let a be Int_position; :: thesis: for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being halt-free Program of
for J being shiftable Program of st I is_closed_on s,P & I is_halting_on s,P & J is_closed_on IExec (I,P,s),P & J is_halting_on IExec (I,P,s),P holds
(IExec ((I ';' J),P,s)) . a = (IExec (J,P,(Initialize (IExec (I,P,s))))) . a
let P be Instruction-Sequence of SCMPDS; :: thesis: for s being 0 -started State of SCMPDS
for I being halt-free Program of
for J being shiftable Program of st I is_closed_on s,P & I is_halting_on s,P & J is_closed_on IExec (I,P,s),P & J is_halting_on IExec (I,P,s),P holds
(IExec ((I ';' J),P,s)) . a = (IExec (J,P,(Initialize (IExec (I,P,s))))) . a
let s be 0 -started State of SCMPDS; :: thesis: for I being halt-free Program of
for J being shiftable Program of st I is_closed_on s,P & I is_halting_on s,P & J is_closed_on IExec (I,P,s),P & J is_halting_on IExec (I,P,s),P holds
(IExec ((I ';' J),P,s)) . a = (IExec (J,P,(Initialize (IExec (I,P,s))))) . a
let I be halt-free Program of ; :: thesis: for J being shiftable Program of st I is_closed_on s,P & I is_halting_on s,P & J is_closed_on IExec (I,P,s),P & J is_halting_on IExec (I,P,s),P holds
(IExec ((I ';' J),P,s)) . a = (IExec (J,P,(Initialize (IExec (I,P,s))))) . a
let J be shiftable Program of ; :: thesis: ( I is_closed_on s,P & I is_halting_on s,P & J is_closed_on IExec (I,P,s),P & J is_halting_on IExec (I,P,s),P implies (IExec ((I ';' J),P,s)) . a = (IExec (J,P,(Initialize (IExec (I,P,s))))) . a )
assume that
A1: I is_closed_on s,P and
A2: I is_halting_on s,P and
A3: J is_closed_on IExec (I,P,s),P and
A4: J is_halting_on IExec (I,P,s),P ; :: thesis: (IExec ((I ';' J),P,s)) . a = (IExec (J,P,(Initialize (IExec (I,P,s))))) . a
A5: not a in dom (Start-At (((IC (IExec (J,P,(Initialize (IExec (I,P,s)))))) + (card I)),SCMPDS)) by SCMPDS_4:18;
IExec ((I ';' J),P,s) = IncIC ((IExec (J,P,(Initialize (IExec (I,P,s))))),(card I)) by A1, A2, A3, A4, Th27;
hence (IExec ((I ';' J),P,s)) . a = (IExec (J,P,(Initialize (IExec (I,P,s))))) . a by A5, FUNCT_4:11; :: thesis: verum