Volume 5, 1993

University of Bialystok

Copyright (c) 1993 Association of Mizar Users

### The abstract of the Mizar article:

### Introduction to Theory of Rearrangement

**by****Yuji Sakai, and****Jaroslaw Kotowicz**- Received May 22, 1993
- MML identifier: REARRAN1

- [ Mizar article, MML identifier index ]

environ vocabulary PARTFUN1, SEQ_1, FUNCT_1, RELAT_1, BOOLE, FINSEQ_1, FINSET_1, CARD_1, ARYTM_1, TARSKI, SQUARE_1, RFINSEQ, RFUNCT_3, RLVECT_1, TDGROUP, FINSEQ_2, FUNCT_3, PROB_1, REARRAN1; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, NUMBERS, XREAL_0, RELAT_1, FUNCT_1, FINSEQ_1, CARD_1, FINSET_1, REAL_1, NAT_1, SQUARE_1, RELSET_1, PARTFUN1, SEQ_1, RFUNCT_1, FINSEQOP, RVSUM_1, TOPREAL1, RFINSEQ, RFUNCT_3; constructors SETWISEO, REAL_1, NAT_1, FINSEQOP, TOPREAL1, RFUNCT_3, MEMBERED, XBOOLE_0; clusters SUBSET_1, FINSET_1, RFUNCT_3, RFINSEQ, RELSET_1, PRELAMB, XREAL_0, FINSEQ_1, ARYTM_3, PARTFUN1, MEMBERED, ZFMISC_1, XBOOLE_0; requirements REAL, NUMERALS, BOOLE, SUBSET, ARITHM; begin reserve n,m,k for Nat, x,y for set, r for Real; definition let D be non empty set, F be PartFunc of D,REAL, r be Real; redefine func r(#)F ->Element of PFuncs(D,REAL); end; definition let IT be FinSequence; attr IT is terms've_same_card_as_number means :: REARRAN1:def 1 for n st 1<=n & n<=len IT for B being finite set st B = IT.n holds card B = n; attr IT is ascending means :: REARRAN1:def 2 for n st 1<=n & n<=len IT - 1 holds IT.n c= IT.(n+1); end; definition let X be set; let IT be FinSequence of X; attr IT is lenght_equal_card_of_set means :: REARRAN1:def 3 ex B being finite set st B = union X & len IT = card B; end; definition let D be non empty finite set; cluster terms've_same_card_as_number ascending lenght_equal_card_of_set FinSequence of bool(D); end; definition let D be non empty finite set; mode RearrangmentGen of D is terms've_same_card_as_number ascending lenght_equal_card_of_set FinSequence of bool(D); end; reserve C,D for non empty finite set, a for FinSequence of bool D; theorem :: REARRAN1:1 for a be FinSequence of bool D holds a is lenght_equal_card_of_set iff len a = card D; theorem :: REARRAN1:2 for a be FinSequence holds a is ascending iff for n,m st n<=m & n in dom a & m in dom a holds a.n c= a.m; theorem :: REARRAN1:3 for a be terms've_same_card_as_number lenght_equal_card_of_set FinSequence of bool D holds a.(len a) = D; theorem :: REARRAN1:4 for a be lenght_equal_card_of_set FinSequence of bool D holds len a <> 0; theorem :: REARRAN1:5 for a be ascending terms've_same_card_as_number FinSequence of bool D holds for n,m holds n in dom a & m in dom a & n<>m implies a.n <> a.m; theorem :: REARRAN1:6 for a be ascending terms've_same_card_as_number FinSequence of bool D holds for n holds 1 <= n & n <= len a - 1 implies a.n <> a.(n+1); theorem :: REARRAN1:7 for a be terms've_same_card_as_number FinSequence of bool D st n in dom a holds a.n <> {}; theorem :: REARRAN1:8 for a be terms've_same_card_as_number FinSequence of bool D st 1<=n & n<=len a - 1 holds a.(n+1) \ a.n <> {}; theorem :: REARRAN1:9 for a be terms've_same_card_as_number lenght_equal_card_of_set FinSequence of bool D ex d be Element of D st a.1 = {d}; theorem :: REARRAN1:10 for a be terms've_same_card_as_number ascending FinSequence of bool D st 1<=n & n<=len a - 1 ex d be Element of D st a.(n+1) \ a.n = {d} & a.(n+1) = a.n \/ {d} & a.(n+1) \ {d} = a.n; definition let D be non empty finite set, A be RearrangmentGen of D; func Co_Gen A -> RearrangmentGen of D means :: REARRAN1:def 4 for m st 1 <= m & m <= len it - 1 holds it.m = D \ A.(len A -m); end; theorem :: REARRAN1:11 for A be RearrangmentGen of D holds Co_Gen(Co_Gen A) = A; theorem :: REARRAN1:12 for F be PartFunc of D,REAL, A be RearrangmentGen of C st F is total & card C = card D holds len MIM FinS(F,D) = len CHI(A,C); definition let D,C be non empty finite set, A be RearrangmentGen of C, F be PartFunc of D,REAL; func Rland (F,A) -> PartFunc of C,REAL equals :: REARRAN1:def 5 Sum (MIM(FinS(F,D)) (#) CHI(A,C)); func Rlor(F,A) -> PartFunc of C,REAL equals :: REARRAN1:def 6 Sum (MIM(FinS(F,D)) (#) CHI(Co_Gen A,C)); end; theorem :: REARRAN1:13 for F be PartFunc of D,REAL, A be RearrangmentGen of C st F is total & card C = card D holds dom Rland(F,A) = C; theorem :: REARRAN1:14 for c be Element of C, F be PartFunc of D,REAL, A be RearrangmentGen of C st F is total & card C = card D holds (c in A.1 implies (MIM(FinS(F,D)) (#) CHI(A,C))#c = MIM(FinS(F,D))) & for n st 1<=n & n<len A & c in A.(n+1) \ A.n holds (MIM(FinS(F,D)) (#) CHI(A,C))#c = (n |-> (0 qua Real)) ^ (MIM(FinS(F,D)/^n)); theorem :: REARRAN1:15 for c be Element of C, F be PartFunc of D,REAL, A be RearrangmentGen of C st F is total & card C = card D holds (c in A.1 implies (Rland(F,A)).c = FinS(F,D).1) & for n st 1<=n & n<len A & c in A.(n+1) \ A.n holds Rland(F,A).c = FinS(F,D).(n+1); theorem :: REARRAN1:16 for F be PartFunc of D,REAL, A be RearrangmentGen of C st F is total & card C = card D holds rng Rland(F,A) = rng FinS(F,D); theorem :: REARRAN1:17 for F be PartFunc of D,REAL, A be RearrangmentGen of C st F is total & card C = card D holds Rland(F,A), FinS(F,D) are_fiberwise_equipotent; theorem :: REARRAN1:18 for F be PartFunc of D,REAL, A be RearrangmentGen of C st F is total & card C = card D holds FinS(Rland(F,A),C) = FinS(F,D); theorem :: REARRAN1:19 for F be PartFunc of D,REAL, A be RearrangmentGen of C st F is total & card C = card D holds Sum (Rland(F,A),C) = Sum (F,D); theorem :: REARRAN1:20 for F be PartFunc of D,REAL, A be RearrangmentGen of C st F is total & card C = card D holds FinS(Rland(F,A) - r,C) = FinS(F-r,D) & Sum (Rland(F,A)-r,C) = Sum (F-r,D); theorem :: REARRAN1:21 for F be PartFunc of D,REAL, A be RearrangmentGen of C st F is total & card C = card D holds dom Rlor(F,A) = C; theorem :: REARRAN1:22 for c be Element of C, F be PartFunc of D,REAL, A be RearrangmentGen of C st F is total & card C = card D holds (c in (Co_Gen A).1 implies (Rlor(F,A)).c = FinS(F,D).1) & for n st 1<=n & n<len (Co_Gen A) & c in (Co_Gen A).(n+1) \ (Co_Gen A).n holds Rlor(F,A).c = FinS(F,D).(n+1); theorem :: REARRAN1:23 for F be PartFunc of D,REAL, A be RearrangmentGen of C st F is total & card C = card D holds rng Rlor(F,A) = rng FinS(F,D); theorem :: REARRAN1:24 for F be PartFunc of D,REAL, A be RearrangmentGen of C st F is total & card C = card D holds Rlor(F,A), FinS(F,D) are_fiberwise_equipotent; theorem :: REARRAN1:25 for F be PartFunc of D,REAL, A be RearrangmentGen of C st F is total & card C = card D holds FinS(Rlor(F,A),C) = FinS(F,D); theorem :: REARRAN1:26 for F be PartFunc of D,REAL, A be RearrangmentGen of C st F is total & card C = card D holds Sum (Rlor(F,A),C) = Sum (F,D); theorem :: REARRAN1:27 for F be PartFunc of D,REAL, A be RearrangmentGen of C st F is total & card C = card D holds FinS((Rlor(F,A)) - r,C) = FinS(F-r,D) & Sum (Rlor(F,A)-r,C) = Sum (F-r,D); theorem :: REARRAN1:28 for F be PartFunc of D,REAL, A be RearrangmentGen of C st F is total & card C = card D holds Rlor(F,A), Rland(F,A) are_fiberwise_equipotent & FinS(Rlor(F,A),C) = FinS(Rland(F,A),C) & Sum (Rlor(F,A),C) = Sum (Rland(F,A),C); theorem :: REARRAN1:29 for F be PartFunc of D,REAL, A be RearrangmentGen of C st F is total & card C = card D holds max+(Rland(F,A) - r), max+(F - r) are_fiberwise_equipotent & FinS(max+(Rland(F,A) - r), C) = FinS(max+(F - r), D) & Sum (max+(Rland(F,A) - r), C) = Sum (max+(F - r), D); theorem :: REARRAN1:30 for F be PartFunc of D,REAL, A be RearrangmentGen of C st F is total & card C = card D holds max-(Rland(F,A) - r), max-(F - r) are_fiberwise_equipotent & FinS(max-(Rland(F,A) - r), C) = FinS(max-(F - r), D) & Sum (max-(Rland(F,A) - r), C) = Sum (max-(F - r), D); theorem :: REARRAN1:31 for F be PartFunc of D,REAL, A be RearrangmentGen of C st F is total & card D = card C holds len FinS(Rland(F,A),C) = card C & 1<=len FinS(Rland(F,A),C); theorem :: REARRAN1:32 for F be PartFunc of D,REAL, A be RearrangmentGen of C st F is total & card D = card C & n in dom A holds FinS(Rland(F,A),C) | n = FinS(Rland(F,A),A.n); theorem :: REARRAN1:33 for F be PartFunc of D,REAL, A be RearrangmentGen of C st F is total & card D = card C holds Rland(F-r,A) = Rland(F,A) - r; theorem :: REARRAN1:34 for F be PartFunc of D,REAL, A be RearrangmentGen of C st F is total & card C = card D holds max+(Rlor(F,A) - r), max+(F - r) are_fiberwise_equipotent & FinS(max+(Rlor(F,A) - r), C) = FinS(max+(F - r), D) & Sum (max+(Rlor(F,A) - r), C) = Sum (max+(F - r), D); theorem :: REARRAN1:35 for F be PartFunc of D,REAL, A be RearrangmentGen of C st F is total & card C = card D holds max-(Rlor(F,A) - r), max-(F - r) are_fiberwise_equipotent & FinS(max-(Rlor(F,A) - r), C) = FinS(max-(F - r), D) & Sum (max-(Rlor(F,A) - r), C) = Sum (max-(F - r), D); theorem :: REARRAN1:36 for F be PartFunc of D,REAL, A be RearrangmentGen of C st F is total & card D = card C holds len FinS(Rlor(F,A),C) = card C & 1<=len FinS(Rlor(F,A),C); theorem :: REARRAN1:37 for F be PartFunc of D,REAL, A be RearrangmentGen of C st F is total & card D = card C & n in dom A holds FinS(Rlor(F,A),C) | n = FinS(Rlor(F,A),(Co_Gen A).n); theorem :: REARRAN1:38 for F be PartFunc of D,REAL, A be RearrangmentGen of C st F is total & card D = card C holds Rlor(F-r,A) = Rlor(F,A) - r; theorem :: REARRAN1:39 for F be PartFunc of D,REAL, A be RearrangmentGen of C st F is total & card D = card C holds Rland(F,A) , F are_fiberwise_equipotent & Rlor (F,A) , F are_fiberwise_equipotent & rng Rland(F,A) = rng F & rng Rlor(F,A) = rng F;

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