Termination w.r.t. Q proof of /tmp/tmpMGXHHz/t005.xml

Termination w.r.t. Q of the given QTRS could be proven:

0 QTRS

↳1 DependencyPairsProof (⇔)

↳2 QDP

↳3 DependencyGraphProof (⇔)

↳4 AND

    ↳5 QDP

        ↳6 QDPOrderProof (⇔)

        ↳7 QDP

        ↳8 QDPOrderProof (⇔)

        ↳9 QDP

        ↳10 PisEmptyProof (⇔)

        ↳11 TRUE

    ↳12 QDP

        ↳13 QDPOrderProof (⇔)

        ↳14 QDP

        ↳15 QDPOrderProof (⇔)

        ↳16 QDP

        ↳17 PisEmptyProof (⇔)

        ↳18 TRUE

(0) Obligation:

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

c(0, x) → x
c(x, c(y, z)) → c(+(x, y), z)
+(0, 0) → 0
+(0, 1) → 1
+(0, 2) → 2
+(0, 3) → 3
+(0, 4) → 4
+(0, 5) → 5
+(0, 6) → 6
+(0, 7) → 7
+(0, 8) → 8
+(0, 9) → 9
+(1, 0) → 1
+(1, 1) → 2
+(1, 2) → 3
+(1, 3) → 4
+(1, 4) → 5
+(1, 5) → 6
+(1, 6) → 7
+(1, 7) → 8
+(1, 8) → 9
+(1, 9) → c(1, 0)
+(2, 0) → 2
+(2, 1) → 3
+(2, 2) → 4
+(2, 3) → 5
+(2, 4) → 6
+(2, 5) → 7
+(2, 6) → 8
+(2, 7) → 9
+(2, 8) → c(1, 0)
+(2, 9) → c(1, 1)
+(3, 0) → 3
+(3, 1) → 4
+(3, 2) → 5
+(3, 3) → 6
+(3, 4) → 7
+(3, 5) → 8
+(3, 6) → 9
+(3, 7) → c(1, 0)
+(3, 8) → c(1, 1)
+(3, 9) → c(1, 2)
+(4, 0) → 4
+(4, 1) → 5
+(4, 2) → 6
+(4, 3) → 7
+(4, 4) → 8
+(4, 5) → 9
+(4, 6) → c(1, 0)
+(4, 7) → c(1, 1)
+(4, 8) → c(1, 2)
+(4, 9) → c(1, 3)
+(5, 0) → 5
+(5, 1) → 6
+(5, 2) → 7
+(5, 3) → 8
+(5, 4) → 9
+(5, 5) → c(1, 0)
+(5, 6) → c(1, 1)
+(5, 7) → c(1, 2)
+(5, 8) → c(1, 3)
+(5, 9) → c(1, 4)
+(6, 0) → 6
+(6, 1) → 7
+(6, 2) → 8
+(6, 3) → 9
+(6, 4) → c(1, 0)
+(6, 5) → c(1, 1)
+(6, 6) → c(1, 2)
+(6, 7) → c(1, 3)
+(6, 8) → c(1, 4)
+(6, 9) → c(1, 5)
+(7, 0) → 7
+(7, 1) → 8
+(7, 2) → 9
+(7, 3) → c(1, 0)
+(7, 4) → c(1, 1)
+(7, 5) → c(1, 2)
+(7, 6) → c(1, 3)
+(7, 7) → c(1, 4)
+(7, 8) → c(1, 5)
+(7, 9) → c(1, 6)
+(8, 0) → 8
+(8, 1) → 9
+(8, 2) → c(1, 0)
+(8, 3) → c(1, 1)
+(8, 4) → c(1, 2)
+(8, 5) → c(1, 3)
+(8, 6) → c(1, 4)
+(8, 7) → c(1, 5)
+(8, 8) → c(1, 6)
+(8, 9) → c(1, 7)
+(9, 0) → 9
+(9, 1) → c(1, 0)
+(9, 2) → c(1, 1)
+(9, 3) → c(1, 2)
+(9, 4) → c(1, 3)
+(9, 5) → c(1, 4)
+(9, 6) → c(1, 5)
+(9, 7) → c(1, 6)
+(9, 8) → c(1, 7)
+(9, 9) → c(1, 8)
+(x, c(y, z)) → c(y, +(x, z))
+(c(x, y), z) → c(x, +(y, z))
*(0, 0) → 0
*(0, 1) → 0
*(0, 2) → 0
*(0, 3) → 0
*(0, 4) → 0
*(0, 5) → 0
*(0, 6) → 0
*(0, 7) → 0
*(0, 8) → 0
*(0, 9) → 0
*(1, 0) → 0
*(1, 1) → 1
*(1, 2) → 2
*(1, 3) → 3
*(1, 4) → 4
*(1, 5) → 5
*(1, 6) → 6
*(1, 7) → 7
*(1, 8) → 8
*(1, 9) → 9
*(2, 0) → 0
*(2, 1) → 2
*(2, 2) → 4
*(2, 3) → 6
*(2, 4) → 8
*(2, 5) → c(1, 0)
*(2, 6) → c(1, 2)
*(2, 7) → c(1, 4)
*(2, 8) → c(1, 6)
*(2, 9) → c(1, 8)
*(3, 0) → 0
*(3, 1) → 3
*(3, 2) → 6
*(3, 3) → 9
*(3, 4) → c(1, 2)
*(3, 5) → c(1, 5)
*(3, 6) → c(1, 8)
*(3, 7) → c(2, 1)
*(3, 8) → c(2, 4)
*(3, 9) → c(2, 7)
*(4, 0) → 0
*(4, 1) → 4
*(4, 2) → 8
*(4, 3) → c(1, 2)
*(4, 4) → c(1, 6)
*(4, 5) → c(2, 0)
*(4, 6) → c(2, 4)
*(4, 7) → c(2, 8)
*(4, 8) → c(3, 2)
*(4, 9) → c(3, 6)
*(5, 0) → 0
*(5, 1) → 5
*(5, 2) → c(1, 0)
*(5, 3) → c(1, 5)
*(5, 4) → c(2, 0)
*(5, 5) → c(2, 5)
*(5, 6) → c(3, 0)
*(5, 7) → c(3, 5)
*(5, 8) → c(4, 0)
*(5, 9) → c(4, 5)
*(6, 0) → 0
*(6, 1) → 6
*(6, 2) → c(1, 2)
*(6, 3) → c(1, 8)
*(6, 4) → c(2, 4)
*(6, 5) → c(3, 0)
*(6, 6) → c(3, 6)
*(6, 7) → c(4, 2)
*(6, 8) → c(4, 8)
*(6, 9) → c(5, 4)
*(7, 0) → 0
*(7, 1) → 7
*(7, 2) → c(1, 4)
*(7, 3) → c(2, 1)
*(7, 4) → c(2, 8)
*(7, 5) → c(3, 5)
*(7, 6) → c(4, 2)
*(7, 7) → c(4, 9)
*(7, 8) → c(5, 6)
*(7, 9) → c(6, 3)
*(8, 0) → 0
*(8, 1) → 8
*(8, 2) → c(1, 8)
*(8, 3) → c(2, 4)
*(8, 4) → c(3, 2)
*(8, 5) → c(4, 0)
*(8, 6) → c(4, 8)
*(8, 7) → c(5, 6)
*(8, 8) → c(6, 4)
*(8, 9) → c(7, 2)
*(9, 0) → 0
*(9, 1) → 9
*(9, 2) → c(1, 8)
*(9, 3) → c(2, 7)
*(9, 4) → c(3, 6)
*(9, 5) → c(4, 5)
*(9, 6) → c(5, 4)
*(9, 7) → c(6, 3)
*(9, 8) → c(7, 2)
*(9, 9) → c(8, 1)
*(x, c(y, z)) → c(*(x, y), *(x, z))
*(c(x, y), z) → c(*(x, z), *(y, z))

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:

C(x, c(y, z)) → C(+(x, y), z)
C(x, c(y, z)) → +¹(x, y)
+¹(1, 9) → C(1, 0)
+¹(2, 8) → C(1, 0)
+¹(2, 9) → C(1, 1)
+¹(3, 7) → C(1, 0)
+¹(3, 8) → C(1, 1)
+¹(3, 9) → C(1, 2)
+¹(4, 6) → C(1, 0)
+¹(4, 7) → C(1, 1)
+¹(4, 8) → C(1, 2)
+¹(4, 9) → C(1, 3)
+¹(5, 5) → C(1, 0)
+¹(5, 6) → C(1, 1)
+¹(5, 7) → C(1, 2)
+¹(5, 8) → C(1, 3)
+¹(5, 9) → C(1, 4)
+¹(6, 4) → C(1, 0)
+¹(6, 5) → C(1, 1)
+¹(6, 6) → C(1, 2)
+¹(6, 7) → C(1, 3)
+¹(6, 8) → C(1, 4)
+¹(6, 9) → C(1, 5)
+¹(7, 3) → C(1, 0)
+¹(7, 4) → C(1, 1)
+¹(7, 5) → C(1, 2)
+¹(7, 6) → C(1, 3)
+¹(7, 7) → C(1, 4)
+¹(7, 8) → C(1, 5)
+¹(7, 9) → C(1, 6)
+¹(8, 2) → C(1, 0)
+¹(8, 3) → C(1, 1)
+¹(8, 4) → C(1, 2)
+¹(8, 5) → C(1, 3)
+¹(8, 6) → C(1, 4)
+¹(8, 7) → C(1, 5)
+¹(8, 8) → C(1, 6)
+¹(8, 9) → C(1, 7)
+¹(9, 1) → C(1, 0)
+¹(9, 2) → C(1, 1)
+¹(9, 3) → C(1, 2)
+¹(9, 4) → C(1, 3)
+¹(9, 5) → C(1, 4)
+¹(9, 6) → C(1, 5)
+¹(9, 7) → C(1, 6)
+¹(9, 8) → C(1, 7)
+¹(9, 9) → C(1, 8)
+¹(x, c(y, z)) → C(y, +(x, z))
+¹(x, c(y, z)) → +¹(x, z)
+¹(c(x, y), z) → C(x, +(y, z))
+¹(c(x, y), z) → +¹(y, z)
*¹(2, 5) → C(1, 0)
*¹(2, 6) → C(1, 2)
*¹(2, 7) → C(1, 4)
*¹(2, 8) → C(1, 6)
*¹(2, 9) → C(1, 8)
*¹(3, 4) → C(1, 2)
*¹(3, 5) → C(1, 5)
*¹(3, 6) → C(1, 8)
*¹(3, 7) → C(2, 1)
*¹(3, 8) → C(2, 4)
*¹(3, 9) → C(2, 7)
*¹(4, 3) → C(1, 2)
*¹(4, 4) → C(1, 6)
*¹(4, 5) → C(2, 0)
*¹(4, 6) → C(2, 4)
*¹(4, 7) → C(2, 8)
*¹(4, 8) → C(3, 2)
*¹(4, 9) → C(3, 6)
*¹(5, 2) → C(1, 0)
*¹(5, 3) → C(1, 5)
*¹(5, 4) → C(2, 0)
*¹(5, 5) → C(2, 5)
*¹(5, 6) → C(3, 0)
*¹(5, 7) → C(3, 5)
*¹(5, 8) → C(4, 0)
*¹(5, 9) → C(4, 5)
*¹(6, 2) → C(1, 2)
*¹(6, 3) → C(1, 8)
*¹(6, 4) → C(2, 4)
*¹(6, 5) → C(3, 0)
*¹(6, 6) → C(3, 6)
*¹(6, 7) → C(4, 2)
*¹(6, 8) → C(4, 8)
*¹(6, 9) → C(5, 4)
*¹(7, 2) → C(1, 4)
*¹(7, 3) → C(2, 1)
*¹(7, 4) → C(2, 8)
*¹(7, 5) → C(3, 5)
*¹(7, 6) → C(4, 2)
*¹(7, 7) → C(4, 9)
*¹(7, 8) → C(5, 6)
*¹(7, 9) → C(6, 3)
*¹(8, 2) → C(1, 8)
*¹(8, 3) → C(2, 4)
*¹(8, 4) → C(3, 2)
*¹(8, 5) → C(4, 0)
*¹(8, 6) → C(4, 8)
*¹(8, 7) → C(5, 6)
*¹(8, 8) → C(6, 4)
*¹(8, 9) → C(7, 2)
*¹(9, 2) → C(1, 8)
*¹(9, 3) → C(2, 7)
*¹(9, 4) → C(3, 6)
*¹(9, 5) → C(4, 5)
*¹(9, 6) → C(5, 4)
*¹(9, 7) → C(6, 3)
*¹(9, 8) → C(7, 2)
*¹(9, 9) → C(8, 1)
*¹(x, c(y, z)) → C(*(x, y), *(x, z))
*¹(x, c(y, z)) → *¹(x, y)
*¹(x, c(y, z)) → *¹(x, z)
*¹(c(x, y), z) → C(*(x, z), *(y, z))
*¹(c(x, y), z) → *¹(x, z)
*¹(c(x, y), z) → *¹(y, z)

The TRS R consists of the following rules:

c(0, x) → x
c(x, c(y, z)) → c(+(x, y), z)
+(0, 0) → 0
+(0, 1) → 1
+(0, 2) → 2
+(0, 3) → 3
+(0, 4) → 4
+(0, 5) → 5
+(0, 6) → 6
+(0, 7) → 7
+(0, 8) → 8
+(0, 9) → 9
+(1, 0) → 1
+(1, 1) → 2
+(1, 2) → 3
+(1, 3) → 4
+(1, 4) → 5
+(1, 5) → 6
+(1, 6) → 7
+(1, 7) → 8
+(1, 8) → 9
+(1, 9) → c(1, 0)
+(2, 0) → 2
+(2, 1) → 3
+(2, 2) → 4
+(2, 3) → 5
+(2, 4) → 6
+(2, 5) → 7
+(2, 6) → 8
+(2, 7) → 9
+(2, 8) → c(1, 0)
+(2, 9) → c(1, 1)
+(3, 0) → 3
+(3, 1) → 4
+(3, 2) → 5
+(3, 3) → 6
+(3, 4) → 7
+(3, 5) → 8
+(3, 6) → 9
+(3, 7) → c(1, 0)
+(3, 8) → c(1, 1)
+(3, 9) → c(1, 2)
+(4, 0) → 4
+(4, 1) → 5
+(4, 2) → 6
+(4, 3) → 7
+(4, 4) → 8
+(4, 5) → 9
+(4, 6) → c(1, 0)
+(4, 7) → c(1, 1)
+(4, 8) → c(1, 2)
+(4, 9) → c(1, 3)
+(5, 0) → 5
+(5, 1) → 6
+(5, 2) → 7
+(5, 3) → 8
+(5, 4) → 9
+(5, 5) → c(1, 0)
+(5, 6) → c(1, 1)
+(5, 7) → c(1, 2)
+(5, 8) → c(1, 3)
+(5, 9) → c(1, 4)
+(6, 0) → 6
+(6, 1) → 7
+(6, 2) → 8
+(6, 3) → 9
+(6, 4) → c(1, 0)
+(6, 5) → c(1, 1)
+(6, 6) → c(1, 2)
+(6, 7) → c(1, 3)
+(6, 8) → c(1, 4)
+(6, 9) → c(1, 5)
+(7, 0) → 7
+(7, 1) → 8
+(7, 2) → 9
+(7, 3) → c(1, 0)
+(7, 4) → c(1, 1)
+(7, 5) → c(1, 2)
+(7, 6) → c(1, 3)
+(7, 7) → c(1, 4)
+(7, 8) → c(1, 5)
+(7, 9) → c(1, 6)
+(8, 0) → 8
+(8, 1) → 9
+(8, 2) → c(1, 0)
+(8, 3) → c(1, 1)
+(8, 4) → c(1, 2)
+(8, 5) → c(1, 3)
+(8, 6) → c(1, 4)
+(8, 7) → c(1, 5)
+(8, 8) → c(1, 6)
+(8, 9) → c(1, 7)
+(9, 0) → 9
+(9, 1) → c(1, 0)
+(9, 2) → c(1, 1)
+(9, 3) → c(1, 2)
+(9, 4) → c(1, 3)
+(9, 5) → c(1, 4)
+(9, 6) → c(1, 5)
+(9, 7) → c(1, 6)
+(9, 8) → c(1, 7)
+(9, 9) → c(1, 8)
+(x, c(y, z)) → c(y, +(x, z))
+(c(x, y), z) → c(x, +(y, z))
*(0, 0) → 0
*(0, 1) → 0
*(0, 2) → 0
*(0, 3) → 0
*(0, 4) → 0
*(0, 5) → 0
*(0, 6) → 0
*(0, 7) → 0
*(0, 8) → 0
*(0, 9) → 0
*(1, 0) → 0
*(1, 1) → 1
*(1, 2) → 2
*(1, 3) → 3
*(1, 4) → 4
*(1, 5) → 5
*(1, 6) → 6
*(1, 7) → 7
*(1, 8) → 8
*(1, 9) → 9
*(2, 0) → 0
*(2, 1) → 2
*(2, 2) → 4
*(2, 3) → 6
*(2, 4) → 8
*(2, 5) → c(1, 0)
*(2, 6) → c(1, 2)
*(2, 7) → c(1, 4)
*(2, 8) → c(1, 6)
*(2, 9) → c(1, 8)
*(3, 0) → 0
*(3, 1) → 3
*(3, 2) → 6
*(3, 3) → 9
*(3, 4) → c(1, 2)
*(3, 5) → c(1, 5)
*(3, 6) → c(1, 8)
*(3, 7) → c(2, 1)
*(3, 8) → c(2, 4)
*(3, 9) → c(2, 7)
*(4, 0) → 0
*(4, 1) → 4
*(4, 2) → 8
*(4, 3) → c(1, 2)
*(4, 4) → c(1, 6)
*(4, 5) → c(2, 0)
*(4, 6) → c(2, 4)
*(4, 7) → c(2, 8)
*(4, 8) → c(3, 2)
*(4, 9) → c(3, 6)
*(5, 0) → 0
*(5, 1) → 5
*(5, 2) → c(1, 0)
*(5, 3) → c(1, 5)
*(5, 4) → c(2, 0)
*(5, 5) → c(2, 5)
*(5, 6) → c(3, 0)
*(5, 7) → c(3, 5)
*(5, 8) → c(4, 0)
*(5, 9) → c(4, 5)
*(6, 0) → 0
*(6, 1) → 6
*(6, 2) → c(1, 2)
*(6, 3) → c(1, 8)
*(6, 4) → c(2, 4)
*(6, 5) → c(3, 0)
*(6, 6) → c(3, 6)
*(6, 7) → c(4, 2)
*(6, 8) → c(4, 8)
*(6, 9) → c(5, 4)
*(7, 0) → 0
*(7, 1) → 7
*(7, 2) → c(1, 4)
*(7, 3) → c(2, 1)
*(7, 4) → c(2, 8)
*(7, 5) → c(3, 5)
*(7, 6) → c(4, 2)
*(7, 7) → c(4, 9)
*(7, 8) → c(5, 6)
*(7, 9) → c(6, 3)
*(8, 0) → 0
*(8, 1) → 8
*(8, 2) → c(1, 8)
*(8, 3) → c(2, 4)
*(8, 4) → c(3, 2)
*(8, 5) → c(4, 0)
*(8, 6) → c(4, 8)
*(8, 7) → c(5, 6)
*(8, 8) → c(6, 4)
*(8, 9) → c(7, 2)
*(9, 0) → 0
*(9, 1) → 9
*(9, 2) → c(1, 8)
*(9, 3) → c(2, 7)
*(9, 4) → c(3, 6)
*(9, 5) → c(4, 5)
*(9, 6) → c(5, 4)
*(9, 7) → c(6, 3)
*(9, 8) → c(7, 2)
*(9, 9) → c(8, 1)
*(x, c(y, z)) → c(*(x, y), *(x, z))
*(c(x, y), z) → c(*(x, z), *(y, z))

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 105 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

C(x, c(y, z)) → +¹(x, y)
+¹(x, c(y, z)) → C(y, +(x, z))
C(x, c(y, z)) → C(+(x, y), z)
+¹(x, c(y, z)) → +¹(x, z)
+¹(c(x, y), z) → C(x, +(y, z))
+¹(c(x, y), z) → +¹(y, z)

The TRS R consists of the following rules:

c(0, x) → x
c(x, c(y, z)) → c(+(x, y), z)
+(0, 0) → 0
+(0, 1) → 1
+(0, 2) → 2
+(0, 3) → 3
+(0, 4) → 4
+(0, 5) → 5
+(0, 6) → 6
+(0, 7) → 7
+(0, 8) → 8
+(0, 9) → 9
+(1, 0) → 1
+(1, 1) → 2
+(1, 2) → 3
+(1, 3) → 4
+(1, 4) → 5
+(1, 5) → 6
+(1, 6) → 7
+(1, 7) → 8
+(1, 8) → 9
+(1, 9) → c(1, 0)
+(2, 0) → 2
+(2, 1) → 3
+(2, 2) → 4
+(2, 3) → 5
+(2, 4) → 6
+(2, 5) → 7
+(2, 6) → 8
+(2, 7) → 9
+(2, 8) → c(1, 0)
+(2, 9) → c(1, 1)
+(3, 0) → 3
+(3, 1) → 4
+(3, 2) → 5
+(3, 3) → 6
+(3, 4) → 7
+(3, 5) → 8
+(3, 6) → 9
+(3, 7) → c(1, 0)
+(3, 8) → c(1, 1)
+(3, 9) → c(1, 2)
+(4, 0) → 4
+(4, 1) → 5
+(4, 2) → 6
+(4, 3) → 7
+(4, 4) → 8
+(4, 5) → 9
+(4, 6) → c(1, 0)
+(4, 7) → c(1, 1)
+(4, 8) → c(1, 2)
+(4, 9) → c(1, 3)
+(5, 0) → 5
+(5, 1) → 6
+(5, 2) → 7
+(5, 3) → 8
+(5, 4) → 9
+(5, 5) → c(1, 0)
+(5, 6) → c(1, 1)
+(5, 7) → c(1, 2)
+(5, 8) → c(1, 3)
+(5, 9) → c(1, 4)
+(6, 0) → 6
+(6, 1) → 7
+(6, 2) → 8
+(6, 3) → 9
+(6, 4) → c(1, 0)
+(6, 5) → c(1, 1)
+(6, 6) → c(1, 2)
+(6, 7) → c(1, 3)
+(6, 8) → c(1, 4)
+(6, 9) → c(1, 5)
+(7, 0) → 7
+(7, 1) → 8
+(7, 2) → 9
+(7, 3) → c(1, 0)
+(7, 4) → c(1, 1)
+(7, 5) → c(1, 2)
+(7, 6) → c(1, 3)
+(7, 7) → c(1, 4)
+(7, 8) → c(1, 5)
+(7, 9) → c(1, 6)
+(8, 0) → 8
+(8, 1) → 9
+(8, 2) → c(1, 0)
+(8, 3) → c(1, 1)
+(8, 4) → c(1, 2)
+(8, 5) → c(1, 3)
+(8, 6) → c(1, 4)
+(8, 7) → c(1, 5)
+(8, 8) → c(1, 6)
+(8, 9) → c(1, 7)
+(9, 0) → 9
+(9, 1) → c(1, 0)
+(9, 2) → c(1, 1)
+(9, 3) → c(1, 2)
+(9, 4) → c(1, 3)
+(9, 5) → c(1, 4)
+(9, 6) → c(1, 5)
+(9, 7) → c(1, 6)
+(9, 8) → c(1, 7)
+(9, 9) → c(1, 8)
+(x, c(y, z)) → c(y, +(x, z))
+(c(x, y), z) → c(x, +(y, z))
*(0, 0) → 0
*(0, 1) → 0
*(0, 2) → 0
*(0, 3) → 0
*(0, 4) → 0
*(0, 5) → 0
*(0, 6) → 0
*(0, 7) → 0
*(0, 8) → 0
*(0, 9) → 0
*(1, 0) → 0
*(1, 1) → 1
*(1, 2) → 2
*(1, 3) → 3
*(1, 4) → 4
*(1, 5) → 5
*(1, 6) → 6
*(1, 7) → 7
*(1, 8) → 8
*(1, 9) → 9
*(2, 0) → 0
*(2, 1) → 2
*(2, 2) → 4
*(2, 3) → 6
*(2, 4) → 8
*(2, 5) → c(1, 0)
*(2, 6) → c(1, 2)
*(2, 7) → c(1, 4)
*(2, 8) → c(1, 6)
*(2, 9) → c(1, 8)
*(3, 0) → 0
*(3, 1) → 3
*(3, 2) → 6
*(3, 3) → 9
*(3, 4) → c(1, 2)
*(3, 5) → c(1, 5)
*(3, 6) → c(1, 8)
*(3, 7) → c(2, 1)
*(3, 8) → c(2, 4)
*(3, 9) → c(2, 7)
*(4, 0) → 0
*(4, 1) → 4
*(4, 2) → 8
*(4, 3) → c(1, 2)
*(4, 4) → c(1, 6)
*(4, 5) → c(2, 0)
*(4, 6) → c(2, 4)
*(4, 7) → c(2, 8)
*(4, 8) → c(3, 2)
*(4, 9) → c(3, 6)
*(5, 0) → 0
*(5, 1) → 5
*(5, 2) → c(1, 0)
*(5, 3) → c(1, 5)
*(5, 4) → c(2, 0)
*(5, 5) → c(2, 5)
*(5, 6) → c(3, 0)
*(5, 7) → c(3, 5)
*(5, 8) → c(4, 0)
*(5, 9) → c(4, 5)
*(6, 0) → 0
*(6, 1) → 6
*(6, 2) → c(1, 2)
*(6, 3) → c(1, 8)
*(6, 4) → c(2, 4)
*(6, 5) → c(3, 0)
*(6, 6) → c(3, 6)
*(6, 7) → c(4, 2)
*(6, 8) → c(4, 8)
*(6, 9) → c(5, 4)
*(7, 0) → 0
*(7, 1) → 7
*(7, 2) → c(1, 4)
*(7, 3) → c(2, 1)
*(7, 4) → c(2, 8)
*(7, 5) → c(3, 5)
*(7, 6) → c(4, 2)
*(7, 7) → c(4, 9)
*(7, 8) → c(5, 6)
*(7, 9) → c(6, 3)
*(8, 0) → 0
*(8, 1) → 8
*(8, 2) → c(1, 8)
*(8, 3) → c(2, 4)
*(8, 4) → c(3, 2)
*(8, 5) → c(4, 0)
*(8, 6) → c(4, 8)
*(8, 7) → c(5, 6)
*(8, 8) → c(6, 4)
*(8, 9) → c(7, 2)
*(9, 0) → 0
*(9, 1) → 9
*(9, 2) → c(1, 8)
*(9, 3) → c(2, 7)
*(9, 4) → c(3, 6)
*(9, 5) → c(4, 5)
*(9, 6) → c(5, 4)
*(9, 7) → c(6, 3)
*(9, 8) → c(7, 2)
*(9, 9) → c(8, 1)
*(x, c(y, z)) → c(*(x, y), *(x, z))
*(c(x, y), z) → c(*(x, z), *(y, z))

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.

C(x, c(y, z)) → +¹(x, y)
+¹(x, c(y, z)) → C(y, +(x, z))
+¹(x, c(y, z)) → +¹(x, z)
+¹(c(x, y), z) → C(x, +(y, z))
+¹(c(x, y), z) → +¹(y, z)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
C(x0, x1, x2) = C(x0)
+¹(x0, x1, x2) = +¹(x0, x1, x2)

Tags:
C has argument tags [1,6,2] and root tag 0
+¹ has argument tags [1,6,5] and root tag 1

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(+(x₁, x₂)) = 1 + x₁ + x₂
POL(+¹(x₁, x₂)) = x₁ + x₂
POL(0) = 0
POL(1) = 0
POL(2) = 0
POL(3) = 0
POL(4) = 0
POL(5) = 0
POL(6) = 0
POL(7) = 0
POL(8) = 1
POL(9) = 1
POL(C(x₁, x₂)) = x₁ + x₂
POL(c(x₁, x₂)) = 1 + x₁ + x₂

The following usable rules [FROCOS05] were oriented:

+(0, 0) → 0
+(0, 1) → 1
+(0, 2) → 2
+(0, 3) → 3
+(0, 4) → 4
+(0, 5) → 5
+(0, 6) → 6
+(0, 7) → 7
+(0, 8) → 8
+(0, 9) → 9
+(1, 0) → 1
+(1, 1) → 2
+(1, 2) → 3
+(1, 3) → 4
+(1, 4) → 5
+(1, 5) → 6
+(1, 6) → 7
+(1, 7) → 8
+(1, 8) → 9
+(1, 9) → c(1, 0)
+(2, 0) → 2
+(2, 1) → 3
+(2, 2) → 4
+(2, 3) → 5
+(2, 4) → 6
+(2, 5) → 7
+(2, 6) → 8
+(2, 7) → 9
+(2, 8) → c(1, 0)
+(2, 9) → c(1, 1)
+(3, 0) → 3
+(3, 1) → 4
+(3, 2) → 5
+(3, 3) → 6
+(3, 4) → 7
+(3, 5) → 8
+(3, 6) → 9
+(3, 7) → c(1, 0)
+(3, 8) → c(1, 1)
+(3, 9) → c(1, 2)
+(4, 0) → 4
+(4, 1) → 5
+(4, 2) → 6
+(4, 3) → 7
+(4, 4) → 8
+(4, 5) → 9
+(4, 6) → c(1, 0)
+(4, 7) → c(1, 1)
+(4, 8) → c(1, 2)
+(4, 9) → c(1, 3)
+(5, 0) → 5
+(5, 1) → 6
+(5, 2) → 7
+(5, 3) → 8
+(5, 4) → 9
+(5, 5) → c(1, 0)
+(5, 6) → c(1, 1)
+(5, 7) → c(1, 2)
+(5, 8) → c(1, 3)
+(5, 9) → c(1, 4)
+(6, 0) → 6
+(6, 1) → 7
+(6, 2) → 8
+(6, 3) → 9
+(6, 4) → c(1, 0)
+(6, 5) → c(1, 1)
+(6, 6) → c(1, 2)
+(6, 7) → c(1, 3)
+(6, 8) → c(1, 4)
+(6, 9) → c(1, 5)
+(7, 0) → 7
+(7, 1) → 8
+(7, 2) → 9
+(7, 3) → c(1, 0)
+(7, 4) → c(1, 1)
+(7, 5) → c(1, 2)
+(7, 6) → c(1, 3)
+(7, 7) → c(1, 4)
+(7, 8) → c(1, 5)
+(7, 9) → c(1, 6)
+(8, 0) → 8
+(8, 1) → 9
+(8, 2) → c(1, 0)
+(8, 3) → c(1, 1)
+(8, 4) → c(1, 2)
+(8, 5) → c(1, 3)
+(8, 6) → c(1, 4)
+(8, 7) → c(1, 5)
+(8, 8) → c(1, 6)
+(8, 9) → c(1, 7)
+(9, 0) → 9
+(9, 1) → c(1, 0)
+(9, 2) → c(1, 1)
+(9, 3) → c(1, 2)
+(9, 4) → c(1, 3)
+(9, 5) → c(1, 4)
+(9, 6) → c(1, 5)
+(9, 7) → c(1, 6)
+(9, 8) → c(1, 7)
+(9, 9) → c(1, 8)
+(x, c(y, z)) → c(y, +(x, z))
+(c(x, y), z) → c(x, +(y, z))
c(x, c(y, z)) → c(+(x, y), z)
c(0, x) → x

(7) Obligation:

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

C(x, c(y, z)) → C(+(x, y), z)

The TRS R consists of the following rules:

c(0, x) → x
c(x, c(y, z)) → c(+(x, y), z)
+(0, 0) → 0
+(0, 1) → 1
+(0, 2) → 2
+(0, 3) → 3
+(0, 4) → 4
+(0, 5) → 5
+(0, 6) → 6
+(0, 7) → 7
+(0, 8) → 8
+(0, 9) → 9
+(1, 0) → 1
+(1, 1) → 2
+(1, 2) → 3
+(1, 3) → 4
+(1, 4) → 5
+(1, 5) → 6
+(1, 6) → 7
+(1, 7) → 8
+(1, 8) → 9
+(1, 9) → c(1, 0)
+(2, 0) → 2
+(2, 1) → 3
+(2, 2) → 4
+(2, 3) → 5
+(2, 4) → 6
+(2, 5) → 7
+(2, 6) → 8
+(2, 7) → 9
+(2, 8) → c(1, 0)
+(2, 9) → c(1, 1)
+(3, 0) → 3
+(3, 1) → 4
+(3, 2) → 5
+(3, 3) → 6
+(3, 4) → 7
+(3, 5) → 8
+(3, 6) → 9
+(3, 7) → c(1, 0)
+(3, 8) → c(1, 1)
+(3, 9) → c(1, 2)
+(4, 0) → 4
+(4, 1) → 5
+(4, 2) → 6
+(4, 3) → 7
+(4, 4) → 8
+(4, 5) → 9
+(4, 6) → c(1, 0)
+(4, 7) → c(1, 1)
+(4, 8) → c(1, 2)
+(4, 9) → c(1, 3)
+(5, 0) → 5
+(5, 1) → 6
+(5, 2) → 7
+(5, 3) → 8
+(5, 4) → 9
+(5, 5) → c(1, 0)
+(5, 6) → c(1, 1)
+(5, 7) → c(1, 2)
+(5, 8) → c(1, 3)
+(5, 9) → c(1, 4)
+(6, 0) → 6
+(6, 1) → 7
+(6, 2) → 8
+(6, 3) → 9
+(6, 4) → c(1, 0)
+(6, 5) → c(1, 1)
+(6, 6) → c(1, 2)
+(6, 7) → c(1, 3)
+(6, 8) → c(1, 4)
+(6, 9) → c(1, 5)
+(7, 0) → 7
+(7, 1) → 8
+(7, 2) → 9
+(7, 3) → c(1, 0)
+(7, 4) → c(1, 1)
+(7, 5) → c(1, 2)
+(7, 6) → c(1, 3)
+(7, 7) → c(1, 4)
+(7, 8) → c(1, 5)
+(7, 9) → c(1, 6)
+(8, 0) → 8
+(8, 1) → 9
+(8, 2) → c(1, 0)
+(8, 3) → c(1, 1)
+(8, 4) → c(1, 2)
+(8, 5) → c(1, 3)
+(8, 6) → c(1, 4)
+(8, 7) → c(1, 5)
+(8, 8) → c(1, 6)
+(8, 9) → c(1, 7)
+(9, 0) → 9
+(9, 1) → c(1, 0)
+(9, 2) → c(1, 1)
+(9, 3) → c(1, 2)
+(9, 4) → c(1, 3)
+(9, 5) → c(1, 4)
+(9, 6) → c(1, 5)
+(9, 7) → c(1, 6)
+(9, 8) → c(1, 7)
+(9, 9) → c(1, 8)
+(x, c(y, z)) → c(y, +(x, z))
+(c(x, y), z) → c(x, +(y, z))
*(0, 0) → 0
*(0, 1) → 0
*(0, 2) → 0
*(0, 3) → 0
*(0, 4) → 0
*(0, 5) → 0
*(0, 6) → 0
*(0, 7) → 0
*(0, 8) → 0
*(0, 9) → 0
*(1, 0) → 0
*(1, 1) → 1
*(1, 2) → 2
*(1, 3) → 3
*(1, 4) → 4
*(1, 5) → 5
*(1, 6) → 6
*(1, 7) → 7
*(1, 8) → 8
*(1, 9) → 9
*(2, 0) → 0
*(2, 1) → 2
*(2, 2) → 4
*(2, 3) → 6
*(2, 4) → 8
*(2, 5) → c(1, 0)
*(2, 6) → c(1, 2)
*(2, 7) → c(1, 4)
*(2, 8) → c(1, 6)
*(2, 9) → c(1, 8)
*(3, 0) → 0
*(3, 1) → 3
*(3, 2) → 6
*(3, 3) → 9
*(3, 4) → c(1, 2)
*(3, 5) → c(1, 5)
*(3, 6) → c(1, 8)
*(3, 7) → c(2, 1)
*(3, 8) → c(2, 4)
*(3, 9) → c(2, 7)
*(4, 0) → 0
*(4, 1) → 4
*(4, 2) → 8
*(4, 3) → c(1, 2)
*(4, 4) → c(1, 6)
*(4, 5) → c(2, 0)
*(4, 6) → c(2, 4)
*(4, 7) → c(2, 8)
*(4, 8) → c(3, 2)
*(4, 9) → c(3, 6)
*(5, 0) → 0
*(5, 1) → 5
*(5, 2) → c(1, 0)
*(5, 3) → c(1, 5)
*(5, 4) → c(2, 0)
*(5, 5) → c(2, 5)
*(5, 6) → c(3, 0)
*(5, 7) → c(3, 5)
*(5, 8) → c(4, 0)
*(5, 9) → c(4, 5)
*(6, 0) → 0
*(6, 1) → 6
*(6, 2) → c(1, 2)
*(6, 3) → c(1, 8)
*(6, 4) → c(2, 4)
*(6, 5) → c(3, 0)
*(6, 6) → c(3, 6)
*(6, 7) → c(4, 2)
*(6, 8) → c(4, 8)
*(6, 9) → c(5, 4)
*(7, 0) → 0
*(7, 1) → 7
*(7, 2) → c(1, 4)
*(7, 3) → c(2, 1)
*(7, 4) → c(2, 8)
*(7, 5) → c(3, 5)
*(7, 6) → c(4, 2)
*(7, 7) → c(4, 9)
*(7, 8) → c(5, 6)
*(7, 9) → c(6, 3)
*(8, 0) → 0
*(8, 1) → 8
*(8, 2) → c(1, 8)
*(8, 3) → c(2, 4)
*(8, 4) → c(3, 2)
*(8, 5) → c(4, 0)
*(8, 6) → c(4, 8)
*(8, 7) → c(5, 6)
*(8, 8) → c(6, 4)
*(8, 9) → c(7, 2)
*(9, 0) → 0
*(9, 1) → 9
*(9, 2) → c(1, 8)
*(9, 3) → c(2, 7)
*(9, 4) → c(3, 6)
*(9, 5) → c(4, 5)
*(9, 6) → c(5, 4)
*(9, 7) → c(6, 3)
*(9, 8) → c(7, 2)
*(9, 9) → c(8, 1)
*(x, c(y, z)) → c(*(x, y), *(x, z))
*(c(x, y), z) → c(*(x, z), *(y, z))

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

(8) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

C(x, c(y, z)) → C(+(x, y), z)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
C(x0, x1, x2) = C(x2)

Tags:
C has argument tags [1,3,1] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(+(x₁, x₂)) = 1
POL(0) = 1
POL(1) = 0
POL(2) = 1
POL(3) = 0
POL(4) = 1
POL(5) = 1
POL(6) = 1
POL(7) = 1
POL(8) = 0
POL(9) = 0
POL(C(x₁, x₂)) = 1
POL(c(x₁, x₂)) = 1 + x₂

The following usable rules [FROCOS05] were oriented: none

(9) Obligation:

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

c(0, x) → x
c(x, c(y, z)) → c(+(x, y), z)
+(0, 0) → 0
+(0, 1) → 1
+(0, 2) → 2
+(0, 3) → 3
+(0, 4) → 4
+(0, 5) → 5
+(0, 6) → 6
+(0, 7) → 7
+(0, 8) → 8
+(0, 9) → 9
+(1, 0) → 1
+(1, 1) → 2
+(1, 2) → 3
+(1, 3) → 4
+(1, 4) → 5
+(1, 5) → 6
+(1, 6) → 7
+(1, 7) → 8
+(1, 8) → 9
+(1, 9) → c(1, 0)
+(2, 0) → 2
+(2, 1) → 3
+(2, 2) → 4
+(2, 3) → 5
+(2, 4) → 6
+(2, 5) → 7
+(2, 6) → 8
+(2, 7) → 9
+(2, 8) → c(1, 0)
+(2, 9) → c(1, 1)
+(3, 0) → 3
+(3, 1) → 4
+(3, 2) → 5
+(3, 3) → 6
+(3, 4) → 7
+(3, 5) → 8
+(3, 6) → 9
+(3, 7) → c(1, 0)
+(3, 8) → c(1, 1)
+(3, 9) → c(1, 2)
+(4, 0) → 4
+(4, 1) → 5
+(4, 2) → 6
+(4, 3) → 7
+(4, 4) → 8
+(4, 5) → 9
+(4, 6) → c(1, 0)
+(4, 7) → c(1, 1)
+(4, 8) → c(1, 2)
+(4, 9) → c(1, 3)
+(5, 0) → 5
+(5, 1) → 6
+(5, 2) → 7
+(5, 3) → 8
+(5, 4) → 9
+(5, 5) → c(1, 0)
+(5, 6) → c(1, 1)
+(5, 7) → c(1, 2)
+(5, 8) → c(1, 3)
+(5, 9) → c(1, 4)
+(6, 0) → 6
+(6, 1) → 7
+(6, 2) → 8
+(6, 3) → 9
+(6, 4) → c(1, 0)
+(6, 5) → c(1, 1)
+(6, 6) → c(1, 2)
+(6, 7) → c(1, 3)
+(6, 8) → c(1, 4)
+(6, 9) → c(1, 5)
+(7, 0) → 7
+(7, 1) → 8
+(7, 2) → 9
+(7, 3) → c(1, 0)
+(7, 4) → c(1, 1)
+(7, 5) → c(1, 2)
+(7, 6) → c(1, 3)
+(7, 7) → c(1, 4)
+(7, 8) → c(1, 5)
+(7, 9) → c(1, 6)
+(8, 0) → 8
+(8, 1) → 9
+(8, 2) → c(1, 0)
+(8, 3) → c(1, 1)
+(8, 4) → c(1, 2)
+(8, 5) → c(1, 3)
+(8, 6) → c(1, 4)
+(8, 7) → c(1, 5)
+(8, 8) → c(1, 6)
+(8, 9) → c(1, 7)
+(9, 0) → 9
+(9, 1) → c(1, 0)
+(9, 2) → c(1, 1)
+(9, 3) → c(1, 2)
+(9, 4) → c(1, 3)
+(9, 5) → c(1, 4)
+(9, 6) → c(1, 5)
+(9, 7) → c(1, 6)
+(9, 8) → c(1, 7)
+(9, 9) → c(1, 8)
+(x, c(y, z)) → c(y, +(x, z))
+(c(x, y), z) → c(x, +(y, z))
*(0, 0) → 0
*(0, 1) → 0
*(0, 2) → 0
*(0, 3) → 0
*(0, 4) → 0
*(0, 5) → 0
*(0, 6) → 0
*(0, 7) → 0
*(0, 8) → 0
*(0, 9) → 0
*(1, 0) → 0
*(1, 1) → 1
*(1, 2) → 2
*(1, 3) → 3
*(1, 4) → 4
*(1, 5) → 5
*(1, 6) → 6
*(1, 7) → 7
*(1, 8) → 8
*(1, 9) → 9
*(2, 0) → 0
*(2, 1) → 2
*(2, 2) → 4
*(2, 3) → 6
*(2, 4) → 8
*(2, 5) → c(1, 0)
*(2, 6) → c(1, 2)
*(2, 7) → c(1, 4)
*(2, 8) → c(1, 6)
*(2, 9) → c(1, 8)
*(3, 0) → 0
*(3, 1) → 3
*(3, 2) → 6
*(3, 3) → 9
*(3, 4) → c(1, 2)
*(3, 5) → c(1, 5)
*(3, 6) → c(1, 8)
*(3, 7) → c(2, 1)
*(3, 8) → c(2, 4)
*(3, 9) → c(2, 7)
*(4, 0) → 0
*(4, 1) → 4
*(4, 2) → 8
*(4, 3) → c(1, 2)
*(4, 4) → c(1, 6)
*(4, 5) → c(2, 0)
*(4, 6) → c(2, 4)
*(4, 7) → c(2, 8)
*(4, 8) → c(3, 2)
*(4, 9) → c(3, 6)
*(5, 0) → 0
*(5, 1) → 5
*(5, 2) → c(1, 0)
*(5, 3) → c(1, 5)
*(5, 4) → c(2, 0)
*(5, 5) → c(2, 5)
*(5, 6) → c(3, 0)
*(5, 7) → c(3, 5)
*(5, 8) → c(4, 0)
*(5, 9) → c(4, 5)
*(6, 0) → 0
*(6, 1) → 6
*(6, 2) → c(1, 2)
*(6, 3) → c(1, 8)
*(6, 4) → c(2, 4)
*(6, 5) → c(3, 0)
*(6, 6) → c(3, 6)
*(6, 7) → c(4, 2)
*(6, 8) → c(4, 8)
*(6, 9) → c(5, 4)
*(7, 0) → 0
*(7, 1) → 7
*(7, 2) → c(1, 4)
*(7, 3) → c(2, 1)
*(7, 4) → c(2, 8)
*(7, 5) → c(3, 5)
*(7, 6) → c(4, 2)
*(7, 7) → c(4, 9)
*(7, 8) → c(5, 6)
*(7, 9) → c(6, 3)
*(8, 0) → 0
*(8, 1) → 8
*(8, 2) → c(1, 8)
*(8, 3) → c(2, 4)
*(8, 4) → c(3, 2)
*(8, 5) → c(4, 0)
*(8, 6) → c(4, 8)
*(8, 7) → c(5, 6)
*(8, 8) → c(6, 4)
*(8, 9) → c(7, 2)
*(9, 0) → 0
*(9, 1) → 9
*(9, 2) → c(1, 8)
*(9, 3) → c(2, 7)
*(9, 4) → c(3, 6)
*(9, 5) → c(4, 5)
*(9, 6) → c(5, 4)
*(9, 7) → c(6, 3)
*(9, 8) → c(7, 2)
*(9, 9) → c(8, 1)
*(x, c(y, z)) → c(*(x, y), *(x, z))
*(c(x, y), z) → c(*(x, z), *(y, z))

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

(10) PisEmptyProof (EQUIVALENT transformation)

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

(11) TRUE

(12) Obligation:

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

*¹(x, c(y, z)) → *¹(x, z)
*¹(x, c(y, z)) → *¹(x, y)
*¹(c(x, y), z) → *¹(x, z)
*¹(c(x, y), z) → *¹(y, z)

The TRS R consists of the following rules:

c(0, x) → x
c(x, c(y, z)) → c(+(x, y), z)
+(0, 0) → 0
+(0, 1) → 1
+(0, 2) → 2
+(0, 3) → 3
+(0, 4) → 4
+(0, 5) → 5
+(0, 6) → 6
+(0, 7) → 7
+(0, 8) → 8
+(0, 9) → 9
+(1, 0) → 1
+(1, 1) → 2
+(1, 2) → 3
+(1, 3) → 4
+(1, 4) → 5
+(1, 5) → 6
+(1, 6) → 7
+(1, 7) → 8
+(1, 8) → 9
+(1, 9) → c(1, 0)
+(2, 0) → 2
+(2, 1) → 3
+(2, 2) → 4
+(2, 3) → 5
+(2, 4) → 6
+(2, 5) → 7
+(2, 6) → 8
+(2, 7) → 9
+(2, 8) → c(1, 0)
+(2, 9) → c(1, 1)
+(3, 0) → 3
+(3, 1) → 4
+(3, 2) → 5
+(3, 3) → 6
+(3, 4) → 7
+(3, 5) → 8
+(3, 6) → 9
+(3, 7) → c(1, 0)
+(3, 8) → c(1, 1)
+(3, 9) → c(1, 2)
+(4, 0) → 4
+(4, 1) → 5
+(4, 2) → 6
+(4, 3) → 7
+(4, 4) → 8
+(4, 5) → 9
+(4, 6) → c(1, 0)
+(4, 7) → c(1, 1)
+(4, 8) → c(1, 2)
+(4, 9) → c(1, 3)
+(5, 0) → 5
+(5, 1) → 6
+(5, 2) → 7
+(5, 3) → 8
+(5, 4) → 9
+(5, 5) → c(1, 0)
+(5, 6) → c(1, 1)
+(5, 7) → c(1, 2)
+(5, 8) → c(1, 3)
+(5, 9) → c(1, 4)
+(6, 0) → 6
+(6, 1) → 7
+(6, 2) → 8
+(6, 3) → 9
+(6, 4) → c(1, 0)
+(6, 5) → c(1, 1)
+(6, 6) → c(1, 2)
+(6, 7) → c(1, 3)
+(6, 8) → c(1, 4)
+(6, 9) → c(1, 5)
+(7, 0) → 7
+(7, 1) → 8
+(7, 2) → 9
+(7, 3) → c(1, 0)
+(7, 4) → c(1, 1)
+(7, 5) → c(1, 2)
+(7, 6) → c(1, 3)
+(7, 7) → c(1, 4)
+(7, 8) → c(1, 5)
+(7, 9) → c(1, 6)
+(8, 0) → 8
+(8, 1) → 9
+(8, 2) → c(1, 0)
+(8, 3) → c(1, 1)
+(8, 4) → c(1, 2)
+(8, 5) → c(1, 3)
+(8, 6) → c(1, 4)
+(8, 7) → c(1, 5)
+(8, 8) → c(1, 6)
+(8, 9) → c(1, 7)
+(9, 0) → 9
+(9, 1) → c(1, 0)
+(9, 2) → c(1, 1)
+(9, 3) → c(1, 2)
+(9, 4) → c(1, 3)
+(9, 5) → c(1, 4)
+(9, 6) → c(1, 5)
+(9, 7) → c(1, 6)
+(9, 8) → c(1, 7)
+(9, 9) → c(1, 8)
+(x, c(y, z)) → c(y, +(x, z))
+(c(x, y), z) → c(x, +(y, z))
*(0, 0) → 0
*(0, 1) → 0
*(0, 2) → 0
*(0, 3) → 0
*(0, 4) → 0
*(0, 5) → 0
*(0, 6) → 0
*(0, 7) → 0
*(0, 8) → 0
*(0, 9) → 0
*(1, 0) → 0
*(1, 1) → 1
*(1, 2) → 2
*(1, 3) → 3
*(1, 4) → 4
*(1, 5) → 5
*(1, 6) → 6
*(1, 7) → 7
*(1, 8) → 8
*(1, 9) → 9
*(2, 0) → 0
*(2, 1) → 2
*(2, 2) → 4
*(2, 3) → 6
*(2, 4) → 8
*(2, 5) → c(1, 0)
*(2, 6) → c(1, 2)
*(2, 7) → c(1, 4)
*(2, 8) → c(1, 6)
*(2, 9) → c(1, 8)
*(3, 0) → 0
*(3, 1) → 3
*(3, 2) → 6
*(3, 3) → 9
*(3, 4) → c(1, 2)
*(3, 5) → c(1, 5)
*(3, 6) → c(1, 8)
*(3, 7) → c(2, 1)
*(3, 8) → c(2, 4)
*(3, 9) → c(2, 7)
*(4, 0) → 0
*(4, 1) → 4
*(4, 2) → 8
*(4, 3) → c(1, 2)
*(4, 4) → c(1, 6)
*(4, 5) → c(2, 0)
*(4, 6) → c(2, 4)
*(4, 7) → c(2, 8)
*(4, 8) → c(3, 2)
*(4, 9) → c(3, 6)
*(5, 0) → 0
*(5, 1) → 5
*(5, 2) → c(1, 0)
*(5, 3) → c(1, 5)
*(5, 4) → c(2, 0)
*(5, 5) → c(2, 5)
*(5, 6) → c(3, 0)
*(5, 7) → c(3, 5)
*(5, 8) → c(4, 0)
*(5, 9) → c(4, 5)
*(6, 0) → 0
*(6, 1) → 6
*(6, 2) → c(1, 2)
*(6, 3) → c(1, 8)
*(6, 4) → c(2, 4)
*(6, 5) → c(3, 0)
*(6, 6) → c(3, 6)
*(6, 7) → c(4, 2)
*(6, 8) → c(4, 8)
*(6, 9) → c(5, 4)
*(7, 0) → 0
*(7, 1) → 7
*(7, 2) → c(1, 4)
*(7, 3) → c(2, 1)
*(7, 4) → c(2, 8)
*(7, 5) → c(3, 5)
*(7, 6) → c(4, 2)
*(7, 7) → c(4, 9)
*(7, 8) → c(5, 6)
*(7, 9) → c(6, 3)
*(8, 0) → 0
*(8, 1) → 8
*(8, 2) → c(1, 8)
*(8, 3) → c(2, 4)
*(8, 4) → c(3, 2)
*(8, 5) → c(4, 0)
*(8, 6) → c(4, 8)
*(8, 7) → c(5, 6)
*(8, 8) → c(6, 4)
*(8, 9) → c(7, 2)
*(9, 0) → 0
*(9, 1) → 9
*(9, 2) → c(1, 8)
*(9, 3) → c(2, 7)
*(9, 4) → c(3, 6)
*(9, 5) → c(4, 5)
*(9, 6) → c(5, 4)
*(9, 7) → c(6, 3)
*(9, 8) → c(7, 2)
*(9, 9) → c(8, 1)
*(x, c(y, z)) → c(*(x, y), *(x, z))
*(c(x, y), z) → c(*(x, z), *(y, z))

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

(13) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

*¹(c(x, y), z) → *¹(x, z)
*¹(c(x, y), z) → *¹(y, z)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
*¹(x0, x1, x2) = *¹(x1)

Tags:
*¹ has argument tags [2,2,1] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(*¹(x₁, x₂)) = 1
POL(c(x₁, x₂)) = 1 + x₁ + x₂

The following usable rules [FROCOS05] were oriented: none

(14) Obligation:

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

*¹(x, c(y, z)) → *¹(x, z)
*¹(x, c(y, z)) → *¹(x, y)

The TRS R consists of the following rules:

c(0, x) → x
c(x, c(y, z)) → c(+(x, y), z)
+(0, 0) → 0
+(0, 1) → 1
+(0, 2) → 2
+(0, 3) → 3
+(0, 4) → 4
+(0, 5) → 5
+(0, 6) → 6
+(0, 7) → 7
+(0, 8) → 8
+(0, 9) → 9
+(1, 0) → 1
+(1, 1) → 2
+(1, 2) → 3
+(1, 3) → 4
+(1, 4) → 5
+(1, 5) → 6
+(1, 6) → 7
+(1, 7) → 8
+(1, 8) → 9
+(1, 9) → c(1, 0)
+(2, 0) → 2
+(2, 1) → 3
+(2, 2) → 4
+(2, 3) → 5
+(2, 4) → 6
+(2, 5) → 7
+(2, 6) → 8
+(2, 7) → 9
+(2, 8) → c(1, 0)
+(2, 9) → c(1, 1)
+(3, 0) → 3
+(3, 1) → 4
+(3, 2) → 5
+(3, 3) → 6
+(3, 4) → 7
+(3, 5) → 8
+(3, 6) → 9
+(3, 7) → c(1, 0)
+(3, 8) → c(1, 1)
+(3, 9) → c(1, 2)
+(4, 0) → 4
+(4, 1) → 5
+(4, 2) → 6
+(4, 3) → 7
+(4, 4) → 8
+(4, 5) → 9
+(4, 6) → c(1, 0)
+(4, 7) → c(1, 1)
+(4, 8) → c(1, 2)
+(4, 9) → c(1, 3)
+(5, 0) → 5
+(5, 1) → 6
+(5, 2) → 7
+(5, 3) → 8
+(5, 4) → 9
+(5, 5) → c(1, 0)
+(5, 6) → c(1, 1)
+(5, 7) → c(1, 2)
+(5, 8) → c(1, 3)
+(5, 9) → c(1, 4)
+(6, 0) → 6
+(6, 1) → 7
+(6, 2) → 8
+(6, 3) → 9
+(6, 4) → c(1, 0)
+(6, 5) → c(1, 1)
+(6, 6) → c(1, 2)
+(6, 7) → c(1, 3)
+(6, 8) → c(1, 4)
+(6, 9) → c(1, 5)
+(7, 0) → 7
+(7, 1) → 8
+(7, 2) → 9
+(7, 3) → c(1, 0)
+(7, 4) → c(1, 1)
+(7, 5) → c(1, 2)
+(7, 6) → c(1, 3)
+(7, 7) → c(1, 4)
+(7, 8) → c(1, 5)
+(7, 9) → c(1, 6)
+(8, 0) → 8
+(8, 1) → 9
+(8, 2) → c(1, 0)
+(8, 3) → c(1, 1)
+(8, 4) → c(1, 2)
+(8, 5) → c(1, 3)
+(8, 6) → c(1, 4)
+(8, 7) → c(1, 5)
+(8, 8) → c(1, 6)
+(8, 9) → c(1, 7)
+(9, 0) → 9
+(9, 1) → c(1, 0)
+(9, 2) → c(1, 1)
+(9, 3) → c(1, 2)
+(9, 4) → c(1, 3)
+(9, 5) → c(1, 4)
+(9, 6) → c(1, 5)
+(9, 7) → c(1, 6)
+(9, 8) → c(1, 7)
+(9, 9) → c(1, 8)
+(x, c(y, z)) → c(y, +(x, z))
+(c(x, y), z) → c(x, +(y, z))
*(0, 0) → 0
*(0, 1) → 0
*(0, 2) → 0
*(0, 3) → 0
*(0, 4) → 0
*(0, 5) → 0
*(0, 6) → 0
*(0, 7) → 0
*(0, 8) → 0
*(0, 9) → 0
*(1, 0) → 0
*(1, 1) → 1
*(1, 2) → 2
*(1, 3) → 3
*(1, 4) → 4
*(1, 5) → 5
*(1, 6) → 6
*(1, 7) → 7
*(1, 8) → 8
*(1, 9) → 9
*(2, 0) → 0
*(2, 1) → 2
*(2, 2) → 4
*(2, 3) → 6
*(2, 4) → 8
*(2, 5) → c(1, 0)
*(2, 6) → c(1, 2)
*(2, 7) → c(1, 4)
*(2, 8) → c(1, 6)
*(2, 9) → c(1, 8)
*(3, 0) → 0
*(3, 1) → 3
*(3, 2) → 6
*(3, 3) → 9
*(3, 4) → c(1, 2)
*(3, 5) → c(1, 5)
*(3, 6) → c(1, 8)
*(3, 7) → c(2, 1)
*(3, 8) → c(2, 4)
*(3, 9) → c(2, 7)
*(4, 0) → 0
*(4, 1) → 4
*(4, 2) → 8
*(4, 3) → c(1, 2)
*(4, 4) → c(1, 6)
*(4, 5) → c(2, 0)
*(4, 6) → c(2, 4)
*(4, 7) → c(2, 8)
*(4, 8) → c(3, 2)
*(4, 9) → c(3, 6)
*(5, 0) → 0
*(5, 1) → 5
*(5, 2) → c(1, 0)
*(5, 3) → c(1, 5)
*(5, 4) → c(2, 0)
*(5, 5) → c(2, 5)
*(5, 6) → c(3, 0)
*(5, 7) → c(3, 5)
*(5, 8) → c(4, 0)
*(5, 9) → c(4, 5)
*(6, 0) → 0
*(6, 1) → 6
*(6, 2) → c(1, 2)
*(6, 3) → c(1, 8)
*(6, 4) → c(2, 4)
*(6, 5) → c(3, 0)
*(6, 6) → c(3, 6)
*(6, 7) → c(4, 2)
*(6, 8) → c(4, 8)
*(6, 9) → c(5, 4)
*(7, 0) → 0
*(7, 1) → 7
*(7, 2) → c(1, 4)
*(7, 3) → c(2, 1)
*(7, 4) → c(2, 8)
*(7, 5) → c(3, 5)
*(7, 6) → c(4, 2)
*(7, 7) → c(4, 9)
*(7, 8) → c(5, 6)
*(7, 9) → c(6, 3)
*(8, 0) → 0
*(8, 1) → 8
*(8, 2) → c(1, 8)
*(8, 3) → c(2, 4)
*(8, 4) → c(3, 2)
*(8, 5) → c(4, 0)
*(8, 6) → c(4, 8)
*(8, 7) → c(5, 6)
*(8, 8) → c(6, 4)
*(8, 9) → c(7, 2)
*(9, 0) → 0
*(9, 1) → 9
*(9, 2) → c(1, 8)
*(9, 3) → c(2, 7)
*(9, 4) → c(3, 6)
*(9, 5) → c(4, 5)
*(9, 6) → c(5, 4)
*(9, 7) → c(6, 3)
*(9, 8) → c(7, 2)
*(9, 9) → c(8, 1)
*(x, c(y, z)) → c(*(x, y), *(x, z))
*(c(x, y), z) → c(*(x, z), *(y, z))

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

(15) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

*¹(x, c(y, z)) → *¹(x, z)
*¹(x, c(y, z)) → *¹(x, y)

The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
*¹(x0, x1, x2) = *¹(x0)

Tags:
*¹ has argument tags [1,1,1] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(*¹(x₁, x₂)) = x₁ + x₂
POL(c(x₁, x₂)) = 1 + x₁ + x₂

The following usable rules [FROCOS05] were oriented: none

(16) Obligation:

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

c(0, x) → x
c(x, c(y, z)) → c(+(x, y), z)
+(0, 0) → 0
+(0, 1) → 1
+(0, 2) → 2
+(0, 3) → 3
+(0, 4) → 4
+(0, 5) → 5
+(0, 6) → 6
+(0, 7) → 7
+(0, 8) → 8
+(0, 9) → 9
+(1, 0) → 1
+(1, 1) → 2
+(1, 2) → 3
+(1, 3) → 4
+(1, 4) → 5
+(1, 5) → 6
+(1, 6) → 7
+(1, 7) → 8
+(1, 8) → 9
+(1, 9) → c(1, 0)
+(2, 0) → 2
+(2, 1) → 3
+(2, 2) → 4
+(2, 3) → 5
+(2, 4) → 6
+(2, 5) → 7
+(2, 6) → 8
+(2, 7) → 9
+(2, 8) → c(1, 0)
+(2, 9) → c(1, 1)
+(3, 0) → 3
+(3, 1) → 4
+(3, 2) → 5
+(3, 3) → 6
+(3, 4) → 7
+(3, 5) → 8
+(3, 6) → 9
+(3, 7) → c(1, 0)
+(3, 8) → c(1, 1)
+(3, 9) → c(1, 2)
+(4, 0) → 4
+(4, 1) → 5
+(4, 2) → 6
+(4, 3) → 7
+(4, 4) → 8
+(4, 5) → 9
+(4, 6) → c(1, 0)
+(4, 7) → c(1, 1)
+(4, 8) → c(1, 2)
+(4, 9) → c(1, 3)
+(5, 0) → 5
+(5, 1) → 6
+(5, 2) → 7
+(5, 3) → 8
+(5, 4) → 9
+(5, 5) → c(1, 0)
+(5, 6) → c(1, 1)
+(5, 7) → c(1, 2)
+(5, 8) → c(1, 3)
+(5, 9) → c(1, 4)
+(6, 0) → 6
+(6, 1) → 7
+(6, 2) → 8
+(6, 3) → 9
+(6, 4) → c(1, 0)
+(6, 5) → c(1, 1)
+(6, 6) → c(1, 2)
+(6, 7) → c(1, 3)
+(6, 8) → c(1, 4)
+(6, 9) → c(1, 5)
+(7, 0) → 7
+(7, 1) → 8
+(7, 2) → 9
+(7, 3) → c(1, 0)
+(7, 4) → c(1, 1)
+(7, 5) → c(1, 2)
+(7, 6) → c(1, 3)
+(7, 7) → c(1, 4)
+(7, 8) → c(1, 5)
+(7, 9) → c(1, 6)
+(8, 0) → 8
+(8, 1) → 9
+(8, 2) → c(1, 0)
+(8, 3) → c(1, 1)
+(8, 4) → c(1, 2)
+(8, 5) → c(1, 3)
+(8, 6) → c(1, 4)
+(8, 7) → c(1, 5)
+(8, 8) → c(1, 6)
+(8, 9) → c(1, 7)
+(9, 0) → 9
+(9, 1) → c(1, 0)
+(9, 2) → c(1, 1)
+(9, 3) → c(1, 2)
+(9, 4) → c(1, 3)
+(9, 5) → c(1, 4)
+(9, 6) → c(1, 5)
+(9, 7) → c(1, 6)
+(9, 8) → c(1, 7)
+(9, 9) → c(1, 8)
+(x, c(y, z)) → c(y, +(x, z))
+(c(x, y), z) → c(x, +(y, z))
*(0, 0) → 0
*(0, 1) → 0
*(0, 2) → 0
*(0, 3) → 0
*(0, 4) → 0
*(0, 5) → 0
*(0, 6) → 0
*(0, 7) → 0
*(0, 8) → 0
*(0, 9) → 0
*(1, 0) → 0
*(1, 1) → 1
*(1, 2) → 2
*(1, 3) → 3
*(1, 4) → 4
*(1, 5) → 5
*(1, 6) → 6
*(1, 7) → 7
*(1, 8) → 8
*(1, 9) → 9
*(2, 0) → 0
*(2, 1) → 2
*(2, 2) → 4
*(2, 3) → 6
*(2, 4) → 8
*(2, 5) → c(1, 0)
*(2, 6) → c(1, 2)
*(2, 7) → c(1, 4)
*(2, 8) → c(1, 6)
*(2, 9) → c(1, 8)
*(3, 0) → 0
*(3, 1) → 3
*(3, 2) → 6
*(3, 3) → 9
*(3, 4) → c(1, 2)
*(3, 5) → c(1, 5)
*(3, 6) → c(1, 8)
*(3, 7) → c(2, 1)
*(3, 8) → c(2, 4)
*(3, 9) → c(2, 7)
*(4, 0) → 0
*(4, 1) → 4
*(4, 2) → 8
*(4, 3) → c(1, 2)
*(4, 4) → c(1, 6)
*(4, 5) → c(2, 0)
*(4, 6) → c(2, 4)
*(4, 7) → c(2, 8)
*(4, 8) → c(3, 2)
*(4, 9) → c(3, 6)
*(5, 0) → 0
*(5, 1) → 5
*(5, 2) → c(1, 0)
*(5, 3) → c(1, 5)
*(5, 4) → c(2, 0)
*(5, 5) → c(2, 5)
*(5, 6) → c(3, 0)
*(5, 7) → c(3, 5)
*(5, 8) → c(4, 0)
*(5, 9) → c(4, 5)
*(6, 0) → 0
*(6, 1) → 6
*(6, 2) → c(1, 2)
*(6, 3) → c(1, 8)
*(6, 4) → c(2, 4)
*(6, 5) → c(3, 0)
*(6, 6) → c(3, 6)
*(6, 7) → c(4, 2)
*(6, 8) → c(4, 8)
*(6, 9) → c(5, 4)
*(7, 0) → 0
*(7, 1) → 7
*(7, 2) → c(1, 4)
*(7, 3) → c(2, 1)
*(7, 4) → c(2, 8)
*(7, 5) → c(3, 5)
*(7, 6) → c(4, 2)
*(7, 7) → c(4, 9)
*(7, 8) → c(5, 6)
*(7, 9) → c(6, 3)
*(8, 0) → 0
*(8, 1) → 8
*(8, 2) → c(1, 8)
*(8, 3) → c(2, 4)
*(8, 4) → c(3, 2)
*(8, 5) → c(4, 0)
*(8, 6) → c(4, 8)
*(8, 7) → c(5, 6)
*(8, 8) → c(6, 4)
*(8, 9) → c(7, 2)
*(9, 0) → 0
*(9, 1) → 9
*(9, 2) → c(1, 8)
*(9, 3) → c(2, 7)
*(9, 4) → c(3, 6)
*(9, 5) → c(4, 5)
*(9, 6) → c(5, 4)
*(9, 7) → c(6, 3)
*(9, 8) → c(7, 2)
*(9, 9) → c(8, 1)
*(x, c(y, z)) → c(*(x, y), *(x, z))
*(c(x, y), z) → c(*(x, z), *(y, z))

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

(17) PisEmptyProof (EQUIVALENT transformation)

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

(0) Obligation:

(1) DependencyPairsProof (EQUIVALENT transformation)

(2) Obligation:

(3) DependencyGraphProof (EQUIVALENT transformation)

(4) Complex Obligation (AND)

(5) Obligation:

(6) QDPOrderProof (EQUIVALENT transformation)

(7) Obligation:

(8) QDPOrderProof (EQUIVALENT transformation)

(9) Obligation:

(10) PisEmptyProof (EQUIVALENT transformation)

(11) TRUE

(12) Obligation:

(13) QDPOrderProof (EQUIVALENT transformation)

(14) Obligation:

(15) QDPOrderProof (EQUIVALENT transformation)

(16) Obligation:

(17) PisEmptyProof (EQUIVALENT transformation)

(18) TRUE