Termination Proof using AProVE

Term Rewriting System R:
[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))
c(0, x) -> x
c(x, c(y, z)) -> c(+(x, y), z)

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Narrowing Transformation

Dependency Pairs:

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

Rules:

+(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(0, x) -> x
c(x, c(y, z)) -> c(+(x, y), z)

Strategy:

innermost

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

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

101 new Dependency Pairs are created:

+'(0, c(y, 0)) -> C(y, 0)
+'(0, c(y, 1)) -> C(y, 1)
+'(0, c(y, 2)) -> C(y, 2)
+'(0, c(y, 3)) -> C(y, 3)
+'(0, c(y, 4)) -> C(y, 4)
+'(0, c(y, 5)) -> C(y, 5)
+'(0, c(y, 6)) -> C(y, 6)
+'(0, c(y, 7)) -> C(y, 7)
+'(0, c(y, 8)) -> C(y, 8)
+'(0, c(y, 9)) -> C(y, 9)
+'(1, c(y, 0)) -> C(y, 1)
+'(1, c(y, 1)) -> C(y, 2)
+'(1, c(y, 2)) -> C(y, 3)
+'(1, c(y, 3)) -> C(y, 4)
+'(1, c(y, 4)) -> C(y, 5)
+'(1, c(y, 5)) -> C(y, 6)
+'(1, c(y, 6)) -> C(y, 7)
+'(1, c(y, 7)) -> C(y, 8)
+'(1, c(y, 8)) -> C(y, 9)
+'(1, c(y, 9)) -> C(y, c(1, 0))
+'(2, c(y, 0)) -> C(y, 2)
+'(2, c(y, 1)) -> C(y, 3)
+'(2, c(y, 2)) -> C(y, 4)
+'(2, c(y, 3)) -> C(y, 5)
+'(2, c(y, 4)) -> C(y, 6)
+'(2, c(y, 5)) -> C(y, 7)
+'(2, c(y, 6)) -> C(y, 8)
+'(2, c(y, 7)) -> C(y, 9)
+'(2, c(y, 8)) -> C(y, c(1, 0))
+'(2, c(y, 9)) -> C(y, c(1, 1))
+'(3, c(y, 0)) -> C(y, 3)
+'(3, c(y, 1)) -> C(y, 4)
+'(3, c(y, 2)) -> C(y, 5)
+'(3, c(y, 3)) -> C(y, 6)
+'(3, c(y, 4)) -> C(y, 7)
+'(3, c(y, 5)) -> C(y, 8)
+'(3, c(y, 6)) -> C(y, 9)
+'(3, c(y, 7)) -> C(y, c(1, 0))
+'(3, c(y, 8)) -> C(y, c(1, 1))
+'(3, c(y, 9)) -> C(y, c(1, 2))
+'(4, c(y, 0)) -> C(y, 4)
+'(4, c(y, 1)) -> C(y, 5)
+'(4, c(y, 2)) -> C(y, 6)
+'(4, c(y, 3)) -> C(y, 7)
+'(4, c(y, 4)) -> C(y, 8)
+'(4, c(y, 5)) -> C(y, 9)
+'(4, c(y, 6)) -> C(y, c(1, 0))
+'(4, c(y, 7)) -> C(y, c(1, 1))
+'(4, c(y, 8)) -> C(y, c(1, 2))
+'(4, c(y, 9)) -> C(y, c(1, 3))
+'(5, c(y, 0)) -> C(y, 5)
+'(5, c(y, 1)) -> C(y, 6)
+'(5, c(y, 2)) -> C(y, 7)
+'(5, c(y, 3)) -> C(y, 8)
+'(5, c(y, 4)) -> C(y, 9)
+'(5, c(y, 5)) -> C(y, c(1, 0))
+'(5, c(y, 6)) -> C(y, c(1, 1))
+'(5, c(y, 7)) -> C(y, c(1, 2))
+'(5, c(y, 8)) -> C(y, c(1, 3))
+'(5, c(y, 9)) -> C(y, c(1, 4))
+'(6, c(y, 0)) -> C(y, 6)
+'(6, c(y, 1)) -> C(y, 7)
+'(6, c(y, 2)) -> C(y, 8)
+'(6, c(y, 3)) -> C(y, 9)
+'(6, c(y, 4)) -> C(y, c(1, 0))
+'(6, c(y, 5)) -> C(y, c(1, 1))
+'(6, c(y, 6)) -> C(y, c(1, 2))
+'(6, c(y, 7)) -> C(y, c(1, 3))
+'(6, c(y, 8)) -> C(y, c(1, 4))
+'(6, c(y, 9)) -> C(y, c(1, 5))
+'(7, c(y, 0)) -> C(y, 7)
+'(7, c(y, 1)) -> C(y, 8)
+'(7, c(y, 2)) -> C(y, 9)
+'(7, c(y, 3)) -> C(y, c(1, 0))
+'(7, c(y, 4)) -> C(y, c(1, 1))
+'(7, c(y, 5)) -> C(y, c(1, 2))
+'(7, c(y, 6)) -> C(y, c(1, 3))
+'(7, c(y, 7)) -> C(y, c(1, 4))
+'(7, c(y, 8)) -> C(y, c(1, 5))
+'(7, c(y, 9)) -> C(y, c(1, 6))
+'(8, c(y, 0)) -> C(y, 8)
+'(8, c(y, 1)) -> C(y, 9)
+'(8, c(y, 2)) -> C(y, c(1, 0))
+'(8, c(y, 3)) -> C(y, c(1, 1))
+'(8, c(y, 4)) -> C(y, c(1, 2))
+'(8, c(y, 5)) -> C(y, c(1, 3))
+'(8, c(y, 6)) -> C(y, c(1, 4))
+'(8, c(y, 7)) -> C(y, c(1, 5))
+'(8, c(y, 8)) -> C(y, c(1, 6))
+'(8, c(y, 9)) -> C(y, c(1, 7))
+'(9, c(y, 0)) -> C(y, 9)
+'(9, c(y, 1)) -> C(y, c(1, 0))
+'(9, c(y, 2)) -> C(y, c(1, 1))
+'(9, c(y, 3)) -> C(y, c(1, 2))
+'(9, c(y, 4)) -> C(y, c(1, 3))
+'(9, c(y, 5)) -> C(y, c(1, 4))
+'(9, c(y, 6)) -> C(y, c(1, 5))
+'(9, c(y, 7)) -> C(y, c(1, 6))
+'(9, c(y, 8)) -> C(y, c(1, 7))
+'(9, c(y, 9)) -> C(y, c(1, 8))
+'(c(x'', y''), c(y, z'')) -> C(y, c(x'', +(y'', z'')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Narrowing Transformation

Dependency Pairs:

+'(c(x'', y''), c(y, z'')) -> C(y, c(x'', +(y'', z'')))
+'(9, c(y, 9)) -> C(y, c(1, 8))
+'(9, c(y, 8)) -> C(y, c(1, 7))
+'(9, c(y, 7)) -> C(y, c(1, 6))
+'(9, c(y, 6)) -> C(y, c(1, 5))
+'(9, c(y, 5)) -> C(y, c(1, 4))
+'(9, c(y, 4)) -> C(y, c(1, 3))
+'(9, c(y, 3)) -> C(y, c(1, 2))
+'(9, c(y, 2)) -> C(y, c(1, 1))
+'(9, c(y, 1)) -> C(y, c(1, 0))
+'(8, c(y, 9)) -> C(y, c(1, 7))
+'(8, c(y, 8)) -> C(y, c(1, 6))
+'(8, c(y, 7)) -> C(y, c(1, 5))
+'(8, c(y, 6)) -> C(y, c(1, 4))
+'(8, c(y, 5)) -> C(y, c(1, 3))
+'(8, c(y, 4)) -> C(y, c(1, 2))
+'(8, c(y, 3)) -> C(y, c(1, 1))
+'(8, c(y, 2)) -> C(y, c(1, 0))
+'(7, c(y, 9)) -> C(y, c(1, 6))
+'(7, c(y, 8)) -> C(y, c(1, 5))
+'(7, c(y, 7)) -> C(y, c(1, 4))
+'(7, c(y, 6)) -> C(y, c(1, 3))
+'(7, c(y, 5)) -> C(y, c(1, 2))
+'(7, c(y, 4)) -> C(y, c(1, 1))
+'(7, c(y, 3)) -> C(y, c(1, 0))
+'(6, c(y, 9)) -> C(y, c(1, 5))
+'(6, c(y, 8)) -> C(y, c(1, 4))
+'(6, c(y, 7)) -> C(y, c(1, 3))
+'(6, c(y, 6)) -> C(y, c(1, 2))
+'(6, c(y, 5)) -> C(y, c(1, 1))
+'(6, c(y, 4)) -> C(y, c(1, 0))
+'(5, c(y, 9)) -> C(y, c(1, 4))
+'(5, c(y, 8)) -> C(y, c(1, 3))
+'(5, c(y, 7)) -> C(y, c(1, 2))
+'(5, c(y, 6)) -> C(y, c(1, 1))
+'(5, c(y, 5)) -> C(y, c(1, 0))
+'(4, c(y, 9)) -> C(y, c(1, 3))
+'(4, c(y, 8)) -> C(y, c(1, 2))
+'(4, c(y, 7)) -> C(y, c(1, 1))
+'(4, c(y, 6)) -> C(y, c(1, 0))
+'(3, c(y, 9)) -> C(y, c(1, 2))
+'(3, c(y, 8)) -> C(y, c(1, 1))
+'(3, c(y, 7)) -> C(y, c(1, 0))
+'(2, c(y, 9)) -> C(y, c(1, 1))
+'(2, c(y, 8)) -> C(y, c(1, 0))
+'(1, c(y, 9)) -> C(y, c(1, 0))
C(x, c(y, z)) -> +'(x, y)
+'(c(x, y), z) -> C(x, +(y, z))
+'(x, c(y, z)) -> +'(x, z)
+'(c(x, y), z) -> +'(y, z)

Rules:

+(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(0, x) -> x
c(x, c(y, z)) -> c(+(x, y), z)

Strategy:

innermost

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

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

101 new Dependency Pairs are created:

+'(c(x, 0), 0) -> C(x, 0)
+'(c(x, 0), 1) -> C(x, 1)
+'(c(x, 0), 2) -> C(x, 2)
+'(c(x, 0), 3) -> C(x, 3)
+'(c(x, 0), 4) -> C(x, 4)
+'(c(x, 0), 5) -> C(x, 5)
+'(c(x, 0), 6) -> C(x, 6)
+'(c(x, 0), 7) -> C(x, 7)
+'(c(x, 0), 8) -> C(x, 8)
+'(c(x, 0), 9) -> C(x, 9)
+'(c(x, 1), 0) -> C(x, 1)
+'(c(x, 1), 1) -> C(x, 2)
+'(c(x, 1), 2) -> C(x, 3)
+'(c(x, 1), 3) -> C(x, 4)
+'(c(x, 1), 4) -> C(x, 5)
+'(c(x, 1), 5) -> C(x, 6)
+'(c(x, 1), 6) -> C(x, 7)
+'(c(x, 1), 7) -> C(x, 8)
+'(c(x, 1), 8) -> C(x, 9)
+'(c(x, 1), 9) -> C(x, c(1, 0))
+'(c(x, 2), 0) -> C(x, 2)
+'(c(x, 2), 1) -> C(x, 3)
+'(c(x, 2), 2) -> C(x, 4)
+'(c(x, 2), 3) -> C(x, 5)
+'(c(x, 2), 4) -> C(x, 6)
+'(c(x, 2), 5) -> C(x, 7)
+'(c(x, 2), 6) -> C(x, 8)
+'(c(x, 2), 7) -> C(x, 9)
+'(c(x, 2), 8) -> C(x, c(1, 0))
+'(c(x, 2), 9) -> C(x, c(1, 1))
+'(c(x, 3), 0) -> C(x, 3)
+'(c(x, 3), 1) -> C(x, 4)
+'(c(x, 3), 2) -> C(x, 5)
+'(c(x, 3), 3) -> C(x, 6)
+'(c(x, 3), 4) -> C(x, 7)
+'(c(x, 3), 5) -> C(x, 8)
+'(c(x, 3), 6) -> C(x, 9)
+'(c(x, 3), 7) -> C(x, c(1, 0))
+'(c(x, 3), 8) -> C(x, c(1, 1))
+'(c(x, 3), 9) -> C(x, c(1, 2))
+'(c(x, 4), 0) -> C(x, 4)
+'(c(x, 4), 1) -> C(x, 5)
+'(c(x, 4), 2) -> C(x, 6)
+'(c(x, 4), 3) -> C(x, 7)
+'(c(x, 4), 4) -> C(x, 8)
+'(c(x, 4), 5) -> C(x, 9)
+'(c(x, 4), 6) -> C(x, c(1, 0))
+'(c(x, 4), 7) -> C(x, c(1, 1))
+'(c(x, 4), 8) -> C(x, c(1, 2))
+'(c(x, 4), 9) -> C(x, c(1, 3))
+'(c(x, 5), 0) -> C(x, 5)
+'(c(x, 5), 1) -> C(x, 6)
+'(c(x, 5), 2) -> C(x, 7)
+'(c(x, 5), 3) -> C(x, 8)
+'(c(x, 5), 4) -> C(x, 9)
+'(c(x, 5), 5) -> C(x, c(1, 0))
+'(c(x, 5), 6) -> C(x, c(1, 1))
+'(c(x, 5), 7) -> C(x, c(1, 2))
+'(c(x, 5), 8) -> C(x, c(1, 3))
+'(c(x, 5), 9) -> C(x, c(1, 4))
+'(c(x, 6), 0) -> C(x, 6)
+'(c(x, 6), 1) -> C(x, 7)
+'(c(x, 6), 2) -> C(x, 8)
+'(c(x, 6), 3) -> C(x, 9)
+'(c(x, 6), 4) -> C(x, c(1, 0))
+'(c(x, 6), 5) -> C(x, c(1, 1))
+'(c(x, 6), 6) -> C(x, c(1, 2))
+'(c(x, 6), 7) -> C(x, c(1, 3))
+'(c(x, 6), 8) -> C(x, c(1, 4))
+'(c(x, 6), 9) -> C(x, c(1, 5))
+'(c(x, 7), 0) -> C(x, 7)
+'(c(x, 7), 1) -> C(x, 8)
+'(c(x, 7), 2) -> C(x, 9)
+'(c(x, 7), 3) -> C(x, c(1, 0))
+'(c(x, 7), 4) -> C(x, c(1, 1))
+'(c(x, 7), 5) -> C(x, c(1, 2))
+'(c(x, 7), 6) -> C(x, c(1, 3))
+'(c(x, 7), 7) -> C(x, c(1, 4))
+'(c(x, 7), 8) -> C(x, c(1, 5))
+'(c(x, 7), 9) -> C(x, c(1, 6))
+'(c(x, 8), 0) -> C(x, 8)
+'(c(x, 8), 1) -> C(x, 9)
+'(c(x, 8), 2) -> C(x, c(1, 0))
+'(c(x, 8), 3) -> C(x, c(1, 1))
+'(c(x, 8), 4) -> C(x, c(1, 2))
+'(c(x, 8), 5) -> C(x, c(1, 3))
+'(c(x, 8), 6) -> C(x, c(1, 4))
+'(c(x, 8), 7) -> C(x, c(1, 5))
+'(c(x, 8), 8) -> C(x, c(1, 6))
+'(c(x, 8), 9) -> C(x, c(1, 7))
+'(c(x, 9), 0) -> C(x, 9)
+'(c(x, 9), 1) -> C(x, c(1, 0))
+'(c(x, 9), 2) -> C(x, c(1, 1))
+'(c(x, 9), 3) -> C(x, c(1, 2))
+'(c(x, 9), 4) -> C(x, c(1, 3))
+'(c(x, 9), 5) -> C(x, c(1, 4))
+'(c(x, 9), 6) -> C(x, c(1, 5))
+'(c(x, 9), 7) -> C(x, c(1, 6))
+'(c(x, 9), 8) -> C(x, c(1, 7))
+'(c(x, 9), 9) -> C(x, c(1, 8))
+'(c(x, y0), c(y'', z'')) -> C(x, c(y'', +(y0, z'')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 3

                 ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

+'(c(x, y0), c(y'', z'')) -> C(x, c(y'', +(y0, z'')))
+'(c(x, 9), 9) -> C(x, c(1, 8))
+'(c(x, 9), 8) -> C(x, c(1, 7))
+'(c(x, 9), 7) -> C(x, c(1, 6))
+'(c(x, 9), 6) -> C(x, c(1, 5))
+'(c(x, 9), 5) -> C(x, c(1, 4))
+'(c(x, 9), 4) -> C(x, c(1, 3))
+'(c(x, 9), 3) -> C(x, c(1, 2))
+'(c(x, 9), 2) -> C(x, c(1, 1))
+'(c(x, 9), 1) -> C(x, c(1, 0))
+'(c(x, 8), 9) -> C(x, c(1, 7))
+'(c(x, 8), 8) -> C(x, c(1, 6))
+'(c(x, 8), 7) -> C(x, c(1, 5))
+'(c(x, 8), 6) -> C(x, c(1, 4))
+'(c(x, 8), 5) -> C(x, c(1, 3))
+'(c(x, 8), 4) -> C(x, c(1, 2))
+'(c(x, 8), 3) -> C(x, c(1, 1))
+'(c(x, 8), 2) -> C(x, c(1, 0))
+'(c(x, 7), 9) -> C(x, c(1, 6))
+'(c(x, 7), 8) -> C(x, c(1, 5))
+'(c(x, 7), 7) -> C(x, c(1, 4))
+'(c(x, 7), 6) -> C(x, c(1, 3))
+'(c(x, 7), 5) -> C(x, c(1, 2))
+'(c(x, 7), 4) -> C(x, c(1, 1))
+'(c(x, 7), 3) -> C(x, c(1, 0))
+'(c(x, 6), 9) -> C(x, c(1, 5))
+'(c(x, 6), 8) -> C(x, c(1, 4))
+'(c(x, 6), 7) -> C(x, c(1, 3))
+'(c(x, 6), 6) -> C(x, c(1, 2))
+'(c(x, 6), 5) -> C(x, c(1, 1))
+'(c(x, 6), 4) -> C(x, c(1, 0))
+'(c(x, 5), 9) -> C(x, c(1, 4))
+'(c(x, 5), 8) -> C(x, c(1, 3))
+'(c(x, 5), 7) -> C(x, c(1, 2))
+'(c(x, 5), 6) -> C(x, c(1, 1))
+'(c(x, 5), 5) -> C(x, c(1, 0))
+'(c(x, 4), 9) -> C(x, c(1, 3))
+'(c(x, 4), 8) -> C(x, c(1, 2))
+'(c(x, 4), 7) -> C(x, c(1, 1))
+'(c(x, 4), 6) -> C(x, c(1, 0))
+'(c(x, 3), 9) -> C(x, c(1, 2))
+'(c(x, 3), 8) -> C(x, c(1, 1))
+'(c(x, 3), 7) -> C(x, c(1, 0))
+'(c(x, 2), 9) -> C(x, c(1, 1))
+'(c(x, 2), 8) -> C(x, c(1, 0))
+'(c(x, 1), 9) -> C(x, c(1, 0))
+'(9, c(y, 9)) -> C(y, c(1, 8))
+'(9, c(y, 8)) -> C(y, c(1, 7))
+'(9, c(y, 7)) -> C(y, c(1, 6))
+'(9, c(y, 6)) -> C(y, c(1, 5))
+'(9, c(y, 5)) -> C(y, c(1, 4))
+'(9, c(y, 4)) -> C(y, c(1, 3))
+'(9, c(y, 3)) -> C(y, c(1, 2))
+'(9, c(y, 2)) -> C(y, c(1, 1))
+'(9, c(y, 1)) -> C(y, c(1, 0))
+'(8, c(y, 9)) -> C(y, c(1, 7))
+'(8, c(y, 8)) -> C(y, c(1, 6))
+'(8, c(y, 7)) -> C(y, c(1, 5))
+'(8, c(y, 6)) -> C(y, c(1, 4))
+'(8, c(y, 5)) -> C(y, c(1, 3))
+'(8, c(y, 4)) -> C(y, c(1, 2))
+'(8, c(y, 3)) -> C(y, c(1, 1))
+'(8, c(y, 2)) -> C(y, c(1, 0))
+'(7, c(y, 9)) -> C(y, c(1, 6))
+'(7, c(y, 8)) -> C(y, c(1, 5))
+'(7, c(y, 7)) -> C(y, c(1, 4))
+'(7, c(y, 6)) -> C(y, c(1, 3))
+'(7, c(y, 5)) -> C(y, c(1, 2))
+'(7, c(y, 4)) -> C(y, c(1, 1))
+'(7, c(y, 3)) -> C(y, c(1, 0))
+'(6, c(y, 9)) -> C(y, c(1, 5))
+'(6, c(y, 8)) -> C(y, c(1, 4))
+'(6, c(y, 7)) -> C(y, c(1, 3))
+'(6, c(y, 6)) -> C(y, c(1, 2))
+'(6, c(y, 5)) -> C(y, c(1, 1))
+'(6, c(y, 4)) -> C(y, c(1, 0))
+'(5, c(y, 9)) -> C(y, c(1, 4))
+'(5, c(y, 8)) -> C(y, c(1, 3))
+'(5, c(y, 7)) -> C(y, c(1, 2))
+'(5, c(y, 6)) -> C(y, c(1, 1))
+'(5, c(y, 5)) -> C(y, c(1, 0))
+'(4, c(y, 9)) -> C(y, c(1, 3))
+'(4, c(y, 8)) -> C(y, c(1, 2))
+'(4, c(y, 7)) -> C(y, c(1, 1))
+'(4, c(y, 6)) -> C(y, c(1, 0))
+'(3, c(y, 9)) -> C(y, c(1, 2))
+'(3, c(y, 8)) -> C(y, c(1, 1))
+'(3, c(y, 7)) -> C(y, c(1, 0))
+'(2, c(y, 9)) -> C(y, c(1, 1))
+'(2, c(y, 8)) -> C(y, c(1, 0))
+'(1, c(y, 9)) -> C(y, c(1, 0))
+'(c(x, y), z) -> +'(y, z)
+'(x, c(y, z)) -> +'(x, z)
C(x, c(y, z)) -> +'(x, y)
+'(c(x'', y''), c(y, z'')) -> C(y, c(x'', +(y'', z'')))

Rules:

+(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(0, x) -> x
c(x, c(y, z)) -> c(+(x, y), z)

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:39 minutes