set O = {0};
reconsider IC1 = 0 as Element of {0} by TARSKI:def 1;
A1:
0 in NAT *
by FINSEQ_1:49;
0 in ((union N) \/ {0}) *
by FINSEQ_1:49;
then
( [1,0,0] in [:NAT,(NAT *),(((union N) \/ {0}) *):] & [0,0,0] in [:NAT,(NAT *),(((union N) \/ {0}) *):] )
by A1, MCART_1:69;
then reconsider ins = {[1,0,0],[0,0,0]} as non empty Subset of [:NAT,(NAT *),(((union N) \/ {0}) *):] by ZFMISC_1:32;
reconsider i = [0,0,0] as Element of ins by TARSKI:def 2;
A2:
dom (0 .--> NAT) = {0}
by FUNCOP_1:13;
A3:
{NAT} c= N \/ {NAT}
by XBOOLE_1:7;
rng (0 .--> NAT) c= N \/ {NAT}
by A3, FUNCOP_1:8;
then reconsider Ok = 0 .--> NAT as Function of {0},(N \/ {NAT}) by A2, RELSET_1:4;
deffunc H1( Element of product Ok) -> set = $1 +* (0 .--> (succ ($1 . 0)));
A4:
now let s be
Element of
product Ok;
H1(s) in product Oknow thus dom (s +* (0 .--> (succ (s . 0)))) =
(dom s) \/ (dom (0 .--> (succ (s . 0))))
by FUNCT_4:def 1
.=
(dom s) \/ {0}
by FUNCOP_1:13
.=
{0} \/ {0}
by PARTFUN1:def 2
.=
dom Ok
by FUNCOP_1:13
;
for o being set st o in dom Ok holds
(s +* (0 .--> (succ (s . 0)))) . o in Ok . olet o be
set ;
( o in dom Ok implies (s +* (0 .--> (succ (s . 0)))) . o in Ok . o )A5:
dom (0 .--> (succ (s . 0))) = {0}
by FUNCOP_1:13;
assume Z:
o in dom Ok
;
(s +* (0 .--> (succ (s . 0)))) . o in Ok . oA8:
o = 0
by Z, TARSKI:def 1;
A10:
Ok . o = NAT
by Z, FUNCOP_1:7;
A12:
(s +* (0 .--> (succ (s . 0)))) . o =
(0 .--> (succ (s . 0))) . o
by A5, Z, FUNCT_4:13
.=
succ (s . 0)
by Z, FUNCOP_1:7
;
0 in dom Ok
by A2, TARSKI:def 1;
then reconsider k =
s . 0 as
Element of
NAT by A8, A10, CARD_3:9;
succ k in NAT
;
hence
(s +* (0 .--> (succ (s . 0)))) . o in Ok . o
by Z, FUNCOP_1:7, A12;
verum end; hence
H1(
s)
in product Ok
by CARD_3:9;
verum end;
consider f being Function of (product Ok),(product Ok) such that
A13:
for s being Element of product Ok holds f . s = H1(s)
from FUNCT_2:sch 8(A4);
set E = ([1,0,0] .--> f) +* ([0,0,0] .--> (id (product Ok)));
A14: dom (([1,0,0] .--> f) +* ([0,0,0] .--> (id (product Ok)))) =
(dom ([1,0,0] .--> f)) \/ (dom ([0,0,0] .--> (id (product Ok))))
by FUNCT_4:def 1
.=
{[1,0,0]} \/ (dom ([0,0,0] .--> (id (product Ok))))
by FUNCOP_1:13
.=
{[1,0,0]} \/ {[0,0,0]}
by FUNCOP_1:13
.=
ins
by ENUMSET1:1
;
A15:
( rng ([1,0,0] .--> f) c= {f} & rng ([0,0,0] .--> (id (product Ok))) c= {(id (product Ok))} )
by FUNCOP_1:13;
A16:
rng (([1,0,0] .--> f) +* ([0,0,0] .--> (id (product Ok)))) c= (rng ([1,0,0] .--> f)) \/ (rng ([0,0,0] .--> (id (product Ok))))
by FUNCT_4:17;
rng (([1,0,0] .--> f) +* ([0,0,0] .--> (id (product Ok)))) c= Funcs ((product Ok),(product Ok))
proof
let e be
set ;
TARSKI:def 3 ( not e in rng (([1,0,0] .--> f) +* ([0,0,0] .--> (id (product Ok)))) or e in Funcs ((product Ok),(product Ok)) )
assume
e in rng (([1,0,0] .--> f) +* ([0,0,0] .--> (id (product Ok))))
;
e in Funcs ((product Ok),(product Ok))
then
(
e in rng ([1,0,0] .--> f) or
e in rng ([0,0,0] .--> (id (product Ok))) )
by A16, XBOOLE_0:def 3;
then
(
e = f or
e = id (product Ok) )
by A15, TARSKI:def 1;
hence
e in Funcs (
(product Ok),
(product Ok))
by FUNCT_2:9;
verum
end;
then reconsider E = ([1,0,0] .--> f) +* ([0,0,0] .--> (id (product Ok))) as Function of ins,(Funcs ((product Ok),(product Ok))) by A14, FUNCT_2:def 1, RELSET_1:4;
set M = AMI-Struct(# {0},IC1,ins,i,Ok,E #);
take
AMI-Struct(# {0},IC1,ins,i,Ok,E #)
; ( the carrier of AMI-Struct(# {0},IC1,ins,i,Ok,E #) = {0} & the ZeroF of AMI-Struct(# {0},IC1,ins,i,Ok,E #) = 0 & the Instructions of AMI-Struct(# {0},IC1,ins,i,Ok,E #) = {[0,0,0],[1,0,0]} & the haltF of AMI-Struct(# {0},IC1,ins,i,Ok,E #) = [0,0,0] & the Object-Kind of AMI-Struct(# {0},IC1,ins,i,Ok,E #) = 0 .--> NAT & ex f being Function of (product the Object-Kind of AMI-Struct(# {0},IC1,ins,i,Ok,E #)),(product the Object-Kind of AMI-Struct(# {0},IC1,ins,i,Ok,E #)) st
( ( for s being Element of product the Object-Kind of AMI-Struct(# {0},IC1,ins,i,Ok,E #) holds f . s = s +* (0 .--> (succ (s . 0))) ) & the Execution of AMI-Struct(# {0},IC1,ins,i,Ok,E #) = ([1,0,0] .--> f) +* ([0,0,0] .--> (id (product the Object-Kind of AMI-Struct(# {0},IC1,ins,i,Ok,E #)))) ) )
thus
the carrier of AMI-Struct(# {0},IC1,ins,i,Ok,E #) = {0}
; ( the ZeroF of AMI-Struct(# {0},IC1,ins,i,Ok,E #) = 0 & the Instructions of AMI-Struct(# {0},IC1,ins,i,Ok,E #) = {[0,0,0],[1,0,0]} & the haltF of AMI-Struct(# {0},IC1,ins,i,Ok,E #) = [0,0,0] & the Object-Kind of AMI-Struct(# {0},IC1,ins,i,Ok,E #) = 0 .--> NAT & ex f being Function of (product the Object-Kind of AMI-Struct(# {0},IC1,ins,i,Ok,E #)),(product the Object-Kind of AMI-Struct(# {0},IC1,ins,i,Ok,E #)) st
( ( for s being Element of product the Object-Kind of AMI-Struct(# {0},IC1,ins,i,Ok,E #) holds f . s = s +* (0 .--> (succ (s . 0))) ) & the Execution of AMI-Struct(# {0},IC1,ins,i,Ok,E #) = ([1,0,0] .--> f) +* ([0,0,0] .--> (id (product the Object-Kind of AMI-Struct(# {0},IC1,ins,i,Ok,E #)))) ) )
thus
the ZeroF of AMI-Struct(# {0},IC1,ins,i,Ok,E #) = 0
; ( the Instructions of AMI-Struct(# {0},IC1,ins,i,Ok,E #) = {[0,0,0],[1,0,0]} & the haltF of AMI-Struct(# {0},IC1,ins,i,Ok,E #) = [0,0,0] & the Object-Kind of AMI-Struct(# {0},IC1,ins,i,Ok,E #) = 0 .--> NAT & ex f being Function of (product the Object-Kind of AMI-Struct(# {0},IC1,ins,i,Ok,E #)),(product the Object-Kind of AMI-Struct(# {0},IC1,ins,i,Ok,E #)) st
( ( for s being Element of product the Object-Kind of AMI-Struct(# {0},IC1,ins,i,Ok,E #) holds f . s = s +* (0 .--> (succ (s . 0))) ) & the Execution of AMI-Struct(# {0},IC1,ins,i,Ok,E #) = ([1,0,0] .--> f) +* ([0,0,0] .--> (id (product the Object-Kind of AMI-Struct(# {0},IC1,ins,i,Ok,E #)))) ) )
thus
the Instructions of AMI-Struct(# {0},IC1,ins,i,Ok,E #) = {[0,0,0],[1,0,0]}
; ( the haltF of AMI-Struct(# {0},IC1,ins,i,Ok,E #) = [0,0,0] & the Object-Kind of AMI-Struct(# {0},IC1,ins,i,Ok,E #) = 0 .--> NAT & ex f being Function of (product the Object-Kind of AMI-Struct(# {0},IC1,ins,i,Ok,E #)),(product the Object-Kind of AMI-Struct(# {0},IC1,ins,i,Ok,E #)) st
( ( for s being Element of product the Object-Kind of AMI-Struct(# {0},IC1,ins,i,Ok,E #) holds f . s = s +* (0 .--> (succ (s . 0))) ) & the Execution of AMI-Struct(# {0},IC1,ins,i,Ok,E #) = ([1,0,0] .--> f) +* ([0,0,0] .--> (id (product the Object-Kind of AMI-Struct(# {0},IC1,ins,i,Ok,E #)))) ) )
thus
the haltF of AMI-Struct(# {0},IC1,ins,i,Ok,E #) = [0,0,0]
; ( the Object-Kind of AMI-Struct(# {0},IC1,ins,i,Ok,E #) = 0 .--> NAT & ex f being Function of (product the Object-Kind of AMI-Struct(# {0},IC1,ins,i,Ok,E #)),(product the Object-Kind of AMI-Struct(# {0},IC1,ins,i,Ok,E #)) st
( ( for s being Element of product the Object-Kind of AMI-Struct(# {0},IC1,ins,i,Ok,E #) holds f . s = s +* (0 .--> (succ (s . 0))) ) & the Execution of AMI-Struct(# {0},IC1,ins,i,Ok,E #) = ([1,0,0] .--> f) +* ([0,0,0] .--> (id (product the Object-Kind of AMI-Struct(# {0},IC1,ins,i,Ok,E #)))) ) )
thus
the Object-Kind of AMI-Struct(# {0},IC1,ins,i,Ok,E #) = 0 .--> NAT
; ex f being Function of (product the Object-Kind of AMI-Struct(# {0},IC1,ins,i,Ok,E #)),(product the Object-Kind of AMI-Struct(# {0},IC1,ins,i,Ok,E #)) st
( ( for s being Element of product the Object-Kind of AMI-Struct(# {0},IC1,ins,i,Ok,E #) holds f . s = s +* (0 .--> (succ (s . 0))) ) & the Execution of AMI-Struct(# {0},IC1,ins,i,Ok,E #) = ([1,0,0] .--> f) +* ([0,0,0] .--> (id (product the Object-Kind of AMI-Struct(# {0},IC1,ins,i,Ok,E #)))) )
reconsider f = f as Function of (product the Object-Kind of AMI-Struct(# {0},IC1,ins,i,Ok,E #)),(product the Object-Kind of AMI-Struct(# {0},IC1,ins,i,Ok,E #)) ;
take
f
; ( ( for s being Element of product the Object-Kind of AMI-Struct(# {0},IC1,ins,i,Ok,E #) holds f . s = s +* (0 .--> (succ (s . 0))) ) & the Execution of AMI-Struct(# {0},IC1,ins,i,Ok,E #) = ([1,0,0] .--> f) +* ([0,0,0] .--> (id (product the Object-Kind of AMI-Struct(# {0},IC1,ins,i,Ok,E #)))) )
thus
for s being Element of product the Object-Kind of AMI-Struct(# {0},IC1,ins,i,Ok,E #) holds f . s = s +* (0 .--> (succ (s . 0)))
by A13; the Execution of AMI-Struct(# {0},IC1,ins,i,Ok,E #) = ([1,0,0] .--> f) +* ([0,0,0] .--> (id (product the Object-Kind of AMI-Struct(# {0},IC1,ins,i,Ok,E #))))
thus
the Execution of AMI-Struct(# {0},IC1,ins,i,Ok,E #) = ([1,0,0] .--> f) +* ([0,0,0] .--> (id (product the Object-Kind of AMI-Struct(# {0},IC1,ins,i,Ok,E #))))
; verum