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:

p1(s1(N)) -> N
+2(N, 0) -> N
+2(s1(N), s1(M)) -> s1(s1(+2(N, M)))
*2(N, 0) -> 0
*2(s1(N), s1(M)) -> s1(+2(N, +2(M, *2(N, M))))
gt2(0, M) -> False
gt2(NzN, 0) -> u_41(is_NzNat1(NzN))
u_41(True) -> True
is_NzNat1(0) -> False
is_NzNat1(s1(N)) -> True
gt2(s1(N), s1(M)) -> gt2(N, M)
lt2(N, M) -> gt2(M, N)
d2(0, N) -> N
d2(s1(N), s1(M)) -> d2(N, M)
quot2(N, NzM) -> u_113(is_NzNat1(NzM), N, NzM)
u_113(True, N, NzM) -> u_13(gt2(N, NzM), N, NzM)
u_13(True, N, NzM) -> s1(quot2(d2(N, NzM), NzM))
quot2(NzM, NzM) -> u_011(is_NzNat1(NzM))
u_011(True) -> s1(0)
quot2(N, NzM) -> u_213(is_NzNat1(NzM), NzM, N)
u_213(True, NzM, N) -> u_21(gt2(NzM, N))
u_21(True) -> 0
gcd2(0, N) -> 0
gcd2(NzM, NzM) -> u_022(is_NzNat1(NzM), NzM)
u_022(True, NzM) -> NzM
gcd2(NzN, NzM) -> u_314(is_NzNat1(NzN), is_NzNat1(NzM), NzN, NzM)
u_314(True, True, NzN, NzM) -> u_33(gt2(NzN, NzM), NzN, NzM)
u_33(True, NzN, NzM) -> gcd2(d2(NzN, NzM), NzM)

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

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

p1(s1(N)) -> N
+2(N, 0) -> N
+2(s1(N), s1(M)) -> s1(s1(+2(N, M)))
*2(N, 0) -> 0
*2(s1(N), s1(M)) -> s1(+2(N, +2(M, *2(N, M))))
gt2(0, M) -> False
gt2(NzN, 0) -> u_41(is_NzNat1(NzN))
u_41(True) -> True
is_NzNat1(0) -> False
is_NzNat1(s1(N)) -> True
gt2(s1(N), s1(M)) -> gt2(N, M)
lt2(N, M) -> gt2(M, N)
d2(0, N) -> N
d2(s1(N), s1(M)) -> d2(N, M)
quot2(N, NzM) -> u_113(is_NzNat1(NzM), N, NzM)
u_113(True, N, NzM) -> u_13(gt2(N, NzM), N, NzM)
u_13(True, N, NzM) -> s1(quot2(d2(N, NzM), NzM))
quot2(NzM, NzM) -> u_011(is_NzNat1(NzM))
u_011(True) -> s1(0)
quot2(N, NzM) -> u_213(is_NzNat1(NzM), NzM, N)
u_213(True, NzM, N) -> u_21(gt2(NzM, N))
u_21(True) -> 0
gcd2(0, N) -> 0
gcd2(NzM, NzM) -> u_022(is_NzNat1(NzM), NzM)
u_022(True, NzM) -> NzM
gcd2(NzN, NzM) -> u_314(is_NzNat1(NzN), is_NzNat1(NzM), NzN, NzM)
u_314(True, True, NzN, NzM) -> u_33(gt2(NzN, NzM), NzN, NzM)
u_33(True, NzN, NzM) -> gcd2(d2(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:

GT2(NzN, 0) -> U_41(is_NzNat1(NzN))
U_33(True, NzN, NzM) -> GCD2(d2(NzN, NzM), NzM)
U_13(True, N, NzM) -> D2(N, NzM)
U_113(True, N, NzM) -> U_13(gt2(N, NzM), N, NzM)
U_213(True, NzM, N) -> GT2(NzM, N)
D2(s1(N), s1(M)) -> D2(N, M)
*12(s1(N), s1(M)) -> +12(N, +2(M, *2(N, M)))
+12(s1(N), s1(M)) -> +12(N, M)
U_213(True, NzM, N) -> U_21(gt2(NzM, N))
GCD2(NzN, NzM) -> IS_NZNAT1(NzM)
LT2(N, M) -> GT2(M, N)
GCD2(NzN, NzM) -> IS_NZNAT1(NzN)
QUOT2(N, NzM) -> U_213(is_NzNat1(NzM), NzM, N)
GCD2(NzM, NzM) -> IS_NZNAT1(NzM)
U_314(True, True, NzN, NzM) -> U_33(gt2(NzN, NzM), NzN, NzM)
GT2(s1(N), s1(M)) -> GT2(N, M)
*12(s1(N), s1(M)) -> *12(N, M)
QUOT2(N, NzM) -> U_113(is_NzNat1(NzM), N, NzM)
GCD2(NzN, NzM) -> U_314(is_NzNat1(NzN), is_NzNat1(NzM), NzN, NzM)
QUOT2(NzM, NzM) -> U_011(is_NzNat1(NzM))
GCD2(NzM, NzM) -> U_022(is_NzNat1(NzM), NzM)
GT2(NzN, 0) -> IS_NZNAT1(NzN)
*12(s1(N), s1(M)) -> +12(M, *2(N, M))
QUOT2(NzM, NzM) -> IS_NZNAT1(NzM)
U_33(True, NzN, NzM) -> D2(NzN, NzM)
U_113(True, N, NzM) -> GT2(N, NzM)
QUOT2(N, NzM) -> IS_NZNAT1(NzM)
U_314(True, True, NzN, NzM) -> GT2(NzN, NzM)
U_13(True, N, NzM) -> QUOT2(d2(N, NzM), NzM)

The TRS R consists of the following rules:

p1(s1(N)) -> N
+2(N, 0) -> N
+2(s1(N), s1(M)) -> s1(s1(+2(N, M)))
*2(N, 0) -> 0
*2(s1(N), s1(M)) -> s1(+2(N, +2(M, *2(N, M))))
gt2(0, M) -> False
gt2(NzN, 0) -> u_41(is_NzNat1(NzN))
u_41(True) -> True
is_NzNat1(0) -> False
is_NzNat1(s1(N)) -> True
gt2(s1(N), s1(M)) -> gt2(N, M)
lt2(N, M) -> gt2(M, N)
d2(0, N) -> N
d2(s1(N), s1(M)) -> d2(N, M)
quot2(N, NzM) -> u_113(is_NzNat1(NzM), N, NzM)
u_113(True, N, NzM) -> u_13(gt2(N, NzM), N, NzM)
u_13(True, N, NzM) -> s1(quot2(d2(N, NzM), NzM))
quot2(NzM, NzM) -> u_011(is_NzNat1(NzM))
u_011(True) -> s1(0)
quot2(N, NzM) -> u_213(is_NzNat1(NzM), NzM, N)
u_213(True, NzM, N) -> u_21(gt2(NzM, N))
u_21(True) -> 0
gcd2(0, N) -> 0
gcd2(NzM, NzM) -> u_022(is_NzNat1(NzM), NzM)
u_022(True, NzM) -> NzM
gcd2(NzN, NzM) -> u_314(is_NzNat1(NzN), is_NzNat1(NzM), NzN, NzM)
u_314(True, True, NzN, NzM) -> u_33(gt2(NzN, NzM), NzN, NzM)
u_33(True, NzN, NzM) -> gcd2(d2(NzN, NzM), NzM)

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof

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

GT2(NzN, 0) -> U_41(is_NzNat1(NzN))
U_33(True, NzN, NzM) -> GCD2(d2(NzN, NzM), NzM)
U_13(True, N, NzM) -> D2(N, NzM)
U_113(True, N, NzM) -> U_13(gt2(N, NzM), N, NzM)
U_213(True, NzM, N) -> GT2(NzM, N)
D2(s1(N), s1(M)) -> D2(N, M)
*12(s1(N), s1(M)) -> +12(N, +2(M, *2(N, M)))
+12(s1(N), s1(M)) -> +12(N, M)
U_213(True, NzM, N) -> U_21(gt2(NzM, N))
GCD2(NzN, NzM) -> IS_NZNAT1(NzM)
LT2(N, M) -> GT2(M, N)
GCD2(NzN, NzM) -> IS_NZNAT1(NzN)
QUOT2(N, NzM) -> U_213(is_NzNat1(NzM), NzM, N)
GCD2(NzM, NzM) -> IS_NZNAT1(NzM)
U_314(True, True, NzN, NzM) -> U_33(gt2(NzN, NzM), NzN, NzM)
GT2(s1(N), s1(M)) -> GT2(N, M)
*12(s1(N), s1(M)) -> *12(N, M)
QUOT2(N, NzM) -> U_113(is_NzNat1(NzM), N, NzM)
GCD2(NzN, NzM) -> U_314(is_NzNat1(NzN), is_NzNat1(NzM), NzN, NzM)
QUOT2(NzM, NzM) -> U_011(is_NzNat1(NzM))
GCD2(NzM, NzM) -> U_022(is_NzNat1(NzM), NzM)
GT2(NzN, 0) -> IS_NZNAT1(NzN)
*12(s1(N), s1(M)) -> +12(M, *2(N, M))
QUOT2(NzM, NzM) -> IS_NZNAT1(NzM)
U_33(True, NzN, NzM) -> D2(NzN, NzM)
U_113(True, N, NzM) -> GT2(N, NzM)
QUOT2(N, NzM) -> IS_NZNAT1(NzM)
U_314(True, True, NzN, NzM) -> GT2(NzN, NzM)
U_13(True, N, NzM) -> QUOT2(d2(N, NzM), NzM)

The TRS R consists of the following rules:

p1(s1(N)) -> N
+2(N, 0) -> N
+2(s1(N), s1(M)) -> s1(s1(+2(N, M)))
*2(N, 0) -> 0
*2(s1(N), s1(M)) -> s1(+2(N, +2(M, *2(N, M))))
gt2(0, M) -> False
gt2(NzN, 0) -> u_41(is_NzNat1(NzN))
u_41(True) -> True
is_NzNat1(0) -> False
is_NzNat1(s1(N)) -> True
gt2(s1(N), s1(M)) -> gt2(N, M)
lt2(N, M) -> gt2(M, N)
d2(0, N) -> N
d2(s1(N), s1(M)) -> d2(N, M)
quot2(N, NzM) -> u_113(is_NzNat1(NzM), N, NzM)
u_113(True, N, NzM) -> u_13(gt2(N, NzM), N, NzM)
u_13(True, N, NzM) -> s1(quot2(d2(N, NzM), NzM))
quot2(NzM, NzM) -> u_011(is_NzNat1(NzM))
u_011(True) -> s1(0)
quot2(N, NzM) -> u_213(is_NzNat1(NzM), NzM, N)
u_213(True, NzM, N) -> u_21(gt2(NzM, N))
u_21(True) -> 0
gcd2(0, N) -> 0
gcd2(NzM, NzM) -> u_022(is_NzNat1(NzM), NzM)
u_022(True, NzM) -> NzM
gcd2(NzN, NzM) -> u_314(is_NzNat1(NzN), is_NzNat1(NzM), NzN, NzM)
u_314(True, True, NzN, NzM) -> u_33(gt2(NzN, NzM), NzN, NzM)
u_33(True, NzN, NzM) -> gcd2(d2(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
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

D2(s1(N), s1(M)) -> D2(N, M)

The TRS R consists of the following rules:

p1(s1(N)) -> N
+2(N, 0) -> N
+2(s1(N), s1(M)) -> s1(s1(+2(N, M)))
*2(N, 0) -> 0
*2(s1(N), s1(M)) -> s1(+2(N, +2(M, *2(N, M))))
gt2(0, M) -> False
gt2(NzN, 0) -> u_41(is_NzNat1(NzN))
u_41(True) -> True
is_NzNat1(0) -> False
is_NzNat1(s1(N)) -> True
gt2(s1(N), s1(M)) -> gt2(N, M)
lt2(N, M) -> gt2(M, N)
d2(0, N) -> N
d2(s1(N), s1(M)) -> d2(N, M)
quot2(N, NzM) -> u_113(is_NzNat1(NzM), N, NzM)
u_113(True, N, NzM) -> u_13(gt2(N, NzM), N, NzM)
u_13(True, N, NzM) -> s1(quot2(d2(N, NzM), NzM))
quot2(NzM, NzM) -> u_011(is_NzNat1(NzM))
u_011(True) -> s1(0)
quot2(N, NzM) -> u_213(is_NzNat1(NzM), NzM, N)
u_213(True, NzM, N) -> u_21(gt2(NzM, N))
u_21(True) -> 0
gcd2(0, N) -> 0
gcd2(NzM, NzM) -> u_022(is_NzNat1(NzM), NzM)
u_022(True, NzM) -> NzM
gcd2(NzN, NzM) -> u_314(is_NzNat1(NzN), is_NzNat1(NzM), NzN, NzM)
u_314(True, True, NzN, NzM) -> u_33(gt2(NzN, NzM), NzN, NzM)
u_33(True, NzN, NzM) -> gcd2(d2(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.


D2(s1(N), s1(M)) -> D2(N, M)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( s1(x1) ) = x1 + 3


POL( D2(x1, x2) ) = 3x2 + 3



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ 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:

p1(s1(N)) -> N
+2(N, 0) -> N
+2(s1(N), s1(M)) -> s1(s1(+2(N, M)))
*2(N, 0) -> 0
*2(s1(N), s1(M)) -> s1(+2(N, +2(M, *2(N, M))))
gt2(0, M) -> False
gt2(NzN, 0) -> u_41(is_NzNat1(NzN))
u_41(True) -> True
is_NzNat1(0) -> False
is_NzNat1(s1(N)) -> True
gt2(s1(N), s1(M)) -> gt2(N, M)
lt2(N, M) -> gt2(M, N)
d2(0, N) -> N
d2(s1(N), s1(M)) -> d2(N, M)
quot2(N, NzM) -> u_113(is_NzNat1(NzM), N, NzM)
u_113(True, N, NzM) -> u_13(gt2(N, NzM), N, NzM)
u_13(True, N, NzM) -> s1(quot2(d2(N, NzM), NzM))
quot2(NzM, NzM) -> u_011(is_NzNat1(NzM))
u_011(True) -> s1(0)
quot2(N, NzM) -> u_213(is_NzNat1(NzM), NzM, N)
u_213(True, NzM, N) -> u_21(gt2(NzM, N))
u_21(True) -> 0
gcd2(0, N) -> 0
gcd2(NzM, NzM) -> u_022(is_NzNat1(NzM), NzM)
u_022(True, NzM) -> NzM
gcd2(NzN, NzM) -> u_314(is_NzNat1(NzN), is_NzNat1(NzM), NzN, NzM)
u_314(True, True, NzN, NzM) -> u_33(gt2(NzN, NzM), NzN, NzM)
u_33(True, NzN, NzM) -> gcd2(d2(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
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

GT2(s1(N), s1(M)) -> GT2(N, M)

The TRS R consists of the following rules:

p1(s1(N)) -> N
+2(N, 0) -> N
+2(s1(N), s1(M)) -> s1(s1(+2(N, M)))
*2(N, 0) -> 0
*2(s1(N), s1(M)) -> s1(+2(N, +2(M, *2(N, M))))
gt2(0, M) -> False
gt2(NzN, 0) -> u_41(is_NzNat1(NzN))
u_41(True) -> True
is_NzNat1(0) -> False
is_NzNat1(s1(N)) -> True
gt2(s1(N), s1(M)) -> gt2(N, M)
lt2(N, M) -> gt2(M, N)
d2(0, N) -> N
d2(s1(N), s1(M)) -> d2(N, M)
quot2(N, NzM) -> u_113(is_NzNat1(NzM), N, NzM)
u_113(True, N, NzM) -> u_13(gt2(N, NzM), N, NzM)
u_13(True, N, NzM) -> s1(quot2(d2(N, NzM), NzM))
quot2(NzM, NzM) -> u_011(is_NzNat1(NzM))
u_011(True) -> s1(0)
quot2(N, NzM) -> u_213(is_NzNat1(NzM), NzM, N)
u_213(True, NzM, N) -> u_21(gt2(NzM, N))
u_21(True) -> 0
gcd2(0, N) -> 0
gcd2(NzM, NzM) -> u_022(is_NzNat1(NzM), NzM)
u_022(True, NzM) -> NzM
gcd2(NzN, NzM) -> u_314(is_NzNat1(NzN), is_NzNat1(NzM), NzN, NzM)
u_314(True, True, NzN, NzM) -> u_33(gt2(NzN, NzM), NzN, NzM)
u_33(True, NzN, NzM) -> gcd2(d2(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.


GT2(s1(N), s1(M)) -> GT2(N, M)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( GT2(x1, x2) ) = 3x2 + 3


POL( s1(x1) ) = x1 + 3



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ 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:

p1(s1(N)) -> N
+2(N, 0) -> N
+2(s1(N), s1(M)) -> s1(s1(+2(N, M)))
*2(N, 0) -> 0
*2(s1(N), s1(M)) -> s1(+2(N, +2(M, *2(N, M))))
gt2(0, M) -> False
gt2(NzN, 0) -> u_41(is_NzNat1(NzN))
u_41(True) -> True
is_NzNat1(0) -> False
is_NzNat1(s1(N)) -> True
gt2(s1(N), s1(M)) -> gt2(N, M)
lt2(N, M) -> gt2(M, N)
d2(0, N) -> N
d2(s1(N), s1(M)) -> d2(N, M)
quot2(N, NzM) -> u_113(is_NzNat1(NzM), N, NzM)
u_113(True, N, NzM) -> u_13(gt2(N, NzM), N, NzM)
u_13(True, N, NzM) -> s1(quot2(d2(N, NzM), NzM))
quot2(NzM, NzM) -> u_011(is_NzNat1(NzM))
u_011(True) -> s1(0)
quot2(N, NzM) -> u_213(is_NzNat1(NzM), NzM, N)
u_213(True, NzM, N) -> u_21(gt2(NzM, N))
u_21(True) -> 0
gcd2(0, N) -> 0
gcd2(NzM, NzM) -> u_022(is_NzNat1(NzM), NzM)
u_022(True, NzM) -> NzM
gcd2(NzN, NzM) -> u_314(is_NzNat1(NzN), is_NzNat1(NzM), NzN, NzM)
u_314(True, True, NzN, NzM) -> u_33(gt2(NzN, NzM), NzN, NzM)
u_33(True, NzN, NzM) -> gcd2(d2(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
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

U_33(True, NzN, NzM) -> GCD2(d2(NzN, NzM), NzM)
U_314(True, True, NzN, NzM) -> U_33(gt2(NzN, NzM), NzN, NzM)
GCD2(NzN, NzM) -> U_314(is_NzNat1(NzN), is_NzNat1(NzM), NzN, NzM)

The TRS R consists of the following rules:

p1(s1(N)) -> N
+2(N, 0) -> N
+2(s1(N), s1(M)) -> s1(s1(+2(N, M)))
*2(N, 0) -> 0
*2(s1(N), s1(M)) -> s1(+2(N, +2(M, *2(N, M))))
gt2(0, M) -> False
gt2(NzN, 0) -> u_41(is_NzNat1(NzN))
u_41(True) -> True
is_NzNat1(0) -> False
is_NzNat1(s1(N)) -> True
gt2(s1(N), s1(M)) -> gt2(N, M)
lt2(N, M) -> gt2(M, N)
d2(0, N) -> N
d2(s1(N), s1(M)) -> d2(N, M)
quot2(N, NzM) -> u_113(is_NzNat1(NzM), N, NzM)
u_113(True, N, NzM) -> u_13(gt2(N, NzM), N, NzM)
u_13(True, N, NzM) -> s1(quot2(d2(N, NzM), NzM))
quot2(NzM, NzM) -> u_011(is_NzNat1(NzM))
u_011(True) -> s1(0)
quot2(N, NzM) -> u_213(is_NzNat1(NzM), NzM, N)
u_213(True, NzM, N) -> u_21(gt2(NzM, N))
u_21(True) -> 0
gcd2(0, N) -> 0
gcd2(NzM, NzM) -> u_022(is_NzNat1(NzM), NzM)
u_022(True, NzM) -> NzM
gcd2(NzN, NzM) -> u_314(is_NzNat1(NzN), is_NzNat1(NzM), NzN, NzM)
u_314(True, True, NzN, NzM) -> u_33(gt2(NzN, NzM), NzN, NzM)
u_33(True, NzN, NzM) -> gcd2(d2(NzN, NzM), NzM)

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

U_113(True, N, NzM) -> U_13(gt2(N, NzM), N, NzM)
QUOT2(N, NzM) -> U_113(is_NzNat1(NzM), N, NzM)
U_13(True, N, NzM) -> QUOT2(d2(N, NzM), NzM)

The TRS R consists of the following rules:

p1(s1(N)) -> N
+2(N, 0) -> N
+2(s1(N), s1(M)) -> s1(s1(+2(N, M)))
*2(N, 0) -> 0
*2(s1(N), s1(M)) -> s1(+2(N, +2(M, *2(N, M))))
gt2(0, M) -> False
gt2(NzN, 0) -> u_41(is_NzNat1(NzN))
u_41(True) -> True
is_NzNat1(0) -> False
is_NzNat1(s1(N)) -> True
gt2(s1(N), s1(M)) -> gt2(N, M)
lt2(N, M) -> gt2(M, N)
d2(0, N) -> N
d2(s1(N), s1(M)) -> d2(N, M)
quot2(N, NzM) -> u_113(is_NzNat1(NzM), N, NzM)
u_113(True, N, NzM) -> u_13(gt2(N, NzM), N, NzM)
u_13(True, N, NzM) -> s1(quot2(d2(N, NzM), NzM))
quot2(NzM, NzM) -> u_011(is_NzNat1(NzM))
u_011(True) -> s1(0)
quot2(N, NzM) -> u_213(is_NzNat1(NzM), NzM, N)
u_213(True, NzM, N) -> u_21(gt2(NzM, N))
u_21(True) -> 0
gcd2(0, N) -> 0
gcd2(NzM, NzM) -> u_022(is_NzNat1(NzM), NzM)
u_022(True, NzM) -> NzM
gcd2(NzN, NzM) -> u_314(is_NzNat1(NzN), is_NzNat1(NzM), NzN, NzM)
u_314(True, True, NzN, NzM) -> u_33(gt2(NzN, NzM), NzN, NzM)
u_33(True, NzN, NzM) -> gcd2(d2(NzN, NzM), NzM)

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP

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

+12(s1(N), s1(M)) -> +12(N, M)

The TRS R consists of the following rules:

p1(s1(N)) -> N
+2(N, 0) -> N
+2(s1(N), s1(M)) -> s1(s1(+2(N, M)))
*2(N, 0) -> 0
*2(s1(N), s1(M)) -> s1(+2(N, +2(M, *2(N, M))))
gt2(0, M) -> False
gt2(NzN, 0) -> u_41(is_NzNat1(NzN))
u_41(True) -> True
is_NzNat1(0) -> False
is_NzNat1(s1(N)) -> True
gt2(s1(N), s1(M)) -> gt2(N, M)
lt2(N, M) -> gt2(M, N)
d2(0, N) -> N
d2(s1(N), s1(M)) -> d2(N, M)
quot2(N, NzM) -> u_113(is_NzNat1(NzM), N, NzM)
u_113(True, N, NzM) -> u_13(gt2(N, NzM), N, NzM)
u_13(True, N, NzM) -> s1(quot2(d2(N, NzM), NzM))
quot2(NzM, NzM) -> u_011(is_NzNat1(NzM))
u_011(True) -> s1(0)
quot2(N, NzM) -> u_213(is_NzNat1(NzM), NzM, N)
u_213(True, NzM, N) -> u_21(gt2(NzM, N))
u_21(True) -> 0
gcd2(0, N) -> 0
gcd2(NzM, NzM) -> u_022(is_NzNat1(NzM), NzM)
u_022(True, NzM) -> NzM
gcd2(NzN, NzM) -> u_314(is_NzNat1(NzN), is_NzNat1(NzM), NzN, NzM)
u_314(True, True, NzN, NzM) -> u_33(gt2(NzN, NzM), NzN, NzM)
u_33(True, NzN, NzM) -> gcd2(d2(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.


+12(s1(N), s1(M)) -> +12(N, M)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( +12(x1, x2) ) = 3x2 + 3


POL( s1(x1) ) = x1 + 3



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ 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:

p1(s1(N)) -> N
+2(N, 0) -> N
+2(s1(N), s1(M)) -> s1(s1(+2(N, M)))
*2(N, 0) -> 0
*2(s1(N), s1(M)) -> s1(+2(N, +2(M, *2(N, M))))
gt2(0, M) -> False
gt2(NzN, 0) -> u_41(is_NzNat1(NzN))
u_41(True) -> True
is_NzNat1(0) -> False
is_NzNat1(s1(N)) -> True
gt2(s1(N), s1(M)) -> gt2(N, M)
lt2(N, M) -> gt2(M, N)
d2(0, N) -> N
d2(s1(N), s1(M)) -> d2(N, M)
quot2(N, NzM) -> u_113(is_NzNat1(NzM), N, NzM)
u_113(True, N, NzM) -> u_13(gt2(N, NzM), N, NzM)
u_13(True, N, NzM) -> s1(quot2(d2(N, NzM), NzM))
quot2(NzM, NzM) -> u_011(is_NzNat1(NzM))
u_011(True) -> s1(0)
quot2(N, NzM) -> u_213(is_NzNat1(NzM), NzM, N)
u_213(True, NzM, N) -> u_21(gt2(NzM, N))
u_21(True) -> 0
gcd2(0, N) -> 0
gcd2(NzM, NzM) -> u_022(is_NzNat1(NzM), NzM)
u_022(True, NzM) -> NzM
gcd2(NzN, NzM) -> u_314(is_NzNat1(NzN), is_NzNat1(NzM), NzN, NzM)
u_314(True, True, NzN, NzM) -> u_33(gt2(NzN, NzM), NzN, NzM)
u_33(True, NzN, NzM) -> gcd2(d2(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
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof

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

*12(s1(N), s1(M)) -> *12(N, M)

The TRS R consists of the following rules:

p1(s1(N)) -> N
+2(N, 0) -> N
+2(s1(N), s1(M)) -> s1(s1(+2(N, M)))
*2(N, 0) -> 0
*2(s1(N), s1(M)) -> s1(+2(N, +2(M, *2(N, M))))
gt2(0, M) -> False
gt2(NzN, 0) -> u_41(is_NzNat1(NzN))
u_41(True) -> True
is_NzNat1(0) -> False
is_NzNat1(s1(N)) -> True
gt2(s1(N), s1(M)) -> gt2(N, M)
lt2(N, M) -> gt2(M, N)
d2(0, N) -> N
d2(s1(N), s1(M)) -> d2(N, M)
quot2(N, NzM) -> u_113(is_NzNat1(NzM), N, NzM)
u_113(True, N, NzM) -> u_13(gt2(N, NzM), N, NzM)
u_13(True, N, NzM) -> s1(quot2(d2(N, NzM), NzM))
quot2(NzM, NzM) -> u_011(is_NzNat1(NzM))
u_011(True) -> s1(0)
quot2(N, NzM) -> u_213(is_NzNat1(NzM), NzM, N)
u_213(True, NzM, N) -> u_21(gt2(NzM, N))
u_21(True) -> 0
gcd2(0, N) -> 0
gcd2(NzM, NzM) -> u_022(is_NzNat1(NzM), NzM)
u_022(True, NzM) -> NzM
gcd2(NzN, NzM) -> u_314(is_NzNat1(NzN), is_NzNat1(NzM), NzN, NzM)
u_314(True, True, NzN, NzM) -> u_33(gt2(NzN, NzM), NzN, NzM)
u_33(True, NzN, NzM) -> gcd2(d2(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.


*12(s1(N), s1(M)) -> *12(N, M)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( *12(x1, x2) ) = 3x2 + 3


POL( s1(x1) ) = x1 + 3



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ 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:

p1(s1(N)) -> N
+2(N, 0) -> N
+2(s1(N), s1(M)) -> s1(s1(+2(N, M)))
*2(N, 0) -> 0
*2(s1(N), s1(M)) -> s1(+2(N, +2(M, *2(N, M))))
gt2(0, M) -> False
gt2(NzN, 0) -> u_41(is_NzNat1(NzN))
u_41(True) -> True
is_NzNat1(0) -> False
is_NzNat1(s1(N)) -> True
gt2(s1(N), s1(M)) -> gt2(N, M)
lt2(N, M) -> gt2(M, N)
d2(0, N) -> N
d2(s1(N), s1(M)) -> d2(N, M)
quot2(N, NzM) -> u_113(is_NzNat1(NzM), N, NzM)
u_113(True, N, NzM) -> u_13(gt2(N, NzM), N, NzM)
u_13(True, N, NzM) -> s1(quot2(d2(N, NzM), NzM))
quot2(NzM, NzM) -> u_011(is_NzNat1(NzM))
u_011(True) -> s1(0)
quot2(N, NzM) -> u_213(is_NzNat1(NzM), NzM, N)
u_213(True, NzM, N) -> u_21(gt2(NzM, N))
u_21(True) -> 0
gcd2(0, N) -> 0
gcd2(NzM, NzM) -> u_022(is_NzNat1(NzM), NzM)
u_022(True, NzM) -> NzM
gcd2(NzN, NzM) -> u_314(is_NzNat1(NzN), is_NzNat1(NzM), NzN, NzM)
u_314(True, True, NzN, NzM) -> u_33(gt2(NzN, NzM), NzN, NzM)
u_33(True, NzN, NzM) -> gcd2(d2(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.