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(* ========================================================================= *)
(* Existence of a (bounded) non-measurable set of reals. *)
(* ========================================================================= *)
needs "Multivariate/realanalysis.ml";;
(* ------------------------------------------------------------------------- *)
(* Classic Vitali proof (positive case simplified via Steinhaus's theorem). *)
(* ------------------------------------------------------------------------- *)
let NON_MEASURABLE_SET = prove
(`?s. real_bounded s /\ ~real_measurable s`,
MAP_EVERY ABBREV_TAC
[`equiv = \x y. &0 <= x /\ x < &1 /\ &0 <= y /\ y < &1 /\ rational(x - y)`;
`(canonize:real->real) = \x. @y. equiv x y`;
`V = IMAGE (canonize:real->real) {x | &0 <= x /\ x < &1}`] THEN
SUBGOAL_THEN `!x. equiv x x <=> &0 <= x /\ x < &1` ASSUME_TAC THENL
[EXPAND_TAC "equiv" THEN REWRITE_TAC[REAL_SUB_REFL; RATIONAL_NUM; CONJ_ACI];
ALL_TAC] THEN
SUBGOAL_THEN `!x y:real. equiv x y ==> equiv y x` ASSUME_TAC THENL
[EXPAND_TAC "equiv" THEN MESON_TAC[RATIONAL_NEG; REAL_NEG_SUB];
ALL_TAC] THEN
SUBGOAL_THEN `!x y z:real. equiv x y /\ equiv y z ==> equiv x z`
ASSUME_TAC THENL
[EXPAND_TAC "equiv" THEN MESON_TAC[RATIONAL_ADD;
REAL_ARITH `x - z:real = (x - y) + (y - z)`];
ALL_TAC] THEN
SUBGOAL_THEN
`!x. &0 <= x /\ x < &1 ==> (equiv:real->real->bool) x (canonize x)`
ASSUME_TAC THENL
[REPEAT STRIP_TAC THEN EXPAND_TAC "canonize" THEN ASM_MESON_TAC[];
ALL_TAC] THEN
SUBGOAL_THEN
`!x y. x IN V /\ y IN V /\ rational(x - y) ==> x = y`
ASSUME_TAC THENL
[EXPAND_TAC "V" THEN
REWRITE_TAC[IMP_CONJ; RIGHT_FORALL_IMP_THM; FORALL_IN_IMAGE] THEN
REWRITE_TAC[IN_ELIM_THM] THEN
X_GEN_TAC `x:real` THEN STRIP_TAC THEN
X_GEN_TAC `y:real` THEN STRIP_TAC THEN STRIP_TAC THEN
EXPAND_TAC "canonize" THEN AP_TERM_TAC THEN
REWRITE_TAC[FUN_EQ_THM] THEN X_GEN_TAC `z:real` THEN
SUBGOAL_THEN `equiv ((canonize:real->real) x) (canonize y) :bool`
(fun th -> MP_TAC th THEN ASM_MESON_TAC[]) THEN
EXPAND_TAC "equiv" THEN REWRITE_TAC[] THEN ASM_MESON_TAC[];
ALL_TAC] THEN
EXISTS_TAC `V:real->bool` THEN CONJ_TAC THENL
[MATCH_MP_TAC REAL_BOUNDED_SUBSET THEN
EXISTS_TAC `real_interval[&0,&1]` THEN
REWRITE_TAC[REAL_BOUNDED_REAL_INTERVAL; SUBSET; IN_REAL_INTERVAL] THEN
ASM SET_TAC[REAL_LT_IMP_LE];
DISCH_TAC] THEN
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [HAS_REAL_MEASURE_MEASURE]) THEN
FIRST_ASSUM(MP_TAC o MATCH_MP REAL_MEASURE_POS_LE) THEN
REWRITE_TAC[REAL_ARITH `&0 <= x <=> &0 < x \/ x = &0`] THEN
DISCH_THEN(DISJ_CASES_THEN2 ASSUME_TAC SUBST1_TAC) THENL
[MP_TAC(ISPEC `V:real->bool` REAL_STEINHAUS) THEN ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC(TAUT `~p ==> p ==> q`) THEN
DISCH_THEN(X_CHOOSE_THEN `d:real` (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
MP_TAC(ISPECL [`d / &2`; `d / &2`] RATIONAL_APPROXIMATION) THEN
ASM_REWRITE_TAC[REAL_HALF; LEFT_IMP_EXISTS_THM] THEN
X_GEN_TAC `q:real` THEN STRIP_TAC THEN
REWRITE_TAC[SUBSET; IN_REAL_INTERVAL; IN_ELIM_THM] THEN
DISCH_THEN(MP_TAC o SPEC `q:real`) THEN REWRITE_TAC[NOT_IMP] THEN
CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
ASM_CASES_TAC `q = &0` THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN
ASM_MESON_TAC[REAL_SUB_0];
REWRITE_TAC[HAS_REAL_MEASURE_0] THEN DISCH_TAC THEN
SUBGOAL_THEN `?r. rational = IMAGE r (:num)` STRIP_ASSUME_TAC THENL
[MATCH_MP_TAC COUNTABLE_AS_IMAGE THEN REWRITE_TAC[COUNTABLE_RATIONAL] THEN
REWRITE_TAC[FUN_EQ_THM; EMPTY] THEN MESON_TAC[RATIONAL_NUM];
ALL_TAC] THEN
MP_TAC(ISPEC `\n. IMAGE (\x. (r:num->real) n + x) V`
REAL_NEGLIGIBLE_COUNTABLE_UNIONS) THEN
ANTS_TAC THENL [ASM_SIMP_TAC[REAL_NEGLIGIBLE_TRANSLATION]; ALL_TAC] THEN
SUBGOAL_THEN `~(real_negligible(real_interval(&0,&1)))` MP_TAC THENL
[SIMP_TAC[GSYM REAL_MEASURABLE_REAL_MEASURE_EQ_0;
REAL_MEASURABLE_REAL_INTERVAL; REAL_MEASURE_REAL_INTERVAL] THEN
REAL_ARITH_TAC;
ALL_TAC] THEN
REWRITE_TAC[CONTRAPOS_THM] THEN
MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_NEGLIGIBLE_SUBSET) THEN
REWRITE_TAC[SUBSET; IN_REAL_INTERVAL] THEN
X_GEN_TAC `x:real` THEN STRIP_TAC THEN
SUBGOAL_THEN `(equiv:real->real->bool) x (canonize x)` MP_TAC THENL
[ASM_MESON_TAC[REAL_LT_IMP_LE]; ALL_TAC] THEN
EXPAND_TAC "equiv" THEN ASM_REWRITE_TAC[] THEN
REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
GEN_REWRITE_TAC LAND_CONV [GSYM IN] THEN
REWRITE_TAC[IN_IMAGE; IN_ELIM_THM; IN_UNIV] THEN
DISCH_THEN(X_CHOOSE_THEN `n:num` (ASSUME_TAC o SYM)) THEN
ONCE_REWRITE_TAC[SIMPLE_IMAGE_GEN] THEN REWRITE_TAC[UNIONS_IMAGE] THEN
REWRITE_TAC[IN_ELIM_THM; IN_IMAGE] THEN
MAP_EVERY EXISTS_TAC [`n:num`; `(canonize:real->real) x`] THEN
ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [REAL_ARITH_TAC; ALL_TAC] THEN
EXPAND_TAC "V" THEN MATCH_MP_TAC FUN_IN_IMAGE THEN
REWRITE_TAC[IN_ELIM_THM] THEN ASM_REAL_ARITH_TAC]);;
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