Termination w.r.t. Q proof of Waldmann_06

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

0 QTRS

↳1 Overlay + Local Confluence (⇔, 0 ms)

↳2 QTRS

↳3 DependencyPairsProof (⇔, 0 ms)

↳4 QDP

↳5 DependencyGraphProof (⇔, 0 ms)

↳6 QDP

↳7 TransformationProof (⇔, 0 ms)

↳8 QDP

↳9 TransformationProof (⇔, 0 ms)

↳10 QDP

↳11 TransformationProof (⇔, 0 ms)

↳12 QDP

↳13 TransformationProof (⇔, 0 ms)

↳14 QDP

↳15 TransformationProof (⇔, 0 ms)

↳16 QDP

↳17 TransformationProof (⇔, 0 ms)

↳18 QDP

↳19 TransformationProof (⇔, 0 ms)

↳20 QDP

↳21 SemLabProof (⇒, 127 ms)

↳22 QDP

↳23 DependencyGraphProof (⇔, 0 ms)

↳24 QDP

↳25 MRRProof (⇔, 0 ms)

↳26 QDP

↳27 DependencyGraphProof (⇔, 0 ms)

↳28 QDP

↳29 MRRProof (⇔, 0 ms)

↳30 QDP

↳31 DependencyGraphProof (⇔, 0 ms)

↳32 TRUE

(0) Obligation:

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

f(x, f(y, a)) → f(f(a, f(h(x), y)), a)

Q is empty.

(1) Overlay + Local Confluence (EQUIVALENT transformation)

The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.

(2) Obligation:

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

f(x, f(y, a)) → f(f(a, f(h(x), y)), a)

The set Q consists of the following terms:

f(x0, f(x1, a))

(3) DependencyPairsProof (EQUIVALENT transformation)

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

(4) Obligation:

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

F(x, f(y, a)) → F(f(a, f(h(x), y)), a)
F(x, f(y, a)) → F(a, f(h(x), y))
F(x, f(y, a)) → F(h(x), y)

The TRS R consists of the following rules:

f(x, f(y, a)) → f(f(a, f(h(x), y)), a)

The set Q consists of the following terms:

f(x0, f(x1, a))

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

(5) DependencyGraphProof (EQUIVALENT transformation)

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

(6) Obligation:

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

F(x, f(y, a)) → F(h(x), y)
F(x, f(y, a)) → F(a, f(h(x), y))

The TRS R consists of the following rules:

f(x, f(y, a)) → f(f(a, f(h(x), y)), a)

The set Q consists of the following terms:

f(x0, f(x1, a))

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

(7) TransformationProof (EQUIVALENT transformation)

By instantiating [LPAR04] the rule F(x, f(y, a)) → F(h(x), y) we obtained the following new rules [LPAR04]:

F(h(z0), f(x1, a)) → F(h(h(z0)), x1) → F(h(z0), f(x1, a)) → F(h(h(z0)), x1)
F(a, f(x1, a)) → F(h(a), x1) → F(a, f(x1, a)) → F(h(a), x1)

(8) Obligation:

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

F(x, f(y, a)) → F(a, f(h(x), y))
F(h(z0), f(x1, a)) → F(h(h(z0)), x1)
F(a, f(x1, a)) → F(h(a), x1)

The TRS R consists of the following rules:

f(x, f(y, a)) → f(f(a, f(h(x), y)), a)

The set Q consists of the following terms:

f(x0, f(x1, a))

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

(9) TransformationProof (EQUIVALENT transformation)

By instantiating [LPAR04] the rule F(x, f(y, a)) → F(a, f(h(x), y)) we obtained the following new rules [LPAR04]:

F(a, f(x1, a)) → F(a, f(h(a), x1)) → F(a, f(x1, a)) → F(a, f(h(a), x1))
F(h(h(z0)), f(x1, a)) → F(a, f(h(h(h(z0))), x1)) → F(h(h(z0)), f(x1, a)) → F(a, f(h(h(h(z0))), x1))
F(h(a), f(x1, a)) → F(a, f(h(h(a)), x1)) → F(h(a), f(x1, a)) → F(a, f(h(h(a)), x1))

(10) Obligation:

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

F(h(z0), f(x1, a)) → F(h(h(z0)), x1)
F(a, f(x1, a)) → F(h(a), x1)
F(a, f(x1, a)) → F(a, f(h(a), x1))
F(h(h(z0)), f(x1, a)) → F(a, f(h(h(h(z0))), x1))
F(h(a), f(x1, a)) → F(a, f(h(h(a)), x1))

The TRS R consists of the following rules:

f(x, f(y, a)) → f(f(a, f(h(x), y)), a)

The set Q consists of the following terms:

f(x0, f(x1, a))

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

(11) TransformationProof (EQUIVALENT transformation)

By instantiating [LPAR04] the rule F(h(z0), f(x1, a)) → F(h(h(z0)), x1) we obtained the following new rules [LPAR04]:

F(h(h(z0)), f(x1, a)) → F(h(h(h(z0))), x1) → F(h(h(z0)), f(x1, a)) → F(h(h(h(z0))), x1)
F(h(a), f(x1, a)) → F(h(h(a)), x1) → F(h(a), f(x1, a)) → F(h(h(a)), x1)

(12) Obligation:

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

F(a, f(x1, a)) → F(h(a), x1)
F(a, f(x1, a)) → F(a, f(h(a), x1))
F(h(h(z0)), f(x1, a)) → F(a, f(h(h(h(z0))), x1))
F(h(a), f(x1, a)) → F(a, f(h(h(a)), x1))
F(h(h(z0)), f(x1, a)) → F(h(h(h(z0))), x1)
F(h(a), f(x1, a)) → F(h(h(a)), x1)

The TRS R consists of the following rules:

f(x, f(y, a)) → f(f(a, f(h(x), y)), a)

The set Q consists of the following terms:

f(x0, f(x1, a))

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

(13) TransformationProof (EQUIVALENT transformation)

By instantiating [LPAR04] the rule F(h(h(z0)), f(x1, a)) → F(a, f(h(h(h(z0))), x1)) we obtained the following new rules [LPAR04]:

F(h(h(h(z0))), f(x1, a)) → F(a, f(h(h(h(h(z0)))), x1)) → F(h(h(h(z0))), f(x1, a)) → F(a, f(h(h(h(h(z0)))), x1))
F(h(h(a)), f(x1, a)) → F(a, f(h(h(h(a))), x1)) → F(h(h(a)), f(x1, a)) → F(a, f(h(h(h(a))), x1))

(14) Obligation:

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

F(a, f(x1, a)) → F(h(a), x1)
F(a, f(x1, a)) → F(a, f(h(a), x1))
F(h(a), f(x1, a)) → F(a, f(h(h(a)), x1))
F(h(h(z0)), f(x1, a)) → F(h(h(h(z0))), x1)
F(h(a), f(x1, a)) → F(h(h(a)), x1)
F(h(h(h(z0))), f(x1, a)) → F(a, f(h(h(h(h(z0)))), x1))
F(h(h(a)), f(x1, a)) → F(a, f(h(h(h(a))), x1))

The TRS R consists of the following rules:

f(x, f(y, a)) → f(f(a, f(h(x), y)), a)

The set Q consists of the following terms:

f(x0, f(x1, a))

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

(15) TransformationProof (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule F(a, f(x1, a)) → F(h(a), x1) we obtained the following new rules [LPAR04]:

F(a, f(f(y_0, a), a)) → F(h(a), f(y_0, a)) → F(a, f(f(y_0, a), a)) → F(h(a), f(y_0, a))

(16) Obligation:

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

F(a, f(x1, a)) → F(a, f(h(a), x1))
F(h(a), f(x1, a)) → F(a, f(h(h(a)), x1))
F(h(h(z0)), f(x1, a)) → F(h(h(h(z0))), x1)
F(h(a), f(x1, a)) → F(h(h(a)), x1)
F(h(h(h(z0))), f(x1, a)) → F(a, f(h(h(h(h(z0)))), x1))
F(h(h(a)), f(x1, a)) → F(a, f(h(h(h(a))), x1))
F(a, f(f(y_0, a), a)) → F(h(a), f(y_0, a))

The TRS R consists of the following rules:

f(x, f(y, a)) → f(f(a, f(h(x), y)), a)

The set Q consists of the following terms:

f(x0, f(x1, a))

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

(17) TransformationProof (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule F(h(h(z0)), f(x1, a)) → F(h(h(h(z0))), x1) we obtained the following new rules [LPAR04]:

F(h(h(x0)), f(f(y_1, a), a)) → F(h(h(h(x0))), f(y_1, a)) → F(h(h(x0)), f(f(y_1, a), a)) → F(h(h(h(x0))), f(y_1, a))

(18) Obligation:

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

F(a, f(x1, a)) → F(a, f(h(a), x1))
F(h(a), f(x1, a)) → F(a, f(h(h(a)), x1))
F(h(a), f(x1, a)) → F(h(h(a)), x1)
F(h(h(h(z0))), f(x1, a)) → F(a, f(h(h(h(h(z0)))), x1))
F(h(h(a)), f(x1, a)) → F(a, f(h(h(h(a))), x1))
F(a, f(f(y_0, a), a)) → F(h(a), f(y_0, a))
F(h(h(x0)), f(f(y_1, a), a)) → F(h(h(h(x0))), f(y_1, a))

The TRS R consists of the following rules:

f(x, f(y, a)) → f(f(a, f(h(x), y)), a)

The set Q consists of the following terms:

f(x0, f(x1, a))

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

(19) TransformationProof (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule F(h(a), f(x1, a)) → F(h(h(a)), x1) we obtained the following new rules [LPAR04]:

F(h(a), f(f(y_0, a), a)) → F(h(h(a)), f(y_0, a)) → F(h(a), f(f(y_0, a), a)) → F(h(h(a)), f(y_0, a))
F(h(a), f(f(f(y_1, a), a), a)) → F(h(h(a)), f(f(y_1, a), a)) → F(h(a), f(f(f(y_1, a), a), a)) → F(h(h(a)), f(f(y_1, a), a))

(20) Obligation:

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

F(a, f(x1, a)) → F(a, f(h(a), x1))
F(h(a), f(x1, a)) → F(a, f(h(h(a)), x1))
F(h(h(h(z0))), f(x1, a)) → F(a, f(h(h(h(h(z0)))), x1))
F(h(h(a)), f(x1, a)) → F(a, f(h(h(h(a))), x1))
F(a, f(f(y_0, a), a)) → F(h(a), f(y_0, a))
F(h(h(x0)), f(f(y_1, a), a)) → F(h(h(h(x0))), f(y_1, a))
F(h(a), f(f(y_0, a), a)) → F(h(h(a)), f(y_0, a))
F(h(a), f(f(f(y_1, a), a), a)) → F(h(h(a)), f(f(y_1, a), a))

The TRS R consists of the following rules:

f(x, f(y, a)) → f(f(a, f(h(x), y)), a)

The set Q consists of the following terms:

f(x0, f(x1, a))

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

(21) SemLabProof (SOUND transformation)

We found the following model for the rules of the TRSs R and P. Interpretation over the domain with elements from 0 to 1.
a: 1
f: 0
h: 0
F: 0
By semantic labelling [SEMLAB] we obtain the following labelled QDP problem.

(22) Obligation:

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

F.1-0(a., f.0-1(x1, a.)) → F.1-0(a., f.0-0(h.1(a.), x1))
F.1-0(a., f.1-1(x1, a.)) → F.1-0(a., f.0-1(h.1(a.), x1))
F.1-0(a., f.0-1(f.0-1(y_0, a.), a.)) → F.0-0(h.1(a.), f.0-1(y_0, a.))
F.1-0(a., f.0-1(f.1-1(y_0, a.), a.)) → F.0-0(h.1(a.), f.1-1(y_0, a.))
F.0-0(h.1(a.), f.0-1(x1, a.)) → F.1-0(a., f.0-0(h.0(h.1(a.)), x1))
F.0-0(h.1(a.), f.1-1(x1, a.)) → F.1-0(a., f.0-1(h.0(h.1(a.)), x1))
F.0-0(h.0(h.0(h.0(z0))), f.0-1(x1, a.)) → F.1-0(a., f.0-0(h.0(h.0(h.0(h.0(z0)))), x1))
F.0-0(h.0(h.0(h.0(z0))), f.1-1(x1, a.)) → F.1-0(a., f.0-1(h.0(h.0(h.0(h.0(z0)))), x1))
F.0-0(h.0(h.0(h.1(z0))), f.0-1(x1, a.)) → F.1-0(a., f.0-0(h.0(h.0(h.0(h.1(z0)))), x1))
F.0-0(h.0(h.0(h.1(z0))), f.1-1(x1, a.)) → F.1-0(a., f.0-1(h.0(h.0(h.0(h.1(z0)))), x1))
F.0-0(h.0(h.1(a.)), f.0-1(x1, a.)) → F.1-0(a., f.0-0(h.0(h.0(h.1(a.))), x1))
F.0-0(h.0(h.1(a.)), f.1-1(x1, a.)) → F.1-0(a., f.0-1(h.0(h.0(h.1(a.))), x1))
F.0-0(h.1(a.), f.0-1(f.0-1(y_0, a.), a.)) → F.0-0(h.0(h.1(a.)), f.0-1(y_0, a.))
F.0-0(h.1(a.), f.0-1(f.1-1(y_0, a.), a.)) → F.0-0(h.0(h.1(a.)), f.1-1(y_0, a.))
F.0-0(h.1(a.), f.0-1(f.0-1(f.0-1(y_1, a.), a.), a.)) → F.0-0(h.0(h.1(a.)), f.0-1(f.0-1(y_1, a.), a.))
F.0-0(h.1(a.), f.0-1(f.0-1(f.1-1(y_1, a.), a.), a.)) → F.0-0(h.0(h.1(a.)), f.0-1(f.1-1(y_1, a.), a.))
F.0-0(h.0(h.0(x0)), f.0-1(f.0-1(y_1, a.), a.)) → F.0-0(h.0(h.0(h.0(x0))), f.0-1(y_1, a.))
F.0-0(h.0(h.0(x0)), f.0-1(f.1-1(y_1, a.), a.)) → F.0-0(h.0(h.0(h.0(x0))), f.1-1(y_1, a.))
F.0-0(h.0(h.1(x0)), f.0-1(f.0-1(y_1, a.), a.)) → F.0-0(h.0(h.0(h.1(x0))), f.0-1(y_1, a.))
F.0-0(h.0(h.1(x0)), f.0-1(f.1-1(y_1, a.), a.)) → F.0-0(h.0(h.0(h.1(x0))), f.1-1(y_1, a.))

The TRS R consists of the following rules:

f.0-0(x, f.0-1(y, a.)) → f.0-1(f.1-0(a., f.0-0(h.0(x), y)), a.)
f.0-0(x, f.1-1(y, a.)) → f.0-1(f.1-0(a., f.0-1(h.0(x), y)), a.)
f.1-0(x, f.0-1(y, a.)) → f.0-1(f.1-0(a., f.0-0(h.1(x), y)), a.)
f.1-0(x, f.1-1(y, a.)) → f.0-1(f.1-0(a., f.0-1(h.1(x), y)), a.)

The set Q consists of the following terms:

f.0-0(x0, f.0-1(x1, a.))
f.0-0(x0, f.1-1(x1, a.))
f.1-0(x0, f.0-1(x1, a.))
f.1-0(x0, f.1-1(x1, a.))

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

(23) DependencyGraphProof (EQUIVALENT transformation)

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

(24) Obligation:

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

F.1-0(a., f.0-1(x1, a.)) → F.1-0(a., f.0-0(h.1(a.), x1))
F.1-0(a., f.0-1(f.0-1(y_0, a.), a.)) → F.0-0(h.1(a.), f.0-1(y_0, a.))
F.0-0(h.1(a.), f.0-1(x1, a.)) → F.1-0(a., f.0-0(h.0(h.1(a.)), x1))
F.0-0(h.1(a.), f.0-1(f.0-1(y_0, a.), a.)) → F.0-0(h.0(h.1(a.)), f.0-1(y_0, a.))
F.0-0(h.0(h.1(a.)), f.0-1(x1, a.)) → F.1-0(a., f.0-0(h.0(h.0(h.1(a.))), x1))
F.0-0(h.0(h.1(x0)), f.0-1(f.0-1(y_1, a.), a.)) → F.0-0(h.0(h.0(h.1(x0))), f.0-1(y_1, a.))
F.0-0(h.0(h.0(h.1(z0))), f.0-1(x1, a.)) → F.1-0(a., f.0-0(h.0(h.0(h.0(h.1(z0)))), x1))
F.0-0(h.0(h.0(x0)), f.0-1(f.0-1(y_1, a.), a.)) → F.0-0(h.0(h.0(h.0(x0))), f.0-1(y_1, a.))
F.0-0(h.0(h.0(h.0(z0))), f.0-1(x1, a.)) → F.1-0(a., f.0-0(h.0(h.0(h.0(h.0(z0)))), x1))
F.0-0(h.0(h.0(x0)), f.0-1(f.1-1(y_1, a.), a.)) → F.0-0(h.0(h.0(h.0(x0))), f.1-1(y_1, a.))
F.0-0(h.0(h.0(h.0(z0))), f.1-1(x1, a.)) → F.1-0(a., f.0-1(h.0(h.0(h.0(h.0(z0)))), x1))
F.0-0(h.0(h.1(x0)), f.0-1(f.1-1(y_1, a.), a.)) → F.0-0(h.0(h.0(h.1(x0))), f.1-1(y_1, a.))
F.0-0(h.0(h.0(h.1(z0))), f.1-1(x1, a.)) → F.1-0(a., f.0-1(h.0(h.0(h.0(h.1(z0)))), x1))
F.0-0(h.1(a.), f.0-1(f.1-1(y_0, a.), a.)) → F.0-0(h.0(h.1(a.)), f.1-1(y_0, a.))
F.0-0(h.0(h.1(a.)), f.1-1(x1, a.)) → F.1-0(a., f.0-1(h.0(h.0(h.1(a.))), x1))
F.0-0(h.1(a.), f.0-1(f.0-1(f.0-1(y_1, a.), a.), a.)) → F.0-0(h.0(h.1(a.)), f.0-1(f.0-1(y_1, a.), a.))
F.0-0(h.1(a.), f.0-1(f.0-1(f.1-1(y_1, a.), a.), a.)) → F.0-0(h.0(h.1(a.)), f.0-1(f.1-1(y_1, a.), a.))

The TRS R consists of the following rules:

f.0-0(x, f.0-1(y, a.)) → f.0-1(f.1-0(a., f.0-0(h.0(x), y)), a.)
f.0-0(x, f.1-1(y, a.)) → f.0-1(f.1-0(a., f.0-1(h.0(x), y)), a.)
f.1-0(x, f.0-1(y, a.)) → f.0-1(f.1-0(a., f.0-0(h.1(x), y)), a.)
f.1-0(x, f.1-1(y, a.)) → f.0-1(f.1-0(a., f.0-1(h.1(x), y)), a.)

The set Q consists of the following terms:

f.0-0(x0, f.0-1(x1, a.))
f.0-0(x0, f.1-1(x1, a.))
f.1-0(x0, f.0-1(x1, a.))
f.1-0(x0, f.1-1(x1, a.))

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

(25) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

F.0-0(h.0(h.0(h.0(z0))), f.1-1(x1, a.)) → F.1-0(a., f.0-1(h.0(h.0(h.0(h.0(z0)))), x1))
F.0-0(h.0(h.0(h.1(z0))), f.1-1(x1, a.)) → F.1-0(a., f.0-1(h.0(h.0(h.0(h.1(z0)))), x1))
F.0-0(h.0(h.1(a.)), f.1-1(x1, a.)) → F.1-0(a., f.0-1(h.0(h.0(h.1(a.))), x1))

Strictly oriented rules of the TRS R:

f.0-0(x, f.1-1(y, a.)) → f.0-1(f.1-0(a., f.0-1(h.0(x), y)), a.)
f.1-0(x, f.1-1(y, a.)) → f.0-1(f.1-0(a., f.0-1(h.1(x), y)), a.)

Used ordering: Polynomial interpretation [POLO]:

POL(F.0-0(x₁, x₂)) = x₁ + x₂
POL(F.1-0(x₁, x₂)) = x₁ + x₂
POL(a.) = 0
POL(f.0-0(x₁, x₂)) = x₁ + x₂
POL(f.0-1(x₁, x₂)) = x₁ + x₂
POL(f.1-0(x₁, x₂)) = x₁ + x₂
POL(f.1-1(x₁, x₂)) = 1 + x₁ + x₂
POL(h.0(x₁)) = x₁
POL(h.1(x₁)) = x₁

(26) Obligation:

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

F.1-0(a., f.0-1(x1, a.)) → F.1-0(a., f.0-0(h.1(a.), x1))
F.1-0(a., f.0-1(f.0-1(y_0, a.), a.)) → F.0-0(h.1(a.), f.0-1(y_0, a.))
F.0-0(h.1(a.), f.0-1(x1, a.)) → F.1-0(a., f.0-0(h.0(h.1(a.)), x1))
F.0-0(h.1(a.), f.0-1(f.0-1(y_0, a.), a.)) → F.0-0(h.0(h.1(a.)), f.0-1(y_0, a.))
F.0-0(h.0(h.1(a.)), f.0-1(x1, a.)) → F.1-0(a., f.0-0(h.0(h.0(h.1(a.))), x1))
F.0-0(h.0(h.1(x0)), f.0-1(f.0-1(y_1, a.), a.)) → F.0-0(h.0(h.0(h.1(x0))), f.0-1(y_1, a.))
F.0-0(h.0(h.0(h.1(z0))), f.0-1(x1, a.)) → F.1-0(a., f.0-0(h.0(h.0(h.0(h.1(z0)))), x1))
F.0-0(h.0(h.0(x0)), f.0-1(f.0-1(y_1, a.), a.)) → F.0-0(h.0(h.0(h.0(x0))), f.0-1(y_1, a.))
F.0-0(h.0(h.0(h.0(z0))), f.0-1(x1, a.)) → F.1-0(a., f.0-0(h.0(h.0(h.0(h.0(z0)))), x1))
F.0-0(h.0(h.0(x0)), f.0-1(f.1-1(y_1, a.), a.)) → F.0-0(h.0(h.0(h.0(x0))), f.1-1(y_1, a.))
F.0-0(h.0(h.1(x0)), f.0-1(f.1-1(y_1, a.), a.)) → F.0-0(h.0(h.0(h.1(x0))), f.1-1(y_1, a.))
F.0-0(h.1(a.), f.0-1(f.1-1(y_0, a.), a.)) → F.0-0(h.0(h.1(a.)), f.1-1(y_0, a.))
F.0-0(h.1(a.), f.0-1(f.0-1(f.0-1(y_1, a.), a.), a.)) → F.0-0(h.0(h.1(a.)), f.0-1(f.0-1(y_1, a.), a.))
F.0-0(h.1(a.), f.0-1(f.0-1(f.1-1(y_1, a.), a.), a.)) → F.0-0(h.0(h.1(a.)), f.0-1(f.1-1(y_1, a.), a.))

The TRS R consists of the following rules:

f.0-0(x, f.0-1(y, a.)) → f.0-1(f.1-0(a., f.0-0(h.0(x), y)), a.)
f.1-0(x, f.0-1(y, a.)) → f.0-1(f.1-0(a., f.0-0(h.1(x), y)), a.)

The set Q consists of the following terms:

f.0-0(x0, f.0-1(x1, a.))
f.0-0(x0, f.1-1(x1, a.))
f.1-0(x0, f.0-1(x1, a.))
f.1-0(x0, f.1-1(x1, a.))

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

(27) DependencyGraphProof (EQUIVALENT transformation)

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

(28) Obligation:

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

F.1-0(a., f.0-1(x1, a.)) → F.1-0(a., f.0-0(h.1(a.), x1))
F.1-0(a., f.0-1(f.0-1(y_0, a.), a.)) → F.0-0(h.1(a.), f.0-1(y_0, a.))
F.0-0(h.1(a.), f.0-1(x1, a.)) → F.1-0(a., f.0-0(h.0(h.1(a.)), x1))
F.0-0(h.1(a.), f.0-1(f.0-1(y_0, a.), a.)) → F.0-0(h.0(h.1(a.)), f.0-1(y_0, a.))
F.0-0(h.0(h.1(a.)), f.0-1(x1, a.)) → F.1-0(a., f.0-0(h.0(h.0(h.1(a.))), x1))
F.0-0(h.0(h.1(x0)), f.0-1(f.0-1(y_1, a.), a.)) → F.0-0(h.0(h.0(h.1(x0))), f.0-1(y_1, a.))
F.0-0(h.0(h.0(h.1(z0))), f.0-1(x1, a.)) → F.1-0(a., f.0-0(h.0(h.0(h.0(h.1(z0)))), x1))
F.0-0(h.0(h.0(x0)), f.0-1(f.0-1(y_1, a.), a.)) → F.0-0(h.0(h.0(h.0(x0))), f.0-1(y_1, a.))
F.0-0(h.0(h.0(h.0(z0))), f.0-1(x1, a.)) → F.1-0(a., f.0-0(h.0(h.0(h.0(h.0(z0)))), x1))
F.0-0(h.1(a.), f.0-1(f.0-1(f.0-1(y_1, a.), a.), a.)) → F.0-0(h.0(h.1(a.)), f.0-1(f.0-1(y_1, a.), a.))
F.0-0(h.1(a.), f.0-1(f.0-1(f.1-1(y_1, a.), a.), a.)) → F.0-0(h.0(h.1(a.)), f.0-1(f.1-1(y_1, a.), a.))

The TRS R consists of the following rules:

f.0-0(x, f.0-1(y, a.)) → f.0-1(f.1-0(a., f.0-0(h.0(x), y)), a.)
f.1-0(x, f.0-1(y, a.)) → f.0-1(f.1-0(a., f.0-0(h.1(x), y)), a.)

The set Q consists of the following terms:

f.0-0(x0, f.0-1(x1, a.))
f.0-0(x0, f.1-1(x1, a.))
f.1-0(x0, f.0-1(x1, a.))
f.1-0(x0, f.1-1(x1, a.))

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

(29) MRRProof (EQUIVALENT transformation)

F.1-0(a., f.0-1(x1, a.)) → F.1-0(a., f.0-0(h.1(a.), x1))
F.0-0(h.1(a.), f.0-1(x1, a.)) → F.1-0(a., f.0-0(h.0(h.1(a.)), x1))
F.0-0(h.1(a.), f.0-1(f.0-1(y_0, a.), a.)) → F.0-0(h.0(h.1(a.)), f.0-1(y_0, a.))
F.0-0(h.0(h.1(a.)), f.0-1(x1, a.)) → F.1-0(a., f.0-0(h.0(h.0(h.1(a.))), x1))
F.0-0(h.0(h.1(x0)), f.0-1(f.0-1(y_1, a.), a.)) → F.0-0(h.0(h.0(h.1(x0))), f.0-1(y_1, a.))
F.0-0(h.0(h.0(h.1(z0))), f.0-1(x1, a.)) → F.1-0(a., f.0-0(h.0(h.0(h.0(h.1(z0)))), x1))
F.0-0(h.0(h.0(x0)), f.0-1(f.0-1(y_1, a.), a.)) → F.0-0(h.0(h.0(h.0(x0))), f.0-1(y_1, a.))
F.0-0(h.0(h.0(h.0(z0))), f.0-1(x1, a.)) → F.1-0(a., f.0-0(h.0(h.0(h.0(h.0(z0)))), x1))
F.0-0(h.1(a.), f.0-1(f.0-1(f.0-1(y_1, a.), a.), a.)) → F.0-0(h.0(h.1(a.)), f.0-1(f.0-1(y_1, a.), a.))
F.0-0(h.1(a.), f.0-1(f.0-1(f.1-1(y_1, a.), a.), a.)) → F.0-0(h.0(h.1(a.)), f.0-1(f.1-1(y_1, a.), a.))

Used ordering: Polynomial interpretation [POLO]:

POL(F.0-0(x₁, x₂)) = 1 + x₁ + x₂
POL(F.1-0(x₁, x₂)) = x₁ + x₂
POL(a.) = 0
POL(f.0-0(x₁, x₂)) = x₁ + x₂
POL(f.0-1(x₁, x₂)) = 1 + x₁ + x₂
POL(f.1-0(x₁, x₂)) = x₁ + x₂
POL(f.1-1(x₁, x₂)) = x₁ + x₂
POL(h.0(x₁)) = x₁
POL(h.1(x₁)) = x₁

(30) Obligation:

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

F.1-0(a., f.0-1(f.0-1(y_0, a.), a.)) → F.0-0(h.1(a.), f.0-1(y_0, a.))

The TRS R consists of the following rules:

f.0-0(x, f.0-1(y, a.)) → f.0-1(f.1-0(a., f.0-0(h.0(x), y)), a.)
f.1-0(x, f.0-1(y, a.)) → f.0-1(f.1-0(a., f.0-0(h.1(x), y)), a.)

The set Q consists of the following terms:

f.0-0(x0, f.0-1(x1, a.))
f.0-0(x0, f.1-1(x1, a.))
f.1-0(x0, f.0-1(x1, a.))
f.1-0(x0, f.1-1(x1, a.))

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

(31) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(0) Obligation:

(1) Overlay + Local Confluence (EQUIVALENT transformation)

(2) Obligation:

(3) DependencyPairsProof (EQUIVALENT transformation)

(4) Obligation:

(5) DependencyGraphProof (EQUIVALENT transformation)

(6) Obligation:

(7) TransformationProof (EQUIVALENT transformation)

(8) Obligation:

(9) TransformationProof (EQUIVALENT transformation)

(10) Obligation:

(11) TransformationProof (EQUIVALENT transformation)

(12) Obligation:

(13) TransformationProof (EQUIVALENT transformation)

(14) Obligation:

(15) TransformationProof (EQUIVALENT transformation)

(16) Obligation:

(17) TransformationProof (EQUIVALENT transformation)

(18) Obligation:

(19) TransformationProof (EQUIVALENT transformation)

(20) Obligation:

(21) SemLabProof (SOUND transformation)

(22) Obligation:

(23) DependencyGraphProof (EQUIVALENT transformation)

(24) Obligation:

(25) MRRProof (EQUIVALENT transformation)

(26) Obligation:

(27) DependencyGraphProof (EQUIVALENT transformation)

(28) Obligation:

(29) MRRProof (EQUIVALENT transformation)

(30) Obligation:

(31) DependencyGraphProof (EQUIVALENT transformation)

(32) TRUE