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

(0) Obligation:

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

:(:(x, y), z) → :(x, :(y, z))
:(+(x, y), z) → +(:(x, z), :(y, z))
:(z, +(x, f(y))) → :(g(z, y), +(x, a))

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

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

(2) Obligation:

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

:¹(:(x, y), z) → :¹(x, :(y, z))
:¹(:(x, y), z) → :¹(y, z)
:¹(+(x, y), z) → :¹(x, z)
:¹(+(x, y), z) → :¹(y, z)
:¹(z, +(x, f(y))) → :¹(g(z, y), +(x, a))

The TRS R consists of the following rules:

:(:(x, y), z) → :(x, :(y, z))
:(+(x, y), z) → +(:(x, z), :(y, z))
:(z, +(x, f(y))) → :(g(z, y), +(x, a))

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

(3) DependencyGraphProof (EQUIVALENT transformation)

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

(4) Obligation:

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

:¹(:(x, y), z) → :¹(y, z)
:¹(:(x, y), z) → :¹(x, :(y, z))
:¹(+(x, y), z) → :¹(x, z)
:¹(+(x, y), z) → :¹(y, z)

The TRS R consists of the following rules:

:(:(x, y), z) → :(x, :(y, z))
:(+(x, y), z) → +(:(x, z), :(y, z))
:(z, +(x, f(y))) → :(g(z, y), +(x, a))

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

(5) QDPSizeChangeProof (EQUIVALENT transformation)

We used the following order and afs together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem.

Order:Homeomorphic Embedding Order

AFS:
+(x1, x2)  =  +(x1, x2)
:(x1, x2)  =  :(x1, x2)

From the DPs we obtained the following set of size-change graphs:

• :¹(:(x, y), z) → :¹(y, z) (allowed arguments on rhs = {1, 2})
  The graph contains the following edges 1 > 1, 2 >= 2

• :¹(:(x, y), z) → :¹(x, :(y, z)) (allowed arguments on rhs = {1})
  The graph contains the following edges 1 > 1

• :¹(+(x, y), z) → :¹(x, z) (allowed arguments on rhs = {1, 2})
  The graph contains the following edges 1 > 1, 2 >= 2

• :¹(+(x, y), z) → :¹(y, z) (allowed arguments on rhs = {1, 2})
  The graph contains the following edges 1 > 1, 2 >= 2

We oriented the following set of usable rules [AAECC05,FROCOS05]. none

(6) TRUE