:: Relocatability
:: by Yasushi Tanaka
::
:: Received June 16, 1994
:: Copyright (c) 1994-2021 Association of Mizar Users

registration

let a, b be Data-Location;

cluster a := b -> ins-loc-free ;

coherence
a := b is ins-loc-free ;

cluster AddTo (a,b) -> ins-loc-free ;

coherence
AddTo (a,b) is ins-loc-free ;

cluster SubFrom (a,b) -> ins-loc-free ;

coherence
SubFrom (a,b) is ins-loc-free ;

cluster MultBy (a,b) -> ins-loc-free ;

coherence
MultBy (a,b) is ins-loc-free ;

cluster Divide (a,b) -> ins-loc-free ;

coherence
Divide (a,b) is ins-loc-free ;

end;

theorem Th1: :: RELOC:1

for k, loc being Nat holds IncAddr ((SCM-goto loc),k) = SCM-goto (loc + k)

proof end;

theorem Th2: :: RELOC:2

for k, loc being Nat
for a being Data-Location holds IncAddr ((a =0_goto loc),k) = a =0_goto (loc + k)

proof end;

theorem Th3: :: RELOC:3

for k, loc being Nat
for a being Data-Location holds IncAddr ((a >0_goto loc),k) = a >0_goto (loc + k)

proof end;

theorem Th4: :: RELOC:4

for I being Instruction of SCM
for k being Element of NAT st not not InsCode I = 0 & ... & not InsCode I = 5 holds
IncAddr (I,k) = I

proof end;

theorem :: RELOC:5

for II, I being Instruction of SCM
for k being Element of NAT st not not InsCode I = 0 & ... & not InsCode I = 5 & IncAddr (II,k) = I holds
II = I

proof end;

registration

cluster SCM -> relocable ;

coherence
SCM is relocable

proof end;

end;

Lm1: for k being Nat
for q being NAT -defined the InstructionsF of SCM -valued finite non halt-free Function
for p being non empty b₂ -autonomic FinPartState of SCM
for s1, s2 being State of SCM st p c= s1 & IncIC (p,k) c= s2 holds
for P1, P2 being Instruction-Sequence of SCM st q c= P1 & Reloc (q,k) c= P2 holds
for i being Nat holds
( (IC (Comput (P1,s1,i))) + k = IC (Comput (P2,s2,i)) & IncAddr ((CurInstr (P1,(Comput (P1,s1,i)))),k) = CurInstr (P2,(Comput (P2,s2,i))) & (Comput (P1,s1,i)) | (dom (DataPart p)) = (Comput (P2,s2,i)) | (dom (DataPart p)) & DataPart (Comput (P1,(s1 +* (DataPart s2)),i)) = DataPart (Comput (P2,s2,i)) )

proof end;

theorem :: RELOC:6

for k being Element of NAT
for q being NAT -defined the InstructionsF of SCM -valued finite non halt-free Function
for p being non empty b₂ -autonomic FinPartState of SCM
for s1, s2 being State of SCM st IC in dom p & p c= s1 & IncIC (p,k) c= s2 holds
for P1, P2 being Instruction-Sequence of SCM st q c= P1 & Reloc (q,k) c= P2 holds
for i being Element of NAT holds (IC (Comput (P1,s1,i))) + k = IC (Comput (P2,s2,i)) by Lm1;

registration

cluster SCM -> relocable1 relocable2 ;

coherence
( SCM is relocable1 & SCM is relocable2 ) by Lm1;

end;

theorem :: RELOC:7

for k being Element of NAT
for q being NAT -defined the InstructionsF of SCM -valued finite non halt-free Function
for p being non empty b₂ -autonomic FinPartState of SCM
for s1, s2, s3 being State of SCM st IC in dom p & p c= s1 & IncIC (p,k) c= s2 & s3 = s1 +* (DataPart s2) holds
for P1, P2 being Instruction-Sequence of SCM st q c= P1 & Reloc (q,k) c= P2 holds
for i being Element of NAT holds DataPart (Comput (P1,s3,i)) = DataPart (Comput (P2,s2,i)) by Lm1;