let IAlph, OAlph be non empty set ; for w1, w2 being FinSequence of IAlph
for tfsm being non empty Mealy-FSM of IAlph,OAlph
for q1t, q2t being State of tfsm st q1t,w1 -leads_to q2t holds
q1t,(w1 ^ w2) -response = (q1t,w1 -response ) ^ (q2t,w2 -response )
let w1, w2 be FinSequence of IAlph; for tfsm being non empty Mealy-FSM of IAlph,OAlph
for q1t, q2t being State of tfsm st q1t,w1 -leads_to q2t holds
q1t,(w1 ^ w2) -response = (q1t,w1 -response ) ^ (q2t,w2 -response )
let tfsm be non empty Mealy-FSM of IAlph,OAlph; for q1t, q2t being State of tfsm st q1t,w1 -leads_to q2t holds
q1t,(w1 ^ w2) -response = (q1t,w1 -response ) ^ (q2t,w2 -response )
let q1t, q2t be State of tfsm; ( q1t,w1 -leads_to q2t implies q1t,(w1 ^ w2) -response = (q1t,w1 -response ) ^ (q2t,w2 -response ) )
set q1w1 = q1t,w1 -response ;
set q2w2 = q2t,w2 -response ;
set q1w1w2 = q1t,(w1 ^ w2) -response ;
set Dq1w1w2a = Del (q1t,w1 -admissible ),((len w1) + 1);
set OF = the OFun of tfsm;
assume A1:
q1t,w1 -leads_to q2t
; q1t,(w1 ^ w2) -response = (q1t,w1 -response ) ^ (q2t,w2 -response )
A2:
now
dom (q1t,w1 -admissible ) = Seg (len (q1t,w1 -admissible ))
by FINSEQ_1:def 3;
then
dom (q1t,w1 -admissible ) = Seg ((len w1) + 1)
by Def2;
then
(len w1) + 1
in dom (q1t,w1 -admissible )
by FINSEQ_1:5;
then consider m being
Nat such that A3:
len (q1t,w1 -admissible ) = m + 1
and A4:
len (Del (q1t,w1 -admissible ),((len w1) + 1)) = m
by FINSEQ_3:113;
A5:
m + 1
= (len w1) + 1
by A3, Def2;
A6:
len (q1t,w1 -response ) = len w1
by Def6;
let k be
Nat;
( 1 <= k & k <= len (q1t,(w1 ^ w2) -response ) implies (q1t,(w1 ^ w2) -response ) . b1 = ((q1t,w1 -response ) ^ (q2t,w2 -response )) . b1 )assume that A7:
1
<= k
and A8:
k <= len (q1t,(w1 ^ w2) -response )
;
(q1t,(w1 ^ w2) -response ) . b1 = ((q1t,w1 -response ) ^ (q2t,w2 -response )) . b1per cases
( ( 1 <= k & k <= len (q1t,w1 -response ) ) or ( (len (q1t,w1 -response )) + 1 <= k & k <= len (q1t,(w1 ^ w2) -response ) ) )
by A7, A8, NAT_1:13;
suppose A10:
( 1
<= k &
k <= len (q1t,w1 -response ) )
;
(q1t,(w1 ^ w2) -response ) . b1 = ((q1t,w1 -response ) ^ (q2t,w2 -response )) . b1then A11:
k <= len w1
by Def6;
then A12:
k in dom w1
by A10, FINSEQ_3:27;
A13:
k in dom (Del (q1t,w1 -admissible ),((len w1) + 1))
by A4, A5, A10, A11, FINSEQ_3:27;
A14:
k < (len w1) + 1
by A11, NAT_1:13;
A15:
k in dom (q1t,w1 -response )
by A10, FINSEQ_3:27;
k <= len (w1 ^ w2)
by A8, Def6;
then
k in dom (w1 ^ w2)
by A7, FINSEQ_3:27;
hence (q1t,(w1 ^ w2) -response ) . k =
the
OFun of
tfsm . [((q1t,(w1 ^ w2) -admissible ) . k),((w1 ^ w2) . k)]
by Def6
.=
the
OFun of
tfsm . [(((Del (q1t,w1 -admissible ),((len w1) + 1)) ^ (q2t,w2 -admissible )) . k),((w1 ^ w2) . k)]
by A1, Th23
.=
the
OFun of
tfsm . [((Del (q1t,w1 -admissible ),((len w1) + 1)) . k),((w1 ^ w2) . k)]
by A13, FINSEQ_1:def 7
.=
the
OFun of
tfsm . [((Del (q1t,w1 -admissible ),((len w1) + 1)) . k),(w1 . k)]
by A12, FINSEQ_1:def 7
.=
the
OFun of
tfsm . [((q1t,w1 -admissible ) . k),(w1 . k)]
by A14, FINSEQ_3:119
.=
(q1t,w1 -response ) . k
by A12, Def6
.=
((q1t,w1 -response ) ^ (q2t,w2 -response )) . k
by A15, FINSEQ_1:def 7
;
verum end; suppose A16:
(
(len (q1t,w1 -response )) + 1
<= k &
k <= len (q1t,(w1 ^ w2) -response ) )
;
(q1t,(w1 ^ w2) -response ) . b1 = ((q1t,w1 -response ) ^ (q2t,w2 -response )) . b1then A17:
((len (q1t,w1 -response )) + 1) - (len (q1t,w1 -response )) <= k - (len (q1t,w1 -response ))
by XREAL_1:11;
then reconsider p =
k - (len (q1t,w1 -response )) as
Element of
NAT by INT_1:16;
A18:
len (q1t,(w1 ^ w2) -response ) =
len (w1 ^ w2)
by Def6
.=
(len w1) + (len w2)
by FINSEQ_1:35
;
then
k <= (len (q1t,w1 -response )) + (len w2)
by A16, Def6;
then
k - (len (q1t,w1 -response )) <= ((len (q1t,w1 -response )) + (len w2)) - (len (q1t,w1 -response ))
by XREAL_1:11;
then A19:
p in dom w2
by A17, FINSEQ_3:27;
A20:
(len (Del (q1t,w1 -admissible ),((len w1) + 1))) + 1
<= k
by A4, A5, A16, Def6;
A21:
(len w1) + 1
<= k
by A16, Def6;
A22:
len (q1t,(w1 ^ w2) -response ) =
len (w1 ^ w2)
by Def6
.=
(len w1) + (len w2)
by FINSEQ_1:35
.=
(len (q1t,w1 -response )) + (len w2)
by Def6
.=
(len (q1t,w1 -response )) + (len (q2t,w2 -response ))
by Def6
;
then A23:
((q1t,w1 -response ) ^ (q2t,w2 -response )) . k =
(q2t,w2 -response ) . p
by A16, FINSEQ_1:36
.=
the
OFun of
tfsm . [((q2t,w2 -admissible ) . p),(w2 . p)]
by A19, Def6
.=
the
OFun of
tfsm . [((q2t,w2 -admissible ) . (k - (len w1))),(w2 . (k - (len (q1t,w1 -response ))))]
by Def6
.=
the
OFun of
tfsm . [((q2t,w2 -admissible ) . (k - (len w1))),(w2 . (k - (len w1)))]
by Def6
;
len w2 <= (len w2) + 1
by NAT_1:11;
then A24:
(len (Del (q1t,w1 -admissible ),((len w1) + 1))) + (len w2) <= (len (Del (q1t,w1 -admissible ),((len w1) + 1))) + ((len w2) + 1)
by XREAL_1:8;
k <= (len (Del (q1t,w1 -admissible ),((len w1) + 1))) + (len w2)
by A4, A5, A6, A16, A22, Def6;
then
k <= (len (Del (q1t,w1 -admissible ),((len w1) + 1))) + ((len w2) + 1)
by A24, XXREAL_0:2;
then A25:
k <= (len (Del (q1t,w1 -admissible ),((len w1) + 1))) + (len (q2t,w2 -admissible ))
by Def2;
k <= len (w1 ^ w2)
by A8, Def6;
then
k in dom (w1 ^ w2)
by A7, FINSEQ_3:27;
then (q1t,(w1 ^ w2) -response ) . k =
the
OFun of
tfsm . [((q1t,(w1 ^ w2) -admissible ) . k),((w1 ^ w2) . k)]
by Def6
.=
the
OFun of
tfsm . [(((Del (q1t,w1 -admissible ),((len w1) + 1)) ^ (q2t,w2 -admissible )) . k),((w1 ^ w2) . k)]
by A1, Th23
.=
the
OFun of
tfsm . [((q2t,w2 -admissible ) . (k - (len (Del (q1t,w1 -admissible ),((len w1) + 1))))),((w1 ^ w2) . k)]
by A20, A25, FINSEQ_1:36
.=
the
OFun of
tfsm . [((q2t,w2 -admissible ) . (k - (len w1))),(w2 . (k - (len w1)))]
by A4, A5, A16, A18, A21, FINSEQ_1:36
;
hence
(q1t,(w1 ^ w2) -response ) . k = ((q1t,w1 -response ) ^ (q2t,w2 -response )) . k
by A23;
verum end; end;
len (q1t,(w1 ^ w2) -response ) =
len (w1 ^ w2)
by Def6
.=
(len w1) + (len w2)
by FINSEQ_1:35
.=
(len (q1t,w1 -response )) + (len w2)
by Def6
.=
(len (q1t,w1 -response )) + (len (q2t,w2 -response ))
by Def6
.=
len ((q1t,w1 -response ) ^ (q2t,w2 -response ))
by FINSEQ_1:35
;
hence
q1t,(w1 ^ w2) -response = (q1t,w1 -response ) ^ (q2t,w2 -response )
by A2, FINSEQ_1:18; verum