:: Concatenation of Finite Sequences Reducing Overlapping Part and an :: Argument of Separators of Sequential Files :: by Hirofumi Fukura and Yatsuka Nakamura :: :: Received March 18, 2004 :: Copyright (c) 2004-2019 Association of Mizar Users :: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland). :: This code can be distributed under the GNU General Public Licence :: version 3.0 or later, or the Creative Commons Attribution-ShareAlike :: License version 3.0 or later, subject to the binding interpretation :: detailed in file COPYING.interpretation. :: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these :: licenses, or see http://www.gnu.org/licenses/gpl.html and :: http://creativecommons.org/licenses/by-sa/3.0/. environ vocabularies FINSEQ_8, ARYTM_1, RELAT_1, FINSEQ_1, FUNCT_1, FINSEQ_5, RFINSEQ, JORDAN3, ARYTM_3, NUMBERS, ORDINAL4, TARSKI, XBOOLE_0, XXREAL_0, CARD_1, SUBSET_1, NAT_1, GRAPH_2; notations TARSKI, XBOOLE_0, SUBSET_1, ORDINAL1, NUMBERS, XCMPLX_0, XXREAL_0, NAT_1, NAT_D, RELAT_1, RFINSEQ, FINSEQ_1, FUNCT_1, FINSEQ_6, FINSEQ_5; constructors SQUARE_1, NAT_1, RFINSEQ, NAT_D, FINSEQ_5, FINSEQ_6, REAL_1, RELSET_1; registrations RELSET_1, XXREAL_0, XREAL_0, NAT_1, FINSEQ_1, ORDINAL1, XBOOLE_0, CARD_1; requirements NUMERALS, REAL, BOOLE, SUBSET, ARITHM; begin theorem :: FINSEQ_8:1 for f,g being FinSequence st len f>=1 holds mid(f^g,1,len f)=f; theorem :: FINSEQ_8:2 for f being FinSequence,i being Nat st i>=len f holds f/^i={}; theorem :: FINSEQ_8:3 for D being non empty set,k1,k2 being Nat holds mid(<*>D,k1,k2)=<*>D; definition let f be FinSequence, k1,k2 be Nat; func smid(f,k1,k2) -> FinSequence equals :: FINSEQ_8:def 1 (f/^(k1-'1))|(k2+1-'k1); end; definition let D be set,f be FinSequence of D,k1,k2 be Nat; redefine func smid(f,k1,k2) -> FinSequence of D; end; theorem :: FINSEQ_8:4 for f being FinSequence,k1,k2 being Nat st k1<=k2 holds smid(f,k1,k2) = mid(f,k1,k2); theorem :: FINSEQ_8:5 for f being FinSequence, k2 being Nat holds smid(f,1,k2)=f|k2; theorem :: FINSEQ_8:6 for f being FinSequence, k2 being Nat st len f<=k2 holds smid(f,1,k2)=f; theorem :: FINSEQ_8:7 for f being FinSequence, k1,k2 being Nat st k1>k2 holds smid(f,k1,k2)={}; theorem :: FINSEQ_8:8 for f being FinSequence, k2 being Nat holds smid(f,0,k2)=smid(f,1,k2+1); theorem :: FINSEQ_8:9 for f,g being FinSequence holds smid(f^g,len f+1,len f+len g)=g; :: Overlapping Part definition let D be non empty set,f,g be FinSequence of D; func ovlpart(f,g) -> FinSequence of D means :: FINSEQ_8:def 2 len it <=len g & it=smid(g,1,len it) & it=smid(f,len f-'len it+1,len f) & for j being Nat st j<=len g & smid(g,1,j)=smid(f,len f-'j+1,len f) holds j<=len it; end; theorem :: FINSEQ_8:10 for D being non empty set,f,g being FinSequence of D holds len ovlpart(f,g)<=len f; :: Concatenation Reducing Overlapping Part definition let D be non empty set,f,g be FinSequence of D; func ovlcon(f,g) -> FinSequence of D equals :: FINSEQ_8:def 3 f^(g/^(len ovlpart(f,g))); end; theorem :: FINSEQ_8:11 for D being non empty set,f,g being FinSequence of D holds ovlcon(f,g)=(f|(len f-'len ovlpart(f,g)))^g; ::Overlapping Left Difference definition let D be non empty set,f,g be FinSequence of D; func ovlldiff(f,g) -> FinSequence of D equals :: FINSEQ_8:def 4 (f|(len f-'len ovlpart(f,g))); end; ::Overlapping Right Difference definition let D be non empty set,f,g be FinSequence of D; func ovlrdiff(f,g) -> FinSequence of D equals :: FINSEQ_8:def 5 (g/^(len ovlpart(f,g))); end; theorem :: FINSEQ_8:12 for D being non empty set,f,g being FinSequence of D holds ovlcon(f,g) =ovlldiff(f,g)^ovlpart(f,g)^ovlrdiff(f,g) & ovlcon(f,g) =ovlldiff(f,g)^(ovlpart(f,g)^ovlrdiff(f,g)); theorem :: FINSEQ_8:13 for D being non empty set,f being FinSequence of D holds ovlcon(f,f)=f & ovlpart(f,f)=f & ovlldiff(f,f)={} & ovlrdiff(f,f)={}; theorem :: FINSEQ_8:14 for D being non empty set,f,g being FinSequence of D holds ovlpart(f^g,g)=g & ovlpart(f,f^g)=f; theorem :: FINSEQ_8:15 for D being non empty set,f,g being FinSequence of D holds len ovlcon(f,g)=len f+len g -len ovlpart(f,g) & len ovlcon(f,g)=len f+len g -'len ovlpart(f,g) & len ovlcon(f,g)=len f+(len g -'len ovlpart(f,g)); theorem :: FINSEQ_8:16 for D being non empty set,f,g being FinSequence of D holds len ovlpart(f,g)<=len f & len ovlpart(f,g)<=len g; definition let D be non empty set,CR be FinSequence of D; pred CR separates_uniquely means :: FINSEQ_8:def 6 for f being FinSequence of D,i,j being Element of NAT st 1<=i & i=len CR; end; theorem :: FINSEQ_8:17 for D being non empty set,CR being FinSequence of D holds CR separates_uniquely iff len ovlpart(CR/^1,CR) =0; definition let f,g be FinSequence, n be Nat; pred g is_substring_of f,n means :: FINSEQ_8:def 7 len g>0 implies ex i being Nat st n<=i & i<=len f & mid(f,i,(i-'1)+len g)=g; end; theorem :: FINSEQ_8:18 for f,g being FinSequence, n,m being Nat st m>=n & g is_substring_of f,m holds g is_substring_of f,n; theorem :: FINSEQ_8:19 for f being FinSequence st 1<= len f holds f is_substring_of f,1; theorem :: FINSEQ_8:20 for f,g being FinSequence st g is_substring_of f,0 holds g is_substring_of f,1; notation let g,f be FinSequence; synonym g is_preposition_of f for g c= f; end; definition let g,f be FinSequence; redefine pred g c= f means :: FINSEQ_8:def 8 len g>0 implies 1<=len f & mid(f,1,len g)=g; end; theorem :: FINSEQ_8:21 for f,g being FinSequence st len g>0 & g is_preposition_of f holds g.1=f.1; definition let f,g be FinSequence; pred g is_postposition_of f means :: FINSEQ_8:def 9 Rev g is_preposition_of Rev f; end; theorem :: FINSEQ_8:22 for f,g being FinSequence st len g=0 holds g is_postposition_of f; theorem :: FINSEQ_8:23 for D being non empty set,f,g being FinSequence of D st g is_postposition_of f holds len g <=len f; theorem :: FINSEQ_8:24 for D being non empty set,f,g being FinSequence of D st g is_postposition_of f holds len g>0 implies len g<=len f & mid(f,(len f+1) -' len g,len f)=g; theorem :: FINSEQ_8:25 for D being non empty set,f,g being FinSequence of D st (len g>0 implies len g<=len f & mid(f,(len f+1) -' len g,len f)=g) holds g is_postposition_of f; theorem :: FINSEQ_8:26 for f,g being FinSequence st 1<=len f & g is_preposition_of f holds g is_substring_of f,1; theorem :: FINSEQ_8:27 for D being non empty set,f,g being FinSequence of D, n being Nat st not g is_substring_of f,n holds for i being Element of NAT st n <= i & 0 < i holds mid(f,i,(i-'1)+len g) <> g; definition let D be non empty set,f,g be FinSequence of D,n be Nat; func instr(n,f,g) -> Element of NAT means :: FINSEQ_8:def 10 (it <> 0 implies n <= it & g is_preposition_of f/^(it-'1) & for j being Element of NAT st j >= n & j > 0 & g is_preposition_of f/^(j-'1) holds j >= it) & (it=0 implies not g is_substring_of f,n ); end; definition let D be non empty set,f,CR be FinSequence of D; func addcr(f,CR) -> FinSequence of D equals :: FINSEQ_8:def 11 ovlcon(f,CR); end; definition let D be non empty set, r,CR be FinSequence of D; pred r is_terminated_by CR means :: FINSEQ_8:def 12 len CR >0 implies len r>=len CR & instr(1,r,CR) = len r + 1 -'len CR; end; theorem :: FINSEQ_8:28 for D being non empty set,f being FinSequence of D holds f is_terminated_by f ; theorem :: FINSEQ_8:29 for f being FinSequence, k1,k2 being Nat st k1 in dom f & k2 in dom f holds smid(f,k1,k2) = (k1,k2)-cut f;