let p be non NAT -defined autonomic FinPartState of ; :: according to AMISTD_5:def 4 :: thesis: for b1 being set holds
( not p c= b1 or for b2 being set holds
( not ProgramPart p c= b2 or for b3 being Element of NAT holds IC (Comput (b2,b1,b3)) in proj1 (ProgramPart p) ) )
let s be State of SCMPDS; :: thesis: ( not p c= s or for b1 being set holds
( not ProgramPart p c= b1 or for b2 being Element of NAT holds IC (Comput (b1,s,b2)) in proj1 (ProgramPart p) ) )
assume A1: p c= s ; :: thesis: for b1 being set holds
( not ProgramPart p c= b1 or for b2 being Element of NAT holds IC (Comput (b1,s,b2)) in proj1 (ProgramPart p) )
let P be the Instructions of SCMPDS -valued ManySortedSet of NAT ; :: thesis: ( not ProgramPart p c= P or for b1 being Element of NAT holds IC (Comput (P,s,b1)) in proj1 (ProgramPart p) )
assume A2: ProgramPart p c= P ; :: thesis: for b1 being Element of NAT holds IC (Comput (P,s,b1)) in proj1 (ProgramPart p)
let i be Element of NAT ; :: thesis: IC (Comput (P,s,i)) in proj1 (ProgramPart p)
set Csi = Comput (P,s,i);
set loc = IC (Comput (P,s,i));
A3: ( IC (Comput (P,s,i)) in dom (ProgramPart p) iff IC (Comput (P,s,i)) in (dom p) /\ NAT ) by RELAT_1:90;
assume not IC (Comput (P,s,i)) in dom (ProgramPart p) ; :: thesis: contradiction
then A4: not IC (Comput (P,s,i)) in dom p by A3, XBOOLE_0:def 4;
set p2 = p +* ((IC (Comput (P,s,i))) .--> (goto 1));
set p1 = p +* ((IC (Comput (P,s,i))) .--> (goto 0));
A5: dom (p +* ((IC (Comput (P,s,i))) .--> (goto 0))) = (dom p) \/ (dom ((IC (Comput (P,s,i))) .--> (goto 0))) by FUNCT_4:def 1;
A6: dom ((IC (Comput (P,s,i))) .--> (goto 1)) = {(IC (Comput (P,s,i)))} by FUNCOP_1:19;
then A7: IC (Comput (P,s,i)) in dom ((IC (Comput (P,s,i))) .--> (goto 1)) by TARSKI:def 1;
A8: dom (p +* ((IC (Comput (P,s,i))) .--> (goto 1))) = (dom p) \/ (dom ((IC (Comput (P,s,i))) .--> (goto 1))) by FUNCT_4:def 1;
then A9: IC (Comput (P,s,i)) in dom (p +* ((IC (Comput (P,s,i))) .--> (goto 1))) by A7, XBOOLE_0:def 3;
consider s2 being State of SCMPDS such that
A10: p +* ((IC (Comput (P,s,i))) .--> (goto 1)) c= s2 by PBOOLE:156;
set Cs2i = Comput ((ProgramPart s2),s2,i);
consider s1 being State of SCMPDS such that
A11: p +* ((IC (Comput (P,s,i))) .--> (goto 0)) c= s1 by PBOOLE:156;
set Cs1i = Comput ((ProgramPart s1),s1,i);
A12: dom ((IC (Comput (P,s,i))) .--> (goto 0)) = {(IC (Comput (P,s,i)))} by FUNCOP_1:19;
then A13: IC (Comput (P,s,i)) in dom ((IC (Comput (P,s,i))) .--> (goto 0)) by TARSKI:def 1;
then A14: IC (Comput (P,s,i)) in dom (p +* ((IC (Comput (P,s,i))) .--> (goto 0))) by A5, XBOOLE_0:def 3;
not p is autonomic
proof
A15: now
let x be set ; :: thesis: ( x in dom p implies p . x = s2 . x )
assume A16: x in dom p ; :: thesis: p . x = s2 . x
dom p misses dom ((IC (Comput (P,s,i))) .--> (goto 1)) by A4, A6, ZFMISC_1:56;
then A17: p . x = (p +* ((IC (Comput (P,s,i))) .--> (goto 1))) . x by A16, FUNCT_4:17;
x in dom (p +* ((IC (Comput (P,s,i))) .--> (goto 1))) by A8, A16, XBOOLE_0:def 3;
hence p . x = s2 . x by A10, A17, GRFUNC_1:8; :: thesis: verum
end;
((IC (Comput (P,s,i))) .--> (goto 1)) . (IC (Comput (P,s,i))) = goto 1 by FUNCOP_1:87;
then (p +* ((IC (Comput (P,s,i))) .--> (goto 1))) . (IC (Comput (P,s,i))) = goto 1 by A7, FUNCT_4:14;
then s2 . (IC (Comput (P,s,i))) = goto 1 by A9, A10, GRFUNC_1:8;
then A18: (Comput ((ProgramPart s2),s2,i)) . (IC (Comput (P,s,i))) = goto 1 by AMI_1:54;
((IC (Comput (P,s,i))) .--> (goto 0)) . (IC (Comput (P,s,i))) = goto 0 by FUNCOP_1:87;
then (p +* ((IC (Comput (P,s,i))) .--> (goto 0))) . (IC (Comput (P,s,i))) = goto 0 by A13, FUNCT_4:14;
then s1 . (IC (Comput (P,s,i))) = goto 0 by A14, A11, GRFUNC_1:8;
then A19: (Comput ((ProgramPart s1),s1,i)) . (IC (Comput (P,s,i))) = goto 0 by AMI_1:54;
take P1 = ProgramPart s1; :: according to EXTPRO_1:def 9 :: thesis: ex b1 being set st
( ProgramPart p c= P1 & ProgramPart p c= b1 & ex b2, b3 being set st
( p c= b2 & p c= b3 & not for b4 being Element of NAT holds (Comput (P1,b2,b4)) | (proj1 (NPP p)) = (Comput (b1,b3,b4)) | (proj1 (NPP p)) ) )
take P2 = ProgramPart s2; :: thesis: ( ProgramPart p c= P1 & ProgramPart p c= P2 & ex b1, b2 being set st
( p c= b1 & p c= b2 & not for b3 being Element of NAT holds (Comput (P1,b1,b3)) | (proj1 (NPP p)) = (Comput (P2,b2,b3)) | (proj1 (NPP p)) ) )
A20: now
let x be set ; :: thesis: ( x in dom p implies p . x = s1 . x )
assume A21: x in dom p ; :: thesis: p . x = s1 . x
dom p misses dom ((IC (Comput (P,s,i))) .--> (goto 0)) by A4, A12, ZFMISC_1:56;
then A22: p . x = (p +* ((IC (Comput (P,s,i))) .--> (goto 0))) . x by A21, FUNCT_4:17;
x in dom (p +* ((IC (Comput (P,s,i))) .--> (goto 0))) by A5, A21, XBOOLE_0:def 3;
hence p . x = s1 . x by A11, A22, GRFUNC_1:8; :: thesis: verum
end;
dom s1 = the carrier of SCMPDS by PARTFUN1:def 4;
then dom p c= dom s1 by RELAT_1:def 18;
then A23: p c= s1 by A20, GRFUNC_1:8;
hence A24: ProgramPart p c= P1 by RELAT_1:105; :: thesis: ( ProgramPart p c= P2 & ex b1, b2 being set st
( p c= b1 & p c= b2 & not for b3 being Element of NAT holds (Comput (P1,b1,b3)) | (proj1 (NPP p)) = (Comput (P2,b2,b3)) | (proj1 (NPP p)) ) )
A25: (Comput ((ProgramPart s1),s1,i)) | (dom (NPP p)) = (Comput (P,s,i)) | (dom (NPP p)) by A1, EXTPRO_1:def 9, A23, A2, A24;
dom s2 = the carrier of SCMPDS by PARTFUN1:def 4;
then dom p c= dom s2 by RELAT_1:def 18;
then A26: p c= s2 by A15, GRFUNC_1:8;
hence A27: ProgramPart p c= P2 by RELAT_1:105; :: thesis: ex b1, b2 being set st
( p c= b1 & p c= b2 & not for b3 being Element of NAT holds (Comput (P1,b1,b3)) | (proj1 (NPP p)) = (Comput (P2,b2,b3)) | (proj1 (NPP p)) )
take s1 ; :: thesis: ex b1 being set st
( p c= s1 & p c= b1 & not for b2 being Element of NAT holds (Comput (P1,s1,b2)) | (proj1 (NPP p)) = (Comput (P2,b1,b2)) | (proj1 (NPP p)) )
take s2 ; :: thesis: ( p c= s1 & p c= s2 & not for b1 being Element of NAT holds (Comput (P1,s1,b1)) | (proj1 (NPP p)) = (Comput (P2,s2,b1)) | (proj1 (NPP p)) )
thus p c= s1 by A23; :: thesis: ( p c= s2 & not for b1 being Element of NAT holds (Comput (P1,s1,b1)) | (proj1 (NPP p)) = (Comput (P2,s2,b1)) | (proj1 (NPP p)) )
thus p c= s2 by A26; :: thesis: not for b1 being Element of NAT holds (Comput (P1,s1,b1)) | (proj1 (NPP p)) = (Comput (P2,s2,b1)) | (proj1 (NPP p))
A28: (Comput ((ProgramPart s1),s1,i)) | (dom (NPP p)) = (Comput ((ProgramPart s2),s2,i)) | (dom (NPP p)) by EXTPRO_1:def 9, A2, A27, A25, A1, A26;
take k = i + 1; :: thesis: not (Comput (P1,s1,k)) | (proj1 (NPP p)) = (Comput (P2,s2,k)) | (proj1 (NPP p))
set Cs2k = Comput ((ProgramPart s2),s2,k);
A29: P2 = ProgramPart (Comput ((ProgramPart s2),s2,i)) by AMI_1:123;
A30: Comput ((ProgramPart s2),s2,k) = Following (P2,(Comput ((ProgramPart s2),s2,i))) by EXTPRO_1:4
.= Exec ((CurInstr ((ProgramPart (Comput ((ProgramPart s2),s2,i))),(Comput ((ProgramPart s2),s2,i)))),(Comput ((ProgramPart s2),s2,i))) by A29 ;
A31: (ProgramPart (Comput ((ProgramPart s2),s2,i))) /. (IC (Comput ((ProgramPart s2),s2,i))) = (Comput ((ProgramPart s2),s2,i)) . (IC (Comput ((ProgramPart s2),s2,i))) by COMPOS_1:38;
A32: IC in dom p by AMISTD_5:6;
A33: (Comput (P,s,i)) . (IC ) = ((Comput (P,s,i)) | (dom (NPP p))) . (IC ) by COMPOS_1:179, FUNCT_1:72, A32;
then (Comput ((ProgramPart s2),s2,i)) . (IC ) = IC (Comput (P,s,i)) by A25, A28, COMPOS_1:179, FUNCT_1:72, A32;
then A34: (Comput ((ProgramPart s2),s2,k)) . (IC ) = ICplusConst ((Comput ((ProgramPart s2),s2,i)),1) by A30, A18, A31, SCMPDS_2:66;
A35: (Comput ((ProgramPart s1),s1,i)) . (IC ) = ((Comput ((ProgramPart s1),s1,i)) | (dom (NPP p))) . (IC ) by COMPOS_1:179, FUNCT_1:72, A32;
then A36: IC (Comput ((ProgramPart s1),s1,i)) = IC (Comput ((ProgramPart s2),s2,i)) by A28, COMPOS_1:179, FUNCT_1:72, A32;
set Cs1k = Comput ((ProgramPart s1),s1,k);
A37: (ProgramPart (Comput ((ProgramPart s1),s1,i))) /. (IC (Comput ((ProgramPart s1),s1,i))) = (Comput ((ProgramPart s1),s1,i)) . (IC (Comput ((ProgramPart s1),s1,i))) by COMPOS_1:38;
A38: P1 = ProgramPart (Comput ((ProgramPart s1),s1,i)) by AMI_1:123;
Comput ((ProgramPart s1),s1,k) = Following (P1,(Comput ((ProgramPart s1),s1,i))) by EXTPRO_1:4
.= Exec ((CurInstr ((ProgramPart (Comput ((ProgramPart s1),s1,i))),(Comput ((ProgramPart s1),s1,i)))),(Comput ((ProgramPart s1),s1,i))) by A38 ;
then A39: (Comput ((ProgramPart s1),s1,k)) . (IC ) = ICplusConst ((Comput ((ProgramPart s1),s1,i)),0) by A19, A25, A35, A33, A37, SCMPDS_2:66;
( ((Comput ((ProgramPart s1),s1,k)) | (dom (NPP p))) . (IC ) = (Comput ((ProgramPart s1),s1,k)) . (IC ) & ((Comput ((ProgramPart s2),s2,k)) | (dom (NPP p))) . (IC ) = (Comput ((ProgramPart s2),s2,k)) . (IC ) ) by FUNCT_1:72, A32, COMPOS_1:179;
hence not (Comput (P1,s1,k)) | (proj1 (NPP p)) = (Comput (P2,s2,k)) | (proj1 (NPP p)) by A34, A36, Th19, A39; :: thesis: verum
end;
hence contradiction ; :: thesis: verum