Volume 4, 1992

University of Bialystok

Copyright (c) 1992 Association of Mizar Users

### The abstract of the Mizar article:

### Reper Algebras

**by****Michal Muzalewski**- Received May 28, 1992
- MML identifier: MIDSP_3

- [ Mizar article, MML identifier index ]

environ vocabulary FINSEQ_1, FUNCT_1, ARYTM_1, RELAT_1, BOOLE, FINSEQ_2, MIDSP_1, BINOP_1, QC_LANG1, PRE_TOPC, ARYTM_3, PARTFUN1, MIDSP_2, VECTSP_1, GROUP_4, RLVECT_1, MIDSP_3; notation TARSKI, XBOOLE_0, SUBSET_1, XCMPLX_0, XREAL_0, FUNCT_1, FUNCT_2, BINOP_1, STRUCT_0, PRE_TOPC, FINSEQ_1, FINSEQ_2, NAT_1, RLVECT_1, MIDSP_1, MIDSP_2; constructors BINOP_1, FINSEQ_2, NAT_1, MIDSP_2, XREAL_0, XBOOLE_0; clusters FINSEQ_2, STRUCT_0, RELSET_1, NAT_1, ZFMISC_1, XBOOLE_0; requirements NUMERALS, SUBSET, BOOLE, ARITHM; begin reserve n,i,j,k,l for Nat; reserve D for non empty set; reserve c,d for Element of D; reserve p,q,q',r for FinSequence of D; theorem :: MIDSP_3:1 len p = j+1+k implies ex q,r,c st len q = j & len r = k & p = q^<*c*>^r; theorem :: MIDSP_3:2 i in Seg(n) implies ex j,k st n = j+1+k & i = j+1; theorem :: MIDSP_3:3 p = q^<*c*>^r & i = len q + 1 implies (for l st 1 <= l & l <= len q holds p.l = q.l) & p.i = c & (for l st i + 1 <= l & l <= len p holds p.l = r.(l-i)); theorem :: MIDSP_3:4 l<=j or l=j+1 or j+2<=l; theorem :: MIDSP_3:5 l in Seg(n)\{i} & i=j+1 implies 1<=l & l<=j or i+1<=l & l<=n; definition let n,i,D,d;let p be Element of (n+1)-tuples_on D; assume i in Seg(n+1); func sub(p,i,d) -> Element of (n+1)-tuples_on D means :: MIDSP_3:def 1 it.i = d & for l st l in (dom p)\{i} holds it.l = p.l; end; ::Section 2: Reper Algebra Structure and its properties begin definition let n; struct(MidStr) ReperAlgebraStr over n (#carrier -> set, MIDPOINT -> BinOp of the carrier, reper -> Function of n-tuples_on the carrier, the carrier#); end; definition let n; let A be non empty set, m be BinOp of A, r be Function of n-tuples_on A,A; cluster ReperAlgebraStr(#A,m,r#) -> non empty; end; definition let n; cluster non empty ReperAlgebraStr over n; end; definition let n; cluster MidSp-like (non empty ReperAlgebraStr over n+2); end; reserve RAS for MidSp-like (non empty ReperAlgebraStr over n+2); reserve a,b,d,pii,p'i for Point of RAS; definition let i,D; mode Tuple of i,D is Element of i-tuples_on D; end; definition let n,RAS,i; mode Tuple of i,RAS is Tuple of i,the carrier of RAS; end; reserve p,q for Tuple of (n+1),RAS; definition let n,RAS,a; redefine func <*a*> -> Tuple of 1,RAS; end; definition let n,RAS,i,j; let p be Tuple of i,RAS, q be Tuple of j,RAS; redefine func p^q -> Tuple of (i+j),RAS; end; theorem :: MIDSP_3:6 <*a*>^p is Tuple of (n+2),RAS; definition let n,RAS,a,p; func *'(a,p) -> Point of RAS equals :: MIDSP_3:def 2 (the reper of RAS).(<*a*>^p); end; definition let n,i,RAS,d,p; func <:p,i,d:> -> Tuple of (n+1),RAS means :: MIDSP_3:def 3 for D for p' being Element of (n+1)-tuples_on D for d' being Element of D st D = the carrier of RAS & p' = p & d' = d holds it = sub(p',i,d'); end; theorem :: MIDSP_3:7 i in Seg(n+1) implies <:p,i,d:>.i = d & for l st l in (dom p)\{i} holds <:p,i,d:>.l = p.l; definition let n; mode Nat of n -> Nat means :: MIDSP_3:def 4 1<=it & it<=n+1; end; reserve m for Nat of n; theorem :: MIDSP_3:8 i is Nat of n iff i in Seg(n+1); canceled; theorem :: MIDSP_3:10 i<=n implies i+1 is Nat of n; theorem :: MIDSP_3:11 (for m holds p.m = q.m) implies p = q; theorem :: MIDSP_3:12 (for l being Nat of n st l=i holds <:p,i,d:>.l = d) & for l,i being Nat of n st l<>i holds <:p,i,d:>.l = p.l; definition let n,D; let p be Element of (n+1)-tuples_on D; let m; redefine func p.m -> Element of D; end; definition let n,RAS; attr RAS is being_invariance means :: MIDSP_3:def 5 for a,b,p,q st (for m holds a@(q.m) = b@(p.m)) holds a@*'(b,q) = b@*'(a,p); synonym RAS is_invariance; end; definition let n,i,RAS; pred RAS has_property_of_zero_in i means :: MIDSP_3:def 6 for a,p holds *'(a,<:p,i,a:>) = a; end; definition let n,i,RAS; pred RAS is_semi_additive_in i means :: MIDSP_3:def 7 for a,pii,p st p.i = pii holds *'(a,<:p,i,a@pii:>) = a@*'(a,p); end; theorem :: MIDSP_3:13 RAS is_semi_additive_in m implies for a,d,p,q st q = <:p,m,d:> holds *'(a,<:p,m,a@d:>) = a@*'(a,q) ; definition let n,i,RAS; pred RAS is_additive_in i means :: MIDSP_3:def 8 for a,pii,p'i,p st p.i = pii holds *'(a,<:p,i,pii@p'i:>) = *'(a,p)@*'(a,<:p,i,p'i:>); end; definition let n,i,RAS; pred RAS is_alternative_in i means :: MIDSP_3:def 9 for a,p,pii st p.i = pii holds *'(a,<:p,i+1,pii:>) = a; end; reserve W for ATLAS of RAS; reserve v for Vector of W; definition let n,RAS,W,i; mode Tuple of i,W is Tuple of i,the carrier of the algebra of W; end; reserve x,y for Tuple of (n+1),W; definition let n,RAS,W,x,i,v; func <:x,i,v:> -> Tuple of (n+1),W means :: MIDSP_3:def 10 for D for x' being Element of (n+1)-tuples_on D for v' being Element of D st D = the carrier of the algebra of W & x' = x & v' = v holds it = sub(x',i,v'); end; theorem :: MIDSP_3:14 i in Seg(n+1) implies <:x,i,v:>.i = v & for l st l in (dom x)\{i} holds <:x,i,v:>.l = x.l; theorem :: MIDSP_3:15 (for l being Nat of n st l=i holds <:x,i,v:>.l = v) & for l,i being Nat of n st l<>i holds <:x,i,v:>.l = x.l; theorem :: MIDSP_3:16 (for m holds x.m = y.m) implies x = y; scheme SeqLambdaD'{n()->Nat,D()->non empty set,F(set)->Element of D()}: ex z being FinSequence of D() st len z = n()+1 & for j being Nat of n() holds z.j = F(j); definition let n,RAS,W,a,x; func (a,x).W -> Tuple of (n+1),RAS means :: MIDSP_3:def 11 it.m = (a,x.m).W; end; definition let n,RAS,W,a,p; func W.(a,p) -> Tuple of (n+1),W means :: MIDSP_3:def 12 it.m = W.(a,p.m); end; theorem :: MIDSP_3:17 W.(a,p) = x iff (a,x).W = p; theorem :: MIDSP_3:18 W.(a,(a,x).W) = x; theorem :: MIDSP_3:19 (a,W.(a,p)).W = p; definition let n,RAS,W,a,x; func Phi(a,x) -> Vector of W equals :: MIDSP_3:def 13 W.(a,*'(a,(a,x).W)); end; theorem :: MIDSP_3:20 W.(a,p) = x & W.(a,b) = v implies (*'(a,p) = b iff Phi(a,x) = v); theorem :: MIDSP_3:21 RAS is_invariance iff for a,b,x holds Phi(a,x) = Phi(b,x); theorem :: MIDSP_3:22 1 in Seg(n+1); canceled; theorem :: MIDSP_3:24 1 is Nat of n; ::Section 3: Reper Algebra and its atlas begin definition let n; mode ReperAlgebra of n -> MidSp-like (non empty ReperAlgebraStr over n+2) means :: MIDSP_3:def 14 it is_invariance; end; reserve RAS for ReperAlgebra of n; reserve a,b,pm,p'm,p''m for Point of RAS; reserve p for Tuple of (n+1),RAS; reserve W for ATLAS of RAS; reserve v for Vector of W; reserve x for Tuple of (n+1),W; theorem :: MIDSP_3:25 Phi(a,x) = Phi(b,x); definition let n,RAS,W,x; func Phi(x) -> Vector of W means :: MIDSP_3:def 15 for a holds it = Phi(a,x); end; theorem :: MIDSP_3:26 W.(a,p) = x & W.(a,b) = v & Phi(x) = v implies *'(a,p) = b; theorem :: MIDSP_3:27 (a,x).W = p & (a,v).W = b & *'(a,p) = b implies Phi(x) = v; theorem :: MIDSP_3:28 W.(a,p) = x & W.(a,b) = v implies W.(a,<:p,m,b:>) = <:x,m,v:>; theorem :: MIDSP_3:29 (a,x).W = p & (a,v).W = b implies (a,<:x,m,v:>).W = <:p,m,b:>; theorem :: MIDSP_3:30 RAS has_property_of_zero_in m iff for x holds Phi(<:x,m,0.W:>) = 0.W; theorem :: MIDSP_3:31 RAS is_semi_additive_in m iff for x holds Phi(<:x,m,Double (x.m):>) = Double Phi(x); theorem :: MIDSP_3:32 RAS has_property_of_zero_in m & RAS is_additive_in m implies RAS is_semi_additive_in m; theorem :: MIDSP_3:33 RAS has_property_of_zero_in m implies (RAS is_additive_in m iff for x,v holds Phi(<:x,m,(x.m)+v:>) = Phi(x) + Phi (<:x,m,v:>)); theorem :: MIDSP_3:34 W.(a,p) = x & m<= n implies W.(a,<:p,m+1,p.m:>) = <:x,m+1,x.m:>; theorem :: MIDSP_3:35 (a,x).W = p & m<=n implies (a,<:x,m+1,x.m:>).W = <:p,m+1,p.m:>; theorem :: MIDSP_3:36 m<=n implies (RAS is_alternative_in m iff for x holds Phi(<:x,m+1,x.m:>) = 0. W );

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