let p be non NAT -defined autonomic FinPartState of ; :: thesis: for s being State of st p c= s holds
for i being Element of NAT holds IC (Computation s,i) in dom (ProgramPart p)
let s be State of ; :: thesis: ( p c= s implies for i being Element of NAT holds IC (Computation s,i) in dom (ProgramPart p) )
assume A1: p c= s ; :: thesis: for i being Element of NAT holds IC (Computation s,i) in dom (ProgramPart p)
let i be Element of NAT ; :: thesis: IC (Computation s,i) in dom (ProgramPart p)
set Csi = Computation s,i;
set loc = IC (Computation s,i);
reconsider ll = IC (Computation s,i) as Element of NAT by ORDINAL1:def 13;
set loc1 = il. (ll + 1);
A2: IC (Computation s,i) in NAT by AMI_1:def 4;
A3: ( IC (Computation s,i) in dom (ProgramPart p) iff IC (Computation s,i) in (dom p) /\ NAT ) by RELAT_1:90;
assume not IC (Computation s,i) in dom (ProgramPart p) ; :: thesis: contradiction
then A4: not IC (Computation s,i) in dom p by A2, A3, XBOOLE_0:def 4;
set p2 = p +* ((IC (Computation s,i)) .--> (goto (il. (ll + 1))));
set p1 = p +* ((IC (Computation s,i)) .--> (goto (IC (Computation s,i))));
A5: dom (p +* ((IC (Computation s,i)) .--> (goto (IC (Computation s,i))))) = (dom p) \/ (dom ((IC (Computation s,i)) .--> (goto (IC (Computation s,i))))) by FUNCT_4:def 1;
A6: dom ((IC (Computation s,i)) .--> (goto (il. (ll + 1)))) = {(IC (Computation s,i))} by FUNCOP_1:19;
then A7: IC (Computation s,i) in dom ((IC (Computation s,i)) .--> (goto (il. (ll + 1)))) by TARSKI:def 1;
A8: dom (p +* ((IC (Computation s,i)) .--> (goto (il. (ll + 1))))) = (dom p) \/ (dom ((IC (Computation s,i)) .--> (goto (il. (ll + 1))))) by FUNCT_4:def 1;
then A9: IC (Computation s,i) in dom (p +* ((IC (Computation s,i)) .--> (goto (il. (ll + 1))))) by A7, XBOOLE_0:def 3;
consider s2 being State of such that
A10: p +* ((IC (Computation s,i)) .--> (goto (il. (ll + 1)))) c= s2 by CARD_3:97;
set Cs2i = Computation s2,i;
consider s1 being State of such that
A11: p +* ((IC (Computation s,i)) .--> (goto (IC (Computation s,i)))) c= s1 by CARD_3:97;
set Cs1i = Computation s1,i;
A12: dom ((IC (Computation s,i)) .--> (goto (IC (Computation s,i)))) = {(IC (Computation s,i))} by FUNCOP_1:19;
then A13: IC (Computation s,i) in dom ((IC (Computation s,i)) .--> (goto (IC (Computation s,i)))) by TARSKI:def 1;
then A14: IC (Computation s,i) in dom (p +* ((IC (Computation s,i)) .--> (goto (IC (Computation s,i))))) 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 (Computation s,i)) .--> (goto (il. (ll + 1)))) by A4, A6, ZFMISC_1:56;
then A17: p . x = (p +* ((IC (Computation s,i)) .--> (goto (il. (ll + 1))))) . x by A16, FUNCT_4:17;
x in dom (p +* ((IC (Computation s,i)) .--> (goto (il. (ll + 1))))) by A8, A16, XBOOLE_0:def 3;
hence p . x = s2 . x by A10, A17, GRFUNC_1:8; :: thesis: verum

end;

((IC (Computation s,i)) .--> (goto (il. (ll + 1)))) . (IC (Computation s,i)) = goto (il. (ll + 1)) by FUNCOP_1:87;
then (p +* ((IC (Computation s,i)) .--> (goto (il. (ll + 1))))) . (IC (Computation s,i)) = goto (il. (ll + 1)) by A7, FUNCT_4:14;
then s2 . (IC (Computation s,i)) = goto (il. (ll + 1)) by A9, A10, GRFUNC_1:8;
then A18: (Computation s2,i) . (IC (Computation s,i)) = goto (il. (ll + 1)) by AMI_1:54;
((IC (Computation s,i)) .--> (goto (IC (Computation s,i)))) . (IC (Computation s,i)) = goto (IC (Computation s,i)) by FUNCOP_1:87;
then (p +* ((IC (Computation s,i)) .--> (goto (IC (Computation s,i))))) . (IC (Computation s,i)) = goto (IC (Computation s,i)) by A13, FUNCT_4:14;
then s1 . (IC (Computation s,i)) = goto (IC (Computation s,i)) by A14, A11, GRFUNC_1:8;
then A19: (Computation s1,i) . (IC (Computation s,i)) = goto (IC (Computation s,i)) by AMI_1:54;
take s1 ; :: according to AMI_1:def 25 :: thesis: ex b₁ being Element of K242(the Object-Kind of SCM ) st
( p c= s1 & p c= b₁ & not for b₂ being Element of K98() holds (Computation s1,b₂) | (dom p) = (Computation b₁,b₂) | (dom p) )
take s2 ; :: thesis: ( p c= s1 & p c= s2 & not for b₁ being Element of K98() holds (Computation s1,b₁) | (dom p) = (Computation s2,b₁) | (dom 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 (Computation s,i)) .--> (goto (IC (Computation s,i)))) by A4, A12, ZFMISC_1:56;
then A22: p . x = (p +* ((IC (Computation s,i)) .--> (goto (IC (Computation s,i))))) . x by A21, FUNCT_4:17;
x in dom (p +* ((IC (Computation s,i)) .--> (goto (IC (Computation s,i))))) 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 SCM by AMI_1:79;
then dom p c= dom s1 by AMI_1:80;
hence A23: p c= s1 by A20, GRFUNC_1:8; :: thesis: ( p c= s2 & not for b₁ being Element of K98() holds (Computation s1,b₁) | (dom p) = (Computation s2,b₁) | (dom p) )
then A24: (Computation s1,i) | (dom p) = (Computation s,i) | (dom p) by A1, AMI_1:def 25;
dom s2 = the carrier of SCM by AMI_1:79;
then dom p c= dom s2 by AMI_1:80;
hence p c= s2 by A15, GRFUNC_1:8; :: thesis: not for b₁ being Element of K98() holds (Computation s1,b₁) | (dom p) = (Computation s2,b₁) | (dom p)
then A25: (Computation s1,i) | (dom p) = (Computation s2,i) | (dom p) by A23, AMI_1:def 25;
take k = i + 1; :: thesis: not (Computation s1,k) | (dom p) = (Computation s2,k) | (dom p)
set Cs1k = Computation s1,k;
A26: Computation s1,k = Following (Computation s1,i) by AMI_1:14
.= Exec (CurInstr (Computation s1,i)),(Computation s1,i) ;
A27: (Computation s,i) . (IC SCM ) = ((Computation s,i) | (dom p)) . (IC SCM ) by Th84, FUNCT_1:72;
then (Computation s1,i) . (IC SCM ) = IC (Computation s,i) by A24, Th84, FUNCT_1:72;
then A28: (Computation s1,k) . (IC SCM ) = IC (Computation s,i) by A26, A19, AMI_3:13;
set Cs2k = Computation s2,k;
A29: Computation s2,k = Following (Computation s2,i) by AMI_1:14
.= Exec (CurInstr (Computation s2,i)),(Computation s2,i) ;
(Computation s2,i) . (IC SCM ) = IC (Computation s,i) by A24, A27, A25, Th84, FUNCT_1:72;
then A30: (Computation s2,k) . (IC SCM ) = il. (ll + 1) by A29, A18, AMI_3:13;
( ((Computation s1,k) | (dom p)) . (IC SCM ) = (Computation s1,k) . (IC SCM ) & ((Computation s2,k) | (dom p)) . (IC SCM ) = (Computation s2,k) . (IC SCM ) ) by Th84, FUNCT_1:72;
hence not (Computation s1,k) | (dom p) = (Computation s2,k) | (dom p) by A28, A30; :: thesis: verum

end;

hence contradiction ; :: thesis: verum