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 n1, n2 being Nat
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 & Following s1,n1 is stable & s2 = (Following s,n1) | the carrier of S2 & Following s2,n2 is stable holds
Following s,(n1 + n2) is stable )
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 n1, n2 being Nat
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 & Following s1,n1 is stable & s2 = (Following s,n1) | the carrier of S2 & Following s2,n2 is stable holds
Following s,(n1 + n2) is stable
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 n1, n2 being Nat
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 & Following s1,n1 is stable & s2 = (Following s,n1) | the carrier of S2 & Following s2,n2 is stable holds
Following s,(n1 + n2) is stable
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 n1, n2 being Nat
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 & Following s1,n1 is stable & s2 = (Following s,n1) | the carrier of S2 & Following s2,n2 is stable holds
Following s,(n1 + n2) is stable
let A be non-empty Circuit of S; :: thesis: ( A1 tolerates A2 & A = A1 +* A2 implies for n1, n2 being Nat
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 & Following s1,n1 is stable & s2 = (Following s,n1) | the carrier of S2 & Following s2,n2 is stable holds
Following s,(n1 + n2) is stable )
assume A2: ( A1 tolerates A2 & A = A1 +* A2 ) ; :: thesis: for n1, n2 being Nat
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 & Following s1,n1 is stable & s2 = (Following s,n1) | the carrier of S2 & Following s2,n2 is stable holds
Following s,(n1 + n2) is stable
let n1, n2 be Nat; :: 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 & Following s1,n1 is stable & s2 = (Following s,n1) | the carrier of S2 & Following s2,n2 is stable holds
Following s,(n1 + n2) is stable
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 & Following s1,n1 is stable & s2 = (Following s,n1) | the carrier of S2 & Following s2,n2 is stable holds
Following s,(n1 + n2) is stable
let s' be State of A1; :: thesis: for s2 being State of A2 st s' = s | the carrier of S1 & Following s',n1 is stable & s2 = (Following s,n1) | the carrier of S2 & Following s2,n2 is stable holds
Following s,(n1 + n2) is stable
let s'' be State of A2; :: thesis: ( s' = s | the carrier of S1 & Following s',n1 is stable & s'' = (Following s,n1) | the carrier of S2 & Following s'',n2 is stable implies Following s,(n1 + n2) is stable )
assume that
A3: ( s' = s | the carrier of S1 & Following s',n1 is stable ) and
A4: ( s'' = (Following s,n1) | the carrier of S2 & Following s'',n2 is stable ) ; :: thesis: Following s,(n1 + n2) is stable
A5: the Sorts of A1 tolerates the Sorts of A2 by A2, CIRCCOMB:def 3;
then reconsider s1 = (Following s,n1) | the carrier of S1, s0 = s | the carrier of S1 as State of A1 by A2, CIRCCOMB:33;
reconsider s2 = (Following s,n1) | the carrier of S2 as State of A2 by A2, A5, CIRCCOMB:33;
A6: ( the carrier of S = the carrier of S1 \/ the carrier of S2 & dom (Following s,(n1 + n2)) = the carrier of S ) by A1, CIRCCOMB:def 2, CIRCUIT1:4;
A7: ( Following (Following s,n1),n2 = Following s,(n1 + n2) & Following (Following s0,n1),n2 = Following s0,(n1 + n2) ) by FACIRC_1:13;
A8: s1 = Following s0,n1 by A1, A2, Th14;
then A9: ( (Following s,(n1 + n2)) | the carrier of S1 = Following s1,n2 & (Following s,(n1 + n2)) | the carrier of S2 = Following s2,n2 ) by A1, A2, A3, A7, Th14, Th19;
then Following (Following s,(n1 + n2)) = (Following (Following s2,n2)) +* (Following (Following s1,n2)) by A1, A2, CIRCCOMB:40
.= (Following s2,n2) +* (Following (Following s1,n2)) by A4, CIRCUIT2:def 6
.= (Following s2,n2) +* (Following s1,(n2 + 1)) by FACIRC_1:12
.= (Following s2,n2) +* s1 by A3, A8, Th3
.= (Following s2,n2) +* (Following s1,n2) by A3, A8, Th3
.= Following s,(n1 + n2) by A6, A9, FUNCT_4:74 ;
hence Following s,(n1 + n2) is stable by CIRCUIT2:def 6; :: thesis: verum