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 ;
A3: 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 Element of 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 Element of 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 A4: ( Bubble-Sort-Algorithm c= P & Bubble-Sort-Algorithm c= Q ) ; :: thesis: for s1, s2 being State of SCM+FSA
for i being Element of 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 Element of 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 Element of 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
A5: Initialized ((fsloc 0) .--> w) c= s1 and
A6: Initialized ((fsloc 0) .--> w) c= s2 and
A7: 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) )
A8: Bubble-Sort-Algorithm c= P by A4;
A9: Bubble-Sort-Algorithm c= Q by A4;
A10: Comput (P,s1,0) = s1 by EXTPRO_1:2;
A11: s1 is 0 -started by A5, MEMSTR_0:17;
A12: s2 is 0 -started by A6, MEMSTR_0:17;
A13: s1 . (intloc 0) = 1 by A5, Th54
.= s2 . (intloc 0) by A6, Th54 ;
A14: s1 . (fsloc 0) = w by A5, Th54
.= s2 . (fsloc 0) by A6, Th54 ;
A15: IC s1 = 0 by A11, MEMSTR_0:def 8
.= IC s2 by A12, MEMSTR_0:def 8 ;
per cases ( 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 ) by A7, NAT_1:34;
suppose A16: 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, A13, EXTPRO_1:2; :: 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 A10, A15, A16, EXTPRO_1:2; :: 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 A10, A14, A16, EXTPRO_1:2; :: thesis: verum
end;
suppose A17: 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 A4, A11, Lm21
.= (Comput (Q,s2,i)) . (intloc 0) by A4, A12, A13, 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 ) = 1 by A4, A11, A17, Lm21
.= (Comput (Q,s2,i)) . (IC ) by A9, A12, 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 A8, A11, A17, Lm21
.= (Comput (Q,s2,i)) . (fsloc 0) by A4, A12, A14, A17, Lm21 ; :: thesis: verum
end;
suppose A18: 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 A8, A11, Lm21
.= (Comput (Q,s2,i)) . (intloc 0) by A4, A12, A13, 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 ) = 2 by A4, A11, A18, Lm21
.= (Comput (Q,s2,i)) . (IC ) by A4, A12, 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 A4, A11, A18, Lm21
.= (Comput (Q,s2,i)) . (fsloc 0) by A4, A12, A14, A18, Lm21 ; :: thesis: verum
end;
suppose A19: 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 A4, A11, Lm21
.= (Comput (Q,s2,i)) . (intloc 0) by A4, A12, A13, 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 ) = 3 by A4, A11, A19, Lm21
.= (Comput (Q,s2,i)) . (IC ) by A4, A12, 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 A8, A11, A19, Lm21
.= (Comput (Q,s2,i)) . (fsloc 0) by A4, A12, A14, A19, Lm21 ; :: thesis: verum
end;
suppose A20: 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 A8, A11, Lm21
.= (Comput (Q,s2,i)) . (intloc 0) by A4, A12, A13, 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 ) = 4 by A4, A11, A20, Lm21
.= (Comput (Q,s2,i)) . (IC ) by A4, A12, 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 A4, A11, A20, Lm21
.= (Comput (Q,s2,i)) . (fsloc 0) by A4, A12, A14, A20, Lm21 ; :: thesis: verum
end;
suppose A21: 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 A8, A11, Lm21
.= (Comput (Q,s2,i)) . (intloc 0) by A4, A12, A13, 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 ) = 5 by A4, A11, A21, Lm21
.= (Comput (Q,s2,i)) . (IC ) by A4, A12, 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 A4, A11, A21, Lm21
.= (Comput (Q,s2,i)) . (fsloc 0) by A4, A12, A14, A21, Lm21 ; :: thesis: verum
end;
suppose A22: 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 A4, A11, Lm21
.= (Comput (Q,s2,i)) . (intloc 0) by A4, A12, A13, 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 ) = 6 by A4, A11, A22, Lm21
.= (Comput (Q,s2,i)) . (IC ) by A4, A12, 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 A4, A11, A22, Lm21
.= (Comput (Q,s2,i)) . (fsloc 0) by A4, A12, A14, A22, Lm21 ; :: thesis: verum
end;
suppose A23: 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 A8, A11, Lm21
.= (Comput (Q,s2,i)) . (intloc 0) by A4, A12, A13, 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 ) = 7 by A4, A11, A23, Lm21
.= (Comput (Q,s2,i)) . (IC ) by A4, A12, 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 A4, A11, A23, Lm21
.= (Comput (Q,s2,i)) . (fsloc 0) by A4, A12, A14, A23, Lm21 ; :: thesis: verum
end;
suppose A24: 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 A4, A11, Lm21
.= (Comput (Q,s2,i)) . (intloc 0) by A4, A12, A13, A24, 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 A4, A11, A24, Lm21
.= (Comput (Q,s2,i)) . (IC ) by A4, A12, A24, 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 A4, A11, A24, Lm21
.= (Comput (Q,s2,i)) . (fsloc 0) by A4, A12, A14, A24, Lm21 ; :: thesis: verum
end;
suppose A25: 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 A4, A11, Lm21
.= (Comput (Q,s2,i)) . (intloc 0) by A4, A12, A13, A25, 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 A4, A11, A25, Lm21
.= (Comput (Q,s2,i)) . (IC ) by A4, A12, A25, 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 A4, A11, A25, Lm21
.= (Comput (Q,s2,i)) . (fsloc 0) by A4, A12, A14, A25, Lm21 ; :: thesis: verum
end;
suppose A26: 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 A4, A11, Lm21
.= (Comput (Q,s2,i)) . (intloc 0) by A4, A12, A13, A26, 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 A4, A11, A26, Lm21
.= (Comput (Q,s2,i)) . (IC ) by A4, A12, A26, 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 A4, A11, A26, Lm21
.= (Comput (Q,s2,i)) . (fsloc 0) by A4, A12, A14, A26, 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 = (UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc Bubble-Sort-Algorithm);
A27: UsedInt*Loc Bubble-Sort-Algorithm = {(fsloc 0)} by Th59;
A28: UsedIntLoc Bubble-Sort-Algorithm = {(intloc 0),(intloc 1),(intloc 2),(intloc 3),(intloc 4),(intloc 5),(intloc 6)} by Th58;
then A29: (UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc Bubble-Sort-Algorithm) = {(fsloc 0),(intloc 0),(intloc 1),(intloc 2),(intloc 3),(intloc 4),(intloc 5),(intloc 6)} by A27, ENUMSET1:22;
A30: for P, Q being Instruction-Sequence of SCM+FSA st Bubble-Sort-Algorithm c= P & Bubble-Sort-Algorithm c= Q holds
for i being Element of 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)) | ((UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc Bubble-Sort-Algorithm)) = (Comput (Q,s2,i)) | ((UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc 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 Element of 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)) | ((UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc Bubble-Sort-Algorithm)) = (Comput (Q,s2,i)) | ((UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc Bubble-Sort-Algorithm)) & (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) ) )
assume A31: ( Bubble-Sort-Algorithm c= P & Bubble-Sort-Algorithm c= Q ) ; :: thesis: for i being Element of 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)) | ((UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc Bubble-Sort-Algorithm)) = (Comput (Q,s2,i)) | ((UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc Bubble-Sort-Algorithm)) & (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) )
let i be Element of 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)) | ((UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc Bubble-Sort-Algorithm)) = (Comput (Q,s2,i)) | ((UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc 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)) | ((UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc Bubble-Sort-Algorithm)) = (Comput (Q,s2,i)) | ((UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc Bubble-Sort-Algorithm)) & (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) ) )
assume that
A33: 11 <= i and
A34: Initialized ((fsloc 0) .--> w) c= s1 and
A35: Initialized ((fsloc 0) .--> w) c= s2 ; :: thesis: ( (Comput (P,s1,i)) | ((UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc Bubble-Sort-Algorithm)) = (Comput (Q,s2,i)) | ((UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc 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);
A36: s1 is 0 -started by A34, MEMSTR_0:17;
A37: s2 is 0 -started by A35, MEMSTR_0:17;
A40: s1 . (intloc 0) = 1 by A34, Th54
.= s2 . (intloc 0) by A35, Th54 ;
A41: s1 . (fsloc 0) = w by A34, Th54
.= s2 . (fsloc 0) by A35, Th54 ;
A42: (UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc Bubble-Sort-Algorithm) c= dom (Comput (P,s1,11)) by Th56;
A43: (UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc Bubble-Sort-Algorithm) c= dom (Comput (Q,s2,11)) by Th56;
now
let x be set ; :: thesis: ( x in (UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc Bubble-Sort-Algorithm) implies (Comput (P,s1,11)) . b1 = (Comput (Q,s2,11)) . b1 )
assume x in (UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc Bubble-Sort-Algorithm) ; :: thesis: (Comput (P,s1,11)) . b1 = (Comput (Q,s2,11)) . b1
then A44: x in {(fsloc 0),(intloc 0),(intloc 1),(intloc 2),(intloc 3),(intloc 4),(intloc 5),(intloc 6)} by A27, A28, 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 A44, ENUMSET1:def 6;
suppose A45: x = fsloc 0 ; :: thesis: (Comput (P,s1,11)) . b1 = (Comput (Q,s2,11)) . b1
hence (Comput (P,s1,11)) . x = s1 . (fsloc 0) by A36, A31, Lm21
.= (Comput (Q,s2,11)) . x by A37, A31, A41, A45, Lm21 ;
:: thesis: verum
end;
suppose A46: x = intloc 0 ; :: thesis: (Comput (P,s1,11)) . b1 = (Comput (Q,s2,11)) . b1
hence (Comput (P,s1,11)) . x = s1 . (intloc 0) by A36, A31, Lm21
.= (Comput (Q,s2,11)) . x by A37, A31, A40, A46, Lm21 ;
:: thesis: verum
end;
suppose A47: 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 A36, A31, Lm21
.= (Comput (Q,s2,11)) . x by A37, A31, A41, A47, Lm21 ;
:: thesis: verum
end;
suppose A48: x = intloc 2 ; :: thesis: (Comput (P,s1,11)) . b1 = (Comput (Q,s2,11)) . b1
hence (Comput (P,s1,11)) . x = s1 . (intloc 0) by A36, A31, Lm21
.= (Comput (Q,s2,11)) . x by A37, A31, A40, A48, Lm21 ;
:: thesis: verum
end;
suppose A49: x = intloc 3 ; :: thesis: (Comput (P,s1,11)) . b1 = (Comput (Q,s2,11)) . b1
hence (Comput (P,s1,11)) . x = s1 . (intloc 0) by A36, A31, Lm21
.= (Comput (Q,s2,11)) . x by A37, A31, A40, A49, Lm21 ;
:: thesis: verum
end;
suppose A50: x = intloc 4 ; :: thesis: (Comput (P,s1,11)) . b1 = (Comput (Q,s2,11)) . b1
hence (Comput (P,s1,11)) . x = s1 . (intloc 0) by A36, A31, Lm21
.= (Comput (Q,s2,11)) . x by A37, A31, A40, A50, Lm21 ;
:: thesis: verum
end;
suppose A51: x = intloc 5 ; :: thesis: (Comput (P,s1,11)) . b1 = (Comput (Q,s2,11)) . b1
hence (Comput (P,s1,11)) . x = s1 . (intloc 0) by A36, A31, Lm21
.= (Comput (Q,s2,11)) . x by A37, A31, A40, A51, Lm21 ;
:: thesis: verum
end;
suppose A52: x = intloc 6 ; :: thesis: (Comput (P,s1,11)) . b1 = (Comput (Q,s2,11)) . b1
hence (Comput (P,s1,11)) . x = s1 . (intloc 0) by A36, A31, Lm21
.= (Comput (Q,s2,11)) . x by A37, A31, A40, A52, Lm21 ;
:: thesis: verum
end;
end;
end;
then A53: (Comput (P,s1,11)) | ((UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc Bubble-Sort-Algorithm)) = (Comput (Q,s2,11)) | ((UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc Bubble-Sort-Algorithm)) by A42, A43, FUNCT_1:95;
A54: (Comput (P,s1,11)) . (IC ) = 11 by A36, A31, Lm21
.= (Comput (Q,s2,11)) . (IC ) by A37, A31, Lm21 ;
A55: for i being Element of NAT holds IC (Comput (P,s1,i)) in dom Bubble-Sort-Algorithm by A34, Th66, A31;
for i being Element of NAT holds IC (Comput (Q,s2,i)) in dom Bubble-Sort-Algorithm by A35, Th66, A31;
hence ( (Comput (P,s1,i)) | ((UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc Bubble-Sort-Algorithm)) = (Comput (Q,s2,i)) | ((UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc Bubble-Sort-Algorithm)) & (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) ) by A33, A53, A54, A55, Th26, A31; :: thesis: verum
end;
set DD = {(intloc 0),(IC ),(fsloc 0)};
A56: dom (Initialized ((fsloc 0) .--> w)) = {(intloc 0),(IC ),(fsloc 0)} by Th42;
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 Element of NAT holds (Comput (P,b1,b3)) | (proj1 (Initialized ((fsloc 0) .--> w))) = (Comput (Q,b2,b3)) | (proj1 (Initialized ((fsloc 0) .--> w))) ) )
assume A57: ( 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 Element of NAT holds (Comput (P,b1,b3)) | (proj1 (Initialized ((fsloc 0) .--> w))) = (Comput (Q,b2,b3)) | (proj1 (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 Element of NAT holds (Comput (P,s1,b1)) | (proj1 (Initialized ((fsloc 0) .--> w))) = (Comput (Q,s2,b1)) | (proj1 (Initialized ((fsloc 0) .--> w))) )
assume that
A58: Initialized ((fsloc 0) .--> w) c= s1 and
A59: Initialized ((fsloc 0) .--> w) c= s2 ; :: thesis: for b1 being Element of NAT holds (Comput (P,s1,b1)) | (proj1 (Initialized ((fsloc 0) .--> w))) = (Comput (Q,s2,b1)) | (proj1 (Initialized ((fsloc 0) .--> w)))
let i be Element of NAT ; :: thesis: (Comput (P,s1,i)) | (proj1 (Initialized ((fsloc 0) .--> w))) = (Comput (Q,s2,i)) | (proj1 (Initialized ((fsloc 0) .--> w)))
set Cs1i = Comput (P,s1,i);
set Cs2i = Comput (Q,s2,i);
Initialized ((fsloc 0) .--> w) = ((fsloc 0) .--> w) +* (Initialize ((intloc 0) .--> 1)) by SCMFSA6A:def 3;
then DataPart (Initialized ((fsloc 0) .--> w)) = (DataPart ((fsloc 0) .--> w)) +* (DataPart (Initialize ((intloc 0) .--> 1))) by FUNCT_4:71
.= (DataPart ((fsloc 0) .--> w)) +* (DataPart ((intloc 0) .--> 1)) by MEMSTR_0:45
.= ((fsloc 0) .--> w) +* (DataPart ((intloc 0) .--> 1)) by MEMSTR_0:7
.= ((fsloc 0) .--> w) +* ((intloc 0) .--> 1) by MEMSTR_0:7 ;
then A67: dom (DataPart (Initialized ((fsloc 0) .--> w))) = (dom ((fsloc 0) .--> w)) \/ (dom ((intloc 0) .--> 1)) by FUNCT_4:def 1
.= {(fsloc 0)} \/ (dom ((intloc 0) .--> 1)) by FUNCOP_1:13
.= {(fsloc 0)} \/ {(intloc 0)} by FUNCOP_1:13
.= {(intloc 0),(fsloc 0)} by ENUMSET1:1 ;
IC in dom (Initialized ((fsloc 0) .--> w)) by A56, ENUMSET1:def 1;
then A68: dom (Initialized ((fsloc 0) .--> w)) = {(IC )} \/ (dom (DataPart (Initialized ((fsloc 0) .--> w)))) by MEMSTR_0:24
.= {(IC ),(intloc 0),(fsloc 0)} by A67, ENUMSET1:2
.= {(intloc 0),(IC ),(fsloc 0)} by ENUMSET1:58 ;
A69: {(intloc 0),(IC ),(fsloc 0)} c= dom (Comput (P,s1,i)) by Th55;
A70: {(intloc 0),(IC ),(fsloc 0)} c= dom (Comput (Q,s2,i)) by Th55;
A71: intloc 0 in (UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc Bubble-Sort-Algorithm) by A29, ENUMSET1:def 6;
A72: fsloc 0 in (UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc Bubble-Sort-Algorithm) by A29, ENUMSET1:def 6;
A73: (UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc Bubble-Sort-Algorithm) c= dom (Comput (P,s1,i)) by Th56;
A74: (UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc Bubble-Sort-Algorithm) c= dom (Comput (Q,s2,i)) by Th56;
A75: ( i > 10 implies 10 + 1 < i + 1 ) by XREAL_1:6;
now
let x be set ; :: thesis: ( x in {(intloc 0),(IC ),(fsloc 0)} implies (Comput (P,s1,i)) . b1 = (Comput (Q,s2,i)) . b1 )
assume A76: 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 A76, ENUMSET1:def 1;
suppose A77: 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 A3, A58, A59, A77, A57; :: thesis: verum
end;
end;
end;
suppose A78: 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 A3, A58, A59, A78, A57; :: thesis: verum
end;
suppose i > 10 ; :: thesis: (Comput (P,s1,i)) . b1 = (Comput (Q,s2,i)) . b1
then 11 <= i by A75, NAT_1:13;
hence (Comput (P,s1,i)) . x = (Comput (Q,s2,i)) . x by A30, A58, A59, A78, A57; :: thesis: verum
end;
end;
end;
suppose A79: 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 A3, A58, A59, A79, A57; :: 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 A68, A69, A70, FUNCT_1:95; :: thesis: verum