set DD = {(intloc 0),(IC ),(fsloc 0)};
let w be FinSequence of INT ; :: thesis: Initialized ((fsloc 0) .--> w) is Insert-Sort-Algorithm -autonomic
set p = Initialized ((fsloc 0) .--> w);
set q = Insert-Sort-Algorithm ;
set UD = {(fsloc 0),(intloc 0),(intloc 1),(intloc 2),(intloc 3),(intloc 4),(intloc 5),(intloc 6)};
set Us = (UsedI*Loc Insert-Sort-Algorithm) \/ (UsedILoc Insert-Sort-Algorithm);
A1: UsedILoc Insert-Sort-Algorithm = {(intloc 0),(intloc 1),(intloc 2),(intloc 3),(intloc 4),(intloc 5),(intloc 6)} by Th18;
A2: UsedI*Loc Insert-Sort-Algorithm = {(fsloc 0)} by Th19;
then A3: (UsedI*Loc Insert-Sort-Algorithm) \/ (UsedILoc Insert-Sort-Algorithm) = {(fsloc 0),(intloc 0),(intloc 1),(intloc 2),(intloc 3),(intloc 4),(intloc 5),(intloc 6)} by A1, ENUMSET1:22;
A4: for i being Nat
for s1, s2 being State of SCM+FSA
for P1, P2 being Instruction-Sequence of SCM+FSA st 11 <= i & Initialized ((fsloc 0) .--> w) c= s1 & Initialized ((fsloc 0) .--> w) c= s2 & Insert-Sort-Algorithm c= P1 & Insert-Sort-Algorithm c= P2 holds
( (Comput (P1,s1,i)) | ((UsedI*Loc Insert-Sort-Algorithm) \/ (UsedILoc Insert-Sort-Algorithm)) = (Comput (P2,s2,i)) | ((UsedI*Loc Insert-Sort-Algorithm) \/ (UsedILoc Insert-Sort-Algorithm)) & (Comput (P1,s1,i)) . (IC ) = (Comput (P2,s2,i)) . (IC ) )

proof

let i be Nat; :: thesis: for s1, s2 being State of SCM+FSA
for P1, P2 being Instruction-Sequence of SCM+FSA st 11 <= i & Initialized ((fsloc 0) .--> w) c= s1 & Initialized ((fsloc 0) .--> w) c= s2 & Insert-Sort-Algorithm c= P1 & Insert-Sort-Algorithm c= P2 holds
( (Comput (P1,s1,i)) | ((UsedI*Loc Insert-Sort-Algorithm) \/ (UsedILoc Insert-Sort-Algorithm)) = (Comput (P2,s2,i)) | ((UsedI*Loc Insert-Sort-Algorithm) \/ (UsedILoc Insert-Sort-Algorithm)) & (Comput (P1,s1,i)) . (IC ) = (Comput (P2,s2,i)) . (IC ) )
let s1, s2 be State of SCM+FSA; :: thesis: for P1, P2 being Instruction-Sequence of SCM+FSA st 11 <= i & Initialized ((fsloc 0) .--> w) c= s1 & Initialized ((fsloc 0) .--> w) c= s2 & Insert-Sort-Algorithm c= P1 & Insert-Sort-Algorithm c= P2 holds
( (Comput (P1,s1,i)) | ((UsedI*Loc Insert-Sort-Algorithm) \/ (UsedILoc Insert-Sort-Algorithm)) = (Comput (P2,s2,i)) | ((UsedI*Loc Insert-Sort-Algorithm) \/ (UsedILoc Insert-Sort-Algorithm)) & (Comput (P1,s1,i)) . (IC ) = (Comput (P2,s2,i)) . (IC ) )
let P1, P2 be Instruction-Sequence of SCM+FSA; :: thesis: ( 11 <= i & Initialized ((fsloc 0) .--> w) c= s1 & Initialized ((fsloc 0) .--> w) c= s2 & Insert-Sort-Algorithm c= P1 & Insert-Sort-Algorithm c= P2 implies ( (Comput (P1,s1,i)) | ((UsedI*Loc Insert-Sort-Algorithm) \/ (UsedILoc Insert-Sort-Algorithm)) = (Comput (P2,s2,i)) | ((UsedI*Loc Insert-Sort-Algorithm) \/ (UsedILoc Insert-Sort-Algorithm)) & (Comput (P1,s1,i)) . (IC ) = (Comput (P2,s2,i)) . (IC ) ) )
assume that
A5: 11 <= i and
A6: Initialized ((fsloc 0) .--> w) c= s1 and
A7: Initialized ((fsloc 0) .--> w) c= s2 and
A8: Insert-Sort-Algorithm c= P1 and
A9: Insert-Sort-Algorithm c= P2 ; :: thesis: ( (Comput (P1,s1,i)) | ((UsedI*Loc Insert-Sort-Algorithm) \/ (UsedILoc Insert-Sort-Algorithm)) = (Comput (P2,s2,i)) | ((UsedI*Loc Insert-Sort-Algorithm) \/ (UsedILoc Insert-Sort-Algorithm)) & (Comput (P1,s1,i)) . (IC ) = (Comput (P2,s2,i)) . (IC ) )
A10: s1 . (intloc 0) = 1 by A6, SCMFSA_M:33
.= s2 . (intloc 0) by A7, SCMFSA_M:33 ;
A11: s2 is 0 -started by A7, MEMSTR_0:17;
set Cs11 = Comput (P1,s1,11);
set Cs21 = Comput (P2,s2,11);
A12: s1 is 0 -started by A6, MEMSTR_0:17;
A13: s1 . (fsloc 0) = w by A6, SCMFSA_M:33
.= s2 . (fsloc 0) by A7, SCMFSA_M:33 ;

A14: now :: thesis: for x being set st x in (UsedI*Loc Insert-Sort-Algorithm) \/ (UsedILoc Insert-Sort-Algorithm) holds
(Comput (P1,s1,11)) . x = (Comput (P2,s2,11)) . x

let x be set ; :: thesis: ( x in (UsedI*Loc Insert-Sort-Algorithm) \/ (UsedILoc Insert-Sort-Algorithm) implies (Comput (P1,s1,11)) . b₁ = (Comput (P2,s2,11)) . b₁ )
assume x in (UsedI*Loc Insert-Sort-Algorithm) \/ (UsedILoc Insert-Sort-Algorithm) ; :: thesis: (Comput (P1,s1,11)) . b₁ = (Comput (P2,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, ENUMSET1:22;

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 A15, ENUMSET1:def 6;

suppose A16: x = fsloc 0 ; :: thesis: (Comput (P1,s1,11)) . b₁ = (Comput (P2,s2,11)) . b₁

hence (Comput (P1,s1,11)) . x = s1 . (fsloc 0) by A12, Lm2, A8
.= (Comput (P2,s2,11)) . x by A11, A13, A16, Lm2, A9 ;
:: thesis: verum

end;

suppose A17: x = intloc 0 ; :: thesis: (Comput (P1,s1,11)) . b₁ = (Comput (P2,s2,11)) . b₁

hence (Comput (P1,s1,11)) . x = s1 . (intloc 0) by A12, Lm2, A8
.= (Comput (P2,s2,11)) . x by A11, A10, A17, Lm2, A9 ;
:: thesis: verum

end;

suppose A18: x = intloc 1 ; :: thesis: (Comput (P1,s1,11)) . b₁ = (Comput (P2,s2,11)) . b₁

hence (Comput (P1,s1,11)) . x = len (s1 . (fsloc 0)) by A12, Lm2, A8
.= (Comput (P2,s2,11)) . x by A11, A13, A18, Lm2, A9 ;
:: thesis: verum

end;

suppose A19: x = intloc 2 ; :: thesis: (Comput (P1,s1,11)) . b₁ = (Comput (P2,s2,11)) . b₁

hence (Comput (P1,s1,11)) . x = s1 . (intloc 0) by A12, Lm2, A8
.= (Comput (P2,s2,11)) . x by A11, A10, A19, Lm2, A9 ;
:: thesis: verum

end;

suppose A20: x = intloc 3 ; :: thesis: (Comput (P1,s1,11)) . b₁ = (Comput (P2,s2,11)) . b₁

hence (Comput (P1,s1,11)) . x = s1 . (intloc 0) by A12, Lm2, A8
.= (Comput (P2,s2,11)) . x by A11, A10, A20, Lm2, A9 ;
:: thesis: verum

end;

suppose A21: x = intloc 4 ; :: thesis: (Comput (P1,s1,11)) . b₁ = (Comput (P2,s2,11)) . b₁

hence (Comput (P1,s1,11)) . x = s1 . (intloc 0) by A12, Lm2, A8
.= (Comput (P2,s2,11)) . x by A11, A10, A21, Lm2, A9 ;
:: thesis: verum

end;

suppose A22: x = intloc 5 ; :: thesis: (Comput (P1,s1,11)) . b₁ = (Comput (P2,s2,11)) . b₁

hence (Comput (P1,s1,11)) . x = s1 . (intloc 0) by A12, Lm2, A8
.= (Comput (P2,s2,11)) . x by A11, A10, A22, Lm2, A9 ;
:: thesis: verum

end;

suppose A23: x = intloc 6 ; :: thesis: (Comput (P1,s1,11)) . b₁ = (Comput (P2,s2,11)) . b₁

hence (Comput (P1,s1,11)) . x = s1 . (intloc 0) by A12, Lm2, A8
.= (Comput (P2,s2,11)) . x by A11, A10, A23, Lm2, A9 ;
:: thesis: verum

end;

end;

end;

A24: for i being Nat holds IC (Comput (P2,s2,i)) in dom Insert-Sort-Algorithm by A7, Th26, A9;
A25: for i being Nat holds IC (Comput (P1,s1,i)) in dom Insert-Sort-Algorithm by A6, Th26, A8;
A26: (UsedI*Loc Insert-Sort-Algorithm) \/ (UsedILoc Insert-Sort-Algorithm) c= dom (Comput (P2,s2,11)) by SCMBSORT:32;
(UsedI*Loc Insert-Sort-Algorithm) \/ (UsedILoc Insert-Sort-Algorithm) c= dom (Comput (P1,s1,11)) by SCMBSORT:32;
then A27: (Comput (P1,s1,11)) | ((UsedI*Loc Insert-Sort-Algorithm) \/ (UsedILoc Insert-Sort-Algorithm)) = (Comput (P2,s2,11)) | ((UsedI*Loc Insert-Sort-Algorithm) \/ (UsedILoc Insert-Sort-Algorithm)) by A26, A14, FUNCT_1:95;
(Comput (P1,s1,11)) . (IC ) = 11 by A12, Lm2, A8
.= (Comput (P2,s2,11)) . (IC ) by A11, Lm2, A9 ;
hence ( (Comput (P1,s1,i)) | ((UsedI*Loc Insert-Sort-Algorithm) \/ (UsedILoc Insert-Sort-Algorithm)) = (Comput (P2,s2,i)) | ((UsedI*Loc Insert-Sort-Algorithm) \/ (UsedILoc Insert-Sort-Algorithm)) & (Comput (P1,s1,i)) . (IC ) = (Comput (P2,s2,i)) . (IC ) ) by A5, A27, A25, A24, A8, A9, SCMBSORT:15; :: thesis: verum

end;

A28: for s1, s2 being State of SCM+FSA
for P1, P2 being Instruction-Sequence of SCM+FSA
for i being Nat st Initialized ((fsloc 0) .--> w) c= s1 & Initialized ((fsloc 0) .--> w) c= s2 & Insert-Sort-Algorithm c= P1 & Insert-Sort-Algorithm c= P2 & i <= 10 holds
( (Comput (P1,s1,i)) . (intloc 0) = (Comput (P2,s2,i)) . (intloc 0) & (Comput (P1,s1,i)) . (IC ) = (Comput (P2,s2,i)) . (IC ) & (Comput (P1,s1,i)) . (fsloc 0) = (Comput (P2,s2,i)) . (fsloc 0) )

proof

let s1, s2 be State of SCM+FSA; :: thesis: for P1, P2 being Instruction-Sequence of SCM+FSA
for i being Nat st Initialized ((fsloc 0) .--> w) c= s1 & Initialized ((fsloc 0) .--> w) c= s2 & Insert-Sort-Algorithm c= P1 & Insert-Sort-Algorithm c= P2 & i <= 10 holds
( (Comput (P1,s1,i)) . (intloc 0) = (Comput (P2,s2,i)) . (intloc 0) & (Comput (P1,s1,i)) . (IC ) = (Comput (P2,s2,i)) . (IC ) & (Comput (P1,s1,i)) . (fsloc 0) = (Comput (P2,s2,i)) . (fsloc 0) )
let P1, P2 be Instruction-Sequence of SCM+FSA; :: thesis: for i being Nat st Initialized ((fsloc 0) .--> w) c= s1 & Initialized ((fsloc 0) .--> w) c= s2 & Insert-Sort-Algorithm c= P1 & Insert-Sort-Algorithm c= P2 & i <= 10 holds
( (Comput (P1,s1,i)) . (intloc 0) = (Comput (P2,s2,i)) . (intloc 0) & (Comput (P1,s1,i)) . (IC ) = (Comput (P2,s2,i)) . (IC ) & (Comput (P1,s1,i)) . (fsloc 0) = (Comput (P2,s2,i)) . (fsloc 0) )
let i be Nat; :: thesis: ( Initialized ((fsloc 0) .--> w) c= s1 & Initialized ((fsloc 0) .--> w) c= s2 & Insert-Sort-Algorithm c= P1 & Insert-Sort-Algorithm c= P2 & i <= 10 implies ( (Comput (P1,s1,i)) . (intloc 0) = (Comput (P2,s2,i)) . (intloc 0) & (Comput (P1,s1,i)) . (IC ) = (Comput (P2,s2,i)) . (IC ) & (Comput (P1,s1,i)) . (fsloc 0) = (Comput (P2,s2,i)) . (fsloc 0) ) )
assume that
A29: Initialized ((fsloc 0) .--> w) c= s1 and
A30: Initialized ((fsloc 0) .--> w) c= s2 and
A31: Insert-Sort-Algorithm c= P1 and
A32: Insert-Sort-Algorithm c= P2 and
A33: i <= 10 ; :: thesis: ( (Comput (P1,s1,i)) . (intloc 0) = (Comput (P2,s2,i)) . (intloc 0) & (Comput (P1,s1,i)) . (IC ) = (Comput (P2,s2,i)) . (IC ) & (Comput (P1,s1,i)) . (fsloc 0) = (Comput (P2,s2,i)) . (fsloc 0) )
A34: s2 is 0 -started by A30, MEMSTR_0:17;
A35: s1 . (fsloc 0) = w by A29, SCMFSA_M:33
.= s2 . (fsloc 0) by A30, SCMFSA_M:33 ;
A36: s1 . (intloc 0) = 1 by A29, SCMFSA_M:33
.= s2 . (intloc 0) by A30, SCMFSA_M:33 ;
A37: s1 is 0 -started by A29, MEMSTR_0:17;
then A38: IC s1 = 0
.= IC s2 by A34 ;
not not i = 0 & ... & not i = 10 by A33;

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 A39: i = 0 ; :: thesis: ( (Comput (P1,s1,i)) . (intloc 0) = (Comput (P2,s2,i)) . (intloc 0) & (Comput (P1,s1,i)) . (IC ) = (Comput (P2,s2,i)) . (IC ) & (Comput (P1,s1,i)) . (fsloc 0) = (Comput (P2,s2,i)) . (fsloc 0) )

hence (Comput (P1,s1,i)) . (intloc 0) = (Comput (P2,s2,i)) . (intloc 0) by A36; :: thesis: ( (Comput (P1,s1,i)) . (IC ) = (Comput (P2,s2,i)) . (IC ) & (Comput (P1,s1,i)) . (fsloc 0) = (Comput (P2,s2,i)) . (fsloc 0) )
thus (Comput (P1,s1,i)) . (IC ) = (Comput (P2,s2,i)) . (IC ) by A38, A39; :: thesis: (Comput (P1,s1,i)) . (fsloc 0) = (Comput (P2,s2,i)) . (fsloc 0)
thus (Comput (P1,s1,i)) . (fsloc 0) = (Comput (P2,s2,i)) . (fsloc 0) by A35, A39; :: thesis: verum

end;

suppose A40: i = 1 ; :: thesis: ( (Comput (P1,s1,i)) . (intloc 0) = (Comput (P2,s2,i)) . (intloc 0) & (Comput (P1,s1,i)) . (IC ) = (Comput (P2,s2,i)) . (IC ) & (Comput (P1,s1,i)) . (fsloc 0) = (Comput (P2,s2,i)) . (fsloc 0) )

hence (Comput (P1,s1,i)) . (intloc 0) = s1 . (intloc 0) by A37, Lm2, A31
.= (Comput (P2,s2,i)) . (intloc 0) by A34, A36, A40, Lm2, A32 ;
:: thesis: ( (Comput (P1,s1,i)) . (IC ) = (Comput (P2,s2,i)) . (IC ) & (Comput (P1,s1,i)) . (fsloc 0) = (Comput (P2,s2,i)) . (fsloc 0) )
thus (Comput (P1,s1,i)) . (IC ) = 1 by A37, A40, Lm2, A31
.= (Comput (P2,s2,i)) . (IC ) by A34, A40, Lm2, A32 ; :: thesis: (Comput (P1,s1,i)) . (fsloc 0) = (Comput (P2,s2,i)) . (fsloc 0)
thus (Comput (P1,s1,i)) . (fsloc 0) = s1 . (fsloc 0) by A37, A40, Lm2, A31
.= (Comput (P2,s2,i)) . (fsloc 0) by A34, A35, A40, Lm2, A32 ; :: thesis: verum

end;

suppose A41: i = 2 ; :: thesis: ( (Comput (P1,s1,i)) . (intloc 0) = (Comput (P2,s2,i)) . (intloc 0) & (Comput (P1,s1,i)) . (IC ) = (Comput (P2,s2,i)) . (IC ) & (Comput (P1,s1,i)) . (fsloc 0) = (Comput (P2,s2,i)) . (fsloc 0) )

hence (Comput (P1,s1,i)) . (intloc 0) = s1 . (intloc 0) by A37, Lm2, A31
.= (Comput (P2,s2,i)) . (intloc 0) by A34, A36, A41, Lm2, A32 ;
:: thesis: ( (Comput (P1,s1,i)) . (IC ) = (Comput (P2,s2,i)) . (IC ) & (Comput (P1,s1,i)) . (fsloc 0) = (Comput (P2,s2,i)) . (fsloc 0) )
thus (Comput (P1,s1,i)) . (IC ) = 2 by A37, A41, Lm2, A31
.= (Comput (P2,s2,i)) . (IC ) by A34, A41, Lm2, A32 ; :: thesis: (Comput (P1,s1,i)) . (fsloc 0) = (Comput (P2,s2,i)) . (fsloc 0)
thus (Comput (P1,s1,i)) . (fsloc 0) = s1 . (fsloc 0) by A37, A41, Lm2, A31
.= (Comput (P2,s2,i)) . (fsloc 0) by A34, A35, A41, Lm2, A32 ; :: thesis: verum

end;

suppose A42: i = 3 ; :: thesis: ( (Comput (P1,s1,i)) . (intloc 0) = (Comput (P2,s2,i)) . (intloc 0) & (Comput (P1,s1,i)) . (IC ) = (Comput (P2,s2,i)) . (IC ) & (Comput (P1,s1,i)) . (fsloc 0) = (Comput (P2,s2,i)) . (fsloc 0) )

hence (Comput (P1,s1,i)) . (intloc 0) = s1 . (intloc 0) by A37, Lm2, A31
.= (Comput (P2,s2,i)) . (intloc 0) by A34, A36, A42, Lm2, A32 ;
:: thesis: ( (Comput (P1,s1,i)) . (IC ) = (Comput (P2,s2,i)) . (IC ) & (Comput (P1,s1,i)) . (fsloc 0) = (Comput (P2,s2,i)) . (fsloc 0) )
thus (Comput (P1,s1,i)) . (IC ) = 3 by A37, A42, Lm2, A31
.= (Comput (P2,s2,i)) . (IC ) by A34, A42, Lm2, A32 ; :: thesis: (Comput (P1,s1,i)) . (fsloc 0) = (Comput (P2,s2,i)) . (fsloc 0)
thus (Comput (P1,s1,i)) . (fsloc 0) = s1 . (fsloc 0) by A37, A42, Lm2, A31
.= (Comput (P2,s2,i)) . (fsloc 0) by A34, A35, A42, Lm2, A32 ; :: thesis: verum

end;

suppose A43: i = 4 ; :: thesis: ( (Comput (P1,s1,i)) . (intloc 0) = (Comput (P2,s2,i)) . (intloc 0) & (Comput (P1,s1,i)) . (IC ) = (Comput (P2,s2,i)) . (IC ) & (Comput (P1,s1,i)) . (fsloc 0) = (Comput (P2,s2,i)) . (fsloc 0) )

hence (Comput (P1,s1,i)) . (intloc 0) = s1 . (intloc 0) by A37, Lm2, A31
.= (Comput (P2,s2,i)) . (intloc 0) by A34, A36, A43, Lm2, A32 ;
:: thesis: ( (Comput (P1,s1,i)) . (IC ) = (Comput (P2,s2,i)) . (IC ) & (Comput (P1,s1,i)) . (fsloc 0) = (Comput (P2,s2,i)) . (fsloc 0) )
thus (Comput (P1,s1,i)) . (IC ) = 4 by A37, A43, Lm2, A31
.= (Comput (P2,s2,i)) . (IC ) by A34, A43, Lm2, A32 ; :: thesis: (Comput (P1,s1,i)) . (fsloc 0) = (Comput (P2,s2,i)) . (fsloc 0)
thus (Comput (P1,s1,i)) . (fsloc 0) = s1 . (fsloc 0) by A37, A43, Lm2, A31
.= (Comput (P2,s2,i)) . (fsloc 0) by A34, A35, A43, Lm2, A32 ; :: thesis: verum

end;

suppose A44: i = 5 ; :: thesis: ( (Comput (P1,s1,i)) . (intloc 0) = (Comput (P2,s2,i)) . (intloc 0) & (Comput (P1,s1,i)) . (IC ) = (Comput (P2,s2,i)) . (IC ) & (Comput (P1,s1,i)) . (fsloc 0) = (Comput (P2,s2,i)) . (fsloc 0) )

hence (Comput (P1,s1,i)) . (intloc 0) = s1 . (intloc 0) by A37, Lm2, A31
.= (Comput (P2,s2,i)) . (intloc 0) by A34, A36, A44, Lm2, A32 ;
:: thesis: ( (Comput (P1,s1,i)) . (IC ) = (Comput (P2,s2,i)) . (IC ) & (Comput (P1,s1,i)) . (fsloc 0) = (Comput (P2,s2,i)) . (fsloc 0) )
thus (Comput (P1,s1,i)) . (IC ) = 5 by A37, A44, Lm2, A31
.= (Comput (P2,s2,i)) . (IC ) by A34, A44, Lm2, A32 ; :: thesis: (Comput (P1,s1,i)) . (fsloc 0) = (Comput (P2,s2,i)) . (fsloc 0)
thus (Comput (P1,s1,i)) . (fsloc 0) = s1 . (fsloc 0) by A37, A44, Lm2, A31
.= (Comput (P2,s2,i)) . (fsloc 0) by A34, A35, A44, Lm2, A32 ; :: thesis: verum

end;

suppose A45: i = 6 ; :: thesis: ( (Comput (P1,s1,i)) . (intloc 0) = (Comput (P2,s2,i)) . (intloc 0) & (Comput (P1,s1,i)) . (IC ) = (Comput (P2,s2,i)) . (IC ) & (Comput (P1,s1,i)) . (fsloc 0) = (Comput (P2,s2,i)) . (fsloc 0) )

hence (Comput (P1,s1,i)) . (intloc 0) = s1 . (intloc 0) by A37, Lm2, A31
.= (Comput (P2,s2,i)) . (intloc 0) by A34, A36, A45, Lm2, A32 ;
:: thesis: ( (Comput (P1,s1,i)) . (IC ) = (Comput (P2,s2,i)) . (IC ) & (Comput (P1,s1,i)) . (fsloc 0) = (Comput (P2,s2,i)) . (fsloc 0) )
thus (Comput (P1,s1,i)) . (IC ) = 6 by A37, A45, Lm2, A31
.= (Comput (P2,s2,i)) . (IC ) by A34, A45, Lm2, A32 ; :: thesis: (Comput (P1,s1,i)) . (fsloc 0) = (Comput (P2,s2,i)) . (fsloc 0)
thus (Comput (P1,s1,i)) . (fsloc 0) = s1 . (fsloc 0) by A37, A45, Lm2, A31
.= (Comput (P2,s2,i)) . (fsloc 0) by A34, A35, A45, Lm2, A32 ; :: thesis: verum

end;

suppose A46: i = 7 ; :: thesis: ( (Comput (P1,s1,i)) . (intloc 0) = (Comput (P2,s2,i)) . (intloc 0) & (Comput (P1,s1,i)) . (IC ) = (Comput (P2,s2,i)) . (IC ) & (Comput (P1,s1,i)) . (fsloc 0) = (Comput (P2,s2,i)) . (fsloc 0) )

hence (Comput (P1,s1,i)) . (intloc 0) = s1 . (intloc 0) by A37, Lm2, A31
.= (Comput (P2,s2,i)) . (intloc 0) by A34, A36, A46, Lm2, A32 ;
:: thesis: ( (Comput (P1,s1,i)) . (IC ) = (Comput (P2,s2,i)) . (IC ) & (Comput (P1,s1,i)) . (fsloc 0) = (Comput (P2,s2,i)) . (fsloc 0) )
thus (Comput (P1,s1,i)) . (IC ) = 7 by A37, A46, Lm2, A31
.= (Comput (P2,s2,i)) . (IC ) by A34, A46, Lm2, A32 ; :: thesis: (Comput (P1,s1,i)) . (fsloc 0) = (Comput (P2,s2,i)) . (fsloc 0)
thus (Comput (P1,s1,i)) . (fsloc 0) = s1 . (fsloc 0) by A37, A46, Lm2, A31
.= (Comput (P2,s2,i)) . (fsloc 0) by A34, A35, A46, Lm2, A32 ; :: thesis: verum

end;

suppose A47: i = 8 ; :: thesis: ( (Comput (P1,s1,i)) . (intloc 0) = (Comput (P2,s2,i)) . (intloc 0) & (Comput (P1,s1,i)) . (IC ) = (Comput (P2,s2,i)) . (IC ) & (Comput (P1,s1,i)) . (fsloc 0) = (Comput (P2,s2,i)) . (fsloc 0) )

hence (Comput (P1,s1,i)) . (intloc 0) = s1 . (intloc 0) by A37, Lm2, A31
.= (Comput (P2,s2,i)) . (intloc 0) by A34, A36, A47, Lm2, A32 ;
:: thesis: ( (Comput (P1,s1,i)) . (IC ) = (Comput (P2,s2,i)) . (IC ) & (Comput (P1,s1,i)) . (fsloc 0) = (Comput (P2,s2,i)) . (fsloc 0) )
thus (Comput (P1,s1,i)) . (IC ) = 8 by A37, A47, Lm2, A31
.= (Comput (P2,s2,i)) . (IC ) by A34, A47, Lm2, A32 ; :: thesis: (Comput (P1,s1,i)) . (fsloc 0) = (Comput (P2,s2,i)) . (fsloc 0)
thus (Comput (P1,s1,i)) . (fsloc 0) = s1 . (fsloc 0) by A37, A47, Lm2, A31
.= (Comput (P2,s2,i)) . (fsloc 0) by A34, A35, A47, Lm2, A32 ; :: thesis: verum

end;

suppose A48: i = 9 ; :: thesis: ( (Comput (P1,s1,i)) . (intloc 0) = (Comput (P2,s2,i)) . (intloc 0) & (Comput (P1,s1,i)) . (IC ) = (Comput (P2,s2,i)) . (IC ) & (Comput (P1,s1,i)) . (fsloc 0) = (Comput (P2,s2,i)) . (fsloc 0) )

hence (Comput (P1,s1,i)) . (intloc 0) = s1 . (intloc 0) by A37, Lm2, A31
.= (Comput (P2,s2,i)) . (intloc 0) by A34, A36, A48, Lm2, A32 ;
:: thesis: ( (Comput (P1,s1,i)) . (IC ) = (Comput (P2,s2,i)) . (IC ) & (Comput (P1,s1,i)) . (fsloc 0) = (Comput (P2,s2,i)) . (fsloc 0) )
thus (Comput (P1,s1,i)) . (IC ) = 9 by A37, A48, Lm2, A31
.= (Comput (P2,s2,i)) . (IC ) by A34, A48, Lm2, A32 ; :: thesis: (Comput (P1,s1,i)) . (fsloc 0) = (Comput (P2,s2,i)) . (fsloc 0)
thus (Comput (P1,s1,i)) . (fsloc 0) = s1 . (fsloc 0) by A37, A48, Lm2, A31
.= (Comput (P2,s2,i)) . (fsloc 0) by A34, A35, A48, Lm2, A32 ; :: thesis: verum

end;

suppose A49: i = 10 ; :: thesis: ( (Comput (P1,s1,i)) . (intloc 0) = (Comput (P2,s2,i)) . (intloc 0) & (Comput (P1,s1,i)) . (IC ) = (Comput (P2,s2,i)) . (IC ) & (Comput (P1,s1,i)) . (fsloc 0) = (Comput (P2,s2,i)) . (fsloc 0) )

hence (Comput (P1,s1,i)) . (intloc 0) = s1 . (intloc 0) by A37, Lm2, A31
.= (Comput (P2,s2,i)) . (intloc 0) by A34, A36, A49, Lm2, A32 ;
:: thesis: ( (Comput (P1,s1,i)) . (IC ) = (Comput (P2,s2,i)) . (IC ) & (Comput (P1,s1,i)) . (fsloc 0) = (Comput (P2,s2,i)) . (fsloc 0) )
thus (Comput (P1,s1,i)) . (IC ) = 10 by A37, A49, Lm2, A31
.= (Comput (P2,s2,i)) . (IC ) by A34, A49, Lm2, A32 ; :: thesis: (Comput (P1,s1,i)) . (fsloc 0) = (Comput (P2,s2,i)) . (fsloc 0)
thus (Comput (P1,s1,i)) . (fsloc 0) = s1 . (fsloc 0) by A37, A49, Lm2, A31
.= (Comput (P2,s2,i)) . (fsloc 0) by A34, A35, A49, Lm2, A32 ; :: thesis: verum

end;

end;

end;

A50: dom (Initialized ((fsloc 0) .--> w)) = {(intloc 0),(IC ),(fsloc 0)} by SCMFSA_M:31;
reconsider ini = (intloc 0) .--> 1 as data-only FinPartState of SCM+FSA ;
for P, Q being Instruction-Sequence of SCM+FSA st Insert-Sort-Algorithm c= P & Insert-Sort-Algorithm c= Q holds
for s1, s2 being State of SCM+FSA st Initialized ((fsloc 0) .--> w) c= s1 & Initialized ((fsloc 0) .--> w) c= s2 holds
for i being Nat holds (Comput (P,s1,i)) | (dom (Initialized ((fsloc 0) .--> w))) = (Comput (Q,s2,i)) | (dom (Initialized ((fsloc 0) .--> w)))

proof

let P1, P2 be Instruction-Sequence of SCM+FSA; :: thesis: ( Insert-Sort-Algorithm c= P1 & Insert-Sort-Algorithm c= P2 implies for s1, s2 being State of SCM+FSA st Initialized ((fsloc 0) .--> w) c= s1 & Initialized ((fsloc 0) .--> w) c= s2 holds
for i being Nat holds (Comput (P1,s1,i)) | (dom (Initialized ((fsloc 0) .--> w))) = (Comput (P2,s2,i)) | (dom (Initialized ((fsloc 0) .--> w))) )
assume that
A51: Insert-Sort-Algorithm c= P1 and
A52: Insert-Sort-Algorithm c= P2 ; :: thesis: for s1, s2 being State of SCM+FSA st Initialized ((fsloc 0) .--> w) c= s1 & Initialized ((fsloc 0) .--> w) c= s2 holds
for i being Nat holds (Comput (P1,s1,i)) | (dom (Initialized ((fsloc 0) .--> w))) = (Comput (P2,s2,i)) | (dom (Initialized ((fsloc 0) .--> w)))
let s1, s2 be State of SCM+FSA; :: thesis: ( Initialized ((fsloc 0) .--> w) c= s1 & Initialized ((fsloc 0) .--> w) c= s2 implies for i being Nat holds (Comput (P1,s1,i)) | (dom (Initialized ((fsloc 0) .--> w))) = (Comput (P2,s2,i)) | (dom (Initialized ((fsloc 0) .--> w))) )
assume that
A53: Initialized ((fsloc 0) .--> w) c= s1 and
A54: Initialized ((fsloc 0) .--> w) c= s2 ; :: thesis: for i being Nat holds (Comput (P1,s1,i)) | (dom (Initialized ((fsloc 0) .--> w))) = (Comput (P2,s2,i)) | (dom (Initialized ((fsloc 0) .--> w)))
let i be Nat; :: thesis: (Comput (P1,s1,i)) | (dom (Initialized ((fsloc 0) .--> w))) = (Comput (P2,s2,i)) | (dom (Initialized ((fsloc 0) .--> w)))
reconsider i = i as Nat ;
set Cs1i = Comput (P1,s1,i);
set Cs2i = Comput (P2,s2,i);
A55: (UsedI*Loc Insert-Sort-Algorithm) \/ (UsedILoc Insert-Sort-Algorithm) c= dom (Comput (P1,s1,i)) by SCMBSORT:32;
A56: fsloc 0 in (UsedI*Loc Insert-Sort-Algorithm) \/ (UsedILoc Insert-Sort-Algorithm) by A3, ENUMSET1:def 6;
A57: (UsedI*Loc Insert-Sort-Algorithm) \/ (UsedILoc Insert-Sort-Algorithm) c= dom (Comput (P2,s2,i)) by SCMBSORT:32;
A58: ( i > 10 implies 10 + 1 < i + 1 ) by XREAL_1:6;
A59: intloc 0 in (UsedI*Loc Insert-Sort-Algorithm) \/ (UsedILoc Insert-Sort-Algorithm) by A3, ENUMSET1:def 6;

A60: now :: thesis: for x being set st x in {(intloc 0),(IC ),(fsloc 0)} holds
(Comput (P1,s1,i)) . x = (Comput (P2,s2,i)) . x

let x be set ; :: thesis: ( x in {(intloc 0),(IC ),(fsloc 0)} implies (Comput (P1,s1,i)) . b₁ = (Comput (P2,s2,i)) . b₁ )
assume A61: x in {(intloc 0),(IC ),(fsloc 0)} ; :: thesis: (Comput (P1,s1,i)) . b₁ = (Comput (P2,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 (P1,s1,i)) . b₁ = (Comput (P2,s2,i)) . b₁

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

suppose i <= 10 ; :: thesis: (Comput (P1,s1,i)) . b₁ = (Comput (P2,s2,i)) . b₁

hence (Comput (P1,s1,i)) . x = (Comput (P2,s2,i)) . x by A28, A53, A54, A62, A51, A52; :: thesis: verum

end;

suppose i > 10 ; :: thesis: (Comput (P1,s1,i)) . b₁ = (Comput (P2,s2,i)) . b₁

then 11 <= i by A58, NAT_1:13;
then (Comput (P1,s1,i)) | ((UsedI*Loc Insert-Sort-Algorithm) \/ (UsedILoc Insert-Sort-Algorithm)) = (Comput (P2,s2,i)) | ((UsedI*Loc Insert-Sort-Algorithm) \/ (UsedILoc Insert-Sort-Algorithm)) by A4, A53, A54, A51, A52;
hence (Comput (P1,s1,i)) . x = (Comput (P2,s2,i)) . x by A59, A55, A57, A62, FUNCT_1:95; :: thesis: verum

end;

end;

end;

suppose A63: x = IC ; :: thesis: (Comput (P1,s1,i)) . b₁ = (Comput (P2,s2,i)) . b₁

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

suppose i <= 10 ; :: thesis: (Comput (P1,s1,i)) . b₁ = (Comput (P2,s2,i)) . b₁

hence (Comput (P1,s1,i)) . x = (Comput (P2,s2,i)) . x by A28, A53, A54, A63, A51, A52; :: thesis: verum

end;

suppose i > 10 ; :: thesis: (Comput (P1,s1,i)) . b₁ = (Comput (P2,s2,i)) . b₁

then 11 <= i by A58, NAT_1:13;
hence (Comput (P1,s1,i)) . x = (Comput (P2,s2,i)) . x by A4, A53, A54, A63, A51, A52; :: thesis: verum

end;

end;

end;

suppose A64: x = fsloc 0 ; :: thesis: (Comput (P1,s1,i)) . b₁ = (Comput (P2,s2,i)) . b₁

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

suppose i <= 10 ; :: thesis: (Comput (P1,s1,i)) . b₁ = (Comput (P2,s2,i)) . b₁

hence (Comput (P1,s1,i)) . x = (Comput (P2,s2,i)) . x by A28, A53, A54, A64, A51, A52; :: thesis: verum

end;

suppose i > 10 ; :: thesis: (Comput (P1,s1,i)) . b₁ = (Comput (P2,s2,i)) . b₁

then 11 <= i by A58, NAT_1:13;
then (Comput (P1,s1,i)) | ((UsedI*Loc Insert-Sort-Algorithm) \/ (UsedILoc Insert-Sort-Algorithm)) = (Comput (P2,s2,i)) | ((UsedI*Loc Insert-Sort-Algorithm) \/ (UsedILoc Insert-Sort-Algorithm)) by A4, A53, A54, A51, A52;
hence (Comput (P1,s1,i)) . x = (Comput (P2,s2,i)) . x by A56, A55, A57, A64, FUNCT_1:95; :: thesis: verum

end;

end;

end;

end;

end;

A65: {(intloc 0),(IC ),(fsloc 0)} c= dom (Comput (P2,s2,i)) by SCMFSA_M:34;
{(intloc 0),(IC ),(fsloc 0)} c= dom (Comput (P1,s1,i)) by SCMFSA_M:34;
then (Comput (P1,s1,i)) | {(intloc 0),(IC ),(fsloc 0)} = (Comput (P2,s2,i)) | {(intloc 0),(IC ),(fsloc 0)} by A65, A60, FUNCT_1:95;
hence (Comput (P1,s1,i)) | (dom (Initialized ((fsloc 0) .--> w))) = (Comput (P2,s2,i)) | (dom (Initialized ((fsloc 0) .--> w))) by A50; :: thesis: verum

end;

hence Initialized ((fsloc 0) .--> w) is Insert-Sort-Algorithm -autonomic ; :: thesis: verum