let x be set ; :: according to EXTPRO_1:def 13 :: thesis: ( not x in proj1 Sorting-Function or ex b₁ being set st
( x = b₁ & (Initialized Bubble-Sort-Algorithm) +* b₁ is Autonomy of Bubble-Sort-Algorithm & Sorting-Function . b₁ c= Result (Bubble-Sort-Algorithm,((Initialized Bubble-Sort-Algorithm) +* b₁)) ) )
assume x in dom Sorting-Function ; :: thesis: ex b₁ being set st
( x = b₁ & (Initialized Bubble-Sort-Algorithm) +* b₁ is Autonomy of Bubble-Sort-Algorithm & Sorting-Function . b₁ c= Result (Bubble-Sort-Algorithm,((Initialized Bubble-Sort-Algorithm) +* b₁)) )
then consider w being FinSequence of INT such that
A1: x = (fsloc 0) .--> w by Th60;
reconsider d = x as FinPartState of SCM+FSA by A1;
set q = Bubble-Sort-Algorithm ;
set p = Initialized Bubble-Sort-Algorithm;
F2: Bubble-Sort-Algorithm c= Initialized Bubble-Sort-Algorithm by SCMFSA6A:26;
A2: dom d = {(fsloc 0)} by A1, FUNCOP_1:19;
take d ; :: thesis: ( x = d & (Initialized Bubble-Sort-Algorithm) +* d is Autonomy of Bubble-Sort-Algorithm & Sorting-Function . d c= Result (Bubble-Sort-Algorithm,((Initialized Bubble-Sort-Algorithm) +* d)) )
thus x = d ; :: thesis: ( (Initialized Bubble-Sort-Algorithm) +* d is Autonomy of Bubble-Sort-Algorithm & Sorting-Function . d c= Result (Bubble-Sort-Algorithm,((Initialized Bubble-Sort-Algorithm) +* d)) )
A3: (Initialized Bubble-Sort-Algorithm) +* d is autonomic by A1, Th68;
A4: dom (Initialized Bubble-Sort-Algorithm) misses dom d by A1, Th46;

now

let t be State of SCM+FSA; :: thesis: ( (Initialized Bubble-Sort-Algorithm) +* d c= t implies ProgramPart t halts_on t )
assume A6: (Initialized Bubble-Sort-Algorithm) +* d c= t ; :: thesis: ProgramPart t halts_on t
set bf = bubble-sort (fsloc 0);
Initialized Bubble-Sort-Algorithm c= (Initialized Bubble-Sort-Algorithm) +* d by A4, FUNCT_4:33;
then Initialized Bubble-Sort-Algorithm c= t by A6, XBOOLE_1:1;
hence ProgramPart t halts_on t by Lm26, EXTPRO_1:def 10; :: thesis: verum

end;

then A7: (Initialized Bubble-Sort-Algorithm) +* d is halting by EXTPRO_1:def 10;
fsloc 0 in FinSeq-Locations by SCMFSA_2:10;
then X1: {(fsloc 0)} c= FinSeq-Locations by ZFMISC_1:37;
FinSeq-Locations misses NAT by SCMFSA_2:14;
X2: dom d misses NAT by XBOOLE_1:63, X1, A2, SCMFSA_2:14;
X3: dom Bubble-Sort-Algorithm c= NAT by RELAT_1:def 18;
then XX: dom d misses dom Bubble-Sort-Algorithm by X2, XBOOLE_1:63;
((Initialized Bubble-Sort-Algorithm) +* d) +* Bubble-Sort-Algorithm = (Initialized Bubble-Sort-Algorithm) +* (d +* Bubble-Sort-Algorithm) by FUNCT_4:15
.= (Initialized Bubble-Sort-Algorithm) +* (Bubble-Sort-Algorithm +* d) by X3, FUNCT_4:36, X2, XBOOLE_1:63
.= ((Initialized Bubble-Sort-Algorithm) +* Bubble-Sort-Algorithm) +* d by FUNCT_4:15
.= (Initialized Bubble-Sort-Algorithm) +* d by F2, FUNCT_4:79 ;
hence KK: (Initialized Bubble-Sort-Algorithm) +* d is Autonomy of Bubble-Sort-Algorithm by A7, A3, EXTPRO_1:def 11; :: thesis: Sorting-Function . d c= Result (Bubble-Sort-Algorithm,((Initialized Bubble-Sort-Algorithm) +* d))
consider z being FinSequence of REAL such that
A8: w,z are_fiberwise_equipotent and
A9: z is non-increasing and
z is FinSequence of INT and
A10: Sorting-Function . d = (fsloc 0) .--> z by A1, Th61;
consider t being State of SCM+FSA such that
A11: (Initialized Bubble-Sort-Algorithm) +* d c= t by PBOOLE:156;
consider u being FinSequence of REAL such that
A12: w,u are_fiberwise_equipotent and
A13: u is non-increasing and
u is FinSequence of INT and
A14: (Result ((ProgramPart t),t)) . (fsloc 0) = u by A1, A11, Th67;
A15: u = z by A8, A9, A12, A13, CLASSES1:84, RFINSEQ:36;
fsloc 0 in the carrier of SCM+FSA ;
then A16: fsloc 0 in dom (Result ((ProgramPart t),t)) by PARTFUN1:def 4;
d c= (Initialized Bubble-Sort-Algorithm) +* d by FUNCT_4:26;
then A17: dom d c= dom ((Initialized Bubble-Sort-Algorithm) +* d) by RELAT_1:25;
A18: dom ((fsloc 0) .--> z) = {(fsloc 0)} by FUNCOP_1:19;
K: dom d misses {(IC SCM+FSA)} by A2, SCMFSA_2:82, ZFMISC_1:17;

now

assume dom d meets NAT ; :: thesis: contradiction
then consider x being set such that
W1: x in dom d and
W2: x in NAT by XBOOLE_0:3;
x = fsloc 0 by W1, A2, TARSKI:def 1;
hence contradiction by W2, SCMFSA_2:85; :: thesis: verum

end;

then dom d misses {(IC SCM+FSA)} \/ NAT by K, XBOOLE_1:70;
then d is data-only by COMPOS_1:def 23;
then dom d misses NAT by COMPOS_1:40;
then LL: ProgramPart ((Initialized Bubble-Sort-Algorithm) +* d) = ProgramPart (Initialized Bubble-Sort-Algorithm) by FUNCT_4:76;
UU: (Initialized Bubble-Sort-Algorithm) +* d is Autonomy of Bubble-Sort-Algorithm by KK;
Bubble-Sort-Algorithm = ProgramPart (Initialized Bubble-Sort-Algorithm) by SCMFSA8C:25;
then Result (Bubble-Sort-Algorithm,((Initialized Bubble-Sort-Algorithm) +* d)) = (Result ((ProgramPart t),t)) | (dom ((Initialized Bubble-Sort-Algorithm) +* d)) by A11, EXTPRO_1:def 12, KK, LL;
hence Sorting-Function . d c= Result (Bubble-Sort-Algorithm,((Initialized Bubble-Sort-Algorithm) +* d)) by A2, A10, A14, A15, A16, A17, A18, FUNCT_4:90, RELAT_1:186; :: thesis: verum