let s1, s2 be State of ; :: thesis: for J being parahalting Program of st J +* (Start-At (insloc 0 )) c= s1 holds
for n being Element of NAT st ProgramPart (Relocated J,n) c= s2 & IC s2 = insloc n & DataPart s1 = DataPart s2 holds
for i being Element of NAT holds
( (IC (Computation s1,i)) + n = IC (Computation s2,i) & IncAddr (CurInstr (Computation s1,i)),n = CurInstr (Computation s2,i) & DataPart (Computation s1,i) = DataPart (Computation s2,i) )
let J be parahalting Program of ; :: thesis: ( J +* (Start-At (insloc 0 )) c= s1 implies for n being Element of NAT st ProgramPart (Relocated J,n) c= s2 & IC s2 = insloc n & DataPart s1 = DataPart s2 holds
for i being Element of NAT holds
( (IC (Computation s1,i)) + n = IC (Computation s2,i) & IncAddr (CurInstr (Computation s1,i)),n = CurInstr (Computation s2,i) & DataPart (Computation s1,i) = DataPart (Computation s2,i) ) )
assume A1: J +* (Start-At (insloc 0 )) c= s1 ; :: thesis: for n being Element of NAT st ProgramPart (Relocated J,n) c= s2 & IC s2 = insloc n & DataPart s1 = DataPart s2 holds
for i being Element of NAT holds
( (IC (Computation s1,i)) + n = IC (Computation s2,i) & IncAddr (CurInstr (Computation s1,i)),n = CurInstr (Computation s2,i) & DataPart (Computation s1,i) = DataPart (Computation s2,i) )
set JAt = J +* (Start-At (insloc 0 ));
A2: insloc 0 in dom J by Th26;
dom J misses dom (Start-At (insloc 0 )) by SF_MASTR:64;
then A3: J c= J +* (Start-At (insloc 0 )) by FUNCT_4:33;
then A4: dom J c= dom (J +* (Start-At (insloc 0 ))) by GRFUNC_1:8;
A5: IC SCM+FSA in dom (J +* (Start-At (insloc 0 ))) by SF_MASTR:65;
then A6: s1 . (IC s1) = s1 . ((J +* (Start-At (insloc 0 ))) . (IC SCM+FSA )) by A1, GRFUNC_1:8
.= s1 . (insloc 0 ) by SF_MASTR:66
.= (J +* (Start-At (insloc 0 ))) . (insloc 0 ) by A1, A4, A2, GRFUNC_1:8
.= J . (insloc 0 ) by A3, A2, GRFUNC_1:8 ;
A7: IC (Computation s1,0 ) = s1 . (IC SCM+FSA ) by AMI_1:13
.= (J +* (Start-At (insloc 0 ))) . (IC SCM+FSA ) by A1, A5, GRFUNC_1:8
.= insloc 0 by SF_MASTR:66 ;
ProgramPart J = J by AMI_1:105;
then A8: insloc 0 in dom (ProgramPart J) by Th26;
let n be Element of NAT ; :: thesis: ( ProgramPart (Relocated J,n) c= s2 & IC s2 = insloc n & DataPart s1 = DataPart s2 implies for i being Element of NAT holds
( (IC (Computation s1,i)) + n = IC (Computation s2,i) & IncAddr (CurInstr (Computation s1,i)),n = CurInstr (Computation s2,i) & DataPart (Computation s1,i) = DataPart (Computation s2,i) ) )
assume that
A9: ProgramPart (Relocated J,n) c= s2 and
A10: IC s2 = insloc n and
A11: DataPart s1 = DataPart s2 ; :: thesis: for i being Element of NAT holds
( (IC (Computation s1,i)) + n = IC (Computation s2,i) & IncAddr (CurInstr (Computation s1,i)),n = CurInstr (Computation s2,i) & DataPart (Computation s1,i) = DataPart (Computation s2,i) )
A12: DataPart (Computation s1,0 ) = DataPart s2 by A11, AMI_1:13
.= DataPart (Computation s2,0 ) by AMI_1:13 ;
defpred S₁[ Element of NAT ] means ( (IC (Computation s1,$1)) + n = IC (Computation s2,$1) & IncAddr (CurInstr (Computation s1,$1)),n = CurInstr (Computation s2,$1) & DataPart (Computation s1,$1) = DataPart (Computation s2,$1) );
A13: for k being Element of NAT st S₁[k] holds
S₁[k + 1]

proof

dom J misses dom (Start-At (insloc 0 )) by SF_MASTR:64;
then A14: J c= J +* (Start-At (insloc 0 )) by FUNCT_4:33;
then A15: dom J c= dom (J +* (Start-At (insloc 0 ))) by GRFUNC_1:8;
let k be Element of NAT ; :: thesis: ( S₁[k] implies S₁[k + 1] )
A16: Computation s1,(k + 1) = Following (Computation s1,k) by AMI_1:14
.= Exec (CurInstr (Computation s1,k)),(Computation s1,k) ;
reconsider l = IC (Computation s1,(k + 1)) as Element of NAT by ORDINAL1:def 13;
reconsider j = CurInstr (Computation s1,(k + 1)) as Instruction of ;
A17: Computation s2,(k + 1) = Following (Computation s2,k) by AMI_1:14
.= Exec (CurInstr (Computation s2,k)),(Computation s2,k) ;
A18: IC (Computation s1,(k + 1)) in dom J by A1, Def2;
assume A19: S₁[k] ; :: thesis: S₁[k + 1]
hence A20: (IC (Computation s1,(k + 1))) + n = IC (Computation s2,(k + 1)) by A16, A17, SCMFSA6A:41; :: thesis: ( IncAddr (CurInstr (Computation s1,(k + 1))),n = CurInstr (Computation s2,(k + 1)) & DataPart (Computation s1,(k + 1)) = DataPart (Computation s2,(k + 1)) )
then ( IC (Computation s2,(k + 1)) in NAT & IC (Computation s2,(k + 1)) in dom (Relocated J,n) ) by A18, AMI_1:def 4, SCMFSA_5:4;
then IC (Computation s2,(k + 1)) in (dom (Relocated J,n)) /\ NAT by XBOOLE_0:def 4;
then A21: IC (Computation s2,(k + 1)) in dom (ProgramPart (Relocated J,n)) by RELAT_1:90;
dom (ProgramPart J) = (dom J) /\ NAT by RELAT_1:90;
then A22: l in dom (ProgramPart J) by A18, XBOOLE_0:def 4;
j = s1 . (IC (Computation s1,(k + 1))) by AMI_1:54
.= (J +* (Start-At (insloc 0 ))) . (IC (Computation s1,(k + 1))) by A1, A15, A18, GRFUNC_1:8
.= J . l by A14, A18, GRFUNC_1:8 ;
hence IncAddr (CurInstr (Computation s1,(k + 1))),n = (Relocated J,n) . (l + n) by A22, SCMFSA_5:7
.= (ProgramPart (Relocated J,n)) . (IC (Computation s2,(k + 1))) by A20, FUNCT_1:72
.= s2 . (IC (Computation s2,(k + 1))) by A9, A21, GRFUNC_1:8
.= CurInstr (Computation s2,(k + 1)) by AMI_1:54 ;
:: thesis: DataPart (Computation s1,(k + 1)) = DataPart (Computation s2,(k + 1))
thus DataPart (Computation s1,(k + 1)) = DataPart (Computation s2,(k + 1)) by A19, A16, A17, SCMFSA6A:41; :: thesis: verum

end;

let i be Element of NAT ; :: thesis: ( (IC (Computation s1,i)) + n = IC (Computation s2,i) & IncAddr (CurInstr (Computation s1,i)),n = CurInstr (Computation s2,i) & DataPart (Computation s1,i) = DataPart (Computation s2,i) )
insloc 0 in dom J by Th26;
then (insloc 0 ) + n in dom (Relocated J,n) by SCMFSA_5:4;
then A23: insloc (0 + n) in dom (ProgramPart (Relocated J,n)) by AMI_1:106;
IncAddr (CurInstr (Computation s1,0 )),n = IncAddr (CurInstr s1),n by AMI_1:13
.= (Relocated J,n) . ((insloc 0 ) + n) by A6, A8, SCMFSA_5:7
.= (ProgramPart (Relocated J,n)) . (insloc n) by FUNCT_1:72
.= CurInstr s2 by A9, A10, A23, GRFUNC_1:8
.= CurInstr (Computation s2,0 ) by AMI_1:13 ;
then A24: S₁[ 0 ] by A10, A7, A12, AMI_1:13;
for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A24, A13);
hence ( (IC (Computation s1,i)) + n = IC (Computation s2,i) & IncAddr (CurInstr (Computation s1,i)),n = CurInstr (Computation s2,i) & DataPart (Computation s1,i) = DataPart (Computation s2,i) ) ; :: thesis: verum