let s1, s2 be State of ; :: thesis: for J being parahalting shiftable Program of st Initialized (stop J) c= s1 holds
for n being Element of NAT st Shift (stop J),n c= s2 & IC s2 = inspos n & DataPart s1 = DataPart s2 holds
for i being Element of NAT holds
( (IC (Computation s1,i)) + n = IC (Computation s2,i) & CurInstr (Computation s1,i) = CurInstr (Computation s2,i) & DataPart (Computation s1,i) = DataPart (Computation s2,i) )
let I be parahalting shiftable Program of ; :: thesis: ( Initialized (stop I) c= s1 implies for n being Element of NAT st Shift (stop I),n c= s2 & IC s2 = inspos n & DataPart s1 = DataPart s2 holds
for i being Element of NAT holds
( (IC (Computation s1,i)) + n = IC (Computation s2,i) & CurInstr (Computation s1,i) = CurInstr (Computation s2,i) & DataPart (Computation s1,i) = DataPart (Computation s2,i) ) )
set SI = stop I;
set II = Initialized (stop I);
assume A1: Initialized (stop I) c= s1 ; :: thesis: for n being Element of NAT st Shift (stop I),n c= s2 & IC s2 = inspos n & DataPart s1 = DataPart s2 holds
for i being Element of NAT holds
( (IC (Computation s1,i)) + n = IC (Computation s2,i) & CurInstr (Computation s1,i) = CurInstr (Computation s2,i) & DataPart (Computation s1,i) = DataPart (Computation s2,i) )
let n be Element of NAT ; :: thesis: ( Shift (stop I),n c= s2 & IC s2 = inspos n & DataPart s1 = DataPart s2 implies for i being Element of NAT holds
( (IC (Computation s1,i)) + n = IC (Computation s2,i) & CurInstr (Computation s1,i) = CurInstr (Computation s2,i) & DataPart (Computation s1,i) = DataPart (Computation s2,i) ) )
assume that
A2: Shift (stop I),n c= s2 and
A3: IC s2 = inspos n and
A4: DataPart s1 = DataPart s2 ; :: thesis: for i being Element of NAT holds
( (IC (Computation s1,i)) + n = IC (Computation s2,i) & CurInstr (Computation s1,i) = CurInstr (Computation s2,i) & DataPart (Computation s1,i) = DataPart (Computation s2,i) )
A5: inspos 0 in dom (stop I) by Th75;
then A6: inspos (0 + n) in dom (Shift (stop I),n) by VALUED_1:25;
defpred S₁[ Nat] means ( (IC (Computation s1,$1)) + n = IC (Computation s2,$1) & CurInstr (Computation s1,$1) = CurInstr (Computation s2,$1) & DataPart (Computation s1,$1) = DataPart (Computation s2,$1) );
dom (stop I) misses dom (Start-At (inspos 0 )) by Th54;
then A7: stop I c= Initialized (stop I) by FUNCT_4:33;
then A8: dom (stop I) c= dom (Initialized (stop I)) by GRFUNC_1:8;
A9: 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] )
assume A10: S₁[k] ; :: thesis: S₁[k + 1]
reconsider m = IC (Computation s1,k) as Element of NAT by ORDINAL1:def 13;
set i = CurInstr (Computation s1,k);
A11: Computation s1,(k + 1) = Following (Computation s1,k) by AMI_1:14
.= Exec (CurInstr (Computation s1,k)),(Computation s1,k) ;
A12: IC (Computation s1,k) = inspos m ;
reconsider l = IC (Computation s1,(k + 1)) as Element of NAT by ORDINAL1:def 13;
A13: IC (Computation s1,(k + 1)) in dom (stop I) by A1, Def9;
then A14: l + n in dom (Shift (stop I),n) by VALUED_1:25;
A15: Computation s2,(k + 1) = Following (Computation s2,k) by AMI_1:14
.= Exec (CurInstr (Computation s2,k)),(Computation s2,k) ;
A16: IC (Computation s1,k) in dom (stop I) by A1, Def9;
A17: CurInstr (Computation s1,k) = s1 . (IC (Computation s1,k)) by AMI_1:54
.= (Initialized (stop I)) . (IC (Computation s1,k)) by A1, A8, A16, GRFUNC_1:8
.= (stop I) . (IC (Computation s1,k)) by A7, A16, GRFUNC_1:8 ;
then A18: ( InsCode (CurInstr (Computation s1,k)) <> 1 & InsCode (CurInstr (Computation s1,k)) <> 3 ) by A16, A12, Def12;
A19: CurInstr (Computation s1,k) valid_at m by A16, A17, A12, Def12;
hence A20: (IC (Computation s1,(k + 1))) + n = IC (Computation s2,(k + 1)) by A10, A11, A15, A12, A18, Th83; :: thesis: ( CurInstr (Computation s1,(k + 1)) = CurInstr (Computation s2,(k + 1)) & DataPart (Computation s1,(k + 1)) = DataPart (Computation s2,(k + 1)) )
CurInstr (Computation s1,(k + 1)) = s1 . l by AMI_1:54
.= (Initialized (stop I)) . l by A1, A8, A13, GRFUNC_1:8
.= (stop I) . l by A7, A13, GRFUNC_1:8 ;
hence CurInstr (Computation s1,(k + 1)) = (Shift (stop I),n) . (IC (Computation s2,(k + 1))) by A20, A13, VALUED_1:def 12
.= s2 . (IC (Computation s2,(k + 1))) by A2, A20, A14, 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 A10, A11, A15, A12, A18, A19, Th83; :: thesis: verum

end;

A21: IC SCMPDS in dom (Initialized (stop I)) by Th7;
then A22: s1 . (IC s1) = s1 . ((Initialized (stop I)) . (IC SCMPDS )) by A1, GRFUNC_1:8
.= s1 . (inspos 0 ) by Th29
.= (Initialized (stop I)) . (inspos 0 ) by A1, A8, A5, GRFUNC_1:8
.= (stop I) . (inspos 0 ) by A7, A5, GRFUNC_1:8 ;
let i be Element of NAT ; :: thesis: ( (IC (Computation s1,i)) + n = IC (Computation s2,i) & CurInstr (Computation s1,i) = CurInstr (Computation s2,i) & DataPart (Computation s1,i) = DataPart (Computation s2,i) )
A23: DataPart (Computation s1,0 ) = DataPart s2 by A4, AMI_1:13
.= DataPart (Computation s2,0 ) by AMI_1:13 ;
A24: IC (Computation s1,0 ) = s1 . (IC SCMPDS ) by AMI_1:13
.= (Initialized (stop I)) . (IC SCMPDS ) by A1, A21, GRFUNC_1:8
.= inspos 0 by Th29 ;
CurInstr (Computation s1,0 ) = CurInstr s1 by AMI_1:13
.= (Shift (stop I),n) . ((inspos 0 ) + n) by A5, A22, VALUED_1:def 12
.= CurInstr s2 by A2, A3, A6, GRFUNC_1:8
.= CurInstr (Computation s2,0 ) by AMI_1:13 ;
then A25: S₁[ 0 ] by A3, A24, A23, AMI_1:13;
for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A25, A9);
hence ( (IC (Computation s1,i)) + n = IC (Computation s2,i) & CurInstr (Computation s1,i) = CurInstr (Computation s2,i) & DataPart (Computation s1,i) = DataPart (Computation s2,i) ) ; :: thesis: verum