YES

Problem 1: 

(VAR b x y)
(THEORY 
(AC * + U))
(RULES
*(0,x) -> 0
*(s(x),s(y)) -> s(+(+(x,y),*(x,y)))
+(0,x) -> x
+(s(x),s(y)) -> s(s(+(x,y)))
U(empty,b) -> b
prod(U(x,y)) -> *(prod(x),prod(y))
prod(empty) -> s(0)
prod(singl(x)) -> x
sum(U(x,y)) -> +(sum(x),sum(y))
sum(empty) -> 0
sum(singl(x)) -> x
)

Problem 1: 

Dependency Pairs Processor:
-> FAxioms:
 *#(*(x3,x4),x5) = *#(x3,*(x4,x5))
 *#(x3,x4) = *#(x4,x3)
 +#(+(x3,x4),x5) = +#(x3,+(x4,x5))
 +#(x3,x4) = +#(x4,x3)
 U#(U(x3,x4),x5) = U#(x3,U(x4,x5))
 U#(x3,x4) = U#(x4,x3)
-> Pairs:
 *#(*(0,x),x3) -> *#(0,x3)
 *#(*(s(x),s(y)),x3) -> *#(s(+(+(x,y),*(x,y))),x3)
 *#(*(s(x),s(y)),x3) -> *#(x,y)
 *#(*(s(x),s(y)),x3) -> +#(+(x,y),*(x,y))
 *#(*(s(x),s(y)),x3) -> +#(x,y)
 *#(s(x),s(y)) -> *#(x,y)
 *#(s(x),s(y)) -> +#(+(x,y),*(x,y))
 *#(s(x),s(y)) -> +#(x,y)
 +#(+(0,x),x3) -> +#(x,x3)
 +#(+(s(x),s(y)),x3) -> +#(s(s(+(x,y))),x3)
 +#(+(s(x),s(y)),x3) -> +#(x,y)
 +#(s(x),s(y)) -> +#(x,y)
 U#(U(empty,b),x3) -> U#(b,x3)
 PROD(U(x,y)) -> *#(prod(x),prod(y))
 PROD(U(x,y)) -> PROD(x)
 PROD(U(x,y)) -> PROD(y)
 SUM(U(x,y)) -> +#(sum(x),sum(y))
 SUM(U(x,y)) -> SUM(x)
 SUM(U(x,y)) -> SUM(y)
-> EAxioms:
 *(*(x3,x4),x5) = *(x3,*(x4,x5))
 *(x3,x4) = *(x4,x3)
 +(+(x3,x4),x5) = +(x3,+(x4,x5))
 +(x3,x4) = +(x4,x3)
 U(U(x3,x4),x5) = U(x3,U(x4,x5))
 U(x3,x4) = U(x4,x3)
-> Rules:
 *(0,x) -> 0
 *(s(x),s(y)) -> s(+(+(x,y),*(x,y)))
 +(0,x) -> x
 +(s(x),s(y)) -> s(s(+(x,y)))
 U(empty,b) -> b
 prod(U(x,y)) -> *(prod(x),prod(y))
 prod(empty) -> s(0)
 prod(singl(x)) -> x
 sum(U(x,y)) -> +(sum(x),sum(y))
 sum(empty) -> 0
 sum(singl(x)) -> x
-> SRules:
 *#(*(x3,x4),x5) -> *#(x3,x4)
 *#(x3,*(x4,x5)) -> *#(x4,x5)
 +#(+(x3,x4),x5) -> +#(x3,x4)
 +#(x3,+(x4,x5)) -> +#(x4,x5)
 U#(U(x3,x4),x5) -> U#(x3,x4)
 U#(x3,U(x4,x5)) -> U#(x4,x5)

Problem 1: 

SCC Processor:
-> FAxioms:
 *#(*(x3,x4),x5) = *#(x3,*(x4,x5))
 *#(x3,x4) = *#(x4,x3)
 +#(+(x3,x4),x5) = +#(x3,+(x4,x5))
 +#(x3,x4) = +#(x4,x3)
 U#(U(x3,x4),x5) = U#(x3,U(x4,x5))
 U#(x3,x4) = U#(x4,x3)
-> Pairs:
 *#(*(0,x),x3) -> *#(0,x3)
 *#(*(s(x),s(y)),x3) -> *#(s(+(+(x,y),*(x,y))),x3)
 *#(*(s(x),s(y)),x3) -> *#(x,y)
 *#(*(s(x),s(y)),x3) -> +#(+(x,y),*(x,y))
 *#(*(s(x),s(y)),x3) -> +#(x,y)
 *#(s(x),s(y)) -> *#(x,y)
 *#(s(x),s(y)) -> +#(+(x,y),*(x,y))
 *#(s(x),s(y)) -> +#(x,y)
 +#(+(0,x),x3) -> +#(x,x3)
 +#(+(s(x),s(y)),x3) -> +#(s(s(+(x,y))),x3)
 +#(+(s(x),s(y)),x3) -> +#(x,y)
 +#(s(x),s(y)) -> +#(x,y)
 U#(U(empty,b),x3) -> U#(b,x3)
 PROD(U(x,y)) -> *#(prod(x),prod(y))
 PROD(U(x,y)) -> PROD(x)
 PROD(U(x,y)) -> PROD(y)
 SUM(U(x,y)) -> +#(sum(x),sum(y))
 SUM(U(x,y)) -> SUM(x)
 SUM(U(x,y)) -> SUM(y)
-> EAxioms:
 *(*(x3,x4),x5) = *(x3,*(x4,x5))
 *(x3,x4) = *(x4,x3)
 +(+(x3,x4),x5) = +(x3,+(x4,x5))
 +(x3,x4) = +(x4,x3)
 U(U(x3,x4),x5) = U(x3,U(x4,x5))
 U(x3,x4) = U(x4,x3)
-> Rules:
 *(0,x) -> 0
 *(s(x),s(y)) -> s(+(+(x,y),*(x,y)))
 +(0,x) -> x
 +(s(x),s(y)) -> s(s(+(x,y)))
 U(empty,b) -> b
 prod(U(x,y)) -> *(prod(x),prod(y))
 prod(empty) -> s(0)
 prod(singl(x)) -> x
 sum(U(x,y)) -> +(sum(x),sum(y))
 sum(empty) -> 0
 sum(singl(x)) -> x
-> SRules:
 *#(*(x3,x4),x5) -> *#(x3,x4)
 *#(x3,*(x4,x5)) -> *#(x4,x5)
 +#(+(x3,x4),x5) -> +#(x3,x4)
 +#(x3,+(x4,x5)) -> +#(x4,x5)
 U#(U(x3,x4),x5) -> U#(x3,x4)
 U#(x3,U(x4,x5)) -> U#(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 U#(U(empty,b),x3) -> U#(b,x3)
-> FAxioms:
 *(*(x3,x4),x5) -> *(x3,*(x4,x5))
 *(x3,x4) -> *(x4,x3)
 +(+(x3,x4),x5) -> +(x3,+(x4,x5))
 +(x3,x4) -> +(x4,x3)
 U(U(x3,x4),x5) -> U(x3,U(x4,x5))
 U(x3,x4) -> U(x4,x3)
 U#(U(x3,x4),x5) -> U#(x3,U(x4,x5))
 U#(x3,x4) -> U#(x4,x3)
-> EAxioms:
 *(*(x3,x4),x5) = *(x3,*(x4,x5))
 *(x3,x4) = *(x4,x3)
 +(+(x3,x4),x5) = +(x3,+(x4,x5))
 +(x3,x4) = +(x4,x3)
 U(U(x3,x4),x5) = U(x3,U(x4,x5))
 U(x3,x4) = U(x4,x3)
->->-> Rules:
 *(0,x) -> 0
 *(s(x),s(y)) -> s(+(+(x,y),*(x,y)))
 +(0,x) -> x
 +(s(x),s(y)) -> s(s(+(x,y)))
 U(empty,b) -> b
 prod(U(x,y)) -> *(prod(x),prod(y))
 prod(empty) -> s(0)
 prod(singl(x)) -> x
 sum(U(x,y)) -> +(sum(x),sum(y))
 sum(empty) -> 0
 sum(singl(x)) -> x
-> SRules:
 U#(U(x3,x4),x5) -> U#(x3,x4)
 U#(x3,U(x4,x5)) -> U#(x4,x5)
->->Cycle:
->->-> Pairs:
 +#(+(0,x),x3) -> +#(x,x3)
 +#(+(s(x),s(y)),x3) -> +#(s(s(+(x,y))),x3)
 +#(+(s(x),s(y)),x3) -> +#(x,y)
 +#(s(x),s(y)) -> +#(x,y)
-> FAxioms:
 *(*(x3,x4),x5) -> *(x3,*(x4,x5))
 *(x3,x4) -> *(x4,x3)
 +(+(x3,x4),x5) -> +(x3,+(x4,x5))
 +(x3,x4) -> +(x4,x3)
 U(U(x3,x4),x5) -> U(x3,U(x4,x5))
 U(x3,x4) -> U(x4,x3)
 +#(+(x3,x4),x5) -> +#(x3,+(x4,x5))
 +#(x3,x4) -> +#(x4,x3)
-> EAxioms:
 *(*(x3,x4),x5) = *(x3,*(x4,x5))
 *(x3,x4) = *(x4,x3)
 +(+(x3,x4),x5) = +(x3,+(x4,x5))
 +(x3,x4) = +(x4,x3)
 U(U(x3,x4),x5) = U(x3,U(x4,x5))
 U(x3,x4) = U(x4,x3)
->->-> Rules:
 *(0,x) -> 0
 *(s(x),s(y)) -> s(+(+(x,y),*(x,y)))
 +(0,x) -> x
 +(s(x),s(y)) -> s(s(+(x,y)))
 U(empty,b) -> b
 prod(U(x,y)) -> *(prod(x),prod(y))
 prod(empty) -> s(0)
 prod(singl(x)) -> x
 sum(U(x,y)) -> +(sum(x),sum(y))
 sum(empty) -> 0
 sum(singl(x)) -> x
-> SRules:
 +#(+(x3,x4),x5) -> +#(x3,x4)
 +#(x3,+(x4,x5)) -> +#(x4,x5)
->->Cycle:
->->-> Pairs:
 SUM(U(x,y)) -> SUM(x)
 SUM(U(x,y)) -> SUM(y)
-> FAxioms:
 *(*(x3,x4),x5) -> *(x3,*(x4,x5))
 *(x3,x4) -> *(x4,x3)
 +(+(x3,x4),x5) -> +(x3,+(x4,x5))
 +(x3,x4) -> +(x4,x3)
 U(U(x3,x4),x5) -> U(x3,U(x4,x5))
 U(x3,x4) -> U(x4,x3)
-> EAxioms:
 *(*(x3,x4),x5) = *(x3,*(x4,x5))
 *(x3,x4) = *(x4,x3)
 +(+(x3,x4),x5) = +(x3,+(x4,x5))
 +(x3,x4) = +(x4,x3)
 U(U(x3,x4),x5) = U(x3,U(x4,x5))
 U(x3,x4) = U(x4,x3)
->->-> Rules:
 *(0,x) -> 0
 *(s(x),s(y)) -> s(+(+(x,y),*(x,y)))
 +(0,x) -> x
 +(s(x),s(y)) -> s(s(+(x,y)))
 U(empty,b) -> b
 prod(U(x,y)) -> *(prod(x),prod(y))
 prod(empty) -> s(0)
 prod(singl(x)) -> x
 sum(U(x,y)) -> +(sum(x),sum(y))
 sum(empty) -> 0
 sum(singl(x)) -> x
-> SRules:
 Empty
->->Cycle:
->->-> Pairs:
 *#(*(0,x),x3) -> *#(0,x3)
 *#(*(s(x),s(y)),x3) -> *#(s(+(+(x,y),*(x,y))),x3)
 *#(*(s(x),s(y)),x3) -> *#(x,y)
 *#(s(x),s(y)) -> *#(x,y)
-> FAxioms:
 *(*(x3,x4),x5) -> *(x3,*(x4,x5))
 *(x3,x4) -> *(x4,x3)
 +(+(x3,x4),x5) -> +(x3,+(x4,x5))
 +(x3,x4) -> +(x4,x3)
 U(U(x3,x4),x5) -> U(x3,U(x4,x5))
 U(x3,x4) -> U(x4,x3)
 *#(*(x3,x4),x5) -> *#(x3,*(x4,x5))
 *#(x3,x4) -> *#(x4,x3)
-> EAxioms:
 *(*(x3,x4),x5) = *(x3,*(x4,x5))
 *(x3,x4) = *(x4,x3)
 +(+(x3,x4),x5) = +(x3,+(x4,x5))
 +(x3,x4) = +(x4,x3)
 U(U(x3,x4),x5) = U(x3,U(x4,x5))
 U(x3,x4) = U(x4,x3)
->->-> Rules:
 *(0,x) -> 0
 *(s(x),s(y)) -> s(+(+(x,y),*(x,y)))
 +(0,x) -> x
 +(s(x),s(y)) -> s(s(+(x,y)))
 U(empty,b) -> b
 prod(U(x,y)) -> *(prod(x),prod(y))
 prod(empty) -> s(0)
 prod(singl(x)) -> x
 sum(U(x,y)) -> +(sum(x),sum(y))
 sum(empty) -> 0
 sum(singl(x)) -> x
-> SRules:
 *#(*(x3,x4),x5) -> *#(x3,x4)
 *#(x3,*(x4,x5)) -> *#(x4,x5)
->->Cycle:
->->-> Pairs:
 PROD(U(x,y)) -> PROD(x)
 PROD(U(x,y)) -> PROD(y)
-> FAxioms:
 *(*(x3,x4),x5) -> *(x3,*(x4,x5))
 *(x3,x4) -> *(x4,x3)
 +(+(x3,x4),x5) -> +(x3,+(x4,x5))
 +(x3,x4) -> +(x4,x3)
 U(U(x3,x4),x5) -> U(x3,U(x4,x5))
 U(x3,x4) -> U(x4,x3)
-> EAxioms:
 *(*(x3,x4),x5) = *(x3,*(x4,x5))
 *(x3,x4) = *(x4,x3)
 +(+(x3,x4),x5) = +(x3,+(x4,x5))
 +(x3,x4) = +(x4,x3)
 U(U(x3,x4),x5) = U(x3,U(x4,x5))
 U(x3,x4) = U(x4,x3)
->->-> Rules:
 *(0,x) -> 0
 *(s(x),s(y)) -> s(+(+(x,y),*(x,y)))
 +(0,x) -> x
 +(s(x),s(y)) -> s(s(+(x,y)))
 U(empty,b) -> b
 prod(U(x,y)) -> *(prod(x),prod(y))
 prod(empty) -> s(0)
 prod(singl(x)) -> x
 sum(U(x,y)) -> +(sum(x),sum(y))
 sum(empty) -> 0
 sum(singl(x)) -> x
-> SRules:
 Empty


The problem is decomposed in 5 subproblems.

Problem 1.1: 

Reduction Pairs Processor:
-> FAxioms:
 U#(U(x3,x4),x5) = U#(x3,U(x4,x5))
 U#(x3,x4) = U#(x4,x3)
-> Pairs:
 U#(U(empty,b),x3) -> U#(b,x3)
-> EAxioms:
 *(*(x3,x4),x5) = *(x3,*(x4,x5))
 *(x3,x4) = *(x4,x3)
 +(+(x3,x4),x5) = +(x3,+(x4,x5))
 +(x3,x4) = +(x4,x3)
 U(U(x3,x4),x5) = U(x3,U(x4,x5))
 U(x3,x4) = U(x4,x3)
-> Usable Equations:
 U(U(x3,x4),x5) = U(x3,U(x4,x5))
 U(x3,x4) = U(x4,x3)
-> Rules:
 *(0,x) -> 0
 *(s(x),s(y)) -> s(+(+(x,y),*(x,y)))
 +(0,x) -> x
 +(s(x),s(y)) -> s(s(+(x,y)))
 U(empty,b) -> b
 prod(U(x,y)) -> *(prod(x),prod(y))
 prod(empty) -> s(0)
 prod(singl(x)) -> x
 sum(U(x,y)) -> +(sum(x),sum(y))
 sum(empty) -> 0
 sum(singl(x)) -> x
-> Usable Rules:
 U(empty,b) -> b
-> SRules:
 U#(U(x3,x4),x5) -> U#(x3,x4)
 U#(x3,U(x4,x5)) -> U#(x4,x5)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[*](X1,X2) = 0
[+](X1,X2) = 0
[U](X1,X2) = X1 + X2
[prod](X) = 0
[sum](X) = 0
[0] = 0
[empty] = 2
[s](X) = 0
[singl](X) = 0
[*#](X1,X2) = 0
[+#](X1,X2) = 0
[U#](X1,X2) = 2.X1 + 2.X2
[PROD](X) = 0
[SUM](X) = 0

Problem 1.1: 

SCC Processor:
-> FAxioms:
 U#(U(x3,x4),x5) = U#(x3,U(x4,x5))
 U#(x3,x4) = U#(x4,x3)
-> Pairs:
 Empty
-> EAxioms:
 *(*(x3,x4),x5) = *(x3,*(x4,x5))
 *(x3,x4) = *(x4,x3)
 +(+(x3,x4),x5) = +(x3,+(x4,x5))
 +(x3,x4) = +(x4,x3)
 U(U(x3,x4),x5) = U(x3,U(x4,x5))
 U(x3,x4) = U(x4,x3)
-> Rules:
 *(0,x) -> 0
 *(s(x),s(y)) -> s(+(+(x,y),*(x,y)))
 +(0,x) -> x
 +(s(x),s(y)) -> s(s(+(x,y)))
 U(empty,b) -> b
 prod(U(x,y)) -> *(prod(x),prod(y))
 prod(empty) -> s(0)
 prod(singl(x)) -> x
 sum(U(x,y)) -> +(sum(x),sum(y))
 sum(empty) -> 0
 sum(singl(x)) -> x
-> SRules:
 U#(U(x3,x4),x5) -> U#(x3,x4)
 U#(x3,U(x4,x5)) -> U#(x4,x5)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Reduction Pairs Processor:
-> FAxioms:
 +#(+(x3,x4),x5) = +#(x3,+(x4,x5))
 +#(x3,x4) = +#(x4,x3)
-> Pairs:
 +#(+(0,x),x3) -> +#(x,x3)
 +#(+(s(x),s(y)),x3) -> +#(s(s(+(x,y))),x3)
 +#(+(s(x),s(y)),x3) -> +#(x,y)
 +#(s(x),s(y)) -> +#(x,y)
-> EAxioms:
 *(*(x3,x4),x5) = *(x3,*(x4,x5))
 *(x3,x4) = *(x4,x3)
 +(+(x3,x4),x5) = +(x3,+(x4,x5))
 +(x3,x4) = +(x4,x3)
 U(U(x3,x4),x5) = U(x3,U(x4,x5))
 U(x3,x4) = U(x4,x3)
-> Usable Equations:
 +(+(x3,x4),x5) = +(x3,+(x4,x5))
 +(x3,x4) = +(x4,x3)
-> Rules:
 *(0,x) -> 0
 *(s(x),s(y)) -> s(+(+(x,y),*(x,y)))
 +(0,x) -> x
 +(s(x),s(y)) -> s(s(+(x,y)))
 U(empty,b) -> b
 prod(U(x,y)) -> *(prod(x),prod(y))
 prod(empty) -> s(0)
 prod(singl(x)) -> x
 sum(U(x,y)) -> +(sum(x),sum(y))
 sum(empty) -> 0
 sum(singl(x)) -> x
-> Usable Rules:
 +(0,x) -> x
 +(s(x),s(y)) -> s(s(+(x,y)))
-> SRules:
 +#(+(x3,x4),x5) -> +#(x3,x4)
 +#(x3,+(x4,x5)) -> +#(x4,x5)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[*](X1,X2) = 0
[+](X1,X2) = X1 + X2 + 2
[U](X1,X2) = 0
[prod](X) = 0
[sum](X) = 0
[0] = 2
[empty] = 0
[s](X) = X + 2
[singl](X) = 0
[*#](X1,X2) = 0
[+#](X1,X2) = 2.X1 + 2.X2
[U#](X1,X2) = 0
[PROD](X) = 0
[SUM](X) = 0

Problem 1.2: 

SCC Processor:
-> FAxioms:
 +#(+(x3,x4),x5) = +#(x3,+(x4,x5))
 +#(x3,x4) = +#(x4,x3)
-> Pairs:
 +#(+(s(x),s(y)),x3) -> +#(s(s(+(x,y))),x3)
 +#(+(s(x),s(y)),x3) -> +#(x,y)
 +#(s(x),s(y)) -> +#(x,y)
-> EAxioms:
 *(*(x3,x4),x5) = *(x3,*(x4,x5))
 *(x3,x4) = *(x4,x3)
 +(+(x3,x4),x5) = +(x3,+(x4,x5))
 +(x3,x4) = +(x4,x3)
 U(U(x3,x4),x5) = U(x3,U(x4,x5))
 U(x3,x4) = U(x4,x3)
-> Rules:
 *(0,x) -> 0
 *(s(x),s(y)) -> s(+(+(x,y),*(x,y)))
 +(0,x) -> x
 +(s(x),s(y)) -> s(s(+(x,y)))
 U(empty,b) -> b
 prod(U(x,y)) -> *(prod(x),prod(y))
 prod(empty) -> s(0)
 prod(singl(x)) -> x
 sum(U(x,y)) -> +(sum(x),sum(y))
 sum(empty) -> 0
 sum(singl(x)) -> x
-> SRules:
 +#(+(x3,x4),x5) -> +#(x3,x4)
 +#(x3,+(x4,x5)) -> +#(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 +#(+(s(x),s(y)),x3) -> +#(s(s(+(x,y))),x3)
 +#(+(s(x),s(y)),x3) -> +#(x,y)
 +#(s(x),s(y)) -> +#(x,y)
-> FAxioms:
 *(*(x3,x4),x5) -> *(x3,*(x4,x5))
 *(x3,x4) -> *(x4,x3)
 +(+(x3,x4),x5) -> +(x3,+(x4,x5))
 +(x3,x4) -> +(x4,x3)
 U(U(x3,x4),x5) -> U(x3,U(x4,x5))
 U(x3,x4) -> U(x4,x3)
 +#(+(x3,x4),x5) -> +#(x3,+(x4,x5))
 +#(x3,x4) -> +#(x4,x3)
-> EAxioms:
 *(*(x3,x4),x5) = *(x3,*(x4,x5))
 *(x3,x4) = *(x4,x3)
 +(+(x3,x4),x5) = +(x3,+(x4,x5))
 +(x3,x4) = +(x4,x3)
 U(U(x3,x4),x5) = U(x3,U(x4,x5))
 U(x3,x4) = U(x4,x3)
->->-> Rules:
 *(0,x) -> 0
 *(s(x),s(y)) -> s(+(+(x,y),*(x,y)))
 +(0,x) -> x
 +(s(x),s(y)) -> s(s(+(x,y)))
 U(empty,b) -> b
 prod(U(x,y)) -> *(prod(x),prod(y))
 prod(empty) -> s(0)
 prod(singl(x)) -> x
 sum(U(x,y)) -> +(sum(x),sum(y))
 sum(empty) -> 0
 sum(singl(x)) -> x
-> SRules:
 +#(+(x3,x4),x5) -> +#(x3,x4)
 +#(x3,+(x4,x5)) -> +#(x4,x5)

Problem 1.2: 

Reduction Pairs Processor:
-> FAxioms:
 +#(+(x3,x4),x5) = +#(x3,+(x4,x5))
 +#(x3,x4) = +#(x4,x3)
-> Pairs:
 +#(+(s(x),s(y)),x3) -> +#(s(s(+(x,y))),x3)
 +#(+(s(x),s(y)),x3) -> +#(x,y)
 +#(s(x),s(y)) -> +#(x,y)
-> EAxioms:
 *(*(x3,x4),x5) = *(x3,*(x4,x5))
 *(x3,x4) = *(x4,x3)
 +(+(x3,x4),x5) = +(x3,+(x4,x5))
 +(x3,x4) = +(x4,x3)
 U(U(x3,x4),x5) = U(x3,U(x4,x5))
 U(x3,x4) = U(x4,x3)
-> Usable Equations:
 +(+(x3,x4),x5) = +(x3,+(x4,x5))
 +(x3,x4) = +(x4,x3)
-> Rules:
 *(0,x) -> 0
 *(s(x),s(y)) -> s(+(+(x,y),*(x,y)))
 +(0,x) -> x
 +(s(x),s(y)) -> s(s(+(x,y)))
 U(empty,b) -> b
 prod(U(x,y)) -> *(prod(x),prod(y))
 prod(empty) -> s(0)
 prod(singl(x)) -> x
 sum(U(x,y)) -> +(sum(x),sum(y))
 sum(empty) -> 0
 sum(singl(x)) -> x
-> Usable Rules:
 +(0,x) -> x
 +(s(x),s(y)) -> s(s(+(x,y)))
-> SRules:
 +#(+(x3,x4),x5) -> +#(x3,x4)
 +#(x3,+(x4,x5)) -> +#(x4,x5)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[*](X1,X2) = 0
[+](X1,X2) = X1 + X2 + 2
[U](X1,X2) = 0
[prod](X) = 0
[sum](X) = 0
[0] = 0
[empty] = 0
[s](X) = X + 2
[singl](X) = 0
[*#](X1,X2) = 0
[+#](X1,X2) = 2.X1 + 2.X2
[U#](X1,X2) = 0
[PROD](X) = 0
[SUM](X) = 0

Problem 1.2: 

SCC Processor:
-> FAxioms:
 +#(+(x3,x4),x5) = +#(x3,+(x4,x5))
 +#(x3,x4) = +#(x4,x3)
-> Pairs:
 +#(+(s(x),s(y)),x3) -> +#(s(s(+(x,y))),x3)
 +#(s(x),s(y)) -> +#(x,y)
-> EAxioms:
 *(*(x3,x4),x5) = *(x3,*(x4,x5))
 *(x3,x4) = *(x4,x3)
 +(+(x3,x4),x5) = +(x3,+(x4,x5))
 +(x3,x4) = +(x4,x3)
 U(U(x3,x4),x5) = U(x3,U(x4,x5))
 U(x3,x4) = U(x4,x3)
-> Rules:
 *(0,x) -> 0
 *(s(x),s(y)) -> s(+(+(x,y),*(x,y)))
 +(0,x) -> x
 +(s(x),s(y)) -> s(s(+(x,y)))
 U(empty,b) -> b
 prod(U(x,y)) -> *(prod(x),prod(y))
 prod(empty) -> s(0)
 prod(singl(x)) -> x
 sum(U(x,y)) -> +(sum(x),sum(y))
 sum(empty) -> 0
 sum(singl(x)) -> x
-> SRules:
 +#(+(x3,x4),x5) -> +#(x3,x4)
 +#(x3,+(x4,x5)) -> +#(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 +#(+(s(x),s(y)),x3) -> +#(s(s(+(x,y))),x3)
 +#(s(x),s(y)) -> +#(x,y)
-> FAxioms:
 *(*(x3,x4),x5) -> *(x3,*(x4,x5))
 *(x3,x4) -> *(x4,x3)
 +(+(x3,x4),x5) -> +(x3,+(x4,x5))
 +(x3,x4) -> +(x4,x3)
 U(U(x3,x4),x5) -> U(x3,U(x4,x5))
 U(x3,x4) -> U(x4,x3)
 +#(+(x3,x4),x5) -> +#(x3,+(x4,x5))
 +#(x3,x4) -> +#(x4,x3)
-> EAxioms:
 *(*(x3,x4),x5) = *(x3,*(x4,x5))
 *(x3,x4) = *(x4,x3)
 +(+(x3,x4),x5) = +(x3,+(x4,x5))
 +(x3,x4) = +(x4,x3)
 U(U(x3,x4),x5) = U(x3,U(x4,x5))
 U(x3,x4) = U(x4,x3)
->->-> Rules:
 *(0,x) -> 0
 *(s(x),s(y)) -> s(+(+(x,y),*(x,y)))
 +(0,x) -> x
 +(s(x),s(y)) -> s(s(+(x,y)))
 U(empty,b) -> b
 prod(U(x,y)) -> *(prod(x),prod(y))
 prod(empty) -> s(0)
 prod(singl(x)) -> x
 sum(U(x,y)) -> +(sum(x),sum(y))
 sum(empty) -> 0
 sum(singl(x)) -> x
-> SRules:
 +#(+(x3,x4),x5) -> +#(x3,x4)
 +#(x3,+(x4,x5)) -> +#(x4,x5)

Problem 1.2: 

Reduction Pairs Processor:
-> FAxioms:
 +#(+(x3,x4),x5) = +#(x3,+(x4,x5))
 +#(x3,x4) = +#(x4,x3)
-> Pairs:
 +#(+(s(x),s(y)),x3) -> +#(s(s(+(x,y))),x3)
 +#(s(x),s(y)) -> +#(x,y)
-> EAxioms:
 *(*(x3,x4),x5) = *(x3,*(x4,x5))
 *(x3,x4) = *(x4,x3)
 +(+(x3,x4),x5) = +(x3,+(x4,x5))
 +(x3,x4) = +(x4,x3)
 U(U(x3,x4),x5) = U(x3,U(x4,x5))
 U(x3,x4) = U(x4,x3)
-> Usable Equations:
 +(+(x3,x4),x5) = +(x3,+(x4,x5))
 +(x3,x4) = +(x4,x3)
-> Rules:
 *(0,x) -> 0
 *(s(x),s(y)) -> s(+(+(x,y),*(x,y)))
 +(0,x) -> x
 +(s(x),s(y)) -> s(s(+(x,y)))
 U(empty,b) -> b
 prod(U(x,y)) -> *(prod(x),prod(y))
 prod(empty) -> s(0)
 prod(singl(x)) -> x
 sum(U(x,y)) -> +(sum(x),sum(y))
 sum(empty) -> 0
 sum(singl(x)) -> x
-> Usable Rules:
 +(0,x) -> x
 +(s(x),s(y)) -> s(s(+(x,y)))
-> SRules:
 +#(+(x3,x4),x5) -> +#(x3,x4)
 +#(x3,+(x4,x5)) -> +#(x4,x5)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[*](X1,X2) = 0
[+](X1,X2) = X1 + X2 + 2
[U](X1,X2) = 0
[prod](X) = 0
[sum](X) = 0
[0] = 0
[empty] = 0
[s](X) = X
[singl](X) = 0
[*#](X1,X2) = 0
[+#](X1,X2) = 2.X1 + 2.X2
[U#](X1,X2) = 0
[PROD](X) = 0
[SUM](X) = 0

Problem 1.2: 

SCC Processor:
-> FAxioms:
 +#(+(x3,x4),x5) = +#(x3,+(x4,x5))
 +#(x3,x4) = +#(x4,x3)
-> Pairs:
 +#(+(s(x),s(y)),x3) -> +#(s(s(+(x,y))),x3)
 +#(s(x),s(y)) -> +#(x,y)
-> EAxioms:
 *(*(x3,x4),x5) = *(x3,*(x4,x5))
 *(x3,x4) = *(x4,x3)
 +(+(x3,x4),x5) = +(x3,+(x4,x5))
 +(x3,x4) = +(x4,x3)
 U(U(x3,x4),x5) = U(x3,U(x4,x5))
 U(x3,x4) = U(x4,x3)
-> Rules:
 *(0,x) -> 0
 *(s(x),s(y)) -> s(+(+(x,y),*(x,y)))
 +(0,x) -> x
 +(s(x),s(y)) -> s(s(+(x,y)))
 U(empty,b) -> b
 prod(U(x,y)) -> *(prod(x),prod(y))
 prod(empty) -> s(0)
 prod(singl(x)) -> x
 sum(U(x,y)) -> +(sum(x),sum(y))
 sum(empty) -> 0
 sum(singl(x)) -> x
-> SRules:
 +#(x3,+(x4,x5)) -> +#(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 +#(+(s(x),s(y)),x3) -> +#(s(s(+(x,y))),x3)
 +#(s(x),s(y)) -> +#(x,y)
-> FAxioms:
 *(*(x3,x4),x5) -> *(x3,*(x4,x5))
 *(x3,x4) -> *(x4,x3)
 +(+(x3,x4),x5) -> +(x3,+(x4,x5))
 +(x3,x4) -> +(x4,x3)
 U(U(x3,x4),x5) -> U(x3,U(x4,x5))
 U(x3,x4) -> U(x4,x3)
 +#(+(x3,x4),x5) -> +#(x3,+(x4,x5))
 +#(x3,x4) -> +#(x4,x3)
-> EAxioms:
 *(*(x3,x4),x5) = *(x3,*(x4,x5))
 *(x3,x4) = *(x4,x3)
 +(+(x3,x4),x5) = +(x3,+(x4,x5))
 +(x3,x4) = +(x4,x3)
 U(U(x3,x4),x5) = U(x3,U(x4,x5))
 U(x3,x4) = U(x4,x3)
->->-> Rules:
 *(0,x) -> 0
 *(s(x),s(y)) -> s(+(+(x,y),*(x,y)))
 +(0,x) -> x
 +(s(x),s(y)) -> s(s(+(x,y)))
 U(empty,b) -> b
 prod(U(x,y)) -> *(prod(x),prod(y))
 prod(empty) -> s(0)
 prod(singl(x)) -> x
 sum(U(x,y)) -> +(sum(x),sum(y))
 sum(empty) -> 0
 sum(singl(x)) -> x
-> SRules:
 +#(x3,+(x4,x5)) -> +#(x4,x5)

Problem 1.2: 

Reduction Pairs Processor:
-> FAxioms:
 +#(+(x3,x4),x5) = +#(x3,+(x4,x5))
 +#(x3,x4) = +#(x4,x3)
-> Pairs:
 +#(+(s(x),s(y)),x3) -> +#(s(s(+(x,y))),x3)
 +#(s(x),s(y)) -> +#(x,y)
-> EAxioms:
 *(*(x3,x4),x5) = *(x3,*(x4,x5))
 *(x3,x4) = *(x4,x3)
 +(+(x3,x4),x5) = +(x3,+(x4,x5))
 +(x3,x4) = +(x4,x3)
 U(U(x3,x4),x5) = U(x3,U(x4,x5))
 U(x3,x4) = U(x4,x3)
-> Usable Equations:
 +(+(x3,x4),x5) = +(x3,+(x4,x5))
 +(x3,x4) = +(x4,x3)
-> Rules:
 *(0,x) -> 0
 *(s(x),s(y)) -> s(+(+(x,y),*(x,y)))
 +(0,x) -> x
 +(s(x),s(y)) -> s(s(+(x,y)))
 U(empty,b) -> b
 prod(U(x,y)) -> *(prod(x),prod(y))
 prod(empty) -> s(0)
 prod(singl(x)) -> x
 sum(U(x,y)) -> +(sum(x),sum(y))
 sum(empty) -> 0
 sum(singl(x)) -> x
-> Usable Rules:
 +(0,x) -> x
 +(s(x),s(y)) -> s(s(+(x,y)))
-> SRules:
 +#(x3,+(x4,x5)) -> +#(x4,x5)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[*](X1,X2) = 0
[+](X1,X2) = X1 + X2 + 2
[U](X1,X2) = 0
[prod](X) = 0
[sum](X) = 0
[0] = 0
[empty] = 0
[s](X) = X + 1
[singl](X) = 0
[*#](X1,X2) = 0
[+#](X1,X2) = X1 + X2
[U#](X1,X2) = 0
[PROD](X) = 0
[SUM](X) = 0

Problem 1.2: 

SCC Processor:
-> FAxioms:
 +#(+(x3,x4),x5) = +#(x3,+(x4,x5))
 +#(x3,x4) = +#(x4,x3)
-> Pairs:
 +#(+(s(x),s(y)),x3) -> +#(s(s(+(x,y))),x3)
-> EAxioms:
 *(*(x3,x4),x5) = *(x3,*(x4,x5))
 *(x3,x4) = *(x4,x3)
 +(+(x3,x4),x5) = +(x3,+(x4,x5))
 +(x3,x4) = +(x4,x3)
 U(U(x3,x4),x5) = U(x3,U(x4,x5))
 U(x3,x4) = U(x4,x3)
-> Rules:
 *(0,x) -> 0
 *(s(x),s(y)) -> s(+(+(x,y),*(x,y)))
 +(0,x) -> x
 +(s(x),s(y)) -> s(s(+(x,y)))
 U(empty,b) -> b
 prod(U(x,y)) -> *(prod(x),prod(y))
 prod(empty) -> s(0)
 prod(singl(x)) -> x
 sum(U(x,y)) -> +(sum(x),sum(y))
 sum(empty) -> 0
 sum(singl(x)) -> x
-> SRules:
 +#(x3,+(x4,x5)) -> +#(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 +#(+(s(x),s(y)),x3) -> +#(s(s(+(x,y))),x3)
-> FAxioms:
 *(*(x3,x4),x5) -> *(x3,*(x4,x5))
 *(x3,x4) -> *(x4,x3)
 +(+(x3,x4),x5) -> +(x3,+(x4,x5))
 +(x3,x4) -> +(x4,x3)
 U(U(x3,x4),x5) -> U(x3,U(x4,x5))
 U(x3,x4) -> U(x4,x3)
 +#(+(x3,x4),x5) -> +#(x3,+(x4,x5))
 +#(x3,x4) -> +#(x4,x3)
-> EAxioms:
 *(*(x3,x4),x5) = *(x3,*(x4,x5))
 *(x3,x4) = *(x4,x3)
 +(+(x3,x4),x5) = +(x3,+(x4,x5))
 +(x3,x4) = +(x4,x3)
 U(U(x3,x4),x5) = U(x3,U(x4,x5))
 U(x3,x4) = U(x4,x3)
->->-> Rules:
 *(0,x) -> 0
 *(s(x),s(y)) -> s(+(+(x,y),*(x,y)))
 +(0,x) -> x
 +(s(x),s(y)) -> s(s(+(x,y)))
 U(empty,b) -> b
 prod(U(x,y)) -> *(prod(x),prod(y))
 prod(empty) -> s(0)
 prod(singl(x)) -> x
 sum(U(x,y)) -> +(sum(x),sum(y))
 sum(empty) -> 0
 sum(singl(x)) -> x
-> SRules:
 +#(x3,+(x4,x5)) -> +#(x4,x5)

Problem 1.2: 

Reduction Pairs Processor:
-> FAxioms:
 +#(+(x3,x4),x5) = +#(x3,+(x4,x5))
 +#(x3,x4) = +#(x4,x3)
-> Pairs:
 +#(+(s(x),s(y)),x3) -> +#(s(s(+(x,y))),x3)
-> EAxioms:
 *(*(x3,x4),x5) = *(x3,*(x4,x5))
 *(x3,x4) = *(x4,x3)
 +(+(x3,x4),x5) = +(x3,+(x4,x5))
 +(x3,x4) = +(x4,x3)
 U(U(x3,x4),x5) = U(x3,U(x4,x5))
 U(x3,x4) = U(x4,x3)
-> Usable Equations:
 +(+(x3,x4),x5) = +(x3,+(x4,x5))
 +(x3,x4) = +(x4,x3)
-> Rules:
 *(0,x) -> 0
 *(s(x),s(y)) -> s(+(+(x,y),*(x,y)))
 +(0,x) -> x
 +(s(x),s(y)) -> s(s(+(x,y)))
 U(empty,b) -> b
 prod(U(x,y)) -> *(prod(x),prod(y))
 prod(empty) -> s(0)
 prod(singl(x)) -> x
 sum(U(x,y)) -> +(sum(x),sum(y))
 sum(empty) -> 0
 sum(singl(x)) -> x
-> Usable Rules:
 +(0,x) -> x
 +(s(x),s(y)) -> s(s(+(x,y)))
-> SRules:
 +#(x3,+(x4,x5)) -> +#(x4,x5)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[*](X1,X2) = 0
[+](X1,X2) = X1 + X2 + 2
[U](X1,X2) = 0
[prod](X) = 0
[sum](X) = 0
[0] = 0
[empty] = 0
[s](X) = 2
[singl](X) = 0
[*#](X1,X2) = 0
[+#](X1,X2) = 2.X1 + 2.X2
[U#](X1,X2) = 0
[PROD](X) = 0
[SUM](X) = 0

Problem 1.2: 

SCC Processor:
-> FAxioms:
 +#(+(x3,x4),x5) = +#(x3,+(x4,x5))
 +#(x3,x4) = +#(x4,x3)
-> Pairs:
 Empty
-> EAxioms:
 *(*(x3,x4),x5) = *(x3,*(x4,x5))
 *(x3,x4) = *(x4,x3)
 +(+(x3,x4),x5) = +(x3,+(x4,x5))
 +(x3,x4) = +(x4,x3)
 U(U(x3,x4),x5) = U(x3,U(x4,x5))
 U(x3,x4) = U(x4,x3)
-> Rules:
 *(0,x) -> 0
 *(s(x),s(y)) -> s(+(+(x,y),*(x,y)))
 +(0,x) -> x
 +(s(x),s(y)) -> s(s(+(x,y)))
 U(empty,b) -> b
 prod(U(x,y)) -> *(prod(x),prod(y))
 prod(empty) -> s(0)
 prod(singl(x)) -> x
 sum(U(x,y)) -> +(sum(x),sum(y))
 sum(empty) -> 0
 sum(singl(x)) -> x
-> SRules:
 +#(x3,+(x4,x5)) -> +#(x4,x5)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Subterm Processor:
-> FAxioms:
 Empty
-> Pairs:
 SUM(U(x,y)) -> SUM(x)
 SUM(U(x,y)) -> SUM(y)
-> EAxioms:
 *(*(x3,x4),x5) = *(x3,*(x4,x5))
 *(x3,x4) = *(x4,x3)
 +(+(x3,x4),x5) = +(x3,+(x4,x5))
 +(x3,x4) = +(x4,x3)
 U(U(x3,x4),x5) = U(x3,U(x4,x5))
 U(x3,x4) = U(x4,x3)
-> Rules:
 *(0,x) -> 0
 *(s(x),s(y)) -> s(+(+(x,y),*(x,y)))
 +(0,x) -> x
 +(s(x),s(y)) -> s(s(+(x,y)))
 U(empty,b) -> b
 prod(U(x,y)) -> *(prod(x),prod(y))
 prod(empty) -> s(0)
 prod(singl(x)) -> x
 sum(U(x,y)) -> +(sum(x),sum(y))
 sum(empty) -> 0
 sum(singl(x)) -> x
-> SRules:
 Empty
->Projection:
 pi(SUM) = [1]

Problem 1.3: 

SCC Processor:
-> FAxioms:
 Empty
-> Pairs:
 Empty
-> EAxioms:
 *(*(x3,x4),x5) = *(x3,*(x4,x5))
 *(x3,x4) = *(x4,x3)
 +(+(x3,x4),x5) = +(x3,+(x4,x5))
 +(x3,x4) = +(x4,x3)
 U(U(x3,x4),x5) = U(x3,U(x4,x5))
 U(x3,x4) = U(x4,x3)
-> Rules:
 *(0,x) -> 0
 *(s(x),s(y)) -> s(+(+(x,y),*(x,y)))
 +(0,x) -> x
 +(s(x),s(y)) -> s(s(+(x,y)))
 U(empty,b) -> b
 prod(U(x,y)) -> *(prod(x),prod(y))
 prod(empty) -> s(0)
 prod(singl(x)) -> x
 sum(U(x,y)) -> +(sum(x),sum(y))
 sum(empty) -> 0
 sum(singl(x)) -> x
-> SRules:
 Empty
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.4: 

Reduction Pairs Processor:
-> FAxioms:
 *#(*(x3,x4),x5) = *#(x3,*(x4,x5))
 *#(x3,x4) = *#(x4,x3)
-> Pairs:
 *#(*(0,x),x3) -> *#(0,x3)
 *#(*(s(x),s(y)),x3) -> *#(s(+(+(x,y),*(x,y))),x3)
 *#(*(s(x),s(y)),x3) -> *#(x,y)
 *#(s(x),s(y)) -> *#(x,y)
-> EAxioms:
 *(*(x3,x4),x5) = *(x3,*(x4,x5))
 *(x3,x4) = *(x4,x3)
 +(+(x3,x4),x5) = +(x3,+(x4,x5))
 +(x3,x4) = +(x4,x3)
 U(U(x3,x4),x5) = U(x3,U(x4,x5))
 U(x3,x4) = U(x4,x3)
-> Usable Equations:
 *(*(x3,x4),x5) = *(x3,*(x4,x5))
 *(x3,x4) = *(x4,x3)
 +(+(x3,x4),x5) = +(x3,+(x4,x5))
 +(x3,x4) = +(x4,x3)
-> Rules:
 *(0,x) -> 0
 *(s(x),s(y)) -> s(+(+(x,y),*(x,y)))
 +(0,x) -> x
 +(s(x),s(y)) -> s(s(+(x,y)))
 U(empty,b) -> b
 prod(U(x,y)) -> *(prod(x),prod(y))
 prod(empty) -> s(0)
 prod(singl(x)) -> x
 sum(U(x,y)) -> +(sum(x),sum(y))
 sum(empty) -> 0
 sum(singl(x)) -> x
-> Usable Rules:
 *(0,x) -> 0
 *(s(x),s(y)) -> s(+(+(x,y),*(x,y)))
 +(0,x) -> x
 +(s(x),s(y)) -> s(s(+(x,y)))
-> SRules:
 *#(*(x3,x4),x5) -> *#(x3,x4)
 *#(x3,*(x4,x5)) -> *#(x4,x5)
->Interpretation type:
 Simple mixed
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 1
->Interpretation:
 
[*](X1,X2) = X1.X2 + X1 + X2
[+](X1,X2) = X1 + X2
[U](X1,X2) = 0
[prod](X) = 0
[sum](X) = 0
[0] = 1
[empty] = 0
[s](X) = X + 1
[singl](X) = 0
[*#](X1,X2) = X1.X2 + X1 + X2
[+#](X1,X2) = 0
[U#](X1,X2) = 0
[PROD](X) = 0
[SUM](X) = 0

Problem 1.4: 

SCC Processor:
-> FAxioms:
 *#(*(x3,x4),x5) = *#(x3,*(x4,x5))
 *#(x3,x4) = *#(x4,x3)
-> Pairs:
 *#(*(0,x),x3) -> *#(0,x3)
 *#(*(s(x),s(y)),x3) -> *#(x,y)
 *#(s(x),s(y)) -> *#(x,y)
-> EAxioms:
 *(*(x3,x4),x5) = *(x3,*(x4,x5))
 *(x3,x4) = *(x4,x3)
 +(+(x3,x4),x5) = +(x3,+(x4,x5))
 +(x3,x4) = +(x4,x3)
 U(U(x3,x4),x5) = U(x3,U(x4,x5))
 U(x3,x4) = U(x4,x3)
-> Rules:
 *(0,x) -> 0
 *(s(x),s(y)) -> s(+(+(x,y),*(x,y)))
 +(0,x) -> x
 +(s(x),s(y)) -> s(s(+(x,y)))
 U(empty,b) -> b
 prod(U(x,y)) -> *(prod(x),prod(y))
 prod(empty) -> s(0)
 prod(singl(x)) -> x
 sum(U(x,y)) -> +(sum(x),sum(y))
 sum(empty) -> 0
 sum(singl(x)) -> x
-> SRules:
 *#(*(x3,x4),x5) -> *#(x3,x4)
 *#(x3,*(x4,x5)) -> *#(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 *#(*(0,x),x3) -> *#(0,x3)
 *#(*(s(x),s(y)),x3) -> *#(x,y)
 *#(s(x),s(y)) -> *#(x,y)
-> FAxioms:
 *(*(x3,x4),x5) -> *(x3,*(x4,x5))
 *(x3,x4) -> *(x4,x3)
 +(+(x3,x4),x5) -> +(x3,+(x4,x5))
 +(x3,x4) -> +(x4,x3)
 U(U(x3,x4),x5) -> U(x3,U(x4,x5))
 U(x3,x4) -> U(x4,x3)
 *#(*(x3,x4),x5) -> *#(x3,*(x4,x5))
 *#(x3,x4) -> *#(x4,x3)
-> EAxioms:
 *(*(x3,x4),x5) = *(x3,*(x4,x5))
 *(x3,x4) = *(x4,x3)
 +(+(x3,x4),x5) = +(x3,+(x4,x5))
 +(x3,x4) = +(x4,x3)
 U(U(x3,x4),x5) = U(x3,U(x4,x5))
 U(x3,x4) = U(x4,x3)
->->-> Rules:
 *(0,x) -> 0
 *(s(x),s(y)) -> s(+(+(x,y),*(x,y)))
 +(0,x) -> x
 +(s(x),s(y)) -> s(s(+(x,y)))
 U(empty,b) -> b
 prod(U(x,y)) -> *(prod(x),prod(y))
 prod(empty) -> s(0)
 prod(singl(x)) -> x
 sum(U(x,y)) -> +(sum(x),sum(y))
 sum(empty) -> 0
 sum(singl(x)) -> x
-> SRules:
 *#(*(x3,x4),x5) -> *#(x3,x4)
 *#(x3,*(x4,x5)) -> *#(x4,x5)

Problem 1.4: 

Reduction Pairs Processor:
-> FAxioms:
 *#(*(x3,x4),x5) = *#(x3,*(x4,x5))
 *#(x3,x4) = *#(x4,x3)
-> Pairs:
 *#(*(0,x),x3) -> *#(0,x3)
 *#(*(s(x),s(y)),x3) -> *#(x,y)
 *#(s(x),s(y)) -> *#(x,y)
-> EAxioms:
 *(*(x3,x4),x5) = *(x3,*(x4,x5))
 *(x3,x4) = *(x4,x3)
 +(+(x3,x4),x5) = +(x3,+(x4,x5))
 +(x3,x4) = +(x4,x3)
 U(U(x3,x4),x5) = U(x3,U(x4,x5))
 U(x3,x4) = U(x4,x3)
-> Usable Equations:
 *(*(x3,x4),x5) = *(x3,*(x4,x5))
 *(x3,x4) = *(x4,x3)
 +(+(x3,x4),x5) = +(x3,+(x4,x5))
 +(x3,x4) = +(x4,x3)
-> Rules:
 *(0,x) -> 0
 *(s(x),s(y)) -> s(+(+(x,y),*(x,y)))
 +(0,x) -> x
 +(s(x),s(y)) -> s(s(+(x,y)))
 U(empty,b) -> b
 prod(U(x,y)) -> *(prod(x),prod(y))
 prod(empty) -> s(0)
 prod(singl(x)) -> x
 sum(U(x,y)) -> +(sum(x),sum(y))
 sum(empty) -> 0
 sum(singl(x)) -> x
-> Usable Rules:
 *(0,x) -> 0
 *(s(x),s(y)) -> s(+(+(x,y),*(x,y)))
 +(0,x) -> x
 +(s(x),s(y)) -> s(s(+(x,y)))
-> SRules:
 *#(*(x3,x4),x5) -> *#(x3,x4)
 *#(x3,*(x4,x5)) -> *#(x4,x5)
->Interpretation type:
 Simple mixed
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 1
->Interpretation:
 
[*](X1,X2) = X1.X2 + X1 + X2
[+](X1,X2) = X1 + X2 + 1
[U](X1,X2) = 0
[prod](X) = 0
[sum](X) = 0
[0] = 1
[empty] = 0
[s](X) = X + 1
[singl](X) = 0
[*#](X1,X2) = X1.X2 + X1 + X2
[+#](X1,X2) = 0
[U#](X1,X2) = 0
[PROD](X) = 0
[SUM](X) = 0

Problem 1.4: 

SCC Processor:
-> FAxioms:
 *#(*(x3,x4),x5) = *#(x3,*(x4,x5))
 *#(x3,x4) = *#(x4,x3)
-> Pairs:
 *#(*(0,x),x3) -> *#(0,x3)
 *#(s(x),s(y)) -> *#(x,y)
-> EAxioms:
 *(*(x3,x4),x5) = *(x3,*(x4,x5))
 *(x3,x4) = *(x4,x3)
 +(+(x3,x4),x5) = +(x3,+(x4,x5))
 +(x3,x4) = +(x4,x3)
 U(U(x3,x4),x5) = U(x3,U(x4,x5))
 U(x3,x4) = U(x4,x3)
-> Rules:
 *(0,x) -> 0
 *(s(x),s(y)) -> s(+(+(x,y),*(x,y)))
 +(0,x) -> x
 +(s(x),s(y)) -> s(s(+(x,y)))
 U(empty,b) -> b
 prod(U(x,y)) -> *(prod(x),prod(y))
 prod(empty) -> s(0)
 prod(singl(x)) -> x
 sum(U(x,y)) -> +(sum(x),sum(y))
 sum(empty) -> 0
 sum(singl(x)) -> x
-> SRules:
 *#(*(x3,x4),x5) -> *#(x3,x4)
 *#(x3,*(x4,x5)) -> *#(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 *#(*(0,x),x3) -> *#(0,x3)
 *#(s(x),s(y)) -> *#(x,y)
-> FAxioms:
 *(*(x3,x4),x5) -> *(x3,*(x4,x5))
 *(x3,x4) -> *(x4,x3)
 +(+(x3,x4),x5) -> +(x3,+(x4,x5))
 +(x3,x4) -> +(x4,x3)
 U(U(x3,x4),x5) -> U(x3,U(x4,x5))
 U(x3,x4) -> U(x4,x3)
 *#(*(x3,x4),x5) -> *#(x3,*(x4,x5))
 *#(x3,x4) -> *#(x4,x3)
-> EAxioms:
 *(*(x3,x4),x5) = *(x3,*(x4,x5))
 *(x3,x4) = *(x4,x3)
 +(+(x3,x4),x5) = +(x3,+(x4,x5))
 +(x3,x4) = +(x4,x3)
 U(U(x3,x4),x5) = U(x3,U(x4,x5))
 U(x3,x4) = U(x4,x3)
->->-> Rules:
 *(0,x) -> 0
 *(s(x),s(y)) -> s(+(+(x,y),*(x,y)))
 +(0,x) -> x
 +(s(x),s(y)) -> s(s(+(x,y)))
 U(empty,b) -> b
 prod(U(x,y)) -> *(prod(x),prod(y))
 prod(empty) -> s(0)
 prod(singl(x)) -> x
 sum(U(x,y)) -> +(sum(x),sum(y))
 sum(empty) -> 0
 sum(singl(x)) -> x
-> SRules:
 *#(*(x3,x4),x5) -> *#(x3,x4)
 *#(x3,*(x4,x5)) -> *#(x4,x5)

Problem 1.4: 

Reduction Pairs Processor:
-> FAxioms:
 *#(*(x3,x4),x5) = *#(x3,*(x4,x5))
 *#(x3,x4) = *#(x4,x3)
-> Pairs:
 *#(*(0,x),x3) -> *#(0,x3)
 *#(s(x),s(y)) -> *#(x,y)
-> EAxioms:
 *(*(x3,x4),x5) = *(x3,*(x4,x5))
 *(x3,x4) = *(x4,x3)
 +(+(x3,x4),x5) = +(x3,+(x4,x5))
 +(x3,x4) = +(x4,x3)
 U(U(x3,x4),x5) = U(x3,U(x4,x5))
 U(x3,x4) = U(x4,x3)
-> Usable Equations:
 *(*(x3,x4),x5) = *(x3,*(x4,x5))
 *(x3,x4) = *(x4,x3)
 +(+(x3,x4),x5) = +(x3,+(x4,x5))
 +(x3,x4) = +(x4,x3)
-> Rules:
 *(0,x) -> 0
 *(s(x),s(y)) -> s(+(+(x,y),*(x,y)))
 +(0,x) -> x
 +(s(x),s(y)) -> s(s(+(x,y)))
 U(empty,b) -> b
 prod(U(x,y)) -> *(prod(x),prod(y))
 prod(empty) -> s(0)
 prod(singl(x)) -> x
 sum(U(x,y)) -> +(sum(x),sum(y))
 sum(empty) -> 0
 sum(singl(x)) -> x
-> Usable Rules:
 *(0,x) -> 0
 *(s(x),s(y)) -> s(+(+(x,y),*(x,y)))
 +(0,x) -> x
 +(s(x),s(y)) -> s(s(+(x,y)))
-> SRules:
 *#(*(x3,x4),x5) -> *#(x3,x4)
 *#(x3,*(x4,x5)) -> *#(x4,x5)
->Interpretation type:
 Simple mixed
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 1
->Interpretation:
 
[*](X1,X2) = X1.X2 + X1 + X2
[+](X1,X2) = X1 + X2 + 1
[U](X1,X2) = 0
[prod](X) = 0
[sum](X) = 0
[0] = 1
[empty] = 0
[s](X) = X + 1
[singl](X) = 0
[*#](X1,X2) = X1.X2 + X1 + X2
[+#](X1,X2) = 0
[U#](X1,X2) = 0
[PROD](X) = 0
[SUM](X) = 0

Problem 1.4: 

SCC Processor:
-> FAxioms:
 *#(*(x3,x4),x5) = *#(x3,*(x4,x5))
 *#(x3,x4) = *#(x4,x3)
-> Pairs:
 *#(*(0,x),x3) -> *#(0,x3)
-> EAxioms:
 *(*(x3,x4),x5) = *(x3,*(x4,x5))
 *(x3,x4) = *(x4,x3)
 +(+(x3,x4),x5) = +(x3,+(x4,x5))
 +(x3,x4) = +(x4,x3)
 U(U(x3,x4),x5) = U(x3,U(x4,x5))
 U(x3,x4) = U(x4,x3)
-> Rules:
 *(0,x) -> 0
 *(s(x),s(y)) -> s(+(+(x,y),*(x,y)))
 +(0,x) -> x
 +(s(x),s(y)) -> s(s(+(x,y)))
 U(empty,b) -> b
 prod(U(x,y)) -> *(prod(x),prod(y))
 prod(empty) -> s(0)
 prod(singl(x)) -> x
 sum(U(x,y)) -> +(sum(x),sum(y))
 sum(empty) -> 0
 sum(singl(x)) -> x
-> SRules:
 *#(*(x3,x4),x5) -> *#(x3,x4)
 *#(x3,*(x4,x5)) -> *#(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 *#(*(0,x),x3) -> *#(0,x3)
-> FAxioms:
 *(*(x3,x4),x5) -> *(x3,*(x4,x5))
 *(x3,x4) -> *(x4,x3)
 +(+(x3,x4),x5) -> +(x3,+(x4,x5))
 +(x3,x4) -> +(x4,x3)
 U(U(x3,x4),x5) -> U(x3,U(x4,x5))
 U(x3,x4) -> U(x4,x3)
 *#(*(x3,x4),x5) -> *#(x3,*(x4,x5))
 *#(x3,x4) -> *#(x4,x3)
-> EAxioms:
 *(*(x3,x4),x5) = *(x3,*(x4,x5))
 *(x3,x4) = *(x4,x3)
 +(+(x3,x4),x5) = +(x3,+(x4,x5))
 +(x3,x4) = +(x4,x3)
 U(U(x3,x4),x5) = U(x3,U(x4,x5))
 U(x3,x4) = U(x4,x3)
->->-> Rules:
 *(0,x) -> 0
 *(s(x),s(y)) -> s(+(+(x,y),*(x,y)))
 +(0,x) -> x
 +(s(x),s(y)) -> s(s(+(x,y)))
 U(empty,b) -> b
 prod(U(x,y)) -> *(prod(x),prod(y))
 prod(empty) -> s(0)
 prod(singl(x)) -> x
 sum(U(x,y)) -> +(sum(x),sum(y))
 sum(empty) -> 0
 sum(singl(x)) -> x
-> SRules:
 *#(*(x3,x4),x5) -> *#(x3,x4)
 *#(x3,*(x4,x5)) -> *#(x4,x5)

Problem 1.4: 

Reduction Pairs Processor:
-> FAxioms:
 *#(*(x3,x4),x5) = *#(x3,*(x4,x5))
 *#(x3,x4) = *#(x4,x3)
-> Pairs:
 *#(*(0,x),x3) -> *#(0,x3)
-> EAxioms:
 *(*(x3,x4),x5) = *(x3,*(x4,x5))
 *(x3,x4) = *(x4,x3)
 +(+(x3,x4),x5) = +(x3,+(x4,x5))
 +(x3,x4) = +(x4,x3)
 U(U(x3,x4),x5) = U(x3,U(x4,x5))
 U(x3,x4) = U(x4,x3)
-> Usable Equations:
 *(*(x3,x4),x5) = *(x3,*(x4,x5))
 *(x3,x4) = *(x4,x3)
 +(+(x3,x4),x5) = +(x3,+(x4,x5))
 +(x3,x4) = +(x4,x3)
-> Rules:
 *(0,x) -> 0
 *(s(x),s(y)) -> s(+(+(x,y),*(x,y)))
 +(0,x) -> x
 +(s(x),s(y)) -> s(s(+(x,y)))
 U(empty,b) -> b
 prod(U(x,y)) -> *(prod(x),prod(y))
 prod(empty) -> s(0)
 prod(singl(x)) -> x
 sum(U(x,y)) -> +(sum(x),sum(y))
 sum(empty) -> 0
 sum(singl(x)) -> x
-> Usable Rules:
 *(0,x) -> 0
 *(s(x),s(y)) -> s(+(+(x,y),*(x,y)))
 +(0,x) -> x
 +(s(x),s(y)) -> s(s(+(x,y)))
-> SRules:
 *#(*(x3,x4),x5) -> *#(x3,x4)
 *#(x3,*(x4,x5)) -> *#(x4,x5)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[*](X1,X2) = X1 + X2 + 2
[+](X1,X2) = X1 + X2
[U](X1,X2) = 0
[prod](X) = 0
[sum](X) = 0
[0] = 2
[empty] = 0
[s](X) = 2
[singl](X) = 0
[*#](X1,X2) = 2.X1 + 2.X2
[+#](X1,X2) = 0
[U#](X1,X2) = 0
[PROD](X) = 0
[SUM](X) = 0

Problem 1.4: 

SCC Processor:
-> FAxioms:
 *#(*(x3,x4),x5) = *#(x3,*(x4,x5))
 *#(x3,x4) = *#(x4,x3)
-> Pairs:
 Empty
-> EAxioms:
 *(*(x3,x4),x5) = *(x3,*(x4,x5))
 *(x3,x4) = *(x4,x3)
 +(+(x3,x4),x5) = +(x3,+(x4,x5))
 +(x3,x4) = +(x4,x3)
 U(U(x3,x4),x5) = U(x3,U(x4,x5))
 U(x3,x4) = U(x4,x3)
-> Rules:
 *(0,x) -> 0
 *(s(x),s(y)) -> s(+(+(x,y),*(x,y)))
 +(0,x) -> x
 +(s(x),s(y)) -> s(s(+(x,y)))
 U(empty,b) -> b
 prod(U(x,y)) -> *(prod(x),prod(y))
 prod(empty) -> s(0)
 prod(singl(x)) -> x
 sum(U(x,y)) -> +(sum(x),sum(y))
 sum(empty) -> 0
 sum(singl(x)) -> x
-> SRules:
 *#(*(x3,x4),x5) -> *#(x3,x4)
 *#(x3,*(x4,x5)) -> *#(x4,x5)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.5: 

Subterm Processor:
-> FAxioms:
 Empty
-> Pairs:
 PROD(U(x,y)) -> PROD(x)
 PROD(U(x,y)) -> PROD(y)
-> EAxioms:
 *(*(x3,x4),x5) = *(x3,*(x4,x5))
 *(x3,x4) = *(x4,x3)
 +(+(x3,x4),x5) = +(x3,+(x4,x5))
 +(x3,x4) = +(x4,x3)
 U(U(x3,x4),x5) = U(x3,U(x4,x5))
 U(x3,x4) = U(x4,x3)
-> Rules:
 *(0,x) -> 0
 *(s(x),s(y)) -> s(+(+(x,y),*(x,y)))
 +(0,x) -> x
 +(s(x),s(y)) -> s(s(+(x,y)))
 U(empty,b) -> b
 prod(U(x,y)) -> *(prod(x),prod(y))
 prod(empty) -> s(0)
 prod(singl(x)) -> x
 sum(U(x,y)) -> +(sum(x),sum(y))
 sum(empty) -> 0
 sum(singl(x)) -> x
-> SRules:
 Empty
->Projection:
 pi(PROD) = [1]

Problem 1.5: 

SCC Processor:
-> FAxioms:
 Empty
-> Pairs:
 Empty
-> EAxioms:
 *(*(x3,x4),x5) = *(x3,*(x4,x5))
 *(x3,x4) = *(x4,x3)
 +(+(x3,x4),x5) = +(x3,+(x4,x5))
 +(x3,x4) = +(x4,x3)
 U(U(x3,x4),x5) = U(x3,U(x4,x5))
 U(x3,x4) = U(x4,x3)
-> Rules:
 *(0,x) -> 0
 *(s(x),s(y)) -> s(+(+(x,y),*(x,y)))
 +(0,x) -> x
 +(s(x),s(y)) -> s(s(+(x,y)))
 U(empty,b) -> b
 prod(U(x,y)) -> *(prod(x),prod(y))
 prod(empty) -> s(0)
 prod(singl(x)) -> x
 sum(U(x,y)) -> +(sum(x),sum(y))
 sum(empty) -> 0
 sum(singl(x)) -> x
-> SRules:
 Empty
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.