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:

g1(A) -> A
g1(B) -> A
g1(B) -> B
g1(C) -> A
g1(C) -> B
g1(C) -> C
foldB2(t, 0) -> t
foldB2(t, s1(n)) -> f2(foldB2(t, n), B)
foldC2(t, 0) -> t
foldC2(t, s1(n)) -> f2(foldC2(t, n), C)
f2(t, x) -> f'2(t, g1(x))
f'2(triple3(a, b, c), C) -> triple3(a, b, s1(c))
f'2(triple3(a, b, c), B) -> f2(triple3(a, b, c), A)
f'2(triple3(a, b, c), A) -> f''1(foldB2(triple3(s1(a), 0, c), b))
f''1(triple3(a, b, c)) -> foldC2(triple3(a, b, 0), c)

Q is empty.


QTRS
  ↳ DependencyPairsProof

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

g1(A) -> A
g1(B) -> A
g1(B) -> B
g1(C) -> A
g1(C) -> B
g1(C) -> C
foldB2(t, 0) -> t
foldB2(t, s1(n)) -> f2(foldB2(t, n), B)
foldC2(t, 0) -> t
foldC2(t, s1(n)) -> f2(foldC2(t, n), C)
f2(t, x) -> f'2(t, g1(x))
f'2(triple3(a, b, c), C) -> triple3(a, b, s1(c))
f'2(triple3(a, b, c), B) -> f2(triple3(a, b, c), A)
f'2(triple3(a, b, c), A) -> f''1(foldB2(triple3(s1(a), 0, c), b))
f''1(triple3(a, b, c)) -> foldC2(triple3(a, b, 0), c)

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:

F'2(triple3(a, b, c), A) -> FOLDB2(triple3(s1(a), 0, c), b)
F2(t, x) -> F'2(t, g1(x))
FOLDC2(t, s1(n)) -> F2(foldC2(t, n), C)
F2(t, x) -> G1(x)
FOLDB2(t, s1(n)) -> FOLDB2(t, n)
F''1(triple3(a, b, c)) -> FOLDC2(triple3(a, b, 0), c)
FOLDC2(t, s1(n)) -> FOLDC2(t, n)
F'2(triple3(a, b, c), A) -> F''1(foldB2(triple3(s1(a), 0, c), b))
F'2(triple3(a, b, c), B) -> F2(triple3(a, b, c), A)
FOLDB2(t, s1(n)) -> F2(foldB2(t, n), B)

The TRS R consists of the following rules:

g1(A) -> A
g1(B) -> A
g1(B) -> B
g1(C) -> A
g1(C) -> B
g1(C) -> C
foldB2(t, 0) -> t
foldB2(t, s1(n)) -> f2(foldB2(t, n), B)
foldC2(t, 0) -> t
foldC2(t, s1(n)) -> f2(foldC2(t, n), C)
f2(t, x) -> f'2(t, g1(x))
f'2(triple3(a, b, c), C) -> triple3(a, b, s1(c))
f'2(triple3(a, b, c), B) -> f2(triple3(a, b, c), A)
f'2(triple3(a, b, c), A) -> f''1(foldB2(triple3(s1(a), 0, c), b))
f''1(triple3(a, b, c)) -> foldC2(triple3(a, b, 0), c)

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:

F'2(triple3(a, b, c), A) -> FOLDB2(triple3(s1(a), 0, c), b)
F2(t, x) -> F'2(t, g1(x))
FOLDC2(t, s1(n)) -> F2(foldC2(t, n), C)
F2(t, x) -> G1(x)
FOLDB2(t, s1(n)) -> FOLDB2(t, n)
F''1(triple3(a, b, c)) -> FOLDC2(triple3(a, b, 0), c)
FOLDC2(t, s1(n)) -> FOLDC2(t, n)
F'2(triple3(a, b, c), A) -> F''1(foldB2(triple3(s1(a), 0, c), b))
F'2(triple3(a, b, c), B) -> F2(triple3(a, b, c), A)
FOLDB2(t, s1(n)) -> F2(foldB2(t, n), B)

The TRS R consists of the following rules:

g1(A) -> A
g1(B) -> A
g1(B) -> B
g1(C) -> A
g1(C) -> B
g1(C) -> C
foldB2(t, 0) -> t
foldB2(t, s1(n)) -> f2(foldB2(t, n), B)
foldC2(t, 0) -> t
foldC2(t, s1(n)) -> f2(foldC2(t, n), C)
f2(t, x) -> f'2(t, g1(x))
f'2(triple3(a, b, c), C) -> triple3(a, b, s1(c))
f'2(triple3(a, b, c), B) -> f2(triple3(a, b, c), A)
f'2(triple3(a, b, c), A) -> f''1(foldB2(triple3(s1(a), 0, c), b))
f''1(triple3(a, b, c)) -> foldC2(triple3(a, b, 0), c)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 1 SCC with 1 less node.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
QDP
          ↳ QDPOrderProof

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

F'2(triple3(a, b, c), A) -> FOLDB2(triple3(s1(a), 0, c), b)
F2(t, x) -> F'2(t, g1(x))
FOLDC2(t, s1(n)) -> F2(foldC2(t, n), C)
FOLDB2(t, s1(n)) -> FOLDB2(t, n)
F''1(triple3(a, b, c)) -> FOLDC2(triple3(a, b, 0), c)
FOLDC2(t, s1(n)) -> FOLDC2(t, n)
F'2(triple3(a, b, c), A) -> F''1(foldB2(triple3(s1(a), 0, c), b))
F'2(triple3(a, b, c), B) -> F2(triple3(a, b, c), A)
FOLDB2(t, s1(n)) -> F2(foldB2(t, n), B)

The TRS R consists of the following rules:

g1(A) -> A
g1(B) -> A
g1(B) -> B
g1(C) -> A
g1(C) -> B
g1(C) -> C
foldB2(t, 0) -> t
foldB2(t, s1(n)) -> f2(foldB2(t, n), B)
foldC2(t, 0) -> t
foldC2(t, s1(n)) -> f2(foldC2(t, n), C)
f2(t, x) -> f'2(t, g1(x))
f'2(triple3(a, b, c), C) -> triple3(a, b, s1(c))
f'2(triple3(a, b, c), B) -> f2(triple3(a, b, c), A)
f'2(triple3(a, b, c), A) -> f''1(foldB2(triple3(s1(a), 0, c), b))
f''1(triple3(a, b, c)) -> foldC2(triple3(a, b, 0), c)

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.


FOLDB2(t, s1(n)) -> FOLDB2(t, n)
FOLDC2(t, s1(n)) -> FOLDC2(t, n)
F'2(triple3(a, b, c), B) -> F2(triple3(a, b, c), A)
The remaining pairs can at least be oriented weakly.

F'2(triple3(a, b, c), A) -> FOLDB2(triple3(s1(a), 0, c), b)
F2(t, x) -> F'2(t, g1(x))
FOLDC2(t, s1(n)) -> F2(foldC2(t, n), C)
F''1(triple3(a, b, c)) -> FOLDC2(triple3(a, b, 0), c)
F'2(triple3(a, b, c), A) -> F''1(foldB2(triple3(s1(a), 0, c), b))
FOLDB2(t, s1(n)) -> F2(foldB2(t, n), B)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( triple3(x1, ..., x3) ) = x2 + x3 + 2


POL( foldC2(x1, x2) ) = x1 + x2


POL( F''1(x1) ) = 2x1


POL( FOLDC2(x1, x2) ) = 2x1 + 2x2


POL( 0 ) = max{0, -2}


POL( F2(x1, x2) ) = 2x1 + 2x2


POL( f''1(x1) ) = x1


POL( F'2(x1, x2) ) = 2x1 + 2x2


POL( g1(x1) ) = x1


POL( C ) = 2


POL( f'2(x1, x2) ) = x1 + x2


POL( FOLDB2(x1, x2) ) = 2x1 + 2x2


POL( foldB2(x1, x2) ) = x1 + x2


POL( A ) = max{0, -2}


POL( f2(x1, x2) ) = x1 + x2


POL( s1(x1) ) = x1 + 2


POL( B ) = 2



The following usable rules [14] were oriented:

g1(C) -> A
foldC2(t, 0) -> t
g1(C) -> B
f'2(triple3(a, b, c), C) -> triple3(a, b, s1(c))
g1(A) -> A
foldB2(t, s1(n)) -> f2(foldB2(t, n), B)
g1(B) -> A
g1(C) -> C
g1(B) -> B
foldB2(t, 0) -> t
f''1(triple3(a, b, c)) -> foldC2(triple3(a, b, 0), c)
foldC2(t, s1(n)) -> f2(foldC2(t, n), C)
f'2(triple3(a, b, c), A) -> f''1(foldB2(triple3(s1(a), 0, c), b))
f'2(triple3(a, b, c), B) -> f2(triple3(a, b, c), A)
f2(t, x) -> f'2(t, g1(x))



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

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

F'2(triple3(a, b, c), A) -> FOLDB2(triple3(s1(a), 0, c), b)
F2(t, x) -> F'2(t, g1(x))
FOLDC2(t, s1(n)) -> F2(foldC2(t, n), C)
F''1(triple3(a, b, c)) -> FOLDC2(triple3(a, b, 0), c)
F'2(triple3(a, b, c), A) -> F''1(foldB2(triple3(s1(a), 0, c), b))
FOLDB2(t, s1(n)) -> F2(foldB2(t, n), B)

The TRS R consists of the following rules:

g1(A) -> A
g1(B) -> A
g1(B) -> B
g1(C) -> A
g1(C) -> B
g1(C) -> C
foldB2(t, 0) -> t
foldB2(t, s1(n)) -> f2(foldB2(t, n), B)
foldC2(t, 0) -> t
foldC2(t, s1(n)) -> f2(foldC2(t, n), C)
f2(t, x) -> f'2(t, g1(x))
f'2(triple3(a, b, c), C) -> triple3(a, b, s1(c))
f'2(triple3(a, b, c), B) -> f2(triple3(a, b, c), A)
f'2(triple3(a, b, c), A) -> f''1(foldB2(triple3(s1(a), 0, c), b))
f''1(triple3(a, b, c)) -> foldC2(triple3(a, b, 0), c)

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.


F'2(triple3(a, b, c), A) -> FOLDB2(triple3(s1(a), 0, c), b)
FOLDB2(t, s1(n)) -> F2(foldB2(t, n), B)
The remaining pairs can at least be oriented weakly.

F2(t, x) -> F'2(t, g1(x))
FOLDC2(t, s1(n)) -> F2(foldC2(t, n), C)
F''1(triple3(a, b, c)) -> FOLDC2(triple3(a, b, 0), c)
F'2(triple3(a, b, c), A) -> F''1(foldB2(triple3(s1(a), 0, c), b))
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( triple3(x1, ..., x3) ) = 2x2 + 1


POL( foldC2(x1, x2) ) = x1


POL( F''1(x1) ) = 2x1


POL( FOLDC2(x1, x2) ) = 2x1


POL( 0 ) = max{0, -1}


POL( F2(x1, x2) ) = 2x1


POL( f''1(x1) ) = x1


POL( F'2(x1, x2) ) = 2x1


POL( g1(x1) ) = max{0, -2}


POL( C ) = 0


POL( f'2(x1, x2) ) = x1 + x2


POL( FOLDB2(x1, x2) ) = max{0, 2x1 + x2 - 1}


POL( foldB2(x1, x2) ) = x1


POL( A ) = max{0, -2}


POL( f2(x1, x2) ) = x1


POL( s1(x1) ) = 2x1 + 2


POL( B ) = 0



The following usable rules [14] were oriented:

g1(B) -> A
g1(C) -> C
g1(C) -> A
foldC2(t, 0) -> t
g1(B) -> B
g1(C) -> B
foldB2(t, 0) -> t
f'2(triple3(a, b, c), C) -> triple3(a, b, s1(c))
g1(A) -> A
f''1(triple3(a, b, c)) -> foldC2(triple3(a, b, 0), c)
foldC2(t, s1(n)) -> f2(foldC2(t, n), C)
f'2(triple3(a, b, c), A) -> f''1(foldB2(triple3(s1(a), 0, c), b))
f'2(triple3(a, b, c), B) -> f2(triple3(a, b, c), A)
f2(t, x) -> f'2(t, g1(x))
foldB2(t, s1(n)) -> f2(foldB2(t, n), B)



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

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

F2(t, x) -> F'2(t, g1(x))
FOLDC2(t, s1(n)) -> F2(foldC2(t, n), C)
F''1(triple3(a, b, c)) -> FOLDC2(triple3(a, b, 0), c)
F'2(triple3(a, b, c), A) -> F''1(foldB2(triple3(s1(a), 0, c), b))

The TRS R consists of the following rules:

g1(A) -> A
g1(B) -> A
g1(B) -> B
g1(C) -> A
g1(C) -> B
g1(C) -> C
foldB2(t, 0) -> t
foldB2(t, s1(n)) -> f2(foldB2(t, n), B)
foldC2(t, 0) -> t
foldC2(t, s1(n)) -> f2(foldC2(t, n), C)
f2(t, x) -> f'2(t, g1(x))
f'2(triple3(a, b, c), C) -> triple3(a, b, s1(c))
f'2(triple3(a, b, c), B) -> f2(triple3(a, b, c), A)
f'2(triple3(a, b, c), A) -> f''1(foldB2(triple3(s1(a), 0, c), b))
f''1(triple3(a, b, c)) -> foldC2(triple3(a, b, 0), c)

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.


FOLDC2(t, s1(n)) -> F2(foldC2(t, n), C)
F'2(triple3(a, b, c), A) -> F''1(foldB2(triple3(s1(a), 0, c), b))
The remaining pairs can at least be oriented weakly.

F2(t, x) -> F'2(t, g1(x))
F''1(triple3(a, b, c)) -> FOLDC2(triple3(a, b, 0), c)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( foldC2(x1, x2) ) = x1 + x2


POL( triple3(x1, ..., x3) ) = 2x2 + x3 + 2


POL( FOLDC2(x1, x2) ) = max{0, 2x1 + 2x2 - 1}


POL( F''1(x1) ) = max{0, 2x1 - 1}


POL( 0 ) = 0


POL( F2(x1, x2) ) = 2x1 + 2


POL( f''1(x1) ) = x1 + 2


POL( F'2(x1, x2) ) = 2x1 + 2


POL( g1(x1) ) = x1


POL( C ) = 2


POL( f'2(x1, x2) ) = x1 + x2


POL( foldB2(x1, x2) ) = x1 + 2x2


POL( A ) = 2


POL( f2(x1, x2) ) = x1 + x2


POL( s1(x1) ) = x1 + 2


POL( B ) = 2



The following usable rules [14] were oriented:

g1(B) -> A
g1(C) -> C
g1(C) -> A
foldC2(t, 0) -> t
g1(B) -> B
g1(C) -> B
foldB2(t, 0) -> t
f'2(triple3(a, b, c), C) -> triple3(a, b, s1(c))
g1(A) -> A
f''1(triple3(a, b, c)) -> foldC2(triple3(a, b, 0), c)
foldC2(t, s1(n)) -> f2(foldC2(t, n), C)
f'2(triple3(a, b, c), A) -> f''1(foldB2(triple3(s1(a), 0, c), b))
f'2(triple3(a, b, c), B) -> f2(triple3(a, b, c), A)
f2(t, x) -> f'2(t, g1(x))
foldB2(t, s1(n)) -> f2(foldB2(t, n), B)



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

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

F2(t, x) -> F'2(t, g1(x))
F''1(triple3(a, b, c)) -> FOLDC2(triple3(a, b, 0), c)

The TRS R consists of the following rules:

g1(A) -> A
g1(B) -> A
g1(B) -> B
g1(C) -> A
g1(C) -> B
g1(C) -> C
foldB2(t, 0) -> t
foldB2(t, s1(n)) -> f2(foldB2(t, n), B)
foldC2(t, 0) -> t
foldC2(t, s1(n)) -> f2(foldC2(t, n), C)
f2(t, x) -> f'2(t, g1(x))
f'2(triple3(a, b, c), C) -> triple3(a, b, s1(c))
f'2(triple3(a, b, c), B) -> f2(triple3(a, b, c), A)
f'2(triple3(a, b, c), A) -> f''1(foldB2(triple3(s1(a), 0, c), b))
f''1(triple3(a, b, c)) -> foldC2(triple3(a, b, 0), c)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 0 SCCs with 2 less nodes.