set GA = GCD-Algorithm ;
defpred S₁[ Element of NAT ] means for s being State of st GCD-Algorithm c= s & IC s = inspos 5 & s . SBP > 0 & s . GBP = 0 & s . (DataLoc (s . SBP ),3) <= $1 & s . (DataLoc (s . SBP ),3) >= 0 & s . (DataLoc (s . SBP ),2) >= s . (DataLoc (s . SBP ),3) holds
ex n being Element of NAT st
( CurInstr (Computation s,n) = return SBP & s . SBP = (Computation s,n) . SBP & (Computation 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) = (Computation s,n) . (intpos j) ) );

now

let s be State of ; :: thesis:
set x = s . (DataLoc (s . SBP ),2);
set y = s . (DataLoc (s . SBP ),3);
assume A1: GCD-Algorithm c= s ; :: thesis:
assume A2: IC s = inspos 5 ; :: thesis:
assume s . SBP > 0 ; :: thesis:
assume s . GBP = 0 ; :: thesis:
assume A3: s . (DataLoc (s . SBP ),3) <= 0 ; :: thesis:
assume A4: s . (DataLoc (s . SBP ),3) >= 0 ; :: thesis:
assume A5: s . (DataLoc (s . SBP ),2) >= s . (DataLoc (s . SBP ),3) ; :: thesis:
A6: Computation s,(1 + 0 ) = Following (Computation s,0 ) by AMI_1:14
.= Following s by AMI_1:13
.= Exec (SBP ,3 <=0_goto 9),s by A1, A2, Lm1 ;
then A7: IC (Computation s,1) = ICplusConst s,9 by A3, SCMPDS_2:68
.= inspos (5 + 9) by A2, SCMPDS_6:23 ;
take n = 1; :: thesis:
thus CurInstr (Computation s,n) = s . (inspos 14) by A7, AMI_1:54
.= return SBP by A1, Lm1 ; :: thesis:
thus (Computation s,n) . SBP = s . SBP by A6, SCMPDS_2:68; :: thesis:
A8: s . (DataLoc (s . SBP ),3) = 0 by A3, A4, XXREAL_0:1;
then A9: abs (s . (DataLoc (s . SBP ),3)) = 0 by ABSVALUE:def 1;
thus (Computation s,n) . (DataLoc (s . SBP ),2) = s . (DataLoc (s . SBP ),2) by A6, SCMPDS_2:68
.= abs (s . (DataLoc (s . SBP ),2)) by A5, A8, ABSVALUE:def 1
.= (abs (s . (DataLoc (s . SBP ),2))) gcd (abs (s . (DataLoc (s . SBP ),3))) by A9, NEWTON:65
.= (s . (DataLoc (s . SBP ),2)) gcd (s . (DataLoc (s . SBP ),3)) by INT_2:51 ; :: thesis:
thus for j being Element of NAT st 1 < j & j <= (s . SBP ) + 1 holds
s . (intpos j) = (Computation s,n) . (intpos j) by A6, SCMPDS_2:68; :: thesis:

end;

then A10: S₁[ 0 ] ;

A11: now

let k be Element of NAT ; :: thesis:
assume A12: S₁[k] ; :: thesis:

now

let s be State of ; :: thesis:
set x = s . (DataLoc (s . SBP ),2);
set y = s . (DataLoc (s . SBP ),3);
set yy = s . (DataLoc (s . SBP ),3);
assume A13: GCD-Algorithm c= s ; :: thesis:
assume A14: IC s = inspos 5 ; :: thesis:
assume A15: s . SBP > 0 ; :: thesis:
assume A16: s . GBP = 0 ; :: thesis:
assume A17: s . (DataLoc (s . SBP ),3) <= k + 1 ; :: thesis:
assume A18: s . (DataLoc (s . SBP ),3) >= 0 ; :: thesis:
assume A19: s . (DataLoc (s . SBP ),2) >= s . (DataLoc (s . SBP ),3) ; :: thesis:
then A20: s . (DataLoc (s . SBP ),2) >= 0 by A18, XXREAL_0:2;
reconsider y = s . (DataLoc (s . SBP ),3) as Element of NAT by A18, INT_1:16;

per cases by A17, NAT_1:8;

suppose y <= k ; :: thesis:

hence ex n being Element of NAT st
( CurInstr (Computation s,n) = return SBP & s . SBP = (Computation s,n) . SBP & (Computation 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) = (Computation s,n) . (intpos j) ) ) by A12, A13, A14, A15, A16, A18, A19; :: thesis:

end;

suppose A21: y = k + 1 ; :: thesis:

then A22: y > 0 by NAT_1:5;
reconsider pn = s . SBP as Element of NAT by A15, INT_1:16;
A23: pn = s . SBP ;
then A24: IC (Computation s,7) = inspos (5 + 7) by A13, A14, A16, A22, Lm4;
A25: Computation s,8 = Exec (goto (- 7)),(Computation s,7) by A13, A14, A16, A22, A23, Lm4;
A26: (Computation s,7) . SBP = pn + 4 by A13, A14, A16, A22, Lm4;
A27: (Computation s,7) . GBP = 0 by A13, A14, A16, A22, A23, Lm4;
A28: (Computation s,7) . (intpos (pn + 7)) = (s . (DataLoc (s . SBP ),2)) mod y by A13, A14, A16, A22, Lm4;
A29: (Computation s,7) . (intpos (pn + 6)) = y by A13, A14, A16, A22, Lm4;
A30: (Computation s,7) . (intpos (pn + 4)) = pn by A13, A14, A16, A22, Lm4;
A31: (Computation s,7) . (intpos (pn + 5)) = inspos 11 by A13, A14, A16, A22, Lm4;
set s8 = Computation s,8;
A32: IC (Computation s,8) = ICplusConst (Computation s,7),(- 7) by A25, SCMPDS_2:66
.= inspos 5 by A24, Th6 ;
A33: GCD-Algorithm c= Computation s,8 by A13, AMI_1:81;
A34: (Computation s,8) . SBP = pn + 4 by A25, A26, SCMPDS_2:66;
A35: 4 <= pn + 4 by NAT_1:11;
then A36: (Computation s,8) . SBP > 0 by A34, XXREAL_0:2;
A37: (Computation s,8) . GBP = 0 by A25, A27, SCMPDS_2:66;
set x1 = (Computation s,8) . (DataLoc ((Computation s,8) . SBP ),2);
set y1 = (Computation s,8) . (DataLoc ((Computation s,8) . SBP ),3);
A38: (Computation s,8) . (DataLoc ((Computation s,8) . SBP ),2) = (Computation s,8) . (intpos ((pn + 4) + 2)) by A34, Th5
.= y by A25, A29, SCMPDS_2:66 ;
A39: (Computation s,8) . (DataLoc ((Computation s,8) . SBP ),3) = (Computation s,8) . (intpos ((pn + 4) + 3)) by A34, Th5
.= (s . (DataLoc (s . SBP ),2)) mod y by A25, A28, SCMPDS_2:66 ;
then A40: (Computation s,8) . (DataLoc ((Computation s,8) . SBP ),3) < y by A21, NAT_1:5, NEWTON:79;
then (Computation s,8) . (DataLoc ((Computation s,8) . SBP ),3) <= k by A21, INT_1:20;
then consider m being Element of NAT such that
A41: CurInstr (Computation (Computation s,8),m) = return SBP and
A42: (Computation s,8) . SBP = (Computation (Computation s,8),m) . SBP and
A43: (Computation (Computation s,8),m) . (DataLoc ((Computation s,8) . SBP ),2) = ((Computation s,8) . (DataLoc ((Computation s,8) . SBP ),2)) gcd ((Computation s,8) . (DataLoc ((Computation s,8) . SBP ),3)) and
A44: for j being Element of NAT st 1 < j & j <= ((Computation s,8) . SBP ) + 1 holds
(Computation s,8) . (intpos j) = (Computation (Computation s,8),m) . (intpos j) by A12, A22, A32, A33, A36, A37, A38, A39, A40, NEWTON:78;
set s9 = Computation s,(m + 8);
A45: (Computation s,8) . SBP = (Computation s,(m + 8)) . SBP by A42, AMI_1:51;
A46: Computation s,(m + (8 + 1)) = Computation s,((m + 8) + 1)
.= Following (Computation s,(m + 8)) by AMI_1:14
.= Exec (return SBP ),(Computation s,(m + 8)) by A41, AMI_1:51 ;
A47: 1 < pn + 4 by A35, XXREAL_0:2;
pn + 4 < ((Computation s,8) . SBP ) + 1 by A34, XREAL_1:31;
then A48: (Computation s,8) . (intpos (pn + 4)) = (Computation (Computation s,8),m) . (intpos (pn + 4)) by A44, A47
.= (Computation s,(m + 8)) . (intpos (pn + 4)) by AMI_1:51 ;
5 <= pn + 5 by NAT_1:11;
then A49: 1 < pn + 5 by XXREAL_0:2;
A50: inspos 11 = (Computation s,8) . (intpos (pn + 5)) by A25, A31, SCMPDS_2:66
.= (Computation (Computation s,8),m) . (intpos (pn + 5)) by A34, A44, A49
.= (Computation s,(m + 8)) . (intpos ((pn + 4) + 1)) by AMI_1:51
.= (Computation s,(m + 8)) . (DataLoc ((Computation s,(m + 8)) . SBP ),RetIC ) by A34, A45, Th5, SCMPDS_1:def 23 ;
A51: IC (Computation s,(m + 9)) = (abs ((Computation s,(m + 8)) . (DataLoc ((Computation s,(m + 8)) . SBP ),RetIC ))) + 2 by A46, SCMPDS_2:70
.= inspos (11 + 2) by A50, Th3 ;
then A52: CurInstr (Computation s,(m + 9)) = s . (inspos 13) by AMI_1:54
.= SBP ,2 := SBP ,6 by A13, Lm1 ;
A53: Computation s,(m + (9 + 1)) = Computation s,((m + 9) + 1)
.= Following (Computation s,(m + 9)) by AMI_1:14
.= Exec (SBP ,2 := SBP ,6),(Computation s,(m + 9)) by A52 ;
A54: (Computation s,(m + 9)) . SBP = (Computation s,(m + 8)) . (DataLoc (pn + 4),RetSP ) by A34, A45, A46, SCMPDS_2:70
.= (Computation s,(m + 8)) . (intpos ((pn + 4) + 0 )) by Th5, SCMPDS_1:def 22
.= pn by A25, A30, A48, SCMPDS_2:66 ;
A55: (Computation s,(m + 9)) . (intpos (pn + 6)) = (Computation s,(m + 8)) . (intpos ((pn + 4) + 2)) by A46, Lm3, SCMPDS_2:70
.= (Computation s,(m + 8)) . (DataLoc ((Computation s,8) . SBP ),2) by A34, Th5
.= ((Computation s,8) . (DataLoc ((Computation s,8) . SBP ),2)) gcd ((Computation s,8) . (DataLoc ((Computation s,8) . SBP ),3)) by A43, AMI_1:51 ;
IC (Computation s,(m + 10)) = Next (IC (Computation s,(m + 9))) by A53, SCMPDS_2:59
.= inspos (13 + 1) by A51 ;
then A56: CurInstr (Computation s,(m + 10)) = s . (inspos 14) by AMI_1:54
.= return SBP by A13, Lm1 ;

hereby :: thesis:

take n = m + 10; :: thesis:
thus CurInstr (Computation s,n) = return SBP by A56; :: thesis:
A57: DataLoc ((Computation s,(m + 9)) . SBP ),2 = intpos (pn + 2) by A54, Th5;
hence (Computation s,n) . SBP = s . SBP by A53, A54, Lm3, SCMPDS_2:59; :: thesis:
thus (Computation s,n) . (DataLoc (s . SBP ),2) = (Computation s,(m + 9)) . (DataLoc pn,6) by A53, A54, SCMPDS_2:59
.= (s . (DataLoc (s . SBP ),3)) gcd ((s . (DataLoc (s . SBP ),2)) mod (s . (DataLoc (s . SBP ),3))) by A38, A39, A55, Th5
.= (s . (DataLoc (s . SBP ),2)) gcd (s . (DataLoc (s . SBP ),3)) by A20, A21, NAT_1:5, NAT_D:30 ; :: thesis:

hereby :: thesis:

let j be Element of NAT ; :: thesis:
assume that
A58: 1 < j and
A59: j <= (s . SBP ) + 1 ; :: thesis:
s . SBP <= (Computation s,8) . SBP by A34, NAT_1:11;
then (s . SBP ) + 1 <= ((Computation s,8) . SBP ) + 1 by XREAL_1:8;
then A60: j <= ((Computation s,8) . SBP ) + 1 by A59, XXREAL_0:2;
A61: (Computation s,(m + 9)) . (intpos j) = (Computation s,(m + 8)) . (intpos j) by A46, A58, AMI_3:52, SCMPDS_2:70
.= (Computation (Computation s,8),m) . (intpos j) by AMI_1:51
.= (Computation s,8) . (intpos j) by A44, A58, A60 ;
A62: pn + 1 < pn + 2 by XREAL_1:8;
(Computation s,7) . (intpos j) = s . (intpos j) by A13, A14, A16, A21, A23, A58, A59, Lm5, NAT_1:5;
hence s . (intpos j) = (Computation s,8) . (intpos j) by A25, SCMPDS_2:66
.= (Computation s,n) . (intpos j) by A53, A57, A59, A61, A62, AMI_3:52, SCMPDS_2:59 ;
:: thesis:

end;

end;

end;

end;

end;

hence S₁[k + 1] ; :: thesis:

end;

A63: for n being Element of NAT holds S₁[n] from NAT_1:sch 1(A10, A11);
let s be State of ; :: thesis:
assume that
A64: GCD-Algorithm c= s and
A65: IC s = inspos 5 and
A66: s . SBP > 0 and
A67: s . GBP = 0 and
A68: s . (DataLoc (s . SBP ),3) >= 0 and
A69: s . (DataLoc (s . SBP ),2) >= s . (DataLoc (s . SBP ),3) ; :: thesis:
reconsider m = s . (DataLoc (s . SBP ),3) as Element of NAT by A68, INT_1:16;
S₁[m] by A63;
hence ex n being Element of NAT st
( CurInstr (Computation s,n) = return SBP & s . SBP = (Computation s,n) . SBP & (Computation 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) = (Computation s,n) . (intpos j) ) ) by A64, A65, A66, A67, A68, A69; :: thesis: