YES

Problem 1: 

(VAR x y z)
(THEORY 
(AC plus times))
(RULES
minus(x,y) -> plus(x,neg(y))
neg(zero(x)) -> zero(neg(x))
neg(0) -> 0
neg(j(x)) -> un(neg(x))
neg(un(x)) -> j(neg(x))
plus(zero(x),zero(y)) -> zero(plus(x,y))
plus(zero(x),j(y)) -> j(plus(x,y))
plus(zero(x),un(y)) -> un(plus(x,y))
plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
plus(un(x),j(y)) -> zero(plus(x,y))
plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
plus(x,0) -> x
times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
times(x,times(0,z)) -> times(0,z)
times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
times(x,un(y)) -> plus(x,zero(times(x,y)))
zero(0) -> 0
)

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:
 MINUS(x,y) -> NEG(y)
 MINUS(x,y) -> PLUS(x,neg(y))
 NEG(zero(x)) -> NEG(x)
 NEG(zero(x)) -> ZERO(neg(x))
 NEG(j(x)) -> NEG(x)
 NEG(un(x)) -> NEG(x)
 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),j(y)),x3) -> PLUS(j(plus(x,y)),x3)
 PLUS(plus(zero(x),j(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(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(j(x),j(y)),x3) -> PLUS(x,plus(y,j(0)))
 PLUS(plus(j(x),j(y)),x3) -> PLUS(y,j(0))
 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
 PLUS(plus(un(x),j(y)),x3) -> ZERO(plus(x,y))
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),zero(y)) -> ZERO(plus(x,y))
 PLUS(zero(x),j(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
 PLUS(j(x),j(y)) -> PLUS(y,j(0))
 PLUS(un(x),j(y)) -> PLUS(x,y)
 PLUS(un(x),j(y)) -> ZERO(plus(x,y))
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
 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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,times(j(y),z)),x3) -> NEG(x)
 TIMES(times(x,times(j(y),z)),x3) -> PLUS(zero(times(x,y)),neg(x))
 TIMES(times(x,times(j(y),z)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
 TIMES(times(x,times(j(y),z)),x3) -> TIMES(times(plus(zero(times(x,y)),neg(x)),z),x3)
 TIMES(times(x,times(j(y),z)),x3) -> TIMES(x,y)
 TIMES(times(x,times(j(y),z)),x3) -> ZERO(times(x,y))
 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,0),x3) -> TIMES(0,x3)
 TIMES(times(x,j(y)),x3) -> NEG(x)
 TIMES(times(x,j(y)),x3) -> PLUS(zero(times(x,y)),neg(x))
 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
 TIMES(times(x,j(y)),x3) -> TIMES(x,y)
 TIMES(times(x,j(y)),x3) -> ZERO(times(x,y))
 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(j(y),z)) -> NEG(x)
 TIMES(x,times(j(y),z)) -> PLUS(zero(times(x,y)),neg(x))
 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
 TIMES(x,times(j(y),z)) -> TIMES(x,y)
 TIMES(x,times(j(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,j(y)) -> NEG(x)
 TIMES(x,j(y)) -> PLUS(zero(times(x,y)),neg(x))
 TIMES(x,j(y)) -> TIMES(x,y)
 TIMES(x,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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:
 MINUS(x,y) -> NEG(y)
 MINUS(x,y) -> PLUS(x,neg(y))
 NEG(zero(x)) -> NEG(x)
 NEG(zero(x)) -> ZERO(neg(x))
 NEG(j(x)) -> NEG(x)
 NEG(un(x)) -> NEG(x)
 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),j(y)),x3) -> PLUS(j(plus(x,y)),x3)
 PLUS(plus(zero(x),j(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(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(j(x),j(y)),x3) -> PLUS(x,plus(y,j(0)))
 PLUS(plus(j(x),j(y)),x3) -> PLUS(y,j(0))
 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
 PLUS(plus(un(x),j(y)),x3) -> ZERO(plus(x,y))
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),zero(y)) -> ZERO(plus(x,y))
 PLUS(zero(x),j(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
 PLUS(j(x),j(y)) -> PLUS(y,j(0))
 PLUS(un(x),j(y)) -> PLUS(x,y)
 PLUS(un(x),j(y)) -> ZERO(plus(x,y))
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
 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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,times(j(y),z)),x3) -> NEG(x)
 TIMES(times(x,times(j(y),z)),x3) -> PLUS(zero(times(x,y)),neg(x))
 TIMES(times(x,times(j(y),z)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
 TIMES(times(x,times(j(y),z)),x3) -> TIMES(times(plus(zero(times(x,y)),neg(x)),z),x3)
 TIMES(times(x,times(j(y),z)),x3) -> TIMES(x,y)
 TIMES(times(x,times(j(y),z)),x3) -> ZERO(times(x,y))
 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,0),x3) -> TIMES(0,x3)
 TIMES(times(x,j(y)),x3) -> NEG(x)
 TIMES(times(x,j(y)),x3) -> PLUS(zero(times(x,y)),neg(x))
 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
 TIMES(times(x,j(y)),x3) -> TIMES(x,y)
 TIMES(times(x,j(y)),x3) -> ZERO(times(x,y))
 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(j(y),z)) -> NEG(x)
 TIMES(x,times(j(y),z)) -> PLUS(zero(times(x,y)),neg(x))
 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
 TIMES(x,times(j(y),z)) -> TIMES(x,y)
 TIMES(x,times(j(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,j(y)) -> NEG(x)
 TIMES(x,j(y)) -> PLUS(zero(times(x,y)),neg(x))
 TIMES(x,j(y)) -> TIMES(x,y)
 TIMES(x,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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),j(y)),x3) -> PLUS(j(plus(x,y)),x3)
 PLUS(plus(zero(x),j(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(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(j(x),j(y)),x3) -> PLUS(x,plus(y,j(0)))
 PLUS(plus(j(x),j(y)),x3) -> PLUS(y,j(0))
 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),j(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
 PLUS(j(x),j(y)) -> PLUS(y,j(0))
 PLUS(un(x),j(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->->Cycle:
->->-> Pairs:
 NEG(zero(x)) -> NEG(x)
 NEG(j(x)) -> NEG(x)
 NEG(un(x)) -> NEG(x)
-> 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)
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> SRules:
 Empty
->->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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,times(j(y),z)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
 TIMES(times(x,times(j(y),z)),x3) -> TIMES(times(plus(zero(times(x,y)),neg(x)),z),x3)
 TIMES(times(x,times(j(y),z)),x3) -> 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,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(x,y)
 TIMES(times(x,0),x3) -> TIMES(0,x3)
 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
 TIMES(times(x,j(y)),x3) -> 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(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
 TIMES(x,times(j(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,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)


The problem is decomposed in 3 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),j(y)),x3) -> PLUS(j(plus(x,y)),x3)
 PLUS(plus(zero(x),j(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(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(j(x),j(y)),x3) -> PLUS(x,plus(y,j(0)))
 PLUS(plus(j(x),j(y)),x3) -> PLUS(y,j(0))
 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),j(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
 PLUS(j(x),j(y)) -> PLUS(y,j(0))
 PLUS(un(x),j(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> Usable Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 zero(0) -> 0
-> 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:
 
[minus](X1,X2) = 0
[neg](X) = 0
[plus](X1,X2) = X1 + X2
[times](X1,X2) = 0
[zero](X) = X + 1
[0] = 0
[j](X) = X + 2
[un](X) = X + 2
[MINUS](X1,X2) = 0
[NEG](X) = 0
[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),j(y)),x3) -> PLUS(j(plus(x,y)),x3)
 PLUS(plus(zero(x),j(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(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(j(x),j(y)),x3) -> PLUS(x,plus(y,j(0)))
 PLUS(plus(j(x),j(y)),x3) -> PLUS(y,j(0))
 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),j(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
 PLUS(j(x),j(y)) -> PLUS(y,j(0))
 PLUS(un(x),j(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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),j(y)),x3) -> PLUS(j(plus(x,y)),x3)
 PLUS(plus(zero(x),j(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(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(j(x),j(y)),x3) -> PLUS(x,plus(y,j(0)))
 PLUS(plus(j(x),j(y)),x3) -> PLUS(y,j(0))
 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),j(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
 PLUS(j(x),j(y)) -> PLUS(y,j(0))
 PLUS(un(x),j(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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),j(y)),x3) -> PLUS(j(plus(x,y)),x3)
 PLUS(plus(zero(x),j(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(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(j(x),j(y)),x3) -> PLUS(x,plus(y,j(0)))
 PLUS(plus(j(x),j(y)),x3) -> PLUS(y,j(0))
 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),j(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
 PLUS(j(x),j(y)) -> PLUS(y,j(0))
 PLUS(un(x),j(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> Usable Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 zero(0) -> 0
-> 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:
 
[minus](X1,X2) = 0
[neg](X) = 0
[plus](X1,X2) = X1 + X2
[times](X1,X2) = 0
[zero](X) = X + 1
[0] = 0
[j](X) = X + 2
[un](X) = X + 2
[MINUS](X1,X2) = 0
[NEG](X) = 0
[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),j(y)),x3) -> PLUS(j(plus(x,y)),x3)
 PLUS(plus(zero(x),j(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(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(j(x),j(y)),x3) -> PLUS(x,plus(y,j(0)))
 PLUS(plus(j(x),j(y)),x3) -> PLUS(y,j(0))
 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),j(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
 PLUS(j(x),j(y)) -> PLUS(y,j(0))
 PLUS(un(x),j(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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),j(y)),x3) -> PLUS(j(plus(x,y)),x3)
 PLUS(plus(zero(x),j(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(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(j(x),j(y)),x3) -> PLUS(x,plus(y,j(0)))
 PLUS(plus(j(x),j(y)),x3) -> PLUS(y,j(0))
 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),j(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
 PLUS(j(x),j(y)) -> PLUS(y,j(0))
 PLUS(un(x),j(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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),j(y)),x3) -> PLUS(j(plus(x,y)),x3)
 PLUS(plus(zero(x),j(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(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(j(x),j(y)),x3) -> PLUS(x,plus(y,j(0)))
 PLUS(plus(j(x),j(y)),x3) -> PLUS(y,j(0))
 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),j(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
 PLUS(j(x),j(y)) -> PLUS(y,j(0))
 PLUS(un(x),j(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> Usable Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 zero(0) -> 0
-> 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:
 
[minus](X1,X2) = 0
[neg](X) = 0
[plus](X1,X2) = X1 + X2
[times](X1,X2) = 0
[zero](X) = X + 2
[0] = 0
[j](X) = X + 2
[un](X) = X + 2
[MINUS](X1,X2) = 0
[NEG](X) = 0
[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),j(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(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(j(x),j(y)),x3) -> PLUS(x,plus(y,j(0)))
 PLUS(plus(j(x),j(y)),x3) -> PLUS(y,j(0))
 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),j(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
 PLUS(j(x),j(y)) -> PLUS(y,j(0))
 PLUS(un(x),j(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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),j(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(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(j(x),j(y)),x3) -> PLUS(x,plus(y,j(0)))
 PLUS(plus(j(x),j(y)),x3) -> PLUS(y,j(0))
 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),j(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
 PLUS(j(x),j(y)) -> PLUS(y,j(0))
 PLUS(un(x),j(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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),j(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(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(j(x),j(y)),x3) -> PLUS(x,plus(y,j(0)))
 PLUS(plus(j(x),j(y)),x3) -> PLUS(y,j(0))
 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),j(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
 PLUS(j(x),j(y)) -> PLUS(y,j(0))
 PLUS(un(x),j(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> Usable Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 zero(0) -> 0
-> 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:
 
[minus](X1,X2) = 0
[neg](X) = 0
[plus](X1,X2) = X1 + X2
[times](X1,X2) = 0
[zero](X) = X
[0] = 0
[j](X) = X + 2
[un](X) = X + 2
[MINUS](X1,X2) = 0
[NEG](X) = 0
[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(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(j(x),j(y)),x3) -> PLUS(x,plus(y,j(0)))
 PLUS(plus(j(x),j(y)),x3) -> PLUS(y,j(0))
 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),j(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
 PLUS(j(x),j(y)) -> PLUS(y,j(0))
 PLUS(un(x),j(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(j(x),j(y)),x3) -> PLUS(x,plus(y,j(0)))
 PLUS(plus(j(x),j(y)),x3) -> PLUS(y,j(0))
 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),j(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
 PLUS(j(x),j(y)) -> PLUS(y,j(0))
 PLUS(un(x),j(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(j(x),j(y)),x3) -> PLUS(x,plus(y,j(0)))
 PLUS(plus(j(x),j(y)),x3) -> PLUS(y,j(0))
 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),j(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
 PLUS(j(x),j(y)) -> PLUS(y,j(0))
 PLUS(un(x),j(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> Usable Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 zero(0) -> 0
-> 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:
 
[minus](X1,X2) = 0
[neg](X) = 0
[plus](X1,X2) = X1 + X2
[times](X1,X2) = 0
[zero](X) = X + 2
[0] = 0
[j](X) = X + 2
[un](X) = X + 2
[MINUS](X1,X2) = 0
[NEG](X) = 0
[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(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(j(x),j(y)),x3) -> PLUS(x,plus(y,j(0)))
 PLUS(plus(j(x),j(y)),x3) -> PLUS(y,j(0))
 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),j(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
 PLUS(j(x),j(y)) -> PLUS(y,j(0))
 PLUS(un(x),j(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(j(x),j(y)),x3) -> PLUS(x,plus(y,j(0)))
 PLUS(plus(j(x),j(y)),x3) -> PLUS(y,j(0))
 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),j(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
 PLUS(j(x),j(y)) -> PLUS(y,j(0))
 PLUS(un(x),j(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(j(x),j(y)),x3) -> PLUS(x,plus(y,j(0)))
 PLUS(plus(j(x),j(y)),x3) -> PLUS(y,j(0))
 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),j(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
 PLUS(j(x),j(y)) -> PLUS(y,j(0))
 PLUS(un(x),j(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> Usable Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 zero(0) -> 0
-> 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:
 
[minus](X1,X2) = 0
[neg](X) = 0
[plus](X1,X2) = X1 + X2
[times](X1,X2) = 0
[zero](X) = X + 1
[0] = 0
[j](X) = X + 2
[un](X) = X + 2
[MINUS](X1,X2) = 0
[NEG](X) = 0
[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(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(j(x),j(y)),x3) -> PLUS(x,plus(y,j(0)))
 PLUS(plus(j(x),j(y)),x3) -> PLUS(y,j(0))
 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),j(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
 PLUS(j(x),j(y)) -> PLUS(y,j(0))
 PLUS(un(x),j(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(j(x),j(y)),x3) -> PLUS(x,plus(y,j(0)))
 PLUS(plus(j(x),j(y)),x3) -> PLUS(y,j(0))
 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),j(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
 PLUS(j(x),j(y)) -> PLUS(y,j(0))
 PLUS(un(x),j(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(j(x),j(y)),x3) -> PLUS(x,plus(y,j(0)))
 PLUS(plus(j(x),j(y)),x3) -> PLUS(y,j(0))
 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),j(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
 PLUS(j(x),j(y)) -> PLUS(y,j(0))
 PLUS(un(x),j(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> Usable Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 zero(0) -> 0
-> 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:
 
[minus](X1,X2) = 0
[neg](X) = 0
[plus](X1,X2) = X1 + X2
[times](X1,X2) = 0
[zero](X) = X
[0] = 0
[j](X) = X + 2
[un](X) = X + 2
[MINUS](X1,X2) = 0
[NEG](X) = 0
[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(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(j(x),j(y)),x3) -> PLUS(y,j(0))
 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),j(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
 PLUS(j(x),j(y)) -> PLUS(y,j(0))
 PLUS(un(x),j(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(j(x),j(y)),x3) -> PLUS(y,j(0))
 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),j(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
 PLUS(j(x),j(y)) -> PLUS(y,j(0))
 PLUS(un(x),j(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(j(x),j(y)),x3) -> PLUS(y,j(0))
 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),j(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
 PLUS(j(x),j(y)) -> PLUS(y,j(0))
 PLUS(un(x),j(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> Usable Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 zero(0) -> 0
-> 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:
 
[minus](X1,X2) = 0
[neg](X) = 0
[plus](X1,X2) = X1 + X2
[times](X1,X2) = 0
[zero](X) = X + 1
[0] = 0
[j](X) = X + 2
[un](X) = X + 2
[MINUS](X1,X2) = 0
[NEG](X) = 0
[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(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),j(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
 PLUS(j(x),j(y)) -> PLUS(y,j(0))
 PLUS(un(x),j(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),j(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
 PLUS(j(x),j(y)) -> PLUS(y,j(0))
 PLUS(un(x),j(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(un(x),j(y)),x3) -> PLUS(zero(plus(x,y)),x3)
 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),j(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
 PLUS(j(x),j(y)) -> PLUS(y,j(0))
 PLUS(un(x),j(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> Usable Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 zero(0) -> 0
-> 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:
 
[minus](X1,X2) = 0
[neg](X) = 0
[plus](X1,X2) = X1 + X2
[times](X1,X2) = 0
[zero](X) = X + 2
[0] = 0
[j](X) = X + 2
[un](X) = X + 2
[MINUS](X1,X2) = 0
[NEG](X) = 0
[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(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),j(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
 PLUS(j(x),j(y)) -> PLUS(y,j(0))
 PLUS(un(x),j(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),j(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
 PLUS(j(x),j(y)) -> PLUS(y,j(0))
 PLUS(un(x),j(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(un(x),j(y)),x3) -> PLUS(x,y)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),j(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
 PLUS(j(x),j(y)) -> PLUS(y,j(0))
 PLUS(un(x),j(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> Usable Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 zero(0) -> 0
-> 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:
 
[minus](X1,X2) = 0
[neg](X) = 0
[plus](X1,X2) = X1 + X2
[times](X1,X2) = 0
[zero](X) = X + 2
[0] = 0
[j](X) = X + 2
[un](X) = X + 2
[MINUS](X1,X2) = 0
[NEG](X) = 0
[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(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),j(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
 PLUS(j(x),j(y)) -> PLUS(y,j(0))
 PLUS(un(x),j(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),j(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
 PLUS(j(x),j(y)) -> PLUS(y,j(0))
 PLUS(un(x),j(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(x,plus(y,un(0)))
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),j(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
 PLUS(j(x),j(y)) -> PLUS(y,j(0))
 PLUS(un(x),j(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> Usable Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 zero(0) -> 0
-> 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:
 
[minus](X1,X2) = 0
[neg](X) = 0
[plus](X1,X2) = X1 + X2
[times](X1,X2) = 0
[zero](X) = X + 2
[0] = 0
[j](X) = X + 1
[un](X) = X + 1
[MINUS](X1,X2) = 0
[NEG](X) = 0
[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(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),j(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
 PLUS(j(x),j(y)) -> PLUS(y,j(0))
 PLUS(un(x),j(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),j(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
 PLUS(j(x),j(y)) -> PLUS(y,j(0))
 PLUS(un(x),j(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(y,un(0))
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),j(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
 PLUS(j(x),j(y)) -> PLUS(y,j(0))
 PLUS(un(x),j(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> Usable Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 zero(0) -> 0
-> 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:
 
[minus](X1,X2) = 0
[neg](X) = 0
[plus](X1,X2) = X1 + X2
[times](X1,X2) = 0
[zero](X) = X + 1
[0] = 0
[j](X) = X + 2
[un](X) = X + 2
[MINUS](X1,X2) = 0
[NEG](X) = 0
[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(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),j(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
 PLUS(j(x),j(y)) -> PLUS(y,j(0))
 PLUS(un(x),j(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),j(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
 PLUS(j(x),j(y)) -> PLUS(y,j(0))
 PLUS(un(x),j(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(zero(x),zero(y)) -> PLUS(x,y)
 PLUS(zero(x),j(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
 PLUS(j(x),j(y)) -> PLUS(y,j(0))
 PLUS(un(x),j(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> Usable Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 zero(0) -> 0
-> 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:
 
[minus](X1,X2) = 0
[neg](X) = 0
[plus](X1,X2) = X1 + X2
[times](X1,X2) = 0
[zero](X) = X + 1
[0] = 0
[j](X) = X + 2
[un](X) = X + 2
[MINUS](X1,X2) = 0
[NEG](X) = 0
[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(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(zero(x),j(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
 PLUS(j(x),j(y)) -> PLUS(y,j(0))
 PLUS(un(x),j(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(zero(x),j(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
 PLUS(j(x),j(y)) -> PLUS(y,j(0))
 PLUS(un(x),j(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(zero(x),j(y)) -> PLUS(x,y)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
 PLUS(j(x),j(y)) -> PLUS(y,j(0))
 PLUS(un(x),j(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> Usable Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 zero(0) -> 0
-> 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:
 
[minus](X1,X2) = 0
[neg](X) = 0
[plus](X1,X2) = X1 + X2
[times](X1,X2) = 0
[zero](X) = X + 2
[0] = 0
[j](X) = X + 1
[un](X) = X + 1
[MINUS](X1,X2) = 0
[NEG](X) = 0
[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(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
 PLUS(j(x),j(y)) -> PLUS(y,j(0))
 PLUS(un(x),j(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
 PLUS(j(x),j(y)) -> PLUS(y,j(0))
 PLUS(un(x),j(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(zero(x),un(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
 PLUS(j(x),j(y)) -> PLUS(y,j(0))
 PLUS(un(x),j(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> Usable Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 zero(0) -> 0
-> 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:
 
[minus](X1,X2) = 0
[neg](X) = 0
[plus](X1,X2) = X1 + X2
[times](X1,X2) = 0
[zero](X) = X + 1
[0] = 0
[j](X) = X + 2
[un](X) = X + 2
[MINUS](X1,X2) = 0
[NEG](X) = 0
[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(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
 PLUS(j(x),j(y)) -> PLUS(y,j(0))
 PLUS(un(x),j(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
 PLUS(j(x),j(y)) -> PLUS(y,j(0))
 PLUS(un(x),j(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(j(x),j(y)) -> PLUS(x,plus(y,j(0)))
 PLUS(j(x),j(y)) -> PLUS(y,j(0))
 PLUS(un(x),j(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> Usable Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 zero(0) -> 0
-> 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:
 
[minus](X1,X2) = 0
[neg](X) = 0
[plus](X1,X2) = X1 + X2
[times](X1,X2) = 0
[zero](X) = X + 2
[0] = 0
[j](X) = X + 2
[un](X) = X + 2
[MINUS](X1,X2) = 0
[NEG](X) = 0
[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(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(j(x),j(y)) -> PLUS(y,j(0))
 PLUS(un(x),j(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(j(x),j(y)) -> PLUS(y,j(0))
 PLUS(un(x),j(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(j(x),j(y)) -> PLUS(y,j(0))
 PLUS(un(x),j(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> Usable Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 zero(0) -> 0
-> 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:
 
[minus](X1,X2) = 0
[neg](X) = 0
[plus](X1,X2) = X1 + X2
[times](X1,X2) = 0
[zero](X) = X + 1
[0] = 0
[j](X) = X + 2
[un](X) = X + 2
[MINUS](X1,X2) = 0
[NEG](X) = 0
[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(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(un(x),j(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(un(x),j(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(un(x),j(y)) -> PLUS(x,y)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> Usable Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 zero(0) -> 0
-> 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:
 
[minus](X1,X2) = 0
[neg](X) = 0
[plus](X1,X2) = X1 + X2
[times](X1,X2) = 0
[zero](X) = X
[0] = 0
[j](X) = X + 2
[un](X) = X + 2
[MINUS](X1,X2) = 0
[NEG](X) = 0
[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(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(un(x),un(y)) -> PLUS(x,plus(y,un(0)))
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> Usable Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 zero(0) -> 0
-> 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:
 
[minus](X1,X2) = 0
[neg](X) = 0
[plus](X1,X2) = X1 + X2
[times](X1,X2) = 0
[zero](X) = X + 2
[0] = 0
[j](X) = X + 2
[un](X) = X + 2
[MINUS](X1,X2) = 0
[NEG](X) = 0
[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(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
 PLUS(un(x),un(y)) -> PLUS(y,un(0))
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> Usable Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 zero(0) -> 0
-> 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:
 
[minus](X1,X2) = 0
[neg](X) = 0
[plus](X1,X2) = X1 + X2
[times](X1,X2) = 0
[zero](X) = X
[0] = 0
[j](X) = X + 2
[un](X) = X + 2
[MINUS](X1,X2) = 0
[NEG](X) = 0
[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(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(x,0),x3) -> PLUS(x,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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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(j(x),j(y)),x3) -> PLUS(un(plus(x,plus(y,j(0)))),x3)
 PLUS(plus(un(x),un(y)),x3) -> PLUS(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(x,0),x3) -> PLUS(x,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)
-> Usable Equations:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
-> Rules:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> Usable Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 zero(0) -> 0
-> 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:
 
[minus](X1,X2) = 0
[neg](X) = 0
[plus](X1,X2) = X1 + X2 + 1
[times](X1,X2) = 0
[zero](X) = 0
[0] = 0
[j](X) = 2
[un](X) = 1
[MINUS](X1,X2) = 0
[NEG](X) = 0
[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(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(x,0),x3) -> PLUS(x,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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(x,0),x3) -> PLUS(x,x3)
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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(j(plus(x,plus(y,un(0)))),x3)
 PLUS(plus(x,0),x3) -> PLUS(x,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)
-> Usable Equations:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
-> Rules:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> Usable Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 zero(0) -> 0
-> 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:
 
[minus](X1,X2) = 0
[neg](X) = 0
[plus](X1,X2) = X1 + X2
[times](X1,X2) = 0
[zero](X) = 2
[0] = 0
[j](X) = 2
[un](X) = 2
[MINUS](X1,X2) = 0
[NEG](X) = 0
[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(x,0),x3) -> PLUS(x,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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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,0),x3) -> PLUS(x,x3)
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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,0),x3) -> PLUS(x,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)
-> Usable Equations:
 plus(plus(x3,x4),x5) = plus(x3,plus(x4,x5))
 plus(x3,x4) = plus(x4,x3)
-> Rules:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> Usable Rules:
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 zero(0) -> 0
-> 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:
 
[minus](X1,X2) = 0
[neg](X) = 0
[plus](X1,X2) = X1 + X2 + 1
[times](X1,X2) = 0
[zero](X) = 2
[0] = 0
[j](X) = 1
[un](X) = 2
[MINUS](X1,X2) = 0
[NEG](X) = 0
[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:
 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> SRules:
 PLUS(plus(x3,x4),x5) -> PLUS(x3,x4)
 PLUS(x3,plus(x4,x5)) -> PLUS(x4,x5)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> FAxioms:
 Empty
-> Pairs:
 NEG(zero(x)) -> NEG(x)
 NEG(j(x)) -> NEG(x)
 NEG(un(x)) -> NEG(x)
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> SRules:
 Empty
->Projection:
 pi(NEG) = [1]

Problem 1.2: 

SCC Processor:
-> FAxioms:
 Empty
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> SRules:
 Empty
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,times(j(y),z)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
 TIMES(times(x,times(j(y),z)),x3) -> TIMES(times(plus(zero(times(x,y)),neg(x)),z),x3)
 TIMES(times(x,times(j(y),z)),x3) -> 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,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(x,y)
 TIMES(times(x,0),x3) -> TIMES(0,x3)
 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
 TIMES(times(x,j(y)),x3) -> 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(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
 TIMES(x,times(j(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,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> Usable Rules:
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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:
 
[minus](X1,X2) = 0
[neg](X) = X
[plus](X1,X2) = X1 + X2
[times](X1,X2) = X1.X2 + X1 + X2
[zero](X) = X + 1
[0] = 0
[j](X) = X + 1
[un](X) = X + 1
[MINUS](X1,X2) = 0
[NEG](X) = 0
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = X1.X2 + X1 + X2
[ZERO](X) = 0

Problem 1.3: 

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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,times(j(y),z)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
 TIMES(times(x,times(j(y),z)),x3) -> TIMES(times(plus(zero(times(x,y)),neg(x)),z),x3)
 TIMES(times(x,times(j(y),z)),x3) -> 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,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(x,y)
 TIMES(times(x,0),x3) -> TIMES(0,x3)
 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
 TIMES(times(x,j(y)),x3) -> 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(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
 TIMES(x,times(j(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,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,times(j(y),z)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
 TIMES(times(x,times(j(y),z)),x3) -> TIMES(times(plus(zero(times(x,y)),neg(x)),z),x3)
 TIMES(times(x,times(j(y),z)),x3) -> 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,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(x,y)
 TIMES(times(x,0),x3) -> TIMES(0,x3)
 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
 TIMES(times(x,j(y)),x3) -> 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(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
 TIMES(x,times(j(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,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)

Problem 1.3: 

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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,times(j(y),z)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
 TIMES(times(x,times(j(y),z)),x3) -> TIMES(times(plus(zero(times(x,y)),neg(x)),z),x3)
 TIMES(times(x,times(j(y),z)),x3) -> 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,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(x,y)
 TIMES(times(x,0),x3) -> TIMES(0,x3)
 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
 TIMES(times(x,j(y)),x3) -> 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(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
 TIMES(x,times(j(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,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> Usable Rules:
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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:
 
[minus](X1,X2) = 0
[neg](X) = X
[plus](X1,X2) = X1 + X2
[times](X1,X2) = X1.X2 + X1 + X2
[zero](X) = X
[0] = 0
[j](X) = X + 1
[un](X) = X + 1
[MINUS](X1,X2) = 0
[NEG](X) = 0
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = X1.X2 + X1 + X2
[ZERO](X) = 0

Problem 1.3: 

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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,times(j(y),z)),x3) -> TIMES(times(plus(zero(times(x,y)),neg(x)),z),x3)
 TIMES(times(x,times(j(y),z)),x3) -> 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,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(x,y)
 TIMES(times(x,0),x3) -> TIMES(0,x3)
 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
 TIMES(times(x,j(y)),x3) -> 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(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
 TIMES(x,times(j(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,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,times(j(y),z)),x3) -> TIMES(times(plus(zero(times(x,y)),neg(x)),z),x3)
 TIMES(times(x,times(j(y),z)),x3) -> 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,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(x,y)
 TIMES(times(x,0),x3) -> TIMES(0,x3)
 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
 TIMES(times(x,j(y)),x3) -> 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(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
 TIMES(x,times(j(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,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)

Problem 1.3: 

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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,times(j(y),z)),x3) -> TIMES(times(plus(zero(times(x,y)),neg(x)),z),x3)
 TIMES(times(x,times(j(y),z)),x3) -> 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,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(x,y)
 TIMES(times(x,0),x3) -> TIMES(0,x3)
 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
 TIMES(times(x,j(y)),x3) -> 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(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
 TIMES(x,times(j(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,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> Usable Rules:
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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:
 
[minus](X1,X2) = 0
[neg](X) = X
[plus](X1,X2) = X1 + X2
[times](X1,X2) = X1.X2 + X1 + X2
[zero](X) = X
[0] = 0
[j](X) = X + 1
[un](X) = X + 1
[MINUS](X1,X2) = 0
[NEG](X) = 0
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = X1.X2 + X1 + X2
[ZERO](X) = 0

Problem 1.3: 

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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,times(j(y),z)),x3) -> 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,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(x,y)
 TIMES(times(x,0),x3) -> TIMES(0,x3)
 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
 TIMES(times(x,j(y)),x3) -> 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(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
 TIMES(x,times(j(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,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,times(j(y),z)),x3) -> 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,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(x,y)
 TIMES(times(x,0),x3) -> TIMES(0,x3)
 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
 TIMES(times(x,j(y)),x3) -> 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(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
 TIMES(x,times(j(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,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)

Problem 1.3: 

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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,times(j(y),z)),x3) -> 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,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(x,y)
 TIMES(times(x,0),x3) -> TIMES(0,x3)
 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
 TIMES(times(x,j(y)),x3) -> 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(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
 TIMES(x,times(j(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,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> Usable Rules:
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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:
 
[minus](X1,X2) = 0
[neg](X) = X
[plus](X1,X2) = X1 + X2
[times](X1,X2) = X1.X2 + X1 + X2
[zero](X) = X + 1
[0] = 0
[j](X) = X + 1
[un](X) = X + 1
[MINUS](X1,X2) = 0
[NEG](X) = 0
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = X1.X2 + X1 + X2
[ZERO](X) = 0

Problem 1.3: 

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(0,z)),x3) -> TIMES(times(0,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,0),x3) -> TIMES(0,x3)
 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
 TIMES(times(x,j(y)),x3) -> 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(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
 TIMES(x,times(j(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,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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(0,z)),x3) -> TIMES(times(0,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,0),x3) -> TIMES(0,x3)
 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
 TIMES(times(x,j(y)),x3) -> 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(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
 TIMES(x,times(j(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,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)

Problem 1.3: 

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(0,z)),x3) -> TIMES(times(0,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,0),x3) -> TIMES(0,x3)
 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
 TIMES(times(x,j(y)),x3) -> 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(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
 TIMES(x,times(j(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,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> Usable Rules:
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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:
 
[minus](X1,X2) = 0
[neg](X) = X + 1
[plus](X1,X2) = X1 + X2
[times](X1,X2) = X1.X2 + X1 + X2
[zero](X) = X
[0] = 0
[j](X) = X + 1
[un](X) = X + 1
[MINUS](X1,X2) = 0
[NEG](X) = 0
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = X1.X2 + X1 + X2
[ZERO](X) = 0

Problem 1.3: 

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(0,z)),x3) -> TIMES(times(0,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,0),x3) -> TIMES(0,x3)
 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
 TIMES(times(x,j(y)),x3) -> 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(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
 TIMES(x,times(j(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,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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(0,z)),x3) -> TIMES(times(0,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,0),x3) -> TIMES(0,x3)
 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
 TIMES(times(x,j(y)),x3) -> 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(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
 TIMES(x,times(j(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,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)

Problem 1.3: 

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(0,z)),x3) -> TIMES(times(0,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,0),x3) -> TIMES(0,x3)
 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
 TIMES(times(x,j(y)),x3) -> 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(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
 TIMES(x,times(j(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,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> Usable Rules:
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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:
 
[minus](X1,X2) = 0
[neg](X) = X + 1
[plus](X1,X2) = X1 + X2
[times](X1,X2) = X1.X2 + X1 + X2
[zero](X) = X
[0] = 0
[j](X) = X + 1
[un](X) = X + 1
[MINUS](X1,X2) = 0
[NEG](X) = 0
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = X1.X2 + X1 + X2
[ZERO](X) = 0

Problem 1.3: 

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(0,z)),x3) -> TIMES(times(0,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,0),x3) -> TIMES(0,x3)
 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
 TIMES(times(x,j(y)),x3) -> 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(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
 TIMES(x,times(j(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,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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(0,z)),x3) -> TIMES(times(0,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,0),x3) -> TIMES(0,x3)
 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
 TIMES(times(x,j(y)),x3) -> 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(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
 TIMES(x,times(j(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,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)

Problem 1.3: 

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(0,z)),x3) -> TIMES(times(0,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,0),x3) -> TIMES(0,x3)
 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
 TIMES(times(x,j(y)),x3) -> 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(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
 TIMES(x,times(j(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,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> Usable Rules:
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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:
 
[minus](X1,X2) = 0
[neg](X) = X
[plus](X1,X2) = X1 + X2
[times](X1,X2) = X1.X2 + X1 + X2
[zero](X) = X
[0] = 0
[j](X) = X + 1
[un](X) = X + 1
[MINUS](X1,X2) = 0
[NEG](X) = 0
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = X1.X2 + X1 + X2
[ZERO](X) = 0

Problem 1.3: 

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(0,z)),x3) -> TIMES(times(0,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,0),x3) -> TIMES(0,x3)
 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
 TIMES(times(x,j(y)),x3) -> 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(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
 TIMES(x,times(j(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,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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(0,z)),x3) -> TIMES(times(0,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,0),x3) -> TIMES(0,x3)
 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
 TIMES(times(x,j(y)),x3) -> 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(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
 TIMES(x,times(j(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,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)

Problem 1.3: 

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(0,z)),x3) -> TIMES(times(0,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,0),x3) -> TIMES(0,x3)
 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
 TIMES(times(x,j(y)),x3) -> 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(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
 TIMES(x,times(j(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,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> Usable Rules:
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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:
 
[minus](X1,X2) = 0
[neg](X) = X
[plus](X1,X2) = X1 + X2
[times](X1,X2) = X1.X2 + X1 + X2
[zero](X) = X + 1
[0] = 0
[j](X) = X + 1
[un](X) = X + 1
[MINUS](X1,X2) = 0
[NEG](X) = 0
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = X1.X2 + X1 + X2
[ZERO](X) = 0

Problem 1.3: 

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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,0),x3) -> TIMES(0,x3)
 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
 TIMES(times(x,j(y)),x3) -> 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(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
 TIMES(x,times(j(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,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,0),x3) -> TIMES(0,x3)
 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
 TIMES(times(x,j(y)),x3) -> 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(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
 TIMES(x,times(j(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,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)

Problem 1.3: 

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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,0),x3) -> TIMES(0,x3)
 TIMES(times(x,j(y)),x3) -> TIMES(plus(zero(times(x,y)),neg(x)),x3)
 TIMES(times(x,j(y)),x3) -> 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(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
 TIMES(x,times(j(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,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> Usable Rules:
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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:
 
[minus](X1,X2) = 0
[neg](X) = X
[plus](X1,X2) = X1 + X2
[times](X1,X2) = X1.X2 + X1 + X2
[zero](X) = X
[0] = 0
[j](X) = X + 1
[un](X) = X + 1
[MINUS](X1,X2) = 0
[NEG](X) = 0
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = X1.X2 + X1 + X2
[ZERO](X) = 0

Problem 1.3: 

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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,0),x3) -> TIMES(0,x3)
 TIMES(times(x,j(y)),x3) -> 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(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
 TIMES(x,times(j(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,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,0),x3) -> TIMES(0,x3)
 TIMES(times(x,j(y)),x3) -> 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(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
 TIMES(x,times(j(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,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)

Problem 1.3: 

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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,0),x3) -> TIMES(0,x3)
 TIMES(times(x,j(y)),x3) -> 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(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(zero(y),z)) -> TIMES(x,y)
 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
 TIMES(x,times(j(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,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> Usable Rules:
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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:
 
[minus](X1,X2) = 0
[neg](X) = X
[plus](X1,X2) = X1 + X2
[times](X1,X2) = X1.X2 + X1 + X2
[zero](X) = X + 1
[0] = 0
[j](X) = X + 1
[un](X) = X + 1
[MINUS](X1,X2) = 0
[NEG](X) = 0
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = X1.X2 + X1 + X2
[ZERO](X) = 0

Problem 1.3: 

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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,0),x3) -> TIMES(0,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(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
 TIMES(x,times(j(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,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,0),x3) -> TIMES(0,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(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
 TIMES(x,times(j(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,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)

Problem 1.3: 

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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,0),x3) -> TIMES(0,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(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
 TIMES(x,times(j(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,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> Usable Rules:
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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:
 
[minus](X1,X2) = 0
[neg](X) = X + 1
[plus](X1,X2) = X1 + X2
[times](X1,X2) = X1.X2 + X1 + X2
[zero](X) = X
[0] = 0
[j](X) = X + 1
[un](X) = X + 1
[MINUS](X1,X2) = 0
[NEG](X) = 0
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = X1.X2 + X1 + X2
[ZERO](X) = 0

Problem 1.3: 

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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,0),x3) -> TIMES(0,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(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
 TIMES(x,times(j(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,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,0),x3) -> TIMES(0,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(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
 TIMES(x,times(j(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,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)

Problem 1.3: 

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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,0),x3) -> TIMES(0,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(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
 TIMES(x,times(j(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,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> Usable Rules:
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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:
 
[minus](X1,X2) = 0
[neg](X) = X + 1
[plus](X1,X2) = X1 + X2
[times](X1,X2) = X1.X2 + X1 + X2
[zero](X) = X
[0] = 0
[j](X) = X + 1
[un](X) = X + 1
[MINUS](X1,X2) = 0
[NEG](X) = 0
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = X1.X2 + X1 + X2
[ZERO](X) = 0

Problem 1.3: 

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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,0),x3) -> TIMES(0,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(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
 TIMES(x,times(j(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,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,0),x3) -> TIMES(0,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(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
 TIMES(x,times(j(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,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)

Problem 1.3: 

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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,0),x3) -> TIMES(0,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(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
 TIMES(x,times(j(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,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> Usable Rules:
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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:
 
[minus](X1,X2) = 0
[neg](X) = X
[plus](X1,X2) = X1 + X2
[times](X1,X2) = X1.X2 + X1 + X2
[zero](X) = X + 1
[0] = 0
[j](X) = X + 1
[un](X) = X + 1
[MINUS](X1,X2) = 0
[NEG](X) = 0
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = X1.X2 + X1 + X2
[ZERO](X) = 0

Problem 1.3: 

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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,0),x3) -> TIMES(0,x3)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
 TIMES(x,times(j(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,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,0),x3) -> TIMES(0,x3)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
 TIMES(x,times(j(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,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)

Problem 1.3: 

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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,0),x3) -> TIMES(0,x3)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(j(y),z)) -> TIMES(plus(zero(times(x,y)),neg(x)),z)
 TIMES(x,times(j(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,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> Usable Rules:
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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:
 
[minus](X1,X2) = 0
[neg](X) = X
[plus](X1,X2) = X1 + X2
[times](X1,X2) = X1.X2 + X1 + X2
[zero](X) = X
[0] = 0
[j](X) = X + 1
[un](X) = X + 1
[MINUS](X1,X2) = 0
[NEG](X) = 0
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = X1.X2 + X1 + X2
[ZERO](X) = 0

Problem 1.3: 

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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,0),x3) -> TIMES(0,x3)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(j(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,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,0),x3) -> TIMES(0,x3)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(j(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,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)

Problem 1.3: 

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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,0),x3) -> TIMES(0,x3)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,times(j(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,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> Usable Rules:
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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:
 
[minus](X1,X2) = 0
[neg](X) = X + 1
[plus](X1,X2) = X1 + X2
[times](X1,X2) = X1.X2 + X1 + X2
[zero](X) = X
[0] = 0
[j](X) = X + 1
[un](X) = X + 1
[MINUS](X1,X2) = 0
[NEG](X) = 0
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = X1.X2 + X1 + X2
[ZERO](X) = 0

Problem 1.3: 

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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,0),x3) -> TIMES(0,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,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,0),x3) -> TIMES(0,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,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)

Problem 1.3: 

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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,0),x3) -> TIMES(0,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,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> Usable Rules:
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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:
 
[minus](X1,X2) = 0
[neg](X) = X + 1
[plus](X1,X2) = X1 + X2
[times](X1,X2) = X1.X2 + X1 + X2
[zero](X) = X
[0] = 0
[j](X) = X + 1
[un](X) = X + 1
[MINUS](X1,X2) = 0
[NEG](X) = 0
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = X1.X2 + X1 + X2
[ZERO](X) = 0

Problem 1.3: 

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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,0),x3) -> TIMES(0,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,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,0),x3) -> TIMES(0,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,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)

Problem 1.3: 

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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,0),x3) -> TIMES(0,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,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> Usable Rules:
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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:
 
[minus](X1,X2) = 0
[neg](X) = X
[plus](X1,X2) = X1 + X2
[times](X1,X2) = X1.X2 + X1 + X2
[zero](X) = X + 1
[0] = 0
[j](X) = X + 1
[un](X) = X + 1
[MINUS](X1,X2) = 0
[NEG](X) = 0
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = X1.X2 + X1 + X2
[ZERO](X) = 0

Problem 1.3: 

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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,0),x3) -> TIMES(0,x3)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,zero(y)) -> TIMES(x,y)
 TIMES(x,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,0),x3) -> TIMES(0,x3)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,zero(y)) -> TIMES(x,y)
 TIMES(x,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)

Problem 1.3: 

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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,0),x3) -> TIMES(0,x3)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,zero(y)) -> TIMES(x,y)
 TIMES(x,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> Usable Rules:
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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:
 
[minus](X1,X2) = 0
[neg](X) = X
[plus](X1,X2) = X1 + X2
[times](X1,X2) = X1.X2 + X1 + X2
[zero](X) = X + 1
[0] = 0
[j](X) = X + 1
[un](X) = X + 1
[MINUS](X1,X2) = 0
[NEG](X) = 0
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = X1.X2 + X1 + X2
[ZERO](X) = 0

Problem 1.3: 

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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,0),x3) -> TIMES(0,x3)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,0),x3) -> TIMES(0,x3)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)

Problem 1.3: 

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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,0),x3) -> TIMES(0,x3)
 TIMES(x,times(zero(y),z)) -> TIMES(zero(times(x,y)),z)
 TIMES(x,j(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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> Usable Rules:
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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:
 
[minus](X1,X2) = 0
[neg](X) = X
[plus](X1,X2) = X1 + X2
[times](X1,X2) = X1.X2 + X1 + X2
[zero](X) = X + 1
[0] = 0
[j](X) = X + 1
[un](X) = X + 1
[MINUS](X1,X2) = 0
[NEG](X) = 0
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = X1.X2 + X1 + X2
[ZERO](X) = 0

Problem 1.3: 

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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,0),x3) -> TIMES(0,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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,0),x3) -> TIMES(0,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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)

Problem 1.3: 

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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,0),x3) -> TIMES(0,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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> Usable Rules:
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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:
 
[minus](X1,X2) = 0
[neg](X) = X + 1
[plus](X1,X2) = X1 + X2
[times](X1,X2) = X1.X2 + X1 + X2
[zero](X) = X
[0] = 0
[j](X) = X + 1
[un](X) = X + 1
[MINUS](X1,X2) = 0
[NEG](X) = 0
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = X1.X2 + X1 + X2
[ZERO](X) = 0

Problem 1.3: 

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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,0),x3) -> TIMES(0,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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,0),x3) -> TIMES(0,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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)

Problem 1.3: 

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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,0),x3) -> TIMES(0,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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> Usable Rules:
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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:
 
[minus](X1,X2) = 0
[neg](X) = 2
[plus](X1,X2) = X1 + X2
[times](X1,X2) = X1 + X2 + 2
[zero](X) = 0
[0] = 0
[j](X) = 2
[un](X) = 2
[MINUS](X1,X2) = 0
[NEG](X) = 0
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = 2.X1 + 2.X2
[ZERO](X) = 0

Problem 1.3: 

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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,0),x3) -> TIMES(0,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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,0),x3) -> TIMES(0,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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)

Problem 1.3: 

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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,0),x3) -> TIMES(0,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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> Usable Rules:
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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:
 
[minus](X1,X2) = 0
[neg](X) = X + 1
[plus](X1,X2) = X1 + X2 + 2
[times](X1,X2) = X1 + X2 + 1
[zero](X) = 0
[0] = 0
[j](X) = 2
[un](X) = 2
[MINUS](X1,X2) = 0
[NEG](X) = 0
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = 2.X1 + 2.X2
[ZERO](X) = 0

Problem 1.3: 

SCC Processor:
-> FAxioms:
 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
 TIMES(x3,x4) = TIMES(x4,x3)
-> Pairs:
 TIMES(times(x,times(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,0),x3) -> TIMES(0,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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,0),x3) -> TIMES(0,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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)

Problem 1.3: 

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(0,z)),x3) -> TIMES(times(0,z),x3)
 TIMES(times(x,zero(y)),x3) -> TIMES(zero(times(x,y)),x3)
 TIMES(times(x,0),x3) -> TIMES(0,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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> Usable Rules:
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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:
 
[minus](X1,X2) = 0
[neg](X) = X + 2
[plus](X1,X2) = X1 + X2 + 2
[times](X1,X2) = X1 + X2 + 2
[zero](X) = 0
[0] = 0
[j](X) = 2
[un](X) = 0
[MINUS](X1,X2) = 0
[NEG](X) = 0
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = 2.X1 + 2.X2
[ZERO](X) = 0

Problem 1.3: 

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,0),x3) -> TIMES(0,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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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,0),x3) -> TIMES(0,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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)

Problem 1.3: 

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,0),x3) -> TIMES(0,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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> Usable Rules:
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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:
 
[minus](X1,X2) = 0
[neg](X) = X + 1
[plus](X1,X2) = X1 + X2 + 2
[times](X1,X2) = X1 + X2 + 2
[zero](X) = 0
[0] = 0
[j](X) = 1
[un](X) = 0
[MINUS](X1,X2) = 0
[NEG](X) = 0
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = X1 + X2
[ZERO](X) = 0

Problem 1.3: 

SCC Processor:
-> FAxioms:
 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
 TIMES(x3,x4) = TIMES(x4,x3)
-> Pairs:
 TIMES(times(x,0),x3) -> TIMES(0,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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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,0),x3) -> TIMES(0,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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)

Problem 1.3: 

Reduction Pairs Processor:
-> FAxioms:
 TIMES(times(x3,x4),x5) = TIMES(x3,times(x4,x5))
 TIMES(x3,x4) = TIMES(x4,x3)
-> Pairs:
 TIMES(times(x,0),x3) -> TIMES(0,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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> Usable Rules:
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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:
 
[minus](X1,X2) = 0
[neg](X) = X
[plus](X1,X2) = X1 + X2 + 2
[times](X1,X2) = X1 + X2 + 2
[zero](X) = 0
[0] = 0
[j](X) = 0
[un](X) = 0
[MINUS](X1,X2) = 0
[NEG](X) = 0
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = X1 + X2
[ZERO](X) = 0

Problem 1.3: 

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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> SRules:
 TIMES(times(x3,x4),x5) -> TIMES(x3,x4)
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)

Problem 1.3: 

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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> Usable Rules:
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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:
 
[minus](X1,X2) = 0
[neg](X) = X
[plus](X1,X2) = X1 + X2 + 2
[times](X1,X2) = X1 + X2 + 2
[zero](X) = 2
[0] = 2
[j](X) = 2
[un](X) = 2
[MINUS](X1,X2) = 0
[NEG](X) = 0
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = 2.X1 + 2.X2
[ZERO](X) = 0

Problem 1.3: 

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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> 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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> SRules:
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)

Problem 1.3: 

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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> Usable Rules:
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> SRules:
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[minus](X1,X2) = 0
[neg](X) = X + 2
[plus](X1,X2) = X1 + X2
[times](X1,X2) = X1 + X2 + 2
[zero](X) = 2
[0] = 2
[j](X) = 2
[un](X) = 2
[MINUS](X1,X2) = 0
[NEG](X) = 0
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = 2.X1 + 2.X2
[ZERO](X) = 0

Problem 1.3: 

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:
 minus(x,y) -> plus(x,neg(y))
 neg(zero(x)) -> zero(neg(x))
 neg(0) -> 0
 neg(j(x)) -> un(neg(x))
 neg(un(x)) -> j(neg(x))
 plus(zero(x),zero(y)) -> zero(plus(x,y))
 plus(zero(x),j(y)) -> j(plus(x,y))
 plus(zero(x),un(y)) -> un(plus(x,y))
 plus(j(x),j(y)) -> un(plus(x,plus(y,j(0))))
 plus(un(x),j(y)) -> zero(plus(x,y))
 plus(un(x),un(y)) -> j(plus(x,plus(y,un(0))))
 plus(x,0) -> x
 times(x,times(zero(y),z)) -> times(zero(times(x,y)),z)
 times(x,times(0,z)) -> times(0,z)
 times(x,times(j(y),z)) -> times(plus(zero(times(x,y)),neg(x)),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,0) -> 0
 times(x,j(y)) -> plus(zero(times(x,y)),neg(x))
 times(x,un(y)) -> plus(x,zero(times(x,y)))
 zero(0) -> 0
-> SRules:
 TIMES(x3,times(x4,x5)) -> TIMES(x4,x5)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.