Termination w.r.t. Q proof of TRS/Cimemaude2.trs

Termination w.r.t. Q of the following Term Rewriting System could not be shown:

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

p(s(N)) → N
+(N, 0) → N
+(s(N), s(M)) → s(s(+(N, M)))
*(N, 0) → 0
*(s(N), s(M)) → s(+(N, +(M, *(N, M))))
gt(0, M) → False
gt(NzN, 0) → u_4(is_NzNat(NzN))
u_4(True) → True
is_NzNat(0) → False
is_NzNat(s(N)) → True
gt(s(N), s(M)) → gt(N, M)
lt(N, M) → gt(M, N)
d(0, N) → N
d(s(N), s(M)) → d(N, M)
quot(N, NzM) → u_11(is_NzNat(NzM), N, NzM)
u_11(True, N, NzM) → u_1(gt(N, NzM), N, NzM)
u_1(True, N, NzM) → s(quot(d(N, NzM), NzM))
quot(NzM, NzM) → u_01(is_NzNat(NzM))
u_01(True) → s(0)
quot(N, NzM) → u_21(is_NzNat(NzM), NzM, N)
u_21(True, NzM, N) → u_2(gt(NzM, N))
u_2(True) → 0
gcd(0, N) → 0
gcd(NzM, NzM) → u_02(is_NzNat(NzM), NzM)
u_02(True, NzM) → NzM
gcd(NzN, NzM) → u_31(is_NzNat(NzN), is_NzNat(NzM), NzN, NzM)
u_31(True, True, NzN, NzM) → u_3(gt(NzN, NzM), NzN, NzM)
u_3(True, NzN, NzM) → gcd(d(NzN, NzM), NzM)

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

p(s(N)) → N
+(N, 0) → N
+(s(N), s(M)) → s(s(+(N, M)))
*(N, 0) → 0
*(s(N), s(M)) → s(+(N, +(M, *(N, M))))
gt(0, M) → False
gt(NzN, 0) → u_4(is_NzNat(NzN))
u_4(True) → True
is_NzNat(0) → False
is_NzNat(s(N)) → True
gt(s(N), s(M)) → gt(N, M)
lt(N, M) → gt(M, N)
d(0, N) → N
d(s(N), s(M)) → d(N, M)
quot(N, NzM) → u_11(is_NzNat(NzM), N, NzM)
u_11(True, N, NzM) → u_1(gt(N, NzM), N, NzM)
u_1(True, N, NzM) → s(quot(d(N, NzM), NzM))
quot(NzM, NzM) → u_01(is_NzNat(NzM))
u_01(True) → s(0)
quot(N, NzM) → u_21(is_NzNat(NzM), NzM, N)
u_21(True, NzM, N) → u_2(gt(NzM, N))
u_2(True) → 0
gcd(0, N) → 0
gcd(NzM, NzM) → u_02(is_NzNat(NzM), NzM)
u_02(True, NzM) → NzM
gcd(NzN, NzM) → u_31(is_NzNat(NzN), is_NzNat(NzM), NzN, NzM)
u_31(True, True, NzN, NzM) → u_3(gt(NzN, NzM), NzN, NzM)
u_3(True, NzN, NzM) → gcd(d(NzN, NzM), NzM)

Q is empty.

Using Dependency Pairs [1,13] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

GT(NzN, 0) → U_4(is_NzNat(NzN))
U_3(True, NzN, NzM) → GCD(d(NzN, NzM), NzM)
U_1(True, N, NzM) → D(N, NzM)
U_11(True, N, NzM) → U_1(gt(N, NzM), N, NzM)
U_21(True, NzM, N) → GT(NzM, N)
D(s(N), s(M)) → D(N, M)
*¹(s(N), s(M)) → +¹(N, +(M, *(N, M)))
+¹(s(N), s(M)) → +¹(N, M)
U_21(True, NzM, N) → U_2(gt(NzM, N))
GCD(NzN, NzM) → IS_NZNAT(NzM)
LT(N, M) → GT(M, N)
GCD(NzN, NzM) → IS_NZNAT(NzN)
QUOT(N, NzM) → U_21(is_NzNat(NzM), NzM, N)
GCD(NzM, NzM) → IS_NZNAT(NzM)
U_31(True, True, NzN, NzM) → U_3(gt(NzN, NzM), NzN, NzM)
GT(s(N), s(M)) → GT(N, M)
*¹(s(N), s(M)) → *¹(N, M)
QUOT(N, NzM) → U_11(is_NzNat(NzM), N, NzM)
GCD(NzN, NzM) → U_31(is_NzNat(NzN), is_NzNat(NzM), NzN, NzM)
QUOT(NzM, NzM) → U_01(is_NzNat(NzM))
GCD(NzM, NzM) → U_02(is_NzNat(NzM), NzM)
GT(NzN, 0) → IS_NZNAT(NzN)
*¹(s(N), s(M)) → +¹(M, *(N, M))
QUOT(NzM, NzM) → IS_NZNAT(NzM)
U_3(True, NzN, NzM) → D(NzN, NzM)
U_11(True, N, NzM) → GT(N, NzM)
QUOT(N, NzM) → IS_NZNAT(NzM)
U_31(True, True, NzN, NzM) → GT(NzN, NzM)
U_1(True, N, NzM) → QUOT(d(N, NzM), NzM)

The TRS R consists of the following rules:

p(s(N)) → N
+(N, 0) → N
+(s(N), s(M)) → s(s(+(N, M)))
*(N, 0) → 0
*(s(N), s(M)) → s(+(N, +(M, *(N, M))))
gt(0, M) → False
gt(NzN, 0) → u_4(is_NzNat(NzN))
u_4(True) → True
is_NzNat(0) → False
is_NzNat(s(N)) → True
gt(s(N), s(M)) → gt(N, M)
lt(N, M) → gt(M, N)
d(0, N) → N
d(s(N), s(M)) → d(N, M)
quot(N, NzM) → u_11(is_NzNat(NzM), N, NzM)
u_11(True, N, NzM) → u_1(gt(N, NzM), N, NzM)
u_1(True, N, NzM) → s(quot(d(N, NzM), NzM))
quot(NzM, NzM) → u_01(is_NzNat(NzM))
u_01(True) → s(0)
quot(N, NzM) → u_21(is_NzNat(NzM), NzM, N)
u_21(True, NzM, N) → u_2(gt(NzM, N))
u_2(True) → 0
gcd(0, N) → 0
gcd(NzM, NzM) → u_02(is_NzNat(NzM), NzM)
u_02(True, NzM) → NzM
gcd(NzN, NzM) → u_31(is_NzNat(NzN), is_NzNat(NzM), NzN, NzM)
u_31(True, True, NzN, NzM) → u_3(gt(NzN, NzM), NzN, NzM)
u_3(True, NzN, NzM) → gcd(d(NzN, NzM), NzM)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

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

GT(NzN, 0) → U_4(is_NzNat(NzN))
U_3(True, NzN, NzM) → GCD(d(NzN, NzM), NzM)
U_1(True, N, NzM) → D(N, NzM)
U_11(True, N, NzM) → U_1(gt(N, NzM), N, NzM)
U_21(True, NzM, N) → GT(NzM, N)
D(s(N), s(M)) → D(N, M)
*¹(s(N), s(M)) → +¹(N, +(M, *(N, M)))
+¹(s(N), s(M)) → +¹(N, M)
U_21(True, NzM, N) → U_2(gt(NzM, N))
GCD(NzN, NzM) → IS_NZNAT(NzM)
LT(N, M) → GT(M, N)
GCD(NzN, NzM) → IS_NZNAT(NzN)
QUOT(N, NzM) → U_21(is_NzNat(NzM), NzM, N)
GCD(NzM, NzM) → IS_NZNAT(NzM)
U_31(True, True, NzN, NzM) → U_3(gt(NzN, NzM), NzN, NzM)
GT(s(N), s(M)) → GT(N, M)
*¹(s(N), s(M)) → *¹(N, M)
QUOT(N, NzM) → U_11(is_NzNat(NzM), N, NzM)
GCD(NzN, NzM) → U_31(is_NzNat(NzN), is_NzNat(NzM), NzN, NzM)
QUOT(NzM, NzM) → U_01(is_NzNat(NzM))
GCD(NzM, NzM) → U_02(is_NzNat(NzM), NzM)
GT(NzN, 0) → IS_NZNAT(NzN)
*¹(s(N), s(M)) → +¹(M, *(N, M))
QUOT(NzM, NzM) → IS_NZNAT(NzM)
U_3(True, NzN, NzM) → D(NzN, NzM)
U_11(True, N, NzM) → GT(N, NzM)
QUOT(N, NzM) → IS_NZNAT(NzM)
U_31(True, True, NzN, NzM) → GT(NzN, NzM)
U_1(True, N, NzM) → QUOT(d(N, NzM), NzM)

The TRS R consists of the following rules:

p(s(N)) → N
+(N, 0) → N
+(s(N), s(M)) → s(s(+(N, M)))
*(N, 0) → 0
*(s(N), s(M)) → s(+(N, +(M, *(N, M))))
gt(0, M) → False
gt(NzN, 0) → u_4(is_NzNat(NzN))
u_4(True) → True
is_NzNat(0) → False
is_NzNat(s(N)) → True
gt(s(N), s(M)) → gt(N, M)
lt(N, M) → gt(M, N)
d(0, N) → N
d(s(N), s(M)) → d(N, M)
quot(N, NzM) → u_11(is_NzNat(NzM), N, NzM)
u_11(True, N, NzM) → u_1(gt(N, NzM), N, NzM)
u_1(True, N, NzM) → s(quot(d(N, NzM), NzM))
quot(NzM, NzM) → u_01(is_NzNat(NzM))
u_01(True) → s(0)
quot(N, NzM) → u_21(is_NzNat(NzM), NzM, N)
u_21(True, NzM, N) → u_2(gt(NzM, N))
u_2(True) → 0
gcd(0, N) → 0
gcd(NzM, NzM) → u_02(is_NzNat(NzM), NzM)
u_02(True, NzM) → NzM
gcd(NzN, NzM) → u_31(is_NzNat(NzN), is_NzNat(NzM), NzN, NzM)
u_31(True, True, NzN, NzM) → u_3(gt(NzN, NzM), NzN, NzM)
u_3(True, NzN, NzM) → gcd(d(NzN, NzM), NzM)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We deleted some edges using various graph approximations

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

        ↳ QDP

          ↳ DependencyGraphProof

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

GT(NzN, 0) → U_4(is_NzNat(NzN))
U_3(True, NzN, NzM) → GCD(d(NzN, NzM), NzM)
U_1(True, N, NzM) → D(N, NzM)
U_11(True, N, NzM) → U_1(gt(N, NzM), N, NzM)
U_21(True, NzM, N) → GT(NzM, N)
D(s(N), s(M)) → D(N, M)
*¹(s(N), s(M)) → +¹(N, +(M, *(N, M)))
U_21(True, NzM, N) → U_2(gt(NzM, N))
+¹(s(N), s(M)) → +¹(N, M)
GCD(NzN, NzM) → IS_NZNAT(NzM)
LT(N, M) → GT(M, N)
GCD(NzN, NzM) → IS_NZNAT(NzN)
GCD(NzM, NzM) → IS_NZNAT(NzM)
QUOT(N, NzM) → U_21(is_NzNat(NzM), NzM, N)
GT(s(N), s(M)) → GT(N, M)
U_31(True, True, NzN, NzM) → U_3(gt(NzN, NzM), NzN, NzM)
*¹(s(N), s(M)) → *¹(N, M)
GCD(NzN, NzM) → U_31(is_NzNat(NzN), is_NzNat(NzM), NzN, NzM)
QUOT(N, NzM) → U_11(is_NzNat(NzM), N, NzM)
GCD(NzM, NzM) → U_02(is_NzNat(NzM), NzM)
QUOT(NzM, NzM) → U_01(is_NzNat(NzM))
*¹(s(N), s(M)) → +¹(M, *(N, M))
GT(NzN, 0) → IS_NZNAT(NzN)
QUOT(NzM, NzM) → IS_NZNAT(NzM)
U_3(True, NzN, NzM) → D(NzN, NzM)
U_11(True, N, NzM) → GT(N, NzM)
QUOT(N, NzM) → IS_NZNAT(NzM)
U_31(True, True, NzN, NzM) → GT(NzN, NzM)
U_1(True, N, NzM) → QUOT(d(N, NzM), NzM)

The TRS R consists of the following rules:

p(s(N)) → N
+(N, 0) → N
+(s(N), s(M)) → s(s(+(N, M)))
*(N, 0) → 0
*(s(N), s(M)) → s(+(N, +(M, *(N, M))))
gt(0, M) → False
gt(NzN, 0) → u_4(is_NzNat(NzN))
u_4(True) → True
is_NzNat(0) → False
is_NzNat(s(N)) → True
gt(s(N), s(M)) → gt(N, M)
lt(N, M) → gt(M, N)
d(0, N) → N
d(s(N), s(M)) → d(N, M)
quot(N, NzM) → u_11(is_NzNat(NzM), N, NzM)
u_11(True, N, NzM) → u_1(gt(N, NzM), N, NzM)
u_1(True, N, NzM) → s(quot(d(N, NzM), NzM))
quot(NzM, NzM) → u_01(is_NzNat(NzM))
u_01(True) → s(0)
quot(N, NzM) → u_21(is_NzNat(NzM), NzM, N)
u_21(True, NzM, N) → u_2(gt(NzM, N))
u_2(True) → 0
gcd(0, N) → 0
gcd(NzM, NzM) → u_02(is_NzNat(NzM), NzM)
u_02(True, NzM) → NzM
gcd(NzN, NzM) → u_31(is_NzNat(NzN), is_NzNat(NzM), NzN, NzM)
u_31(True, True, NzN, NzM) → u_3(gt(NzN, NzM), NzN, NzM)
u_3(True, NzN, NzM) → gcd(d(NzN, NzM), NzM)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 6 SCCs with 19 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ QDPOrderProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

D(s(N), s(M)) → D(N, M)

The TRS R consists of the following rules:

p(s(N)) → N
+(N, 0) → N
+(s(N), s(M)) → s(s(+(N, M)))
*(N, 0) → 0
*(s(N), s(M)) → s(+(N, +(M, *(N, M))))
gt(0, M) → False
gt(NzN, 0) → u_4(is_NzNat(NzN))
u_4(True) → True
is_NzNat(0) → False
is_NzNat(s(N)) → True
gt(s(N), s(M)) → gt(N, M)
lt(N, M) → gt(M, N)
d(0, N) → N
d(s(N), s(M)) → d(N, M)
quot(N, NzM) → u_11(is_NzNat(NzM), N, NzM)
u_11(True, N, NzM) → u_1(gt(N, NzM), N, NzM)
u_1(True, N, NzM) → s(quot(d(N, NzM), NzM))
quot(NzM, NzM) → u_01(is_NzNat(NzM))
u_01(True) → s(0)
quot(N, NzM) → u_21(is_NzNat(NzM), NzM, N)
u_21(True, NzM, N) → u_2(gt(NzM, N))
u_2(True) → 0
gcd(0, N) → 0
gcd(NzM, NzM) → u_02(is_NzNat(NzM), NzM)
u_02(True, NzM) → NzM
gcd(NzN, NzM) → u_31(is_NzNat(NzN), is_NzNat(NzM), NzN, NzM)
u_31(True, True, NzN, NzM) → u_3(gt(NzN, NzM), NzN, NzM)
u_3(True, NzN, NzM) → gcd(d(NzN, NzM), NzM)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

D(s(N), s(M)) → D(N, M)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
D(x1, x2) = D(x1)
s(x1) = s(x1)

Lexicographic path order with status [19].
Precedence:

s₁ > D₁

Status:

D₁: [1]
s₁: multiset

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ PisEmptyProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

p(s(N)) → N
+(N, 0) → N
+(s(N), s(M)) → s(s(+(N, M)))
*(N, 0) → 0
*(s(N), s(M)) → s(+(N, +(M, *(N, M))))
gt(0, M) → False
gt(NzN, 0) → u_4(is_NzNat(NzN))
u_4(True) → True
is_NzNat(0) → False
is_NzNat(s(N)) → True
gt(s(N), s(M)) → gt(N, M)
lt(N, M) → gt(M, N)
d(0, N) → N
d(s(N), s(M)) → d(N, M)
quot(N, NzM) → u_11(is_NzNat(NzM), N, NzM)
u_11(True, N, NzM) → u_1(gt(N, NzM), N, NzM)
u_1(True, N, NzM) → s(quot(d(N, NzM), NzM))
quot(NzM, NzM) → u_01(is_NzNat(NzM))
u_01(True) → s(0)
quot(N, NzM) → u_21(is_NzNat(NzM), NzM, N)
u_21(True, NzM, N) → u_2(gt(NzM, N))
u_2(True) → 0
gcd(0, N) → 0
gcd(NzM, NzM) → u_02(is_NzNat(NzM), NzM)
u_02(True, NzM) → NzM
gcd(NzN, NzM) → u_31(is_NzNat(NzN), is_NzNat(NzM), NzN, NzM)
u_31(True, True, NzN, NzM) → u_3(gt(NzN, NzM), NzN, NzM)
u_3(True, NzN, NzM) → gcd(d(NzN, NzM), NzM)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ QDPOrderProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

GT(s(N), s(M)) → GT(N, M)

The TRS R consists of the following rules:

p(s(N)) → N
+(N, 0) → N
+(s(N), s(M)) → s(s(+(N, M)))
*(N, 0) → 0
*(s(N), s(M)) → s(+(N, +(M, *(N, M))))
gt(0, M) → False
gt(NzN, 0) → u_4(is_NzNat(NzN))
u_4(True) → True
is_NzNat(0) → False
is_NzNat(s(N)) → True
gt(s(N), s(M)) → gt(N, M)
lt(N, M) → gt(M, N)
d(0, N) → N
d(s(N), s(M)) → d(N, M)
quot(N, NzM) → u_11(is_NzNat(NzM), N, NzM)
u_11(True, N, NzM) → u_1(gt(N, NzM), N, NzM)
u_1(True, N, NzM) → s(quot(d(N, NzM), NzM))
quot(NzM, NzM) → u_01(is_NzNat(NzM))
u_01(True) → s(0)
quot(N, NzM) → u_21(is_NzNat(NzM), NzM, N)
u_21(True, NzM, N) → u_2(gt(NzM, N))
u_2(True) → 0
gcd(0, N) → 0
gcd(NzM, NzM) → u_02(is_NzNat(NzM), NzM)
u_02(True, NzM) → NzM
gcd(NzN, NzM) → u_31(is_NzNat(NzN), is_NzNat(NzM), NzN, NzM)
u_31(True, True, NzN, NzM) → u_3(gt(NzN, NzM), NzN, NzM)
u_3(True, NzN, NzM) → gcd(d(NzN, NzM), NzM)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

GT(s(N), s(M)) → GT(N, M)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
GT(x1, x2) = GT(x1)
s(x1) = s(x1)

Lexicographic path order with status [19].
Precedence:

s₁ > GT₁

Status:

GT₁: [1]
s₁: multiset

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ PisEmptyProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

p(s(N)) → N
+(N, 0) → N
+(s(N), s(M)) → s(s(+(N, M)))
*(N, 0) → 0
*(s(N), s(M)) → s(+(N, +(M, *(N, M))))
gt(0, M) → False
gt(NzN, 0) → u_4(is_NzNat(NzN))
u_4(True) → True
is_NzNat(0) → False
is_NzNat(s(N)) → True
gt(s(N), s(M)) → gt(N, M)
lt(N, M) → gt(M, N)
d(0, N) → N
d(s(N), s(M)) → d(N, M)
quot(N, NzM) → u_11(is_NzNat(NzM), N, NzM)
u_11(True, N, NzM) → u_1(gt(N, NzM), N, NzM)
u_1(True, N, NzM) → s(quot(d(N, NzM), NzM))
quot(NzM, NzM) → u_01(is_NzNat(NzM))
u_01(True) → s(0)
quot(N, NzM) → u_21(is_NzNat(NzM), NzM, N)
u_21(True, NzM, N) → u_2(gt(NzM, N))
u_2(True) → 0
gcd(0, N) → 0
gcd(NzM, NzM) → u_02(is_NzNat(NzM), NzM)
u_02(True, NzM) → NzM
gcd(NzN, NzM) → u_31(is_NzNat(NzN), is_NzNat(NzM), NzN, NzM)
u_31(True, True, NzN, NzM) → u_3(gt(NzN, NzM), NzN, NzM)
u_3(True, NzN, NzM) → gcd(d(NzN, NzM), NzM)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

U_3(True, NzN, NzM) → GCD(d(NzN, NzM), NzM)
U_31(True, True, NzN, NzM) → U_3(gt(NzN, NzM), NzN, NzM)
GCD(NzN, NzM) → U_31(is_NzNat(NzN), is_NzNat(NzM), NzN, NzM)

The TRS R consists of the following rules:

p(s(N)) → N
+(N, 0) → N
+(s(N), s(M)) → s(s(+(N, M)))
*(N, 0) → 0
*(s(N), s(M)) → s(+(N, +(M, *(N, M))))
gt(0, M) → False
gt(NzN, 0) → u_4(is_NzNat(NzN))
u_4(True) → True
is_NzNat(0) → False
is_NzNat(s(N)) → True
gt(s(N), s(M)) → gt(N, M)
lt(N, M) → gt(M, N)
d(0, N) → N
d(s(N), s(M)) → d(N, M)
quot(N, NzM) → u_11(is_NzNat(NzM), N, NzM)
u_11(True, N, NzM) → u_1(gt(N, NzM), N, NzM)
u_1(True, N, NzM) → s(quot(d(N, NzM), NzM))
quot(NzM, NzM) → u_01(is_NzNat(NzM))
u_01(True) → s(0)
quot(N, NzM) → u_21(is_NzNat(NzM), NzM, N)
u_21(True, NzM, N) → u_2(gt(NzM, N))
u_2(True) → 0
gcd(0, N) → 0
gcd(NzM, NzM) → u_02(is_NzNat(NzM), NzM)
u_02(True, NzM) → NzM
gcd(NzN, NzM) → u_31(is_NzNat(NzN), is_NzNat(NzM), NzN, NzM)
u_31(True, True, NzN, NzM) → u_3(gt(NzN, NzM), NzN, NzM)
u_3(True, NzN, NzM) → gcd(d(NzN, NzM), NzM)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

U_11(True, N, NzM) → U_1(gt(N, NzM), N, NzM)
QUOT(N, NzM) → U_11(is_NzNat(NzM), N, NzM)
U_1(True, N, NzM) → QUOT(d(N, NzM), NzM)

The TRS R consists of the following rules:

p(s(N)) → N
+(N, 0) → N
+(s(N), s(M)) → s(s(+(N, M)))
*(N, 0) → 0
*(s(N), s(M)) → s(+(N, +(M, *(N, M))))
gt(0, M) → False
gt(NzN, 0) → u_4(is_NzNat(NzN))
u_4(True) → True
is_NzNat(0) → False
is_NzNat(s(N)) → True
gt(s(N), s(M)) → gt(N, M)
lt(N, M) → gt(M, N)
d(0, N) → N
d(s(N), s(M)) → d(N, M)
quot(N, NzM) → u_11(is_NzNat(NzM), N, NzM)
u_11(True, N, NzM) → u_1(gt(N, NzM), N, NzM)
u_1(True, N, NzM) → s(quot(d(N, NzM), NzM))
quot(NzM, NzM) → u_01(is_NzNat(NzM))
u_01(True) → s(0)
quot(N, NzM) → u_21(is_NzNat(NzM), NzM, N)
u_21(True, NzM, N) → u_2(gt(NzM, N))
u_2(True) → 0
gcd(0, N) → 0
gcd(NzM, NzM) → u_02(is_NzNat(NzM), NzM)
u_02(True, NzM) → NzM
gcd(NzN, NzM) → u_31(is_NzNat(NzN), is_NzNat(NzM), NzN, NzM)
u_31(True, True, NzN, NzM) → u_3(gt(NzN, NzM), NzN, NzM)
u_3(True, NzN, NzM) → gcd(d(NzN, NzM), NzM)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ QDPOrderProof

              ↳ QDP

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

+¹(s(N), s(M)) → +¹(N, M)

The TRS R consists of the following rules:

p(s(N)) → N
+(N, 0) → N
+(s(N), s(M)) → s(s(+(N, M)))
*(N, 0) → 0
*(s(N), s(M)) → s(+(N, +(M, *(N, M))))
gt(0, M) → False
gt(NzN, 0) → u_4(is_NzNat(NzN))
u_4(True) → True
is_NzNat(0) → False
is_NzNat(s(N)) → True
gt(s(N), s(M)) → gt(N, M)
lt(N, M) → gt(M, N)
d(0, N) → N
d(s(N), s(M)) → d(N, M)
quot(N, NzM) → u_11(is_NzNat(NzM), N, NzM)
u_11(True, N, NzM) → u_1(gt(N, NzM), N, NzM)
u_1(True, N, NzM) → s(quot(d(N, NzM), NzM))
quot(NzM, NzM) → u_01(is_NzNat(NzM))
u_01(True) → s(0)
quot(N, NzM) → u_21(is_NzNat(NzM), NzM, N)
u_21(True, NzM, N) → u_2(gt(NzM, N))
u_2(True) → 0
gcd(0, N) → 0
gcd(NzM, NzM) → u_02(is_NzNat(NzM), NzM)
u_02(True, NzM) → NzM
gcd(NzN, NzM) → u_31(is_NzNat(NzN), is_NzNat(NzM), NzN, NzM)
u_31(True, True, NzN, NzM) → u_3(gt(NzN, NzM), NzN, NzM)
u_3(True, NzN, NzM) → gcd(d(NzN, NzM), NzM)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

+¹(s(N), s(M)) → +¹(N, M)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
+¹(x1, x2) = +¹(x1)
s(x1) = s(x1)

Lexicographic path order with status [19].
Precedence:

s₁ > +^1₁

Status:

+^1₁: [1]
s₁: multiset

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ PisEmptyProof

              ↳ QDP

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

p(s(N)) → N
+(N, 0) → N
+(s(N), s(M)) → s(s(+(N, M)))
*(N, 0) → 0
*(s(N), s(M)) → s(+(N, +(M, *(N, M))))
gt(0, M) → False
gt(NzN, 0) → u_4(is_NzNat(NzN))
u_4(True) → True
is_NzNat(0) → False
is_NzNat(s(N)) → True
gt(s(N), s(M)) → gt(N, M)
lt(N, M) → gt(M, N)
d(0, N) → N
d(s(N), s(M)) → d(N, M)
quot(N, NzM) → u_11(is_NzNat(NzM), N, NzM)
u_11(True, N, NzM) → u_1(gt(N, NzM), N, NzM)
u_1(True, N, NzM) → s(quot(d(N, NzM), NzM))
quot(NzM, NzM) → u_01(is_NzNat(NzM))
u_01(True) → s(0)
quot(N, NzM) → u_21(is_NzNat(NzM), NzM, N)
u_21(True, NzM, N) → u_2(gt(NzM, N))
u_2(True) → 0
gcd(0, N) → 0
gcd(NzM, NzM) → u_02(is_NzNat(NzM), NzM)
u_02(True, NzM) → NzM
gcd(NzN, NzM) → u_31(is_NzNat(NzN), is_NzNat(NzM), NzN, NzM)
u_31(True, True, NzN, NzM) → u_3(gt(NzN, NzM), NzN, NzM)
u_3(True, NzN, NzM) → gcd(d(NzN, NzM), NzM)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ QDPOrderProof

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

*¹(s(N), s(M)) → *¹(N, M)

The TRS R consists of the following rules:

p(s(N)) → N
+(N, 0) → N
+(s(N), s(M)) → s(s(+(N, M)))
*(N, 0) → 0
*(s(N), s(M)) → s(+(N, +(M, *(N, M))))
gt(0, M) → False
gt(NzN, 0) → u_4(is_NzNat(NzN))
u_4(True) → True
is_NzNat(0) → False
is_NzNat(s(N)) → True
gt(s(N), s(M)) → gt(N, M)
lt(N, M) → gt(M, N)
d(0, N) → N
d(s(N), s(M)) → d(N, M)
quot(N, NzM) → u_11(is_NzNat(NzM), N, NzM)
u_11(True, N, NzM) → u_1(gt(N, NzM), N, NzM)
u_1(True, N, NzM) → s(quot(d(N, NzM), NzM))
quot(NzM, NzM) → u_01(is_NzNat(NzM))
u_01(True) → s(0)
quot(N, NzM) → u_21(is_NzNat(NzM), NzM, N)
u_21(True, NzM, N) → u_2(gt(NzM, N))
u_2(True) → 0
gcd(0, N) → 0
gcd(NzM, NzM) → u_02(is_NzNat(NzM), NzM)
u_02(True, NzM) → NzM
gcd(NzN, NzM) → u_31(is_NzNat(NzN), is_NzNat(NzM), NzN, NzM)
u_31(True, True, NzN, NzM) → u_3(gt(NzN, NzM), NzN, NzM)
u_3(True, NzN, NzM) → gcd(d(NzN, NzM), NzM)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

*¹(s(N), s(M)) → *¹(N, M)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
*¹(x1, x2) = *¹(x1)
s(x1) = s(x1)

Lexicographic path order with status [19].
Precedence:

s₁ > *^1₁

Status:

*^1₁: [1]
s₁: multiset

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ PisEmptyProof

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

p(s(N)) → N
+(N, 0) → N
+(s(N), s(M)) → s(s(+(N, M)))
*(N, 0) → 0
*(s(N), s(M)) → s(+(N, +(M, *(N, M))))
gt(0, M) → False
gt(NzN, 0) → u_4(is_NzNat(NzN))
u_4(True) → True
is_NzNat(0) → False
is_NzNat(s(N)) → True
gt(s(N), s(M)) → gt(N, M)
lt(N, M) → gt(M, N)
d(0, N) → N
d(s(N), s(M)) → d(N, M)
quot(N, NzM) → u_11(is_NzNat(NzM), N, NzM)
u_11(True, N, NzM) → u_1(gt(N, NzM), N, NzM)
u_1(True, N, NzM) → s(quot(d(N, NzM), NzM))
quot(NzM, NzM) → u_01(is_NzNat(NzM))
u_01(True) → s(0)
quot(N, NzM) → u_21(is_NzNat(NzM), NzM, N)
u_21(True, NzM, N) → u_2(gt(NzM, N))
u_2(True) → 0
gcd(0, N) → 0
gcd(NzM, NzM) → u_02(is_NzNat(NzM), NzM)
u_02(True, NzM) → NzM
gcd(NzN, NzM) → u_31(is_NzNat(NzN), is_NzNat(NzM), NzN, NzM)
u_31(True, True, NzN, NzM) → u_3(gt(NzN, NzM), NzN, NzM)
u_3(True, NzN, NzM) → gcd(d(NzN, NzM), NzM)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.