Termination w.r.t. Q of the following Term Rewriting System could be proven:
Q restricted rewrite system:
The TRS R consists of the following rules:
start(i) → busy(F, closed, stop, false, false, false, i)
busy(BF, d, stop, b1, b2, b3, i) → incorrect
busy(FS, d, stop, b1, b2, b3, i) → incorrect
busy(fl, open, up, b1, b2, b3, i) → incorrect
busy(fl, open, down, b1, b2, b3, i) → incorrect
busy(B, closed, stop, false, false, false, empty) → correct
busy(F, closed, stop, false, false, false, empty) → correct
busy(S, closed, stop, false, false, false, empty) → correct
busy(B, closed, stop, false, false, false, newbuttons(i1, i2, i3, i)) → idle(B, closed, stop, false, false, false, newbuttons(i1, i2, i3, i))
busy(F, closed, stop, false, false, false, newbuttons(i1, i2, i3, i)) → idle(F, closed, stop, false, false, false, newbuttons(i1, i2, i3, i))
busy(S, closed, stop, false, false, false, newbuttons(i1, i2, i3, i)) → idle(S, closed, stop, false, false, false, newbuttons(i1, i2, i3, i))
busy(B, open, stop, false, b2, b3, i) → idle(B, closed, stop, false, b2, b3, i)
busy(F, open, stop, b1, false, b3, i) → idle(F, closed, stop, b1, false, b3, i)
busy(S, open, stop, b1, b2, false, i) → idle(S, closed, stop, b1, b2, false, i)
busy(B, d, stop, true, b2, b3, i) → idle(B, open, stop, false, b2, b3, i)
busy(F, d, stop, b1, true, b3, i) → idle(F, open, stop, b1, false, b3, i)
busy(S, d, stop, b1, b2, true, i) → idle(S, open, stop, b1, b2, false, i)
busy(B, closed, down, b1, b2, b3, i) → idle(B, closed, stop, b1, b2, b3, i)
busy(S, closed, up, b1, b2, b3, i) → idle(S, closed, stop, b1, b2, b3, i)
busy(B, closed, up, true, b2, b3, i) → idle(B, closed, stop, true, b2, b3, i)
busy(F, closed, up, b1, true, b3, i) → idle(F, closed, stop, b1, true, b3, i)
busy(F, closed, down, b1, true, b3, i) → idle(F, closed, stop, b1, true, b3, i)
busy(S, closed, down, b1, b2, true, i) → idle(S, closed, stop, b1, b2, true, i)
busy(B, closed, up, false, b2, b3, i) → idle(BF, closed, up, false, b2, b3, i)
busy(F, closed, up, b1, false, b3, i) → idle(FS, closed, up, b1, false, b3, i)
busy(F, closed, down, b1, false, b3, i) → idle(BF, closed, down, b1, false, b3, i)
busy(S, closed, down, b1, b2, false, i) → idle(FS, closed, down, b1, b2, false, i)
busy(BF, closed, up, b1, b2, b3, i) → idle(F, closed, up, b1, b2, b3, i)
busy(BF, closed, down, b1, b2, b3, i) → idle(B, closed, down, b1, b2, b3, i)
busy(FS, closed, up, b1, b2, b3, i) → idle(S, closed, up, b1, b2, b3, i)
busy(FS, closed, down, b1, b2, b3, i) → idle(F, closed, down, b1, b2, b3, i)
busy(B, closed, stop, false, true, b3, i) → idle(B, closed, up, false, true, b3, i)
busy(B, closed, stop, false, false, true, i) → idle(B, closed, up, false, false, true, i)
busy(F, closed, stop, true, false, b3, i) → idle(F, closed, down, true, false, b3, i)
busy(F, closed, stop, false, false, true, i) → idle(F, closed, up, false, false, true, i)
busy(S, closed, stop, b1, true, false, i) → idle(S, closed, down, b1, true, false, i)
busy(S, closed, stop, true, false, false, i) → idle(S, closed, down, true, false, false, i)
idle(fl, d, m, b1, b2, b3, empty) → busy(fl, d, m, b1, b2, b3, empty)
idle(fl, d, m, b1, b2, b3, newbuttons(i1, i2, i3, i)) → busy(fl, d, m, or(b1, i1), or(b2, i2), or(b3, i3), i)
or(true, b) → true
or(false, b) → b
Q is empty.
↳ QTRS
↳ RRRPoloQTRSProof
Q restricted rewrite system:
The TRS R consists of the following rules:
start(i) → busy(F, closed, stop, false, false, false, i)
busy(BF, d, stop, b1, b2, b3, i) → incorrect
busy(FS, d, stop, b1, b2, b3, i) → incorrect
busy(fl, open, up, b1, b2, b3, i) → incorrect
busy(fl, open, down, b1, b2, b3, i) → incorrect
busy(B, closed, stop, false, false, false, empty) → correct
busy(F, closed, stop, false, false, false, empty) → correct
busy(S, closed, stop, false, false, false, empty) → correct
busy(B, closed, stop, false, false, false, newbuttons(i1, i2, i3, i)) → idle(B, closed, stop, false, false, false, newbuttons(i1, i2, i3, i))
busy(F, closed, stop, false, false, false, newbuttons(i1, i2, i3, i)) → idle(F, closed, stop, false, false, false, newbuttons(i1, i2, i3, i))
busy(S, closed, stop, false, false, false, newbuttons(i1, i2, i3, i)) → idle(S, closed, stop, false, false, false, newbuttons(i1, i2, i3, i))
busy(B, open, stop, false, b2, b3, i) → idle(B, closed, stop, false, b2, b3, i)
busy(F, open, stop, b1, false, b3, i) → idle(F, closed, stop, b1, false, b3, i)
busy(S, open, stop, b1, b2, false, i) → idle(S, closed, stop, b1, b2, false, i)
busy(B, d, stop, true, b2, b3, i) → idle(B, open, stop, false, b2, b3, i)
busy(F, d, stop, b1, true, b3, i) → idle(F, open, stop, b1, false, b3, i)
busy(S, d, stop, b1, b2, true, i) → idle(S, open, stop, b1, b2, false, i)
busy(B, closed, down, b1, b2, b3, i) → idle(B, closed, stop, b1, b2, b3, i)
busy(S, closed, up, b1, b2, b3, i) → idle(S, closed, stop, b1, b2, b3, i)
busy(B, closed, up, true, b2, b3, i) → idle(B, closed, stop, true, b2, b3, i)
busy(F, closed, up, b1, true, b3, i) → idle(F, closed, stop, b1, true, b3, i)
busy(F, closed, down, b1, true, b3, i) → idle(F, closed, stop, b1, true, b3, i)
busy(S, closed, down, b1, b2, true, i) → idle(S, closed, stop, b1, b2, true, i)
busy(B, closed, up, false, b2, b3, i) → idle(BF, closed, up, false, b2, b3, i)
busy(F, closed, up, b1, false, b3, i) → idle(FS, closed, up, b1, false, b3, i)
busy(F, closed, down, b1, false, b3, i) → idle(BF, closed, down, b1, false, b3, i)
busy(S, closed, down, b1, b2, false, i) → idle(FS, closed, down, b1, b2, false, i)
busy(BF, closed, up, b1, b2, b3, i) → idle(F, closed, up, b1, b2, b3, i)
busy(BF, closed, down, b1, b2, b3, i) → idle(B, closed, down, b1, b2, b3, i)
busy(FS, closed, up, b1, b2, b3, i) → idle(S, closed, up, b1, b2, b3, i)
busy(FS, closed, down, b1, b2, b3, i) → idle(F, closed, down, b1, b2, b3, i)
busy(B, closed, stop, false, true, b3, i) → idle(B, closed, up, false, true, b3, i)
busy(B, closed, stop, false, false, true, i) → idle(B, closed, up, false, false, true, i)
busy(F, closed, stop, true, false, b3, i) → idle(F, closed, down, true, false, b3, i)
busy(F, closed, stop, false, false, true, i) → idle(F, closed, up, false, false, true, i)
busy(S, closed, stop, b1, true, false, i) → idle(S, closed, down, b1, true, false, i)
busy(S, closed, stop, true, false, false, i) → idle(S, closed, down, true, false, false, i)
idle(fl, d, m, b1, b2, b3, empty) → busy(fl, d, m, b1, b2, b3, empty)
idle(fl, d, m, b1, b2, b3, newbuttons(i1, i2, i3, i)) → busy(fl, d, m, or(b1, i1), or(b2, i2), or(b3, i3), i)
or(true, b) → true
or(false, b) → b
Q is empty.
The following Q TRS is given: Q restricted rewrite system:
The TRS R consists of the following rules:
start(i) → busy(F, closed, stop, false, false, false, i)
busy(BF, d, stop, b1, b2, b3, i) → incorrect
busy(FS, d, stop, b1, b2, b3, i) → incorrect
busy(fl, open, up, b1, b2, b3, i) → incorrect
busy(fl, open, down, b1, b2, b3, i) → incorrect
busy(B, closed, stop, false, false, false, empty) → correct
busy(F, closed, stop, false, false, false, empty) → correct
busy(S, closed, stop, false, false, false, empty) → correct
busy(B, closed, stop, false, false, false, newbuttons(i1, i2, i3, i)) → idle(B, closed, stop, false, false, false, newbuttons(i1, i2, i3, i))
busy(F, closed, stop, false, false, false, newbuttons(i1, i2, i3, i)) → idle(F, closed, stop, false, false, false, newbuttons(i1, i2, i3, i))
busy(S, closed, stop, false, false, false, newbuttons(i1, i2, i3, i)) → idle(S, closed, stop, false, false, false, newbuttons(i1, i2, i3, i))
busy(B, open, stop, false, b2, b3, i) → idle(B, closed, stop, false, b2, b3, i)
busy(F, open, stop, b1, false, b3, i) → idle(F, closed, stop, b1, false, b3, i)
busy(S, open, stop, b1, b2, false, i) → idle(S, closed, stop, b1, b2, false, i)
busy(B, d, stop, true, b2, b3, i) → idle(B, open, stop, false, b2, b3, i)
busy(F, d, stop, b1, true, b3, i) → idle(F, open, stop, b1, false, b3, i)
busy(S, d, stop, b1, b2, true, i) → idle(S, open, stop, b1, b2, false, i)
busy(B, closed, down, b1, b2, b3, i) → idle(B, closed, stop, b1, b2, b3, i)
busy(S, closed, up, b1, b2, b3, i) → idle(S, closed, stop, b1, b2, b3, i)
busy(B, closed, up, true, b2, b3, i) → idle(B, closed, stop, true, b2, b3, i)
busy(F, closed, up, b1, true, b3, i) → idle(F, closed, stop, b1, true, b3, i)
busy(F, closed, down, b1, true, b3, i) → idle(F, closed, stop, b1, true, b3, i)
busy(S, closed, down, b1, b2, true, i) → idle(S, closed, stop, b1, b2, true, i)
busy(B, closed, up, false, b2, b3, i) → idle(BF, closed, up, false, b2, b3, i)
busy(F, closed, up, b1, false, b3, i) → idle(FS, closed, up, b1, false, b3, i)
busy(F, closed, down, b1, false, b3, i) → idle(BF, closed, down, b1, false, b3, i)
busy(S, closed, down, b1, b2, false, i) → idle(FS, closed, down, b1, b2, false, i)
busy(BF, closed, up, b1, b2, b3, i) → idle(F, closed, up, b1, b2, b3, i)
busy(BF, closed, down, b1, b2, b3, i) → idle(B, closed, down, b1, b2, b3, i)
busy(FS, closed, up, b1, b2, b3, i) → idle(S, closed, up, b1, b2, b3, i)
busy(FS, closed, down, b1, b2, b3, i) → idle(F, closed, down, b1, b2, b3, i)
busy(B, closed, stop, false, true, b3, i) → idle(B, closed, up, false, true, b3, i)
busy(B, closed, stop, false, false, true, i) → idle(B, closed, up, false, false, true, i)
busy(F, closed, stop, true, false, b3, i) → idle(F, closed, down, true, false, b3, i)
busy(F, closed, stop, false, false, true, i) → idle(F, closed, up, false, false, true, i)
busy(S, closed, stop, b1, true, false, i) → idle(S, closed, down, b1, true, false, i)
busy(S, closed, stop, true, false, false, i) → idle(S, closed, down, true, false, false, i)
idle(fl, d, m, b1, b2, b3, empty) → busy(fl, d, m, b1, b2, b3, empty)
idle(fl, d, m, b1, b2, b3, newbuttons(i1, i2, i3, i)) → busy(fl, d, m, or(b1, i1), or(b2, i2), or(b3, i3), i)
or(true, b) → true
or(false, b) → b
Q is empty.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:
start(i) → busy(F, closed, stop, false, false, false, i)
busy(fl, open, up, b1, b2, b3, i) → incorrect
busy(fl, open, down, b1, b2, b3, i) → incorrect
busy(B, closed, stop, false, false, false, newbuttons(i1, i2, i3, i)) → idle(B, closed, stop, false, false, false, newbuttons(i1, i2, i3, i))
busy(F, closed, stop, false, false, false, newbuttons(i1, i2, i3, i)) → idle(F, closed, stop, false, false, false, newbuttons(i1, i2, i3, i))
busy(S, closed, stop, false, false, false, newbuttons(i1, i2, i3, i)) → idle(S, closed, stop, false, false, false, newbuttons(i1, i2, i3, i))
busy(B, open, stop, false, b2, b3, i) → idle(B, closed, stop, false, b2, b3, i)
busy(F, open, stop, b1, false, b3, i) → idle(F, closed, stop, b1, false, b3, i)
busy(S, open, stop, b1, b2, false, i) → idle(S, closed, stop, b1, b2, false, i)
idle(fl, d, m, b1, b2, b3, newbuttons(i1, i2, i3, i)) → busy(fl, d, m, or(b1, i1), or(b2, i2), or(b3, i3), i)
Used ordering:
Polynomial interpretation [25]:
POL(B) = 0
POL(BF) = 0
POL(F) = 0
POL(FS) = 0
POL(S) = 0
POL(busy(x1, x2, x3, x4, x5, x6, x7)) = 2·x1 + x2 + 2·x3 + 2·x4 + 2·x5 + 2·x6 + 2·x7
POL(closed) = 0
POL(correct) = 0
POL(down) = 0
POL(empty) = 0
POL(false) = 0
POL(idle(x1, x2, x3, x4, x5, x6, x7)) = 2·x1 + x2 + 2·x3 + 2·x4 + 2·x5 + 2·x6 + x7
POL(incorrect) = 0
POL(newbuttons(x1, x2, x3, x4)) = 1 + 2·x1 + 2·x2 + 2·x3 + 2·x4
POL(open) = 2
POL(or(x1, x2)) = x1 + x2
POL(start(x1)) = 1 + 2·x1
POL(stop) = 0
POL(true) = 1
POL(up) = 0
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
Q restricted rewrite system:
The TRS R consists of the following rules:
busy(BF, d, stop, b1, b2, b3, i) → incorrect
busy(FS, d, stop, b1, b2, b3, i) → incorrect
busy(B, closed, stop, false, false, false, empty) → correct
busy(F, closed, stop, false, false, false, empty) → correct
busy(S, closed, stop, false, false, false, empty) → correct
busy(B, d, stop, true, b2, b3, i) → idle(B, open, stop, false, b2, b3, i)
busy(F, d, stop, b1, true, b3, i) → idle(F, open, stop, b1, false, b3, i)
busy(S, d, stop, b1, b2, true, i) → idle(S, open, stop, b1, b2, false, i)
busy(B, closed, down, b1, b2, b3, i) → idle(B, closed, stop, b1, b2, b3, i)
busy(S, closed, up, b1, b2, b3, i) → idle(S, closed, stop, b1, b2, b3, i)
busy(B, closed, up, true, b2, b3, i) → idle(B, closed, stop, true, b2, b3, i)
busy(F, closed, up, b1, true, b3, i) → idle(F, closed, stop, b1, true, b3, i)
busy(F, closed, down, b1, true, b3, i) → idle(F, closed, stop, b1, true, b3, i)
busy(S, closed, down, b1, b2, true, i) → idle(S, closed, stop, b1, b2, true, i)
busy(B, closed, up, false, b2, b3, i) → idle(BF, closed, up, false, b2, b3, i)
busy(F, closed, up, b1, false, b3, i) → idle(FS, closed, up, b1, false, b3, i)
busy(F, closed, down, b1, false, b3, i) → idle(BF, closed, down, b1, false, b3, i)
busy(S, closed, down, b1, b2, false, i) → idle(FS, closed, down, b1, b2, false, i)
busy(BF, closed, up, b1, b2, b3, i) → idle(F, closed, up, b1, b2, b3, i)
busy(BF, closed, down, b1, b2, b3, i) → idle(B, closed, down, b1, b2, b3, i)
busy(FS, closed, up, b1, b2, b3, i) → idle(S, closed, up, b1, b2, b3, i)
busy(FS, closed, down, b1, b2, b3, i) → idle(F, closed, down, b1, b2, b3, i)
busy(B, closed, stop, false, true, b3, i) → idle(B, closed, up, false, true, b3, i)
busy(B, closed, stop, false, false, true, i) → idle(B, closed, up, false, false, true, i)
busy(F, closed, stop, true, false, b3, i) → idle(F, closed, down, true, false, b3, i)
busy(F, closed, stop, false, false, true, i) → idle(F, closed, up, false, false, true, i)
busy(S, closed, stop, b1, true, false, i) → idle(S, closed, down, b1, true, false, i)
busy(S, closed, stop, true, false, false, i) → idle(S, closed, down, true, false, false, i)
idle(fl, d, m, b1, b2, b3, empty) → busy(fl, d, m, b1, b2, b3, empty)
or(true, b) → true
or(false, b) → b
Q is empty.
The following Q TRS is given: Q restricted rewrite system:
The TRS R consists of the following rules:
busy(BF, d, stop, b1, b2, b3, i) → incorrect
busy(FS, d, stop, b1, b2, b3, i) → incorrect
busy(B, closed, stop, false, false, false, empty) → correct
busy(F, closed, stop, false, false, false, empty) → correct
busy(S, closed, stop, false, false, false, empty) → correct
busy(B, d, stop, true, b2, b3, i) → idle(B, open, stop, false, b2, b3, i)
busy(F, d, stop, b1, true, b3, i) → idle(F, open, stop, b1, false, b3, i)
busy(S, d, stop, b1, b2, true, i) → idle(S, open, stop, b1, b2, false, i)
busy(B, closed, down, b1, b2, b3, i) → idle(B, closed, stop, b1, b2, b3, i)
busy(S, closed, up, b1, b2, b3, i) → idle(S, closed, stop, b1, b2, b3, i)
busy(B, closed, up, true, b2, b3, i) → idle(B, closed, stop, true, b2, b3, i)
busy(F, closed, up, b1, true, b3, i) → idle(F, closed, stop, b1, true, b3, i)
busy(F, closed, down, b1, true, b3, i) → idle(F, closed, stop, b1, true, b3, i)
busy(S, closed, down, b1, b2, true, i) → idle(S, closed, stop, b1, b2, true, i)
busy(B, closed, up, false, b2, b3, i) → idle(BF, closed, up, false, b2, b3, i)
busy(F, closed, up, b1, false, b3, i) → idle(FS, closed, up, b1, false, b3, i)
busy(F, closed, down, b1, false, b3, i) → idle(BF, closed, down, b1, false, b3, i)
busy(S, closed, down, b1, b2, false, i) → idle(FS, closed, down, b1, b2, false, i)
busy(BF, closed, up, b1, b2, b3, i) → idle(F, closed, up, b1, b2, b3, i)
busy(BF, closed, down, b1, b2, b3, i) → idle(B, closed, down, b1, b2, b3, i)
busy(FS, closed, up, b1, b2, b3, i) → idle(S, closed, up, b1, b2, b3, i)
busy(FS, closed, down, b1, b2, b3, i) → idle(F, closed, down, b1, b2, b3, i)
busy(B, closed, stop, false, true, b3, i) → idle(B, closed, up, false, true, b3, i)
busy(B, closed, stop, false, false, true, i) → idle(B, closed, up, false, false, true, i)
busy(F, closed, stop, true, false, b3, i) → idle(F, closed, down, true, false, b3, i)
busy(F, closed, stop, false, false, true, i) → idle(F, closed, up, false, false, true, i)
busy(S, closed, stop, b1, true, false, i) → idle(S, closed, down, b1, true, false, i)
busy(S, closed, stop, true, false, false, i) → idle(S, closed, down, true, false, false, i)
idle(fl, d, m, b1, b2, b3, empty) → busy(fl, d, m, b1, b2, b3, empty)
or(true, b) → true
or(false, b) → b
Q is empty.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:
or(true, b) → true
or(false, b) → b
Used ordering:
Polynomial interpretation [25]:
POL(B) = 0
POL(BF) = 0
POL(F) = 0
POL(FS) = 0
POL(S) = 0
POL(busy(x1, x2, x3, x4, x5, x6, x7)) = 2·x1 + 2·x2 + x3 + x4 + 2·x5 + 2·x6 + x7
POL(closed) = 0
POL(correct) = 0
POL(down) = 0
POL(empty) = 0
POL(false) = 0
POL(idle(x1, x2, x3, x4, x5, x6, x7)) = 2·x1 + 2·x2 + x3 + x4 + 2·x5 + 2·x6 + x7
POL(incorrect) = 0
POL(open) = 0
POL(or(x1, x2)) = 1 + x1 + x2
POL(stop) = 0
POL(true) = 0
POL(up) = 0
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
Q restricted rewrite system:
The TRS R consists of the following rules:
busy(BF, d, stop, b1, b2, b3, i) → incorrect
busy(FS, d, stop, b1, b2, b3, i) → incorrect
busy(B, closed, stop, false, false, false, empty) → correct
busy(F, closed, stop, false, false, false, empty) → correct
busy(S, closed, stop, false, false, false, empty) → correct
busy(B, d, stop, true, b2, b3, i) → idle(B, open, stop, false, b2, b3, i)
busy(F, d, stop, b1, true, b3, i) → idle(F, open, stop, b1, false, b3, i)
busy(S, d, stop, b1, b2, true, i) → idle(S, open, stop, b1, b2, false, i)
busy(B, closed, down, b1, b2, b3, i) → idle(B, closed, stop, b1, b2, b3, i)
busy(S, closed, up, b1, b2, b3, i) → idle(S, closed, stop, b1, b2, b3, i)
busy(B, closed, up, true, b2, b3, i) → idle(B, closed, stop, true, b2, b3, i)
busy(F, closed, up, b1, true, b3, i) → idle(F, closed, stop, b1, true, b3, i)
busy(F, closed, down, b1, true, b3, i) → idle(F, closed, stop, b1, true, b3, i)
busy(S, closed, down, b1, b2, true, i) → idle(S, closed, stop, b1, b2, true, i)
busy(B, closed, up, false, b2, b3, i) → idle(BF, closed, up, false, b2, b3, i)
busy(F, closed, up, b1, false, b3, i) → idle(FS, closed, up, b1, false, b3, i)
busy(F, closed, down, b1, false, b3, i) → idle(BF, closed, down, b1, false, b3, i)
busy(S, closed, down, b1, b2, false, i) → idle(FS, closed, down, b1, b2, false, i)
busy(BF, closed, up, b1, b2, b3, i) → idle(F, closed, up, b1, b2, b3, i)
busy(BF, closed, down, b1, b2, b3, i) → idle(B, closed, down, b1, b2, b3, i)
busy(FS, closed, up, b1, b2, b3, i) → idle(S, closed, up, b1, b2, b3, i)
busy(FS, closed, down, b1, b2, b3, i) → idle(F, closed, down, b1, b2, b3, i)
busy(B, closed, stop, false, true, b3, i) → idle(B, closed, up, false, true, b3, i)
busy(B, closed, stop, false, false, true, i) → idle(B, closed, up, false, false, true, i)
busy(F, closed, stop, true, false, b3, i) → idle(F, closed, down, true, false, b3, i)
busy(F, closed, stop, false, false, true, i) → idle(F, closed, up, false, false, true, i)
busy(S, closed, stop, b1, true, false, i) → idle(S, closed, down, b1, true, false, i)
busy(S, closed, stop, true, false, false, i) → idle(S, closed, down, true, false, false, i)
idle(fl, d, m, b1, b2, b3, empty) → busy(fl, d, m, b1, b2, b3, empty)
Q is empty.
The following Q TRS is given: Q restricted rewrite system:
The TRS R consists of the following rules:
busy(BF, d, stop, b1, b2, b3, i) → incorrect
busy(FS, d, stop, b1, b2, b3, i) → incorrect
busy(B, closed, stop, false, false, false, empty) → correct
busy(F, closed, stop, false, false, false, empty) → correct
busy(S, closed, stop, false, false, false, empty) → correct
busy(B, d, stop, true, b2, b3, i) → idle(B, open, stop, false, b2, b3, i)
busy(F, d, stop, b1, true, b3, i) → idle(F, open, stop, b1, false, b3, i)
busy(S, d, stop, b1, b2, true, i) → idle(S, open, stop, b1, b2, false, i)
busy(B, closed, down, b1, b2, b3, i) → idle(B, closed, stop, b1, b2, b3, i)
busy(S, closed, up, b1, b2, b3, i) → idle(S, closed, stop, b1, b2, b3, i)
busy(B, closed, up, true, b2, b3, i) → idle(B, closed, stop, true, b2, b3, i)
busy(F, closed, up, b1, true, b3, i) → idle(F, closed, stop, b1, true, b3, i)
busy(F, closed, down, b1, true, b3, i) → idle(F, closed, stop, b1, true, b3, i)
busy(S, closed, down, b1, b2, true, i) → idle(S, closed, stop, b1, b2, true, i)
busy(B, closed, up, false, b2, b3, i) → idle(BF, closed, up, false, b2, b3, i)
busy(F, closed, up, b1, false, b3, i) → idle(FS, closed, up, b1, false, b3, i)
busy(F, closed, down, b1, false, b3, i) → idle(BF, closed, down, b1, false, b3, i)
busy(S, closed, down, b1, b2, false, i) → idle(FS, closed, down, b1, b2, false, i)
busy(BF, closed, up, b1, b2, b3, i) → idle(F, closed, up, b1, b2, b3, i)
busy(BF, closed, down, b1, b2, b3, i) → idle(B, closed, down, b1, b2, b3, i)
busy(FS, closed, up, b1, b2, b3, i) → idle(S, closed, up, b1, b2, b3, i)
busy(FS, closed, down, b1, b2, b3, i) → idle(F, closed, down, b1, b2, b3, i)
busy(B, closed, stop, false, true, b3, i) → idle(B, closed, up, false, true, b3, i)
busy(B, closed, stop, false, false, true, i) → idle(B, closed, up, false, false, true, i)
busy(F, closed, stop, true, false, b3, i) → idle(F, closed, down, true, false, b3, i)
busy(F, closed, stop, false, false, true, i) → idle(F, closed, up, false, false, true, i)
busy(S, closed, stop, b1, true, false, i) → idle(S, closed, down, b1, true, false, i)
busy(S, closed, stop, true, false, false, i) → idle(S, closed, down, true, false, false, i)
idle(fl, d, m, b1, b2, b3, empty) → busy(fl, d, m, b1, b2, b3, empty)
Q is empty.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:
busy(B, closed, stop, false, false, false, empty) → correct
busy(F, closed, stop, false, false, false, empty) → correct
busy(S, closed, stop, false, false, false, empty) → correct
busy(B, d, stop, true, b2, b3, i) → idle(B, open, stop, false, b2, b3, i)
busy(F, d, stop, b1, true, b3, i) → idle(F, open, stop, b1, false, b3, i)
busy(S, d, stop, b1, b2, true, i) → idle(S, open, stop, b1, b2, false, i)
Used ordering:
Polynomial interpretation [25]:
POL(B) = 0
POL(BF) = 0
POL(F) = 0
POL(FS) = 0
POL(S) = 0
POL(busy(x1, x2, x3, x4, x5, x6, x7)) = 2·x1 + 2·x2 + x3 + x4 + x5 + x6 + x7
POL(closed) = 0
POL(correct) = 2
POL(down) = 0
POL(empty) = 0
POL(false) = 1
POL(idle(x1, x2, x3, x4, x5, x6, x7)) = 2·x1 + 2·x2 + x3 + x4 + x5 + x6 + x7
POL(incorrect) = 0
POL(open) = 0
POL(stop) = 0
POL(true) = 2
POL(up) = 0
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
Q restricted rewrite system:
The TRS R consists of the following rules:
busy(BF, d, stop, b1, b2, b3, i) → incorrect
busy(FS, d, stop, b1, b2, b3, i) → incorrect
busy(B, closed, down, b1, b2, b3, i) → idle(B, closed, stop, b1, b2, b3, i)
busy(S, closed, up, b1, b2, b3, i) → idle(S, closed, stop, b1, b2, b3, i)
busy(B, closed, up, true, b2, b3, i) → idle(B, closed, stop, true, b2, b3, i)
busy(F, closed, up, b1, true, b3, i) → idle(F, closed, stop, b1, true, b3, i)
busy(F, closed, down, b1, true, b3, i) → idle(F, closed, stop, b1, true, b3, i)
busy(S, closed, down, b1, b2, true, i) → idle(S, closed, stop, b1, b2, true, i)
busy(B, closed, up, false, b2, b3, i) → idle(BF, closed, up, false, b2, b3, i)
busy(F, closed, up, b1, false, b3, i) → idle(FS, closed, up, b1, false, b3, i)
busy(F, closed, down, b1, false, b3, i) → idle(BF, closed, down, b1, false, b3, i)
busy(S, closed, down, b1, b2, false, i) → idle(FS, closed, down, b1, b2, false, i)
busy(BF, closed, up, b1, b2, b3, i) → idle(F, closed, up, b1, b2, b3, i)
busy(BF, closed, down, b1, b2, b3, i) → idle(B, closed, down, b1, b2, b3, i)
busy(FS, closed, up, b1, b2, b3, i) → idle(S, closed, up, b1, b2, b3, i)
busy(FS, closed, down, b1, b2, b3, i) → idle(F, closed, down, b1, b2, b3, i)
busy(B, closed, stop, false, true, b3, i) → idle(B, closed, up, false, true, b3, i)
busy(B, closed, stop, false, false, true, i) → idle(B, closed, up, false, false, true, i)
busy(F, closed, stop, true, false, b3, i) → idle(F, closed, down, true, false, b3, i)
busy(F, closed, stop, false, false, true, i) → idle(F, closed, up, false, false, true, i)
busy(S, closed, stop, b1, true, false, i) → idle(S, closed, down, b1, true, false, i)
busy(S, closed, stop, true, false, false, i) → idle(S, closed, down, true, false, false, i)
idle(fl, d, m, b1, b2, b3, empty) → busy(fl, d, m, b1, b2, b3, empty)
Q is empty.
The following Q TRS is given: Q restricted rewrite system:
The TRS R consists of the following rules:
busy(BF, d, stop, b1, b2, b3, i) → incorrect
busy(FS, d, stop, b1, b2, b3, i) → incorrect
busy(B, closed, down, b1, b2, b3, i) → idle(B, closed, stop, b1, b2, b3, i)
busy(S, closed, up, b1, b2, b3, i) → idle(S, closed, stop, b1, b2, b3, i)
busy(B, closed, up, true, b2, b3, i) → idle(B, closed, stop, true, b2, b3, i)
busy(F, closed, up, b1, true, b3, i) → idle(F, closed, stop, b1, true, b3, i)
busy(F, closed, down, b1, true, b3, i) → idle(F, closed, stop, b1, true, b3, i)
busy(S, closed, down, b1, b2, true, i) → idle(S, closed, stop, b1, b2, true, i)
busy(B, closed, up, false, b2, b3, i) → idle(BF, closed, up, false, b2, b3, i)
busy(F, closed, up, b1, false, b3, i) → idle(FS, closed, up, b1, false, b3, i)
busy(F, closed, down, b1, false, b3, i) → idle(BF, closed, down, b1, false, b3, i)
busy(S, closed, down, b1, b2, false, i) → idle(FS, closed, down, b1, b2, false, i)
busy(BF, closed, up, b1, b2, b3, i) → idle(F, closed, up, b1, b2, b3, i)
busy(BF, closed, down, b1, b2, b3, i) → idle(B, closed, down, b1, b2, b3, i)
busy(FS, closed, up, b1, b2, b3, i) → idle(S, closed, up, b1, b2, b3, i)
busy(FS, closed, down, b1, b2, b3, i) → idle(F, closed, down, b1, b2, b3, i)
busy(B, closed, stop, false, true, b3, i) → idle(B, closed, up, false, true, b3, i)
busy(B, closed, stop, false, false, true, i) → idle(B, closed, up, false, false, true, i)
busy(F, closed, stop, true, false, b3, i) → idle(F, closed, down, true, false, b3, i)
busy(F, closed, stop, false, false, true, i) → idle(F, closed, up, false, false, true, i)
busy(S, closed, stop, b1, true, false, i) → idle(S, closed, down, b1, true, false, i)
busy(S, closed, stop, true, false, false, i) → idle(S, closed, down, true, false, false, i)
idle(fl, d, m, b1, b2, b3, empty) → busy(fl, d, m, b1, b2, b3, empty)
Q is empty.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:
busy(BF, d, stop, b1, b2, b3, i) → incorrect
busy(FS, d, stop, b1, b2, b3, i) → incorrect
Used ordering:
Polynomial interpretation [25]:
POL(B) = 0
POL(BF) = 0
POL(F) = 0
POL(FS) = 0
POL(S) = 0
POL(busy(x1, x2, x3, x4, x5, x6, x7)) = 2 + x1 + x2 + x3 + x4 + x5 + x6 + x7
POL(closed) = 0
POL(down) = 0
POL(empty) = 0
POL(false) = 0
POL(idle(x1, x2, x3, x4, x5, x6, x7)) = 2 + x1 + x2 + x3 + x4 + x5 + x6 + x7
POL(incorrect) = 1
POL(stop) = 0
POL(true) = 0
POL(up) = 0
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ AAECC Innermost
Q restricted rewrite system:
The TRS R consists of the following rules:
busy(B, closed, down, b1, b2, b3, i) → idle(B, closed, stop, b1, b2, b3, i)
busy(S, closed, up, b1, b2, b3, i) → idle(S, closed, stop, b1, b2, b3, i)
busy(B, closed, up, true, b2, b3, i) → idle(B, closed, stop, true, b2, b3, i)
busy(F, closed, up, b1, true, b3, i) → idle(F, closed, stop, b1, true, b3, i)
busy(F, closed, down, b1, true, b3, i) → idle(F, closed, stop, b1, true, b3, i)
busy(S, closed, down, b1, b2, true, i) → idle(S, closed, stop, b1, b2, true, i)
busy(B, closed, up, false, b2, b3, i) → idle(BF, closed, up, false, b2, b3, i)
busy(F, closed, up, b1, false, b3, i) → idle(FS, closed, up, b1, false, b3, i)
busy(F, closed, down, b1, false, b3, i) → idle(BF, closed, down, b1, false, b3, i)
busy(S, closed, down, b1, b2, false, i) → idle(FS, closed, down, b1, b2, false, i)
busy(BF, closed, up, b1, b2, b3, i) → idle(F, closed, up, b1, b2, b3, i)
busy(BF, closed, down, b1, b2, b3, i) → idle(B, closed, down, b1, b2, b3, i)
busy(FS, closed, up, b1, b2, b3, i) → idle(S, closed, up, b1, b2, b3, i)
busy(FS, closed, down, b1, b2, b3, i) → idle(F, closed, down, b1, b2, b3, i)
busy(B, closed, stop, false, true, b3, i) → idle(B, closed, up, false, true, b3, i)
busy(B, closed, stop, false, false, true, i) → idle(B, closed, up, false, false, true, i)
busy(F, closed, stop, true, false, b3, i) → idle(F, closed, down, true, false, b3, i)
busy(F, closed, stop, false, false, true, i) → idle(F, closed, up, false, false, true, i)
busy(S, closed, stop, b1, true, false, i) → idle(S, closed, down, b1, true, false, i)
busy(S, closed, stop, true, false, false, i) → idle(S, closed, down, true, false, false, i)
idle(fl, d, m, b1, b2, b3, empty) → busy(fl, d, m, b1, b2, b3, empty)
Q is empty.
We have applied [19,8] to switch to innermost. The TRS R 1 is none
The TRS R 2 is
busy(B, closed, down, b1, b2, b3, i) → idle(B, closed, stop, b1, b2, b3, i)
busy(S, closed, up, b1, b2, b3, i) → idle(S, closed, stop, b1, b2, b3, i)
busy(B, closed, up, true, b2, b3, i) → idle(B, closed, stop, true, b2, b3, i)
busy(F, closed, up, b1, true, b3, i) → idle(F, closed, stop, b1, true, b3, i)
busy(F, closed, down, b1, true, b3, i) → idle(F, closed, stop, b1, true, b3, i)
busy(S, closed, down, b1, b2, true, i) → idle(S, closed, stop, b1, b2, true, i)
busy(B, closed, up, false, b2, b3, i) → idle(BF, closed, up, false, b2, b3, i)
busy(F, closed, up, b1, false, b3, i) → idle(FS, closed, up, b1, false, b3, i)
busy(F, closed, down, b1, false, b3, i) → idle(BF, closed, down, b1, false, b3, i)
busy(S, closed, down, b1, b2, false, i) → idle(FS, closed, down, b1, b2, false, i)
busy(BF, closed, up, b1, b2, b3, i) → idle(F, closed, up, b1, b2, b3, i)
busy(BF, closed, down, b1, b2, b3, i) → idle(B, closed, down, b1, b2, b3, i)
busy(FS, closed, up, b1, b2, b3, i) → idle(S, closed, up, b1, b2, b3, i)
busy(FS, closed, down, b1, b2, b3, i) → idle(F, closed, down, b1, b2, b3, i)
busy(B, closed, stop, false, true, b3, i) → idle(B, closed, up, false, true, b3, i)
busy(B, closed, stop, false, false, true, i) → idle(B, closed, up, false, false, true, i)
busy(F, closed, stop, true, false, b3, i) → idle(F, closed, down, true, false, b3, i)
busy(F, closed, stop, false, false, true, i) → idle(F, closed, up, false, false, true, i)
busy(S, closed, stop, b1, true, false, i) → idle(S, closed, down, b1, true, false, i)
busy(S, closed, stop, true, false, false, i) → idle(S, closed, down, true, false, false, i)
idle(fl, d, m, b1, b2, b3, empty) → busy(fl, d, m, b1, b2, b3, empty)
The signature Sigma is {busy, idle}
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ AAECC Innermost
↳ QTRS
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
busy(B, closed, down, b1, b2, b3, i) → idle(B, closed, stop, b1, b2, b3, i)
busy(S, closed, up, b1, b2, b3, i) → idle(S, closed, stop, b1, b2, b3, i)
busy(B, closed, up, true, b2, b3, i) → idle(B, closed, stop, true, b2, b3, i)
busy(F, closed, up, b1, true, b3, i) → idle(F, closed, stop, b1, true, b3, i)
busy(F, closed, down, b1, true, b3, i) → idle(F, closed, stop, b1, true, b3, i)
busy(S, closed, down, b1, b2, true, i) → idle(S, closed, stop, b1, b2, true, i)
busy(B, closed, up, false, b2, b3, i) → idle(BF, closed, up, false, b2, b3, i)
busy(F, closed, up, b1, false, b3, i) → idle(FS, closed, up, b1, false, b3, i)
busy(F, closed, down, b1, false, b3, i) → idle(BF, closed, down, b1, false, b3, i)
busy(S, closed, down, b1, b2, false, i) → idle(FS, closed, down, b1, b2, false, i)
busy(BF, closed, up, b1, b2, b3, i) → idle(F, closed, up, b1, b2, b3, i)
busy(BF, closed, down, b1, b2, b3, i) → idle(B, closed, down, b1, b2, b3, i)
busy(FS, closed, up, b1, b2, b3, i) → idle(S, closed, up, b1, b2, b3, i)
busy(FS, closed, down, b1, b2, b3, i) → idle(F, closed, down, b1, b2, b3, i)
busy(B, closed, stop, false, true, b3, i) → idle(B, closed, up, false, true, b3, i)
busy(B, closed, stop, false, false, true, i) → idle(B, closed, up, false, false, true, i)
busy(F, closed, stop, true, false, b3, i) → idle(F, closed, down, true, false, b3, i)
busy(F, closed, stop, false, false, true, i) → idle(F, closed, up, false, false, true, i)
busy(S, closed, stop, b1, true, false, i) → idle(S, closed, down, b1, true, false, i)
busy(S, closed, stop, true, false, false, i) → idle(S, closed, down, true, false, false, i)
idle(fl, d, m, b1, b2, b3, empty) → busy(fl, d, m, b1, b2, b3, empty)
The set Q consists of the following terms:
busy(B, closed, down, x0, x1, x2, x3)
busy(S, closed, up, x0, x1, x2, x3)
busy(B, closed, up, true, x0, x1, x2)
busy(F, closed, up, x0, true, x1, x2)
busy(F, closed, down, x0, true, x1, x2)
busy(S, closed, down, x0, x1, true, x2)
busy(B, closed, up, false, x0, x1, x2)
busy(F, closed, up, x0, false, x1, x2)
busy(F, closed, down, x0, false, x1, x2)
busy(S, closed, down, x0, x1, false, x2)
busy(BF, closed, up, x0, x1, x2, x3)
busy(BF, closed, down, x0, x1, x2, x3)
busy(FS, closed, up, x0, x1, x2, x3)
busy(FS, closed, down, x0, x1, x2, x3)
busy(B, closed, stop, false, true, x0, x1)
busy(B, closed, stop, false, false, true, x0)
busy(F, closed, stop, true, false, x0, x1)
busy(F, closed, stop, false, false, true, x0)
busy(S, closed, stop, x0, true, false, x1)
busy(S, closed, stop, true, false, false, x0)
idle(x0, x1, x2, x3, x4, x5, empty)
Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:
BUSY(B, closed, stop, false, false, true, i) → IDLE(B, closed, up, false, false, true, i)
BUSY(B, closed, up, false, b2, b3, i) → IDLE(BF, closed, up, false, b2, b3, i)
BUSY(F, closed, up, b1, true, b3, i) → IDLE(F, closed, stop, b1, true, b3, i)
BUSY(FS, closed, up, b1, b2, b3, i) → IDLE(S, closed, up, b1, b2, b3, i)
BUSY(FS, closed, down, b1, b2, b3, i) → IDLE(F, closed, down, b1, b2, b3, i)
BUSY(S, closed, down, b1, b2, false, i) → IDLE(FS, closed, down, b1, b2, false, i)
BUSY(BF, closed, up, b1, b2, b3, i) → IDLE(F, closed, up, b1, b2, b3, i)
BUSY(F, closed, down, b1, true, b3, i) → IDLE(F, closed, stop, b1, true, b3, i)
BUSY(S, closed, stop, b1, true, false, i) → IDLE(S, closed, down, b1, true, false, i)
BUSY(S, closed, down, b1, b2, true, i) → IDLE(S, closed, stop, b1, b2, true, i)
BUSY(B, closed, up, true, b2, b3, i) → IDLE(B, closed, stop, true, b2, b3, i)
IDLE(fl, d, m, b1, b2, b3, empty) → BUSY(fl, d, m, b1, b2, b3, empty)
BUSY(B, closed, stop, false, true, b3, i) → IDLE(B, closed, up, false, true, b3, i)
BUSY(F, closed, stop, false, false, true, i) → IDLE(F, closed, up, false, false, true, i)
BUSY(B, closed, down, b1, b2, b3, i) → IDLE(B, closed, stop, b1, b2, b3, i)
BUSY(S, closed, stop, true, false, false, i) → IDLE(S, closed, down, true, false, false, i)
BUSY(F, closed, stop, true, false, b3, i) → IDLE(F, closed, down, true, false, b3, i)
BUSY(F, closed, down, b1, false, b3, i) → IDLE(BF, closed, down, b1, false, b3, i)
BUSY(S, closed, up, b1, b2, b3, i) → IDLE(S, closed, stop, b1, b2, b3, i)
BUSY(F, closed, up, b1, false, b3, i) → IDLE(FS, closed, up, b1, false, b3, i)
BUSY(BF, closed, down, b1, b2, b3, i) → IDLE(B, closed, down, b1, b2, b3, i)
The TRS R consists of the following rules:
busy(B, closed, down, b1, b2, b3, i) → idle(B, closed, stop, b1, b2, b3, i)
busy(S, closed, up, b1, b2, b3, i) → idle(S, closed, stop, b1, b2, b3, i)
busy(B, closed, up, true, b2, b3, i) → idle(B, closed, stop, true, b2, b3, i)
busy(F, closed, up, b1, true, b3, i) → idle(F, closed, stop, b1, true, b3, i)
busy(F, closed, down, b1, true, b3, i) → idle(F, closed, stop, b1, true, b3, i)
busy(S, closed, down, b1, b2, true, i) → idle(S, closed, stop, b1, b2, true, i)
busy(B, closed, up, false, b2, b3, i) → idle(BF, closed, up, false, b2, b3, i)
busy(F, closed, up, b1, false, b3, i) → idle(FS, closed, up, b1, false, b3, i)
busy(F, closed, down, b1, false, b3, i) → idle(BF, closed, down, b1, false, b3, i)
busy(S, closed, down, b1, b2, false, i) → idle(FS, closed, down, b1, b2, false, i)
busy(BF, closed, up, b1, b2, b3, i) → idle(F, closed, up, b1, b2, b3, i)
busy(BF, closed, down, b1, b2, b3, i) → idle(B, closed, down, b1, b2, b3, i)
busy(FS, closed, up, b1, b2, b3, i) → idle(S, closed, up, b1, b2, b3, i)
busy(FS, closed, down, b1, b2, b3, i) → idle(F, closed, down, b1, b2, b3, i)
busy(B, closed, stop, false, true, b3, i) → idle(B, closed, up, false, true, b3, i)
busy(B, closed, stop, false, false, true, i) → idle(B, closed, up, false, false, true, i)
busy(F, closed, stop, true, false, b3, i) → idle(F, closed, down, true, false, b3, i)
busy(F, closed, stop, false, false, true, i) → idle(F, closed, up, false, false, true, i)
busy(S, closed, stop, b1, true, false, i) → idle(S, closed, down, b1, true, false, i)
busy(S, closed, stop, true, false, false, i) → idle(S, closed, down, true, false, false, i)
idle(fl, d, m, b1, b2, b3, empty) → busy(fl, d, m, b1, b2, b3, empty)
The set Q consists of the following terms:
busy(B, closed, down, x0, x1, x2, x3)
busy(S, closed, up, x0, x1, x2, x3)
busy(B, closed, up, true, x0, x1, x2)
busy(F, closed, up, x0, true, x1, x2)
busy(F, closed, down, x0, true, x1, x2)
busy(S, closed, down, x0, x1, true, x2)
busy(B, closed, up, false, x0, x1, x2)
busy(F, closed, up, x0, false, x1, x2)
busy(F, closed, down, x0, false, x1, x2)
busy(S, closed, down, x0, x1, false, x2)
busy(BF, closed, up, x0, x1, x2, x3)
busy(BF, closed, down, x0, x1, x2, x3)
busy(FS, closed, up, x0, x1, x2, x3)
busy(FS, closed, down, x0, x1, x2, x3)
busy(B, closed, stop, false, true, x0, x1)
busy(B, closed, stop, false, false, true, x0)
busy(F, closed, stop, true, false, x0, x1)
busy(F, closed, stop, false, false, true, x0)
busy(S, closed, stop, x0, true, false, x1)
busy(S, closed, stop, true, false, false, x0)
idle(x0, x1, x2, x3, x4, x5, empty)
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ AAECC Innermost
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
BUSY(B, closed, stop, false, false, true, i) → IDLE(B, closed, up, false, false, true, i)
BUSY(B, closed, up, false, b2, b3, i) → IDLE(BF, closed, up, false, b2, b3, i)
BUSY(F, closed, up, b1, true, b3, i) → IDLE(F, closed, stop, b1, true, b3, i)
BUSY(FS, closed, up, b1, b2, b3, i) → IDLE(S, closed, up, b1, b2, b3, i)
BUSY(FS, closed, down, b1, b2, b3, i) → IDLE(F, closed, down, b1, b2, b3, i)
BUSY(S, closed, down, b1, b2, false, i) → IDLE(FS, closed, down, b1, b2, false, i)
BUSY(BF, closed, up, b1, b2, b3, i) → IDLE(F, closed, up, b1, b2, b3, i)
BUSY(F, closed, down, b1, true, b3, i) → IDLE(F, closed, stop, b1, true, b3, i)
BUSY(S, closed, stop, b1, true, false, i) → IDLE(S, closed, down, b1, true, false, i)
BUSY(S, closed, down, b1, b2, true, i) → IDLE(S, closed, stop, b1, b2, true, i)
BUSY(B, closed, up, true, b2, b3, i) → IDLE(B, closed, stop, true, b2, b3, i)
IDLE(fl, d, m, b1, b2, b3, empty) → BUSY(fl, d, m, b1, b2, b3, empty)
BUSY(B, closed, stop, false, true, b3, i) → IDLE(B, closed, up, false, true, b3, i)
BUSY(F, closed, stop, false, false, true, i) → IDLE(F, closed, up, false, false, true, i)
BUSY(B, closed, down, b1, b2, b3, i) → IDLE(B, closed, stop, b1, b2, b3, i)
BUSY(S, closed, stop, true, false, false, i) → IDLE(S, closed, down, true, false, false, i)
BUSY(F, closed, stop, true, false, b3, i) → IDLE(F, closed, down, true, false, b3, i)
BUSY(F, closed, down, b1, false, b3, i) → IDLE(BF, closed, down, b1, false, b3, i)
BUSY(S, closed, up, b1, b2, b3, i) → IDLE(S, closed, stop, b1, b2, b3, i)
BUSY(F, closed, up, b1, false, b3, i) → IDLE(FS, closed, up, b1, false, b3, i)
BUSY(BF, closed, down, b1, b2, b3, i) → IDLE(B, closed, down, b1, b2, b3, i)
The TRS R consists of the following rules:
busy(B, closed, down, b1, b2, b3, i) → idle(B, closed, stop, b1, b2, b3, i)
busy(S, closed, up, b1, b2, b3, i) → idle(S, closed, stop, b1, b2, b3, i)
busy(B, closed, up, true, b2, b3, i) → idle(B, closed, stop, true, b2, b3, i)
busy(F, closed, up, b1, true, b3, i) → idle(F, closed, stop, b1, true, b3, i)
busy(F, closed, down, b1, true, b3, i) → idle(F, closed, stop, b1, true, b3, i)
busy(S, closed, down, b1, b2, true, i) → idle(S, closed, stop, b1, b2, true, i)
busy(B, closed, up, false, b2, b3, i) → idle(BF, closed, up, false, b2, b3, i)
busy(F, closed, up, b1, false, b3, i) → idle(FS, closed, up, b1, false, b3, i)
busy(F, closed, down, b1, false, b3, i) → idle(BF, closed, down, b1, false, b3, i)
busy(S, closed, down, b1, b2, false, i) → idle(FS, closed, down, b1, b2, false, i)
busy(BF, closed, up, b1, b2, b3, i) → idle(F, closed, up, b1, b2, b3, i)
busy(BF, closed, down, b1, b2, b3, i) → idle(B, closed, down, b1, b2, b3, i)
busy(FS, closed, up, b1, b2, b3, i) → idle(S, closed, up, b1, b2, b3, i)
busy(FS, closed, down, b1, b2, b3, i) → idle(F, closed, down, b1, b2, b3, i)
busy(B, closed, stop, false, true, b3, i) → idle(B, closed, up, false, true, b3, i)
busy(B, closed, stop, false, false, true, i) → idle(B, closed, up, false, false, true, i)
busy(F, closed, stop, true, false, b3, i) → idle(F, closed, down, true, false, b3, i)
busy(F, closed, stop, false, false, true, i) → idle(F, closed, up, false, false, true, i)
busy(S, closed, stop, b1, true, false, i) → idle(S, closed, down, b1, true, false, i)
busy(S, closed, stop, true, false, false, i) → idle(S, closed, down, true, false, false, i)
idle(fl, d, m, b1, b2, b3, empty) → busy(fl, d, m, b1, b2, b3, empty)
The set Q consists of the following terms:
busy(B, closed, down, x0, x1, x2, x3)
busy(S, closed, up, x0, x1, x2, x3)
busy(B, closed, up, true, x0, x1, x2)
busy(F, closed, up, x0, true, x1, x2)
busy(F, closed, down, x0, true, x1, x2)
busy(S, closed, down, x0, x1, true, x2)
busy(B, closed, up, false, x0, x1, x2)
busy(F, closed, up, x0, false, x1, x2)
busy(F, closed, down, x0, false, x1, x2)
busy(S, closed, down, x0, x1, false, x2)
busy(BF, closed, up, x0, x1, x2, x3)
busy(BF, closed, down, x0, x1, x2, x3)
busy(FS, closed, up, x0, x1, x2, x3)
busy(FS, closed, down, x0, x1, x2, x3)
busy(B, closed, stop, false, true, x0, x1)
busy(B, closed, stop, false, false, true, x0)
busy(F, closed, stop, true, false, x0, x1)
busy(F, closed, stop, false, false, true, x0)
busy(S, closed, stop, x0, true, false, x1)
busy(S, closed, stop, true, false, false, x0)
idle(x0, x1, x2, x3, x4, x5, empty)
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ AAECC Innermost
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
Q DP problem:
The TRS P consists of the following rules:
BUSY(B, closed, stop, false, false, true, i) → IDLE(B, closed, up, false, false, true, i)
BUSY(B, closed, up, false, b2, b3, i) → IDLE(BF, closed, up, false, b2, b3, i)
BUSY(F, closed, up, b1, true, b3, i) → IDLE(F, closed, stop, b1, true, b3, i)
BUSY(FS, closed, up, b1, b2, b3, i) → IDLE(S, closed, up, b1, b2, b3, i)
BUSY(FS, closed, down, b1, b2, b3, i) → IDLE(F, closed, down, b1, b2, b3, i)
BUSY(S, closed, down, b1, b2, false, i) → IDLE(FS, closed, down, b1, b2, false, i)
BUSY(BF, closed, up, b1, b2, b3, i) → IDLE(F, closed, up, b1, b2, b3, i)
BUSY(F, closed, down, b1, true, b3, i) → IDLE(F, closed, stop, b1, true, b3, i)
BUSY(S, closed, stop, b1, true, false, i) → IDLE(S, closed, down, b1, true, false, i)
BUSY(S, closed, down, b1, b2, true, i) → IDLE(S, closed, stop, b1, b2, true, i)
BUSY(B, closed, up, true, b2, b3, i) → IDLE(B, closed, stop, true, b2, b3, i)
IDLE(fl, d, m, b1, b2, b3, empty) → BUSY(fl, d, m, b1, b2, b3, empty)
BUSY(B, closed, stop, false, true, b3, i) → IDLE(B, closed, up, false, true, b3, i)
BUSY(F, closed, stop, false, false, true, i) → IDLE(F, closed, up, false, false, true, i)
BUSY(B, closed, down, b1, b2, b3, i) → IDLE(B, closed, stop, b1, b2, b3, i)
BUSY(S, closed, stop, true, false, false, i) → IDLE(S, closed, down, true, false, false, i)
BUSY(F, closed, stop, true, false, b3, i) → IDLE(F, closed, down, true, false, b3, i)
BUSY(F, closed, down, b1, false, b3, i) → IDLE(BF, closed, down, b1, false, b3, i)
BUSY(S, closed, up, b1, b2, b3, i) → IDLE(S, closed, stop, b1, b2, b3, i)
BUSY(F, closed, up, b1, false, b3, i) → IDLE(FS, closed, up, b1, false, b3, i)
BUSY(BF, closed, down, b1, b2, b3, i) → IDLE(B, closed, down, b1, b2, b3, i)
R is empty.
The set Q consists of the following terms:
busy(B, closed, down, x0, x1, x2, x3)
busy(S, closed, up, x0, x1, x2, x3)
busy(B, closed, up, true, x0, x1, x2)
busy(F, closed, up, x0, true, x1, x2)
busy(F, closed, down, x0, true, x1, x2)
busy(S, closed, down, x0, x1, true, x2)
busy(B, closed, up, false, x0, x1, x2)
busy(F, closed, up, x0, false, x1, x2)
busy(F, closed, down, x0, false, x1, x2)
busy(S, closed, down, x0, x1, false, x2)
busy(BF, closed, up, x0, x1, x2, x3)
busy(BF, closed, down, x0, x1, x2, x3)
busy(FS, closed, up, x0, x1, x2, x3)
busy(FS, closed, down, x0, x1, x2, x3)
busy(B, closed, stop, false, true, x0, x1)
busy(B, closed, stop, false, false, true, x0)
busy(F, closed, stop, true, false, x0, x1)
busy(F, closed, stop, false, false, true, x0)
busy(S, closed, stop, x0, true, false, x1)
busy(S, closed, stop, true, false, false, x0)
idle(x0, x1, x2, x3, x4, x5, empty)
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
busy(B, closed, down, x0, x1, x2, x3)
busy(S, closed, up, x0, x1, x2, x3)
busy(B, closed, up, true, x0, x1, x2)
busy(F, closed, up, x0, true, x1, x2)
busy(F, closed, down, x0, true, x1, x2)
busy(S, closed, down, x0, x1, true, x2)
busy(B, closed, up, false, x0, x1, x2)
busy(F, closed, up, x0, false, x1, x2)
busy(F, closed, down, x0, false, x1, x2)
busy(S, closed, down, x0, x1, false, x2)
busy(BF, closed, up, x0, x1, x2, x3)
busy(BF, closed, down, x0, x1, x2, x3)
busy(FS, closed, up, x0, x1, x2, x3)
busy(FS, closed, down, x0, x1, x2, x3)
busy(B, closed, stop, false, true, x0, x1)
busy(B, closed, stop, false, false, true, x0)
busy(F, closed, stop, true, false, x0, x1)
busy(F, closed, stop, false, false, true, x0)
busy(S, closed, stop, x0, true, false, x1)
busy(S, closed, stop, true, false, false, x0)
idle(x0, x1, x2, x3, x4, x5, empty)
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ AAECC Innermost
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
Q DP problem:
The TRS P consists of the following rules:
BUSY(B, closed, stop, false, false, true, i) → IDLE(B, closed, up, false, false, true, i)
BUSY(B, closed, up, false, b2, b3, i) → IDLE(BF, closed, up, false, b2, b3, i)
BUSY(F, closed, up, b1, true, b3, i) → IDLE(F, closed, stop, b1, true, b3, i)
BUSY(FS, closed, up, b1, b2, b3, i) → IDLE(S, closed, up, b1, b2, b3, i)
BUSY(FS, closed, down, b1, b2, b3, i) → IDLE(F, closed, down, b1, b2, b3, i)
BUSY(S, closed, down, b1, b2, false, i) → IDLE(FS, closed, down, b1, b2, false, i)
BUSY(BF, closed, up, b1, b2, b3, i) → IDLE(F, closed, up, b1, b2, b3, i)
BUSY(F, closed, down, b1, true, b3, i) → IDLE(F, closed, stop, b1, true, b3, i)
BUSY(S, closed, stop, b1, true, false, i) → IDLE(S, closed, down, b1, true, false, i)
BUSY(S, closed, down, b1, b2, true, i) → IDLE(S, closed, stop, b1, b2, true, i)
BUSY(B, closed, up, true, b2, b3, i) → IDLE(B, closed, stop, true, b2, b3, i)
IDLE(fl, d, m, b1, b2, b3, empty) → BUSY(fl, d, m, b1, b2, b3, empty)
BUSY(B, closed, stop, false, true, b3, i) → IDLE(B, closed, up, false, true, b3, i)
BUSY(F, closed, stop, false, false, true, i) → IDLE(F, closed, up, false, false, true, i)
BUSY(B, closed, down, b1, b2, b3, i) → IDLE(B, closed, stop, b1, b2, b3, i)
BUSY(F, closed, stop, true, false, b3, i) → IDLE(F, closed, down, true, false, b3, i)
BUSY(S, closed, stop, true, false, false, i) → IDLE(S, closed, down, true, false, false, i)
BUSY(F, closed, down, b1, false, b3, i) → IDLE(BF, closed, down, b1, false, b3, i)
BUSY(S, closed, up, b1, b2, b3, i) → IDLE(S, closed, stop, b1, b2, b3, i)
BUSY(F, closed, up, b1, false, b3, i) → IDLE(FS, closed, up, b1, false, b3, i)
BUSY(BF, closed, down, b1, b2, b3, i) → IDLE(B, closed, down, b1, b2, b3, i)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule BUSY(B, closed, stop, false, false, true, i) → IDLE(B, closed, up, false, false, true, i) we obtained the following new rules:
BUSY(B, closed, stop, false, false, true, empty) → IDLE(B, closed, up, false, false, true, empty)
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ AAECC Innermost
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
Q DP problem:
The TRS P consists of the following rules:
BUSY(B, closed, up, false, b2, b3, i) → IDLE(BF, closed, up, false, b2, b3, i)
BUSY(FS, closed, up, b1, b2, b3, i) → IDLE(S, closed, up, b1, b2, b3, i)
BUSY(F, closed, up, b1, true, b3, i) → IDLE(F, closed, stop, b1, true, b3, i)
BUSY(FS, closed, down, b1, b2, b3, i) → IDLE(F, closed, down, b1, b2, b3, i)
BUSY(BF, closed, up, b1, b2, b3, i) → IDLE(F, closed, up, b1, b2, b3, i)
BUSY(S, closed, down, b1, b2, false, i) → IDLE(FS, closed, down, b1, b2, false, i)
BUSY(B, closed, stop, false, false, true, empty) → IDLE(B, closed, up, false, false, true, empty)
BUSY(F, closed, down, b1, true, b3, i) → IDLE(F, closed, stop, b1, true, b3, i)
BUSY(S, closed, stop, b1, true, false, i) → IDLE(S, closed, down, b1, true, false, i)
BUSY(S, closed, down, b1, b2, true, i) → IDLE(S, closed, stop, b1, b2, true, i)
BUSY(B, closed, up, true, b2, b3, i) → IDLE(B, closed, stop, true, b2, b3, i)
IDLE(fl, d, m, b1, b2, b3, empty) → BUSY(fl, d, m, b1, b2, b3, empty)
BUSY(B, closed, stop, false, true, b3, i) → IDLE(B, closed, up, false, true, b3, i)
BUSY(F, closed, stop, false, false, true, i) → IDLE(F, closed, up, false, false, true, i)
BUSY(B, closed, down, b1, b2, b3, i) → IDLE(B, closed, stop, b1, b2, b3, i)
BUSY(S, closed, stop, true, false, false, i) → IDLE(S, closed, down, true, false, false, i)
BUSY(F, closed, stop, true, false, b3, i) → IDLE(F, closed, down, true, false, b3, i)
BUSY(F, closed, down, b1, false, b3, i) → IDLE(BF, closed, down, b1, false, b3, i)
BUSY(S, closed, up, b1, b2, b3, i) → IDLE(S, closed, stop, b1, b2, b3, i)
BUSY(F, closed, up, b1, false, b3, i) → IDLE(FS, closed, up, b1, false, b3, i)
BUSY(BF, closed, down, b1, b2, b3, i) → IDLE(B, closed, down, b1, b2, b3, i)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule BUSY(B, closed, up, false, b2, b3, i) → IDLE(BF, closed, up, false, b2, b3, i) we obtained the following new rules:
BUSY(B, closed, up, false, z4, z5, empty) → IDLE(BF, closed, up, false, z4, z5, empty)
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ AAECC Innermost
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
Q DP problem:
The TRS P consists of the following rules:
BUSY(F, closed, up, b1, true, b3, i) → IDLE(F, closed, stop, b1, true, b3, i)
BUSY(FS, closed, up, b1, b2, b3, i) → IDLE(S, closed, up, b1, b2, b3, i)
BUSY(FS, closed, down, b1, b2, b3, i) → IDLE(F, closed, down, b1, b2, b3, i)
BUSY(S, closed, down, b1, b2, false, i) → IDLE(FS, closed, down, b1, b2, false, i)
BUSY(BF, closed, up, b1, b2, b3, i) → IDLE(F, closed, up, b1, b2, b3, i)
BUSY(B, closed, stop, false, false, true, empty) → IDLE(B, closed, up, false, false, true, empty)
BUSY(F, closed, down, b1, true, b3, i) → IDLE(F, closed, stop, b1, true, b3, i)
BUSY(S, closed, stop, b1, true, false, i) → IDLE(S, closed, down, b1, true, false, i)
BUSY(S, closed, down, b1, b2, true, i) → IDLE(S, closed, stop, b1, b2, true, i)
BUSY(B, closed, up, true, b2, b3, i) → IDLE(B, closed, stop, true, b2, b3, i)
IDLE(fl, d, m, b1, b2, b3, empty) → BUSY(fl, d, m, b1, b2, b3, empty)
BUSY(B, closed, stop, false, true, b3, i) → IDLE(B, closed, up, false, true, b3, i)
BUSY(F, closed, stop, false, false, true, i) → IDLE(F, closed, up, false, false, true, i)
BUSY(B, closed, down, b1, b2, b3, i) → IDLE(B, closed, stop, b1, b2, b3, i)
BUSY(F, closed, stop, true, false, b3, i) → IDLE(F, closed, down, true, false, b3, i)
BUSY(S, closed, stop, true, false, false, i) → IDLE(S, closed, down, true, false, false, i)
BUSY(F, closed, down, b1, false, b3, i) → IDLE(BF, closed, down, b1, false, b3, i)
BUSY(B, closed, up, false, z4, z5, empty) → IDLE(BF, closed, up, false, z4, z5, empty)
BUSY(S, closed, up, b1, b2, b3, i) → IDLE(S, closed, stop, b1, b2, b3, i)
BUSY(F, closed, up, b1, false, b3, i) → IDLE(FS, closed, up, b1, false, b3, i)
BUSY(BF, closed, down, b1, b2, b3, i) → IDLE(B, closed, down, b1, b2, b3, i)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule BUSY(F, closed, up, b1, true, b3, i) → IDLE(F, closed, stop, b1, true, b3, i) we obtained the following new rules:
BUSY(F, closed, up, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty)
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ AAECC Innermost
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
Q DP problem:
The TRS P consists of the following rules:
BUSY(FS, closed, up, b1, b2, b3, i) → IDLE(S, closed, up, b1, b2, b3, i)
BUSY(FS, closed, down, b1, b2, b3, i) → IDLE(F, closed, down, b1, b2, b3, i)
BUSY(BF, closed, up, b1, b2, b3, i) → IDLE(F, closed, up, b1, b2, b3, i)
BUSY(S, closed, down, b1, b2, false, i) → IDLE(FS, closed, down, b1, b2, false, i)
BUSY(B, closed, stop, false, false, true, empty) → IDLE(B, closed, up, false, false, true, empty)
BUSY(F, closed, down, b1, true, b3, i) → IDLE(F, closed, stop, b1, true, b3, i)
BUSY(S, closed, stop, b1, true, false, i) → IDLE(S, closed, down, b1, true, false, i)
BUSY(S, closed, down, b1, b2, true, i) → IDLE(S, closed, stop, b1, b2, true, i)
BUSY(B, closed, up, true, b2, b3, i) → IDLE(B, closed, stop, true, b2, b3, i)
BUSY(F, closed, up, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty)
IDLE(fl, d, m, b1, b2, b3, empty) → BUSY(fl, d, m, b1, b2, b3, empty)
BUSY(B, closed, stop, false, true, b3, i) → IDLE(B, closed, up, false, true, b3, i)
BUSY(F, closed, stop, false, false, true, i) → IDLE(F, closed, up, false, false, true, i)
BUSY(B, closed, down, b1, b2, b3, i) → IDLE(B, closed, stop, b1, b2, b3, i)
BUSY(S, closed, stop, true, false, false, i) → IDLE(S, closed, down, true, false, false, i)
BUSY(F, closed, stop, true, false, b3, i) → IDLE(F, closed, down, true, false, b3, i)
BUSY(F, closed, down, b1, false, b3, i) → IDLE(BF, closed, down, b1, false, b3, i)
BUSY(B, closed, up, false, z4, z5, empty) → IDLE(BF, closed, up, false, z4, z5, empty)
BUSY(S, closed, up, b1, b2, b3, i) → IDLE(S, closed, stop, b1, b2, b3, i)
BUSY(F, closed, up, b1, false, b3, i) → IDLE(FS, closed, up, b1, false, b3, i)
BUSY(BF, closed, down, b1, b2, b3, i) → IDLE(B, closed, down, b1, b2, b3, i)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule BUSY(FS, closed, up, b1, b2, b3, i) → IDLE(S, closed, up, b1, b2, b3, i) we obtained the following new rules:
BUSY(FS, closed, up, z3, z4, z5, empty) → IDLE(S, closed, up, z3, z4, z5, empty)
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ AAECC Innermost
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
Q DP problem:
The TRS P consists of the following rules:
BUSY(FS, closed, down, b1, b2, b3, i) → IDLE(F, closed, down, b1, b2, b3, i)
BUSY(S, closed, down, b1, b2, false, i) → IDLE(FS, closed, down, b1, b2, false, i)
BUSY(BF, closed, up, b1, b2, b3, i) → IDLE(F, closed, up, b1, b2, b3, i)
BUSY(B, closed, stop, false, false, true, empty) → IDLE(B, closed, up, false, false, true, empty)
BUSY(F, closed, down, b1, true, b3, i) → IDLE(F, closed, stop, b1, true, b3, i)
BUSY(S, closed, stop, b1, true, false, i) → IDLE(S, closed, down, b1, true, false, i)
BUSY(S, closed, down, b1, b2, true, i) → IDLE(S, closed, stop, b1, b2, true, i)
BUSY(B, closed, up, true, b2, b3, i) → IDLE(B, closed, stop, true, b2, b3, i)
IDLE(fl, d, m, b1, b2, b3, empty) → BUSY(fl, d, m, b1, b2, b3, empty)
BUSY(F, closed, up, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty)
BUSY(B, closed, stop, false, true, b3, i) → IDLE(B, closed, up, false, true, b3, i)
BUSY(F, closed, stop, false, false, true, i) → IDLE(F, closed, up, false, false, true, i)
BUSY(B, closed, down, b1, b2, b3, i) → IDLE(B, closed, stop, b1, b2, b3, i)
BUSY(F, closed, stop, true, false, b3, i) → IDLE(F, closed, down, true, false, b3, i)
BUSY(S, closed, stop, true, false, false, i) → IDLE(S, closed, down, true, false, false, i)
BUSY(FS, closed, up, z3, z4, z5, empty) → IDLE(S, closed, up, z3, z4, z5, empty)
BUSY(F, closed, down, b1, false, b3, i) → IDLE(BF, closed, down, b1, false, b3, i)
BUSY(B, closed, up, false, z4, z5, empty) → IDLE(BF, closed, up, false, z4, z5, empty)
BUSY(S, closed, up, b1, b2, b3, i) → IDLE(S, closed, stop, b1, b2, b3, i)
BUSY(F, closed, up, b1, false, b3, i) → IDLE(FS, closed, up, b1, false, b3, i)
BUSY(BF, closed, down, b1, b2, b3, i) → IDLE(B, closed, down, b1, b2, b3, i)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule BUSY(FS, closed, down, b1, b2, b3, i) → IDLE(F, closed, down, b1, b2, b3, i) we obtained the following new rules:
BUSY(FS, closed, down, z3, z4, z5, empty) → IDLE(F, closed, down, z3, z4, z5, empty)
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ AAECC Innermost
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
Q DP problem:
The TRS P consists of the following rules:
BUSY(BF, closed, up, b1, b2, b3, i) → IDLE(F, closed, up, b1, b2, b3, i)
BUSY(S, closed, down, b1, b2, false, i) → IDLE(FS, closed, down, b1, b2, false, i)
BUSY(B, closed, stop, false, false, true, empty) → IDLE(B, closed, up, false, false, true, empty)
BUSY(F, closed, down, b1, true, b3, i) → IDLE(F, closed, stop, b1, true, b3, i)
BUSY(S, closed, stop, b1, true, false, i) → IDLE(S, closed, down, b1, true, false, i)
BUSY(S, closed, down, b1, b2, true, i) → IDLE(S, closed, stop, b1, b2, true, i)
BUSY(B, closed, up, true, b2, b3, i) → IDLE(B, closed, stop, true, b2, b3, i)
BUSY(F, closed, up, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty)
IDLE(fl, d, m, b1, b2, b3, empty) → BUSY(fl, d, m, b1, b2, b3, empty)
BUSY(B, closed, stop, false, true, b3, i) → IDLE(B, closed, up, false, true, b3, i)
BUSY(F, closed, stop, false, false, true, i) → IDLE(F, closed, up, false, false, true, i)
BUSY(B, closed, down, b1, b2, b3, i) → IDLE(B, closed, stop, b1, b2, b3, i)
BUSY(S, closed, stop, true, false, false, i) → IDLE(S, closed, down, true, false, false, i)
BUSY(F, closed, stop, true, false, b3, i) → IDLE(F, closed, down, true, false, b3, i)
BUSY(F, closed, down, b1, false, b3, i) → IDLE(BF, closed, down, b1, false, b3, i)
BUSY(FS, closed, up, z3, z4, z5, empty) → IDLE(S, closed, up, z3, z4, z5, empty)
BUSY(B, closed, up, false, z4, z5, empty) → IDLE(BF, closed, up, false, z4, z5, empty)
BUSY(S, closed, up, b1, b2, b3, i) → IDLE(S, closed, stop, b1, b2, b3, i)
BUSY(F, closed, up, b1, false, b3, i) → IDLE(FS, closed, up, b1, false, b3, i)
BUSY(FS, closed, down, z3, z4, z5, empty) → IDLE(F, closed, down, z3, z4, z5, empty)
BUSY(BF, closed, down, b1, b2, b3, i) → IDLE(B, closed, down, b1, b2, b3, i)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule BUSY(S, closed, down, b1, b2, false, i) → IDLE(FS, closed, down, b1, b2, false, i) we obtained the following new rules:
BUSY(S, closed, down, z3, z4, false, empty) → IDLE(FS, closed, down, z3, z4, false, empty)
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ AAECC Innermost
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
Q DP problem:
The TRS P consists of the following rules:
BUSY(BF, closed, up, b1, b2, b3, i) → IDLE(F, closed, up, b1, b2, b3, i)
BUSY(B, closed, stop, false, false, true, empty) → IDLE(B, closed, up, false, false, true, empty)
BUSY(S, closed, down, z3, z4, false, empty) → IDLE(FS, closed, down, z3, z4, false, empty)
BUSY(F, closed, down, b1, true, b3, i) → IDLE(F, closed, stop, b1, true, b3, i)
BUSY(S, closed, stop, b1, true, false, i) → IDLE(S, closed, down, b1, true, false, i)
BUSY(S, closed, down, b1, b2, true, i) → IDLE(S, closed, stop, b1, b2, true, i)
BUSY(B, closed, up, true, b2, b3, i) → IDLE(B, closed, stop, true, b2, b3, i)
IDLE(fl, d, m, b1, b2, b3, empty) → BUSY(fl, d, m, b1, b2, b3, empty)
BUSY(F, closed, up, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty)
BUSY(B, closed, stop, false, true, b3, i) → IDLE(B, closed, up, false, true, b3, i)
BUSY(F, closed, stop, false, false, true, i) → IDLE(F, closed, up, false, false, true, i)
BUSY(B, closed, down, b1, b2, b3, i) → IDLE(B, closed, stop, b1, b2, b3, i)
BUSY(F, closed, stop, true, false, b3, i) → IDLE(F, closed, down, true, false, b3, i)
BUSY(S, closed, stop, true, false, false, i) → IDLE(S, closed, down, true, false, false, i)
BUSY(FS, closed, up, z3, z4, z5, empty) → IDLE(S, closed, up, z3, z4, z5, empty)
BUSY(F, closed, down, b1, false, b3, i) → IDLE(BF, closed, down, b1, false, b3, i)
BUSY(B, closed, up, false, z4, z5, empty) → IDLE(BF, closed, up, false, z4, z5, empty)
BUSY(S, closed, up, b1, b2, b3, i) → IDLE(S, closed, stop, b1, b2, b3, i)
BUSY(F, closed, up, b1, false, b3, i) → IDLE(FS, closed, up, b1, false, b3, i)
BUSY(BF, closed, down, b1, b2, b3, i) → IDLE(B, closed, down, b1, b2, b3, i)
BUSY(FS, closed, down, z3, z4, z5, empty) → IDLE(F, closed, down, z3, z4, z5, empty)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule BUSY(BF, closed, up, b1, b2, b3, i) → IDLE(F, closed, up, b1, b2, b3, i) we obtained the following new rules:
BUSY(BF, closed, up, z3, z4, z5, empty) → IDLE(F, closed, up, z3, z4, z5, empty)
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ AAECC Innermost
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
Q DP problem:
The TRS P consists of the following rules:
BUSY(BF, closed, up, z3, z4, z5, empty) → IDLE(F, closed, up, z3, z4, z5, empty)
BUSY(B, closed, stop, false, false, true, empty) → IDLE(B, closed, up, false, false, true, empty)
BUSY(S, closed, down, z3, z4, false, empty) → IDLE(FS, closed, down, z3, z4, false, empty)
BUSY(F, closed, down, b1, true, b3, i) → IDLE(F, closed, stop, b1, true, b3, i)
BUSY(S, closed, stop, b1, true, false, i) → IDLE(S, closed, down, b1, true, false, i)
BUSY(S, closed, down, b1, b2, true, i) → IDLE(S, closed, stop, b1, b2, true, i)
BUSY(B, closed, up, true, b2, b3, i) → IDLE(B, closed, stop, true, b2, b3, i)
BUSY(F, closed, up, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty)
IDLE(fl, d, m, b1, b2, b3, empty) → BUSY(fl, d, m, b1, b2, b3, empty)
BUSY(B, closed, stop, false, true, b3, i) → IDLE(B, closed, up, false, true, b3, i)
BUSY(F, closed, stop, false, false, true, i) → IDLE(F, closed, up, false, false, true, i)
BUSY(B, closed, down, b1, b2, b3, i) → IDLE(B, closed, stop, b1, b2, b3, i)
BUSY(S, closed, stop, true, false, false, i) → IDLE(S, closed, down, true, false, false, i)
BUSY(F, closed, stop, true, false, b3, i) → IDLE(F, closed, down, true, false, b3, i)
BUSY(F, closed, down, b1, false, b3, i) → IDLE(BF, closed, down, b1, false, b3, i)
BUSY(FS, closed, up, z3, z4, z5, empty) → IDLE(S, closed, up, z3, z4, z5, empty)
BUSY(B, closed, up, false, z4, z5, empty) → IDLE(BF, closed, up, false, z4, z5, empty)
BUSY(S, closed, up, b1, b2, b3, i) → IDLE(S, closed, stop, b1, b2, b3, i)
BUSY(F, closed, up, b1, false, b3, i) → IDLE(FS, closed, up, b1, false, b3, i)
BUSY(FS, closed, down, z3, z4, z5, empty) → IDLE(F, closed, down, z3, z4, z5, empty)
BUSY(BF, closed, down, b1, b2, b3, i) → IDLE(B, closed, down, b1, b2, b3, i)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule BUSY(F, closed, down, b1, true, b3, i) → IDLE(F, closed, stop, b1, true, b3, i) we obtained the following new rules:
BUSY(F, closed, down, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty)
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ AAECC Innermost
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
Q DP problem:
The TRS P consists of the following rules:
BUSY(BF, closed, up, z3, z4, z5, empty) → IDLE(F, closed, up, z3, z4, z5, empty)
BUSY(B, closed, stop, false, false, true, empty) → IDLE(B, closed, up, false, false, true, empty)
BUSY(S, closed, down, z3, z4, false, empty) → IDLE(FS, closed, down, z3, z4, false, empty)
BUSY(S, closed, stop, b1, true, false, i) → IDLE(S, closed, down, b1, true, false, i)
BUSY(S, closed, down, b1, b2, true, i) → IDLE(S, closed, stop, b1, b2, true, i)
BUSY(B, closed, up, true, b2, b3, i) → IDLE(B, closed, stop, true, b2, b3, i)
IDLE(fl, d, m, b1, b2, b3, empty) → BUSY(fl, d, m, b1, b2, b3, empty)
BUSY(F, closed, up, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty)
BUSY(B, closed, stop, false, true, b3, i) → IDLE(B, closed, up, false, true, b3, i)
BUSY(F, closed, stop, false, false, true, i) → IDLE(F, closed, up, false, false, true, i)
BUSY(B, closed, down, b1, b2, b3, i) → IDLE(B, closed, stop, b1, b2, b3, i)
BUSY(F, closed, stop, true, false, b3, i) → IDLE(F, closed, down, true, false, b3, i)
BUSY(S, closed, stop, true, false, false, i) → IDLE(S, closed, down, true, false, false, i)
BUSY(FS, closed, up, z3, z4, z5, empty) → IDLE(S, closed, up, z3, z4, z5, empty)
BUSY(F, closed, down, b1, false, b3, i) → IDLE(BF, closed, down, b1, false, b3, i)
BUSY(B, closed, up, false, z4, z5, empty) → IDLE(BF, closed, up, false, z4, z5, empty)
BUSY(S, closed, up, b1, b2, b3, i) → IDLE(S, closed, stop, b1, b2, b3, i)
BUSY(F, closed, up, b1, false, b3, i) → IDLE(FS, closed, up, b1, false, b3, i)
BUSY(BF, closed, down, b1, b2, b3, i) → IDLE(B, closed, down, b1, b2, b3, i)
BUSY(FS, closed, down, z3, z4, z5, empty) → IDLE(F, closed, down, z3, z4, z5, empty)
BUSY(F, closed, down, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule BUSY(S, closed, stop, b1, true, false, i) → IDLE(S, closed, down, b1, true, false, i) we obtained the following new rules:
BUSY(S, closed, stop, z3, true, false, empty) → IDLE(S, closed, down, z3, true, false, empty)
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ AAECC Innermost
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
Q DP problem:
The TRS P consists of the following rules:
BUSY(BF, closed, up, z3, z4, z5, empty) → IDLE(F, closed, up, z3, z4, z5, empty)
BUSY(S, closed, stop, z3, true, false, empty) → IDLE(S, closed, down, z3, true, false, empty)
BUSY(B, closed, stop, false, false, true, empty) → IDLE(B, closed, up, false, false, true, empty)
BUSY(S, closed, down, z3, z4, false, empty) → IDLE(FS, closed, down, z3, z4, false, empty)
BUSY(S, closed, down, b1, b2, true, i) → IDLE(S, closed, stop, b1, b2, true, i)
BUSY(B, closed, up, true, b2, b3, i) → IDLE(B, closed, stop, true, b2, b3, i)
BUSY(F, closed, up, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty)
IDLE(fl, d, m, b1, b2, b3, empty) → BUSY(fl, d, m, b1, b2, b3, empty)
BUSY(B, closed, stop, false, true, b3, i) → IDLE(B, closed, up, false, true, b3, i)
BUSY(F, closed, stop, false, false, true, i) → IDLE(F, closed, up, false, false, true, i)
BUSY(B, closed, down, b1, b2, b3, i) → IDLE(B, closed, stop, b1, b2, b3, i)
BUSY(S, closed, stop, true, false, false, i) → IDLE(S, closed, down, true, false, false, i)
BUSY(F, closed, stop, true, false, b3, i) → IDLE(F, closed, down, true, false, b3, i)
BUSY(F, closed, down, b1, false, b3, i) → IDLE(BF, closed, down, b1, false, b3, i)
BUSY(FS, closed, up, z3, z4, z5, empty) → IDLE(S, closed, up, z3, z4, z5, empty)
BUSY(B, closed, up, false, z4, z5, empty) → IDLE(BF, closed, up, false, z4, z5, empty)
BUSY(S, closed, up, b1, b2, b3, i) → IDLE(S, closed, stop, b1, b2, b3, i)
BUSY(F, closed, up, b1, false, b3, i) → IDLE(FS, closed, up, b1, false, b3, i)
BUSY(FS, closed, down, z3, z4, z5, empty) → IDLE(F, closed, down, z3, z4, z5, empty)
BUSY(BF, closed, down, b1, b2, b3, i) → IDLE(B, closed, down, b1, b2, b3, i)
BUSY(F, closed, down, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule BUSY(S, closed, down, b1, b2, true, i) → IDLE(S, closed, stop, b1, b2, true, i) we obtained the following new rules:
BUSY(S, closed, down, z3, z4, true, empty) → IDLE(S, closed, stop, z3, z4, true, empty)
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ AAECC Innermost
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
Q DP problem:
The TRS P consists of the following rules:
BUSY(S, closed, down, z3, z4, true, empty) → IDLE(S, closed, stop, z3, z4, true, empty)
BUSY(BF, closed, up, z3, z4, z5, empty) → IDLE(F, closed, up, z3, z4, z5, empty)
BUSY(S, closed, stop, z3, true, false, empty) → IDLE(S, closed, down, z3, true, false, empty)
BUSY(B, closed, stop, false, false, true, empty) → IDLE(B, closed, up, false, false, true, empty)
BUSY(S, closed, down, z3, z4, false, empty) → IDLE(FS, closed, down, z3, z4, false, empty)
BUSY(B, closed, up, true, b2, b3, i) → IDLE(B, closed, stop, true, b2, b3, i)
IDLE(fl, d, m, b1, b2, b3, empty) → BUSY(fl, d, m, b1, b2, b3, empty)
BUSY(F, closed, up, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty)
BUSY(B, closed, stop, false, true, b3, i) → IDLE(B, closed, up, false, true, b3, i)
BUSY(F, closed, stop, false, false, true, i) → IDLE(F, closed, up, false, false, true, i)
BUSY(B, closed, down, b1, b2, b3, i) → IDLE(B, closed, stop, b1, b2, b3, i)
BUSY(F, closed, stop, true, false, b3, i) → IDLE(F, closed, down, true, false, b3, i)
BUSY(S, closed, stop, true, false, false, i) → IDLE(S, closed, down, true, false, false, i)
BUSY(FS, closed, up, z3, z4, z5, empty) → IDLE(S, closed, up, z3, z4, z5, empty)
BUSY(F, closed, down, b1, false, b3, i) → IDLE(BF, closed, down, b1, false, b3, i)
BUSY(B, closed, up, false, z4, z5, empty) → IDLE(BF, closed, up, false, z4, z5, empty)
BUSY(S, closed, up, b1, b2, b3, i) → IDLE(S, closed, stop, b1, b2, b3, i)
BUSY(F, closed, up, b1, false, b3, i) → IDLE(FS, closed, up, b1, false, b3, i)
BUSY(BF, closed, down, b1, b2, b3, i) → IDLE(B, closed, down, b1, b2, b3, i)
BUSY(FS, closed, down, z3, z4, z5, empty) → IDLE(F, closed, down, z3, z4, z5, empty)
BUSY(F, closed, down, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule BUSY(B, closed, up, true, b2, b3, i) → IDLE(B, closed, stop, true, b2, b3, i) we obtained the following new rules:
BUSY(B, closed, up, true, z4, z5, empty) → IDLE(B, closed, stop, true, z4, z5, empty)
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ AAECC Innermost
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
Q DP problem:
The TRS P consists of the following rules:
BUSY(S, closed, down, z3, z4, true, empty) → IDLE(S, closed, stop, z3, z4, true, empty)
BUSY(BF, closed, up, z3, z4, z5, empty) → IDLE(F, closed, up, z3, z4, z5, empty)
BUSY(B, closed, up, true, z4, z5, empty) → IDLE(B, closed, stop, true, z4, z5, empty)
BUSY(S, closed, stop, z3, true, false, empty) → IDLE(S, closed, down, z3, true, false, empty)
BUSY(B, closed, stop, false, false, true, empty) → IDLE(B, closed, up, false, false, true, empty)
BUSY(S, closed, down, z3, z4, false, empty) → IDLE(FS, closed, down, z3, z4, false, empty)
BUSY(F, closed, up, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty)
IDLE(fl, d, m, b1, b2, b3, empty) → BUSY(fl, d, m, b1, b2, b3, empty)
BUSY(B, closed, stop, false, true, b3, i) → IDLE(B, closed, up, false, true, b3, i)
BUSY(F, closed, stop, false, false, true, i) → IDLE(F, closed, up, false, false, true, i)
BUSY(B, closed, down, b1, b2, b3, i) → IDLE(B, closed, stop, b1, b2, b3, i)
BUSY(S, closed, stop, true, false, false, i) → IDLE(S, closed, down, true, false, false, i)
BUSY(F, closed, stop, true, false, b3, i) → IDLE(F, closed, down, true, false, b3, i)
BUSY(F, closed, down, b1, false, b3, i) → IDLE(BF, closed, down, b1, false, b3, i)
BUSY(FS, closed, up, z3, z4, z5, empty) → IDLE(S, closed, up, z3, z4, z5, empty)
BUSY(B, closed, up, false, z4, z5, empty) → IDLE(BF, closed, up, false, z4, z5, empty)
BUSY(S, closed, up, b1, b2, b3, i) → IDLE(S, closed, stop, b1, b2, b3, i)
BUSY(F, closed, up, b1, false, b3, i) → IDLE(FS, closed, up, b1, false, b3, i)
BUSY(FS, closed, down, z3, z4, z5, empty) → IDLE(F, closed, down, z3, z4, z5, empty)
BUSY(BF, closed, down, b1, b2, b3, i) → IDLE(B, closed, down, b1, b2, b3, i)
BUSY(F, closed, down, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule IDLE(fl, d, m, b1, b2, b3, empty) → BUSY(fl, d, m, b1, b2, b3, empty) we obtained the following new rules:
IDLE(F, closed, down, z0, z1, z2, empty) → BUSY(F, closed, down, z0, z1, z2, empty)
IDLE(FS, closed, down, z0, z1, false, empty) → BUSY(FS, closed, down, z0, z1, false, empty)
IDLE(F, closed, down, true, false, z0, empty) → BUSY(F, closed, down, true, false, z0, empty)
IDLE(S, closed, stop, z0, z1, true, empty) → BUSY(S, closed, stop, z0, z1, true, empty)
IDLE(S, closed, stop, z0, z1, z2, empty) → BUSY(S, closed, stop, z0, z1, z2, empty)
IDLE(BF, closed, down, z0, false, z1, empty) → BUSY(BF, closed, down, z0, false, z1, empty)
IDLE(F, closed, stop, z0, true, z1, empty) → BUSY(F, closed, stop, z0, true, z1, empty)
IDLE(B, closed, stop, true, z0, z1, empty) → BUSY(B, closed, stop, true, z0, z1, empty)
IDLE(F, closed, up, false, false, true, empty) → BUSY(F, closed, up, false, false, true, empty)
IDLE(S, closed, down, true, false, false, empty) → BUSY(S, closed, down, true, false, false, empty)
IDLE(B, closed, down, z0, z1, z2, empty) → BUSY(B, closed, down, z0, z1, z2, empty)
IDLE(F, closed, up, z0, z1, z2, empty) → BUSY(F, closed, up, z0, z1, z2, empty)
IDLE(B, closed, up, false, false, true, empty) → BUSY(B, closed, up, false, false, true, empty)
IDLE(S, closed, down, z0, true, false, empty) → BUSY(S, closed, down, z0, true, false, empty)
IDLE(B, closed, up, false, true, z0, empty) → BUSY(B, closed, up, false, true, z0, empty)
IDLE(B, closed, stop, z0, z1, z2, empty) → BUSY(B, closed, stop, z0, z1, z2, empty)
IDLE(FS, closed, up, z0, false, z1, empty) → BUSY(FS, closed, up, z0, false, z1, empty)
IDLE(BF, closed, up, false, z0, z1, empty) → BUSY(BF, closed, up, false, z0, z1, empty)
IDLE(S, closed, up, z0, z1, z2, empty) → BUSY(S, closed, up, z0, z1, z2, empty)
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ AAECC Innermost
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
BUSY(BF, closed, up, z3, z4, z5, empty) → IDLE(F, closed, up, z3, z4, z5, empty)
BUSY(B, closed, up, true, z4, z5, empty) → IDLE(B, closed, stop, true, z4, z5, empty)
BUSY(S, closed, stop, z3, true, false, empty) → IDLE(S, closed, down, z3, true, false, empty)
IDLE(S, closed, stop, z0, z1, true, empty) → BUSY(S, closed, stop, z0, z1, true, empty)
IDLE(F, closed, stop, z0, true, z1, empty) → BUSY(F, closed, stop, z0, true, z1, empty)
IDLE(BF, closed, down, z0, false, z1, empty) → BUSY(BF, closed, down, z0, false, z1, empty)
IDLE(S, closed, down, true, false, false, empty) → BUSY(S, closed, down, true, false, false, empty)
BUSY(F, closed, up, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty)
IDLE(B, closed, stop, z0, z1, z2, empty) → BUSY(B, closed, stop, z0, z1, z2, empty)
IDLE(FS, closed, up, z0, false, z1, empty) → BUSY(FS, closed, up, z0, false, z1, empty)
BUSY(B, closed, down, b1, b2, b3, i) → IDLE(B, closed, stop, b1, b2, b3, i)
BUSY(S, closed, stop, true, false, false, i) → IDLE(S, closed, down, true, false, false, i)
BUSY(F, closed, stop, true, false, b3, i) → IDLE(F, closed, down, true, false, b3, i)
IDLE(S, closed, up, z0, z1, z2, empty) → BUSY(S, closed, up, z0, z1, z2, empty)
BUSY(S, closed, up, b1, b2, b3, i) → IDLE(S, closed, stop, b1, b2, b3, i)
BUSY(BF, closed, down, b1, b2, b3, i) → IDLE(B, closed, down, b1, b2, b3, i)
BUSY(F, closed, down, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty)
IDLE(F, closed, down, z0, z1, z2, empty) → BUSY(F, closed, down, z0, z1, z2, empty)
BUSY(S, closed, down, z3, z4, true, empty) → IDLE(S, closed, stop, z3, z4, true, empty)
IDLE(FS, closed, down, z0, z1, false, empty) → BUSY(FS, closed, down, z0, z1, false, empty)
IDLE(F, closed, down, true, false, z0, empty) → BUSY(F, closed, down, true, false, z0, empty)
BUSY(B, closed, stop, false, false, true, empty) → IDLE(B, closed, up, false, false, true, empty)
IDLE(S, closed, stop, z0, z1, z2, empty) → BUSY(S, closed, stop, z0, z1, z2, empty)
BUSY(S, closed, down, z3, z4, false, empty) → IDLE(FS, closed, down, z3, z4, false, empty)
IDLE(F, closed, up, false, false, true, empty) → BUSY(F, closed, up, false, false, true, empty)
IDLE(B, closed, stop, true, z0, z1, empty) → BUSY(B, closed, stop, true, z0, z1, empty)
IDLE(B, closed, down, z0, z1, z2, empty) → BUSY(B, closed, down, z0, z1, z2, empty)
IDLE(B, closed, up, false, false, true, empty) → BUSY(B, closed, up, false, false, true, empty)
IDLE(F, closed, up, z0, z1, z2, empty) → BUSY(F, closed, up, z0, z1, z2, empty)
IDLE(B, closed, up, false, true, z0, empty) → BUSY(B, closed, up, false, true, z0, empty)
IDLE(S, closed, down, z0, true, false, empty) → BUSY(S, closed, down, z0, true, false, empty)
BUSY(B, closed, stop, false, true, b3, i) → IDLE(B, closed, up, false, true, b3, i)
BUSY(F, closed, stop, false, false, true, i) → IDLE(F, closed, up, false, false, true, i)
BUSY(F, closed, down, b1, false, b3, i) → IDLE(BF, closed, down, b1, false, b3, i)
BUSY(FS, closed, up, z3, z4, z5, empty) → IDLE(S, closed, up, z3, z4, z5, empty)
IDLE(BF, closed, up, false, z0, z1, empty) → BUSY(BF, closed, up, false, z0, z1, empty)
BUSY(B, closed, up, false, z4, z5, empty) → IDLE(BF, closed, up, false, z4, z5, empty)
BUSY(F, closed, up, b1, false, b3, i) → IDLE(FS, closed, up, b1, false, b3, i)
BUSY(FS, closed, down, z3, z4, z5, empty) → IDLE(F, closed, down, z3, z4, z5, empty)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 9 less nodes.
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ AAECC Innermost
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
Q DP problem:
The TRS P consists of the following rules:
BUSY(BF, closed, up, z3, z4, z5, empty) → IDLE(F, closed, up, z3, z4, z5, empty)
BUSY(S, closed, stop, z3, true, false, empty) → IDLE(S, closed, down, z3, true, false, empty)
IDLE(BF, closed, down, z0, false, z1, empty) → BUSY(BF, closed, down, z0, false, z1, empty)
IDLE(S, closed, down, true, false, false, empty) → BUSY(S, closed, down, true, false, false, empty)
IDLE(B, closed, stop, z0, z1, z2, empty) → BUSY(B, closed, stop, z0, z1, z2, empty)
IDLE(FS, closed, up, z0, false, z1, empty) → BUSY(FS, closed, up, z0, false, z1, empty)
BUSY(B, closed, down, b1, b2, b3, i) → IDLE(B, closed, stop, b1, b2, b3, i)
BUSY(S, closed, stop, true, false, false, i) → IDLE(S, closed, down, true, false, false, i)
BUSY(S, closed, up, b1, b2, b3, i) → IDLE(S, closed, stop, b1, b2, b3, i)
IDLE(S, closed, up, z0, z1, z2, empty) → BUSY(S, closed, up, z0, z1, z2, empty)
BUSY(BF, closed, down, b1, b2, b3, i) → IDLE(B, closed, down, b1, b2, b3, i)
IDLE(F, closed, down, z0, z1, z2, empty) → BUSY(F, closed, down, z0, z1, z2, empty)
IDLE(FS, closed, down, z0, z1, false, empty) → BUSY(FS, closed, down, z0, z1, false, empty)
IDLE(F, closed, down, true, false, z0, empty) → BUSY(F, closed, down, true, false, z0, empty)
BUSY(B, closed, stop, false, false, true, empty) → IDLE(B, closed, up, false, false, true, empty)
IDLE(S, closed, stop, z0, z1, z2, empty) → BUSY(S, closed, stop, z0, z1, z2, empty)
BUSY(S, closed, down, z3, z4, false, empty) → IDLE(FS, closed, down, z3, z4, false, empty)
IDLE(F, closed, up, false, false, true, empty) → BUSY(F, closed, up, false, false, true, empty)
IDLE(B, closed, down, z0, z1, z2, empty) → BUSY(B, closed, down, z0, z1, z2, empty)
IDLE(F, closed, up, z0, z1, z2, empty) → BUSY(F, closed, up, z0, z1, z2, empty)
IDLE(B, closed, up, false, false, true, empty) → BUSY(B, closed, up, false, false, true, empty)
IDLE(S, closed, down, z0, true, false, empty) → BUSY(S, closed, down, z0, true, false, empty)
IDLE(B, closed, up, false, true, z0, empty) → BUSY(B, closed, up, false, true, z0, empty)
BUSY(B, closed, stop, false, true, b3, i) → IDLE(B, closed, up, false, true, b3, i)
BUSY(FS, closed, up, z3, z4, z5, empty) → IDLE(S, closed, up, z3, z4, z5, empty)
BUSY(F, closed, down, b1, false, b3, i) → IDLE(BF, closed, down, b1, false, b3, i)
IDLE(BF, closed, up, false, z0, z1, empty) → BUSY(BF, closed, up, false, z0, z1, empty)
BUSY(B, closed, up, false, z4, z5, empty) → IDLE(BF, closed, up, false, z4, z5, empty)
BUSY(F, closed, up, b1, false, b3, i) → IDLE(FS, closed, up, b1, false, b3, i)
BUSY(FS, closed, down, z3, z4, z5, empty) → IDLE(F, closed, down, z3, z4, z5, empty)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule BUSY(B, closed, up, false, z4, z5, empty) → IDLE(BF, closed, up, false, z4, z5, empty) we obtained the following new rules:
BUSY(B, closed, up, false, false, true, empty) → IDLE(BF, closed, up, false, false, true, empty)
BUSY(B, closed, up, false, true, z0, empty) → IDLE(BF, closed, up, false, true, z0, empty)
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ AAECC Innermost
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
Q DP problem:
The TRS P consists of the following rules:
BUSY(BF, closed, up, z3, z4, z5, empty) → IDLE(F, closed, up, z3, z4, z5, empty)
BUSY(S, closed, stop, z3, true, false, empty) → IDLE(S, closed, down, z3, true, false, empty)
IDLE(BF, closed, down, z0, false, z1, empty) → BUSY(BF, closed, down, z0, false, z1, empty)
IDLE(S, closed, down, true, false, false, empty) → BUSY(S, closed, down, true, false, false, empty)
BUSY(B, closed, up, false, true, z0, empty) → IDLE(BF, closed, up, false, true, z0, empty)
IDLE(B, closed, stop, z0, z1, z2, empty) → BUSY(B, closed, stop, z0, z1, z2, empty)
IDLE(FS, closed, up, z0, false, z1, empty) → BUSY(FS, closed, up, z0, false, z1, empty)
BUSY(B, closed, down, b1, b2, b3, i) → IDLE(B, closed, stop, b1, b2, b3, i)
BUSY(S, closed, stop, true, false, false, i) → IDLE(S, closed, down, true, false, false, i)
IDLE(S, closed, up, z0, z1, z2, empty) → BUSY(S, closed, up, z0, z1, z2, empty)
BUSY(S, closed, up, b1, b2, b3, i) → IDLE(S, closed, stop, b1, b2, b3, i)
BUSY(BF, closed, down, b1, b2, b3, i) → IDLE(B, closed, down, b1, b2, b3, i)
BUSY(B, closed, up, false, false, true, empty) → IDLE(BF, closed, up, false, false, true, empty)
IDLE(F, closed, down, z0, z1, z2, empty) → BUSY(F, closed, down, z0, z1, z2, empty)
IDLE(FS, closed, down, z0, z1, false, empty) → BUSY(FS, closed, down, z0, z1, false, empty)
IDLE(F, closed, down, true, false, z0, empty) → BUSY(F, closed, down, true, false, z0, empty)
BUSY(B, closed, stop, false, false, true, empty) → IDLE(B, closed, up, false, false, true, empty)
IDLE(S, closed, stop, z0, z1, z2, empty) → BUSY(S, closed, stop, z0, z1, z2, empty)
BUSY(S, closed, down, z3, z4, false, empty) → IDLE(FS, closed, down, z3, z4, false, empty)
IDLE(F, closed, up, false, false, true, empty) → BUSY(F, closed, up, false, false, true, empty)
IDLE(B, closed, down, z0, z1, z2, empty) → BUSY(B, closed, down, z0, z1, z2, empty)
IDLE(B, closed, up, false, false, true, empty) → BUSY(B, closed, up, false, false, true, empty)
IDLE(F, closed, up, z0, z1, z2, empty) → BUSY(F, closed, up, z0, z1, z2, empty)
BUSY(B, closed, stop, false, true, b3, i) → IDLE(B, closed, up, false, true, b3, i)
IDLE(B, closed, up, false, true, z0, empty) → BUSY(B, closed, up, false, true, z0, empty)
IDLE(S, closed, down, z0, true, false, empty) → BUSY(S, closed, down, z0, true, false, empty)
BUSY(F, closed, down, b1, false, b3, i) → IDLE(BF, closed, down, b1, false, b3, i)
BUSY(FS, closed, up, z3, z4, z5, empty) → IDLE(S, closed, up, z3, z4, z5, empty)
IDLE(BF, closed, up, false, z0, z1, empty) → BUSY(BF, closed, up, false, z0, z1, empty)
BUSY(F, closed, up, b1, false, b3, i) → IDLE(FS, closed, up, b1, false, b3, i)
BUSY(FS, closed, down, z3, z4, z5, empty) → IDLE(F, closed, down, z3, z4, z5, empty)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule IDLE(BF, closed, up, false, z0, z1, empty) → BUSY(BF, closed, up, false, z0, z1, empty) we obtained the following new rules:
IDLE(BF, closed, up, false, false, true, empty) → BUSY(BF, closed, up, false, false, true, empty)
IDLE(BF, closed, up, false, true, z0, empty) → BUSY(BF, closed, up, false, true, z0, empty)
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ AAECC Innermost
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
Q DP problem:
The TRS P consists of the following rules:
BUSY(BF, closed, up, z3, z4, z5, empty) → IDLE(F, closed, up, z3, z4, z5, empty)
BUSY(S, closed, stop, z3, true, false, empty) → IDLE(S, closed, down, z3, true, false, empty)
IDLE(BF, closed, up, false, true, z0, empty) → BUSY(BF, closed, up, false, true, z0, empty)
IDLE(BF, closed, down, z0, false, z1, empty) → BUSY(BF, closed, down, z0, false, z1, empty)
IDLE(S, closed, down, true, false, false, empty) → BUSY(S, closed, down, true, false, false, empty)
IDLE(BF, closed, up, false, false, true, empty) → BUSY(BF, closed, up, false, false, true, empty)
IDLE(B, closed, stop, z0, z1, z2, empty) → BUSY(B, closed, stop, z0, z1, z2, empty)
BUSY(B, closed, up, false, true, z0, empty) → IDLE(BF, closed, up, false, true, z0, empty)
IDLE(FS, closed, up, z0, false, z1, empty) → BUSY(FS, closed, up, z0, false, z1, empty)
BUSY(B, closed, down, b1, b2, b3, i) → IDLE(B, closed, stop, b1, b2, b3, i)
BUSY(S, closed, stop, true, false, false, i) → IDLE(S, closed, down, true, false, false, i)
BUSY(S, closed, up, b1, b2, b3, i) → IDLE(S, closed, stop, b1, b2, b3, i)
IDLE(S, closed, up, z0, z1, z2, empty) → BUSY(S, closed, up, z0, z1, z2, empty)
BUSY(BF, closed, down, b1, b2, b3, i) → IDLE(B, closed, down, b1, b2, b3, i)
IDLE(F, closed, down, z0, z1, z2, empty) → BUSY(F, closed, down, z0, z1, z2, empty)
BUSY(B, closed, up, false, false, true, empty) → IDLE(BF, closed, up, false, false, true, empty)
IDLE(FS, closed, down, z0, z1, false, empty) → BUSY(FS, closed, down, z0, z1, false, empty)
IDLE(F, closed, down, true, false, z0, empty) → BUSY(F, closed, down, true, false, z0, empty)
BUSY(B, closed, stop, false, false, true, empty) → IDLE(B, closed, up, false, false, true, empty)
IDLE(S, closed, stop, z0, z1, z2, empty) → BUSY(S, closed, stop, z0, z1, z2, empty)
BUSY(S, closed, down, z3, z4, false, empty) → IDLE(FS, closed, down, z3, z4, false, empty)
IDLE(F, closed, up, false, false, true, empty) → BUSY(F, closed, up, false, false, true, empty)
IDLE(B, closed, down, z0, z1, z2, empty) → BUSY(B, closed, down, z0, z1, z2, empty)
IDLE(F, closed, up, z0, z1, z2, empty) → BUSY(F, closed, up, z0, z1, z2, empty)
IDLE(B, closed, up, false, false, true, empty) → BUSY(B, closed, up, false, false, true, empty)
IDLE(S, closed, down, z0, true, false, empty) → BUSY(S, closed, down, z0, true, false, empty)
IDLE(B, closed, up, false, true, z0, empty) → BUSY(B, closed, up, false, true, z0, empty)
BUSY(B, closed, stop, false, true, b3, i) → IDLE(B, closed, up, false, true, b3, i)
BUSY(FS, closed, up, z3, z4, z5, empty) → IDLE(S, closed, up, z3, z4, z5, empty)
BUSY(F, closed, down, b1, false, b3, i) → IDLE(BF, closed, down, b1, false, b3, i)
BUSY(F, closed, up, b1, false, b3, i) → IDLE(FS, closed, up, b1, false, b3, i)
BUSY(FS, closed, down, z3, z4, z5, empty) → IDLE(F, closed, down, z3, z4, z5, empty)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule BUSY(BF, closed, up, z3, z4, z5, empty) → IDLE(F, closed, up, z3, z4, z5, empty) we obtained the following new rules:
BUSY(BF, closed, up, false, true, z0, empty) → IDLE(F, closed, up, false, true, z0, empty)
BUSY(BF, closed, up, false, false, true, empty) → IDLE(F, closed, up, false, false, true, empty)
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ AAECC Innermost
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
Q DP problem:
The TRS P consists of the following rules:
BUSY(S, closed, stop, z3, true, false, empty) → IDLE(S, closed, down, z3, true, false, empty)
BUSY(BF, closed, up, false, true, z0, empty) → IDLE(F, closed, up, false, true, z0, empty)
IDLE(BF, closed, up, false, true, z0, empty) → BUSY(BF, closed, up, false, true, z0, empty)
IDLE(BF, closed, down, z0, false, z1, empty) → BUSY(BF, closed, down, z0, false, z1, empty)
IDLE(S, closed, down, true, false, false, empty) → BUSY(S, closed, down, true, false, false, empty)
BUSY(B, closed, up, false, true, z0, empty) → IDLE(BF, closed, up, false, true, z0, empty)
IDLE(B, closed, stop, z0, z1, z2, empty) → BUSY(B, closed, stop, z0, z1, z2, empty)
IDLE(BF, closed, up, false, false, true, empty) → BUSY(BF, closed, up, false, false, true, empty)
IDLE(FS, closed, up, z0, false, z1, empty) → BUSY(FS, closed, up, z0, false, z1, empty)
BUSY(B, closed, down, b1, b2, b3, i) → IDLE(B, closed, stop, b1, b2, b3, i)
BUSY(S, closed, stop, true, false, false, i) → IDLE(S, closed, down, true, false, false, i)
BUSY(BF, closed, up, false, false, true, empty) → IDLE(F, closed, up, false, false, true, empty)
IDLE(S, closed, up, z0, z1, z2, empty) → BUSY(S, closed, up, z0, z1, z2, empty)
BUSY(S, closed, up, b1, b2, b3, i) → IDLE(S, closed, stop, b1, b2, b3, i)
BUSY(BF, closed, down, b1, b2, b3, i) → IDLE(B, closed, down, b1, b2, b3, i)
BUSY(B, closed, up, false, false, true, empty) → IDLE(BF, closed, up, false, false, true, empty)
IDLE(F, closed, down, z0, z1, z2, empty) → BUSY(F, closed, down, z0, z1, z2, empty)
IDLE(FS, closed, down, z0, z1, false, empty) → BUSY(FS, closed, down, z0, z1, false, empty)
IDLE(F, closed, down, true, false, z0, empty) → BUSY(F, closed, down, true, false, z0, empty)
BUSY(B, closed, stop, false, false, true, empty) → IDLE(B, closed, up, false, false, true, empty)
IDLE(S, closed, stop, z0, z1, z2, empty) → BUSY(S, closed, stop, z0, z1, z2, empty)
BUSY(S, closed, down, z3, z4, false, empty) → IDLE(FS, closed, down, z3, z4, false, empty)
IDLE(F, closed, up, false, false, true, empty) → BUSY(F, closed, up, false, false, true, empty)
IDLE(B, closed, down, z0, z1, z2, empty) → BUSY(B, closed, down, z0, z1, z2, empty)
IDLE(B, closed, up, false, false, true, empty) → BUSY(B, closed, up, false, false, true, empty)
IDLE(F, closed, up, z0, z1, z2, empty) → BUSY(F, closed, up, z0, z1, z2, empty)
BUSY(B, closed, stop, false, true, b3, i) → IDLE(B, closed, up, false, true, b3, i)
IDLE(B, closed, up, false, true, z0, empty) → BUSY(B, closed, up, false, true, z0, empty)
IDLE(S, closed, down, z0, true, false, empty) → BUSY(S, closed, down, z0, true, false, empty)
BUSY(F, closed, down, b1, false, b3, i) → IDLE(BF, closed, down, b1, false, b3, i)
BUSY(FS, closed, up, z3, z4, z5, empty) → IDLE(S, closed, up, z3, z4, z5, empty)
BUSY(F, closed, up, b1, false, b3, i) → IDLE(FS, closed, up, b1, false, b3, i)
BUSY(FS, closed, down, z3, z4, z5, empty) → IDLE(F, closed, down, z3, z4, z5, empty)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule BUSY(F, closed, up, b1, false, b3, i) → IDLE(FS, closed, up, b1, false, b3, i) we obtained the following new rules:
BUSY(F, closed, up, z0, false, z2, empty) → IDLE(FS, closed, up, z0, false, z2, empty)
BUSY(F, closed, up, false, false, true, empty) → IDLE(FS, closed, up, false, false, true, empty)
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ AAECC Innermost
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
Q DP problem:
The TRS P consists of the following rules:
BUSY(S, closed, stop, z3, true, false, empty) → IDLE(S, closed, down, z3, true, false, empty)
BUSY(BF, closed, up, false, true, z0, empty) → IDLE(F, closed, up, false, true, z0, empty)
IDLE(BF, closed, up, false, true, z0, empty) → BUSY(BF, closed, up, false, true, z0, empty)
IDLE(BF, closed, down, z0, false, z1, empty) → BUSY(BF, closed, down, z0, false, z1, empty)
IDLE(S, closed, down, true, false, false, empty) → BUSY(S, closed, down, true, false, false, empty)
BUSY(F, closed, up, z0, false, z2, empty) → IDLE(FS, closed, up, z0, false, z2, empty)
IDLE(BF, closed, up, false, false, true, empty) → BUSY(BF, closed, up, false, false, true, empty)
IDLE(B, closed, stop, z0, z1, z2, empty) → BUSY(B, closed, stop, z0, z1, z2, empty)
BUSY(B, closed, up, false, true, z0, empty) → IDLE(BF, closed, up, false, true, z0, empty)
IDLE(FS, closed, up, z0, false, z1, empty) → BUSY(FS, closed, up, z0, false, z1, empty)
BUSY(B, closed, down, b1, b2, b3, i) → IDLE(B, closed, stop, b1, b2, b3, i)
BUSY(S, closed, stop, true, false, false, i) → IDLE(S, closed, down, true, false, false, i)
BUSY(BF, closed, up, false, false, true, empty) → IDLE(F, closed, up, false, false, true, empty)
BUSY(S, closed, up, b1, b2, b3, i) → IDLE(S, closed, stop, b1, b2, b3, i)
IDLE(S, closed, up, z0, z1, z2, empty) → BUSY(S, closed, up, z0, z1, z2, empty)
BUSY(BF, closed, down, b1, b2, b3, i) → IDLE(B, closed, down, b1, b2, b3, i)
IDLE(F, closed, down, z0, z1, z2, empty) → BUSY(F, closed, down, z0, z1, z2, empty)
BUSY(B, closed, up, false, false, true, empty) → IDLE(BF, closed, up, false, false, true, empty)
IDLE(FS, closed, down, z0, z1, false, empty) → BUSY(FS, closed, down, z0, z1, false, empty)
IDLE(F, closed, down, true, false, z0, empty) → BUSY(F, closed, down, true, false, z0, empty)
BUSY(B, closed, stop, false, false, true, empty) → IDLE(B, closed, up, false, false, true, empty)
IDLE(S, closed, stop, z0, z1, z2, empty) → BUSY(S, closed, stop, z0, z1, z2, empty)
BUSY(S, closed, down, z3, z4, false, empty) → IDLE(FS, closed, down, z3, z4, false, empty)
IDLE(F, closed, up, false, false, true, empty) → BUSY(F, closed, up, false, false, true, empty)
IDLE(B, closed, down, z0, z1, z2, empty) → BUSY(B, closed, down, z0, z1, z2, empty)
IDLE(F, closed, up, z0, z1, z2, empty) → BUSY(F, closed, up, z0, z1, z2, empty)
IDLE(B, closed, up, false, false, true, empty) → BUSY(B, closed, up, false, false, true, empty)
IDLE(S, closed, down, z0, true, false, empty) → BUSY(S, closed, down, z0, true, false, empty)
IDLE(B, closed, up, false, true, z0, empty) → BUSY(B, closed, up, false, true, z0, empty)
BUSY(B, closed, stop, false, true, b3, i) → IDLE(B, closed, up, false, true, b3, i)
BUSY(FS, closed, up, z3, z4, z5, empty) → IDLE(S, closed, up, z3, z4, z5, empty)
BUSY(F, closed, down, b1, false, b3, i) → IDLE(BF, closed, down, b1, false, b3, i)
BUSY(F, closed, up, false, false, true, empty) → IDLE(FS, closed, up, false, false, true, empty)
BUSY(FS, closed, down, z3, z4, z5, empty) → IDLE(F, closed, down, z3, z4, z5, empty)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule BUSY(FS, closed, up, z3, z4, z5, empty) → IDLE(S, closed, up, z3, z4, z5, empty) we obtained the following new rules:
BUSY(FS, closed, up, z0, false, z1, empty) → IDLE(S, closed, up, z0, false, z1, empty)
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ AAECC Innermost
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
Q DP problem:
The TRS P consists of the following rules:
BUSY(S, closed, stop, z3, true, false, empty) → IDLE(S, closed, down, z3, true, false, empty)
BUSY(FS, closed, up, z0, false, z1, empty) → IDLE(S, closed, up, z0, false, z1, empty)
BUSY(BF, closed, up, false, true, z0, empty) → IDLE(F, closed, up, false, true, z0, empty)
IDLE(BF, closed, up, false, true, z0, empty) → BUSY(BF, closed, up, false, true, z0, empty)
IDLE(BF, closed, down, z0, false, z1, empty) → BUSY(BF, closed, down, z0, false, z1, empty)
IDLE(S, closed, down, true, false, false, empty) → BUSY(S, closed, down, true, false, false, empty)
BUSY(B, closed, up, false, true, z0, empty) → IDLE(BF, closed, up, false, true, z0, empty)
IDLE(B, closed, stop, z0, z1, z2, empty) → BUSY(B, closed, stop, z0, z1, z2, empty)
IDLE(BF, closed, up, false, false, true, empty) → BUSY(BF, closed, up, false, false, true, empty)
BUSY(F, closed, up, z0, false, z2, empty) → IDLE(FS, closed, up, z0, false, z2, empty)
IDLE(FS, closed, up, z0, false, z1, empty) → BUSY(FS, closed, up, z0, false, z1, empty)
BUSY(B, closed, down, b1, b2, b3, i) → IDLE(B, closed, stop, b1, b2, b3, i)
BUSY(S, closed, stop, true, false, false, i) → IDLE(S, closed, down, true, false, false, i)
BUSY(BF, closed, up, false, false, true, empty) → IDLE(F, closed, up, false, false, true, empty)
IDLE(S, closed, up, z0, z1, z2, empty) → BUSY(S, closed, up, z0, z1, z2, empty)
BUSY(S, closed, up, b1, b2, b3, i) → IDLE(S, closed, stop, b1, b2, b3, i)
BUSY(BF, closed, down, b1, b2, b3, i) → IDLE(B, closed, down, b1, b2, b3, i)
BUSY(B, closed, up, false, false, true, empty) → IDLE(BF, closed, up, false, false, true, empty)
IDLE(F, closed, down, z0, z1, z2, empty) → BUSY(F, closed, down, z0, z1, z2, empty)
IDLE(FS, closed, down, z0, z1, false, empty) → BUSY(FS, closed, down, z0, z1, false, empty)
IDLE(F, closed, down, true, false, z0, empty) → BUSY(F, closed, down, true, false, z0, empty)
BUSY(B, closed, stop, false, false, true, empty) → IDLE(B, closed, up, false, false, true, empty)
IDLE(S, closed, stop, z0, z1, z2, empty) → BUSY(S, closed, stop, z0, z1, z2, empty)
BUSY(S, closed, down, z3, z4, false, empty) → IDLE(FS, closed, down, z3, z4, false, empty)
IDLE(F, closed, up, false, false, true, empty) → BUSY(F, closed, up, false, false, true, empty)
IDLE(B, closed, down, z0, z1, z2, empty) → BUSY(B, closed, down, z0, z1, z2, empty)
IDLE(B, closed, up, false, false, true, empty) → BUSY(B, closed, up, false, false, true, empty)
IDLE(F, closed, up, z0, z1, z2, empty) → BUSY(F, closed, up, z0, z1, z2, empty)
BUSY(B, closed, stop, false, true, b3, i) → IDLE(B, closed, up, false, true, b3, i)
IDLE(B, closed, up, false, true, z0, empty) → BUSY(B, closed, up, false, true, z0, empty)
IDLE(S, closed, down, z0, true, false, empty) → BUSY(S, closed, down, z0, true, false, empty)
BUSY(F, closed, down, b1, false, b3, i) → IDLE(BF, closed, down, b1, false, b3, i)
BUSY(F, closed, up, false, false, true, empty) → IDLE(FS, closed, up, false, false, true, empty)
BUSY(FS, closed, down, z3, z4, z5, empty) → IDLE(F, closed, down, z3, z4, z5, empty)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule IDLE(S, closed, up, z0, z1, z2, empty) → BUSY(S, closed, up, z0, z1, z2, empty) we obtained the following new rules:
IDLE(S, closed, up, z0, false, z1, empty) → BUSY(S, closed, up, z0, false, z1, empty)
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ AAECC Innermost
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
Q DP problem:
The TRS P consists of the following rules:
BUSY(S, closed, stop, z3, true, false, empty) → IDLE(S, closed, down, z3, true, false, empty)
BUSY(FS, closed, up, z0, false, z1, empty) → IDLE(S, closed, up, z0, false, z1, empty)
BUSY(BF, closed, up, false, true, z0, empty) → IDLE(F, closed, up, false, true, z0, empty)
IDLE(BF, closed, up, false, true, z0, empty) → BUSY(BF, closed, up, false, true, z0, empty)
IDLE(BF, closed, down, z0, false, z1, empty) → BUSY(BF, closed, down, z0, false, z1, empty)
IDLE(S, closed, down, true, false, false, empty) → BUSY(S, closed, down, true, false, false, empty)
BUSY(F, closed, up, z0, false, z2, empty) → IDLE(FS, closed, up, z0, false, z2, empty)
IDLE(BF, closed, up, false, false, true, empty) → BUSY(BF, closed, up, false, false, true, empty)
IDLE(B, closed, stop, z0, z1, z2, empty) → BUSY(B, closed, stop, z0, z1, z2, empty)
BUSY(B, closed, up, false, true, z0, empty) → IDLE(BF, closed, up, false, true, z0, empty)
IDLE(FS, closed, up, z0, false, z1, empty) → BUSY(FS, closed, up, z0, false, z1, empty)
BUSY(B, closed, down, b1, b2, b3, i) → IDLE(B, closed, stop, b1, b2, b3, i)
BUSY(S, closed, stop, true, false, false, i) → IDLE(S, closed, down, true, false, false, i)
BUSY(BF, closed, up, false, false, true, empty) → IDLE(F, closed, up, false, false, true, empty)
BUSY(S, closed, up, b1, b2, b3, i) → IDLE(S, closed, stop, b1, b2, b3, i)
BUSY(BF, closed, down, b1, b2, b3, i) → IDLE(B, closed, down, b1, b2, b3, i)
IDLE(F, closed, down, z0, z1, z2, empty) → BUSY(F, closed, down, z0, z1, z2, empty)
BUSY(B, closed, up, false, false, true, empty) → IDLE(BF, closed, up, false, false, true, empty)
IDLE(FS, closed, down, z0, z1, false, empty) → BUSY(FS, closed, down, z0, z1, false, empty)
IDLE(F, closed, down, true, false, z0, empty) → BUSY(F, closed, down, true, false, z0, empty)
BUSY(B, closed, stop, false, false, true, empty) → IDLE(B, closed, up, false, false, true, empty)
IDLE(S, closed, stop, z0, z1, z2, empty) → BUSY(S, closed, stop, z0, z1, z2, empty)
BUSY(S, closed, down, z3, z4, false, empty) → IDLE(FS, closed, down, z3, z4, false, empty)
IDLE(S, closed, up, z0, false, z1, empty) → BUSY(S, closed, up, z0, false, z1, empty)
IDLE(F, closed, up, false, false, true, empty) → BUSY(F, closed, up, false, false, true, empty)
IDLE(B, closed, down, z0, z1, z2, empty) → BUSY(B, closed, down, z0, z1, z2, empty)
IDLE(F, closed, up, z0, z1, z2, empty) → BUSY(F, closed, up, z0, z1, z2, empty)
IDLE(B, closed, up, false, false, true, empty) → BUSY(B, closed, up, false, false, true, empty)
IDLE(S, closed, down, z0, true, false, empty) → BUSY(S, closed, down, z0, true, false, empty)
IDLE(B, closed, up, false, true, z0, empty) → BUSY(B, closed, up, false, true, z0, empty)
BUSY(B, closed, stop, false, true, b3, i) → IDLE(B, closed, up, false, true, b3, i)
BUSY(F, closed, down, b1, false, b3, i) → IDLE(BF, closed, down, b1, false, b3, i)
BUSY(F, closed, up, false, false, true, empty) → IDLE(FS, closed, up, false, false, true, empty)
BUSY(FS, closed, down, z3, z4, z5, empty) → IDLE(F, closed, down, z3, z4, z5, empty)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule BUSY(S, closed, up, b1, b2, b3, i) → IDLE(S, closed, stop, b1, b2, b3, i) we obtained the following new rules:
BUSY(S, closed, up, z0, false, z1, empty) → IDLE(S, closed, stop, z0, false, z1, empty)
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ AAECC Innermost
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
Q DP problem:
The TRS P consists of the following rules:
BUSY(S, closed, stop, z3, true, false, empty) → IDLE(S, closed, down, z3, true, false, empty)
BUSY(FS, closed, up, z0, false, z1, empty) → IDLE(S, closed, up, z0, false, z1, empty)
BUSY(BF, closed, up, false, true, z0, empty) → IDLE(F, closed, up, false, true, z0, empty)
IDLE(BF, closed, up, false, true, z0, empty) → BUSY(BF, closed, up, false, true, z0, empty)
IDLE(BF, closed, down, z0, false, z1, empty) → BUSY(BF, closed, down, z0, false, z1, empty)
IDLE(S, closed, down, true, false, false, empty) → BUSY(S, closed, down, true, false, false, empty)
BUSY(B, closed, up, false, true, z0, empty) → IDLE(BF, closed, up, false, true, z0, empty)
IDLE(B, closed, stop, z0, z1, z2, empty) → BUSY(B, closed, stop, z0, z1, z2, empty)
IDLE(BF, closed, up, false, false, true, empty) → BUSY(BF, closed, up, false, false, true, empty)
BUSY(F, closed, up, z0, false, z2, empty) → IDLE(FS, closed, up, z0, false, z2, empty)
IDLE(FS, closed, up, z0, false, z1, empty) → BUSY(FS, closed, up, z0, false, z1, empty)
BUSY(S, closed, up, z0, false, z1, empty) → IDLE(S, closed, stop, z0, false, z1, empty)
BUSY(B, closed, down, b1, b2, b3, i) → IDLE(B, closed, stop, b1, b2, b3, i)
BUSY(S, closed, stop, true, false, false, i) → IDLE(S, closed, down, true, false, false, i)
BUSY(BF, closed, up, false, false, true, empty) → IDLE(F, closed, up, false, false, true, empty)
BUSY(BF, closed, down, b1, b2, b3, i) → IDLE(B, closed, down, b1, b2, b3, i)
BUSY(B, closed, up, false, false, true, empty) → IDLE(BF, closed, up, false, false, true, empty)
IDLE(F, closed, down, z0, z1, z2, empty) → BUSY(F, closed, down, z0, z1, z2, empty)
IDLE(FS, closed, down, z0, z1, false, empty) → BUSY(FS, closed, down, z0, z1, false, empty)
IDLE(F, closed, down, true, false, z0, empty) → BUSY(F, closed, down, true, false, z0, empty)
BUSY(B, closed, stop, false, false, true, empty) → IDLE(B, closed, up, false, false, true, empty)
IDLE(S, closed, stop, z0, z1, z2, empty) → BUSY(S, closed, stop, z0, z1, z2, empty)
BUSY(S, closed, down, z3, z4, false, empty) → IDLE(FS, closed, down, z3, z4, false, empty)
IDLE(S, closed, up, z0, false, z1, empty) → BUSY(S, closed, up, z0, false, z1, empty)
IDLE(F, closed, up, false, false, true, empty) → BUSY(F, closed, up, false, false, true, empty)
IDLE(B, closed, down, z0, z1, z2, empty) → BUSY(B, closed, down, z0, z1, z2, empty)
IDLE(B, closed, up, false, false, true, empty) → BUSY(B, closed, up, false, false, true, empty)
IDLE(F, closed, up, z0, z1, z2, empty) → BUSY(F, closed, up, z0, z1, z2, empty)
BUSY(B, closed, stop, false, true, b3, i) → IDLE(B, closed, up, false, true, b3, i)
IDLE(B, closed, up, false, true, z0, empty) → BUSY(B, closed, up, false, true, z0, empty)
IDLE(S, closed, down, z0, true, false, empty) → BUSY(S, closed, down, z0, true, false, empty)
BUSY(F, closed, down, b1, false, b3, i) → IDLE(BF, closed, down, b1, false, b3, i)
BUSY(F, closed, up, false, false, true, empty) → IDLE(FS, closed, up, false, false, true, empty)
BUSY(FS, closed, down, z3, z4, z5, empty) → IDLE(F, closed, down, z3, z4, z5, empty)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule IDLE(S, closed, stop, z0, z1, z2, empty) → BUSY(S, closed, stop, z0, z1, z2, empty) we obtained the following new rules:
IDLE(S, closed, stop, z0, false, z1, empty) → BUSY(S, closed, stop, z0, false, z1, empty)
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ AAECC Innermost
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
BUSY(S, closed, stop, z3, true, false, empty) → IDLE(S, closed, down, z3, true, false, empty)
BUSY(FS, closed, up, z0, false, z1, empty) → IDLE(S, closed, up, z0, false, z1, empty)
BUSY(BF, closed, up, false, true, z0, empty) → IDLE(F, closed, up, false, true, z0, empty)
IDLE(BF, closed, up, false, true, z0, empty) → BUSY(BF, closed, up, false, true, z0, empty)
IDLE(BF, closed, down, z0, false, z1, empty) → BUSY(BF, closed, down, z0, false, z1, empty)
IDLE(S, closed, down, true, false, false, empty) → BUSY(S, closed, down, true, false, false, empty)
BUSY(F, closed, up, z0, false, z2, empty) → IDLE(FS, closed, up, z0, false, z2, empty)
IDLE(BF, closed, up, false, false, true, empty) → BUSY(BF, closed, up, false, false, true, empty)
IDLE(B, closed, stop, z0, z1, z2, empty) → BUSY(B, closed, stop, z0, z1, z2, empty)
BUSY(B, closed, up, false, true, z0, empty) → IDLE(BF, closed, up, false, true, z0, empty)
IDLE(FS, closed, up, z0, false, z1, empty) → BUSY(FS, closed, up, z0, false, z1, empty)
BUSY(B, closed, down, b1, b2, b3, i) → IDLE(B, closed, stop, b1, b2, b3, i)
BUSY(S, closed, up, z0, false, z1, empty) → IDLE(S, closed, stop, z0, false, z1, empty)
BUSY(S, closed, stop, true, false, false, i) → IDLE(S, closed, down, true, false, false, i)
BUSY(BF, closed, up, false, false, true, empty) → IDLE(F, closed, up, false, false, true, empty)
BUSY(BF, closed, down, b1, b2, b3, i) → IDLE(B, closed, down, b1, b2, b3, i)
IDLE(F, closed, down, z0, z1, z2, empty) → BUSY(F, closed, down, z0, z1, z2, empty)
BUSY(B, closed, up, false, false, true, empty) → IDLE(BF, closed, up, false, false, true, empty)
IDLE(FS, closed, down, z0, z1, false, empty) → BUSY(FS, closed, down, z0, z1, false, empty)
IDLE(F, closed, down, true, false, z0, empty) → BUSY(F, closed, down, true, false, z0, empty)
BUSY(B, closed, stop, false, false, true, empty) → IDLE(B, closed, up, false, false, true, empty)
BUSY(S, closed, down, z3, z4, false, empty) → IDLE(FS, closed, down, z3, z4, false, empty)
IDLE(S, closed, up, z0, false, z1, empty) → BUSY(S, closed, up, z0, false, z1, empty)
IDLE(F, closed, up, false, false, true, empty) → BUSY(F, closed, up, false, false, true, empty)
IDLE(B, closed, down, z0, z1, z2, empty) → BUSY(B, closed, down, z0, z1, z2, empty)
IDLE(F, closed, up, z0, z1, z2, empty) → BUSY(F, closed, up, z0, z1, z2, empty)
IDLE(B, closed, up, false, false, true, empty) → BUSY(B, closed, up, false, false, true, empty)
IDLE(S, closed, stop, z0, false, z1, empty) → BUSY(S, closed, stop, z0, false, z1, empty)
IDLE(S, closed, down, z0, true, false, empty) → BUSY(S, closed, down, z0, true, false, empty)
IDLE(B, closed, up, false, true, z0, empty) → BUSY(B, closed, up, false, true, z0, empty)
BUSY(B, closed, stop, false, true, b3, i) → IDLE(B, closed, up, false, true, b3, i)
BUSY(F, closed, down, b1, false, b3, i) → IDLE(BF, closed, down, b1, false, b3, i)
BUSY(F, closed, up, false, false, true, empty) → IDLE(FS, closed, up, false, false, true, empty)
BUSY(FS, closed, down, z3, z4, z5, empty) → IDLE(F, closed, down, z3, z4, z5, empty)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 2 less nodes.
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ AAECC Innermost
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
Q DP problem:
The TRS P consists of the following rules:
BUSY(FS, closed, up, z0, false, z1, empty) → IDLE(S, closed, up, z0, false, z1, empty)
BUSY(BF, closed, up, false, true, z0, empty) → IDLE(F, closed, up, false, true, z0, empty)
IDLE(BF, closed, up, false, true, z0, empty) → BUSY(BF, closed, up, false, true, z0, empty)
IDLE(BF, closed, down, z0, false, z1, empty) → BUSY(BF, closed, down, z0, false, z1, empty)
IDLE(S, closed, down, true, false, false, empty) → BUSY(S, closed, down, true, false, false, empty)
IDLE(BF, closed, up, false, false, true, empty) → BUSY(BF, closed, up, false, false, true, empty)
BUSY(F, closed, up, z0, false, z2, empty) → IDLE(FS, closed, up, z0, false, z2, empty)
BUSY(B, closed, up, false, true, z0, empty) → IDLE(BF, closed, up, false, true, z0, empty)
IDLE(B, closed, stop, z0, z1, z2, empty) → BUSY(B, closed, stop, z0, z1, z2, empty)
IDLE(FS, closed, up, z0, false, z1, empty) → BUSY(FS, closed, up, z0, false, z1, empty)
BUSY(S, closed, up, z0, false, z1, empty) → IDLE(S, closed, stop, z0, false, z1, empty)
BUSY(B, closed, down, b1, b2, b3, i) → IDLE(B, closed, stop, b1, b2, b3, i)
BUSY(S, closed, stop, true, false, false, i) → IDLE(S, closed, down, true, false, false, i)
BUSY(BF, closed, up, false, false, true, empty) → IDLE(F, closed, up, false, false, true, empty)
BUSY(BF, closed, down, b1, b2, b3, i) → IDLE(B, closed, down, b1, b2, b3, i)
BUSY(B, closed, up, false, false, true, empty) → IDLE(BF, closed, up, false, false, true, empty)
IDLE(F, closed, down, z0, z1, z2, empty) → BUSY(F, closed, down, z0, z1, z2, empty)
IDLE(FS, closed, down, z0, z1, false, empty) → BUSY(FS, closed, down, z0, z1, false, empty)
IDLE(F, closed, down, true, false, z0, empty) → BUSY(F, closed, down, true, false, z0, empty)
BUSY(B, closed, stop, false, false, true, empty) → IDLE(B, closed, up, false, false, true, empty)
BUSY(S, closed, down, z3, z4, false, empty) → IDLE(FS, closed, down, z3, z4, false, empty)
IDLE(S, closed, up, z0, false, z1, empty) → BUSY(S, closed, up, z0, false, z1, empty)
IDLE(F, closed, up, false, false, true, empty) → BUSY(F, closed, up, false, false, true, empty)
IDLE(B, closed, down, z0, z1, z2, empty) → BUSY(B, closed, down, z0, z1, z2, empty)
IDLE(B, closed, up, false, false, true, empty) → BUSY(B, closed, up, false, false, true, empty)
IDLE(F, closed, up, z0, z1, z2, empty) → BUSY(F, closed, up, z0, z1, z2, empty)
IDLE(S, closed, stop, z0, false, z1, empty) → BUSY(S, closed, stop, z0, false, z1, empty)
BUSY(B, closed, stop, false, true, b3, i) → IDLE(B, closed, up, false, true, b3, i)
IDLE(B, closed, up, false, true, z0, empty) → BUSY(B, closed, up, false, true, z0, empty)
BUSY(F, closed, down, b1, false, b3, i) → IDLE(BF, closed, down, b1, false, b3, i)
BUSY(F, closed, up, false, false, true, empty) → IDLE(FS, closed, up, false, false, true, empty)
BUSY(FS, closed, down, z3, z4, z5, empty) → IDLE(F, closed, down, z3, z4, z5, empty)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule IDLE(F, closed, up, z0, z1, z2, empty) → BUSY(F, closed, up, z0, z1, z2, empty) we obtained the following new rules:
IDLE(F, closed, up, false, false, true, empty) → BUSY(F, closed, up, false, false, true, empty)
IDLE(F, closed, up, false, true, z0, empty) → BUSY(F, closed, up, false, true, z0, empty)
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ AAECC Innermost
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
BUSY(FS, closed, up, z0, false, z1, empty) → IDLE(S, closed, up, z0, false, z1, empty)
BUSY(BF, closed, up, false, true, z0, empty) → IDLE(F, closed, up, false, true, z0, empty)
IDLE(BF, closed, up, false, true, z0, empty) → BUSY(BF, closed, up, false, true, z0, empty)
IDLE(BF, closed, down, z0, false, z1, empty) → BUSY(BF, closed, down, z0, false, z1, empty)
IDLE(S, closed, down, true, false, false, empty) → BUSY(S, closed, down, true, false, false, empty)
IDLE(B, closed, stop, z0, z1, z2, empty) → BUSY(B, closed, stop, z0, z1, z2, empty)
BUSY(B, closed, up, false, true, z0, empty) → IDLE(BF, closed, up, false, true, z0, empty)
BUSY(F, closed, up, z0, false, z2, empty) → IDLE(FS, closed, up, z0, false, z2, empty)
IDLE(BF, closed, up, false, false, true, empty) → BUSY(BF, closed, up, false, false, true, empty)
IDLE(FS, closed, up, z0, false, z1, empty) → BUSY(FS, closed, up, z0, false, z1, empty)
BUSY(B, closed, down, b1, b2, b3, i) → IDLE(B, closed, stop, b1, b2, b3, i)
BUSY(S, closed, up, z0, false, z1, empty) → IDLE(S, closed, stop, z0, false, z1, empty)
BUSY(S, closed, stop, true, false, false, i) → IDLE(S, closed, down, true, false, false, i)
BUSY(BF, closed, up, false, false, true, empty) → IDLE(F, closed, up, false, false, true, empty)
BUSY(BF, closed, down, b1, b2, b3, i) → IDLE(B, closed, down, b1, b2, b3, i)
IDLE(F, closed, up, false, true, z0, empty) → BUSY(F, closed, up, false, true, z0, empty)
IDLE(F, closed, down, z0, z1, z2, empty) → BUSY(F, closed, down, z0, z1, z2, empty)
BUSY(B, closed, up, false, false, true, empty) → IDLE(BF, closed, up, false, false, true, empty)
IDLE(FS, closed, down, z0, z1, false, empty) → BUSY(FS, closed, down, z0, z1, false, empty)
IDLE(F, closed, down, true, false, z0, empty) → BUSY(F, closed, down, true, false, z0, empty)
BUSY(B, closed, stop, false, false, true, empty) → IDLE(B, closed, up, false, false, true, empty)
BUSY(S, closed, down, z3, z4, false, empty) → IDLE(FS, closed, down, z3, z4, false, empty)
IDLE(S, closed, up, z0, false, z1, empty) → BUSY(S, closed, up, z0, false, z1, empty)
IDLE(F, closed, up, false, false, true, empty) → BUSY(F, closed, up, false, false, true, empty)
IDLE(B, closed, down, z0, z1, z2, empty) → BUSY(B, closed, down, z0, z1, z2, empty)
IDLE(B, closed, up, false, false, true, empty) → BUSY(B, closed, up, false, false, true, empty)
IDLE(S, closed, stop, z0, false, z1, empty) → BUSY(S, closed, stop, z0, false, z1, empty)
IDLE(B, closed, up, false, true, z0, empty) → BUSY(B, closed, up, false, true, z0, empty)
BUSY(B, closed, stop, false, true, b3, i) → IDLE(B, closed, up, false, true, b3, i)
BUSY(F, closed, down, b1, false, b3, i) → IDLE(BF, closed, down, b1, false, b3, i)
BUSY(F, closed, up, false, false, true, empty) → IDLE(FS, closed, up, false, false, true, empty)
BUSY(FS, closed, down, z3, z4, z5, empty) → IDLE(F, closed, down, z3, z4, z5, empty)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 6 less nodes.
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ AAECC Innermost
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
Q DP problem:
The TRS P consists of the following rules:
BUSY(FS, closed, up, z0, false, z1, empty) → IDLE(S, closed, up, z0, false, z1, empty)
IDLE(BF, closed, down, z0, false, z1, empty) → BUSY(BF, closed, down, z0, false, z1, empty)
IDLE(S, closed, down, true, false, false, empty) → BUSY(S, closed, down, true, false, false, empty)
BUSY(F, closed, up, z0, false, z2, empty) → IDLE(FS, closed, up, z0, false, z2, empty)
IDLE(BF, closed, up, false, false, true, empty) → BUSY(BF, closed, up, false, false, true, empty)
IDLE(B, closed, stop, z0, z1, z2, empty) → BUSY(B, closed, stop, z0, z1, z2, empty)
IDLE(FS, closed, up, z0, false, z1, empty) → BUSY(FS, closed, up, z0, false, z1, empty)
BUSY(S, closed, up, z0, false, z1, empty) → IDLE(S, closed, stop, z0, false, z1, empty)
BUSY(B, closed, down, b1, b2, b3, i) → IDLE(B, closed, stop, b1, b2, b3, i)
BUSY(S, closed, stop, true, false, false, i) → IDLE(S, closed, down, true, false, false, i)
BUSY(BF, closed, up, false, false, true, empty) → IDLE(F, closed, up, false, false, true, empty)
BUSY(BF, closed, down, b1, b2, b3, i) → IDLE(B, closed, down, b1, b2, b3, i)
BUSY(B, closed, up, false, false, true, empty) → IDLE(BF, closed, up, false, false, true, empty)
IDLE(F, closed, down, z0, z1, z2, empty) → BUSY(F, closed, down, z0, z1, z2, empty)
IDLE(FS, closed, down, z0, z1, false, empty) → BUSY(FS, closed, down, z0, z1, false, empty)
IDLE(F, closed, down, true, false, z0, empty) → BUSY(F, closed, down, true, false, z0, empty)
BUSY(B, closed, stop, false, false, true, empty) → IDLE(B, closed, up, false, false, true, empty)
BUSY(S, closed, down, z3, z4, false, empty) → IDLE(FS, closed, down, z3, z4, false, empty)
IDLE(S, closed, up, z0, false, z1, empty) → BUSY(S, closed, up, z0, false, z1, empty)
IDLE(F, closed, up, false, false, true, empty) → BUSY(F, closed, up, false, false, true, empty)
IDLE(B, closed, down, z0, z1, z2, empty) → BUSY(B, closed, down, z0, z1, z2, empty)
IDLE(B, closed, up, false, false, true, empty) → BUSY(B, closed, up, false, false, true, empty)
IDLE(S, closed, stop, z0, false, z1, empty) → BUSY(S, closed, stop, z0, false, z1, empty)
BUSY(F, closed, down, b1, false, b3, i) → IDLE(BF, closed, down, b1, false, b3, i)
BUSY(F, closed, up, false, false, true, empty) → IDLE(FS, closed, up, false, false, true, empty)
BUSY(FS, closed, down, z3, z4, z5, empty) → IDLE(F, closed, down, z3, z4, z5, empty)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule BUSY(F, closed, up, z0, false, z2, empty) → IDLE(FS, closed, up, z0, false, z2, empty) we obtained the following new rules:
BUSY(F, closed, up, false, false, true, empty) → IDLE(FS, closed, up, false, false, true, empty)
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ AAECC Innermost
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
Q DP problem:
The TRS P consists of the following rules:
BUSY(FS, closed, up, z0, false, z1, empty) → IDLE(S, closed, up, z0, false, z1, empty)
IDLE(BF, closed, down, z0, false, z1, empty) → BUSY(BF, closed, down, z0, false, z1, empty)
IDLE(S, closed, down, true, false, false, empty) → BUSY(S, closed, down, true, false, false, empty)
IDLE(B, closed, stop, z0, z1, z2, empty) → BUSY(B, closed, stop, z0, z1, z2, empty)
IDLE(BF, closed, up, false, false, true, empty) → BUSY(BF, closed, up, false, false, true, empty)
IDLE(FS, closed, up, z0, false, z1, empty) → BUSY(FS, closed, up, z0, false, z1, empty)
BUSY(B, closed, down, b1, b2, b3, i) → IDLE(B, closed, stop, b1, b2, b3, i)
BUSY(S, closed, up, z0, false, z1, empty) → IDLE(S, closed, stop, z0, false, z1, empty)
BUSY(S, closed, stop, true, false, false, i) → IDLE(S, closed, down, true, false, false, i)
BUSY(BF, closed, up, false, false, true, empty) → IDLE(F, closed, up, false, false, true, empty)
BUSY(BF, closed, down, b1, b2, b3, i) → IDLE(B, closed, down, b1, b2, b3, i)
IDLE(F, closed, down, z0, z1, z2, empty) → BUSY(F, closed, down, z0, z1, z2, empty)
BUSY(B, closed, up, false, false, true, empty) → IDLE(BF, closed, up, false, false, true, empty)
IDLE(FS, closed, down, z0, z1, false, empty) → BUSY(FS, closed, down, z0, z1, false, empty)
IDLE(F, closed, down, true, false, z0, empty) → BUSY(F, closed, down, true, false, z0, empty)
BUSY(B, closed, stop, false, false, true, empty) → IDLE(B, closed, up, false, false, true, empty)
BUSY(S, closed, down, z3, z4, false, empty) → IDLE(FS, closed, down, z3, z4, false, empty)
IDLE(S, closed, up, z0, false, z1, empty) → BUSY(S, closed, up, z0, false, z1, empty)
IDLE(F, closed, up, false, false, true, empty) → BUSY(F, closed, up, false, false, true, empty)
IDLE(B, closed, down, z0, z1, z2, empty) → BUSY(B, closed, down, z0, z1, z2, empty)
IDLE(B, closed, up, false, false, true, empty) → BUSY(B, closed, up, false, false, true, empty)
IDLE(S, closed, stop, z0, false, z1, empty) → BUSY(S, closed, stop, z0, false, z1, empty)
BUSY(F, closed, down, b1, false, b3, i) → IDLE(BF, closed, down, b1, false, b3, i)
BUSY(F, closed, up, false, false, true, empty) → IDLE(FS, closed, up, false, false, true, empty)
BUSY(FS, closed, down, z3, z4, z5, empty) → IDLE(F, closed, down, z3, z4, z5, empty)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule IDLE(FS, closed, up, z0, false, z1, empty) → BUSY(FS, closed, up, z0, false, z1, empty) we obtained the following new rules:
IDLE(FS, closed, up, false, false, true, empty) → BUSY(FS, closed, up, false, false, true, empty)
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ AAECC Innermost
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
Q DP problem:
The TRS P consists of the following rules:
BUSY(FS, closed, up, z0, false, z1, empty) → IDLE(S, closed, up, z0, false, z1, empty)
IDLE(BF, closed, down, z0, false, z1, empty) → BUSY(BF, closed, down, z0, false, z1, empty)
IDLE(S, closed, down, true, false, false, empty) → BUSY(S, closed, down, true, false, false, empty)
IDLE(BF, closed, up, false, false, true, empty) → BUSY(BF, closed, up, false, false, true, empty)
IDLE(B, closed, stop, z0, z1, z2, empty) → BUSY(B, closed, stop, z0, z1, z2, empty)
BUSY(S, closed, up, z0, false, z1, empty) → IDLE(S, closed, stop, z0, false, z1, empty)
BUSY(B, closed, down, b1, b2, b3, i) → IDLE(B, closed, stop, b1, b2, b3, i)
BUSY(S, closed, stop, true, false, false, i) → IDLE(S, closed, down, true, false, false, i)
IDLE(FS, closed, up, false, false, true, empty) → BUSY(FS, closed, up, false, false, true, empty)
BUSY(BF, closed, up, false, false, true, empty) → IDLE(F, closed, up, false, false, true, empty)
BUSY(BF, closed, down, b1, b2, b3, i) → IDLE(B, closed, down, b1, b2, b3, i)
BUSY(B, closed, up, false, false, true, empty) → IDLE(BF, closed, up, false, false, true, empty)
IDLE(F, closed, down, z0, z1, z2, empty) → BUSY(F, closed, down, z0, z1, z2, empty)
IDLE(FS, closed, down, z0, z1, false, empty) → BUSY(FS, closed, down, z0, z1, false, empty)
IDLE(F, closed, down, true, false, z0, empty) → BUSY(F, closed, down, true, false, z0, empty)
BUSY(B, closed, stop, false, false, true, empty) → IDLE(B, closed, up, false, false, true, empty)
BUSY(S, closed, down, z3, z4, false, empty) → IDLE(FS, closed, down, z3, z4, false, empty)
IDLE(S, closed, up, z0, false, z1, empty) → BUSY(S, closed, up, z0, false, z1, empty)
IDLE(F, closed, up, false, false, true, empty) → BUSY(F, closed, up, false, false, true, empty)
IDLE(B, closed, down, z0, z1, z2, empty) → BUSY(B, closed, down, z0, z1, z2, empty)
IDLE(B, closed, up, false, false, true, empty) → BUSY(B, closed, up, false, false, true, empty)
IDLE(S, closed, stop, z0, false, z1, empty) → BUSY(S, closed, stop, z0, false, z1, empty)
BUSY(F, closed, down, b1, false, b3, i) → IDLE(BF, closed, down, b1, false, b3, i)
BUSY(F, closed, up, false, false, true, empty) → IDLE(FS, closed, up, false, false, true, empty)
BUSY(FS, closed, down, z3, z4, z5, empty) → IDLE(F, closed, down, z3, z4, z5, empty)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule BUSY(FS, closed, up, z0, false, z1, empty) → IDLE(S, closed, up, z0, false, z1, empty) we obtained the following new rules:
BUSY(FS, closed, up, false, false, true, empty) → IDLE(S, closed, up, false, false, true, empty)
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ AAECC Innermost
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
Q DP problem:
The TRS P consists of the following rules:
BUSY(FS, closed, up, false, false, true, empty) → IDLE(S, closed, up, false, false, true, empty)
IDLE(BF, closed, down, z0, false, z1, empty) → BUSY(BF, closed, down, z0, false, z1, empty)
IDLE(S, closed, down, true, false, false, empty) → BUSY(S, closed, down, true, false, false, empty)
IDLE(B, closed, stop, z0, z1, z2, empty) → BUSY(B, closed, stop, z0, z1, z2, empty)
IDLE(BF, closed, up, false, false, true, empty) → BUSY(BF, closed, up, false, false, true, empty)
BUSY(B, closed, down, b1, b2, b3, i) → IDLE(B, closed, stop, b1, b2, b3, i)
BUSY(S, closed, up, z0, false, z1, empty) → IDLE(S, closed, stop, z0, false, z1, empty)
BUSY(S, closed, stop, true, false, false, i) → IDLE(S, closed, down, true, false, false, i)
IDLE(FS, closed, up, false, false, true, empty) → BUSY(FS, closed, up, false, false, true, empty)
BUSY(BF, closed, up, false, false, true, empty) → IDLE(F, closed, up, false, false, true, empty)
BUSY(BF, closed, down, b1, b2, b3, i) → IDLE(B, closed, down, b1, b2, b3, i)
IDLE(F, closed, down, z0, z1, z2, empty) → BUSY(F, closed, down, z0, z1, z2, empty)
BUSY(B, closed, up, false, false, true, empty) → IDLE(BF, closed, up, false, false, true, empty)
IDLE(FS, closed, down, z0, z1, false, empty) → BUSY(FS, closed, down, z0, z1, false, empty)
IDLE(F, closed, down, true, false, z0, empty) → BUSY(F, closed, down, true, false, z0, empty)
BUSY(B, closed, stop, false, false, true, empty) → IDLE(B, closed, up, false, false, true, empty)
BUSY(S, closed, down, z3, z4, false, empty) → IDLE(FS, closed, down, z3, z4, false, empty)
IDLE(S, closed, up, z0, false, z1, empty) → BUSY(S, closed, up, z0, false, z1, empty)
IDLE(F, closed, up, false, false, true, empty) → BUSY(F, closed, up, false, false, true, empty)
IDLE(B, closed, down, z0, z1, z2, empty) → BUSY(B, closed, down, z0, z1, z2, empty)
IDLE(B, closed, up, false, false, true, empty) → BUSY(B, closed, up, false, false, true, empty)
IDLE(S, closed, stop, z0, false, z1, empty) → BUSY(S, closed, stop, z0, false, z1, empty)
BUSY(F, closed, down, b1, false, b3, i) → IDLE(BF, closed, down, b1, false, b3, i)
BUSY(F, closed, up, false, false, true, empty) → IDLE(FS, closed, up, false, false, true, empty)
BUSY(FS, closed, down, z3, z4, z5, empty) → IDLE(F, closed, down, z3, z4, z5, empty)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule IDLE(S, closed, up, z0, false, z1, empty) → BUSY(S, closed, up, z0, false, z1, empty) we obtained the following new rules:
IDLE(S, closed, up, false, false, true, empty) → BUSY(S, closed, up, false, false, true, empty)
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ AAECC Innermost
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
Q DP problem:
The TRS P consists of the following rules:
BUSY(FS, closed, up, false, false, true, empty) → IDLE(S, closed, up, false, false, true, empty)
IDLE(BF, closed, down, z0, false, z1, empty) → BUSY(BF, closed, down, z0, false, z1, empty)
IDLE(S, closed, down, true, false, false, empty) → BUSY(S, closed, down, true, false, false, empty)
IDLE(BF, closed, up, false, false, true, empty) → BUSY(BF, closed, up, false, false, true, empty)
IDLE(B, closed, stop, z0, z1, z2, empty) → BUSY(B, closed, stop, z0, z1, z2, empty)
BUSY(S, closed, up, z0, false, z1, empty) → IDLE(S, closed, stop, z0, false, z1, empty)
BUSY(B, closed, down, b1, b2, b3, i) → IDLE(B, closed, stop, b1, b2, b3, i)
BUSY(S, closed, stop, true, false, false, i) → IDLE(S, closed, down, true, false, false, i)
IDLE(FS, closed, up, false, false, true, empty) → BUSY(FS, closed, up, false, false, true, empty)
BUSY(BF, closed, up, false, false, true, empty) → IDLE(F, closed, up, false, false, true, empty)
BUSY(BF, closed, down, b1, b2, b3, i) → IDLE(B, closed, down, b1, b2, b3, i)
BUSY(B, closed, up, false, false, true, empty) → IDLE(BF, closed, up, false, false, true, empty)
IDLE(F, closed, down, z0, z1, z2, empty) → BUSY(F, closed, down, z0, z1, z2, empty)
IDLE(FS, closed, down, z0, z1, false, empty) → BUSY(FS, closed, down, z0, z1, false, empty)
IDLE(S, closed, up, false, false, true, empty) → BUSY(S, closed, up, false, false, true, empty)
IDLE(F, closed, down, true, false, z0, empty) → BUSY(F, closed, down, true, false, z0, empty)
BUSY(B, closed, stop, false, false, true, empty) → IDLE(B, closed, up, false, false, true, empty)
BUSY(S, closed, down, z3, z4, false, empty) → IDLE(FS, closed, down, z3, z4, false, empty)
IDLE(F, closed, up, false, false, true, empty) → BUSY(F, closed, up, false, false, true, empty)
IDLE(B, closed, down, z0, z1, z2, empty) → BUSY(B, closed, down, z0, z1, z2, empty)
IDLE(B, closed, up, false, false, true, empty) → BUSY(B, closed, up, false, false, true, empty)
IDLE(S, closed, stop, z0, false, z1, empty) → BUSY(S, closed, stop, z0, false, z1, empty)
BUSY(F, closed, down, b1, false, b3, i) → IDLE(BF, closed, down, b1, false, b3, i)
BUSY(F, closed, up, false, false, true, empty) → IDLE(FS, closed, up, false, false, true, empty)
BUSY(FS, closed, down, z3, z4, z5, empty) → IDLE(F, closed, down, z3, z4, z5, empty)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule BUSY(S, closed, up, z0, false, z1, empty) → IDLE(S, closed, stop, z0, false, z1, empty) we obtained the following new rules:
BUSY(S, closed, up, false, false, true, empty) → IDLE(S, closed, stop, false, false, true, empty)
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ AAECC Innermost
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
Q DP problem:
The TRS P consists of the following rules:
BUSY(FS, closed, up, false, false, true, empty) → IDLE(S, closed, up, false, false, true, empty)
IDLE(BF, closed, down, z0, false, z1, empty) → BUSY(BF, closed, down, z0, false, z1, empty)
IDLE(S, closed, down, true, false, false, empty) → BUSY(S, closed, down, true, false, false, empty)
BUSY(S, closed, up, false, false, true, empty) → IDLE(S, closed, stop, false, false, true, empty)
IDLE(B, closed, stop, z0, z1, z2, empty) → BUSY(B, closed, stop, z0, z1, z2, empty)
IDLE(BF, closed, up, false, false, true, empty) → BUSY(BF, closed, up, false, false, true, empty)
BUSY(B, closed, down, b1, b2, b3, i) → IDLE(B, closed, stop, b1, b2, b3, i)
BUSY(S, closed, stop, true, false, false, i) → IDLE(S, closed, down, true, false, false, i)
IDLE(FS, closed, up, false, false, true, empty) → BUSY(FS, closed, up, false, false, true, empty)
BUSY(BF, closed, up, false, false, true, empty) → IDLE(F, closed, up, false, false, true, empty)
BUSY(BF, closed, down, b1, b2, b3, i) → IDLE(B, closed, down, b1, b2, b3, i)
IDLE(F, closed, down, z0, z1, z2, empty) → BUSY(F, closed, down, z0, z1, z2, empty)
BUSY(B, closed, up, false, false, true, empty) → IDLE(BF, closed, up, false, false, true, empty)
IDLE(FS, closed, down, z0, z1, false, empty) → BUSY(FS, closed, down, z0, z1, false, empty)
IDLE(S, closed, up, false, false, true, empty) → BUSY(S, closed, up, false, false, true, empty)
IDLE(F, closed, down, true, false, z0, empty) → BUSY(F, closed, down, true, false, z0, empty)
BUSY(B, closed, stop, false, false, true, empty) → IDLE(B, closed, up, false, false, true, empty)
BUSY(S, closed, down, z3, z4, false, empty) → IDLE(FS, closed, down, z3, z4, false, empty)
IDLE(F, closed, up, false, false, true, empty) → BUSY(F, closed, up, false, false, true, empty)
IDLE(B, closed, down, z0, z1, z2, empty) → BUSY(B, closed, down, z0, z1, z2, empty)
IDLE(B, closed, up, false, false, true, empty) → BUSY(B, closed, up, false, false, true, empty)
IDLE(S, closed, stop, z0, false, z1, empty) → BUSY(S, closed, stop, z0, false, z1, empty)
BUSY(F, closed, down, b1, false, b3, i) → IDLE(BF, closed, down, b1, false, b3, i)
BUSY(F, closed, up, false, false, true, empty) → IDLE(FS, closed, up, false, false, true, empty)
BUSY(FS, closed, down, z3, z4, z5, empty) → IDLE(F, closed, down, z3, z4, z5, empty)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule IDLE(S, closed, stop, z0, false, z1, empty) → BUSY(S, closed, stop, z0, false, z1, empty) we obtained the following new rules:
IDLE(S, closed, stop, false, false, true, empty) → BUSY(S, closed, stop, false, false, true, empty)
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ RRRPoloQTRSProof
↳ QTRS
↳ AAECC Innermost
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
BUSY(FS, closed, up, false, false, true, empty) → IDLE(S, closed, up, false, false, true, empty)
IDLE(BF, closed, down, z0, false, z1, empty) → BUSY(BF, closed, down, z0, false, z1, empty)
IDLE(S, closed, down, true, false, false, empty) → BUSY(S, closed, down, true, false, false, empty)
BUSY(S, closed, up, false, false, true, empty) → IDLE(S, closed, stop, false, false, true, empty)
IDLE(BF, closed, up, false, false, true, empty) → BUSY(BF, closed, up, false, false, true, empty)
IDLE(B, closed, stop, z0, z1, z2, empty) → BUSY(B, closed, stop, z0, z1, z2, empty)
BUSY(B, closed, down, b1, b2, b3, i) → IDLE(B, closed, stop, b1, b2, b3, i)
BUSY(S, closed, stop, true, false, false, i) → IDLE(S, closed, down, true, false, false, i)
IDLE(S, closed, stop, false, false, true, empty) → BUSY(S, closed, stop, false, false, true, empty)
IDLE(FS, closed, up, false, false, true, empty) → BUSY(FS, closed, up, false, false, true, empty)
BUSY(BF, closed, up, false, false, true, empty) → IDLE(F, closed, up, false, false, true, empty)
BUSY(BF, closed, down, b1, b2, b3, i) → IDLE(B, closed, down, b1, b2, b3, i)
BUSY(B, closed, up, false, false, true, empty) → IDLE(BF, closed, up, false, false, true, empty)
IDLE(F, closed, down, z0, z1, z2, empty) → BUSY(F, closed, down, z0, z1, z2, empty)
IDLE(FS, closed, down, z0, z1, false, empty) → BUSY(FS, closed, down, z0, z1, false, empty)
IDLE(S, closed, up, false, false, true, empty) → BUSY(S, closed, up, false, false, true, empty)
IDLE(F, closed, down, true, false, z0, empty) → BUSY(F, closed, down, true, false, z0, empty)
BUSY(B, closed, stop, false, false, true, empty) → IDLE(B, closed, up, false, false, true, empty)
BUSY(S, closed, down, z3, z4, false, empty) → IDLE(FS, closed, down, z3, z4, false, empty)
IDLE(F, closed, up, false, false, true, empty) → BUSY(F, closed, up, false, false, true, empty)
IDLE(B, closed, down, z0, z1, z2, empty) → BUSY(B, closed, down, z0, z1, z2, empty)
IDLE(B, closed, up, false, false, true, empty) → BUSY(B, closed, up, false, false, true, empty)
BUSY(F, closed, down, b1, false, b3, i) → IDLE(BF, closed, down, b1, false, b3, i)
BUSY(F, closed, up, false, false, true, empty) → IDLE(FS, closed, up, false, false, true, empty)
BUSY(FS, closed, down, z3, z4, z5, empty) → IDLE(F, closed, down, z3, z4, z5, empty)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 25 less nodes.