let S1, S2, S be non empty non void Circuit-like ManySortedSign ; :: thesis: ( InputVertices S1 misses InnerVertices S2 & 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 s being State of A
for s1 being State of A1 st s1 = s | the carrier of S1 & s1 is stabilizing holds
for s2 being State of A2 st s2 = (Following (s,(stabilization-time s1))) | the carrier of S2 & s2 is stabilizing holds
(Result s) | the carrier of S1 = Result s1 )
assume A1: ( InputVertices S1 misses InnerVertices S2 & S = S1 +* S2 ) ; :: thesis: 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 s being State of A
for s1 being State of A1 st s1 = s | the carrier of S1 & s1 is stabilizing holds
for s2 being State of A2 st s2 = (Following (s,(stabilization-time s1))) | the carrier of S2 & s2 is stabilizing holds
(Result s) | the carrier of S1 = Result s1
let A1 be non-empty Circuit of S1; :: thesis: 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 s being State of A
for s1 being State of A1 st s1 = s | the carrier of S1 & s1 is stabilizing holds
for s2 being State of A2 st s2 = (Following (s,(stabilization-time s1))) | the carrier of S2 & s2 is stabilizing holds
(Result s) | the carrier of S1 = Result s1
let A2 be non-empty Circuit of S2; :: thesis: for A being non-empty Circuit of S st A1 tolerates A2 & A = A1 +* A2 holds
for s being State of A
for s1 being State of A1 st s1 = s | the carrier of S1 & s1 is stabilizing holds
for s2 being State of A2 st s2 = (Following (s,(stabilization-time s1))) | the carrier of S2 & s2 is stabilizing holds
(Result s) | the carrier of S1 = Result s1
let A be non-empty Circuit of S; :: thesis: ( A1 tolerates A2 & A = A1 +* A2 implies for s being State of A
for s1 being State of A1 st s1 = s | the carrier of S1 & s1 is stabilizing holds
for s2 being State of A2 st s2 = (Following (s,(stabilization-time s1))) | the carrier of S2 & s2 is stabilizing holds
(Result s) | the carrier of S1 = Result s1 )
assume A2: ( A1 tolerates A2 & A = A1 +* A2 ) ; :: thesis: for s being State of A
for s1 being State of A1 st s1 = s | the carrier of S1 & s1 is stabilizing holds
for s2 being State of A2 st s2 = (Following (s,(stabilization-time s1))) | the carrier of S2 & s2 is stabilizing holds
(Result s) | the carrier of S1 = Result s1
let s be State of A; :: thesis: for s1 being State of A1 st s1 = s | the carrier of S1 & s1 is stabilizing holds
for s2 being State of A2 st s2 = (Following (s,(stabilization-time s1))) | the carrier of S2 & s2 is stabilizing holds
(Result s) | the carrier of S1 = Result s1
let s1 be State of A1; :: thesis: ( s1 = s | the carrier of S1 & s1 is stabilizing implies for s2 being State of A2 st s2 = (Following (s,(stabilization-time s1))) | the carrier of S2 & s2 is stabilizing holds
(Result s) | the carrier of S1 = Result s1 )
assume that
A3: s1 = s | the carrier of S1 and
A4: s1 is stabilizing ; :: thesis: for s2 being State of A2 st s2 = (Following (s,(stabilization-time s1))) | the carrier of S2 & s2 is stabilizing holds
(Result s) | the carrier of S1 = Result s1
let s2 be State of A2; :: thesis: ( s2 = (Following (s,(stabilization-time s1))) | the carrier of S2 & s2 is stabilizing implies (Result s) | the carrier of S1 = Result s1 )
assume A5: ( s2 = (Following (s,(stabilization-time s1))) | the carrier of S2 & s2 is stabilizing ) ; :: thesis: (Result s) | the carrier of S1 = Result s1
A6: stabilization-time s = (stabilization-time s1) + (stabilization-time s2) by A1, A2, A3, A4, A5, Th10;
thus (Result s) | the carrier of S1 = (Following (s,(stabilization-time s))) | the carrier of S1 by A1, A2, A3, A4, A5, Th2, Th9
.= Following (s1,(stabilization-time s)) by A1, A2, A3, CIRCCMB2:13
.= Result s1 by A4, A6, Th5, NAT_1:11 ; :: thesis: verum