(* ========================================================================= *) (* 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]);;