let n be Element of NAT ; :: thesis: for R being good Ring
for s being State of (SCM R) st not R is trivial holds
for p being non NAT -defined autonomic FinPartState of st p c= s holds
IC (Comput ((ProgramPart s),s,n)) in dom (ProgramPart p)
let R be good Ring; :: thesis: for s being State of (SCM R) st not R is trivial holds
for p being non NAT -defined autonomic FinPartState of st p c= s holds
IC (Comput ((ProgramPart s),s,n)) in dom (ProgramPart p)
let s be State of (SCM R); :: thesis: ( not R is trivial implies for p being non NAT -defined autonomic FinPartState of st p c= s holds
IC (Comput ((ProgramPart s),s,n)) in dom (ProgramPart p) )
assume A1: not R is trivial ; :: thesis: for p being non NAT -defined autonomic FinPartState of st p c= s holds
IC (Comput ((ProgramPart s),s,n)) in dom (ProgramPart p)
set Csi = Comput ((ProgramPart s),s,n);
let p be non NAT -defined autonomic FinPartState of ; :: thesis: ( p c= s implies IC (Comput ((ProgramPart s),s,n)) in dom (ProgramPart p) )
assume A2: p c= s ; :: thesis: IC (Comput ((ProgramPart s),s,n)) in dom (ProgramPart p)
set loc = IC (Comput ((ProgramPart s),s,n));
consider ll being natural number such that
A3: IC (Comput ((ProgramPart s),s,n)) = ll ;
set loc1 = ll + 1;
A4: IC (Comput ((ProgramPart s),s,n)) <> ll + 1 by A3;
set p2 = p +* ((IC (Comput ((ProgramPart s),s,n))) .--> (goto ((ll + 1),R)));
A5: ( IC (Comput ((ProgramPart s),s,n)) in dom (ProgramPart p) iff IC (Comput ((ProgramPart s),s,n)) in (dom p) /\ NAT ) by RELAT_1:90;
set p1 = p +* ((IC (Comput ((ProgramPart s),s,n))) .--> (goto ((IC (Comput ((ProgramPart s),s,n))),R)));
A7: dom ((IC (Comput ((ProgramPart s),s,n))) .--> (goto ((IC (Comput ((ProgramPart s),s,n))),R))) = {(IC (Comput ((ProgramPart s),s,n)))} by FUNCOP_1:19;
then A8: IC (Comput ((ProgramPart s),s,n)) in dom ((IC (Comput ((ProgramPart s),s,n))) .--> (goto ((IC (Comput ((ProgramPart s),s,n))),R))) by TARSKI:def 1;
assume not IC (Comput ((ProgramPart s),s,n)) in dom (ProgramPart p) ; :: thesis: contradiction
then A9: not IC (Comput ((ProgramPart s),s,n)) in dom p by A5, XBOOLE_0:def 4;
consider s2 being State of (SCM R) such that
A10: p +* ((IC (Comput ((ProgramPart s),s,n))) .--> (goto ((ll + 1),R))) c= s2 by PBOOLE:156;
set Cs2i = Comput ((ProgramPart s2),s2,n);
consider s1 being State of (SCM R) such that
A11: p +* ((IC (Comput ((ProgramPart s),s,n))) .--> (goto ((IC (Comput ((ProgramPart s),s,n))),R))) c= s1 by PBOOLE:156;
set Cs1i = Comput ((ProgramPart s1),s1,n);
A12: dom ((IC (Comput ((ProgramPart s),s,n))) .--> (goto ((ll + 1),R))) = {(IC (Comput ((ProgramPart s),s,n)))} by FUNCOP_1:19;
then A13: IC (Comput ((ProgramPart s),s,n)) in dom ((IC (Comput ((ProgramPart s),s,n))) .--> (goto ((ll + 1),R))) by TARSKI:def 1;
A14: dom (p +* ((IC (Comput ((ProgramPart s),s,n))) .--> (goto ((ll + 1),R)))) = (dom p) \/ (dom ((IC (Comput ((ProgramPart s),s,n))) .--> (goto ((ll + 1),R)))) by FUNCT_4:def 1;
then A15: IC (Comput ((ProgramPart s),s,n)) in dom (p +* ((IC (Comput ((ProgramPart s),s,n))) .--> (goto ((ll + 1),R)))) by A13, XBOOLE_0:def 3;
A16: dom (p +* ((IC (Comput ((ProgramPart s),s,n))) .--> (goto ((IC (Comput ((ProgramPart s),s,n))),R)))) = (dom p) \/ (dom ((IC (Comput ((ProgramPart s),s,n))) .--> (goto ((IC (Comput ((ProgramPart s),s,n))),R)))) by FUNCT_4:def 1;
then A17: IC (Comput ((ProgramPart s),s,n)) in dom (p +* ((IC (Comput ((ProgramPart s),s,n))) .--> (goto ((IC (Comput ((ProgramPart s),s,n))),R)))) by A8, XBOOLE_0:def 3;
not p is autonomic

proof

A18: now

let x be set ; :: thesis: ( x in dom p implies p . x = s2 . x )
assume A19: x in dom p ; :: thesis: p . x = s2 . x
dom p misses dom ((IC (Comput ((ProgramPart s),s,n))) .--> (goto ((ll + 1),R))) by A9, A12, ZFMISC_1:56;
then A20: p . x = (p +* ((IC (Comput ((ProgramPart s),s,n))) .--> (goto ((ll + 1),R)))) . x by A19, FUNCT_4:17;
x in dom (p +* ((IC (Comput ((ProgramPart s),s,n))) .--> (goto ((ll + 1),R)))) by A14, A19, XBOOLE_0:def 3;
hence p . x = s2 . x by A10, A20, GRFUNC_1:8; :: thesis: verum

end;

((IC (Comput ((ProgramPart s),s,n))) .--> (goto ((ll + 1),R))) . (IC (Comput ((ProgramPart s),s,n))) = goto ((ll + 1),R) by FUNCOP_1:87;
then (p +* ((IC (Comput ((ProgramPart s),s,n))) .--> (goto ((ll + 1),R)))) . (IC (Comput ((ProgramPart s),s,n))) = goto ((ll + 1),R) by A13, FUNCT_4:14;
then s2 . (IC (Comput ((ProgramPart s),s,n))) = goto ((ll + 1),R) by A15, A10, GRFUNC_1:8;
then A21: (Comput ((ProgramPart s2),s2,n)) . (IC (Comput ((ProgramPart s),s,n))) = goto ((ll + 1),R) by AMI_1:54;
((IC (Comput ((ProgramPart s),s,n))) .--> (goto ((IC (Comput ((ProgramPart s),s,n))),R))) . (IC (Comput ((ProgramPart s),s,n))) = goto ((IC (Comput ((ProgramPart s),s,n))),R) by FUNCOP_1:87;
then (p +* ((IC (Comput ((ProgramPart s),s,n))) .--> (goto ((IC (Comput ((ProgramPart s),s,n))),R)))) . (IC (Comput ((ProgramPart s),s,n))) = goto ((IC (Comput ((ProgramPart s),s,n))),R) by A8, FUNCT_4:14;
then s1 . (IC (Comput ((ProgramPart s),s,n))) = goto ((IC (Comput ((ProgramPart s),s,n))),R) by A17, A11, GRFUNC_1:8;
then A22: (Comput ((ProgramPart s1),s1,n)) . (IC (Comput ((ProgramPart s),s,n))) = goto ((IC (Comput ((ProgramPart s),s,n))),R) by AMI_1:54;
take s1 ; :: according to EXTPRO_1:def 9 :: thesis: ex b₁ being set st
( p c= s1 & p c= b₁ & not for b₂ being Element of NAT holds (Comput ((ProgramPart s1),s1,b₂)) | (proj1 p) = (Comput ((ProgramPart b₁),b₁,b₂)) | (proj1 p) )
take s2 ; :: thesis: ( p c= s1 & p c= s2 & not for b₁ being Element of NAT holds (Comput ((ProgramPart s1),s1,b₁)) | (proj1 p) = (Comput ((ProgramPart s2),s2,b₁)) | (proj1 p) )

A23: now

let x be set ; :: thesis: ( x in dom p implies p . x = s1 . x )
assume A24: x in dom p ; :: thesis: p . x = s1 . x
dom p misses dom ((IC (Comput ((ProgramPart s),s,n))) .--> (goto ((IC (Comput ((ProgramPart s),s,n))),R))) by A9, A7, ZFMISC_1:56;
then A25: p . x = (p +* ((IC (Comput ((ProgramPart s),s,n))) .--> (goto ((IC (Comput ((ProgramPart s),s,n))),R)))) . x by A24, FUNCT_4:17;
x in dom (p +* ((IC (Comput ((ProgramPart s),s,n))) .--> (goto ((IC (Comput ((ProgramPart s),s,n))),R)))) by A16, A24, XBOOLE_0:def 3;
hence p . x = s1 . x by A11, A25, GRFUNC_1:8; :: thesis: verum

end;

dom s1 = the carrier of (SCM R) by PARTFUN1:def 4;
then dom p c= dom s1 by RELAT_1:def 18;
hence A26: p c= s1 by A23, GRFUNC_1:8; :: thesis: ( p c= s2 & not for b₁ being Element of NAT holds (Comput ((ProgramPart s1),s1,b₁)) | (proj1 p) = (Comput ((ProgramPart s2),s2,b₁)) | (proj1 p) )
then A27: (Comput ((ProgramPart s1),s1,n)) | (dom p) = (Comput ((ProgramPart s),s,n)) | (dom p) by A2, EXTPRO_1:def 9;
dom s2 = the carrier of (SCM R) by PARTFUN1:def 4;
then dom p c= dom s2 by RELAT_1:def 18;
hence p c= s2 by A18, GRFUNC_1:8; :: thesis: not for b₁ being Element of NAT holds (Comput ((ProgramPart s1),s1,b₁)) | (proj1 p) = (Comput ((ProgramPart s2),s2,b₁)) | (proj1 p)
then A28: (Comput ((ProgramPart s1),s1,n)) | (dom p) = (Comput ((ProgramPart s2),s2,n)) | (dom p) by A26, EXTPRO_1:def 9;
take k = n + 1; :: thesis: not (Comput ((ProgramPart s1),s1,k)) | (proj1 p) = (Comput ((ProgramPart s2),s2,k)) | (proj1 p)
set Cs1k = Comput ((ProgramPart s1),s1,k);
A29: Comput ((ProgramPart s1),s1,k) = Following ((ProgramPart s1),(Comput ((ProgramPart s1),s1,n))) by EXTPRO_1:4
.= Exec ((CurInstr ((ProgramPart (Comput ((ProgramPart s1),s1,n))),(Comput ((ProgramPart s1),s1,n)))),(Comput ((ProgramPart s1),s1,n))) by AMI_1:123 ;
Y: (ProgramPart (Comput ((ProgramPart s1),s1,n))) /. (IC (Comput ((ProgramPart s),s,n))) = (Comput ((ProgramPart s1),s1,n)) . (IC (Comput ((ProgramPart s),s,n))) by COMPOS_1:38;
A30: (Comput ((ProgramPart s),s,n)) . (IC (SCM R)) = ((Comput ((ProgramPart s),s,n)) | (dom p)) . (IC (SCM R)) by A1, Th25, FUNCT_1:72;
then (Comput ((ProgramPart s1),s1,n)) . (IC (SCM R)) = IC (Comput ((ProgramPart s),s,n)) by A1, A27, Th25, FUNCT_1:72;
then A31: (Comput ((ProgramPart s1),s1,k)) . (IC (SCM R)) = IC (Comput ((ProgramPart s),s,n)) by A29, A22, Y, SCMRING2:17;
set Cs2k = Comput ((ProgramPart s2),s2,k);
A32: Comput ((ProgramPart s2),s2,k) = Following ((ProgramPart s2),(Comput ((ProgramPart s2),s2,n))) by EXTPRO_1:4
.= Exec ((CurInstr ((ProgramPart (Comput ((ProgramPart s2),s2,n))),(Comput ((ProgramPart s2),s2,n)))),(Comput ((ProgramPart s2),s2,n))) by AMI_1:123 ;
Y: (ProgramPart (Comput ((ProgramPart s2),s2,n))) /. (IC (Comput ((ProgramPart s),s,n))) = (Comput ((ProgramPart s2),s2,n)) . (IC (Comput ((ProgramPart s),s,n))) by COMPOS_1:38;
(Comput ((ProgramPart s2),s2,n)) . (IC (SCM R)) = IC (Comput ((ProgramPart s),s,n)) by A1, A27, A30, A28, Th25, FUNCT_1:72;
then A33: (Comput ((ProgramPart s2),s2,k)) . (IC (SCM R)) = ll + 1 by A32, A21, Y, SCMRING2:17;
((Comput ((ProgramPart s1),s1,k)) | (dom p)) . (IC (SCM R)) = (Comput ((ProgramPart s1),s1,k)) . (IC (SCM R)) by A1, Th25, FUNCT_1:72;
hence (Comput ((ProgramPart s1),s1,k)) | (dom p) <> (Comput ((ProgramPart s2),s2,k)) | (dom p) by A1, A4, A31, A33, Th25, FUNCT_1:72; :: thesis: verum

end;

hence contradiction ; :: thesis: verum