let P be Instruction-Sequence of SCMPDS; :: thesis: for s being State of SCMPDS
for I being Program of
for J being halt-free parahalting shiftable Program of
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 holds
(IExec ((if=0 (a,k1,I,J)),P,(Initialize s))) . b = (IExec (J,P,(Initialize s))) . b
let s be State of SCMPDS; :: thesis: for I being Program of
for J being halt-free parahalting shiftable Program of
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 holds
(IExec ((if=0 (a,k1,I,J)),P,(Initialize s))) . b = (IExec (J,P,(Initialize s))) . b
let I be Program of ; :: thesis: for J being halt-free parahalting shiftable Program of
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 holds
(IExec ((if=0 (a,k1,I,J)),P,(Initialize s))) . b = (IExec (J,P,(Initialize s))) . b
let J be halt-free parahalting shiftable Program of ; :: thesis: for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 holds
(IExec ((if=0 (a,k1,I,J)),P,(Initialize s))) . b = (IExec (J,P,(Initialize s))) . b
let a, b be Int_position; :: thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) <> 0 holds
(IExec ((if=0 (a,k1,I,J)),P,(Initialize s))) . b = (IExec (J,P,(Initialize s))) . b
let k1 be Integer; :: thesis: ( s . (DataLoc ((s . a),k1)) <> 0 implies (IExec ((if=0 (a,k1,I,J)),P,(Initialize s))) . b = (IExec (J,P,(Initialize s))) . b )
assume A1: s . (DataLoc ((s . a),k1)) <> 0 ; :: thesis: (IExec ((if=0 (a,k1,I,J)),P,(Initialize s))) . b = (IExec (J,P,(Initialize s))) . b
A2: not b in dom (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by SCMPDS_4:18;
( J is_closed_on s,P & J is_halting_on s,P ) by Th11, Th12;
then IExec ((if=0 (a,k1,I,J)),P,(Initialize s)) = (IExec (J,P,(Initialize s))) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by A1, Th32;
hence (IExec ((if=0 (a,k1,I,J)),P,(Initialize s))) . b = (IExec (J,P,(Initialize s))) . b by A2, FUNCT_4:11; :: thesis: verum