Volume 5, 1993

University of Bialystok

Copyright (c) 1993 Association of Mizar Users

### The abstract of the Mizar article:

### Development of Terminology for \bf SCM

**by****Grzegorz Bancerek, and****Piotr Rudnicki**- Received October 8, 1993
- MML identifier: SCM_1

- [ Mizar article, MML identifier index ]

environ vocabulary AMI_3, INT_1, FINSEQ_1, AMI_1, AMI_2, FUNCT_1, RELAT_1, ARYTM_1, NAT_1, MCART_1, SCM_1; notation TARSKI, XBOOLE_0, SUBSET_1, NUMBERS, XCMPLX_0, XREAL_0, NAT_1, INT_1, MCART_1, RELAT_1, FUNCT_1, FUNCT_2, FINSEQ_1, STRUCT_0, AMI_1, AMI_2, AMI_3; constructors NAT_1, AMI_2, AMI_3, MEMBERED, XBOOLE_0; clusters AMI_1, AMI_3, INT_1, XBOOLE_0, RELSET_1, FINSEQ_1, FRAENKEL, XREAL_0, MEMBERED, ZFMISC_1, NUMBERS, ORDINAL2; requirements NUMERALS, REAL, SUBSET, BOOLE, ARITHM; begin definition let i be Integer; redefine func <*i*> -> FinSequence of INT; end; theorem :: SCM_1:1 for s being State of SCM holds IC s = s.0 & CurInstr s = s.(s.0); theorem :: SCM_1:2 for s being State of SCM, k being Nat holds CurInstr (Computation s).k = s.(IC (Computation s).k) & CurInstr (Computation s).k = s.((Computation s).k.0); theorem :: SCM_1:3 for s being State of SCM st ex k being Nat st s.(IC (Computation s).k) = halt SCM holds s is halting; theorem :: SCM_1:4 for s being State of SCM, k being Nat st s.(IC (Computation s).k) = halt SCM holds (Result s) = (Computation s).k; canceled 2; theorem :: SCM_1:7 for n, m being Nat holds IC SCM <> il.n & IC SCM <> dl.n & il.n <> dl.m; definition let I be FinSequence of the Instructions of SCM, D be FinSequence of INT, il, ps, ds be Nat; mode State-consisting of il, ps, ds, I, D -> State of SCM means :: SCM_1:def 1 IC it = il.il & (for k being Nat st k < len I holds it.il.(ps+k) = I.(k+1)) & (for k being Nat st k < len D holds it.dl.(ds+k) = D.(k+1)); end; theorem :: SCM_1:8 for x1, x2, x3, x4 being set, p being FinSequence st p = <*x1*>^<*x2*>^<*x3*>^<*x4*> holds len p = 4 & p.1 = x1 & p.2 = x2 & p.3 = x3 & p.4 = x4; theorem :: SCM_1:9 for x1, x2, x3, x4, x5 being set, p being FinSequence st p = <*x1*>^<*x2*>^<*x3*>^<*x4*>^<*x5*> holds len p = 5 & p.1 = x1 & p.2 = x2 & p.3 = x3 & p.4 = x4 & p.5 = x5; theorem :: SCM_1:10 for x1, x2, x3, x4, x5, x6 being set, p being FinSequence st p = <*x1*>^<*x2*>^<*x3*>^<*x4*>^<*x5*>^<*x6*> holds len p = 6 & p.1 = x1 & p.2 = x2 & p.3 = x3 & p.4 = x4 & p.5 = x5 & p.6 = x6; theorem :: SCM_1:11 for x1, x2, x3, x4, x5, x6, x7 being set, p being FinSequence st p = <*x1*>^<*x2*>^<*x3*>^<*x4*>^<*x5*>^<*x6*>^<*x7*> holds len p = 7 & p.1 = x1 & p.2 = x2 & p.3 = x3 & p.4 = x4 & p.5 = x5 & p.6 = x6 & p.7 = x7; theorem :: SCM_1:12 for x1,x2,x3,x4, x5, x6, x7, x8 being set, p being FinSequence st p = <*x1*>^<*x2*>^<*x3*>^<*x4*>^<*x5*>^<*x6*>^<*x7*>^<*x8*> holds len p = 8 & p.1 = x1 & p.2 = x2 & p.3 = x3 & p.4 = x4 & p.5 = x5 & p.6 = x6 & p.7 = x7 & p.8 = x8; theorem :: SCM_1:13 for x1,x2,x3,x4,x5,x6,x7, x8, x9 being set, p being FinSequence st p = <*x1*>^<*x2*>^<*x3*>^<*x4*>^<*x5*>^<*x6*>^<*x7*>^<*x8*>^<*x9*> holds len p = 9 & p.1 = x1 & p.2 = x2 & p.3 = x3 & p.4 = x4 & p.5 = x5 & p.6 = x6 & p.7 = x7 & p.8 = x8 & p.9 = x9; theorem :: SCM_1:14 for I1, I2, I3, I4, I5, I6, I7, I8, I9 being Instruction of SCM, i1, i2, i3, i4 being Integer, il being Nat, s being State-consisting of il, 0, 0, <*I1*>^<*I2*>^<*I3*>^<*I4*>^<*I5*>^<*I6*>^<*I7*>^<*I8*>^<*I9*>, <*i1*>^<*i2*>^<*i3*>^<*i4*> holds IC s = il.il & s.il.0 = I1 & s.il.1 = I2 & s.il.2 = I3 & s.il.3 = I4 & s.il.4 = I5 & s.il.5 = I6 & s.il.6 = I7 & s.il.7 = I8 & s.il.8 = I9 & s.dl.0 = i1 & s.dl.1 = i2 & s.dl.2 = i3 & s.dl.3 = i4; theorem :: SCM_1:15 for I1, I2 being Instruction of SCM, i1, i2 being Integer, il being Nat, s being State-consisting of il, 0, 0, <*I1*>^<*I2*>, <*i1*>^<*i2*> holds IC s = il.il & s.il.0 = I1 & s.il.1 = I2 & s.dl.0 = i1 & s.dl.1 = i2; definition let N be with_non-empty_elements set; let S be halting IC-Ins-separated definite (non empty non void AMI-Struct over N); let s be State of S such that s is halting; func Complexity s -> Nat means :: SCM_1:def 2 CurInstr((Computation s).it) = halt S & for k being Nat st CurInstr((Computation s).k) = halt S holds it <= k; synonym LifeSpan s; end; theorem :: SCM_1:16 for s being State of SCM, k being Nat holds s.(IC (Computation s).k) <> halt SCM & s.(IC (Computation s).(k+1)) = halt SCM iff Complexity s = k+1 & s is halting; theorem :: SCM_1:17 for s being State of SCM, k being Nat st IC (Computation s).k <> IC (Computation s).(k+1) & s.(IC (Computation s).(k+1)) = halt SCM holds Complexity s = k+1; theorem :: SCM_1:18 for k, n being Nat, s being State of SCM, a, b being Data-Location st IC (Computation s).k = il.n & s.il.n = a := b holds IC (Computation s).(k+1) = il.(n+1) & (Computation s).(k+1).a = (Computation s).k.b & for d being Data-Location st d <> a holds (Computation s).(k+1).d = (Computation s).k.d; theorem :: SCM_1:19 for k, n being Nat, s being State of SCM, a, b being Data-Location st IC (Computation s).k = il.n & s.il.n = AddTo(a,b) holds IC (Computation s).(k+1) = il.(n+1) & (Computation s).(k+1).a = (Computation s).k.a+(Computation s).k.b & for d being Data-Location st d <> a holds (Computation s).(k+1).d = (Computation s).k.d; theorem :: SCM_1:20 for k, n being Nat, s being State of SCM, a, b being Data-Location st IC (Computation s).k = il.n & s.il.n = SubFrom(a,b) holds IC (Computation s).(k+1) = il.(n+1) & (Computation s).(k+1).a = (Computation s).k.a-(Computation s).k.b & for d being Data-Location st d <> a holds (Computation s).(k+1).d = (Computation s).k.d; theorem :: SCM_1:21 for k, n being Nat, s being State of SCM, a, b being Data-Location st IC (Computation s).k = il.n & s.il.n = MultBy(a,b) holds IC (Computation s).(k+1) = il.(n+1) & (Computation s).(k+1).a = (Computation s).k.a*(Computation s).k.b & for d being Data-Location st d <> a holds (Computation s).(k+1).d = (Computation s).k.d; theorem :: SCM_1:22 for k, n being Nat, s being State of SCM, a, b being Data-Location st IC (Computation s).k = il.n & s.il.n = Divide(a,b) & a<>b holds IC (Computation s).(k+1) = il.(n+1) & (Computation s).(k+1).a = (Computation s).k.a div (Computation s).k.b & (Computation s).(k+1).b = (Computation s).k.a mod (Computation s).k.b & for d being Data-Location st d <> a & d <> b holds (Computation s).(k+1).d = (Computation s).k.d; theorem :: SCM_1:23 for k, n being Nat, s being State of SCM, il being Instruction-Location of SCM st IC (Computation s).k = il.n & s.il.n = goto il holds IC (Computation s).(k+1) = il & for d being Data-Location holds (Computation s).(k+1).d = (Computation s).k.d; theorem :: SCM_1:24 for k, n being Nat, s being State of SCM, a being Data-Location, il being Instruction-Location of SCM st IC (Computation s).k = il.n & s.il.n = a =0_goto il holds ((Computation s).k.a = 0 implies IC (Computation s).(k+1) = il) & ((Computation s).k.a <>0 implies IC (Computation s).(k+1) = il.(n+1)) & for d being Data-Location holds (Computation s).(k+1).d = (Computation s).k.d; theorem :: SCM_1:25 for k, n being Nat, s being State of SCM, a being Data-Location, il being Instruction-Location of SCM st IC (Computation s).k = il.n & s.il.n = a >0_goto il holds ((Computation s).k.a > 0 implies IC (Computation s).(k+1) = il) & ((Computation s).k.a <= 0 implies IC (Computation s).(k+1) = il.(n+1)) & for d being Data-Location holds (Computation s).(k+1).d = (Computation s).k.d; theorem :: SCM_1:26 (halt SCM)`1 = 0 & (for a, b being Data-Location holds (a := b)`1 = 1) & (for a, b being Data-Location holds (AddTo(a,b))`1 = 2) & (for a, b being Data-Location holds (SubFrom(a,b))`1 = 3) & (for a, b being Data-Location holds (MultBy(a,b))`1 = 4) & (for a, b being Data-Location holds (Divide(a,b))`1 = 5) & (for i being Instruction-Location of SCM holds (goto i)`1 = 6) & (for a being Data-Location, i being Instruction-Location of SCM holds (a =0_goto i)`1 = 7) & (for a being Data-Location, i being Instruction-Location of SCM holds (a >0_goto i)`1 = 8); theorem :: SCM_1:27 for N being non empty with_non-empty_elements set, S be IC-Ins-separated definite halting (non empty non void AMI-Struct over N), s being State of S, m being Nat holds s is halting iff (Computation s).m is halting; theorem :: SCM_1:28 for s1, s2 being State of SCM, k, c being Nat st s2 = (Computation s1).k & Complexity s2 = c & s2 is halting & 0 < c holds Complexity s1 = k+c; theorem :: SCM_1:29 for s1, s2 being State of SCM, k being Nat st s2 = (Computation s1).k & s2 is halting holds Result s2 = Result s1; theorem :: SCM_1:30 for I1, I2, I3, I4, I5, I6, I7, I8, I9 being Instruction of SCM, i1, i2, i3, i4 being Integer, il being Nat, s being State of SCM st IC s = il.il & s.il.0 = I1 & s.il.1 = I2 & s.il.2 = I3 & s.il.3 = I4 & s.il.4 = I5 & s.il.5 = I6 & s.il.6 = I7 & s.il.7 = I8 & s.il.8 = I9 & s.dl.0 = i1 & s.dl.1 = i2 & s.dl.2 = i3 & s.dl.3 = i4 holds s is State-consisting of il, 0, 0, <*I1*>^<*I2*>^<*I3*>^<*I4*>^<*I5*>^<*I6*>^<*I7*>^<*I8*>^<*I9*>, <*i1*>^<*i2*>^<*i3*>^<*i4*>; :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :: Empty program :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: theorem :: SCM_1:31 for s being State-consisting of 0, 0, 0, <*halt SCM*>, <*>INT holds s is halting & Complexity s = 0 & Result s = s; :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :: Assignment :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: theorem :: SCM_1:32 for i1, i2 being Integer, s being State-consisting of 0, 0, 0, <*dl.0 := dl.1*>^<*halt SCM*>, <*i1*>^<*i2*> holds s is halting & Complexity s = 1 & (Result s).dl.0 = i2 & for d being Data-Location st d<>dl.0 holds (Result s).d = s.d; :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :: Adding two integers :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: theorem :: SCM_1:33 for i1, i2 being Integer, s being State-consisting of 0, 0, 0, <*AddTo(dl.0,dl.1)*>^<*halt SCM*>, <*i1*>^<*i2*> holds s is halting & Complexity s = 1 & (Result s).dl.0 = i1 + i2 & for d being Data-Location st d<>dl.0 holds (Result s).d = s.d; :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :: Subtracting two integers :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: theorem :: SCM_1:34 for i1, i2 being Integer, s being State-consisting of 0, 0, 0, <*SubFrom(dl.0,dl.1)*>^<*halt SCM*>, <*i1*>^<*i2*> holds s is halting & Complexity s = 1 & (Result s).dl.0 = i1 - i2 & for d being Data-Location st d<>dl.0 holds (Result s).d = s.d; :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :: Multiplying two integers :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: theorem :: SCM_1:35 for i1, i2 being Integer, s being State-consisting of 0, 0, 0, <*MultBy(dl.0,dl.1)*>^<*halt SCM*>, <*i1*>^<*i2*> holds s is halting & Complexity s = 1 & (Result s).dl.0 = i1 * i2 & for d being Data-Location st d<>dl.0 holds (Result s).d = s.d; :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :: Dividing two integers :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: theorem :: SCM_1:36 for i1, i2 being Integer, s being State-consisting of 0, 0, 0, <*Divide(dl.0,dl.1)*>^<*halt SCM*>, <*i1*>^<*i2*> holds s is halting & Complexity s = 1 & (Result s).dl.0 = i1 div i2 & (Result s).dl.1 = i1 mod i2 & :: (i2 <> 0 implies abs((Result s).dl.1) < abs i2 for d being Data-Location st d<>dl.0 & d<>dl.1 holds (Result s).d = s.d; :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :: Unconditional jump :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: theorem :: SCM_1:37 for i1, i2 being Integer, s being State-consisting of 0, 0, 0, <*goto il.1*>^<*halt SCM*>, <*i1*>^<*i2*> holds s is halting & Complexity s = 1 & for d being Data-Location holds (Result s).d = s.d; :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :: Jump at zero :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: theorem :: SCM_1:38 for i1, i2 being Integer, s being State-consisting of 0, 0, 0, <*dl.0 =0_goto il.1*>^<*halt SCM*>, <*i1*>^<*i2*> holds s is halting & Complexity s = 1 & for d being Data-Location holds (Result s).d = s.d; :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :: Jump at greater than zero :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: theorem :: SCM_1:39 for i1, i2 being Integer, s being State-consisting of 0, 0, 0, <*dl.0 >0_goto il.1*>^<*halt SCM*>, <*i1*>^<*i2*> holds s is halting & Complexity s = 1 & for d being Data-Location holds (Result s).d = s.d;

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