let P be Instruction-Sequence of SCMPDS; :: thesis:
set a = GBP ;
let s be 0 -started State of SCMPDS; :: thesis:
let m4, md be Element of NAT ; :: thesis:
assume that
A1: s . GBP = 0 and
A2: s . (intpos 5) > 0 and
A3: s . (intpos 4) = m4 + (s . (intpos 5)) and
A4: m4 >= 8 and
A5: s . (intpos 2) = md and
A6: md >= 8 ; :: thesis:
s . (intpos 5) >= 1 + 0 by A2, INT_1:7;
then reconsider m5 = (s . (intpos 5)) - 1 as Element of NAT by INT_1:3, XREAL_1:48;
A7: s . (intpos 5) = m5 + 1 ;
defpred S₁[ Element of NAT ] means for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . GBP = 0 & t . (intpos 5) = $1 + 1 & t . (intpos 4) = m4 + (t . (intpos 5)) & t . (intpos 2) = md holds
( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . GBP = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 5) = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE 0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) ) );

A8: now

let k be Element of NAT ; :: thesis:
assume A9: S₁[k] ; :: thesis:
thus S₁[k + 1] :: thesis:

proof

set me = m4 + ((k + 1) + 1);
let t be 0 -started State of SCMPDS; :: thesis:
let Q be Instruction-Sequence of SCMPDS; :: thesis:
set b = DataLoc ((t . GBP),5);
assume that
A10: t . GBP = 0 and
A11: t . (intpos 5) = (k + 1) + 1 and
A12: t . (intpos 4) = m4 + (t . (intpos 5)) and
A13: t . (intpos 2) = md ; :: thesis:
set It = IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t);
set IT = Q;
set It0 = Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t));
A23: m4 + ((k + 1) + 1) >= m4 by NAT_1:11;
then A24: m4 + ((k + 1) + 1) >= 8 by A4, XXREAL_0:2;
then A25: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 1) = t . (intpos 1) by A6, A10, A11, A12, A13, Lm8;
A26: ( t . (intpos md) >= t . (intpos (m4 + ((k + 1) + 1))) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 5) = 0 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 4) = t . (intpos 4) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 7) = t . (intpos 5) ) ) by A6, A10, A11, A12, A13, A24, Lm8;
A27: DataLoc ((t . GBP),5) = intpos (0 + 5) by A10, SCMP_GCD:1;

A28: now

let v be 0 -started State of SCMPDS; :: thesis:
let V be Instruction-Sequence of SCMPDS; :: thesis:
assume that
A29: v . (intpos 4) >= m4 + (v . (DataLoc ((t . GBP),5))) and
A30: v . (intpos 2) = t . (intpos 2) and
A31: v . GBP = t . GBP and
A32: v . (DataLoc ((t . GBP),5)) > 0 ; :: thesis:
A33: m4 + (v . (DataLoc ((t . GBP),5))) > m4 + 0 by A32, XREAL_1:6;
then reconsider ME = v . (intpos 4) as Element of NAT by A29, INT_1:3;
v . (intpos 4) >= m4 by A29, A33, XXREAL_0:2;
then A41: ME >= 8 by A4, XXREAL_0:2;
then A42: ( v . (intpos md) < v . (intpos ME) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 5) = (v . (intpos 5)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) = (v . (intpos 4)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 7) = (v . (intpos 5)) - 1 ) ) by A6, A10, A13, A30, A31, Lm8;
thus (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . GBP = v . GBP by A6, A10, A13, A30, A31, A41, Lm8; :: thesis:
thus ( (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_closed_on v,V & (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_halting_on v,V ) by SCMPDS_6:20, SCMPDS_6:21; :: thesis:
A43: ( v . (intpos md) >= v . (intpos ME) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 5) = 0 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) = v . (intpos 4) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 7) = v . (intpos 5) ) ) by A6, A10, A13, A30, A31, A41, Lm8;

hereby :: thesis:

per cases ;

suppose A44: v . (intpos md) < v . (intpos ME) ; :: thesis:

hence (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) by A27, A42, XREAL_1:146; :: thesis:
(m4 + (v . (DataLoc ((t . GBP),5)))) - 1 = m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5))) by A27, A42, A44;
hence (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5))) by A29, A42, A44, XREAL_1:9; :: thesis:

end;

suppose A45: v . (intpos md) >= v . (intpos ME) ; :: thesis:

hence (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) by A6, A10, A13, A27, A30, A31, A32, A41, Lm8; :: thesis:
m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5))) < m4 + (v . (DataLoc ((t . GBP),5))) by A27, A32, A43, A45, XREAL_1:6;
hence (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5))) by A29, A43, A45, XXREAL_0:2; :: thesis:

end;

thus (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) by A6, A10, A13, A30, A31, A41, Lm8; :: thesis:

end;

A46: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 3) = t . (intpos 3) by A6, A10, A11, A12, A13, A24, Lm8;
A47: ( t . (intpos md) < t . (intpos (m4 + ((k + 1) + 1))) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 5) = (t . (intpos 5)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 4) = (t . (intpos 4)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 7) = (t . (intpos 5)) - 1 ) ) by A6, A10, A11, A12, A13, A24, Lm8;
A48: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . GBP = 0 by A6, A10, A11, A12, A13, A24, Lm8;
then B48: (Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))) . GBP = 0 by SCMPDS_5:15;
A49: DataLoc (((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . GBP),5) = intpos (0 + 5) by A48, SCMP_GCD:1;
A50: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 2) = t . (intpos 2) by A6, A10, A11, A12, A13, A24, Lm8;
then B50: (Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))) . (intpos 2) = t . (intpos 2) by SCMPDS_5:15;

per cases ;

suppose A51: t . (intpos md) < t . (intpos (m4 + ((k + 1) + 1))) ; :: thesis:

then A52: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 4) = m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 5)) by A12, A47;
B52: (Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))) . (intpos 4) = m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 5)) by A12, A47, A51, SCMPDS_5:15
.= m4 + ((Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))) . (intpos 5)) by SCMPDS_5:15 ;
YY: (Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))) . (intpos 5) = (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 5) by SCMPDS_5:15;
Y1: (Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))) . (intpos 1) = (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 1) by SCMPDS_5:15;
Y3: (Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))) . (intpos 3) = (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 3) by SCMPDS_5:15;
Y4: (Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))) . (intpos 4) = (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 4) by SCMPDS_5:15;
(IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 5) = k + 1 by A11, A47, A51;
then consider mE being Element of NAT such that
A53: mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,(Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))))) . (intpos 7) and
A54: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,(Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))))) . (intpos 4) = m4 + mE and
A55: mE <= (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 5) and
A56: for i being Element of NAT st m4 + mE 0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,(Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))))) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,(Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))))) . (intpos i) and
A57: ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,(Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))))) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,(Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))))) . (intpos (m4 + mE)) ) by A9, A12, A13, A47, A26, B48, B50, B52, YY;
A58: IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t) = IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,(Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)))) by A11, A12, A27, A28, Th5;
hence (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . GBP = 0 by A9, A11, A12, A13, A47, A26, A52, YY, B50, B48, Y4; :: thesis:
thus (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) by A9, A11, A12, A13, A25, A47, A26, A52, A58, YY, B50, B48, Y1, Y4; :: thesis:
thus (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 5) = 0 by A9, A11, A12, A13, A47, A26, A52, A58, YY, B50, B48, Y4; :: thesis:
thus (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) by A9, A11, A12, A13, A47, A26, A52, A58, YY, B50, B48, Y4; :: thesis:
thus (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) by A9, A11, A12, A13, A46, A47, A26, A52, A58, YY, B50, B48, Y3, Y4; :: thesis:

A59: now

let i be Element of NAT ; :: thesis:
assume A61: i >= 8 ; :: thesis:
hence (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = (Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))) . (intpos i) by A9, A11, A12, A13, A47, A26, A52, A58, YY, B50, B48, Y4
.= (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos i) by SCMPDS_5:15
.= t . (intpos i) by A6, A10, A11, A12, A13, A24, A61, Lm8 ;
:: thesis:

end;

hereby :: thesis:

let i be Element of NAT ; :: thesis:
assume that
A63: m4 + mE < i and
A64: i <= t . (intpos 4) ; :: thesis:

per cases ;

suppose A65: i = t . (intpos 4) ; :: thesis:

(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + ((k + 1) + 1))) = t . (intpos (m4 + ((k + 1) + 1))) by A4, A23, A59, XXREAL_0:2;
hence (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) by A6, A11, A12, A51, A59, A65; :: thesis:

end;

suppose i <> t . (intpos 4) ; :: thesis:

then i < t . (intpos 4) by A64, XXREAL_0:1;
then i + 1 <= t . (intpos 4) by INT_1:7;
then i <= (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 4) by A62, XREAL_1:19;
hence (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) by A58, A56, A63; :: thesis:

end;

thus ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) by A58, A57; :: thesis:

end;

suppose A66: t . (intpos md) >= t . (intpos (m4 + ((k + 1) + 1))) ; :: thesis:

A67: now

X1: (Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))) . GBP = (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . GBP by SCMPDS_5:15;
X2: (Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))) . (DataLoc (((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . GBP),5)) = (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (DataLoc (((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . GBP),5)) by SCMPDS_5:15;
let x be Int_position ; :: thesis:
thus (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . x = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,(Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))))) . x by A11, A12, A27, A28, Th5
.= (Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))) . x by A26, A49, A66, X1, X2, SCMPDS_8:23
.= (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . x by SCMPDS_5:15 ; :: thesis:

end;

A68: now

let i be Element of NAT ; :: thesis:
assume A70: i >= 8 ; :: thesis:
thus (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos i) by A67
.= t . (intpos i) by A6, A10, A11, A12, A13, A24, A70, Lm8 ; :: thesis:

end;

hence for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ; :: thesis:
take ME = (k + 1) + 1; :: thesis:
thus (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) = ME by A11, A26, A66, A67; :: thesis:
thus (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + ME by A11, A12, A26, A66, A67; :: thesis:
thus ME <= t . (intpos 5) by A11; :: thesis:
thus for i being Element of NAT st m4 + ME 0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) by A11, A12; :: thesis:
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + ((k + 1) + 1))) = t . (intpos (m4 + ((k + 1) + 1))) by A4, A23, A68, XXREAL_0:2;
hence ( ME = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + ME)) ) by A6, A66, A68; :: thesis:

end;

A118: S₁[ 0 ]

proof

set me = m4 + 1;
let t be 0 -started State of SCMPDS; :: thesis:
let Q be Instruction-Sequence of SCMPDS; :: thesis:
set It = IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t);
set IT = Q;
set b = DataLoc ((t . GBP),5);
assume that
A80: t . GBP = 0 and
A81: t . (intpos 5) = 0 + 1 and
A82: t . (intpos 4) = m4 + (t . (intpos 5)) and
A83: t . (intpos 2) = md ; :: thesis:
A84: m4 + 1 >= m4 by NAT_1:11;
then A85: m4 + 1 >= 8 by A4, XXREAL_0:2;
then A86: ( t . (intpos md) < t . (intpos (m4 + 1)) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 5) = (t . (intpos 5)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 4) = (t . (intpos 4)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 7) = (t . (intpos 5)) - 1 ) ) by A6, A80, A81, A82, A83, Lm8;
A87: ( t . (intpos md) >= t . (intpos (m4 + 1)) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 5) = 0 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 4) = t . (intpos 4) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 7) = t . (intpos 5) ) ) by A6, A80, A81, A82, A83, A85, Lm8;
A88: DataLoc ((t . GBP),5) = intpos (0 + 5) by A80, SCMP_GCD:1;

A89: now

let v be 0 -started State of SCMPDS; :: thesis:
let V be Instruction-Sequence of SCMPDS; :: thesis:
assume that
A90: v . (intpos 4) >= m4 + (v . (DataLoc ((t . GBP),5))) and
A91: v . (intpos 2) = t . (intpos 2) and
A92: v . GBP = t . GBP and
A93: v . (DataLoc ((t . GBP),5)) > 0 ; :: thesis:
A94: m4 + (v . (DataLoc ((t . GBP),5))) > m4 + 0 by A93, XREAL_1:6;
then reconsider ME = v . (intpos 4) as Element of NAT by A90, INT_1:3;
v . (intpos 4) >= m4 by A90, A94, XXREAL_0:2;
then A102: ME >= 8 by A4, XXREAL_0:2;
then A103: ( v . (intpos md) < v . (intpos ME) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 5) = (v . (intpos 5)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) = (v . (intpos 4)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 7) = (v . (intpos 5)) - 1 ) ) by A6, A80, A83, A91, A92, Lm8;
thus (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . GBP = v . GBP by A6, A80, A83, A91, A92, A102, Lm8; :: thesis:
thus ( (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_closed_on v,V & (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_halting_on v,V ) by SCMPDS_6:20, SCMPDS_6:21; :: thesis:
A104: ( v . (intpos md) >= v . (intpos ME) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 5) = 0 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) = v . (intpos 4) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 7) = v . (intpos 5) ) ) by A6, A80, A83, A91, A92, A102, Lm8;

hereby :: thesis:

per cases ;

suppose A105: v . (intpos md) < v . (intpos ME) ; :: thesis:

hence (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) by A88, A103, XREAL_1:146; :: thesis:
(m4 + (v . (DataLoc ((t . GBP),5)))) - 1 = m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5))) by A88, A103, A105;
hence (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5))) by A90, A103, A105, XREAL_1:9; :: thesis:

end;

suppose A106: v . (intpos md) >= v . (intpos ME) ; :: thesis:

hence (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) by A6, A80, A83, A88, A91, A92, A93, A102, Lm8; :: thesis:
m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5))) < m4 + (v . (DataLoc ((t . GBP),5))) by A88, A93, A104, A106, XREAL_1:6;
hence (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5))) by A90, A104, A106, XXREAL_0:2; :: thesis:

end;

A107: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . GBP = 0 by A6, A80, A81, A82, A83, A85, Lm8;
then A108: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (DataLoc (((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . GBP),5)) = 0 by A81, A86, A87, SCMP_GCD:1;

A109: now

X1: (Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))) . GBP = (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . GBP by SCMPDS_5:15;
X2: (Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))) . (DataLoc (((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . GBP),5)) = (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (DataLoc (((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . GBP),5)) by SCMPDS_5:15;
let x be Int_position ; :: thesis:
thus (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . x = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,(Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))))) . x by A81, A82, A88, A89, Th5
.= (Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))) . x by A108, X1, X2, SCMPDS_8:23
.= (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . x by SCMPDS_5:15 ; :: thesis:

end;

hence (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . GBP = 0 by A107; :: thesis:
(IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 1) = t . (intpos 1) by A6, A80, A81, A82, A83, A85, Lm8;
hence (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) by A109; :: thesis:
thus (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 5) = 0 by A81, A86, A87, A109; :: thesis:
(IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 2) = t . (intpos 2) by A6, A80, A81, A82, A83, A85, Lm8;
hence (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) by A109; :: thesis:
(IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 3) = t . (intpos 3) by A6, A80, A81, A82, A83, A85, Lm8;
hence (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) by A109; :: thesis:

A110: now

let i be Element of NAT ; :: thesis:
assume A111: i >= 8 ; :: thesis:
thus (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos i) by A109
.= t . (intpos i) by A6, A80, A81, A82, A83, A85, A111, Lm8 ; :: thesis:

end;

per cases ;

suppose A114: t . (intpos md) < t . (intpos (m4 + 1)) ; :: thesis:

take ME = 0 ; :: thesis:
thus (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) = ME by A81, A86, A109, A114; :: thesis:
thus (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + ME by A81, A82, A86, A109, A114; :: thesis:
thus ME <= t . (intpos 5) by A81; :: thesis:

hereby :: thesis:

let i be Element of NAT ; :: thesis:
assume that
A115: m4 + ME 0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) by A6, A110, A113, A114; :: thesis:

end;

thus ( ME = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + ME)) ) ; :: thesis:

end;

suppose A117: t . (intpos md) >= t . (intpos (m4 + 1)) ; :: thesis:

take ME = 1; :: thesis:
thus (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) = ME by A81, A87, A109, A117; :: thesis:
thus (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + ME by A81, A82, A87, A109, A117; :: thesis:
thus ME <= t . (intpos 5) by A81; :: thesis:
thus for i being Element of NAT st m4 + ME 0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) by A81, A82; :: thesis:
thus ( ME = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + ME)) ) by A6, A110, A113, A117; :: thesis:

end;

for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A118, A8);
hence ( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . GBP = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 5) = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 3) = s . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos i) = s . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 4) = m4 + mE & mE <= s . (intpos 5) & ( for i being Element of NAT st m4 + mE 0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos (m4 + mE)) ) ) ) by A1, A3, A5, A7; :: thesis: