Volume 8, 1996

University of Bialystok

Copyright (c) 1996 Association of Mizar Users

### The abstract of the Mizar article:

### Some Multi Instructions Defined by Sequence of Instructions of \SCMFSA

**by****Noriko Asamoto**- Received April 24, 1996
- MML identifier: SCMFSA_7

- [ Mizar article, MML identifier index ]

environ vocabulary AMI_3, FINSET_1, AMI_1, SCMFSA_2, FINSEQ_1, FUNCT_4, RELAT_1, ABSVALUE, ARYTM_1, INT_1, TARSKI, BOOLE, FUNCT_1, DTCONSTR, BINOP_1, FINSOP_1, SETWISEO, CARD_1, FINSEQ_2, AMI_2, SCMFSA_7, ORDINAL2, FINSEQ_4, ARYTM; notation TARSKI, XBOOLE_0, SUBSET_1, ORDINAL1, NUMBERS, XCMPLX_0, XREAL_0, REAL_1, NAT_1, ABSVALUE, BINARITH, ORDINAL2, INT_1, RELAT_1, BINOP_1, CARD_1, SETWISEO, FINSEQ_1, FINSET_1, FINSEQ_4, DTCONSTR, FINSOP_1, FUNCT_1, FUNCT_7, STRUCT_0, AMI_1, SCMFSA_2; constructors WELLORD2, REAL_1, BINARITH, BINOP_1, FINSOP_1, FUNCT_7, SETWISEO, AMI_2, SCMFSA_2, FINSEQ_4, MEMBERED; clusters XBOOLE_0, AMI_1, RELSET_1, PRELAMB, FINSEQ_5, FINSEQ_6, SCMFSA_2, INT_1, DTCONSTR, FINSEQ_1, NAT_1, FRAENKEL, XREAL_0, MEMBERED, ORDINAL2; requirements REAL, NUMERALS, BOOLE, SUBSET, ARITHM; begin reserve m for Nat; definition cluster -> finite FinPartState of SCM+FSA; end; definition let p be FinSequence, x,y be set; cluster p +*(x,y) -> FinSequence-like; end; theorem :: SCMFSA_7:1 for k being natural number holds abs(k) = k; theorem :: SCMFSA_7:2 for a,b,c being Nat st a >= c & b >= c & a -' c = b -' c holds a = b; theorem :: SCMFSA_7:3 for a,b being natural number st a >= b holds a -' b = a - b; theorem :: SCMFSA_7:4 for a,b being Integer st a < b holds a <= b - 1; scheme CardMono''{ A() -> set, D() -> non empty set, G(set) -> set }: A(),{ G(d) where d is Element of D() : d in A() } are_equipotent provided A() c= D() and for d1,d2 being Element of D() st d1 in A() & d2 in A() & G(d1) = G(d2) holds d1 = d2; theorem :: SCMFSA_7:5 for p1,p2,q being FinSequence st p1 c= q & p2 c= q & len p1 = len p2 holds p1 = p2; canceled 2; theorem :: SCMFSA_7:8 for p,q being FinSequence st p c= q holds len p <= len q; theorem :: SCMFSA_7:9 for p,q being FinSequence, i being Nat st 1 <= i & i <= len p holds (p ^ q).i = p.i; theorem :: SCMFSA_7:10 for p,q being FinSequence, i being Nat st 1 <= i & i <= len q holds (p ^ q).(len p + i) = q.i; canceled; theorem :: SCMFSA_7:12 for p being FinSequence st p <> {} holds len p in dom p; theorem :: SCMFSA_7:13 for D being set holds FlattenSeq <*>(D*) = <*>D; theorem :: SCMFSA_7:14 for D being set, F,G be FinSequence of D* holds FlattenSeq (F ^ G) = FlattenSeq F ^ FlattenSeq G; theorem :: SCMFSA_7:15 for D being set, p,q be Element of D* holds FlattenSeq <* p,q *> = p ^ q; theorem :: SCMFSA_7:16 for D being set, p,q,r be Element of D* holds FlattenSeq <* p,q,r *> = p ^ q ^ r; theorem :: SCMFSA_7:17 for D being non empty set, p,q being FinSequence of D st p c= q holds ex p' being FinSequence of D st p ^ p' = q; theorem :: SCMFSA_7:18 for D being non empty set, p,q being FinSequence of D, i being Nat st p c= q & 1 <= i & i <= len p holds q.i = p.i; theorem :: SCMFSA_7:19 for D being set, F,G be FinSequence of D* holds F c= G implies FlattenSeq F c= FlattenSeq G; theorem :: SCMFSA_7:20 for p being FinSequence holds p | Seg 0 = {}; theorem :: SCMFSA_7:21 for f,g being FinSequence holds f | Seg 0 = g | Seg 0; theorem :: SCMFSA_7:22 for D being non empty set, x being Element of D holds <* x *> is FinSequence of D; theorem :: SCMFSA_7:23 for D being set, p,q being FinSequence of D holds p ^ q is FinSequence of D; definition let f be FinSequence of the Instructions of SCM+FSA; func Load f->FinPartState of SCM+FSA means :: SCMFSA_7:def 1 dom it = {insloc (m-'1): m in dom f} & for k being Nat st insloc k in dom it holds it.insloc k = f/.(k+1); end; canceled; theorem :: SCMFSA_7:25 for f being FinSequence of the Instructions of SCM+FSA holds card Load f = len f; theorem :: SCMFSA_7:26 for p being FinSequence of the Instructions of SCM+FSA, k being Nat holds insloc k in dom Load p iff k + 1 in dom p; theorem :: SCMFSA_7:27 for k,n being Nat holds k < n iff 0 < k + 1 & k + 1 <= n; theorem :: SCMFSA_7:28 for k, n being Nat holds k < n iff 1 <= k + 1 & k + 1 <= n; theorem :: SCMFSA_7:29 for p being FinSequence of the Instructions of SCM+FSA, k being Nat holds insloc k in dom Load p iff k < len p; theorem :: SCMFSA_7:30 for f being non empty FinSequence of the Instructions of SCM+FSA holds 1 in dom f & insloc 0 in dom Load f; theorem :: SCMFSA_7:31 for p,q being FinSequence of the Instructions of SCM+FSA holds Load p c= Load (p ^ q); theorem :: SCMFSA_7:32 for p,q being FinSequence of the Instructions of SCM+FSA holds p c= q implies Load p c= Load q; definition let a be Int-Location; let k be Integer; func a := k -> FinPartState of SCM+FSA means :: SCMFSA_7:def 2 ex k1 being Nat st k1 + 1 = k & it = Load(<* a:= intloc 0 *> ^ ( k1 |-> AddTo(a,intloc 0) ) ^ <* halt SCM+FSA *> ) if k > 0 otherwise ex k1 being Nat st k1 + k = 1 & it = Load(<* a:= intloc 0 *> ^ ( k1 |-> SubFrom(a,intloc 0) ) ^ <* halt SCM+FSA *> ); end; definition let a be Int-Location; let k be Integer; func aSeq(a,k) -> FinSequence of the Instructions of SCM+FSA means :: SCMFSA_7:def 3 ex k1 being Nat st k1 + 1 = k & it = <* a:= intloc 0 *> ^ (k1 |-> AddTo(a,intloc 0)) if k > 0 otherwise ex k1 being Nat st k1 + k = 1 & it = <* a:= intloc 0 *> ^ (k1 |-> SubFrom(a,intloc 0)); end; theorem :: SCMFSA_7:33 for a being Int-Location, k being Integer holds a:=k = Load (aSeq(a,k) ^ <* halt SCM+FSA *>); definition let f be FinSeq-Location; let p be FinSequence of INT; func aSeq(f,p) -> FinSequence of the Instructions of SCM+FSA means :: SCMFSA_7:def 4 ex pp being FinSequence of (the Instructions of SCM+FSA)* st len pp = len p & (for k being Nat st 1 <= k & k <= len p holds ex i being Integer st i = p.k & pp.k = (aSeq(intloc 1,k) ^ aSeq(intloc 2,i) ^ <* (f,intloc 1):=intloc 2 *>)) & it = FlattenSeq pp; end; definition let f be FinSeq-Location; let p be FinSequence of INT; func f := p -> FinPartState of SCM+FSA equals :: SCMFSA_7:def 5 Load (aSeq(intloc 1,len p) ^ <* f:=<0,...,0>intloc 1 *> ^ aSeq(f,p) ^ <* halt SCM+FSA *> ); end; theorem :: SCMFSA_7:34 for a being Int-Location holds a:=1 = Load ( <* a:= intloc 0 *> ^ <* halt SCM+FSA *> ); theorem :: SCMFSA_7:35 for a being Int-Location holds a:=0 = Load (<* a:= intloc 0 *>^<*SubFrom(a,intloc 0)*>^<*halt SCM+FSA*>); theorem :: SCMFSA_7:36 for s being State of SCM+FSA st s.intloc 0 = 1 for c0 being Nat st IC s = insloc c0 for a being Int-Location, k being Integer st a <> intloc 0 & (for c being Nat st c in dom aSeq(a,k) holds aSeq(a,k).c = s.insloc (c0 + c -' 1)) holds (for i being Nat st i <= len aSeq(a,k) holds IC (Computation s).i = insloc (c0 + i) & (for b being Int-Location st b <> a holds (Computation s).i.b = s.b) & (for f being FinSeq-Location holds (Computation s).i.f = s.f)) & (Computation s).(len aSeq(a,k)).a = k; theorem :: SCMFSA_7:37 for s being State of SCM+FSA st IC s = insloc 0 & s.intloc 0 = 1 for a being Int-Location for k being Integer st Load aSeq(a,k) c= s & a<>intloc 0 holds (for i being Nat st i <= len aSeq(a,k) holds IC (Computation s).i = insloc i & (for b being Int-Location st b <> a holds (Computation s).i.b = s.b) & (for f being FinSeq-Location holds (Computation s).i.f = s.f)) & (Computation s).(len aSeq(a,k)).a = k; :: Users' guide theorem :: SCMFSA_7:38 for s being State of SCM+FSA st IC s = insloc 0 & s.intloc 0 = 1 for a being Int-Location, k being Integer st a:=k c= s & a<>intloc 0 holds s is halting & (Result s).a = k & (for b being Int-Location st b <> a holds (Result s).b = s.b) & (for f being FinSeq-Location holds (Result s).f = s.f); theorem :: SCMFSA_7:39 for s being State of SCM+FSA st IC s = insloc 0 & s.intloc 0 = 1 for f being FinSeq-Location, p being FinSequence of INT st f:=p c= s holds s is halting & (Result s).f = p & (for b being Int-Location st b <> intloc 1 & b <> intloc 2 holds (Result s).b = s.b) & (for g being FinSeq-Location st g <> f holds (Result s).g = s.g);

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