let s1, s2 be State of SCM+FSA ; :: thesis: for J being InitHalting Program of SCM+FSA st Initialized J c= s1 holds
for n being Element of NAT st ProgramPart (Relocated J,n) c= s2 & IC s2 = n & DataPart s1 = DataPart s2 holds
for i being Element of NAT holds
( (IC (Comput (ProgramPart s1),s1,i)) + n = IC (Comput (ProgramPart s2),s2,i) & IncAddr (CurInstr (ProgramPart s1),(Comput (ProgramPart s1),s1,i)),n = CurInstr (ProgramPart s2),(Comput (ProgramPart s2),s2,i) & DataPart (Comput (ProgramPart s1),s1,i) = DataPart (Comput (ProgramPart s2),s2,i) )
let J be InitHalting Program of SCM+FSA ; :: thesis: ( Initialized J c= s1 implies for n being Element of NAT st ProgramPart (Relocated J,n) c= s2 & IC s2 = n & DataPart s1 = DataPart s2 holds
for i being Element of NAT holds
( (IC (Comput (ProgramPart s1),s1,i)) + n = IC (Comput (ProgramPart s2),s2,i) & IncAddr (CurInstr (ProgramPart s1),(Comput (ProgramPart s1),s1,i)),n = CurInstr (ProgramPart s2),(Comput (ProgramPart s2),s2,i) & DataPart (Comput (ProgramPart s1),s1,i) = DataPart (Comput (ProgramPart s2),s2,i) ) )
set JAt = Initialized J;
assume A1: Initialized J c= s1 ; :: thesis: for n being Element of NAT st ProgramPart (Relocated J,n) c= s2 & IC s2 = n & DataPart s1 = DataPart s2 holds
for i being Element of NAT holds
( (IC (Comput (ProgramPart s1),s1,i)) + n = IC (Comput (ProgramPart s2),s2,i) & IncAddr (CurInstr (ProgramPart s1),(Comput (ProgramPart s1),s1,i)),n = CurInstr (ProgramPart s2),(Comput (ProgramPart s2),s2,i) & DataPart (Comput (ProgramPart s1),s1,i) = DataPart (Comput (ProgramPart s2),s2,i) )
let n be Element of NAT ; :: thesis: ( ProgramPart (Relocated J,n) c= s2 & IC s2 = n & DataPart s1 = DataPart s2 implies for i being Element of NAT holds
( (IC (Comput (ProgramPart s1),s1,i)) + n = IC (Comput (ProgramPart s2),s2,i) & IncAddr (CurInstr (ProgramPart s1),(Comput (ProgramPart s1),s1,i)),n = CurInstr (ProgramPart s2),(Comput (ProgramPart s2),s2,i) & DataPart (Comput (ProgramPart s1),s1,i) = DataPart (Comput (ProgramPart s2),s2,i) ) )
assume that
A2: ProgramPart (Relocated J,n) c= s2 and
A3: IC s2 = n and
A4: DataPart s1 = DataPart s2 ; :: thesis: for i being Element of NAT holds
( (IC (Comput (ProgramPart s1),s1,i)) + n = IC (Comput (ProgramPart s2),s2,i) & IncAddr (CurInstr (ProgramPart s1),(Comput (ProgramPart s1),s1,i)),n = CurInstr (ProgramPart s2),(Comput (ProgramPart s2),s2,i) & DataPart (Comput (ProgramPart s1),s1,i) = DataPart (Comput (ProgramPart s2),s2,i) )
A5: DataPart (Comput (ProgramPart s1),s1,0 ) = DataPart s2 by A4, AMI_1:13
.= DataPart (Comput (ProgramPart s2),s2,0 ) by AMI_1:13 ;
defpred S₁[ Nat] means ( (IC (Comput (ProgramPart s1),s1,$1)) + n = IC (Comput (ProgramPart s2),s2,$1) & IncAddr (CurInstr (ProgramPart s1),(Comput (ProgramPart s1),s1,$1)),n = CurInstr (ProgramPart s2),(Comput (ProgramPart s2),s2,$1) & DataPart (Comput (ProgramPart s1),s1,$1) = DataPart (Comput (ProgramPart s2),s2,$1) );
A6: J c= Initialized J by SCMFSA6A:26;
then A7: dom J c= dom (Initialized J) by GRFUNC_1:8;
A8: for k being Element of NAT st S₁[k] holds
S₁[k + 1]

proof

let k be Element of NAT ; :: thesis: ( S₁[k] implies S₁[k + 1] )
A9: Comput (ProgramPart s1),s1,(k + 1) = Following (ProgramPart s1),(Comput (ProgramPart s1),s1,k) by AMI_1:14
.= Exec (CurInstr (ProgramPart s1),(Comput (ProgramPart s1),s1,k)),(Comput (ProgramPart s1),s1,k) ;
reconsider l = IC (Comput (ProgramPart s1),s1,(k + 1)) as Element of NAT ;
reconsider j = CurInstr (ProgramPart (Comput (ProgramPart s1),s1,(k + 1))),(Comput (ProgramPart s1),s1,(k + 1)) as Instruction of SCM+FSA ;
Y: (ProgramPart (Comput (ProgramPart s1),s1,(k + 1))) /. (IC (Comput (ProgramPart s1),s1,(k + 1))) = (Comput (ProgramPart s1),s1,(k + 1)) . (IC (Comput (ProgramPart s1),s1,(k + 1))) by COMPOS_1:38;
A10: Comput (ProgramPart s2),s2,(k + 1) = Following (ProgramPart s2),(Comput (ProgramPart s2),s2,k) by AMI_1:14
.= Exec (CurInstr (ProgramPart s2),(Comput (ProgramPart s2),s2,k)),(Comput (ProgramPart s2),s2,k) ;
A11: IC (Comput (ProgramPart s1),s1,(k + 1)) in dom J by A1, Def1;
assume A12: S₁[k] ; :: thesis: S₁[k + 1]
hence A13: (IC (Comput (ProgramPart s1),s1,(k + 1))) + n = IC (Comput (ProgramPart s2),s2,(k + 1)) by A9, A10, SCMFSA6A:41; :: thesis: ( IncAddr (CurInstr (ProgramPart s1),(Comput (ProgramPart s1),s1,(k + 1))),n = CurInstr (ProgramPart s2),(Comput (ProgramPart s2),s2,(k + 1)) & DataPart (Comput (ProgramPart s1),s1,(k + 1)) = DataPart (Comput (ProgramPart s2),s2,(k + 1)) )
then ( IC (Comput (ProgramPart s2),s2,(k + 1)) in NAT & IC (Comput (ProgramPart s2),s2,(k + 1)) in dom (Relocated J,n) ) by A11, AMISTD_2:71;
then IC (Comput (ProgramPart s2),s2,(k + 1)) in (dom (Relocated J,n)) /\ NAT by XBOOLE_0:def 4;
then A14: IC (Comput (ProgramPart s2),s2,(k + 1)) in dom (ProgramPart (Relocated J,n)) by RELAT_1:90;
dom (ProgramPart J) = (dom J) /\ NAT by RELAT_1:90;
then A15: l in dom (ProgramPart J) by A11, XBOOLE_0:def 4;
Z: (ProgramPart (Comput (ProgramPart s2),s2,(k + 1))) /. (IC (Comput (ProgramPart s2),s2,(k + 1))) = (Comput (ProgramPart s2),s2,(k + 1)) . (IC (Comput (ProgramPart s2),s2,(k + 1))) by COMPOS_1:38;
TX1: ProgramPart s1 = ProgramPart (Comput (ProgramPart s1),s1,(k + 1)) by AMI_1:123;
TX2: ProgramPart s2 = ProgramPart (Comput (ProgramPart s2),s2,(k + 1)) by AMI_1:123;
j = s1 . (IC (Comput (ProgramPart s1),s1,(k + 1))) by Y, AMI_1:54
.= (Initialized J) . (IC (Comput (ProgramPart s1),s1,(k + 1))) by A1, A7, A11, GRFUNC_1:8
.= J . l by A6, A11, GRFUNC_1:8 ;
hence IncAddr (CurInstr (ProgramPart s1),(Comput (ProgramPart s1),s1,(k + 1))),n = (Relocated J,n) . (l + n) by A15, TX1, AMISTD_2:76
.= (ProgramPart (Relocated J,n)) . (IC (Comput (ProgramPart s2),s2,(k + 1))) by A13, FUNCT_1:72
.= s2 . (IC (Comput (ProgramPart s2),s2,(k + 1))) by A2, A14, GRFUNC_1:8
.= CurInstr (ProgramPart s2),(Comput (ProgramPart s2),s2,(k + 1)) by Z, TX2, AMI_1:54 ;
:: thesis: DataPart (Comput (ProgramPart s1),s1,(k + 1)) = DataPart (Comput (ProgramPart s2),s2,(k + 1))
thus DataPart (Comput (ProgramPart s1),s1,(k + 1)) = DataPart (Comput (ProgramPart s2),s2,(k + 1)) by A12, A9, A10, SCMFSA6A:41; :: thesis: verum

end;

ProgramPart J = J by RELAT_1:209;
then A16: 0 in dom (ProgramPart J) by AFINSQ_1:69;
A17: 0 in dom J by AFINSQ_1:69;
A18: IC SCM+FSA in dom (Initialized J) by SCMFSA6A:24;
then A19: s1 . (IC s1) = s1 . ((Initialized J) . (IC SCM+FSA )) by A1, GRFUNC_1:8
.= s1 . 0 by SCMFSA6A:46
.= (Initialized J) . 0 by A1, A7, A17, GRFUNC_1:8
.= J . 0 by A6, A17, GRFUNC_1:8 ;
let i be Element of NAT ; :: thesis: ( (IC (Comput (ProgramPart s1),s1,i)) + n = IC (Comput (ProgramPart s2),s2,i) & IncAddr (CurInstr (ProgramPart s1),(Comput (ProgramPart s1),s1,i)),n = CurInstr (ProgramPart s2),(Comput (ProgramPart s2),s2,i) & DataPart (Comput (ProgramPart s1),s1,i) = DataPart (Comput (ProgramPart s2),s2,i) )
0 in dom J by AFINSQ_1:69;
then 0 + n in dom (Relocated J,n) by AMISTD_2:71;
then A20: 0 + n in dom (ProgramPart (Relocated J,n)) by COMPOS_1:16;
A21: IC (Comput (ProgramPart s1),s1,0 ) = s1 . (IC SCM+FSA ) by AMI_1:13
.= (Initialized J) . (IC SCM+FSA ) by A1, A18, GRFUNC_1:8
.= 0 by SCMFSA6A:46 ;
u: Comput (ProgramPart s1),s1,0 = s1 by AMI_1:13;
v: Comput (ProgramPart s2),s2,0 = s2 by AMI_1:13;
Y: (ProgramPart s1) /. (IC s1) = s1 . (IC s1) by COMPOS_1:38;
Z: (ProgramPart s2) /. (IC s2) = s2 . (IC s2) by COMPOS_1:38;
IncAddr (CurInstr (ProgramPart s1),(Comput (ProgramPart s1),s1,0 )),n = IncAddr (CurInstr (ProgramPart s1),s1),n by u
.= (Relocated J,n) . (0 + n) by A19, A16, Y, AMISTD_2:76
.= (ProgramPart (Relocated J,n)) . n by FUNCT_1:72
.= CurInstr (ProgramPart s2),s2 by A2, A3, A20, Z, GRFUNC_1:8
.= CurInstr (ProgramPart s2),(Comput (ProgramPart s2),s2,0 ) by v ;
then A22: S₁[ 0 ] by A3, A21, A5, AMI_1:13;
for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A22, A8);
hence ( (IC (Comput (ProgramPart s1),s1,i)) + n = IC (Comput (ProgramPart s2),s2,i) & IncAddr (CurInstr (ProgramPart s1),(Comput (ProgramPart s1),s1,i)),n = CurInstr (ProgramPart s2),(Comput (ProgramPart s2),s2,i) & DataPart (Comput (ProgramPart s1),s1,i) = DataPart (Comput (ProgramPart s2),s2,i) ) ; :: thesis: verum