Term Rewriting System R:
[x, y, z]
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))

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

C(x, c(y, z)) -> C(+(x, y), z)
C(x, c(y, z)) -> +'(x, y)
+'(1, 9) -> C(1, 0)
+'(2, 8) -> C(1, 0)
+'(2, 9) -> C(1, 1)
+'(3, 7) -> C(1, 0)
+'(3, 8) -> C(1, 1)
+'(3, 9) -> C(1, 2)
+'(4, 6) -> C(1, 0)
+'(4, 7) -> C(1, 1)
+'(4, 8) -> C(1, 2)
+'(4, 9) -> C(1, 3)
+'(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, 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, 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, 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, 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))
+'(x, c(y, z)) -> +'(x, z)
+'(c(x, y), z) -> C(x, +(y, z))
+'(c(x, y), z) -> +'(y, z)
*'(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, 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, 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, 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, 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, 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, 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, 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))
*'(x, c(y, z)) -> *'(x, y)
*'(x, c(y, z)) -> *'(x, z)
*'(c(x, y), z) -> C(*(x, z), *(y, z))
*'(c(x, y), z) -> *'(x, z)
*'(c(x, y), z) -> *'(y, z)

Furthermore, R contains two SCCs.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Argument Filtering and Ordering`
`       →DP Problem 2`
`         ↳AFS`

Dependency Pairs:

+'(c(x, y), z) -> +'(y, z)
+'(c(x, y), z) -> C(x, +(y, z))
+'(x, c(y, z)) -> +'(x, z)
+'(x, c(y, z)) -> C(y, +(x, z))
C(x, c(y, z)) -> +'(x, y)

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))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

+'(c(x, y), z) -> +'(y, z)
+'(x, c(y, z)) -> +'(x, z)
C(x, c(y, z)) -> +'(x, y)

The following usable rules for innermost w.r.t. to the AFS can be oriented:

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))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(4) =  0 POL(8) =  0 POL(5) =  0 POL(2) =  0 POL(6) =  0 POL(c(x1, x2)) =  1 + x1 + x2 POL(C(x1, x2)) =  x1 + x2 POL(0) =  0 POL(1) =  0 POL(9) =  0 POL(3) =  0 POL(+(x1, x2)) =  1 + x1 + x2 POL(+'(x1, x2)) =  x1 + x2 POL(7) =  0

resulting in one new DP problem.
Used Argument Filtering System:
+'(x1, x2) -> +'(x1, x2)
c(x1, x2) -> c(x1, x2)
C(x1, x2) -> C(x1, x2)
+(x1, x2) -> +(x1, x2)

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`           →DP Problem 3`
`             ↳Dependency Graph`
`       →DP Problem 2`
`         ↳AFS`

Dependency Pairs:

+'(c(x, y), z) -> C(x, +(y, z))
+'(x, c(y, z)) -> C(y, +(x, z))

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))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳Argument Filtering and Ordering`

Dependency Pairs:

*'(c(x, y), z) -> *'(y, z)
*'(c(x, y), z) -> *'(x, z)
*'(x, c(y, z)) -> *'(x, z)
*'(x, c(y, z)) -> *'(x, y)

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))

Strategy:

innermost

The following dependency pairs can be strictly oriented:

*'(c(x, y), z) -> *'(y, z)
*'(c(x, y), z) -> *'(x, z)
*'(x, c(y, z)) -> *'(x, z)
*'(x, c(y, z)) -> *'(x, y)

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(c(x1, x2)) =  1 + x1 + x2 POL(*'(x1, x2)) =  x1 + x2

resulting in one new DP problem.
Used Argument Filtering System:
*'(x1, x2) -> *'(x1, x2)
c(x1, x2) -> c(x1, x2)

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳AFS`
`           →DP Problem 4`
`             ↳Dependency Graph`

Dependency Pair:

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))

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:08 minutes