:: Computation in { \bf SCM_FSA }
:: by Andrzej Trybulec and Yatsuka Nakamura
::
:: Received February 7, 1996
:: Copyright (c) 1996 Association of Mizar Users

theorem :: SCMFSA_3:1

not IC SCM+FSA in Int-Locations

proof end;

theorem :: SCMFSA_3:2

not IC SCM+FSA in FinSeq-Locations

proof end;

theorem :: SCMFSA_3:3

for i being Instruction of SCM+FSA
for I being Instruction of SCM st i = I holds
for s being State of SCM+FSA
for S being State of SCM st S = (s | the carrier of SCM ) +* (NAT --> I) holds
Exec i,s = (s +* (Exec I,S)) +* (s | NAT ) by SCMFSA_2:75;

theorem Th4: :: 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 )}) holds
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 )})

proof end;

theorem :: SCMFSA_3:5

canceled;

theorem :: SCMFSA_3:6

canceled;

theorem :: SCMFSA_3:7

canceled;

theorem :: SCMFSA_3:8

canceled;

theorem :: SCMFSA_3:9

canceled;

theorem :: SCMFSA_3:10

for i being Instruction of SCM+FSA
for s being State of SCM+FSA
for p being preProgram of SCM+FSA holds Exec i,(s +* p) = (Exec i,s) +* p

proof end;

theorem :: SCMFSA_3:11

for s being State of SCM+FSA
for iloc being Instruction-Location of SCM+FSA
for a being Int-Location holds s . a = (s +* (Start-At iloc)) . a

proof end;

theorem :: SCMFSA_3:12

for s being State of SCM+FSA
for iloc being Instruction-Location of SCM+FSA
for a being FinSeq-Location holds s . a = (s +* (Start-At iloc)) . a

proof end;

definition

let la be Int-Location ;
let a be Integer;
:: original: .-->
redefine func la .--> a -> FinPartState of SCM+FSA ;
coherence
la .--> a is FinPartState of SCM+FSA

proof end;

end;

theorem :: SCMFSA_3:13

canceled;

theorem Th14: :: SCMFSA_3:14

for p being autonomic FinPartState of SCM+FSA st DataPart p <> {} holds
IC SCM+FSA in dom p

proof end;

registration

cluster autonomic non programmed Element of sproduct the Object-Kind of SCM+FSA ;
existence
ex b₁ being FinPartState of SCM+FSA st
( b₁ is autonomic & not b₁ is programmed )

proof end;

end;

theorem Th15: :: SCMFSA_3:15

for p being autonomic non programmed FinPartState of SCM+FSA holds IC SCM+FSA in dom p

proof end;

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

proof end;

theorem Th17: :: SCMFSA_3:17

for p being autonomic non programmed FinPartState of SCM+FSA
for s being State of SCM+FSA st p c= s holds
for i being Element of NAT holds IC (Computation s,i) in dom (ProgramPart p)

proof end;

theorem Th18: :: SCMFSA_3:18

for p being autonomic non programmed FinPartState of SCM+FSA
for s1, s2 being State of SCM+FSA st p c= s1 & p c= s2 holds
for i being Element of NAT holds
( IC (Computation s1,i) = IC (Computation s2,i) & CurInstr (Computation s1,i) = CurInstr (Computation s2,i) )

proof end;

theorem :: SCMFSA_3:19

for p being autonomic non programmed FinPartState of SCM+FSA
for s1, s2 being State of SCM+FSA st p c= s1 & p c= s2 holds
for i being Element of NAT
for 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

proof end;

theorem :: SCMFSA_3:20

for p being autonomic non programmed FinPartState of SCM+FSA
for s1, s2 being State of SCM+FSA st p c= s1 & p c= s2 holds
for i being Element of NAT
for 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)

proof end;

theorem :: SCMFSA_3:21

for p being autonomic non programmed FinPartState of SCM+FSA
for s1, s2 being State of SCM+FSA st p c= s1 & p c= s2 holds
for i being Element of NAT
for 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)

proof end;

theorem :: SCMFSA_3:22

for p being autonomic non programmed FinPartState of SCM+FSA
for s1, s2 being State of SCM+FSA st p c= s1 & p c= s2 holds
for i being Element of NAT
for 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)

proof end;

theorem :: SCMFSA_3:23

for p being autonomic non programmed FinPartState of SCM+FSA
for s1, s2 being State of SCM+FSA st p c= s1 & p c= s2 holds
for i being Element of NAT
for 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)

proof end;

theorem :: SCMFSA_3:24

for p being autonomic non programmed FinPartState of SCM+FSA
for s1, s2 being State of SCM+FSA st p c= s1 & p c= s2 holds
for i being Element of NAT
for 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)

proof end;

theorem :: SCMFSA_3:25

for p being autonomic non programmed FinPartState of SCM+FSA
for s1, s2 being State of SCM+FSA st p c= s1 & p c= s2 holds
for i being Element of NAT
for da being Int-Location
for 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 )

proof end;

theorem :: SCMFSA_3:26

for p being autonomic non programmed FinPartState of SCM+FSA
for s1, s2 being State of SCM+FSA st p c= s1 & p c= s2 holds
for i being Element of NAT
for da being Int-Location
for 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 )

proof end;

theorem :: SCMFSA_3:27

for p being autonomic non programmed FinPartState of SCM+FSA
for s1, s2 being State of SCM+FSA st p c= s1 & p c= s2 holds
for i being Element of NAT
for da, db being Int-Location
for f being FinSeq-Location st CurInstr (Computation s1,i) = da := f,db & da in dom p holds
for k1, k2 being Element of 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

proof end;

theorem :: SCMFSA_3:28

for p being autonomic non programmed FinPartState of SCM+FSA
for s1, s2 being State of SCM+FSA st p c= s1 & p c= s2 holds
for i being Element of NAT
for da, db being Int-Location
for f being FinSeq-Location st CurInstr (Computation s1,i) = f,db := da & f in dom p holds
for k1, k2 being Element of 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)

proof end;

theorem :: SCMFSA_3:29

for p being autonomic non programmed FinPartState of SCM+FSA
for s1, s2 being State of SCM+FSA st p c= s1 & p c= s2 holds
for i being Element of NAT
for da being Int-Location
for 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)

proof end;

theorem :: SCMFSA_3:30

for p being autonomic non programmed FinPartState of SCM+FSA
for s1, s2 being State of SCM+FSA st p c= s1 & p c= s2 holds
for i being Element of NAT
for da being Int-Location
for f being FinSeq-Location st CurInstr (Computation s1,i) = f :=<0,...,0> da & f in dom p holds
for k1, k2 being Element of NAT st k1 = abs ((Computation s1,i) . da) & k2 = abs ((Computation s2,i) . da) holds
k1 |-> 0 = k2 |-> 0

proof end;