Termination w.r.t. Q proof of HirokawaMiddeldorp_04

Termination w.r.t. Q of the given QTRS could be proven:

0 QTRS

↳1 QTRSRRRProof (⇔, 135 ms)

↳2 QTRS

↳3 QTRSRRRProof (⇔, 43 ms)

↳4 QTRS

↳5 QTRSRRRProof (⇔, 0 ms)

↳6 QTRS

↳7 QTRSRRRProof (⇔, 8 ms)

↳8 QTRS

↳9 AAECC Innermost (⇔, 0 ms)

↳10 QTRS

↳11 DependencyPairsProof (⇔, 0 ms)

↳12 QDP

↳13 UsableRulesProof (⇔, 0 ms)

↳14 QDP

↳15 QReductionProof (⇔, 0 ms)

↳16 QDP

↳17 TransformationProof (⇔, 0 ms)

↳18 QDP

↳19 TransformationProof (⇔, 0 ms)

↳20 QDP

↳21 TransformationProof (⇔, 0 ms)

↳22 QDP

↳23 TransformationProof (⇔, 0 ms)

↳24 QDP

↳25 TransformationProof (⇔, 0 ms)

↳26 QDP

↳27 TransformationProof (⇔, 0 ms)

↳28 QDP

↳29 TransformationProof (⇔, 0 ms)

↳30 QDP

↳31 TransformationProof (⇔, 0 ms)

↳32 QDP

↳33 TransformationProof (⇔, 0 ms)

↳34 QDP

↳35 TransformationProof (⇔, 0 ms)

↳36 QDP

↳37 TransformationProof (⇔, 0 ms)

↳38 QDP

↳39 TransformationProof (⇔, 0 ms)

↳40 QDP

↳41 TransformationProof (⇔, 0 ms)

↳42 QDP

↳43 TransformationProof (⇔, 0 ms)

↳44 QDP

↳45 TransformationProof (⇔, 0 ms)

↳46 QDP

↳47 TransformationProof (⇔, 0 ms)

↳48 QDP

↳49 TransformationProof (⇔, 0 ms)

↳50 QDP

↳51 TransformationProof (⇔, 0 ms)

↳52 QDP

↳53 TransformationProof (⇔, 0 ms)

↳54 QDP

↳55 TransformationProof (⇔, 0 ms)

↳56 QDP

↳57 TransformationProof (⇔, 0 ms)

↳58 QDP

↳59 DependencyGraphProof (⇔, 0 ms)

↳60 QDP

↳61 TransformationProof (⇔, 0 ms)

↳62 QDP

↳63 TransformationProof (⇔, 0 ms)

↳64 QDP

↳65 TransformationProof (⇔, 0 ms)

↳66 QDP

↳67 TransformationProof (⇔, 0 ms)

↳68 QDP

↳69 DependencyGraphProof (⇔, 0 ms)

↳70 QDP

↳71 TransformationProof (⇔, 0 ms)

↳72 QDP

↳73 TransformationProof (⇔, 0 ms)

↳74 QDP

↳75 TransformationProof (⇔, 0 ms)

↳76 QDP

↳77 TransformationProof (⇔, 0 ms)

↳78 QDP

↳79 TransformationProof (⇔, 0 ms)

↳80 QDP

↳81 TransformationProof (⇔, 0 ms)

↳82 QDP

↳83 DependencyGraphProof (⇔, 0 ms)

↳84 TRUE

(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) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(B) = 0
POL(BF) = 0
POL(F) = 0
POL(FS) = 0
POL(S) = 0
POL(busy(x₁, x₂, x₃, x₄, x₅, x₆, x₇)) = 1 + 2·x₁ + x₂ + x₃ + 2·x₄ + 2·x₅ + 2·x₆ + x₇
POL(closed) = 0
POL(correct) = 1
POL(down) = 0
POL(empty) = 1
POL(false) = 0
POL(idle(x₁, x₂, x₃, x₄, x₅, x₆, x₇)) = 1 + 2·x₁ + x₂ + 2·x₃ + 2·x₄ + 2·x₅ + 2·x₆ + x₇
POL(incorrect) = 0
POL(newbuttons(x₁, x₂, x₃, x₄)) = 2 + 2·x₁ + 2·x₂ + 2·x₃ + 2·x₄
POL(open) = 2
POL(or(x₁, x₂)) = x₁ + x₂
POL(start(x₁)) = 1 + 2·x₁
POL(stop) = 0
POL(true) = 1
POL(up) = 0

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

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, open, stop, false, b2, b3, i) → idle(B, closed, stop, false, b2, b3, i)
busy(F, open, stop, b1, false, b3, i) → idle(F, closed, stop, b1, false, b3, i)
busy(S, open, stop, b1, b2, false, i) → idle(S, closed, stop, b1, b2, false, i)
idle(fl, d, m, b1, b2, b3, newbuttons(i1, i2, i3, i)) → busy(fl, d, m, or(b1, i1), or(b2, i2), or(b3, i3), i)

(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(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, d, stop, true, b2, b3, i) → idle(B, open, stop, false, b2, b3, i)
busy(F, d, stop, b1, true, b3, i) → idle(F, open, stop, b1, false, b3, i)
busy(S, d, stop, b1, b2, true, i) → idle(S, open, stop, b1, b2, false, i)
busy(B, closed, down, b1, b2, b3, i) → idle(B, closed, stop, b1, b2, b3, i)
busy(S, closed, up, b1, b2, b3, i) → idle(S, closed, stop, b1, b2, b3, i)
busy(B, closed, up, true, b2, b3, i) → idle(B, closed, stop, true, b2, b3, i)
busy(F, closed, up, b1, true, b3, i) → idle(F, closed, stop, b1, true, b3, i)
busy(F, closed, down, b1, true, b3, i) → idle(F, closed, stop, b1, true, b3, i)
busy(S, closed, down, b1, b2, true, i) → idle(S, closed, stop, b1, b2, true, i)
busy(B, closed, up, false, b2, b3, i) → idle(BF, closed, up, false, b2, b3, i)
busy(F, closed, up, b1, false, b3, i) → idle(FS, closed, up, b1, false, b3, i)
busy(F, closed, down, b1, false, b3, i) → idle(BF, closed, down, b1, false, b3, i)
busy(S, closed, down, b1, b2, false, i) → idle(FS, closed, down, b1, b2, false, i)
busy(BF, closed, up, b1, b2, b3, i) → idle(F, closed, up, b1, b2, b3, i)
busy(BF, closed, down, b1, b2, b3, i) → idle(B, closed, down, b1, b2, b3, i)
busy(FS, closed, up, b1, b2, b3, i) → idle(S, closed, up, b1, b2, b3, i)
busy(FS, closed, down, b1, b2, b3, i) → idle(F, closed, down, b1, b2, b3, i)
busy(B, closed, stop, false, true, b3, i) → idle(B, closed, up, false, true, b3, i)
busy(B, closed, stop, false, false, true, i) → idle(B, closed, up, false, false, true, i)
busy(F, closed, stop, true, false, b3, i) → idle(F, closed, down, true, false, b3, i)
busy(F, closed, stop, false, false, true, i) → idle(F, closed, up, false, false, true, i)
busy(S, closed, stop, b1, true, false, i) → idle(S, closed, down, b1, true, false, i)
busy(S, closed, stop, true, false, false, i) → idle(S, closed, down, true, false, false, i)
idle(fl, d, m, b1, b2, b3, empty) → busy(fl, d, m, b1, b2, b3, empty)
or(true, b) → true
or(false, b) → b

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(B) = 1
POL(BF) = 1
POL(F) = 1
POL(FS) = 1
POL(S) = 1
POL(busy(x₁, x₂, x₃, x₄, x₅, x₆, x₇)) = 2·x₁ + x₂ + x₃ + x₄ + x₅ + x₆ + x₇
POL(closed) = 0
POL(down) = 0
POL(empty) = 1
POL(false) = 0
POL(idle(x₁, x₂, x₃, x₄, x₅, x₆, x₇)) = 2·x₁ + x₂ + x₃ + x₄ + x₅ + x₆ + x₇
POL(newbuttons(x₁, x₂, x₃, x₄)) = 2 + x₁ + 2·x₂ + x₃ + x₄
POL(open) = 1
POL(or(x₁, x₂)) = 1 + 2·x₁ + x₂
POL(start(x₁)) = 2 + 2·x₁
POL(stop) = 0
POL(true) = 2
POL(up) = 0

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

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)
or(true, b) → true
or(false, b) → b

(4) Obligation:

Q restricted rewrite system:
The TRS R 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, closed, down, b1, b2, b3, i) → idle(B, closed, stop, b1, b2, b3, i)
busy(S, closed, up, b1, b2, b3, i) → idle(S, closed, stop, b1, b2, b3, i)
busy(B, closed, up, true, b2, b3, i) → idle(B, closed, stop, true, b2, b3, i)
busy(F, closed, up, b1, true, b3, i) → idle(F, closed, stop, b1, true, b3, i)
busy(F, closed, down, b1, true, b3, i) → idle(F, closed, stop, b1, true, b3, i)
busy(S, closed, down, b1, b2, true, i) → idle(S, closed, stop, b1, b2, true, i)
busy(B, closed, up, false, b2, b3, i) → idle(BF, closed, up, false, b2, b3, i)
busy(F, closed, up, b1, false, b3, i) → idle(FS, closed, up, b1, false, b3, i)
busy(F, closed, down, b1, false, b3, i) → idle(BF, closed, down, b1, false, b3, i)
busy(S, closed, down, b1, b2, false, i) → idle(FS, closed, down, b1, b2, false, i)
busy(BF, closed, up, b1, b2, b3, i) → idle(F, closed, up, b1, b2, b3, i)
busy(BF, closed, down, b1, b2, b3, i) → idle(B, closed, down, b1, b2, b3, i)
busy(FS, closed, up, b1, b2, b3, i) → idle(S, closed, up, b1, b2, b3, i)
busy(FS, closed, down, b1, b2, b3, i) → idle(F, closed, down, b1, b2, b3, i)
busy(B, closed, stop, false, true, b3, i) → idle(B, closed, up, false, true, b3, i)
busy(B, closed, stop, false, false, true, i) → idle(B, closed, up, false, false, true, i)
busy(F, closed, stop, true, false, b3, i) → idle(F, closed, down, true, false, b3, i)
busy(F, closed, stop, false, false, true, i) → idle(F, closed, up, false, false, true, i)
busy(S, closed, stop, b1, true, false, i) → idle(S, closed, down, b1, true, false, i)
busy(S, closed, stop, true, false, false, i) → idle(S, closed, down, true, false, false, i)
idle(fl, d, m, b1, b2, b3, empty) → busy(fl, d, m, b1, b2, b3, empty)

Q is empty.

(5) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(B) = 0
POL(BF) = 0
POL(F) = 0
POL(FS) = 0
POL(S) = 0
POL(busy(x₁, x₂, x₃, x₄, x₅, x₆, x₇)) = 2 + x₁ + x₂ + x₃ + x₄ + x₅ + x₆ + 2·x₇
POL(closed) = 0
POL(down) = 0
POL(empty) = 0
POL(false) = 0
POL(idle(x₁, x₂, x₃, x₄, x₅, x₆, x₇)) = 2 + 2·x₁ + x₂ + x₃ + x₄ + x₅ + x₆ + x₇
POL(newbuttons(x₁, x₂, x₃, x₄)) = 1 + x₁ + 2·x₂ + x₃ + x₄
POL(start(x₁)) = 2 + 2·x₁
POL(stop) = 0
POL(true) = 0
POL(up) = 0

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

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))

(6) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

start(i) → busy(F, closed, stop, false, false, false, i)
busy(B, closed, down, b1, b2, b3, i) → idle(B, closed, stop, b1, b2, b3, i)
busy(S, closed, up, b1, b2, b3, i) → idle(S, closed, stop, b1, b2, b3, i)
busy(B, closed, up, true, b2, b3, i) → idle(B, closed, stop, true, b2, b3, i)
busy(F, closed, up, b1, true, b3, i) → idle(F, closed, stop, b1, true, b3, i)
busy(F, closed, down, b1, true, b3, i) → idle(F, closed, stop, b1, true, b3, i)
busy(S, closed, down, b1, b2, true, i) → idle(S, closed, stop, b1, b2, true, i)
busy(B, closed, up, false, b2, b3, i) → idle(BF, closed, up, false, b2, b3, i)
busy(F, closed, up, b1, false, b3, i) → idle(FS, closed, up, b1, false, b3, i)
busy(F, closed, down, b1, false, b3, i) → idle(BF, closed, down, b1, false, b3, i)
busy(S, closed, down, b1, b2, false, i) → idle(FS, closed, down, b1, b2, false, i)
busy(BF, closed, up, b1, b2, b3, i) → idle(F, closed, up, b1, b2, b3, i)
busy(BF, closed, down, b1, b2, b3, i) → idle(B, closed, down, b1, b2, b3, i)
busy(FS, closed, up, b1, b2, b3, i) → idle(S, closed, up, b1, b2, b3, i)
busy(FS, closed, down, b1, b2, b3, i) → idle(F, closed, down, b1, b2, b3, i)
busy(B, closed, stop, false, true, b3, i) → idle(B, closed, up, false, true, b3, i)
busy(B, closed, stop, false, false, true, i) → idle(B, closed, up, false, false, true, i)
busy(F, closed, stop, true, false, b3, i) → idle(F, closed, down, true, false, b3, i)
busy(F, closed, stop, false, false, true, i) → idle(F, closed, up, false, false, true, i)
busy(S, closed, stop, b1, true, false, i) → idle(S, closed, down, b1, true, false, i)
busy(S, closed, stop, true, false, false, i) → idle(S, closed, down, true, false, false, i)
idle(fl, d, m, b1, b2, b3, empty) → busy(fl, d, m, b1, b2, b3, empty)

Q is empty.

(7) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(B) = 0
POL(BF) = 0
POL(F) = 0
POL(FS) = 0
POL(S) = 0
POL(busy(x₁, x₂, x₃, x₄, x₅, x₆, x₇)) = x₁ + x₂ + x₃ + 2·x₄ + 2·x₅ + x₆ + 2·x₇
POL(closed) = 1
POL(down) = 0
POL(empty) = 1
POL(false) = 0
POL(idle(x₁, x₂, x₃, x₄, x₅, x₆, x₇)) = 2·x₁ + x₂ + 2·x₃ + 2·x₄ + 2·x₅ + x₆ + 2·x₇
POL(start(x₁)) = 2 + 2·x₁
POL(stop) = 0
POL(true) = 0
POL(up) = 0

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

start(i) → busy(F, closed, stop, false, false, false, i)

(8) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

busy(B, closed, down, b1, b2, b3, i) → idle(B, closed, stop, b1, b2, b3, i)
busy(S, closed, up, b1, b2, b3, i) → idle(S, closed, stop, b1, b2, b3, i)
busy(B, closed, up, true, b2, b3, i) → idle(B, closed, stop, true, b2, b3, i)
busy(F, closed, up, b1, true, b3, i) → idle(F, closed, stop, b1, true, b3, i)
busy(F, closed, down, b1, true, b3, i) → idle(F, closed, stop, b1, true, b3, i)
busy(S, closed, down, b1, b2, true, i) → idle(S, closed, stop, b1, b2, true, i)
busy(B, closed, up, false, b2, b3, i) → idle(BF, closed, up, false, b2, b3, i)
busy(F, closed, up, b1, false, b3, i) → idle(FS, closed, up, b1, false, b3, i)
busy(F, closed, down, b1, false, b3, i) → idle(BF, closed, down, b1, false, b3, i)
busy(S, closed, down, b1, b2, false, i) → idle(FS, closed, down, b1, b2, false, i)
busy(BF, closed, up, b1, b2, b3, i) → idle(F, closed, up, b1, b2, b3, i)
busy(BF, closed, down, b1, b2, b3, i) → idle(B, closed, down, b1, b2, b3, i)
busy(FS, closed, up, b1, b2, b3, i) → idle(S, closed, up, b1, b2, b3, i)
busy(FS, closed, down, b1, b2, b3, i) → idle(F, closed, down, b1, b2, b3, i)
busy(B, closed, stop, false, true, b3, i) → idle(B, closed, up, false, true, b3, i)
busy(B, closed, stop, false, false, true, i) → idle(B, closed, up, false, false, true, i)
busy(F, closed, stop, true, false, b3, i) → idle(F, closed, down, true, false, b3, i)
busy(F, closed, stop, false, false, true, i) → idle(F, closed, up, false, false, true, i)
busy(S, closed, stop, b1, true, false, i) → idle(S, closed, down, b1, true, false, i)
busy(S, closed, stop, true, false, false, i) → idle(S, closed, down, true, false, false, i)
idle(fl, d, m, b1, b2, b3, empty) → busy(fl, d, m, b1, b2, b3, empty)

Q is empty.

(9) AAECC Innermost (EQUIVALENT transformation)

We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is none

The TRS R 2 is

busy(B, closed, down, b1, b2, b3, i) → idle(B, closed, stop, b1, b2, b3, i)
busy(S, closed, up, b1, b2, b3, i) → idle(S, closed, stop, b1, b2, b3, i)
busy(B, closed, up, true, b2, b3, i) → idle(B, closed, stop, true, b2, b3, i)
busy(F, closed, up, b1, true, b3, i) → idle(F, closed, stop, b1, true, b3, i)
busy(F, closed, down, b1, true, b3, i) → idle(F, closed, stop, b1, true, b3, i)
busy(S, closed, down, b1, b2, true, i) → idle(S, closed, stop, b1, b2, true, i)
busy(B, closed, up, false, b2, b3, i) → idle(BF, closed, up, false, b2, b3, i)
busy(F, closed, up, b1, false, b3, i) → idle(FS, closed, up, b1, false, b3, i)
busy(F, closed, down, b1, false, b3, i) → idle(BF, closed, down, b1, false, b3, i)
busy(S, closed, down, b1, b2, false, i) → idle(FS, closed, down, b1, b2, false, i)
busy(BF, closed, up, b1, b2, b3, i) → idle(F, closed, up, b1, b2, b3, i)
busy(BF, closed, down, b1, b2, b3, i) → idle(B, closed, down, b1, b2, b3, i)
busy(FS, closed, up, b1, b2, b3, i) → idle(S, closed, up, b1, b2, b3, i)
busy(FS, closed, down, b1, b2, b3, i) → idle(F, closed, down, b1, b2, b3, i)
busy(B, closed, stop, false, true, b3, i) → idle(B, closed, up, false, true, b3, i)
busy(B, closed, stop, false, false, true, i) → idle(B, closed, up, false, false, true, i)
busy(F, closed, stop, true, false, b3, i) → idle(F, closed, down, true, false, b3, i)
busy(F, closed, stop, false, false, true, i) → idle(F, closed, up, false, false, true, i)
busy(S, closed, stop, b1, true, false, i) → idle(S, closed, down, b1, true, false, i)
busy(S, closed, stop, true, false, false, i) → idle(S, closed, down, true, false, false, i)
idle(fl, d, m, b1, b2, b3, empty) → busy(fl, d, m, b1, b2, b3, empty)

The signature Sigma is {busy, idle}

(10) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

busy(B, closed, down, b1, b2, b3, i) → idle(B, closed, stop, b1, b2, b3, i)
busy(S, closed, up, b1, b2, b3, i) → idle(S, closed, stop, b1, b2, b3, i)
busy(B, closed, up, true, b2, b3, i) → idle(B, closed, stop, true, b2, b3, i)
busy(F, closed, up, b1, true, b3, i) → idle(F, closed, stop, b1, true, b3, i)
busy(F, closed, down, b1, true, b3, i) → idle(F, closed, stop, b1, true, b3, i)
busy(S, closed, down, b1, b2, true, i) → idle(S, closed, stop, b1, b2, true, i)
busy(B, closed, up, false, b2, b3, i) → idle(BF, closed, up, false, b2, b3, i)
busy(F, closed, up, b1, false, b3, i) → idle(FS, closed, up, b1, false, b3, i)
busy(F, closed, down, b1, false, b3, i) → idle(BF, closed, down, b1, false, b3, i)
busy(S, closed, down, b1, b2, false, i) → idle(FS, closed, down, b1, b2, false, i)
busy(BF, closed, up, b1, b2, b3, i) → idle(F, closed, up, b1, b2, b3, i)
busy(BF, closed, down, b1, b2, b3, i) → idle(B, closed, down, b1, b2, b3, i)
busy(FS, closed, up, b1, b2, b3, i) → idle(S, closed, up, b1, b2, b3, i)
busy(FS, closed, down, b1, b2, b3, i) → idle(F, closed, down, b1, b2, b3, i)
busy(B, closed, stop, false, true, b3, i) → idle(B, closed, up, false, true, b3, i)
busy(B, closed, stop, false, false, true, i) → idle(B, closed, up, false, false, true, i)
busy(F, closed, stop, true, false, b3, i) → idle(F, closed, down, true, false, b3, i)
busy(F, closed, stop, false, false, true, i) → idle(F, closed, up, false, false, true, i)
busy(S, closed, stop, b1, true, false, i) → idle(S, closed, down, b1, true, false, i)
busy(S, closed, stop, true, false, false, i) → idle(S, closed, down, true, false, false, i)
idle(fl, d, m, b1, b2, b3, empty) → busy(fl, d, m, b1, b2, b3, empty)

The set Q consists of the following terms:

busy(B, closed, down, x0, x1, x2, x3)
busy(S, closed, up, x0, x1, x2, x3)
busy(B, closed, up, true, x0, x1, x2)
busy(F, closed, up, x0, true, x1, x2)
busy(F, closed, down, x0, true, x1, x2)
busy(S, closed, down, x0, x1, true, x2)
busy(B, closed, up, false, x0, x1, x2)
busy(F, closed, up, x0, false, x1, x2)
busy(F, closed, down, x0, false, x1, x2)
busy(S, closed, down, x0, x1, false, x2)
busy(BF, closed, up, x0, x1, x2, x3)
busy(BF, closed, down, x0, x1, x2, x3)
busy(FS, closed, up, x0, x1, x2, x3)
busy(FS, closed, down, x0, x1, x2, x3)
busy(B, closed, stop, false, true, x0, x1)
busy(B, closed, stop, false, false, true, x0)
busy(F, closed, stop, true, false, x0, x1)
busy(F, closed, stop, false, false, true, x0)
busy(S, closed, stop, x0, true, false, x1)
busy(S, closed, stop, true, false, false, x0)
idle(x0, x1, x2, x3, x4, x5, empty)

(11) DependencyPairsProof (EQUIVALENT transformation)

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

(12) Obligation:

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

BUSY(B, closed, down, b1, b2, b3, i) → IDLE(B, closed, stop, b1, b2, b3, i)
BUSY(S, closed, up, b1, b2, b3, i) → IDLE(S, closed, stop, b1, b2, b3, i)
BUSY(B, closed, up, true, b2, b3, i) → IDLE(B, closed, stop, true, b2, b3, i)
BUSY(F, closed, up, b1, true, b3, i) → IDLE(F, closed, stop, b1, true, b3, i)
BUSY(F, closed, down, b1, true, b3, i) → IDLE(F, closed, stop, b1, true, b3, i)
BUSY(S, closed, down, b1, b2, true, i) → IDLE(S, closed, stop, b1, b2, true, i)
BUSY(B, closed, up, false, b2, b3, i) → IDLE(BF, closed, up, false, b2, b3, i)
BUSY(F, closed, up, b1, false, b3, i) → IDLE(FS, closed, up, b1, false, b3, i)
BUSY(F, closed, down, b1, false, b3, i) → IDLE(BF, closed, down, b1, false, b3, i)
BUSY(S, closed, down, b1, b2, false, i) → IDLE(FS, closed, down, b1, b2, false, i)
BUSY(BF, closed, up, b1, b2, b3, i) → IDLE(F, closed, up, b1, b2, b3, i)
BUSY(BF, closed, down, b1, b2, b3, i) → IDLE(B, closed, down, b1, b2, b3, i)
BUSY(FS, closed, up, b1, b2, b3, i) → IDLE(S, closed, up, b1, b2, b3, i)
BUSY(FS, closed, down, b1, b2, b3, i) → IDLE(F, closed, down, b1, b2, b3, i)
BUSY(B, closed, stop, false, true, b3, i) → IDLE(B, closed, up, false, true, b3, i)
BUSY(B, closed, stop, false, false, true, i) → IDLE(B, closed, up, false, false, true, i)
BUSY(F, closed, stop, true, false, b3, i) → IDLE(F, closed, down, true, false, b3, i)
BUSY(F, closed, stop, false, false, true, i) → IDLE(F, closed, up, false, false, true, i)
BUSY(S, closed, stop, b1, true, false, i) → IDLE(S, closed, down, b1, true, false, i)
BUSY(S, closed, stop, true, false, false, i) → IDLE(S, closed, down, true, false, false, i)
IDLE(fl, d, m, b1, b2, b3, empty) → BUSY(fl, d, m, b1, b2, b3, empty)

The TRS R consists of the following rules:

busy(B, closed, down, b1, b2, b3, i) → idle(B, closed, stop, b1, b2, b3, i)
busy(S, closed, up, b1, b2, b3, i) → idle(S, closed, stop, b1, b2, b3, i)
busy(B, closed, up, true, b2, b3, i) → idle(B, closed, stop, true, b2, b3, i)
busy(F, closed, up, b1, true, b3, i) → idle(F, closed, stop, b1, true, b3, i)
busy(F, closed, down, b1, true, b3, i) → idle(F, closed, stop, b1, true, b3, i)
busy(S, closed, down, b1, b2, true, i) → idle(S, closed, stop, b1, b2, true, i)
busy(B, closed, up, false, b2, b3, i) → idle(BF, closed, up, false, b2, b3, i)
busy(F, closed, up, b1, false, b3, i) → idle(FS, closed, up, b1, false, b3, i)
busy(F, closed, down, b1, false, b3, i) → idle(BF, closed, down, b1, false, b3, i)
busy(S, closed, down, b1, b2, false, i) → idle(FS, closed, down, b1, b2, false, i)
busy(BF, closed, up, b1, b2, b3, i) → idle(F, closed, up, b1, b2, b3, i)
busy(BF, closed, down, b1, b2, b3, i) → idle(B, closed, down, b1, b2, b3, i)
busy(FS, closed, up, b1, b2, b3, i) → idle(S, closed, up, b1, b2, b3, i)
busy(FS, closed, down, b1, b2, b3, i) → idle(F, closed, down, b1, b2, b3, i)
busy(B, closed, stop, false, true, b3, i) → idle(B, closed, up, false, true, b3, i)
busy(B, closed, stop, false, false, true, i) → idle(B, closed, up, false, false, true, i)
busy(F, closed, stop, true, false, b3, i) → idle(F, closed, down, true, false, b3, i)
busy(F, closed, stop, false, false, true, i) → idle(F, closed, up, false, false, true, i)
busy(S, closed, stop, b1, true, false, i) → idle(S, closed, down, b1, true, false, i)
busy(S, closed, stop, true, false, false, i) → idle(S, closed, down, true, false, false, i)
idle(fl, d, m, b1, b2, b3, empty) → busy(fl, d, m, b1, b2, b3, empty)

The set Q consists of the following terms:

busy(B, closed, down, x0, x1, x2, x3)
busy(S, closed, up, x0, x1, x2, x3)
busy(B, closed, up, true, x0, x1, x2)
busy(F, closed, up, x0, true, x1, x2)
busy(F, closed, down, x0, true, x1, x2)
busy(S, closed, down, x0, x1, true, x2)
busy(B, closed, up, false, x0, x1, x2)
busy(F, closed, up, x0, false, x1, x2)
busy(F, closed, down, x0, false, x1, x2)
busy(S, closed, down, x0, x1, false, x2)
busy(BF, closed, up, x0, x1, x2, x3)
busy(BF, closed, down, x0, x1, x2, x3)
busy(FS, closed, up, x0, x1, x2, x3)
busy(FS, closed, down, x0, x1, x2, x3)
busy(B, closed, stop, false, true, x0, x1)
busy(B, closed, stop, false, false, true, x0)
busy(F, closed, stop, true, false, x0, x1)
busy(F, closed, stop, false, false, true, x0)
busy(S, closed, stop, x0, true, false, x1)
busy(S, closed, stop, true, false, false, x0)
idle(x0, x1, x2, x3, x4, x5, empty)

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

(13) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(14) Obligation:

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

BUSY(B, closed, down, b1, b2, b3, i) → IDLE(B, closed, stop, b1, b2, b3, i)
BUSY(S, closed, up, b1, b2, b3, i) → IDLE(S, closed, stop, b1, b2, b3, i)
BUSY(B, closed, up, true, b2, b3, i) → IDLE(B, closed, stop, true, b2, b3, i)
BUSY(F, closed, up, b1, true, b3, i) → IDLE(F, closed, stop, b1, true, b3, i)
BUSY(F, closed, down, b1, true, b3, i) → IDLE(F, closed, stop, b1, true, b3, i)
BUSY(S, closed, down, b1, b2, true, i) → IDLE(S, closed, stop, b1, b2, true, i)
BUSY(B, closed, up, false, b2, b3, i) → IDLE(BF, closed, up, false, b2, b3, i)
BUSY(F, closed, up, b1, false, b3, i) → IDLE(FS, closed, up, b1, false, b3, i)
BUSY(F, closed, down, b1, false, b3, i) → IDLE(BF, closed, down, b1, false, b3, i)
BUSY(S, closed, down, b1, b2, false, i) → IDLE(FS, closed, down, b1, b2, false, i)
BUSY(BF, closed, up, b1, b2, b3, i) → IDLE(F, closed, up, b1, b2, b3, i)
BUSY(BF, closed, down, b1, b2, b3, i) → IDLE(B, closed, down, b1, b2, b3, i)
BUSY(FS, closed, up, b1, b2, b3, i) → IDLE(S, closed, up, b1, b2, b3, i)
BUSY(FS, closed, down, b1, b2, b3, i) → IDLE(F, closed, down, b1, b2, b3, i)
BUSY(B, closed, stop, false, true, b3, i) → IDLE(B, closed, up, false, true, b3, i)
BUSY(B, closed, stop, false, false, true, i) → IDLE(B, closed, up, false, false, true, i)
BUSY(F, closed, stop, true, false, b3, i) → IDLE(F, closed, down, true, false, b3, i)
BUSY(F, closed, stop, false, false, true, i) → IDLE(F, closed, up, false, false, true, i)
BUSY(S, closed, stop, b1, true, false, i) → IDLE(S, closed, down, b1, true, false, i)
BUSY(S, closed, stop, true, false, false, i) → IDLE(S, closed, down, true, false, false, i)
IDLE(fl, d, m, b1, b2, b3, empty) → BUSY(fl, d, m, b1, b2, b3, empty)

R is empty.
The set Q consists of the following terms:

busy(B, closed, down, x0, x1, x2, x3)
busy(S, closed, up, x0, x1, x2, x3)
busy(B, closed, up, true, x0, x1, x2)
busy(F, closed, up, x0, true, x1, x2)
busy(F, closed, down, x0, true, x1, x2)
busy(S, closed, down, x0, x1, true, x2)
busy(B, closed, up, false, x0, x1, x2)
busy(F, closed, up, x0, false, x1, x2)
busy(F, closed, down, x0, false, x1, x2)
busy(S, closed, down, x0, x1, false, x2)
busy(BF, closed, up, x0, x1, x2, x3)
busy(BF, closed, down, x0, x1, x2, x3)
busy(FS, closed, up, x0, x1, x2, x3)
busy(FS, closed, down, x0, x1, x2, x3)
busy(B, closed, stop, false, true, x0, x1)
busy(B, closed, stop, false, false, true, x0)
busy(F, closed, stop, true, false, x0, x1)
busy(F, closed, stop, false, false, true, x0)
busy(S, closed, stop, x0, true, false, x1)
busy(S, closed, stop, true, false, false, x0)
idle(x0, x1, x2, x3, x4, x5, empty)

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

(15) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

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)

(16) Obligation:

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

BUSY(B, closed, down, b1, b2, b3, i) → IDLE(B, closed, stop, b1, b2, b3, i)
BUSY(S, closed, up, b1, b2, b3, i) → IDLE(S, closed, stop, b1, b2, b3, i)
BUSY(B, closed, up, true, b2, b3, i) → IDLE(B, closed, stop, true, b2, b3, i)
BUSY(F, closed, up, b1, true, b3, i) → IDLE(F, closed, stop, b1, true, b3, i)
BUSY(F, closed, down, b1, true, b3, i) → IDLE(F, closed, stop, b1, true, b3, i)
BUSY(S, closed, down, b1, b2, true, i) → IDLE(S, closed, stop, b1, b2, true, i)
BUSY(B, closed, up, false, b2, b3, i) → IDLE(BF, closed, up, false, b2, b3, i)
BUSY(F, closed, up, b1, false, b3, i) → IDLE(FS, closed, up, b1, false, b3, i)
BUSY(F, closed, down, b1, false, b3, i) → IDLE(BF, closed, down, b1, false, b3, i)
BUSY(S, closed, down, b1, b2, false, i) → IDLE(FS, closed, down, b1, b2, false, i)
BUSY(BF, closed, up, b1, b2, b3, i) → IDLE(F, closed, up, b1, b2, b3, i)
BUSY(BF, closed, down, b1, b2, b3, i) → IDLE(B, closed, down, b1, b2, b3, i)
BUSY(FS, closed, up, b1, b2, b3, i) → IDLE(S, closed, up, b1, b2, b3, i)
BUSY(FS, closed, down, b1, b2, b3, i) → IDLE(F, closed, down, b1, b2, b3, i)
BUSY(B, closed, stop, false, true, b3, i) → IDLE(B, closed, up, false, true, b3, i)
BUSY(B, closed, stop, false, false, true, i) → IDLE(B, closed, up, false, false, true, i)
BUSY(F, closed, stop, true, false, b3, i) → IDLE(F, closed, down, true, false, b3, i)
BUSY(F, closed, stop, false, false, true, i) → IDLE(F, closed, up, false, false, true, i)
BUSY(S, closed, stop, b1, true, false, i) → IDLE(S, closed, down, b1, true, false, i)
BUSY(S, closed, stop, true, false, false, i) → IDLE(S, closed, down, true, false, false, i)
IDLE(fl, d, m, b1, b2, b3, empty) → BUSY(fl, d, m, b1, b2, b3, empty)

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

(17) TransformationProof (EQUIVALENT transformation)

By instantiating [LPAR04] the rule BUSY(B, closed, down, b1, b2, b3, i) → IDLE(B, closed, stop, b1, b2, b3, i) we obtained the following new rules [LPAR04]:

BUSY(B, closed, down, z3, z4, z5, empty) → IDLE(B, closed, stop, z3, z4, z5, empty) → BUSY(B, closed, down, z3, z4, z5, empty) → IDLE(B, closed, stop, z3, z4, z5, empty)

(18) Obligation:

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

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)
BUSY(B, closed, down, z3, z4, z5, empty) → IDLE(B, closed, stop, z3, z4, z5, empty)

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

(19) TransformationProof (EQUIVALENT transformation)

By instantiating [LPAR04] the rule BUSY(S, closed, up, b1, b2, b3, i) → IDLE(S, closed, stop, b1, b2, b3, i) we obtained the following new rules [LPAR04]:

BUSY(S, closed, up, z3, z4, z5, empty) → IDLE(S, closed, stop, z3, z4, z5, empty) → BUSY(S, closed, up, z3, z4, z5, empty) → IDLE(S, closed, stop, z3, z4, z5, empty)

(20) Obligation:

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

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)
BUSY(B, closed, down, z3, z4, z5, empty) → IDLE(B, closed, stop, z3, z4, z5, empty)
BUSY(S, closed, up, z3, z4, z5, empty) → IDLE(S, closed, stop, z3, z4, z5, empty)

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

(21) TransformationProof (EQUIVALENT transformation)

By instantiating [LPAR04] the rule BUSY(B, closed, up, true, b2, b3, i) → IDLE(B, closed, stop, true, b2, b3, i) we obtained the following new rules [LPAR04]:

BUSY(B, closed, up, true, z4, z5, empty) → IDLE(B, closed, stop, true, z4, z5, empty) → BUSY(B, closed, up, true, z4, z5, empty) → IDLE(B, closed, stop, true, z4, z5, empty)

(22) Obligation:

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

BUSY(F, closed, up, b1, true, b3, i) → IDLE(F, closed, stop, b1, true, b3, i)
BUSY(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)
BUSY(B, closed, down, z3, z4, z5, empty) → IDLE(B, closed, stop, z3, z4, z5, empty)
BUSY(S, closed, up, z3, z4, z5, empty) → IDLE(S, closed, stop, z3, z4, z5, empty)
BUSY(B, closed, up, true, z4, z5, empty) → IDLE(B, closed, stop, true, z4, z5, empty)

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

(23) TransformationProof (EQUIVALENT transformation)

By instantiating [LPAR04] the rule BUSY(F, closed, up, b1, true, b3, i) → IDLE(F, closed, stop, b1, true, b3, i) we obtained the following new rules [LPAR04]:

BUSY(F, closed, up, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty) → BUSY(F, closed, up, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty)

(24) Obligation:

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

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)
BUSY(B, closed, down, z3, z4, z5, empty) → IDLE(B, closed, stop, z3, z4, z5, empty)
BUSY(S, closed, up, z3, z4, z5, empty) → IDLE(S, closed, stop, z3, z4, z5, empty)
BUSY(B, closed, up, true, z4, z5, empty) → IDLE(B, closed, stop, true, z4, z5, empty)
BUSY(F, closed, up, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty)

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

(25) TransformationProof (EQUIVALENT transformation)

By instantiating [LPAR04] the rule BUSY(F, closed, down, b1, true, b3, i) → IDLE(F, closed, stop, b1, true, b3, i) we obtained the following new rules [LPAR04]:

BUSY(F, closed, down, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty) → BUSY(F, closed, down, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty)

(26) Obligation:

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

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)
BUSY(B, closed, down, z3, z4, z5, empty) → IDLE(B, closed, stop, z3, z4, z5, empty)
BUSY(S, closed, up, z3, z4, z5, empty) → IDLE(S, closed, stop, z3, z4, z5, empty)
BUSY(B, closed, up, true, z4, z5, empty) → IDLE(B, closed, stop, true, z4, z5, empty)
BUSY(F, closed, up, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty)
BUSY(F, closed, down, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty)

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

(27) TransformationProof (EQUIVALENT transformation)

By instantiating [LPAR04] the rule BUSY(S, closed, down, b1, b2, true, i) → IDLE(S, closed, stop, b1, b2, true, i) we obtained the following new rules [LPAR04]:

BUSY(S, closed, down, z3, z4, true, empty) → IDLE(S, closed, stop, z3, z4, true, empty) → BUSY(S, closed, down, z3, z4, true, empty) → IDLE(S, closed, stop, z3, z4, true, empty)

(28) Obligation:

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

BUSY(B, closed, up, false, b2, b3, i) → IDLE(BF, closed, up, false, b2, b3, i)
BUSY(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)
BUSY(B, closed, down, z3, z4, z5, empty) → IDLE(B, closed, stop, z3, z4, z5, empty)
BUSY(S, closed, up, z3, z4, z5, empty) → IDLE(S, closed, stop, z3, z4, z5, empty)
BUSY(B, closed, up, true, z4, z5, empty) → IDLE(B, closed, stop, true, z4, z5, empty)
BUSY(F, closed, up, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty)
BUSY(F, closed, down, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty)
BUSY(S, closed, down, z3, z4, true, empty) → IDLE(S, closed, stop, z3, z4, true, empty)

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

(29) TransformationProof (EQUIVALENT transformation)

By instantiating [LPAR04] the rule BUSY(B, closed, up, false, b2, b3, i) → IDLE(BF, closed, up, false, b2, b3, i) we obtained the following new rules [LPAR04]:

BUSY(B, closed, up, false, z4, z5, empty) → IDLE(BF, closed, up, false, z4, z5, empty) → BUSY(B, closed, up, false, z4, z5, empty) → IDLE(BF, closed, up, false, z4, z5, empty)

(30) Obligation:

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

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)
BUSY(B, closed, down, z3, z4, z5, empty) → IDLE(B, closed, stop, z3, z4, z5, empty)
BUSY(S, closed, up, z3, z4, z5, empty) → IDLE(S, closed, stop, z3, z4, z5, empty)
BUSY(B, closed, up, true, z4, z5, empty) → IDLE(B, closed, stop, true, z4, z5, empty)
BUSY(F, closed, up, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty)
BUSY(F, closed, down, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty)
BUSY(S, closed, down, z3, z4, true, empty) → IDLE(S, closed, stop, z3, z4, true, empty)
BUSY(B, closed, up, false, z4, z5, empty) → IDLE(BF, closed, up, false, z4, z5, empty)

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

(31) TransformationProof (EQUIVALENT transformation)

By instantiating [LPAR04] the rule BUSY(F, closed, up, b1, false, b3, i) → IDLE(FS, closed, up, b1, false, b3, i) we obtained the following new rules [LPAR04]:

BUSY(F, closed, up, z3, false, z5, empty) → IDLE(FS, closed, up, z3, false, z5, empty) → BUSY(F, closed, up, z3, false, z5, empty) → IDLE(FS, closed, up, z3, false, z5, empty)

(32) Obligation:

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

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)
BUSY(B, closed, down, z3, z4, z5, empty) → IDLE(B, closed, stop, z3, z4, z5, empty)
BUSY(S, closed, up, z3, z4, z5, empty) → IDLE(S, closed, stop, z3, z4, z5, empty)
BUSY(B, closed, up, true, z4, z5, empty) → IDLE(B, closed, stop, true, z4, z5, empty)
BUSY(F, closed, up, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty)
BUSY(F, closed, down, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty)
BUSY(S, closed, down, z3, z4, true, empty) → IDLE(S, closed, stop, z3, z4, true, empty)
BUSY(B, closed, up, false, z4, z5, empty) → IDLE(BF, closed, up, false, z4, z5, empty)
BUSY(F, closed, up, z3, false, z5, empty) → IDLE(FS, closed, up, z3, false, z5, empty)

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

(33) TransformationProof (EQUIVALENT transformation)

By instantiating [LPAR04] the rule BUSY(F, closed, down, b1, false, b3, i) → IDLE(BF, closed, down, b1, false, b3, i) we obtained the following new rules [LPAR04]:

BUSY(F, closed, down, z3, false, z5, empty) → IDLE(BF, closed, down, z3, false, z5, empty) → BUSY(F, closed, down, z3, false, z5, empty) → IDLE(BF, closed, down, z3, false, z5, empty)

(34) Obligation:

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

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)
BUSY(B, closed, down, z3, z4, z5, empty) → IDLE(B, closed, stop, z3, z4, z5, empty)
BUSY(S, closed, up, z3, z4, z5, empty) → IDLE(S, closed, stop, z3, z4, z5, empty)
BUSY(B, closed, up, true, z4, z5, empty) → IDLE(B, closed, stop, true, z4, z5, empty)
BUSY(F, closed, up, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty)
BUSY(F, closed, down, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty)
BUSY(S, closed, down, z3, z4, true, empty) → IDLE(S, closed, stop, z3, z4, true, empty)
BUSY(B, closed, up, false, z4, z5, empty) → IDLE(BF, closed, up, false, z4, z5, empty)
BUSY(F, closed, up, z3, false, z5, empty) → IDLE(FS, closed, up, z3, false, z5, empty)
BUSY(F, closed, down, z3, false, z5, empty) → IDLE(BF, closed, down, z3, false, z5, empty)

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

(35) TransformationProof (EQUIVALENT transformation)

By instantiating [LPAR04] the rule BUSY(S, closed, down, b1, b2, false, i) → IDLE(FS, closed, down, b1, b2, false, i) we obtained the following new rules [LPAR04]:

BUSY(S, closed, down, z3, z4, false, empty) → IDLE(FS, closed, down, z3, z4, false, empty) → BUSY(S, closed, down, z3, z4, false, empty) → IDLE(FS, closed, down, z3, z4, false, empty)

(36) Obligation:

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

BUSY(BF, closed, up, b1, b2, b3, i) → IDLE(F, closed, up, b1, b2, b3, i)
BUSY(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)
BUSY(B, closed, down, z3, z4, z5, empty) → IDLE(B, closed, stop, z3, z4, z5, empty)
BUSY(S, closed, up, z3, z4, z5, empty) → IDLE(S, closed, stop, z3, z4, z5, empty)
BUSY(B, closed, up, true, z4, z5, empty) → IDLE(B, closed, stop, true, z4, z5, empty)
BUSY(F, closed, up, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty)
BUSY(F, closed, down, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty)
BUSY(S, closed, down, z3, z4, true, empty) → IDLE(S, closed, stop, z3, z4, true, empty)
BUSY(B, closed, up, false, z4, z5, empty) → IDLE(BF, closed, up, false, z4, z5, empty)
BUSY(F, closed, up, z3, false, z5, empty) → IDLE(FS, closed, up, z3, false, z5, empty)
BUSY(F, closed, down, z3, false, z5, empty) → IDLE(BF, closed, down, z3, false, z5, empty)
BUSY(S, closed, down, z3, z4, false, empty) → IDLE(FS, closed, down, z3, z4, false, empty)

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

(37) TransformationProof (EQUIVALENT transformation)

By instantiating [LPAR04] the rule BUSY(BF, closed, up, b1, b2, b3, i) → IDLE(F, closed, up, b1, b2, b3, i) we obtained the following new rules [LPAR04]:

BUSY(BF, closed, up, z3, z4, z5, empty) → IDLE(F, closed, up, z3, z4, z5, empty) → BUSY(BF, closed, up, z3, z4, z5, empty) → IDLE(F, closed, up, z3, z4, z5, empty)

(38) Obligation:

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

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)
BUSY(B, closed, down, z3, z4, z5, empty) → IDLE(B, closed, stop, z3, z4, z5, empty)
BUSY(S, closed, up, z3, z4, z5, empty) → IDLE(S, closed, stop, z3, z4, z5, empty)
BUSY(B, closed, up, true, z4, z5, empty) → IDLE(B, closed, stop, true, z4, z5, empty)
BUSY(F, closed, up, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty)
BUSY(F, closed, down, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty)
BUSY(S, closed, down, z3, z4, true, empty) → IDLE(S, closed, stop, z3, z4, true, empty)
BUSY(B, closed, up, false, z4, z5, empty) → IDLE(BF, closed, up, false, z4, z5, empty)
BUSY(F, closed, up, z3, false, z5, empty) → IDLE(FS, closed, up, z3, false, z5, empty)
BUSY(F, closed, down, z3, false, z5, empty) → IDLE(BF, closed, down, z3, false, z5, empty)
BUSY(S, closed, down, z3, z4, false, empty) → IDLE(FS, closed, down, z3, z4, false, empty)
BUSY(BF, closed, up, z3, z4, z5, empty) → IDLE(F, closed, up, z3, z4, z5, empty)

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

(39) TransformationProof (EQUIVALENT transformation)

By instantiating [LPAR04] the rule BUSY(BF, closed, down, b1, b2, b3, i) → IDLE(B, closed, down, b1, b2, b3, i) we obtained the following new rules [LPAR04]:

BUSY(BF, closed, down, z3, z4, z5, empty) → IDLE(B, closed, down, z3, z4, z5, empty) → BUSY(BF, closed, down, z3, z4, z5, empty) → IDLE(B, closed, down, z3, z4, z5, empty)

(40) Obligation:

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

BUSY(FS, closed, up, b1, b2, b3, i) → IDLE(S, closed, up, b1, b2, b3, i)
BUSY(FS, closed, down, b1, b2, b3, i) → IDLE(F, closed, down, b1, b2, b3, i)
BUSY(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)
BUSY(B, closed, down, z3, z4, z5, empty) → IDLE(B, closed, stop, z3, z4, z5, empty)
BUSY(S, closed, up, z3, z4, z5, empty) → IDLE(S, closed, stop, z3, z4, z5, empty)
BUSY(B, closed, up, true, z4, z5, empty) → IDLE(B, closed, stop, true, z4, z5, empty)
BUSY(F, closed, up, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty)
BUSY(F, closed, down, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty)
BUSY(S, closed, down, z3, z4, true, empty) → IDLE(S, closed, stop, z3, z4, true, empty)
BUSY(B, closed, up, false, z4, z5, empty) → IDLE(BF, closed, up, false, z4, z5, empty)
BUSY(F, closed, up, z3, false, z5, empty) → IDLE(FS, closed, up, z3, false, z5, empty)
BUSY(F, closed, down, z3, false, z5, empty) → IDLE(BF, closed, down, z3, false, z5, empty)
BUSY(S, closed, down, z3, z4, false, empty) → IDLE(FS, closed, down, z3, z4, false, empty)
BUSY(BF, closed, up, z3, z4, z5, empty) → IDLE(F, closed, up, z3, z4, z5, empty)
BUSY(BF, closed, down, z3, z4, z5, empty) → IDLE(B, closed, down, z3, z4, z5, empty)

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

(41) TransformationProof (EQUIVALENT transformation)

By instantiating [LPAR04] the rule BUSY(FS, closed, up, b1, b2, b3, i) → IDLE(S, closed, up, b1, b2, b3, i) we obtained the following new rules [LPAR04]:

BUSY(FS, closed, up, z3, z4, z5, empty) → IDLE(S, closed, up, z3, z4, z5, empty) → BUSY(FS, closed, up, z3, z4, z5, empty) → IDLE(S, closed, up, z3, z4, z5, empty)

(42) Obligation:

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

BUSY(FS, closed, down, b1, b2, b3, i) → IDLE(F, closed, down, b1, b2, b3, i)
BUSY(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)
BUSY(B, closed, down, z3, z4, z5, empty) → IDLE(B, closed, stop, z3, z4, z5, empty)
BUSY(S, closed, up, z3, z4, z5, empty) → IDLE(S, closed, stop, z3, z4, z5, empty)
BUSY(B, closed, up, true, z4, z5, empty) → IDLE(B, closed, stop, true, z4, z5, empty)
BUSY(F, closed, up, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty)
BUSY(F, closed, down, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty)
BUSY(S, closed, down, z3, z4, true, empty) → IDLE(S, closed, stop, z3, z4, true, empty)
BUSY(B, closed, up, false, z4, z5, empty) → IDLE(BF, closed, up, false, z4, z5, empty)
BUSY(F, closed, up, z3, false, z5, empty) → IDLE(FS, closed, up, z3, false, z5, empty)
BUSY(F, closed, down, z3, false, z5, empty) → IDLE(BF, closed, down, z3, false, z5, empty)
BUSY(S, closed, down, z3, z4, false, empty) → IDLE(FS, closed, down, z3, z4, false, empty)
BUSY(BF, closed, up, z3, z4, z5, empty) → IDLE(F, closed, up, z3, z4, z5, empty)
BUSY(BF, closed, down, z3, z4, z5, empty) → IDLE(B, closed, down, z3, z4, z5, empty)
BUSY(FS, closed, up, z3, z4, z5, empty) → IDLE(S, closed, up, z3, z4, z5, empty)

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

(43) TransformationProof (EQUIVALENT transformation)

By instantiating [LPAR04] the rule BUSY(FS, closed, down, b1, b2, b3, i) → IDLE(F, closed, down, b1, b2, b3, i) we obtained the following new rules [LPAR04]:

BUSY(FS, closed, down, z3, z4, z5, empty) → IDLE(F, closed, down, z3, z4, z5, empty) → BUSY(FS, closed, down, z3, z4, z5, empty) → IDLE(F, closed, down, z3, z4, z5, empty)

(44) Obligation:

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

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)
BUSY(B, closed, down, z3, z4, z5, empty) → IDLE(B, closed, stop, z3, z4, z5, empty)
BUSY(S, closed, up, z3, z4, z5, empty) → IDLE(S, closed, stop, z3, z4, z5, empty)
BUSY(B, closed, up, true, z4, z5, empty) → IDLE(B, closed, stop, true, z4, z5, empty)
BUSY(F, closed, up, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty)
BUSY(F, closed, down, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty)
BUSY(S, closed, down, z3, z4, true, empty) → IDLE(S, closed, stop, z3, z4, true, empty)
BUSY(B, closed, up, false, z4, z5, empty) → IDLE(BF, closed, up, false, z4, z5, empty)
BUSY(F, closed, up, z3, false, z5, empty) → IDLE(FS, closed, up, z3, false, z5, empty)
BUSY(F, closed, down, z3, false, z5, empty) → IDLE(BF, closed, down, z3, false, z5, empty)
BUSY(S, closed, down, z3, z4, false, empty) → IDLE(FS, closed, down, z3, z4, false, empty)
BUSY(BF, closed, up, z3, z4, z5, empty) → IDLE(F, closed, up, z3, z4, z5, empty)
BUSY(BF, closed, down, z3, z4, z5, empty) → IDLE(B, closed, down, z3, z4, z5, empty)
BUSY(FS, closed, up, z3, z4, z5, empty) → IDLE(S, closed, up, z3, z4, z5, empty)
BUSY(FS, closed, down, z3, z4, z5, empty) → IDLE(F, closed, down, z3, z4, z5, empty)

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

(45) TransformationProof (EQUIVALENT transformation)

By instantiating [LPAR04] the rule BUSY(B, closed, stop, false, true, b3, i) → IDLE(B, closed, up, false, true, b3, i) we obtained the following new rules [LPAR04]:

BUSY(B, closed, stop, false, true, z5, empty) → IDLE(B, closed, up, false, true, z5, empty) → BUSY(B, closed, stop, false, true, z5, empty) → IDLE(B, closed, up, false, true, z5, empty)

(46) Obligation:

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

BUSY(B, closed, stop, false, false, true, i) → IDLE(B, closed, up, false, false, true, i)
BUSY(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)
BUSY(B, closed, down, z3, z4, z5, empty) → IDLE(B, closed, stop, z3, z4, z5, empty)
BUSY(S, closed, up, z3, z4, z5, empty) → IDLE(S, closed, stop, z3, z4, z5, empty)
BUSY(B, closed, up, true, z4, z5, empty) → IDLE(B, closed, stop, true, z4, z5, empty)
BUSY(F, closed, up, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty)
BUSY(F, closed, down, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty)
BUSY(S, closed, down, z3, z4, true, empty) → IDLE(S, closed, stop, z3, z4, true, empty)
BUSY(B, closed, up, false, z4, z5, empty) → IDLE(BF, closed, up, false, z4, z5, empty)
BUSY(F, closed, up, z3, false, z5, empty) → IDLE(FS, closed, up, z3, false, z5, empty)
BUSY(F, closed, down, z3, false, z5, empty) → IDLE(BF, closed, down, z3, false, z5, empty)
BUSY(S, closed, down, z3, z4, false, empty) → IDLE(FS, closed, down, z3, z4, false, empty)
BUSY(BF, closed, up, z3, z4, z5, empty) → IDLE(F, closed, up, z3, z4, z5, empty)
BUSY(BF, closed, down, z3, z4, z5, empty) → IDLE(B, closed, down, z3, z4, z5, empty)
BUSY(FS, closed, up, z3, z4, z5, empty) → IDLE(S, closed, up, z3, z4, z5, empty)
BUSY(FS, closed, down, z3, z4, z5, empty) → IDLE(F, closed, down, z3, z4, z5, empty)
BUSY(B, closed, stop, false, true, z5, empty) → IDLE(B, closed, up, false, true, z5, empty)

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

(47) TransformationProof (EQUIVALENT transformation)

By instantiating [LPAR04] the rule BUSY(B, closed, stop, false, false, true, i) → IDLE(B, closed, up, false, false, true, i) we obtained the following new rules [LPAR04]:

BUSY(B, closed, stop, false, false, true, empty) → IDLE(B, closed, up, false, false, true, empty) → BUSY(B, closed, stop, false, false, true, empty) → IDLE(B, closed, up, false, false, true, empty)

(48) Obligation:

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

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)
BUSY(B, closed, down, z3, z4, z5, empty) → IDLE(B, closed, stop, z3, z4, z5, empty)
BUSY(S, closed, up, z3, z4, z5, empty) → IDLE(S, closed, stop, z3, z4, z5, empty)
BUSY(B, closed, up, true, z4, z5, empty) → IDLE(B, closed, stop, true, z4, z5, empty)
BUSY(F, closed, up, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty)
BUSY(F, closed, down, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty)
BUSY(S, closed, down, z3, z4, true, empty) → IDLE(S, closed, stop, z3, z4, true, empty)
BUSY(B, closed, up, false, z4, z5, empty) → IDLE(BF, closed, up, false, z4, z5, empty)
BUSY(F, closed, up, z3, false, z5, empty) → IDLE(FS, closed, up, z3, false, z5, empty)
BUSY(F, closed, down, z3, false, z5, empty) → IDLE(BF, closed, down, z3, false, z5, empty)
BUSY(S, closed, down, z3, z4, false, empty) → IDLE(FS, closed, down, z3, z4, false, empty)
BUSY(BF, closed, up, z3, z4, z5, empty) → IDLE(F, closed, up, z3, z4, z5, empty)
BUSY(BF, closed, down, z3, z4, z5, empty) → IDLE(B, closed, down, z3, z4, z5, empty)
BUSY(FS, closed, up, z3, z4, z5, empty) → IDLE(S, closed, up, z3, z4, z5, empty)
BUSY(FS, closed, down, z3, z4, z5, empty) → IDLE(F, closed, down, z3, z4, z5, empty)
BUSY(B, closed, stop, false, true, z5, empty) → IDLE(B, closed, up, false, true, z5, empty)
BUSY(B, closed, stop, false, false, true, empty) → IDLE(B, closed, up, false, false, true, empty)

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

(49) TransformationProof (EQUIVALENT transformation)

By instantiating [LPAR04] the rule BUSY(F, closed, stop, true, false, b3, i) → IDLE(F, closed, down, true, false, b3, i) we obtained the following new rules [LPAR04]:

BUSY(F, closed, stop, true, false, z5, empty) → IDLE(F, closed, down, true, false, z5, empty) → BUSY(F, closed, stop, true, false, z5, empty) → IDLE(F, closed, down, true, false, z5, empty)

(50) Obligation:

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

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)
BUSY(B, closed, down, z3, z4, z5, empty) → IDLE(B, closed, stop, z3, z4, z5, empty)
BUSY(S, closed, up, z3, z4, z5, empty) → IDLE(S, closed, stop, z3, z4, z5, empty)
BUSY(B, closed, up, true, z4, z5, empty) → IDLE(B, closed, stop, true, z4, z5, empty)
BUSY(F, closed, up, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty)
BUSY(F, closed, down, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty)
BUSY(S, closed, down, z3, z4, true, empty) → IDLE(S, closed, stop, z3, z4, true, empty)
BUSY(B, closed, up, false, z4, z5, empty) → IDLE(BF, closed, up, false, z4, z5, empty)
BUSY(F, closed, up, z3, false, z5, empty) → IDLE(FS, closed, up, z3, false, z5, empty)
BUSY(F, closed, down, z3, false, z5, empty) → IDLE(BF, closed, down, z3, false, z5, empty)
BUSY(S, closed, down, z3, z4, false, empty) → IDLE(FS, closed, down, z3, z4, false, empty)
BUSY(BF, closed, up, z3, z4, z5, empty) → IDLE(F, closed, up, z3, z4, z5, empty)
BUSY(BF, closed, down, z3, z4, z5, empty) → IDLE(B, closed, down, z3, z4, z5, empty)
BUSY(FS, closed, up, z3, z4, z5, empty) → IDLE(S, closed, up, z3, z4, z5, empty)
BUSY(FS, closed, down, z3, z4, z5, empty) → IDLE(F, closed, down, z3, z4, z5, empty)
BUSY(B, closed, stop, false, true, z5, empty) → IDLE(B, closed, up, false, true, z5, empty)
BUSY(B, closed, stop, false, false, true, empty) → IDLE(B, closed, up, false, false, true, empty)
BUSY(F, closed, stop, true, false, z5, empty) → IDLE(F, closed, down, true, false, z5, empty)

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

(51) TransformationProof (EQUIVALENT transformation)

By instantiating [LPAR04] the rule BUSY(F, closed, stop, false, false, true, i) → IDLE(F, closed, up, false, false, true, i) we obtained the following new rules [LPAR04]:

BUSY(F, closed, stop, false, false, true, empty) → IDLE(F, closed, up, false, false, true, empty) → BUSY(F, closed, stop, false, false, true, empty) → IDLE(F, closed, up, false, false, true, empty)

(52) Obligation:

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

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)
BUSY(B, closed, down, z3, z4, z5, empty) → IDLE(B, closed, stop, z3, z4, z5, empty)
BUSY(S, closed, up, z3, z4, z5, empty) → IDLE(S, closed, stop, z3, z4, z5, empty)
BUSY(B, closed, up, true, z4, z5, empty) → IDLE(B, closed, stop, true, z4, z5, empty)
BUSY(F, closed, up, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty)
BUSY(F, closed, down, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty)
BUSY(S, closed, down, z3, z4, true, empty) → IDLE(S, closed, stop, z3, z4, true, empty)
BUSY(B, closed, up, false, z4, z5, empty) → IDLE(BF, closed, up, false, z4, z5, empty)
BUSY(F, closed, up, z3, false, z5, empty) → IDLE(FS, closed, up, z3, false, z5, empty)
BUSY(F, closed, down, z3, false, z5, empty) → IDLE(BF, closed, down, z3, false, z5, empty)
BUSY(S, closed, down, z3, z4, false, empty) → IDLE(FS, closed, down, z3, z4, false, empty)
BUSY(BF, closed, up, z3, z4, z5, empty) → IDLE(F, closed, up, z3, z4, z5, empty)
BUSY(BF, closed, down, z3, z4, z5, empty) → IDLE(B, closed, down, z3, z4, z5, empty)
BUSY(FS, closed, up, z3, z4, z5, empty) → IDLE(S, closed, up, z3, z4, z5, empty)
BUSY(FS, closed, down, z3, z4, z5, empty) → IDLE(F, closed, down, z3, z4, z5, empty)
BUSY(B, closed, stop, false, true, z5, empty) → IDLE(B, closed, up, false, true, z5, empty)
BUSY(B, closed, stop, false, false, true, empty) → IDLE(B, closed, up, false, false, true, empty)
BUSY(F, closed, stop, true, false, z5, empty) → IDLE(F, closed, down, true, false, z5, empty)
BUSY(F, closed, stop, false, false, true, empty) → IDLE(F, closed, up, false, false, true, empty)

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

(53) TransformationProof (EQUIVALENT transformation)

By instantiating [LPAR04] the rule BUSY(S, closed, stop, b1, true, false, i) → IDLE(S, closed, down, b1, true, false, i) we obtained the following new rules [LPAR04]:

BUSY(S, closed, stop, z3, true, false, empty) → IDLE(S, closed, down, z3, true, false, empty) → BUSY(S, closed, stop, z3, true, false, empty) → IDLE(S, closed, down, z3, true, false, empty)

(54) Obligation:

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

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)
BUSY(B, closed, down, z3, z4, z5, empty) → IDLE(B, closed, stop, z3, z4, z5, empty)
BUSY(S, closed, up, z3, z4, z5, empty) → IDLE(S, closed, stop, z3, z4, z5, empty)
BUSY(B, closed, up, true, z4, z5, empty) → IDLE(B, closed, stop, true, z4, z5, empty)
BUSY(F, closed, up, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty)
BUSY(F, closed, down, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty)
BUSY(S, closed, down, z3, z4, true, empty) → IDLE(S, closed, stop, z3, z4, true, empty)
BUSY(B, closed, up, false, z4, z5, empty) → IDLE(BF, closed, up, false, z4, z5, empty)
BUSY(F, closed, up, z3, false, z5, empty) → IDLE(FS, closed, up, z3, false, z5, empty)
BUSY(F, closed, down, z3, false, z5, empty) → IDLE(BF, closed, down, z3, false, z5, empty)
BUSY(S, closed, down, z3, z4, false, empty) → IDLE(FS, closed, down, z3, z4, false, empty)
BUSY(BF, closed, up, z3, z4, z5, empty) → IDLE(F, closed, up, z3, z4, z5, empty)
BUSY(BF, closed, down, z3, z4, z5, empty) → IDLE(B, closed, down, z3, z4, z5, empty)
BUSY(FS, closed, up, z3, z4, z5, empty) → IDLE(S, closed, up, z3, z4, z5, empty)
BUSY(FS, closed, down, z3, z4, z5, empty) → IDLE(F, closed, down, z3, z4, z5, empty)
BUSY(B, closed, stop, false, true, z5, empty) → IDLE(B, closed, up, false, true, z5, empty)
BUSY(B, closed, stop, false, false, true, empty) → IDLE(B, closed, up, false, false, true, empty)
BUSY(F, closed, stop, true, false, z5, empty) → IDLE(F, closed, down, true, false, z5, empty)
BUSY(F, closed, stop, false, false, true, empty) → IDLE(F, closed, up, false, false, true, empty)
BUSY(S, closed, stop, z3, true, false, empty) → IDLE(S, closed, down, z3, true, false, empty)

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

(55) TransformationProof (EQUIVALENT transformation)

By instantiating [LPAR04] the rule BUSY(S, closed, stop, true, false, false, i) → IDLE(S, closed, down, true, false, false, i) we obtained the following new rules [LPAR04]:

BUSY(S, closed, stop, true, false, false, empty) → IDLE(S, closed, down, true, false, false, empty) → BUSY(S, closed, stop, true, false, false, empty) → IDLE(S, closed, down, true, false, false, empty)

(56) 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, z3, z4, z5, empty) → IDLE(B, closed, stop, z3, z4, z5, empty)
BUSY(S, closed, up, z3, z4, z5, empty) → IDLE(S, closed, stop, z3, z4, z5, empty)
BUSY(B, closed, up, true, z4, z5, empty) → IDLE(B, closed, stop, true, z4, z5, empty)
BUSY(F, closed, up, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty)
BUSY(F, closed, down, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty)
BUSY(S, closed, down, z3, z4, true, empty) → IDLE(S, closed, stop, z3, z4, true, empty)
BUSY(B, closed, up, false, z4, z5, empty) → IDLE(BF, closed, up, false, z4, z5, empty)
BUSY(F, closed, up, z3, false, z5, empty) → IDLE(FS, closed, up, z3, false, z5, empty)
BUSY(F, closed, down, z3, false, z5, empty) → IDLE(BF, closed, down, z3, false, z5, empty)
BUSY(S, closed, down, z3, z4, false, empty) → IDLE(FS, closed, down, z3, z4, false, empty)
BUSY(BF, closed, up, z3, z4, z5, empty) → IDLE(F, closed, up, z3, z4, z5, empty)
BUSY(BF, closed, down, z3, z4, z5, empty) → IDLE(B, closed, down, z3, z4, z5, empty)
BUSY(FS, closed, up, z3, z4, z5, empty) → IDLE(S, closed, up, z3, z4, z5, empty)
BUSY(FS, closed, down, z3, z4, z5, empty) → IDLE(F, closed, down, z3, z4, z5, empty)
BUSY(B, closed, stop, false, true, z5, empty) → IDLE(B, closed, up, false, true, z5, empty)
BUSY(B, closed, stop, false, false, true, empty) → IDLE(B, closed, up, false, false, true, empty)
BUSY(F, closed, stop, true, false, z5, empty) → IDLE(F, closed, down, true, false, z5, empty)
BUSY(F, closed, stop, false, false, true, empty) → IDLE(F, closed, up, false, false, true, empty)
BUSY(S, closed, stop, z3, true, false, empty) → IDLE(S, closed, down, z3, true, false, empty)
BUSY(S, closed, stop, true, false, false, empty) → IDLE(S, closed, down, true, false, false, empty)

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

(57) TransformationProof (EQUIVALENT transformation)

By instantiating [LPAR04] the rule IDLE(fl, d, m, b1, b2, b3, empty) → BUSY(fl, d, m, b1, b2, b3, empty) we obtained the following new rules [LPAR04]:

IDLE(B, closed, stop, z0, z1, z2, empty) → BUSY(B, closed, stop, z0, z1, z2, empty) → IDLE(B, closed, stop, z0, z1, z2, empty) → BUSY(B, closed, stop, z0, z1, z2, empty)
IDLE(S, closed, stop, z0, z1, z2, empty) → BUSY(S, closed, stop, z0, z1, z2, empty) → IDLE(S, closed, stop, z0, z1, z2, empty) → BUSY(S, closed, stop, z0, z1, z2, empty)
IDLE(B, closed, stop, true, z0, z1, empty) → BUSY(B, closed, stop, true, z0, z1, empty) → IDLE(B, closed, stop, true, z0, z1, empty) → BUSY(B, closed, stop, true, z0, z1, empty)
IDLE(F, closed, stop, z0, true, z1, empty) → BUSY(F, closed, stop, z0, true, z1, empty) → IDLE(F, closed, stop, z0, true, z1, empty) → BUSY(F, closed, stop, z0, true, z1, empty)
IDLE(S, closed, stop, z0, z1, true, empty) → BUSY(S, closed, stop, z0, z1, true, empty) → IDLE(S, closed, stop, z0, z1, true, empty) → BUSY(S, closed, stop, z0, z1, true, empty)
IDLE(BF, closed, up, false, z0, z1, empty) → BUSY(BF, closed, up, false, z0, z1, empty) → IDLE(BF, closed, up, false, z0, z1, empty) → BUSY(BF, closed, up, false, z0, z1, empty)
IDLE(FS, closed, up, z0, false, z1, empty) → BUSY(FS, closed, up, z0, false, z1, empty) → IDLE(FS, closed, up, z0, false, z1, empty) → BUSY(FS, closed, up, z0, false, z1, empty)
IDLE(BF, closed, down, z0, false, z1, empty) → BUSY(BF, closed, down, z0, false, z1, empty) → IDLE(BF, closed, down, z0, false, z1, empty) → BUSY(BF, closed, down, z0, false, z1, empty)
IDLE(FS, closed, down, z0, z1, false, empty) → BUSY(FS, closed, down, z0, z1, false, empty) → IDLE(FS, closed, down, z0, z1, false, empty) → BUSY(FS, closed, down, z0, z1, false, empty)
IDLE(F, closed, up, z0, z1, z2, empty) → BUSY(F, closed, up, z0, z1, z2, empty) → IDLE(F, closed, up, z0, z1, z2, empty) → BUSY(F, closed, up, z0, z1, z2, empty)
IDLE(B, closed, down, z0, z1, z2, empty) → BUSY(B, closed, down, z0, z1, z2, empty) → IDLE(B, closed, down, z0, z1, z2, empty) → BUSY(B, closed, down, z0, z1, z2, empty)
IDLE(S, closed, up, z0, z1, z2, empty) → BUSY(S, closed, up, z0, z1, z2, empty) → IDLE(S, closed, up, z0, z1, z2, empty) → BUSY(S, closed, up, z0, z1, z2, empty)
IDLE(F, closed, down, z0, z1, z2, empty) → BUSY(F, closed, down, z0, z1, z2, empty) → IDLE(F, closed, down, z0, z1, z2, empty) → BUSY(F, closed, down, z0, z1, z2, empty)
IDLE(B, closed, up, false, true, z0, empty) → BUSY(B, closed, up, false, true, z0, empty) → IDLE(B, closed, up, false, true, z0, empty) → BUSY(B, closed, up, false, true, z0, empty)
IDLE(B, closed, up, false, false, true, empty) → BUSY(B, closed, up, false, false, true, empty) → IDLE(B, closed, up, false, false, true, empty) → BUSY(B, closed, up, false, false, true, empty)
IDLE(F, closed, down, true, false, z0, empty) → BUSY(F, closed, down, true, false, z0, empty) → IDLE(F, closed, down, true, false, z0, empty) → BUSY(F, closed, down, true, false, z0, empty)
IDLE(F, closed, up, false, false, true, empty) → BUSY(F, closed, up, false, false, true, empty) → IDLE(F, closed, up, false, false, true, empty) → BUSY(F, closed, up, false, false, true, empty)
IDLE(S, closed, down, z0, true, false, empty) → BUSY(S, closed, down, z0, true, false, empty) → IDLE(S, closed, down, z0, true, false, empty) → BUSY(S, closed, down, z0, true, false, empty)
IDLE(S, closed, down, true, false, false, empty) → BUSY(S, closed, down, true, false, false, empty) → IDLE(S, closed, down, true, false, false, empty) → BUSY(S, closed, down, true, false, false, empty)

(58) Obligation:

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

BUSY(B, closed, down, z3, z4, z5, empty) → IDLE(B, closed, stop, z3, z4, z5, empty)
BUSY(S, closed, up, z3, z4, z5, empty) → IDLE(S, closed, stop, z3, z4, z5, empty)
BUSY(B, closed, up, true, z4, z5, empty) → IDLE(B, closed, stop, true, z4, z5, empty)
BUSY(F, closed, up, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty)
BUSY(F, closed, down, z3, true, z5, empty) → IDLE(F, closed, stop, z3, true, z5, empty)
BUSY(S, closed, down, z3, z4, true, empty) → IDLE(S, closed, stop, z3, z4, true, empty)
BUSY(B, closed, up, false, z4, z5, empty) → IDLE(BF, closed, up, false, z4, z5, empty)
BUSY(F, closed, up, z3, false, z5, empty) → IDLE(FS, closed, up, z3, false, z5, empty)
BUSY(F, closed, down, z3, false, z5, empty) → IDLE(BF, closed, down, z3, false, z5, empty)
BUSY(S, closed, down, z3, z4, false, empty) → IDLE(FS, closed, down, z3, z4, false, empty)
BUSY(BF, closed, up, z3, z4, z5, empty) → IDLE(F, closed, up, z3, z4, z5, empty)
BUSY(BF, closed, down, z3, z4, z5, empty) → IDLE(B, closed, down, z3, z4, z5, empty)
BUSY(FS, closed, up, z3, z4, z5, empty) → IDLE(S, closed, up, z3, z4, z5, empty)
BUSY(FS, closed, down, z3, z4, z5, empty) → IDLE(F, closed, down, z3, z4, z5, empty)
BUSY(B, closed, stop, false, true, z5, empty) → IDLE(B, closed, up, false, true, z5, empty)
BUSY(B, closed, stop, false, false, true, empty) → IDLE(B, closed, up, false, false, true, empty)
BUSY(F, closed, stop, true, false, z5, empty) → IDLE(F, closed, down, true, false, z5, empty)
BUSY(F, closed, stop, false, false, true, empty) → IDLE(F, closed, up, false, false, true, empty)
BUSY(S, closed, stop, z3, true, false, empty) → IDLE(S, closed, down, z3, true, false, empty)
BUSY(S, closed, stop, true, false, false, empty) → IDLE(S, closed, down, true, false, false, empty)
IDLE(B, closed, stop, z0, z1, z2, empty) → BUSY(B, closed, stop, z0, z1, z2, empty)
IDLE(S, closed, stop, z0, z1, z2, empty) → BUSY(S, closed, stop, z0, z1, z2, empty)
IDLE(B, closed, stop, true, z0, z1, empty) → BUSY(B, closed, stop, true, z0, z1, empty)
IDLE(F, closed, stop, z0, true, z1, empty) → BUSY(F, closed, stop, z0, true, z1, empty)
IDLE(S, closed, stop, z0, z1, true, empty) → BUSY(S, closed, stop, z0, z1, true, empty)
IDLE(BF, closed, up, false, z0, z1, empty) → BUSY(BF, closed, up, false, z0, z1, empty)
IDLE(FS, closed, up, z0, false, z1, empty) → BUSY(FS, closed, up, z0, false, z1, empty)
IDLE(BF, closed, down, z0, false, z1, empty) → BUSY(BF, closed, down, z0, false, z1, empty)
IDLE(FS, closed, down, z0, z1, false, empty) → BUSY(FS, closed, down, z0, z1, false, empty)
IDLE(F, closed, up, z0, z1, z2, empty) → BUSY(F, closed, up, z0, z1, z2, empty)
IDLE(B, closed, down, z0, z1, z2, empty) → BUSY(B, closed, down, z0, z1, z2, empty)
IDLE(S, closed, up, z0, z1, z2, empty) → BUSY(S, closed, up, z0, z1, z2, empty)
IDLE(F, closed, down, z0, z1, z2, empty) → BUSY(F, closed, down, z0, z1, z2, empty)
IDLE(B, closed, up, false, true, z0, empty) → BUSY(B, closed, up, false, true, z0, empty)
IDLE(B, closed, up, false, false, true, empty) → BUSY(B, closed, up, false, false, true, empty)
IDLE(F, closed, down, true, false, z0, empty) → BUSY(F, closed, down, true, false, z0, empty)
IDLE(F, closed, up, false, false, true, empty) → BUSY(F, closed, up, false, false, true, empty)
IDLE(S, closed, down, z0, true, false, empty) → BUSY(S, closed, down, z0, true, false, empty)
IDLE(S, closed, down, true, false, false, empty) → BUSY(S, closed, down, true, false, false, empty)

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

(59) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 9 less nodes.

(60) Obligation:

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

IDLE(B, closed, stop, z0, z1, z2, empty) → BUSY(B, closed, stop, z0, z1, z2, empty)
BUSY(B, closed, stop, false, true, z5, empty) → IDLE(B, closed, up, false, true, z5, empty)
IDLE(B, closed, up, false, true, z0, empty) → BUSY(B, closed, up, false, true, z0, empty)
BUSY(B, closed, up, false, z4, z5, empty) → IDLE(BF, closed, up, false, z4, z5, empty)
IDLE(BF, closed, up, false, z0, z1, empty) → BUSY(BF, closed, up, false, z0, z1, empty)
BUSY(BF, closed, up, z3, z4, z5, empty) → IDLE(F, closed, up, z3, z4, z5, empty)
IDLE(F, closed, up, z0, z1, z2, empty) → BUSY(F, closed, up, z0, z1, z2, empty)
BUSY(F, closed, up, z3, false, z5, empty) → IDLE(FS, closed, up, z3, false, z5, empty)
IDLE(FS, closed, up, z0, false, z1, empty) → BUSY(FS, closed, up, z0, false, z1, empty)
BUSY(FS, closed, up, z3, z4, z5, empty) → IDLE(S, closed, up, z3, z4, z5, empty)
IDLE(S, closed, up, z0, z1, z2, empty) → BUSY(S, closed, up, z0, z1, z2, empty)
BUSY(S, closed, up, z3, z4, z5, empty) → IDLE(S, closed, stop, z3, z4, z5, empty)
IDLE(S, closed, stop, z0, z1, z2, empty) → BUSY(S, closed, stop, z0, z1, z2, empty)
BUSY(S, closed, stop, z3, true, false, empty) → IDLE(S, closed, down, z3, true, false, empty)
IDLE(S, closed, down, z0, true, false, empty) → BUSY(S, closed, down, z0, true, false, empty)
BUSY(S, closed, down, z3, z4, false, empty) → IDLE(FS, closed, down, z3, z4, false, empty)
IDLE(FS, closed, down, z0, z1, false, empty) → BUSY(FS, closed, down, z0, z1, false, empty)
BUSY(FS, closed, down, z3, z4, z5, empty) → IDLE(F, closed, down, z3, z4, z5, empty)
IDLE(F, closed, down, z0, z1, z2, empty) → BUSY(F, closed, down, z0, z1, z2, empty)
BUSY(F, closed, down, z3, false, z5, empty) → IDLE(BF, closed, down, z3, false, z5, empty)
IDLE(BF, closed, down, z0, false, z1, empty) → BUSY(BF, closed, down, z0, false, z1, empty)
BUSY(BF, closed, down, z3, z4, z5, empty) → IDLE(B, closed, down, z3, z4, z5, empty)
IDLE(B, closed, down, z0, z1, z2, empty) → BUSY(B, closed, down, z0, z1, z2, empty)
BUSY(B, closed, down, z3, z4, z5, empty) → IDLE(B, closed, stop, z3, z4, z5, empty)
IDLE(F, closed, down, true, false, z0, empty) → BUSY(F, closed, down, true, false, z0, empty)
BUSY(S, closed, stop, true, false, false, empty) → IDLE(S, closed, down, true, false, false, empty)
IDLE(S, closed, down, true, false, false, empty) → BUSY(S, closed, down, true, false, false, empty)
IDLE(F, closed, up, false, false, true, empty) → BUSY(F, closed, up, false, false, true, empty)
BUSY(B, closed, stop, false, false, true, empty) → IDLE(B, closed, up, false, false, true, empty)
IDLE(B, closed, up, false, false, true, empty) → BUSY(B, closed, up, false, false, true, empty)

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

(61) TransformationProof (EQUIVALENT transformation)

By instantiating [LPAR04] the rule BUSY(B, closed, up, false, z4, z5, empty) → IDLE(BF, closed, up, false, z4, z5, empty) we obtained the following new rules [LPAR04]:

BUSY(B, closed, up, false, true, z0, empty) → IDLE(BF, closed, up, false, true, z0, empty) → BUSY(B, closed, up, false, true, z0, empty) → IDLE(BF, closed, up, false, true, z0, empty)
BUSY(B, closed, up, false, false, true, empty) → IDLE(BF, closed, up, false, false, true, empty) → BUSY(B, closed, up, false, false, true, empty) → IDLE(BF, closed, up, false, false, true, empty)

(62) Obligation:

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

IDLE(B, closed, stop, z0, z1, z2, empty) → BUSY(B, closed, stop, z0, z1, z2, empty)
BUSY(B, closed, stop, false, true, z5, empty) → IDLE(B, closed, up, false, true, z5, empty)
IDLE(B, closed, up, false, true, z0, empty) → BUSY(B, closed, up, false, true, z0, empty)
IDLE(BF, closed, up, false, z0, z1, empty) → BUSY(BF, closed, up, false, z0, z1, empty)
BUSY(BF, closed, up, z3, z4, z5, empty) → IDLE(F, closed, up, z3, z4, z5, empty)
IDLE(F, closed, up, z0, z1, z2, empty) → BUSY(F, closed, up, z0, z1, z2, empty)
BUSY(F, closed, up, z3, false, z5, empty) → IDLE(FS, closed, up, z3, false, z5, empty)
IDLE(FS, closed, up, z0, false, z1, empty) → BUSY(FS, closed, up, z0, false, z1, empty)
BUSY(FS, closed, up, z3, z4, z5, empty) → IDLE(S, closed, up, z3, z4, z5, empty)
IDLE(S, closed, up, z0, z1, z2, empty) → BUSY(S, closed, up, z0, z1, z2, empty)
BUSY(S, closed, up, z3, z4, z5, empty) → IDLE(S, closed, stop, z3, z4, z5, empty)
IDLE(S, closed, stop, z0, z1, z2, empty) → BUSY(S, closed, stop, z0, z1, z2, empty)
BUSY(S, closed, stop, z3, true, false, empty) → IDLE(S, closed, down, z3, true, false, empty)
IDLE(S, closed, down, z0, true, false, empty) → BUSY(S, closed, down, z0, true, false, empty)
BUSY(S, closed, down, z3, z4, false, empty) → IDLE(FS, closed, down, z3, z4, false, empty)
IDLE(FS, closed, down, z0, z1, false, empty) → BUSY(FS, closed, down, z0, z1, false, empty)
BUSY(FS, closed, down, z3, z4, z5, empty) → IDLE(F, closed, down, z3, z4, z5, empty)
IDLE(F, closed, down, z0, z1, z2, empty) → BUSY(F, closed, down, z0, z1, z2, empty)
BUSY(F, closed, down, z3, false, z5, empty) → IDLE(BF, closed, down, z3, false, z5, empty)
IDLE(BF, closed, down, z0, false, z1, empty) → BUSY(BF, closed, down, z0, false, z1, empty)
BUSY(BF, closed, down, z3, z4, z5, empty) → IDLE(B, closed, down, z3, z4, z5, empty)
IDLE(B, closed, down, z0, z1, z2, empty) → BUSY(B, closed, down, z0, z1, z2, empty)
BUSY(B, closed, down, z3, z4, z5, empty) → IDLE(B, closed, stop, z3, z4, z5, empty)
IDLE(F, closed, down, true, false, z0, empty) → BUSY(F, closed, down, true, false, z0, empty)
BUSY(S, closed, stop, true, false, false, empty) → IDLE(S, closed, down, true, false, false, empty)
IDLE(S, closed, down, true, false, false, empty) → BUSY(S, closed, down, true, false, false, empty)
IDLE(F, closed, up, false, false, true, empty) → BUSY(F, closed, up, false, false, true, empty)
BUSY(B, closed, stop, false, false, true, empty) → IDLE(B, closed, up, false, false, true, empty)
IDLE(B, closed, up, false, false, true, empty) → BUSY(B, closed, up, false, false, true, empty)
BUSY(B, closed, up, false, true, z0, empty) → IDLE(BF, closed, up, false, true, z0, empty)
BUSY(B, closed, up, false, false, true, empty) → IDLE(BF, closed, up, false, false, true, empty)

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

(63) TransformationProof (EQUIVALENT transformation)

By instantiating [LPAR04] the rule IDLE(BF, closed, up, false, z0, z1, empty) → BUSY(BF, closed, up, false, z0, z1, empty) we obtained the following new rules [LPAR04]:

IDLE(BF, closed, up, false, true, z0, empty) → BUSY(BF, closed, up, false, true, z0, empty) → IDLE(BF, closed, up, false, true, z0, empty) → BUSY(BF, closed, up, false, true, z0, empty)
IDLE(BF, closed, up, false, false, true, empty) → BUSY(BF, closed, up, false, false, true, empty) → IDLE(BF, closed, up, false, false, true, empty) → BUSY(BF, closed, up, false, false, true, empty)

(64) Obligation:

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

IDLE(B, closed, stop, z0, z1, z2, empty) → BUSY(B, closed, stop, z0, z1, z2, empty)
BUSY(B, closed, stop, false, true, z5, empty) → IDLE(B, closed, up, false, true, z5, empty)
IDLE(B, closed, up, false, true, z0, empty) → BUSY(B, closed, up, false, true, z0, empty)
BUSY(BF, closed, up, z3, z4, z5, empty) → IDLE(F, closed, up, z3, z4, z5, empty)
IDLE(F, closed, up, z0, z1, z2, empty) → BUSY(F, closed, up, z0, z1, z2, empty)
BUSY(F, closed, up, z3, false, z5, empty) → IDLE(FS, closed, up, z3, false, z5, empty)
IDLE(FS, closed, up, z0, false, z1, empty) → BUSY(FS, closed, up, z0, false, z1, empty)
BUSY(FS, closed, up, z3, z4, z5, empty) → IDLE(S, closed, up, z3, z4, z5, empty)
IDLE(S, closed, up, z0, z1, z2, empty) → BUSY(S, closed, up, z0, z1, z2, empty)
BUSY(S, closed, up, z3, z4, z5, empty) → IDLE(S, closed, stop, z3, z4, z5, empty)
IDLE(S, closed, stop, z0, z1, z2, empty) → BUSY(S, closed, stop, z0, z1, z2, empty)
BUSY(S, closed, stop, z3, true, false, empty) → IDLE(S, closed, down, z3, true, false, empty)
IDLE(S, closed, down, z0, true, false, empty) → BUSY(S, closed, down, z0, true, false, empty)
BUSY(S, closed, down, z3, z4, false, empty) → IDLE(FS, closed, down, z3, z4, false, empty)
IDLE(FS, closed, down, z0, z1, false, empty) → BUSY(FS, closed, down, z0, z1, false, empty)
BUSY(FS, closed, down, z3, z4, z5, empty) → IDLE(F, closed, down, z3, z4, z5, empty)
IDLE(F, closed, down, z0, z1, z2, empty) → BUSY(F, closed, down, z0, z1, z2, empty)
BUSY(F, closed, down, z3, false, z5, empty) → IDLE(BF, closed, down, z3, false, z5, empty)
IDLE(BF, closed, down, z0, false, z1, empty) → BUSY(BF, closed, down, z0, false, z1, empty)
BUSY(BF, closed, down, z3, z4, z5, empty) → IDLE(B, closed, down, z3, z4, z5, empty)
IDLE(B, closed, down, z0, z1, z2, empty) → BUSY(B, closed, down, z0, z1, z2, empty)
BUSY(B, closed, down, z3, z4, z5, empty) → IDLE(B, closed, stop, z3, z4, z5, empty)
IDLE(F, closed, down, true, false, z0, empty) → BUSY(F, closed, down, true, false, z0, empty)
BUSY(S, closed, stop, true, false, false, empty) → IDLE(S, closed, down, true, false, false, empty)
IDLE(S, closed, down, true, false, false, empty) → BUSY(S, closed, down, true, false, false, empty)
IDLE(F, closed, up, false, false, true, empty) → BUSY(F, closed, up, false, false, true, empty)
BUSY(B, closed, stop, false, false, true, empty) → IDLE(B, closed, up, false, false, true, empty)
IDLE(B, closed, up, false, false, true, empty) → BUSY(B, closed, up, false, false, true, empty)
BUSY(B, closed, up, false, true, z0, empty) → IDLE(BF, closed, up, false, true, z0, empty)
BUSY(B, closed, up, false, false, true, empty) → IDLE(BF, closed, up, false, false, true, empty)
IDLE(BF, closed, up, false, true, z0, empty) → BUSY(BF, closed, up, false, true, z0, empty)
IDLE(BF, closed, up, false, false, true, empty) → BUSY(BF, closed, up, false, false, true, empty)

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

(65) TransformationProof (EQUIVALENT transformation)

By instantiating [LPAR04] the rule BUSY(BF, closed, up, z3, z4, z5, empty) → IDLE(F, closed, up, z3, z4, z5, empty) we obtained the following new rules [LPAR04]:

BUSY(BF, closed, up, false, true, z0, empty) → IDLE(F, closed, up, false, true, z0, empty) → BUSY(BF, closed, up, false, true, z0, empty) → IDLE(F, closed, up, false, true, z0, empty)
BUSY(BF, closed, up, false, false, true, empty) → IDLE(F, closed, up, false, false, true, empty) → BUSY(BF, closed, up, false, false, true, empty) → IDLE(F, closed, up, false, false, true, empty)

(66) Obligation:

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

IDLE(B, closed, stop, z0, z1, z2, empty) → BUSY(B, closed, stop, z0, z1, z2, empty)
BUSY(B, closed, stop, false, true, z5, empty) → IDLE(B, closed, up, false, true, z5, empty)
IDLE(B, closed, up, false, true, z0, empty) → BUSY(B, closed, up, false, true, z0, empty)
IDLE(F, closed, up, z0, z1, z2, empty) → BUSY(F, closed, up, z0, z1, z2, empty)
BUSY(F, closed, up, z3, false, z5, empty) → IDLE(FS, closed, up, z3, false, z5, empty)
IDLE(FS, closed, up, z0, false, z1, empty) → BUSY(FS, closed, up, z0, false, z1, empty)
BUSY(FS, closed, up, z3, z4, z5, empty) → IDLE(S, closed, up, z3, z4, z5, empty)
IDLE(S, closed, up, z0, z1, z2, empty) → BUSY(S, closed, up, z0, z1, z2, empty)
BUSY(S, closed, up, z3, z4, z5, empty) → IDLE(S, closed, stop, z3, z4, z5, empty)
IDLE(S, closed, stop, z0, z1, z2, empty) → BUSY(S, closed, stop, z0, z1, z2, empty)
BUSY(S, closed, stop, z3, true, false, empty) → IDLE(S, closed, down, z3, true, false, empty)
IDLE(S, closed, down, z0, true, false, empty) → BUSY(S, closed, down, z0, true, false, empty)
BUSY(S, closed, down, z3, z4, false, empty) → IDLE(FS, closed, down, z3, z4, false, empty)
IDLE(FS, closed, down, z0, z1, false, empty) → BUSY(FS, closed, down, z0, z1, false, empty)
BUSY(FS, closed, down, z3, z4, z5, empty) → IDLE(F, closed, down, z3, z4, z5, empty)
IDLE(F, closed, down, z0, z1, z2, empty) → BUSY(F, closed, down, z0, z1, z2, empty)
BUSY(F, closed, down, z3, false, z5, empty) → IDLE(BF, closed, down, z3, false, z5, empty)
IDLE(BF, closed, down, z0, false, z1, empty) → BUSY(BF, closed, down, z0, false, z1, empty)
BUSY(BF, closed, down, z3, z4, z5, empty) → IDLE(B, closed, down, z3, z4, z5, empty)
IDLE(B, closed, down, z0, z1, z2, empty) → BUSY(B, closed, down, z0, z1, z2, empty)
BUSY(B, closed, down, z3, z4, z5, empty) → IDLE(B, closed, stop, z3, z4, z5, empty)
IDLE(F, closed, down, true, false, z0, empty) → BUSY(F, closed, down, true, false, z0, empty)
BUSY(S, closed, stop, true, false, false, empty) → IDLE(S, closed, down, true, false, false, empty)
IDLE(S, closed, down, true, false, false, empty) → BUSY(S, closed, down, true, false, false, empty)
IDLE(F, closed, up, false, false, true, empty) → BUSY(F, closed, up, false, false, true, empty)
BUSY(B, closed, stop, false, false, true, empty) → IDLE(B, closed, up, false, false, true, empty)
IDLE(B, closed, up, false, false, true, empty) → BUSY(B, closed, up, false, false, true, empty)
BUSY(B, closed, up, false, true, z0, empty) → IDLE(BF, closed, up, false, true, z0, empty)
BUSY(B, closed, up, false, false, true, empty) → IDLE(BF, closed, up, false, false, true, empty)
IDLE(BF, closed, up, false, true, z0, empty) → BUSY(BF, closed, up, false, true, z0, empty)
IDLE(BF, closed, up, false, false, true, empty) → BUSY(BF, closed, up, false, false, true, empty)
BUSY(BF, closed, up, false, true, z0, empty) → IDLE(F, closed, up, false, true, z0, empty)
BUSY(BF, closed, up, false, false, true, empty) → IDLE(F, closed, up, false, false, true, empty)

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

(67) TransformationProof (EQUIVALENT transformation)

By instantiating [LPAR04] the rule IDLE(F, closed, up, z0, z1, z2, empty) → BUSY(F, closed, up, z0, z1, z2, empty) we obtained the following new rules [LPAR04]:

IDLE(F, closed, up, false, true, z0, empty) → BUSY(F, closed, up, false, true, z0, empty) → IDLE(F, closed, up, false, true, z0, empty) → BUSY(F, closed, up, false, true, z0, empty)
IDLE(F, closed, up, false, false, true, empty) → BUSY(F, closed, up, false, false, true, empty) → IDLE(F, closed, up, false, false, true, empty) → BUSY(F, closed, up, false, false, true, empty)

(68) Obligation:

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

IDLE(B, closed, stop, z0, z1, z2, empty) → BUSY(B, closed, stop, z0, z1, z2, empty)
BUSY(B, closed, stop, false, true, z5, empty) → IDLE(B, closed, up, false, true, z5, empty)
IDLE(B, closed, up, false, true, z0, empty) → BUSY(B, closed, up, false, true, z0, empty)
BUSY(F, closed, up, z3, false, z5, empty) → IDLE(FS, closed, up, z3, false, z5, empty)
IDLE(FS, closed, up, z0, false, z1, empty) → BUSY(FS, closed, up, z0, false, z1, empty)
BUSY(FS, closed, up, z3, z4, z5, empty) → IDLE(S, closed, up, z3, z4, z5, empty)
IDLE(S, closed, up, z0, z1, z2, empty) → BUSY(S, closed, up, z0, z1, z2, empty)
BUSY(S, closed, up, z3, z4, z5, empty) → IDLE(S, closed, stop, z3, z4, z5, empty)
IDLE(S, closed, stop, z0, z1, z2, empty) → BUSY(S, closed, stop, z0, z1, z2, empty)
BUSY(S, closed, stop, z3, true, false, empty) → IDLE(S, closed, down, z3, true, false, empty)
IDLE(S, closed, down, z0, true, false, empty) → BUSY(S, closed, down, z0, true, false, empty)
BUSY(S, closed, down, z3, z4, false, empty) → IDLE(FS, closed, down, z3, z4, false, empty)
IDLE(FS, closed, down, z0, z1, false, empty) → BUSY(FS, closed, down, z0, z1, false, empty)
BUSY(FS, closed, down, z3, z4, z5, empty) → IDLE(F, closed, down, z3, z4, z5, empty)
IDLE(F, closed, down, z0, z1, z2, empty) → BUSY(F, closed, down, z0, z1, z2, empty)
BUSY(F, closed, down, z3, false, z5, empty) → IDLE(BF, closed, down, z3, false, z5, empty)
IDLE(BF, closed, down, z0, false, z1, empty) → BUSY(BF, closed, down, z0, false, z1, empty)
BUSY(BF, closed, down, z3, z4, z5, empty) → IDLE(B, closed, down, z3, z4, z5, empty)
IDLE(B, closed, down, z0, z1, z2, empty) → BUSY(B, closed, down, z0, z1, z2, empty)
BUSY(B, closed, down, z3, z4, z5, empty) → IDLE(B, closed, stop, z3, z4, z5, empty)
IDLE(F, closed, down, true, false, z0, empty) → BUSY(F, closed, down, true, false, z0, empty)
BUSY(S, closed, stop, true, false, false, empty) → IDLE(S, closed, down, true, false, false, empty)
IDLE(S, closed, down, true, false, false, empty) → BUSY(S, closed, down, true, false, false, empty)
IDLE(F, closed, up, false, false, true, empty) → BUSY(F, closed, up, false, false, true, empty)
BUSY(B, closed, stop, false, false, true, empty) → IDLE(B, closed, up, false, false, true, empty)
IDLE(B, closed, up, false, false, true, empty) → BUSY(B, closed, up, false, false, true, empty)
BUSY(B, closed, up, false, true, z0, empty) → IDLE(BF, closed, up, false, true, z0, empty)
BUSY(B, closed, up, false, false, true, empty) → IDLE(BF, closed, up, false, false, true, empty)
IDLE(BF, closed, up, false, true, z0, empty) → BUSY(BF, closed, up, false, true, z0, empty)
IDLE(BF, closed, up, false, false, true, empty) → BUSY(BF, closed, up, false, false, true, empty)
BUSY(BF, closed, up, false, true, z0, empty) → IDLE(F, closed, up, false, true, z0, empty)
BUSY(BF, closed, up, false, false, true, empty) → IDLE(F, closed, up, false, false, true, empty)
IDLE(F, closed, up, false, true, z0, empty) → BUSY(F, closed, up, false, true, z0, empty)

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

(69) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 6 less nodes.

(70) Obligation:

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

BUSY(B, closed, stop, false, false, true, empty) → IDLE(B, closed, up, false, false, true, empty)
IDLE(B, closed, up, false, false, true, empty) → BUSY(B, closed, up, false, false, true, empty)
BUSY(B, closed, up, false, false, true, empty) → IDLE(BF, closed, up, false, false, true, empty)
IDLE(BF, closed, up, false, false, true, empty) → BUSY(BF, closed, up, false, false, true, empty)
BUSY(BF, closed, up, false, false, true, empty) → IDLE(F, closed, up, false, false, true, empty)
IDLE(F, closed, up, false, false, true, empty) → BUSY(F, closed, up, false, false, true, empty)
BUSY(F, closed, up, z3, false, z5, empty) → IDLE(FS, closed, up, z3, false, z5, empty)
IDLE(FS, closed, up, z0, false, z1, empty) → BUSY(FS, closed, up, z0, false, z1, empty)
BUSY(FS, closed, up, z3, z4, z5, empty) → IDLE(S, closed, up, z3, z4, z5, empty)
IDLE(S, closed, up, z0, z1, z2, empty) → BUSY(S, closed, up, z0, z1, z2, empty)
BUSY(S, closed, up, z3, z4, z5, empty) → IDLE(S, closed, stop, z3, z4, z5, empty)
IDLE(S, closed, stop, z0, z1, z2, empty) → BUSY(S, closed, stop, z0, z1, z2, empty)
BUSY(S, closed, stop, z3, true, false, empty) → IDLE(S, closed, down, z3, true, false, empty)
IDLE(S, closed, down, z0, true, false, empty) → BUSY(S, closed, down, z0, true, false, empty)
BUSY(S, closed, down, z3, z4, false, empty) → IDLE(FS, closed, down, z3, z4, false, empty)
IDLE(FS, closed, down, z0, z1, false, empty) → BUSY(FS, closed, down, z0, z1, false, empty)
BUSY(FS, closed, down, z3, z4, z5, empty) → IDLE(F, closed, down, z3, z4, z5, empty)
IDLE(F, closed, down, z0, z1, z2, empty) → BUSY(F, closed, down, z0, z1, z2, empty)
BUSY(F, closed, down, z3, false, z5, empty) → IDLE(BF, closed, down, z3, false, z5, empty)
IDLE(BF, closed, down, z0, false, z1, empty) → BUSY(BF, closed, down, z0, false, z1, empty)
BUSY(BF, closed, down, z3, z4, z5, empty) → IDLE(B, closed, down, z3, z4, z5, empty)
IDLE(B, closed, down, z0, z1, z2, empty) → BUSY(B, closed, down, z0, z1, z2, empty)
BUSY(B, closed, down, z3, z4, z5, empty) → IDLE(B, closed, stop, z3, z4, z5, empty)
IDLE(B, closed, stop, z0, z1, z2, empty) → BUSY(B, closed, stop, z0, z1, z2, empty)
IDLE(F, closed, down, true, false, z0, empty) → BUSY(F, closed, down, true, false, z0, empty)
BUSY(S, closed, stop, true, false, false, empty) → IDLE(S, closed, down, true, false, false, empty)
IDLE(S, closed, down, true, false, false, empty) → BUSY(S, closed, down, true, false, false, empty)

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

(71) TransformationProof (EQUIVALENT transformation)

By instantiating [LPAR04] the rule BUSY(F, closed, up, z3, false, z5, empty) → IDLE(FS, closed, up, z3, false, z5, empty) we obtained the following new rules [LPAR04]:

BUSY(F, closed, up, false, false, true, empty) → IDLE(FS, closed, up, false, false, true, empty) → BUSY(F, closed, up, false, false, true, empty) → IDLE(FS, closed, up, false, false, true, empty)

(72) Obligation:

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

BUSY(B, closed, stop, false, false, true, empty) → IDLE(B, closed, up, false, false, true, empty)
IDLE(B, closed, up, false, false, true, empty) → BUSY(B, closed, up, false, false, true, empty)
BUSY(B, closed, up, false, false, true, empty) → IDLE(BF, closed, up, false, false, true, empty)
IDLE(BF, closed, up, false, false, true, empty) → BUSY(BF, closed, up, false, false, true, empty)
BUSY(BF, closed, up, false, false, true, empty) → IDLE(F, closed, up, false, false, true, empty)
IDLE(F, closed, up, false, false, true, empty) → BUSY(F, closed, up, false, false, true, empty)
IDLE(FS, closed, up, z0, false, z1, empty) → BUSY(FS, closed, up, z0, false, z1, empty)
BUSY(FS, closed, up, z3, z4, z5, empty) → IDLE(S, closed, up, z3, z4, z5, empty)
IDLE(S, closed, up, z0, z1, z2, empty) → BUSY(S, closed, up, z0, z1, z2, empty)
BUSY(S, closed, up, z3, z4, z5, empty) → IDLE(S, closed, stop, z3, z4, z5, empty)
IDLE(S, closed, stop, z0, z1, z2, empty) → BUSY(S, closed, stop, z0, z1, z2, empty)
BUSY(S, closed, stop, z3, true, false, empty) → IDLE(S, closed, down, z3, true, false, empty)
IDLE(S, closed, down, z0, true, false, empty) → BUSY(S, closed, down, z0, true, false, empty)
BUSY(S, closed, down, z3, z4, false, empty) → IDLE(FS, closed, down, z3, z4, false, empty)
IDLE(FS, closed, down, z0, z1, false, empty) → BUSY(FS, closed, down, z0, z1, false, empty)
BUSY(FS, closed, down, z3, z4, z5, empty) → IDLE(F, closed, down, z3, z4, z5, empty)
IDLE(F, closed, down, z0, z1, z2, empty) → BUSY(F, closed, down, z0, z1, z2, empty)
BUSY(F, closed, down, z3, false, z5, empty) → IDLE(BF, closed, down, z3, false, z5, empty)
IDLE(BF, closed, down, z0, false, z1, empty) → BUSY(BF, closed, down, z0, false, z1, empty)
BUSY(BF, closed, down, z3, z4, z5, empty) → IDLE(B, closed, down, z3, z4, z5, empty)
IDLE(B, closed, down, z0, z1, z2, empty) → BUSY(B, closed, down, z0, z1, z2, empty)
BUSY(B, closed, down, z3, z4, z5, empty) → IDLE(B, closed, stop, z3, z4, z5, empty)
IDLE(B, closed, stop, z0, z1, z2, empty) → BUSY(B, closed, stop, z0, z1, z2, empty)
IDLE(F, closed, down, true, false, z0, empty) → BUSY(F, closed, down, true, false, z0, empty)
BUSY(S, closed, stop, true, false, false, empty) → IDLE(S, closed, down, true, false, false, empty)
IDLE(S, closed, down, true, false, false, empty) → BUSY(S, closed, down, true, false, false, empty)
BUSY(F, closed, up, false, false, true, empty) → IDLE(FS, closed, up, false, false, true, empty)

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

(73) TransformationProof (EQUIVALENT transformation)

By instantiating [LPAR04] the rule IDLE(FS, closed, up, z0, false, z1, empty) → BUSY(FS, closed, up, z0, false, z1, empty) we obtained the following new rules [LPAR04]:

IDLE(FS, closed, up, false, false, true, empty) → BUSY(FS, closed, up, false, false, true, empty) → IDLE(FS, closed, up, false, false, true, empty) → BUSY(FS, closed, up, false, false, true, empty)

(74) Obligation:

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

BUSY(B, closed, stop, false, false, true, empty) → IDLE(B, closed, up, false, false, true, empty)
IDLE(B, closed, up, false, false, true, empty) → BUSY(B, closed, up, false, false, true, empty)
BUSY(B, closed, up, false, false, true, empty) → IDLE(BF, closed, up, false, false, true, empty)
IDLE(BF, closed, up, false, false, true, empty) → BUSY(BF, closed, up, false, false, true, empty)
BUSY(BF, closed, up, false, false, true, empty) → IDLE(F, closed, up, false, false, true, empty)
IDLE(F, closed, up, false, false, true, empty) → BUSY(F, closed, up, false, false, true, empty)
BUSY(FS, closed, up, z3, z4, z5, empty) → IDLE(S, closed, up, z3, z4, z5, empty)
IDLE(S, closed, up, z0, z1, z2, empty) → BUSY(S, closed, up, z0, z1, z2, empty)
BUSY(S, closed, up, z3, z4, z5, empty) → IDLE(S, closed, stop, z3, z4, z5, empty)
IDLE(S, closed, stop, z0, z1, z2, empty) → BUSY(S, closed, stop, z0, z1, z2, empty)
BUSY(S, closed, stop, z3, true, false, empty) → IDLE(S, closed, down, z3, true, false, empty)
IDLE(S, closed, down, z0, true, false, empty) → BUSY(S, closed, down, z0, true, false, empty)
BUSY(S, closed, down, z3, z4, false, empty) → IDLE(FS, closed, down, z3, z4, false, empty)
IDLE(FS, closed, down, z0, z1, false, empty) → BUSY(FS, closed, down, z0, z1, false, empty)
BUSY(FS, closed, down, z3, z4, z5, empty) → IDLE(F, closed, down, z3, z4, z5, empty)
IDLE(F, closed, down, z0, z1, z2, empty) → BUSY(F, closed, down, z0, z1, z2, empty)
BUSY(F, closed, down, z3, false, z5, empty) → IDLE(BF, closed, down, z3, false, z5, empty)
IDLE(BF, closed, down, z0, false, z1, empty) → BUSY(BF, closed, down, z0, false, z1, empty)
BUSY(BF, closed, down, z3, z4, z5, empty) → IDLE(B, closed, down, z3, z4, z5, empty)
IDLE(B, closed, down, z0, z1, z2, empty) → BUSY(B, closed, down, z0, z1, z2, empty)
BUSY(B, closed, down, z3, z4, z5, empty) → IDLE(B, closed, stop, z3, z4, z5, empty)
IDLE(B, closed, stop, z0, z1, z2, empty) → BUSY(B, closed, stop, z0, z1, z2, empty)
IDLE(F, closed, down, true, false, z0, empty) → BUSY(F, closed, down, true, false, z0, empty)
BUSY(S, closed, stop, true, false, false, empty) → IDLE(S, closed, down, true, false, false, empty)
IDLE(S, closed, down, true, false, false, empty) → BUSY(S, closed, down, true, false, false, empty)
BUSY(F, closed, up, false, false, true, empty) → IDLE(FS, closed, up, false, false, true, empty)
IDLE(FS, closed, up, false, false, true, empty) → BUSY(FS, closed, up, false, false, true, empty)

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

(75) TransformationProof (EQUIVALENT transformation)

By instantiating [LPAR04] the rule BUSY(FS, closed, up, z3, z4, z5, empty) → IDLE(S, closed, up, z3, z4, z5, empty) we obtained the following new rules [LPAR04]:

BUSY(FS, closed, up, false, false, true, empty) → IDLE(S, closed, up, false, false, true, empty) → BUSY(FS, closed, up, false, false, true, empty) → IDLE(S, closed, up, false, false, true, empty)

(76) Obligation:

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

BUSY(B, closed, stop, false, false, true, empty) → IDLE(B, closed, up, false, false, true, empty)
IDLE(B, closed, up, false, false, true, empty) → BUSY(B, closed, up, false, false, true, empty)
BUSY(B, closed, up, false, false, true, empty) → IDLE(BF, closed, up, false, false, true, empty)
IDLE(BF, closed, up, false, false, true, empty) → BUSY(BF, closed, up, false, false, true, empty)
BUSY(BF, closed, up, false, false, true, empty) → IDLE(F, closed, up, false, false, true, empty)
IDLE(F, closed, up, false, false, true, empty) → BUSY(F, closed, up, false, false, true, empty)
IDLE(S, closed, up, z0, z1, z2, empty) → BUSY(S, closed, up, z0, z1, z2, empty)
BUSY(S, closed, up, z3, z4, z5, empty) → IDLE(S, closed, stop, z3, z4, z5, empty)
IDLE(S, closed, stop, z0, z1, z2, empty) → BUSY(S, closed, stop, z0, z1, z2, empty)
BUSY(S, closed, stop, z3, true, false, empty) → IDLE(S, closed, down, z3, true, false, empty)
IDLE(S, closed, down, z0, true, false, empty) → BUSY(S, closed, down, z0, true, false, empty)
BUSY(S, closed, down, z3, z4, false, empty) → IDLE(FS, closed, down, z3, z4, false, empty)
IDLE(FS, closed, down, z0, z1, false, empty) → BUSY(FS, closed, down, z0, z1, false, empty)
BUSY(FS, closed, down, z3, z4, z5, empty) → IDLE(F, closed, down, z3, z4, z5, empty)
IDLE(F, closed, down, z0, z1, z2, empty) → BUSY(F, closed, down, z0, z1, z2, empty)
BUSY(F, closed, down, z3, false, z5, empty) → IDLE(BF, closed, down, z3, false, z5, empty)
IDLE(BF, closed, down, z0, false, z1, empty) → BUSY(BF, closed, down, z0, false, z1, empty)
BUSY(BF, closed, down, z3, z4, z5, empty) → IDLE(B, closed, down, z3, z4, z5, empty)
IDLE(B, closed, down, z0, z1, z2, empty) → BUSY(B, closed, down, z0, z1, z2, empty)
BUSY(B, closed, down, z3, z4, z5, empty) → IDLE(B, closed, stop, z3, z4, z5, empty)
IDLE(B, closed, stop, z0, z1, z2, empty) → BUSY(B, closed, stop, z0, z1, z2, empty)
IDLE(F, closed, down, true, false, z0, empty) → BUSY(F, closed, down, true, false, z0, empty)
BUSY(S, closed, stop, true, false, false, empty) → IDLE(S, closed, down, true, false, false, empty)
IDLE(S, closed, down, true, false, false, empty) → BUSY(S, closed, down, true, false, false, empty)
BUSY(F, closed, up, false, false, true, empty) → IDLE(FS, closed, up, false, false, true, empty)
IDLE(FS, closed, up, false, false, true, empty) → BUSY(FS, closed, up, false, false, true, empty)
BUSY(FS, closed, up, false, false, true, empty) → IDLE(S, closed, up, false, false, true, empty)

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

(77) TransformationProof (EQUIVALENT transformation)

By instantiating [LPAR04] the rule IDLE(S, closed, up, z0, z1, z2, empty) → BUSY(S, closed, up, z0, z1, z2, empty) we obtained the following new rules [LPAR04]:

IDLE(S, closed, up, false, false, true, empty) → BUSY(S, closed, up, false, false, true, empty) → IDLE(S, closed, up, false, false, true, empty) → BUSY(S, closed, up, false, false, true, empty)

(78) Obligation:

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

BUSY(B, closed, stop, false, false, true, empty) → IDLE(B, closed, up, false, false, true, empty)
IDLE(B, closed, up, false, false, true, empty) → BUSY(B, closed, up, false, false, true, empty)
BUSY(B, closed, up, false, false, true, empty) → IDLE(BF, closed, up, false, false, true, empty)
IDLE(BF, closed, up, false, false, true, empty) → BUSY(BF, closed, up, false, false, true, empty)
BUSY(BF, closed, up, false, false, true, empty) → IDLE(F, closed, up, false, false, true, empty)
IDLE(F, closed, up, false, false, true, empty) → BUSY(F, closed, up, false, false, true, empty)
BUSY(S, closed, up, z3, z4, z5, empty) → IDLE(S, closed, stop, z3, z4, z5, empty)
IDLE(S, closed, stop, z0, z1, z2, empty) → BUSY(S, closed, stop, z0, z1, z2, empty)
BUSY(S, closed, stop, z3, true, false, empty) → IDLE(S, closed, down, z3, true, false, empty)
IDLE(S, closed, down, z0, true, false, empty) → BUSY(S, closed, down, z0, true, false, empty)
BUSY(S, closed, down, z3, z4, false, empty) → IDLE(FS, closed, down, z3, z4, false, empty)
IDLE(FS, closed, down, z0, z1, false, empty) → BUSY(FS, closed, down, z0, z1, false, empty)
BUSY(FS, closed, down, z3, z4, z5, empty) → IDLE(F, closed, down, z3, z4, z5, empty)
IDLE(F, closed, down, z0, z1, z2, empty) → BUSY(F, closed, down, z0, z1, z2, empty)
BUSY(F, closed, down, z3, false, z5, empty) → IDLE(BF, closed, down, z3, false, z5, empty)
IDLE(BF, closed, down, z0, false, z1, empty) → BUSY(BF, closed, down, z0, false, z1, empty)
BUSY(BF, closed, down, z3, z4, z5, empty) → IDLE(B, closed, down, z3, z4, z5, empty)
IDLE(B, closed, down, z0, z1, z2, empty) → BUSY(B, closed, down, z0, z1, z2, empty)
BUSY(B, closed, down, z3, z4, z5, empty) → IDLE(B, closed, stop, z3, z4, z5, empty)
IDLE(B, closed, stop, z0, z1, z2, empty) → BUSY(B, closed, stop, z0, z1, z2, empty)
IDLE(F, closed, down, true, false, z0, empty) → BUSY(F, closed, down, true, false, z0, empty)
BUSY(S, closed, stop, true, false, false, empty) → IDLE(S, closed, down, true, false, false, empty)
IDLE(S, closed, down, true, false, false, empty) → BUSY(S, closed, down, true, false, false, empty)
BUSY(F, closed, up, false, false, true, empty) → IDLE(FS, closed, up, false, false, true, empty)
IDLE(FS, closed, up, false, false, true, empty) → BUSY(FS, closed, up, false, false, true, empty)
BUSY(FS, closed, up, false, false, true, empty) → IDLE(S, closed, up, false, false, true, empty)
IDLE(S, closed, up, false, false, true, empty) → BUSY(S, closed, up, false, false, true, empty)

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

(79) TransformationProof (EQUIVALENT transformation)

By instantiating [LPAR04] the rule BUSY(S, closed, up, z3, z4, z5, empty) → IDLE(S, closed, stop, z3, z4, z5, empty) we obtained the following new rules [LPAR04]:

BUSY(S, closed, up, false, false, true, empty) → IDLE(S, closed, stop, false, false, true, empty) → BUSY(S, closed, up, false, false, true, empty) → IDLE(S, closed, stop, false, false, true, empty)

(80) Obligation:

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

BUSY(B, closed, stop, false, false, true, empty) → IDLE(B, closed, up, false, false, true, empty)
IDLE(B, closed, up, false, false, true, empty) → BUSY(B, closed, up, false, false, true, empty)
BUSY(B, closed, up, false, false, true, empty) → IDLE(BF, closed, up, false, false, true, empty)
IDLE(BF, closed, up, false, false, true, empty) → BUSY(BF, closed, up, false, false, true, empty)
BUSY(BF, closed, up, false, false, true, empty) → IDLE(F, closed, up, false, false, true, empty)
IDLE(F, closed, up, false, false, true, empty) → BUSY(F, closed, up, false, false, true, empty)
IDLE(S, closed, stop, z0, z1, z2, empty) → BUSY(S, closed, stop, z0, z1, z2, empty)
BUSY(S, closed, stop, z3, true, false, empty) → IDLE(S, closed, down, z3, true, false, empty)
IDLE(S, closed, down, z0, true, false, empty) → BUSY(S, closed, down, z0, true, false, empty)
BUSY(S, closed, down, z3, z4, false, empty) → IDLE(FS, closed, down, z3, z4, false, empty)
IDLE(FS, closed, down, z0, z1, false, empty) → BUSY(FS, closed, down, z0, z1, false, empty)
BUSY(FS, closed, down, z3, z4, z5, empty) → IDLE(F, closed, down, z3, z4, z5, empty)
IDLE(F, closed, down, z0, z1, z2, empty) → BUSY(F, closed, down, z0, z1, z2, empty)
BUSY(F, closed, down, z3, false, z5, empty) → IDLE(BF, closed, down, z3, false, z5, empty)
IDLE(BF, closed, down, z0, false, z1, empty) → BUSY(BF, closed, down, z0, false, z1, empty)
BUSY(BF, closed, down, z3, z4, z5, empty) → IDLE(B, closed, down, z3, z4, z5, empty)
IDLE(B, closed, down, z0, z1, z2, empty) → BUSY(B, closed, down, z0, z1, z2, empty)
BUSY(B, closed, down, z3, z4, z5, empty) → IDLE(B, closed, stop, z3, z4, z5, empty)
IDLE(B, closed, stop, z0, z1, z2, empty) → BUSY(B, closed, stop, z0, z1, z2, empty)
IDLE(F, closed, down, true, false, z0, empty) → BUSY(F, closed, down, true, false, z0, empty)
BUSY(S, closed, stop, true, false, false, empty) → IDLE(S, closed, down, true, false, false, empty)
IDLE(S, closed, down, true, false, false, empty) → BUSY(S, closed, down, true, false, false, empty)
BUSY(F, closed, up, false, false, true, empty) → IDLE(FS, closed, up, false, false, true, empty)
IDLE(FS, closed, up, false, false, true, empty) → BUSY(FS, closed, up, false, false, true, empty)
BUSY(FS, closed, up, false, false, true, empty) → IDLE(S, closed, up, false, false, true, empty)
IDLE(S, closed, up, false, false, true, empty) → BUSY(S, closed, up, false, false, true, empty)
BUSY(S, closed, up, false, false, true, empty) → IDLE(S, closed, stop, false, false, true, empty)

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

(81) TransformationProof (EQUIVALENT transformation)

By instantiating [LPAR04] the rule IDLE(S, closed, stop, z0, z1, z2, empty) → BUSY(S, closed, stop, z0, z1, z2, empty) we obtained the following new rules [LPAR04]:

IDLE(S, closed, stop, false, false, true, empty) → BUSY(S, closed, stop, false, false, true, empty) → IDLE(S, closed, stop, false, false, true, empty) → BUSY(S, closed, stop, false, false, true, empty)

(82) Obligation:

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

BUSY(B, closed, stop, false, false, true, empty) → IDLE(B, closed, up, false, false, true, empty)
IDLE(B, closed, up, false, false, true, empty) → BUSY(B, closed, up, false, false, true, empty)
BUSY(B, closed, up, false, false, true, empty) → IDLE(BF, closed, up, false, false, true, empty)
IDLE(BF, closed, up, false, false, true, empty) → BUSY(BF, closed, up, false, false, true, empty)
BUSY(BF, closed, up, false, false, true, empty) → IDLE(F, closed, up, false, false, true, empty)
IDLE(F, closed, up, false, false, true, empty) → BUSY(F, closed, up, false, false, true, empty)
BUSY(S, closed, stop, z3, true, false, empty) → IDLE(S, closed, down, z3, true, false, empty)
IDLE(S, closed, down, z0, true, false, empty) → BUSY(S, closed, down, z0, true, false, empty)
BUSY(S, closed, down, z3, z4, false, empty) → IDLE(FS, closed, down, z3, z4, false, empty)
IDLE(FS, closed, down, z0, z1, false, empty) → BUSY(FS, closed, down, z0, z1, false, empty)
BUSY(FS, closed, down, z3, z4, z5, empty) → IDLE(F, closed, down, z3, z4, z5, empty)
IDLE(F, closed, down, z0, z1, z2, empty) → BUSY(F, closed, down, z0, z1, z2, empty)
BUSY(F, closed, down, z3, false, z5, empty) → IDLE(BF, closed, down, z3, false, z5, empty)
IDLE(BF, closed, down, z0, false, z1, empty) → BUSY(BF, closed, down, z0, false, z1, empty)
BUSY(BF, closed, down, z3, z4, z5, empty) → IDLE(B, closed, down, z3, z4, z5, empty)
IDLE(B, closed, down, z0, z1, z2, empty) → BUSY(B, closed, down, z0, z1, z2, empty)
BUSY(B, closed, down, z3, z4, z5, empty) → IDLE(B, closed, stop, z3, z4, z5, empty)
IDLE(B, closed, stop, z0, z1, z2, empty) → BUSY(B, closed, stop, z0, z1, z2, empty)
IDLE(F, closed, down, true, false, z0, empty) → BUSY(F, closed, down, true, false, z0, empty)
BUSY(S, closed, stop, true, false, false, empty) → IDLE(S, closed, down, true, false, false, empty)
IDLE(S, closed, down, true, false, false, empty) → BUSY(S, closed, down, true, false, false, empty)
BUSY(F, closed, up, false, false, true, empty) → IDLE(FS, closed, up, false, false, true, empty)
IDLE(FS, closed, up, false, false, true, empty) → BUSY(FS, closed, up, false, false, true, empty)
BUSY(FS, closed, up, false, false, true, empty) → IDLE(S, closed, up, false, false, true, empty)
IDLE(S, closed, up, false, false, true, empty) → BUSY(S, closed, up, false, false, true, empty)
BUSY(S, closed, up, false, false, true, empty) → IDLE(S, closed, stop, false, false, true, empty)
IDLE(S, closed, stop, false, false, true, empty) → BUSY(S, closed, stop, false, false, true, empty)

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

(83) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 27 less nodes.