Termination w.r.t. Q of the following Term Rewriting System could be proven:

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

plus(0, Y) → Y
plus(s(X), Y) → s(plus(X, Y))
min(X, 0) → X
min(s(X), s(Y)) → min(X, Y)
min(min(X, Y), Z) → min(X, plus(Y, Z))
quot(0, s(Y)) → 0
quot(s(X), s(Y)) → s(quot(min(X, Y), s(Y)))

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

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

plus(0, Y) → Y
plus(s(X), Y) → s(plus(X, Y))
min(X, 0) → X
min(s(X), s(Y)) → min(X, Y)
min(min(X, Y), Z) → min(X, plus(Y, Z))
quot(0, s(Y)) → 0
quot(s(X), s(Y)) → s(quot(min(X, Y), s(Y)))

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:

PLUS(s(X), Y) → PLUS(X, Y)
MIN(min(X, Y), Z) → PLUS(Y, Z)
MIN(s(X), s(Y)) → MIN(X, Y)
QUOT(s(X), s(Y)) → MIN(X, Y)
MIN(min(X, Y), Z) → MIN(X, plus(Y, Z))
QUOT(s(X), s(Y)) → QUOT(min(X, Y), s(Y))

The TRS R consists of the following rules:

plus(0, Y) → Y
plus(s(X), Y) → s(plus(X, Y))
min(X, 0) → X
min(s(X), s(Y)) → min(X, Y)
min(min(X, Y), Z) → min(X, plus(Y, Z))
quot(0, s(Y)) → 0
quot(s(X), s(Y)) → s(quot(min(X, Y), s(Y)))

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:

PLUS(s(X), Y) → PLUS(X, Y)
MIN(min(X, Y), Z) → PLUS(Y, Z)
MIN(s(X), s(Y)) → MIN(X, Y)
QUOT(s(X), s(Y)) → MIN(X, Y)
MIN(min(X, Y), Z) → MIN(X, plus(Y, Z))
QUOT(s(X), s(Y)) → QUOT(min(X, Y), s(Y))

The TRS R consists of the following rules:

plus(0, Y) → Y
plus(s(X), Y) → s(plus(X, Y))
min(X, 0) → X
min(s(X), s(Y)) → min(X, Y)
min(min(X, Y), Z) → min(X, plus(Y, Z))
quot(0, s(Y)) → 0
quot(s(X), s(Y)) → s(quot(min(X, Y), s(Y)))

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:

PLUS(s(X), Y) → PLUS(X, Y)
MIN(min(X, Y), Z) → PLUS(Y, Z)
MIN(s(X), s(Y)) → MIN(X, Y)
QUOT(s(X), s(Y)) → MIN(X, Y)
MIN(min(X, Y), Z) → MIN(X, plus(Y, Z))
QUOT(s(X), s(Y)) → QUOT(min(X, Y), s(Y))

The TRS R consists of the following rules:

plus(0, Y) → Y
plus(s(X), Y) → s(plus(X, Y))
min(X, 0) → X
min(s(X), s(Y)) → min(X, Y)
min(min(X, Y), Z) → min(X, plus(Y, Z))
quot(0, s(Y)) → 0
quot(s(X), s(Y)) → s(quot(min(X, Y), s(Y)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 3 SCCs with 2 less nodes.

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

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

PLUS(s(X), Y) → PLUS(X, Y)

The TRS R consists of the following rules:

plus(0, Y) → Y
plus(s(X), Y) → s(plus(X, Y))
min(X, 0) → X
min(s(X), s(Y)) → min(X, Y)
min(min(X, Y), Z) → min(X, plus(Y, Z))
quot(0, s(Y)) → 0
quot(s(X), s(Y)) → s(quot(min(X, Y), s(Y)))

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.


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

Recursive Path Order [2].
Precedence:
trivial

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ EdgeDeletionProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ PisEmptyProof
              ↳ QDP
              ↳ QDP

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

plus(0, Y) → Y
plus(s(X), Y) → s(plus(X, Y))
min(X, 0) → X
min(s(X), s(Y)) → min(X, Y)
min(min(X, Y), Z) → min(X, plus(Y, Z))
quot(0, s(Y)) → 0
quot(s(X), s(Y)) → s(quot(min(X, Y), s(Y)))

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

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

MIN(s(X), s(Y)) → MIN(X, Y)
MIN(min(X, Y), Z) → MIN(X, plus(Y, Z))

The TRS R consists of the following rules:

plus(0, Y) → Y
plus(s(X), Y) → s(plus(X, Y))
min(X, 0) → X
min(s(X), s(Y)) → min(X, Y)
min(min(X, Y), Z) → min(X, plus(Y, Z))
quot(0, s(Y)) → 0
quot(s(X), s(Y)) → s(quot(min(X, Y), s(Y)))

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.


MIN(min(X, Y), Z) → MIN(X, plus(Y, Z))
The remaining pairs can at least be oriented weakly.

MIN(s(X), s(Y)) → MIN(X, Y)
Used ordering: Combined order from the following AFS and order.
MIN(x1, x2)  =  x1
s(x1)  =  x1
min(x1, x2)  =  min(x1)

Recursive Path Order [2].
Precedence:
trivial

The following usable rules [14] were oriented: none



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

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

MIN(s(X), s(Y)) → MIN(X, Y)

The TRS R consists of the following rules:

plus(0, Y) → Y
plus(s(X), Y) → s(plus(X, Y))
min(X, 0) → X
min(s(X), s(Y)) → min(X, Y)
min(min(X, Y), Z) → min(X, plus(Y, Z))
quot(0, s(Y)) → 0
quot(s(X), s(Y)) → s(quot(min(X, Y), s(Y)))

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.


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

Recursive Path Order [2].
Precedence:
trivial

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ EdgeDeletionProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ PisEmptyProof
              ↳ QDP

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

plus(0, Y) → Y
plus(s(X), Y) → s(plus(X, Y))
min(X, 0) → X
min(s(X), s(Y)) → min(X, Y)
min(min(X, Y), Z) → min(X, plus(Y, Z))
quot(0, s(Y)) → 0
quot(s(X), s(Y)) → s(quot(min(X, Y), s(Y)))

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
                ↳ QDPOrderProof

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

QUOT(s(X), s(Y)) → QUOT(min(X, Y), s(Y))

The TRS R consists of the following rules:

plus(0, Y) → Y
plus(s(X), Y) → s(plus(X, Y))
min(X, 0) → X
min(s(X), s(Y)) → min(X, Y)
min(min(X, Y), Z) → min(X, plus(Y, Z))
quot(0, s(Y)) → 0
quot(s(X), s(Y)) → s(quot(min(X, Y), s(Y)))

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.


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

Recursive Path Order [2].
Precedence:
trivial

The following usable rules [14] were oriented:

min(X, 0) → X
min(min(X, Y), Z) → min(X, plus(Y, Z))
min(s(X), s(Y)) → min(X, Y)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ EdgeDeletionProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ PisEmptyProof

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

plus(0, Y) → Y
plus(s(X), Y) → s(plus(X, Y))
min(X, 0) → X
min(s(X), s(Y)) → min(X, Y)
min(min(X, Y), Z) → min(X, plus(Y, Z))
quot(0, s(Y)) → 0
quot(s(X), s(Y)) → s(quot(min(X, Y), s(Y)))

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.