let P be Instruction-Sequence of SCMPDS; :: thesis: for s being 0 -started State of SCMPDS
for I being halt-free parahalting shiftable Program of SCMPDS
for J being shiftable Program of SCMPDS
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,s)) . b = (IExec (I,P,s)) . b
let s be 0 -started State of SCMPDS; :: thesis: for I being halt-free parahalting shiftable Program of SCMPDS
for J being shiftable Program of SCMPDS
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,s)) . b = (IExec (I,P,s)) . b
let I be halt-free parahalting shiftable Program of SCMPDS; :: thesis: for J being shiftable Program of SCMPDS
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,s)) . b = (IExec (I,P,s)) . b
let J be shiftable Program of SCMPDS; :: 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,s)) . b = (IExec (I,P,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,s)) . b = (IExec (I,P,s)) . b
let k1 be Integer; :: thesis: ( s . (DataLoc ((s . a),k1)) = 0 implies (IExec ((if=0 (a,k1,I,J)),P,s)) . b = (IExec (I,P,s)) . b )
assume A1: s . (DataLoc ((s . a),k1)) = 0 ; :: thesis: (IExec ((if=0 (a,k1,I,J)),P,s)) . b = (IExec (I,P,s)) . b
A2: not b in dom (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by SCMPDS_4:18;
( I is_closed_on s,P & I is_halting_on s,P ) by Th34, Th35;
then IExec ((if=0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by A1, Th54;
hence (IExec ((if=0 (a,k1,I,J)),P,s)) . b = (IExec (I,P,s)) . b by A2, FUNCT_4:11; :: thesis: verum