let S1, S2 be non empty non void Circuit-like ManySortedSign ; :: thesis: ( InputVertices S1 misses InnerVertices S2 implies for S being non empty non void Circuit-like ManySortedSign st S = S1 +* S2 holds
for A1 being non-empty Circuit of S1
for A2 being non-empty Circuit of S2 st A1 tolerates A2 holds
for A being non-empty Circuit of S st 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
s is stabilizing )
assume A1: InputVertices S1 misses InnerVertices S2 ; :: thesis: for S being non empty non void Circuit-like ManySortedSign st S = S1 +* S2 holds
for A1 being non-empty Circuit of S1
for A2 being non-empty Circuit of S2 st A1 tolerates A2 holds
for A being non-empty Circuit of S st 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
s is stabilizing
let S be non empty non void Circuit-like ManySortedSign ; :: thesis: ( S = S1 +* S2 implies for A1 being non-empty Circuit of S1
for A2 being non-empty Circuit of S2 st A1 tolerates A2 holds
for A being non-empty Circuit of S st 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
s is stabilizing )
assume A2: S = S1 +* S2 ; :: thesis: for A1 being non-empty Circuit of S1
for A2 being non-empty Circuit of S2 st A1 tolerates A2 holds
for A being non-empty Circuit of S st 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
s is stabilizing
let A1 be non-empty Circuit of S1; :: thesis: for A2 being non-empty Circuit of S2 st A1 tolerates A2 holds
for A being non-empty Circuit of S st 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
s is stabilizing
let A2 be non-empty Circuit of S2; :: thesis: ( A1 tolerates A2 implies for A being non-empty Circuit of S st 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
s is stabilizing )
assume A3: A1 tolerates A2 ; :: thesis: for A being non-empty Circuit of S st 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
s is stabilizing
let A be non-empty Circuit of S; :: thesis: ( 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
s is stabilizing )
assume A4: 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
s is stabilizing
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
s is stabilizing
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
s is stabilizing )
assume that
A5: s1 = s | the carrier of S1 and
A6: 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
s is stabilizing
set n1 = stabilization-time s1;
A7: Following (s1,(stabilization-time s1)) is stable by A6, Def5;
let s2 be State of A2; :: thesis: ( s2 = (Following (s,(stabilization-time s1))) | the carrier of S2 & s2 is stabilizing implies s is stabilizing )
assume that
A8: s2 = (Following (s,(stabilization-time s1))) | the carrier of S2 and
A9: s2 is stabilizing ; :: thesis: s is stabilizing
set n2 = stabilization-time s2;
Following (s2,(stabilization-time s2)) is stable by A9, Def5;
then Following (s,((stabilization-time s1) + (stabilization-time s2))) is stable by A1, A2, A3, A4, A5, A7, A8, CIRCCMB2:19;
hence s is stabilizing ; :: thesis: verum