:: Darboux's Theorem :: by Noboru Endou , Katsumi Wasaki and Yasunari Shidama :: :: Received December 7, 1999 :: Copyright (c) 1999-2012 Association of Mizar Users begin Lm1: for j being Element of NAT holds (j -' j) + 1 = 1 proofend; Lm2: for n being Element of NAT st 1 <= n & n <= 2 & not n = 1 holds n = 2 proofend; definition let A be non empty closed_interval Subset of REAL; let D be Division of A; func delta D -> Real equals :: INTEGRA3:def 1 max (rng (upper_volume ((chi (A,A)),D))); correctness coherence max (rng (upper_volume ((chi (A,A)),D))) is Real; by XREAL_0:def_1; end; :: deftheorem defines delta INTEGRA3:def_1_:_ for A being non empty closed_interval Subset of REAL for D being Division of A holds delta D = max (rng (upper_volume ((chi (A,A)),D))); definition let A be non empty closed_interval Subset of REAL; let T be DivSequence of A; func delta T -> Real_Sequence means :Def2: :: INTEGRA3:def 2 for i being Element of NAT holds it . i = delta (T . i); existence ex b1 being Real_Sequence st for i being Element of NAT holds b1 . i = delta (T . i) proofend; uniqueness for b1, b2 being Real_Sequence st ( for i being Element of NAT holds b1 . i = delta (T . i) ) & ( for i being Element of NAT holds b2 . i = delta (T . i) ) holds b1 = b2 proofend; end; :: deftheorem Def2 defines delta INTEGRA3:def_2_:_ for A being non empty closed_interval Subset of REAL for T being DivSequence of A for b3 being Real_Sequence holds ( b3 = delta T iff for i being Element of NAT holds b3 . i = delta (T . i) ); theorem :: INTEGRA3:1 for A being non empty closed_interval Subset of REAL for D1, D2 being Division of A st D1 <= D2 holds delta D1 >= delta D2 proofend; theorem Th2: :: INTEGRA3:2 for A being non empty closed_interval Subset of REAL for D being Division of A st vol A <> 0 holds ex i being Element of NAT st ( i in dom D & vol (divset (D,i)) > 0 ) proofend; theorem Th3: :: INTEGRA3:3 for x being Real for A being non empty closed_interval Subset of REAL for D being Division of A st x in A holds ex j being Element of NAT st ( j in dom D & x in divset (D,j) ) proofend; theorem Th4: :: INTEGRA3:4 for A being non empty closed_interval Subset of REAL for D1, D2 being Division of A ex D being Division of A st ( D1 <= D & D2 <= D & rng D = (rng D1) \/ (rng D2) ) proofend; theorem Th5: :: INTEGRA3:5 for A being non empty closed_interval Subset of REAL for D1, D being Division of A st delta D1 < min (rng (upper_volume ((chi (A,A)),D))) holds for x, y being Real for i being Element of NAT st i in dom D1 & x in (rng D) /\ (divset (D1,i)) & y in (rng D) /\ (divset (D1,i)) holds x = y proofend; theorem Th6: :: INTEGRA3:6 for p, q being FinSequence of REAL st rng p = rng q & p is increasing & q is increasing holds p = q proofend; theorem Th7: :: INTEGRA3:7 for i, j being Element of NAT for A being non empty closed_interval Subset of REAL for D, D1 being Division of A st D <= D1 & i in dom D & j in dom D & i <= j holds ( indx (D1,D,i) <= indx (D1,D,j) & indx (D1,D,i) in dom D1 ) proofend; theorem Th8: :: INTEGRA3:8 for i, j being Element of NAT for A being non empty closed_interval Subset of REAL for D, D1 being Division of A st D <= D1 & i in dom D & j in dom D & i < j holds indx (D1,D,i) < indx (D1,D,j) proofend; theorem Th9: :: INTEGRA3:9 for A being non empty closed_interval Subset of REAL for D being Division of A holds delta D >= 0 proofend; Lm3: for A being non empty closed_interval Subset of REAL for g being Function of A,REAL st g | A is bounded holds upper_bound (rng g) >= lower_bound (rng g) proofend; Lm4: for A, B being non empty closed_interval Subset of REAL for f being Function of A,REAL st f | A is bounded & B c= A holds ( lower_bound (rng (f | B)) >= lower_bound (rng f) & lower_bound (rng f) <= upper_bound (rng (f | B)) & upper_bound (rng (f | B)) <= upper_bound (rng f) & lower_bound (rng (f | B)) <= upper_bound (rng f) ) proofend; Lm5: for j being Element of NAT for A being non empty closed_interval Subset of REAL for D1 being Division of A st j in dom D1 holds vol (divset (D1,j)) <= delta D1 proofend; Lm6: for x being Real for A being non empty closed_interval Subset of REAL for D1, D2 being Division of A for j1 being Element of NAT st j1 = (len D1) - 1 & x in divset (D1,(len D1)) & len D1 >= 2 & D1 <= D2 & rng D2 = (rng D1) \/ {x} holds rng (D2 | (indx (D2,D1,j1))) = rng (D1 | j1) proofend; theorem Th10: :: INTEGRA3:10 for x being Real for A being non empty closed_interval Subset of REAL for D1, D2 being Division of A for g being Function of A,REAL st x in divset (D1,(len D1)) & len D1 >= 2 & D1 <= D2 & rng D2 = (rng D1) \/ {x} & g | A is bounded holds (Sum (lower_volume (g,D2))) - (Sum (lower_volume (g,D1))) <= ((upper_bound (rng g)) - (lower_bound (rng g))) * (delta D1) proofend; theorem Th11: :: INTEGRA3:11 for x being Real for A being non empty closed_interval Subset of REAL for D1, D2 being Division of A for g being Function of A,REAL st x in divset (D1,(len D1)) & len D1 >= 2 & D1 <= D2 & rng D2 = (rng D1) \/ {x} & g | A is bounded holds (Sum (upper_volume (g,D1))) - (Sum (upper_volume (g,D2))) <= ((upper_bound (rng g)) - (lower_bound (rng g))) * (delta D1) proofend; Lm7: for y being Real for A being non empty closed_interval Subset of REAL for f being Function of A,REAL st vol A <> 0 & y in rng (lower_sum_set f) holds ex D being Division of A st ( D in dom (lower_sum_set f) & y = (lower_sum_set f) . D & D . 1 > lower_bound A ) proofend; theorem Th12: :: INTEGRA3:12 for r being Real for i, j being Element of NAT for A being non empty closed_interval Subset of REAL for D being Division of A st i in dom D & j in dom D & i <= j & r < (mid (D,i,j)) . 1 holds ex B being non empty closed_interval Subset of REAL st ( r = lower_bound B & upper_bound B = (mid (D,i,j)) . (len (mid (D,i,j))) & mid (D,i,j) is Division of B ) proofend; Lm8: for A being non empty closed_interval Subset of REAL for D1 being Division of A st vol A <> 0 & len D1 = 1 holds <*(lower_bound A)*> ^ D1 is non empty increasing FinSequence of REAL proofend; Lm9: for A being non empty closed_interval Subset of REAL for D2 being Division of A st lower_bound A < D2 . 1 holds <*(lower_bound A)*> ^ D2 is non empty increasing FinSequence of REAL proofend; Lm10: for A being non empty closed_interval Subset of REAL for D1 being Division of A for f being Function of A,REAL for MD1 being Division of A st MD1 = <*(lower_bound A)*> ^ D1 holds ( ( for i being Element of NAT st i in Seg (len D1) holds divset (MD1,(i + 1)) = divset (D1,i) ) & upper_volume (f,D1) = (upper_volume (f,MD1)) /^ 1 & lower_volume (f,D1) = (lower_volume (f,MD1)) /^ 1 ) proofend; Lm11: for A being non empty closed_interval Subset of REAL for D2, MD2 being Division of A st MD2 = <*(lower_bound A)*> ^ D2 holds vol (divset (MD2,1)) = 0 proofend; Lm12: for A being non empty closed_interval Subset of REAL for D1, MD1 being Division of A st MD1 = <*(lower_bound A)*> ^ D1 holds delta MD1 = delta D1 proofend; theorem Th13: :: INTEGRA3:13 for x being Real for A being non empty closed_interval Subset of REAL for D1, D2 being Division of A for f being Function of A,REAL st x in divset (D1,(len D1)) & vol A <> 0 & D1 <= D2 & rng D2 = (rng D1) \/ {x} & f | A is bounded & x > lower_bound A holds (Sum (lower_volume (f,D2))) - (Sum (lower_volume (f,D1))) <= ((upper_bound (rng f)) - (lower_bound (rng f))) * (delta D1) proofend; theorem Th14: :: INTEGRA3:14 for x being Real for A being non empty closed_interval Subset of REAL for D1, D2 being Division of A for f being Function of A,REAL st x in divset (D1,(len D1)) & vol A <> 0 & D1 <= D2 & rng D2 = (rng D1) \/ {x} & f | A is bounded & x > lower_bound A holds (Sum (upper_volume (f,D1))) - (Sum (upper_volume (f,D2))) <= ((upper_bound (rng f)) - (lower_bound (rng f))) * (delta D1) proofend; theorem Th15: :: INTEGRA3:15 for r being Real for i, j being Element of NAT for A being non empty closed_interval Subset of REAL for D1, D2 being Division of A st i in dom D1 & j in dom D1 & i <= j & D1 <= D2 & r < (mid (D2,(indx (D2,D1,i)),(indx (D2,D1,j)))) . 1 holds ex B being non empty closed_interval Subset of REAL ex MD1, MD2 being Division of B st ( r = lower_bound B & upper_bound B = MD2 . (len MD2) & upper_bound B = MD1 . (len MD1) & MD1 <= MD2 & MD1 = mid (D1,i,j) & MD2 = mid (D2,(indx (D2,D1,i)),(indx (D2,D1,j))) ) proofend; theorem Th16: :: INTEGRA3:16 for x being Real for A being non empty closed_interval Subset of REAL for D being Division of A st x in rng D holds ( D . 1 <= x & x <= D . (len D) ) proofend; theorem Th17: :: INTEGRA3:17 for p being FinSequence of REAL for i, j, k being Element of NAT st p is increasing & i in dom p & j in dom p & k in dom p & p . i <= p . k & p . k <= p . j holds p . k in rng (mid (p,i,j)) proofend; theorem Th18: :: INTEGRA3:18 for i being Element of NAT for A being non empty closed_interval Subset of REAL for D being Division of A for f being Function of A,REAL st f | A is bounded & i in dom D holds lower_bound (rng (f | (divset (D,i)))) <= upper_bound (rng f) proofend; theorem Th19: :: INTEGRA3:19 for i being Element of NAT for A being non empty closed_interval Subset of REAL for D being Division of A for f being Function of A,REAL st f | A is bounded & i in dom D holds upper_bound (rng (f | (divset (D,i)))) >= lower_bound (rng f) proofend; begin :: [WP: ] Darboux's_Theorem theorem :: INTEGRA3:20 for A being non empty closed_interval Subset of REAL for f being Function of A,REAL for T being DivSequence of A st f | A is bounded & delta T is 0 -convergent & delta T is non-zero & vol A <> 0 holds ( lower_sum (f,T) is convergent & lim (lower_sum (f,T)) = lower_integral f ) proofend; theorem :: INTEGRA3:21 for A being non empty closed_interval Subset of REAL for f being Function of A,REAL for T being DivSequence of A st f | A is bounded & delta T is 0 -convergent & delta T is non-zero & vol A <> 0 holds ( upper_sum (f,T) is convergent & lim (upper_sum (f,T)) = upper_integral f ) proofend;