set DD = {(intloc 0),(IC SCM+FSA),(fsloc 0)};
let w be FinSequence of INT ; :: thesis: (Initialized Insert-Sort-Algorithm) +* ((fsloc 0) .--> w) is autonomic
set p = (Initialized Insert-Sort-Algorithm) +* ((fsloc 0) .--> w);
set q = Insert-Sort-Algorithm ;
X: (Initialized Insert-Sort-Algorithm) +* ((fsloc 0) .--> w) is 0 -started by SCMBSORT:47;
set UD = {(fsloc 0),(intloc 0),(intloc 1),(intloc 2),(intloc 3),(intloc 4),(intloc 5),(intloc 6)};
set Us = (UsedInt*Loc Insert-Sort-Algorithm) \/ (UsedIntLoc Insert-Sort-Algorithm);
A1: UsedIntLoc Insert-Sort-Algorithm = {(intloc 0),(intloc 1),(intloc 2),(intloc 3),(intloc 4),(intloc 5),(intloc 6)} by Th38;
A2: UsedInt*Loc Insert-Sort-Algorithm = {(fsloc 0)} by Th39;
then A3: (UsedInt*Loc Insert-Sort-Algorithm) \/ (UsedIntLoc Insert-Sort-Algorithm) = {(fsloc 0),(intloc 0),(intloc 1),(intloc 2),(intloc 3),(intloc 4),(intloc 5),(intloc 6)} by A1, ENUMSET1:62;
A4: for i being Element of NAT
for s1, s2 being State of SCM+FSA st 11 <= i & (Initialized Insert-Sort-Algorithm) +* ((fsloc 0) .--> w) c= s1 & (Initialized Insert-Sort-Algorithm) +* ((fsloc 0) .--> w) c= s2 holds
( (Comput ((ProgramPart s1),s1,i)) | ((UsedInt*Loc Insert-Sort-Algorithm) \/ (UsedIntLoc Insert-Sort-Algorithm)) = (Comput ((ProgramPart s2),s2,i)) | ((UsedInt*Loc Insert-Sort-Algorithm) \/ (UsedIntLoc Insert-Sort-Algorithm)) & (Comput ((ProgramPart s1),s1,i)) . (IC SCM+FSA) = (Comput ((ProgramPart s2),s2,i)) . (IC SCM+FSA) )

proof

let i be Element of NAT ; :: thesis: for s1, s2 being State of SCM+FSA st 11 <= i & (Initialized Insert-Sort-Algorithm) +* ((fsloc 0) .--> w) c= s1 & (Initialized Insert-Sort-Algorithm) +* ((fsloc 0) .--> w) c= s2 holds
( (Comput ((ProgramPart s1),s1,i)) | ((UsedInt*Loc Insert-Sort-Algorithm) \/ (UsedIntLoc Insert-Sort-Algorithm)) = (Comput ((ProgramPart s2),s2,i)) | ((UsedInt*Loc Insert-Sort-Algorithm) \/ (UsedIntLoc Insert-Sort-Algorithm)) & (Comput ((ProgramPart s1),s1,i)) . (IC SCM+FSA) = (Comput ((ProgramPart s2),s2,i)) . (IC SCM+FSA) )
let s1, s2 be State of SCM+FSA; :: thesis: ( 11 <= i & (Initialized Insert-Sort-Algorithm) +* ((fsloc 0) .--> w) c= s1 & (Initialized Insert-Sort-Algorithm) +* ((fsloc 0) .--> w) c= s2 implies ( (Comput ((ProgramPart s1),s1,i)) | ((UsedInt*Loc Insert-Sort-Algorithm) \/ (UsedIntLoc Insert-Sort-Algorithm)) = (Comput ((ProgramPart s2),s2,i)) | ((UsedInt*Loc Insert-Sort-Algorithm) \/ (UsedIntLoc Insert-Sort-Algorithm)) & (Comput ((ProgramPart s1),s1,i)) . (IC SCM+FSA) = (Comput ((ProgramPart s2),s2,i)) . (IC SCM+FSA) ) )
assume that
A5: 11 <= i and
A6: (Initialized Insert-Sort-Algorithm) +* ((fsloc 0) .--> w) c= s1 and
A7: (Initialized Insert-Sort-Algorithm) +* ((fsloc 0) .--> w) c= s2 ; :: thesis: ( (Comput ((ProgramPart s1),s1,i)) | ((UsedInt*Loc Insert-Sort-Algorithm) \/ (UsedIntLoc Insert-Sort-Algorithm)) = (Comput ((ProgramPart s2),s2,i)) | ((UsedInt*Loc Insert-Sort-Algorithm) \/ (UsedIntLoc Insert-Sort-Algorithm)) & (Comput ((ProgramPart s1),s1,i)) . (IC SCM+FSA) = (Comput ((ProgramPart s2),s2,i)) . (IC SCM+FSA) )
A8: s1 . (intloc 0) = 1 by A6, SCMBSORT:54
.= s2 . (intloc 0) by A7, SCMBSORT:54 ;
A9: s2 is 0 -started by A7, X, COMPOS_1:8;
A10: Insert-Sort-Algorithm c= s2 by A7, SCMBSORT:53;
set Cs11 = Comput ((ProgramPart s1),s1,11);
set Cs21 = Comput ((ProgramPart s2),s2,11);
A11: s1 is 0 -started by A6, X, COMPOS_1:8;
A12: Insert-Sort-Algorithm c= s1 by A6, SCMBSORT:53;
A13: s1 . (fsloc 0) = w by A6, SCMBSORT:54
.= s2 . (fsloc 0) by A7, SCMBSORT:54 ;

A14: now

let x be set ; :: thesis: ( x in (UsedInt*Loc Insert-Sort-Algorithm) \/ (UsedIntLoc Insert-Sort-Algorithm) implies (Comput ((ProgramPart s1),s1,11)) . b₁ = (Comput ((ProgramPart s2),s2,11)) . b₁ )
assume x in (UsedInt*Loc Insert-Sort-Algorithm) \/ (UsedIntLoc Insert-Sort-Algorithm) ; :: thesis: (Comput ((ProgramPart s1),s1,11)) . b₁ = (Comput ((ProgramPart s2),s2,11)) . b₁
then A15: x in {(fsloc 0),(intloc 0),(intloc 1),(intloc 2),(intloc 3),(intloc 4),(intloc 5),(intloc 6)} by A2, A1, ENUMSET1:62;

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 A15, ENUMSET1:def 6;

suppose A16: x = fsloc 0 ; :: thesis: (Comput ((ProgramPart s1),s1,11)) . b₁ = (Comput ((ProgramPart s2),s2,11)) . b₁

hence (Comput ((ProgramPart s1),s1,11)) . x = s1 . (fsloc 0) by A11, A12, Lm2
.= (Comput ((ProgramPart s2),s2,11)) . x by A9, A10, A13, A16, Lm2 ;
:: thesis: verum

end;

suppose A17: x = intloc 0 ; :: thesis: (Comput ((ProgramPart s1),s1,11)) . b₁ = (Comput ((ProgramPart s2),s2,11)) . b₁

hence (Comput ((ProgramPart s1),s1,11)) . x = s1 . (intloc 0) by A11, A12, Lm2
.= (Comput ((ProgramPart s2),s2,11)) . x by A9, A10, A8, A17, Lm2 ;
:: thesis: verum

end;

suppose A18: x = intloc 1 ; :: thesis: (Comput ((ProgramPart s1),s1,11)) . b₁ = (Comput ((ProgramPart s2),s2,11)) . b₁

hence (Comput ((ProgramPart s1),s1,11)) . x = len (s1 . (fsloc 0)) by A11, A12, Lm2
.= (Comput ((ProgramPart s2),s2,11)) . x by A9, A10, A13, A18, Lm2 ;
:: thesis: verum

end;

suppose A19: x = intloc 2 ; :: thesis: (Comput ((ProgramPart s1),s1,11)) . b₁ = (Comput ((ProgramPart s2),s2,11)) . b₁

hence (Comput ((ProgramPart s1),s1,11)) . x = s1 . (intloc 0) by A11, A12, Lm2
.= (Comput ((ProgramPart s2),s2,11)) . x by A9, A10, A8, A19, Lm2 ;
:: thesis: verum

end;

suppose A20: x = intloc 3 ; :: thesis: (Comput ((ProgramPart s1),s1,11)) . b₁ = (Comput ((ProgramPart s2),s2,11)) . b₁

hence (Comput ((ProgramPart s1),s1,11)) . x = s1 . (intloc 0) by A11, A12, Lm2
.= (Comput ((ProgramPart s2),s2,11)) . x by A9, A10, A8, A20, Lm2 ;
:: thesis: verum

end;

suppose A21: x = intloc 4 ; :: thesis: (Comput ((ProgramPart s1),s1,11)) . b₁ = (Comput ((ProgramPart s2),s2,11)) . b₁

hence (Comput ((ProgramPart s1),s1,11)) . x = s1 . (intloc 0) by A11, A12, Lm2
.= (Comput ((ProgramPart s2),s2,11)) . x by A9, A10, A8, A21, Lm2 ;
:: thesis: verum

end;

suppose A22: x = intloc 5 ; :: thesis: (Comput ((ProgramPart s1),s1,11)) . b₁ = (Comput ((ProgramPart s2),s2,11)) . b₁

hence (Comput ((ProgramPart s1),s1,11)) . x = s1 . (intloc 0) by A11, A12, Lm2
.= (Comput ((ProgramPart s2),s2,11)) . x by A9, A10, A8, A22, Lm2 ;
:: thesis: verum

end;

suppose A23: x = intloc 6 ; :: thesis: (Comput ((ProgramPart s1),s1,11)) . b₁ = (Comput ((ProgramPart s2),s2,11)) . b₁

hence (Comput ((ProgramPart s1),s1,11)) . x = s1 . (intloc 0) by A11, A12, Lm2
.= (Comput ((ProgramPart s2),s2,11)) . x by A9, A10, A8, A23, Lm2 ;
:: thesis: verum

end;

end;

end;

A24: for i being Element of NAT holds IC (Comput ((ProgramPart s2),s2,i)) in dom Insert-Sort-Algorithm by A7, Th46;
A25: for i being Element of NAT holds IC (Comput ((ProgramPart s1),s1,i)) in dom Insert-Sort-Algorithm by A6, Th46;
A26: (UsedInt*Loc Insert-Sort-Algorithm) \/ (UsedIntLoc Insert-Sort-Algorithm) c= dom (Comput ((ProgramPart s2),s2,11)) by SCMBSORT:56;
(UsedInt*Loc Insert-Sort-Algorithm) \/ (UsedIntLoc Insert-Sort-Algorithm) c= dom (Comput ((ProgramPart s1),s1,11)) by SCMBSORT:56;
then A27: (Comput ((ProgramPart s1),s1,11)) | ((UsedInt*Loc Insert-Sort-Algorithm) \/ (UsedIntLoc Insert-Sort-Algorithm)) = (Comput ((ProgramPart s2),s2,11)) | ((UsedInt*Loc Insert-Sort-Algorithm) \/ (UsedIntLoc Insert-Sort-Algorithm)) by A26, A14, FUNCT_1:165;
(Comput ((ProgramPart s1),s1,11)) . (IC SCM+FSA) = 11 by A11, A12, Lm2
.= (Comput ((ProgramPart s2),s2,11)) . (IC SCM+FSA) by A9, A10, Lm2 ;
hence ( (Comput ((ProgramPart s1),s1,i)) | ((UsedInt*Loc Insert-Sort-Algorithm) \/ (UsedIntLoc Insert-Sort-Algorithm)) = (Comput ((ProgramPart s2),s2,i)) | ((UsedInt*Loc Insert-Sort-Algorithm) \/ (UsedIntLoc Insert-Sort-Algorithm)) & (Comput ((ProgramPart s1),s1,i)) . (IC SCM+FSA) = (Comput ((ProgramPart s2),s2,i)) . (IC SCM+FSA) ) by A5, A12, A10, A27, A25, A24, SCMBSORT:26; :: thesis: verum

end;

A28: for s1, s2 being State of SCM+FSA
for i being Element of NAT st (Initialized Insert-Sort-Algorithm) +* ((fsloc 0) .--> w) c= s1 & (Initialized Insert-Sort-Algorithm) +* ((fsloc 0) .--> w) c= s2 & i <= 10 holds
( (Comput ((ProgramPart s1),s1,i)) . (intloc 0) = (Comput ((ProgramPart s2),s2,i)) . (intloc 0) & (Comput ((ProgramPart s1),s1,i)) . (IC SCM+FSA) = (Comput ((ProgramPart s2),s2,i)) . (IC SCM+FSA) & (Comput ((ProgramPart s1),s1,i)) . (fsloc 0) = (Comput ((ProgramPart s2),s2,i)) . (fsloc 0) )

proof

let s1, s2 be State of SCM+FSA; :: thesis: for i being Element of NAT st (Initialized Insert-Sort-Algorithm) +* ((fsloc 0) .--> w) c= s1 & (Initialized Insert-Sort-Algorithm) +* ((fsloc 0) .--> w) c= s2 & i <= 10 holds
( (Comput ((ProgramPart s1),s1,i)) . (intloc 0) = (Comput ((ProgramPart s2),s2,i)) . (intloc 0) & (Comput ((ProgramPart s1),s1,i)) . (IC SCM+FSA) = (Comput ((ProgramPart s2),s2,i)) . (IC SCM+FSA) & (Comput ((ProgramPart s1),s1,i)) . (fsloc 0) = (Comput ((ProgramPart s2),s2,i)) . (fsloc 0) )
let i be Element of NAT ; :: thesis: ( (Initialized Insert-Sort-Algorithm) +* ((fsloc 0) .--> w) c= s1 & (Initialized Insert-Sort-Algorithm) +* ((fsloc 0) .--> w) c= s2 & i <= 10 implies ( (Comput ((ProgramPart s1),s1,i)) . (intloc 0) = (Comput ((ProgramPart s2),s2,i)) . (intloc 0) & (Comput ((ProgramPart s1),s1,i)) . (IC SCM+FSA) = (Comput ((ProgramPart s2),s2,i)) . (IC SCM+FSA) & (Comput ((ProgramPart s1),s1,i)) . (fsloc 0) = (Comput ((ProgramPart s2),s2,i)) . (fsloc 0) ) )
assume that
A29: (Initialized Insert-Sort-Algorithm) +* ((fsloc 0) .--> w) c= s1 and
A30: (Initialized Insert-Sort-Algorithm) +* ((fsloc 0) .--> w) c= s2 and
A31: i <= 10 ; :: thesis: ( (Comput ((ProgramPart s1),s1,i)) . (intloc 0) = (Comput ((ProgramPart s2),s2,i)) . (intloc 0) & (Comput ((ProgramPart s1),s1,i)) . (IC SCM+FSA) = (Comput ((ProgramPart s2),s2,i)) . (IC SCM+FSA) & (Comput ((ProgramPart s1),s1,i)) . (fsloc 0) = (Comput ((ProgramPart s2),s2,i)) . (fsloc 0) )
A32: s2 is 0 -started by A30, X, COMPOS_1:8;
A33: s1 . (fsloc 0) = w by A29, SCMBSORT:54
.= s2 . (fsloc 0) by A30, SCMBSORT:54 ;
A34: s1 . (intloc 0) = 1 by A29, SCMBSORT:54
.= s2 . (intloc 0) by A30, SCMBSORT:54 ;
A35: Comput ((ProgramPart s1),s1,0) = s1 by EXTPRO_1:3;
A36: Insert-Sort-Algorithm c= s2 by A30, SCMBSORT:53;
A37: s1 is 0 -started by A29, X, COMPOS_1:8;
then A38: IC s1 = 0 by COMPOS_1:def 16
.= IC s2 by A32, COMPOS_1:def 16 ;
A39: Insert-Sort-Algorithm c= s1 by A29, SCMBSORT:53;

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 A31, NAT_1:35;

suppose A40: i = 0 ; :: thesis: ( (Comput ((ProgramPart s1),s1,i)) . (intloc 0) = (Comput ((ProgramPart s2),s2,i)) . (intloc 0) & (Comput ((ProgramPart s1),s1,i)) . (IC SCM+FSA) = (Comput ((ProgramPart s2),s2,i)) . (IC SCM+FSA) & (Comput ((ProgramPart s1),s1,i)) . (fsloc 0) = (Comput ((ProgramPart s2),s2,i)) . (fsloc 0) )

hence (Comput ((ProgramPart s1),s1,i)) . (intloc 0) = (Comput ((ProgramPart s2),s2,i)) . (intloc 0) by A35, A34, EXTPRO_1:3; :: thesis: ( (Comput ((ProgramPart s1),s1,i)) . (IC SCM+FSA) = (Comput ((ProgramPart s2),s2,i)) . (IC SCM+FSA) & (Comput ((ProgramPart s1),s1,i)) . (fsloc 0) = (Comput ((ProgramPart s2),s2,i)) . (fsloc 0) )
thus (Comput ((ProgramPart s1),s1,i)) . (IC SCM+FSA) = (Comput ((ProgramPart s2),s2,i)) . (IC SCM+FSA) by A35, A38, A40, EXTPRO_1:3; :: thesis: (Comput ((ProgramPart s1),s1,i)) . (fsloc 0) = (Comput ((ProgramPart s2),s2,i)) . (fsloc 0)
thus (Comput ((ProgramPart s1),s1,i)) . (fsloc 0) = (Comput ((ProgramPart s2),s2,i)) . (fsloc 0) by A35, A33, A40, EXTPRO_1:3; :: thesis: verum

end;

suppose A41: i = 1 ; :: thesis: ( (Comput ((ProgramPart s1),s1,i)) . (intloc 0) = (Comput ((ProgramPart s2),s2,i)) . (intloc 0) & (Comput ((ProgramPart s1),s1,i)) . (IC SCM+FSA) = (Comput ((ProgramPart s2),s2,i)) . (IC SCM+FSA) & (Comput ((ProgramPart s1),s1,i)) . (fsloc 0) = (Comput ((ProgramPart s2),s2,i)) . (fsloc 0) )

hence (Comput ((ProgramPart s1),s1,i)) . (intloc 0) = s1 . (intloc 0) by A37, A39, Lm2
.= (Comput ((ProgramPart s2),s2,i)) . (intloc 0) by A32, A36, A34, A41, Lm2 ;
:: thesis: ( (Comput ((ProgramPart s1),s1,i)) . (IC SCM+FSA) = (Comput ((ProgramPart s2),s2,i)) . (IC SCM+FSA) & (Comput ((ProgramPart s1),s1,i)) . (fsloc 0) = (Comput ((ProgramPart s2),s2,i)) . (fsloc 0) )
thus (Comput ((ProgramPart s1),s1,i)) . (IC SCM+FSA) = 1 by A37, A39, A41, Lm2
.= (Comput ((ProgramPart s2),s2,i)) . (IC SCM+FSA) by A32, A36, A41, Lm2 ; :: thesis: (Comput ((ProgramPart s1),s1,i)) . (fsloc 0) = (Comput ((ProgramPart s2),s2,i)) . (fsloc 0)
thus (Comput ((ProgramPart s1),s1,i)) . (fsloc 0) = s1 . (fsloc 0) by A37, A39, A41, Lm2
.= (Comput ((ProgramPart s2),s2,i)) . (fsloc 0) by A32, A36, A33, A41, Lm2 ; :: thesis: verum

end;

suppose A42: i = 2 ; :: thesis: ( (Comput ((ProgramPart s1),s1,i)) . (intloc 0) = (Comput ((ProgramPart s2),s2,i)) . (intloc 0) & (Comput ((ProgramPart s1),s1,i)) . (IC SCM+FSA) = (Comput ((ProgramPart s2),s2,i)) . (IC SCM+FSA) & (Comput ((ProgramPart s1),s1,i)) . (fsloc 0) = (Comput ((ProgramPart s2),s2,i)) . (fsloc 0) )

hence (Comput ((ProgramPart s1),s1,i)) . (intloc 0) = s1 . (intloc 0) by A37, A39, Lm2
.= (Comput ((ProgramPart s2),s2,i)) . (intloc 0) by A32, A36, A34, A42, Lm2 ;
:: thesis: ( (Comput ((ProgramPart s1),s1,i)) . (IC SCM+FSA) = (Comput ((ProgramPart s2),s2,i)) . (IC SCM+FSA) & (Comput ((ProgramPart s1),s1,i)) . (fsloc 0) = (Comput ((ProgramPart s2),s2,i)) . (fsloc 0) )
thus (Comput ((ProgramPart s1),s1,i)) . (IC SCM+FSA) = 2 by A37, A39, A42, Lm2
.= (Comput ((ProgramPart s2),s2,i)) . (IC SCM+FSA) by A32, A36, A42, Lm2 ; :: thesis: (Comput ((ProgramPart s1),s1,i)) . (fsloc 0) = (Comput ((ProgramPart s2),s2,i)) . (fsloc 0)
thus (Comput ((ProgramPart s1),s1,i)) . (fsloc 0) = s1 . (fsloc 0) by A37, A39, A42, Lm2
.= (Comput ((ProgramPart s2),s2,i)) . (fsloc 0) by A32, A36, A33, A42, Lm2 ; :: thesis: verum

end;

suppose A43: i = 3 ; :: thesis: ( (Comput ((ProgramPart s1),s1,i)) . (intloc 0) = (Comput ((ProgramPart s2),s2,i)) . (intloc 0) & (Comput ((ProgramPart s1),s1,i)) . (IC SCM+FSA) = (Comput ((ProgramPart s2),s2,i)) . (IC SCM+FSA) & (Comput ((ProgramPart s1),s1,i)) . (fsloc 0) = (Comput ((ProgramPart s2),s2,i)) . (fsloc 0) )

hence (Comput ((ProgramPart s1),s1,i)) . (intloc 0) = s1 . (intloc 0) by A37, A39, Lm2
.= (Comput ((ProgramPart s2),s2,i)) . (intloc 0) by A32, A36, A34, A43, Lm2 ;
:: thesis: ( (Comput ((ProgramPart s1),s1,i)) . (IC SCM+FSA) = (Comput ((ProgramPart s2),s2,i)) . (IC SCM+FSA) & (Comput ((ProgramPart s1),s1,i)) . (fsloc 0) = (Comput ((ProgramPart s2),s2,i)) . (fsloc 0) )
thus (Comput ((ProgramPart s1),s1,i)) . (IC SCM+FSA) = 3 by A37, A39, A43, Lm2
.= (Comput ((ProgramPart s2),s2,i)) . (IC SCM+FSA) by A32, A36, A43, Lm2 ; :: thesis: (Comput ((ProgramPart s1),s1,i)) . (fsloc 0) = (Comput ((ProgramPart s2),s2,i)) . (fsloc 0)
thus (Comput ((ProgramPart s1),s1,i)) . (fsloc 0) = s1 . (fsloc 0) by A37, A39, A43, Lm2
.= (Comput ((ProgramPart s2),s2,i)) . (fsloc 0) by A32, A36, A33, A43, Lm2 ; :: thesis: verum

end;

suppose A44: i = 4 ; :: thesis: ( (Comput ((ProgramPart s1),s1,i)) . (intloc 0) = (Comput ((ProgramPart s2),s2,i)) . (intloc 0) & (Comput ((ProgramPart s1),s1,i)) . (IC SCM+FSA) = (Comput ((ProgramPart s2),s2,i)) . (IC SCM+FSA) & (Comput ((ProgramPart s1),s1,i)) . (fsloc 0) = (Comput ((ProgramPart s2),s2,i)) . (fsloc 0) )

hence (Comput ((ProgramPart s1),s1,i)) . (intloc 0) = s1 . (intloc 0) by A37, A39, Lm2
.= (Comput ((ProgramPart s2),s2,i)) . (intloc 0) by A32, A36, A34, A44, Lm2 ;
:: thesis: ( (Comput ((ProgramPart s1),s1,i)) . (IC SCM+FSA) = (Comput ((ProgramPart s2),s2,i)) . (IC SCM+FSA) & (Comput ((ProgramPart s1),s1,i)) . (fsloc 0) = (Comput ((ProgramPart s2),s2,i)) . (fsloc 0) )
thus (Comput ((ProgramPart s1),s1,i)) . (IC SCM+FSA) = 4 by A37, A39, A44, Lm2
.= (Comput ((ProgramPart s2),s2,i)) . (IC SCM+FSA) by A32, A36, A44, Lm2 ; :: thesis: (Comput ((ProgramPart s1),s1,i)) . (fsloc 0) = (Comput ((ProgramPart s2),s2,i)) . (fsloc 0)
thus (Comput ((ProgramPart s1),s1,i)) . (fsloc 0) = s1 . (fsloc 0) by A37, A39, A44, Lm2
.= (Comput ((ProgramPart s2),s2,i)) . (fsloc 0) by A32, A36, A33, A44, Lm2 ; :: thesis: verum

end;

suppose A45: i = 5 ; :: thesis: ( (Comput ((ProgramPart s1),s1,i)) . (intloc 0) = (Comput ((ProgramPart s2),s2,i)) . (intloc 0) & (Comput ((ProgramPart s1),s1,i)) . (IC SCM+FSA) = (Comput ((ProgramPart s2),s2,i)) . (IC SCM+FSA) & (Comput ((ProgramPart s1),s1,i)) . (fsloc 0) = (Comput ((ProgramPart s2),s2,i)) . (fsloc 0) )

hence (Comput ((ProgramPart s1),s1,i)) . (intloc 0) = s1 . (intloc 0) by A37, A39, Lm2
.= (Comput ((ProgramPart s2),s2,i)) . (intloc 0) by A32, A36, A34, A45, Lm2 ;
:: thesis: ( (Comput ((ProgramPart s1),s1,i)) . (IC SCM+FSA) = (Comput ((ProgramPart s2),s2,i)) . (IC SCM+FSA) & (Comput ((ProgramPart s1),s1,i)) . (fsloc 0) = (Comput ((ProgramPart s2),s2,i)) . (fsloc 0) )
thus (Comput ((ProgramPart s1),s1,i)) . (IC SCM+FSA) = 5 by A37, A39, A45, Lm2
.= (Comput ((ProgramPart s2),s2,i)) . (IC SCM+FSA) by A32, A36, A45, Lm2 ; :: thesis: (Comput ((ProgramPart s1),s1,i)) . (fsloc 0) = (Comput ((ProgramPart s2),s2,i)) . (fsloc 0)
thus (Comput ((ProgramPart s1),s1,i)) . (fsloc 0) = s1 . (fsloc 0) by A37, A39, A45, Lm2
.= (Comput ((ProgramPart s2),s2,i)) . (fsloc 0) by A32, A36, A33, A45, Lm2 ; :: thesis: verum

end;

suppose A46: i = 6 ; :: thesis: ( (Comput ((ProgramPart s1),s1,i)) . (intloc 0) = (Comput ((ProgramPart s2),s2,i)) . (intloc 0) & (Comput ((ProgramPart s1),s1,i)) . (IC SCM+FSA) = (Comput ((ProgramPart s2),s2,i)) . (IC SCM+FSA) & (Comput ((ProgramPart s1),s1,i)) . (fsloc 0) = (Comput ((ProgramPart s2),s2,i)) . (fsloc 0) )

hence (Comput ((ProgramPart s1),s1,i)) . (intloc 0) = s1 . (intloc 0) by A37, A39, Lm2
.= (Comput ((ProgramPart s2),s2,i)) . (intloc 0) by A32, A36, A34, A46, Lm2 ;
:: thesis: ( (Comput ((ProgramPart s1),s1,i)) . (IC SCM+FSA) = (Comput ((ProgramPart s2),s2,i)) . (IC SCM+FSA) & (Comput ((ProgramPart s1),s1,i)) . (fsloc 0) = (Comput ((ProgramPart s2),s2,i)) . (fsloc 0) )
thus (Comput ((ProgramPart s1),s1,i)) . (IC SCM+FSA) = 6 by A37, A39, A46, Lm2
.= (Comput ((ProgramPart s2),s2,i)) . (IC SCM+FSA) by A32, A36, A46, Lm2 ; :: thesis: (Comput ((ProgramPart s1),s1,i)) . (fsloc 0) = (Comput ((ProgramPart s2),s2,i)) . (fsloc 0)
thus (Comput ((ProgramPart s1),s1,i)) . (fsloc 0) = s1 . (fsloc 0) by A37, A39, A46, Lm2
.= (Comput ((ProgramPart s2),s2,i)) . (fsloc 0) by A32, A36, A33, A46, Lm2 ; :: thesis: verum

end;

suppose A47: i = 7 ; :: thesis: ( (Comput ((ProgramPart s1),s1,i)) . (intloc 0) = (Comput ((ProgramPart s2),s2,i)) . (intloc 0) & (Comput ((ProgramPart s1),s1,i)) . (IC SCM+FSA) = (Comput ((ProgramPart s2),s2,i)) . (IC SCM+FSA) & (Comput ((ProgramPart s1),s1,i)) . (fsloc 0) = (Comput ((ProgramPart s2),s2,i)) . (fsloc 0) )

hence (Comput ((ProgramPart s1),s1,i)) . (intloc 0) = s1 . (intloc 0) by A37, A39, Lm2
.= (Comput ((ProgramPart s2),s2,i)) . (intloc 0) by A32, A36, A34, A47, Lm2 ;
:: thesis: ( (Comput ((ProgramPart s1),s1,i)) . (IC SCM+FSA) = (Comput ((ProgramPart s2),s2,i)) . (IC SCM+FSA) & (Comput ((ProgramPart s1),s1,i)) . (fsloc 0) = (Comput ((ProgramPart s2),s2,i)) . (fsloc 0) )
thus (Comput ((ProgramPart s1),s1,i)) . (IC SCM+FSA) = 7 by A37, A39, A47, Lm2
.= (Comput ((ProgramPart s2),s2,i)) . (IC SCM+FSA) by A32, A36, A47, Lm2 ; :: thesis: (Comput ((ProgramPart s1),s1,i)) . (fsloc 0) = (Comput ((ProgramPart s2),s2,i)) . (fsloc 0)
thus (Comput ((ProgramPart s1),s1,i)) . (fsloc 0) = s1 . (fsloc 0) by A37, A39, A47, Lm2
.= (Comput ((ProgramPart s2),s2,i)) . (fsloc 0) by A32, A36, A33, A47, Lm2 ; :: thesis: verum

end;

suppose A48: i = 8 ; :: thesis: ( (Comput ((ProgramPart s1),s1,i)) . (intloc 0) = (Comput ((ProgramPart s2),s2,i)) . (intloc 0) & (Comput ((ProgramPart s1),s1,i)) . (IC SCM+FSA) = (Comput ((ProgramPart s2),s2,i)) . (IC SCM+FSA) & (Comput ((ProgramPart s1),s1,i)) . (fsloc 0) = (Comput ((ProgramPart s2),s2,i)) . (fsloc 0) )

hence (Comput ((ProgramPart s1),s1,i)) . (intloc 0) = s1 . (intloc 0) by A37, A39, Lm2
.= (Comput ((ProgramPart s2),s2,i)) . (intloc 0) by A32, A36, A34, A48, Lm2 ;
:: thesis: ( (Comput ((ProgramPart s1),s1,i)) . (IC SCM+FSA) = (Comput ((ProgramPart s2),s2,i)) . (IC SCM+FSA) & (Comput ((ProgramPart s1),s1,i)) . (fsloc 0) = (Comput ((ProgramPart s2),s2,i)) . (fsloc 0) )
thus (Comput ((ProgramPart s1),s1,i)) . (IC SCM+FSA) = 8 by A37, A39, A48, Lm2
.= (Comput ((ProgramPart s2),s2,i)) . (IC SCM+FSA) by A32, A36, A48, Lm2 ; :: thesis: (Comput ((ProgramPart s1),s1,i)) . (fsloc 0) = (Comput ((ProgramPart s2),s2,i)) . (fsloc 0)
thus (Comput ((ProgramPart s1),s1,i)) . (fsloc 0) = s1 . (fsloc 0) by A37, A39, A48, Lm2
.= (Comput ((ProgramPart s2),s2,i)) . (fsloc 0) by A32, A36, A33, A48, Lm2 ; :: thesis: verum

end;

suppose A49: i = 9 ; :: thesis: ( (Comput ((ProgramPart s1),s1,i)) . (intloc 0) = (Comput ((ProgramPart s2),s2,i)) . (intloc 0) & (Comput ((ProgramPart s1),s1,i)) . (IC SCM+FSA) = (Comput ((ProgramPart s2),s2,i)) . (IC SCM+FSA) & (Comput ((ProgramPart s1),s1,i)) . (fsloc 0) = (Comput ((ProgramPart s2),s2,i)) . (fsloc 0) )

hence (Comput ((ProgramPart s1),s1,i)) . (intloc 0) = s1 . (intloc 0) by A37, A39, Lm2
.= (Comput ((ProgramPart s2),s2,i)) . (intloc 0) by A32, A36, A34, A49, Lm2 ;
:: thesis: ( (Comput ((ProgramPart s1),s1,i)) . (IC SCM+FSA) = (Comput ((ProgramPart s2),s2,i)) . (IC SCM+FSA) & (Comput ((ProgramPart s1),s1,i)) . (fsloc 0) = (Comput ((ProgramPart s2),s2,i)) . (fsloc 0) )
thus (Comput ((ProgramPart s1),s1,i)) . (IC SCM+FSA) = 9 by A37, A39, A49, Lm2
.= (Comput ((ProgramPart s2),s2,i)) . (IC SCM+FSA) by A32, A36, A49, Lm2 ; :: thesis: (Comput ((ProgramPart s1),s1,i)) . (fsloc 0) = (Comput ((ProgramPart s2),s2,i)) . (fsloc 0)
thus (Comput ((ProgramPart s1),s1,i)) . (fsloc 0) = s1 . (fsloc 0) by A37, A39, A49, Lm2
.= (Comput ((ProgramPart s2),s2,i)) . (fsloc 0) by A32, A36, A33, A49, Lm2 ; :: thesis: verum

end;

suppose A50: i = 10 ; :: thesis: ( (Comput ((ProgramPart s1),s1,i)) . (intloc 0) = (Comput ((ProgramPart s2),s2,i)) . (intloc 0) & (Comput ((ProgramPart s1),s1,i)) . (IC SCM+FSA) = (Comput ((ProgramPart s2),s2,i)) . (IC SCM+FSA) & (Comput ((ProgramPart s1),s1,i)) . (fsloc 0) = (Comput ((ProgramPart s2),s2,i)) . (fsloc 0) )

hence (Comput ((ProgramPart s1),s1,i)) . (intloc 0) = s1 . (intloc 0) by A37, A39, Lm2
.= (Comput ((ProgramPart s2),s2,i)) . (intloc 0) by A32, A36, A34, A50, Lm2 ;
:: thesis: ( (Comput ((ProgramPart s1),s1,i)) . (IC SCM+FSA) = (Comput ((ProgramPart s2),s2,i)) . (IC SCM+FSA) & (Comput ((ProgramPart s1),s1,i)) . (fsloc 0) = (Comput ((ProgramPart s2),s2,i)) . (fsloc 0) )
thus (Comput ((ProgramPart s1),s1,i)) . (IC SCM+FSA) = 10 by A37, A39, A50, Lm2
.= (Comput ((ProgramPart s2),s2,i)) . (IC SCM+FSA) by A32, A36, A50, Lm2 ; :: thesis: (Comput ((ProgramPart s1),s1,i)) . (fsloc 0) = (Comput ((ProgramPart s2),s2,i)) . (fsloc 0)
thus (Comput ((ProgramPart s1),s1,i)) . (fsloc 0) = s1 . (fsloc 0) by A37, A39, A50, Lm2
.= (Comput ((ProgramPart s2),s2,i)) . (fsloc 0) by A32, A36, A33, A50, Lm2 ; :: thesis: verum

end;

end;

end;

A51: dom ((Initialized Insert-Sort-Algorithm) +* ((fsloc 0) .--> w)) = (dom Insert-Sort-Algorithm) \/ {(intloc 0),(IC SCM+FSA),(fsloc 0)} by SCMBSORT:42;

now

let s1, s2 be State of SCM+FSA; :: thesis: for i being Element of NAT st (Initialized Insert-Sort-Algorithm) +* ((fsloc 0) .--> w) c= s1 & (Initialized Insert-Sort-Algorithm) +* ((fsloc 0) .--> w) c= s2 holds
(Comput ((ProgramPart s1),s1,i)) | (dom ((Initialized Insert-Sort-Algorithm) +* ((fsloc 0) .--> w))) = (Comput ((ProgramPart s2),s2,i)) | (dom ((Initialized Insert-Sort-Algorithm) +* ((fsloc 0) .--> w)))
let i be Element of NAT ; :: thesis: ( (Initialized Insert-Sort-Algorithm) +* ((fsloc 0) .--> w) c= s1 & (Initialized Insert-Sort-Algorithm) +* ((fsloc 0) .--> w) c= s2 implies (Comput ((ProgramPart s1),s1,i)) | (dom ((Initialized Insert-Sort-Algorithm) +* ((fsloc 0) .--> w))) = (Comput ((ProgramPart s2),s2,i)) | (dom ((Initialized Insert-Sort-Algorithm) +* ((fsloc 0) .--> w))) )
assume that
A52: (Initialized Insert-Sort-Algorithm) +* ((fsloc 0) .--> w) c= s1 and
A53: (Initialized Insert-Sort-Algorithm) +* ((fsloc 0) .--> w) c= s2 ; :: thesis: (Comput ((ProgramPart s1),s1,i)) | (dom ((Initialized Insert-Sort-Algorithm) +* ((fsloc 0) .--> w))) = (Comput ((ProgramPart s2),s2,i)) | (dom ((Initialized Insert-Sort-Algorithm) +* ((fsloc 0) .--> w)))
set Cs1i = Comput ((ProgramPart s1),s1,i);
set Cs2i = Comput ((ProgramPart s2),s2,i);
A54: (UsedInt*Loc Insert-Sort-Algorithm) \/ (UsedIntLoc Insert-Sort-Algorithm) c= dom (Comput ((ProgramPart s1),s1,i)) by SCMBSORT:56;
A55: Insert-Sort-Algorithm c= s2 by A53, SCMBSORT:53;
Insert-Sort-Algorithm c= s1 by A52, SCMBSORT:53;
then A56: (Comput ((ProgramPart s1),s1,i)) | (dom Insert-Sort-Algorithm) = (Comput ((ProgramPart s2),s2,i)) | (dom Insert-Sort-Algorithm) by A55, AMI_1:124;
A57: fsloc 0 in (UsedInt*Loc Insert-Sort-Algorithm) \/ (UsedIntLoc Insert-Sort-Algorithm) by A3, ENUMSET1:def 6;
A58: (UsedInt*Loc Insert-Sort-Algorithm) \/ (UsedIntLoc Insert-Sort-Algorithm) c= dom (Comput ((ProgramPart s2),s2,i)) by SCMBSORT:56;
A59: ( i > 10 implies 10 + 1 < i + 1 ) by XREAL_1:8;
A60: intloc 0 in (UsedInt*Loc Insert-Sort-Algorithm) \/ (UsedIntLoc Insert-Sort-Algorithm) by A3, ENUMSET1:def 6;

A61: now

let x be set ; :: thesis: ( x in {(intloc 0),(IC SCM+FSA),(fsloc 0)} implies (Comput ((ProgramPart s1),s1,i)) . b₁ = (Comput ((ProgramPart s2),s2,i)) . b₁ )
assume A62: x in {(intloc 0),(IC SCM+FSA),(fsloc 0)} ; :: thesis: (Comput ((ProgramPart s1),s1,i)) . b₁ = (Comput ((ProgramPart s2),s2,i)) . b₁

per cases ( x = intloc 0 or x = IC SCM+FSA or x = fsloc 0 ) by A62, ENUMSET1:def 1;

suppose A63: x = intloc 0 ; :: thesis: (Comput ((ProgramPart s1),s1,i)) . b₁ = (Comput ((ProgramPart s2),s2,i)) . b₁

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

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

hence (Comput ((ProgramPart s1),s1,i)) . x = (Comput ((ProgramPart s2),s2,i)) . x by A28, A52, A53, A63; :: thesis: verum

end;

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

then 11 <= i by A59, NAT_1:13;
then (Comput ((ProgramPart s1),s1,i)) | ((UsedInt*Loc Insert-Sort-Algorithm) \/ (UsedIntLoc Insert-Sort-Algorithm)) = (Comput ((ProgramPart s2),s2,i)) | ((UsedInt*Loc Insert-Sort-Algorithm) \/ (UsedIntLoc Insert-Sort-Algorithm)) by A4, A52, A53;
hence (Comput ((ProgramPart s1),s1,i)) . x = (Comput ((ProgramPart s2),s2,i)) . x by A60, A54, A58, A63, FUNCT_1:165; :: thesis: verum

end;

end;

end;

suppose A64: x = IC SCM+FSA ; :: thesis: (Comput ((ProgramPart s1),s1,i)) . b₁ = (Comput ((ProgramPart s2),s2,i)) . b₁

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

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

hence (Comput ((ProgramPart s1),s1,i)) . x = (Comput ((ProgramPart s2),s2,i)) . x by A28, A52, A53, A64; :: thesis: verum

end;

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

then 11 <= i by A59, NAT_1:13;
hence (Comput ((ProgramPart s1),s1,i)) . x = (Comput ((ProgramPart s2),s2,i)) . x by A4, A52, A53, A64; :: thesis: verum

end;

end;

end;

suppose A65: x = fsloc 0 ; :: thesis: (Comput ((ProgramPart s1),s1,i)) . b₁ = (Comput ((ProgramPart s2),s2,i)) . b₁

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

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

hence (Comput ((ProgramPart s1),s1,i)) . x = (Comput ((ProgramPart s2),s2,i)) . x by A28, A52, A53, A65; :: thesis: verum

end;

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

then 11 <= i by A59, NAT_1:13;
then (Comput ((ProgramPart s1),s1,i)) | ((UsedInt*Loc Insert-Sort-Algorithm) \/ (UsedIntLoc Insert-Sort-Algorithm)) = (Comput ((ProgramPart s2),s2,i)) | ((UsedInt*Loc Insert-Sort-Algorithm) \/ (UsedIntLoc Insert-Sort-Algorithm)) by A4, A52, A53;
hence (Comput ((ProgramPart s1),s1,i)) . x = (Comput ((ProgramPart s2),s2,i)) . x by A57, A54, A58, A65, FUNCT_1:165; :: thesis: verum

end;

end;

end;

end;

end;

A66: {(intloc 0),(IC SCM+FSA),(fsloc 0)} c= dom (Comput ((ProgramPart s2),s2,i)) by SCMBSORT:55;
{(intloc 0),(IC SCM+FSA),(fsloc 0)} c= dom (Comput ((ProgramPart s1),s1,i)) by SCMBSORT:55;
then (Comput ((ProgramPart s1),s1,i)) | {(intloc 0),(IC SCM+FSA),(fsloc 0)} = (Comput ((ProgramPart s2),s2,i)) | {(intloc 0),(IC SCM+FSA),(fsloc 0)} by A66, A61, FUNCT_1:165;
hence (Comput ((ProgramPart s1),s1,i)) | (dom ((Initialized Insert-Sort-Algorithm) +* ((fsloc 0) .--> w))) = (Comput ((ProgramPart s2),s2,i)) | (dom ((Initialized Insert-Sort-Algorithm) +* ((fsloc 0) .--> w))) by A51, A56, RELAT_1:185; :: thesis: verum

end;

then for s1, s2 being State of SCM+FSA st (Initialized Insert-Sort-Algorithm) +* ((fsloc 0) .--> w) c= s1 & (Initialized Insert-Sort-Algorithm) +* ((fsloc 0) .--> w) c= s2 holds
for i being Element of NAT holds (Comput ((ProgramPart s1),s1,i)) | (dom ((Initialized Insert-Sort-Algorithm) +* ((fsloc 0) .--> w))) = (Comput ((ProgramPart s2),s2,i)) | (dom ((Initialized Insert-Sort-Algorithm) +* ((fsloc 0) .--> w))) ;
hence (Initialized Insert-Sort-Algorithm) +* ((fsloc 0) .--> w) is autonomic by EXTPRO_1:def 9; :: thesis: verum