let w be FinSequence of INT ; :: thesis: Initialized ((fsloc 0) .--> w) is Bubble-Sort-Algorithm -autonomic
set p = Initialized ((fsloc 0) .--> w);
set q = Bubble-Sort-Algorithm ;
A1: for P, Q being Instruction-Sequence of SCM+FSA st Bubble-Sort-Algorithm c= P & Bubble-Sort-Algorithm c= Q holds
for s1, s2 being State of SCM+FSA
for i being Nat st Initialized ((fsloc 0) .--> w) c= s1 & Initialized ((fsloc 0) .--> w) c= s2 & i <= 10 holds
( (Comput (P,s1,i)) . (intloc 0) = (Comput (Q,s2,i)) . (intloc 0) & (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) )

proof

let P, Q be Instruction-Sequence of SCM+FSA; :: thesis: ( Bubble-Sort-Algorithm c= P & Bubble-Sort-Algorithm c= Q implies for s1, s2 being State of SCM+FSA
for i being Nat st Initialized ((fsloc 0) .--> w) c= s1 & Initialized ((fsloc 0) .--> w) c= s2 & i <= 10 holds
( (Comput (P,s1,i)) . (intloc 0) = (Comput (Q,s2,i)) . (intloc 0) & (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) ) )
assume A2: ( Bubble-Sort-Algorithm c= P & Bubble-Sort-Algorithm c= Q ) ; :: thesis: for s1, s2 being State of SCM+FSA
for i being Nat st Initialized ((fsloc 0) .--> w) c= s1 & Initialized ((fsloc 0) .--> w) c= s2 & i <= 10 holds
( (Comput (P,s1,i)) . (intloc 0) = (Comput (Q,s2,i)) . (intloc 0) & (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) )
let s1, s2 be State of SCM+FSA; :: thesis: for i being Nat st Initialized ((fsloc 0) .--> w) c= s1 & Initialized ((fsloc 0) .--> w) c= s2 & i <= 10 holds
( (Comput (P,s1,i)) . (intloc 0) = (Comput (Q,s2,i)) . (intloc 0) & (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) )
let i be Nat; :: thesis: ( Initialized ((fsloc 0) .--> w) c= s1 & Initialized ((fsloc 0) .--> w) c= s2 & i <= 10 implies ( (Comput (P,s1,i)) . (intloc 0) = (Comput (Q,s2,i)) . (intloc 0) & (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) ) )
assume that
A3: Initialized ((fsloc 0) .--> w) c= s1 and
A4: Initialized ((fsloc 0) .--> w) c= s2 and
A5: i <= 10 ; :: thesis: ( (Comput (P,s1,i)) . (intloc 0) = (Comput (Q,s2,i)) . (intloc 0) & (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) )
A6: Bubble-Sort-Algorithm c= P by A2;
A7: Bubble-Sort-Algorithm c= Q by A2;
A8: s1 is 0 -started by A3, MEMSTR_0:17;
A9: s2 is 0 -started by A4, MEMSTR_0:17;
A10: s1 . (intloc 0) = 1 by A3, SCMFSA_M:33
.= s2 . (intloc 0) by A4, SCMFSA_M:33 ;
A11: s1 . (fsloc 0) = w by A3, SCMFSA_M:33
.= s2 . (fsloc 0) by A4, SCMFSA_M:33 ;
A12: IC s1 = 0 by A8
.= IC s2 by A9 ;
not not i = 0 & ... & not i = 10 by A5;

per cases then ( i = 0 or i = 1 or i = 2 or i = 3 or i = 4 or i = 5 or i = 6 or i = 7 or i = 8 or i = 9 or i = 10 ) ;

suppose A13: i = 0 ; :: thesis: ( (Comput (P,s1,i)) . (intloc 0) = (Comput (Q,s2,i)) . (intloc 0) & (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) )

hence (Comput (P,s1,i)) . (intloc 0) = (Comput (Q,s2,i)) . (intloc 0) by A10; :: thesis: ( (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) )
thus (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) by A12, A13; :: thesis: (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0)
thus (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) by A11, A13; :: thesis: verum

end;

suppose A14: i = 1 ; :: thesis: ( (Comput (P,s1,i)) . (intloc 0) = (Comput (Q,s2,i)) . (intloc 0) & (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) )

hence (Comput (P,s1,i)) . (intloc 0) = s1 . (intloc 0) by A2, A8, Lm21
.= (Comput (Q,s2,i)) . (intloc 0) by A2, A9, A10, A14, Lm21 ;
:: thesis: ( (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) )
thus (Comput (P,s1,i)) . (IC ) = 1 by A2, A8, A14, Lm21
.= (Comput (Q,s2,i)) . (IC ) by A7, A9, A14, Lm21 ; :: thesis: (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0)
thus (Comput (P,s1,i)) . (fsloc 0) = s1 . (fsloc 0) by A6, A8, A14, Lm21
.= (Comput (Q,s2,i)) . (fsloc 0) by A2, A9, A11, A14, Lm21 ; :: thesis: verum

end;

suppose A15: i = 2 ; :: thesis: ( (Comput (P,s1,i)) . (intloc 0) = (Comput (Q,s2,i)) . (intloc 0) & (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) )

hence (Comput (P,s1,i)) . (intloc 0) = s1 . (intloc 0) by A6, A8, Lm21
.= (Comput (Q,s2,i)) . (intloc 0) by A2, A9, A10, A15, Lm21 ;
:: thesis: ( (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) )
thus (Comput (P,s1,i)) . (IC ) = 2 by A2, A8, A15, Lm21
.= (Comput (Q,s2,i)) . (IC ) by A2, A9, A15, Lm21 ; :: thesis: (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0)
thus (Comput (P,s1,i)) . (fsloc 0) = s1 . (fsloc 0) by A2, A8, A15, Lm21
.= (Comput (Q,s2,i)) . (fsloc 0) by A2, A9, A11, A15, Lm21 ; :: thesis: verum

end;

suppose A16: i = 3 ; :: thesis: ( (Comput (P,s1,i)) . (intloc 0) = (Comput (Q,s2,i)) . (intloc 0) & (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) )

hence (Comput (P,s1,i)) . (intloc 0) = s1 . (intloc 0) by A2, A8, Lm21
.= (Comput (Q,s2,i)) . (intloc 0) by A2, A9, A10, A16, Lm21 ;
:: thesis: ( (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) )
thus (Comput (P,s1,i)) . (IC ) = 3 by A2, A8, A16, Lm21
.= (Comput (Q,s2,i)) . (IC ) by A2, A9, A16, Lm21 ; :: thesis: (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0)
thus (Comput (P,s1,i)) . (fsloc 0) = s1 . (fsloc 0) by A6, A8, A16, Lm21
.= (Comput (Q,s2,i)) . (fsloc 0) by A2, A9, A11, A16, Lm21 ; :: thesis: verum

end;

suppose A17: i = 4 ; :: thesis: ( (Comput (P,s1,i)) . (intloc 0) = (Comput (Q,s2,i)) . (intloc 0) & (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) )

hence (Comput (P,s1,i)) . (intloc 0) = s1 . (intloc 0) by A6, A8, Lm21
.= (Comput (Q,s2,i)) . (intloc 0) by A2, A9, A10, A17, Lm21 ;
:: thesis: ( (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) )
thus (Comput (P,s1,i)) . (IC ) = 4 by A2, A8, A17, Lm21
.= (Comput (Q,s2,i)) . (IC ) by A2, A9, A17, Lm21 ; :: thesis: (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0)
thus (Comput (P,s1,i)) . (fsloc 0) = s1 . (fsloc 0) by A2, A8, A17, Lm21
.= (Comput (Q,s2,i)) . (fsloc 0) by A2, A9, A11, A17, Lm21 ; :: thesis: verum

end;

suppose A18: i = 5 ; :: thesis: ( (Comput (P,s1,i)) . (intloc 0) = (Comput (Q,s2,i)) . (intloc 0) & (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) )

hence (Comput (P,s1,i)) . (intloc 0) = s1 . (intloc 0) by A6, A8, Lm21
.= (Comput (Q,s2,i)) . (intloc 0) by A2, A9, A10, A18, Lm21 ;
:: thesis: ( (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) )
thus (Comput (P,s1,i)) . (IC ) = 5 by A2, A8, A18, Lm21
.= (Comput (Q,s2,i)) . (IC ) by A2, A9, A18, Lm21 ; :: thesis: (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0)
thus (Comput (P,s1,i)) . (fsloc 0) = s1 . (fsloc 0) by A2, A8, A18, Lm21
.= (Comput (Q,s2,i)) . (fsloc 0) by A2, A9, A11, A18, Lm21 ; :: thesis: verum

end;

suppose A19: i = 6 ; :: thesis: ( (Comput (P,s1,i)) . (intloc 0) = (Comput (Q,s2,i)) . (intloc 0) & (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) )

hence (Comput (P,s1,i)) . (intloc 0) = s1 . (intloc 0) by A2, A8, Lm21
.= (Comput (Q,s2,i)) . (intloc 0) by A2, A9, A10, A19, Lm21 ;
:: thesis: ( (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) )
thus (Comput (P,s1,i)) . (IC ) = 6 by A2, A8, A19, Lm21
.= (Comput (Q,s2,i)) . (IC ) by A2, A9, A19, Lm21 ; :: thesis: (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0)
thus (Comput (P,s1,i)) . (fsloc 0) = s1 . (fsloc 0) by A2, A8, A19, Lm21
.= (Comput (Q,s2,i)) . (fsloc 0) by A2, A9, A11, A19, Lm21 ; :: thesis: verum

end;

suppose A20: i = 7 ; :: thesis: ( (Comput (P,s1,i)) . (intloc 0) = (Comput (Q,s2,i)) . (intloc 0) & (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) )

hence (Comput (P,s1,i)) . (intloc 0) = s1 . (intloc 0) by A6, A8, Lm21
.= (Comput (Q,s2,i)) . (intloc 0) by A2, A9, A10, A20, Lm21 ;
:: thesis: ( (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) )
thus (Comput (P,s1,i)) . (IC ) = 7 by A2, A8, A20, Lm21
.= (Comput (Q,s2,i)) . (IC ) by A2, A9, A20, Lm21 ; :: thesis: (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0)
thus (Comput (P,s1,i)) . (fsloc 0) = s1 . (fsloc 0) by A2, A8, A20, Lm21
.= (Comput (Q,s2,i)) . (fsloc 0) by A2, A9, A11, A20, Lm21 ; :: thesis: verum

end;

suppose A21: i = 8 ; :: thesis: ( (Comput (P,s1,i)) . (intloc 0) = (Comput (Q,s2,i)) . (intloc 0) & (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) )

hence (Comput (P,s1,i)) . (intloc 0) = s1 . (intloc 0) by A2, A8, Lm21
.= (Comput (Q,s2,i)) . (intloc 0) by A2, A9, A10, A21, Lm21 ;
:: thesis: ( (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) )
thus (Comput (P,s1,i)) . (IC ) = 8 by A2, A8, A21, Lm21
.= (Comput (Q,s2,i)) . (IC ) by A2, A9, A21, Lm21 ; :: thesis: (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0)
thus (Comput (P,s1,i)) . (fsloc 0) = s1 . (fsloc 0) by A2, A8, A21, Lm21
.= (Comput (Q,s2,i)) . (fsloc 0) by A2, A9, A11, A21, Lm21 ; :: thesis: verum

end;

suppose A22: i = 9 ; :: thesis: ( (Comput (P,s1,i)) . (intloc 0) = (Comput (Q,s2,i)) . (intloc 0) & (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) )

hence (Comput (P,s1,i)) . (intloc 0) = s1 . (intloc 0) by A2, A8, Lm21
.= (Comput (Q,s2,i)) . (intloc 0) by A2, A9, A10, A22, Lm21 ;
:: thesis: ( (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) )
thus (Comput (P,s1,i)) . (IC ) = 9 by A2, A8, A22, Lm21
.= (Comput (Q,s2,i)) . (IC ) by A2, A9, A22, Lm21 ; :: thesis: (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0)
thus (Comput (P,s1,i)) . (fsloc 0) = s1 . (fsloc 0) by A2, A8, A22, Lm21
.= (Comput (Q,s2,i)) . (fsloc 0) by A2, A9, A11, A22, Lm21 ; :: thesis: verum

end;

suppose A23: i = 10 ; :: thesis: ( (Comput (P,s1,i)) . (intloc 0) = (Comput (Q,s2,i)) . (intloc 0) & (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) )

hence (Comput (P,s1,i)) . (intloc 0) = s1 . (intloc 0) by A2, A8, Lm21
.= (Comput (Q,s2,i)) . (intloc 0) by A2, A9, A10, A23, Lm21 ;
:: thesis: ( (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) )
thus (Comput (P,s1,i)) . (IC ) = 10 by A2, A8, A23, Lm21
.= (Comput (Q,s2,i)) . (IC ) by A2, A9, A23, Lm21 ; :: thesis: (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0)
thus (Comput (P,s1,i)) . (fsloc 0) = s1 . (fsloc 0) by A2, A8, A23, Lm21
.= (Comput (Q,s2,i)) . (fsloc 0) by A2, A9, A11, A23, Lm21 ; :: thesis: verum

end;

end;

end;

set UD = {(fsloc 0),(intloc 0),(intloc 1),(intloc 2),(intloc 3),(intloc 4),(intloc 5),(intloc 6)};
set Us = (UsedI*Loc Bubble-Sort-Algorithm) \/ (UsedILoc Bubble-Sort-Algorithm);
A24: UsedI*Loc Bubble-Sort-Algorithm = {(fsloc 0)} by Th22;
A25: UsedILoc Bubble-Sort-Algorithm = {(intloc 0),(intloc 1),(intloc 2),(intloc 3),(intloc 4),(intloc 5),(intloc 6)} by Th21;
then A26: (UsedI*Loc Bubble-Sort-Algorithm) \/ (UsedILoc Bubble-Sort-Algorithm) = {(fsloc 0),(intloc 0),(intloc 1),(intloc 2),(intloc 3),(intloc 4),(intloc 5),(intloc 6)} by A24, ENUMSET1:22;
A27: for P, Q being Instruction-Sequence of SCM+FSA st Bubble-Sort-Algorithm c= P & Bubble-Sort-Algorithm c= Q holds
for i being Nat
for s1, s2 being State of SCM+FSA st 11 <= i & Initialized ((fsloc 0) .--> w) c= s1 & Initialized ((fsloc 0) .--> w) c= s2 holds
( (Comput (P,s1,i)) | ((UsedI*Loc Bubble-Sort-Algorithm) \/ (UsedILoc Bubble-Sort-Algorithm)) = (Comput (Q,s2,i)) | ((UsedI*Loc Bubble-Sort-Algorithm) \/ (UsedILoc Bubble-Sort-Algorithm)) & (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) )

proof

let P, Q be Instruction-Sequence of SCM+FSA; :: thesis: ( Bubble-Sort-Algorithm c= P & Bubble-Sort-Algorithm c= Q implies for i being Nat
for s1, s2 being State of SCM+FSA st 11 <= i & Initialized ((fsloc 0) .--> w) c= s1 & Initialized ((fsloc 0) .--> w) c= s2 holds
( (Comput (P,s1,i)) | ((UsedI*Loc Bubble-Sort-Algorithm) \/ (UsedILoc Bubble-Sort-Algorithm)) = (Comput (Q,s2,i)) | ((UsedI*Loc Bubble-Sort-Algorithm) \/ (UsedILoc Bubble-Sort-Algorithm)) & (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) ) )
assume A28: ( Bubble-Sort-Algorithm c= P & Bubble-Sort-Algorithm c= Q ) ; :: thesis: for i being Nat
for s1, s2 being State of SCM+FSA st 11 <= i & Initialized ((fsloc 0) .--> w) c= s1 & Initialized ((fsloc 0) .--> w) c= s2 holds
( (Comput (P,s1,i)) | ((UsedI*Loc Bubble-Sort-Algorithm) \/ (UsedILoc Bubble-Sort-Algorithm)) = (Comput (Q,s2,i)) | ((UsedI*Loc Bubble-Sort-Algorithm) \/ (UsedILoc Bubble-Sort-Algorithm)) & (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) )
let i be Nat; :: thesis: for s1, s2 being State of SCM+FSA st 11 <= i & Initialized ((fsloc 0) .--> w) c= s1 & Initialized ((fsloc 0) .--> w) c= s2 holds
( (Comput (P,s1,i)) | ((UsedI*Loc Bubble-Sort-Algorithm) \/ (UsedILoc Bubble-Sort-Algorithm)) = (Comput (Q,s2,i)) | ((UsedI*Loc Bubble-Sort-Algorithm) \/ (UsedILoc Bubble-Sort-Algorithm)) & (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) )
let s1, s2 be State of SCM+FSA; :: thesis: ( 11 <= i & Initialized ((fsloc 0) .--> w) c= s1 & Initialized ((fsloc 0) .--> w) c= s2 implies ( (Comput (P,s1,i)) | ((UsedI*Loc Bubble-Sort-Algorithm) \/ (UsedILoc Bubble-Sort-Algorithm)) = (Comput (Q,s2,i)) | ((UsedI*Loc Bubble-Sort-Algorithm) \/ (UsedILoc Bubble-Sort-Algorithm)) & (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) ) )
assume that
A29: 11 <= i and
A30: Initialized ((fsloc 0) .--> w) c= s1 and
A31: Initialized ((fsloc 0) .--> w) c= s2 ; :: thesis: ( (Comput (P,s1,i)) | ((UsedI*Loc Bubble-Sort-Algorithm) \/ (UsedILoc Bubble-Sort-Algorithm)) = (Comput (Q,s2,i)) | ((UsedI*Loc Bubble-Sort-Algorithm) \/ (UsedILoc Bubble-Sort-Algorithm)) & (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) )
set Cs11 = Comput (P,s1,11);
set Cs21 = Comput (Q,s2,11);
A32: s1 is 0 -started by A30, MEMSTR_0:17;
A33: s2 is 0 -started by A31, MEMSTR_0:17;
A34: s1 . (intloc 0) = 1 by A30, SCMFSA_M:33
.= s2 . (intloc 0) by A31, SCMFSA_M:33 ;
A35: s1 . (fsloc 0) = w by A30, SCMFSA_M:33
.= s2 . (fsloc 0) by A31, SCMFSA_M:33 ;
A36: (UsedI*Loc Bubble-Sort-Algorithm) \/ (UsedILoc Bubble-Sort-Algorithm) c= dom (Comput (P,s1,11)) by Th19;
A37: (UsedI*Loc Bubble-Sort-Algorithm) \/ (UsedILoc Bubble-Sort-Algorithm) c= dom (Comput (Q,s2,11)) by Th19;

now :: thesis: for x being set st x in (UsedI*Loc Bubble-Sort-Algorithm) \/ (UsedILoc Bubble-Sort-Algorithm) holds
(Comput (P,s1,11)) . x = (Comput (Q,s2,11)) . x

let x be set ; :: thesis: ( x in (UsedI*Loc Bubble-Sort-Algorithm) \/ (UsedILoc Bubble-Sort-Algorithm) implies (Comput (P,s1,11)) . b₁ = (Comput (Q,s2,11)) . b₁ )
assume x in (UsedI*Loc Bubble-Sort-Algorithm) \/ (UsedILoc Bubble-Sort-Algorithm) ; :: thesis: (Comput (P,s1,11)) . b₁ = (Comput (Q,s2,11)) . b₁
then A38: x in {(fsloc 0),(intloc 0),(intloc 1),(intloc 2),(intloc 3),(intloc 4),(intloc 5),(intloc 6)} by A24, A25, ENUMSET1:22;

per cases ( x = fsloc 0 or x = intloc 0 or x = intloc 1 or x = intloc 2 or x = intloc 3 or x = intloc 4 or x = intloc 5 or x = intloc 6 ) by A38, ENUMSET1:def 6;

suppose A39: x = fsloc 0 ; :: thesis: (Comput (P,s1,11)) . b₁ = (Comput (Q,s2,11)) . b₁

hence (Comput (P,s1,11)) . x = s1 . (fsloc 0) by A32, A28, Lm21
.= (Comput (Q,s2,11)) . x by A33, A28, A35, A39, Lm21 ;
:: thesis: verum

end;

suppose A40: x = intloc 0 ; :: thesis: (Comput (P,s1,11)) . b₁ = (Comput (Q,s2,11)) . b₁

hence (Comput (P,s1,11)) . x = s1 . (intloc 0) by A32, A28, Lm21
.= (Comput (Q,s2,11)) . x by A33, A28, A34, A40, Lm21 ;
:: thesis: verum

end;

suppose A41: x = intloc 1 ; :: thesis: (Comput (P,s1,11)) . b₁ = (Comput (Q,s2,11)) . b₁

hence (Comput (P,s1,11)) . x = len (s1 . (fsloc 0)) by A32, A28, Lm21
.= (Comput (Q,s2,11)) . x by A33, A28, A35, A41, Lm21 ;
:: thesis: verum

end;

suppose A42: x = intloc 2 ; :: thesis: (Comput (P,s1,11)) . b₁ = (Comput (Q,s2,11)) . b₁

hence (Comput (P,s1,11)) . x = s1 . (intloc 0) by A32, A28, Lm21
.= (Comput (Q,s2,11)) . x by A33, A28, A34, A42, Lm21 ;
:: thesis: verum

end;

suppose A43: x = intloc 3 ; :: thesis: (Comput (P,s1,11)) . b₁ = (Comput (Q,s2,11)) . b₁

hence (Comput (P,s1,11)) . x = s1 . (intloc 0) by A32, A28, Lm21
.= (Comput (Q,s2,11)) . x by A33, A28, A34, A43, Lm21 ;
:: thesis: verum

end;

suppose A44: x = intloc 4 ; :: thesis: (Comput (P,s1,11)) . b₁ = (Comput (Q,s2,11)) . b₁

hence (Comput (P,s1,11)) . x = s1 . (intloc 0) by A32, A28, Lm21
.= (Comput (Q,s2,11)) . x by A33, A28, A34, A44, Lm21 ;
:: thesis: verum

end;

suppose A45: x = intloc 5 ; :: thesis: (Comput (P,s1,11)) . b₁ = (Comput (Q,s2,11)) . b₁

hence (Comput (P,s1,11)) . x = s1 . (intloc 0) by A32, A28, Lm21
.= (Comput (Q,s2,11)) . x by A33, A28, A34, A45, Lm21 ;
:: thesis: verum

end;

suppose A46: x = intloc 6 ; :: thesis: (Comput (P,s1,11)) . b₁ = (Comput (Q,s2,11)) . b₁

hence (Comput (P,s1,11)) . x = s1 . (intloc 0) by A32, A28, Lm21
.= (Comput (Q,s2,11)) . x by A33, A28, A34, A46, Lm21 ;
:: thesis: verum

end;

end;

end;

then A47: (Comput (P,s1,11)) | ((UsedI*Loc Bubble-Sort-Algorithm) \/ (UsedILoc Bubble-Sort-Algorithm)) = (Comput (Q,s2,11)) | ((UsedI*Loc Bubble-Sort-Algorithm) \/ (UsedILoc Bubble-Sort-Algorithm)) by A36, A37, FUNCT_1:95;
A48: (Comput (P,s1,11)) . (IC ) = 11 by A32, A28, Lm21
.= (Comput (Q,s2,11)) . (IC ) by A33, A28, Lm21 ;
A49: for i being Nat holds IC (Comput (P,s1,i)) in dom Bubble-Sort-Algorithm by A30, Th27, A28;
for i being Nat holds IC (Comput (Q,s2,i)) in dom Bubble-Sort-Algorithm by A31, Th27, A28;
hence ( (Comput (P,s1,i)) | ((UsedI*Loc Bubble-Sort-Algorithm) \/ (UsedILoc Bubble-Sort-Algorithm)) = (Comput (Q,s2,i)) | ((UsedI*Loc Bubble-Sort-Algorithm) \/ (UsedILoc Bubble-Sort-Algorithm)) & (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) ) by A29, A47, A48, A49, Th14, A28; :: thesis: verum

end;

set DD = {(intloc 0),(IC ),(fsloc 0)};
let P, Q be Instruction-Sequence of SCM+FSA; :: according to EXTPRO_1:def 10 :: thesis: ( not Bubble-Sort-Algorithm c= P or not Bubble-Sort-Algorithm c= Q or for b₁, b₂ being set holds
( not Initialized ((fsloc 0) .--> w) c= b₁ or not Initialized ((fsloc 0) .--> w) c= b₂ or for b₃ being set holds (Comput (P,b₁,b₃)) | (dom (Initialized ((fsloc 0) .--> w))) = (Comput (Q,b₂,b₃)) | (dom (Initialized ((fsloc 0) .--> w))) ) )
assume A50: ( Bubble-Sort-Algorithm c= P & Bubble-Sort-Algorithm c= Q ) ; :: thesis: for b₁, b₂ being set holds
( not Initialized ((fsloc 0) .--> w) c= b₁ or not Initialized ((fsloc 0) .--> w) c= b₂ or for b₃ being set holds (Comput (P,b₁,b₃)) | (dom (Initialized ((fsloc 0) .--> w))) = (Comput (Q,b₂,b₃)) | (dom (Initialized ((fsloc 0) .--> w))) )
let s1, s2 be State of SCM+FSA; :: thesis: ( not Initialized ((fsloc 0) .--> w) c= s1 or not Initialized ((fsloc 0) .--> w) c= s2 or for b₁ being set holds (Comput (P,s1,b₁)) | (dom (Initialized ((fsloc 0) .--> w))) = (Comput (Q,s2,b₁)) | (dom (Initialized ((fsloc 0) .--> w))) )
assume that
A51: Initialized ((fsloc 0) .--> w) c= s1 and
A52: Initialized ((fsloc 0) .--> w) c= s2 ; :: thesis: for b₁ being set holds (Comput (P,s1,b₁)) | (dom (Initialized ((fsloc 0) .--> w))) = (Comput (Q,s2,b₁)) | (dom (Initialized ((fsloc 0) .--> w)))
let i be Nat; :: thesis: (Comput (P,s1,i)) | (dom (Initialized ((fsloc 0) .--> w))) = (Comput (Q,s2,i)) | (dom (Initialized ((fsloc 0) .--> w)))
reconsider i = i as Nat ;
set Cs1i = Comput (P,s1,i);
set Cs2i = Comput (Q,s2,i);
A53: dom (Initialized ((fsloc 0) .--> w)) = {(intloc 0),(IC ),(fsloc 0)} by SCMFSA_M:31;
A54: {(intloc 0),(IC ),(fsloc 0)} c= dom (Comput (P,s1,i)) by SCMFSA_M:34;
A55: {(intloc 0),(IC ),(fsloc 0)} c= dom (Comput (Q,s2,i)) by SCMFSA_M:34;
A56: intloc 0 in (UsedI*Loc Bubble-Sort-Algorithm) \/ (UsedILoc Bubble-Sort-Algorithm) by A26, ENUMSET1:def 6;
A57: fsloc 0 in (UsedI*Loc Bubble-Sort-Algorithm) \/ (UsedILoc Bubble-Sort-Algorithm) by A26, ENUMSET1:def 6;
A58: (UsedI*Loc Bubble-Sort-Algorithm) \/ (UsedILoc Bubble-Sort-Algorithm) c= dom (Comput (P,s1,i)) by Th19;
A59: (UsedI*Loc Bubble-Sort-Algorithm) \/ (UsedILoc Bubble-Sort-Algorithm) c= dom (Comput (Q,s2,i)) by Th19;
A60: ( i > 10 implies 10 + 1 < i + 1 ) by XREAL_1:6;

now :: thesis: for x being set st x in {(intloc 0),(IC ),(fsloc 0)} holds
(Comput (P,s1,i)) . x = (Comput (Q,s2,i)) . x

let x be set ; :: thesis: ( x in {(intloc 0),(IC ),(fsloc 0)} implies (Comput (P,s1,i)) . b₁ = (Comput (Q,s2,i)) . b₁ )
assume A61: x in {(intloc 0),(IC ),(fsloc 0)} ; :: thesis: (Comput (P,s1,i)) . b₁ = (Comput (Q,s2,i)) . b₁

per cases ( x = intloc 0 or x = IC or x = fsloc 0 ) by A61, ENUMSET1:def 1;

suppose A62: x = intloc 0 ; :: thesis: (Comput (P,s1,i)) . b₁ = (Comput (Q,s2,i)) . b₁

per cases ( i <= 10 or i > 10 ) ;

suppose i <= 10 ; :: thesis: (Comput (P,s1,i)) . b₁ = (Comput (Q,s2,i)) . b₁

hence (Comput (P,s1,i)) . x = (Comput (Q,s2,i)) . x by A1, A51, A52, A62, A50; :: thesis: verum

end;

suppose i > 10 ; :: thesis: (Comput (P,s1,i)) . b₁ = (Comput (Q,s2,i)) . b₁

then 11 <= i by A60, NAT_1:13;
then (Comput (P,s1,i)) | ((UsedI*Loc Bubble-Sort-Algorithm) \/ (UsedILoc Bubble-Sort-Algorithm)) = (Comput (Q,s2,i)) | ((UsedI*Loc Bubble-Sort-Algorithm) \/ (UsedILoc Bubble-Sort-Algorithm)) by A27, A51, A52, A50;
hence (Comput (P,s1,i)) . x = (Comput (Q,s2,i)) . x by A56, A58, A59, A62, FUNCT_1:95; :: thesis: verum

end;

end;

end;

suppose A63: x = IC ; :: thesis: (Comput (P,s1,i)) . b₁ = (Comput (Q,s2,i)) . b₁

per cases ( i <= 10 or i > 10 ) ;

suppose i <= 10 ; :: thesis: (Comput (P,s1,i)) . b₁ = (Comput (Q,s2,i)) . b₁

hence (Comput (P,s1,i)) . x = (Comput (Q,s2,i)) . x by A1, A51, A52, A63, A50; :: thesis: verum

end;

suppose i > 10 ; :: thesis: (Comput (P,s1,i)) . b₁ = (Comput (Q,s2,i)) . b₁

then 11 <= i by A60, NAT_1:13;
hence (Comput (P,s1,i)) . x = (Comput (Q,s2,i)) . x by A27, A51, A52, A63, A50; :: thesis: verum

end;

end;

end;

suppose A64: x = fsloc 0 ; :: thesis: (Comput (P,s1,i)) . b₁ = (Comput (Q,s2,i)) . b₁

per cases ( i <= 10 or i > 10 ) ;

suppose i <= 10 ; :: thesis: (Comput (P,s1,i)) . b₁ = (Comput (Q,s2,i)) . b₁

hence (Comput (P,s1,i)) . x = (Comput (Q,s2,i)) . x by A1, A51, A52, A64, A50; :: thesis: verum

end;

suppose i > 10 ; :: thesis: (Comput (P,s1,i)) . b₁ = (Comput (Q,s2,i)) . b₁

then 11 <= i by A60, NAT_1:13;
then (Comput (P,s1,i)) | ((UsedI*Loc Bubble-Sort-Algorithm) \/ (UsedILoc Bubble-Sort-Algorithm)) = (Comput (Q,s2,i)) | ((UsedI*Loc Bubble-Sort-Algorithm) \/ (UsedILoc Bubble-Sort-Algorithm)) by A27, A51, A52, A50;
hence (Comput (P,s1,i)) . x = (Comput (Q,s2,i)) . x by A57, A58, A59, A64, FUNCT_1:95; :: thesis: verum

end;

end;

end;

end;

end;

hence (Comput (P,s1,i)) | (dom (Initialized ((fsloc 0) .--> w))) = (Comput (Q,s2,i)) | (dom (Initialized ((fsloc 0) .--> w))) by A53, A54, A55, FUNCT_1:95; :: thesis: verum