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:

D(t) → 1
D(constant) → 0
D(+(x, y)) → +(D(x), D(y))
D(*(x, y)) → +(*(y, D(x)), *(x, D(y)))
D(-(x, y)) → -(D(x), D(y))

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:

D¹(+(x, y)) → D¹(x)
D¹(+(x, y)) → D¹(y)
D¹(*(x, y)) → D¹(x)
D¹(*(x, y)) → D¹(y)
D¹(-(x, y)) → D¹(x)
D¹(-(x, y)) → D¹(y)

The TRS R consists of the following rules:

D(t) → 1
D(constant) → 0
D(+(x, y)) → +(D(x), D(y))
D(*(x, y)) → +(*(y, D(x)), *(x, D(y)))
D(-(x, y)) → -(D(x), D(y))

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

(3) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

D¹(+(x, y)) → D¹(x)
D¹(+(x, y)) → D¹(y)
D¹(*(x, y)) → D¹(x)
D¹(*(x, y)) → D¹(y)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
D¹(x0, x1)  =  D¹(x0, x1)

Tags:
D¹ has argument tags [1,0] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
D¹(x1)  =  D¹
+(x1, x2)  =  +(x1, x2)
*(x1, x2)  =  *(x1, x2)
-(x1, x2)  =  -(x1, x2)

Recursive path order with status [RPO].
Quasi-Precedence:

[D^1, +₂, *₂]

Status:

D^1: multiset
+₂: multiset
*₂: multiset
-₂: multiset

The following usable rules [FROCOS05] were oriented: none

(4) Obligation:

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

D¹(-(x, y)) → D¹(x)
D¹(-(x, y)) → D¹(y)

The TRS R consists of the following rules:

D(t) → 1
D(constant) → 0
D(+(x, y)) → +(D(x), D(y))
D(*(x, y)) → +(*(y, D(x)), *(x, D(y)))
D(-(x, y)) → -(D(x), D(y))

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

(5) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

D¹(-(x, y)) → D¹(x)
D¹(-(x, y)) → D¹(y)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
D¹(x0, x1)  =  D¹(x0, x1)

Tags:
D¹ has argument tags [1,0] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
D¹(x1)  =  D¹
-(x1, x2)  =  -(x1, x2)

Recursive path order with status [RPO].
Quasi-Precedence:

[D^1, -₂]

Status:

D^1: multiset
-₂: multiset

The following usable rules [FROCOS05] were oriented: none

(6) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

D(t) → 1
D(constant) → 0
D(+(x, y)) → +(D(x), D(y))
D(*(x, y)) → +(*(y, D(x)), *(x, D(y)))
D(-(x, y)) → -(D(x), D(y))

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

(7) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(8) TRUE