let P be Instruction-Sequence of SCMPDS; :: thesis:
set b1 = DataLoc (0,1);
reconsider Pt = Partition as halt-free shiftable Program of ;
let s be 0 -started State of SCMPDS; :: thesis:
let p0, n be Nat; :: thesis:
A1: Initialize s = s by MEMSTR_0:44;
set pn = p0 + n;
set l1 = (GBP,2) := (SBP,((p0 + n) + 1));
set l2 = SubFrom (GBP,2,SBP,(p0 + n));
set l3 = (GBP,2) := (SBP,(p0 + n));
set l4 = (GBP,4) := (SBP,((p0 + n) + 1));
set l5 = (SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1));
set l6 = (SBP,((p0 + n) + 1)) := (GBP,4);
set l7 = (SBP,((p0 + n) + 2)) := (GBP,4);
set l8 = AddTo (SBP,((p0 + n) + 1),(- 1));
set l9 = AddTo (SBP,((p0 + n) + 2),1);
set l0 = AddTo (GBP,1,2);
set lb = Load (AddTo (GBP,1,(- 2)));
set L5 = ((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2));
set TR = ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)));
set IF4 = if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))));
set WB4 = (((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))));
set WH4 = while>0 (GBP,1,((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))));
set m1 = GBP := 0;
set m2 = SBP := 1;
set m3 = (SBP,(p0 + n)) := (p0 + 1);
set m4 = (SBP,((p0 + n) + 1)) := (p0 + n);
set M4 = (((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n));
set s1 = IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s);
set a = GBP ;
set P1 = P;
A2: 7 + n >= 7 by NAT_1:11;
A3: card (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))) = (card ((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1)))) + 1 by SCMP_GCD:4
.= ((card (((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1))))) + 1) + 1 by SCMP_GCD:4
.= (4 + 1) + 1 by Th4 ;
A4: card ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))) = (card (((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n))))) + (card (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))) by AFINSQ_1:17
.= 2 + (card (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))) by SCMP_GCD:5
.= 2 + (((card (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))))) + (card (Load (AddTo (GBP,1,(- 2)))))) + 2) by SCMPDS_6:65
.= 2 + (((card (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))))) + 1) + 2) by COMPOS_1:54
.= 2 + ((((card ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt)) + (card (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))))) + 1) + 2) by AFINSQ_1:17
.= 2 + (((((card (((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1))))) + (card Pt)) + (card (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))))) + 1) + 2) by AFINSQ_1:17
.= 2 + ((((2 + 38) + (card (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))))) + 1) + 2) by Th9, SCMP_GCD:5
.= 51 by A3 ;
thus card (QuickSort (n,p0)) = (card ((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n)))) + (card (while>0 (GBP,1,((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))))))) by AFINSQ_1:17
.= 4 + (card (while>0 (GBP,1,((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))))))) by Th4
.= 4 + (51 + 2) by A4, SCMPDS_8:17
.= 57 ; :: thesis:
assume A5: p0 >= 7 ; :: thesis:
then p0 + n >= 7 + n by XREAL_1:6;
then A6: p0 + n >= 7 by A2, XXREAL_0:2;
A7: for t being State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for m1 being Nat st t . GBP = 0 & t . SBP = m1 holds
( (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,(Initialize t))) . (intpos 2) = t . (intpos (m1 + (p0 + n))) & (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,(Initialize t))) . (intpos 4) = t . (intpos ((m1 + (p0 + n)) + 1)) & ( for i being Nat st i <> 2 & i <> 4 holds
(IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,(Initialize t))) . (intpos i) = t . (intpos i) ) )

let t be State of SCMPDS; :: thesis:
let Q be Instruction-Sequence of SCMPDS; :: thesis:
let m1 be Nat; :: thesis:
set t0 = Initialize t;
set t2 = IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,(Initialize t));
set t3 = Exec (((GBP,2) := (SBP,(p0 + n))),(Initialize t));
set mp = m1 + (p0 + n);
set Q0 = Q;
assume that
A8: t . GBP = 0 and
A9: t . SBP = m1 ; :: thesis:
A10: (Initialize t) . GBP = 0 by A8, SCMPDS_5:15;
then A11: DataLoc (((Initialize t) . GBP),2) = intpos (0 + 2) by SCMP_GCD:1;
then (Exec (((GBP,2) := (SBP,(p0 + n))),(Initialize t))) . GBP = 0 by A10, AMI_3:10, SCMPDS_2:47;
then A12: DataLoc (((Exec (((GBP,2) := (SBP,(p0 + n))),(Initialize t))) . GBP),4) = intpos (0 + 4) by SCMP_GCD:1;
A13: (Initialize t) . SBP = m1 by A9, SCMPDS_5:15;
then A14: (Exec (((GBP,2) := (SBP,(p0 + n))),(Initialize t))) . SBP = m1 by A11, AMI_3:10, SCMPDS_2:47;
m1 + (p0 + n) >= 0 + 7 by A6, XREAL_1:7;
then (m1 + (p0 + n)) + 1 >= 7 + 1 by XREAL_1:6;
then (m1 + (p0 + n)) + 1 > 2 by XXREAL_0:2;
then A15: (Exec (((GBP,2) := (SBP,(p0 + n))),(Initialize t))) . (intpos ((m1 + (p0 + n)) + 1)) = (Initialize t) . (intpos ((m1 + (p0 + n)) + 1)) by A11, AMI_3:10, SCMPDS_2:47
.= t . (intpos ((m1 + (p0 + n)) + 1)) by SCMPDS_5:15 ;
A16: (Exec (((GBP,2) := (SBP,(p0 + n))),(Initialize t))) . (intpos 2) = (Initialize t) . (DataLoc (((Initialize t) . SBP),(p0 + n))) by A11, SCMPDS_2:47
.= (Initialize t) . (intpos (m1 + (p0 + n))) by A13, SCMP_GCD:1
.= t . (intpos (m1 + (p0 + n))) by SCMPDS_5:15 ;
thus (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,(Initialize t))) . (intpos 2) = (Exec (((GBP,4) := (SBP,((p0 + n) + 1))),(Exec (((GBP,2) := (SBP,(p0 + n))),(Initialize t))))) . (intpos 2) by SCMPDS_5:42
.= t . (intpos (m1 + (p0 + n))) by A16, A12, AMI_3:10, SCMPDS_2:47 ; :: thesis:
thus (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,(Initialize t))) . (intpos 4) = (Exec (((GBP,4) := (SBP,((p0 + n) + 1))),(Exec (((GBP,2) := (SBP,(p0 + n))),(Initialize t))))) . (intpos 4) by SCMPDS_5:42
.= (Exec (((GBP,2) := (SBP,(p0 + n))),(Initialize t))) . (DataLoc (((Exec (((GBP,2) := (SBP,(p0 + n))),(Initialize t))) . SBP),((p0 + n) + 1))) by A12, SCMPDS_2:47
.= t . (intpos ((m1 + (p0 + n)) + 1)) by A14, A15, SCMP_GCD:1 ; :: thesis:

hereby :: thesis:

let i be Nat; :: thesis:
assume that
A17: i <> 2 and
A18: i <> 4 ; :: thesis:
thus (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,(Initialize t))) . (intpos i) = (Exec (((GBP,4) := (SBP,((p0 + n) + 1))),(Exec (((GBP,2) := (SBP,(p0 + n))),(Initialize t))))) . (intpos i) by SCMPDS_5:42
.= (Exec (((GBP,2) := (SBP,(p0 + n))),(Initialize t))) . (intpos i) by A12, A18, AMI_3:10, SCMPDS_2:47
.= (Initialize t) . (intpos i) by A11, A17, AMI_3:10, SCMPDS_2:47
.= t . (intpos i) by SCMPDS_5:15 ; :: thesis:

end;

A19: for t being State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for m, m1 being Nat st t . GBP = 0 & t . SBP = m1 & m1 = m + 1 holds
( (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . GBP = 0 & (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . SBP = m1 + 2 & (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos (m1 + (p0 + n))) = t . (intpos (m1 + (p0 + n))) & (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 1)) = (t . (intpos 4)) - 1 & (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 2)) = (t . (intpos 4)) + 1 & (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 3)) = t . (intpos ((m1 + (p0 + n)) + 1)) & ( for i being Nat st i >= 8 & i < m1 + (p0 + n) holds
(IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos i) = t . (intpos i) ) )

proof

let t be State of SCMPDS; :: thesis:
let Q be Instruction-Sequence of SCMPDS; :: thesis:
let m, m1 be Nat; :: thesis:
assume that
A20: t . GBP = 0 and
A21: t . SBP = m1 and
A22: m1 = m + 1 ; :: thesis:
set t0 = Initialize t;
set t1 = IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t));
set Q0 = Q;
set t2 = IExec (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))),Q,(Initialize t));
set t3 = IExec ((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))),Q,(Initialize t));
set t4 = IExec (((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))),Q,(Initialize t));
set t5 = IExec ((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))),Q,(Initialize t));
set t6 = Exec (((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))),(Initialize t));
set mp = m1 + (p0 + n);
A23: (m1 + (p0 + n)) + 3 <> (m1 + (p0 + n)) + 1 ;
m1 >= 1 by A22, NAT_1:11;
then A24: m1 + (p0 + n) >= 1 + 7 by A6, XREAL_1:7;
then A25: (m1 + (p0 + n)) + 1 >= 8 + 1 by XREAL_1:6;
then A26: (m1 + (p0 + n)) + 1 > 4 by XXREAL_0:2;
A27: (m1 + (p0 + n)) + 1 > 1 by A25, XXREAL_0:2;
A28: (m1 + (p0 + n)) + 0 <> (m1 + (p0 + n)) + 2 ;
A29: m1 + (p0 + n) > 1 by A24, XXREAL_0:2;
A30: (m1 + (p0 + n)) + 1 <> (m1 + (p0 + n)) + 2 ;
A31: (Initialize t) . SBP = m1 by A21, SCMPDS_5:15;
then A32: DataLoc (((Initialize t) . SBP),((p0 + n) + 3)) = intpos ((m1 + (p0 + n)) + 3) by SCMP_GCD:1;

A33: now :: thesis:

let i be Nat; :: thesis:
assume i <> (m1 + (p0 + n)) + 3 ; :: thesis:
hence (Exec (((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))),(Initialize t))) . (intpos i) = (Initialize t) . (intpos i) by A32, AMI_3:10, SCMPDS_2:47
.= t . (intpos i) by SCMPDS_5:15 ;
:: thesis:

end;

(m1 + (p0 + n)) + 0 <> (m1 + (p0 + n)) + 3 ;
then A34: (Exec (((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))),(Initialize t))) . (intpos (m1 + (p0 + n))) = t . (intpos (m1 + (p0 + n))) by A33;
A35: (m1 + (p0 + n)) + 1 <> (m1 + (p0 + n)) + 2 ;
A36: (m1 + (p0 + n)) + 3 <> (m1 + (p0 + n)) + 2 ;
A37: (m1 + (p0 + n)) + 3 >= 8 + 3 by A24, XREAL_1:6;
then A38: (m1 + (p0 + n)) + 3 > 1 by XXREAL_0:2;
(m1 + (p0 + n)) + 3 > 1 by A37, XXREAL_0:2;
then A39: (Exec (((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))),(Initialize t))) . SBP = m1 by A21, A33;
then A40: DataLoc (((Exec (((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))),(Initialize t))) . SBP),((p0 + n) + 1)) = intpos ((m1 + (p0 + n)) + 1) by SCMP_GCD:1;

A41: now :: thesis:

let i be Nat; :: thesis:
assume A42: i <> (m1 + (p0 + n)) + 1 ; :: thesis:
thus (IExec ((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))),Q,(Initialize t))) . (intpos i) = (Exec (((SBP,((p0 + n) + 1)) := (GBP,4)),(Exec (((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))),(Initialize t))))) . (intpos i) by SCMPDS_5:42
.= (Exec (((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))),(Initialize t))) . (intpos i) by A40, A42, AMI_3:10, SCMPDS_2:47 ; :: thesis:

end;

(m1 + (p0 + n)) + 1 > 1 by A25, XXREAL_0:2;
then A43: (IExec ((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))),Q,(Initialize t))) . SBP = m1 by A39, A41;
then A44: DataLoc (((IExec ((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))),Q,(Initialize t))) . SBP),((p0 + n) + 2)) = intpos ((m1 + (p0 + n)) + 2) by SCMP_GCD:1;

A45: now :: thesis:

let i be Nat; :: thesis:
assume A46: i <> (m1 + (p0 + n)) + 2 ; :: thesis:
thus (IExec (((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))),Q,(Initialize t))) . (intpos i) = (Exec (((SBP,((p0 + n) + 2)) := (GBP,4)),(IExec ((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))),Q,(Initialize t))))) . (intpos i) by SCMPDS_5:41
.= (IExec ((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))),Q,(Initialize t))) . (intpos i) by A44, A46, AMI_3:10, SCMPDS_2:47 ; :: thesis:

end;

(Exec (((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))),(Initialize t))) . (intpos ((m1 + (p0 + n)) + 3)) = (Initialize t) . (DataLoc (((Initialize t) . SBP),((p0 + n) + 1))) by A32, SCMPDS_2:47
.= (Initialize t) . (intpos (m1 + ((p0 + n) + 1))) by A31, SCMP_GCD:1
.= t . (intpos ((m1 + (p0 + n)) + 1)) by SCMPDS_5:15 ;
then (IExec ((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 3)) = t . (intpos ((m1 + (p0 + n)) + 1)) by A41, A23;
then A47: (IExec (((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 3)) = t . (intpos ((m1 + (p0 + n)) + 1)) by A45, A36;
A48: (m1 + (p0 + n)) + 3 > 4 by A37, XXREAL_0:2;
then A49: (Exec (((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))),(Initialize t))) . (intpos 4) = t . (intpos 4) by A33;
A50: (m1 + (p0 + n)) + 2 >= 8 + 2 by A24, XREAL_1:6;
then A51: (m1 + (p0 + n)) + 2 > 1 by XXREAL_0:2;
(m1 + (p0 + n)) + 2 > 1 by A50, XXREAL_0:2;
then A52: (IExec (((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))),Q,(Initialize t))) . SBP = m1 by A43, A45;
then A53: DataLoc (((IExec (((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))),Q,(Initialize t))) . SBP),((p0 + n) + 1)) = intpos ((m1 + (p0 + n)) + 1) by SCMP_GCD:1;

A54: now :: thesis:

let i be Nat; :: thesis:
assume A55: i <> (m1 + (p0 + n)) + 1 ; :: thesis:
thus (IExec ((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))),Q,(Initialize t))) . (intpos i) = (Exec ((AddTo (SBP,((p0 + n) + 1),(- 1))),(IExec (((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))),Q,(Initialize t))))) . (intpos i) by SCMPDS_5:41
.= (IExec (((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))),Q,(Initialize t))) . (intpos i) by A53, A55, AMI_3:10, SCMPDS_2:48 ; :: thesis:

end;

(m1 + (p0 + n)) + 1 > 1 by A25, XXREAL_0:2;
then A56: (IExec ((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))),Q,(Initialize t))) . SBP = m1 by A52, A54;
then A57: DataLoc (((IExec ((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))),Q,(Initialize t))) . SBP),((p0 + n) + 2)) = intpos ((m1 + (p0 + n)) + 2) by SCMP_GCD:1;

A58: now :: thesis:

let i be Nat; :: thesis:
assume A59: i <> (m1 + (p0 + n)) + 2 ; :: thesis:
thus (IExec (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))),Q,(Initialize t))) . (intpos i) = (Exec ((AddTo (SBP,((p0 + n) + 2),1)),(IExec ((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))),Q,(Initialize t))))) . (intpos i) by SCMPDS_5:41
.= (IExec ((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))),Q,(Initialize t))) . (intpos i) by A57, A59, AMI_3:10, SCMPDS_2:48 ; :: thesis:

end;

A60: (Exec (((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))),(Initialize t))) . GBP = 0 by A20, A33;
then A61: (IExec ((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))),Q,(Initialize t))) . GBP = 0 by A41;
then (IExec (((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))),Q,(Initialize t))) . GBP = 0 by A45;
then (IExec ((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))),Q,(Initialize t))) . GBP = 0 by A54;
then A62: (IExec (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))),Q,(Initialize t))) . GBP = 0 by A58;
then A63: DataLoc (((IExec (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))),Q,(Initialize t))) . GBP),1) = intpos (0 + 1) by SCMP_GCD:1;
(m1 + (p0 + n)) + 2 > 1 by A50, XXREAL_0:2;
then A64: (IExec (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))),Q,(Initialize t))) . SBP = m1 by A56, A58;

A65: now :: thesis:

let i be Nat; :: thesis:
assume i <> 1 ; :: thesis:
then A66: intpos i <> DataLoc (((IExec (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))),Q,(Initialize t))) . GBP),1) by A62, AMI_3:10, SCMP_GCD:1;
thus (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos i) = (Exec ((AddTo (GBP,1,2)),(IExec (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))),Q,(Initialize t))))) . (intpos i) by SCMPDS_5:41
.= (IExec (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))),Q,(Initialize t))) . (intpos i) by A66, SCMPDS_2:48 ; :: thesis:

end;

hence (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . GBP = 0 by A62; :: thesis:
A67: (m1 + (p0 + n)) + 0 <> (m1 + (p0 + n)) + 2 ;
(m1 + (p0 + n)) + 0 <> (m1 + (p0 + n)) + 1 ;
then (IExec ((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))),Q,(Initialize t))) . (intpos (m1 + (p0 + n))) = t . (intpos (m1 + (p0 + n))) by A34, A41;
then A68: (IExec (((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))),Q,(Initialize t))) . (intpos (m1 + (p0 + n))) = t . (intpos (m1 + (p0 + n))) by A45, A67;
thus (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . SBP = (Exec ((AddTo (GBP,1,2)),(IExec (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))),Q,(Initialize t))))) . SBP by SCMPDS_5:41
.= m1 + 2 by A64, A63, SCMPDS_2:48 ; :: thesis:
(m1 + (p0 + n)) + 0 <> (m1 + (p0 + n)) + 1 ;
then (IExec ((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))),Q,(Initialize t))) . (intpos (m1 + (p0 + n))) = t . (intpos (m1 + (p0 + n))) by A68, A54;
then (IExec (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))),Q,(Initialize t))) . (intpos (m1 + (p0 + n))) = t . (intpos (m1 + (p0 + n))) by A58, A28;
hence (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos (m1 + (p0 + n))) = t . (intpos (m1 + (p0 + n))) by A65, A29; :: thesis:
A69: (m1 + (p0 + n)) + 1 <> (m1 + (p0 + n)) + 2 ;
(IExec (((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 2)) = (Exec (((SBP,((p0 + n) + 2)) := (GBP,4)),(IExec ((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))),Q,(Initialize t))))) . (intpos ((m1 + (p0 + n)) + 2)) by SCMPDS_5:41
.= (IExec ((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))),Q,(Initialize t))) . (DataLoc (((IExec ((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))),Q,(Initialize t))) . GBP),4)) by A44, SCMPDS_2:47
.= (IExec ((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))),Q,(Initialize t))) . (intpos (0 + 4)) by A61, SCMP_GCD:1
.= t . (intpos 4) by A49, A41, A26 ;
then A70: (IExec ((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 2)) = t . (intpos 4) by A54, A30;
(IExec ((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 1)) = (Exec (((SBP,((p0 + n) + 1)) := (GBP,4)),(Exec (((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))),(Initialize t))))) . (intpos ((m1 + (p0 + n)) + 1)) by SCMPDS_5:42
.= (Exec (((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))),(Initialize t))) . (DataLoc (((Exec (((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))),(Initialize t))) . GBP),4)) by A40, SCMPDS_2:47
.= (Exec (((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))),(Initialize t))) . (intpos (0 + 4)) by A60, SCMP_GCD:1
.= t . (intpos 4) by A33, A48 ;
then A71: (IExec (((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 1)) = t . (intpos 4) by A45, A69;
(IExec ((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 1)) = (Exec ((AddTo (SBP,((p0 + n) + 1),(- 1))),(IExec (((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))),Q,(Initialize t))))) . (intpos ((m1 + (p0 + n)) + 1)) by SCMPDS_5:41
.= (t . (intpos 4)) + (- 1) by A71, A53, SCMPDS_2:48
.= (t . (intpos 4)) - 1 ;
then (IExec (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 1)) = (t . (intpos 4)) - 1 by A58, A35;
hence (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 1)) = (t . (intpos 4)) - 1 by A65, A27; :: thesis:
(IExec (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 2)) = (Exec ((AddTo (SBP,((p0 + n) + 2),1)),(IExec ((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))),Q,(Initialize t))))) . (intpos ((m1 + (p0 + n)) + 2)) by SCMPDS_5:41
.= (t . (intpos 4)) + 1 by A70, A57, SCMPDS_2:48 ;
hence (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 2)) = (t . (intpos 4)) + 1 by A65, A51; :: thesis:
A72: (m1 + (p0 + n)) + 3 <> (m1 + (p0 + n)) + 2 ;
(m1 + (p0 + n)) + 3 <> (m1 + (p0 + n)) + 1 ;
then (IExec ((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 3)) = t . (intpos ((m1 + (p0 + n)) + 1)) by A47, A54;
then (IExec (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 3)) = t . (intpos ((m1 + (p0 + n)) + 1)) by A58, A72;
hence (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 3)) = t . (intpos ((m1 + (p0 + n)) + 1)) by A65, A38; :: thesis:

hereby :: thesis:

A73: m1 + (p0 + n) <= (m1 + (p0 + n)) + 3 by NAT_1:11;
A74: m1 + (p0 + n) <= (m1 + (p0 + n)) + 2 by NAT_1:11;
A75: m1 + (p0 + n) <= (m1 + (p0 + n)) + 1 by NAT_1:11;
let i be Nat; :: thesis:
assume that
A76: i >= 8 and
A77: i < m1 + (p0 + n) ; :: thesis:
i > 1 by A76, XXREAL_0:2;
hence (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos i) = (IExec (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))),Q,(Initialize t))) . (intpos i) by A65
.= (IExec ((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))),Q,(Initialize t))) . (intpos i) by A58, A77, A74
.= (IExec (((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))),Q,(Initialize t))) . (intpos i) by A54, A77, A75
.= (IExec ((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))),Q,(Initialize t))) . (intpos i) by A45, A77, A74
.= (Exec (((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))),(Initialize t))) . (intpos i) by A41, A77, A75
.= t . (intpos i) by A33, A77, A73 ;
:: thesis:

end;

A78: for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for m, m1, md being Nat
for n4 being Integer
for f1, f2 being FinSequence of INT st t . GBP = 0 & t . SBP = m1 & m1 = m + 1 & t . (intpos (m1 + (p0 + n))) = md & md >= p0 + 1 & n4 = t . (intpos ((m1 + (p0 + n)) + 1)) & n4 - md > 0 & n4 <= p0 + n & f1 is_FinSequence_on t,p0 & len f1 = n & f2 is_FinSequence_on IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t),p0 & len f2 = n holds
( (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . GBP = 0 & (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos 1) = m1 + 2 & md = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos (m1 + (p0 + n))) & n4 = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 3)) & ( for j being Nat st 1 <= j & j < m1 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((p0 + n) + j)) = t . (intpos ((p0 + n) + j)) ) & f1,f2 are_fiberwise_equipotent & ex m4 being Nat st
( md <= m4 & m4 <= n4 & m4 - 1 = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 1)) & m4 + 1 = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 2)) & ( for i being Nat st md <= i & i < m4 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos m4) >= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos i) ) & ( for i being Nat st m4 < i & i <= n4 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos m4) <= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos i) ) & ( for j being Nat st ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos j) = t . (intpos j) ) ) )

proof

let t be 0 -started State of SCMPDS; :: thesis:
let Q be Instruction-Sequence of SCMPDS; :: thesis:
let m, m1, md be Nat; :: thesis:
let n4 be Integer; :: thesis:
let f1, f2 be FinSequence of INT ; :: thesis:
A79: Initialize t = t by MEMSTR_0:44;
set t1 = IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t);
set lPt = (((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt;
set t2 = IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t);
set Q2 = Q;
set t4 = IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t);
set mp = m1 + (p0 + n);
set Q4 = Q;
A80: (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . GBP = (Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))) . GBP by SCMPDS_5:15;
A81: (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos 2) = (Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))) . (intpos 2) by SCMPDS_5:15;
A82: (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos 4) = (Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))) . (intpos 4) by SCMPDS_5:15;
A83: (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos 1) = (Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))) . (intpos 1) by SCMPDS_5:15;
set tp = IExec (Pt,Q,(Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))));
assume that
A84: t . GBP = 0 and
A85: t . SBP = m1 and
A86: m1 = m + 1 and
A87: t . (intpos (m1 + (p0 + n))) = md and
A88: md >= p0 + 1 and
A89: n4 = t . (intpos ((m1 + (p0 + n)) + 1)) and
A90: n4 - md > 0 and
A91: n4 <= p0 + n ; :: thesis:
A92: (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . GBP = 0 by A7, A84, A85, A79;
assume that
A93: f1 is_FinSequence_on t,p0 and
A94: len f1 = n ; :: thesis:
A95: p0 + 1 >= 7 + 1 by A5, XREAL_1:6;

A96: now :: thesis:

let i be Nat; :: thesis:
assume that
A97: 1 <= i and
A98: i <= len f1 ; :: thesis:
p0 + i >= p0 + 1 by A97, XREAL_1:6;
then A99: p0 + i >= 8 by A95, XXREAL_0:2;
then A100: p0 + i > 2 by XXREAL_0:2;
A101: p0 + i > 4 by A99, XXREAL_0:2;
thus f1 . i = t . (intpos (p0 + i)) by A93, A97, A98
.= (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos (p0 + i)) by A7, A84, A85, A100, A101, A79 ; :: thesis:

end;

A102: (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos 4) = n4 by A7, A84, A85, A89, A79;
A103: (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos 4) = t . (intpos ((m1 + (p0 + n)) + 1)) by A7, A84, A85, A79;
A104: (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos 2) = t . (intpos (m1 + (p0 + n))) by A7, A84, A85, A79;
then Pt is_closed_on Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)),Q by A5, A87, A88, A92, Th10, A80, A81;
then A105: Pt is_closed_on IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t),Q by SCMPDS_6:125;
Pt is_halting_on Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)),Q by A5, A87, A88, A104, A92, Th10, A80, A81;
then A106: Pt is_halting_on IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t),Q by SCMPDS_6:126;
then A107: (((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt is_closed_on t,Q by A105, A79, SCPISORT:9;
assume that
A108: f2 is_FinSequence_on IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t),p0 and
A109: len f2 = n ; :: thesis:
consider f4 being FinSequence of INT such that
A110: len f4 = n and
A111: for i being Nat st 1 <= i & i <= len f4 holds
f4 . i = (IExec (Pt,Q,(Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))))) . (intpos (p0 + i)) by SCPISORT:1;
A112: f1 is_FinSequence_on Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)),p0

proof

let i be Nat; :: according to SCPISORT:def 1 :: thesis:
assume ( 1 <= i & i <= len f1 ) ; :: thesis:
then f1 . i = (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos (p0 + i)) by A96;
hence f1 . i = (Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))) . (intpos (p0 + i)) by SCMPDS_5:15; :: thesis:

end;

A113: f4 is_FinSequence_on IExec (Pt,Q,(Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)))),p0 by A111;
then consider m4 being Nat such that
A114: m4 = (IExec (Pt,Q,(Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))))) . (intpos 4) and
A115: md <= m4 and
A116: m4 <= (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos 4) and
A117: for i being Nat st md <= i & i < m4 holds
(IExec (Pt,Q,(Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))))) . (intpos m4) >= (IExec (Pt,Q,(Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))))) . (intpos i) and
A118: for i being Nat st m4 < i & i <= (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos 4) holds
(IExec (Pt,Q,(Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))))) . (intpos m4) <= (IExec (Pt,Q,(Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))))) . (intpos i) and
A119: for i being Nat st i >= p0 + 1 & ( i < (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos 2) or i > (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos 4) ) holds
(IExec (Pt,Q,(Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))))) . (intpos i) = (Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))) . (intpos i) by A5, A87, A88, A89, A90, A91, A94, A104, A103, A92, A112, A110, Th11, A80, A81, A82;
(IExec (Pt,Q,(Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))))) . GBP = 0 by A5, A87, A88, A89, A90, A91, A94, A104, A103, A92, A112, A110, A113, Th11, A80, A81, A82;
then A120: (IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t)) . GBP = 0 by A105, A106, SCPISORT:7;
A121: (((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt is_halting_on t,Q by A105, A106, A79, SCPISORT:9;
(IExec (Pt,Q,(Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))))) . (intpos 1) = (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos 1) by A5, A87, A88, A89, A90, A91, A94, A104, A103, A92, A112, A110, A113, Th11, A80, A81, A82, A83;
then A122: (IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t)) . SBP = (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos 1) by A105, A106, SCPISORT:7
.= m1 by A7, A84, A85, A79 ;
then A123: (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize (IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t))))) . (intpos (m1 + (p0 + n))) = (IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t)) . (intpos (m1 + (p0 + n))) by A19, A86, A120;
(IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize (IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t))))) . GBP = 0 by A19, A86, A120, A122;
hence (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . GBP = 0 by A107, A121, SCPISORT:6; :: thesis:
(IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize (IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t))))) . SBP = m1 + 2 by A19, A86, A120, A122;
hence (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos 1) = m1 + 2 by A107, A121, SCPISORT:6; :: thesis:
A124: (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize (IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t))))) . (intpos ((m1 + (p0 + n)) + 3)) = (IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t)) . (intpos ((m1 + (p0 + n)) + 1)) by A19, A86, A120, A122;
A125: (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize (IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t))))) . (intpos ((m1 + (p0 + n)) + 2)) = ((IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t)) . (intpos 4)) + 1 by A19, A86, A120, A122;
A126: (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize (IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t))))) . (intpos ((m1 + (p0 + n)) + 1)) = ((IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t)) . (intpos 4)) - 1 by A19, A86, A120, A122;
A127: 1 + (p0 + n) > p0 + n by XREAL_1:29;
A128: m1 >= 1 by A86, NAT_1:11;
then m1 + (p0 + n) >= 1 + (p0 + n) by XREAL_1:6;
then A129: m1 + (p0 + n) > p0 + n by A127, XXREAL_0:2;
then A130: m1 + (p0 + n) > (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos 4) by A91, A102, XXREAL_0:2;
then A131: m4 < m1 + (p0 + n) by A116, XXREAL_0:2;
A132: dom f2 = Seg n by A109, FINSEQ_1:def 3;

A133: now :: thesis:

let i be Nat; :: thesis:
reconsider a = i as Nat ;
assume A134: i in dom f2 ; :: thesis:
then A135: 1 <= i by A132, FINSEQ_1:1;
then p0 + i >= p0 + 1 by XREAL_1:6;
then A136: p0 + i >= 8 by A95, XXREAL_0:2;
A137: i <= n by A132, A134, FINSEQ_1:1;
then p0 + i <= p0 + n by XREAL_1:6;
then A138: p0 + i < m1 + (p0 + n) by A129, XXREAL_0:2;
thus f2 . i = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos (p0 + a)) by A108, A109, A135, A137
.= (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize (IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t))))) . (intpos (p0 + a)) by A107, A121, SCPISORT:6
.= (IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t)) . (intpos (p0 + a)) by A19, A86, A120, A122, A136, A138
.= (IExec (Pt,Q,(Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))))) . (intpos (p0 + a)) by A105, A106, SCPISORT:7
.= f4 . i by A110, A111, A135, A137 ; :: thesis:

end;

A139: m1 + (p0 + n) >= 1 + 7 by A6, A128, XREAL_1:7;
then A140: m1 + (p0 + n) > 4 by XXREAL_0:2;
A141: (m1 + (p0 + n)) + 1 > m1 + (p0 + n) by XREAL_1:29;
then A142: (m1 + (p0 + n)) + 1 > (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos 4) by A130, XXREAL_0:2;
A143: p0 + n >= p0 by NAT_1:11;
then A144: m1 + (p0 + n) >= p0 + 1 by A128, XREAL_1:7;
then A145: (m1 + (p0 + n)) + 1 >= p0 + 1 by A141, XXREAL_0:2;
A146: (m1 + (p0 + n)) + 1 >= 8 + 1 by A139, XREAL_1:6;
then A147: (m1 + (p0 + n)) + 1 > 4 by XXREAL_0:2;
A148: (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos (m1 + (p0 + n))) = (Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))) . (intpos (m1 + (p0 + n))) by SCMPDS_5:15;
A149: (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 1)) = (Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))) . (intpos ((m1 + (p0 + n)) + 1)) by SCMPDS_5:15;
m1 + (p0 + n) > 2 by A139, XXREAL_0:2;
hence md = (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos (m1 + (p0 + n))) by A7, A84, A85, A87, A140, A79
.= (IExec (Pt,Q,(Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))))) . (intpos (m1 + (p0 + n))) by A119, A144, A130, A148
.= (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize (IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t))))) . (intpos (m1 + (p0 + n))) by A105, A106, A123, SCPISORT:7
.= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos (m1 + (p0 + n))) by A107, A121, SCPISORT:6 ;
:: thesis:
(m1 + (p0 + n)) + 1 > 2 by A146, XXREAL_0:2;
hence n4 = (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 1)) by A7, A84, A85, A89, A147, A79
.= (IExec (Pt,Q,(Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))))) . (intpos ((m1 + (p0 + n)) + 1)) by A119, A145, A142, A149
.= (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize (IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t))))) . (intpos ((m1 + (p0 + n)) + 3)) by A105, A106, A124, SCPISORT:7
.= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 3)) by A107, A121, SCPISORT:6 ;
:: thesis:

hereby :: thesis:

let j be Nat; :: thesis:
assume that
A150: 1 <= j and
A151: j < m1 ; :: thesis:
A152: (p0 + n) + j < m1 + (p0 + n) by A151, XREAL_1:6;
(p0 + n) + j >= (p0 + n) + 1 by A150, XREAL_1:6;
then (p0 + n) + j > p0 + n by A127, XXREAL_0:2;
then A153: (p0 + n) + j > (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos 4) by A91, A102, XXREAL_0:2;
A154: (p0 + n) + j >= p0 + 1 by A143, A150, XREAL_1:7;
A155: (p0 + n) + j >= 1 + 7 by A6, A150, XREAL_1:7;
then A156: (p0 + n) + j > 2 by XXREAL_0:2;
A157: (p0 + n) + j > 4 by A155, XXREAL_0:2;
A158: (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos ((p0 + n) + j)) = (Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))) . (intpos ((p0 + n) + j)) by SCMPDS_5:15;
thus (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((p0 + n) + j)) = (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize (IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t))))) . (intpos ((p0 + n) + j)) by A107, A121, SCPISORT:6
.= (IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t)) . (intpos ((p0 + n) + j)) by A19, A86, A120, A122, A155, A152
.= (IExec (Pt,Q,(Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))))) . (intpos ((p0 + n) + j)) by A105, A106, SCPISORT:7
.= (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos ((p0 + n) + j)) by A119, A154, A153, A158
.= t . (intpos ((p0 + n) + j)) by A7, A84, A85, A156, A157, A79 ; :: thesis:

end;

f1,f4 are_fiberwise_equipotent by A5, A87, A88, A89, A90, A91, A94, A104, A103, A92, A112, A110, A113, Th11, A80, A81, A82;
hence f1,f2 are_fiberwise_equipotent by A109, A110, A133, FINSEQ_2:9; :: thesis:
take m4 ; :: thesis:
thus ( md <= m4 & m4 <= n4 ) by A7, A84, A85, A89, A115, A116, A79; :: thesis:
thus m4 - 1 = ((IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t)) . (intpos 4)) - 1 by A114, A105, A106, SCPISORT:7
.= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 1)) by A107, A121, A126, SCPISORT:6 ; :: thesis:
thus m4 + 1 = ((IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t)) . (intpos 4)) + 1 by A114, A105, A106, SCPISORT:7
.= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 2)) by A107, A121, A125, SCPISORT:6 ; :: thesis:
A159: md >= 8 by A88, A95, XXREAL_0:2;
then A160: m4 >= 8 by A115, XXREAL_0:2;

hereby :: thesis:

A161: (IExec (Pt,Q,(Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))))) . (intpos m4) = (IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t)) . (intpos m4) by A105, A106, SCPISORT:7
.= (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize (IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t))))) . (intpos m4) by A19, A86, A120, A122, A131, A160
.= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos m4) by A107, A121, SCPISORT:6 ;
let i be Nat; :: thesis:
assume that
A162: md <= i and
A163: i < m4 ; :: thesis:
A164: i < m1 + (p0 + n) by A131, A163, XXREAL_0:2;
A165: i >= 8 by A159, A162, XXREAL_0:2;
(IExec (Pt,Q,(Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))))) . (intpos i) = (IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t)) . (intpos i) by A105, A106, SCPISORT:7
.= (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize (IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t))))) . (intpos i) by A19, A86, A120, A122, A164, A165
.= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos i) by A107, A121, SCPISORT:6 ;
hence (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos m4) >= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos i) by A117, A162, A163, A161; :: thesis:

end;

hereby :: thesis:

A166: (IExec (Pt,Q,(Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))))) . (intpos m4) = (IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t)) . (intpos m4) by A105, A106, SCPISORT:7
.= (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize (IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t))))) . (intpos m4) by A19, A86, A120, A122, A131, A160
.= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos m4) by A107, A121, SCPISORT:6 ;
let i be Nat; :: thesis:
assume that
A167: m4 < i and
A168: i <= n4 ; :: thesis:
A169: i < m1 + (p0 + n) by A89, A103, A130, A168, XXREAL_0:2;
i >= md by A115, A167, XXREAL_0:2;
then A170: i >= 8 by A159, XXREAL_0:2;
(IExec (Pt,Q,(Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))))) . (intpos i) = (IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t)) . (intpos i) by A105, A106, SCPISORT:7
.= (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize (IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t))))) . (intpos i) by A19, A86, A120, A122, A169, A170
.= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos i) by A107, A121, SCPISORT:6 ;
hence (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos m4) <= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos i) by A102, A118, A167, A168, A166; :: thesis:

end;

hereby :: thesis:

let j be Nat; :: thesis:
assume A171: ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) ; :: thesis:

A172: now :: thesis:

per cases by A171;

suppose A173: ( p0 + 1 <= j & j < md ) ; :: thesis:

hence ( j >= p0 + 1 & ( j < (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos 2) or j > (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos 4) ) ) by A7, A84, A85, A87, A79; :: thesis:
md < m1 + (p0 + n) by A115, A131, XXREAL_0:2;
hence j < m1 + (p0 + n) by A173, XXREAL_0:2; :: thesis:

end;

suppose A174: ( n4 < j & j <= p0 + n ) ; :: thesis:

n4 >= md by A102, A115, A116, XXREAL_0:2;
then j >= md by A174, XXREAL_0:2;
hence j >= p0 + 1 by A88, XXREAL_0:2; :: thesis:
thus ( j < (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos 2) or j > (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos 4) ) by A7, A84, A85, A89, A174, A79; :: thesis:
thus j < m1 + (p0 + n) by A129, A174, XXREAL_0:2; :: thesis:

end;

then A175: j >= 8 by A95, XXREAL_0:2;
then A176: j > 2 by XXREAL_0:2;
A177: j > 4 by A175, XXREAL_0:2;
A178: (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos j) = (Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))) . (intpos j) by SCMPDS_5:15;
thus (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos j) = (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize (IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t))))) . (intpos j) by A107, A121, SCPISORT:6
.= (IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t)) . (intpos j) by A19, A86, A120, A122, A172, A175
.= (IExec (Pt,Q,(Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))))) . (intpos j) by A105, A106, SCPISORT:7
.= (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos j) by A119, A172, A178
.= t . (intpos j) by A7, A84, A85, A176, A177, A79 ; :: thesis:

end;

A179: for t being State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for m being Nat st t . GBP = 0 & t . SBP = m holds
( (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))) . (intpos 2) = (t . (intpos ((m + (p0 + n)) + 1))) - (t . (intpos (m + (p0 + n)))) & ( for i being Nat st i <> 2 holds
(IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))) . (intpos i) = t . (intpos i) ) )

proof

let t be State of SCMPDS; :: thesis:
let Q be Instruction-Sequence of SCMPDS; :: thesis:
let m be Nat; :: thesis:
assume that
A180: t . GBP = 0 and
A181: t . SBP = m ; :: thesis:
set t0 = Initialize t;
set t1 = IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t));
set t2 = Exec (((GBP,2) := (SBP,((p0 + n) + 1))),(Initialize t));
set mp = m + (p0 + n);
A182: (Initialize t) . GBP = 0 by A180, SCMPDS_5:15;
then A183: DataLoc (((Initialize t) . GBP),2) = intpos (0 + 2) by SCMP_GCD:1;
then (Exec (((GBP,2) := (SBP,((p0 + n) + 1))),(Initialize t))) . GBP = 0 by A182, AMI_3:10, SCMPDS_2:47;
then A184: DataLoc (((Exec (((GBP,2) := (SBP,((p0 + n) + 1))),(Initialize t))) . GBP),2) = intpos (0 + 2) by SCMP_GCD:1;
A185: (Initialize t) . SBP = m by A181, SCMPDS_5:15;
then A186: (Exec (((GBP,2) := (SBP,((p0 + n) + 1))),(Initialize t))) . SBP = m by A183, AMI_3:10, SCMPDS_2:47;
m + (p0 + n) >= 0 + 7 by A6, XREAL_1:7;
then m + (p0 + n) > 2 by XXREAL_0:2;
then A187: (Exec (((GBP,2) := (SBP,((p0 + n) + 1))),(Initialize t))) . (intpos (m + (p0 + n))) = (Initialize t) . (intpos (m + (p0 + n))) by A183, AMI_3:10, SCMPDS_2:47
.= t . (intpos (m + (p0 + n))) by SCMPDS_5:15 ;
A188: (Exec (((GBP,2) := (SBP,((p0 + n) + 1))),(Initialize t))) . (intpos 2) = (Initialize t) . (DataLoc (((Initialize t) . SBP),((p0 + n) + 1))) by A183, SCMPDS_2:47
.= (Initialize t) . (intpos (m + ((p0 + n) + 1))) by A185, SCMP_GCD:1
.= t . (intpos ((m + (p0 + n)) + 1)) by SCMPDS_5:15 ;
thus (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))) . (intpos 2) = (Exec ((SubFrom (GBP,2,SBP,(p0 + n))),(Exec (((GBP,2) := (SBP,((p0 + n) + 1))),(Initialize t))))) . (intpos 2) by SCMPDS_5:42
.= ((Exec (((GBP,2) := (SBP,((p0 + n) + 1))),(Initialize t))) . (intpos 2)) - ((Exec (((GBP,2) := (SBP,((p0 + n) + 1))),(Initialize t))) . (DataLoc (((Exec (((GBP,2) := (SBP,((p0 + n) + 1))),(Initialize t))) . SBP),(p0 + n)))) by A184, SCMPDS_2:50
.= (t . (intpos ((m + (p0 + n)) + 1))) - (t . (intpos (m + (p0 + n)))) by A186, A188, A187, SCMP_GCD:1 ; :: thesis:

hereby :: thesis:

let i be Nat; :: thesis:
assume A189: i <> 2 ; :: thesis:
thus (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))) . (intpos i) = (Exec ((SubFrom (GBP,2,SBP,(p0 + n))),(Exec (((GBP,2) := (SBP,((p0 + n) + 1))),(Initialize t))))) . (intpos i) by SCMPDS_5:42
.= (Exec (((GBP,2) := (SBP,((p0 + n) + 1))),(Initialize t))) . (intpos i) by A184, A189, AMI_3:10, SCMPDS_2:50
.= (Initialize t) . (intpos i) by A183, A189, AMI_3:10, SCMPDS_2:47
.= t . (intpos i) by SCMPDS_5:15 ; :: thesis:

end;

A190: (p0 + n) + 1 >= 7 + 1 by A6, XREAL_1:6;
A191: for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for m being Nat st t . GBP = 0 & t . SBP = m & t . (intpos ((m + (p0 + n)) + 1)) <= t . (intpos (m + (p0 + n))) holds
( (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . GBP = 0 & (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos 1) = m - 2 & ( for j being Nat st 1 <= j & j < m holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((p0 + n) + j)) = t . (intpos ((p0 + n) + j)) ) & ( for j being Nat st 1 <= j & j <= n holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) = t . (intpos (p0 + j)) ) )

proof

let t be 0 -started State of SCMPDS; :: thesis:
let Q be Instruction-Sequence of SCMPDS; :: thesis:
let m be Nat; :: thesis:
set mp = m + (p0 + n);
set t1 = IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t);
set Q1 = Q;
A192: Initialize t = t by MEMSTR_0:44;
assume that
A193: t . GBP = 0 and
A194: t . SBP = m and
A195: t . (intpos ((m + (p0 + n)) + 1)) <= t . (intpos (m + (p0 + n))) ; :: thesis:
A196: (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . (intpos 1) = m by A179, A193, A194, A192;
A197: (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . (intpos 2) = (t . (intpos ((m + (p0 + n)) + 1))) - (t . (intpos (m + (p0 + n)))) by A179, A193, A194, A192;
then A198: (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . (intpos 2) <= 0 by A195, XREAL_1:47;
set li = AddTo (GBP,1,(- 2));
set t0 = Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t));
A199: Load (AddTo (GBP,1,(- 2))) is_halting_on IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t),Q by SCMPDS_6:21;
A200: (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . GBP = 0 by A179, A193, A194, A192;
then A201: DataLoc (((IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . GBP),2) = intpos (0 + 2) by SCMP_GCD:1;
A202: (Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))) . GBP = 0 by A200, SCMPDS_5:15;
then A203: DataLoc (((Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))) . GBP),1) = intpos (0 + 1) by SCMP_GCD:1;
A204: Load (AddTo (GBP,1,(- 2))) is_closed_on IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t),Q by SCMPDS_6:20;
then A205: if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))) is_halting_on IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t),Q by A201, A198, A199, SCMPDS_6:69;
A206: if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))) is_closed_on IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t),Q by A201, A198, A204, A199, SCMPDS_6:69;

A207: now :: thesis:

let x be Int_position; :: thesis:
A208: (Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))) . GBP = (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . GBP by SCMPDS_5:15;
A209: (Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))) . (DataLoc (((IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . GBP),2)) = (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . (DataLoc (((IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . GBP),2)) by SCMPDS_5:15;
thus (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . x = (IExec ((if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . x by A206, A205, SCPISORT:7
.= (IExec ((Load (AddTo (GBP,1,(- 2)))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . x by A195, A197, A201, A208, A209, SCMPDS_6:74, XREAL_1:47
.= (Exec ((AddTo (GBP,1,(- 2))),(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . x by SCMPDS_5:40 ; :: thesis:

end;

A210: now :: thesis:

let i be Nat; :: thesis:
assume i <> 1 ; :: thesis:
then A211: intpos i <> DataLoc (((Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))) . GBP),1) by A202, AMI_3:10, SCMP_GCD:1;
thus (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i) = (Exec ((AddTo (GBP,1,(- 2))),(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . (intpos i) by A207
.= (Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))) . (intpos i) by A211, SCMPDS_2:48
.= (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . (intpos i) by SCMPDS_5:15 ; :: thesis:

end;

hence (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . GBP = 0 by A200; :: thesis:
thus (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos 1) = (Exec ((AddTo (GBP,1,(- 2))),(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . (intpos 1) by A207
.= ((Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))) . (intpos 1)) + (- 2) by A203, SCMPDS_2:48
.= ((Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))) . (intpos 1)) - 2
.= m - 2 by A196, SCMPDS_5:15 ; :: thesis:

hereby :: thesis:

let j be Nat; :: thesis:
assume that
A212: 1 <= j and
j < m ; :: thesis:
(p0 + n) + j >= (p0 + n) + 1 by A212, XREAL_1:6;
then A213: (p0 + n) + j >= 8 by A190, XXREAL_0:2;
then A214: (p0 + n) + j > 2 by XXREAL_0:2;
(p0 + n) + j > 1 by A213, XXREAL_0:2;
hence (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((p0 + n) + j)) = (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . (intpos ((p0 + n) + j)) by A210
.= t . (intpos ((p0 + n) + j)) by A179, A193, A194, A214, A192 ;
:: thesis:

end;

hereby :: thesis:

let j be Nat; :: thesis:
assume that
A215: 1 <= j and
j <= n ; :: thesis:
A216: p0 + 1 >= 7 + 1 by A5, XREAL_1:6;
p0 + j >= p0 + 1 by A215, XREAL_1:6;
then A217: p0 + j >= 8 by A216, XXREAL_0:2;
then A218: p0 + j > 2 by XXREAL_0:2;
p0 + j > 1 by A217, XXREAL_0:2;
hence (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) = (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . (intpos (p0 + j)) by A210
.= t . (intpos (p0 + j)) by A179, A193, A194, A218, A192 ;
:: thesis:

end;

A219: for t being State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for m1, md being Nat st t . GBP = 0 & t . SBP = m1 & t . (intpos (m1 + (p0 + n))) = md & md >= p0 + 1 & (t . (intpos ((m1 + (p0 + n)) + 1))) - md > 0 holds
( ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))) is_closed_on t,Q & ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))) is_halting_on t,Q )

proof

let t be State of SCMPDS; :: thesis:
let Q be Instruction-Sequence of SCMPDS; :: thesis:
let m1, md be Nat; :: thesis:
set lPt = (((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt;
set t2 = IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,(Initialize t));
set mp = m1 + (p0 + n);
set Q2 = Q;
A220: (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,(Initialize t))) . GBP = (Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,(Initialize t)))) . GBP by SCMPDS_5:15;
A221: (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,(Initialize t))) . (intpos 2) = (Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,(Initialize t)))) . (intpos 2) by SCMPDS_5:15;
assume that
A222: t . GBP = 0 and
A223: t . SBP = m1 and
A224: t . (intpos (m1 + (p0 + n))) = md and
A225: md >= p0 + 1 and
(t . (intpos ((m1 + (p0 + n)) + 1))) - md > 0 ; :: thesis:
A226: (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,(Initialize t))) . GBP = 0 by A7, A222, A223;
A227: (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,(Initialize t))) . (intpos 2) = t . (intpos (m1 + (p0 + n))) by A7, A222, A223;
then Pt is_halting_on Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,(Initialize t))),Q by A5, A224, A225, A226, Th10, A220, A221;
then A228: Pt is_halting_on IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,(Initialize t)),Q by SCMPDS_6:126;
Pt is_closed_on Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,(Initialize t))),Q by A5, A224, A225, A227, A226, Th10, A220, A221;
then A229: Pt is_closed_on IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,(Initialize t)),Q by SCMPDS_6:125;
then A230: (((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt is_halting_on t,Q by A228, SCPISORT:9;
(((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt is_closed_on t,Q by A229, A228, SCPISORT:9;
hence ( ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))) is_closed_on t,Q & ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))) is_halting_on t,Q ) by A230, SCPISORT:8; :: thesis:

end;

A231: for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for m, m1, md being Nat
for n4 being Integer
for f1, f2 being FinSequence of INT st t . GBP = 0 & t . SBP = m1 & m1 = m + 1 & t . (intpos (m1 + (p0 + n))) = md & md >= p0 + 1 & n4 = t . (intpos ((m1 + (p0 + n)) + 1)) & n4 - md > 0 & n4 <= p0 + n & f1 is_FinSequence_on t,p0 & len f1 = n & f2 is_FinSequence_on IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t),p0 & len f2 = n holds
( (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . GBP = 0 & (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos 1) = m1 + 2 & md = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (m1 + (p0 + n))) & n4 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((m1 + (p0 + n)) + 3)) & ( for j being Nat st 1 <= j & j < m1 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((p0 + n) + j)) = t . (intpos ((p0 + n) + j)) ) & f1,f2 are_fiberwise_equipotent & ex m4 being Nat st
( md <= m4 & m4 <= n4 & m4 - 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((m1 + (p0 + n)) + 1)) & m4 + 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((m1 + (p0 + n)) + 2)) & ( for i being Nat st md <= i & i < m4 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) >= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i) ) & ( for i being Nat st m4 < i & i <= n4 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) <= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i) ) & ( for j being Nat st ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos j) = t . (intpos j) ) ) )

proof

now :: thesis:

let i be Nat; :: thesis:
assume that
A247: 1 <= i and
A248: i <= len f1 ; :: thesis:
p0 + i >= p0 + 1 by A247, XREAL_1:6;
then p0 + i >= 8 by A246, XXREAL_0:2;
then A249: p0 + i > 2 by XXREAL_0:2;
thus f1 . i = t . (intpos (p0 + i)) by A244, A247, A248
.= (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . (intpos (p0 + i)) by A179, A233, A234, A249, A232 ; :: thesis:

end;

then A250: f1 is_FinSequence_on IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t),p0 ;
(m1 + (p0 + n)) + 1 >= 7 + 1 by A242, XREAL_1:6;
then (m1 + (p0 + n)) + 1 > 2 by XXREAL_0:2;
then A251: n4 = (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 1)) by A179, A233, A234, A238, A232;
A252: (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . GBP = 0 by A179, A233, A234, A232;
then A253: DataLoc (((IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . GBP),2) = intpos (0 + 2) by SCMP_GCD:1;
A254: (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . SBP = m1 by A179, A233, A234, A232;
then A255: ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))) is_halting_on IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t),Q by A219, A237, A239, A243, A251, A252;
A256: ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))) is_halting_on Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)),Q

proof

Q +* (stop (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))))) halts_on Initialize (Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))) by A255, SCMPDS_6:def 3;
hence ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))) is_halting_on Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)),Q by SCMPDS_6:def 3; :: thesis:

end;

A257: (Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))) . GBP = (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . GBP by SCMPDS_5:15;
A258: (Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))) . (DataLoc (((IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . GBP),2)) = (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . (DataLoc (((IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . GBP),2)) by SCMPDS_5:15;
A259: ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))) is_closed_on IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t),Q by A219, A237, A239, A254, A243, A251, A252;
A260: ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))) is_closed_on Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)),Q

proof

for k being Nat holds IC (Comput ((Q +* (stop (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))))),(Initialize (Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)))),k)) in dom (stop (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))))) by A259, SCMPDS_6:def 2;
hence ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))) is_closed_on Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)),Q by SCMPDS_6:def 2; :: thesis:

end;

then A261: if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))) is_halting_on Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)),Q by A253, A241, A257, A258, A256, SCMPDS_6:68;
A262: if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))) is_halting_on IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t),Q

proof

Q +* (stop (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))) halts_on Initialize (Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))) by A261, SCMPDS_6:def 3;
hence if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))) is_halting_on IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t),Q by SCMPDS_6:def 3; :: thesis:

end;

assume that
A263: f2 is_FinSequence_on IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t),p0 and
A264: len f2 = n ; :: thesis:
A265: if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))) is_closed_on Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)),Q by A253, A241, A257, A258, A256, A260, SCMPDS_6:68;
A266: if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))) is_closed_on IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t),Q

proof

for k being Nat holds IC (Comput ((Q +* (stop (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))))),(Initialize (Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)))),k)) in dom (stop (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))) by A265, SCMPDS_6:def 2;
hence if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))) is_closed_on IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t),Q by SCMPDS_6:def 2; :: thesis:

end;

A267: now :: thesis:

let x be Int_position; :: thesis:
thus (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . x = (IExec ((if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . x by A266, A262, SCPISORT:7
.= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . x by A253, A241, Th1, A257, A258, A260, A256 ; :: thesis:

end;

now :: thesis:

let i be Nat; :: thesis:
assume that
A268: 1 <= i and
A269: i <= len f2 ; :: thesis:
thus f2 . i = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + i)) by A263, A268, A269
.= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . (intpos (p0 + i)) by A267 ; :: thesis:

end;

then A270: f2 is_FinSequence_on IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)))),p0 ;
A271: (Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))) . SBP = (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . SBP by SCMPDS_5:15;
A272: (Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))) . GBP = (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . GBP by SCMPDS_5:15;
A273: (Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))) . (intpos (m1 + (p0 + n))) = (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . (intpos (m1 + (p0 + n))) by SCMPDS_5:15;
A274: (Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))) . (intpos ((m1 + (p0 + n)) + 1)) = (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 1)) by SCMPDS_5:15;
A275: f1 is_FinSequence_on Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)),p0

proof

let i be Nat; :: according to SCPISORT:def 1 :: thesis:
assume ( 1 <= i & i <= len f1 ) ; :: thesis:
then f1 . i = (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . (intpos (p0 + i)) by A250;
hence f1 . i = (Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))) . (intpos (p0 + i)) by SCMPDS_5:15; :: thesis:

end;

(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . GBP = 0 by A78, A235, A237, A239, A240, A245, A264, A254, A243, A251, A252, A270, A271, A272, A273, A274, A275;
hence (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . GBP = 0 by A267; :: thesis:
consider m4 being Nat such that
A276: md <= m4 and
A277: m4 <= n4 and
A278: m4 - 1 = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . (intpos ((m1 + (p0 + n)) + 1)) and
A279: m4 + 1 = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . (intpos ((m1 + (p0 + n)) + 2)) and
A280: for i being Nat st md <= i & i < m4 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . (intpos m4) >= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . (intpos i) and
A281: for i being Nat st m4 < i & i <= n4 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . (intpos m4) <= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . (intpos i) and
A282: for j being Nat st ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . (intpos j) = (Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))) . (intpos j) by A78, A235, A237, A239, A240, A245, A264, A254, A243, A251, A252, A270, A271, A272, A273, A274, A275;
A283: for j being Nat st ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . (intpos j) = (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . (intpos j)

proof

let j be Nat; :: thesis:
assume ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) ; :: thesis:
then (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . (intpos j) = (Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))) . (intpos j) by A282;
hence (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . (intpos j) = (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . (intpos j) by SCMPDS_5:15; :: thesis:

end;

hereby :: thesis:

let j be Nat; :: thesis:
assume that
A284: 1 <= j and
A285: j < m1 ; :: thesis:
(p0 + n) + j >= (p0 + n) + 1 by A284, XREAL_1:6;
then (p0 + n) + j >= 8 by A190, XXREAL_0:2;
then A286: (p0 + n) + j > 2 by XXREAL_0:2;
thus (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((p0 + n) + j)) = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . (intpos ((p0 + n) + j)) by A267
.= (Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))) . (intpos ((p0 + n) + j)) by A78, A235, A237, A239, A240, A245, A264, A254, A243, A251, A252, A270, A284, A285, A271, A272, A273, A274, A275
.= (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . (intpos ((p0 + n) + j)) by SCMPDS_5:15
.= t . (intpos ((p0 + n) + j)) by A179, A233, A234, A286, A232 ; :: thesis:

end;

thus f1,f2 are_fiberwise_equipotent by A78, A235, A237, A239, A240, A245, A264, A254, A243, A251, A252, A270, A271, A272, A273, A274, A275; :: thesis:
take m4 ; :: thesis:
thus ( md <= m4 & m4 <= n4 ) by A276, A277; :: thesis:
thus m4 - 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((m1 + (p0 + n)) + 1)) by A267, A278; :: thesis:
thus m4 + 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((m1 + (p0 + n)) + 2)) by A267, A279; :: thesis:

hereby :: thesis:

let i be Nat; :: thesis:
assume that
A287: md <= i and
A288: i < m4 ; :: thesis:
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . (intpos m4) >= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . (intpos i) by A280, A287, A288;
then (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) >= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . (intpos i) by A267;
hence (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) >= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i) by A267; :: thesis:

end;

hereby :: thesis:

let i be Nat; :: thesis:
assume that
A289: m4 < i and
A290: i <= n4 ; :: thesis:
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . (intpos m4) <= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . (intpos i) by A281, A289, A290;
then (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) <= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . (intpos i) by A267;
hence (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) <= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i) by A267; :: thesis:

end;

hereby :: thesis:

let j be Nat; :: thesis:
assume A291: ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) ; :: thesis:

now :: thesis:

per cases by A291;

suppose ( p0 + 1 <= j & j < md ) ; :: thesis:

hence p0 + 1 <= j ; :: thesis:

end;

suppose ( n4 < j & j <= p0 + n ) ; :: thesis:

then j >= m4 by A277, XXREAL_0:2;
then j >= md by A276, XXREAL_0:2;
hence j >= p0 + 1 by A237, XXREAL_0:2; :: thesis:

end;

then j >= 8 by A246, XXREAL_0:2;
then A292: j > 2 by XXREAL_0:2;
thus (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos j) = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . (intpos j) by A267
.= (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . (intpos j) by A283, A291
.= t . (intpos j) by A179, A233, A234, A292, A232 ; :: thesis:

end;

consider f being FinSequence of INT such that
A293: len f = n and
A294: f is_FinSequence_on s,p0 by SCPISORT:2;
A295: (p0 + n) + 2 >= 7 + 2 by A6, XREAL_1:6;
A296: ( (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)) . GBP = 0 & (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)) . (intpos 1) = 1 & (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)) . (intpos ((p0 + n) + 1)) = p0 + 1 & (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)) . (intpos ((p0 + n) + 2)) = p0 + n & ( for i being Nat st i >= 8 & i <= p0 + n holds
(IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)) . (intpos i) = s . (intpos i) ) )

proof

set t2 = IExec ((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))),P,s);
set t3 = IExec (((GBP := 0) ';' (SBP := 1)),P,s);
set t4 = Exec ((GBP := 0),s);
A297: (Exec ((GBP := 0),s)) . GBP = 0 by SCMPDS_2:45;
A298: (IExec (((GBP := 0) ';' (SBP := 1)),P,s)) . GBP = (Exec ((SBP := 1),(Exec ((GBP := 0),s)))) . GBP by SCMPDS_5:42
.= 0 by A297, SCMPDS_2:45, SCMP_GCD:3 ;
A299: (IExec (((GBP := 0) ';' (SBP := 1)),P,s)) . SBP = (Exec ((SBP := 1),(Exec ((GBP := 0),s)))) . SBP by SCMPDS_5:42
.= 1 by SCMPDS_2:45 ;
then A300: DataLoc (((IExec (((GBP := 0) ';' (SBP := 1)),P,s)) . SBP),(p0 + n)) = intpos ((p0 + n) + 1) by SCMP_GCD:1;
A301: (IExec ((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))),P,s)) . GBP = (Exec (((SBP,(p0 + n)) := (p0 + 1)),(IExec (((GBP := 0) ';' (SBP := 1)),P,s)))) . GBP by SCMPDS_5:41
.= 0 by A298, A300, AMI_3:10, SCMPDS_2:46 ;
A302: (IExec ((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))),P,s)) . (intpos ((p0 + n) + 1)) = (Exec (((SBP,(p0 + n)) := (p0 + 1)),(IExec (((GBP := 0) ';' (SBP := 1)),P,s)))) . (intpos ((p0 + n) + 1)) by SCMPDS_5:41
.= p0 + 1 by A300, SCMPDS_2:46 ;
A303: (p0 + n) + 2 > 1 by A295, XXREAL_0:2;
A304: (p0 + n) + 1 > 1 by A190, XXREAL_0:2;
A305: (IExec ((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))),P,s)) . SBP = (Exec (((SBP,(p0 + n)) := (p0 + 1)),(IExec (((GBP := 0) ';' (SBP := 1)),P,s)))) . SBP by SCMPDS_5:41
.= 1 by A299, A300, A304, AMI_3:10, SCMPDS_2:46 ;
then A306: DataLoc (((IExec ((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))),P,s)) . SBP),((p0 + n) + 1)) = intpos ((p0 + n) + (1 + 1)) by SCMP_GCD:1;
thus (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)) . GBP = (Exec (((SBP,((p0 + n) + 1)) := (p0 + n)),(IExec ((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))),P,s)))) . GBP by SCMPDS_5:41
.= 0 by A301, A306, AMI_3:10, SCMPDS_2:46 ; :: thesis:
thus (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)) . (intpos 1) = (Exec (((SBP,((p0 + n) + 1)) := (p0 + n)),(IExec ((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))),P,s)))) . SBP by SCMPDS_5:41
.= 1 by A305, A306, A303, AMI_3:10, SCMPDS_2:46 ; :: thesis:
A307: (p0 + n) + 2 > (p0 + n) + 1 by XREAL_1:6;
thus (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)) . (intpos ((p0 + n) + 1)) = (Exec (((SBP,((p0 + n) + 1)) := (p0 + n)),(IExec ((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))),P,s)))) . (intpos ((p0 + n) + 1)) by SCMPDS_5:41
.= p0 + 1 by A302, A306, A307, AMI_3:10, SCMPDS_2:46 ; :: thesis:
thus (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)) . (intpos ((p0 + n) + 2)) = (Exec (((SBP,((p0 + n) + 1)) := (p0 + n)),(IExec ((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))),P,s)))) . (intpos ((p0 + n) + 2)) by SCMPDS_5:41
.= p0 + n by A306, SCMPDS_2:46 ; :: thesis:
A308: for i being Nat st i >= 8 holds
(Exec ((GBP := 0),s)) . (intpos i) = s . (intpos i) by AMI_3:10, SCMPDS_2:45;

A309: now :: thesis:

let i be Nat; :: thesis:
assume A310: i >= 8 ; :: thesis:
then A311: i > 1 by XXREAL_0:2;
thus (IExec (((GBP := 0) ';' (SBP := 1)),P,s)) . (intpos i) = (Exec ((SBP := 1),(Exec ((GBP := 0),s)))) . (intpos i) by SCMPDS_5:42
.= (Exec ((GBP := 0),s)) . (intpos i) by A311, AMI_3:10, SCMPDS_2:45
.= s . (intpos i) by A308, A310 ; :: thesis:

end;

A312: now :: thesis:

let i be Nat; :: thesis:
assume that
A313: i >= 8 and
A314: i <= p0 + n ; :: thesis:
A315: (p0 + n) + 1 > p0 + n by XREAL_1:29;
thus (IExec ((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))),P,s)) . (intpos i) = (Exec (((SBP,(p0 + n)) := (p0 + 1)),(IExec (((GBP := 0) ';' (SBP := 1)),P,s)))) . (intpos i) by SCMPDS_5:41
.= (IExec (((GBP := 0) ';' (SBP := 1)),P,s)) . (intpos i) by A300, A314, A315, AMI_3:10, SCMPDS_2:46
.= s . (intpos i) by A309, A313 ; :: thesis:

end;

A316: (p0 + n) + 2 > (p0 + n) + 0 by XREAL_1:6;
let i be Nat; :: thesis:
assume that
A317: i >= 8 and
A318: i <= p0 + n ; :: thesis:
thus (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)) . (intpos i) = (Exec (((SBP,((p0 + n) + 1)) := (p0 + n)),(IExec ((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))),P,s)))) . (intpos i) by SCMPDS_5:41
.= (IExec ((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))),P,s)) . (intpos i) by A306, A318, A316, AMI_3:10, SCMPDS_2:46
.= s . (intpos i) by A312, A317, A318 ; :: thesis:

end;

now :: thesis:

A319: p0 + 1 >= 7 + 1 by A5, XREAL_1:6;
let i be Nat; :: thesis:
assume that
A320: 1 <= i and
A321: i <= len f ; :: thesis:
A322: p0 + i <= p0 + n by A293, A321, XREAL_1:6;
p0 + i >= p0 + 1 by A320, XREAL_1:6;
then A323: p0 + i >= 8 by A319, XXREAL_0:2;
thus f . i = s . (intpos (p0 + i)) by A294, A320, A321
.= (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)) . (intpos (p0 + i)) by A296, A323, A322 ; :: thesis:

end;

then A324: f is_FinSequence_on IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s),p0 ;
consider g being FinSequence of INT such that
A325: len g = n and
A326: g is_FinSequence_on IExec ((QuickSort (n,p0)),P,s),p0 by SCPISORT:2;
A327: DataLoc (0,1) = intpos (0 + 1) by SCMP_GCD:1;
A328: for t being State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for m, md being Nat st t . GBP = 0 & t . SBP = m & t . (intpos (m + (p0 + n))) = md & ( md >= p0 + 1 or (t . (intpos ((m + (p0 + n)) + 1))) - md <= 0 ) holds
( (((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))) is_closed_on t,Q & (((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))) is_halting_on t,Q )

proof

let t be State of SCMPDS; :: thesis:
let Q be Instruction-Sequence of SCMPDS; :: thesis:
let m, md be Nat; :: thesis:
set mp = m + (p0 + n);
set t1 = IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t));
set Q1 = Q;
assume that
A329: t . GBP = 0 and
A330: t . SBP = m and
A331: t . (intpos (m + (p0 + n))) = md and
A332: ( md >= p0 + 1 or (t . (intpos ((m + (p0 + n)) + 1))) - md <= 0 ) ; :: thesis:
A333: (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))) . (intpos 2) = (t . (intpos ((m + (p0 + n)) + 1))) - (t . (intpos (m + (p0 + n)))) by A179, A329, A330;
A334: (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))) . GBP = 0 by A179, A329, A330;
then A335: DataLoc (((IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))) . GBP),2) = intpos (0 + 2) by SCMP_GCD:1;

per cases ) . (DataLoc (((IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))) . GBP),2)) > 0 or (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))) . (DataLoc (((IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))) . GBP),2)) <= 0 ) ;

suppose A336: (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))) . (DataLoc (((IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))) . GBP),2)) > 0 ; :: thesis:

A337: m + (p0 + n) >= 0 + 7 by A6, XREAL_1:7;
then (m + (p0 + n)) + 1 >= 7 + 1 by XREAL_1:6;
then (m + (p0 + n)) + 1 > 2 by XXREAL_0:2;
then A338: ((IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))) . (intpos ((m + (p0 + n)) + 1))) - md > 0 by A179, A329, A330, A331, A333, A335, A336;
A339: (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))) . SBP = m by A179, A329, A330;
m + (p0 + n) > 2 by A337, XXREAL_0:2;
then A340: (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))) . (intpos (m + (p0 + n))) = md by A179, A329, A330, A331;
then A341: ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))) is_halting_on IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t)),Q by A219, A179, A329, A330, A331, A332, A334, A335, A336, A339, A338;
A342: (Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t)))) . (DataLoc (((IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))) . GBP),2)) = (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))) . (DataLoc (((IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))) . GBP),2)) by SCMPDS_5:15;
A343: (Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t)))) . GBP = (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))) . GBP by SCMPDS_5:15;
A344: ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))) is_closed_on IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t)),Q by A219, A179, A329, A330, A331, A332, A334, A335, A336, A339, A340, A338;
A345: ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))) is_halting_on Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))),Q

proof

Q +* (stop (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))))) halts_on Initialize (Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t)))) by A341, SCMPDS_6:def 3;
hence ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))) is_halting_on Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))),Q by SCMPDS_6:def 3; :: thesis:

end;

A346: ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))) is_closed_on Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))),Q

proof

for k being Nat holds IC (Comput ((Q +* (stop (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))))),(Initialize (Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))))),k)) in dom (stop (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))))) by A344, SCMPDS_6:def 2;
hence ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))) is_closed_on Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))),Q by SCMPDS_6:def 2; :: thesis:

end;

then A347: if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))) is_halting_on Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))),Q by A336, A345, A342, A343, SCMPDS_6:68;
A348: if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))) is_halting_on IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t)),Q

proof

Q +* (stop (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))) halts_on Initialize (Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t)))) by A347, SCMPDS_6:def 3;
hence if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))) is_halting_on IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t)),Q by SCMPDS_6:def 3; :: thesis:

end;

A349: if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))) is_closed_on Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))),Q by A336, A346, A345, A342, A343, SCMPDS_6:68;
if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))) is_closed_on IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t)),Q

proof

for k being Nat holds IC (Comput ((Q +* (stop (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))))),(Initialize (Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))))),k)) in dom (stop (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))) by A349, SCMPDS_6:def 2;
hence if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))) is_closed_on IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t)),Q by SCMPDS_6:def 2; :: thesis:

end;

suppose A350: (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))) . (DataLoc (((IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))) . GBP),2)) <= 0 ; :: thesis:

A351: Load (AddTo (GBP,1,(- 2))) is_halting_on IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t)),Q by SCMPDS_6:21;
A352: Load (AddTo (GBP,1,(- 2))) is_closed_on IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t)),Q by SCMPDS_6:20;
then A353: if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))) is_halting_on IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t)),Q by A350, A351, SCMPDS_6:69;
if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))) is_closed_on IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t)),Q by A350, A352, A351, SCMPDS_6:69;
hence ( (((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))) is_closed_on t,Q & (((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))) is_halting_on t,Q ) by A353, SCPISORT:9; :: thesis:

end;

A354: now :: thesis:

let t be 0 -started State of SCMPDS; :: thesis:
let Q be Instruction-Sequence of SCMPDS; :: thesis:
let f1, f2 be FinSequence of INT ; :: thesis:
let k1, k2, y1, yn be Nat; :: thesis:
set mm = (2 * k1) + 1;
set md = p0 + y1;
set n4 = p0 + yn;
assume that
A355: t . GBP = 0 and
A356: (2 * k1) + 1 = t . (DataLoc (0,1)) and
A357: k2 = ((p0 + n) + (2 * k1)) + 1 and
A358: p0 + y1 = t . (intpos k2) and
A359: p0 + yn = t . (intpos (k2 + 1)) and
A360: ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) ; :: thesis:
set mp = ((2 * k1) + 1) + (p0 + n);
A361: p0 + y1 = t . (intpos (((2 * k1) + 1) + (p0 + n))) by A357, A358;

now :: thesis:

per cases by A360;

case ( 1 <= y1 & yn <= n ) ; :: thesis:

hence p0 + 1 <= p0 + y1 by XREAL_1:6; :: thesis:

end;

case y1 >= yn ; :: thesis:

then p0 + y1 >= t . (intpos ((((2 * k1) + 1) + (p0 + n)) + 1)) by A357, A359, XREAL_1:6;
hence (t . (intpos ((((2 * k1) + 1) + (p0 + n)) + 1))) - (p0 + y1) <= 0 by XREAL_1:47; :: thesis:

end;

hence ( (((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))) is_closed_on t,Q & (((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))) is_halting_on t,Q ) by A328, A327, A355, A356, A361; :: thesis:
consider f3 being FinSequence of INT such that
A362: len f3 = n and
A363: for i being Nat st 1 <= i & i <= len f3 holds
f3 . i = t . (intpos (p0 + i)) by SCPISORT:1;
consider f4 being FinSequence of INT such that
A364: len f4 = n and
A365: for i being Nat st 1 <= i & i <= len f4 holds
f4 . i = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + i)) by SCPISORT:1;
A366: f4 is_FinSequence_on IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t),p0 by A365;
A367: f3 is_FinSequence_on t,p0 by A363;

hereby :: thesis:

per cases ;

suppose t . (intpos ((((2 * k1) + 1) + (p0 + n)) + 1)) <= t . (intpos (((2 * k1) + 1) + (p0 + n))) ; :: thesis:

hence ( (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . GBP = t . GBP & ( for j being Nat st 1 <= j & j < (2 * k1) + 1 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((p0 + n) + j)) = t . (intpos ((p0 + n) + j)) ) ) by A191, A327, A355, A356; :: thesis:

end;

suppose A368: t . (intpos ((((2 * k1) + 1) + (p0 + n)) + 1)) > t . (intpos (((2 * k1) + 1) + (p0 + n))) ; :: thesis:

then A369: (p0 + yn) - (p0 + y1) > 0 by A357, A358, A359, XREAL_1:50;
A370: p0 + yn <= p0 + n by A357, A358, A359, A360, A368, XREAL_1:6;
p0 + 1 <= p0 + y1 by A357, A358, A359, A360, A368, XREAL_1:6;
hence ( (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . GBP = t . GBP & ( for j being Nat st 1 <= j & j < (2 * k1) + 1 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((p0 + n) + j)) = t . (intpos ((p0 + n) + j)) ) ) by A231, A327, A355, A356, A357, A359, A361, A362, A367, A364, A366, A369, A370; :: thesis:

end;

hereby :: thesis:

assume y1 >= yn ; :: thesis:
then A371: t . (intpos ((((2 * k1) + 1) + (p0 + n)) + 1)) <= t . (intpos (((2 * k1) + 1) + (p0 + n))) by A357, A358, A359, XREAL_1:6;
hence (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (DataLoc (0,1)) = ((2 * k1) + 1) - 2 by A191, A327, A355, A356
.= (2 * k1) - 1 ;
:: thesis:
thus for j being Nat st 1 <= j & j <= n holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) = t . (intpos (p0 + j)) by A191, A327, A355, A356, A371; :: thesis:

end;

A372: k2 = ((2 * k1) + 1) + (p0 + n) by A357;
set s = Initialize (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s));
set a = GBP ;
set c = 0 ;
set m = p0;
set i = 1;
set I = (((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))));
set P = P;
set pn = p0 + n;
set m1 = (p0 + n) + 1;
thus ( y1 < yn implies ( (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (DataLoc (0,1)) = (2 * k1) + 3 & ( for j being Nat st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) = t . (intpos (p0 + j)) ) & ex ym being Nat st
( y1 <= ym & ym <= yn & p0 + y1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos k2) & (p0 + ym) - 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (k2 + 1)) & (p0 + ym) + 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (k2 + 2)) & p0 + yn = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (k2 + 3)) & ( for j being Nat st y1 <= j & j < ym holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) <= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + ym)) ) & ( for j being Nat st ym < j & j <= yn holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) >= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + ym)) ) ) ) ) :: thesis:

proof

assume A373: y1 < yn ; :: thesis:
then A374: p0 + 1 <= p0 + y1 by A360, XREAL_1:6;
p0 + yn > p0 + y1 by A373, XREAL_1:6;
then A375: (p0 + yn) - (p0 + y1) > 0 by XREAL_1:50;
A376: p0 + yn <= p0 + n by A360, A373, XREAL_1:6;
hence (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (DataLoc (0,1)) = ((2 * k1) + 1) + 2 by A231, A327, A355, A356, A357, A359, A361, A362, A367, A364, A366, A375, A374
.= (2 * k1) + 3 ;
:: thesis:
consider m4 being Nat such that
A377: p0 + y1 <= m4 and
A378: m4 <= p0 + yn and
A379: m4 - 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((((2 * k1) + 1) + (p0 + n)) + 1)) and
A380: m4 + 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((((2 * k1) + 1) + (p0 + n)) + 2)) and
A381: for i being Nat st p0 + y1 <= i & i < m4 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) >= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i) and
A382: for i being Nat st m4 < i & i <= p0 + yn holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) <= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i) and
A383: for j being Nat st ( ( p0 + 1 <= j & j < p0 + y1 ) or ( p0 + yn < j & j <= p0 + n ) ) holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos j) = t . (intpos j) by A231, A327, A355, A356, A357, A358, A359, A362, A367, A364, A366, A375, A374, A376;

hereby :: thesis:

let j be Nat; :: thesis:
assume ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) ; :: thesis:
then ( ( p0 + 1 <= p0 + j & p0 + j < p0 + y1 ) or ( p0 + yn < p0 + j & p0 + j <= p0 + n ) ) by XREAL_1:6;
hence (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) = t . (intpos (p0 + j)) by A383; :: thesis:

end;

m4 - p0 >= y1 by A377, XREAL_1:19;
then reconsider ym = m4 - p0 as Element of NAT by INT_1:3;
take ym ; :: thesis:
m4 = p0 + ym ;
hence ( y1 <= ym & ym <= yn ) by A377, A378, XREAL_1:6; :: thesis:
thus p0 + y1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos k2) by A231, A327, A355, A356, A358, A359, A372, A362, A367, A364, A366, A375, A374, A376; :: thesis:
thus (p0 + ym) - 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (k2 + 1)) by A357, A379; :: thesis:
thus (p0 + ym) + 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (k2 + 2)) by A357, A380; :: thesis:
thus p0 + yn = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (k2 + 3)) by A231, A327, A355, A356, A358, A359, A372, A362, A367, A364, A366, A375, A374, A376; :: thesis:

hereby :: thesis:

let j be Nat; :: thesis:
assume that
A384: y1 <= j and
A385: j < ym ; :: thesis:
A386: p0 + j < p0 + ym by A385, XREAL_1:6;
p0 + y1 <= p0 + j by A384, XREAL_1:6;
hence (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) <= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + ym)) by A381, A386; :: thesis:

end;

hereby :: thesis:

let j be Nat; :: thesis:
assume that
A387: ym < j and
A388: j <= yn ; :: thesis:
A389: p0 + j <= p0 + yn by A388, XREAL_1:6;
p0 + ym < p0 + j by A387, XREAL_1:6;
hence (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) >= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + ym)) by A382, A389; :: thesis:

end;

hereby :: thesis:

assume that
A390: f1 is_FinSequence_on t,p0 and
A391: f2 is_FinSequence_on IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t),p0 and
A392: len f1 = n and
A393: len f2 = n ; :: thesis:

per cases ;

suppose A394: t . (intpos ((((2 * k1) + 1) + (p0 + n)) + 1)) <= t . (intpos (((2 * k1) + 1) + (p0 + n))) ; :: thesis:

A395: dom f1 = Seg n by A392, FINSEQ_1:def 3;

now :: thesis:

let i be Nat; :: thesis:
reconsider a = i as Nat ;
assume A396: i in dom f1 ; :: thesis:
then A397: 1 <= i by A395, FINSEQ_1:1;
A398: i <= n by A395, A396, FINSEQ_1:1;
hence f1 . i = t . (intpos (p0 + a)) by A390, A392, A397
.= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + a)) by A191, A327, A355, A356, A394, A397, A398
.= f2 . i by A391, A393, A397, A398 ;
:: thesis:

end;

hence f1,f2 are_fiberwise_equipotent by A392, A393, FINSEQ_2:9; :: thesis:

end;

suppose A399: t . (intpos ((((2 * k1) + 1) + (p0 + n)) + 1)) > t . (intpos (((2 * k1) + 1) + (p0 + n))) ; :: thesis:

then A400: (p0 + yn) - (p0 + y1) > 0 by A357, A358, A359, XREAL_1:50;
A401: p0 + yn <= p0 + n by A357, A358, A359, A360, A399, XREAL_1:6;
p0 + 1 <= p0 + y1 by A357, A358, A359, A360, A399, XREAL_1:6;
hence f1,f2 are_fiberwise_equipotent by A231, A327, A355, A356, A357, A359, A361, A390, A391, A392, A393, A400, A401; :: thesis:

end;

A402: (Initialize (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s))) . GBP = (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)) . GBP by SCMPDS_5:15;
A403: (Initialize (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s))) . (intpos ((p0 + n) + 1)) = (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)) . (intpos ((p0 + n) + 1)) by SCMPDS_5:15;
A404: (Initialize (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s))) . (intpos (((p0 + n) + 1) + 1)) = (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)) . (intpos (((p0 + n) + 1) + 1)) by SCMPDS_5:15;
A405: (Initialize (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s))) . (DataLoc (0,1)) = (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)) . (DataLoc (0,1)) by SCMPDS_5:15;
A406: while>0 (GBP,1,((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))))) is_closed_on Initialize (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)),P by A327, A354, A296, Lm5, A402, A403, A404, A405;
A407: while>0 (GBP,1,((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))))) is_closed_on IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s),P

proof

for k being Nat holds IC (Comput ((P +* (stop (while>0 (GBP,1,((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))))))),(Initialize (Initialize (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)))),k)) in dom (stop (while>0 (GBP,1,((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))))))) by A406, SCMPDS_6:def 2;
hence while>0 (GBP,1,((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))))) is_closed_on IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s),P by SCMPDS_6:def 2; :: thesis:

end;

A408: while>0 (GBP,1,((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))))) is_halting_on Initialize (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)),P by A327, A354, A296, Lm5, A402, A403, A404, A405;
A409: while>0 (GBP,1,((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))))) is_halting_on IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s),P

proof

P +* (stop (while>0 (GBP,1,((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))))))) halts_on Initialize (Initialize (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s))) by A408, SCMPDS_6:def 3;
hence while>0 (GBP,1,((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))))) is_halting_on IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s),P by SCMPDS_6:def 3; :: thesis:

end;

hence QuickSort (n,p0) is_halting_on s,P by A407, A1, SCPISORT:9; :: thesis:
take f ; :: thesis:
take g ; :: thesis:
thus ( len f = n & f is_FinSequence_on s,p0 & len g = n & g is_FinSequence_on IExec ((QuickSort (n,p0)),P,s),p0 ) by A293, A294, A325, A326; :: thesis:

now :: thesis:

let i be Nat; :: thesis:
assume that
A410: 1 <= i and
A411: i <= len g ; :: thesis:
thus g . i = (IExec ((((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))) ';' (while>0 (GBP,1,((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))))))),P,s)) . (intpos (p0 + i)) by A326, A410, A411
.= (IExec ((while>0 (GBP,1,((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))))),P,(Initialize (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s))))) . (intpos (p0 + i)) by A409, A407, SCPISORT:7 ; :: thesis:

end;

then A412: g is_FinSequence_on IExec ((while>0 (GBP,1,((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))))),P,(Initialize (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)))),p0 ;
A413: (Initialize (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s))) . GBP = (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)) . GBP by SCMPDS_5:15;
A414: (Initialize (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s))) . (DataLoc (0,1)) = (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)) . (DataLoc (0,1)) by SCMPDS_5:15;
A415: (Initialize (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s))) . (intpos ((p0 + n) + 1)) = (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)) . (intpos ((p0 + n) + 1)) by SCMPDS_5:15;
A416: (Initialize (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s))) . (intpos (((p0 + n) + 1) + 1)) = (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)) . (intpos (((p0 + n) + 1) + 1)) by SCMPDS_5:15;
f is_FinSequence_on Initialize (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)),p0

proof

let i be Nat; :: according to SCPISORT:def 1 :: thesis:
assume ( 1 <= i & i <= len f ) ; :: thesis:
then f . i = (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)) . (intpos (p0 + i)) by A324;
hence f . i = (Initialize (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s))) . (intpos (p0 + i)) by SCMPDS_5:15; :: thesis:

end;

hence ( f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n ) by Lm3, A325, A293, A412, A414, A327, A296, A413, A415, A416, A354; :: thesis: