Termination w.r.t. Q of the given QTRS could not be shown:

(0) Obligation:

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.

(1) AAECC Innermost (EQUIVALENT transformation)

We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is

or(true, b) → true
or(false, b) → b

The TRS R 2 is

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)

The signature Sigma is {start, busy, incorrect, correct, idle}

(2) Obligation:

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

The set Q consists of the following terms:

start(x0)
busy(BF, x0, stop, x1, x2, x3, x4)
busy(FS, x0, stop, x1, x2, x3, x4)
busy(x0, open, up, x1, x2, x3, x4)
busy(x0, open, down, x1, x2, x3, x4)
busy(B, closed, stop, false, false, false, empty)
busy(F, closed, stop, false, false, false, empty)
busy(S, closed, stop, false, false, false, empty)
busy(B, closed, stop, false, false, false, newbuttons(x0, x1, x2, x3))
busy(F, closed, stop, false, false, false, newbuttons(x0, x1, x2, x3))
busy(S, closed, stop, false, false, false, newbuttons(x0, x1, x2, x3))
busy(B, open, stop, false, x0, x1, x2)
busy(F, open, stop, x0, false, x1, x2)
busy(S, open, stop, x0, x1, false, x2)
busy(B, x0, stop, true, x1, x2, x3)
busy(F, x0, stop, x1, true, x2, x3)
busy(S, x0, stop, x1, x2, true, x3)
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)
idle(x0, x1, x2, x3, x4, x5, newbuttons(x6, x7, x8, x9))
or(true, x0)
or(false, x0)

(3) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(4) Obligation:

Q DP problem:
The TRS P consists of the following rules:

START(i) → BUSY(F, closed, stop, false, false, false, i)
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)
IDLE(fl, d, m, b1, b2, b3, newbuttons(i1, i2, i3, i)) → OR(b1, i1)
IDLE(fl, d, m, b1, b2, b3, newbuttons(i1, i2, i3, i)) → OR(b2, i2)
IDLE(fl, d, m, b1, b2, b3, newbuttons(i1, i2, i3, i)) → OR(b3, i3)

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

The set Q consists of the following terms:

start(x0)
busy(BF, x0, stop, x1, x2, x3, x4)
busy(FS, x0, stop, x1, x2, x3, x4)
busy(x0, open, up, x1, x2, x3, x4)
busy(x0, open, down, x1, x2, x3, x4)
busy(B, closed, stop, false, false, false, empty)
busy(F, closed, stop, false, false, false, empty)
busy(S, closed, stop, false, false, false, empty)
busy(B, closed, stop, false, false, false, newbuttons(x0, x1, x2, x3))
busy(F, closed, stop, false, false, false, newbuttons(x0, x1, x2, x3))
busy(S, closed, stop, false, false, false, newbuttons(x0, x1, x2, x3))
busy(B, open, stop, false, x0, x1, x2)
busy(F, open, stop, x0, false, x1, x2)
busy(S, open, stop, x0, x1, false, x2)
busy(B, x0, stop, true, x1, x2, x3)
busy(F, x0, stop, x1, true, x2, x3)
busy(S, x0, stop, x1, x2, true, x3)
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)
idle(x0, x1, x2, x3, x4, x5, newbuttons(x6, x7, x8, x9))
or(true, x0)
or(false, x0)

We have to consider all minimal (P,Q,R)-chains.

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes.

(6) Obligation:

Q DP problem:
The TRS P consists of the following rules:

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)
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)
IDLE(fl, d, m, b1, b2, b3, empty) → BUSY(fl, d, m, b1, b2, b3, empty)
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)

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

The set Q consists of the following terms:

start(x0)
busy(BF, x0, stop, x1, x2, x3, x4)
busy(FS, x0, stop, x1, x2, x3, x4)
busy(x0, open, up, x1, x2, x3, x4)
busy(x0, open, down, x1, x2, x3, x4)
busy(B, closed, stop, false, false, false, empty)
busy(F, closed, stop, false, false, false, empty)
busy(S, closed, stop, false, false, false, empty)
busy(B, closed, stop, false, false, false, newbuttons(x0, x1, x2, x3))
busy(F, closed, stop, false, false, false, newbuttons(x0, x1, x2, x3))
busy(S, closed, stop, false, false, false, newbuttons(x0, x1, x2, x3))
busy(B, open, stop, false, x0, x1, x2)
busy(F, open, stop, x0, false, x1, x2)
busy(S, open, stop, x0, x1, false, x2)
busy(B, x0, stop, true, x1, x2, x3)
busy(F, x0, stop, x1, true, x2, x3)
busy(S, x0, stop, x1, x2, true, x3)
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)
idle(x0, x1, x2, x3, x4, x5, newbuttons(x6, x7, x8, x9))
or(true, x0)
or(false, x0)

We have to consider all minimal (P,Q,R)-chains.

(7) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
IDLE(x1, x2, x3, x4, x5, x6, x7)  =  x7
newbuttons(x1, x2, x3, x4)  =  newbuttons(x4)
BUSY(x1, x2, x3, x4, x5, x6, x7)  =  x7
or(x1, x2)  =  or(x1, x2)
B  =  B
closed  =  closed
stop  =  stop
false  =  false
F  =  F
S  =  S
open  =  open
empty  =  empty
true  =  true
down  =  down
up  =  up
BF  =  BF
FS  =  FS
start(x1)  =  start(x1)
busy(x1, x2, x3, x4, x5, x6, x7)  =  x7
incorrect  =  incorrect
correct  =  correct
idle(x1, x2, x3, x4, x5, x6, x7)  =  x7

Recursive path order with status [RPO].
Precedence:

or₂ > true > incorrect
stop > closed > down > B > newbuttons₁ > false > true > incorrect
stop > closed > down > B > newbuttons₁ > false > BF > incorrect
stop > closed > down > FS > F > false > true > incorrect
stop > closed > down > FS > F > false > BF > incorrect
stop > closed > up > B > newbuttons₁ > false > true > incorrect
stop > closed > up > B > newbuttons₁ > false > BF > incorrect
stop > closed > up > FS > F > false > true > incorrect
stop > closed > up > FS > F > false > BF > incorrect
stop > closed > correct > incorrect
stop > S > down > B > newbuttons₁ > false > true > incorrect
stop > S > down > B > newbuttons₁ > false > BF > incorrect
stop > S > down > FS > F > false > true > incorrect
stop > S > down > FS > F > false > BF > incorrect
stop > S > correct > incorrect
open > closed > down > B > newbuttons₁ > false > true > incorrect
open > closed > down > B > newbuttons₁ > false > BF > incorrect
open > closed > down > FS > F > false > true > incorrect
open > closed > down > FS > F > false > BF > incorrect
open > closed > up > B > newbuttons₁ > false > true > incorrect
open > closed > up > B > newbuttons₁ > false > BF > incorrect
open > closed > up > FS > F > false > true > incorrect
open > closed > up > FS > F > false > BF > incorrect
open > closed > correct > incorrect
open > S > down > B > newbuttons₁ > false > true > incorrect
open > S > down > B > newbuttons₁ > false > BF > incorrect
open > S > down > FS > F > false > true > incorrect
open > S > down > FS > F > false > BF > incorrect
open > S > correct > incorrect
empty > correct > incorrect
start₁ > incorrect

Status:

newbuttons₁: multiset
or₂: multiset
B: multiset
closed: multiset
stop: multiset
false: multiset
F: multiset
S: multiset
open: multiset
empty: multiset
true: multiset
down: multiset
up: multiset
BF: multiset
FS: multiset
start₁: multiset
incorrect: multiset
correct: multiset

The following usable rules [FROCOS05] were oriented:

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

(8) Obligation:

Q DP problem:
The TRS P consists of the following rules:

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)
IDLE(fl, d, m, b1, b2, b3, empty) → BUSY(fl, d, m, b1, b2, b3, empty)
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)

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

The set Q consists of the following terms:

start(x0)
busy(BF, x0, stop, x1, x2, x3, x4)
busy(FS, x0, stop, x1, x2, x3, x4)
busy(x0, open, up, x1, x2, x3, x4)
busy(x0, open, down, x1, x2, x3, x4)
busy(B, closed, stop, false, false, false, empty)
busy(F, closed, stop, false, false, false, empty)
busy(S, closed, stop, false, false, false, empty)
busy(B, closed, stop, false, false, false, newbuttons(x0, x1, x2, x3))
busy(F, closed, stop, false, false, false, newbuttons(x0, x1, x2, x3))
busy(S, closed, stop, false, false, false, newbuttons(x0, x1, x2, x3))
busy(B, open, stop, false, x0, x1, x2)
busy(F, open, stop, x0, false, x1, x2)
busy(S, open, stop, x0, x1, false, x2)
busy(B, x0, stop, true, x1, x2, x3)
busy(F, x0, stop, x1, true, x2, x3)
busy(S, x0, stop, x1, x2, true, x3)
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)
idle(x0, x1, x2, x3, x4, x5, newbuttons(x6, x7, x8, x9))
or(true, x0)
or(false, x0)

We have to consider all minimal (P,Q,R)-chains.

(9) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.

(10) Obligation:

Q DP problem:
The TRS P consists of the following rules:

IDLE(fl, d, m, b1, b2, b3, empty) → BUSY(fl, d, m, b1, b2, b3, empty)
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)

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

The set Q consists of the following terms:

start(x0)
busy(BF, x0, stop, x1, x2, x3, x4)
busy(FS, x0, stop, x1, x2, x3, x4)
busy(x0, open, up, x1, x2, x3, x4)
busy(x0, open, down, x1, x2, x3, x4)
busy(B, closed, stop, false, false, false, empty)
busy(F, closed, stop, false, false, false, empty)
busy(S, closed, stop, false, false, false, empty)
busy(B, closed, stop, false, false, false, newbuttons(x0, x1, x2, x3))
busy(F, closed, stop, false, false, false, newbuttons(x0, x1, x2, x3))
busy(S, closed, stop, false, false, false, newbuttons(x0, x1, x2, x3))
busy(B, open, stop, false, x0, x1, x2)
busy(F, open, stop, x0, false, x1, x2)
busy(S, open, stop, x0, x1, false, x2)
busy(B, x0, stop, true, x1, x2, x3)
busy(F, x0, stop, x1, true, x2, x3)
busy(S, x0, stop, x1, x2, true, x3)
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)
idle(x0, x1, x2, x3, x4, x5, newbuttons(x6, x7, x8, x9))
or(true, x0)
or(false, x0)

We have to consider all minimal (P,Q,R)-chains.

(11) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

BUSY(B, d, stop, true, b2, b3, i) → IDLE(B, open, stop, false, b2, b3, i)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
IDLE(x1, x2, x3, x4, x5, x6, x7)  =  x4
empty  =  empty
BUSY(x1, x2, x3, x4, x5, x6, x7)  =  x4
B  =  B
open  =  open
stop  =  stop
false  =  false
closed  =  closed
F  =  F
S  =  S
true  =  true
down  =  down
up  =  up
BF  =  BF
FS  =  FS
start(x1)  =  start(x1)
busy(x1, x2, x3, x4, x5, x6, x7)  =  x7
incorrect  =  incorrect
correct  =  correct
newbuttons(x1, x2, x3, x4)  =  newbuttons(x1, x2, x3, x4)
idle(x1, x2, x3, x4, x5, x6, x7)  =  x7
or(x1, x2)  =  or(x1, x2)

Recursive path order with status [RPO].
Precedence:

empty > correct > incorrect
F > true > stop > down > BF > B > open > closed > incorrect
F > true > stop > down > BF > B > up > closed > incorrect
F > true > stop > down > BF > B > correct > incorrect
F > true > stop > newbuttons₄ > closed > incorrect
F > true > stop > newbuttons₄ > or₂ > incorrect
F > true > false > down > BF > B > open > closed > incorrect
F > true > false > down > BF > B > up > closed > incorrect
F > true > false > down > BF > B > correct > incorrect
F > true > false > FS > up > closed > incorrect
F > true > false > newbuttons₄ > closed > incorrect
F > true > false > newbuttons₄ > or₂ > incorrect
F > true > S > down > BF > B > open > closed > incorrect
F > true > S > down > BF > B > up > closed > incorrect
F > true > S > down > BF > B > correct > incorrect
F > true > S > FS > up > closed > incorrect
F > true > S > newbuttons₄ > closed > incorrect
F > true > S > newbuttons₄ > or₂ > incorrect
start₁ > closed > incorrect

Status:

empty: multiset
B: multiset
open: multiset
stop: multiset
false: multiset
closed: multiset
F: multiset
S: multiset
true: multiset
down: multiset
up: multiset
BF: multiset
FS: multiset
start₁: multiset
incorrect: multiset
correct: multiset
newbuttons₄: multiset
or₂: multiset

The following usable rules [FROCOS05] were oriented:

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

(12) Obligation:

Q DP problem:
The TRS P consists of the following rules:

IDLE(fl, d, m, b1, b2, b3, empty) → BUSY(fl, d, m, b1, b2, b3, empty)
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(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)

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

The set Q consists of the following terms:

start(x0)
busy(BF, x0, stop, x1, x2, x3, x4)
busy(FS, x0, stop, x1, x2, x3, x4)
busy(x0, open, up, x1, x2, x3, x4)
busy(x0, open, down, x1, x2, x3, x4)
busy(B, closed, stop, false, false, false, empty)
busy(F, closed, stop, false, false, false, empty)
busy(S, closed, stop, false, false, false, empty)
busy(B, closed, stop, false, false, false, newbuttons(x0, x1, x2, x3))
busy(F, closed, stop, false, false, false, newbuttons(x0, x1, x2, x3))
busy(S, closed, stop, false, false, false, newbuttons(x0, x1, x2, x3))
busy(B, open, stop, false, x0, x1, x2)
busy(F, open, stop, x0, false, x1, x2)
busy(S, open, stop, x0, x1, false, x2)
busy(B, x0, stop, true, x1, x2, x3)
busy(F, x0, stop, x1, true, x2, x3)
busy(S, x0, stop, x1, x2, true, x3)
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)
idle(x0, x1, x2, x3, x4, x5, newbuttons(x6, x7, x8, x9))
or(true, x0)
or(false, x0)

We have to consider all minimal (P,Q,R)-chains.

(13) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

BUSY(F, d, stop, b1, true, b3, i) → IDLE(F, open, stop, b1, false, b3, i)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
IDLE(x1, x2, x3, x4, x5, x6, x7)  =  x5
empty  =  empty
BUSY(x1, x2, x3, x4, x5, x6, x7)  =  x5
B  =  B
open  =  open
stop  =  stop
false  =  false
closed  =  closed
F  =  F
S  =  S
true  =  true
down  =  down
up  =  up
BF  =  BF
FS  =  FS
start(x1)  =  x1
busy(x1, x2, x3, x4, x5, x6, x7)  =  x7
incorrect  =  incorrect
correct  =  correct
newbuttons(x1, x2, x3, x4)  =  newbuttons(x1, x2, x3, x4)
idle(x1, x2, x3, x4, x5, x6, x7)  =  x7
or(x1, x2)  =  or(x1, x2)

Recursive path order with status [RPO].
Precedence:

empty > correct > incorrect
stop > B > closed > true > open > incorrect
stop > B > closed > true > false > F > incorrect
stop > B > closed > BF > F > incorrect
stop > B > closed > newbuttons₄ > false > F > incorrect
stop > S > closed > true > open > incorrect
stop > S > closed > true > false > F > incorrect
stop > S > closed > BF > F > incorrect
stop > S > closed > newbuttons₄ > false > F > incorrect
stop > S > correct > incorrect
stop > up > closed > true > open > incorrect
stop > up > closed > true > false > F > incorrect
stop > up > closed > BF > F > incorrect
stop > up > closed > newbuttons₄ > false > F > incorrect
down > B > closed > true > open > incorrect
down > B > closed > true > false > F > incorrect
down > B > closed > BF > F > incorrect
down > B > closed > newbuttons₄ > false > F > incorrect
down > S > closed > true > open > incorrect
down > S > closed > true > false > F > incorrect
down > S > closed > BF > F > incorrect
down > S > closed > newbuttons₄ > false > F > incorrect
down > S > correct > incorrect
FS > S > closed > true > open > incorrect
FS > S > closed > true > false > F > incorrect
FS > S > closed > BF > F > incorrect
FS > S > closed > newbuttons₄ > false > F > incorrect
FS > S > correct > incorrect
FS > up > closed > true > open > incorrect
FS > up > closed > true > false > F > incorrect
FS > up > closed > BF > F > incorrect
FS > up > closed > newbuttons₄ > false > F > incorrect
or₂ > true > open > incorrect
or₂ > true > false > F > incorrect

Status:

empty: multiset
B: multiset
open: multiset
stop: multiset
false: multiset
closed: multiset
F: multiset
S: multiset
true: multiset
down: multiset
up: multiset
BF: multiset
FS: multiset
incorrect: multiset
correct: multiset
newbuttons₄: multiset
or₂: multiset

The following usable rules [FROCOS05] were oriented:

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

(14) Obligation:

Q DP problem:
The TRS P consists of the following rules:

IDLE(fl, d, m, b1, b2, b3, empty) → BUSY(fl, d, m, b1, b2, b3, empty)
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(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)

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

The set Q consists of the following terms:

start(x0)
busy(BF, x0, stop, x1, x2, x3, x4)
busy(FS, x0, stop, x1, x2, x3, x4)
busy(x0, open, up, x1, x2, x3, x4)
busy(x0, open, down, x1, x2, x3, x4)
busy(B, closed, stop, false, false, false, empty)
busy(F, closed, stop, false, false, false, empty)
busy(S, closed, stop, false, false, false, empty)
busy(B, closed, stop, false, false, false, newbuttons(x0, x1, x2, x3))
busy(F, closed, stop, false, false, false, newbuttons(x0, x1, x2, x3))
busy(S, closed, stop, false, false, false, newbuttons(x0, x1, x2, x3))
busy(B, open, stop, false, x0, x1, x2)
busy(F, open, stop, x0, false, x1, x2)
busy(S, open, stop, x0, x1, false, x2)
busy(B, x0, stop, true, x1, x2, x3)
busy(F, x0, stop, x1, true, x2, x3)
busy(S, x0, stop, x1, x2, true, x3)
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)
idle(x0, x1, x2, x3, x4, x5, newbuttons(x6, x7, x8, x9))
or(true, x0)
or(false, x0)

We have to consider all minimal (P,Q,R)-chains.

(15) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

BUSY(S, d, stop, b1, b2, true, i) → IDLE(S, open, stop, b1, b2, false, i)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
IDLE(x1, x2, x3, x4, x5, x6, x7)  =  x6
empty  =  empty
BUSY(x1, x2, x3, x4, x5, x6, x7)  =  x6
B  =  B
open  =  open
stop  =  stop
false  =  false
closed  =  closed
F  =  F
S  =  S
true  =  true
down  =  down
up  =  up
BF  =  BF
FS  =  FS
start(x1)  =  x1
busy(x1, x2, x3, x4, x5, x6, x7)  =  x7
incorrect  =  incorrect
correct  =  correct
newbuttons(x1, x2, x3, x4)  =  x4
idle(x1, x2, x3, x4, x5, x6, x7)  =  x7
or(x1, x2)  =  or(x1, x2)

Recursive path order with status [RPO].
Precedence:

empty > correct > incorrect
open > closed > BF > down > stop > F > true > B > incorrect
open > closed > BF > down > stop > F > true > false > incorrect
open > closed > BF > down > stop > F > true > S > incorrect
open > closed > correct > incorrect
up > FS > closed > BF > down > stop > F > true > B > incorrect
up > FS > closed > BF > down > stop > F > true > false > incorrect
up > FS > closed > BF > down > stop > F > true > S > incorrect
up > FS > closed > correct > incorrect
or₂ > true > B > incorrect
or₂ > true > false > incorrect
or₂ > true > S > incorrect

Status:

empty: multiset
B: multiset
open: multiset
stop: multiset
false: multiset
closed: multiset
F: multiset
S: multiset
true: multiset
down: multiset
up: multiset
BF: multiset
FS: multiset
incorrect: multiset
correct: multiset
or₂: multiset

The following usable rules [FROCOS05] were oriented:

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

(16) Obligation:

Q DP problem:
The TRS P consists of the following rules:

IDLE(fl, d, m, b1, b2, b3, empty) → BUSY(fl, d, m, b1, b2, b3, empty)
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, 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)

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

The set Q consists of the following terms:

start(x0)
busy(BF, x0, stop, x1, x2, x3, x4)
busy(FS, x0, stop, x1, x2, x3, x4)
busy(x0, open, up, x1, x2, x3, x4)
busy(x0, open, down, x1, x2, x3, x4)
busy(B, closed, stop, false, false, false, empty)
busy(F, closed, stop, false, false, false, empty)
busy(S, closed, stop, false, false, false, empty)
busy(B, closed, stop, false, false, false, newbuttons(x0, x1, x2, x3))
busy(F, closed, stop, false, false, false, newbuttons(x0, x1, x2, x3))
busy(S, closed, stop, false, false, false, newbuttons(x0, x1, x2, x3))
busy(B, open, stop, false, x0, x1, x2)
busy(F, open, stop, x0, false, x1, x2)
busy(S, open, stop, x0, x1, false, x2)
busy(B, x0, stop, true, x1, x2, x3)
busy(F, x0, stop, x1, true, x2, x3)
busy(S, x0, stop, x1, x2, true, x3)
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)
idle(x0, x1, x2, x3, x4, x5, newbuttons(x6, x7, x8, x9))
or(true, x0)
or(false, x0)

We have to consider all minimal (P,Q,R)-chains.

(17) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
IDLE(x1, x2, x3, x4, x5, x6, x7)  =  x2
empty  =  empty
BUSY(x1, x2, x3, x4, x5, x6, x7)  =  x2
B  =  B
open  =  open
stop  =  stop
false  =  false
closed  =  closed
F  =  F
S  =  S
down  =  down
up  =  up
true  =  true
BF  =  BF
FS  =  FS
start(x1)  =  start(x1)
busy(x1, x2, x3, x4, x5, x6, x7)  =  x7
incorrect  =  incorrect
correct  =  correct
newbuttons(x1, x2, x3, x4)  =  x4
idle(x1, x2, x3, x4, x5, x6, x7)  =  x7
or(x1, x2)  =  or(x1, x2)

Recursive path order with status [RPO].
Precedence:

empty > correct > incorrect
stop > F > open > B > false > up > incorrect
stop > F > open > B > false > true > incorrect
stop > F > open > B > BF > closed > up > incorrect
stop > F > open > B > BF > closed > true > incorrect
stop > F > open > S > false > up > incorrect
stop > F > open > S > false > true > incorrect
stop > F > open > S > closed > up > incorrect
stop > F > open > S > closed > true > incorrect
stop > F > down > B > false > up > incorrect
stop > F > down > B > false > true > incorrect
stop > F > down > B > BF > closed > up > incorrect
stop > F > down > B > BF > closed > true > incorrect
stop > F > down > FS > S > false > up > incorrect
stop > F > down > FS > S > false > true > incorrect
stop > F > down > FS > S > closed > up > incorrect
stop > F > down > FS > S > closed > true > incorrect
stop > F > correct > incorrect
start₁ > false > up > incorrect
start₁ > false > true > incorrect
start₁ > closed > up > incorrect
start₁ > closed > true > incorrect
or₂ > true > incorrect

Status:

empty: multiset
B: multiset
open: multiset
stop: multiset
false: multiset
closed: multiset
F: multiset
S: multiset
down: multiset
up: multiset
true: multiset
BF: multiset
FS: multiset
start₁: multiset
incorrect: multiset
correct: multiset
or₂: multiset

The following usable rules [FROCOS05] were oriented:

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

(18) Obligation:

Q DP problem:
The TRS P consists of the following rules:

IDLE(fl, d, m, b1, b2, b3, empty) → BUSY(fl, d, m, b1, b2, b3, empty)
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)

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

The set Q consists of the following terms:

start(x0)
busy(BF, x0, stop, x1, x2, x3, x4)
busy(FS, x0, stop, x1, x2, x3, x4)
busy(x0, open, up, x1, x2, x3, x4)
busy(x0, open, down, x1, x2, x3, x4)
busy(B, closed, stop, false, false, false, empty)
busy(F, closed, stop, false, false, false, empty)
busy(S, closed, stop, false, false, false, empty)
busy(B, closed, stop, false, false, false, newbuttons(x0, x1, x2, x3))
busy(F, closed, stop, false, false, false, newbuttons(x0, x1, x2, x3))
busy(S, closed, stop, false, false, false, newbuttons(x0, x1, x2, x3))
busy(B, open, stop, false, x0, x1, x2)
busy(F, open, stop, x0, false, x1, x2)
busy(S, open, stop, x0, x1, false, x2)
busy(B, x0, stop, true, x1, x2, x3)
busy(F, x0, stop, x1, true, x2, x3)
busy(S, x0, stop, x1, x2, true, x3)
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)
idle(x0, x1, x2, x3, x4, x5, newbuttons(x6, x7, x8, x9))
or(true, x0)
or(false, x0)

We have to consider all minimal (P,Q,R)-chains.