YES

Problem 1: 

(VAR x y)
(THEORY 
(AC plus times))
(RULES
0(S) -> S
minus(x,y) -> plus(opp(y),x)
opp(0(x)) -> 0(opp(x))
opp(1(x)) -> j(opp(x))
opp(S) -> S
opp(j(x)) -> 1(opp(x))
plus(0(x),0(y)) -> 0(plus(x,y))
plus(0(x),1(y)) -> 1(plus(x,y))
plus(0(x),j(y)) -> j(plus(x,y))
plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
plus(1(x),j(y)) -> 0(plus(x,y))
plus(S,x) -> x
plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
times(0(x),y) -> 0(times(x,y))
times(1(x),y) -> plus(0(times(x,y)),y)
times(S,x) -> S
times(j(x),y) -> minus(0(times(x,y)),y)
)

Problem 1: 

Dependency Pairs Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
 TIMES(x2,x3) = TIMES(x3,x2)
-> Pairs:
 MINUS(x,y) -> OPP(y)
 MINUS(x,y) -> PLUS(opp(y),x)
 OPP(0(x)) -> 0#(opp(x))
 OPP(0(x)) -> OPP(x)
 OPP(1(x)) -> OPP(x)
 OPP(j(x)) -> OPP(x)
 PLUS(0(x),0(y)) -> 0#(plus(x,y))
 PLUS(0(x),0(y)) -> PLUS(x,y)
 PLUS(0(x),1(y)) -> PLUS(x,y)
 PLUS(0(x),j(y)) -> PLUS(x,y)
 PLUS(plus(0(x),0(y)),x2) -> 0#(plus(x,y))
 PLUS(plus(0(x),0(y)),x2) -> PLUS(0(plus(x,y)),x2)
 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y)
 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2)
 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(0(x),j(y)),x2) -> PLUS(j(plus(x,y)),x2)
 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),j(y)),x2) -> 0#(plus(x,y))
 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
 PLUS(1(x),1(y)) -> PLUS(x,y)
 PLUS(1(x),j(y)) -> 0#(plus(x,y))
 PLUS(1(x),j(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
 TIMES(0(x),y) -> 0#(times(x,y))
 TIMES(0(x),y) -> TIMES(x,y)
 TIMES(times(0(x),y),x2) -> 0#(times(x,y))
 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
 TIMES(times(0(x),y),x2) -> TIMES(x,y)
 TIMES(times(1(x),y),x2) -> 0#(times(x,y))
 TIMES(times(1(x),y),x2) -> PLUS(0(times(x,y)),y)
 TIMES(times(1(x),y),x2) -> TIMES(plus(0(times(x,y)),y),x2)
 TIMES(times(1(x),y),x2) -> TIMES(x,y)
 TIMES(times(S,x),x2) -> TIMES(S,x2)
 TIMES(times(j(x),y),x2) -> 0#(times(x,y))
 TIMES(times(j(x),y),x2) -> MINUS(0(times(x,y)),y)
 TIMES(times(j(x),y),x2) -> TIMES(minus(0(times(x,y)),y),x2)
 TIMES(times(j(x),y),x2) -> TIMES(x,y)
 TIMES(1(x),y) -> 0#(times(x,y))
 TIMES(1(x),y) -> PLUS(0(times(x,y)),y)
 TIMES(1(x),y) -> TIMES(x,y)
 TIMES(j(x),y) -> 0#(times(x,y))
 TIMES(j(x),y) -> MINUS(0(times(x,y)),y)
 TIMES(j(x),y) -> TIMES(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)

Problem 1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
 TIMES(x2,x3) = TIMES(x3,x2)
-> Pairs:
 MINUS(x,y) -> OPP(y)
 MINUS(x,y) -> PLUS(opp(y),x)
 OPP(0(x)) -> 0#(opp(x))
 OPP(0(x)) -> OPP(x)
 OPP(1(x)) -> OPP(x)
 OPP(j(x)) -> OPP(x)
 PLUS(0(x),0(y)) -> 0#(plus(x,y))
 PLUS(0(x),0(y)) -> PLUS(x,y)
 PLUS(0(x),1(y)) -> PLUS(x,y)
 PLUS(0(x),j(y)) -> PLUS(x,y)
 PLUS(plus(0(x),0(y)),x2) -> 0#(plus(x,y))
 PLUS(plus(0(x),0(y)),x2) -> PLUS(0(plus(x,y)),x2)
 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y)
 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2)
 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(0(x),j(y)),x2) -> PLUS(j(plus(x,y)),x2)
 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),j(y)),x2) -> 0#(plus(x,y))
 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
 PLUS(1(x),1(y)) -> PLUS(x,y)
 PLUS(1(x),j(y)) -> 0#(plus(x,y))
 PLUS(1(x),j(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
 TIMES(0(x),y) -> 0#(times(x,y))
 TIMES(0(x),y) -> TIMES(x,y)
 TIMES(times(0(x),y),x2) -> 0#(times(x,y))
 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
 TIMES(times(0(x),y),x2) -> TIMES(x,y)
 TIMES(times(1(x),y),x2) -> 0#(times(x,y))
 TIMES(times(1(x),y),x2) -> PLUS(0(times(x,y)),y)
 TIMES(times(1(x),y),x2) -> TIMES(plus(0(times(x,y)),y),x2)
 TIMES(times(1(x),y),x2) -> TIMES(x,y)
 TIMES(times(S,x),x2) -> TIMES(S,x2)
 TIMES(times(j(x),y),x2) -> 0#(times(x,y))
 TIMES(times(j(x),y),x2) -> MINUS(0(times(x,y)),y)
 TIMES(times(j(x),y),x2) -> TIMES(minus(0(times(x,y)),y),x2)
 TIMES(times(j(x),y),x2) -> TIMES(x,y)
 TIMES(1(x),y) -> 0#(times(x,y))
 TIMES(1(x),y) -> PLUS(0(times(x,y)),y)
 TIMES(1(x),y) -> TIMES(x,y)
 TIMES(j(x),y) -> 0#(times(x,y))
 TIMES(j(x),y) -> MINUS(0(times(x,y)),y)
 TIMES(j(x),y) -> TIMES(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(0(x),0(y)) -> PLUS(x,y)
 PLUS(0(x),1(y)) -> PLUS(x,y)
 PLUS(0(x),j(y)) -> PLUS(x,y)
 PLUS(plus(0(x),0(y)),x2) -> PLUS(0(plus(x,y)),x2)
 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y)
 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2)
 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(0(x),j(y)),x2) -> PLUS(j(plus(x,y)),x2)
 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
 PLUS(1(x),1(y)) -> PLUS(x,y)
 PLUS(1(x),j(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> FAxioms:
 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
 plus(x2,x3) -> plus(x3,x2)
 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
 times(x2,x3) -> times(x3,x2)
 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) -> PLUS(x3,x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
->->-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->->Cycle:
->->-> Pairs:
 OPP(0(x)) -> OPP(x)
 OPP(1(x)) -> OPP(x)
 OPP(j(x)) -> OPP(x)
-> FAxioms:
 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
 plus(x2,x3) -> plus(x3,x2)
 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
 times(x2,x3) -> times(x3,x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
->->-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 Empty
->->Cycle:
->->-> Pairs:
 TIMES(0(x),y) -> TIMES(x,y)
 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
 TIMES(times(0(x),y),x2) -> TIMES(x,y)
 TIMES(times(1(x),y),x2) -> TIMES(plus(0(times(x,y)),y),x2)
 TIMES(times(1(x),y),x2) -> TIMES(x,y)
 TIMES(times(S,x),x2) -> TIMES(S,x2)
 TIMES(times(j(x),y),x2) -> TIMES(minus(0(times(x,y)),y),x2)
 TIMES(times(j(x),y),x2) -> TIMES(x,y)
 TIMES(1(x),y) -> TIMES(x,y)
 TIMES(j(x),y) -> TIMES(x,y)
-> FAxioms:
 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
 plus(x2,x3) -> plus(x3,x2)
 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
 times(x2,x3) -> times(x3,x2)
 TIMES(times(x2,x3),x4) -> TIMES(x2,times(x3,x4))
 TIMES(x2,x3) -> TIMES(x3,x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
->->-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)


The problem is decomposed in 3 subproblems.

Problem 1.1: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(0(x),0(y)) -> PLUS(x,y)
 PLUS(0(x),1(y)) -> PLUS(x,y)
 PLUS(0(x),j(y)) -> PLUS(x,y)
 PLUS(plus(0(x),0(y)),x2) -> PLUS(0(plus(x,y)),x2)
 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y)
 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2)
 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(0(x),j(y)),x2) -> PLUS(j(plus(x,y)),x2)
 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
 PLUS(1(x),1(y)) -> PLUS(x,y)
 PLUS(1(x),j(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Usable Equations:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> Usable Rules:
 0(S) -> S
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = X + 2
[minus](X1,X2) = 0
[opp](X) = 0
[plus](X1,X2) = X1 + X2
[times](X1,X2) = 0
[1](X) = X + 2
[S] = 0
[j](X) = X + 2
[0#](X) = 0
[MINUS](X1,X2) = 0
[OPP](X) = 0
[PLUS](X1,X2) = 2.X1 + 2.X2
[TIMES](X1,X2) = 0

Problem 1.1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(0(x),1(y)) -> PLUS(x,y)
 PLUS(0(x),j(y)) -> PLUS(x,y)
 PLUS(plus(0(x),0(y)),x2) -> PLUS(0(plus(x,y)),x2)
 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y)
 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2)
 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(0(x),j(y)),x2) -> PLUS(j(plus(x,y)),x2)
 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
 PLUS(1(x),1(y)) -> PLUS(x,y)
 PLUS(1(x),j(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(0(x),1(y)) -> PLUS(x,y)
 PLUS(0(x),j(y)) -> PLUS(x,y)
 PLUS(plus(0(x),0(y)),x2) -> PLUS(0(plus(x,y)),x2)
 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y)
 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2)
 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(0(x),j(y)),x2) -> PLUS(j(plus(x,y)),x2)
 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
 PLUS(1(x),1(y)) -> PLUS(x,y)
 PLUS(1(x),j(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> FAxioms:
 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
 plus(x2,x3) -> plus(x3,x2)
 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
 times(x2,x3) -> times(x3,x2)
 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) -> PLUS(x3,x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
->->-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)

Problem 1.1: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(0(x),1(y)) -> PLUS(x,y)
 PLUS(0(x),j(y)) -> PLUS(x,y)
 PLUS(plus(0(x),0(y)),x2) -> PLUS(0(plus(x,y)),x2)
 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y)
 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2)
 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(0(x),j(y)),x2) -> PLUS(j(plus(x,y)),x2)
 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
 PLUS(1(x),1(y)) -> PLUS(x,y)
 PLUS(1(x),j(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Usable Equations:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> Usable Rules:
 0(S) -> S
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = X + 2
[minus](X1,X2) = 0
[opp](X) = 0
[plus](X1,X2) = X1 + X2
[times](X1,X2) = 0
[1](X) = X + 2
[S] = 0
[j](X) = X + 2
[0#](X) = 0
[MINUS](X1,X2) = 0
[OPP](X) = 0
[PLUS](X1,X2) = 2.X1 + 2.X2
[TIMES](X1,X2) = 0

Problem 1.1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(0(x),j(y)) -> PLUS(x,y)
 PLUS(plus(0(x),0(y)),x2) -> PLUS(0(plus(x,y)),x2)
 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y)
 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2)
 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(0(x),j(y)),x2) -> PLUS(j(plus(x,y)),x2)
 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
 PLUS(1(x),1(y)) -> PLUS(x,y)
 PLUS(1(x),j(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(0(x),j(y)) -> PLUS(x,y)
 PLUS(plus(0(x),0(y)),x2) -> PLUS(0(plus(x,y)),x2)
 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y)
 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2)
 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(0(x),j(y)),x2) -> PLUS(j(plus(x,y)),x2)
 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
 PLUS(1(x),1(y)) -> PLUS(x,y)
 PLUS(1(x),j(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> FAxioms:
 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
 plus(x2,x3) -> plus(x3,x2)
 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
 times(x2,x3) -> times(x3,x2)
 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) -> PLUS(x3,x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
->->-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)

Problem 1.1: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(0(x),j(y)) -> PLUS(x,y)
 PLUS(plus(0(x),0(y)),x2) -> PLUS(0(plus(x,y)),x2)
 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y)
 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2)
 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(0(x),j(y)),x2) -> PLUS(j(plus(x,y)),x2)
 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
 PLUS(1(x),1(y)) -> PLUS(x,y)
 PLUS(1(x),j(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Usable Equations:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> Usable Rules:
 0(S) -> S
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = X
[minus](X1,X2) = 0
[opp](X) = 0
[plus](X1,X2) = X1 + X2
[times](X1,X2) = 0
[1](X) = X + 2
[S] = 0
[j](X) = X + 2
[0#](X) = 0
[MINUS](X1,X2) = 0
[OPP](X) = 0
[PLUS](X1,X2) = 2.X1 + 2.X2
[TIMES](X1,X2) = 0

Problem 1.1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(plus(0(x),0(y)),x2) -> PLUS(0(plus(x,y)),x2)
 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y)
 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2)
 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(0(x),j(y)),x2) -> PLUS(j(plus(x,y)),x2)
 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
 PLUS(1(x),1(y)) -> PLUS(x,y)
 PLUS(1(x),j(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(0(x),0(y)),x2) -> PLUS(0(plus(x,y)),x2)
 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y)
 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2)
 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(0(x),j(y)),x2) -> PLUS(j(plus(x,y)),x2)
 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
 PLUS(1(x),1(y)) -> PLUS(x,y)
 PLUS(1(x),j(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> FAxioms:
 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
 plus(x2,x3) -> plus(x3,x2)
 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
 times(x2,x3) -> times(x3,x2)
 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) -> PLUS(x3,x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
->->-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)

Problem 1.1: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(plus(0(x),0(y)),x2) -> PLUS(0(plus(x,y)),x2)
 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y)
 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2)
 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(0(x),j(y)),x2) -> PLUS(j(plus(x,y)),x2)
 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
 PLUS(1(x),1(y)) -> PLUS(x,y)
 PLUS(1(x),j(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Usable Equations:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> Usable Rules:
 0(S) -> S
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = X + 1
[minus](X1,X2) = 0
[opp](X) = 0
[plus](X1,X2) = X1 + X2
[times](X1,X2) = 0
[1](X) = X + 2
[S] = 0
[j](X) = X + 2
[0#](X) = 0
[MINUS](X1,X2) = 0
[OPP](X) = 0
[PLUS](X1,X2) = 2.X1 + 2.X2
[TIMES](X1,X2) = 0

Problem 1.1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y)
 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2)
 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(0(x),j(y)),x2) -> PLUS(j(plus(x,y)),x2)
 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
 PLUS(1(x),1(y)) -> PLUS(x,y)
 PLUS(1(x),j(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y)
 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2)
 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(0(x),j(y)),x2) -> PLUS(j(plus(x,y)),x2)
 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
 PLUS(1(x),1(y)) -> PLUS(x,y)
 PLUS(1(x),j(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> FAxioms:
 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
 plus(x2,x3) -> plus(x3,x2)
 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
 times(x2,x3) -> times(x3,x2)
 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) -> PLUS(x3,x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
->->-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)

Problem 1.1: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(plus(0(x),0(y)),x2) -> PLUS(x,y)
 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2)
 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(0(x),j(y)),x2) -> PLUS(j(plus(x,y)),x2)
 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
 PLUS(1(x),1(y)) -> PLUS(x,y)
 PLUS(1(x),j(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Usable Equations:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> Usable Rules:
 0(S) -> S
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = X + 2
[minus](X1,X2) = 0
[opp](X) = 0
[plus](X1,X2) = X1 + X2
[times](X1,X2) = 0
[1](X) = X + 2
[S] = 0
[j](X) = X + 2
[0#](X) = 0
[MINUS](X1,X2) = 0
[OPP](X) = 0
[PLUS](X1,X2) = 2.X1 + 2.X2
[TIMES](X1,X2) = 0

Problem 1.1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2)
 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(0(x),j(y)),x2) -> PLUS(j(plus(x,y)),x2)
 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
 PLUS(1(x),1(y)) -> PLUS(x,y)
 PLUS(1(x),j(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2)
 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(0(x),j(y)),x2) -> PLUS(j(plus(x,y)),x2)
 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
 PLUS(1(x),1(y)) -> PLUS(x,y)
 PLUS(1(x),j(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> FAxioms:
 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
 plus(x2,x3) -> plus(x3,x2)
 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
 times(x2,x3) -> times(x3,x2)
 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) -> PLUS(x3,x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
->->-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)

Problem 1.1: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(plus(0(x),1(y)),x2) -> PLUS(1(plus(x,y)),x2)
 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(0(x),j(y)),x2) -> PLUS(j(plus(x,y)),x2)
 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
 PLUS(1(x),1(y)) -> PLUS(x,y)
 PLUS(1(x),j(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Usable Equations:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> Usable Rules:
 0(S) -> S
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = X + 2
[minus](X1,X2) = 0
[opp](X) = 0
[plus](X1,X2) = X1 + X2
[times](X1,X2) = 0
[1](X) = X + 2
[S] = 0
[j](X) = X + 2
[0#](X) = 0
[MINUS](X1,X2) = 0
[OPP](X) = 0
[PLUS](X1,X2) = 2.X1 + 2.X2
[TIMES](X1,X2) = 0

Problem 1.1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(0(x),j(y)),x2) -> PLUS(j(plus(x,y)),x2)
 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
 PLUS(1(x),1(y)) -> PLUS(x,y)
 PLUS(1(x),j(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(0(x),j(y)),x2) -> PLUS(j(plus(x,y)),x2)
 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
 PLUS(1(x),1(y)) -> PLUS(x,y)
 PLUS(1(x),j(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> FAxioms:
 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
 plus(x2,x3) -> plus(x3,x2)
 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
 times(x2,x3) -> times(x3,x2)
 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) -> PLUS(x3,x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
->->-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)

Problem 1.1: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(plus(0(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(0(x),j(y)),x2) -> PLUS(j(plus(x,y)),x2)
 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
 PLUS(1(x),1(y)) -> PLUS(x,y)
 PLUS(1(x),j(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Usable Equations:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> Usable Rules:
 0(S) -> S
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = X + 1
[minus](X1,X2) = 0
[opp](X) = 0
[plus](X1,X2) = X1 + X2
[times](X1,X2) = 0
[1](X) = X + 2
[S] = 0
[j](X) = X + 2
[0#](X) = 0
[MINUS](X1,X2) = 0
[OPP](X) = 0
[PLUS](X1,X2) = 2.X1 + 2.X2
[TIMES](X1,X2) = 0

Problem 1.1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(plus(0(x),j(y)),x2) -> PLUS(j(plus(x,y)),x2)
 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
 PLUS(1(x),1(y)) -> PLUS(x,y)
 PLUS(1(x),j(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(0(x),j(y)),x2) -> PLUS(j(plus(x,y)),x2)
 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
 PLUS(1(x),1(y)) -> PLUS(x,y)
 PLUS(1(x),j(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> FAxioms:
 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
 plus(x2,x3) -> plus(x3,x2)
 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
 times(x2,x3) -> times(x3,x2)
 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) -> PLUS(x3,x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
->->-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)

Problem 1.1: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(plus(0(x),j(y)),x2) -> PLUS(j(plus(x,y)),x2)
 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
 PLUS(1(x),1(y)) -> PLUS(x,y)
 PLUS(1(x),j(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Usable Equations:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> Usable Rules:
 0(S) -> S
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = X + 1
[minus](X1,X2) = 0
[opp](X) = 0
[plus](X1,X2) = X1 + X2
[times](X1,X2) = 0
[1](X) = X + 2
[S] = 0
[j](X) = X + 2
[0#](X) = 0
[MINUS](X1,X2) = 0
[OPP](X) = 0
[PLUS](X1,X2) = 2.X1 + 2.X2
[TIMES](X1,X2) = 0

Problem 1.1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
 PLUS(1(x),1(y)) -> PLUS(x,y)
 PLUS(1(x),j(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
 PLUS(1(x),1(y)) -> PLUS(x,y)
 PLUS(1(x),j(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> FAxioms:
 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
 plus(x2,x3) -> plus(x3,x2)
 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
 times(x2,x3) -> times(x3,x2)
 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) -> PLUS(x3,x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
->->-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)

Problem 1.1: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(plus(0(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
 PLUS(1(x),1(y)) -> PLUS(x,y)
 PLUS(1(x),j(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Usable Equations:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> Usable Rules:
 0(S) -> S
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = X
[minus](X1,X2) = 0
[opp](X) = 0
[plus](X1,X2) = X1 + X2
[times](X1,X2) = 0
[1](X) = X + 2
[S] = 0
[j](X) = X + 2
[0#](X) = 0
[MINUS](X1,X2) = 0
[OPP](X) = 0
[PLUS](X1,X2) = 2.X1 + 2.X2
[TIMES](X1,X2) = 0

Problem 1.1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
 PLUS(1(x),1(y)) -> PLUS(x,y)
 PLUS(1(x),j(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
 PLUS(1(x),1(y)) -> PLUS(x,y)
 PLUS(1(x),j(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> FAxioms:
 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
 plus(x2,x3) -> plus(x3,x2)
 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
 times(x2,x3) -> times(x3,x2)
 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) -> PLUS(x3,x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
->->-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)

Problem 1.1: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(plus(1(x),1(y)),x2) -> PLUS(1(S),plus(x,y))
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
 PLUS(1(x),1(y)) -> PLUS(x,y)
 PLUS(1(x),j(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Usable Equations:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> Usable Rules:
 0(S) -> S
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = X + 1
[minus](X1,X2) = 0
[opp](X) = 0
[plus](X1,X2) = X1 + X2
[times](X1,X2) = 0
[1](X) = X + 2
[S] = 0
[j](X) = X + 2
[0#](X) = 0
[MINUS](X1,X2) = 0
[OPP](X) = 0
[PLUS](X1,X2) = 2.X1 + 2.X2
[TIMES](X1,X2) = 0

Problem 1.1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
 PLUS(1(x),1(y)) -> PLUS(x,y)
 PLUS(1(x),j(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
 PLUS(1(x),1(y)) -> PLUS(x,y)
 PLUS(1(x),j(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> FAxioms:
 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
 plus(x2,x3) -> plus(x3,x2)
 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
 times(x2,x3) -> times(x3,x2)
 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) -> PLUS(x3,x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
->->-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)

Problem 1.1: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(1(x),1(y)),x2) -> PLUS(x,y)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
 PLUS(1(x),1(y)) -> PLUS(x,y)
 PLUS(1(x),j(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Usable Equations:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> Usable Rules:
 0(S) -> S
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = X + 1
[minus](X1,X2) = 0
[opp](X) = 0
[plus](X1,X2) = X1 + X2
[times](X1,X2) = 0
[1](X) = X + 2
[S] = 0
[j](X) = X + 2
[0#](X) = 0
[MINUS](X1,X2) = 0
[OPP](X) = 0
[PLUS](X1,X2) = 2.X1 + 2.X2
[TIMES](X1,X2) = 0

Problem 1.1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
 PLUS(1(x),1(y)) -> PLUS(x,y)
 PLUS(1(x),j(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
 PLUS(1(x),1(y)) -> PLUS(x,y)
 PLUS(1(x),j(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> FAxioms:
 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
 plus(x2,x3) -> plus(x3,x2)
 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
 times(x2,x3) -> times(x3,x2)
 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) -> PLUS(x3,x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
->->-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)

Problem 1.1: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(0(plus(x,y)),x2)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
 PLUS(1(x),1(y)) -> PLUS(x,y)
 PLUS(1(x),j(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Usable Equations:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> Usable Rules:
 0(S) -> S
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = X + 2
[minus](X1,X2) = 0
[opp](X) = 0
[plus](X1,X2) = X1 + X2
[times](X1,X2) = 0
[1](X) = X + 2
[S] = 0
[j](X) = X + 2
[0#](X) = 0
[MINUS](X1,X2) = 0
[OPP](X) = 0
[PLUS](X1,X2) = 2.X1 + 2.X2
[TIMES](X1,X2) = 0

Problem 1.1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
 PLUS(1(x),1(y)) -> PLUS(x,y)
 PLUS(1(x),j(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
 PLUS(1(x),1(y)) -> PLUS(x,y)
 PLUS(1(x),j(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> FAxioms:
 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
 plus(x2,x3) -> plus(x3,x2)
 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
 times(x2,x3) -> times(x3,x2)
 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) -> PLUS(x3,x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
->->-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)

Problem 1.1: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(1(x),j(y)),x2) -> PLUS(x,y)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
 PLUS(1(x),1(y)) -> PLUS(x,y)
 PLUS(1(x),j(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Usable Equations:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> Usable Rules:
 0(S) -> S
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = X
[minus](X1,X2) = 0
[opp](X) = 0
[plus](X1,X2) = X1 + X2
[times](X1,X2) = 0
[1](X) = X + 2
[S] = 0
[j](X) = X + 2
[0#](X) = 0
[MINUS](X1,X2) = 0
[OPP](X) = 0
[PLUS](X1,X2) = 2.X1 + 2.X2
[TIMES](X1,X2) = 0

Problem 1.1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
 PLUS(1(x),1(y)) -> PLUS(x,y)
 PLUS(1(x),j(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
 PLUS(1(x),1(y)) -> PLUS(x,y)
 PLUS(1(x),j(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> FAxioms:
 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
 plus(x2,x3) -> plus(x3,x2)
 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
 times(x2,x3) -> times(x3,x2)
 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) -> PLUS(x3,x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
->->-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)

Problem 1.1: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(j(S),plus(x,y))
 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
 PLUS(1(x),1(y)) -> PLUS(x,y)
 PLUS(1(x),j(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Usable Equations:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> Usable Rules:
 0(S) -> S
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = X + 2
[minus](X1,X2) = 0
[opp](X) = 0
[plus](X1,X2) = X1 + X2
[times](X1,X2) = 0
[1](X) = X + 2
[S] = 0
[j](X) = X + 2
[0#](X) = 0
[MINUS](X1,X2) = 0
[OPP](X) = 0
[PLUS](X1,X2) = 2.X1 + 2.X2
[TIMES](X1,X2) = 0

Problem 1.1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
 PLUS(1(x),1(y)) -> PLUS(x,y)
 PLUS(1(x),j(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
 PLUS(1(x),1(y)) -> PLUS(x,y)
 PLUS(1(x),j(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> FAxioms:
 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
 plus(x2,x3) -> plus(x3,x2)
 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
 times(x2,x3) -> times(x3,x2)
 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) -> PLUS(x3,x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
->->-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)

Problem 1.1: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(x,y)
 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
 PLUS(1(x),1(y)) -> PLUS(x,y)
 PLUS(1(x),j(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Usable Equations:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> Usable Rules:
 0(S) -> S
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = X + 2
[minus](X1,X2) = 0
[opp](X) = 0
[plus](X1,X2) = X1 + X2
[times](X1,X2) = 0
[1](X) = X + 2
[S] = 0
[j](X) = X + 2
[0#](X) = 0
[MINUS](X1,X2) = 0
[OPP](X) = 0
[PLUS](X1,X2) = 2.X1 + 2.X2
[TIMES](X1,X2) = 0

Problem 1.1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
 PLUS(1(x),1(y)) -> PLUS(x,y)
 PLUS(1(x),j(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
 PLUS(1(x),1(y)) -> PLUS(x,y)
 PLUS(1(x),j(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> FAxioms:
 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
 plus(x2,x3) -> plus(x3,x2)
 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
 times(x2,x3) -> times(x3,x2)
 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) -> PLUS(x3,x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
->->-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)

Problem 1.1: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(1(x),1(y)) -> PLUS(1(S),plus(x,y))
 PLUS(1(x),1(y)) -> PLUS(x,y)
 PLUS(1(x),j(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Usable Equations:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> Usable Rules:
 0(S) -> S
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = X
[minus](X1,X2) = 0
[opp](X) = 0
[plus](X1,X2) = X1 + X2
[times](X1,X2) = 0
[1](X) = X + 2
[S] = 0
[j](X) = X + 2
[0#](X) = 0
[MINUS](X1,X2) = 0
[OPP](X) = 0
[PLUS](X1,X2) = 2.X1 + 2.X2
[TIMES](X1,X2) = 0

Problem 1.1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(1(x),1(y)) -> PLUS(x,y)
 PLUS(1(x),j(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(1(x),1(y)) -> PLUS(x,y)
 PLUS(1(x),j(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> FAxioms:
 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
 plus(x2,x3) -> plus(x3,x2)
 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
 times(x2,x3) -> times(x3,x2)
 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) -> PLUS(x3,x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
->->-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)

Problem 1.1: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(1(x),1(y)) -> PLUS(x,y)
 PLUS(1(x),j(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Usable Equations:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> Usable Rules:
 0(S) -> S
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = X + 1
[minus](X1,X2) = 0
[opp](X) = 0
[plus](X1,X2) = X1 + X2
[times](X1,X2) = 0
[1](X) = X + 2
[S] = 0
[j](X) = X + 2
[0#](X) = 0
[MINUS](X1,X2) = 0
[OPP](X) = 0
[PLUS](X1,X2) = 2.X1 + 2.X2
[TIMES](X1,X2) = 0

Problem 1.1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(1(x),j(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(1(x),j(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> FAxioms:
 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
 plus(x2,x3) -> plus(x3,x2)
 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
 times(x2,x3) -> times(x3,x2)
 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) -> PLUS(x3,x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
->->-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)

Problem 1.1: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(1(x),j(y)) -> PLUS(x,y)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Usable Equations:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> Usable Rules:
 0(S) -> S
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = X + 2
[minus](X1,X2) = 0
[opp](X) = 0
[plus](X1,X2) = X1 + X2
[times](X1,X2) = 0
[1](X) = X + 2
[S] = 0
[j](X) = X + 2
[0#](X) = 0
[MINUS](X1,X2) = 0
[OPP](X) = 0
[PLUS](X1,X2) = 2.X1 + 2.X2
[TIMES](X1,X2) = 0

Problem 1.1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> FAxioms:
 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
 plus(x2,x3) -> plus(x3,x2)
 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
 times(x2,x3) -> times(x3,x2)
 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) -> PLUS(x3,x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
->->-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)

Problem 1.1: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(j(x),j(y)) -> PLUS(j(S),plus(x,y))
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Usable Equations:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> Usable Rules:
 0(S) -> S
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = X + 2
[minus](X1,X2) = 0
[opp](X) = 0
[plus](X1,X2) = X1 + X2
[times](X1,X2) = 0
[1](X) = X + 2
[S] = 0
[j](X) = X + 2
[0#](X) = 0
[MINUS](X1,X2) = 0
[OPP](X) = 0
[PLUS](X1,X2) = 2.X1 + 2.X2
[TIMES](X1,X2) = 0

Problem 1.1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> FAxioms:
 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
 plus(x2,x3) -> plus(x3,x2)
 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
 times(x2,x3) -> times(x3,x2)
 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) -> PLUS(x3,x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
->->-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)

Problem 1.1: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
 PLUS(j(x),j(y)) -> PLUS(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Usable Equations:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> Usable Rules:
 0(S) -> S
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = X + 1
[minus](X1,X2) = 0
[opp](X) = 0
[plus](X1,X2) = X1 + X2
[times](X1,X2) = 0
[1](X) = X + 2
[S] = 0
[j](X) = X + 2
[0#](X) = 0
[MINUS](X1,X2) = 0
[OPP](X) = 0
[PLUS](X1,X2) = 2.X1 + 2.X2
[TIMES](X1,X2) = 0

Problem 1.1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
-> FAxioms:
 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
 plus(x2,x3) -> plus(x3,x2)
 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
 times(x2,x3) -> times(x3,x2)
 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) -> PLUS(x3,x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
->->-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)

Problem 1.1: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(plus(1(x),1(y)),x2) -> PLUS(j(plus(1(S),plus(x,y))),x2)
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Usable Equations:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> Usable Rules:
 0(S) -> S
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 2
[minus](X1,X2) = 0
[opp](X) = 0
[plus](X1,X2) = X1 + X2 + 2
[times](X1,X2) = 0
[1](X) = 2
[S] = 0
[j](X) = 0
[0#](X) = 0
[MINUS](X1,X2) = 0
[OPP](X) = 0
[PLUS](X1,X2) = 2.X1 + 2.X2
[TIMES](X1,X2) = 0

Problem 1.1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
-> FAxioms:
 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
 plus(x2,x3) -> plus(x3,x2)
 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
 times(x2,x3) -> times(x3,x2)
 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) -> PLUS(x3,x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
->->-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)

Problem 1.1: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(plus(S,x),x2) -> PLUS(x,x2)
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Usable Equations:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> Usable Rules:
 0(S) -> S
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 0
[minus](X1,X2) = 0
[opp](X) = 0
[plus](X1,X2) = X1 + X2 + 2
[times](X1,X2) = 0
[1](X) = 0
[S] = 0
[j](X) = 1
[0#](X) = 0
[MINUS](X1,X2) = 0
[OPP](X) = 0
[PLUS](X1,X2) = X1 + X2
[TIMES](X1,X2) = 0

Problem 1.1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
-> FAxioms:
 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
 plus(x2,x3) -> plus(x3,x2)
 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
 times(x2,x3) -> times(x3,x2)
 PLUS(plus(x2,x3),x4) -> PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) -> PLUS(x3,x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
->->-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)

Problem 1.1: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 PLUS(plus(j(x),j(y)),x2) -> PLUS(1(plus(j(S),plus(x,y))),x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Usable Equations:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> Usable Rules:
 0(S) -> S
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 2
[minus](X1,X2) = 0
[opp](X) = 0
[plus](X1,X2) = X1 + X2 + 2
[times](X1,X2) = 0
[1](X) = 1
[S] = 0
[j](X) = 0
[0#](X) = 0
[MINUS](X1,X2) = 0
[OPP](X) = 0
[PLUS](X1,X2) = 2.X1 + 2.X2
[TIMES](X1,X2) = 0

Problem 1.1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x2,x3),x4) = PLUS(x2,plus(x3,x4))
 PLUS(x2,x3) = PLUS(x3,x2)
-> Pairs:
 Empty
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 PLUS(plus(x2,x3),x4) -> PLUS(x2,x3)
 PLUS(x2,plus(x3,x4)) -> PLUS(x3,x4)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> FAxioms:
 Empty
-> Pairs:
 OPP(0(x)) -> OPP(x)
 OPP(1(x)) -> OPP(x)
 OPP(j(x)) -> OPP(x)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 Empty
->Projection:
 pi(OPP) = [1]

Problem 1.2: 

SCC Processor:
-> FAxioms:
 Empty
-> Pairs:
 Empty
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 Empty
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Reduction Pairs Processor:
-> FAxioms:
 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
 TIMES(x2,x3) = TIMES(x3,x2)
-> Pairs:
 TIMES(0(x),y) -> TIMES(x,y)
 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
 TIMES(times(0(x),y),x2) -> TIMES(x,y)
 TIMES(times(1(x),y),x2) -> TIMES(plus(0(times(x,y)),y),x2)
 TIMES(times(1(x),y),x2) -> TIMES(x,y)
 TIMES(times(S,x),x2) -> TIMES(S,x2)
 TIMES(times(j(x),y),x2) -> TIMES(minus(0(times(x,y)),y),x2)
 TIMES(times(j(x),y),x2) -> TIMES(x,y)
 TIMES(1(x),y) -> TIMES(x,y)
 TIMES(j(x),y) -> TIMES(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Usable Equations:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> Usable Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
->Interpretation type:
 Simple mixed
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 1
->Interpretation:
 
[0](X) = X + 1
[minus](X1,X2) = X1 + X2
[opp](X) = X
[plus](X1,X2) = X1 + X2
[times](X1,X2) = X1.X2 + X1 + X2
[1](X) = X + 1
[S] = 0
[j](X) = X + 1
[0#](X) = 0
[MINUS](X1,X2) = 0
[OPP](X) = 0
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = X1.X2 + X1 + X2

Problem 1.3: 

SCC Processor:
-> FAxioms:
 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
 TIMES(x2,x3) = TIMES(x3,x2)
-> Pairs:
 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
 TIMES(times(0(x),y),x2) -> TIMES(x,y)
 TIMES(times(1(x),y),x2) -> TIMES(plus(0(times(x,y)),y),x2)
 TIMES(times(1(x),y),x2) -> TIMES(x,y)
 TIMES(times(S,x),x2) -> TIMES(S,x2)
 TIMES(times(j(x),y),x2) -> TIMES(minus(0(times(x,y)),y),x2)
 TIMES(times(j(x),y),x2) -> TIMES(x,y)
 TIMES(1(x),y) -> TIMES(x,y)
 TIMES(j(x),y) -> TIMES(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
 TIMES(times(0(x),y),x2) -> TIMES(x,y)
 TIMES(times(1(x),y),x2) -> TIMES(plus(0(times(x,y)),y),x2)
 TIMES(times(1(x),y),x2) -> TIMES(x,y)
 TIMES(times(S,x),x2) -> TIMES(S,x2)
 TIMES(times(j(x),y),x2) -> TIMES(minus(0(times(x,y)),y),x2)
 TIMES(times(j(x),y),x2) -> TIMES(x,y)
 TIMES(1(x),y) -> TIMES(x,y)
 TIMES(j(x),y) -> TIMES(x,y)
-> FAxioms:
 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
 plus(x2,x3) -> plus(x3,x2)
 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
 times(x2,x3) -> times(x3,x2)
 TIMES(times(x2,x3),x4) -> TIMES(x2,times(x3,x4))
 TIMES(x2,x3) -> TIMES(x3,x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
->->-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)

Problem 1.3: 

Reduction Pairs Processor:
-> FAxioms:
 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
 TIMES(x2,x3) = TIMES(x3,x2)
-> Pairs:
 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
 TIMES(times(0(x),y),x2) -> TIMES(x,y)
 TIMES(times(1(x),y),x2) -> TIMES(plus(0(times(x,y)),y),x2)
 TIMES(times(1(x),y),x2) -> TIMES(x,y)
 TIMES(times(S,x),x2) -> TIMES(S,x2)
 TIMES(times(j(x),y),x2) -> TIMES(minus(0(times(x,y)),y),x2)
 TIMES(times(j(x),y),x2) -> TIMES(x,y)
 TIMES(1(x),y) -> TIMES(x,y)
 TIMES(j(x),y) -> TIMES(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Usable Equations:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> Usable Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
->Interpretation type:
 Simple mixed
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 1
->Interpretation:
 
[0](X) = X + 1
[minus](X1,X2) = X1 + X2
[opp](X) = X
[plus](X1,X2) = X1 + X2
[times](X1,X2) = X1.X2 + X1 + X2
[1](X) = X + 1
[S] = 0
[j](X) = X + 1
[0#](X) = 0
[MINUS](X1,X2) = 0
[OPP](X) = 0
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = X1.X2 + X1 + X2

Problem 1.3: 

SCC Processor:
-> FAxioms:
 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
 TIMES(x2,x3) = TIMES(x3,x2)
-> Pairs:
 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
 TIMES(times(1(x),y),x2) -> TIMES(plus(0(times(x,y)),y),x2)
 TIMES(times(1(x),y),x2) -> TIMES(x,y)
 TIMES(times(S,x),x2) -> TIMES(S,x2)
 TIMES(times(j(x),y),x2) -> TIMES(minus(0(times(x,y)),y),x2)
 TIMES(times(j(x),y),x2) -> TIMES(x,y)
 TIMES(1(x),y) -> TIMES(x,y)
 TIMES(j(x),y) -> TIMES(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
 TIMES(times(1(x),y),x2) -> TIMES(plus(0(times(x,y)),y),x2)
 TIMES(times(1(x),y),x2) -> TIMES(x,y)
 TIMES(times(S,x),x2) -> TIMES(S,x2)
 TIMES(times(j(x),y),x2) -> TIMES(minus(0(times(x,y)),y),x2)
 TIMES(times(j(x),y),x2) -> TIMES(x,y)
 TIMES(1(x),y) -> TIMES(x,y)
 TIMES(j(x),y) -> TIMES(x,y)
-> FAxioms:
 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
 plus(x2,x3) -> plus(x3,x2)
 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
 times(x2,x3) -> times(x3,x2)
 TIMES(times(x2,x3),x4) -> TIMES(x2,times(x3,x4))
 TIMES(x2,x3) -> TIMES(x3,x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
->->-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)

Problem 1.3: 

Reduction Pairs Processor:
-> FAxioms:
 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
 TIMES(x2,x3) = TIMES(x3,x2)
-> Pairs:
 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
 TIMES(times(1(x),y),x2) -> TIMES(plus(0(times(x,y)),y),x2)
 TIMES(times(1(x),y),x2) -> TIMES(x,y)
 TIMES(times(S,x),x2) -> TIMES(S,x2)
 TIMES(times(j(x),y),x2) -> TIMES(minus(0(times(x,y)),y),x2)
 TIMES(times(j(x),y),x2) -> TIMES(x,y)
 TIMES(1(x),y) -> TIMES(x,y)
 TIMES(j(x),y) -> TIMES(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Usable Equations:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> Usable Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
->Interpretation type:
 Simple mixed
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 1
->Interpretation:
 
[0](X) = X
[minus](X1,X2) = X1 + X2 + 1
[opp](X) = X + 1
[plus](X1,X2) = X1 + X2
[times](X1,X2) = X1.X2 + X1 + X2
[1](X) = X + 1
[S] = 0
[j](X) = X + 1
[0#](X) = 0
[MINUS](X1,X2) = 0
[OPP](X) = 0
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = X1.X2 + X1 + X2

Problem 1.3: 

SCC Processor:
-> FAxioms:
 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
 TIMES(x2,x3) = TIMES(x3,x2)
-> Pairs:
 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
 TIMES(times(1(x),y),x2) -> TIMES(x,y)
 TIMES(times(S,x),x2) -> TIMES(S,x2)
 TIMES(times(j(x),y),x2) -> TIMES(minus(0(times(x,y)),y),x2)
 TIMES(times(j(x),y),x2) -> TIMES(x,y)
 TIMES(1(x),y) -> TIMES(x,y)
 TIMES(j(x),y) -> TIMES(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
 TIMES(times(1(x),y),x2) -> TIMES(x,y)
 TIMES(times(S,x),x2) -> TIMES(S,x2)
 TIMES(times(j(x),y),x2) -> TIMES(minus(0(times(x,y)),y),x2)
 TIMES(times(j(x),y),x2) -> TIMES(x,y)
 TIMES(1(x),y) -> TIMES(x,y)
 TIMES(j(x),y) -> TIMES(x,y)
-> FAxioms:
 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
 plus(x2,x3) -> plus(x3,x2)
 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
 times(x2,x3) -> times(x3,x2)
 TIMES(times(x2,x3),x4) -> TIMES(x2,times(x3,x4))
 TIMES(x2,x3) -> TIMES(x3,x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
->->-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)

Problem 1.3: 

Reduction Pairs Processor:
-> FAxioms:
 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
 TIMES(x2,x3) = TIMES(x3,x2)
-> Pairs:
 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
 TIMES(times(1(x),y),x2) -> TIMES(x,y)
 TIMES(times(S,x),x2) -> TIMES(S,x2)
 TIMES(times(j(x),y),x2) -> TIMES(minus(0(times(x,y)),y),x2)
 TIMES(times(j(x),y),x2) -> TIMES(x,y)
 TIMES(1(x),y) -> TIMES(x,y)
 TIMES(j(x),y) -> TIMES(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Usable Equations:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> Usable Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
->Interpretation type:
 Simple mixed
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 1
->Interpretation:
 
[0](X) = X + 1
[minus](X1,X2) = X1 + X2
[opp](X) = X
[plus](X1,X2) = X1 + X2
[times](X1,X2) = X1.X2 + X1 + X2
[1](X) = X + 1
[S] = 0
[j](X) = X + 1
[0#](X) = 0
[MINUS](X1,X2) = 0
[OPP](X) = 0
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = X1.X2 + X1 + X2

Problem 1.3: 

SCC Processor:
-> FAxioms:
 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
 TIMES(x2,x3) = TIMES(x3,x2)
-> Pairs:
 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
 TIMES(times(S,x),x2) -> TIMES(S,x2)
 TIMES(times(j(x),y),x2) -> TIMES(minus(0(times(x,y)),y),x2)
 TIMES(times(j(x),y),x2) -> TIMES(x,y)
 TIMES(1(x),y) -> TIMES(x,y)
 TIMES(j(x),y) -> TIMES(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
 TIMES(times(S,x),x2) -> TIMES(S,x2)
 TIMES(times(j(x),y),x2) -> TIMES(minus(0(times(x,y)),y),x2)
 TIMES(times(j(x),y),x2) -> TIMES(x,y)
 TIMES(1(x),y) -> TIMES(x,y)
 TIMES(j(x),y) -> TIMES(x,y)
-> FAxioms:
 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
 plus(x2,x3) -> plus(x3,x2)
 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
 times(x2,x3) -> times(x3,x2)
 TIMES(times(x2,x3),x4) -> TIMES(x2,times(x3,x4))
 TIMES(x2,x3) -> TIMES(x3,x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
->->-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)

Problem 1.3: 

Reduction Pairs Processor:
-> FAxioms:
 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
 TIMES(x2,x3) = TIMES(x3,x2)
-> Pairs:
 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
 TIMES(times(S,x),x2) -> TIMES(S,x2)
 TIMES(times(j(x),y),x2) -> TIMES(minus(0(times(x,y)),y),x2)
 TIMES(times(j(x),y),x2) -> TIMES(x,y)
 TIMES(1(x),y) -> TIMES(x,y)
 TIMES(j(x),y) -> TIMES(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Usable Equations:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> Usable Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
->Interpretation type:
 Simple mixed
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 1
->Interpretation:
 
[0](X) = X
[minus](X1,X2) = X1 + X2
[opp](X) = X
[plus](X1,X2) = X1 + X2
[times](X1,X2) = X1.X2 + X1 + X2
[1](X) = X + 1
[S] = 0
[j](X) = X + 1
[0#](X) = 0
[MINUS](X1,X2) = 0
[OPP](X) = 0
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = X1.X2 + X1 + X2

Problem 1.3: 

SCC Processor:
-> FAxioms:
 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
 TIMES(x2,x3) = TIMES(x3,x2)
-> Pairs:
 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
 TIMES(times(S,x),x2) -> TIMES(S,x2)
 TIMES(times(j(x),y),x2) -> TIMES(x,y)
 TIMES(1(x),y) -> TIMES(x,y)
 TIMES(j(x),y) -> TIMES(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
 TIMES(times(S,x),x2) -> TIMES(S,x2)
 TIMES(times(j(x),y),x2) -> TIMES(x,y)
 TIMES(1(x),y) -> TIMES(x,y)
 TIMES(j(x),y) -> TIMES(x,y)
-> FAxioms:
 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
 plus(x2,x3) -> plus(x3,x2)
 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
 times(x2,x3) -> times(x3,x2)
 TIMES(times(x2,x3),x4) -> TIMES(x2,times(x3,x4))
 TIMES(x2,x3) -> TIMES(x3,x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
->->-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)

Problem 1.3: 

Reduction Pairs Processor:
-> FAxioms:
 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
 TIMES(x2,x3) = TIMES(x3,x2)
-> Pairs:
 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
 TIMES(times(S,x),x2) -> TIMES(S,x2)
 TIMES(times(j(x),y),x2) -> TIMES(x,y)
 TIMES(1(x),y) -> TIMES(x,y)
 TIMES(j(x),y) -> TIMES(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Usable Equations:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> Usable Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
->Interpretation type:
 Simple mixed
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 1
->Interpretation:
 
[0](X) = X
[minus](X1,X2) = X1 + X2 + 1
[opp](X) = X + 1
[plus](X1,X2) = X1 + X2
[times](X1,X2) = X1.X2 + X1 + X2
[1](X) = X + 1
[S] = 0
[j](X) = X + 1
[0#](X) = 0
[MINUS](X1,X2) = 0
[OPP](X) = 0
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = X1.X2 + X1 + X2

Problem 1.3: 

SCC Processor:
-> FAxioms:
 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
 TIMES(x2,x3) = TIMES(x3,x2)
-> Pairs:
 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
 TIMES(times(S,x),x2) -> TIMES(S,x2)
 TIMES(1(x),y) -> TIMES(x,y)
 TIMES(j(x),y) -> TIMES(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
 TIMES(times(S,x),x2) -> TIMES(S,x2)
 TIMES(1(x),y) -> TIMES(x,y)
 TIMES(j(x),y) -> TIMES(x,y)
-> FAxioms:
 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
 plus(x2,x3) -> plus(x3,x2)
 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
 times(x2,x3) -> times(x3,x2)
 TIMES(times(x2,x3),x4) -> TIMES(x2,times(x3,x4))
 TIMES(x2,x3) -> TIMES(x3,x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
->->-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)

Problem 1.3: 

Reduction Pairs Processor:
-> FAxioms:
 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
 TIMES(x2,x3) = TIMES(x3,x2)
-> Pairs:
 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
 TIMES(times(S,x),x2) -> TIMES(S,x2)
 TIMES(1(x),y) -> TIMES(x,y)
 TIMES(j(x),y) -> TIMES(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Usable Equations:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> Usable Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
->Interpretation type:
 Simple mixed
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 1
->Interpretation:
 
[0](X) = X + 1
[minus](X1,X2) = X1 + X2
[opp](X) = X
[plus](X1,X2) = X1 + X2
[times](X1,X2) = X1.X2 + X1 + X2
[1](X) = X + 1
[S] = 0
[j](X) = X + 1
[0#](X) = 0
[MINUS](X1,X2) = 0
[OPP](X) = 0
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = X1.X2 + X1 + X2

Problem 1.3: 

SCC Processor:
-> FAxioms:
 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
 TIMES(x2,x3) = TIMES(x3,x2)
-> Pairs:
 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
 TIMES(times(S,x),x2) -> TIMES(S,x2)
 TIMES(j(x),y) -> TIMES(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
 TIMES(times(S,x),x2) -> TIMES(S,x2)
 TIMES(j(x),y) -> TIMES(x,y)
-> FAxioms:
 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
 plus(x2,x3) -> plus(x3,x2)
 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
 times(x2,x3) -> times(x3,x2)
 TIMES(times(x2,x3),x4) -> TIMES(x2,times(x3,x4))
 TIMES(x2,x3) -> TIMES(x3,x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
->->-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)

Problem 1.3: 

Reduction Pairs Processor:
-> FAxioms:
 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
 TIMES(x2,x3) = TIMES(x3,x2)
-> Pairs:
 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
 TIMES(times(S,x),x2) -> TIMES(S,x2)
 TIMES(j(x),y) -> TIMES(x,y)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Usable Equations:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> Usable Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
->Interpretation type:
 Simple mixed
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 1
->Interpretation:
 
[0](X) = X + 1
[minus](X1,X2) = X1 + X2
[opp](X) = X
[plus](X1,X2) = X1 + X2
[times](X1,X2) = X1.X2 + X1 + X2
[1](X) = X + 1
[S] = 0
[j](X) = X + 1
[0#](X) = 0
[MINUS](X1,X2) = 0
[OPP](X) = 0
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = X1.X2 + X1 + X2

Problem 1.3: 

SCC Processor:
-> FAxioms:
 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
 TIMES(x2,x3) = TIMES(x3,x2)
-> Pairs:
 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
 TIMES(times(S,x),x2) -> TIMES(S,x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
 TIMES(times(S,x),x2) -> TIMES(S,x2)
-> FAxioms:
 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
 plus(x2,x3) -> plus(x3,x2)
 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
 times(x2,x3) -> times(x3,x2)
 TIMES(times(x2,x3),x4) -> TIMES(x2,times(x3,x4))
 TIMES(x2,x3) -> TIMES(x3,x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
->->-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)

Problem 1.3: 

Reduction Pairs Processor:
-> FAxioms:
 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
 TIMES(x2,x3) = TIMES(x3,x2)
-> Pairs:
 TIMES(times(0(x),y),x2) -> TIMES(0(times(x,y)),x2)
 TIMES(times(S,x),x2) -> TIMES(S,x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Usable Equations:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> Usable Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 1
[minus](X1,X2) = 2.X1 + X2 + 1
[opp](X) = X
[plus](X1,X2) = X1 + X2 + 1
[times](X1,X2) = X1 + X2 + 2
[1](X) = 1
[S] = 1
[j](X) = 1
[0#](X) = 0
[MINUS](X1,X2) = 0
[OPP](X) = 0
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = 2.X1 + 2.X2

Problem 1.3: 

SCC Processor:
-> FAxioms:
 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
 TIMES(x2,x3) = TIMES(x3,x2)
-> Pairs:
 TIMES(times(S,x),x2) -> TIMES(S,x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 TIMES(times(S,x),x2) -> TIMES(S,x2)
-> FAxioms:
 plus(plus(x2,x3),x4) -> plus(x2,plus(x3,x4))
 plus(x2,x3) -> plus(x3,x2)
 times(times(x2,x3),x4) -> times(x2,times(x3,x4))
 times(x2,x3) -> times(x3,x2)
 TIMES(times(x2,x3),x4) -> TIMES(x2,times(x3,x4))
 TIMES(x2,x3) -> TIMES(x3,x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
->->-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)

Problem 1.3: 

Reduction Pairs Processor:
-> FAxioms:
 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
 TIMES(x2,x3) = TIMES(x3,x2)
-> Pairs:
 TIMES(times(S,x),x2) -> TIMES(S,x2)
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Usable Equations:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> Usable Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0](X) = 1
[minus](X1,X2) = X1 + X2 + 2
[opp](X) = X + 1
[plus](X1,X2) = X1 + X2 + 1
[times](X1,X2) = X1 + X2 + 1
[1](X) = 1
[S] = 1
[j](X) = 2
[0#](X) = 0
[MINUS](X1,X2) = 0
[OPP](X) = 0
[PLUS](X1,X2) = 0
[TIMES](X1,X2) = 2.X1 + 2.X2

Problem 1.3: 

SCC Processor:
-> FAxioms:
 TIMES(times(x2,x3),x4) = TIMES(x2,times(x3,x4))
 TIMES(x2,x3) = TIMES(x3,x2)
-> Pairs:
 Empty
-> EAxioms:
 plus(plus(x2,x3),x4) = plus(x2,plus(x3,x4))
 plus(x2,x3) = plus(x3,x2)
 times(times(x2,x3),x4) = times(x2,times(x3,x4))
 times(x2,x3) = times(x3,x2)
-> Rules:
 0(S) -> S
 minus(x,y) -> plus(opp(y),x)
 opp(0(x)) -> 0(opp(x))
 opp(1(x)) -> j(opp(x))
 opp(S) -> S
 opp(j(x)) -> 1(opp(x))
 plus(0(x),0(y)) -> 0(plus(x,y))
 plus(0(x),1(y)) -> 1(plus(x,y))
 plus(0(x),j(y)) -> j(plus(x,y))
 plus(1(x),1(y)) -> j(plus(1(S),plus(x,y)))
 plus(1(x),j(y)) -> 0(plus(x,y))
 plus(S,x) -> x
 plus(j(x),j(y)) -> 1(plus(j(S),plus(x,y)))
 times(0(x),y) -> 0(times(x,y))
 times(1(x),y) -> plus(0(times(x,y)),y)
 times(S,x) -> S
 times(j(x),y) -> minus(0(times(x,y)),y)
-> SRules:
 TIMES(times(x2,x3),x4) -> TIMES(x2,x3)
 TIMES(x2,times(x3,x4)) -> TIMES(x3,x4)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.