let S1, S2, S be non empty non void Circuit-like ManySortedSign ; ( InputVertices S1 misses InnerVertices S2 & InputVertices S2 misses InnerVertices S1 & S = S1 +* S2 implies for A1 being non-empty Circuit of S1
for A2 being non-empty Circuit of S2
for A being non-empty Circuit of S st A1 tolerates A2 & A = A1 +* A2 holds
for n1, n2 being Nat
for s being State of A
for s1 being State of A1 st s1 = s | the carrier of S1 holds
for s2 being State of A2 st s2 = s | the carrier of S2 & Following (s1,n1) is stable & Following (s2,n2) is stable holds
Following (s,(max (n1,n2))) is stable )
assume that
A1:
InputVertices S1 misses InnerVertices S2
and
A2:
InputVertices S2 misses InnerVertices S1
and
A3:
S = S1 +* S2
; for A1 being non-empty Circuit of S1
for A2 being non-empty Circuit of S2
for A being non-empty Circuit of S st A1 tolerates A2 & A = A1 +* A2 holds
for n1, n2 being Nat
for s being State of A
for s1 being State of A1 st s1 = s | the carrier of S1 holds
for s2 being State of A2 st s2 = s | the carrier of S2 & Following (s1,n1) is stable & Following (s2,n2) is stable holds
Following (s,(max (n1,n2))) is stable
let A1 be non-empty Circuit of S1; for A2 being non-empty Circuit of S2
for A being non-empty Circuit of S st A1 tolerates A2 & A = A1 +* A2 holds
for n1, n2 being Nat
for s being State of A
for s1 being State of A1 st s1 = s | the carrier of S1 holds
for s2 being State of A2 st s2 = s | the carrier of S2 & Following (s1,n1) is stable & Following (s2,n2) is stable holds
Following (s,(max (n1,n2))) is stable
let A2 be non-empty Circuit of S2; for A being non-empty Circuit of S st A1 tolerates A2 & A = A1 +* A2 holds
for n1, n2 being Nat
for s being State of A
for s1 being State of A1 st s1 = s | the carrier of S1 holds
for s2 being State of A2 st s2 = s | the carrier of S2 & Following (s1,n1) is stable & Following (s2,n2) is stable holds
Following (s,(max (n1,n2))) is stable
let A be non-empty Circuit of S; ( A1 tolerates A2 & A = A1 +* A2 implies for n1, n2 being Nat
for s being State of A
for s1 being State of A1 st s1 = s | the carrier of S1 holds
for s2 being State of A2 st s2 = s | the carrier of S2 & Following (s1,n1) is stable & Following (s2,n2) is stable holds
Following (s,(max (n1,n2))) is stable )
assume that
A4:
A1 tolerates A2
and
A5:
A = A1 +* A2
; for n1, n2 being Nat
for s being State of A
for s1 being State of A1 st s1 = s | the carrier of S1 holds
for s2 being State of A2 st s2 = s | the carrier of S2 & Following (s1,n1) is stable & Following (s2,n2) is stable holds
Following (s,(max (n1,n2))) is stable
let n1, n2 be Nat; for s being State of A
for s1 being State of A1 st s1 = s | the carrier of S1 holds
for s2 being State of A2 st s2 = s | the carrier of S2 & Following (s1,n1) is stable & Following (s2,n2) is stable holds
Following (s,(max (n1,n2))) is stable
let s be State of A; for s1 being State of A1 st s1 = s | the carrier of S1 holds
for s2 being State of A2 st s2 = s | the carrier of S2 & Following (s1,n1) is stable & Following (s2,n2) is stable holds
Following (s,(max (n1,n2))) is stable
set n = max (n1,n2);
let s0 be State of A1; ( s0 = s | the carrier of S1 implies for s2 being State of A2 st s2 = s | the carrier of S2 & Following (s0,n1) is stable & Following (s2,n2) is stable holds
Following (s,(max (n1,n2))) is stable )
assume A6:
s0 = s | the carrier of S1
; for s2 being State of A2 st s2 = s | the carrier of S2 & Following (s0,n1) is stable & Following (s2,n2) is stable holds
Following (s,(max (n1,n2))) is stable
A7:
(Following (s,(max (n1,n2)))) | the carrier of S1 = Following (s0,(max (n1,n2)))
by A1, A3, A4, A5, A6, Th13;
S1 tolerates S2
by A4, CIRCCOMB:def 3;
then A8:
S1 +* S2 = S2 +* S1
by CIRCCOMB:5;
let s3 be State of A2; ( s3 = s | the carrier of S2 & Following (s0,n1) is stable & Following (s3,n2) is stable implies Following (s,(max (n1,n2))) is stable )
assume that
A9:
s3 = s | the carrier of S2
and
A10:
Following (s0,n1) is stable
and
A11:
Following (s3,n2) is stable
; Following (s,(max (n1,n2))) is stable
A1 +* A2 = A2 +* A1
by A4, CIRCCOMB:22;
then A12:
(Following (s,(max (n1,n2)))) | the carrier of S2 = Following (s3,(max (n1,n2)))
by A2, A3, A4, A5, A9, A8, Th13, CIRCCOMB:19;
A13:
Following (s3,(max (n1,n2))) is stable
by A11, Th4, XXREAL_0:25;
A14:
Following (s0,(max (n1,n2))) is stable
by A10, Th4, XXREAL_0:25;
thus Following (s,(max (n1,n2))) =
(Following (s0,(max (n1,n2)))) +* (Following (s3,(max (n1,n2))))
by A1, A2, A3, A4, A5, A6, A9, Th21
.=
(Following (Following (s0,(max (n1,n2))))) +* (Following (s3,(max (n1,n2))))
by A14
.=
(Following (Following (s0,(max (n1,n2))))) +* (Following (Following (s3,(max (n1,n2)))))
by A13
.=
Following (Following (s,(max (n1,n2))))
by A2, A3, A4, A5, A7, A12, CIRCCOMB:32
; CIRCUIT2:def 6 verum