let s be State of SCMPDS; :: thesis: ( GCD-Algorithm c= s & IC s = 5 & s . SBP > 0 & s . GBP = 0 & s . (DataLoc ((s . SBP),3)) >= 0 & s . (DataLoc ((s . SBP),2)) >= 0 implies ex n being Element of NAT st
( CurInstr ((ProgramPart (Comput ((ProgramPart s),s,n))),(Comput ((ProgramPart s),s,n))) = return SBP & s . SBP = (Comput ((ProgramPart s),s,n)) . SBP & (Comput ((ProgramPart s),s,n)) . (DataLoc ((s . SBP),2)) = (s . (DataLoc ((s . SBP),2))) gcd (s . (DataLoc ((s . SBP),3))) & ( for j being Element of NAT st 1 < j & j <= (s . SBP) + 1 holds
s . (intpos j) = (Comput ((ProgramPart s),s,n)) . (intpos j) ) ) )
set GA = GCD-Algorithm ;
set x = s . (DataLoc ((s . SBP),2));
set y = s . (DataLoc ((s . SBP),3));
set yy = s . (DataLoc ((s . SBP),3));
assume that
A1: GCD-Algorithm c= s and
A2: IC s = 5 and
A3: s . SBP > 0 and
A4: s . GBP = 0 and
A5: s . (DataLoc ((s . SBP),3)) >= 0 and
A6: s . (DataLoc ((s . SBP),2)) >= 0 ; :: thesis: ex n being Element of NAT st
( CurInstr ((ProgramPart (Comput ((ProgramPart s),s,n))),(Comput ((ProgramPart s),s,n))) = return SBP & s . SBP = (Comput ((ProgramPart s),s,n)) . SBP & (Comput ((ProgramPart s),s,n)) . (DataLoc ((s . SBP),2)) = (s . (DataLoc ((s . SBP),2))) gcd (s . (DataLoc ((s . SBP),3))) & ( for j being Element of NAT st 1 < j & j <= (s . SBP) + 1 holds
s . (intpos j) = (Comput ((ProgramPart s),s,n)) . (intpos j) ) )

per cases ( s . (DataLoc ((s . SBP),2)) >= s . (DataLoc ((s . SBP),3)) or s . (DataLoc ((s . SBP),2)) < s . (DataLoc ((s . SBP),3)) ) ;

suppose s . (DataLoc ((s . SBP),2)) >= s . (DataLoc ((s . SBP),3)) ; :: thesis: ex n being Element of NAT st
( CurInstr ((ProgramPart (Comput ((ProgramPart s),s,n))),(Comput ((ProgramPart s),s,n))) = return SBP & s . SBP = (Comput ((ProgramPart s),s,n)) . SBP & (Comput ((ProgramPart s),s,n)) . (DataLoc ((s . SBP),2)) = (s . (DataLoc ((s . SBP),2))) gcd (s . (DataLoc ((s . SBP),3))) & ( for j being Element of NAT st 1 < j & j <= (s . SBP) + 1 holds
s . (intpos j) = (Comput ((ProgramPart s),s,n)) . (intpos j) ) )

hence ex n being Element of NAT st
( CurInstr ((ProgramPart (Comput ((ProgramPart s),s,n))),(Comput ((ProgramPart s),s,n))) = return SBP & s . SBP = (Comput ((ProgramPart s),s,n)) . SBP & (Comput ((ProgramPart s),s,n)) . (DataLoc ((s . SBP),2)) = (s . (DataLoc ((s . SBP),2))) gcd (s . (DataLoc ((s . SBP),3))) & ( for j being Element of NAT st 1 < j & j <= (s . SBP) + 1 holds
s . (intpos j) = (Comput ((ProgramPart s),s,n)) . (intpos j) ) ) by A1, A2, A3, A4, A5, Th16; :: thesis: verum

end;

suppose s . (DataLoc ((s . SBP),2)) < s . (DataLoc ((s . SBP),3)) ; :: thesis: ex n being Element of NAT st
( CurInstr ((ProgramPart (Comput ((ProgramPart s),s,n))),(Comput ((ProgramPart s),s,n))) = return SBP & s . SBP = (Comput ((ProgramPart s),s,n)) . SBP & (Comput ((ProgramPart s),s,n)) . (DataLoc ((s . SBP),2)) = (s . (DataLoc ((s . SBP),2))) gcd (s . (DataLoc ((s . SBP),3))) & ( for j being Element of NAT st 1 < j & j <= (s . SBP) + 1 holds
s . (intpos j) = (Comput ((ProgramPart s),s,n)) . (intpos j) ) )

then A7: s . (DataLoc ((s . SBP),3)) > 0 by A6, XXREAL_0:2;
reconsider y = s . (DataLoc ((s . SBP),3)) as Element of NAT by A5, INT_1:16;
reconsider pn = s . SBP as Element of NAT by A3, INT_1:16;
A8: pn = s . SBP ;
then A9: IC (Comput ((ProgramPart s),s,7)) = 5 + 7 by A1, A2, A4, A7, Lm4;
A10: Comput ((ProgramPart s),s,8) = Exec ((goto (- 7)),(Comput ((ProgramPart s),s,7))) by A1, A2, A4, A7, A8, Lm4;
A11: (Comput ((ProgramPart s),s,7)) . SBP = pn + 4 by A1, A2, A4, A7, Lm4;
A12: (Comput ((ProgramPart s),s,7)) . GBP = 0 by A1, A2, A4, A7, A8, Lm4;
A13: (Comput ((ProgramPart s),s,7)) . (intpos (pn + 7)) = (s . (DataLoc ((s . SBP),2))) mod y by A1, A2, A4, A7, Lm4;
A14: (Comput ((ProgramPart s),s,7)) . (intpos (pn + 6)) = y by A1, A2, A4, A7, Lm4;
A15: (Comput ((ProgramPart s),s,7)) . (intpos (pn + 4)) = pn by A1, A2, A4, A7, Lm4;
A16: (Comput ((ProgramPart s),s,7)) . (intpos (pn + 5)) = 11 by A1, A2, A4, A7, Lm4;
set s8 = Comput ((ProgramPart s),s,8);
A17: IC (Comput ((ProgramPart s),s,8)) = ICplusConst ((Comput ((ProgramPart s),s,7)),(- 7)) by A10, SCMPDS_2:66
.= 5 by A9, Th6 ;
A18: GCD-Algorithm c= Comput ((ProgramPart s),s,8) by A1, AMI_1:81;
A19: (Comput ((ProgramPart s),s,8)) . SBP = pn + 4 by A10, A11, SCMPDS_2:66;
A20: 4 <= pn + 4 by NAT_1:11;
then A21: (Comput ((ProgramPart s),s,8)) . SBP > 0 by A19, XXREAL_0:2;
A22: (Comput ((ProgramPart s),s,8)) . GBP = 0 by A10, A12, SCMPDS_2:66;
set x1 = (Comput ((ProgramPart s),s,8)) . (DataLoc (((Comput ((ProgramPart s),s,8)) . SBP),2));
set y1 = (Comput ((ProgramPart s),s,8)) . (DataLoc (((Comput ((ProgramPart s),s,8)) . SBP),3));
A23: (Comput ((ProgramPart s),s,8)) . (DataLoc (((Comput ((ProgramPart s),s,8)) . SBP),2)) = (Comput ((ProgramPart s),s,8)) . (intpos ((pn + 4) + 2)) by A19, Th5
.= y by A10, A14, SCMPDS_2:66 ;
A24: (Comput ((ProgramPart s),s,8)) . (DataLoc (((Comput ((ProgramPart s),s,8)) . SBP),3)) = (Comput ((ProgramPart s),s,8)) . (intpos ((pn + 4) + 3)) by A19, Th5
.= (s . (DataLoc ((s . SBP),2))) mod y by A10, A13, SCMPDS_2:66 ;
then (Comput ((ProgramPart s),s,8)) . (DataLoc (((Comput ((ProgramPart s),s,8)) . SBP),3)) < y by A7, NEWTON:79;
then consider m being Element of NAT such that
A25: CurInstr ((ProgramPart (Comput ((ProgramPart (Comput ((ProgramPart s),s,8))),(Comput ((ProgramPart s),s,8)),m))),(Comput ((ProgramPart (Comput ((ProgramPart s),s,8))),(Comput ((ProgramPart s),s,8)),m))) = return SBP and
A26: (Comput ((ProgramPart s),s,8)) . SBP = (Comput ((ProgramPart (Comput ((ProgramPart s),s,8))),(Comput ((ProgramPart s),s,8)),m)) . SBP and
A27: (Comput ((ProgramPart (Comput ((ProgramPart s),s,8))),(Comput ((ProgramPart s),s,8)),m)) . (DataLoc (((Comput ((ProgramPart s),s,8)) . SBP),2)) = ((Comput ((ProgramPart s),s,8)) . (DataLoc (((Comput ((ProgramPart s),s,8)) . SBP),2))) gcd ((Comput ((ProgramPart s),s,8)) . (DataLoc (((Comput ((ProgramPart s),s,8)) . SBP),3))) and
A28: for j being Element of NAT st 1 < j & j <= ((Comput ((ProgramPart s),s,8)) . SBP) + 1 holds
(Comput ((ProgramPart s),s,8)) . (intpos j) = (Comput ((ProgramPart (Comput ((ProgramPart s),s,8))),(Comput ((ProgramPart s),s,8)),m)) . (intpos j) by A5, A17, A18, A21, A22, A23, A24, Th16, NEWTON:78;
set s9 = Comput ((ProgramPart s),s,(m + 8));
T9: ProgramPart s = ProgramPart (Comput ((ProgramPart s),s,8)) by AMI_1:123;
then A29: (Comput ((ProgramPart s),s,8)) . SBP = (Comput ((ProgramPart s),s,(m + 8))) . SBP by A26, EXTPRO_1:5;
T: ProgramPart s = ProgramPart (Comput ((ProgramPart s),s,8)) by AMI_1:123;
x: Comput ((ProgramPart s),s,(m + 8)) = Comput ((ProgramPart s),(Comput ((ProgramPart s),s,8)),m) by EXTPRO_1:5;
S: ProgramPart s = ProgramPart (Comput ((ProgramPart s),s,(m + 8))) by AMI_1:123;
A30: Comput ((ProgramPart s),s,(m + (8 + 1))) = Comput ((ProgramPart s),s,((m + 8) + 1))
.= Following ((ProgramPart s),(Comput ((ProgramPart s),s,(m + 8)))) by EXTPRO_1:4
.= Exec ((return SBP),(Comput ((ProgramPart s),s,(m + 8)))) by A25, x, T, S ;
A31: 1 < pn + 4 by A20, XXREAL_0:2;
pn + 4 < ((Comput ((ProgramPart s),s,8)) . SBP) + 1 by A19, XREAL_1:31;
then A32: (Comput ((ProgramPart s),s,8)) . (intpos (pn + 4)) = (Comput ((ProgramPart (Comput ((ProgramPart s),s,8))),(Comput ((ProgramPart s),s,8)),m)) . (intpos (pn + 4)) by A28, A31
.= (Comput ((ProgramPart s),s,(m + 8))) . (intpos (pn + 4)) by T9, EXTPRO_1:5 ;
5 <= pn + 5 by NAT_1:11;
then A33: 1 < pn + 5 by XXREAL_0:2;
A34: 11 = (Comput ((ProgramPart s),s,8)) . (intpos (pn + 5)) by A10, A16, SCMPDS_2:66
.= (Comput ((ProgramPart (Comput ((ProgramPart s),s,8))),(Comput ((ProgramPart s),s,8)),m)) . (intpos (pn + 5)) by A19, A28, A33
.= (Comput ((ProgramPart s),s,(m + 8))) . (intpos ((pn + 4) + 1)) by T9, EXTPRO_1:5
.= (Comput ((ProgramPart s),s,(m + 8))) . (DataLoc (((Comput ((ProgramPart s),s,(m + 8))) . SBP),RetIC)) by A19, A29, Th5, SCMPDS_1:def 23 ;
Y: (ProgramPart (Comput ((ProgramPart s),s,(m + 9)))) /. (IC (Comput ((ProgramPart s),s,(m + 9)))) = (Comput ((ProgramPart s),s,(m + 9))) . (IC (Comput ((ProgramPart s),s,(m + 9)))) by COMPOS_1:38;
A35: IC (Comput ((ProgramPart s),s,(m + 9))) = (abs ((Comput ((ProgramPart s),s,(m + 8))) . (DataLoc (((Comput ((ProgramPart s),s,(m + 8))) . SBP),RetIC)))) + 2 by A30, SCMPDS_2:70
.= 11 + 2 by A34, ABSVALUE:46 ;
then A36: CurInstr ((ProgramPart (Comput ((ProgramPart s),s,(m + 9)))),(Comput ((ProgramPart s),s,(m + 9)))) = s . 13 by Y, AMI_1:54
.= (SBP,2) := (SBP,6) by A1, Lm1 ;
T: ProgramPart s = ProgramPart (Comput ((ProgramPart s),s,(m + 9))) by AMI_1:123;
A37: Comput ((ProgramPart s),s,(m + (9 + 1))) = Comput ((ProgramPart s),s,((m + 9) + 1))
.= Following ((ProgramPart s),(Comput ((ProgramPart s),s,(m + 9)))) by EXTPRO_1:4
.= Exec (((SBP,2) := (SBP,6)),(Comput ((ProgramPart s),s,(m + 9)))) by A36, T ;
A38: (Comput ((ProgramPart s),s,(m + 9))) . SBP = (Comput ((ProgramPart s),s,(m + 8))) . (DataLoc ((pn + 4),RetSP)) by A19, A29, A30, SCMPDS_2:70
.= (Comput ((ProgramPart s),s,(m + 8))) . (intpos ((pn + 4) + 0)) by Th5, SCMPDS_1:def 22
.= pn by A10, A15, A32, SCMPDS_2:66 ;
A39: (Comput ((ProgramPart s),s,(m + 9))) . (intpos (pn + 6)) = (Comput ((ProgramPart s),s,(m + 8))) . (intpos ((pn + 4) + 2)) by A30, Lm3, SCMPDS_2:70
.= (Comput ((ProgramPart s),s,(m + 8))) . (DataLoc (((Comput ((ProgramPart s),s,8)) . SBP),2)) by A19, Th5
.= ((Comput ((ProgramPart s),s,8)) . (DataLoc (((Comput ((ProgramPart s),s,8)) . SBP),2))) gcd ((Comput ((ProgramPart s),s,8)) . (DataLoc (((Comput ((ProgramPart s),s,8)) . SBP),3))) by A27, T9, EXTPRO_1:5 ;
Y: (ProgramPart (Comput ((ProgramPart s),s,(m + 10)))) /. (IC (Comput ((ProgramPart s),s,(m + 10)))) = (Comput ((ProgramPart s),s,(m + 10))) . (IC (Comput ((ProgramPart s),s,(m + 10)))) by COMPOS_1:38;
IC (Comput ((ProgramPart s),s,(m + 10))) = succ (IC (Comput ((ProgramPart s),s,(m + 9)))) by A37, SCMPDS_2:59
.= 13 + 1 by A35 ;
then A40: CurInstr ((ProgramPart (Comput ((ProgramPart s),s,(m + 10)))),(Comput ((ProgramPart s),s,(m + 10)))) = s . 14 by Y, AMI_1:54
.= return SBP by A1, Lm1 ;

hereby :: thesis: verum

take n = m + 10; :: thesis: ( CurInstr ((ProgramPart (Comput ((ProgramPart s),s,n))),(Comput ((ProgramPart s),s,n))) = return SBP & (Comput ((ProgramPart s),s,n)) . SBP = s . SBP & (Comput ((ProgramPart s),s,n)) . (DataLoc ((s . SBP),2)) = (s . (DataLoc ((s . SBP),2))) gcd (s . (DataLoc ((s . SBP),3))) & ( for j being Element of NAT st 1 < j & j <= (s . SBP) + 1 holds
s . (intpos j) = (Comput ((ProgramPart s),s,n)) . (intpos j) ) )
thus CurInstr ((ProgramPart (Comput ((ProgramPart s),s,n))),(Comput ((ProgramPart s),s,n))) = return SBP by A40; :: thesis: ( (Comput ((ProgramPart s),s,n)) . SBP = s . SBP & (Comput ((ProgramPart s),s,n)) . (DataLoc ((s . SBP),2)) = (s . (DataLoc ((s . SBP),2))) gcd (s . (DataLoc ((s . SBP),3))) & ( for j being Element of NAT st 1 < j & j <= (s . SBP) + 1 holds
s . (intpos j) = (Comput ((ProgramPart s),s,n)) . (intpos j) ) )
A41: DataLoc (((Comput ((ProgramPart s),s,(m + 9))) . SBP),2) = intpos (pn + 2) by A38, Th5;
hence (Comput ((ProgramPart s),s,n)) . SBP = s . SBP by A37, A38, Lm3, SCMPDS_2:59; :: thesis: ( (Comput ((ProgramPart s),s,n)) . (DataLoc ((s . SBP),2)) = (s . (DataLoc ((s . SBP),2))) gcd (s . (DataLoc ((s . SBP),3))) & ( for j being Element of NAT st 1 < j & j <= (s . SBP) + 1 holds
s . (intpos j) = (Comput ((ProgramPart s),s,n)) . (intpos j) ) )
thus (Comput ((ProgramPart s),s,n)) . (DataLoc ((s . SBP),2)) = (Comput ((ProgramPart s),s,(m + 9))) . (DataLoc (pn,6)) by A37, A38, SCMPDS_2:59
.= (s . (DataLoc ((s . SBP),3))) gcd ((s . (DataLoc ((s . SBP),2))) mod (s . (DataLoc ((s . SBP),3)))) by A23, A24, A39, Th5
.= (s . (DataLoc ((s . SBP),2))) gcd (s . (DataLoc ((s . SBP),3))) by A6, A7, NAT_D:30 ; :: thesis: for j being Element of NAT st 1 < j & j <= (s . SBP) + 1 holds
s . (intpos j) = (Comput ((ProgramPart s),s,n)) . (intpos j)

hereby :: thesis: verum

let j be Element of NAT ; :: thesis: ( 1 < j & j <= (s . SBP) + 1 implies s . (intpos j) = (Comput ((ProgramPart s),s,n)) . (intpos j) )
assume that
A42: 1 < j and
A43: j <= (s . SBP) + 1 ; :: thesis: s . (intpos j) = (Comput ((ProgramPart s),s,n)) . (intpos j)
s . SBP <= (Comput ((ProgramPart s),s,8)) . SBP by A19, NAT_1:11;
then (s . SBP) + 1 <= ((Comput ((ProgramPart s),s,8)) . SBP) + 1 by XREAL_1:8;
then A44: j <= ((Comput ((ProgramPart s),s,8)) . SBP) + 1 by A43, XXREAL_0:2;
A45: (Comput ((ProgramPart s),s,(m + 9))) . (intpos j) = (Comput ((ProgramPart s),s,(m + 8))) . (intpos j) by A30, A42, AMI_3:52, SCMPDS_2:70
.= (Comput ((ProgramPart (Comput ((ProgramPart s),s,8))),(Comput ((ProgramPart s),s,8)),m)) . (intpos j) by T9, EXTPRO_1:5
.= (Comput ((ProgramPart s),s,8)) . (intpos j) by A28, A42, A44 ;
A46: pn + 1 < pn + 2 by XREAL_1:8;
(Comput ((ProgramPart s),s,7)) . (intpos j) = s . (intpos j) by A1, A2, A4, A7, A8, A42, A43, Lm5;
hence s . (intpos j) = (Comput ((ProgramPart s),s,8)) . (intpos j) by A10, SCMPDS_2:66
.= (Comput ((ProgramPart s),s,n)) . (intpos j) by A37, A41, A43, A45, A46, AMI_3:52, SCMPDS_2:59 ;
:: thesis: verum

end;

end;

end;

end;