YES

Problem 1: 

(VAR x y z)
(THEORY 
(AC plus times))
(RULES
plus(zero(x),zero(y)) -> zero(plus(x,y))
plus(zero(x),un(y)) -> un(plus(x,y))
plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
plus(x,S) -> x
times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
times(x,times(S,z)) -> times(S,z)
times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
times(x,zero(y)) -> zero(times(x,y))
times(x,S) -> S
times(x,un(y)) -> plus(x,zero(times(x,y)))
zero(S) -> S
)

Problem 1: 

Dependency Pairs Processor:
-> FAxioms:
 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) = PLUS(x4,x3)
 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
 TIMES(x3,x4) = TIMES(x4,x3)
-> Pairs:
 PLUS(plus(zero(x),zero(y)),x3) -> PLUS(zero(plus(x,y)),x3)
 PLUS(plus(zero(x),zero(y)),x3) -> PLUS(x,y)
 PLUS(plus(zero(x),zero(y)),x3) -> ZERO(plus(x,y))
 PLUS(plus(zero(x),un(y)),x3) -> PLUS(un(plus(x,y)),x3)
 PLUS(plus(zero(x),un(y)),x3) -> PLUS(x,y)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(zero(plus(x,plus(y,un(S)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(S)))
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(S))
 PLUS(plus(un(x),un(y)),x3) -> ZERO(plus(x,plus(y,un(S))))
 PLUS(plus(x,S),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),zero(y)) -> ZERO(plus(x,y))
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S)))
 PLUS(un(x),un(y)) -> PLUS(y,un(S))
 PLUS(un(x),un(y)) -> ZERO(plus(x,plus(y,un(S))))
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(x,y)
 TIMES(times(x,times(zero(y),z)),x3) -> ZERO(times(x,y))
 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3)
 TIMES(times(x,times(un(y),z)),x3) -> PLUS(x,zero(times(x,y)))
 TIMES(times(x,times(un(y),z)),x3) -> TIMES(plus(x,zero(times(x,y))),z)
 TIMES(times(x,times(un(y),z)),x3) -> TIMES(times(plus(x,zero(times(x,y))),z),x3)
 TIMES(times(x,times(un(y),z)),x3) -> TIMES(x,y)
 TIMES(times(x,times(un(y),z)),x3) -> ZERO(times(x,y))
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(x,y)
 TIMES(times(x,zero(y)),x3) -> ZERO(times(x,y))
 TIMES(times(x,S),x3) -> TIMES(S,x3)
 TIMES(times(x,un(y)),x3) -> PLUS(x,zero(times(x,y)))
 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3)
 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
 TIMES(times(x,un(y)),x3) -> ZERO(times(x,y))
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
 TIMES(x,times(zero(y),z)) -> ZERO(times(x,y))
 TIMES(x,times(un(y),z)) -> PLUS(x,zero(times(x,y)))
 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
 TIMES(x,times(un(y),z)) -> TIMES(x,y)
 TIMES(x,times(un(y),z)) -> ZERO(times(x,y))
 TIMES(x,zero(y)) -> TIMES(x,y)
 TIMES(x,zero(y)) -> ZERO(times(x,y))
 TIMES(x,un(y)) -> PLUS(x,zero(times(x,y)))
 TIMES(x,un(y)) -> TIMES(x,y)
 TIMES(x,un(y)) -> ZERO(times(x,y))
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)

Problem 1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) = PLUS(x4,x3)
 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
 TIMES(x3,x4) = TIMES(x4,x3)
-> Pairs:
 PLUS(plus(zero(x),zero(y)),x3) -> PLUS(zero(plus(x,y)),x3)
 PLUS(plus(zero(x),zero(y)),x3) -> PLUS(x,y)
 PLUS(plus(zero(x),zero(y)),x3) -> ZERO(plus(x,y))
 PLUS(plus(zero(x),un(y)),x3) -> PLUS(un(plus(x,y)),x3)
 PLUS(plus(zero(x),un(y)),x3) -> PLUS(x,y)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(zero(plus(x,plus(y,un(S)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(S)))
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(S))
 PLUS(plus(un(x),un(y)),x3) -> ZERO(plus(x,plus(y,un(S))))
 PLUS(plus(x,S),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),zero(y)) -> ZERO(plus(x,y))
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S)))
 PLUS(un(x),un(y)) -> PLUS(y,un(S))
 PLUS(un(x),un(y)) -> ZERO(plus(x,plus(y,un(S))))
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(x,y)
 TIMES(times(x,times(zero(y),z)),x3) -> ZERO(times(x,y))
 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3)
 TIMES(times(x,times(un(y),z)),x3) -> PLUS(x,zero(times(x,y)))
 TIMES(times(x,times(un(y),z)),x3) -> TIMES(plus(x,zero(times(x,y))),z)
 TIMES(times(x,times(un(y),z)),x3) -> TIMES(times(plus(x,zero(times(x,y))),z),x3)
 TIMES(times(x,times(un(y),z)),x3) -> TIMES(x,y)
 TIMES(times(x,times(un(y),z)),x3) -> ZERO(times(x,y))
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(x,y)
 TIMES(times(x,zero(y)),x3) -> ZERO(times(x,y))
 TIMES(times(x,S),x3) -> TIMES(S,x3)
 TIMES(times(x,un(y)),x3) -> PLUS(x,zero(times(x,y)))
 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3)
 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
 TIMES(times(x,un(y)),x3) -> ZERO(times(x,y))
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
 TIMES(x,times(zero(y),z)) -> ZERO(times(x,y))
 TIMES(x,times(un(y),z)) -> PLUS(x,zero(times(x,y)))
 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
 TIMES(x,times(un(y),z)) -> TIMES(x,y)
 TIMES(x,times(un(y),z)) -> ZERO(times(x,y))
 TIMES(x,zero(y)) -> TIMES(x,y)
 TIMES(x,zero(y)) -> ZERO(times(x,y))
 TIMES(x,un(y)) -> PLUS(x,zero(times(x,y)))
 TIMES(x,un(y)) -> TIMES(x,y)
 TIMES(x,un(y)) -> ZERO(times(x,y))
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(zero(x),zero(y)),x3) -> PLUS(zero(plus(x,y)),x3)
 PLUS(plus(zero(x),zero(y)),x3) -> PLUS(x,y)
 PLUS(plus(zero(x),un(y)),x3) -> PLUS(un(plus(x,y)),x3)
 PLUS(plus(zero(x),un(y)),x3) -> PLUS(x,y)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(zero(plus(x,plus(y,un(S)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(S)))
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(S))
 PLUS(plus(x,S),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S)))
 PLUS(un(x),un(y)) -> PLUS(y,un(S))
-> FAxioms:
 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
 plus(x3,x4) -> plus(x4,x3)
 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
 times(x3,x4) -> times(x4,x3)
 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) -> PLUS(x4,x3)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
->->-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->->Cycle:
->->-> Pairs:
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(x,y)
 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3)
 TIMES(times(x,times(un(y),z)),x3) -> TIMES(plus(x,zero(times(x,y))),z)
 TIMES(times(x,times(un(y),z)),x3) -> TIMES(times(plus(x,zero(times(x,y))),z),x3)
 TIMES(times(x,times(un(y),z)),x3) -> TIMES(x,y)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(x,y)
 TIMES(times(x,S),x3) -> TIMES(S,x3)
 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3)
 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
 TIMES(x,times(un(y),z)) -> TIMES(x,y)
 TIMES(x,zero(y)) -> TIMES(x,y)
 TIMES(x,un(y)) -> TIMES(x,y)
-> FAxioms:
 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
 plus(x3,x4) -> plus(x4,x3)
 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
 times(x3,x4) -> times(x4,x3)
 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
 TIMES(x3,x4) -> TIMES(x4,x3)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
->->-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) = PLUS(x4,x3)
-> Pairs:
 PLUS(plus(zero(x),zero(y)),x3) -> PLUS(zero(plus(x,y)),x3)
 PLUS(plus(zero(x),zero(y)),x3) -> PLUS(x,y)
 PLUS(plus(zero(x),un(y)),x3) -> PLUS(un(plus(x,y)),x3)
 PLUS(plus(zero(x),un(y)),x3) -> PLUS(x,y)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(zero(plus(x,plus(y,un(S)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(S)))
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(S))
 PLUS(plus(x,S),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S)))
 PLUS(un(x),un(y)) -> PLUS(y,un(S))
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Usable Equations:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> Usable Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 zero(S) -> S
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[plus](X1,X2) = X1 + X2
[times](X1,X2) = 0
[zero](X) = X + 1
[S] = 0
[un](X) = X + 2
[PLUS](X1,X2) = 2.X1 + 2.X2
[TIMES](X1,X2) = 0
[ZERO](X) = 0

Problem 1.1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) = PLUS(x4,x3)
-> Pairs:
 PLUS(plus(zero(x),zero(y)),x3) -> PLUS(x,y)
 PLUS(plus(zero(x),un(y)),x3) -> PLUS(un(plus(x,y)),x3)
 PLUS(plus(zero(x),un(y)),x3) -> PLUS(x,y)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(zero(plus(x,plus(y,un(S)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(S)))
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(S))
 PLUS(plus(x,S),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S)))
 PLUS(un(x),un(y)) -> PLUS(y,un(S))
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(zero(x),zero(y)),x3) -> PLUS(x,y)
 PLUS(plus(zero(x),un(y)),x3) -> PLUS(un(plus(x,y)),x3)
 PLUS(plus(zero(x),un(y)),x3) -> PLUS(x,y)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(zero(plus(x,plus(y,un(S)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(S)))
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(S))
 PLUS(plus(x,S),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S)))
 PLUS(un(x),un(y)) -> PLUS(y,un(S))
-> FAxioms:
 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
 plus(x3,x4) -> plus(x4,x3)
 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
 times(x3,x4) -> times(x4,x3)
 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) -> PLUS(x4,x3)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
->->-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)

Problem 1.1: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) = PLUS(x4,x3)
-> Pairs:
 PLUS(plus(zero(x),zero(y)),x3) -> PLUS(x,y)
 PLUS(plus(zero(x),un(y)),x3) -> PLUS(un(plus(x,y)),x3)
 PLUS(plus(zero(x),un(y)),x3) -> PLUS(x,y)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(zero(plus(x,plus(y,un(S)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(S)))
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(S))
 PLUS(plus(x,S),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S)))
 PLUS(un(x),un(y)) -> PLUS(y,un(S))
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Usable Equations:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> Usable Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 zero(S) -> S
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[plus](X1,X2) = X1 + X2 + 2
[times](X1,X2) = 0
[zero](X) = X
[S] = 0
[un](X) = X + 2
[PLUS](X1,X2) = 2.X1 + 2.X2
[TIMES](X1,X2) = 0
[ZERO](X) = 0

Problem 1.1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) = PLUS(x4,x3)
-> Pairs:
 PLUS(plus(zero(x),un(y)),x3) -> PLUS(un(plus(x,y)),x3)
 PLUS(plus(zero(x),un(y)),x3) -> PLUS(x,y)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(zero(plus(x,plus(y,un(S)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(S)))
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(S))
 PLUS(plus(x,S),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S)))
 PLUS(un(x),un(y)) -> PLUS(y,un(S))
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(zero(x),un(y)),x3) -> PLUS(un(plus(x,y)),x3)
 PLUS(plus(zero(x),un(y)),x3) -> PLUS(x,y)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(zero(plus(x,plus(y,un(S)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(S)))
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(S))
 PLUS(plus(x,S),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S)))
 PLUS(un(x),un(y)) -> PLUS(y,un(S))
-> FAxioms:
 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
 plus(x3,x4) -> plus(x4,x3)
 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
 times(x3,x4) -> times(x4,x3)
 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) -> PLUS(x4,x3)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
->->-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)

Problem 1.1: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) = PLUS(x4,x3)
-> Pairs:
 PLUS(plus(zero(x),un(y)),x3) -> PLUS(un(plus(x,y)),x3)
 PLUS(plus(zero(x),un(y)),x3) -> PLUS(x,y)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(zero(plus(x,plus(y,un(S)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(S)))
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(S))
 PLUS(plus(x,S),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S)))
 PLUS(un(x),un(y)) -> PLUS(y,un(S))
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Usable Equations:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> Usable Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 zero(S) -> S
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[plus](X1,X2) = X1 + X2 + 1
[times](X1,X2) = 0
[zero](X) = X + 1
[S] = 0
[un](X) = X + 2
[PLUS](X1,X2) = 2.X1 + 2.X2
[TIMES](X1,X2) = 0
[ZERO](X) = 0

Problem 1.1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) = PLUS(x4,x3)
-> Pairs:
 PLUS(plus(zero(x),un(y)),x3) -> PLUS(x,y)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(zero(plus(x,plus(y,un(S)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(S)))
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(S))
 PLUS(plus(x,S),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S)))
 PLUS(un(x),un(y)) -> PLUS(y,un(S))
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(zero(x),un(y)),x3) -> PLUS(x,y)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(zero(plus(x,plus(y,un(S)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(S)))
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(S))
 PLUS(plus(x,S),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S)))
 PLUS(un(x),un(y)) -> PLUS(y,un(S))
-> FAxioms:
 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
 plus(x3,x4) -> plus(x4,x3)
 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
 times(x3,x4) -> times(x4,x3)
 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) -> PLUS(x4,x3)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
->->-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)

Problem 1.1: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) = PLUS(x4,x3)
-> Pairs:
 PLUS(plus(zero(x),un(y)),x3) -> PLUS(x,y)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(zero(plus(x,plus(y,un(S)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(S)))
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(S))
 PLUS(plus(x,S),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S)))
 PLUS(un(x),un(y)) -> PLUS(y,un(S))
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Usable Equations:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> Usable Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 zero(S) -> S
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[plus](X1,X2) = X1 + X2 + 2
[times](X1,X2) = 0
[zero](X) = X
[S] = 0
[un](X) = X + 2
[PLUS](X1,X2) = 2.X1 + 2.X2
[TIMES](X1,X2) = 0
[ZERO](X) = 0

Problem 1.1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) = PLUS(x4,x3)
-> Pairs:
 PLUS(plus(un(x),un(y)),x3) -> PLUS(zero(plus(x,plus(y,un(S)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(S)))
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(S))
 PLUS(plus(x,S),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S)))
 PLUS(un(x),un(y)) -> PLUS(y,un(S))
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(un(x),un(y)),x3) -> PLUS(zero(plus(x,plus(y,un(S)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(S)))
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(S))
 PLUS(plus(x,S),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S)))
 PLUS(un(x),un(y)) -> PLUS(y,un(S))
-> FAxioms:
 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
 plus(x3,x4) -> plus(x4,x3)
 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
 times(x3,x4) -> times(x4,x3)
 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) -> PLUS(x4,x3)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
->->-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)

Problem 1.1: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) = PLUS(x4,x3)
-> Pairs:
 PLUS(plus(un(x),un(y)),x3) -> PLUS(zero(plus(x,plus(y,un(S)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(S)))
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(S))
 PLUS(plus(x,S),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S)))
 PLUS(un(x),un(y)) -> PLUS(y,un(S))
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Usable Equations:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> Usable Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 zero(S) -> S
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[plus](X1,X2) = X1 + X2 + 1
[times](X1,X2) = 0
[zero](X) = X
[S] = 0
[un](X) = X + 2
[PLUS](X1,X2) = 2.X1 + 2.X2
[TIMES](X1,X2) = 0
[ZERO](X) = 0

Problem 1.1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) = PLUS(x4,x3)
-> Pairs:
 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(S)))
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(S))
 PLUS(plus(x,S),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S)))
 PLUS(un(x),un(y)) -> PLUS(y,un(S))
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(S)))
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(S))
 PLUS(plus(x,S),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S)))
 PLUS(un(x),un(y)) -> PLUS(y,un(S))
-> FAxioms:
 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
 plus(x3,x4) -> plus(x4,x3)
 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
 times(x3,x4) -> times(x4,x3)
 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) -> PLUS(x4,x3)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
->->-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)

Problem 1.1: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) = PLUS(x4,x3)
-> Pairs:
 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(S)))
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(S))
 PLUS(plus(x,S),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S)))
 PLUS(un(x),un(y)) -> PLUS(y,un(S))
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Usable Equations:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> Usable Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 zero(S) -> S
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[plus](X1,X2) = X1 + X2 + 1
[times](X1,X2) = 0
[zero](X) = X
[S] = 1
[un](X) = X + 2
[PLUS](X1,X2) = X1 + X2
[TIMES](X1,X2) = 0
[ZERO](X) = 0

Problem 1.1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) = PLUS(x4,x3)
-> Pairs:
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(S))
 PLUS(plus(x,S),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S)))
 PLUS(un(x),un(y)) -> PLUS(y,un(S))
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(S))
 PLUS(plus(x,S),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S)))
 PLUS(un(x),un(y)) -> PLUS(y,un(S))
-> FAxioms:
 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
 plus(x3,x4) -> plus(x4,x3)
 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
 times(x3,x4) -> times(x4,x3)
 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) -> PLUS(x4,x3)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
->->-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)

Problem 1.1: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) = PLUS(x4,x3)
-> Pairs:
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(S))
 PLUS(plus(x,S),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S)))
 PLUS(un(x),un(y)) -> PLUS(y,un(S))
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Usable Equations:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> Usable Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 zero(S) -> S
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[plus](X1,X2) = X1 + X2 + 1
[times](X1,X2) = 0
[zero](X) = X
[S] = 1
[un](X) = X + 2
[PLUS](X1,X2) = X1 + X2
[TIMES](X1,X2) = 0
[ZERO](X) = 0

Problem 1.1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) = PLUS(x4,x3)
-> Pairs:
 PLUS(plus(x,S),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S)))
 PLUS(un(x),un(y)) -> PLUS(y,un(S))
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(x,S),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S)))
 PLUS(un(x),un(y)) -> PLUS(y,un(S))
-> FAxioms:
 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
 plus(x3,x4) -> plus(x4,x3)
 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
 times(x3,x4) -> times(x4,x3)
 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) -> PLUS(x4,x3)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
->->-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)

Problem 1.1: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) = PLUS(x4,x3)
-> Pairs:
 PLUS(plus(x,S),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S)))
 PLUS(un(x),un(y)) -> PLUS(y,un(S))
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Usable Equations:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> Usable Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 zero(S) -> S
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[plus](X1,X2) = X1 + X2 + 1
[times](X1,X2) = 0
[zero](X) = X
[S] = 1
[un](X) = X + 2
[PLUS](X1,X2) = X1 + X2
[TIMES](X1,X2) = 0
[ZERO](X) = 0

Problem 1.1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) = PLUS(x4,x3)
-> Pairs:
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S)))
 PLUS(un(x),un(y)) -> PLUS(y,un(S))
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S)))
 PLUS(un(x),un(y)) -> PLUS(y,un(S))
-> FAxioms:
 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
 plus(x3,x4) -> plus(x4,x3)
 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
 times(x3,x4) -> times(x4,x3)
 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) -> PLUS(x4,x3)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
->->-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)

Problem 1.1: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) = PLUS(x4,x3)
-> Pairs:
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S)))
 PLUS(un(x),un(y)) -> PLUS(y,un(S))
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Usable Equations:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> Usable Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 zero(S) -> S
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[plus](X1,X2) = X1 + X2 + 1
[times](X1,X2) = 0
[zero](X) = X
[S] = 1
[un](X) = X + 2
[PLUS](X1,X2) = X1 + X2
[TIMES](X1,X2) = 0
[ZERO](X) = 0

Problem 1.1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) = PLUS(x4,x3)
-> Pairs:
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S)))
 PLUS(un(x),un(y)) -> PLUS(y,un(S))
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S)))
 PLUS(un(x),un(y)) -> PLUS(y,un(S))
-> FAxioms:
 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
 plus(x3,x4) -> plus(x4,x3)
 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
 times(x3,x4) -> times(x4,x3)
 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) -> PLUS(x4,x3)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
->->-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)

Problem 1.1: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) = PLUS(x4,x3)
-> Pairs:
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S)))
 PLUS(un(x),un(y)) -> PLUS(y,un(S))
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Usable Equations:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> Usable Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 zero(S) -> S
-> SRules:
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[plus](X1,X2) = X1 + X2
[times](X1,X2) = 0
[zero](X) = X + 1
[S] = 1
[un](X) = X + 2
[PLUS](X1,X2) = X1 + X2
[TIMES](X1,X2) = 0
[ZERO](X) = 0

Problem 1.1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) = PLUS(x4,x3)
-> Pairs:
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S)))
 PLUS(un(x),un(y)) -> PLUS(y,un(S))
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S)))
 PLUS(un(x),un(y)) -> PLUS(y,un(S))
-> FAxioms:
 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
 plus(x3,x4) -> plus(x4,x3)
 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
 times(x3,x4) -> times(x4,x3)
 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) -> PLUS(x4,x3)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
->->-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)

Problem 1.1: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) = PLUS(x4,x3)
-> Pairs:
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S)))
 PLUS(un(x),un(y)) -> PLUS(y,un(S))
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Usable Equations:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> Usable Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 zero(S) -> S
-> SRules:
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[plus](X1,X2) = X1 + X2
[times](X1,X2) = 0
[zero](X) = X + 1
[S] = 1
[un](X) = X + 2
[PLUS](X1,X2) = X1 + X2
[TIMES](X1,X2) = 0
[ZERO](X) = 0

Problem 1.1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) = PLUS(x4,x3)
-> Pairs:
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S)))
 PLUS(un(x),un(y)) -> PLUS(y,un(S))
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S)))
 PLUS(un(x),un(y)) -> PLUS(y,un(S))
-> FAxioms:
 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
 plus(x3,x4) -> plus(x4,x3)
 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
 times(x3,x4) -> times(x4,x3)
 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) -> PLUS(x4,x3)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
->->-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)

Problem 1.1: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) = PLUS(x4,x3)
-> Pairs:
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(S)))
 PLUS(un(x),un(y)) -> PLUS(y,un(S))
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Usable Equations:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> Usable Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 zero(S) -> S
-> SRules:
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[plus](X1,X2) = X1 + X2
[times](X1,X2) = 0
[zero](X) = 2.X
[S] = 0
[un](X) = 2.X + 2
[PLUS](X1,X2) = 2.X1 + 2.X2
[TIMES](X1,X2) = 0
[ZERO](X) = 0

Problem 1.1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) = PLUS(x4,x3)
-> Pairs:
 PLUS(un(x),un(y)) -> PLUS(y,un(S))
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(un(x),un(y)) -> PLUS(y,un(S))
-> FAxioms:
 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
 plus(x3,x4) -> plus(x4,x3)
 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
 times(x3,x4) -> times(x4,x3)
 PLUS(plus(x3,x4),x5) -> PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) -> PLUS(x4,x3)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
->->-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)

Problem 1.1: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) = PLUS(x4,x3)
-> Pairs:
 PLUS(un(x),un(y)) -> PLUS(y,un(S))
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Usable Equations:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> Usable Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 zero(S) -> S
-> SRules:
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[plus](X1,X2) = X1 + X2
[times](X1,X2) = 0
[zero](X) = 2.X
[S] = 0
[un](X) = 2.X + 2
[PLUS](X1,X2) = 2.X1 + 2.X2
[TIMES](X1,X2) = 0
[ZERO](X) = 0

Problem 1.1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x3,x4),x5) = PLUS(x3,plus(x4,x5))
 PLUS(x3,x4) = PLUS(x4,x3)
-> Pairs:
 Empty
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Reduction Pairs Processor:
-> FAxioms:
 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
 TIMES(x3,x4) = TIMES(x4,x3)
-> Pairs:
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(x,y)
 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3)
 TIMES(times(x,times(un(y),z)),x3) -> TIMES(plus(x,zero(times(x,y))),z)
 TIMES(times(x,times(un(y),z)),x3) -> TIMES(times(plus(x,zero(times(x,y))),z),x3)
 TIMES(times(x,times(un(y),z)),x3) -> TIMES(x,y)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(x,y)
 TIMES(times(x,S),x3) -> TIMES(S,x3)
 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3)
 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
 TIMES(x,times(un(y),z)) -> TIMES(x,y)
 TIMES(x,zero(y)) -> TIMES(x,y)
 TIMES(x,un(y)) -> TIMES(x,y)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Usable Equations:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> Usable Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
->Interpretation type:
 Simple mixed
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 1
->Interpretation:
 
[plus](X1,X2) = X1 + X2
[times](X1,X2) = X1.X2 + X1 + X2
[zero](X) = X + 1
[S] = 0
[un](X) = X + 1
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = X1.X2 + X1 + X2
[ZERO](X) = 0

Problem 1.2: 

SCC Processor:
-> FAxioms:
 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
 TIMES(x3,x4) = TIMES(x4,x3)
-> Pairs:
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3)
 TIMES(times(x,times(un(y),z)),x3) -> TIMES(plus(x,zero(times(x,y))),z)
 TIMES(times(x,times(un(y),z)),x3) -> TIMES(times(plus(x,zero(times(x,y))),z),x3)
 TIMES(times(x,times(un(y),z)),x3) -> TIMES(x,y)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(x,y)
 TIMES(times(x,S),x3) -> TIMES(S,x3)
 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3)
 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
 TIMES(x,times(un(y),z)) -> TIMES(x,y)
 TIMES(x,zero(y)) -> TIMES(x,y)
 TIMES(x,un(y)) -> TIMES(x,y)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3)
 TIMES(times(x,times(un(y),z)),x3) -> TIMES(plus(x,zero(times(x,y))),z)
 TIMES(times(x,times(un(y),z)),x3) -> TIMES(times(plus(x,zero(times(x,y))),z),x3)
 TIMES(times(x,times(un(y),z)),x3) -> TIMES(x,y)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(x,y)
 TIMES(times(x,S),x3) -> TIMES(S,x3)
 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3)
 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
 TIMES(x,times(un(y),z)) -> TIMES(x,y)
 TIMES(x,zero(y)) -> TIMES(x,y)
 TIMES(x,un(y)) -> TIMES(x,y)
-> FAxioms:
 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
 plus(x3,x4) -> plus(x4,x3)
 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
 times(x3,x4) -> times(x4,x3)
 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
 TIMES(x3,x4) -> TIMES(x4,x3)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
->->-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)

Problem 1.2: 

Reduction Pairs Processor:
-> FAxioms:
 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
 TIMES(x3,x4) = TIMES(x4,x3)
-> Pairs:
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3)
 TIMES(times(x,times(un(y),z)),x3) -> TIMES(plus(x,zero(times(x,y))),z)
 TIMES(times(x,times(un(y),z)),x3) -> TIMES(times(plus(x,zero(times(x,y))),z),x3)
 TIMES(times(x,times(un(y),z)),x3) -> TIMES(x,y)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(x,y)
 TIMES(times(x,S),x3) -> TIMES(S,x3)
 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3)
 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
 TIMES(x,times(un(y),z)) -> TIMES(x,y)
 TIMES(x,zero(y)) -> TIMES(x,y)
 TIMES(x,un(y)) -> TIMES(x,y)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Usable Equations:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> Usable Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
->Interpretation type:
 Simple mixed
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 1
->Interpretation:
 
[plus](X1,X2) = X1 + X2
[times](X1,X2) = X1.X2 + X1 + X2
[zero](X) = X
[S] = 1
[un](X) = X + 1
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = X1.X2 + X1 + X2
[ZERO](X) = 0

Problem 1.2: 

SCC Processor:
-> FAxioms:
 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
 TIMES(x3,x4) = TIMES(x4,x3)
-> Pairs:
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3)
 TIMES(times(x,times(un(y),z)),x3) -> TIMES(times(plus(x,zero(times(x,y))),z),x3)
 TIMES(times(x,times(un(y),z)),x3) -> TIMES(x,y)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(x,y)
 TIMES(times(x,S),x3) -> TIMES(S,x3)
 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3)
 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
 TIMES(x,times(un(y),z)) -> TIMES(x,y)
 TIMES(x,zero(y)) -> TIMES(x,y)
 TIMES(x,un(y)) -> TIMES(x,y)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3)
 TIMES(times(x,times(un(y),z)),x3) -> TIMES(times(plus(x,zero(times(x,y))),z),x3)
 TIMES(times(x,times(un(y),z)),x3) -> TIMES(x,y)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(x,y)
 TIMES(times(x,S),x3) -> TIMES(S,x3)
 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3)
 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
 TIMES(x,times(un(y),z)) -> TIMES(x,y)
 TIMES(x,zero(y)) -> TIMES(x,y)
 TIMES(x,un(y)) -> TIMES(x,y)
-> FAxioms:
 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
 plus(x3,x4) -> plus(x4,x3)
 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
 times(x3,x4) -> times(x4,x3)
 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
 TIMES(x3,x4) -> TIMES(x4,x3)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
->->-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)

Problem 1.2: 

Reduction Pairs Processor:
-> FAxioms:
 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
 TIMES(x3,x4) = TIMES(x4,x3)
-> Pairs:
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3)
 TIMES(times(x,times(un(y),z)),x3) -> TIMES(times(plus(x,zero(times(x,y))),z),x3)
 TIMES(times(x,times(un(y),z)),x3) -> TIMES(x,y)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(x,y)
 TIMES(times(x,S),x3) -> TIMES(S,x3)
 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3)
 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
 TIMES(x,times(un(y),z)) -> TIMES(x,y)
 TIMES(x,zero(y)) -> TIMES(x,y)
 TIMES(x,un(y)) -> TIMES(x,y)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Usable Equations:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> Usable Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
->Interpretation type:
 Simple mixed
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 1
->Interpretation:
 
[plus](X1,X2) = X1 + X2
[times](X1,X2) = X1.X2 + X1 + X2
[zero](X) = X
[S] = 1
[un](X) = X + 1
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = X1.X2 + X1 + X2
[ZERO](X) = 0

Problem 1.2: 

SCC Processor:
-> FAxioms:
 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
 TIMES(x3,x4) = TIMES(x4,x3)
-> Pairs:
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3)
 TIMES(times(x,times(un(y),z)),x3) -> TIMES(x,y)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(x,y)
 TIMES(times(x,S),x3) -> TIMES(S,x3)
 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3)
 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
 TIMES(x,times(un(y),z)) -> TIMES(x,y)
 TIMES(x,zero(y)) -> TIMES(x,y)
 TIMES(x,un(y)) -> TIMES(x,y)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3)
 TIMES(times(x,times(un(y),z)),x3) -> TIMES(x,y)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(x,y)
 TIMES(times(x,S),x3) -> TIMES(S,x3)
 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3)
 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
 TIMES(x,times(un(y),z)) -> TIMES(x,y)
 TIMES(x,zero(y)) -> TIMES(x,y)
 TIMES(x,un(y)) -> TIMES(x,y)
-> FAxioms:
 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
 plus(x3,x4) -> plus(x4,x3)
 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
 times(x3,x4) -> times(x4,x3)
 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
 TIMES(x3,x4) -> TIMES(x4,x3)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
->->-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)

Problem 1.2: 

Reduction Pairs Processor:
-> FAxioms:
 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
 TIMES(x3,x4) = TIMES(x4,x3)
-> Pairs:
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3)
 TIMES(times(x,times(un(y),z)),x3) -> TIMES(x,y)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(x,y)
 TIMES(times(x,S),x3) -> TIMES(S,x3)
 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3)
 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
 TIMES(x,times(un(y),z)) -> TIMES(x,y)
 TIMES(x,zero(y)) -> TIMES(x,y)
 TIMES(x,un(y)) -> TIMES(x,y)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Usable Equations:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> Usable Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
->Interpretation type:
 Simple mixed
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 1
->Interpretation:
 
[plus](X1,X2) = X1 + X2
[times](X1,X2) = X1.X2 + X1 + X2
[zero](X) = X
[S] = 1
[un](X) = X + 1
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = X1.X2 + X1 + X2
[ZERO](X) = 0

Problem 1.2: 

SCC Processor:
-> FAxioms:
 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
 TIMES(x3,x4) = TIMES(x4,x3)
-> Pairs:
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(x,y)
 TIMES(times(x,S),x3) -> TIMES(S,x3)
 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3)
 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
 TIMES(x,times(un(y),z)) -> TIMES(x,y)
 TIMES(x,zero(y)) -> TIMES(x,y)
 TIMES(x,un(y)) -> TIMES(x,y)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(x,y)
 TIMES(times(x,S),x3) -> TIMES(S,x3)
 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3)
 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
 TIMES(x,times(un(y),z)) -> TIMES(x,y)
 TIMES(x,zero(y)) -> TIMES(x,y)
 TIMES(x,un(y)) -> TIMES(x,y)
-> FAxioms:
 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
 plus(x3,x4) -> plus(x4,x3)
 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
 times(x3,x4) -> times(x4,x3)
 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
 TIMES(x3,x4) -> TIMES(x4,x3)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
->->-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)

Problem 1.2: 

Reduction Pairs Processor:
-> FAxioms:
 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
 TIMES(x3,x4) = TIMES(x4,x3)
-> Pairs:
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(x,y)
 TIMES(times(x,S),x3) -> TIMES(S,x3)
 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3)
 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
 TIMES(x,times(un(y),z)) -> TIMES(x,y)
 TIMES(x,zero(y)) -> TIMES(x,y)
 TIMES(x,un(y)) -> TIMES(x,y)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Usable Equations:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> Usable Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
->Interpretation type:
 Simple mixed
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 1
->Interpretation:
 
[plus](X1,X2) = X1 + X2
[times](X1,X2) = X1.X2 + X1 + X2
[zero](X) = X + 1
[S] = 0
[un](X) = X + 1
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = X1.X2 + X1 + X2
[ZERO](X) = 0

Problem 1.2: 

SCC Processor:
-> FAxioms:
 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
 TIMES(x3,x4) = TIMES(x4,x3)
-> Pairs:
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,S),x3) -> TIMES(S,x3)
 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3)
 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
 TIMES(x,times(un(y),z)) -> TIMES(x,y)
 TIMES(x,zero(y)) -> TIMES(x,y)
 TIMES(x,un(y)) -> TIMES(x,y)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,S),x3) -> TIMES(S,x3)
 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3)
 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
 TIMES(x,times(un(y),z)) -> TIMES(x,y)
 TIMES(x,zero(y)) -> TIMES(x,y)
 TIMES(x,un(y)) -> TIMES(x,y)
-> FAxioms:
 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
 plus(x3,x4) -> plus(x4,x3)
 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
 times(x3,x4) -> times(x4,x3)
 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
 TIMES(x3,x4) -> TIMES(x4,x3)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
->->-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)

Problem 1.2: 

Reduction Pairs Processor:
-> FAxioms:
 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
 TIMES(x3,x4) = TIMES(x4,x3)
-> Pairs:
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,S),x3) -> TIMES(S,x3)
 TIMES(times(x,un(y)),x3) -> TIMES(plus(x,zero(times(x,y))),x3)
 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
 TIMES(x,times(un(y),z)) -> TIMES(x,y)
 TIMES(x,zero(y)) -> TIMES(x,y)
 TIMES(x,un(y)) -> TIMES(x,y)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Usable Equations:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> Usable Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
->Interpretation type:
 Simple mixed
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 1
->Interpretation:
 
[plus](X1,X2) = X1 + X2
[times](X1,X2) = X1.X2 + X1 + X2
[zero](X) = X
[S] = 1
[un](X) = X + 1
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = X1.X2 + X1 + X2
[ZERO](X) = 0

Problem 1.2: 

SCC Processor:
-> FAxioms:
 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
 TIMES(x3,x4) = TIMES(x4,x3)
-> Pairs:
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,S),x3) -> TIMES(S,x3)
 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
 TIMES(x,times(un(y),z)) -> TIMES(x,y)
 TIMES(x,zero(y)) -> TIMES(x,y)
 TIMES(x,un(y)) -> TIMES(x,y)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,S),x3) -> TIMES(S,x3)
 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
 TIMES(x,times(un(y),z)) -> TIMES(x,y)
 TIMES(x,zero(y)) -> TIMES(x,y)
 TIMES(x,un(y)) -> TIMES(x,y)
-> FAxioms:
 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
 plus(x3,x4) -> plus(x4,x3)
 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
 times(x3,x4) -> times(x4,x3)
 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
 TIMES(x3,x4) -> TIMES(x4,x3)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
->->-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)

Problem 1.2: 

Reduction Pairs Processor:
-> FAxioms:
 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
 TIMES(x3,x4) = TIMES(x4,x3)
-> Pairs:
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,S),x3) -> TIMES(S,x3)
 TIMES(times(x,un(y)),x3) -> TIMES(x,y)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
 TIMES(x,times(un(y),z)) -> TIMES(x,y)
 TIMES(x,zero(y)) -> TIMES(x,y)
 TIMES(x,un(y)) -> TIMES(x,y)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Usable Equations:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> Usable Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
->Interpretation type:
 Simple mixed
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 1
->Interpretation:
 
[plus](X1,X2) = X1 + X2 + 1
[times](X1,X2) = X1.X2 + X1 + X2
[zero](X) = X
[S] = 0
[un](X) = X + 1
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = X1.X2 + X1 + X2
[ZERO](X) = 0

Problem 1.2: 

SCC Processor:
-> FAxioms:
 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
 TIMES(x3,x4) = TIMES(x4,x3)
-> Pairs:
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,S),x3) -> TIMES(S,x3)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
 TIMES(x,times(un(y),z)) -> TIMES(x,y)
 TIMES(x,zero(y)) -> TIMES(x,y)
 TIMES(x,un(y)) -> TIMES(x,y)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,S),x3) -> TIMES(S,x3)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
 TIMES(x,times(un(y),z)) -> TIMES(x,y)
 TIMES(x,zero(y)) -> TIMES(x,y)
 TIMES(x,un(y)) -> TIMES(x,y)
-> FAxioms:
 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
 plus(x3,x4) -> plus(x4,x3)
 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
 times(x3,x4) -> times(x4,x3)
 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
 TIMES(x3,x4) -> TIMES(x4,x3)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
->->-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)

Problem 1.2: 

Reduction Pairs Processor:
-> FAxioms:
 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
 TIMES(x3,x4) = TIMES(x4,x3)
-> Pairs:
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,S),x3) -> TIMES(S,x3)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
 TIMES(x,times(un(y),z)) -> TIMES(x,y)
 TIMES(x,zero(y)) -> TIMES(x,y)
 TIMES(x,un(y)) -> TIMES(x,y)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Usable Equations:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> Usable Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
->Interpretation type:
 Simple mixed
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 1
->Interpretation:
 
[plus](X1,X2) = X1 + X2
[times](X1,X2) = X1.X2 + X1 + X2
[zero](X) = X + 1
[S] = 0
[un](X) = X + 1
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = X1.X2 + X1 + X2
[ZERO](X) = 0

Problem 1.2: 

SCC Processor:
-> FAxioms:
 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
 TIMES(x3,x4) = TIMES(x4,x3)
-> Pairs:
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,S),x3) -> TIMES(S,x3)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
 TIMES(x,times(un(y),z)) -> TIMES(x,y)
 TIMES(x,zero(y)) -> TIMES(x,y)
 TIMES(x,un(y)) -> TIMES(x,y)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,S),x3) -> TIMES(S,x3)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
 TIMES(x,times(un(y),z)) -> TIMES(x,y)
 TIMES(x,zero(y)) -> TIMES(x,y)
 TIMES(x,un(y)) -> TIMES(x,y)
-> FAxioms:
 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
 plus(x3,x4) -> plus(x4,x3)
 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
 times(x3,x4) -> times(x4,x3)
 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
 TIMES(x3,x4) -> TIMES(x4,x3)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
->->-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)

Problem 1.2: 

Reduction Pairs Processor:
-> FAxioms:
 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
 TIMES(x3,x4) = TIMES(x4,x3)
-> Pairs:
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,S),x3) -> TIMES(S,x3)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(un(y),z)) -> TIMES(plus(x,zero(times(x,y))),z)
 TIMES(x,times(un(y),z)) -> TIMES(x,y)
 TIMES(x,zero(y)) -> TIMES(x,y)
 TIMES(x,un(y)) -> TIMES(x,y)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Usable Equations:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> Usable Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
->Interpretation type:
 Simple mixed
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 1
->Interpretation:
 
[plus](X1,X2) = X1 + X2
[times](X1,X2) = X1.X2 + X1 + X2
[zero](X) = X
[S] = 1
[un](X) = X + 1
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = X1.X2 + X1 + X2
[ZERO](X) = 0

Problem 1.2: 

SCC Processor:
-> FAxioms:
 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
 TIMES(x3,x4) = TIMES(x4,x3)
-> Pairs:
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,S),x3) -> TIMES(S,x3)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(un(y),z)) -> TIMES(x,y)
 TIMES(x,zero(y)) -> TIMES(x,y)
 TIMES(x,un(y)) -> TIMES(x,y)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,S),x3) -> TIMES(S,x3)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(un(y),z)) -> TIMES(x,y)
 TIMES(x,zero(y)) -> TIMES(x,y)
 TIMES(x,un(y)) -> TIMES(x,y)
-> FAxioms:
 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
 plus(x3,x4) -> plus(x4,x3)
 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
 times(x3,x4) -> times(x4,x3)
 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
 TIMES(x3,x4) -> TIMES(x4,x3)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
->->-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)

Problem 1.2: 

Reduction Pairs Processor:
-> FAxioms:
 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
 TIMES(x3,x4) = TIMES(x4,x3)
-> Pairs:
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,S),x3) -> TIMES(S,x3)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(un(y),z)) -> TIMES(x,y)
 TIMES(x,zero(y)) -> TIMES(x,y)
 TIMES(x,un(y)) -> TIMES(x,y)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Usable Equations:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> Usable Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
->Interpretation type:
 Simple mixed
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 1
->Interpretation:
 
[plus](X1,X2) = X1 + X2
[times](X1,X2) = X1.X2 + X1 + X2
[zero](X) = X + 1
[S] = 0
[un](X) = X + 1
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = X1.X2 + X1 + X2
[ZERO](X) = 0

Problem 1.2: 

SCC Processor:
-> FAxioms:
 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
 TIMES(x3,x4) = TIMES(x4,x3)
-> Pairs:
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,S),x3) -> TIMES(S,x3)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,zero(y)) -> TIMES(x,y)
 TIMES(x,un(y)) -> TIMES(x,y)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,S),x3) -> TIMES(S,x3)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,zero(y)) -> TIMES(x,y)
 TIMES(x,un(y)) -> TIMES(x,y)
-> FAxioms:
 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
 plus(x3,x4) -> plus(x4,x3)
 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
 times(x3,x4) -> times(x4,x3)
 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
 TIMES(x3,x4) -> TIMES(x4,x3)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
->->-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)

Problem 1.2: 

Reduction Pairs Processor:
-> FAxioms:
 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
 TIMES(x3,x4) = TIMES(x4,x3)
-> Pairs:
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,S),x3) -> TIMES(S,x3)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,zero(y)) -> TIMES(x,y)
 TIMES(x,un(y)) -> TIMES(x,y)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Usable Equations:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> Usable Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
->Interpretation type:
 Simple mixed
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 1
->Interpretation:
 
[plus](X1,X2) = X1 + X2
[times](X1,X2) = X1.X2 + X1 + X2
[zero](X) = X + 1
[S] = 0
[un](X) = X + 1
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = X1.X2 + X1 + X2
[ZERO](X) = 0

Problem 1.2: 

SCC Processor:
-> FAxioms:
 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
 TIMES(x3,x4) = TIMES(x4,x3)
-> Pairs:
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,S),x3) -> TIMES(S,x3)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,un(y)) -> TIMES(x,y)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,S),x3) -> TIMES(S,x3)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,un(y)) -> TIMES(x,y)
-> FAxioms:
 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
 plus(x3,x4) -> plus(x4,x3)
 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
 times(x3,x4) -> times(x4,x3)
 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
 TIMES(x3,x4) -> TIMES(x4,x3)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
->->-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)

Problem 1.2: 

Reduction Pairs Processor:
-> FAxioms:
 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
 TIMES(x3,x4) = TIMES(x4,x3)
-> Pairs:
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,S),x3) -> TIMES(S,x3)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,un(y)) -> TIMES(x,y)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Usable Equations:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> Usable Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
->Interpretation type:
 Simple mixed
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 1
->Interpretation:
 
[plus](X1,X2) = X1 + X2
[times](X1,X2) = X1.X2 + X1 + X2
[zero](X) = X
[S] = 1
[un](X) = X + 1
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = X1.X2 + X1 + X2
[ZERO](X) = 0

Problem 1.2: 

SCC Processor:
-> FAxioms:
 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
 TIMES(x3,x4) = TIMES(x4,x3)
-> Pairs:
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,S),x3) -> TIMES(S,x3)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,S),x3) -> TIMES(S,x3)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
-> FAxioms:
 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
 plus(x3,x4) -> plus(x4,x3)
 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
 times(x3,x4) -> times(x4,x3)
 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
 TIMES(x3,x4) -> TIMES(x4,x3)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
->->-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)

Problem 1.2: 

Reduction Pairs Processor:
-> FAxioms:
 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
 TIMES(x3,x4) = TIMES(x4,x3)
-> Pairs:
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(times(zero(times(x,y)),z),x3)
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,S),x3) -> TIMES(S,x3)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Usable Equations:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> Usable Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[plus](X1,X2) = X1 + X2
[times](X1,X2) = X1 + X2 + 1
[zero](X) = 0
[S] = 0
[un](X) = 0
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = 2.X1 + 2.X2
[ZERO](X) = 0

Problem 1.2: 

SCC Processor:
-> FAxioms:
 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
 TIMES(x3,x4) = TIMES(x4,x3)
-> Pairs:
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,S),x3) -> TIMES(S,x3)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,S),x3) -> TIMES(S,x3)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
-> FAxioms:
 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
 plus(x3,x4) -> plus(x4,x3)
 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
 times(x3,x4) -> times(x4,x3)
 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
 TIMES(x3,x4) -> TIMES(x4,x3)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
->->-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)

Problem 1.2: 

Reduction Pairs Processor:
-> FAxioms:
 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
 TIMES(x3,x4) = TIMES(x4,x3)
-> Pairs:
 TIMES(times(x,times(zero(y),z)),x3) -> TIMES(zero(times(x,y)),z)
 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,S),x3) -> TIMES(S,x3)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Usable Equations:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> Usable Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[plus](X1,X2) = X1 + X2 + 2
[times](X1,X2) = X1 + X2 + 2
[zero](X) = 0
[S] = 0
[un](X) = 1
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = 2.X1 + 2.X2
[ZERO](X) = 0

Problem 1.2: 

SCC Processor:
-> FAxioms:
 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
 TIMES(x3,x4) = TIMES(x4,x3)
-> Pairs:
 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,S),x3) -> TIMES(S,x3)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,S),x3) -> TIMES(S,x3)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
-> FAxioms:
 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
 plus(x3,x4) -> plus(x4,x3)
 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
 times(x3,x4) -> times(x4,x3)
 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
 TIMES(x3,x4) -> TIMES(x4,x3)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
->->-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)

Problem 1.2: 

Reduction Pairs Processor:
-> FAxioms:
 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
 TIMES(x3,x4) = TIMES(x4,x3)
-> Pairs:
 TIMES(times(x,times(S,z)),x3) -> TIMES(times(S,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,S),x3) -> TIMES(S,x3)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Usable Equations:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> Usable Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[plus](X1,X2) = X1 + X2 + 1
[times](X1,X2) = X1 + X2 + 2
[zero](X) = 2
[S] = 0
[un](X) = 2
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = 2.X1 + 2.X2
[ZERO](X) = 0

Problem 1.2: 

SCC Processor:
-> FAxioms:
 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
 TIMES(x3,x4) = TIMES(x4,x3)
-> Pairs:
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,S),x3) -> TIMES(S,x3)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,S),x3) -> TIMES(S,x3)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
-> FAxioms:
 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
 plus(x3,x4) -> plus(x4,x3)
 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
 times(x3,x4) -> times(x4,x3)
 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
 TIMES(x3,x4) -> TIMES(x4,x3)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
->->-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)

Problem 1.2: 

Reduction Pairs Processor:
-> FAxioms:
 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
 TIMES(x3,x4) = TIMES(x4,x3)
-> Pairs:
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,S),x3) -> TIMES(S,x3)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Usable Equations:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> Usable Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[plus](X1,X2) = X1 + X2
[times](X1,X2) = X1 + X2 + 2
[zero](X) = 0
[S] = 0
[un](X) = 0
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = 2.X1 + 2.X2
[ZERO](X) = 0

Problem 1.2: 

SCC Processor:
-> FAxioms:
 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
 TIMES(x3,x4) = TIMES(x4,x3)
-> Pairs:
 TIMES(times(x,S),x3) -> TIMES(S,x3)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 TIMES(times(x,S),x3) -> TIMES(S,x3)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
-> FAxioms:
 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
 plus(x3,x4) -> plus(x4,x3)
 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
 times(x3,x4) -> times(x4,x3)
 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
 TIMES(x3,x4) -> TIMES(x4,x3)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
->->-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)

Problem 1.2: 

Reduction Pairs Processor:
-> FAxioms:
 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
 TIMES(x3,x4) = TIMES(x4,x3)
-> Pairs:
 TIMES(times(x,S),x3) -> TIMES(S,x3)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Usable Equations:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> Usable Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[plus](X1,X2) = X1 + X2 + 1
[times](X1,X2) = X1 + X2 + 1
[zero](X) = 0
[S] = 0
[un](X) = 0
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = 2.X1 + 2.X2
[ZERO](X) = 0

Problem 1.2: 

SCC Processor:
-> FAxioms:
 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
 TIMES(x3,x4) = TIMES(x4,x3)
-> Pairs:
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
-> FAxioms:
 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
 plus(x3,x4) -> plus(x4,x3)
 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
 times(x3,x4) -> times(x4,x3)
 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
 TIMES(x3,x4) -> TIMES(x4,x3)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
->->-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)

Problem 1.2: 

Reduction Pairs Processor:
-> FAxioms:
 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
 TIMES(x3,x4) = TIMES(x4,x3)
-> Pairs:
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Usable Equations:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> Usable Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[plus](X1,X2) = X1 + X2 + 1
[times](X1,X2) = X1 + X2 + 2
[zero](X) = 1
[S] = 1
[un](X) = 0
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = 2.X1 + 2.X2
[ZERO](X) = 0

Problem 1.2: 

SCC Processor:
-> FAxioms:
 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
 TIMES(x3,x4) = TIMES(x4,x3)
-> Pairs:
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
-> FAxioms:
 plus(plus(x3,x4),x5) -> plus(x3,plus(x4,x5))
 plus(x3,x4) -> plus(x4,x3)
 times(times(x3,x4),x5) -> times(x3,times(x4,x5))
 times(x3,x4) -> times(x4,x3)
 TIMES(times(x3,x4),x5) -> TIMES(x3,times(x4,x5))
 TIMES(x3,x4) -> TIMES(x4,x3)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
->->-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)

Problem 1.2: 

Reduction Pairs Processor:
-> FAxioms:
 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
 TIMES(x3,x4) = TIMES(x4,x3)
-> Pairs:
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Usable Equations:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> Usable Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[plus](X1,X2) = X1 + X2 + 1
[times](X1,X2) = X1 + X2 + 2
[zero](X) = 2
[S] = 2
[un](X) = 1
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = 2.X1 + 2.X2
[ZERO](X) = 0

Problem 1.2: 

SCC Processor:
-> FAxioms:
 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
 TIMES(x3,x4) = TIMES(x4,x3)
-> Pairs:
 Empty
-> EAxioms:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
 times(times(x3,x4),x5) = times(x3,times(x4,x5))
 times(x3,x4) = times(x4,x3)
-> Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(un(x),un(y)) -> zero(plus(x,plus(y,un(S))))
 plus(x,S) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(S,z)) -> times(S,z)
 times(x,times(un(y),z)) -> times(plus(x,zero(times(x,y))),z)
 times(x,zero(y)) -> zero(times(x,y))
 times(x,S) -> S
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(S) -> S
-> SRules:
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.