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:

plus(plus(X, Y), Z) → plus(X, plus(Y, Z))
times(X, s(Y)) → plus(X, times(Y, X))

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:

PLUS(plus(X, Y), Z) → PLUS(X, plus(Y, Z))
PLUS(plus(X, Y), Z) → PLUS(Y, Z)
TIMES(X, s(Y)) → PLUS(X, times(Y, X))
TIMES(X, s(Y)) → TIMES(Y, X)

The TRS R consists of the following rules:

plus(plus(X, Y), Z) → plus(X, plus(Y, Z))
times(X, s(Y)) → plus(X, times(Y, X))

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 2 SCCs with 1 less node.

(4) Complex Obligation (AND)

(5) Obligation:

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

PLUS(plus(X, Y), Z) → PLUS(Y, Z)
PLUS(plus(X, Y), Z) → PLUS(X, plus(Y, Z))

The TRS R consists of the following rules:

plus(plus(X, Y), Z) → plus(X, plus(Y, Z))
times(X, s(Y)) → plus(X, times(Y, X))

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

(6) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PLUS(plus(X, Y), Z) → PLUS(Y, Z)
PLUS(plus(X, Y), Z) → PLUS(X, plus(Y, Z))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PLUS(x1, x2)  =  x1
plus(x1, x2)  =  plus(x1, x2)
times(x1, x2)  =  times(x1, x2)
s(x1)  =  s(x1)

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

times₂ > [plus₂, s₁]

Status:

plus₂: [1,2]
times₂: multiset
s₁: multiset

The following usable rules [FROCOS05] were oriented:

plus(plus(X, Y), Z) → plus(X, plus(Y, Z))
times(X, s(Y)) → plus(X, times(Y, X))

(7) Obligation:

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

plus(plus(X, Y), Z) → plus(X, plus(Y, Z))
times(X, s(Y)) → plus(X, times(Y, X))

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

(8) PisEmptyProof (EQUIVALENT transformation)

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

(9) TRUE

(10) Obligation:

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

TIMES(X, s(Y)) → TIMES(Y, X)

The TRS R consists of the following rules:

plus(plus(X, Y), Z) → plus(X, plus(Y, Z))
times(X, s(Y)) → plus(X, times(Y, X))

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

(11) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TIMES(X, s(Y)) → TIMES(Y, X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
TIMES(x1, x2)  =  TIMES(x1, x2)
s(x1)  =  s(x1)
plus(x1, x2)  =  plus
times(x1, x2)  =  times(x1, x2)

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

[TIMES₂, s₁] > [plus, times₂]

Status:

TIMES₂: multiset
s₁: multiset
plus: multiset
times₂: multiset

The following usable rules [FROCOS05] were oriented:

plus(plus(X, Y), Z) → plus(X, plus(Y, Z))
times(X, s(Y)) → plus(X, times(Y, X))

(12) Obligation:

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

plus(plus(X, Y), Z) → plus(X, plus(Y, Z))
times(X, s(Y)) → plus(X, times(Y, X))

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

(13) PisEmptyProof (EQUIVALENT transformation)

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

(14) TRUE