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 that
A1: InputVertices S1 misses InnerVertices S2 and
A2: 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 that
A3: A1 tolerates A2 and
A4: 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 s9 be State of A1; :: thesis: for s2 being State of A2 st s9 = s | the carrier of S1 & Following s9,n1 is stable & s2 = (Following s,n1) | the carrier of S2 & Following s2,n2 is stable holds
Following s,(n1 + n2) is stable
let s99 be State of A2; :: thesis: ( s9 = s | the carrier of S1 & Following s9,n1 is stable & s99 = (Following s,n1) | the carrier of S2 & Following s99,n2 is stable implies Following s,(n1 + n2) is stable )
assume that
A5: ( s9 = s | the carrier of S1 & Following s9,n1 is stable ) and
A6: ( s99 = (Following s,n1) | the carrier of S2 & Following s99,n2 is stable ) ; :: thesis: Following s,(n1 + n2) is stable
A7: the Sorts of A1 tolerates the Sorts of A2 by A3, CIRCCOMB:def 3;
then reconsider s1 = (Following s,n1) | the carrier of S1, s0 = s | the carrier of S1 as State of A1 by A4, CIRCCOMB:33;
A8: Following (Following s,n1),n2 = Following s,(n1 + n2) by FACIRC_1:13;
then A9: (Following s,(n1 + n2)) | the carrier of S1 = Following s1,n2 by A1, A2, A3, A4, Th14;
reconsider s2 = (Following s,n1) | the carrier of S2 as State of A2 by A4, A7, CIRCCOMB:33;
A10: dom (Following s,(n1 + n2)) = the carrier of S by CIRCUIT1:4;
A11: the carrier of S = the carrier of S1 \/ the carrier of S2 by A2, CIRCCOMB:def 2;
A12: s1 = Following s0,n1 by A1, A2, A3, A4, Th14;
then A13: (Following s,(n1 + n2)) | the carrier of S2 = Following s2,n2 by A1, A2, A3, A4, A5, A8, Th19;
then Following (Following s,(n1 + n2)) = (Following (Following s2,n2)) +* (Following (Following s1,n2)) by A1, A2, A3, A4, A9, CIRCCOMB:40
.= (Following s2,n2) +* (Following (Following s1,n2)) by A6, CIRCUIT2:def 6
.= (Following s2,n2) +* (Following s1,(n2 + 1)) by FACIRC_1:12
.= (Following s2,n2) +* s1 by A5, A12, Th3
.= (Following s2,n2) +* (Following s1,n2) by A5, A12, Th3
.= Following s,(n1 + n2) by A11, A10, A9, A13, FUNCT_4:74 ;
hence Following s,(n1 + n2) is stable by CIRCUIT2:def 6; :: thesis: verum