let IAlph be non empty set ; for fsm being non empty FSM of IAlph
for w1, w2 being FinSequence of IAlph
for q1, q2 being State of fsm st q1,w1 -leads_to q2 holds
q1,(w1 ^ w2) -admissible = (Del (q1,w1 -admissible ),((len w1) + 1)) ^ (q2,w2 -admissible )
let fsm be non empty FSM of IAlph; for w1, w2 being FinSequence of IAlph
for q1, q2 being State of fsm st q1,w1 -leads_to q2 holds
q1,(w1 ^ w2) -admissible = (Del (q1,w1 -admissible ),((len w1) + 1)) ^ (q2,w2 -admissible )
let w1, w2 be FinSequence of IAlph; for q1, q2 being State of fsm st q1,w1 -leads_to q2 holds
q1,(w1 ^ w2) -admissible = (Del (q1,w1 -admissible ),((len w1) + 1)) ^ (q2,w2 -admissible )
let q1, q2 be State of fsm; ( q1,w1 -leads_to q2 implies q1,(w1 ^ w2) -admissible = (Del (q1,w1 -admissible ),((len w1) + 1)) ^ (q2,w2 -admissible ) )
set q1w = q1,(w1 ^ w2) -admissible ;
set q1w1 = q1,w1 -admissible ;
set q2w2 = q2,w2 -admissible ;
set Dw1 = Del (q1,w1 -admissible ),((len w1) + 1);
A1:
len (q1,w1 -admissible ) = (len w1) + 1
by Def2;
( len (q1,w1 -admissible ) = (len w1) + 1 & dom (q1,w1 -admissible ) = Seg (len (q1,w1 -admissible )) )
by Def2, FINSEQ_1:def 3;
then
(len w1) + 1 in dom (q1,w1 -admissible )
by FINSEQ_1:5;
then A2:
ex m being Nat st
( len (q1,w1 -admissible ) = m + 1 & len (Del (q1,w1 -admissible ),((len w1) + 1)) = m )
by FINSEQ_3:113;
A3: len (q1,(w1 ^ w2) -admissible ) =
(len (w1 ^ w2)) + 1
by Def2
.=
((len w1) + (len w2)) + 1
by FINSEQ_1:35
.=
(len (Del (q1,w1 -admissible ),((len w1) + 1))) + ((len w2) + 1)
by A2, A1
.=
(len (Del (q1,w1 -admissible ),((len w1) + 1))) + (len (q2,w2 -admissible ))
by Def2
.=
len ((Del (q1,w1 -admissible ),((len w1) + 1)) ^ (q2,w2 -admissible ))
by FINSEQ_1:35
;
assume A4:
q1,w1 -leads_to q2
; q1,(w1 ^ w2) -admissible = (Del (q1,w1 -admissible ),((len w1) + 1)) ^ (q2,w2 -admissible )
now let k be
Nat;
( 1 <= k & k <= len (q1,(w1 ^ w2) -admissible ) implies (q1,(w1 ^ w2) -admissible ) . b1 = ((Del (q1,w1 -admissible ),((len w1) + 1)) ^ (q2,w2 -admissible )) . b1 )assume A5:
( 1
<= k &
k <= len (q1,(w1 ^ w2) -admissible ) )
;
(q1,(w1 ^ w2) -admissible ) . b1 = ((Del (q1,w1 -admissible ),((len w1) + 1)) ^ (q2,w2 -admissible )) . b1per cases
( ( 1 <= k & k <= len w1 ) or ( (len w1) + 1 <= k & k <= len (q1,(w1 ^ w2) -admissible ) ) )
by A5, NAT_1:13;
suppose A6:
( 1
<= k &
k <= len w1 )
;
(q1,(w1 ^ w2) -admissible ) . b1 = ((Del (q1,w1 -admissible ),((len w1) + 1)) ^ (q2,w2 -admissible )) . b1then A7:
k < (len w1) + 1
by NAT_1:13;
A8:
k in dom (Del (q1,w1 -admissible ),((len w1) + 1))
by A2, A1, A6, FINSEQ_3:27;
k in NAT
by ORDINAL1:def 13;
hence (q1,(w1 ^ w2) -admissible ) . k =
(q1,w1 -admissible ) . k
by A6, Th20
.=
(Del (q1,w1 -admissible ),((len w1) + 1)) . k
by A7, FINSEQ_3:119
.=
((Del (q1,w1 -admissible ),((len w1) + 1)) ^ (q2,w2 -admissible )) . k
by A8, FINSEQ_1:def 7
;
verum end; suppose A10:
(
(len w1) + 1
<= k &
k <= len (q1,(w1 ^ w2) -admissible ) )
;
(q1,(w1 ^ w2) -admissible ) . b1 = ((Del (q1,w1 -admissible ),((len w1) + 1)) ^ (q2,w2 -admissible )) . b1then
k <= (len (Del (q1,w1 -admissible ),((len w1) + 1))) + (len (q2,w2 -admissible ))
by A3, FINSEQ_1:35;
then A11:
((Del (q1,w1 -admissible ),((len w1) + 1)) ^ (q2,w2 -admissible )) . k = (q2,w2 -admissible ) . (k - (len w1))
by A2, A1, A10, FINSEQ_1:36;
((len w1) + 1) - (len w1) <= k - (len w1)
by A10, XREAL_1:11;
then reconsider i =
k - (len w1) as
Element of
NAT by INT_1:16;
A12:
k = (len w1) + i
;
len (q1,(w1 ^ w2) -admissible ) = (len (w1 ^ w2)) + 1
by Def2;
then
k <= ((len w1) + (len w2)) + 1
by A10, FINSEQ_1:35;
then
k <= (len w1) + ((len w2) + 1)
;
then A13:
i <= (len w2) + 1
by A12, XREAL_1:8;
1
<= i
by A10, A12, XREAL_1:8;
hence
(q1,(w1 ^ w2) -admissible ) . k = ((Del (q1,w1 -admissible ),((len w1) + 1)) ^ (q2,w2 -admissible )) . k
by A4, A12, A13, A11, Th22;
verum end; end;
hence
q1,(w1 ^ w2) -admissible = (Del (q1,w1 -admissible ),((len w1) + 1)) ^ (q2,w2 -admissible )
by A3, FINSEQ_1:18; verum