTER] THEN
GEN_TAC THEN DISCH_TAC THEN REWRITE_TAC[
real_div] THEN
MATCH_MP_TAC
REAL_LE_RMUL THEN
SIMP_TAC[
REAL_LE_INV_EQ;
REAL_POW_LE;
REAL_POS; REAL_OF_NUM_LE] THEN
GEN_REWRITE_TAC RAND_CONV [ARITH_RULE `2 = 2
EXP 1`] THEN
GEN_REWRITE_TAC (RAND_CONV o RAND_CONV) [GSYM DIMINDEX_1] THEN
MATCH_MP_TAC
TAGGED_PARTIAL_DIVISION_COMMON_TAGS THEN
ASM_MESON_TAC[
tagged_division_of];
ALL_TAC] THEN
REWRITE_TAC[
real_div;
SUM_LMUL; REAL_ARITH `&2 * x <= &4 <=> x <= &2`;
REAL_INV_POW] THEN
SUBGOAL_THEN
`(\x:real^1. inv (&2) pow n x) = (\n. inv(&2) pow n) o n`
SUBST1_TAC THENL [REWRITE_TAC[
o_DEF]; ALL_TAC] THEN
W(MP_TAC o PART_MATCH (rand o rand)
SUM_IMAGE o lhand o snd) THEN
ANTS_TAC THENL [ASM SET_TAC[]; DISCH_THEN(SUBST1_TAC o SYM)] THEN
SUBGOAL_THEN
`?m.
IMAGE (n:real^1->num)
(
IMAGE (r:num->real^1) t
INTER
IMAGE (FST:real^1#(real^1->bool)->real^1) p)
SUBSET 0..m`
STRIP_ASSUME_TAC THENL
[REWRITE_TAC[
SUBSET;
IN_NUMSEG;
LE_0] THEN
MATCH_MP_TAC
UPPER_BOUND_FINITE_SET THEN
ASM_SIMP_TAC[
FINITE_IMAGE;
FINITE_INTER];
ALL_TAC] THEN
MATCH_MP_TAC
REAL_LE_TRANS THEN
EXISTS_TAC `sum(0..m) (\n. inv(&2) pow n)` THEN CONJ_TAC THENL
[MATCH_MP_TAC
SUM_SUBSET THEN
ASM_SIMP_TAC[
FINITE_IMAGE;
FINITE_INTER;
FINITE_NUMSEG] THEN
SIMP_TAC[
REAL_LE_INV_EQ;
REAL_POW_LE;
REAL_POS] THEN ASM SET_TAC[];
REWRITE_TAC[
SUM_GP;
LT;
SUB_0] THEN
CONV_TAC REAL_RAT_REDUCE_CONV THEN
REWRITE_TAC[REAL_ARITH `(&1 - x) / (&1 / &2) <= &2 <=> &0 <= x`] THEN
MATCH_MP_TAC
REAL_POW_LE THEN CONV_TAC REAL_RAT_REDUCE_CONV];
MP_TAC(ISPECL [`\x:real^1. x`; `p:(real^1#(real^1->bool))->bool`;
`a:real^1`; `b:real^1`]
ADDITIVE_TAGGED_DIVISION_1) THEN
ASM_SIMP_TAC[] THEN DISCH_THEN(MP_TAC o AP_TERM `drop`) THEN
ASM_SIMP_TAC[
DROP_VSUM;
DROP_SUB] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN
REWRITE_TAC[GSYM
SUM_LMUL] THEN MATCH_MP_TAC
REAL_LE_TRANS THEN
EXISTS_TAC
`sum {(x:real^1,k:real^1->bool) |
(x,k)
IN p /\ ~(x
IN IMAGE r (t:num->bool))}
(\x. e / &2 * (drop o
(\(x,k).
interval_upperbound k -
interval_lowerbound k)) x)` THEN
CONJ_TAC THENL
[MATCH_MP_TAC
VSUM_NORM_LE THEN ASM_REWRITE_TAC[
FORALL_IN_GSPEC] THEN
SIMP_TAC[
o_DEF] THEN
REWRITE_TAC[NORM_ARITH `norm(a - (b - c):real^N) = norm(b - c - a)`] THEN
MAP_EVERY X_GEN_TAC [`x:real^1`; `k:real^1->bool`] THEN STRIP_TAC THEN
SUBGOAL_THEN `?u v:real^1. k = interval[u,v] /\ x
IN interval[u,v]`
MP_TAC THENL [ASM_MESON_TAC[
TAGGED_DIVISION_OF]; ALL_TAC] THEN
DISCH_THEN(REPEAT_TCL CHOOSE_THEN
(CONJUNCTS_THEN2 SUBST_ALL_TAC MP_TAC)) THEN
REWRITE_TAC[
IN_INTERVAL_1] THEN DISCH_THEN(fun th ->
ASSUME_TAC th THEN MP_TAC(MATCH_MP
REAL_LE_TRANS th)) THEN
SIMP_TAC[
CONTENT_1;
INTERVAL_LOWERBOUND_1;
INTERVAL_UPPERBOUND_1] THEN
DISCH_TAC THEN REMOVE_THEN "U" (MP_TAC o SPEC `x:real^1`) THEN
ASM_REWRITE_TAC[] THEN
SUBGOAL_THEN `interval[u:real^1,v]
SUBSET interval[a,b]` ASSUME_TAC THENL
[ASM_MESON_TAC[
TAGGED_DIVISION_OF]; ALL_TAC] THEN
ANTS_TAC THENL [ASM_MESON_TAC[
SUBSET;
IN_INTERVAL_1]; ALL_TAC] THEN
DISCH_THEN(fun th ->
MP_TAC(ISPEC `u:real^1` th) THEN MP_TAC(ISPEC `v:real^1` th)) THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [fine]) THEN
DISCH_THEN(MP_TAC o SPECL [`x:real^1`; `interval[u:real^1,v]`]) THEN
ASM_REWRITE_TAC[
SUBSET;
IN_BALL] THEN
DISCH_THEN(fun th ->
MP_TAC(ISPEC `u:real^1` th) THEN MP_TAC(ISPEC `v:real^1` th)) THEN
ASM_REWRITE_TAC[dist;
ENDS_IN_INTERVAL;
INTERVAL_NE_EMPTY_1] THEN
ASM_SIMP_TAC[
REAL_LT_IMP_LE;
NORM_SUB] THEN DISCH_TAC THEN DISCH_TAC THEN
REPEAT(ANTS_TAC THENL
[ASM_MESON_TAC[
ENDS_IN_INTERVAL;
SUBSET;
INTERVAL_NE_EMPTY_1;
REAL_LT_IMP_LE];
ONCE_REWRITE_TAC[TAUT `p ==> q ==> r <=> q ==> p ==> r`]]) THEN
REWRITE_TAC[
NORM_1;
DROP_SUB] THEN
ASM_SIMP_TAC[REAL_ARITH `a <= b ==> abs(a - b) = b - a`;
REAL_ARITH `b <= a ==> abs(a - b) = a - b`] THEN
REWRITE_TAC[
REAL_SUB_LDISTRIB] THEN MATCH_MP_TAC(NORM_ARITH
`x - y:real^N = z ==> norm(x) <= c - b
==> norm(y) <= b - a ==> norm(z) <= c - a`) THEN
VECTOR_ARITH_TAC;
MATCH_MP_TAC
SUM_SUBSET_SIMPLE THEN ASM_REWRITE_TAC[] THEN
CONJ_TAC THENL [ASM SET_TAC[]; REWRITE_TAC[
FORALL_PAIR_THM]] THEN
REWRITE_TAC[
IN_DIFF;
IN_ELIM_PAIR_THM] THEN
MAP_EVERY X_GEN_TAC [`x:real^1`; `k:real^1->bool`] THEN STRIP_TAC THEN
SUBGOAL_THEN `?u v:real^1. k = interval[u,v] /\ x
IN interval[u,v]`
MP_TAC THENL [ASM_MESON_TAC[
TAGGED_DIVISION_OF]; ALL_TAC] THEN
DISCH_THEN(REPEAT_TCL CHOOSE_THEN
(CONJUNCTS_THEN2 SUBST_ALL_TAC MP_TAC)) THEN
REWRITE_TAC[
IN_INTERVAL_1;
o_THM] THEN
DISCH_THEN(MP_TAC o MATCH_MP
REAL_LE_TRANS) THEN
SIMP_TAC[
INTERVAL_LOWERBOUND_1;
INTERVAL_UPPERBOUND_1] THEN
REPEAT STRIP_TAC THEN MATCH_MP_TAC
REAL_LE_MUL THEN
ASM_REWRITE_TAC[
DROP_SUB] THEN ASM_REAL_ARITH_TAC]]);;
let EQUIINTEGRABLE_ON_SPLIT = prove
(`!fs:(real^M->real^N)->bool k a b c.
fs
equiintegrable_on (interval[a,b]
INTER {x | x$k <= c}) /\
fs
equiintegrable_on (interval[a,b]
INTER {x | x$k >= c}) /\
1 <= k /\ k <= dimindex(:M)
==> fs
equiintegrable_on (interval[a,b])`,
let lemma1 = prove
(`(!x k. (x,k) IN {x,f k | P x k} ==> Q x k) <=>
(!x k. P x k ==> Q x (f k))`,
REWRITE_TAC[IN_ELIM_THM; PAIR_EQ] THEN
SET_TAC[]) in
let lemma2 = prove
(`!f:B->B s:(A#B)->bool.
FINITE s ==> FINITE {x,f k | (x,k) IN s /\ P x k}`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC FINITE_SUBSET THEN
EXISTS_TAC `IMAGE (\(x:A,k:B). x,(f k:B)) s` THEN
ASM_SIMP_TAC[FINITE_IMAGE] THEN
REWRITE_TAC[SUBSET; FORALL_PAIR_THM; lemma1; IN_IMAGE] THEN
REWRITE_TAC[EXISTS_PAIR_THM; PAIR_EQ] THEN MESON_TAC[]) in
let lemma3 = prove
(`!f:real^M->real^N g:(real^M->bool)->(real^M->bool) p.
FINITE p
==> vsum {x,g k |x,k| (x,k) IN p /\ ~(g k = {})}
(\(x,k). content k % f x) =
vsum (IMAGE (\(x,k). x,g k) p) (\(x,k). content k % f x)`,
REPEAT STRIP_TAC THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC VSUM_SUPERSET THEN
ASM_SIMP_TAC[FINITE_IMAGE; lemma2] THEN
REWRITE_TAC[IMP_CONJ; FORALL_IN_IMAGE] THEN
REWRITE_TAC[FORALL_PAIR_THM; SUBSET; IN_IMAGE; EXISTS_PAIR_THM] THEN
REWRITE_TAC[IN_ELIM_THM; PAIR_EQ; VECTOR_MUL_EQ_0] THEN
MESON_TAC[CONTENT_EMPTY]) in
let lemma4 = prove
(`(\(x,l). content (g l) % f x) =
(\(x,l). content l % f x) o (\(x,l). x,g l)`,
REWRITE_TAC[FUN_EQ_THM; o_THM; FORALL_PAIR_THM]) in
REPEAT GEN_TAC THEN
ASM_CASES_TAC `1 <= k /\ k <= dimindex(:M)` THEN ASM_REWRITE_TAC[] THEN
REWRITE_TAC[equiintegrable_on] THEN
MATCH_MP_TAC(TAUT
`(a /\ b ==> c) /\ (a /\ b /\ c ==> a' /\ b' ==> c')
==> (a /\ a') /\ (b /\ b') ==> c /\ c'`) THEN
CONJ_TAC THENL
[REWRITE_TAC[integrable_on] THEN ASM MESON_TAC[HAS_INTEGRAL_SPLIT];
STRIP_TAC] THEN
SUBGOAL_THEN
`!f:real^M->real^N.
f IN fs
==> integral (interval[a,b]) f =
integral (interval [a,b] INTER {x | x$k <= c}) f +
integral (interval [a,b] INTER {x | x$k >= c}) f`
(fun th -> SIMP_TAC[th])
THENL
[REPEAT STRIP_TAC THEN MATCH_MP_TAC INTEGRAL_UNIQUE THEN
MATCH_MP_TAC HAS_INTEGRAL_SPLIT THEN
MAP_EVERY EXISTS_TAC [`k:num`; `c:real`] THEN
ASM_SIMP_TAC[GSYM HAS_INTEGRAL_INTEGRAL];
ALL_TAC] THEN
DISCH_TAC THEN X_GEN_TAC `e:real` THEN STRIP_TAC THEN
FIRST_X_ASSUM(CONJUNCTS_THEN2 (MP_TAC o SPEC `e / &2`) STRIP_ASSUME_TAC) THEN
FIRST_X_ASSUM(MP_TAC o SPEC `e / &2`) THEN ASM_REWRITE_TAC[REAL_HALF] THEN
DISCH_THEN(X_CHOOSE_THEN `d2:real^M->real^M->bool`
(CONJUNCTS_THEN2 ASSUME_TAC (LABEL_TAC "I2"))) THEN
DISCH_THEN(X_CHOOSE_THEN `d1:real^M->real^M->bool`
(CONJUNCTS_THEN2 ASSUME_TAC (LABEL_TAC "I1"))) THEN
EXISTS_TAC `\x. if x$k = c then (d1(x:real^M) INTER d2(x)):real^M->bool
else ball(x,abs(x$k - c)) INTER d1(x) INTER d2(x)` THEN
CONJ_TAC THENL
[REWRITE_TAC[gauge] THEN GEN_TAC THEN
RULE_ASSUM_TAC(REWRITE_RULE[gauge]) THEN COND_CASES_TAC THEN
ASM_SIMP_TAC[OPEN_INTER; IN_INTER; OPEN_BALL; IN_BALL] THEN
ASM_REWRITE_TAC[DIST_REFL; GSYM REAL_ABS_NZ; REAL_SUB_0];
ALL_TAC] THEN
X_GEN_TAC `f:real^M->real^N` THEN
X_GEN_TAC `p:(real^M#(real^M->bool))->bool` THEN STRIP_TAC THEN
SUBGOAL_THEN
`(!x:real^M kk. (x,kk) IN p /\ ~(kk INTER {x:real^M | x$k <= c} = {})
==> x$k <= c) /\
(!x:real^M kk. (x,kk) IN p /\ ~(kk INTER {x:real^M | x$k >= c} = {})
==> x$k >= c)`
STRIP_ASSUME_TAC THENL
[CONJ_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [fine]) THEN
MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `x:real^M` THEN
MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `kk:real^M->bool` THEN
DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[REAL_LE_REFL; real_ge] THEN DISCH_THEN
(MP_TAC o MATCH_MP (SET_RULE `k SUBSET (a INTER b) ==> k SUBSET a`)) THEN
REWRITE_TAC[SUBSET; IN_BALL; dist] THEN DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
DISCH_THEN(X_CHOOSE_THEN `u:real^M` MP_TAC) THEN
REWRITE_TAC[IN_INTER; IN_ELIM_THM] THEN REPEAT STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `u:real^M`) THEN ASM_REWRITE_TAC[] THEN
ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN
REWRITE_TAC[REAL_NOT_LE; REAL_NOT_LT] THEN STRIP_TAC THEN
MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `abs((x - u:real^M)$k)` THEN
ASM_SIMP_TAC[COMPONENT_LE_NORM] THEN
ASM_SIMP_TAC[VECTOR_SUB_COMPONENT] THEN
ASM_REAL_ARITH_TAC;
ALL_TAC] THEN
REMOVE_THEN "I2" (MP_TAC o SPEC
`{(x:real^M,kk INTER {x:real^M | x$k >= c}) |x,kk|
(x,kk) IN p /\ ~(kk INTER {x:real^M | x$k >= c} = {})}` o
SPEC `f:real^M->real^N`) THEN
REMOVE_THEN "I1" (MP_TAC o SPEC
`{(x:real^M,kk INTER {x:real^M | x$k <= c}) |x,kk|
(x,kk) IN p /\ ~(kk INTER {x:real^M | x$k <= c} = {})}` o
SPEC `f:real^M->real^N`) THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(TAUT
`(a /\ b) /\ (a' /\ b' ==> c) ==> (a ==> a') ==> (b ==> b') ==> c`) THEN
CONJ_TAC THENL
[FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [TAGGED_DIVISION_OF]) THEN
REWRITE_TAC[TAGGED_DIVISION_OF] THEN
REPEAT(MATCH_MP_TAC(TAUT
`(a ==> (a' /\ a'')) /\ (b ==> (b' /\ d) /\ (b'' /\ e))
==> a /\ b ==> ((a' /\ b') /\ d) /\ ((a'' /\ b'') /\ e)`) THEN
CONJ_TAC) THEN
REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
REWRITE_TAC[lemma1] THEN REWRITE_TAC[IMP_IMP] THENL
[SIMP_TAC[lemma2];
REWRITE_TAC[AND_FORALL_THM] THEN
MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `x:real^M` THEN
MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `kk:real^M->bool` THEN
DISCH_THEN(fun th -> CONJ_TAC THEN STRIP_TAC THEN MP_TAC th) THEN
(ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_AND THEN CONJ_TAC THENL
[SIMP_TAC[IN_INTER; IN_ELIM_THM] THEN ASM_MESON_TAC[]; ALL_TAC]) THEN
(MATCH_MP_TAC MONO_AND THEN CONJ_TAC THENL [SET_TAC[]; ALL_TAC]) THEN
ASM_MESON_TAC[INTERVAL_SPLIT];
DISCH_THEN(fun th -> CONJ_TAC THEN MP_TAC th) THEN
(REPEAT(MATCH_MP_TAC MONO_FORALL THEN GEN_TAC) THEN
DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_SIMP_TAC[] THEN
REPEAT(MATCH_MP_TAC MONO_FORALL THEN GEN_TAC) THEN
DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_SIMP_TAC[] THEN
ANTS_TAC THENL [ASM_MESON_TAC[PAIR_EQ]; ALL_TAC] THEN
MATCH_MP_TAC(SET_RULE
`s SUBSET s' /\ t SUBSET t'
==> s' INTER t' = {} ==> s INTER t = {}`) THEN
CONJ_TAC THEN MATCH_MP_TAC SUBSET_INTERIOR THEN SET_TAC[]);
ALL_TAC] THEN
MATCH_MP_TAC(TAUT `(a ==> b /\ c) /\ d /\ e
==> (a ==> (b /\ d) /\ (c /\ e))`) THEN
CONJ_TAC THENL
[DISCH_THEN(fun th -> CONJ_TAC THEN MP_TAC th) THEN
DISCH_THEN(SUBST1_TAC o SYM) THEN REWRITE_TAC[INTER_UNIONS] THEN
ONCE_REWRITE_TAC[EXTENSION] THEN REWRITE_TAC[IN_UNIONS] THEN
X_GEN_TAC `x:real^M` THEN AP_TERM_TAC THEN
GEN_REWRITE_TAC I [FUN_EQ_THM] THEN X_GEN_TAC `kk:real^M->bool` THEN
REWRITE_TAC[IN_ELIM_THM; PAIR_EQ] THEN MESON_TAC[NOT_IN_EMPTY];
ALL_TAC] THEN
CONJ_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [fine]) THEN
REWRITE_TAC[fine; lemma1] THEN
REPEAT(MATCH_MP_TAC MONO_FORALL THEN GEN_TAC) THEN
DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN
ASM_SIMP_TAC[] THEN SET_TAC[];
ALL_TAC] THEN
DISCH_THEN(MP_TAC o MATCH_MP (REAL_ARITH
`x < e / &2 /\ y < e / &2 ==> x + y < e`)) THEN
DISCH_THEN(MP_TAC o MATCH_MP NORM_TRIANGLE_LT) THEN
MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN AP_TERM_TAC THEN
REWRITE_TAC[VECTOR_ARITH
`(a - i) + (b - j) = c - (i + j) <=> a + b = c:real^N`] THEN
FIRST_ASSUM(ASSUME_TAC o MATCH_MP TAGGED_DIVISION_OF_FINITE) THEN
MATCH_MP_TAC EQ_TRANS THEN
EXISTS_TAC
`vsum p (\(x,l). content (l INTER {x:real^M | x$k <= c}) %
(f:real^M->real^N) x) +
vsum p (\(x,l). content (l INTER {x:real^M | x$k >= c}) %
(f:real^M->real^N) x)` THEN
CONJ_TAC THENL
[ALL_TAC;
ASM_SIMP_TAC[GSYM VSUM_ADD] THEN MATCH_MP_TAC VSUM_EQ THEN
REWRITE_TAC[FORALL_PAIR_THM; GSYM VECTOR_ADD_RDISTRIB] THEN
MAP_EVERY X_GEN_TAC [`x:real^M`; `l:real^M->bool`] THEN
DISCH_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [TAGGED_DIVISION_OF]) THEN
DISCH_THEN(MP_TAC o SPECL [`x:real^M`; `l:real^M->bool`] o
el 1 o CONJUNCTS) THEN
ASM_REWRITE_TAC[] THEN STRIP_TAC THEN
ASM_SIMP_TAC[GSYM CONTENT_SPLIT]] THEN
ASM_SIMP_TAC[lemma3] THEN BINOP_TAC THEN
(GEN_REWRITE_TAC (RAND_CONV o RAND_CONV) [lemma4] THEN
MATCH_MP_TAC VSUM_IMAGE_NONZERO THEN ASM_REWRITE_TAC[FORALL_PAIR_THM] THEN
REWRITE_TAC[PAIR_EQ] THEN
ASM_MESON_TAC[TAGGED_DIVISION_SPLIT_LEFT_INJ; VECTOR_MUL_LZERO;
TAGGED_DIVISION_SPLIT_RIGHT_INJ]));;
let EQUIINTEGRABLE_LIMIT = prove
(`!f g:real^M->real^N a b.
{f n | n
IN (:num)}
equiintegrable_on interval[a,b] /\
(!x. x
IN interval[a,b] ==> ((\n. f n x) --> g x) sequentially)
==> g
integrable_on interval[a,b] /\
((\n. integral(interval[a,b]) (f n)) --> integral(interval[a,b]) g)
sequentially`,
REPEAT GEN_TAC THEN STRIP_TAC THEN
ASM_CASES_TAC `content(interval[a:real^M,b]) = &0` THEN
ASM_SIMP_TAC[
INTEGRABLE_ON_NULL;
INTEGRAL_NULL;
LIM_CONST] THEN
SUBGOAL_THEN `cauchy (\n. integral(interval[a,b]) (f n :real^M->real^N))`
MP_TAC THENL
[REWRITE_TAC[cauchy] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
equiintegrable_on]) THEN
REWRITE_TAC[
IMP_CONJ;
RIGHT_FORALL_IMP_THM;
FORALL_IN_GSPEC;
IN_UNIV] THEN
DISCH_TAC THEN REWRITE_TAC[IMP_IMP; GSYM
CONJ_ASSOC] 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 `d:real^M->real^M->bool` STRIP_ASSUME_TAC) THEN
FIRST_ASSUM(MP_TAC o MATCH_MP
FINE_DIVISION_EXISTS) THEN
DISCH_THEN(MP_TAC o SPECL [`a:real^M`; `b:real^M`]) THEN
DISCH_THEN(X_CHOOSE_THEN `p:(real^M#(real^M->bool))->bool`
STRIP_ASSUME_TAC) THEN
FIRST_ASSUM(ASSUME_TAC o MATCH_MP
TAGGED_DIVISION_OF_FINITE) THEN
FIRST_X_ASSUM(MP_TAC o GEN `n:num` o SPECL
[`n:num`; `p:(real^M#(real^M->bool))->bool`]) THEN
ASM_REWRITE_TAC[] THEN DISCH_TAC THEN SUBGOAL_THEN
`cauchy (\n. vsum p (\(x,k:real^M->bool).
content k % (f:num->real^M->real^N) n x))`
MP_TAC THENL
[MATCH_MP_TAC
CONVERGENT_IMP_CAUCHY THEN
EXISTS_TAC `vsum p (\(x,k:real^M->bool).
content k % (g:real^M->real^N) x)` THEN
MATCH_MP_TAC
(REWRITE_RULE[
LAMBDA_PAIR_THM]
(REWRITE_RULE[
FORALL_PAIR_THM]
(ISPEC `\(x:real^M,k:real^M->bool) (n:num).
content k % (f n x:real^N)`
LIM_VSUM))) THEN
ASM_REWRITE_TAC[] THEN
MAP_EVERY X_GEN_TAC [`x:real^M`; `k:real^M->bool`] THEN DISCH_TAC THEN
MATCH_MP_TAC
LIM_CMUL THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
TAGGED_DIVISION_OF]) THEN
ASM_SIMP_TAC[
SUBSET] THEN ASM_MESON_TAC[];
REWRITE_TAC[cauchy] THEN DISCH_THEN(MP_TAC o SPEC `e / &3`) THEN
ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
REWRITE_TAC[IMP_IMP;
RIGHT_IMP_FORALL_THM;
GE] THEN
MATCH_MP_TAC
MONO_EXISTS THEN X_GEN_TAC `N:num` THEN
MATCH_MP_TAC
MONO_FORALL THEN X_GEN_TAC `m:num` THEN
MATCH_MP_TAC
MONO_FORALL THEN X_GEN_TAC `n:num` THEN
ASM_CASES_TAC `N:num <= m /\ N <= n` THEN ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC(NORM_ARITH
`norm(sm - gm:real^N) < e / &3 /\ norm(sn - gn) < e / &3
==> dist(sm,sn) < e / &3 ==> dist(gm,gn) < e`) THEN
ASM_REWRITE_TAC[]];
REWRITE_TAC[GSYM
CONVERGENT_EQ_CAUCHY] THEN
DISCH_THEN(X_CHOOSE_TAC `l:real^N`) THEN
SUBGOAL_THEN `((g:real^M->real^N)
has_integral l) (interval[a,b])`
(fun th -> ASM_MESON_TAC[th;
integrable_on;
INTEGRAL_UNIQUE]) THEN
REWRITE_TAC[
has_integral] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
equiintegrable_on]) THEN
REWRITE_TAC[
IMP_CONJ;
RIGHT_FORALL_IMP_THM;
FORALL_IN_GSPEC;
IN_UNIV] THEN
DISCH_TAC THEN REWRITE_TAC[IMP_IMP; GSYM
CONJ_ASSOC] 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 `d:real^M->real^M->bool` THEN
STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
X_GEN_TAC `p:(real^M#(real^M->bool))->bool` THEN STRIP_TAC THEN
MATCH_MP_TAC(REAL_ARITH `&0 < e /\ x <= e / &2 ==> x < e`) THEN
ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC(ISPEC `sequentially`
LIM_NORM_UBOUND) THEN
EXISTS_TAC `\n:num. vsum p (\(x,k:real^M->bool). content k % f n x) -
integral (interval [a,b]) (f n :real^M->real^N)` THEN
ASM_SIMP_TAC[
TRIVIAL_LIMIT_SEQUENTIALLY;
REAL_LT_IMP_LE] THEN
REWRITE_TAC[
EVENTUALLY_TRUE] THEN
MATCH_MP_TAC
LIM_SUB THEN ASM_REWRITE_TAC[] THEN
FIRST_ASSUM(ASSUME_TAC o MATCH_MP
TAGGED_DIVISION_OF_FINITE) THEN
MATCH_MP_TAC
(REWRITE_RULE[
LAMBDA_PAIR_THM]
(REWRITE_RULE[
FORALL_PAIR_THM]
(ISPEC `\(x:real^M,k:real^M->bool) (n:num).
content k % (f n x:real^N)`
LIM_VSUM))) THEN
ASM_REWRITE_TAC[] THEN
MAP_EVERY X_GEN_TAC [`x:real^M`; `k:real^M->bool`] THEN DISCH_TAC THEN
MATCH_MP_TAC
LIM_CMUL THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
TAGGED_DIVISION_OF]) THEN
ASM_SIMP_TAC[
SUBSET] THEN ASM_MESON_TAC[]]);;
let EQUIINTEGRABLE_UNION = prove
(`!fs:(real^M->real^N)->bool gs s.
fs
equiintegrable_on s /\ gs
equiintegrable_on s
==> (fs
UNION gs)
equiintegrable_on s`,
REPEAT GEN_TAC THEN REWRITE_TAC[
equiintegrable_on;
IN_UNION] THEN
REWRITE_TAC[TAUT `a \/ b ==> c <=> (a ==> c) /\ (b ==> c)`] THEN
REWRITE_TAC[
FORALL_AND_THM] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
X_GEN_TAC `e:real` THEN DISCH_TAC THEN
REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `e:real`)) THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_THEN `d1:real^M->real^M->bool` STRIP_ASSUME_TAC) THEN
DISCH_THEN(X_CHOOSE_THEN `d2:real^M->real^M->bool` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `\x. (d1:real^M->real^M->bool) x
INTER d2 x` THEN
ASM_SIMP_TAC[
GAUGE_INTER;
FINE_INTER] THEN
REPEAT STRIP_TAC THEN ASM_SIMP_TAC[]);;
let EQUIINTEGRABLE_EQ = prove
(`!fs gs:(real^M->real^N)->bool s.
fs
equiintegrable_on s /\
(!g. g
IN gs ==> ?f. f
IN fs /\ (!x. x
IN s ==> f x = g x))
==> gs
equiintegrable_on s`,
REPEAT GEN_TAC THEN REWRITE_TAC[
equiintegrable_on] THEN
DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC (LABEL_TAC "*")) THEN
CONJ_TAC THENL
[X_GEN_TAC `g:real^M->real^N` THEN DISCH_TAC THEN
REMOVE_THEN "*" (MP_TAC o SPEC `g:real^M->real^N`) THEN
ASM_REWRITE_TAC[
LEFT_IMP_EXISTS_THM] THEN
X_GEN_TAC `f:real^M->real^N` THEN STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `f:real^M->real^N`) THEN
ASM_MESON_TAC[
INTEGRABLE_SPIKE;
IN_DIFF;
NEGLIGIBLE_EMPTY];
X_GEN_TAC `e:real` THEN DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `e:real`) THEN ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC
MONO_EXISTS THEN X_GEN_TAC `d:real^M->real^M->bool` THEN
STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
MAP_EVERY X_GEN_TAC
[`g:real^M->real^N`;`p:(real^M#(real^M->bool))->bool`] THEN
STRIP_TAC THEN REMOVE_THEN "*" (MP_TAC o SPEC `g:real^M->real^N`) THEN
ASM_REWRITE_TAC[
LEFT_IMP_EXISTS_THM] THEN
X_GEN_TAC `f:real^M->real^N` THEN STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPECL
[`f:real^M->real^N`;`p:(real^M#(real^M->bool))->bool`]) THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(MESON[]
`x:real^N = y /\ a = b ==> norm(x - a) < e ==> norm(y - b) < e`) THEN
CONJ_TAC THENL
[MATCH_MP_TAC
VSUM_EQ THEN REWRITE_TAC[
FORALL_PAIR_THM] THEN
RULE_ASSUM_TAC(REWRITE_RULE[
TAGGED_DIVISION_OF;
SUBSET]) THEN
ASM_MESON_TAC[];
ASM_MESON_TAC[
INTEGRAL_EQ]]]);;
let EQUIINTEGRABLE_ADD = prove
(`!fs:(real^M->real^N)->bool gs s.
fs
equiintegrable_on s /\ gs
equiintegrable_on s
==> {(\x. f x + g x) | f
IN fs /\ g
IN gs}
equiintegrable_on s`,
REPEAT GEN_TAC THEN
SIMP_TAC[
equiintegrable_on;
INTEGRABLE_ADD;
FORALL_IN_GSPEC] THEN
DISCH_THEN(CONJUNCTS_THEN2
(CONJUNCTS_THEN2 ASSUME_TAC (LABEL_TAC "f"))
(CONJUNCTS_THEN2 ASSUME_TAC (LABEL_TAC "g"))) THEN
X_GEN_TAC `e:real` THEN DISCH_TAC THEN
REWRITE_TAC[
IMP_CONJ;
RIGHT_FORALL_IMP_THM;
FORALL_IN_GSPEC] THEN
ASM_SIMP_TAC[
RIGHT_IMP_FORALL_THM;
INTEGRAL_ADD; IMP_IMP] THEN
REMOVE_THEN "g" (MP_TAC o SPEC `e / &2`) THEN
REMOVE_THEN "f" (MP_TAC o SPEC `e / &2`) THEN
ASM_REWRITE_TAC[
REAL_HALF] THEN
DISCH_THEN(X_CHOOSE_THEN `d1:real^M->real^M->bool`
(CONJUNCTS_THEN2 ASSUME_TAC (LABEL_TAC "f"))) THEN
DISCH_THEN(X_CHOOSE_THEN `d2:real^M->real^M->bool`
(CONJUNCTS_THEN2 ASSUME_TAC (LABEL_TAC "g"))) THEN
EXISTS_TAC `\x. (d1:real^M->real^M->bool) x
INTER d2 x` THEN
ASM_SIMP_TAC[
GAUGE_INTER;
FINE_INTER] THEN
MAP_EVERY X_GEN_TAC [`f:real^M->real^N`; `g:real^M->real^N`;
`p:(real^M#(real^M->bool))->bool`] THEN STRIP_TAC THEN
REMOVE_THEN "g" (MP_TAC o SPECL
[`g:real^M->real^N`; `p:(real^M#(real^M->bool))->bool`]) THEN
REMOVE_THEN "f" (MP_TAC o SPECL
[`f:real^M->real^N`; `p:(real^M#(real^M->bool))->bool`]) THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(NORM_ARITH
`s + s' = t
==> norm(s - i) < e / &2 ==> norm(s' - i') < e / &2
==> norm(t - (i + i')) < e`) THEN
FIRST_ASSUM(ASSUME_TAC o MATCH_MP
TAGGED_DIVISION_OF_FINITE) THEN
ASM_SIMP_TAC[GSYM
VSUM_ADD] THEN
REWRITE_TAC[
LAMBDA_PAIR_THM;
VECTOR_ADD_LDISTRIB]);;
let EQUIINTEGRABLE_UNIFORM_LIMIT = prove
(`!fs:(real^M->real^N)->bool a b.
fs
equiintegrable_on interval[a,b]
==> {g | !e. &0 < e
==> ?f. f
IN fs /\
!x. x
IN interval[a,b] ==> norm(g x - f x) < e}
equiintegrable_on interval[a,b]`,
REPEAT STRIP_TAC THEN
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
equiintegrable_on]) THEN
REWRITE_TAC[
equiintegrable_on;
IN_ELIM_THM] THEN REPEAT GEN_TAC THEN
STRIP_TAC THEN CONJ_TAC THENL
[ASM_MESON_TAC[
INTEGRABLE_UNIFORM_LIMIT;
REAL_LT_IMP_LE]; ALL_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 `d:real^M->real^M->bool` THEN
STRIP_TAC THEN ASM_REWRITE_TAC[] THEN MAP_EVERY X_GEN_TAC
[`g:real^M->real^N`;`p:(real^M#(real^M->bool))->bool`] THEN
STRIP_TAC THEN
FIRST_ASSUM(ASSUME_TAC o MATCH_MP
TAGGED_DIVISION_OF_FINITE) THEN
SUBGOAL_THEN `(g:real^M->real^N)
integrable_on interval[a,b]`
ASSUME_TAC THENL
[ASM_MESON_TAC[
INTEGRABLE_UNIFORM_LIMIT;
REAL_LT_IMP_LE]; ALL_TAC] THEN
FIRST_X_ASSUM(MP_TAC o GEN `n:num` o SPEC `inv(&n + &1)`) THEN
REWRITE_TAC[
REAL_LT_INV_EQ; REAL_ARITH `&0 < &n + &1`] THEN
REWRITE_TAC[
SKOLEM_THM;
FORALL_AND_THM;
LEFT_IMP_EXISTS_THM] THEN
X_GEN_TAC `f:num->real^M->real^N` THEN STRIP_TAC THEN
SUBGOAL_THEN
`!x. x
IN interval[a,b]
==> ((\n. f n x) --> (g:real^M->real^N) x) sequentially`
ASSUME_TAC THENL
[X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN
REWRITE_TAC[
LIM_SEQUENTIALLY] THEN X_GEN_TAC `k:real` THEN DISCH_TAC THEN
MP_TAC(SPEC `k:real`
REAL_ARCH_INV) THEN ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC
MONO_EXISTS THEN X_GEN_TAC `N:num` THEN
STRIP_TAC THEN X_GEN_TAC `n:num` THEN DISCH_TAC THEN
REWRITE_TAC[NORM_ARITH `dist(a:real^N,b) = norm(b - a)`] THEN
MATCH_MP_TAC
REAL_LT_TRANS THEN EXISTS_TAC `inv(&n + &1)` THEN
ASM_SIMP_TAC[] THEN
MATCH_MP_TAC
REAL_LET_TRANS THEN EXISTS_TAC `inv(&N)` THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC
REAL_LE_INV2 THEN
REWRITE_TAC[REAL_OF_NUM_ADD; REAL_OF_NUM_LE;
REAL_OF_NUM_LT] THEN
ASM_ARITH_TAC;
ALL_TAC] THEN
MP_TAC(ISPECL [`f:num->real^M->real^N`; `g:real^M->real^N`;
`a:real^M`; `b:real^M`]
EQUIINTEGRABLE_LIMIT) THEN
ANTS_TAC THENL
[ASM_REWRITE_TAC[] THEN MATCH_MP_TAC
EQUIINTEGRABLE_SUBSET THEN
EXISTS_TAC `fs:(real^M->real^N)->bool` THEN ASM SET_TAC[];
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (LABEL_TAC "*"))] THEN
SUBGOAL_THEN
`((\n. vsum p (\(x,k:real^M->bool).
content k % (f:num->real^M->real^N) n x)) -->
vsum p (\(x,k). content k % g x)) sequentially`
(LABEL_TAC "+")
THENL
[MATCH_MP_TAC
(REWRITE_RULE[
LAMBDA_PAIR_THM]
(REWRITE_RULE[
FORALL_PAIR_THM]
(ISPEC `\(x:real^M,k:real^M->bool) (n:num).
content k % (f n x:real^N)`
LIM_VSUM))) THEN
ASM_REWRITE_TAC[] THEN
MAP_EVERY X_GEN_TAC [`x:real^M`; `k:real^M->bool`] THEN DISCH_TAC THEN
MATCH_MP_TAC
LIM_CMUL THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
TAGGED_DIVISION_OF]) THEN
ASM_SIMP_TAC[
SUBSET] THEN ASM_MESON_TAC[];
ALL_TAC] THEN
REMOVE_THEN "*" (MP_TAC o REWRITE_RULE[
LIM_SEQUENTIALLY]) THEN
DISCH_THEN(MP_TAC o SPEC `e / &4`) THEN
ANTS_TAC THENL [ASM_REAL_ARITH_TAC; REWRITE_TAC[dist]] THEN
DISCH_THEN(X_CHOOSE_THEN `N1:num` (LABEL_TAC "*")) THEN
REMOVE_THEN "+" (MP_TAC o REWRITE_RULE[
LIM_SEQUENTIALLY]) THEN
DISCH_THEN(MP_TAC o SPEC `e / &4`) THEN
ANTS_TAC THENL [ASM_REAL_ARITH_TAC; REWRITE_TAC[dist]] THEN
DISCH_THEN(X_CHOOSE_THEN `N2:num` (LABEL_TAC "+")) THEN
SUBGOAL_THEN `?n:num. N1 <= n /\ N2 <= n` STRIP_ASSUME_TAC THENL
[EXISTS_TAC `N1 + N2:num` THEN ARITH_TAC; ALL_TAC] THEN
REMOVE_THEN "*" (MP_TAC o SPEC `n:num`) THEN
REMOVE_THEN "+" (MP_TAC o SPEC `n:num`) THEN
FIRST_X_ASSUM(MP_TAC o SPECL
[`(f:num->real^M->real^N) n`;`p:(real^M#(real^M->bool))->bool`]) THEN
ASM_REWRITE_TAC[] THEN NORM_ARITH_TAC);;
let EQUIINTEGRABLE_REFLECT = prove
(`!fs:(real^M->real^N)->bool a b.
fs
equiintegrable_on interval[a,b]
==> {(\x. f(--x)) | f
IN fs}
equiintegrable_on interval[--b,--a]`,
let lemma = prove
(`(!x k. (x,k) IN IMAGE (\(x,k). f x k,g x k) s ==> Q x k) <=>
(!x k. (x,k) IN s ==> Q (f x k) (g x k))`,
REWRITE_TAC[IN_IMAGE; PAIR_EQ; EXISTS_PAIR_THM] THEN SET_TAC[]) in
REPEAT GEN_TAC THEN REWRITE_TAC[equiintegrable_on] THEN
REWRITE_TAC[RIGHT_FORALL_IMP_THM; IMP_CONJ; FORALL_IN_GSPEC] THEN
DISCH_TAC THEN DISCH_TAC THEN
ASM_REWRITE_TAC[INTEGRABLE_REFLECT; INTEGRAL_REFLECT] THEN
X_GEN_TAC `e:real` THEN DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `e:real`) THEN ASM_REWRITE_TAC[] THEN
FIRST_X_ASSUM(K ALL_TAC o check (is_forall o concl)) THEN
DISCH_THEN(X_CHOOSE_THEN `d:real^M->real^M->bool` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `\x. IMAGE (--) ((d:real^M->real^M->bool) (--x))` THEN
CONJ_TAC THENL
[FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [gauge]) THEN
SIMP_TAC[gauge; OPEN_NEGATIONS] THEN DISCH_TAC THEN
GEN_TAC THEN GEN_REWRITE_TAC LAND_CONV [GSYM VECTOR_NEG_NEG] THEN
MATCH_MP_TAC FUN_IN_IMAGE THEN ASM_REWRITE_TAC[];
ALL_TAC] THEN
X_GEN_TAC `f:real^M->real^N` THEN DISCH_TAC THEN
X_GEN_TAC `p:real^M#(real^M->bool)->bool` THEN REPEAT DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `f:real^M->real^N`) THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN(MP_TAC o SPEC
`IMAGE (\(x,k). (--x:real^M,IMAGE (--) (k:real^M->bool))) p`) THEN
ANTS_TAC THENL
[FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [TAGGED_DIVISION_OF]) THEN
REWRITE_TAC[TAGGED_DIVISION_OF] THEN
STRIP_TAC THEN ASM_SIMP_TAC[FINITE_IMAGE] THEN
REWRITE_TAC[RIGHT_FORALL_IMP_THM; IMP_CONJ; lemma] THEN
REPEAT CONJ_TAC THENL
[MAP_EVERY X_GEN_TAC [`x:real^M`; `k:real^M->bool`] THEN DISCH_TAC THEN
ASM_SIMP_TAC[FUN_IN_IMAGE] THEN CONJ_TAC THENL
[REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN
ONCE_REWRITE_TAC[GSYM IN_INTERVAL_REFLECT] THEN
ASM_SIMP_TAC[VECTOR_NEG_NEG; GSYM SUBSET] THEN ASM_MESON_TAC[];
REWRITE_TAC[EXTENSION; IN_IMAGE] THEN
REWRITE_TAC[VECTOR_ARITH `x:real^N = --y <=> --x = y`] THEN
ONCE_REWRITE_TAC[GSYM IN_INTERVAL_REFLECT] THEN
REWRITE_TAC[UNWIND_THM1] THEN
SUBGOAL_THEN `?u v:real^M. k = interval[u,v]`
(REPEAT_TCL CHOOSE_THEN SUBST_ALL_TAC)
THENL [ASM_MESON_TAC[TAGGED_DIVISION_OF]; ALL_TAC] THEN
ASM_MESON_TAC[VECTOR_NEG_NEG]];
MAP_EVERY X_GEN_TAC [`x:real^M`; `k:real^M->bool`] THEN DISCH_TAC THEN
MAP_EVERY X_GEN_TAC [`y:real^M`; `l:real^M->bool`] THEN DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPECL
[`x:real^M`; `k:real^M->bool`;
`y:real^M`; `l:real^M->bool`]) THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_IMP THEN
CONJ_TAC THENL [MESON_TAC[PAIR_EQ]; ALL_TAC] THEN
REWRITE_TAC[INTERIOR_NEGATIONS] THEN
MATCH_MP_TAC(SET_RULE
`(!x. f(f x) = x)
==> s INTER t = {} ==> IMAGE f s INTER IMAGE f t = {}`) THEN
REWRITE_TAC[VECTOR_NEG_NEG];
GEN_REWRITE_TAC I [EXTENSION] THEN
ONCE_REWRITE_TAC[GSYM IN_INTERVAL_REFLECT] THEN
FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN X_GEN_TAC `y:real^M` THEN
REWRITE_TAC[IN_UNIONS; IN_ELIM_THM] THEN
MATCH_MP_TAC(MESON[]
`!f. (!x. f(f x) = x) /\ (!x. P x <=> Q(f x))
==> ((?x. P x) <=> (?x. Q x))`) THEN
EXISTS_TAC `IMAGE ((--):real^M->real^M)` THEN CONJ_TAC THENL
[REWRITE_TAC[GSYM IMAGE_o; o_DEF; VECTOR_NEG_NEG; IMAGE_ID];
ALL_TAC] THEN
X_GEN_TAC `t:real^M->bool` THEN BINOP_TAC THENL
[REWRITE_TAC[IN_IMAGE; EXISTS_PAIR_THM; PAIR_EQ] THEN
SUBGOAL_THEN `!k:real^M->bool. IMAGE (--) (IMAGE (--) k) = k`
MP_TAC THENL
[REWRITE_TAC[GSYM IMAGE_o; o_DEF; VECTOR_NEG_NEG; IMAGE_ID];
MESON_TAC[]];
MATCH_MP_TAC(SET_RULE
`(!x. f(f x) = x) ==> (y IN s <=> f y IN IMAGE f s)`) THEN
REWRITE_TAC[VECTOR_NEG_NEG]]];
ANTS_TAC THENL
[FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [fine]) THEN
REWRITE_TAC[fine; lemma] THEN
REPEAT(MATCH_MP_TAC MONO_FORALL THEN GEN_TAC) THEN
MATCH_MP_TAC MONO_IMP THEN REWRITE_TAC[] THEN
MATCH_MP_TAC(SET_RULE
`(!x. f(f x) = x) ==> k SUBSET IMAGE f s ==> IMAGE f k SUBSET s`) THEN
REWRITE_TAC[VECTOR_NEG_NEG];
ALL_TAC] THEN
MATCH_MP_TAC(NORM_ARITH
`x:real^N = y ==> norm(x - i) < e ==> norm(y - i) < e`) THEN
W(MP_TAC o PART_MATCH (lhs o rand) VSUM_IMAGE o lhs o snd) THEN
FIRST_ASSUM(ASSUME_TAC o MATCH_MP TAGGED_DIVISION_OF_FINITE) THEN
ASM_REWRITE_TAC[] THEN ANTS_TAC THENL
[MATCH_MP_TAC(MESON[]
`(!x. f(f x) = x)
==> !x y. x IN p /\ y IN p /\ f x = f y ==> x = y`) THEN
REWRITE_TAC[FORALL_PAIR_THM; GSYM IMAGE_o; o_DEF; VECTOR_NEG_NEG;
IMAGE_ID];
DISCH_THEN SUBST1_TAC THEN MATCH_MP_TAC VSUM_EQ THEN
REWRITE_TAC[FORALL_PAIR_THM; o_THM] THEN
MAP_EVERY X_GEN_TAC [`x:real^M`; `k:real^M->bool`] THEN DISCH_TAC THEN
SUBGOAL_THEN `?u v:real^M. k = interval[u,v]`
(REPEAT_TCL CHOOSE_THEN SUBST_ALL_TAC)
THENL [ASM_MESON_TAC[TAGGED_DIVISION_OF]; ALL_TAC] THEN
AP_THM_TAC THEN AP_TERM_TAC THEN
SUBGOAL_THEN `(--):real^M->real^M = (\x. --(&1) % x + vec 0)` SUBST1_TAC
THENL [REWRITE_TAC[FUN_EQ_THM] THEN VECTOR_ARITH_TAC; ALL_TAC] THEN
REWRITE_TAC[CONTENT_IMAGE_AFFINITY_INTERVAL; REAL_ABS_NEG] THEN
REWRITE_TAC[REAL_POW_ONE; REAL_MUL_LID; REAL_ABS_NUM]]]);;
let SUM_CONTENT_AREA_OVER_THIN_DIVISION = prove
(`!d a b:real^M s i c.
d
division_of s /\ s
SUBSET interval[a,b] /\
1 <= i /\ i <= dimindex(:M) /\ a$i <= c /\ c <= b$i /\
(!k. k
IN d ==> ~(k
INTER {x | x$i = c} = {}))
==> (b$i - a$i) *
sum d (\k. content k /
(
interval_upperbound k$i -
interval_lowerbound k$i))
<= &2 * content(interval[a,b])`,
let lemma0 = prove
(`!k:real^M->bool i.
1 <= i /\ i <= dimindex(:M)
==> content k / (interval_upperbound k$i - interval_lowerbound k$i) =
if content k = &0 then &0
else product ((1..dimindex(:M)) DELETE i)
(\j. interval_upperbound k$j -
interval_lowerbound k$j)`,
REPEAT STRIP_TAC THEN COND_CASES_TAC THEN
ASM_REWRITE_TAC[real_div; REAL_MUL_LZERO] THEN
REWRITE_TAC[content] THEN
COND_CASES_TAC THENL [ASM_MESON_TAC[CONTENT_EMPTY]; ALL_TAC] THEN
SUBGOAL_THEN
`1..dimindex(:M) = i INSERT ((1..dimindex(:M)) DELETE i)`
MP_TAC THENL
[REWRITE_TAC[SET_RULE `s = x INSERT (s DELETE x) <=> x IN s`] THEN
ASM_REWRITE_TAC[IN_NUMSEG];
DISCH_THEN(fun th -> GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV)
[th])] THEN
ASM_SIMP_TAC[PRODUCT_CLAUSES; IN_NUMSEG; FINITE_DELETE; FINITE_NUMSEG;
IN_DELETE] THEN
MATCH_MP_TAC(REAL_FIELD `~(y = &0) ==> (y * x) * inv y = x`) THEN
DISCH_TAC THEN
UNDISCH_TAC `~(content(k:real^M->bool) = &0)` THEN
ASM_REWRITE_TAC[content; PRODUCT_EQ_0_NUMSEG] THEN ASM_MESON_TAC[])
and lemma1 = prove
(`!d a b:real^M s i.
d division_of s /\ s SUBSET interval[a,b] /\
1 <= i /\ i <= dimindex(:M) /\
((!k. k IN d
==> ~(content k = &0) /\ ~(k INTER {x | x$i = a$i} = {})) \/
(!k. k IN d
==> ~(content k = &0) /\ ~(k INTER {x | x$i = b$i} = {})))
==> (b$i - a$i) *
sum d (\k. content k /
(interval_upperbound k$i - interval_lowerbound k$i))
<= content(interval[a,b])`,
REPEAT GEN_TAC THEN DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC) THEN
FIRST_ASSUM(ASSUME_TAC o MATCH_MP DIVISION_OF_FINITE) THEN
ABBREV_TAC
`extend =
\k:real^M->bool. interval
[(lambda j. if j = i then (a:real^M)$i
else interval_lowerbound k$j):real^M,
(lambda j. if j = i then (b:real^M)$i
else interval_upperbound k$j)]` THEN
SUBGOAL_THEN `!k. k IN d ==> k SUBSET interval[a:real^M,b]`
ASSUME_TAC THENL
[RULE_ASSUM_TAC(REWRITE_RULE[division_of]) THEN ASM SET_TAC[];
ALL_TAC] THEN
SUBGOAL_THEN `!k:real^M->bool. k IN d ==> ~(k = {})` ASSUME_TAC THENL
[ASM_MESON_TAC[division_of]; ALL_TAC] THEN
SUBGOAL_THEN
`(!k. k IN d ==> ~((extend:(real^M->bool)->(real^M->bool)) k = {})) /\
(!k. k IN d ==> extend k SUBSET interval[a,b])`
STRIP_ASSUME_TAC THENL
[FIRST_ASSUM(fun th -> REWRITE_TAC[MATCH_MP FORALL_IN_DIVISION th]) THEN
CONJ_TAC THEN MAP_EVERY X_GEN_TAC [`u:real^M`; `v:real^M`] THEN
(DISCH_TAC THEN EXPAND_TAC "extend" THEN
SUBGOAL_THEN `interval[u:real^M,v] SUBSET interval[a,b]` MP_TAC THENL
[ASM_SIMP_TAC[]; ALL_TAC] THEN
SUBGOAL_THEN `~(interval[u:real^M,v] = {})` MP_TAC THENL
[ASM_SIMP_TAC[]; ALL_TAC] THEN
SIMP_TAC[SUBSET_INTERVAL; INTERVAL_NE_EMPTY; LAMBDA_BETA;
INTERVAL_LOWERBOUND; INTERVAL_UPPERBOUND] THEN
MESON_TAC[REAL_LE_TRANS; REAL_LE_REFL]);
ALL_TAC] THEN
SUBGOAL_THEN
`!k1 k2. k1 IN d /\ k2 IN d /\ ~(k1 = k2)
==> interior((extend:(real^M->bool)->(real^M->bool)) k1) INTER
interior(extend k2) = {}`
ASSUME_TAC THENL
[REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
FIRST_ASSUM(fun th -> REWRITE_TAC[MATCH_MP FORALL_IN_DIVISION th]) THEN
MAP_EVERY X_GEN_TAC [`u:real^M`; `v:real^M`] THEN DISCH_TAC THEN
MAP_EVERY X_GEN_TAC [`w:real^M`; `z:real^M`] THEN DISCH_TAC THEN
DISCH_TAC THEN
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [division_of]) THEN
DISCH_THEN(MP_TAC o el 2 o CONJUNCTS) THEN DISCH_THEN(MP_TAC o SPECL
[`interval[u:real^M,v]`; `interval[w:real^M,z]`]) THEN
ASM_REWRITE_TAC[INTERIOR_CLOSED_INTERVAL] THEN
ONCE_REWRITE_TAC[GSYM CONTRAPOS_THM] THEN
REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_INTER] THEN
EXPAND_TAC "extend" THEN
SIMP_TAC[INTERIOR_CLOSED_INTERVAL; IN_INTERVAL; LAMBDA_BETA] THEN
SUBGOAL_THEN `~(interval[u:real^M,v] = {}) /\
~(interval[w:real^M,z] = {})`
MP_TAC THENL [ASM_SIMP_TAC[]; ALL_TAC] THEN
SIMP_TAC[SUBSET_INTERVAL; INTERVAL_NE_EMPTY; LAMBDA_BETA;
INTERVAL_LOWERBOUND; INTERVAL_UPPERBOUND] THEN
STRIP_TAC THEN DISCH_THEN(X_CHOOSE_THEN `x:real^M` MP_TAC) THEN
MP_TAC(MESON[]
`(!P. (!j:num. P j) <=> P i /\ (!j. ~(j = i) ==> P j))`) THEN
DISCH_THEN(fun th -> GEN_REWRITE_TAC
(LAND_CONV o ONCE_DEPTH_CONV) [th]) THEN
ASM_SIMP_TAC[IMP_IMP] THEN STRIP_TAC THEN
FIRST_X_ASSUM(DISJ_CASES_THEN
(fun th -> MP_TAC(SPEC `interval[u:real^M,v]` th) THEN
MP_TAC(SPEC `interval[w:real^M,z]` th))) THEN
ASM_REWRITE_TAC[CONTENT_EQ_0_INTERIOR; INTERIOR_CLOSED_INTERVAL] THEN
REWRITE_TAC[IMP_CONJ; GSYM MEMBER_NOT_EMPTY; IN_INTER; IN_ELIM_THM] THEN
REWRITE_TAC[IN_INTERVAL; LEFT_IMP_EXISTS_THM] THEN
X_GEN_TAC `q:real^M` THEN STRIP_TAC THEN
X_GEN_TAC `r:real^M` THEN STRIP_TAC THEN
X_GEN_TAC `s:real^M` THEN STRIP_TAC THEN
X_GEN_TAC `t:real^M` THEN STRIP_TAC THENL
[EXISTS_TAC `(lambda j. if j = i then min ((q:real^M)$i) ((s:real^M)$i)
else (x:real^M)$j):real^M`;
EXISTS_TAC `(lambda j. if j = i then max ((q:real^M)$i) ((s:real^M)$i)
else (x:real^M)$j):real^M`] THEN
(SIMP_TAC[AND_FORALL_THM; LAMBDA_BETA] THEN X_GEN_TAC `j:num` THEN
ASM_CASES_TAC `j:num = i` THEN ASM_SIMP_TAC[] THEN
UNDISCH_THEN `j:num = i` SUBST_ALL_TAC THEN
SUBGOAL_THEN `interval[u:real^M,v] SUBSET interval[a,b] /\
interval[w:real^M,z] SUBSET interval[a,b]`
MP_TAC THENL [ASM_SIMP_TAC[]; ALL_TAC] THEN
SUBGOAL_THEN `~(interval[u:real^M,v] = {}) /\
~(interval[w:real^M,z] = {})`
MP_TAC THENL [ASM_SIMP_TAC[]; ALL_TAC] THEN
SIMP_TAC[INTERVAL_NE_EMPTY; SUBSET_INTERVAL] THEN
REPEAT STRIP_TAC THEN
REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `i:num`)) THEN ASM_REWRITE_TAC[] THEN
ASM_REAL_ARITH_TAC);
ALL_TAC] THEN
MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC
`sum (IMAGE (extend:(real^M->bool)->(real^M->bool)) d) content` THEN
CONJ_TAC THENL
[W(MP_TAC o PART_MATCH (lhs o rand) SUM_IMAGE_NONZERO o rand o snd) THEN
ANTS_TAC THENL
[ASM_REWRITE_TAC[] THEN
MAP_EVERY X_GEN_TAC [`k1:real^M->bool`; `k2:real^M->bool`] THEN
STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL
[`k1:real^M->bool`; `k2:real^M->bool`]) THEN
ASM_REWRITE_TAC[INTER_IDEMPOT] THEN
EXPAND_TAC "extend" THEN REWRITE_TAC[CONTENT_EQ_0_INTERIOR];
DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[GSYM SUM_LMUL] THEN
MATCH_MP_TAC SUM_LE THEN ASM_REWRITE_TAC[] THEN
FIRST_ASSUM(fun th -> REWRITE_TAC[MATCH_MP FORALL_IN_DIVISION th]) THEN
MAP_EVERY X_GEN_TAC [`u:real^M`; `v:real^M`] THEN DISCH_TAC THEN
ASM_CASES_TAC `content(interval[u:real^M,v]) = &0` THENL
[ASM_REWRITE_TAC[real_div; REAL_MUL_LZERO; REAL_MUL_RZERO; o_THM] THEN
EXPAND_TAC "extend" THEN REWRITE_TAC[CONTENT_POS_LE];
ALL_TAC] THEN
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM CONTENT_LT_NZ]) THEN
DISCH_THEN(fun th -> ASSUME_TAC th THEN MP_TAC th) THEN
REWRITE_TAC[CONTENT_POS_LT_EQ] THEN STRIP_TAC THEN
ASM_SIMP_TAC[INTERVAL_LOWERBOUND; INTERVAL_UPPERBOUND;
REAL_LT_IMP_LE; real_div; REAL_MUL_ASSOC] THEN
ASM_SIMP_TAC[GSYM real_div; REAL_LE_LDIV_EQ; REAL_SUB_LT] THEN
SUBGOAL_THEN
`~((extend:(real^M->bool)->(real^M->bool)) (interval[u,v]) = {})`
MP_TAC THENL [ASM_SIMP_TAC[]; ALL_TAC] THEN
EXPAND_TAC "extend" THEN ASM_SIMP_TAC[content; o_THM] THEN
ASM_SIMP_TAC[INTERVAL_NE_EMPTY; INTERVAL_LOWERBOUND;
INTERVAL_UPPERBOUND; REAL_LT_IMP_LE] THEN
DISCH_THEN(K ALL_TAC) THEN
SUBGOAL_THEN
`1..dimindex(:M) = i INSERT ((1..dimindex(:M)) DELETE i)`
SUBST1_TAC THENL
[MATCH_MP_TAC(SET_RULE `x IN s ==> s = x INSERT (s DELETE x)`) THEN
ASM_REWRITE_TAC[IN_NUMSEG];
ALL_TAC] THEN
SIMP_TAC[PRODUCT_CLAUSES; FINITE_NUMSEG; FINITE_DELETE] THEN
ASM_SIMP_TAC[IN_DELETE; IN_NUMSEG; LAMBDA_BETA] THEN
MATCH_MP_TAC REAL_EQ_IMP_LE THEN MATCH_MP_TAC(REAL_RING
`x:real = y ==> ab * uv * x = (ab * y) * uv`) THEN
MATCH_MP_TAC PRODUCT_EQ THEN
SIMP_TAC[IN_DELETE; IN_NUMSEG; LAMBDA_BETA]];
MATCH_MP_TAC SUBADDITIVE_CONTENT_DIVISION THEN EXISTS_TAC
`UNIONS (IMAGE (extend:(real^M->bool)->(real^M->bool)) d)` THEN
ASM_SIMP_TAC[UNIONS_SUBSET; division_of; FINITE_IMAGE] THEN
REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_IMAGE] THEN
FIRST_ASSUM(fun th -> REWRITE_TAC[MATCH_MP FORALL_IN_DIVISION th]) THEN
REPEAT CONJ_TAC THEN MAP_EVERY X_GEN_TAC [`u:real^M`; `v:real^M`] THEN
DISCH_TAC THENL
[CONJ_TAC THENL [ASM SET_TAC[]; ASM_SIMP_TAC[]] THEN
EXPAND_TAC "extend" THEN REWRITE_TAC[] THEN MESON_TAC[];
ASM_MESON_TAC[];
ASM_SIMP_TAC[]]]) in
REPEAT STRIP_TAC THEN
ASM_CASES_TAC `content(interval[a:real^M,b]) = &0` THENL
[MATCH_MP_TAC(REAL_ARITH `x = &0 /\ &0 <= y ==> x <= &2 * y`) THEN
REWRITE_TAC[CONTENT_POS_LE; REAL_ENTIRE] THEN DISJ2_TAC THEN
MATCH_MP_TAC SUM_EQ_0 THEN X_GEN_TAC `k:real^M->bool` THEN
DISCH_TAC THEN REWRITE_TAC[real_div; REAL_ENTIRE] THEN DISJ1_TAC THEN
MATCH_MP_TAC CONTENT_0_SUBSET THEN
MAP_EVERY EXISTS_TAC [`a:real^M`; `b:real^M`] THEN
ASM_MESON_TAC[division_of; SUBSET_TRANS];
ALL_TAC] THEN
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM CONTENT_LT_NZ]) THEN
DISCH_THEN(fun th -> ASSUME_TAC th THEN MP_TAC th) THEN
REWRITE_TAC[CONTENT_POS_LT_EQ] THEN STRIP_TAC THEN
FIRST_ASSUM(ASSUME_TAC o MATCH_MP DIVISION_OF_FINITE) THEN
MP_TAC(ISPECL
[`{k | k IN {l INTER {x | x$i <= c} | l |
l IN d /\ ~(l INTER {x:real^M | x$i <= c} = {})} /\
~(content k = &0)}`;
`a:real^M`;
`(lambda j. if j = i then c else (b:real^M)$j):real^M`;
`UNIONS {k | k IN {l INTER {x | x$i <= c} | l |
l IN d /\ ~(l INTER {x:real^M | x$i <= c} = {})} /\
~(content k = &0)}`;
`i:num`] lemma1) THEN
MP_TAC(ISPECL
[`{k | k IN {l INTER {x | x$i >= c} | l |
l IN d /\ ~(l INTER {x:real^M | x$i >= c} = {})} /\
~(content k = &0)}`;
`(lambda j. if j = i then c else (a:real^M)$j):real^M`;
`b:real^M`;
`UNIONS {k | k IN {l INTER {x | x$i >= c} | l |
l IN d /\ ~(l INTER {x:real^M | x$i >= c} = {})} /\
~(content k = &0)}`;
`i:num`] lemma1) THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(TAUT
`(p1 /\ p2) /\ (q1 /\ q2 ==> r) ==> (p2 ==> q2) ==> (p1 ==> q1) ==> r`) THEN
CONJ_TAC THENL
[CONJ_TAC THEN
(REPEAT CONJ_TAC THENL
[REWRITE_TAC[division_of] THEN CONJ_TAC THENL
[MATCH_MP_TAC FINITE_RESTRICT THEN
ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN
MATCH_MP_TAC FINITE_IMAGE THEN MATCH_MP_TAC FINITE_RESTRICT THEN
ASM_REWRITE_TAC[];
ALL_TAC] THEN
REWRITE_TAC[FORALL_IN_GSPEC; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN
CONJ_TAC THENL
[FIRST_ASSUM(fun th -> REWRITE_TAC[MATCH_MP FORALL_IN_DIVISION th]) THEN
MAP_EVERY X_GEN_TAC [`u:real^M`; `v:real^M`] THEN
REPEAT DISCH_TAC THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL
[REWRITE_TAC[IN_ELIM_THM; SUBSET; IN_UNIONS] THEN ASM_MESON_TAC[];
ASM_SIMP_TAC[INTERVAL_SPLIT] THEN MESON_TAC[]];
X_GEN_TAC `k:real^M->bool` THEN REPEAT DISCH_TAC THEN
X_GEN_TAC `l:real^M->bool` THEN REPEAT DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [division_of]) THEN
DISCH_THEN(MP_TAC o SPECL [`k:real^M->bool`; `l:real^M->bool`] o
el 2 o CONJUNCTS) THEN
ANTS_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
MATCH_MP_TAC(SET_RULE
`s SUBSET s' /\ t SUBSET t'
==> s' INTER t' = {} ==> s INTER t = {}`) THEN
CONJ_TAC THEN MATCH_MP_TAC SUBSET_INTERIOR THEN SET_TAC[]];
REWRITE_TAC[UNIONS_SUBSET; FORALL_IN_GSPEC; IMP_CONJ] THEN
X_GEN_TAC `k:real^M->bool` THEN REPEAT DISCH_TAC THEN
SUBGOAL_THEN `k SUBSET interval[a:real^M,b]` MP_TAC THENL
[ASM_MESON_TAC[division_of; SUBSET_TRANS]; ALL_TAC] THEN
MATCH_MP_TAC(SET_RULE
`i INTER h SUBSET j ==> k SUBSET i ==> k INTER h SUBSET j`) THEN
ASM_SIMP_TAC[INTERVAL_SPLIT; SUBSET_INTERVAL; LAMBDA_BETA] THEN
MATCH_MP_TAC MONO_FORALL THEN GEN_TAC THEN MATCH_MP_TAC MONO_IMP THEN
REAL_ARITH_TAC;
ALL_TAC])
THENL [DISJ2_TAC; DISJ1_TAC] THEN
REWRITE_TAC[FORALL_IN_GSPEC; IMP_CONJ] THEN
ASM_SIMP_TAC[LAMBDA_BETA; real_ge] THEN ASM SET_TAC[REAL_LE_REFL];
ASM_SIMP_TAC[LAMBDA_BETA]] THEN
SUBGOAL_THEN
`sum {k | k IN
{ l INTER {x | x$i <= c} | l |
l IN d /\ ~(l INTER {x:real^M | x$i <= c} = {})} /\
~(content k = &0)}
(\k. content k /
(interval_upperbound k$i - interval_lowerbound k$i)) =
sum d ((\k. content k /
(interval_upperbound k$i - interval_lowerbound k$i)) o
(\k. k INTER {x | x$i <= c})) /\
sum {k | k IN
{ l INTER {x | x$i >= c} | l |
l IN d /\ ~(l INTER {x:real^M | x$i >= c} = {})} /\
~(content k = &0)}
(\k. content k /
(interval_upperbound k$i - interval_lowerbound k$i)) =
sum d ((\k. content k /
(interval_upperbound k$i - interval_lowerbound k$i)) o
(\k. k INTER {x | x$i >= c}))`
(CONJUNCTS_THEN SUBST1_TAC) THENL
[CONJ_TAC THEN
(W(MP_TAC o PART_MATCH (rand o rand) SUM_IMAGE_NONZERO o rand o snd) THEN
ASM_REWRITE_TAC[] THEN ANTS_TAC THENL
[MAP_EVERY X_GEN_TAC [`k:real^M->bool`; `l:real^M->bool`] THEN
STRIP_TAC THEN
REWRITE_TAC[real_div; REAL_ENTIRE] THEN DISJ1_TAC THEN
(MATCH_MP_TAC DIVISION_SPLIT_LEFT_INJ ORELSE
MATCH_MP_TAC DIVISION_SPLIT_RIGHT_INJ) THEN ASM_MESON_TAC[];
DISCH_THEN(SUBST1_TAC o SYM) THEN CONV_TAC SYM_CONV THEN
MATCH_MP_TAC SUM_SUPERSET THEN CONJ_TAC THENL [SET_TAC[]; ALL_TAC] THEN
GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP (SET_RULE
`x IN IMAGE f d /\
~(x IN {x | x IN {f y |y| y IN d /\ ~(f y = a)} /\ ~P x})
==> (!y. f y = a ==> P(f y)) ==> P x`)) THEN
SIMP_TAC[CONTENT_EMPTY; real_div; REAL_MUL_LZERO]]);
ALL_TAC] THEN
MAP_EVERY (fun (t,tac) ->
ASM_CASES_TAC t THENL
[FIRST_X_ASSUM SUBST_ALL_TAC THEN DISCH_THEN(MP_TAC o tac) THEN
MATCH_MP_TAC(REAL_ARITH `x = y /\ a <= b ==> x <= a ==> y <= b`) THEN
CONJ_TAC THENL
[AP_TERM_TAC THEN MATCH_MP_TAC SUM_EQ THEN
X_GEN_TAC `k:real^M->bool` THEN DISCH_TAC THEN
PURE_REWRITE_TAC[o_THM] THEN AP_TERM_TAC THEN
REWRITE_TAC[real_ge; SET_RULE
`k INTER {x | P x} = k <=> (!x. x IN k ==> P x)`] THEN
X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN
SUBGOAL_THEN `x IN interval[a:real^M,b]` MP_TAC THENL
[ASM_MESON_TAC[SUBSET; division_of]; ALL_TAC] THEN
ASM_SIMP_TAC[IN_INTERVAL];
MATCH_MP_TAC(REAL_ARITH `&0 <= y /\ x <= y ==> x <= &2 * y`) THEN
REWRITE_TAC[CONTENT_POS_LE] THEN MATCH_MP_TAC CONTENT_SUBSET THEN
SIMP_TAC[SUBSET_INTERVAL; LAMBDA_BETA] THEN
MESON_TAC[REAL_LE_REFL]];
ALL_TAC])
[`c = (a:real^M)$i`,CONJUNCT2; `c = (b:real^M)$i`,CONJUNCT1] THEN
SUBGOAL_THEN `(a:real^M)$i < c /\ c < (b:real^M)$i` STRIP_ASSUME_TAC THENL
[ASM_REAL_ARITH_TAC; ALL_TAC] THEN
ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
ASM_SIMP_TAC[GSYM REAL_LE_RDIV_EQ; REAL_SUB_LT] THEN
REWRITE_TAC[REAL_ARITH `(x * &2) / y = &2 * x / y`] THEN
MATCH_MP_TAC(REAL_ARITH
`s <= s1 + s2 /\ c1 = c /\ c2 = c
==> s1 <= c1 /\ s2 <= c2 ==> s <= &2 * c`) THEN
CONJ_TAC THENL
[ASM_SIMP_TAC[GSYM SUM_ADD] THEN MATCH_MP_TAC SUM_LE THEN
ASM_SIMP_TAC[lemma0] THEN
FIRST_ASSUM(fun th -> REWRITE_TAC[MATCH_MP FORALL_IN_DIVISION th]) THEN
MAP_EVERY X_GEN_TAC [`u:real^M`; `v:real^M`] THEN DISCH_TAC THEN
SUBGOAL_THEN
`~(interval[u:real^M,v] = {}) /\ interval[u,v] SUBSET interval[a,b]`
MP_TAC THENL [ASM_MESON_TAC[division_of; SUBSET_TRANS]; ALL_TAC] THEN
SIMP_TAC[INTERVAL_NE_EMPTY; SUBSET_INTERVAL; IMP_CONJ] THEN
REPEAT STRIP_TAC THEN REWRITE_TAC[o_THM] THEN
MATCH_MP_TAC(REAL_ARITH
`&0 <= x /\ c1 + c2 = c /\
(~(c1 = &0) ==> x1 = x) /\ (~(c2 = &0) ==> x2 = x)
==> (if c = &0 then &0 else x) <=
(if c1 = &0 then &0 else x1) +
(if c2 = &0 then &0 else x2)`) THEN
ASM_SIMP_TAC[GSYM CONTENT_SPLIT] THEN
ASM_SIMP_TAC[INTERVAL_SPLIT; CONTENT_POS_LE] THEN CONJ_TAC THENL
[MATCH_MP_TAC PRODUCT_POS_LE THEN
ASM_SIMP_TAC[FINITE_DELETE; FINITE_NUMSEG; IN_DELETE; IN_NUMSEG;
INTERVAL_UPPERBOUND; INTERVAL_LOWERBOUND; REAL_SUB_LE];
REWRITE_TAC[CONTENT_EQ_0; REAL_NOT_LE; MESON[]
`~(?i. P i /\ Q i /\ R i) <=> (!i. P i /\ Q i ==> ~R i)`] THEN
SIMP_TAC[INTERVAL_UPPERBOUND; INTERVAL_LOWERBOUND; REAL_LT_IMP_LE] THEN
REPEAT STRIP_TAC THEN MATCH_MP_TAC PRODUCT_EQ THEN
ASM_SIMP_TAC[INTERVAL_UPPERBOUND; INTERVAL_LOWERBOUND; IN_DELETE;
IN_NUMSEG; LAMBDA_BETA]];
SUBGOAL_THEN
`~(interval[a,b] = {}) /\
~(interval[a:real^M,(lambda j. if j = i then c else b$j)] = {}) /\
~(interval[(lambda j. if j = i then c else a$j):real^M,b] = {})`
MP_TAC THENL
[SIMP_TAC[INTERVAL_NE_EMPTY; LAMBDA_BETA] THEN
ASM_MESON_TAC[REAL_LT_IMP_LE; REAL_LE_REFL];
ALL_TAC] THEN
SIMP_TAC[content] THEN
SIMP_TAC[INTERVAL_NE_EMPTY; INTERVAL_UPPERBOUND; INTERVAL_LOWERBOUND] THEN
STRIP_TAC THEN
SUBGOAL_THEN
`1..dimindex(:M) = i INSERT ((1..dimindex(:M)) DELETE i)`
SUBST1_TAC THENL
[MATCH_MP_TAC(SET_RULE `x IN s ==> s = x INSERT (s DELETE x)`) THEN
ASM_REWRITE_TAC[IN_NUMSEG];
ALL_TAC] THEN
SIMP_TAC[PRODUCT_CLAUSES; FINITE_NUMSEG; FINITE_DELETE] THEN
ASM_SIMP_TAC[IN_DELETE; IN_NUMSEG; LAMBDA_BETA] THEN
CONJ_TAC THEN MATCH_MP_TAC(REAL_FIELD
`y < x /\ z < w /\ a = b
==> ((x - y) * a) / (x - y) = ((w - z) * b) / (w - z)`) THEN
ASM_SIMP_TAC[] THEN MATCH_MP_TAC PRODUCT_EQ THEN
SIMP_TAC[IN_DELETE; IN_NUMSEG; LAMBDA_BETA]]);;
let BOUNDED_EQUIINTEGRAL_OVER_THIN_TAGGED_PARTIAL_DIVISION = prove
(`!fs f:real^M->real^N a b e.
fs
equiintegrable_on interval[a,b] /\ f
IN fs /\
(!h x. h
IN fs /\ x
IN interval[a,b] ==> norm(h x) <= norm(f x)) /\
&0 < e
==> ?d. gauge d /\
!c i p h. c
IN interval[a,b] /\ 1 <= i /\ i <= dimindex(:M) /\
p
tagged_partial_division_of interval[a,b] /\
d fine p /\
h
IN fs /\
(!x k. (x,k)
IN p ==> ~(k
INTER {x | x$i = c$i} = {}))
==> sum p(\(x,k). norm(integral k h)) < e`,
REPEAT STRIP_TAC THEN
ASM_CASES_TAC `content(interval[a:real^M,b]) = &0` THENL
[EXISTS_TAC `\x:real^M. ball(x,&1)` THEN REWRITE_TAC[
GAUGE_TRIVIAL] THEN
REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH
`&0 < e ==> x = &0 ==> x < e`)) THEN
MATCH_MP_TAC
SUM_EQ_0 THEN REWRITE_TAC[
FORALL_PAIR_THM] THEN
GEN_TAC THEN X_GEN_TAC `k:real^M->bool` THEN DISCH_TAC THEN
SUBGOAL_THEN
`?u v:real^M. k = interval[u,v] /\ interval[u,v]
SUBSET interval[a,b]`
STRIP_ASSUME_TAC THENL
[ASM_MESON_TAC[
tagged_partial_division_of]; ALL_TAC] THEN
ASM_REWRITE_TAC[
NORM_EQ_0] THEN MATCH_MP_TAC
INTEGRAL_NULL THEN
ASM_MESON_TAC[
CONTENT_0_SUBSET];
ALL_TAC] THEN
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM
CONTENT_LT_NZ]) THEN
DISCH_THEN(fun th -> ASSUME_TAC th THEN MP_TAC th) THEN
REWRITE_TAC[
CONTENT_POS_LT_EQ] THEN STRIP_TAC THEN
SUBGOAL_THEN
`?d. gauge d /\
!p h. p
tagged_partial_division_of interval [a,b] /\
d fine p /\ (h:real^M->real^N)
IN fs
==> sum p (\(x,k). norm(content k % h x - integral k h)) <
e / &2`
(X_CHOOSE_THEN `g0:real^M->real^M->bool` STRIP_ASSUME_TAC)
THENL
[FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
equiintegrable_on]) THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (MP_TAC o SPEC
`e / &5 / (&(dimindex(:N)) + &1)`)) THEN
ASM_SIMP_TAC[
REAL_LT_DIV; REAL_ARITH `&0 < &5 /\ &0 < &n + &1`] THEN
MATCH_MP_TAC
MONO_EXISTS THEN X_GEN_TAC `g:real^M->real^M->bool` THEN
STRIP_TAC THEN ASM_REWRITE_TAC[] THEN MAP_EVERY X_GEN_TAC
[`p:(real^M#(real^M->bool))->bool`; `h:real^M->real^N`] THEN
STRIP_TAC THEN
MP_TAC(ISPECL [`h:real^M->real^N`; `a:real^M`; `b:real^M`;
`g:real^M->real^M->bool`; `e / &5 / (&(dimindex(:N)) + &1)`]
HENSTOCK_LEMMA_PART2) THEN
ASM_SIMP_TAC[
REAL_LT_DIV; REAL_ARITH `&0 < &5 /\ &0 < &n + &1`] THEN
DISCH_THEN(MP_TAC o SPEC `p:(real^M#(real^M->bool))->bool`) THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(REAL_ARITH
`a < b ==> x <= a ==> x < b`) THEN
REWRITE_TAC[REAL_ARITH `&2 * d * e / &5 / (d + &1) =
(e * &2 / &5 * d) / (d + &1)`] THEN
SIMP_TAC[
REAL_LT_LDIV_EQ; REAL_ARITH `&0 < &n + &1`] THEN
MATCH_MP_TAC(REAL_ARITH
`&0 < e /\ &0 < e * d ==> e * &2 / &5 * d < e / &2 * (d + &1)`) THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC
REAL_LT_MUL THEN
ASM_SIMP_TAC[
REAL_OF_NUM_LT;
LE_1;
DIMINDEX_GE_1];
ALL_TAC] THEN
ABBREV_TAC
`g:real^M->real^M->bool =
\x. g0(x)
INTER
ball(x,(e / &8 / (norm(f x:real^N) + &1)) *
inf(
IMAGE (\m. b$m - a$m) (1..dimindex(:M))) /
content(interval[a:real^M,b]))` THEN
SUBGOAL_THEN `gauge(g:real^M->real^M->bool)` ASSUME_TAC THENL
[EXPAND_TAC "g" THEN MATCH_MP_TAC
GAUGE_INTER THEN ASM_REWRITE_TAC[] THEN
REWRITE_TAC[gauge;
OPEN_BALL;
CENTRE_IN_BALL] THEN
X_GEN_TAC `x:real^M` THEN MATCH_MP_TAC
REAL_LT_MUL THEN
ASM_SIMP_TAC[
REAL_LT_DIV; NORM_ARITH
`&0 < &8 /\ &0 < norm(x:real^N) + &1`] THEN
MATCH_MP_TAC
REAL_LT_DIV THEN ASM_REWRITE_TAC[] THEN
W(MP_TAC o PART_MATCH (lhand o rand)
REAL_LT_INF_FINITE o snd) THEN
ANTS_TAC THENL
[ASM_SIMP_TAC[
FINITE_IMAGE;
FINITE_NUMSEG;
FINITE_RESTRICT] THEN
SIMP_TAC[
IMAGE_EQ_EMPTY;
NUMSEG_EMPTY;
GSYM
NOT_LE;
DIMINDEX_GE_1] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [
CART_EQ]) THEN
REWRITE_TAC[
EXTENSION;
IN_ELIM_THM;
NOT_IN_EMPTY;
IN_NUMSEG] THEN
MESON_TAC[];
DISCH_THEN SUBST1_TAC THEN
REWRITE_TAC[
FORALL_IN_IMAGE] THEN
ASM_SIMP_TAC[
REAL_SUB_LT;
IN_NUMSEG; GSYM
REAL_ABS_NZ;
REAL_SUB_0;
IN_ELIM_THM]];
ALL_TAC] THEN
EXISTS_TAC `g:real^M->real^M->bool` THEN ASM_REWRITE_TAC[] THEN
MAP_EVERY X_GEN_TAC
[`c:real^M`; `i:num`; `p:(real^M#(real^M->bool))->bool`;
`h:real^M->real^N`] THEN
STRIP_TAC THEN
SUBGOAL_THEN
`interval[c:real^M,b]
SUBSET interval[a,b]`
ASSUME_TAC THENL
[UNDISCH_TAC `c
IN interval[a:real^M,b]` THEN
SIMP_TAC[
IN_INTERVAL;
SUBSET_INTERVAL;
REAL_LE_REFL];
ALL_TAC] THEN
SUBGOAL_THEN `
FINITE(p:(real^M#(real^M->bool))->bool)` ASSUME_TAC THENL
[ASM_MESON_TAC[
tagged_partial_division_of]; ALL_TAC] THEN
MP_TAC(ASSUME `(g:real^M->real^M->bool) fine p`) THEN
EXPAND_TAC "g" THEN REWRITE_TAC[
FINE_INTER] THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (LABEL_TAC "F")) THEN
FIRST_X_ASSUM(MP_TAC o SPEC `p:(real^M#(real^M->bool))->bool`) THEN
DISCH_THEN(MP_TAC o SPEC `h:real^M->real^N`) THEN
ANTS_TAC THENL
[ASM_MESON_TAC[
TAGGED_PARTIAL_DIVISION_OF_SUBSET]; ALL_TAC] THEN
MATCH_MP_TAC(REAL_ARITH
`x - y <= e / &2 ==> y < e / &2 ==> x < e`) THEN
ASM_SIMP_TAC[GSYM
SUM_SUB] THEN
ONCE_REWRITE_TAC[
LAMBDA_PAIR_THM] THEN REWRITE_TAC[] THEN
MATCH_MP_TAC
REAL_LE_TRANS THEN
EXISTS_TAC
`sum p (\(x:real^M,k:real^M->bool). norm(content k % h x:real^N))` THEN
CONJ_TAC THENL
[MATCH_MP_TAC
SUM_LE THEN ASM_REWRITE_TAC[
FORALL_PAIR_THM] THEN
REWRITE_TAC[NORM_ARITH `norm y - norm(x - y:real^N) <= norm x`];
ALL_TAC] THEN
MATCH_MP_TAC
REAL_LE_TRANS THEN
EXISTS_TAC
`sum p (\(x:real^M,k).
e / &4 * (b$i - a$i) / content(interval[a:real^M,b]) *
content(k:real^M->bool) /
(
interval_upperbound k$i -
interval_lowerbound k$i))` THEN
CONJ_TAC THENL
[MATCH_MP_TAC
SUM_LE THEN ASM_REWRITE_TAC[
FORALL_PAIR_THM] THEN
MAP_EVERY X_GEN_TAC [`x:real^M`; `k:real^M->bool`] THEN DISCH_TAC THEN
ASM_CASES_TAC `content(k:real^M->bool) = &0` THENL
[ASM_REWRITE_TAC[
real_div;
REAL_MUL_LZERO;
VECTOR_MUL_LZERO;
NORM_0;
REAL_MUL_RZERO;
REAL_LE_REFL];
ALL_TAC] THEN
REWRITE_TAC[REAL_ARITH `a * b * content k / d = content k * (a * b) / d`;
NORM_MUL] THEN
SUBGOAL_THEN `&0 < content(k:real^M->bool)` ASSUME_TAC THENL
[ASM_MESON_TAC[
CONTENT_LT_NZ;
tagged_partial_division_of]; ALL_TAC] THEN
ASM_SIMP_TAC[
real_abs;
REAL_LT_IMP_LE;
REAL_LE_LMUL_EQ] THEN
MATCH_MP_TAC(REAL_ARITH `x + &1 <= y ==> x <= y`) THEN
SUBGOAL_THEN `?u v. k = interval[u:real^M,v]` MP_TAC THENL
[ASM_MESON_TAC[
tagged_partial_division_of]; ALL_TAC] THEN
DISCH_THEN(REPEAT_TCL CHOOSE_THEN SUBST_ALL_TAC) THEN
MP_TAC(ISPECL [`u:real^M`; `v:real^M`]
CONTENT_POS_LT_EQ) THEN
ASM_SIMP_TAC[
INTERVAL_UPPERBOUND;
INTERVAL_LOWERBOUND;
REAL_LT_IMP_LE] THEN
DISCH_TAC THEN
W(MP_TAC o PART_MATCH (lhand o rand)
REAL_LE_RDIV_EQ o snd) THEN
ASM_SIMP_TAC[
REAL_SUB_LT] THEN DISCH_THEN SUBST1_TAC THEN
GEN_REWRITE_TAC LAND_CONV [REAL_MUL_SYM] THEN
SIMP_TAC[GSYM
REAL_LE_RDIV_EQ; NORM_ARITH `&0 < norm(x:real^N) + &1`] THEN
REMOVE_THEN "F" MP_TAC THEN REWRITE_TAC[fine] THEN
DISCH_THEN(MP_TAC o SPECL [`x:real^M`; `interval[u:real^M,v]`]) THEN
ASM_REWRITE_TAC[
SUBSET] THEN
DISCH_THEN(fun th -> MP_TAC(SPEC `v:real^M` th) THEN
MP_TAC(SPEC `u:real^M` th)) THEN
ASM_SIMP_TAC[
INTERVAL_NE_EMPTY;
REAL_LT_IMP_LE;
ENDS_IN_INTERVAL] THEN
REWRITE_TAC[
IN_BALL; IMP_IMP] THEN
MATCH_MP_TAC(NORM_ARITH
`abs(vi - ui) <= norm(v - u:real^N) /\ &2 * a <= b
==> dist(x,u) < a /\ dist(x,v) < a ==> vi - ui <= b`) THEN
ASM_SIMP_TAC[GSYM
VECTOR_SUB_COMPONENT;
COMPONENT_LE_NORM] THEN
REWRITE_TAC[REAL_ARITH `&2 * e / &8 / x * y = e / &4 * y / x`] THEN
REWRITE_TAC[
real_div; GSYM REAL_MUL_ASSOC] THEN
MATCH_MP_TAC
REAL_LE_LMUL THEN ASM_SIMP_TAC[
REAL_LT_IMP_LE] THEN
MATCH_MP_TAC
REAL_LE_LMUL THEN CONV_TAC REAL_RAT_REDUCE_CONV THEN
ONCE_REWRITE_TAC[REAL_ARITH `a * inv b * inv c:real = (a / c) / b`] THEN
ASM_SIMP_TAC[
REAL_LE_DIV2_EQ] THEN
MATCH_MP_TAC(REAL_ARITH `abs x <= e ==> x <= e`) THEN
REWRITE_TAC[
real_div;
REAL_ABS_MUL] THEN MATCH_MP_TAC
REAL_LE_MUL2 THEN
REWRITE_TAC[
REAL_ABS_POS] THEN CONJ_TAC THENL
[MATCH_MP_TAC(REAL_ARITH `&0 <= x /\ x <= y ==> abs x <= y`) THEN
SIMP_TAC[
REAL_INF_LE_FINITE;
FINITE_IMAGE;
IMAGE_EQ_EMPTY;
FINITE_NUMSEG;
NUMSEG_EMPTY;
NOT_LT;
DIMINDEX_GE_1;
REAL_LE_INF_FINITE] THEN
REWRITE_TAC[
FORALL_IN_IMAGE;
EXISTS_IN_IMAGE;
IN_NUMSEG] THEN
ASM_SIMP_TAC[
VECTOR_SUB_COMPONENT;
REAL_LE_REFL;
REAL_SUB_LE;
REAL_LT_IMP_LE] THEN
ASM_MESON_TAC[
REAL_LE_REFL];
REWRITE_TAC[
REAL_ABS_INV] THEN MATCH_MP_TAC
REAL_LE_INV2 THEN
CONJ_TAC THENL [NORM_ARITH_TAC; ALL_TAC] THEN
MATCH_MP_TAC(REAL_ARITH `x <= y ==> x + &1 <= abs(y + &1)`) THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN
ASM_MESON_TAC[
tagged_partial_division_of;
SUBSET]];
ALL_TAC] THEN
FIRST_ASSUM(MP_TAC o MATCH_MP
TAGGED_PARTIAL_DIVISION_OF_UNION_SELF) THEN
DISCH_THEN(MP_TAC o MATCH_MP (REWRITE_RULE[
IMP_CONJ]
SUM_OVER_TAGGED_DIVISION_LEMMA)) THEN
DISCH_THEN(fun th ->
W(MP_TAC o PART_MATCH (lhs o rand) th o lhand o snd)) THEN
REWRITE_TAC[] THEN ANTS_TAC THENL
[SIMP_TAC[
real_div;
REAL_MUL_LZERO;
REAL_MUL_RZERO];
DISCH_THEN SUBST1_TAC] THEN
REWRITE_TAC[
SUM_LMUL; REAL_ARITH
`e / &4 * ba / c * s <= e / &2 <=> e * (ba * s) / c <= e * &2`] THEN
ASM_SIMP_TAC[
REAL_LE_LMUL_EQ] THEN ASM_SIMP_TAC[
REAL_LE_LDIV_EQ] THEN
MATCH_MP_TAC
SUM_CONTENT_AREA_OVER_THIN_DIVISION THEN
EXISTS_TAC `
UNIONS(
IMAGE SND (p:(real^M#(real^M->bool))->bool))` THEN
EXISTS_TAC `(c:real^M)$i` THEN
RULE_ASSUM_TAC(REWRITE_RULE[
IN_INTERVAL]) THEN ASM_SIMP_TAC[] THEN
REPEAT CONJ_TAC THENL
[MATCH_MP_TAC
DIVISION_OF_TAGGED_DIVISION THEN
ASM_MESON_TAC[
TAGGED_PARTIAL_DIVISION_OF_UNION_SELF];
REWRITE_TAC[
UNIONS_SUBSET;
FORALL_IN_IMAGE;
FORALL_PAIR_THM] THEN
ASM_MESON_TAC[
tagged_partial_division_of];
ASM_REWRITE_TAC[
FORALL_IN_IMAGE;
FORALL_PAIR_THM]]);;
let EQUIINTEGRABLE_HALFSPACE_RESTRICTIONS_LE = prove
(`!fs f:real^M->real^N a b.
fs
equiintegrable_on interval[a,b] /\ f
IN fs /\
(!h x. h
IN fs /\ x
IN interval[a,b] ==> norm(h x) <= norm(f x))
==> { (\x. if x$i <= c then h x else vec 0) |
i
IN 1..dimindex(:M) /\ c
IN (:real) /\ h
IN fs }
equiintegrable_on interval[a,b]`,
let lemma = prove
(`(!x k. (x,k) IN IMAGE (\(x,k). f x k,g x k) s ==> Q x k) <=>
(!x k. (x,k) IN s ==> Q (f x k) (g x k))`,
REWRITE_TAC[IN_IMAGE; PAIR_EQ; EXISTS_PAIR_THM] THEN SET_TAC[]) in
REPEAT STRIP_TAC THEN
ASM_CASES_TAC `content(interval[a:real^M,b]) = &0` THEN
ASM_SIMP_TAC[EQUIINTEGRABLE_ON_NULL] THEN
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM CONTENT_LT_NZ]) THEN
DISCH_THEN(fun th -> ASSUME_TAC th THEN MP_TAC th) THEN
REWRITE_TAC[CONTENT_POS_LT_EQ] THEN STRIP_TAC THEN
REWRITE_TAC[equiintegrable_on] THEN
REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_GSPEC] THEN
REWRITE_TAC[IN_UNIV; IMP_IMP; GSYM CONJ_ASSOC; RIGHT_IMP_FORALL_THM;
IN_NUMSEG] THEN
FIRST_ASSUM(ASSUME_TAC o CONJUNCT1 o REWRITE_RULE[equiintegrable_on]) THEN
MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN CONJ_TAC THENL
[REPEAT GEN_TAC THEN
ONCE_REWRITE_TAC[SET_RULE `x$i <= c <=> x IN {x:real^N | x$i <= c}`] THEN
REWRITE_TAC[INTEGRABLE_RESTRICT_INTER] THEN
ONCE_REWRITE_TAC[INTER_COMM] THEN SIMP_TAC[INTERVAL_SPLIT] THEN
REPEAT STRIP_TAC THEN MATCH_MP_TAC INTEGRABLE_ON_SUBINTERVAL THEN
EXISTS_TAC `interval[a:real^M,b]` THEN ASM_SIMP_TAC[] THEN
SIMP_TAC[SUBSET_INTERVAL; LAMBDA_BETA; REAL_LE_REFL] THEN
REPEAT STRIP_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
REAL_ARITH_TAC;
DISCH_TAC] THEN
X_GEN_TAC `e:real` THEN DISCH_TAC THEN
MP_TAC(ISPECL [`fs:(real^M->real^N)->bool`; `f:real^M->real^N`;
`a:real^M`; `b:real^M`; `e / &12`]
BOUNDED_EQUIINTEGRAL_OVER_THIN_TAGGED_PARTIAL_DIVISION) THEN
ASM_SIMP_TAC[REAL_LT_DIV; REAL_ARITH `&0 < &12`] THEN
DISCH_THEN(X_CHOOSE_THEN `g0:real^M->real^M->bool` STRIP_ASSUME_TAC) THEN
SUBGOAL_THEN
`?d. gauge d /\
!p h. p tagged_partial_division_of interval [a,b] /\
d fine p /\ (h:real^M->real^N) IN fs
==> sum p (\(x,k). norm(content k % h x - integral k h)) <
e / &3`
(X_CHOOSE_THEN `g1:real^M->real^M->bool` STRIP_ASSUME_TAC)
THENL
[FIRST_ASSUM(MP_TAC o CONJUNCT2 o REWRITE_RULE[equiintegrable_on]) THEN
DISCH_THEN(MP_TAC o SPEC `e / &7 / (&(dimindex(:N)) + &1)`) THEN
ASM_SIMP_TAC[REAL_LT_DIV; REAL_ARITH `&0 < &7 /\ &0 < &n + &1`] THEN
MATCH_MP_TAC MONO_EXISTS THEN
X_GEN_TAC `d:real^M->real^M->bool` THEN STRIP_TAC THEN
ASM_REWRITE_TAC[] THEN
MAP_EVERY X_GEN_TAC
[`p:(real^M#(real^M->bool))->bool`; `h:real^M->real^N`] THEN
STRIP_TAC THEN
MP_TAC(ISPECL [`h:real^M->real^N`; `a:real^M`; `b:real^M`;
`d:real^M->real^M->bool`; `e / &7 / (&(dimindex(:N)) + &1)`]
HENSTOCK_LEMMA_PART2) THEN
ASM_SIMP_TAC[REAL_LT_DIV; REAL_ARITH `&0 < &7 /\ &0 < &n + &1`] THEN
DISCH_THEN(MP_TAC o SPEC `p:(real^M#(real^M->bool))->bool`) THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(REAL_ARITH
`a < b ==> x <= a ==> x < b`) THEN
REWRITE_TAC[REAL_ARITH `&2 * d * e / &7 / (d + &1) =
(e * &2 / &7 * d) / (d + &1)`] THEN
SIMP_TAC[REAL_LT_LDIV_EQ; REAL_ARITH `&0 < &n + &1`] THEN
MATCH_MP_TAC(REAL_ARITH
`&0 < e /\ &0 < e * d ==> e * &2 / &7 * d < e / &3 * (d + &1)`) THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC REAL_LT_MUL THEN
ASM_SIMP_TAC[REAL_OF_NUM_LT; LE_1; DIMINDEX_GE_1];
ALL_TAC] THEN
EXISTS_TAC `\x. (g0:real^M->real^M->bool) x INTER g1 x` THEN
ASM_SIMP_TAC[GAUGE_INTER; FINE_INTER] THEN
X_GEN_TAC `i:num` THEN
ASM_CASES_TAC `1 <= i` THEN ASM_REWRITE_TAC[] THEN
ASM_CASES_TAC `i <= dimindex(:M)` THEN ASM_REWRITE_TAC[] THEN
MP_TAC(MESON[]
`!P. ((!c. (a:real^M)$i <= c /\ c <= (b:real^M)$i ==> P c) ==> (!c. P c)) /\
(!c. (a:real^M)$i <= c /\ c <= (b:real^M)$i ==> P c)
==> !c. P c`) THEN
DISCH_THEN MATCH_MP_TAC THEN CONJ_TAC THENL
[DISCH_THEN(LABEL_TAC "*") THEN
X_GEN_TAC `c:real` THEN
ASM_CASES_TAC `(a:real^M)$i <= c /\ c <= (b:real^M)$i` THENL
[REMOVE_THEN "*" MATCH_MP_TAC THEN ASM_REWRITE_TAC[]; ALL_TAC] THEN
REMOVE_THEN "*" (MP_TAC o SPEC `(b:real^M)$i`) THEN
ASM_SIMP_TAC[REAL_LT_IMP_LE; REAL_LE_REFL] THEN
MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `h:real^M->real^N` THEN
MATCH_MP_TAC MONO_FORALL THEN
X_GEN_TAC `p:real^M#(real^M->bool)->bool` THEN
DISCH_THEN(fun th -> STRIP_TAC THEN MP_TAC th) THEN ASM_REWRITE_TAC[] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [DE_MORGAN_THM]) THEN
REWRITE_TAC[REAL_NOT_LE] THEN STRIP_TAC THENL
[DISCH_TAC THEN MATCH_MP_TAC(NORM_ARITH
`x:real^N = vec 0 /\ y = vec 0 /\ &0 < e ==> norm(x - y) < e`) THEN
ASM_REWRITE_TAC[] THEN CONJ_TAC THENL
[MATCH_MP_TAC VSUM_EQ_0 THEN REWRITE_TAC[FORALL_PAIR_THM] THEN
MAP_EVERY X_GEN_TAC [`x:real^M`; `k:real^M->bool`] THEN DISCH_TAC THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[VECTOR_MUL_RZERO] THEN
SUBGOAL_THEN `(x:real^M) IN interval[a,b]` MP_TAC THENL
[ASM_MESON_TAC[TAGGED_DIVISION_OF; SUBSET]; ALL_TAC] THEN
REWRITE_TAC[IN_INTERVAL] THEN
DISCH_THEN(MP_TAC o SPEC `i:num`) THEN ASM_REWRITE_TAC[] THEN
ASM_REAL_ARITH_TAC;
MATCH_MP_TAC EQ_TRANS THEN
EXISTS_TAC `integral(interval[a,b]) ((\x. vec 0):real^M->real^N)` THEN
CONJ_TAC THENL [ALL_TAC; REWRITE_TAC[INTEGRAL_0]] THEN
MATCH_MP_TAC INTEGRAL_EQ THEN REWRITE_TAC[] THEN GEN_TAC THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[IN_INTERVAL] THEN
DISCH_THEN(MP_TAC o SPEC `i:num`) THEN ASM_REWRITE_TAC[] THEN
ASM_REAL_ARITH_TAC];
MATCH_MP_TAC(NORM_ARITH
`x:real^N = y /\ w = z ==> norm(x - w) < e ==> norm(y - z) < e`) THEN
CONJ_TAC THENL
[MATCH_MP_TAC VSUM_EQ THEN REWRITE_TAC[FORALL_PAIR_THM] THEN
MAP_EVERY X_GEN_TAC [`x:real^M`; `k:real^M->bool`] THEN DISCH_TAC THEN
SUBGOAL_THEN `(x:real^M) IN interval[a,b]` MP_TAC THENL
[ASM_MESON_TAC[TAGGED_DIVISION_OF; SUBSET]; ALL_TAC] THEN
REWRITE_TAC[IN_INTERVAL] THEN
DISCH_THEN(MP_TAC o SPEC `i:num`) THEN ASM_REWRITE_TAC[] THEN
STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[VECTOR_MUL_RZERO] THEN
ASM_REAL_ARITH_TAC;
MATCH_MP_TAC INTEGRAL_EQ THEN REWRITE_TAC[] THEN GEN_TAC THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[IN_INTERVAL] THEN
DISCH_THEN(MP_TAC o SPEC `i:num`) THEN ASM_REWRITE_TAC[] THEN
ASM_REAL_ARITH_TAC]];
ALL_TAC] THEN
X_GEN_TAC `c:real` THEN DISCH_TAC THEN
MAP_EVERY X_GEN_TAC [`h:real^M->real^N`;
`p:(real^M#(real^M->bool))->bool`] THEN STRIP_TAC THEN
ABBREV_TAC
`q:(real^M#(real^M->bool))->bool =
{(x,k) | (x,k) IN p /\ ~(k INTER {x | x$i <= c} = {})}` THEN
MP_TAC(ISPECL
[`\x. if x$i <= c then (h:real^M->real^N) x else vec 0`;
`a:real^M`; `b:real^M`; `p:(real^M#(real^M->bool))->bool`]
INTEGRAL_COMBINE_TAGGED_DIVISION_TOPDOWN) THEN
ASM_SIMP_TAC[] THEN DISCH_THEN SUBST1_TAC THEN
SUBGOAL_THEN `FINITE(p:(real^M#(real^M->bool))->bool)` ASSUME_TAC THENL
[ASM_MESON_TAC[TAGGED_DIVISION_OF]; ALL_TAC] THEN
SUBGOAL_THEN `q SUBSET (p:(real^M#(real^M->bool))->bool)` ASSUME_TAC THENL
[EXPAND_TAC "q" THEN SIMP_TAC[SUBSET; FORALL_PAIR_THM; IN_ELIM_PAIR_THM];
ALL_TAC] THEN
SUBGOAL_THEN `FINITE(q:(real^M#(real^M->bool))->bool)` ASSUME_TAC THENL
[ASM_MESON_TAC[FINITE_SUBSET]; ALL_TAC] THEN
ASM_SIMP_TAC[GSYM VSUM_SUB] THEN ONCE_REWRITE_TAC[LAMBDA_PAIR_THM] THEN
REWRITE_TAC[] THEN
SUBGOAL_THEN `q tagged_partial_division_of interval[a:real^M,b] /\
g0 fine q /\ g1 fine q`
STRIP_ASSUME_TAC THENL
[ASM_MESON_TAC[TAGGED_PARTIAL_DIVISION_SUBSET; tagged_division_of;
FINE_SUBSET];
ALL_TAC] THEN
MATCH_MP_TAC(MESON[] `!q. vsum p s = vsum q s /\ norm(vsum q s) < e
==> norm(vsum p s:real^N) < e`) THEN
EXISTS_TAC `q:(real^M#(real^M->bool))->bool` THEN CONJ_TAC THENL
[MATCH_MP_TAC VSUM_SUPERSET THEN ASM_REWRITE_TAC[FORALL_PAIR_THM] THEN
EXPAND_TAC "q" THEN REWRITE_TAC[IN_ELIM_PAIR_THM] THEN
MAP_EVERY X_GEN_TAC [`x:real^M`; `k:real^M->bool`] THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ASM_REWRITE_TAC[] THEN
SUBGOAL_THEN `(x:real^M) IN k` ASSUME_TAC THENL
[ASM_MESON_TAC[TAGGED_DIVISION_OF]; ALL_TAC] THEN
DISCH_THEN(fun th -> ASSUME_TAC th THEN MP_TAC th) THEN
REWRITE_TAC[EXTENSION] THEN DISCH_THEN(MP_TAC o SPEC `x:real^M`) THEN
REWRITE_TAC[IN_INTER; NOT_IN_EMPTY] THEN ASM_REWRITE_TAC[] THEN
REWRITE_TAC[IN_ELIM_THM] THEN DISCH_TAC THEN
ASM_REWRITE_TAC[VECTOR_MUL_RZERO] THEN
REWRITE_TAC[VECTOR_NEG_EQ_0; VECTOR_SUB_LZERO] THEN
MATCH_MP_TAC EQ_TRANS THEN
EXISTS_TAC `integral k ((\x. vec 0):real^M->real^N)` THEN
CONJ_TAC THENL [ALL_TAC; REWRITE_TAC[INTEGRAL_0]] THEN
MATCH_MP_TAC INTEGRAL_EQ THEN ASM SET_TAC[];
ALL_TAC] THEN
SUBGOAL_THEN
`norm(vsum q (\(x,k). content k % h x - integral k (h:real^M->real^N)))
< e / &3`
MP_TAC THENL
[MATCH_MP_TAC REAL_LET_TRANS THEN EXISTS_TAC `sum q
(\(x,k). norm(content k % h x - integral k (h:real^M->real^N)))` THEN
ASM_SIMP_TAC[] THEN MATCH_MP_TAC VSUM_NORM_LE THEN
ASM_REWRITE_TAC[FORALL_PAIR_THM; REAL_LE_REFL];
ALL_TAC] THEN
MATCH_MP_TAC(NORM_ARITH
`norm(x - y:real^N) <= &2 * e / &3
==> norm(x) < e / &3 ==> norm(y) < e`) THEN
ASM_SIMP_TAC[GSYM VSUM_SUB] THEN ONCE_REWRITE_TAC[LAMBDA_PAIR_THM] THEN
REWRITE_TAC[] THEN
ABBREV_TAC
`r:(real^M#(real^M->bool))->bool =
{(x,k) | (x,k) IN q /\ ~(k SUBSET {x | x$i <= c})}` THEN
SUBGOAL_THEN `r SUBSET (q:(real^M#(real^M->bool))->bool)` ASSUME_TAC THENL
[EXPAND_TAC "r" THEN SIMP_TAC[SUBSET; FORALL_PAIR_THM; IN_ELIM_PAIR_THM];
ALL_TAC] THEN
SUBGOAL_THEN `FINITE(r:(real^M#(real^M->bool))->bool)` ASSUME_TAC THENL
[ASM_MESON_TAC[FINITE_SUBSET]; ALL_TAC] THEN
SUBGOAL_THEN `r tagged_partial_division_of interval[a:real^M,b] /\
g0 fine r /\ g1 fine r`
STRIP_ASSUME_TAC THENL
[ASM_MESON_TAC[TAGGED_PARTIAL_DIVISION_SUBSET; FINE_SUBSET];
ALL_TAC] THEN
MATCH_MP_TAC(MESON[] `!r. vsum q s = vsum r s /\ norm(vsum r s) <= e
==> norm(vsum q s:real^N) <= e`) THEN
EXISTS_TAC `r:(real^M#(real^M->bool))->bool` THEN CONJ_TAC THENL
[MATCH_MP_TAC VSUM_SUPERSET THEN ASM_REWRITE_TAC[FORALL_PAIR_THM] THEN
EXPAND_TAC "r" THEN REWRITE_TAC[IN_ELIM_PAIR_THM] THEN
MAP_EVERY X_GEN_TAC [`x:real^M`; `k:real^M->bool`] THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN ASM_REWRITE_TAC[] THEN
SUBGOAL_THEN `(x:real^M) IN k` ASSUME_TAC THENL
[ASM_MESON_TAC[tagged_partial_division_of]; ALL_TAC] THEN
DISCH_THEN(fun th -> ASSUME_TAC th THEN MP_TAC th) THEN
REWRITE_TAC[SUBSET] THEN DISCH_THEN(MP_TAC o SPEC `x:real^M`) THEN
ASM_REWRITE_TAC[IN_ELIM_THM] THEN DISCH_TAC THEN ASM_REWRITE_TAC[] THEN
REWRITE_TAC[VECTOR_ARITH `c - i - (c - j):real^N = j - i`] THEN
REWRITE_TAC[VECTOR_SUB_EQ] THEN MATCH_MP_TAC INTEGRAL_EQ THEN
ASM SET_TAC[];
ALL_TAC] THEN
W(MP_TAC o PART_MATCH (lhand o rand) VSUM_NORM o lhand o snd) THEN
ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN
ONCE_REWRITE_TAC[LAMBDA_PAIR_THM] THEN REWRITE_TAC[] THEN
MAP_EVERY ABBREV_TAC
[`s:(real^M#(real^M->bool))->bool =
{(x,k) | (x,k) IN r /\ x IN {x | x$i <= c}}`;
`t:(real^M#(real^M->bool))->bool =
{(x,k) | (x,k) IN r /\ ~(x IN {x | x$i <= c})}`] THEN
SUBGOAL_THEN
`(s:(real^M#(real^M->bool))->bool) SUBSET r /\
(t:(real^M#(real^M->bool))->bool) SUBSET r`
STRIP_ASSUME_TAC THENL
[MAP_EVERY EXPAND_TAC ["s";
let INDEFINITE_INTEGRAL_CONTINUOUS = prove
(`!f:real^M->real^N a b c d e.
f
integrable_on interval[a,b] /\
c
IN interval[a,b] /\ d
IN interval[a,b] /\ &0 < e
==> ?k. &0 < k /\
!c' d'. c'
IN interval[a,b] /\
d'
IN interval[a,b] /\
norm(c' - c) <= k /\ norm(d' - d) <= k
==> norm(integral(interval[c',d']) f -
integral(interval[c,d]) f) < e`,
REPEAT STRIP_TAC THEN
ONCE_REWRITE_TAC[MESON[] `(?k. P k /\ Q k) <=> ~(!k. P k ==> ~Q k)`] THEN
DISCH_THEN(MP_TAC o GEN `n:num` o SPEC `inv(&n + &1)`) THEN
PURE_REWRITE_TAC[
NOT_FORALL_THM; NOT_IMP] THEN
REWRITE_TAC[
REAL_LT_INV_EQ; REAL_ARITH `&0 < &n + &1`;
SKOLEM_THM] THEN
REWRITE_TAC[
NOT_EXISTS_THM;
REAL_NOT_LT; GSYM
CONJ_ASSOC] THEN
MAP_EVERY X_GEN_TAC [`u:num->real^M`; `v:num->real^M`] THEN
REWRITE_TAC[
FORALL_AND_THM] THEN STRIP_TAC THEN
ABBREV_TAC
`k:real^M->bool =
UNIONS (
IMAGE (\i. {x | x$i = (c:real^M)$i}
UNION {x | x$i = (d:real^M)$i})
(1..dimindex(:M)))` THEN
SUBGOAL_THEN `negligible(k:real^M->bool)` ASSUME_TAC THENL
[EXPAND_TAC "k" THEN MATCH_MP_TAC
NEGLIGIBLE_UNIONS THEN
SIMP_TAC[
FINITE_IMAGE;
FINITE_NUMSEG;
FORALL_IN_IMAGE] THEN
X_GEN_TAC `i:num` THEN REWRITE_TAC[
IN_NUMSEG] THEN STRIP_TAC THEN
ASM_SIMP_TAC[
NEGLIGIBLE_UNION;
NEGLIGIBLE_STANDARD_HYPERPLANE];
ALL_TAC] THEN
MP_TAC(ISPECL
[`\n:num x. if x
IN interval[u n,v n] then
if x
IN k then vec 0 else (f:real^M->real^N) x
else vec 0`;
`\x. if x
IN interval[c,d] then
if x
IN k then vec 0 else (f:real^M->real^N) x
else vec 0`;
`a:real^M`; `b:real^M`]
EQUIINTEGRABLE_LIMIT) THEN
REWRITE_TAC[NOT_IMP] THEN REPEAT CONJ_TAC THENL
[SUBGOAL_THEN
`(\x. if x
IN k then vec 0 else (f:real^M->real^N) x)
integrable_on interval[a,b]`
MP_TAC THENL
[UNDISCH_TAC `(f:real^M->real^N)
integrable_on interval[a,b]` THEN
MATCH_MP_TAC
INTEGRABLE_SPIKE THEN EXISTS_TAC `k:real^M->bool` THEN
ASM_REWRITE_TAC[] THEN SET_TAC[];
ALL_TAC] THEN
DISCH_THEN(MP_TAC o MATCH_MP
EQUIINTEGRABLE_CLOSED_INTERVAL_RESTRICTIONS) THEN
MATCH_MP_TAC(REWRITE_RULE[
IMP_CONJ_ALT]
EQUIINTEGRABLE_SUBSET) THEN
REWRITE_TAC[
SUBSET;
FORALL_IN_GSPEC;
IN_UNIV] THEN
REWRITE_TAC[
IN_ELIM_THM] THEN
X_GEN_TAC `n:num` THEN MAP_EVERY EXISTS_TAC
[`(u:num->real^M) n`; `(v:num->real^M) n`] THEN
REWRITE_TAC[];
X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN
ASM_CASES_TAC `(x:real^M)
IN k` THEN
ASM_REWRITE_TAC[
COND_ID;
LIM_CONST] THEN MATCH_MP_TAC
LIM_EVENTUALLY THEN
REWRITE_TAC[
EVENTUALLY_SEQUENTIALLY] THEN
MP_TAC(SPEC `inf (
IMAGE (\i. min (abs((x:real^M)$i - (c:real^M)$i))
(abs((x:real^M)$i - (d:real^M)$i)))
(1..dimindex(:M)))`
REAL_ARCH_INV) THEN
SIMP_TAC[
REAL_LT_INF_FINITE;
FINITE_IMAGE;
IMAGE_EQ_EMPTY;
FINITE_NUMSEG;
NUMSEG_EMPTY;
NOT_LT;
DIMINDEX_GE_1] THEN
ASM_SIMP_TAC[
FORALL_IN_IMAGE;
REAL_LT_MIN;
IN_NUMSEG] THEN
UNDISCH_TAC `~((x:real^M)
IN k)` THEN EXPAND_TAC "k" THEN
REWRITE_TAC[
UNIONS_IMAGE;
IN_ELIM_THM;
NOT_EXISTS_THM] THEN
REWRITE_TAC[
IN_NUMSEG; SET_RULE
`~(p /\ x
IN (s
UNION t)) <=> p ==> ~(x
IN s) /\ ~(x
IN t)`] THEN
REWRITE_TAC[
IN_ELIM_THM; REAL_ARITH `&0 < abs(x - y) <=> ~(x = y)`] THEN
DISCH_TAC THEN ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC
MONO_EXISTS THEN X_GEN_TAC `N:num` THEN STRIP_TAC THEN
X_GEN_TAC `n:num` THEN DISCH_TAC THEN
SUBGOAL_THEN `x
IN interval[(u:num->real^M) n,v n] <=> x
IN interval[c,d]`
(fun th -> REWRITE_TAC[th]) THEN
REWRITE_TAC[
IN_INTERVAL] THEN AP_TERM_TAC THEN
GEN_REWRITE_TAC I [
FUN_EQ_THM] THEN X_GEN_TAC `i:num` THEN
ASM_CASES_TAC `1 <= i /\ i <= dimindex(:M)` THEN ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC(REAL_ARITH
`!N n. abs(u - c) <= n /\ abs(v - d) <= n /\
N < abs(x - c) /\ N < abs(x - d) /\ n <= N
==> (u <= x /\ x <= v <=> c <= x /\ x <= d)`) THEN
MAP_EVERY EXISTS_TAC [`inv(&N)`; `inv(&n + &1)`] THEN
ASM_SIMP_TAC[] THEN REPEAT (CONJ_TAC THENL
[ASM_MESON_TAC[
REAL_LE_TRANS;
COMPONENT_LE_NORM;
VECTOR_SUB_COMPONENT];
ALL_TAC]) THEN
MATCH_MP_TAC
REAL_LE_INV2 THEN
REWRITE_TAC[REAL_OF_NUM_ADD; REAL_OF_NUM_LE;
REAL_OF_NUM_LT] THEN
ASM_ARITH_TAC;
DISCH_THEN(MP_TAC o CONJUNCT2) THEN
REWRITE_TAC[
INTEGRAL_RESTRICT_INTER] THEN
SUBGOAL_THEN
`interval[c:real^M,d]
INTER interval[a,b] = interval[c,d] /\
!n:num. interval[u n,v n]
INTER interval[a,b] = interval[u n,v n]`
(fun th -> SIMP_TAC[th])
THENL
[REWRITE_TAC[SET_RULE `s
INTER t = s <=> s
SUBSET t`] THEN
REWRITE_TAC[
SUBSET_INTERVAL] THEN
RULE_ASSUM_TAC(REWRITE_RULE[
IN_INTERVAL]) THEN ASM_MESON_TAC[];
ALL_TAC] THEN
REWRITE_TAC[
LIM_SEQUENTIALLY] THEN
DISCH_THEN(MP_TAC o SPEC `e:real`) THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_THEN `N:num` (MP_TAC o SPEC `N:num`)) THEN
REWRITE_TAC[
LE_REFL;
REAL_NOT_LT] THEN
FIRST_ASSUM(fun th -> MP_TAC(SPEC `N:num` th) THEN MATCH_MP_TAC
(NORM_ARITH `x = a /\ y = b ==> e <= norm(x - y) ==> e <= dist(a,b)`)) THEN
CONJ_TAC THEN MATCH_MP_TAC
INTEGRAL_SPIKE THEN
EXISTS_TAC `k:real^M->bool` THEN ASM_SIMP_TAC[
IN_DIFF]]);;
let SECOND_MEAN_VALUE_THEOREM_FULL = prove
(`!f:real^1->real^1 g a b.
~(interval[a,b] = {}) /\
f
integrable_on interval [a,b] /\
(!x y. x
IN interval[a,b] /\ y
IN interval[a,b] /\ drop x <= drop y
==> g x <= g y)
==> ?c. c
IN interval [a,b] /\
((\x. g x % f x)
has_integral
(g(a) % integral (interval[a,c]) f +
g(b) % integral (interval[c,b]) f)) (interval[a,b])`,
let lemma1 = prove
(`!f:real->real s.
(!x. x IN s ==> &0 <= f x /\ f x <= &1)
==> (!n x. x IN s /\ ~(n = 0)
==> abs(f x -
sum(1..n) (\k. if &k / &n <= f(x)
then inv(&n) else &0)) < inv(&n))`,
REPEAT STRIP_TAC THEN
SUBGOAL_THEN `?m. floor(&n * (f:real->real) x) = &m` CHOOSE_TAC THENL
[MATCH_MP_TAC FLOOR_POS THEN ASM_SIMP_TAC[REAL_LE_MUL; REAL_POS];
ALL_TAC] THEN
SUBGOAL_THEN `!k. &k / &n <= (f:real->real) x <=> k <= m` ASSUME_TAC THENL
[REWRITE_TAC[GSYM REAL_OF_NUM_LE] THEN
FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN
ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_OF_NUM_LT; LE_1] THEN
SIMP_TAC[REAL_LE_FLOOR; INTEGER_CLOSED; REAL_MUL_SYM];
ALL_TAC] THEN
ASM_REWRITE_TAC[GSYM SUM_RESTRICT_SET] THEN
FIRST_X_ASSUM(MP_TAC o SPEC `n + 1`) THEN
REWRITE_TAC[GSYM REAL_OF_NUM_ADD; real_div; REAL_ADD_RDISTRIB] THEN
ASM_SIMP_TAC[REAL_MUL_RINV; REAL_MUL_LID; REAL_OF_NUM_EQ] THEN
ASM_SIMP_TAC[REAL_ARITH `y <= &1 /\ &0 < i ==> ~(&1 + i <= y)`;
REAL_LT_INV_EQ; REAL_OF_NUM_LT; LE_1; NOT_LE] THEN
SIMP_TAC[IN_NUMSEG; ARITH_RULE
`m < n + 1 ==> ((1 <= k /\ k <= n) /\ k <= m <=> 1 <= k /\ k <= m)`] THEN
DISCH_TAC THEN REWRITE_TAC[GSYM numseg; SUM_CONST_NUMSEG; ADD_SUB] THEN
MATCH_MP_TAC REAL_LT_LCANCEL_IMP THEN EXISTS_TAC `abs(&n)` THEN
REWRITE_TAC[GSYM REAL_ABS_MUL] THEN
ASM_SIMP_TAC[REAL_ABS_NUM; REAL_MUL_RINV; REAL_OF_NUM_EQ] THEN
ASM_SIMP_TAC[REAL_OF_NUM_LT; LE_1; REAL_SUB_LDISTRIB; GSYM real_div] THEN
ASM_SIMP_TAC[REAL_DIV_LMUL; REAL_OF_NUM_EQ] THEN
MATCH_MP_TAC(REAL_ARITH `f <= x /\ x < f + &1 ==> abs(x - f) < &1`) THEN
FIRST_X_ASSUM(SUBST1_TAC o SYM) THEN REWRITE_TAC[FLOOR]) in
let lemma2 = prove
(`!f:real^1->real^N g a b.
f integrable_on interval[a,b] /\
(!x y. drop x <= drop y ==> g(x) <= g(y))
==> {(\x. if c <= g(x) then f x else vec 0) | c IN (:real)}
equiintegrable_on interval[a,b]`,
REPEAT STRIP_TAC THEN
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM EQUIINTEGRABLE_ON_SING]) THEN
DISCH_THEN(fun th ->
MP_TAC(SPEC `f:real^1->real^N` (MATCH_MP (REWRITE_RULE[IMP_CONJ]
EQUIINTEGRABLE_HALFSPACE_RESTRICTIONS_GE) th)) THEN
MP_TAC(SPEC `f:real^1->real^N` (MATCH_MP (REWRITE_RULE[IMP_CONJ]
EQUIINTEGRABLE_HALFSPACE_RESTRICTIONS_GT) th)) THEN
MP_TAC th) THEN
SIMP_TAC[IN_SING; REAL_LE_REFL] THEN
SUBGOAL_THEN `{(\x. vec 0):real^1->real^N} equiintegrable_on interval[a,b]`
MP_TAC THENL
[REWRITE_TAC[EQUIINTEGRABLE_ON_SING; INTEGRABLE_CONST]; ALL_TAC] THEN
REPEAT(ONCE_REWRITE_TAC[IMP_IMP] THEN
DISCH_THEN(MP_TAC o MATCH_MP EQUIINTEGRABLE_UNION)) THEN
REWRITE_TAC[NUMSEG_SING; DIMINDEX_1; IN_SING] THEN
REWRITE_TAC[SET_RULE `{m i c h | i = 1 /\ c IN (:real) /\ h = f} =
{m 1 c f | c IN (:real)}`] THEN
MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] EQUIINTEGRABLE_SUBSET) THEN
REWRITE_TAC[SUBSET; FORALL_IN_GSPEC; IN_UNIV] THEN
X_GEN_TAC `y:real` THEN
ASM_CASES_TAC `!x. y <= (g:real^1->real) x` THENL
[ASM_REWRITE_TAC[ETA_AX; IN_UNION; IN_SING]; ALL_TAC] THEN
ASM_CASES_TAC `!x. ~(y <= (g:real^1->real) x)` THENL
[ASM_REWRITE_TAC[ETA_AX; IN_UNION; IN_SING]; ALL_TAC] THEN
MP_TAC(ISPEC `IMAGE drop {x | y <= (g:real^1->real) x}` INF) THEN
REWRITE_TAC[FORALL_IN_IMAGE; IN_ELIM_THM; IMAGE_EQ_EMPTY] THEN
ANTS_TAC THENL
[ASM_REWRITE_TAC[EXTENSION; IN_ELIM_THM; NOT_IN_EMPTY] THEN
ASM_MESON_TAC[REAL_LE_TRANS; REAL_LE_TOTAL];
STRIP_TAC THEN REWRITE_TAC[real_gt; real_ge]] THEN
REWRITE_TAC[IN_UNION; GSYM DISJ_ASSOC] THEN
ASM_CASES_TAC `y <= g(lift(inf(IMAGE drop {x | y <= g x})))` THENL
[REPEAT DISJ2_TAC; REPLICATE_TAC 2 DISJ2_TAC THEN DISJ1_TAC] THEN
REWRITE_TAC[IN_ELIM_THM] THEN
EXISTS_TAC `inf(IMAGE drop {x | y <= g x})` THEN
REWRITE_TAC[FUN_EQ_THM] THEN
MATCH_MP_TAC(MESON[]
`(!x. P x <=> Q x)
==> !x. (if P x then f x else b) = (if Q x then f x else b)`) THEN
X_GEN_TAC `x:real^1` THEN REWRITE_TAC[GSYM REAL_NOT_LE; GSYM drop] THEN
ASM_MESON_TAC[REAL_LE_TOTAL; REAL_LT_ANTISYM; REAL_LE_TRANS; LIFT_DROP]) in
let lemma3 = prove
(`!f:real^1->real^N g a b.
f integrable_on interval[a,b] /\
(!x y. drop x <= drop y ==> g(x) <= g(y))
==> {(\x. vsum (1..n)
(\k. if &k / &n <= g x then inv(&n) % f(x) else vec 0)) |
~(n = 0)}
equiintegrable_on interval[a,b]`,
REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o
MATCH_MP lemma2) THEN
DISCH_THEN(MP_TAC o MATCH_MP
(INST_TYPE [`:num`,`:A`] EQUIINTEGRABLE_SUM)) THEN
MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] EQUIINTEGRABLE_SUBSET) THEN
REWRITE_TAC[SUBSET; FORALL_IN_GSPEC; IN_UNIV] THEN X_GEN_TAC `n:num` THEN
DISCH_TAC THEN REWRITE_TAC[IN_ELIM_THM] THEN
MAP_EVERY EXISTS_TAC [`1..n`; `\k:num. inv(&n)`;
`\k x. if &k / &n <= g x then (f:real^1->real^N) x else vec 0`] THEN
ASM_SIMP_TAC[SUM_CONST_NUMSEG; ADD_SUB; REAL_MUL_RINV; REAL_OF_NUM_EQ] THEN
REWRITE_TAC[FINITE_NUMSEG; COND_RAND; COND_RATOR; VECTOR_MUL_RZERO] THEN
X_GEN_TAC `k:num` THEN
REWRITE_TAC[IN_NUMSEG; REAL_LE_INV_EQ; REAL_POS] THEN STRIP_TAC THEN
EXISTS_TAC `&k / &n` THEN REWRITE_TAC[]) in
let lemma4 = prove
(`!f:real^1->real^1 g a b.
~(interval[a,b] = {}) /\
f integrable_on interval[a,b] /\
(!x y. drop x <= drop y ==> g(x) <= g(y)) /\
(!x. x IN interval[a,b] ==> &0 <= g x /\ g x <= &1)
==> (\x. g(x) % f(x)) integrable_on interval[a,b] /\
?c. c IN interval[a,b] /\
integral (interval[a,b]) (\x. g(x) % f(x)) =
integral (interval[c,b]) f`,
REPEAT GEN_TAC THEN STRIP_TAC THEN
SUBGOAL_THEN
`?m M. IMAGE (\x. integral (interval[x,b]) (f:real^1->real^1))
(interval[a,b]) = interval[m,M]`
STRIP_ASSUME_TAC THENL
[REWRITE_TAC[GSYM CONNECTED_COMPACT_INTERVAL_1] THEN CONJ_TAC THENL
[MATCH_MP_TAC CONNECTED_CONTINUOUS_IMAGE;
MATCH_MP_TAC COMPACT_CONTINUOUS_IMAGE] THEN
ASM_SIMP_TAC[INDEFINITE_INTEGRAL_CONTINUOUS_LEFT; CONVEX_CONNECTED;
CONVEX_INTERVAL; COMPACT_INTERVAL];
ALL_TAC] THEN
MP_TAC(ISPECL[`f:real^1->real^1`; `g:real^1->real`; `a:real^1`; `b:real^1`]
lemma3) THEN
ASM_REWRITE_TAC[] THEN DISCH_TAC THEN
SUBGOAL_THEN
`!n. ?c. c IN interval[a,b] /\
integral (interval[c,b]) (f:real^1->real^1) =
integral (interval[a,b])
(\x. vsum (1..n)
(\k. if &k / &n <= g x then inv(&n) % f x else vec 0))`
MP_TAC THENL
[X_GEN_TAC `n:num` THEN ASM_CASES_TAC `n = 0` THENL
[ASM_REWRITE_TAC[VSUM_CLAUSES_NUMSEG; ARITH_EQ; INTEGRAL_0] THEN
EXISTS_TAC `b:real^1` THEN ASM_REWRITE_TAC[ENDS_IN_INTERVAL] THEN
SIMP_TAC[INTEGRAL_NULL; CONTENT_EQ_0_1; REAL_LE_REFL];
ALL_TAC] THEN
MP_TAC(ISPECL [`f:real^1->real^1`; `g:real^1->real`;
`a:real^1`; `b:real^1`] lemma2) THEN
ASM_REWRITE_TAC[equiintegrable_on; FORALL_IN_GSPEC; IN_UNIV] THEN
DISCH_THEN(ASSUME_TAC o CONJUNCT1) THEN
REWRITE_TAC[MESON[VECTOR_MUL_RZERO]
`(if p then a % x else vec 0:real^1) =
a % (if p then x else vec 0)`] THEN
ASM_SIMP_TAC[VSUM_LMUL; INTEGRAL_CMUL; INTEGRABLE_VSUM; ETA_AX;
FINITE_NUMSEG; INTEGRAL_VSUM] THEN
SUBGOAL_THEN
`!y:real. ?d:real^1.
d IN interval[a,b] /\
integral (interval[a,b]) (\x. if y <= g x then f x else vec 0) =
integral (interval[d,b]) (f:real^1->real^1)`
MP_TAC THENL
[X_GEN_TAC `y:real` THEN
SUBGOAL_THEN
`{x | y <= g x} = {} \/
{x | y <= g x} = (:real^1) \/
(?a. {x | y <= g x} = {x | a <= drop x}) \/
(?a. {x | y <= g x} = {x | a < drop x})`
MP_TAC THENL
[MATCH_MP_TAC(TAUT `(~a /\ ~b ==> c \/ d) ==> a \/ b \/ c \/ d`) THEN
DISCH_TAC THEN
MP_TAC(ISPEC `IMAGE drop {x | y <= (g:real^1->real) x}` INF) THEN
ASM_REWRITE_TAC[FORALL_IN_IMAGE; IN_ELIM_THM; IMAGE_EQ_EMPTY] THEN
ANTS_TAC THENL
[FIRST_ASSUM(MP_TAC o CONJUNCT2) THEN
REWRITE_TAC[EXTENSION; IN_ELIM_THM; IN_UNIV] THEN
ASM_MESON_TAC[REAL_LE_TRANS; REAL_LE_TOTAL];
STRIP_TAC] THEN
ASM_CASES_TAC `y <= g(lift(inf(IMAGE drop {x | y <= g x})))` THENL
[DISJ1_TAC; DISJ2_TAC] THEN
REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN
EXISTS_TAC `inf(IMAGE drop {x | y <= g x})` THEN
REWRITE_TAC[FUN_EQ_THM] THEN
X_GEN_TAC `x:real^1` THEN
REWRITE_TAC[GSYM REAL_NOT_LE; GSYM drop] THEN
ASM_MESON_TAC[REAL_LE_TOTAL; REAL_LT_ANTISYM;
REAL_LE_TRANS; LIFT_DROP];
REWRITE_TAC[EXTENSION; IN_UNIV; NOT_IN_EMPTY; IN_ELIM_THM] THEN
DISCH_THEN(DISJ_CASES_THEN2 ASSUME_TAC MP_TAC) THENL
[EXISTS_TAC `b:real^1` THEN ASM_REWRITE_TAC[] THEN
SIMP_TAC[INTEGRAL_NULL; CONTENT_EQ_0_1; REAL_LE_REFL] THEN
ASM_REWRITE_TAC[ENDS_IN_INTERVAL; INTEGRAL_0];
ALL_TAC] THEN
DISCH_THEN(DISJ_CASES_THEN2 ASSUME_TAC MP_TAC) THENL
[EXISTS_TAC `a:real^1` THEN
ASM_REWRITE_TAC[ETA_AX; ENDS_IN_INTERVAL];
ALL_TAC] THEN
GEN_REWRITE_TAC LAND_CONV [OR_EXISTS_THM] THEN
REWRITE_TAC[EXISTS_DROP] THEN
DISCH_THEN(X_CHOOSE_THEN `d:real^1` ASSUME_TAC) THEN
ASM_CASES_TAC `drop d < drop a` THENL
[EXISTS_TAC `a:real^1` THEN
ASM_REWRITE_TAC[ETA_AX; ENDS_IN_INTERVAL] THEN
MATCH_MP_TAC INTEGRAL_EQ THEN
REWRITE_TAC[IN_DIFF; IN_INTERVAL_1; NOT_IN_EMPTY] THEN
GEN_TAC THEN COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
UNDISCH_TAC `~(y <= (g:real^1->real) x)` THEN
FIRST_X_ASSUM DISJ_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
ASM_REAL_ARITH_TAC;
ALL_TAC] THEN
ASM_CASES_TAC `drop b < drop d` THENL
[EXISTS_TAC `b:real^1` THEN
SIMP_TAC[INTEGRAL_NULL; CONTENT_EQ_0_1; REAL_LE_REFL] THEN
ASM_REWRITE_TAC[ENDS_IN_INTERVAL; INTEGRAL_0] THEN
MATCH_MP_TAC INTEGRAL_EQ_0 THEN REWRITE_TAC[IN_INTERVAL_1] THEN
REPEAT STRIP_TAC THEN
COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
UNDISCH_TAC `y <= (g:real^1->real) x` THEN
FIRST_X_ASSUM DISJ_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
ASM_REAL_ARITH_TAC;
ALL_TAC] THEN
EXISTS_TAC `d:real^1` THEN
ASM_REWRITE_TAC[IN_INTERVAL_1; GSYM REAL_NOT_LT] THEN
ONCE_REWRITE_TAC[SET_RULE
`~((g:real^1->real) x < y) <=> x IN {x | ~(g x < y)}`] THEN
REWRITE_TAC[INTEGRAL_RESTRICT_INTER] THEN
MATCH_MP_TAC INTEGRAL_SPIKE_SET THEN
MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `{d:real^1}` THEN
REWRITE_TAC[NEGLIGIBLE_SING; REAL_NOT_LT; SUBSET] THEN GEN_TAC THEN
REWRITE_TAC[SUBSET; IN_UNION; IN_INTER; IN_DIFF; IN_INTERVAL_1;
IN_ELIM_THM; IN_SING; GSYM DROP_EQ] THEN
FIRST_X_ASSUM DISJ_CASES_TAC THEN ASM_REWRITE_TAC[] THEN
ASM_REAL_ARITH_TAC];
DISCH_THEN(MP_TAC o GEN `k:num` o SPEC `&k / &n`) THEN
REWRITE_TAC[SKOLEM_THM; FORALL_AND_THM; LEFT_IMP_EXISTS_THM] THEN
X_GEN_TAC `d:num->real^1` THEN STRIP_TAC THEN
FIRST_ASSUM(MP_TAC o MATCH_MP (SET_RULE
`IMAGE f s = t ==> !y. y IN t ==> ?x. x IN s /\ f x = y`)) THEN
REWRITE_TAC[GSYM VSUM_LMUL] THEN DISCH_THEN MATCH_MP_TAC THEN
MATCH_MP_TAC(REWRITE_RULE[CONVEX_INDEXED]
(CONJUNCT1(SPEC_ALL CONVEX_INTERVAL))) THEN
REWRITE_TAC[SUM_CONST_NUMSEG; ADD_SUB; REAL_LE_INV_EQ; REAL_POS] THEN
ASM_SIMP_TAC[REAL_MUL_RINV; REAL_OF_NUM_EQ] THEN ASM SET_TAC[]];
REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM; FORALL_AND_THM] THEN
X_GEN_TAC `c:num->real^1` THEN DISCH_THEN(STRIP_ASSUME_TAC o GSYM)] THEN
SUBGOAL_THEN `compact(interval[a:real^1,b])` MP_TAC THENL
[REWRITE_TAC[COMPACT_INTERVAL]; REWRITE_TAC[compact]] THEN
DISCH_THEN(MP_TAC o SPEC `c:num->real^1`) THEN
ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC
[`d:real^1`; `s:num->num`] THEN STRIP_TAC THEN
MP_TAC(ISPECL
[`\n:num x. vsum (1..(s n))
(\k. if &k / &(s n) <= g x
then inv(&(s n)) % (f:real^1->real^1) x
else vec 0)`;
`\x. g x % (f:real^1->real^1) x`; `a:real^1`; `b:real^1`]
EQUIINTEGRABLE_LIMIT) THEN
ASM_REWRITE_TAC[] THEN ANTS_TAC THENL
[CONJ_TAC THENL
[MATCH_MP_TAC EQUIINTEGRABLE_SUBSET THEN
EXISTS_TAC
`{\x. vsum(1..0) (\k. if &k / &0 <= g x
then inv(&0) % (f:real^1->real^1) x else vec 0)}
UNION
{\x. vsum (1..n)
(\k. if &k / &n <= g x then inv (&n) % f x else vec 0)
| ~(n = 0)}` THEN
CONJ_TAC THENL
[MATCH_MP_TAC EQUIINTEGRABLE_UNION THEN ASM_REWRITE_TAC[] THEN
REWRITE_TAC[EQUIINTEGRABLE_ON_SING; VSUM_CLAUSES_NUMSEG;
ARITH_EQ] THEN
REWRITE_TAC[INTEGRABLE_0];
REWRITE_TAC[SUBSET; FORALL_IN_GSPEC; IN_UNIV; IN_UNION] THEN
REWRITE_TAC[IN_ELIM_THM; IN_SING] THEN
X_GEN_TAC `n:num` THEN ASM_CASES_TAC `(s:num->num) n = 0` THEN
ASM_REWRITE_TAC[] THEN DISJ2_TAC THEN
EXISTS_TAC `(s:num->num) n` THEN ASM_REWRITE_TAC[]];
X_GEN_TAC `x:real^1` THEN DISCH_TAC THEN REWRITE_TAC[] THEN
ONCE_REWRITE_TAC[MESON[VECTOR_MUL_LZERO]
`(if p then a % x else vec 0) = (if p then a else &0) % x`] THEN
REWRITE_TAC[VSUM_RMUL] THEN MATCH_MP_TAC LIM_VMUL THEN
REWRITE_TAC[LIM_SEQUENTIALLY; o_DEF; DIST_LIFT] THEN
X_GEN_TAC `e:real` THEN DISCH_TAC THEN
MP_TAC(ISPEC `e:real` REAL_ARCH_INV) THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN
X_GEN_TAC `N:num` THEN STRIP_TAC THEN X_GEN_TAC `n:num` THEN
DISCH_TAC THEN
ONCE_REWRITE_TAC[REAL_ABS_SUB] THEN
MATCH_MP_TAC REAL_LT_TRANS THEN EXISTS_TAC `inv(&n)` THEN
CONJ_TAC THENL
[MP_TAC(ISPECL
[`(g:real^1->real) o lift`; `IMAGE drop (interval[a,b])`]
lemma1) THEN
ASM_REWRITE_TAC[FORALL_IN_IMAGE; o_DEF; LIFT_DROP; IMP_CONJ;
RIGHT_FORALL_IMP_THM] THEN
REWRITE_TAC[IMP_IMP] THEN DISCH_TAC THEN
MATCH_MP_TAC REAL_LTE_TRANS THEN
EXISTS_TAC `inv(&((s:num->num) n))` THEN CONJ_TAC THENL
[FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[];
MATCH_MP_TAC REAL_LE_INV2 THEN
REWRITE_TAC[REAL_OF_NUM_LE; REAL_OF_NUM_LT]] THEN
FIRST_ASSUM(MP_TAC o SPEC `n:num` o MATCH_MP MONOTONE_BIGGER) THEN
ASM_ARITH_TAC;
MATCH_MP_TAC REAL_LET_TRANS THEN EXISTS_TAC `inv(&N)` THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC REAL_LE_INV2 THEN
REWRITE_TAC[REAL_OF_NUM_LE; REAL_OF_NUM_LT] THEN ASM_ARITH_TAC]];
STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
EXISTS_TAC `d:real^1` THEN ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC(ISPEC `sequentially` LIM_UNIQUE) THEN
EXISTS_TAC `\n. integral (interval [c((s:num->num) n),b])
(f:real^1->real^1)` THEN
ASM_REWRITE_TAC[TRIVIAL_LIMIT_SEQUENTIALLY] THEN
MP_TAC(ISPECL [`f:real^1->real^1`; `a:real^1`; `b:real^1`]
INDEFINITE_INTEGRAL_CONTINUOUS_LEFT) THEN
ASM_REWRITE_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN] THEN
DISCH_THEN(MP_TAC o SPEC `d:real^1`) THEN ASM_REWRITE_TAC[] THEN
REWRITE_TAC[CONTINUOUS_WITHIN_SEQUENTIALLY] THEN
DISCH_THEN(MP_TAC o SPEC `(c:num->real^1) o (s:num->num)`) THEN
ASM_REWRITE_TAC[] THEN ASM_REWRITE_TAC[o_DEF]]) in
REPEAT GEN_TAC THEN STRIP_TAC THEN
SUBGOAL_THEN `(g:real^1->real) a <= g b` MP_TAC THENL
[FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[ENDS_IN_INTERVAL] THEN
ASM_MESON_TAC[INTERVAL_EQ_EMPTY_1; REAL_LET_TOTAL];
ALL_TAC] THEN
REWRITE_TAC[REAL_LE_LT] THEN STRIP_TAC THENL
[ALL_TAC;
SUBGOAL_THEN
`!x. x IN interval[a,b] ==> g(x) % (f:real^1->real^1)(x) = g(a) % f x`
ASSUME_TAC THENL
[X_GEN_TAC `x:real^1` THEN
REWRITE_TAC[IN_INTERVAL_1] THEN STRIP_TAC THEN
AP_THM_TAC THEN AP_TERM_TAC THEN
RULE_ASSUM_TAC(REWRITE_RULE
[IN_INTERVAL_1; INTERVAL_EQ_EMPTY_1; REAL_NOT_LT]) THEN
ASM_MESON_TAC[REAL_LE_ANTISYM; REAL_LE_TRANS; REAL_LE_TOTAL];
ALL_TAC] THEN
EXISTS_TAC `a:real^1` THEN ASM_REWRITE_TAC[ENDS_IN_INTERVAL] THEN
MATCH_MP_TAC HAS_INTEGRAL_EQ THEN
EXISTS_TAC `\x. g(a:real^1) % (f:real^1->real^1) x` THEN
ASM_SIMP_TAC[INTEGRAL_NULL; CONTENT_EQ_0_1; REAL_LE_REFL] THEN
ASM_SIMP_TAC[INTEGRAL_CMUL; VECTOR_MUL_RZERO; VECTOR_ADD_LID] THEN
MATCH_MP_TAC HAS_INTEGRAL_CMUL THEN
ASM_REWRITE_TAC[GSYM HAS_INTEGRAL_INTEGRAL]] THEN
MP_TAC(ISPECL
[`f:real^1->real^1`;
`\x. if drop x < drop a then &0
else if drop b < drop x then &1
else (g(x) - g(a)) / (g(b) - g(a))`;
`a:real^1`; `b:real^1`]
lemma4) THEN ASM_REWRITE_TAC[] THEN
ANTS_TAC THENL
[CONJ_TAC THEN
REPEAT GEN_TAC THEN
REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[REAL_POS; REAL_LE_REFL]) THEN
TRY ASM_REAL_ARITH_TAC THEN
ASM_SIMP_TAC[IN_INTERVAL_1; REAL_LE_DIV2_EQ; REAL_SUB_LT] THEN
ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_LE_RDIV_EQ; REAL_SUB_LT] THEN
ASM_REWRITE_TAC[REAL_MUL_LZERO; REAL_MUL_LID; REAL_SUB_LE;
REAL_ARITH `x - a <= y - a <=> x <= y`] THEN
REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
REWRITE_TAC[IN_INTERVAL_1] THEN ASM_REAL_ARITH_TAC;
ALL_TAC] THEN
REWRITE_TAC[RIGHT_AND_EXISTS_THM] THEN
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `c:real^1` THEN
ONCE_REWRITE_TAC[TAUT `a /\ b /\ c ==> d <=> b ==> a /\ c ==> d`] THEN
DISCH_TAC THEN ASM_REWRITE_TAC[GSYM HAS_INTEGRAL_INTEGRABLE_INTEGRAL] THEN
DISCH_THEN(MP_TAC o SPEC `(g:real^1->real) b - g a` o
MATCH_MP HAS_INTEGRAL_CMUL) THEN
FIRST_ASSUM(MP_TAC o MATCH_MP INTEGRABLE_INTEGRAL) THEN
DISCH_THEN(MP_TAC o SPEC `(g:real^1->real)(a)` o
MATCH_MP HAS_INTEGRAL_CMUL) THEN REWRITE_TAC[IMP_IMP] THEN
DISCH_THEN(MP_TAC o MATCH_MP HAS_INTEGRAL_ADD) THEN
MP_TAC(ISPECL [`f:real^1->real^1`; `a:real^1`; `b:real^1`; `c:real^1`]
INTEGRAL_COMBINE) THEN
ANTS_TAC THENL [ASM_MESON_TAC[IN_INTERVAL_1]; ALL_TAC] THEN
DISCH_THEN(SUBST1_TAC o SYM) THEN
REWRITE_TAC[VECTOR_ARITH
`ga % (i1 + i2) + (gb - ga) % i2:real^N = ga % i1 + gb % i2`] THEN
MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] HAS_INTEGRAL_EQ) THEN
X_GEN_TAC `x:real^1` THEN REWRITE_TAC[IN_INTERVAL_1] THEN STRIP_TAC THEN
ASM_SIMP_TAC[GSYM REAL_NOT_LE; VECTOR_MUL_ASSOC] THEN
ASM_SIMP_TAC[REAL_DIV_LMUL; REAL_LT_IMP_NZ; REAL_SUB_LT] THEN
VECTOR_ARITH_TAC);;
let SECOND_MEAN_VALUE_THEOREM = prove
(`!f:real^1->real^1 g a b.
~(interval[a,b] = {}) /\
f
integrable_on interval [a,b] /\
(!x y. x
IN interval[a,b] /\ y
IN interval[a,b] /\ drop x <= drop y
==> g x <= g y)
==> ?c. c
IN interval [a,b] /\
integral (interval[a,b]) (\x. g x % f x) =
g(a) % integral (interval[a,c]) f +
g(b) % integral (interval[c,b]) f`,
let SECOND_MEAN_VALUE_THEOREM_GEN_FULL = prove
(`!f:real^1->real^1 g a b u v.
~(interval[a,b] = {}) /\ f
integrable_on interval [a,b] /\
(!x. x
IN interval(a,b) ==> u <= g x /\ g x <= v) /\
(!x y. x
IN interval[a,b] /\ y
IN interval[a,b] /\ drop x <= drop y
==> g x <= g y)
==> ?c. c
IN interval [a,b] /\
((\x. g x % f x)
has_integral
(u % integral (interval[a,c]) f +
v % integral (interval[c,b]) f)) (interval[a,b])`,
let SECOND_MEAN_VALUE_THEOREM_GEN = prove
(`!f:real^1->real^1 g a b u v.
~(interval[a,b] = {}) /\ f
integrable_on interval [a,b] /\
(!x. x
IN interval(a,b) ==> u <= g x /\ g x <= v) /\
(!x y. x
IN interval[a,b] /\ y
IN interval[a,b] /\ drop x <= drop y
==> g x <= g y)
==> ?c. c
IN interval [a,b] /\
integral (interval[a,b]) (\x. g x % f x) =
u % integral (interval[a,c]) f +
v % integral (interval[c,b]) f`,
let SECOND_MEAN_VALUE_THEOREM_BONNET = prove
(`!f:real^1->real^1 g a b.
~(interval[a,b] = {}) /\ f
integrable_on interval[a,b] /\
(!x. x
IN interval[a,b] ==> &0 <= g x) /\
(!x y. x
IN interval[a,b] /\ y
IN interval[a,b] /\ drop x <= drop y
==> g x <= g y)
==> ?c. c
IN interval [a,b] /\
integral (interval[a,b]) (\x. g x % f x) =
g(b) % integral (interval[c,b]) f`,
let INTEGRABLE_INCREASING_PRODUCT_UNIV = prove
(`!f:real^1->real^N g B.
f
integrable_on (:real^1) /\
(!x y. drop x <= drop y ==> g x <= g y) /\
(!x. abs(g x) <= B)
==> (\x. g x % f x)
integrable_on (:real^1)`,
let lemma = prove
(`!f:real^1->real^1 g B.
f integrable_on (:real^1) /\
(!x y. drop x <= drop y ==> g x <= g y) /\
(!x. abs(g x) <= B)
==> (\x. g x % f x) integrable_on (:real^1)`,
REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[INTEGRABLE_ALT_SUBSET] THEN
REWRITE_TAC[IN_UNIV; ETA_AX] THEN STRIP_TAC THEN
MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN CONJ_TAC THENL
[REPEAT GEN_TAC THEN MATCH_MP_TAC INTEGRABLE_INCREASING_PRODUCT THEN
ASM_SIMP_TAC[];
DISCH_TAC] THEN
X_GEN_TAC `e:real` THEN DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `e / (&8 * abs B + &8)`) THEN
ASM_SIMP_TAC[REAL_LT_DIV; REAL_ARITH `&0 < &8 * abs B + &8`] THEN
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `C:real` THEN STRIP_TAC THEN
ASM_REWRITE_TAC[] THEN
SUBGOAL_THEN `~(ball(vec 0:real^1,C) = {})` ASSUME_TAC THENL
[ASM_REWRITE_TAC[BALL_EQ_EMPTY; REAL_NOT_LE]; ALL_TAC] THEN
MAP_EVERY X_GEN_TAC [`a:real^1`; `b:real^1`; `c:real^1`; `d:real^1`] THEN
STRIP_TAC THEN SUBGOAL_THEN
`~(interval[a:real^1,b] = {}) /\ ~(interval[c:real^1,d] = {})`
MP_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
REWRITE_TAC[INTERVAL_EQ_EMPTY_1; REAL_NOT_LT] THEN STRIP_TAC THEN
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [SUBSET_INTERVAL_1]) THEN
ASM_REWRITE_TAC[GSYM REAL_NOT_LE] THEN STRIP_TAC THEN
MP_TAC(ISPECL [`\x. g x % (f:real^1->real^1) x`;
`c:real^1`; `b:real^1`; `a:real^1`] INTEGRAL_COMBINE) THEN
MP_TAC(ISPECL [`\x. g x % (f:real^1->real^1) x`;
`c:real^1`; `d:real^1`; `b:real^1`] INTEGRAL_COMBINE) THEN
ASM_REWRITE_TAC[] THEN
ANTS_TAC THENL [ASM_REAL_ARITH_TAC; DISCH_THEN(SUBST1_TAC o SYM)] THEN
DISCH_THEN(SUBST1_TAC o SYM) THEN
REWRITE_TAC[REAL_NOT_LE; NORM_ARITH
`norm(ab - ((ca + ab) + bd):real^1) = norm(ca + bd)`] THEN
MP_TAC(ISPECL[`f:real^1->real^1`; `g:real^1->real`; `c:real^1`; `a:real^1`]
SECOND_MEAN_VALUE_THEOREM) THEN
ASM_SIMP_TAC[INTERVAL_EQ_EMPTY_1; REAL_NOT_LT] THEN
DISCH_THEN(X_CHOOSE_THEN `u:real^1` STRIP_ASSUME_TAC) THEN
MP_TAC(ISPECL[`f:real^1->real^1`; `g:real^1->real`; `b:real^1`; `d:real^1`]
SECOND_MEAN_VALUE_THEOREM) THEN
ASM_SIMP_TAC[INTERVAL_EQ_EMPTY_1; REAL_NOT_LT] THEN
DISCH_THEN(X_CHOOSE_THEN `v:real^1` STRIP_ASSUME_TAC) THEN
ASM_REWRITE_TAC[] THEN
SUBGOAL_THEN
`!x y. drop y <= drop a
==> norm(integral (interval[x,y]) (f:real^1->real^1))
< e / (&4 * abs B + &4)`
(LABEL_TAC "L")
THENL
[REPEAT STRIP_TAC THEN
ASM_CASES_TAC `drop x <= drop y` THENL
[FIRST_X_ASSUM(fun th ->
MP_TAC(SPECL[`a:real^1`; `b:real^1`; `y:real^1`; `b:real^1`] th) THEN
MP_TAC(SPECL[`a:real^1`; `b:real^1`; `x:real^1`; `b:real^1`] th)) THEN
ASM_REWRITE_TAC[SUBSET_INTERVAL_1; REAL_LE_REFL] THEN
ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
MP_TAC(ISPECL [`f:real^1->real^1`; `x:real^1`; `b:real^1`; `y:real^1`]
INTEGRAL_COMBINE) THEN
ASM_REWRITE_TAC[] THEN ANTS_TAC THENL
[ASM_ARITH_TAC; DISCH_THEN(SUBST1_TAC o SYM)] THEN
MATCH_MP_TAC(NORM_ARITH
`&2 * d = e
==> norm(ab - (xy + yb)) < d
==> norm(ab - yb) < d
==> norm(xy:real^1) < e`) THEN
CONV_TAC REAL_FIELD;
SUBGOAL_THEN `interval[x:real^1,y] = {}` SUBST1_TAC THENL
[REWRITE_TAC[INTERVAL_EQ_EMPTY_1] THEN ASM_REAL_ARITH_TAC;
REWRITE_TAC[INTEGRAL_EMPTY; NORM_0] THEN
MATCH_MP_TAC REAL_LT_DIV THEN ASM_REAL_ARITH_TAC]];
ALL_TAC] THEN
SUBGOAL_THEN
`!x y. drop b <= drop x
==> norm(integral (interval[x,y]) (f:real^1->real^1))
< e / (&4 * abs B + &4)`
(LABEL_TAC "R")
THENL
[REPEAT STRIP_TAC THEN
ASM_CASES_TAC `drop x <= drop y` THENL
[FIRST_X_ASSUM(fun th ->
MP_TAC(SPECL[`a:real^1`; `b:real^1`; `a:real^1`; `x:real^1`] th) THEN
MP_TAC(SPECL[`a:real^1`; `b:real^1`; `a:real^1`; `y:real^1`] th)) THEN
ASM_REWRITE_TAC[SUBSET_INTERVAL_1; REAL_LE_REFL] THEN
ANTS_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
MP_TAC(ISPECL [`f:real^1->real^1`; `a:real^1`; `y:real^1`; `x:real^1`]
INTEGRAL_COMBINE) THEN
ASM_REWRITE_TAC[] THEN ANTS_TAC THENL
[ASM_ARITH_TAC; DISCH_THEN(SUBST1_TAC o SYM)] THEN
MATCH_MP_TAC(NORM_ARITH
`&2 * d = e
==> norm(ab - (ax + xy)) < d
==> norm(ab - ax) < d
==> norm(xy:real^1) < e`) THEN
CONV_TAC REAL_FIELD;
SUBGOAL_THEN `interval[x:real^1,y] = {}` SUBST1_TAC THENL
[REWRITE_TAC[INTERVAL_EQ_EMPTY_1] THEN ASM_REAL_ARITH_TAC;
REWRITE_TAC[INTEGRAL_EMPTY; NORM_0] THEN
MATCH_MP_TAC REAL_LT_DIV THEN ASM_REAL_ARITH_TAC]];
ALL_TAC] THEN
RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1]) THEN
MATCH_MP_TAC REAL_LET_TRANS THEN
EXISTS_TAC `&4 * B * e / (&4 * abs B + &4)` THEN CONJ_TAC THENL
[MATCH_MP_TAC(NORM_ARITH
`(norm a <= e /\ norm b <= e) /\ (norm c <= e /\ norm d <= e)
==> norm((a + b) + (c + d):real^1) <= &4 * e`) THEN
REWRITE_TAC[NORM_MUL] THEN CONJ_TAC THENL
[CONJ_TAC THEN MATCH_MP_TAC REAL_LE_MUL2 THEN
ASM_REWRITE_TAC[NORM_POS_LE; REAL_ABS_POS] THEN
MATCH_MP_TAC REAL_LT_IMP_LE THEN
REMOVE_THEN "L" MATCH_MP_TAC THEN ASM_REAL_ARITH_TAC;
CONJ_TAC THEN MATCH_MP_TAC REAL_LE_MUL2 THEN
ASM_REWRITE_TAC[NORM_POS_LE; REAL_ABS_POS] THEN
MATCH_MP_TAC REAL_LT_IMP_LE THEN
REMOVE_THEN "R" MATCH_MP_TAC THEN ASM_REAL_ARITH_TAC];
REWRITE_TAC[REAL_ARITH
`&4 * B * e / y < e <=> e * (&4 * B) / y < e * &1`] THEN
ASM_SIMP_TAC[REAL_LT_LMUL_EQ; REAL_LT_LDIV_EQ;
REAL_ARITH `&0 < &4 * abs B + &4`] THEN
REAL_ARITH_TAC]) in
GEN_TAC THEN ONCE_REWRITE_TAC[INTEGRABLE_COMPONENTWISE] THEN
REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN
REWRITE_TAC[VECTOR_MUL_COMPONENT; LIFT_CMUL] THEN
MATCH_MP_TAC lemma THEN EXISTS_TAC `B:real` THEN ASM_SIMP_TAC[]);;
let HAS_BOUNDED_VARIATION_DARBOUX_STRICT = prove
(`!f a b.
f
has_bounded_variation_on interval[a,b] <=>
?g h. (!x y. x
IN interval[a,b] /\ y
IN interval[a,b] /\ drop x < drop y
==> drop(g x) < drop(g y)) /\
(!x y. x
IN interval[a,b] /\ y
IN interval[a,b] /\ drop x < drop y
==> drop(h x) < drop(h y)) /\
(!x. f x = g x - h x)`,
REPEAT GEN_TAC THEN REWRITE_TAC[
HAS_BOUNDED_VARIATION_DARBOUX] THEN
EQ_TAC THEN REWRITE_TAC[
LEFT_IMP_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC [`g:real^1->real^1`; `h:real^1->real^1`] THEN
STRIP_TAC THENL
[MAP_EVERY EXISTS_TAC [`\x:real^1. g x + x`; `\x:real^1. h x + x`] THEN
ASM_REWRITE_TAC[VECTOR_ARITH `(a + x) - (b + x):real^1 = a - b`] THEN
REPEAT STRIP_TAC THEN REWRITE_TAC[
DROP_ADD] THEN
MATCH_MP_TAC
REAL_LET_ADD2 THEN ASM_REWRITE_TAC[] THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_SIMP_TAC[
REAL_LT_IMP_LE];
MAP_EVERY EXISTS_TAC [`g:real^1->real^1`; `h:real^1->real^1`] THEN
ASM_REWRITE_TAC[
REAL_LE_LT;
DROP_EQ] THEN ASM_MESON_TAC[]]);;
let INCREASING_LEFT_LIMIT_1 = prove
(`!f a b c.
(!x y. x
IN interval[a,b] /\ y
IN interval[a,b] /\ drop x <= drop y
==> drop(f x) <= drop(f y)) /\
c
IN interval[a,b]
==> ?l. (f --> l) (at c within interval[a,c])`,
REPEAT STRIP_TAC THEN EXISTS_TAC
`lift(sup {drop(f x) | x
IN interval[a,b] /\ drop x < drop c})` THEN
ONCE_REWRITE_TAC[
SIMPLE_IMAGE_GEN] THEN REWRITE_TAC[
LIM_WITHIN] THEN
REWRITE_TAC[
DIST_REAL; GSYM drop] THEN
ASM_CASES_TAC `{x | x
IN interval[a,b] /\ drop x < drop c} = {}` THENL
[GEN_TAC THEN DISCH_TAC THEN EXISTS_TAC `&1` THEN
REWRITE_TAC[
REAL_LT_01] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
EXTENSION]) THEN
MATCH_MP_TAC
MONO_FORALL THEN GEN_TAC THEN MATCH_MP_TAC(TAUT
`(a ==> ~b) ==> a ==> b ==> c`) THEN
REWRITE_TAC[
NOT_IN_EMPTY;
IN_ELIM_THM;
IN_INTERVAL_1] THEN
RULE_ASSUM_TAC(REWRITE_RULE[
IN_INTERVAL_1]) THEN ASM_REAL_ARITH_TAC;
ALL_TAC] THEN
X_GEN_TAC `e:real` THEN DISCH_TAC THEN
MP_TAC(ISPEC `{drop(f x) | x
IN interval[a,b] /\ drop x < drop c}`
SUP) THEN
ASM_REWRITE_TAC[
FORALL_IN_GSPEC] THEN ANTS_TAC THENL
[CONJ_TAC THENL
[ONCE_REWRITE_TAC[
SIMPLE_IMAGE_GEN] THEN ASM_SIMP_TAC[
IMAGE_EQ_EMPTY];
EXISTS_TAC `drop(f(b:real^1))` THEN REPEAT STRIP_TAC THEN
FIRST_X_ASSUM MATCH_MP_TAC THEN REWRITE_TAC[
IN_INTERVAL_1] THEN
RULE_ASSUM_TAC(REWRITE_RULE[
IN_INTERVAL_1]) THEN ASM_REAL_ARITH_TAC];
ONCE_REWRITE_TAC[
SIMPLE_IMAGE_GEN] THEN REWRITE_TAC[
IMAGE_ID] THEN
ABBREV_TAC `s = sup (
IMAGE (\x. drop(f x))
{x | x
IN interval[a,b] /\ drop x < drop c})` THEN
REWRITE_TAC[
LIFT_DROP] THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (MP_TAC o SPEC `s - e:real`)) THEN
ASM_SIMP_TAC[REAL_ARITH `&0 < e ==> ~(s <= s - e)`;
NOT_FORALL_THM] THEN
REWRITE_TAC[NOT_IMP;
REAL_NOT_LE;
IN_INTERVAL_1] THEN
DISCH_THEN(X_CHOOSE_THEN `d:real^1` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `drop c - drop d` THEN
RULE_ASSUM_TAC(REWRITE_RULE[
IN_INTERVAL_1]) THEN
CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
X_GEN_TAC `x:real^1` THEN STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPECL [`d:real^1`; `x:real^1`]) THEN
FIRST_X_ASSUM(MP_TAC o SPEC `x:real^1`) THEN ASM_REAL_ARITH_TAC]);;
let DECREASING_LEFT_LIMIT_1 = prove
(`!f a b c.
(!x y. x
IN interval[a,b] /\ y
IN interval[a,b] /\ drop x <= drop y
==> drop(f y) <= drop(f x)) /\
c
IN interval[a,b]
==> ?l. (f --> l) (at c within interval[a,c])`,
let INCREASING_RIGHT_LIMIT_1 = prove
(`!f a b c.
(!x y. x
IN interval[a,b] /\ y
IN interval[a,b] /\ drop x <= drop y
==> drop(f x) <= drop(f y)) /\
c
IN interval[a,b]
==> ?l. (f --> l) (at c within interval[c,b])`,
let DECREASING_RIGHT_LIMIT_1 = prove
(`!f a b c.
(!x y. x
IN interval[a,b] /\ y
IN interval[a,b] /\ drop x <= drop y
==> drop(f y) <= drop(f x)) /\
c
IN interval[a,b]
==> ?l. (f --> l) (at c within interval[c,b])`,
let VECTOR_VARIATION_CONTINUOUS_LEFT = prove
(`!f:real^1->real^1 a b c.
f
has_bounded_variation_on interval[a,b] /\ c
IN interval[a,b]
==> ((\x. lift(
vector_variation(interval[a,x]) f))
continuous (at c within interval[a,c]) <=>
f continuous (at c within interval[a,c]))`,
REPEAT STRIP_TAC THEN EQ_TAC THENL
[REWRITE_TAC[
continuous_within] THEN
REWRITE_TAC[
DIST_LIFT;
IN_ELIM_THM;
DIST_REAL; GSYM drop] 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
MATCH_MP_TAC
MONO_EXISTS THEN X_GEN_TAC `d:real` THEN STRIP_TAC THEN
ASM_REWRITE_TAC[] THEN X_GEN_TAC `x:real^1` THEN STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `x:real^1`) THEN ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC(REWRITE_RULE[
IMP_CONJ]
REAL_LET_TRANS) THEN
REWRITE_TAC[GSYM
DROP_SUB] THEN
MP_TAC(ISPECL [`f:real^1->real^1`; `a:real^1`; `c:real^1`; `x:real^1`]
VECTOR_VARIATION_COMBINE) THEN
ANTS_TAC THENL
[RULE_ASSUM_TAC(REWRITE_RULE[
IN_INTERVAL_1]) THEN
REPEAT(CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC]) THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[
IMP_CONJ]
HAS_BOUNDED_VARIATION_ON_SUBSET)) THEN
REWRITE_TAC[
SUBSET_INTERVAL_1] THEN ASM_REAL_ARITH_TAC;
ALL_TAC] THEN
DISCH_THEN(SUBST1_TAC o SYM) THEN
REWRITE_TAC[REAL_ARITH `abs(a - (a + b)) = abs b`] THEN
REWRITE_TAC[drop; GSYM
NORM_REAL] THEN
MATCH_MP_TAC(REAL_ARITH `x <= a ==> x <= abs a`) THEN
ONCE_REWRITE_TAC[
NORM_SUB] THEN
MATCH_MP_TAC
VECTOR_VARIATION_GE_NORM_FUNCTION THEN CONJ_TAC THENL
[FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[
IMP_CONJ]
HAS_BOUNDED_VARIATION_ON_SUBSET));
REWRITE_TAC[
SEGMENT_1] THEN COND_CASES_TAC] THEN
REWRITE_TAC[
SUBSET_INTERVAL_1] THEN
RULE_ASSUM_TAC(REWRITE_RULE[
IN_INTERVAL_1]) THEN ASM_REAL_ARITH_TAC;
ALL_TAC] THEN
DISCH_TAC THEN ASM_CASES_TAC `c
limit_point_of interval[a:real^1,c]` THENL
[ALL_TAC;
ASM_REWRITE_TAC[
CONTINUOUS_WITHIN;
LIM;
TRIVIAL_LIMIT_WITHIN]] THEN
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
HAS_BOUNDED_VARIATION_DARBOUX]) THEN
REWRITE_TAC[
LEFT_IMP_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC [`g:real^1->real^1`; `h:real^1->real^1`] THEN
STRIP_TAC THEN
MP_TAC(ISPECL [`h:real^1->real^1`; `a:real^1`; `b:real^1`; `c:real^1`]
INCREASING_LEFT_LIMIT_1) THEN
MP_TAC(ISPECL [`g:real^1->real^1`; `a:real^1`; `b:real^1`; `c:real^1`]
INCREASING_LEFT_LIMIT_1) THEN
ASM_REWRITE_TAC[
LEFT_IMP_EXISTS_THM] THEN
X_GEN_TAC `gc:real^1` THEN DISCH_TAC THEN
X_GEN_TAC `hc:real^1` THEN DISCH_TAC THEN
ABBREV_TAC `k = gc - (g:real^1->real^1) c` THEN
SUBGOAL_THEN `hc - (h:real^1->real^1) c = k` ASSUME_TAC THENL
[EXPAND_TAC "k" THEN
ONCE_REWRITE_TAC[VECTOR_ARITH
`hc' - hc:real^1 = gc' - gc <=> gc' - hc' = gc - hc`] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
CONTINUOUS_WITHIN]) THEN
ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC(REWRITE_RULE[TAUT `a /\ b /\ c ==> d <=> a /\ b ==> c ==> d`]
LIM_UNIQUE) THEN
ASM_REWRITE_TAC[
TRIVIAL_LIMIT_WITHIN] THEN
GEN_REWRITE_TAC (RATOR_CONV o LAND_CONV) [GSYM ETA_AX] THEN
ASM_SIMP_TAC[
LIM_SUB];
ALL_TAC] THEN
MAP_EVERY ABBREV_TAC
[`g':real^1->real^1 = \x. if drop c <= drop x then g(x) + k else g(x)`;
`h':real^1->real^1 = \x. if drop c <= drop x then h(x) + k else h(x)`] THEN
SUBGOAL_THEN
`(!x y. x
IN interval[a,c] /\ y
IN interval[a,c] /\ drop x <= drop y
==> drop(g' x) <= drop(g' y)) /\
(!x y. x
IN interval[a,c] /\ y
IN interval[a,c] /\ drop x <= drop y
==> drop(h' x) <= drop(h' y))`
STRIP_ASSUME_TAC THENL
[MAP_EVERY EXPAND_TAC ["g'";
let VECTOR_VARIATION_CONTINUOUS_RIGHT = prove
(`!f:real^1->real^1 a b c.
f
has_bounded_variation_on interval[a,b] /\ c
IN interval[a,b]
==> ((\x. lift(
vector_variation(interval[a,x]) f))
continuous (at c within interval[c,b]) <=>
f continuous (at c within interval[c,b]))`,
REPEAT STRIP_TAC THEN EQ_TAC THENL
[REWRITE_TAC[
continuous_within] THEN
REWRITE_TAC[
DIST_LIFT;
IN_ELIM_THM;
DIST_REAL; GSYM drop] 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
MATCH_MP_TAC
MONO_EXISTS THEN X_GEN_TAC `d:real` THEN STRIP_TAC THEN
ASM_REWRITE_TAC[] THEN X_GEN_TAC `x:real^1` THEN STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `x:real^1`) THEN ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC(REWRITE_RULE[
IMP_CONJ]
REAL_LET_TRANS) THEN
REWRITE_TAC[GSYM
DROP_SUB] THEN
MP_TAC(ISPECL [`f:real^1->real^1`; `a:real^1`; `x:real^1`; `c:real^1`]
VECTOR_VARIATION_COMBINE) THEN
ANTS_TAC THENL
[RULE_ASSUM_TAC(REWRITE_RULE[
IN_INTERVAL_1]) THEN
REPEAT(CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC]) THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[
IMP_CONJ]
HAS_BOUNDED_VARIATION_ON_SUBSET)) THEN
REWRITE_TAC[
SUBSET_INTERVAL_1] THEN ASM_REAL_ARITH_TAC;
ALL_TAC] THEN
DISCH_THEN(SUBST1_TAC o SYM) THEN
REWRITE_TAC[REAL_ARITH `abs((a + b) - a) = abs b`] THEN
REWRITE_TAC[drop; GSYM
NORM_REAL] THEN
MATCH_MP_TAC(REAL_ARITH `x <= a ==> x <= abs a`) THEN
MATCH_MP_TAC
VECTOR_VARIATION_GE_NORM_FUNCTION THEN CONJ_TAC THENL
[FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[
IMP_CONJ]
HAS_BOUNDED_VARIATION_ON_SUBSET));
REWRITE_TAC[
SEGMENT_1] THEN COND_CASES_TAC] THEN
REWRITE_TAC[
SUBSET_INTERVAL_1] THEN
RULE_ASSUM_TAC(REWRITE_RULE[
IN_INTERVAL_1]) THEN ASM_REAL_ARITH_TAC;
ALL_TAC] THEN
DISCH_TAC THEN ASM_CASES_TAC `c
limit_point_of interval[c:real^1,b]` THENL
[ALL_TAC;
ASM_REWRITE_TAC[
CONTINUOUS_WITHIN;
LIM;
TRIVIAL_LIMIT_WITHIN]] THEN
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
HAS_BOUNDED_VARIATION_DARBOUX]) THEN
REWRITE_TAC[
LEFT_IMP_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC [`g:real^1->real^1`; `h:real^1->real^1`] THEN
STRIP_TAC THEN
MP_TAC(ISPECL [`h:real^1->real^1`; `a:real^1`; `b:real^1`; `c:real^1`]
INCREASING_RIGHT_LIMIT_1) THEN
MP_TAC(ISPECL [`g:real^1->real^1`; `a:real^1`; `b:real^1`; `c:real^1`]
INCREASING_RIGHT_LIMIT_1) THEN
ASM_REWRITE_TAC[
LEFT_IMP_EXISTS_THM] THEN
X_GEN_TAC `gc:real^1` THEN DISCH_TAC THEN
X_GEN_TAC `hc:real^1` THEN DISCH_TAC THEN
ABBREV_TAC `k = gc - (g:real^1->real^1) c` THEN
SUBGOAL_THEN `hc - (h:real^1->real^1) c = k` ASSUME_TAC THENL
[EXPAND_TAC "k" THEN
ONCE_REWRITE_TAC[VECTOR_ARITH
`hc' - hc:real^1 = gc' - gc <=> gc' - hc' = gc - hc`] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [
CONTINUOUS_WITHIN]) THEN
ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC(REWRITE_RULE[TAUT `a /\ b /\ c ==> d <=> a /\ b ==> c ==> d`]
LIM_UNIQUE) THEN
ASM_REWRITE_TAC[
TRIVIAL_LIMIT_WITHIN] THEN
GEN_REWRITE_TAC (RATOR_CONV o LAND_CONV) [GSYM ETA_AX] THEN
ASM_SIMP_TAC[
LIM_SUB];
ALL_TAC] THEN
MAP_EVERY ABBREV_TAC
[`g':real^1->real^1 = \x. if drop x <= drop c then g(x) + k else g(x)`;
`h':real^1->real^1 = \x. if drop x <= drop c then h(x) + k else h(x)`] THEN
SUBGOAL_THEN
`(!x y. x
IN interval[c,b] /\ y
IN interval[c,b] /\ drop x <= drop y
==> drop(g' x) <= drop(g' y)) /\
(!x y. x
IN interval[c,b] /\ y
IN interval[c,b] /\ drop x <= drop y
==> drop(h' x) <= drop(h' y))`
STRIP_ASSUME_TAC THENL
[MAP_EVERY EXPAND_TAC ["g'";
let HAS_BOUNDED_VARIATION_DARBOUX_STRONG = prove
(`!f a b.
f
has_bounded_variation_on interval[a,b]
==> ?g h. (!x. f x = g x - h x) /\
(!x y. x
IN interval[a,b] /\ y
IN interval[a,b] /\
drop x <= drop y
==> drop(g x) <= drop(g y)) /\
(!x y. x
IN interval[a,b] /\ y
IN interval[a,b] /\
drop x <= drop y
==> drop(h x) <= drop(h y)) /\
(!x y. x
IN interval[a,b] /\ y
IN interval[a,b] /\
drop x < drop y
==> drop(g x) < drop(g y)) /\
(!x y. x
IN interval[a,b] /\ y
IN interval[a,b] /\
drop x < drop y
==> drop(h x) < drop(h y)) /\
(!x. x
IN interval[a,b] /\
f continuous (at x within interval[a,x])
==> g continuous (at x within interval[a,x]) /\
h continuous (at x within interval[a,x])) /\
(!x. x
IN interval[a,b] /\
f continuous (at x within interval[x,b])
==> g continuous (at x within interval[x,b]) /\
h continuous (at x within interval[x,b])) /\
(!x. x
IN interval[a,b] /\
f continuous (at x within interval[a,b])
==> g continuous (at x within interval[a,b]) /\
h continuous (at x within interval[a,b]))`,
let HAS_BOUNDED_VARIATION_COUNTABLE_DISCONTINUITIES = prove
(`!f:real^1->real^1 a b.
f
has_bounded_variation_on interval[a,b]
==>
COUNTABLE {x | x
IN interval[a,b] /\ ~(f continuous at x)}`,
SUBGOAL_THEN
`!f a b.
(!x y. x
IN interval[a,b] /\ y
IN interval[a,b] /\ drop x <= drop y
==> drop(f x) <= drop(f y))
==>
COUNTABLE {x | x
IN interval[a,b] /\ ~(f continuous at x)}`
ASSUME_TAC THENL
[ALL_TAC;
REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o
GEN_REWRITE_RULE I [
HAS_BOUNDED_VARIATION_DARBOUX]) THEN
REWRITE_TAC[
LEFT_IMP_EXISTS_THM] THEN
MAP_EVERY X_GEN_TAC [`g:real^1->real^1`; `h:real^1->real^1`] THEN
STRIP_TAC THEN FIRST_X_ASSUM(fun th ->
MP_TAC(ISPECL [`g:real^1->real^1`; `a:real^1`; `b:real^1`] th) THEN
MP_TAC(ISPECL [`h:real^1->real^1`; `a:real^1`; `b:real^1`] th)) THEN
ASM_REWRITE_TAC[IMP_IMP; GSYM
COUNTABLE_UNION] THEN
MATCH_MP_TAC(REWRITE_RULE[
IMP_CONJ_ALT]
COUNTABLE_SUBSET) THEN
REWRITE_TAC[
SUBSET;
IN_UNION;
IN_ELIM_THM] THEN GEN_TAC THEN
MATCH_MP_TAC(TAUT
`(p /\ q ==> r) ==> a /\ ~r ==> a /\ ~p \/ a /\ ~q`) THEN
GEN_REWRITE_TAC (RAND_CONV o LAND_CONV) [GSYM ETA_AX] THEN
ASM_SIMP_TAC[
CONTINUOUS_SUB]] THEN
REPEAT STRIP_TAC THEN ASM_CASES_TAC `interval[a:real^1,b] = {}` THEN
ASM_REWRITE_TAC[
NOT_IN_EMPTY;
EMPTY_GSPEC;
COUNTABLE_EMPTY] THEN
RULE_ASSUM_TAC(REWRITE_RULE[
INTERVAL_EQ_EMPTY_1;
REAL_NOT_LT]) THEN
ASM_SIMP_TAC[
CLOSED_OPEN_INTERVAL_1] THEN
MATCH_MP_TAC
COUNTABLE_SUBSET THEN EXISTS_TAC
`a
INSERT b
INSERT
{x | x
IN interval(a,b) /\ ~((f:real^1->real^1) continuous at x)}` THEN
CONJ_TAC THENL [REWRITE_TAC[
COUNTABLE_INSERT]; SET_TAC[]] THEN
SUBGOAL_THEN
`(!c:real^1. c
IN interval(a,b) ==> c
limit_point_of interval[a,c]) /\
(!c:real^1. c
IN interval(a,b) ==> c
limit_point_of interval[c,b])`
STRIP_ASSUME_TAC THENL
[SIMP_TAC[
IN_INTERVAL_1;
REAL_LE_REFL;
LIMPT_OF_CONVEX;
CONVEX_INTERVAL;
REAL_LT_IMP_LE] THEN
REWRITE_TAC[GSYM
INTERVAL_SING; GSYM
SUBSET_ANTISYM_EQ] THEN
REWRITE_TAC[
SUBSET_INTERVAL_1] THEN REAL_ARITH_TAC;
ALL_TAC] THEN
MP_TAC(ISPECL [`f:real^1->real^1`; `a:real^1`; `b:real^1`]
INCREASING_LEFT_LIMIT_1) THEN
ASM_REWRITE_TAC[
RIGHT_IMP_EXISTS_THM;
SKOLEM_THM] THEN
DISCH_THEN(X_CHOOSE_THEN `l:real^1->real^1` (LABEL_TAC "l")) THEN
MP_TAC(ISPECL [`f:real^1->real^1`; `a:real^1`; `b:real^1`]
INCREASING_RIGHT_LIMIT_1) THEN
ASM_REWRITE_TAC[
RIGHT_IMP_EXISTS_THM;
SKOLEM_THM] THEN
DISCH_THEN(X_CHOOSE_THEN `r:real^1->real^1` (LABEL_TAC "r")) THEN
SUBGOAL_THEN
`!c. c
IN interval(a:real^1,b)
==> drop(l c) <= drop(f c) /\ drop(f c) <= drop(r c)`
ASSUME_TAC THENL
[REPEAT STRIP_TAC THENL
[MATCH_MP_TAC(ISPEC `at c within interval[a:real^1,c]`
LIM_DROP_UBOUND);
MATCH_MP_TAC(ISPEC `at c within interval[c:real^1,b]`
LIM_DROP_LBOUND)] THEN
EXISTS_TAC `f:real^1->real^1` THEN
ASM_SIMP_TAC[REWRITE_RULE[
SUBSET]
INTERVAL_OPEN_SUBSET_CLOSED;
TRIVIAL_LIMIT_WITHIN;
EVENTUALLY_WITHIN] THEN
EXISTS_TAC `&1` THEN REWRITE_TAC[
REAL_LT_01;
IN_INTERVAL_1] THEN
REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
REWRITE_TAC[
IN_INTERVAL_1] THEN
RULE_ASSUM_TAC(REWRITE_RULE[
IN_INTERVAL_1]) THEN ASM_REAL_ARITH_TAC;
ALL_TAC] THEN
SUBGOAL_THEN
`(!c x. c
IN interval(a:real^1,b) /\ x
IN interval[a,b] /\ drop x < drop c
==> drop(f x) <= drop(l c)) /\
(!c x. c
IN interval(a:real^1,b) /\ x
IN interval[a,b] /\ drop c < drop x
==> drop(r c) <= drop(f x))`
STRIP_ASSUME_TAC THENL
[REPEAT STRIP_TAC THENL
[MATCH_MP_TAC(ISPEC `at c within interval[a:real^1,c]`
LIM_DROP_LBOUND);
MATCH_MP_TAC(ISPEC `at c within interval[c:real^1,b]`
LIM_DROP_UBOUND)] THEN
EXISTS_TAC `f:real^1->real^1` THEN
ASM_SIMP_TAC[REWRITE_RULE[
SUBSET]
INTERVAL_OPEN_SUBSET_CLOSED;
TRIVIAL_LIMIT_WITHIN;
EVENTUALLY_WITHIN]
THENL
[EXISTS_TAC `drop c - drop x`; EXISTS_TAC `drop x - drop c`] THEN
ASM_REWRITE_TAC[
REAL_SUB_LT] THEN
X_GEN_TAC `y:real^1` THEN
REWRITE_TAC[
IN_INTERVAL_1;
IN_ELIM_THM;
DIST_REAL; GSYM drop] THEN
STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
ASM_REWRITE_TAC[
IN_INTERVAL_1] THEN
RULE_ASSUM_TAC(REWRITE_RULE[
IN_INTERVAL_1]) THEN ASM_REAL_ARITH_TAC;
ALL_TAC] THEN
REWRITE_TAC[
COUNTABLE;
ge_c] THEN
TRANS_TAC
CARD_LE_TRANS `rational` THEN
GEN_REWRITE_TAC RAND_CONV [GSYM
ge_c] THEN
REWRITE_TAC[
COUNTABLE_RATIONAL; GSYM
COUNTABLE;
le_c] THEN
SUBGOAL_THEN
`!c. c
IN interval(a,b) /\ ~((f:real^1->real^1) continuous at c)
==> drop(l(c:real^1)) < drop(r c)`
ASSUME_TAC THENL
[REPEAT STRIP_TAC THEN REWRITE_TAC[
REAL_LT_LE] THEN
CONJ_TAC THENL [ASM_MESON_TAC[
REAL_LE_TRANS]; ALL_TAC] THEN
REWRITE_TAC[
DROP_EQ] THEN DISCH_TAC THEN
SUBGOAL_THEN `l c = (f:real^1->real^1) c /\ r c = f c` ASSUME_TAC THENL
[ASM_MESON_TAC[REAL_LE_ANTISYM;
DROP_EQ]; ALL_TAC] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [
CONTINUOUS_AT]) THEN
REWRITE_TAC[] THEN
SUBGOAL_THEN
`((f:real^1->real^1) --> f c) (at c within interval(a,b))`
MP_TAC THENL
[ALL_TAC; ASM_SIMP_TAC[
OPEN_INTERVAL;
LIM_WITHIN_OPEN]] THEN
MATCH_MP_TAC
LIM_WITHIN_SUBSET THEN
EXISTS_TAC `interval[a:real^1,c]
UNION interval[c,b]` THEN
REWRITE_TAC[
LIM_WITHIN_UNION] THEN CONJ_TAC THENL
[ASM_MESON_TAC[REWRITE_RULE[
SUBSET]
INTERVAL_OPEN_SUBSET_CLOSED];
REWRITE_TAC[
SUBSET;
IN_UNION;
IN_INTERVAL_1] THEN REAL_ARITH_TAC];
ALL_TAC] THEN
SUBGOAL_THEN
`!c. c
IN interval(a,b) /\ ~((f:real^1->real^1) continuous at c)
==> ?q. rational q /\ drop(l c) < q /\ q < drop(r c)`
MP_TAC THENL
[REPEAT STRIP_TAC THEN
SUBGOAL_THEN `drop(l(c:real^1)) < drop(r c)` ASSUME_TAC THENL
[ASM_MESON_TAC[]; ALL_TAC] THEN
MP_TAC(ISPECL [`(drop(l(c:real^1)) + drop(r c)) / &2`;
`(drop(r c) - drop(l(c:real^1))) / &2`]
RATIONAL_APPROXIMATION) THEN
ASM_REWRITE_TAC[
REAL_HALF;
REAL_SUB_LT] THEN
MATCH_MP_TAC
MONO_EXISTS THEN SIMP_TAC[] THEN REAL_ARITH_TAC;
ALL_TAC] THEN
GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [
RIGHT_IMP_EXISTS_THM] THEN
REWRITE_TAC[
SKOLEM_THM;
IN_ELIM_THM;
IN_INTERVAL_1] THEN
MATCH_MP_TAC
MONO_EXISTS THEN X_GEN_TAC `q:real^1->real` THEN
SIMP_TAC[
IN] THEN DISCH_THEN(LABEL_TAC "*") THEN
MATCH_MP_TAC(MESON[
REAL_LE_TOTAL]
`(!x y. P x y ==> P y x) /\ (!x y. drop x <= drop y ==> P x y)
==> !x y. P x y`) THEN
CONJ_TAC THENL [MESON_TAC[]; ALL_TAC] THEN
MAP_EVERY X_GEN_TAC [`x:real^1`; `y:real^1`] THEN
REWRITE_TAC[
REAL_LE_LT;
DROP_EQ] THEN
ASM_CASES_TAC `x:real^1 = y` THEN ASM_REWRITE_TAC[] THEN
REPEAT STRIP_TAC THEN
SUBGOAL_THEN `q(x:real^1) < q(y)` MP_TAC THENL
[ALL_TAC; ASM_REWRITE_TAC[
REAL_LT_REFL]] THEN
MATCH_MP_TAC
REAL_LTE_TRANS THEN EXISTS_TAC `drop(r(x:real^1))` THEN
ASM_SIMP_TAC[
REAL_LT_IMP_LE] THEN
MATCH_MP_TAC
REAL_LE_TRANS THEN EXISTS_TAC `drop(l(y:real^1))` THEN
ASM_SIMP_TAC[
REAL_LT_IMP_LE] THEN
MATCH_MP_TAC
REAL_LE_TRANS THEN
EXISTS_TAC `drop(f(inv(&2) % (x + y):real^1))` THEN
CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
ASM_REWRITE_TAC[
IN_INTERVAL_1;
DROP_CMUL;
DROP_ADD] THEN
ASM_REAL_ARITH_TAC);;
let INTEGRAL_PASTECART_CONTINUOUS = prove
(`!f:real^(M,N)finite_sum->real^P a b c d.
f
continuous_on interval[pastecart a c,pastecart b d]
==> integral (interval[pastecart a c,pastecart b d]) f =
integral (interval[a,b])
(\x. integral (interval[c,d])
(\y. f(pastecart x y)))`,
let lemma1 = prove
(`!(f:real^(M,N)finite_sum->real^P) a b c d x.
f continuous_on interval [pastecart a c,pastecart b d] /\
x IN interval[a,b]
==> (\y. (f:real^(M,N)finite_sum->real^P) (pastecart x y))
integrable_on interval[c,d]`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC INTEGRABLE_CONTINUOUS THEN
GEN_REWRITE_TAC LAND_CONV [GSYM o_DEF] THEN
MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN
SIMP_TAC[CONTINUOUS_ON_PASTECART; CONTINUOUS_ON_CONST;
CONTINUOUS_ON_ID] THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ]
CONTINUOUS_ON_SUBSET)) THEN
ASM_SIMP_TAC[SUBSET; FORALL_IN_IMAGE; GSYM PCROSS_INTERVAL; PCROSS;
IN_ELIM_PASTECART_THM]) in
let lemma2 = prove
(`!(f:real^(M,N)finite_sum->real^P) a b c d.
f continuous_on interval [pastecart a c,pastecart b d]
==> (\x. integral (interval [c,d]) (\y. f (pastecart x y))) integrable_on
interval [a,b]`,
REPEAT STRIP_TAC THEN
FIRST_ASSUM(ASSUME_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] lemma1)) THEN
MATCH_MP_TAC INTEGRABLE_CONTINUOUS THEN REWRITE_TAC[continuous_on] THEN
X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN
X_GEN_TAC `e:real` THEN DISCH_TAC THEN
ASM_SIMP_TAC[dist; GSYM INTEGRAL_SUB] THEN
ASM_CASES_TAC `content(interval[c:real^N,d]) = &0` THENL
[ASM_SIMP_TAC[INTEGRAL_NULL; NORM_0] THEN MESON_TAC[REAL_LT_01];
ALL_TAC] THEN
FIRST_ASSUM(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ]
COMPACT_UNIFORMLY_CONTINUOUS)) THEN
REWRITE_TAC[COMPACT_INTERVAL; uniformly_continuous_on] THEN
RULE_ASSUM_TAC(REWRITE_RULE[GSYM CONTENT_LT_NZ]) THEN
DISCH_THEN(MP_TAC o SPEC `e / &2 / content(interval[c:real^N,d])`) THEN
ASM_SIMP_TAC[REAL_LT_DIV; REAL_HALF; FORALL_PASTECART] THEN
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:real` THEN
REWRITE_TAC[GSYM PCROSS_INTERVAL; PCROSS; dist;
IN_ELIM_PASTECART_THM] THEN
STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
X_GEN_TAC `x':real^M` THEN DISCH_TAC THEN
MATCH_MP_TAC(REAL_ARITH `&0 < e /\ x <= e / &2 ==> x < e`) THEN
ASM_REWRITE_TAC[] THEN
SUBGOAL_THEN
`e / &2 = e / &2 / content(interval[c:real^N,d]) * content(interval[c,d])`
SUBST1_TAC THENL
[UNDISCH_TAC `&0 < content(interval[c:real^N,d])` THEN
CONV_TAC REAL_FIELD;
MATCH_MP_TAC HAS_INTEGRAL_BOUND THEN
EXISTS_TAC `\y. (f:real^(M,N)finite_sum->real^P) (pastecart x' y) -
(f:real^(M,N)finite_sum->real^P) (pastecart x y)` THEN
ASM_SIMP_TAC[REAL_LT_IMP_LE; REAL_HALF; REAL_LT_DIV;
GSYM HAS_INTEGRAL_INTEGRAL; INTEGRABLE_SUB; lemma1] THEN
REPEAT STRIP_TAC THEN
MATCH_MP_TAC REAL_LT_IMP_LE THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
ASM_SIMP_TAC[NORM_PASTECART; PASTECART_SUB; VECTOR_SUB_REFL; NORM_0] THEN
CONV_TAC REAL_RAT_REDUCE_CONV THEN
ASM_SIMP_TAC[REAL_ADD_RID; POW_2_SQRT; NORM_POS_LE]]) in
let lemma3 = prove
(`!f:real^(M,N)finite_sum->real^P e s.
&0 < e /\ f continuous_on s
==> operative(/\)
(\k. !a b c d.
interval[pastecart a c,pastecart b d] SUBSET k /\
interval[pastecart a c,pastecart b d] SUBSET s
==> norm(integral (interval[pastecart a c,pastecart b d]) f -
integral (interval[a,b])
(\x. integral (interval[c,d])
(\y. f(pastecart x y))))
<= e * content(interval[pastecart a c,pastecart b d]))`,
REPEAT STRIP_TAC THEN REWRITE_TAC[operative; NEUTRAL_AND] THEN
CONJ_TAC THEN MAP_EVERY X_GEN_TAC
[`A:real^(M,N)finite_sum`;`B:real^(M,N)finite_sum`] THENL
[DISCH_TAC THEN
MAP_EVERY X_GEN_TAC [`a:real^M`; `b:real^M`; `c:real^N`; `d:real^N`] THEN
DISCH_TAC THEN
SUBGOAL_THEN
`content(interval[pastecart (a:real^M) (c:real^N),pastecart b d]) = &0`
(fun th -> ASSUME_TAC th THEN MP_TAC th)
THENL [ASM_MESON_TAC[CONTENT_0_SUBSET]; ALL_TAC] THEN
REWRITE_TAC[CONTENT_PASTECART; REAL_ENTIRE] THEN STRIP_TAC THEN
ASM_SIMP_TAC[INTEGRAL_NULL; REAL_MUL_LZERO; REAL_MUL_RZERO;
VECTOR_SUB_REFL; NORM_0; REAL_LE_REFL; INTEGRAL_0];
MAP_EVERY X_GEN_TAC [`l:real`; `k:num`] THEN STRIP_TAC THEN EQ_TAC THENL
[REWRITE_TAC[AND_FORALL_THM] THEN
REPEAT(MATCH_MP_TAC MONO_FORALL THEN GEN_TAC) THEN
MATCH_MP_TAC(TAUT
`(P1 ==> P) /\ (P2 ==> P)
==> (P ==> Q) ==> (P1 ==> Q) /\ (P2 ==> Q)`) THEN
SET_TAC[];
DISCH_TAC]] THEN
MAP_EVERY X_GEN_TAC [`a:real^M`; `b:real^M`; `c:real^N`; `d:real^N`] THEN
STRIP_TAC THEN RULE_ASSUM_TAC(REWRITE_RULE[DIMINDEX_FINITE_SUM]) THEN
ASM_CASES_TAC `k <= dimindex(:M)` THENL
[FIRST_X_ASSUM(CONJUNCTS_THEN2
(MP_TAC o SPECL
[`a:real^M`;
`(lambda i. if i = k then min (b$k) l else (b:real^M)$i):real^M`;
`c:real^N`; `d:real^N`])
(MP_TAC o SPECL
[`(lambda i. if i = k then max (a$k) l else (a:real^M)$i):real^M`;
`b:real^M`; `c:real^N`; `d:real^N`])) THEN
ASM_SIMP_TAC[GSYM INTERVAL_SPLIT; GSYM PCROSS_INTERVAL; PCROSS] THEN
SUBGOAL_THEN
`!P Q. { pastecart (x:real^M) (y:real^N) |
x IN interval[a,b] INTER {x | P (x$k)} /\ Q y} =
{pastecart x y | x IN interval[a,b] /\ Q y} INTER {x | P (x$k)}`
(fun th -> REWRITE_TAC[th])
THENL
[REWRITE_TAC[EXTENSION; FORALL_PASTECART; IN_ELIM_PASTECART_THM;
IN_INTER] THEN
ASM_SIMP_TAC[pastecart; IN_ELIM_THM; LAMBDA_BETA; DIMINDEX_FINITE_SUM;
ARITH_RULE `i:num <= m ==> i <= m + n`] THEN
MESON_TAC[];
ALL_TAC] THEN
REWRITE_TAC[REWRITE_RULE[PCROSS] PCROSS_INTERVAL] THEN
ASM_SIMP_TAC[SET_RULE `s SUBSET t ==> s INTER u SUBSET t INTER u`] THEN
ASM_SIMP_TAC[SET_RULE `s SUBSET t ==> s INTER u SUBSET t`] THEN
MATCH_MP_TAC(NORM_ARITH
`y = y1 + y2 /\ x = x1 + x2 /\ e = e1 + e2
==> norm(y2 - x2:real^N) <= e2 ==> norm(y1 - x1) <= e1
==> norm(y - x) <= e`) THEN
REPEAT CONJ_TAC THENL
[MATCH_MP_TAC(SIMP_RULE[GSYM INTERVAL_SPLIT] INTEGRAL_SPLIT) THEN
ASM_SIMP_TAC[DIMINDEX_FINITE_SUM;
ARITH_RULE `i:num <= m ==> i <= m + n`] THEN
ASM_MESON_TAC[INTEGRABLE_CONTINUOUS; CONTINUOUS_ON_SUBSET];
MATCH_MP_TAC(SIMP_RULE[GSYM INTERVAL_SPLIT] INTEGRAL_SPLIT) THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC lemma2 THEN
ASM_MESON_TAC[CONTINUOUS_ON_SUBSET];
REWRITE_TAC[GSYM REAL_ADD_LDISTRIB] THEN AP_TERM_TAC THEN
MATCH_MP_TAC CONTENT_SPLIT THEN
ASM_REWRITE_TAC[DIMINDEX_FINITE_SUM]];
FIRST_X_ASSUM(CONJUNCTS_THEN2
(MP_TAC o SPECL
[`a:real^M`; `b:real^M`; `c:real^N`;
`(lambda i. if i = k - dimindex(:M)
then min (d$(k - dimindex(:M))) l
else (d:real^N)$i):real^N`])
(MP_TAC o SPECL
[`a:real^M`; `b:real^M`;
`(lambda i. if i = k - dimindex(:M)
then max (c$(k - dimindex(:M))) l
else (c:real^N)$i):real^N`;
`d:real^N`])) THEN
ASM_SIMP_TAC[GSYM INTERVAL_SPLIT; GSYM PCROSS_INTERVAL; PCROSS;
ARITH_RULE `~(i <= m) ==> 1 <= i - m`;
ARITH_RULE `~(i <= m) /\ i:num <= m + n ==> i - m <= n`] THEN
SUBGOAL_THEN
`!P Q. { pastecart (x:real^M) (y:real^N) |
P x /\ y IN interval[c,d] INTER {y | Q (y$(k - dimindex(:M)))}} =
{pastecart x y | P x /\ y IN interval[c,d]} INTER
{x | Q (x$k)}`
(fun th -> REWRITE_TAC[th])
THENL
[REWRITE_TAC[EXTENSION; FORALL_PASTECART; IN_ELIM_PASTECART_THM;
IN_INTER] THEN
ASM_SIMP_TAC[pastecart; IN_ELIM_THM; LAMBDA_BETA;
DIMINDEX_FINITE_SUM] THEN
MESON_TAC[];
ALL_TAC] THEN
REWRITE_TAC[REWRITE_RULE[PCROSS] PCROSS_INTERVAL] THEN
ASM_SIMP_TAC[SET_RULE `s SUBSET t ==> s INTER u SUBSET t INTER u`] THEN
ASM_SIMP_TAC[SET_RULE `s SUBSET t ==> s INTER u SUBSET t`] THEN
MATCH_MP_TAC(NORM_ARITH
`y = y1 + y2 /\ x = x1 + x2 /\ e = e1 + e2
==> norm(y2 - x2:real^N) <= e2 ==> norm(y1 - x1) <= e1
==> norm(y - x) <= e`) THEN
REPEAT CONJ_TAC THENL
[MATCH_MP_TAC(SIMP_RULE[GSYM INTERVAL_SPLIT] INTEGRAL_SPLIT) THEN
ASM_SIMP_TAC[DIMINDEX_FINITE_SUM;
ARITH_RULE `i:num <= m ==> i <= m + n`] THEN
ASM_MESON_TAC[INTEGRABLE_CONTINUOUS; CONTINUOUS_ON_SUBSET];
W(MP_TAC o PART_MATCH (rand o rand) INTEGRAL_ADD o rand o snd) THEN
REWRITE_TAC[] THEN ANTS_TAC THENL
[ASM_SIMP_TAC[INTERVAL_SPLIT; DIMINDEX_FINITE_SUM;
ARITH_RULE `~(i <= m) ==> 1 <= i - m`;
ARITH_RULE `~(i <= m) /\ i:num <= m + n ==> i - m <= n`] THEN
CONJ_TAC THEN MATCH_MP_TAC lemma2 THEN
MATCH_MP_TAC CONTINUOUS_ON_SUBSET THEN
EXISTS_TAC `s:real^(M,N)finite_sum->bool` THEN
ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM(MATCH_MP_TAC o
MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] SUBSET_TRANS)) THEN
REWRITE_TAC[SUBSET_INTERVAL] THEN DISCH_THEN(K ALL_TAC) THEN
ASM_SIMP_TAC[pastecart; LAMBDA_BETA; DIMINDEX_FINITE_SUM;
ARITH_RULE `~(i <= m) ==> 1 <= i - m`;
ARITH_RULE `~(i <= m) /\ i:num <= m + n ==> i - m <= n`] THEN
REPEAT STRIP_TAC THEN
REPEAT(COND_CASES_TAC THEN ASM_REWRITE_TAC[REAL_LE_REFL]) THEN
REAL_ARITH_TAC;
DISCH_THEN(SUBST1_TAC o SYM) THEN MATCH_MP_TAC INTEGRAL_EQ THEN
X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN REWRITE_TAC[] THEN
MATCH_MP_TAC(SIMP_RULE[GSYM INTERVAL_SPLIT] INTEGRAL_SPLIT) THEN
CONJ_TAC THENL [ALL_TAC; ASM_ARITH_TAC] THEN
MATCH_MP_TAC lemma1 THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET]];
REWRITE_TAC[GSYM REAL_ADD_LDISTRIB] THEN AP_TERM_TAC THEN
MATCH_MP_TAC CONTENT_SPLIT THEN
ASM_REWRITE_TAC[DIMINDEX_FINITE_SUM]]]) in
REPEAT STRIP_TAC THEN ASM_CASES_TAC
`content(interval[pastecart(a:real^M) (c:real^N),pastecart b d]) = &0` THEN
ASM_SIMP_TAC[INTEGRAL_NULL] THEN
RULE_ASSUM_TAC(REWRITE_RULE[CONTENT_PASTECART]) THEN
RULE_ASSUM_TAC(REWRITE_RULE[REAL_ENTIRE; DE_MORGAN_THM]) THENL
[FIRST_X_ASSUM DISJ_CASES_TAC THEN
ASM_SIMP_TAC[INTEGRAL_NULL; INTEGRAL_0];
ALL_TAC] THEN
SUBGOAL_THEN
`&0 < content(interval[pastecart(a:real^M) (c:real^N),pastecart b d])`
ASSUME_TAC THENL
[ASM_REWRITE_TAC[CONTENT_LT_NZ; CONTENT_PASTECART; REAL_ENTIRE];
ALL_TAC] THEN
ONCE_REWRITE_TAC[GSYM VECTOR_SUB_EQ] THEN
ONCE_REWRITE_TAC[GSYM NORM_EQ_0] THEN
MATCH_MP_TAC(MESON[REAL_ARITH
`~(x = &0) ==> &0 < abs x / &2 /\ ~(abs x <= abs x / &2)`]
`(!e. &0 < e ==> abs x <= e) ==> x = &0`) THEN
X_GEN_TAC `e:real` THEN DISCH_TAC THEN REWRITE_TAC[REAL_ABS_NORM] THEN
MP_TAC(ISPECL
[`f:real^(M,N)finite_sum->real^P`;
`e / content(interval[pastecart(a:real^M) (c:real^N),pastecart b d])`;
`interval[pastecart(a:real^M) (c:real^N),pastecart b d]`]
lemma3) THEN
ASM_SIMP_TAC[REAL_LT_DIV] THEN DISCH_THEN(MP_TAC o MATCH_MP
(REWRITE_RULE[IMP_CONJ] OPERATIVE_DIVISION_AND)) THEN
REWRITE_TAC[] THEN
FIRST_ASSUM(MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ]
COMPACT_UNIFORMLY_CONTINUOUS)) THEN
REWRITE_TAC[COMPACT_INTERVAL; uniformly_continuous_on] THEN
DISCH_THEN(MP_TAC o SPEC
`e / &2 /
content(interval[pastecart(a:real^M) (c:real^N),pastecart b d])`) THEN
ASM_SIMP_TAC[dist; REAL_HALF; REAL_LT_DIV] THEN
DISCH_THEN(X_CHOOSE_THEN `k:real` STRIP_ASSUME_TAC) THEN
SUBGOAL_THEN
`?p. p tagged_division_of
interval[pastecart(a:real^M) (c:real^N),pastecart b d] /\
(\x. ball(x,k)) fine p`
STRIP_ASSUME_TAC THENL
[ASM_MESON_TAC[FINE_DIVISION_EXISTS; GAUGE_BALL]; ALL_TAC] THEN
DISCH_THEN(MP_TAC o SPEC
`IMAGE SND (p:real^(M,N)finite_sum#(real^(M,N)finite_sum->bool)->bool)`) THEN
DISCH_THEN(MP_TAC o SPECL
[`pastecart(a:real^M) (c:real^N)`; `pastecart(b:real^M) (d:real^N)`]) THEN
ASM_SIMP_TAC[DIVISION_OF_TAGGED_DIVISION] THEN
REWRITE_TAC[FORALL_IN_IMAGE; FORALL_PAIR_THM] THEN
MATCH_MP_TAC(TAUT `(b ==> c) /\ a ==> (a <=> b) ==> c`) THEN CONJ_TAC THENL
[DISCH_THEN(MP_TAC o SPECL
[`a:real^M`; `b:real^M`; `c:real^N`; `d:real^N`]) THEN
ASM_SIMP_TAC[REAL_DIV_RMUL; REAL_LT_IMP_NZ; SUBSET_REFL];
ALL_TAC] THEN
MAP_EVERY X_GEN_TAC
[`t:real^(M,N)finite_sum`; `l:real^(M,N)finite_sum->bool`] THEN
DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [TAGGED_DIVISION_OF]) THEN
DISCH_THEN(MP_TAC o SPECL
[`t:real^(M,N)finite_sum`; `l:real^(M,N)finite_sum->bool`] o
el 1 o CONJUNCTS) THEN
ASM_REWRITE_TAC[] THEN
DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC) THEN
ASM_REWRITE_TAC[] THEN
MAP_EVERY X_GEN_TAC [`u:real^M`; `v:real^M`; `w:real^N`; `z:real^N`] THEN
STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [fine]) THEN
REWRITE_TAC[SUBSET; IN_BALL; dist] THEN DISCH_TAC THEN
MP_TAC(ISPECL
[`u:real^M`; `v:real^M`; `w:real^N`; `z:real^N`;
`(f:real^(M,N)finite_sum->real^P) t`]
INTEGRAL_PASTECART_CONST) THEN
MATCH_MP_TAC(NORM_ARITH
`norm(x - x') <= e / &2 /\ norm(y - y') <= e / &2
==> x':real^P = y' ==> norm(x - y) <= e`) THEN
REWRITE_TAC[REAL_ARITH `(e / c * d) / &2 = e / &2 / c * d`] THEN
CONJ_TAC THENL
[MATCH_MP_TAC HAS_INTEGRAL_BOUND THEN
EXISTS_TAC `\y. (f:real^(M,N)finite_sum->real^P) y - f t` THEN
ASM_SIMP_TAC[REAL_HALF; REAL_LT_DIV; REAL_LT_IMP_LE] THEN CONJ_TAC THENL
[MATCH_MP_TAC HAS_INTEGRAL_SUB THEN
REWRITE_TAC[GSYM HAS_INTEGRAL_INTEGRAL; INTEGRABLE_CONST] THEN
ASM_MESON_TAC[INTEGRABLE_CONTINUOUS; INTEGRABLE_ON_SUBINTERVAL];
X_GEN_TAC `y:real^(M,N)finite_sum` THEN DISCH_TAC THEN
MATCH_MP_TAC REAL_LT_IMP_LE THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
MATCH_MP_TAC(TAUT `(a /\ b) /\ (a /\ b ==> c) ==> a /\ b /\ c`) THEN
CONJ_TAC THENL [ASM SET_TAC[]; ASM_MESON_TAC[NORM_SUB; SUBSET]]];
GEN_REWRITE_TAC (RAND_CONV o RAND_CONV) [CONTENT_PASTECART] THEN
ONCE_REWRITE_TAC[REAL_ARITH `a * b * c:real = (a * c) * b`] THEN
MATCH_MP_TAC HAS_INTEGRAL_BOUND THEN EXISTS_TAC
`\x. integral (interval [w,z]) (\y. f (pastecart x y)) -
integral (interval [w,z])
(\y. (f:real^(M,N)finite_sum->real^P) t)` THEN
REPEAT CONJ_TAC THENL
[MATCH_MP_TAC REAL_LE_MUL THEN REWRITE_TAC[CONTENT_POS_LE] THEN
ASM_SIMP_TAC[REAL_HALF; REAL_LT_DIV; REAL_LT_IMP_LE];
MATCH_MP_TAC HAS_INTEGRAL_SUB THEN
REWRITE_TAC[GSYM HAS_INTEGRAL_INTEGRAL; INTEGRABLE_CONST] THEN
MATCH_MP_TAC lemma2 THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET];
X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN REWRITE_TAC[] THEN
MATCH_MP_TAC HAS_INTEGRAL_BOUND THEN EXISTS_TAC
`\y. (f:real^(M,N)finite_sum->real^P) (pastecart x y) - f t` THEN
ASM_SIMP_TAC[REAL_HALF; REAL_LT_DIV; REAL_LT_IMP_LE] THEN
CONJ_TAC THENL
[MATCH_MP_TAC HAS_INTEGRAL_SUB THEN
REWRITE_TAC[GSYM HAS_INTEGRAL_INTEGRAL; INTEGRABLE_CONST] THEN
MATCH_MP_TAC lemma1 THEN ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; SUBSET];
X_GEN_TAC `s:real^N` THEN DISCH_TAC THEN
MATCH_MP_TAC REAL_LT_IMP_LE THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
REPEAT CONJ_TAC THENL
[ASM SET_TAC[];
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
`s SUBSET t ==> x IN s ==> x IN t`)) THEN
ASM_REWRITE_TAC[GSYM PCROSS_INTERVAL; PCROSS;
IN_ELIM_PASTECART_THM];
RULE_ASSUM_TAC(REWRITE_RULE[IMP_IMP; RIGHT_IMP_FORALL_THM]) THEN
ONCE_REWRITE_TAC[NORM_SUB] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
EXISTS_TAC `l:real^(M,N)finite_sum->bool` THEN
ASM_REWRITE_TAC[] THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (SET_RULE
`s SUBSET t ==> x IN s ==> x IN t`)) THEN
ASM_REWRITE_TAC[GSYM PCROSS_INTERVAL; PCROSS;
IN_ELIM_PASTECART_THM]]]]]);;
let INTEGRAL_SWAP_CONTINUOUS = prove
(`!f:real^M->real^N->real^P a b c d.
(\z. f (fstcart z) (sndcart z))
continuous_on interval[pastecart a c,pastecart b d]
==> integral (interval[a,b]) (\x. integral (interval[c,d]) (f x)) =
integral (interval[c,d])
(\y. integral (interval[a,b]) (\x. f x y))`,