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.

`   R`
`     ↳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.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polynomial Ordering`

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

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)

Additionally, the following usable rules for innermost can be 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(0, x) -> x
c(x, c(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(3) =  0 POL(9) =  0 POL(+(x1, x2)) =  1 + x1 + x2 POL(+'(x1, x2)) =  x1 + x2 POL(7) =  0

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`           →DP Problem 2`
`             ↳Dependency Graph`

Dependency Pairs:

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

Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:02 minutes