(* ========================================================================= *)
(* Brunn-Minkowski theorem and related results.                              *)
(* ========================================================================= *)

needs "Multivariate/realanalysis.ml";;
needs "Library/agm.ml";;

(* ------------------------------------------------------------------------- *)
(* First, the special case of a box.                                         *)
(* ------------------------------------------------------------------------- *)

let BRUNN_MINKOWSKI_INTERVAL = 
prove (`!a b c d:real^N. ~(interval[a,b] = {}) /\ ~(interval[c,d] = {}) ==> root (dimindex(:N)) (measure {x + y | x IN interval[a,b] /\ y IN interval[c,d]}) >= root (dimindex(:N)) (measure(interval[a,b])) + root (dimindex(:N)) (measure(interval[c,d]))`,
REPEAT STRIP_TAC THEN ASM_SIMP_TAC[SUMS_INTERVALS; real_ge] THEN RULE_ASSUM_TAC(REWRITE_RULE[INTERVAL_NE_EMPTY]) THEN ASM_CASES_TAC `measure(interval[a:real^N,b]) = &0` THENL [ASM_SIMP_TAC[ROOT_0; DIMINDEX_GE_1; LE_1; REAL_ADD_LID; ROOT_MONO_LE_EQ; MEASURE_POS_LE; MEASURABLE_INTERVAL] THEN ASM_SIMP_TAC[MEASURE_INTERVAL; CONTENT_CLOSED_INTERVAL; VECTOR_ADD_COMPONENT; REAL_ARITH `a <= b /\ c <= d ==> a + c <= b + d`] THEN MATCH_MP_TAC PRODUCT_LE_NUMSEG THEN X_GEN_TAC `i:num` THEN DISCH_TAC THEN REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `i:num`)) THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC; FIRST_X_ASSUM(MP_TAC o check (is_neg o concl)) THEN SIMP_TAC[MEASURABLE_MEASURE_EQ_0; MEASURABLE_INTERVAL] THEN REWRITE_TAC[NEGLIGIBLE_INTERVAL; INTERVAL_NE_EMPTY] THEN STRIP_TAC] THEN ASM_CASES_TAC `measure(interval[c:real^N,d]) = &0` THENL [ASM_SIMP_TAC[ROOT_0; DIMINDEX_GE_1; LE_1; REAL_ADD_RID; ROOT_MONO_LE_EQ; MEASURE_POS_LE; MEASURABLE_INTERVAL] THEN ASM_SIMP_TAC[MEASURE_INTERVAL; CONTENT_CLOSED_INTERVAL; VECTOR_ADD_COMPONENT; REAL_ARITH `a <= b /\ c <= d ==> a + c <= b + d`] THEN MATCH_MP_TAC PRODUCT_LE_NUMSEG THEN X_GEN_TAC `i:num` THEN DISCH_TAC THEN REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `i:num`)) THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC; FIRST_X_ASSUM(MP_TAC o check (is_neg o concl)) THEN SIMP_TAC[MEASURABLE_MEASURE_EQ_0; MEASURABLE_INTERVAL] THEN REWRITE_TAC[NEGLIGIBLE_INTERVAL; INTERVAL_NE_EMPTY] THEN STRIP_TAC] THEN ASM_SIMP_TAC[MEASURE_INTERVAL; CONTENT_CLOSED_INTERVAL; VECTOR_ADD_COMPONENT; REAL_ARITH `a <= b /\ c <= d ==> a + c <= b + d`] THEN GEN_REWRITE_TAC RAND_CONV [GSYM REAL_MUL_LID] THEN W(MP_TAC o PART_MATCH (rand o rand) REAL_LE_LDIV_EQ o snd) THEN ANTS_TAC THENL [MATCH_MP_TAC ROOT_POS_LT THEN ASM_SIMP_TAC[GSYM REAL_LE_LDIV_EQ; PRODUCT_POS_LT_NUMSEG; IN_NUMSEG; REAL_ARITH `a < b /\ c < d ==> &0 < (b + d) - (a + c)`; DIMINDEX_GE_1; LE_1]; DISCH_THEN(SUBST1_TAC o SYM)] THEN REWRITE_TAC[real_div; REAL_ADD_RDISTRIB] THEN REWRITE_TAC[GSYM real_div] THEN W(MP_TAC o PART_MATCH (lhand o rand) (GSYM REAL_ROOT_DIV) o lhand o lhand o snd) THEN W(MP_TAC o PART_MATCH (lhand o rand) (GSYM REAL_ROOT_DIV) o rand o lhand o rand o snd) THEN ASM_SIMP_TAC[DIMINDEX_NONZERO; PRODUCT_POS_LE_NUMSEG; IN_NUMSEG; REAL_SUB_LE; REAL_ARITH `a < b /\ c < d ==> a + c <= b + d`] THEN REPEAT(DISCH_THEN(K ALL_TAC)) THEN REWRITE_TAC[GSYM PRODUCT_DIV_NUMSEG] THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `sum (1..dimindex(:N)) (\i. ((b:real^N)$i - (a:real^N)$i) / ((b$i + d$i) - (a$i + c$i))) / &(dimindex(:N)) + sum (1..dimindex(:N)) (\i. ((d:real^N)$i - (c:real^N)$i) / ((b$i + d$i) - (a$i + c$i))) / &(dimindex(:N))` THEN CONJ_TAC THENL [MATCH_MP_TAC REAL_LE_ADD2 THEN CONJ_TAC THEN W(MP_TAC o PART_MATCH (lhand o rand) REAL_ROOT_LE o snd) THEN (ANTS_TAC THENL [REWRITE_TAC[DIMINDEX_NONZERO] THEN CONJ_TAC THENL [MATCH_MP_TAC PRODUCT_POS_LE_NUMSEG; MATCH_MP_TAC REAL_LE_DIV THEN REWRITE_TAC[REAL_POS] THEN MATCH_MP_TAC SUM_POS_LE_NUMSEG]; DISCH_THEN SUBST1_TAC THEN MATCH_MP_TAC AGM THEN REWRITE_TAC[DIMINDEX_GE_1]] THEN X_GEN_TAC `i:num` THEN DISCH_TAC THEN REWRITE_TAC[REAL_SUB_LE] THEN MATCH_MP_TAC REAL_LE_DIV THEN REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `i:num`)) THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC); REWRITE_TAC[real_div; GSYM REAL_ADD_RDISTRIB] THEN REWRITE_TAC[GSYM SUM_ADD_NUMSEG] THEN ASM_SIMP_TAC[REAL_FIELD `a < b /\ c < d ==> (b - a) * inv((b + d) - (a + c)) + (d - c) * inv((b + d) - (a + c)) = &1`] THEN REWRITE_TAC[SUM_CONST_NUMSEG; ADD_SUB] THEN ASM_SIMP_TAC[REAL_MUL_RID; REAL_MUL_RINV; REAL_LE_REFL; REAL_OF_NUM_EQ; DIMINDEX_NONZERO]]);;
(* ------------------------------------------------------------------------- *) (* Now for a finite union of boxes. *) (* ------------------------------------------------------------------------- *)
let BRUNN_MINKOWSKI_ELEMENTARY = 
prove (`!s t:real^N->bool. (s = {} <=> t = {}) /\ (?d. d division_of s) /\ (?d. d division_of t) ==> root (dimindex(:N)) (measure {x + y | x IN s /\ y IN t}) >= root (dimindex(:N)) (measure s) + root (dimindex(:N)) (measure t)`,
REWRITE_TAC[LEFT_AND_EXISTS_THM] THEN REWRITE_TAC[RIGHT_AND_EXISTS_THM; LEFT_IMP_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`s:real^N->bool`; `t:real^N->bool`; `d1:(real^N->bool)->bool`; `d2:(real^N->bool)->bool`] THEN ASM_CASES_TAC `t:real^N->bool = {}` THENL [ASM_SIMP_TAC[NOT_IN_EMPTY; SET_RULE `{f x y |x,y| F} = {}`] THEN SIMP_TAC[MEASURE_EMPTY; ROOT_0; DIMINDEX_NONZERO] THEN STRIP_TAC THEN REAL_ARITH_TAC; ASM_REWRITE_TAC[] THEN POP_ASSUM MP_TAC THEN ONCE_REWRITE_TAC[TAUT `p ==> q /\ r ==> s <=> q /\ p /\ r ==> s`]] THEN X_CHOOSE_THEN `n:num` MP_TAC (ISPEC `CARD(d1:(real^N->bool)->bool) + CARD(d2:(real^N->bool)->bool)` (GSYM EXISTS_REFL)) THEN REWRITE_TAC[IMP_IMP; GSYM CONJ_ASSOC] THEN MAP_EVERY (fun t -> SPEC_TAC(t,t)) [`t:real^N->bool`; `s:real^N->bool`; `d2:(real^N->bool)->bool`; `d1:(real^N->bool)->bool`; `n:num`] THEN MATCH_MP_TAC num_WF THEN X_GEN_TAC `n:num` THEN REWRITE_TAC[MESON[] `(!m. m < n ==> !a b c d. f a b = m /\ stuff a b c d ==> other a b c d) <=> (!a b c d. f a b:num < n /\ stuff a b c d ==> other a b c d)`] THEN DISCH_TAC THEN MATCH_MP_TAC(MESON[] `(!d d' s s'. P d d' s s' ==> P d' d s' s) /\ (!d d' s s'. ~(2 <= CARD d) /\ ~(2 <= CARD d') ==> P d d' s s') /\ (!d d' s s'. negligible s ==> P d d' s s') /\ (!d d' s s'. 2 <= CARD d /\ ~(negligible s) /\ ~(negligible s') ==> P d d' s s') ==> !d d' s s'. P d d' s s'`) THEN REPEAT CONJ_TAC THENL [REPEAT GEN_TAC THEN MATCH_MP_TAC MONO_IMP THEN CONJ_TAC THENL [REWRITE_TAC[ADD_SYM; CONJ_ACI]; ALL_TAC] THEN MATCH_MP_TAC(REAL_ARITH `x = y ==> x >= a + b ==> y >= b + a`) THEN AP_TERM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[EXTENSION; IN_ELIM_THM] THEN MESON_TAC[VECTOR_ADD_SYM]; REPEAT GEN_TAC THEN ASM_CASES_TAC `FINITE(d1:(real^N->bool)->bool) /\ FINITE(d2:(real^N->bool)->bool)` THENL [ALL_TAC; REWRITE_TAC[division_of] THEN ASM_MESON_TAC[]] THEN ASM_SIMP_TAC[CARD_EQ_0; ARITH_RULE `~(2 <= n) <=> n = 0 \/ n = 1`] THEN ASM_CASES_TAC `d1:(real^N->bool)->bool = {}` THENL [ASM_REWRITE_TAC[EMPTY_DIVISION_OF] THEN MESON_TAC[]; ALL_TAC] THEN ASM_CASES_TAC `d2:(real^N->bool)->bool = {}` THENL [ASM_REWRITE_TAC[EMPTY_DIVISION_OF] THEN MESON_TAC[]; ALL_TAC] THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THEN SUBGOAL_THEN `(d1:(real^N->bool)->bool) HAS_SIZE 1 /\ (d2:(real^N->bool)->bool) HAS_SIZE 1` MP_TAC THENL [ASM_REWRITE_TAC[HAS_SIZE]; ALL_TAC] THEN CONV_TAC(LAND_CONV(BINOP_CONV HAS_SIZE_CONV)) THEN DISCH_THEN(CONJUNCTS_THEN2 (X_CHOOSE_THEN `u:real^N->bool` SUBST_ALL_TAC) (X_CHOOSE_THEN `v:real^N->bool` SUBST_ALL_TAC)) THEN STRIP_TAC THEN RULE_ASSUM_TAC(REWRITE_RULE[division_of; UNIONS_1; IN_SING]) THEN REPEAT(FIRST_X_ASSUM (CONJUNCTS_THEN2 MP_TAC (SUBST_ALL_TAC o SYM o CONJUNCT2) o CONJUNCT2)) THEN REWRITE_TAC[FORALL_UNWIND_THM2] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC BRUNN_MINKOWSKI_INTERVAL THEN ASM_MESON_TAC[]; REPEAT STRIP_TAC THEN SUBGOAL_THEN `measure(s:real^N->bool) = &0` SUBST1_TAC THENL [ASM_SIMP_TAC[MEASURE_EQ_0]; ALL_TAC] THEN SIMP_TAC[ROOT_0; DIMINDEX_NONZERO; REAL_ADD_LID; real_ge] THEN MATCH_MP_TAC ROOT_MONO_LE THEN REWRITE_TAC[DIMINDEX_NONZERO] THEN CONJ_TAC THENL [MATCH_MP_TAC MEASURE_POS_LE THEN ASM_MESON_TAC[MEASURABLE_ELEMENTARY]; ALL_TAC] THEN SUBGOAL_THEN `?a:real^N. a IN s` STRIP_ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `measure(IMAGE (\x:real^N. a + x) t)` THEN CONJ_TAC THENL [REWRITE_TAC[MEASURE_TRANSLATION; REAL_LE_REFL]; ALL_TAC] THEN MATCH_MP_TAC MEASURE_SUBSET THEN REWRITE_TAC[MEASURABLE_TRANSLATION_EQ] THEN CONJ_TAC THENL [ASM_MESON_TAC[MEASURABLE_ELEMENTARY]; ALL_TAC] THEN REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_ELIM_THM] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC MEASURABLE_COMPACT THEN MATCH_MP_TAC COMPACT_SUMS THEN ASM_MESON_TAC[ELEMENTARY_COMPACT]; REWRITE_TAC[IMP_IMP; GSYM CONJ_ASSOC]] THEN SUBGOAL_THEN `!d1 d2 s t i j k. CARD d1 + CARD d2 = n /\ 1 <= k /\ k <= dimindex(:N) /\ ~(i = j) /\ i IN d1 /\ i SUBSET {x:real^N | x$k <= &0} /\ j IN d1 /\ j SUBSET {x | x$k >= &0} /\ ~(negligible i) /\ ~(negligible j) /\ ~(s = {}) /\ ~(t = {}) /\ ~(negligible s) /\ ~(negligible t) /\ d1 division_of s /\ d2 division_of t ==> root(dimindex (:N)) (measure {x + y | x IN s /\ y IN t}) >= root(dimindex (:N)) (measure s) + root(dimindex (:N)) (measure t)` MP_TAC THENL [ALL_TAC; POP_ASSUM(LABEL_TAC "*") THEN DISCH_THEN(LABEL_TAC "+") THEN REPEAT STRIP_TAC THEN ASM_CASES_TAC `?i:real^N->bool. i IN d1 /\ interior i = {}` THENL [REMOVE_THEN "+" (K ALL_TAC) THEN REMOVE_THEN "*" MP_TAC THEN DISCH_THEN(MP_TAC o SPECL [`{i:real^N->bool | i IN d1 /\ ~(interior i = {})}`; `d2:(real^N->bool)->bool`; `UNIONS {i:real^N->bool | i IN d1 /\ ~(interior i = {})}`; `t:real^N->bool`]) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [REPEAT CONJ_TAC THENL [EXPAND_TAC "n" THEN REWRITE_TAC[LT_ADD_RCANCEL] THEN MATCH_MP_TAC CARD_PSUBSET THEN CONJ_TAC THENL [ASM SET_TAC[]; ASM_MESON_TAC[DIVISION_OF_FINITE]]; DISCH_TAC THEN SUBGOAL_THEN `negligible(UNIONS {i | i IN d1 /\ ~(interior i = {})} UNION UNIONS {i:real^N->bool | i IN d1 /\ interior i = {}})` MP_TAC THENL [ASM_REWRITE_TAC[UNION_EMPTY] THEN MATCH_MP_TAC NEGLIGIBLE_UNIONS THEN CONJ_TAC THENL [MATCH_MP_TAC FINITE_RESTRICT THEN ASM_MESON_TAC[DIVISION_OF_FINITE]; REWRITE_TAC[IN_ELIM_THM; IMP_CONJ] THEN FIRST_ASSUM(fun th -> GEN_REWRITE_TAC I [MATCH_MP FORALL_IN_DIVISION th]) THEN REWRITE_TAC[INTERIOR_CLOSED_INTERVAL; NEGLIGIBLE_INTERVAL]]; REWRITE_TAC[GSYM UNIONS_UNION; SET_RULE `{x | x IN s /\ ~Q x} UNION {x | x IN s /\ Q x} = s`] THEN RULE_ASSUM_TAC(REWRITE_RULE[division_of]) THEN ASM_REWRITE_TAC[]]; MATCH_MP_TAC DIVISION_OF_SUBSET THEN EXISTS_TAC `d1:(real^N->bool)->bool` THEN CONJ_TAC THENL [ASM_MESON_TAC[division_of]; SET_TAC[]]]; MATCH_MP_TAC(REAL_ARITH `c' <= c /\ a' = a ==> c' >= a' + b ==> c >= a + b`) THEN CONJ_TAC THENL [MATCH_MP_TAC ROOT_MONO_LE THEN REWRITE_TAC[DIMINDEX_NONZERO] THEN CONJ_TAC THENL [MATCH_MP_TAC MEASURE_POS_LE THEN MATCH_MP_TAC MEASURABLE_COMPACT THEN MATCH_MP_TAC COMPACT_SUMS THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[ELEMENTARY_COMPACT]] THEN MATCH_MP_TAC COMPACT_UNIONS THEN RULE_ASSUM_TAC(REWRITE_RULE[division_of]) THEN ASM_SIMP_TAC[FINITE_RESTRICT; IN_ELIM_THM] THEN ASM_MESON_TAC[COMPACT_INTERVAL]; ALL_TAC] THEN MATCH_MP_TAC MEASURE_SUBSET THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC MEASURABLE_COMPACT THEN MATCH_MP_TAC COMPACT_SUMS THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[ELEMENTARY_COMPACT]] THEN MATCH_MP_TAC COMPACT_UNIONS THEN RULE_ASSUM_TAC(REWRITE_RULE[division_of]) THEN ASM_SIMP_TAC[FINITE_RESTRICT; IN_ELIM_THM] THEN ASM_MESON_TAC[COMPACT_INTERVAL]; MATCH_MP_TAC MEASURABLE_COMPACT THEN MATCH_MP_TAC COMPACT_SUMS THEN ASM_MESON_TAC[ELEMENTARY_COMPACT]; MATCH_MP_TAC(SET_RULE `s' SUBSET s ==> {f x y | x IN s' /\ y IN t} SUBSET {f x y | x IN s /\ y IN t}`) THEN SUBGOAL_THEN `s:real^N->bool = UNIONS d1` SUBST1_TAC THENL [ASM_MESON_TAC[division_of]; MATCH_MP_TAC SUBSET_UNIONS THEN SET_TAC[]]]; AP_TERM_TAC THEN MATCH_MP_TAC MEASURE_NEGLIGIBLE_SYMDIFF THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `UNIONS {i:real^N->bool | i IN d1 /\ interior i = {}}` THEN CONJ_TAC THENL [MATCH_MP_TAC NEGLIGIBLE_UNIONS THEN CONJ_TAC THENL [MATCH_MP_TAC FINITE_RESTRICT THEN ASM_MESON_TAC[DIVISION_OF_FINITE]; REWRITE_TAC[IN_ELIM_THM; IMP_CONJ] THEN FIRST_ASSUM(fun th -> GEN_REWRITE_TAC I [MATCH_MP FORALL_IN_DIVISION th]) THEN REWRITE_TAC[INTERIOR_CLOSED_INTERVAL; NEGLIGIBLE_INTERVAL]]; MATCH_MP_TAC(SET_RULE `s' UNION s'' = s ==> (s' DIFF s) UNION (s DIFF s') SUBSET s''`) THEN REWRITE_TAC[GSYM UNIONS_UNION; SET_RULE `{x | x IN s /\ ~Q x} UNION {x | x IN s /\ Q x} = s`] THEN RULE_ASSUM_TAC(REWRITE_RULE[division_of]) THEN ASM_REWRITE_TAC[]]]]; FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [NOT_EXISTS_THM]) THEN REWRITE_TAC[TAUT `~(a /\ b) <=> a ==> ~b`] THEN DISCH_TAC THEN REMOVE_THEN "*" (K ALL_TAC) THEN SUBGOAL_THEN `?d:(real^N->bool)->bool. d SUBSET d1 /\ d HAS_SIZE 2` MP_TAC THENL [FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP CHOOSE_SUBSET o MATCH_MP DIVISION_OF_FINITE) THEN ASM_REWRITE_TAC[]; ALL_TAC] THEN CONV_TAC(LAND_CONV(ONCE_DEPTH_CONV HAS_SIZE_CONV)) THEN REWRITE_TAC[RIGHT_AND_EXISTS_THM] THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN DISCH_THEN(X_CHOOSE_THEN `i:real^N->bool` MP_TAC) THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN DISCH_THEN(X_CHOOSE_THEN `j:real^N->bool` MP_TAC) THEN ONCE_REWRITE_TAC[TAUT `p /\ q /\ r <=> r /\ p /\ q`] THEN REWRITE_TAC[UNWIND_THM2; INSERT_SUBSET; EMPTY_SUBSET] THEN STRIP_TAC THEN MP_TAC(ASSUME `d1 division_of (s:real^N->bool)`) THEN REWRITE_TAC[division_of] THEN DISCH_THEN(MP_TAC o el 2 o CONJUNCTS) THEN DISCH_THEN(MP_TAC o SPECL [`i:real^N->bool`; `j:real^N->bool`]) THEN ASM_REWRITE_TAC[] THEN SUBGOAL_THEN `?u v w z. i = interval[u:real^N,v] /\ j = interval[w:real^N,z]` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[DIVISION_OF]; ALL_TAC] THEN ASM_REWRITE_TAC[GSYM INTERIOR_INTER; INTER_INTERVAL] THEN REWRITE_TAC[INTERIOR_CLOSED_INTERVAL; INTERVAL_EQ_EMPTY] THEN DISCH_THEN(X_CHOOSE_THEN `k:num` MP_TAC) THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN SIMP_TAC[LAMBDA_BETA; ASSUME `1 <= k`; ASSUME `k <= dimindex(:N)`] THEN GEN_REWRITE_TAC LAND_CONV [REAL_LE_BETWEEN] THEN DISCH_THEN(X_CHOOSE_THEN `a:real` MP_TAC) THEN DISCH_THEN(MP_TAC o MATCH_MP (REAL_ARITH `min v z <= a /\ a <= max u w ==> u < v /\ w < z ==> u <= a /\ v <= a /\ a <= w /\ a <= z \/ w <= a /\ z <= a /\ a <= u /\ a <= v`)) THEN ANTS_TAC THENL [UNDISCH_TAC `!i:real^N->bool. i IN d1 ==> ~(interior i = {})` THEN DISCH_THEN(fun th -> MP_TAC(ISPEC `interval[u:real^N,v]` th) THEN MP_TAC(ISPEC `interval[w:real^N,z]` th)) THEN REWRITE_TAC[INTERIOR_CLOSED_INTERVAL; INTERVAL_NE_EMPTY] THEN ASM_MESON_TAC[]; ALL_TAC] THEN STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPECL [`IMAGE (IMAGE (\x:real^N. x - a % basis k)) d1`; `d2:(real^N->bool)->bool`; `IMAGE (\x:real^N. x - a % basis k) s`; `t:real^N->bool`]) THENL [DISCH_THEN(MP_TAC o SPECL [`IMAGE (\x:real^N. x - a % basis k) i`; `IMAGE (\x:real^N. x - a % basis k) j`; `k:num`]); DISCH_THEN(MP_TAC o SPECL [`IMAGE (\x:real^N. x - a % basis k) j`; `IMAGE (\x:real^N. x - a % basis k) i`; `k:num`])] THEN (ASM_REWRITE_TAC[] THEN ANTS_TAC THEN REPEAT CONJ_TAC THENL [EXPAND_TAC "n" THEN AP_THM_TAC THEN AP_TERM_TAC THEN MATCH_MP_TAC CARD_IMAGE_INJ THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[DIVISION_OF_FINITE]] THEN MATCH_MP_TAC(MESON[] `(!x y. Q x y ==> R x y) ==> (!x y. P x /\ P y /\ Q x y ==> R x y)`) THEN REWRITE_TAC[INJECTIVE_IMAGE] THEN VECTOR_ARITH_TAC; MATCH_MP_TAC(SET_RULE `(!x y. f x = f y ==> x = y) /\ ~(s = t) ==> ~(IMAGE f s = IMAGE f t)`) THEN REWRITE_TAC[VECTOR_ARITH `x - a:real^N = y - a <=> x = y`] THEN ASM_MESON_TAC[]; MATCH_MP_TAC FUN_IN_IMAGE THEN ASM_MESON_TAC[]; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_ELIM_THM] THEN ASM_SIMP_TAC[VECTOR_SUB_COMPONENT; VECTOR_MUL_COMPONENT; BASIS_COMPONENT; REAL_MUL_RID; REAL_LE_SUB_RADD; REAL_ADD_LID] THEN REWRITE_TAC[IN_INTERVAL] THEN ASM_MESON_TAC[REAL_LE_TRANS]; MATCH_MP_TAC FUN_IN_IMAGE THEN ASM_MESON_TAC[]; REWRITE_TAC[SUBSET; FORALL_IN_IMAGE; IN_ELIM_THM; real_ge] THEN ASM_SIMP_TAC[VECTOR_SUB_COMPONENT; VECTOR_MUL_COMPONENT; BASIS_COMPONENT; REAL_MUL_RID; REAL_LE_SUB_LADD; REAL_ADD_LID] THEN REWRITE_TAC[IN_INTERVAL] THEN ASM_MESON_TAC[REAL_LE_TRANS]; REWRITE_TAC[VECTOR_ARITH `x - a:real^N = --a + x`] THEN ASM_REWRITE_TAC[NEGLIGIBLE_TRANSLATION_EQ] THEN ASM_MESON_TAC[NEGLIGIBLE_INTERVAL; INTERIOR_CLOSED_INTERVAL]; REWRITE_TAC[VECTOR_ARITH `x - a:real^N = --a + x`] THEN ASM_REWRITE_TAC[NEGLIGIBLE_TRANSLATION_EQ] THEN ASM_MESON_TAC[NEGLIGIBLE_INTERVAL; INTERIOR_CLOSED_INTERVAL]; ASM_REWRITE_TAC[IMAGE_EQ_EMPTY]; REWRITE_TAC[VECTOR_ARITH `x - a:real^N = --a + x`] THEN ASM_REWRITE_TAC[NEGLIGIBLE_TRANSLATION_EQ]; REWRITE_TAC[VECTOR_ARITH `x - a:real^N = --a + x`] THEN ASM_REWRITE_TAC[DIVISION_OF_TRANSLATION]; MATCH_MP_TAC(REAL_ARITH `a = a' /\ b = b' ==> a >= b + c ==> a' >= b' + c`) THEN REWRITE_TAC[VECTOR_ARITH `x - a:real^N = --a + x`] THEN REWRITE_TAC[MEASURE_TRANSLATION] THEN REWRITE_TAC[GSYM VECTOR_ADD_ASSOC; SET_RULE `{f x y | x IN IMAGE g s /\ y IN t} = {f (g x) y | x IN s /\ y IN t}`] THEN REWRITE_TAC[SET_RULE `{a + x + y:real^N | x IN s /\ y IN t} = IMAGE (\z. a + z) {x + y | x IN s /\ y IN t}`] THEN REWRITE_TAC[MEASURE_TRANSLATION]])]] THEN SUBGOAL_THEN `!d1 d2 s t i j k. CARD d1 + CARD d2 = n /\ 1 <= k /\ k <= dimindex(:N) /\ ~(i = j) /\ i IN d1 /\ i SUBSET {x:real^N | x$k <= &0} /\ ~(negligible i) /\ j IN d1 /\ j SUBSET {x | x$k >= &0} /\ ~(negligible j) /\ measure(t INTER {x | x$k <= &0}) / measure t = measure(s INTER {x | x$k <= &0}) / measure s /\ measure(t INTER {x | x$k >= &0}) / measure t = measure(s INTER {x | x$k >= &0}) / measure s /\ ~(s = {}) /\ ~(t = {}) /\ ~(negligible s) /\ ~(negligible t) /\ d1 division_of s /\ d2 division_of t ==> root(dimindex (:N)) (measure {x + y | x IN s /\ y IN t}) >= root(dimindex (:N)) (measure s) + root(dimindex (:N)) (measure t)` MP_TAC THENL [ALL_TAC; POP_ASSUM(K ALL_TAC) THEN DISCH_TAC THEN REPEAT STRIP_TAC THEN SUBGOAL_THEN `&0 < measure(s:real^N->bool) /\ &0 < measure(t:real^N->bool)` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[MEASURABLE_MEASURE_POS_LT; MEASURABLE_ELEMENTARY]; ALL_TAC] THEN SUBGOAL_THEN `?a. measure(t INTER {x:real^N | x$k <= a}) / measure t = measure(s INTER {x:real^N | x$k <= &0}) / measure s` STRIP_ASSUME_TAC THENL [SUBGOAL_THEN `&0 <= measure(s INTER {x:real^N | x$k <= &0}) / measure s /\ measure(s INTER {x:real^N | x$k <= &0}) / measure s <= &1` MP_TAC THENL [ASM_SIMP_TAC[REAL_LE_RDIV_EQ; REAL_LE_LDIV_EQ; REAL_MUL_LZERO] THEN CONJ_TAC THENL [MATCH_MP_TAC MEASURE_POS_LE; REWRITE_TAC[REAL_MUL_LID] THEN MATCH_MP_TAC MEASURE_SUBSET THEN REPEAT CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[MEASURABLE_ELEMENTARY]; SET_TAC[]]] THEN MATCH_MP_TAC MEASURABLE_COMPACT THEN MATCH_MP_TAC COMPACT_INTER_CLOSED THEN REWRITE_TAC[CLOSED_HALFSPACE_COMPONENT_LE] THEN ASM_MESON_TAC[ELEMENTARY_COMPACT]; SPEC_TAC(`measure(s INTER {x:real^N | x$k <= &0}) / measure s`, `u:real`)] THEN X_GEN_TAC `u:real` THEN STRIP_TAC THEN SUBGOAL_THEN `?b:real. &0 < b /\ !x:real^N. x IN t ==> abs(x$k) <= b` STRIP_ASSUME_TAC THENL [SUBGOAL_THEN `bounded(t:real^N->bool)` MP_TAC THENL [ASM_MESON_TAC[ELEMENTARY_BOUNDED]; REWRITE_TAC[BOUNDED_POS]] THEN MATCH_MP_TAC MONO_EXISTS THEN ASM_MESON_TAC[COMPONENT_LE_NORM; REAL_LE_TRANS]; ALL_TAC] THEN SUBGOAL_THEN `?a. a IN real_interval[--b,b] /\ measure (t INTER {x:real^N | x$k <= a}) / measure t = u` (fun th -> MESON_TAC[th]) THEN MATCH_MP_TAC REAL_IVT_INCREASING THEN REPEAT CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `&0 <= u ==> x = &0 ==> x <= u`)) THEN REWRITE_TAC[real_div; REAL_ENTIRE] THEN DISJ1_TAC THEN MATCH_MP_TAC MEASURE_EQ_0 THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `{x:real^N | x$k = --b}` THEN ASM_SIMP_TAC[NEGLIGIBLE_STANDARD_HYPERPLANE] THEN SIMP_TAC[SUBSET; IN_INTER; IN_ELIM_THM; GSYM REAL_LE_ANTISYM] THEN ASM_MESON_TAC[REAL_ARITH `abs x <= b ==> --b <= x`]; FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH `u <= &1 ==> x = &1 ==> u <= x`)) THEN ASM_SIMP_TAC[REAL_EQ_LDIV_EQ; REAL_MUL_LID] THEN AP_TERM_TAC THEN REWRITE_TAC[EXTENSION; IN_INTER; IN_ELIM_THM] THEN ASM_MESON_TAC[REAL_ARITH `abs x <= b ==> x <= b`]] THEN REWRITE_TAC[real_div] THEN MATCH_MP_TAC REAL_CONTINUOUS_ON_RMUL THEN REWRITE_TAC[REAL_CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN] THEN GEN_TAC THEN DISCH_TAC THEN MATCH_MP_TAC REAL_CONTINUOUS_ATREAL_WITHINREAL THEN MATCH_MP_TAC REAL_CONTINUOUS_MEASURE_IN_HALFSPACE_LE THEN ASM_MESON_TAC[MEASURABLE_ELEMENTARY]; ALL_TAC] THEN FIRST_X_ASSUM(MP_TAC o SPECL [`d1:(real^N->bool)->bool`; `IMAGE (IMAGE (\x:real^N. x - a % basis k)) d2`; `s:real^N->bool`; `IMAGE (\x:real^N. x - a % basis k) t`; `i:real^N->bool`; `j:real^N->bool`; `k:num`]) THEN ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[VECTOR_ARITH `x - a:real^N = --a + x`] THEN ASM_REWRITE_TAC[NEGLIGIBLE_TRANSLATION_EQ; DIVISION_OF_TRANSLATION] THEN ASM_REWRITE_TAC[IMAGE_EQ_EMPTY] THEN ANTS_TAC THENL [CONJ_TAC THENL [EXPAND_TAC "n" THEN AP_TERM_TAC THEN MATCH_MP_TAC CARD_IMAGE_INJ THEN CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[DIVISION_OF_FINITE]] THEN MATCH_MP_TAC(MESON[] `(!x y. Q x y ==> R x y) ==> (!x y. P x /\ P y /\ Q x y ==> R x y)`) THEN REWRITE_TAC[INJECTIVE_IMAGE] THEN VECTOR_ARITH_TAC; ALL_TAC] THEN SUBGOAL_THEN `IMAGE (\x. --(a % basis k) + x) t INTER {x:real^N | x$k >= &0} = IMAGE (\x. --(a % basis k) + x) (t INTER {x | x$k >= a}) /\ IMAGE (\x. --(a % basis k) + x) t INTER {x:real^N | x$k <= &0} = IMAGE (\x. --(a % basis k) + x) (t INTER {x | x$k <= a})` (CONJUNCTS_THEN SUBST1_TAC) THENL [CONJ_TAC THEN MATCH_MP_TAC(SET_RULE `!g. (!x. f(x) IN s' <=> x IN s) /\ (!x. g(f x) = x) ==> IMAGE f t INTER s' = IMAGE f (t INTER s)`) THEN ASM_SIMP_TAC[IN_ELIM_THM; VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT; BASIS_COMPONENT; VECTOR_NEG_COMPONENT] THEN EXISTS_TAC `\x:real^N. a % basis k + x` THEN (CONJ_TAC THENL [REAL_ARITH_TAC; VECTOR_ARITH_TAC]); ALL_TAC] THEN REWRITE_TAC[MEASURE_TRANSLATION] THEN MATCH_MP_TAC(REAL_FIELD `&0 < s /\ &0 < t /\ t' / t = s' / s /\ s' + s'' = s /\ t' + t'' = t ==> t' / t = s' / s /\ t'' / t = s'' / s`) THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THEN MATCH_MP_TAC MEASURE_NEGLIGIBLE_UNION_EQ THEN (REPEAT CONJ_TAC THENL [MATCH_MP_TAC MEASURABLE_COMPACT THEN MATCH_MP_TAC COMPACT_INTER_CLOSED THEN REWRITE_TAC[CLOSED_HALFSPACE_COMPONENT_LE; CLOSED_HALFSPACE_COMPONENT_GE] THEN ASM_MESON_TAC[ELEMENTARY_COMPACT]; MATCH_MP_TAC MEASURABLE_COMPACT THEN MATCH_MP_TAC COMPACT_INTER_CLOSED THEN REWRITE_TAC[CLOSED_HALFSPACE_COMPONENT_LE; CLOSED_HALFSPACE_COMPONENT_GE] THEN ASM_MESON_TAC[ELEMENTARY_COMPACT]; MATCH_MP_TAC(SET_RULE `(!x. P x \/ Q x) ==> s INTER {x | P x} UNION s INTER {x | Q x} = s`) THEN REAL_ARITH_TAC; REWRITE_TAC[SET_RULE `(t INTER {x | P x}) INTER (t INTER {x | Q x}) = t INTER {x | P x /\ Q x}`] THEN MATCH_MP_TAC(MESON[NEGLIGIBLE_SUBSET; INTER_SUBSET] `negligible t ==> negligible(s INTER t)`) THEN REWRITE_TAC[REAL_ARITH `x <= a /\ x >= a <=> x = a`] THEN ASM_SIMP_TAC[NEGLIGIBLE_STANDARD_HYPERPLANE]]); REWRITE_TAC[MEASURE_TRANSLATION] THEN MATCH_MP_TAC(REAL_ARITH `a' = a ==> a' >= b ==> a >= b`) THEN AP_TERM_TAC THEN REWRITE_TAC[SET_RULE `{f x y | x IN s /\ y IN IMAGE g t} = {f x (g y) | x IN s /\ y IN t}`] THEN ONCE_REWRITE_TAC[VECTOR_ARITH `x + a + y:real^N = a + x + y`] THEN REWRITE_TAC[SET_RULE `{a + x + y:real^N | x IN s /\ y IN t} = IMAGE (\z. a + z) {x + y | x IN s /\ y IN t}`] THEN REWRITE_TAC[MEASURE_TRANSLATION]]] THEN REPEAT STRIP_TAC THEN REWRITE_TAC[real_ge] THEN SUBGOAL_THEN `measurable(s:real^N->bool) /\ measurable(t:real^N->bool)` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[MEASURABLE_ELEMENTARY]; ALL_TAC] THEN SUBGOAL_THEN `measurable {x + y:real^N | x IN s /\ y IN t}` ASSUME_TAC THENL [MATCH_MP_TAC MEASURABLE_COMPACT THEN MATCH_MP_TAC COMPACT_SUMS THEN ASM_MESON_TAC[ELEMENTARY_COMPACT]; ALL_TAC] THEN SUBGOAL_THEN `&0 < measure(s:real^N->bool) /\ &0 < measure(t:real^N->bool)` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[MEASURABLE_MEASURE_POS_LT]; ALL_TAC] THEN SUBGOAL_THEN `FINITE(d1:(real^N->bool)->bool) /\ FINITE(d2:(real^N->bool)->bool)` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[DIVISION_OF_FINITE]; ALL_TAC] THEN W(MP_TAC o PART_MATCH (lhs o rand) REAL_LE_ROOT o snd) THEN ASM_SIMP_TAC[DIMINDEX_NONZERO; MEASURE_POS_LE; ROOT_POS_LE; REAL_LE_ADD] THEN DISCH_THEN SUBST1_TAC THEN ABBREV_TAC `dl = {l INTER {x:real^N | x$k <= &0} |l| l IN d1 DELETE j /\ ~(l INTER {x | x$k <= &0} = {})}` THEN ABBREV_TAC `dr = {l INTER {x:real^N | x$k >= &0} |l| l IN d1 DELETE i /\ ~(l INTER {x | x$k >= &0} = {})}` THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `measure {x + y:real^N | x IN UNIONS dl /\ y IN (t INTER {x | x$k <= &0})} + measure {x + y | x IN UNIONS dr /\ y IN (t INTER {x | x$k >= &0})}` THEN CONJ_TAC THENL [ALL_TAC; MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `measure {x + y:real^N | x IN (s INTER {x | x$k <= &0}) /\ y IN (t INTER {x | x$k <= &0})} + measure {x + y:real^N | x IN (s INTER {x | x$k >= &0}) /\ y IN (t INTER {x | x$k >= &0})}` THEN CONJ_TAC THENL [MATCH_MP_TAC REAL_LE_ADD2 THEN CONJ_TAC THEN (MATCH_MP_TAC MEASURE_SUBSET THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC MEASURABLE_COMPACT THEN MATCH_MP_TAC COMPACT_SUMS THEN CONJ_TAC THENL [MATCH_MP_TAC COMPACT_UNIONS THEN MAP_EVERY EXPAND_TAC ["dl";
"dr"] THEN ONCE_REWRITE_TAC[SET_RULE `{f x |x| x IN s /\ ~(f x = a)} = {t | t IN IMAGE f s /\ ~(t = a)}`] THEN ASM_SIMP_TAC[IN_DELETE; FINITE_DELETE; FINITE_IMAGE; FINITE_RESTRICT; IMP_CONJ] THEN REWRITE_TAC[IN_ELIM_THM; IMP_CONJ; FORALL_IN_IMAGE; IN_DELETE] THEN FIRST_ASSUM(fun th -> GEN_REWRITE_TAC I [MATCH_MP FORALL_IN_DIVISION th]) THEN ASM_SIMP_TAC[INTERVAL_SPLIT; COMPACT_INTERVAL]; MATCH_MP_TAC COMPACT_INTER_CLOSED THEN REWRITE_TAC[CLOSED_HALFSPACE_COMPONENT_LE; CLOSED_HALFSPACE_COMPONENT_GE] THEN ASM_MESON_TAC[ELEMENTARY_COMPACT]]; MATCH_MP_TAC MEASURABLE_COMPACT THEN MATCH_MP_TAC COMPACT_SUMS THEN CONJ_TAC THEN MATCH_MP_TAC COMPACT_INTER_CLOSED THEN REWRITE_TAC[CLOSED_HALFSPACE_COMPONENT_LE; CLOSED_HALFSPACE_COMPONENT_GE] THEN ASM_MESON_TAC[ELEMENTARY_COMPACT]; MATCH_MP_TAC(SET_RULE `s SUBSET s' ==> {x + y:real^N | x IN s /\ y IN t} SUBSET {x + y:real^N | x IN s' /\ y IN t}`) THEN MAP_EVERY EXPAND_TAC ["dl"; "dr"] THEN REWRITE_TAC[SUBSET; FORALL_IN_UNIONS] THEN REWRITE_TAC[FORALL_IN_GSPEC; IMP_CONJ; RIGHT_FORALL_IMP_THM] THEN SIMP_TAC[IN_DELETE; IN_INTER; IN_ELIM_THM] THEN RULE_ASSUM_TAC(REWRITE_RULE[division_of]) THEN ASM_MESON_TAC[IN_UNIONS]]); ALL_TAC] THEN SUBGOAL_THEN `s = (s INTER {x:real^N | x$k <= &0}) UNION (s INTER {x | x$k >= &0}) /\ t = (t INTER {x:real^N | x$k <= &0}) UNION (t INTER {x | x$k >= &0})` MP_TAC THENL [CONJ_TAC THEN MATCH_MP_TAC(SET_RULE `(!x. P x \/ Q x) ==> s = (s INTER {x | P x}) UNION (s INTER {x | Q x})`) THEN REAL_ARITH_TAC; DISCH_THEN(fun th -> GEN_REWRITE_TAC (RAND_CONV o ONCE_DEPTH_CONV) [th])] THEN W(MP_TAC o PART_MATCH (rand o rand) MEASURE_NEGLIGIBLE_UNION o lhand o snd) THEN ANTS_TAC THENL [REPEAT(CONJ_TAC THENL [MATCH_MP_TAC MEASURABLE_COMPACT THEN MATCH_MP_TAC COMPACT_SUMS THEN CONJ_TAC THEN MATCH_MP_TAC COMPACT_INTER_CLOSED THEN REWRITE_TAC[CLOSED_HALFSPACE_COMPONENT_LE; CLOSED_HALFSPACE_COMPONENT_GE] THEN ASM_MESON_TAC[ELEMENTARY_COMPACT]; ALL_TAC]) THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `{x:real^N | x$k = &0}` THEN ASM_SIMP_TAC[NEGLIGIBLE_STANDARD_HYPERPLANE] THEN REWRITE_TAC[GSYM REAL_LE_ANTISYM] THEN MATCH_MP_TAC(SET_RULE `s SUBSET {x | P x} /\ t SUBSET {x | Q x} ==> (s INTER t) SUBSET {x | P x /\ Q x}`) THEN REWRITE_TAC[SUBSET; FORALL_IN_GSPEC] THEN SIMP_TAC[IN_INTER; IN_ELIM_THM; REAL_LE_ADD; VECTOR_ADD_COMPONENT; real_ge; REAL_ARITH `x <= &0 /\ y <= &0 ==> x + y <= &0`]; DISCH_THEN(SUBST1_TAC o SYM)] THEN MATCH_MP_TAC MEASURE_SUBSET THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC MEASURABLE_UNION THEN CONJ_TAC THEN MATCH_MP_TAC MEASURABLE_COMPACT THEN MATCH_MP_TAC COMPACT_SUMS THEN CONJ_TAC THEN MATCH_MP_TAC COMPACT_INTER_CLOSED THEN REWRITE_TAC[CLOSED_HALFSPACE_COMPONENT_LE; CLOSED_HALFSPACE_COMPONENT_GE] THEN ASM_MESON_TAC[ELEMENTARY_COMPACT]; MATCH_MP_TAC MEASURABLE_COMPACT THEN MATCH_MP_TAC COMPACT_SUMS THEN CONJ_TAC THEN MATCH_MP_TAC COMPACT_UNION THEN CONJ_TAC THEN MATCH_MP_TAC COMPACT_INTER_CLOSED THEN REWRITE_TAC[CLOSED_HALFSPACE_COMPONENT_LE; CLOSED_HALFSPACE_COMPONENT_GE] THEN ASM_MESON_TAC[ELEMENTARY_COMPACT]; SET_TAC[]]] THEN SUBGOAL_THEN `&0 < measure(s INTER {x:real^N | x$k <= &0}) /\ &0 < measure(s INTER {x:real^N | x$k >= &0})` STRIP_ASSUME_TAC THENL [ASM_SIMP_TAC[MEASURABLE_MEASURE_POS_LT; MEASURABLE_INTER_HALFSPACE_LE; MEASURABLE_INTER_HALFSPACE_GE] THEN CONJ_TAC THENL [UNDISCH_TAC `~negligible(i:real^N->bool)`; UNDISCH_TAC `~negligible(j:real^N->bool)`] THEN REWRITE_TAC[CONTRAPOS_THM] THEN MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] NEGLIGIBLE_SUBSET) THEN ASM_REWRITE_TAC[SUBSET_INTER] THEN UNDISCH_TAC `d1 division_of (s:real^N->bool)` THEN REWRITE_TAC[division_of] THEN ASM SET_TAC[]; ALL_TAC] THEN FIRST_X_ASSUM(fun th -> MP_TAC(SPECL [`dl:(real^N->bool)->bool`; `{l INTER {x:real^N | x$k <= &0} |l| l IN d2 /\ ~(l INTER {x | x$k <= &0} = {})}`; `UNIONS dl :real^N->bool`; `t INTER {x:real^N | x$k <= &0}`] th) THEN MP_TAC(SPECL [`dr:(real^N->bool)->bool`; `{l INTER {x:real^N | x$k >= &0} |l| l IN d2 /\ ~(l INTER {x | x$k >= &0} = {})}`; `UNIONS dr :real^N->bool`; `t INTER {x:real^N | x$k >= &0}`] th)) THEN ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [REPEAT CONJ_TAC THENL [EXPAND_TAC "n" THEN MATCH_MP_TAC LTE_ADD2 THEN MAP_EVERY EXPAND_TAC ["dl"; "dr"] THEN ONCE_REWRITE_TAC[SET_RULE `{f x |x| x IN s /\ ~(f x = a)} = {t | t IN IMAGE f s /\ ~(t = a)}`] THEN CONJ_TAC THENL [MATCH_MP_TAC LET_TRANS THEN EXISTS_TAC `CARD(d1 DELETE (i:real^N->bool))` THEN CONJ_TAC THENL [ALL_TAC; MATCH_MP_TAC CARD_PSUBSET THEN ASM SET_TAC[]]; ALL_TAC] THEN MATCH_MP_TAC(ARITH_RULE `CARD {x | x IN IMAGE f s /\ P x} <= CARD(IMAGE f s) /\ CARD(IMAGE f s) <= CARD s ==> CARD {x | x IN IMAGE f s /\ P x} <= CARD s`) THEN ASM_SIMP_TAC[CARD_IMAGE_LE; FINITE_DELETE] THEN MATCH_MP_TAC CARD_SUBSET THEN ASM_SIMP_TAC[FINITE_IMAGE; FINITE_DELETE] THEN SET_TAC[]; REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_UNIONS] THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN EXISTS_TAC `j INTER {x:real^N | x$k >= &0}` THEN REWRITE_TAC[RIGHT_EXISTS_AND_THM; MEMBER_NOT_EMPTY] THEN MAP_EVERY EXPAND_TAC ["dl"; "dr"] THEN ONCE_REWRITE_TAC[SET_RULE `{f x |x| x IN s /\ ~(f x = a)} = {t | t IN IMAGE f s /\ ~(t = a)}`] THEN REWRITE_TAC[IN_ELIM_THM; IN_IMAGE; IN_DELETE; GSYM CONJ_ASSOC] THEN CONJ_TAC THENL [EXISTS_TAC `j:real^N->bool` THEN ASM_REWRITE_TAC[]; MATCH_MP_TAC(SET_RULE `~(s = {}) /\ s SUBSET t ==> ~(s INTER t = {})`) THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[division_of]]; DISCH_TAC THEN UNDISCH_TAC `measure (t INTER {x:real^N | x$k >= &0}) / measure t = measure (s INTER {x:real^N | x$k >= &0}) / measure s` THEN ASM_SIMP_TAC[MEASURE_EMPTY; REAL_EQ_RDIV_EQ] THEN REWRITE_TAC[real_div; REAL_MUL_LZERO] THEN CONV_TAC(RAND_CONV SYM_CONV) THEN ASM_SIMP_TAC[REAL_LT_IMP_NZ]; REWRITE_TAC[division_of] THEN MAP_EVERY EXPAND_TAC ["dl"; "dr"] THEN ONCE_REWRITE_TAC[SET_RULE `{f x |x| x IN s /\ ~(f x = a)} = {t | t IN IMAGE f s /\ ~(t = a)}`] THEN ASM_SIMP_TAC[FINITE_RESTRICT; FINITE_IMAGE; FINITE_DELETE] THEN REWRITE_TAC[IN_ELIM_THM; IMP_CONJ; FORALL_IN_IMAGE; IN_DELETE] THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM; FORALL_IN_IMAGE; IN_DELETE] THEN CONJ_TAC THENL [FIRST_ASSUM(fun th -> GEN_REWRITE_TAC I [MATCH_MP FORALL_IN_DIVISION th]) THEN REPEAT GEN_TAC THEN REPEAT DISCH_TAC THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ALL_TAC; ASM_SIMP_TAC[INTERVAL_SPLIT] THEN MESON_TAC[]] THEN MATCH_MP_TAC(SET_RULE `x IN s ==> x SUBSET UNIONS s`) THEN ASM SET_TAC[]; REPEAT STRIP_TAC THEN MATCH_MP_TAC(SET_RULE `interior(s) INTER interior(s') = {} /\ interior(s INTER t) SUBSET interior s /\ interior(s' INTER t) SUBSET interior s' ==> interior(s INTER t) INTER interior(s' INTER t) = {}`) THEN SIMP_TAC[SUBSET_INTERIOR; INTER_SUBSET] THEN ASM_MESON_TAC[division_of]]; REWRITE_TAC[division_of] THEN MAP_EVERY EXPAND_TAC ["dl"; "dr"] THEN ONCE_REWRITE_TAC[SET_RULE `{f x |x| x IN s /\ ~(f x = a)} = {t | t IN IMAGE f s /\ ~(t = a)}`] THEN ASM_SIMP_TAC[FINITE_RESTRICT; FINITE_IMAGE; FINITE_DELETE] THEN REWRITE_TAC[IN_ELIM_THM; IMP_CONJ; FORALL_IN_IMAGE; IN_DELETE] THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM; FORALL_IN_IMAGE; IN_DELETE] THEN REPEAT CONJ_TAC THENL [FIRST_ASSUM(fun th -> GEN_REWRITE_TAC I [MATCH_MP FORALL_IN_DIVISION th]) THEN REPEAT GEN_TAC THEN REPEAT DISCH_TAC THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ALL_TAC; ASM_SIMP_TAC[INTERVAL_SPLIT] THEN MESON_TAC[]] THEN MATCH_MP_TAC(SET_RULE `s SUBSET t ==> s INTER u SUBSET t INTER u`) THEN RULE_ASSUM_TAC(REWRITE_RULE[division_of]) THEN ASM_MESON_TAC[]; REPEAT STRIP_TAC THEN MATCH_MP_TAC(SET_RULE `interior(s) INTER interior(s') = {} /\ interior(s INTER t) SUBSET interior s /\ interior(s' INTER t) SUBSET interior s' ==> interior(s INTER t) INTER interior(s' INTER t) = {}`) THEN SIMP_TAC[SUBSET_INTERIOR; INTER_SUBSET] THEN ASM_MESON_TAC[division_of]; REWRITE_TAC[SET_RULE `{x | x IN s /\ ~(x = a)} = s DELETE a`] THEN GEN_REWRITE_TAC LAND_CONV [SET_RULE `s = {} UNION s`] THEN REWRITE_TAC[GSYM UNIONS_INSERT] THEN REWRITE_TAC[SET_RULE `x INSERT (s DELETE x) = x INSERT s`] THEN REWRITE_TAC[UNIONS_INSERT; UNION_EMPTY] THEN REWRITE_TAC[GSYM SIMPLE_IMAGE; GSYM INTER_UNIONS] THEN AP_THM_TAC THEN AP_TERM_TAC THEN ASM_MESON_TAC[division_of]]]; ALL_TAC] THEN ONCE_REWRITE_TAC[TAUT `p ==> q ==> r <=> q ==> p ==> r`] THEN ANTS_TAC THENL [REPEAT CONJ_TAC THENL [EXPAND_TAC "n" THEN MATCH_MP_TAC LTE_ADD2 THEN MAP_EVERY EXPAND_TAC ["dl"; "dr"] THEN ONCE_REWRITE_TAC[SET_RULE `{f x |x| x IN s /\ ~(f x = a)} = {t | t IN IMAGE f s /\ ~(t = a)}`] THEN CONJ_TAC THENL [MATCH_MP_TAC LET_TRANS THEN EXISTS_TAC `CARD(d1 DELETE (j:real^N->bool))` THEN CONJ_TAC THENL [ALL_TAC; MATCH_MP_TAC CARD_PSUBSET THEN ASM SET_TAC[]]; ALL_TAC] THEN MATCH_MP_TAC(ARITH_RULE `CARD {x | x IN IMAGE f s /\ P x} <= CARD(IMAGE f s) /\ CARD(IMAGE f s) <= CARD s ==> CARD {x | x IN IMAGE f s /\ P x} <= CARD s`) THEN ASM_SIMP_TAC[CARD_IMAGE_LE; FINITE_DELETE] THEN MATCH_MP_TAC CARD_SUBSET THEN ASM_SIMP_TAC[FINITE_IMAGE; FINITE_DELETE] THEN SET_TAC[]; REWRITE_TAC[GSYM MEMBER_NOT_EMPTY; IN_UNIONS] THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN EXISTS_TAC `i INTER {x:real^N | x$k <= &0}` THEN REWRITE_TAC[RIGHT_EXISTS_AND_THM; MEMBER_NOT_EMPTY] THEN MAP_EVERY EXPAND_TAC ["dl"; "dr"] THEN ONCE_REWRITE_TAC[SET_RULE `{f x |x| x IN s /\ ~(f x = a)} = {t | t IN IMAGE f s /\ ~(t = a)}`] THEN REWRITE_TAC[IN_ELIM_THM; IN_IMAGE; IN_DELETE; GSYM CONJ_ASSOC] THEN CONJ_TAC THENL [EXISTS_TAC `i:real^N->bool` THEN ASM_REWRITE_TAC[]; MATCH_MP_TAC(SET_RULE `~(s = {}) /\ s SUBSET t ==> ~(s INTER t = {})`) THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[division_of]]; DISCH_TAC THEN UNDISCH_TAC `measure (t INTER {x:real^N | x$k <= &0}) / measure t = measure (s INTER {x:real^N | x$k <= &0}) / measure s` THEN ASM_SIMP_TAC[MEASURE_EMPTY; REAL_EQ_RDIV_EQ] THEN REWRITE_TAC[real_div; REAL_MUL_LZERO] THEN CONV_TAC(RAND_CONV SYM_CONV) THEN ASM_SIMP_TAC[REAL_LT_IMP_NZ]; REWRITE_TAC[division_of] THEN MAP_EVERY EXPAND_TAC ["dl"; "dr"] THEN ONCE_REWRITE_TAC[SET_RULE `{f x |x| x IN s /\ ~(f x = a)} = {t | t IN IMAGE f s /\ ~(t = a)}`] THEN ASM_SIMP_TAC[FINITE_RESTRICT; FINITE_IMAGE; FINITE_DELETE] THEN REWRITE_TAC[IN_ELIM_THM; IMP_CONJ; FORALL_IN_IMAGE; IN_DELETE] THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM; FORALL_IN_IMAGE; IN_DELETE] THEN CONJ_TAC THENL [FIRST_ASSUM(fun th -> GEN_REWRITE_TAC I [MATCH_MP FORALL_IN_DIVISION th]) THEN REPEAT GEN_TAC THEN REPEAT DISCH_TAC THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ALL_TAC; ASM_SIMP_TAC[INTERVAL_SPLIT] THEN MESON_TAC[]] THEN MATCH_MP_TAC(SET_RULE `x IN s ==> x SUBSET UNIONS s`) THEN ASM SET_TAC[]; REPEAT STRIP_TAC THEN MATCH_MP_TAC(SET_RULE `interior(s) INTER interior(s') = {} /\ interior(s INTER t) SUBSET interior s /\ interior(s' INTER t) SUBSET interior s' ==> interior(s INTER t) INTER interior(s' INTER t) = {}`) THEN SIMP_TAC[SUBSET_INTERIOR; INTER_SUBSET] THEN ASM_MESON_TAC[division_of]]; REWRITE_TAC[division_of] THEN MAP_EVERY EXPAND_TAC ["dl"; "dr"] THEN ONCE_REWRITE_TAC[SET_RULE `{f x |x| x IN s /\ ~(f x = a)} = {t | t IN IMAGE f s /\ ~(t = a)}`] THEN ASM_SIMP_TAC[FINITE_RESTRICT; FINITE_IMAGE; FINITE_DELETE] THEN REWRITE_TAC[IN_ELIM_THM; IMP_CONJ; FORALL_IN_IMAGE; IN_DELETE] THEN REWRITE_TAC[RIGHT_FORALL_IMP_THM; FORALL_IN_IMAGE; IN_DELETE] THEN REPEAT CONJ_TAC THENL [FIRST_ASSUM(fun th -> GEN_REWRITE_TAC I [MATCH_MP FORALL_IN_DIVISION th]) THEN REPEAT GEN_TAC THEN REPEAT DISCH_TAC THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ALL_TAC; ASM_SIMP_TAC[INTERVAL_SPLIT] THEN MESON_TAC[]] THEN MATCH_MP_TAC(SET_RULE `s SUBSET t ==> s INTER u SUBSET t INTER u`) THEN RULE_ASSUM_TAC(REWRITE_RULE[division_of]) THEN ASM_MESON_TAC[]; REPEAT STRIP_TAC THEN MATCH_MP_TAC(SET_RULE `interior(s) INTER interior(s') = {} /\ interior(s INTER t) SUBSET interior s /\ interior(s' INTER t) SUBSET interior s' ==> interior(s INTER t) INTER interior(s' INTER t) = {}`) THEN SIMP_TAC[SUBSET_INTERIOR; INTER_SUBSET] THEN ASM_MESON_TAC[division_of]; REWRITE_TAC[SET_RULE `{x | x IN s /\ ~(x = a)} = s DELETE a`] THEN GEN_REWRITE_TAC LAND_CONV [SET_RULE `s = {} UNION s`] THEN REWRITE_TAC[GSYM UNIONS_INSERT] THEN REWRITE_TAC[SET_RULE `x INSERT (s DELETE x) = x INSERT s`] THEN REWRITE_TAC[UNIONS_INSERT; UNION_EMPTY] THEN REWRITE_TAC[GSYM SIMPLE_IMAGE; GSYM INTER_UNIONS] THEN AP_THM_TAC THEN AP_TERM_TAC THEN ASM_MESON_TAC[division_of]]]; ALL_TAC] THEN REWRITE_TAC[real_ge; IMP_IMP] THEN SUBGOAL_THEN `compact(UNIONS dl:real^N->bool) /\ compact(UNIONS dr:real^N->bool)` STRIP_ASSUME_TAC THENL [CONJ_TAC THEN MATCH_MP_TAC COMPACT_UNIONS THEN MAP_EVERY EXPAND_TAC ["dl"; "dr"] THEN ONCE_REWRITE_TAC[SET_RULE `{f x |x| x IN s /\ ~(f x = a)} = {t | t IN IMAGE f s /\ ~(t = a)}`] THEN ASM_SIMP_TAC[FINITE_RESTRICT; FINITE_IMAGE; FINITE_DELETE] THEN REWRITE_TAC[IMP_CONJ; IN_ELIM_THM; FORALL_IN_IMAGE; IN_DELETE] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC COMPACT_INTER_CLOSED THEN REWRITE_TAC[CLOSED_HALFSPACE_COMPONENT_LE; CLOSED_HALFSPACE_COMPONENT_GE] THEN ASM_MESON_TAC[division_of; COMPACT_INTERVAL]; ALL_TAC] THEN SUBGOAL_THEN `measurable(UNIONS dl:real^N->bool) /\ measurable(UNIONS dr:real^N->bool)` STRIP_ASSUME_TAC THENL [ASM_SIMP_TAC[MEASURABLE_COMPACT]; ALL_TAC] THEN SUBGOAL_THEN `measurable { x + y:real^N | x IN UNIONS dl /\ y IN t INTER {x | x$k <= &0}} /\ measurable { x + y:real^N | x IN UNIONS dr /\ y IN t INTER {x | &0 <= x$k}}` STRIP_ASSUME_TAC THENL [CONJ_TAC THEN MATCH_MP_TAC MEASURABLE_COMPACT THEN MATCH_MP_TAC COMPACT_SUMS THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC COMPACT_INTER_CLOSED THEN REWRITE_TAC[REAL_ARITH `&0 <= x <=> x >= &0`; CLOSED_HALFSPACE_COMPONENT_LE; CLOSED_HALFSPACE_COMPONENT_GE] THEN ASM_MESON_TAC[ELEMENTARY_COMPACT]; ALL_TAC] THEN ASM (CONV_TAC o GEN_SIMPLIFY_CONV TOP_DEPTH_SQCONV (basic_ss []) 5) [REAL_LE_ROOT; DIMINDEX_NONZERO; REAL_LE_ADD; ROOT_POS_LE; MEASURE_POS_LE; MEASURABLE_INTER_HALFSPACE_LE; MEASURABLE_INTER_HALFSPACE_GE; REAL_ARITH `&0 <= x <=> x >= &0`] THEN MATCH_MP_TAC(REAL_ARITH `x <= a' + b' ==> a' <= a /\ b' <= b ==> x <= a + b`) THEN SUBGOAL_THEN `measure(UNIONS dl :real^N->bool) = measure(s INTER {x:real^N | x$k <= &0}) /\ measure(UNIONS dr :real^N->bool) = measure(s INTER {x:real^N | x$k >= &0})` (CONJUNCTS_THEN SUBST1_TAC) THENL [MAP_EVERY EXPAND_TAC ["dl"; "dr"] THEN ONCE_REWRITE_TAC[SET_RULE `{f x |x| x IN s /\ ~(f x = a)} = {t | t IN IMAGE f s /\ ~(t = a)}`] THEN REWRITE_TAC[SET_RULE `{x | x IN s /\ ~(x = a)} = s DELETE a`] THEN CONJ_TAC THEN GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [SET_RULE `s = {} UNION s`] THEN REWRITE_TAC[GSYM UNIONS_INSERT] THEN REWRITE_TAC[SET_RULE `x INSERT (s DELETE x) = x INSERT s`] THEN REWRITE_TAC[UNIONS_INSERT; UNION_EMPTY] THEN REWRITE_TAC[GSYM SIMPLE_IMAGE; GSYM INTER_UNIONS] THEN MATCH_MP_TAC MEASURE_NEGLIGIBLE_SYMDIFF THEN MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `{x:real^N | x$k = &0}` THEN ASM_SIMP_TAC[NEGLIGIBLE_STANDARD_HYPERPLANE] THEN MATCH_MP_TAC(SET_RULE `s SUBSET t /\ t DIFF s SUBSET u ==> (s DIFF t UNION t DIFF s) SUBSET u`) THEN REWRITE_TAC[SET_RULE `s INTER u DIFF t INTER u = (s DIFF t) INTER u`] THEN (SUBGOAL_THEN `s:real^N->bool = UNIONS d1` SUBST1_TAC THENL [ASM_MESON_TAC[division_of]; ALL_TAC] THEN CONJ_TAC THENL [SET_TAC[]; ALL_TAC]) THEN REWRITE_TAC[GSYM REAL_LE_ANTISYM; real_ge; SET_RULE `{x | P x /\ Q x} = {x | P x} INTER {x | Q x}`] THENL [MATCH_MP_TAC(SET_RULE `s SUBSET t ==> s INTER u SUBSET u INTER t`); MATCH_MP_TAC(SET_RULE `s SUBSET t ==> s INTER u SUBSET t INTER u`)] THEN RULE_ASSUM_TAC(REWRITE_RULE[real_ge]) THEN FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ_ALT] SUBSET_TRANS)) THEN SET_TAC[]; ALL_TAC] THEN SUBGOAL_THEN `root (dimindex (:N)) (measure (s INTER {x:real^N | x$k <= &0})) + root (dimindex (:N)) (measure (t INTER {x:real^N | x$k <= &0})) = root (dimindex (:N)) (measure (s INTER {x | x$k <= &0})) * (&1 + root (dimindex (:N)) (measure (t INTER {x | x$k <= &0})) / root (dimindex (:N)) (measure (s INTER {x | x$k <= &0}))) /\ root (dimindex (:N)) (measure (s INTER {x:real^N | x$k >= &0})) + root (dimindex (:N)) (measure (t INTER {x:real^N | x$k >= &0})) = root (dimindex (:N)) (measure (s INTER {x | x$k >= &0})) * (&1 + root (dimindex (:N)) (measure (t INTER {x | x$k >= &0})) / root (dimindex (:N)) (measure (s INTER {x | x$k >= &0})))` (CONJUNCTS_THEN SUBST1_TAC) THENL [CONJ_TAC THEN MATCH_MP_TAC(REAL_FIELD `&0 < s ==> s + t = s * (&1 + t / s)`) THEN ASM_SIMP_TAC[ROOT_POS_LT; DIMINDEX_NONZERO]; ALL_TAC] THEN ASM_SIMP_TAC[DIMINDEX_NONZERO; GSYM REAL_ROOT_DIV; MEASURE_POS_LE; MEASURABLE_INTER_HALFSPACE_LE; MEASURABLE_INTER_HALFSPACE_GE] THEN SUBGOAL_THEN `measure(t INTER {x:real^N | x$k <= &0}) / measure(s INTER {x:real^N | x$k <= &0}) = measure t / measure s /\ measure(t INTER {x:real^N | x$k >= &0}) / measure(s INTER {x:real^N | x$k >= &0}) = measure t / measure s` (CONJUNCTS_THEN SUBST1_TAC) THENL [MATCH_MP_TAC(REAL_FIELD `tn / t = sn / s /\ tp / t = sp / s /\ &0 < sp /\ &0 < sn /\ &0 < s /\ &0 < t ==> tn / sn = t / s /\ tp / sp = t / s`) THEN ASM_REWRITE_TAC[]; ALL_TAC] THEN REWRITE_TAC[GSYM REAL_ADD_RDISTRIB; REAL_POW_MUL] THEN ASM_SIMP_TAC[REAL_POW_ROOT; DIMINDEX_NONZERO; MEASURE_POS_LE; MEASURABLE_INTER_HALFSPACE_LE; MEASURABLE_INTER_HALFSPACE_GE] THEN SUBGOAL_THEN `measure (s INTER {x | x$k <= &0}) + measure (s INTER {x | x$k >= &0}) = root (dimindex(:N)) (measure(s:real^N->bool)) pow (dimindex(:N))` SUBST1_TAC THENL [ASM_SIMP_TAC[REAL_POW_ROOT; DIMINDEX_NONZERO; REAL_LT_IMP_LE] THEN MATCH_MP_TAC MEASURE_NEGLIGIBLE_UNION_EQ THEN ASM_SIMP_TAC[MEASURABLE_INTER_HALFSPACE_LE; MEASURABLE_INTER_HALFSPACE_GE] THEN CONJ_TAC THENL [REWRITE_TAC[EXTENSION; IN_UNION; IN_INTER; IN_ELIM_THM] THEN X_GEN_TAC `x:real^N` THEN ASM_CASES_TAC `(x:real^N) IN s` THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC; MATCH_MP_TAC NEGLIGIBLE_SUBSET THEN EXISTS_TAC `{x:real^N | x$k = &0}` THEN ASM_SIMP_TAC[NEGLIGIBLE_STANDARD_HYPERPLANE] THEN REWRITE_TAC[GSYM REAL_LE_ANTISYM; real_ge] THEN SET_TAC[]]; ASM_SIMP_TAC[GSYM REAL_ROOT_MUL; MEASURE_POS_LE; DIMINDEX_NONZERO; REAL_LE_DIV; GSYM REAL_POW_MUL; REAL_ADD_LDISTRIB; REAL_MUL_RID] THEN ASM_SIMP_TAC[REAL_DIV_LMUL; REAL_LT_IMP_NZ; REAL_LE_REFL]]);; (* ------------------------------------------------------------------------- *) (* Now for open sets. *) (* ------------------------------------------------------------------------- *)
let BRUNN_MINKOWSKI_OPEN = 
prove (`!s t:real^N->bool. (s = {} <=> t = {}) /\ bounded s /\ open s /\ bounded t /\ open t ==> root (dimindex(:N)) (measure {x + y | x IN s /\ y IN t}) >= root (dimindex(:N)) (measure s) + root (dimindex(:N)) (measure t)`,
REPEAT GEN_TAC THEN ASM_CASES_TAC `t:real^N->bool = {}` THEN ASM_SIMP_TAC[SET_RULE `{x + y:real^N | x IN {} /\ y IN {}} = {}`; REAL_LE_REFL; MEASURE_EMPTY; ROOT_0; DIMINDEX_NONZERO; real_ge; REAL_ADD_LID] THEN STRIP_TAC THEN MATCH_MP_TAC(ISPEC `atreal(&0) within {e | &0 <= e}` REALLIM_UBOUND) THEN EXISTS_TAC `\e. root (dimindex(:N)) (measure(s:real^N->bool) - e) + root (dimindex(:N)) (measure(t:real^N->bool) - e)` THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC REALLIM_TRANSFORM_EVENTUALLY THEN EXISTS_TAC `\e. (measure(s:real^N->bool) - e) rpow (inv(&(dimindex(:N)))) + (measure(t:real^N->bool) - e) rpow (inv(&(dimindex(:N))))` THEN REWRITE_TAC[] THEN CONJ_TAC THENL [REWRITE_TAC[EVENTUALLY_WITHINREAL] THEN EXISTS_TAC `min (measure(s:real^N->bool)) (measure(t:real^N->bool))` THEN ASM_SIMP_TAC[REAL_LT_MIN; IN_ELIM_THM; REAL_SUB_RZERO; MEASURE_OPEN_POS_LT] THEN REPEAT STRIP_TAC THEN BINOP_TAC THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC REAL_ROOT_RPOW THEN REWRITE_TAC[DIMINDEX_NONZERO] THEN ASM_REAL_ARITH_TAC; ASM_SIMP_TAC[REAL_ROOT_RPOW; MEASURE_OPEN_POS_LT; DIMINDEX_NONZERO; REAL_LT_IMP_LE] THEN MATCH_MP_TAC REALLIM_ADD THEN CONJ_TAC THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [REAL_ARITH `measure(s:real^N->bool) = measure s - &0`] THEN REWRITE_TAC[GSYM REAL_CONTINUOUS_WITHINREAL] THEN REWRITE_TAC[GSYM(REWRITE_CONV [o_DEF] `(\x. x rpow y) o (\e. s - e)`)] THEN MATCH_MP_TAC REAL_CONTINUOUS_WITHINREAL_COMPOSE THEN ASM_SIMP_TAC[REAL_CONTINUOUS_SUB; REAL_CONTINUOUS_WITHIN_ID; REAL_CONTINUOUS_CONST] THEN MATCH_MP_TAC REAL_CONTINUOUS_WITHIN_RPOW THEN REWRITE_TAC[REAL_LE_INV_EQ; REAL_POS]]; W(MP_TAC o PART_MATCH (lhand o rand) TRIVIAL_LIMIT_WITHIN_REALINTERVAL o rand o snd) THEN ANTS_TAC THENL [REWRITE_TAC[is_realinterval; IN_ELIM_THM] THEN REAL_ARITH_TAC; DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[EXTENSION; IN_SING] THEN DISCH_THEN(MP_TAC o SPEC `&1`) THEN REWRITE_TAC[IN_ELIM_THM] THEN REAL_ARITH_TAC]; REWRITE_TAC[EVENTUALLY_WITHINREAL] THEN EXISTS_TAC `min (measure(s:real^N->bool)) (measure(t:real^N->bool))` THEN ASM_SIMP_TAC[REAL_LT_MIN; IN_ELIM_THM; REAL_SUB_RZERO; MEASURE_OPEN_POS_LT] THEN X_GEN_TAC `e:real` THEN REWRITE_TAC[real_abs] THEN ASM_CASES_TAC `&0 <= e` THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THEN MAP_EVERY (fun l -> MP_TAC(ISPECL l OPEN_MEASURABLE_INNER_DIVISION)) [[`t:real^N->bool`; `e:real`]; [`s:real^N->bool`; `e:real`]] THEN ASM_SIMP_TAC[MEASURABLE_OPEN; GSYM REAL_LT_SUB_RADD] THEN DISCH_THEN(X_CHOOSE_THEN `D:(real^N->bool)->bool` STRIP_ASSUME_TAC) THEN DISCH_THEN(X_CHOOSE_THEN `E:(real^N->bool)->bool` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`UNIONS D:real^N->bool`; `UNIONS E:real^N->bool`] BRUNN_MINKOWSKI_ELEMENTARY) THEN ANTS_TAC THENL [ASM_MESON_TAC[MEASURE_EMPTY; REAL_ARITH `e < s ==> ~(s - e < &0)`]; ALL_TAC] THEN MATCH_MP_TAC(REAL_ARITH `s1 <= r1 /\ s2 <= r2 /\ rs <= s ==> rs >= r1 + r2 ==> s1 + s2 <= s`) THEN REPEAT CONJ_TAC THEN MATCH_MP_TAC ROOT_MONO_LE THEN ASM_SIMP_TAC[DIMINDEX_NONZERO; REAL_SUB_LE; REAL_LT_IMP_LE] THEN SUBGOAL_THEN `measurable {x + y :real^N | x IN UNIONS D /\ y IN UNIONS E}` ASSUME_TAC THENL [MATCH_MP_TAC MEASURABLE_COMPACT THEN MATCH_MP_TAC COMPACT_SUMS THEN ASM_MESON_TAC[ELEMENTARY_COMPACT]; ALL_TAC] THEN CONJ_TAC THENL [MATCH_MP_TAC MEASURE_POS_LE; MATCH_MP_TAC MEASURE_SUBSET] THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN MATCH_MP_TAC MEASURABLE_OPEN THEN ASM_SIMP_TAC[BOUNDED_SUMS; OPEN_SUMS]]);;
(* ------------------------------------------------------------------------- *) (* Now for convex sets. *) (* ------------------------------------------------------------------------- *)
let BRUNN_MINKOWSKI_CONVEX = 
prove (`!s t:real^N->bool. (s = {} <=> t = {}) /\ bounded s /\ convex s /\ bounded t /\ convex t ==> root (dimindex(:N)) (measure {x + y | x IN s /\ y IN t}) >= root (dimindex(:N)) (measure s) + root (dimindex(:N)) (measure t)`,
REPEAT GEN_TAC THEN ASM_CASES_TAC `t:real^N->bool = {}` THEN ASM_SIMP_TAC[BRUNN_MINKOWSKI_OPEN; OPEN_EMPTY] THEN STRIP_TAC THEN ASM_SIMP_TAC[GSYM MEASURE_INTERIOR; NEGLIGIBLE_CONVEX_FRONTIER; real_ge] THEN ASM_CASES_TAC `interior s:real^N->bool = {}` THENL [ASM_SIMP_TAC[MEASURE_EMPTY; ROOT_0; DIMINDEX_NONZERO; REAL_ADD_LID] THEN MATCH_MP_TAC ROOT_MONO_LE THEN ASM_SIMP_TAC[DIMINDEX_NONZERO; MEASURE_POS_LE; MEASURABLE_INTERIOR] THEN SUBGOAL_THEN `?a:real^N. a IN s` STRIP_ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `measure(IMAGE (\x:real^N. a + x) t)` THEN CONJ_TAC THENL [ASM_SIMP_TAC[MEASURE_TRANSLATION; MEASURE_INTERIOR; NEGLIGIBLE_CONVEX_FRONTIER; REAL_LE_REFL]; MATCH_MP_TAC MEASURE_SUBSET THEN ASM_SIMP_TAC[MEASURABLE_TRANSLATION_EQ; MEASURABLE_CONVEX; CONVEX_SUMS; BOUNDED_SUMS] THEN ASM SET_TAC[]]; ALL_TAC] THEN ASM_CASES_TAC `interior t:real^N->bool = {}` THENL [ASM_SIMP_TAC[MEASURE_EMPTY; ROOT_0; DIMINDEX_NONZERO; REAL_ADD_RID] THEN MATCH_MP_TAC ROOT_MONO_LE THEN ASM_SIMP_TAC[DIMINDEX_NONZERO; MEASURE_POS_LE; MEASURABLE_INTERIOR] THEN SUBGOAL_THEN `?a:real^N. a IN t` STRIP_ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `measure(IMAGE (\x:real^N. a + x) s)` THEN CONJ_TAC THENL [ASM_SIMP_TAC[MEASURE_TRANSLATION; MEASURE_INTERIOR; NEGLIGIBLE_CONVEX_FRONTIER; REAL_LE_REFL]; MATCH_MP_TAC MEASURE_SUBSET THEN ASM_SIMP_TAC[MEASURABLE_TRANSLATION_EQ; MEASURABLE_CONVEX; CONVEX_SUMS; BOUNDED_SUMS] THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [VECTOR_ADD_SYM] THEN ASM SET_TAC[]]; ALL_TAC] THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `root (dimindex (:N)) (measure {x + y:real^N | x IN interior s /\ y IN interior t})` THEN ASM_SIMP_TAC[GSYM real_ge; BRUNN_MINKOWSKI_OPEN; BOUNDED_INTERIOR; OPEN_INTERIOR] THEN REWRITE_TAC[real_ge] THEN MATCH_MP_TAC ROOT_MONO_LE THEN REWRITE_TAC[DIMINDEX_NONZERO] THEN CONJ_TAC THENL [MATCH_MP_TAC MEASURE_POS_LE; MATCH_MP_TAC MEASURE_SUBSET] THEN ASM_SIMP_TAC[DIMINDEX_NONZERO; MEASURE_POS_LE; BOUNDED_SUMS; MEASURABLE_CONVEX; BOUNDED_INTERIOR; CONVEX_SUMS; CONVEX_INTERIOR] THEN MATCH_MP_TAC(SET_RULE `s' SUBSET s /\ t' SUBSET t ==> {x + y:real^N | x IN s' /\ y IN t'} SUBSET {x + y | x IN s /\ y IN t}`) THEN REWRITE_TAC[INTERIOR_SUBSET]);;
(* ------------------------------------------------------------------------- *) (* Now for compact sets. *) (* ------------------------------------------------------------------------- *)
let INTERS_SUMS_CLOSED_BALL_SEQUENTIAL = 
prove (`!s:real^N->bool. closed s ==> INTERS {{x + d | x IN s /\ d IN ball(vec 0,inv(&n + &1))} | n IN (:num)} = s`,
REPEAT STRIP_TAC THEN GEN_REWRITE_TAC I [EXTENSION] THEN REWRITE_TAC[INTERS_GSPEC; IN_ELIM_THM; IN_UNIV] THEN X_GEN_TAC `x:real^N` THEN EQ_TAC THENL [ASM_CASES_TAC `(x:real^N) IN s` THEN ASM_REWRITE_TAC[] THEN MP_TAC(ISPECL [`s:real^N->bool`; `x:real^N`] SEPARATE_POINT_CLOSED) THEN ASM_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN X_GEN_TAC `d:real` THEN DISCH_THEN(CONJUNCTS_THEN2 MP_TAC ASSUME_TAC) THEN GEN_REWRITE_TAC LAND_CONV [REAL_ARCH_INV] THEN DISCH_THEN(X_CHOOSE_THEN `n:num` STRIP_ASSUME_TAC) THEN DISCH_THEN(MP_TAC o SPEC `n:num`) THEN REWRITE_TAC[NOT_EXISTS_THM] THEN MAP_EVERY X_GEN_TAC [`y:real^N`; `e:real^N`] THEN REWRITE_TAC[IN_BALL_0] THEN STRIP_TAC THEN FIRST_X_ASSUM SUBST_ALL_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `y:real^N`) THEN ASM_REWRITE_TAC[NORM_ARITH `dist(y + e:real^N,y) = norm e`] THEN SUBGOAL_THEN `inv(&n + &1) <= inv(&n)` MP_TAC THENL [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; ASM_REAL_ARITH_TAC]; DISCH_TAC THEN X_GEN_TAC `n:num` THEN MAP_EVERY EXISTS_TAC [`x:real^N`; `vec 0:real^N`] THEN ASM_REWRITE_TAC[CENTRE_IN_BALL; VECTOR_ADD_RID; REAL_LT_INV_EQ] THEN REAL_ARITH_TAC]);;
let MEASURE_SUMS_COMPACT_EPSILON = 
prove (`!s:real^N->bool. compact s ==> ((\e. measure {x + d | x IN s /\ d IN ball(vec 0,e)}) ---> measure s) (atreal (&0) within {e | &0 <= e})`,
REPEAT STRIP_TAC THEN MP_TAC(ISPEC `\n. {x + d:real^N | x IN s /\ d IN ball(vec 0,inv(&n + &1))}` HAS_MEASURE_NESTED_INTERS) THEN ASM_SIMP_TAC[INTERS_SUMS_CLOSED_BALL_SEQUENTIAL; COMPACT_IMP_CLOSED] THEN ANTS_TAC THENL [ASM_SIMP_TAC[MEASURABLE_OPEN; BOUNDED_SUMS; OPEN_SUMS; COMPACT_IMP_BOUNDED; BOUNDED_BALL; OPEN_BALL] THEN GEN_TAC THEN MATCH_MP_TAC(SET_RULE `t' SUBSET t ==> {x + y:real^N | x IN s /\ y IN t'} SUBSET {x + y | x IN s /\ y IN t}`) THEN MATCH_MP_TAC SUBSET_BALL THEN MATCH_MP_TAC REAL_LE_INV2 THEN REWRITE_TAC[REAL_OF_NUM_ADD; REAL_OF_NUM_LE; REAL_OF_NUM_LT] THEN ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[GSYM(REWRITE_RULE[o_DEF] TENDSTO_REAL)] THEN REWRITE_TAC[REALLIM_SEQUENTIALLY; REALLIM_WITHINREAL] THEN STRIP_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 EXISTS_TAC `inv(&N + &1)` THEN ASM_REWRITE_TAC[REAL_LT_INV_EQ] THEN CONJ_TAC THENL [REAL_ARITH_TAC; ALL_TAC] THEN X_GEN_TAC `d:real` THEN REWRITE_TAC[IN_ELIM_THM; REAL_SUB_RZERO] THEN ASM_CASES_TAC `abs d = d` THENL [FIRST_X_ASSUM SUBST1_TAC THEN STRIP_TAC; ASM_REAL_ARITH_TAC] THEN FIRST_X_ASSUM(MP_TAC o SPEC `N:num`) THEN REWRITE_TAC[LE_REFL] THEN MATCH_MP_TAC(REAL_ARITH `m <= m1 /\ m1 <= m2 ==> abs(m2 - m) < e ==> abs(m1 - m) < e`) THEN CONJ_TAC THEN MATCH_MP_TAC MEASURE_SUBSET THEN ASM_SIMP_TAC[MEASURABLE_OPEN; BOUNDED_SUMS; OPEN_SUMS; COMPACT_IMP_BOUNDED; BOUNDED_BALL; OPEN_BALL] THENL [REWRITE_TAC[SUBSET; IN_ELIM_THM] THEN X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN MAP_EVERY EXISTS_TAC [`x:real^N`; `vec 0:real^N`] THEN ASM_REWRITE_TAC[CENTRE_IN_BALL; VECTOR_ADD_RID]; MATCH_MP_TAC(SET_RULE `t' SUBSET t ==> {x + y:real^N | x IN s /\ y IN t'} SUBSET {x + y | x IN s /\ y IN t}`) THEN MATCH_MP_TAC SUBSET_BALL THEN ASM_SIMP_TAC[REAL_LT_IMP_LE]]);;
let BRUNN_MINKOWSKI_COMPACT = 
prove (`!s t:real^N->bool. (s = {} <=> t = {}) /\ compact s /\ compact t ==> root (dimindex(:N)) (measure {x + y | x IN s /\ y IN t}) >= root (dimindex(:N)) (measure s) + root (dimindex(:N)) (measure t)`,
let lemma1 = prove
   (`{ x + y:real^N | x IN {x + d | x IN s /\ d IN ball(vec 0,e)} /\
                      y IN {y + d | y IN t /\ d IN ball(vec 0,e)}} =
     { z + d | z IN {x + y | x IN s /\ y IN t} /\ d IN ball(vec 0,&2 * e) }`,
    MATCH_MP_TAC SUBSET_ANTISYM THEN
    REWRITE_TAC[SUBSET; FORALL_IN_GSPEC; RIGHT_FORALL_IMP_THM; IMP_CONJ] THEN
    REWRITE_TAC[IN_ELIM_THM; IN_BALL_0] THEN CONJ_TAC THENL
     [X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
      X_GEN_TAC `d:real^N` THEN DISCH_TAC THEN
      X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN
      X_GEN_TAC `k:real^N` THEN DISCH_TAC THEN
      EXISTS_TAC `x + y:real^N` THEN EXISTS_TAC `d + k:real^N` THEN
      CONJ_TAC THENL [ALL_TAC; VECTOR_ARITH_TAC] THEN
      ASM_SIMP_TAC[NORM_ARITH
       `norm(d:real^N) < e /\ norm(k) < e ==> norm(d + k) < &2 * e`] THEN
      EXISTS_TAC `x:real^N` THEN EXISTS_TAC `y:real^N` THEN
      ASM_REWRITE_TAC[] THEN VECTOR_ARITH_TAC;
      X_GEN_TAC `x:real^N` THEN DISCH_TAC THEN
      X_GEN_TAC `y:real^N` THEN DISCH_TAC THEN
      X_GEN_TAC `d:real^N` THEN DISCH_TAC THEN
      EXISTS_TAC `x + inv(&2) % d:real^N` THEN
      EXISTS_TAC `y + inv(&2) % d:real^N` THEN
      CONJ_TAC THENL [ALL_TAC; VECTOR_ARITH_TAC] THEN CONJ_TAC THENL
       [EXISTS_TAC `x:real^N`; EXISTS_TAC `y:real^N`] THEN
      EXISTS_TAC `inv(&2) % d:real^N` THEN
      ASM_REWRITE_TAC[NORM_MUL] THEN ASM_REAL_ARITH_TAC])
  and lemma2 = prove
   (`(f ---> l) (atreal (&0) within {e | &0 <= e})
     ==> ((\e. f(&2 * e)) ---> l) (atreal (&0) within {e | &0 <= e})`,
    REWRITE_TAC[REALLIM_WITHINREAL] 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
    REWRITE_TAC[IN_ELIM_THM; REAL_SUB_RZERO] THEN
    DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN
    EXISTS_TAC `d / &2` THEN ASM_REWRITE_TAC[REAL_HALF] THEN
    REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN
    ASM_REAL_ARITH_TAC) in
  REPEAT GEN_TAC THEN ASM_CASES_TAC `t:real^N->bool = {}` THEN
  ASM_SIMP_TAC[BRUNN_MINKOWSKI_OPEN; OPEN_EMPTY; BOUNDED_EMPTY] THEN
  STRIP_TAC THEN REWRITE_TAC[real_ge] THEN
  MATCH_MP_TAC(ISPEC `atreal (&0) within {e | &0 <= e}` REALLIM_LE) THEN
  EXISTS_TAC
   `\e. root (dimindex(:N))
             (measure {x + d:real^N | x IN s /\ d IN ball(vec 0,e)}) +
        root (dimindex(:N))
             (measure {x + d:real^N | x IN t /\ d IN ball(vec 0,e)})` THEN
  EXISTS_TAC
   `\e. root (dimindex(:N))
             (measure { x + y:real^N |
                   x IN {x + d | x IN s /\ d IN ball(vec 0,e)} /\
                   y IN {y + d | y IN t /\ d IN ball(vec 0,e)}})` THEN
  REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL
   [ASM (CONV_TAC o GEN_SIMPLIFY_CONV TOP_DEPTH_SQCONV (basic_ss []) 5)
     [REAL_ROOT_RPOW; DIMINDEX_NONZERO; MEASURE_POS_LE; MEASURABLE_COMPACT;
      MEASURABLE_OPEN; BOUNDED_SUMS; OPEN_SUMS;
      COMPACT_IMP_BOUNDED; BOUNDED_BALL; OPEN_BALL] THEN
    MATCH_MP_TAC REALLIM_ADD THEN CONJ_TAC THEN MATCH_MP_TAC REALLIM_RPOW THEN
    REWRITE_TAC[REAL_LE_INV_EQ; REAL_POS] THEN
    MATCH_MP_TAC MEASURE_SUMS_COMPACT_EPSILON THEN ASM_REWRITE_TAC[];
    REWRITE_TAC[lemma1] THEN MATCH_MP_TAC lemma2 THEN
    ASM (CONV_TAC o GEN_SIMPLIFY_CONV TOP_DEPTH_SQCONV (basic_ss []) 6)
     [REAL_ROOT_RPOW; DIMINDEX_NONZERO; MEASURE_POS_LE; MEASURABLE_COMPACT;
      MEASURABLE_OPEN; BOUNDED_SUMS; OPEN_SUMS; COMPACT_SUMS;
      COMPACT_IMP_BOUNDED; BOUNDED_BALL; OPEN_BALL] THEN
    MATCH_MP_TAC REALLIM_RPOW THEN
    REWRITE_TAC[REAL_LE_INV_EQ; REAL_POS] THEN
    MATCH_MP_TAC MEASURE_SUMS_COMPACT_EPSILON THEN
    ASM_SIMP_TAC[COMPACT_SUMS];
    W(MP_TAC o PART_MATCH (lhand o rand) TRIVIAL_LIMIT_WITHIN_REALINTERVAL o
      rand o snd) THEN
    ANTS_TAC THENL
     [REWRITE_TAC[is_realinterval; IN_ELIM_THM] THEN REAL_ARITH_TAC;
      DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[EXTENSION; IN_SING] THEN
      DISCH_THEN(MP_TAC o SPEC `&1`) THEN REWRITE_TAC[IN_ELIM_THM] THEN
      REAL_ARITH_TAC];
    MATCH_MP_TAC ALWAYS_EVENTUALLY THEN X_GEN_TAC `e:real` THEN
    REWRITE_TAC[GSYM real_ge] THEN
    MATCH_MP_TAC BRUNN_MINKOWSKI_OPEN THEN
    SIMP_TAC[OPEN_SUMS; OPEN_BALL] THEN
    ASM_SIMP_TAC[BOUNDED_SUMS; BOUNDED_BALL; COMPACT_IMP_BOUNDED] THEN
    ASM SET_TAC[]]);;
(* ------------------------------------------------------------------------- *) (* Finally, for an arbitrary measurable set. In this general case, the *) (* measurability of the sum-set is needed as an additional hypothesis. *) (* ------------------------------------------------------------------------- *)
let BRUNN_MINKOWSKI_MEASURABLE = 
prove (`!s t:real^N->bool. (s = {} <=> t = {}) /\ measurable s /\ measurable t /\ measurable {x + y | x IN s /\ y IN t} ==> root (dimindex(:N)) (measure {x + y | x IN s /\ y IN t}) >= root (dimindex(:N)) (measure s) + root (dimindex(:N)) (measure t)`,
REPEAT GEN_TAC THEN ASM_CASES_TAC `t:real^N->bool = {}` THEN ASM_SIMP_TAC[BRUNN_MINKOWSKI_OPEN; OPEN_EMPTY; BOUNDED_EMPTY] THEN STRIP_TAC THEN REWRITE_TAC[real_ge] THEN ASM_CASES_TAC `measure(s:real^N->bool) = &0` THENL [ASM_SIMP_TAC[ROOT_0; DIMINDEX_NONZERO; REAL_ADD_LID] THEN MATCH_MP_TAC ROOT_MONO_LE THEN ASM_SIMP_TAC[DIMINDEX_NONZERO; MEASURE_POS_LE] THEN SUBGOAL_THEN `?a:real^N. a IN s` STRIP_ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `measure(IMAGE (\x:real^N. a + x) t)` THEN CONJ_TAC THENL [ASM_SIMP_TAC[MEASURE_TRANSLATION; REAL_LE_REFL]; MATCH_MP_TAC MEASURE_SUBSET THEN ASM_REWRITE_TAC[MEASURABLE_TRANSLATION_EQ] THEN ASM SET_TAC[]]; ALL_TAC] THEN ASM_CASES_TAC `measure(t:real^N->bool) = &0` THENL [ASM_SIMP_TAC[ROOT_0; DIMINDEX_NONZERO; REAL_ADD_RID] THEN MATCH_MP_TAC ROOT_MONO_LE THEN ASM_SIMP_TAC[DIMINDEX_NONZERO; MEASURE_POS_LE] THEN SUBGOAL_THEN `?a:real^N. a IN t` STRIP_ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `measure(IMAGE (\x:real^N. a + x) s)` THEN CONJ_TAC THENL [ASM_SIMP_TAC[MEASURE_TRANSLATION; REAL_LE_REFL]; MATCH_MP_TAC MEASURE_SUBSET THEN ASM_REWRITE_TAC[MEASURABLE_TRANSLATION_EQ] THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [VECTOR_ADD_SYM] THEN ASM SET_TAC[]]; ALL_TAC] THEN SUBGOAL_THEN `&0 < measure(s:real^N->bool) /\ &0 < measure(t:real^N->bool)` STRIP_ASSUME_TAC THENL [ASM_MESON_TAC[MEASURABLE_MEASURE_POS_LT; MEASURABLE_MEASURE_EQ_0]; ALL_TAC] THEN MATCH_MP_TAC(ISPEC `atreal(&0) within {e | &0 <= e}` REALLIM_UBOUND) THEN EXISTS_TAC `\e. root (dimindex(:N)) (measure(s:real^N->bool) - e) + root (dimindex(:N)) (measure(t:real^N->bool) - e)` THEN REPEAT CONJ_TAC THENL [MATCH_MP_TAC REALLIM_TRANSFORM_EVENTUALLY THEN EXISTS_TAC `\e. (measure(s:real^N->bool) - e) rpow (inv(&(dimindex(:N)))) + (measure(t:real^N->bool) - e) rpow (inv(&(dimindex(:N))))` THEN REWRITE_TAC[] THEN CONJ_TAC THENL [REWRITE_TAC[EVENTUALLY_WITHINREAL] THEN EXISTS_TAC `min (measure(s:real^N->bool)) (measure(t:real^N->bool))` THEN ASM_SIMP_TAC[REAL_LT_MIN; IN_ELIM_THM; REAL_SUB_RZERO; MEASURE_OPEN_POS_LT] THEN REPEAT STRIP_TAC THEN BINOP_TAC THEN CONV_TAC SYM_CONV THEN MATCH_MP_TAC REAL_ROOT_RPOW THEN REWRITE_TAC[DIMINDEX_NONZERO] THEN ASM_REAL_ARITH_TAC; ASM_SIMP_TAC[REAL_ROOT_RPOW; MEASURE_OPEN_POS_LT; DIMINDEX_NONZERO; REAL_LT_IMP_LE] THEN MATCH_MP_TAC REALLIM_ADD THEN CONJ_TAC THEN GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [REAL_ARITH `measure(s:real^N->bool) = measure s - &0`] THEN REWRITE_TAC[GSYM REAL_CONTINUOUS_WITHINREAL] THEN REWRITE_TAC[GSYM(REWRITE_CONV [o_DEF] `(\x. x rpow y) o (\e. s - e)`)] THEN MATCH_MP_TAC REAL_CONTINUOUS_WITHINREAL_COMPOSE THEN ASM_SIMP_TAC[REAL_CONTINUOUS_SUB; REAL_CONTINUOUS_WITHIN_ID; REAL_CONTINUOUS_CONST] THEN MATCH_MP_TAC REAL_CONTINUOUS_WITHIN_RPOW THEN REWRITE_TAC[REAL_LE_INV_EQ; REAL_POS]]; W(MP_TAC o PART_MATCH (lhand o rand) TRIVIAL_LIMIT_WITHIN_REALINTERVAL o rand o snd) THEN ANTS_TAC THENL [REWRITE_TAC[is_realinterval; IN_ELIM_THM] THEN REAL_ARITH_TAC; DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[EXTENSION; IN_SING] THEN DISCH_THEN(MP_TAC o SPEC `&1`) THEN REWRITE_TAC[IN_ELIM_THM] THEN REAL_ARITH_TAC]; REWRITE_TAC[EVENTUALLY_WITHINREAL] THEN EXISTS_TAC `min (measure(s:real^N->bool)) (measure(t:real^N->bool))` THEN ASM_SIMP_TAC[REAL_LT_MIN; IN_ELIM_THM; REAL_SUB_RZERO] THEN X_GEN_TAC `e:real` THEN REWRITE_TAC[real_abs] THEN ASM_CASES_TAC `&0 <= e` THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THEN MAP_EVERY (fun l -> MP_TAC(ISPECL l MEASURABLE_INNER_COMPACT)) [[`t:real^N->bool`; `e:real`]; [`s:real^N->bool`; `e:real`]] THEN ASM_SIMP_TAC[MEASURABLE_OPEN; GSYM REAL_LT_SUB_RADD] THEN DISCH_THEN(X_CHOOSE_THEN `s':real^N->bool` STRIP_ASSUME_TAC) THEN DISCH_THEN(X_CHOOSE_THEN `t':real^N->bool` STRIP_ASSUME_TAC) THEN MP_TAC(ISPECL [`s':real^N->bool`; `t':real^N->bool`] BRUNN_MINKOWSKI_COMPACT) THEN ANTS_TAC THENL [ASM_MESON_TAC[MEASURE_EMPTY; REAL_ARITH `e < s ==> ~(s - e < &0)`]; ALL_TAC] THEN MATCH_MP_TAC(REAL_ARITH `s1 <= r1 /\ s2 <= r2 /\ rs <= s ==> rs >= r1 + r2 ==> s1 + s2 <= s`) THEN REPEAT CONJ_TAC THEN MATCH_MP_TAC ROOT_MONO_LE THEN ASM_SIMP_TAC[DIMINDEX_NONZERO; REAL_SUB_LE; REAL_LT_IMP_LE] THEN ASM_SIMP_TAC[MEASURE_POS_LE; COMPACT_SUMS; MEASURABLE_COMPACT] THEN MATCH_MP_TAC MEASURE_SUBSET THEN ASM_SIMP_TAC[MEASURE_POS_LE; COMPACT_SUMS; MEASURABLE_COMPACT] THEN ASM SET_TAC[]]);;