Volume 8, 1996

University of Bialystok

Copyright (c) 1996 Association of Mizar Users

### The abstract of the Mizar article:

### Computation in \SCMFSA

**by****Andrzej Trybulec, and****Yatsuka Nakamura**- Received February 7, 1996
- MML identifier: SCMFSA_3

- [ Mizar article, MML identifier index ]

environ vocabulary SCMFSA_2, AMI_1, INT_1, AMI_3, RELAT_1, FUNCT_4, FUNCOP_1, AMI_2, BOOLE, FUNCT_1, AMI_5, ABSVALUE, FINSEQ_1, FINSEQ_2, CARD_3, CAT_1, FINSET_1, ARYTM_1, NAT_1, FINSEQ_4; notation TARSKI, XBOOLE_0, SUBSET_1, XCMPLX_0, XREAL_0, CARD_3, ABSVALUE, RELAT_1, FUNCT_1, FUNCT_4, INT_1, NAT_1, STRUCT_0, CQC_LANG, FINSET_1, FINSEQ_1, FINSEQ_2, FINSEQ_4, CAT_3, AMI_1, AMI_2, AMI_3, AMI_5, FUNCT_7, SCMFSA_2; constructors NAT_1, AMI_5, SCMFSA_1, FUNCT_7, SCMFSA_2, FINSEQ_4, CAT_3, MEMBERED; clusters AMI_1, AMI_3, INT_1, FUNCT_1, RELSET_1, SCMFSA_2, FINSEQ_5, FINSEQ_1, FRAENKEL, ORDINAL2, NUMBERS; requirements NUMERALS, REAL, SUBSET, BOOLE, ARITHM; begin reserve k for Nat, da,db for Int-Location, fa for FinSeq-Location; theorem :: SCMFSA_3:1 not IC SCM+FSA in Int-Locations; theorem :: SCMFSA_3:2 not IC SCM+FSA in FinSeq-Locations; theorem :: SCMFSA_3:3 for i being Instruction of SCM+FSA, I being Instruction of SCM st i = I for s being State of SCM+FSA, S being State of SCM st S = s|(the carrier of SCM) +* ((the Instruction-Locations of SCM) --> I) holds Exec(i,s) = s +*Exec(I,S) +* s|the Instruction-Locations of SCM+FSA; theorem :: SCMFSA_3:4 for s1,s2 being State of SCM+FSA st (s1 | (Int-Locations \/ FinSeq-Locations \/ {IC SCM+FSA})) = (s2 | (Int-Locations \/ FinSeq-Locations \/ {IC SCM+FSA})) for l being Instruction of SCM+FSA holds Exec (l,s1) | (Int-Locations \/ FinSeq-Locations \/ {IC SCM+FSA}) = Exec (l,s2) | (Int-Locations \/ FinSeq-Locations \/ {IC SCM+FSA}); theorem :: SCMFSA_3:5 for N being with_non-empty_elements set for S being steady-programmed (non empty non void AMI-Struct over N) for i being Instruction of S, s being State of S holds Exec (i, s) | the Instruction-Locations of S = s | the Instruction-Locations of S; begin :: Finite partial states of SCM+FSA theorem :: SCMFSA_3:6 for p being FinPartState of SCM+FSA holds DataPart p = p | (Int-Locations \/ FinSeq-Locations); theorem :: SCMFSA_3:7 for p being FinPartState of SCM+FSA holds p is data-only iff dom p c= Int-Locations \/ FinSeq-Locations; theorem :: SCMFSA_3:8 for p being FinPartState of SCM+FSA holds dom DataPart p c= Int-Locations \/ FinSeq-Locations; theorem :: SCMFSA_3:9 for p being FinPartState of SCM+FSA holds dom ProgramPart p c= the Instruction-Locations of SCM+FSA; theorem :: SCMFSA_3:10 for i being Instruction of SCM+FSA, s being State of SCM+FSA, p being programmed FinPartState of SCM+FSA holds Exec (i, s +* p) = Exec (i,s) +* p; theorem :: SCMFSA_3:11 for s being State of SCM+FSA, iloc being Instruction-Location of SCM+FSA, a being Int-Location holds s.a = (s +* Start-At iloc).a; theorem :: SCMFSA_3:12 for s being State of SCM+FSA, iloc being Instruction-Location of SCM+FSA, a being FinSeq-Location holds s.a = (s +* Start-At iloc).a; theorem :: SCMFSA_3:13 for s, t being State of SCM+FSA holds s +* t|(Int-Locations \/ FinSeq-Locations) is State of SCM+FSA; begin :: Autonomic finite partial states of SCM+FSA definition let la be Int-Location; let a be Integer; redefine func la .--> a -> FinPartState of SCM+FSA; end; theorem :: SCMFSA_3:14 for p being autonomic FinPartState of SCM+FSA st DataPart p <> {} holds IC SCM+FSA in dom p; definition cluster autonomic non programmed FinPartState of SCM+FSA; end; theorem :: SCMFSA_3:15 for p being autonomic non programmed FinPartState of SCM+FSA holds IC SCM+FSA in dom p; theorem :: SCMFSA_3:16 for p being autonomic FinPartState of SCM+FSA st IC SCM+FSA in dom p holds IC p in dom p; theorem :: SCMFSA_3:17 for p being autonomic non programmed FinPartState of SCM+FSA, s being State of SCM+FSA st p c= s for i being Nat holds IC (Computation s).i in dom ProgramPart(p); theorem :: SCMFSA_3:18 for p being autonomic non programmed FinPartState of SCM+FSA, s1, s2 being State of SCM+FSA st p c= s1 & p c= s2 for i being Nat holds IC (Computation s1).i = IC (Computation s2).i & CurInstr ((Computation s1).i) = CurInstr ((Computation s2).i); theorem :: SCMFSA_3:19 for p being autonomic non programmed FinPartState of SCM+FSA, s1, s2 being State of SCM+FSA st p c= s1 & p c= s2 for i being Nat, da, db being Int-Location st CurInstr ((Computation s1).i) = da := db & da in dom p holds (Computation s1).i.db = (Computation s2).i.db; theorem :: SCMFSA_3:20 for p being autonomic non programmed FinPartState of SCM+FSA, s1, s2 being State of SCM+FSA st p c= s1 & p c= s2 for i being Nat, da, db being Int-Location st CurInstr ((Computation s1).i) = AddTo(da, db) & da in dom p holds (Computation s1).i.da + (Computation s1).i.db = (Computation s2).i.da + (Computation s2).i.db; theorem :: SCMFSA_3:21 for p being autonomic non programmed FinPartState of SCM+FSA, s1, s2 being State of SCM+FSA st p c= s1 & p c= s2 for i being Nat, da, db being Int-Location st CurInstr ((Computation s1).i) = SubFrom(da, db) & da in dom p holds (Computation s1).i.da - (Computation s1).i.db = (Computation s2).i.da - (Computation s2).i.db; theorem :: SCMFSA_3:22 for p being autonomic non programmed FinPartState of SCM+FSA, s1, s2 being State of SCM+FSA st p c= s1 & p c= s2 for i being Nat, da, db being Int-Location st CurInstr ((Computation s1).i) = MultBy(da, db) & da in dom p holds (Computation s1).i.da * (Computation s1).i.db = (Computation s2).i.da * (Computation s2).i.db; theorem :: SCMFSA_3:23 for p being autonomic non programmed FinPartState of SCM+FSA, s1, s2 being State of SCM+FSA st p c= s1 & p c= s2 for i being Nat, da, db being Int-Location st CurInstr ((Computation s1).i) = Divide(da, db) & da in dom p & da <> db holds (Computation s1).i.da div (Computation s1).i.db = (Computation s2).i.da div (Computation s2).i.db; theorem :: SCMFSA_3:24 for p being autonomic non programmed FinPartState of SCM+FSA, s1, s2 being State of SCM+FSA st p c= s1 & p c= s2 for i being Nat, da, db being Int-Location st CurInstr ((Computation s1).i) = Divide(da, db) & db in dom p & da <> db holds (Computation s1).i.da mod (Computation s1).i.db = (Computation s2).i.da mod (Computation s2).i.db; theorem :: SCMFSA_3:25 for p being autonomic non programmed FinPartState of SCM+FSA, s1, s2 being State of SCM+FSA st p c= s1 & p c= s2 for i being Nat, da being Int-Location, loc being Instruction-Location of SCM+FSA st CurInstr ((Computation s1).i) = da=0_goto loc & loc <> Next (IC (Computation s1).i) holds ((Computation s1).i.da = 0 iff (Computation s2).i.da = 0); theorem :: SCMFSA_3:26 for p being autonomic non programmed FinPartState of SCM+FSA, s1, s2 being State of SCM+FSA st p c= s1 & p c= s2 for i being Nat, da being Int-Location, loc being Instruction-Location of SCM+FSA st CurInstr ((Computation s1).i) = da>0_goto loc & loc <> Next (IC (Computation s1).i) holds ((Computation s1).i.da > 0 iff (Computation s2).i.da > 0); theorem :: SCMFSA_3:27 for p being autonomic non programmed FinPartState of SCM+FSA, s1, s2 being State of SCM+FSA st p c= s1 & p c= s2 for i being Nat, da, db being Int-Location, f being FinSeq-Location st CurInstr ((Computation s1).i) = da := (f,db) & da in dom p for k1,k2 being Nat st k1 = abs((Computation s1).i.db) & k2 = abs((Computation s2).i.db) holds ((Computation s1).i.f)/.k1 = ((Computation s2).i.f)/.k2; theorem :: SCMFSA_3:28 for p being autonomic non programmed FinPartState of SCM+FSA, s1, s2 being State of SCM+FSA st p c= s1 & p c= s2 for i being Nat, da, db being Int-Location, f being FinSeq-Location st CurInstr ((Computation s1).i) = (f,db):=da & f in dom p for k1,k2 being Nat st k1 = abs((Computation s1).i.db) & k2 = abs((Computation s2).i.db) holds (Computation s1).i.f+*(k1,(Computation s1).i.da) = (Computation s2).i.f+*(k2,(Computation s2).i.da); theorem :: SCMFSA_3:29 for p being autonomic non programmed FinPartState of SCM+FSA, s1, s2 being State of SCM+FSA st p c= s1 & p c= s2 for i being Nat, da being Int-Location, f being FinSeq-Location st CurInstr ((Computation s1).i) = da :=len f & da in dom p holds len((Computation s1).i.f) = len((Computation s2).i.f); theorem :: SCMFSA_3:30 for p being autonomic non programmed FinPartState of SCM+FSA, s1, s2 being State of SCM+FSA st p c= s1 & p c= s2 for i being Nat, da being Int-Location, f being FinSeq-Location st CurInstr ((Computation s1).i) = f:=<0,...,0>da & f in dom p for k1,k2 being Nat st k1 = abs((Computation s1).i.da) & k2 = abs((Computation s2).i.da) holds k1 |-> 0 = k2 |-> 0;

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