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 s1 being State of A1
for s2 being State of A2
for s being State of A st s1 = s | the carrier of S1 & s2 = s | the carrier of S2 & s1 is stable holds
for n being Nat holds (Following s,n) | the carrier of S2 = Following s2,n )
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 s1 being State of A1
for s2 being State of A2
for s being State of A st s1 = s | the carrier of S1 & s2 = s | the carrier of S2 & s1 is stable holds
for n being Nat holds (Following s,n) | the carrier of S2 = Following s2,n
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 s1 being State of A1
for s2 being State of A2
for s being State of A st s1 = s | the carrier of S1 & s2 = s | the carrier of S2 & s1 is stable holds
for n being Nat holds (Following s,n) | the carrier of S2 = Following s2,n
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 s1 being State of A1
for s2 being State of A2
for s being State of A st s1 = s | the carrier of S1 & s2 = s | the carrier of S2 & s1 is stable holds
for n being Nat holds (Following s,n) | the carrier of S2 = Following s2,n
let A be non-empty Circuit of S; :: thesis: ( A1 tolerates A2 & A = A1 +* A2 implies for s1 being State of A1
for s2 being State of A2
for s being State of A st s1 = s | the carrier of S1 & s2 = s | the carrier of S2 & s1 is stable holds
for n being Nat holds (Following s,n) | the carrier of S2 = Following s2,n )
assume that
A2: A1 tolerates A2 and
A3: A = A1 +* A2 ; :: thesis: for s1 being State of A1
for s2 being State of A2
for s being State of A st s1 = s | the carrier of S1 & s2 = s | the carrier of S2 & s1 is stable holds
for n being Nat holds (Following s,n) | the carrier of S2 = Following s2,n
let s1 be State of A1; :: thesis: for s2 being State of A2
for s being State of A st s1 = s | the carrier of S1 & s2 = s | the carrier of S2 & s1 is stable holds
for n being Nat holds (Following s,n) | the carrier of S2 = Following s2,n
let s2 be State of A2; :: thesis: for s being State of A st s1 = s | the carrier of S1 & s2 = s | the carrier of S2 & s1 is stable holds
for n being Nat holds (Following s,n) | the carrier of S2 = Following s2,n
let s be State of A; :: thesis: ( s1 = s | the carrier of S1 & s2 = s | the carrier of S2 & s1 is stable implies for n being Nat holds (Following s,n) | the carrier of S2 = Following s2,n )
assume that
A4: s1 = s | the carrier of S1 and
A5: s2 = s | the carrier of S2 and
A6: s1 is stable ; :: thesis: for n being Nat holds (Following s,n) | the carrier of S2 = Following s2,n
defpred S3[ Nat] means (Following s,$1) | the carrier of S2 = Following s2,$1;
A7: now
let n be Nat; :: thesis: ( S3[n] implies S3[n + 1] )
A8: ( Following s,(n + 1) = Following (Following s,n) & Following (Following s2,n) = Following s2,(n + 1) ) by FACIRC_1:12;
the Sorts of A1 tolerates the Sorts of A2 by A2, CIRCCOMB:def 3;
then reconsider Fs1 = (Following s,n) | the carrier of S1 as State of A1 by A3, CIRCCOMB:33;
Following s1,n = Fs1 by A1, A2, A3, A4, Th14;
then A9: Fs1 is stable by A6, Th3;
assume S3[n] ; :: thesis: S3[n + 1]
hence S3[n + 1] by A1, A2, A3, A8, A9, Th16; :: thesis: verum
end;
(Following s,0 ) | the carrier of S2 = s2 by A5, FACIRC_1:11
.= Following s2,0 by FACIRC_1:11 ;
then A10: S3[ 0 ] ;
thus for n being Nat holds S3[n] from NAT_1:sch 2(A10, A7); :: thesis: verum