YES

Problem 1: 

(VAR v_NonEmpty:S x:S y:S z:S)
(RULES
+(c(x:S,y:S),z:S) -> c(x:S,+(y:S,z:S))
+(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:S,c(y:S,z:S)) -> c(y:S,+(x:S,z:S))
c(0,x:S) -> x:S
c(x:S,c(y:S,z:S)) -> c(+(x:S,y:S),z:S)
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 +#(c(x:S,y:S),z:S) -> +#(y:S,z:S)
 +#(c(x:S,y:S),z:S) -> C(x:S,+(y:S,z:S))
 +#(x:S,c(y:S,z:S)) -> +#(x:S,z:S)
 +#(x:S,c(y:S,z:S)) -> C(y:S,+(x:S,z:S))
 C(x:S,c(y:S,z:S)) -> +#(x:S,y:S)
 C(x:S,c(y:S,z:S)) -> C(+(x:S,y:S),z:S)
-> Rules:
 +(c(x:S,y:S),z:S) -> c(x:S,+(y:S,z:S))
 +(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:S,c(y:S,z:S)) -> c(y:S,+(x:S,z:S))
 c(0,x:S) -> x:S
 c(x:S,c(y:S,z:S)) -> c(+(x:S,y:S),z:S)

Problem 1: 

SCC Processor:
-> Pairs:
 +#(c(x:S,y:S),z:S) -> +#(y:S,z:S)
 +#(c(x:S,y:S),z:S) -> C(x:S,+(y:S,z:S))
 +#(x:S,c(y:S,z:S)) -> +#(x:S,z:S)
 +#(x:S,c(y:S,z:S)) -> C(y:S,+(x:S,z:S))
 C(x:S,c(y:S,z:S)) -> +#(x:S,y:S)
 C(x:S,c(y:S,z:S)) -> C(+(x:S,y:S),z:S)
-> Rules:
 +(c(x:S,y:S),z:S) -> c(x:S,+(y:S,z:S))
 +(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:S,c(y:S,z:S)) -> c(y:S,+(x:S,z:S))
 c(0,x:S) -> x:S
 c(x:S,c(y:S,z:S)) -> c(+(x:S,y:S),z:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 +#(c(x:S,y:S),z:S) -> +#(y:S,z:S)
 +#(c(x:S,y:S),z:S) -> C(x:S,+(y:S,z:S))
 +#(x:S,c(y:S,z:S)) -> +#(x:S,z:S)
 +#(x:S,c(y:S,z:S)) -> C(y:S,+(x:S,z:S))
 C(x:S,c(y:S,z:S)) -> +#(x:S,y:S)
 C(x:S,c(y:S,z:S)) -> C(+(x:S,y:S),z:S)
->->-> Rules:
 +(c(x:S,y:S),z:S) -> c(x:S,+(y:S,z:S))
 +(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:S,c(y:S,z:S)) -> c(y:S,+(x:S,z:S))
 c(0,x:S) -> x:S
 c(x:S,c(y:S,z:S)) -> c(+(x:S,y:S),z:S)

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 +#(c(x:S,y:S),z:S) -> +#(y:S,z:S)
 +#(c(x:S,y:S),z:S) -> C(x:S,+(y:S,z:S))
 +#(x:S,c(y:S,z:S)) -> +#(x:S,z:S)
 +#(x:S,c(y:S,z:S)) -> C(y:S,+(x:S,z:S))
 C(x:S,c(y:S,z:S)) -> +#(x:S,y:S)
 C(x:S,c(y:S,z:S)) -> C(+(x:S,y:S),z:S)
-> Rules:
 +(c(x:S,y:S),z:S) -> c(x:S,+(y:S,z:S))
 +(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:S,c(y:S,z:S)) -> c(y:S,+(x:S,z:S))
 c(0,x:S) -> x:S
 c(x:S,c(y:S,z:S)) -> c(+(x:S,y:S),z:S)
-> Usable rules:
 +(c(x:S,y:S),z:S) -> c(x:S,+(y:S,z:S))
 +(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:S,c(y:S,z:S)) -> c(y:S,+(x:S,z:S))
 c(0,x:S) -> x:S
 c(x:S,c(y:S,z:S)) -> c(+(x:S,y:S),z:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[+](X1,X2) = X1 + X2 + 2
[c](X1,X2) = X1 + X2 + 2
[0] = 2
[1] = 0
[2] = 0
[3] = 0
[4] = 2
[5] = 2
[6] = 0
[7] = 2
[8] = 2
[9] = 2
[+#](X1,X2) = 2.X1 + 2.X2 + 2
[C](X1,X2) = 2.X1 + 2.X2 + 2

Problem 1: 

SCC Processor:
-> Pairs:
 +#(c(x:S,y:S),z:S) -> C(x:S,+(y:S,z:S))
 +#(x:S,c(y:S,z:S)) -> +#(x:S,z:S)
 +#(x:S,c(y:S,z:S)) -> C(y:S,+(x:S,z:S))
 C(x:S,c(y:S,z:S)) -> +#(x:S,y:S)
 C(x:S,c(y:S,z:S)) -> C(+(x:S,y:S),z:S)
-> Rules:
 +(c(x:S,y:S),z:S) -> c(x:S,+(y:S,z:S))
 +(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:S,c(y:S,z:S)) -> c(y:S,+(x:S,z:S))
 c(0,x:S) -> x:S
 c(x:S,c(y:S,z:S)) -> c(+(x:S,y:S),z:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 +#(c(x:S,y:S),z:S) -> C(x:S,+(y:S,z:S))
 +#(x:S,c(y:S,z:S)) -> +#(x:S,z:S)
 +#(x:S,c(y:S,z:S)) -> C(y:S,+(x:S,z:S))
 C(x:S,c(y:S,z:S)) -> +#(x:S,y:S)
 C(x:S,c(y:S,z:S)) -> C(+(x:S,y:S),z:S)
->->-> Rules:
 +(c(x:S,y:S),z:S) -> c(x:S,+(y:S,z:S))
 +(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:S,c(y:S,z:S)) -> c(y:S,+(x:S,z:S))
 c(0,x:S) -> x:S
 c(x:S,c(y:S,z:S)) -> c(+(x:S,y:S),z:S)

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 +#(c(x:S,y:S),z:S) -> C(x:S,+(y:S,z:S))
 +#(x:S,c(y:S,z:S)) -> +#(x:S,z:S)
 +#(x:S,c(y:S,z:S)) -> C(y:S,+(x:S,z:S))
 C(x:S,c(y:S,z:S)) -> +#(x:S,y:S)
 C(x:S,c(y:S,z:S)) -> C(+(x:S,y:S),z:S)
-> Rules:
 +(c(x:S,y:S),z:S) -> c(x:S,+(y:S,z:S))
 +(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:S,c(y:S,z:S)) -> c(y:S,+(x:S,z:S))
 c(0,x:S) -> x:S
 c(x:S,c(y:S,z:S)) -> c(+(x:S,y:S),z:S)
-> Usable rules:
 +(c(x:S,y:S),z:S) -> c(x:S,+(y:S,z:S))
 +(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:S,c(y:S,z:S)) -> c(y:S,+(x:S,z:S))
 c(0,x:S) -> x:S
 c(x:S,c(y:S,z:S)) -> c(+(x:S,y:S),z:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[+](X1,X2) = X1 + X2 + 2
[c](X1,X2) = X1 + X2 + 2
[0] = 1
[1] = 0
[2] = 0
[3] = 1
[4] = 0
[5] = 1
[6] = 1
[7] = 0
[8] = 2
[9] = 2
[+#](X1,X2) = 2.X1 + 2.X2 + 2
[C](X1,X2) = 2.X1 + 2.X2 + 1

Problem 1: 

SCC Processor:
-> Pairs:
 +#(x:S,c(y:S,z:S)) -> +#(x:S,z:S)
 +#(x:S,c(y:S,z:S)) -> C(y:S,+(x:S,z:S))
 C(x:S,c(y:S,z:S)) -> +#(x:S,y:S)
 C(x:S,c(y:S,z:S)) -> C(+(x:S,y:S),z:S)
-> Rules:
 +(c(x:S,y:S),z:S) -> c(x:S,+(y:S,z:S))
 +(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:S,c(y:S,z:S)) -> c(y:S,+(x:S,z:S))
 c(0,x:S) -> x:S
 c(x:S,c(y:S,z:S)) -> c(+(x:S,y:S),z:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 +#(x:S,c(y:S,z:S)) -> +#(x:S,z:S)
 +#(x:S,c(y:S,z:S)) -> C(y:S,+(x:S,z:S))
 C(x:S,c(y:S,z:S)) -> +#(x:S,y:S)
 C(x:S,c(y:S,z:S)) -> C(+(x:S,y:S),z:S)
->->-> Rules:
 +(c(x:S,y:S),z:S) -> c(x:S,+(y:S,z:S))
 +(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:S,c(y:S,z:S)) -> c(y:S,+(x:S,z:S))
 c(0,x:S) -> x:S
 c(x:S,c(y:S,z:S)) -> c(+(x:S,y:S),z:S)

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 +#(x:S,c(y:S,z:S)) -> +#(x:S,z:S)
 +#(x:S,c(y:S,z:S)) -> C(y:S,+(x:S,z:S))
 C(x:S,c(y:S,z:S)) -> +#(x:S,y:S)
 C(x:S,c(y:S,z:S)) -> C(+(x:S,y:S),z:S)
-> Rules:
 +(c(x:S,y:S),z:S) -> c(x:S,+(y:S,z:S))
 +(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:S,c(y:S,z:S)) -> c(y:S,+(x:S,z:S))
 c(0,x:S) -> x:S
 c(x:S,c(y:S,z:S)) -> c(+(x:S,y:S),z:S)
-> Usable rules:
 +(c(x:S,y:S),z:S) -> c(x:S,+(y:S,z:S))
 +(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:S,c(y:S,z:S)) -> c(y:S,+(x:S,z:S))
 c(0,x:S) -> x:S
 c(x:S,c(y:S,z:S)) -> c(+(x:S,y:S),z:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[+](X1,X2) = X1 + X2 + 2
[c](X1,X2) = X1 + X2 + 2
[0] = 1
[1] = 1
[2] = 1
[3] = 0
[4] = 1
[5] = 1
[6] = 2
[7] = 2
[8] = 2
[9] = 2
[+#](X1,X2) = 2.X1 + 2.X2 + 2
[C](X1,X2) = 2.X1 + 2.X2

Problem 1: 

SCC Processor:
-> Pairs:
 +#(x:S,c(y:S,z:S)) -> C(y:S,+(x:S,z:S))
 C(x:S,c(y:S,z:S)) -> +#(x:S,y:S)
 C(x:S,c(y:S,z:S)) -> C(+(x:S,y:S),z:S)
-> Rules:
 +(c(x:S,y:S),z:S) -> c(x:S,+(y:S,z:S))
 +(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:S,c(y:S,z:S)) -> c(y:S,+(x:S,z:S))
 c(0,x:S) -> x:S
 c(x:S,c(y:S,z:S)) -> c(+(x:S,y:S),z:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 +#(x:S,c(y:S,z:S)) -> C(y:S,+(x:S,z:S))
 C(x:S,c(y:S,z:S)) -> +#(x:S,y:S)
 C(x:S,c(y:S,z:S)) -> C(+(x:S,y:S),z:S)
->->-> Rules:
 +(c(x:S,y:S),z:S) -> c(x:S,+(y:S,z:S))
 +(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:S,c(y:S,z:S)) -> c(y:S,+(x:S,z:S))
 c(0,x:S) -> x:S
 c(x:S,c(y:S,z:S)) -> c(+(x:S,y:S),z:S)

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 +#(x:S,c(y:S,z:S)) -> C(y:S,+(x:S,z:S))
 C(x:S,c(y:S,z:S)) -> +#(x:S,y:S)
 C(x:S,c(y:S,z:S)) -> C(+(x:S,y:S),z:S)
-> Rules:
 +(c(x:S,y:S),z:S) -> c(x:S,+(y:S,z:S))
 +(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:S,c(y:S,z:S)) -> c(y:S,+(x:S,z:S))
 c(0,x:S) -> x:S
 c(x:S,c(y:S,z:S)) -> c(+(x:S,y:S),z:S)
-> Usable rules:
 +(c(x:S,y:S),z:S) -> c(x:S,+(y:S,z:S))
 +(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:S,c(y:S,z:S)) -> c(y:S,+(x:S,z:S))
 c(0,x:S) -> x:S
 c(x:S,c(y:S,z:S)) -> c(+(x:S,y:S),z:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[+](X1,X2) = X1 + X2 + 1
[c](X1,X2) = 2.X1 + X2 + 2
[0] = 1
[1] = 0
[2] = 1
[3] = 2
[4] = 2
[5] = 2
[6] = 1
[7] = 2
[8] = 2
[9] = 2
[+#](X1,X2) = 2.X1 + X2 + 2
[C](X1,X2) = 2.X1 + X2 + 2

Problem 1: 

SCC Processor:
-> Pairs:
 C(x:S,c(y:S,z:S)) -> +#(x:S,y:S)
 C(x:S,c(y:S,z:S)) -> C(+(x:S,y:S),z:S)
-> Rules:
 +(c(x:S,y:S),z:S) -> c(x:S,+(y:S,z:S))
 +(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:S,c(y:S,z:S)) -> c(y:S,+(x:S,z:S))
 c(0,x:S) -> x:S
 c(x:S,c(y:S,z:S)) -> c(+(x:S,y:S),z:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 C(x:S,c(y:S,z:S)) -> C(+(x:S,y:S),z:S)
->->-> Rules:
 +(c(x:S,y:S),z:S) -> c(x:S,+(y:S,z:S))
 +(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:S,c(y:S,z:S)) -> c(y:S,+(x:S,z:S))
 c(0,x:S) -> x:S
 c(x:S,c(y:S,z:S)) -> c(+(x:S,y:S),z:S)

Problem 1: 

Subterm Processor:
-> Pairs:
 C(x:S,c(y:S,z:S)) -> C(+(x:S,y:S),z:S)
-> Rules:
 +(c(x:S,y:S),z:S) -> c(x:S,+(y:S,z:S))
 +(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:S,c(y:S,z:S)) -> c(y:S,+(x:S,z:S))
 c(0,x:S) -> x:S
 c(x:S,c(y:S,z:S)) -> c(+(x:S,y:S),z:S)
->Projection:
 pi(C) = 2

Problem 1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 +(c(x:S,y:S),z:S) -> c(x:S,+(y:S,z:S))
 +(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:S,c(y:S,z:S)) -> c(y:S,+(x:S,z:S))
 c(0,x:S) -> x:S
 c(x:S,c(y:S,z:S)) -> c(+(x:S,y:S),z:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.