let S1, S2 be non empty non void Circuit-like ManySortedSign ; :: thesis: ( InputVertices S1 misses InnerVertices S2 & InputVertices S2 misses InnerVertices S1 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
for s2 being State of A2 st s1 = s | the carrier of S1 & s2 = s | the carrier of S2 & s1 is stabilizing & s2 is stabilizing holds
s is stabilizing )
assume A1: ( InputVertices S1 misses InnerVertices S2 & InputVertices S2 misses InnerVertices S1 ) ; :: 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
for s2 being State of A2 st s1 = s | the carrier of S1 & s2 = s | the carrier of S2 & s1 is stabilizing & 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
for s2 being State of A2 st s1 = s | the carrier of S1 & s2 = s | the carrier of S2 & s1 is stabilizing & 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
for s2 being State of A2 st s1 = s | the carrier of S1 & s2 = s | the carrier of S2 & s1 is stabilizing & 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
for s2 being State of A2 st s1 = s | the carrier of S1 & s2 = s | the carrier of S2 & s1 is stabilizing & 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
for s2 being State of A2 st s1 = s | the carrier of S1 & s2 = s | the carrier of S2 & s1 is stabilizing & 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
for s2 being State of A2 st s1 = s | the carrier of S1 & s2 = s | the carrier of S2 & s1 is stabilizing & 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
for s2 being State of A2 st s1 = s | the carrier of S1 & s2 = s | the carrier of S2 & s1 is stabilizing & 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
for s2 being State of A2 st s1 = s | the carrier of S1 & s2 = s | the carrier of S2 & s1 is stabilizing & s2 is stabilizing holds
s is stabilizing
let s be State of A; :: thesis: for s1 being State of A1
for s2 being State of A2 st s1 = s | the carrier of S1 & s2 = s | the carrier of S2 & s1 is stabilizing & s2 is stabilizing holds
s is stabilizing
let s1 be State of A1; :: thesis: for s2 being State of A2 st s1 = s | the carrier of S1 & s2 = s | the carrier of S2 & s1 is stabilizing & s2 is stabilizing holds
s is stabilizing
let s2 be State of A2; :: thesis: ( s1 = s | the carrier of S1 & s2 = s | the carrier of S2 & s1 is stabilizing & s2 is stabilizing implies s is stabilizing )
assume A5: ( s1 = s | the carrier of S1 & s2 = s | the carrier of S2 & s1 is stabilizing & s2 is stabilizing ) ; :: thesis: s is stabilizing
then consider n1 being Element of NAT such that
A6: Following s1,n1 is stable by Def1;
consider n2 being Element of NAT such that
A7: Following s2,n2 is stable by A5, Def1;
Following s,(max n1,n2) is stable by A1, A2, A3, A4, A5, A6, A7, CIRCCMB2:23;
hence s is stabilizing by Def1; :: thesis: verum