let s be State of SCMPDS ; :: thesis: for I being halt-free parahalting 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),s) . b = (IExec I,s) . b
let I be halt-free parahalting 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),s) . b = (IExec I,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),s) . b = (IExec I,s) . b
let k1 be Integer; :: thesis: ( s . (DataLoc (s . a),k1) = 0 implies (IExec (if=0 a,k1,I),s) . b = (IExec I,s) . b )
assume A1: s . (DataLoc (s . a),k1) = 0 ; :: thesis: (IExec (if=0 a,k1,I),s) . b = (IExec I,s) . b
A2: not b in dom (Start-At ((card I) + 1),SCMPDS ) by SCMPDS_4:59;
( I is_closed_on s & I is_halting_on s ) by Th34, Th35;
then IExec (if=0 a,k1,I),s = (IExec I,s) +* (Start-At ((card I) + 1),SCMPDS ) by A1, Th64;
hence (IExec (if=0 a,k1,I),s) . b = (IExec I,s) . b by A2, FUNCT_4:12; :: thesis: verum