let S1, S2, S be non empty non void Circuit-like ManySortedSign ; :: thesis: ( InputVertices S1 misses InnerVertices S2 & InputVertices S2 misses InnerVertices S1 & 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 n being Nat
for s being State of A
for s1 being State of A1 st s1 = s | the carrier of S1 holds
for s2 being State of A2 st s2 = s | the carrier of S2 & ( not Following (s1,n) is stable or not Following (s2,n) is stable ) holds
not Following (s,n) is stable )
assume that
A1: InputVertices S1 misses InnerVertices S2 and
A2: InputVertices S2 misses InnerVertices S1 and
A3: 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 n being Nat
for s being State of A
for s1 being State of A1 st s1 = s | the carrier of S1 holds
for s2 being State of A2 st s2 = s | the carrier of S2 & ( not Following (s1,n) is stable or not Following (s2,n) is stable ) holds
not Following (s,n) 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 n being Nat
for s being State of A
for s1 being State of A1 st s1 = s | the carrier of S1 holds
for s2 being State of A2 st s2 = s | the carrier of S2 & ( not Following (s1,n) is stable or not Following (s2,n) is stable ) holds
not Following (s,n) 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 n being Nat
for s being State of A
for s1 being State of A1 st s1 = s | the carrier of S1 holds
for s2 being State of A2 st s2 = s | the carrier of S2 & ( not Following (s1,n) is stable or not Following (s2,n) is stable ) holds
not Following (s,n) is stable
let A be non-empty Circuit of S; :: thesis: ( A1 tolerates A2 & A = A1 +* A2 implies for n being Nat
for s being State of A
for s1 being State of A1 st s1 = s | the carrier of S1 holds
for s2 being State of A2 st s2 = s | the carrier of S2 & ( not Following (s1,n) is stable or not Following (s2,n) is stable ) holds
not Following (s,n) is stable )
assume A4: ( A1 tolerates A2 & A = A1 +* A2 ) ; :: thesis: for n being Nat
for s being State of A
for s1 being State of A1 st s1 = s | the carrier of S1 holds
for s2 being State of A2 st s2 = s | the carrier of S2 & ( not Following (s1,n) is stable or not Following (s2,n) is stable ) holds
not Following (s,n) is stable
let n be Nat; :: thesis: for s being State of A
for s1 being State of A1 st s1 = s | the carrier of S1 holds
for s2 being State of A2 st s2 = s | the carrier of S2 & ( not Following (s1,n) is stable or not Following (s2,n) is stable ) holds
not Following (s,n) is stable
let s be State of A; :: thesis: for s1 being State of A1 st s1 = s | the carrier of S1 holds
for s2 being State of A2 st s2 = s | the carrier of S2 & ( not Following (s1,n) is stable or not Following (s2,n) is stable ) holds
not Following (s,n) is stable
let s0 be State of A1; :: thesis: ( s0 = s | the carrier of S1 implies for s2 being State of A2 st s2 = s | the carrier of S2 & ( not Following (s0,n) is stable or not Following (s2,n) is stable ) holds
not Following (s,n) is stable )
assume s0 = s | the carrier of S1 ; :: thesis: for s2 being State of A2 st s2 = s | the carrier of S2 & ( not Following (s0,n) is stable or not Following (s2,n) is stable ) holds
not Following (s,n) is stable
then A5: (Following (s,n)) | the carrier of S1 = Following (s0,n) by A1, A3, A4, Th13;
let s3 be State of A2; :: thesis: ( s3 = s | the carrier of S2 & ( not Following (s0,n) is stable or not Following (s3,n) is stable ) implies not Following (s,n) is stable )
assume that
A6: s3 = s | the carrier of S2 and
A7: ( not Following (s0,n) is stable or not Following (s3,n) is stable ) ; :: thesis: not Following (s,n) is stable
(Following (s,n)) | the carrier of S2 = Following (s3,n) by A2, A3, A4, A6, Th14;
hence not Following (s,n) is stable by A3, A4, A7, A5, Th17; :: thesis: verum